The Heat Capacity of Superfluid ⁺He in the Presence of a Constant Heat Flux Near Tλ Citation Harter, Alexa Welsh (2001) The Heat Capacity of Superfluid ⁺He in the Presence of a Constant Heat Flux Near Tλ. Dissertation (Ph.D.), California Institute of Technology. https://resolver.caltech.edu/CaltechTHESIS:07252014-090646708 Abstract We present the first experimental evidence that the heat capacity of superfluid 4 He, at temperatures very close to the lambda transition temperature, T λ ,is enhanced by a constant heat flux, Q. The heat capacity at constant Q, C Q ,is predicted to diverge at a temperature T c (Q) < T λ at which superflow becomes unstable. In agreement with previous measurements, we find that dissipation enters our cell at a temperature, T DAS (Q),below the theoretical value, T c (Q). Our measurements of C Q were taken using the discrete pulse method at fourteen different heat flux values in the range 1µW/cm 2 ≤ Q≤ 4µW /cm 2 . The excess heat capacity ∆C Q we measure has the predicted scaling behavior as a function of T and Q:∆C Q • t α ∝ (Q/Q c ) 2 , where Q c T) ~ t 2ν is the critical heat current that results from the inversion of the equation for T c (Q). We find that if the theoretical value of T c ( Q) is correct, then ∆C Q is considerably larger than anticipated. On the other hand,if T c (Q)≈ T DAS (Q),then ∆C Q is the same magnitude as the theoretically predicted enhancement. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Physics, superfluid ⁺He ,constant heat Tλ Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Goodstein, David L. Thesis Committee: Goodstein, David L. (chair) Cross, Michael Clifford (co-chair) Eisenstein, James P. Yeh, Nai-Chang Defense Date: 20 July 2000 Record Number: CaltechTHESIS:07252014-090646708 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:07252014-090646708 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 8606 Collection: CaltechTHESIS Deposited By: Dan Anguka Deposited On: 25 Jul 2014 17:34 Last Modified: 12 Sep 2022 22:20 Thesis Files Preview PDF - Final Version See Usage Policy. 30MB Repository Staff Only: item control page

The Heat Capacity of Superfluid 4 He in the Presence of a Constant Heat Flux Near T>. T hC'sis by Alexa Welsh Ha rter Iu Partial FulfillmC'nt of t lw Req uin·uH'nts for t hr DcgrC'C' of Doctor of P hilosophy California Inst it ute' of Tcdmolog~' P asadrna, Califor nia 2001 (DC'fC'nded .Jul:v 20, 2000 ) ll © 2001 Al<'xa \Y<>lsh Ha rl cr .\11 H ig hts RcscrYcd Ill Acknowledgements Thc·rc a r<' a numbcr of p<•o plc• wit ho nt wh o m t his thesis cou ld n ot ha\'1' l><'<'n compld <•d . 1 ow<' a great d eal to Ill.\. LIH•sis a d vis or, D avid Goodstein, w ho u ot ou ly sh a r ed his sci<•n t ifir ideas. enthus ias m , and uncommo n ahilit~· t o sort o u t t he <'SS<'nt ial, b ut also d <'m o ns trat.ed mnch pati<'llC<' and und<'rs t a nding . I wonld like• to t ha n k Talso C hui , 11i cbard LC'<>, a nd P <' Le r Day, who taug ht me a num b er of <'XP<'r illH'Htal IC'chuiq ues a nd provid ed iuval ua hl<• a ssis tall(:C' in the labor atory. I a m iJH.lebL<'d a lso to PPIPr \\'<:> ichma n , who gave Ill<' cous i<kra hl<' ins ig ht into m~· <'X p <'rin H'nt a nd p roofr<'ad m y tlH'sis w ith a thro ris t·s carr fnl c·~·c•. B ill \\'ebc r, m y fe llow s ub- bas<'HWut d w<'ller. was a tren H'Hdo us sc ien tific sou ud ing board . frie nd . a nd t eam m al<' durin p, m.Y da~·s a t Calteeh . f\ fike Li lly and Kal~·a ni S ukhatm<' haw m~· g rat i t ud <' fo r o ff<' ri ng Ill<' t IH'ir eo11 ns0L com m is<' rat io n . fri <• ndsh i p. a nd m ost im port a nt ly fo r making Sllr<' I ba t I wa s prop erly caffC'ina t <'d. Hoiau Egnor , .\ntlwuy L<'ou a rdo. a ud r.lich ael ~ lurphy provided m e w ith ho m es away from home during the wc'ek o f m .Y defens<'. X inka i VVu a nd A ndrew C h at t o d <'S<'rY<' 111.\' tha n ks fo r t a kin g ov<'r this proj c•ct so that. tudik<' o th<'rs , I can cut tlw cord w ith o u t r<'g r<'l. T he n ' a rc a llUUlbcr of o t.hcr me mbe rs o f t.he low t en qJ<'ral u r<' couulluHit.~· wh o provided helpful adv ice and assis t a iiC<' ov<'r t 11<' y<'ars. A m o ng t he m a r<' !l ob D11nean , 1\il s Asplund , 1\:c'ith Sdnvr1h , .Jo hn Hartman , Dav id R ow<', Yury l\Tu kharsk.v, D av<' P NlrSOll , .J o ln1 Li pa, R udol pli Ha nssm a nu , a nd G uc ut er A hlers . .l\ l y pa rents d eserve' a g reat ck a l o f CT('(Ii I fo r no t b elieving m y fo urth grade t rad 1<'r w ltr n siH' t o ld t ll<'m I wo uld JH' \'<'r h<' good a t m at h a nd for gi ,·ing nH' n mch <'n<·ouragc'IIH' nt a lo ng t lw way. I C\111 a lso grat e ful to R <'ina ldo Tiom <in , who l<•ft Anu Arbor l o j o iu IIH' i11 P asad e na. cud ured m.v crazy II o urs, b u t m ost o f al L was a lwa~'S t lw r e fo r Ill <'. ). l .Y firs t few yea rs a t Caltech wo uld not have b ee n t he sam <' w it ho ut To 111 l\lurphy. B ill \\.<'IH'r. L a ura G rego . C hris Fassnach t. a ud Art hur Street. w ho hel p Pd kec>p me a wa ke• a ll(! ins pired during m a n:v lat e nig ht pro blem sessions. I ow<' the m , H ideo l\ Iahuchi , 13abra A kma l, and o thers a dc•ht of g rat.itnd<' . .A nd th a nks t o al l wh o jo in C'C I Ill<' o n 1u.v variou s a d v<'nt u n•s a nd a e Li vi Li<'s: ra ndom po Ll ucks a nd cookfests, Iale uigh t l'xcurs ions t.o collect brui ses on Lhe ice slopes of \1\'ri gh t.wood , <'XJ><'d i t.i ou s t.o 1\ la nnno t.h , bikiug at til<' crack of dawn thro ug h God 's co nn try, t hawing o ut at th<' Yo no m o no. g<•tting resctH'd h~· B ~ IvV t<'s t drivers in tlH' m iddle of a n as t <'roid shower in t!H' A ng<>l<'s .\latio ua l Fon's l , a nd pl ay ing soccer with the Phys. R <'v. 1'\<'tters. I would a lso like t o aeknowkdge Ri<· ha rd L<'e for p rep a rin g a f<·w figu n·s t hat in sonH' fo r Ill or a no ther into t his t h esis ami fo r taking n o tes with m etic ulous d e t a il : a nd Xinka i \\'u for l<:> ndi ng a h a nd wit It d ata a n a lysis. c'v<'nlu all~· got in corporal <'d 1\' Abstract \Ve prcsC'nt thC' firs t exp erim ental c·,·idf'ncC' that thC' heat eapacit? of supcrflnicl '1Hc·, at tcm JwraturC's Vf'ry c lose to t lw lambda transition tC'mpC'raturC', T>., is C'nhanccd by a constan t h0at flu x, Q. T h f' lwa t capacit_\· at constant Q, Cq, is preclictC'd to di vNge at a temperature 7'c.(Q) < T>.. at wh ich s uperftow becomE's uustable. In agreement wiLh previous mcas urelllcnLs. we fiud Lh at dissipation f'Hters our cell at a tempcratur<>, Tr)As(Q), below the theoretical value. T,_.(Q). Our measurements of CQ were taken us iug t h C' discrete pulse method at fourtecu d ifferent heat. flux val ucs iu t he range' 1 Jlv\)cm 2 ::; Q ::; 4 ft\\'/cm 2 . The excess h eat capacity .6.CQ we m easu re' has the prc•clicted scaling helJa,·ior as a function ofT aud Q: .6.CQ · f 0 ex: (Q/Qcf, where' Qc(T) rv t"l•/ is t h e critical hc•at c n rr<'nt that rC's ults from tlw inwrsion of t hC' <'qnation for Tc(Q). \i\'C' find that if the thC'orC't.ical YahJC' of Tc(Q) is c·orr<'ct, thC'n .6.CQ is con siclcrahl.v largC'r than anticipated. O n t h e othf'r hand , if Tr(Q) ~ TnA s(Q), the n .6.Cq is t.hC' samC' uJagn it udf' as thC' tlworf'tica ll y preclictC'd enhanccm<'nt. v Contents Acknowledgements 111 Abstract IV 1 1 Introduction l.J Supt:'rfluidity and t h<' two-fluid model 3 1.:2 Ileat transport in supNfluid 4 1-Ie . 8 1.3 The static superfl uid transition 8 1.4 Hf'at flow and the supcrft uid transition 10 1.4.1 T he depressed temperat ure of s upcrftuicl hreakdowu 11 1.4.2 HeaL capacity divergenc<' . . . . . . . . . . . . . . . 13 T lH' exponents r<'lated t o tlw supcrftuid transit ion uodC'r lwat fl ow 16 1.5.1 T h<' cxponPnt of t h<' transit ion tempC'raturC' clC'prC'ssion 16 1.5.2 T hc exponent of t hc h<'at capacity diverg<'ncc .. . .. 17 Prcclictions for tlw magnitude of t he heat capacity <'nhancement 18 1.5 1.6 1.7 2 1.6.1 T he dPprcssion of (J 8 vvith ll ' . 18 1.6.2 T hf' calcul aLiou of Cw 19 1.6.3 T h<J calc ulation of CQ 21 T he nature of the supcrfiuid trausitiou under heat flow 26 The Superfluid Transition in the Q-T Plane 27 2.1 Introductiou . . . . . . . . . . . . . . . . . . 28 2. 2 Noncq uili briutu critical phenomena aud scaling . 31 2.2.1 T hennodyuamic formalism . 32 2.2.2 Nouequilibrium scaling . . . 34 2.3 Tlw HOllt'quilibrinm s uperfluid-uormal iut erfac<' 41 VI 3 2.3.1 Interface statics . . . . 41 2.3.:2 Transition from thermodynamic to intcrfac<' sLate 45 2.3.3 Interface· dy ua mics . . . 47 2.4 Thr self-organiz<'ci critical stat.r 50 2.5 Sprcifi c hrat at constant lwat curTf'nt 54 2.5.1 E uha uccd spec ific heat . . . . 34 2.5.2 The supc rfluid breakdown temperature 55 2.5.3 Experimeu tal m easurements 58 Apparatus 65 3. 1 Limitations on the experimental d 0sigu . . . . . 65 3. 1.1 Cc•JI h0ig ht: gravity and finite size 0ff0cts 65 3. 1.2 TIH'rmom<'t<'r pl<tcemf'nt , specific heat f'rror, and tempera t nre instability: the singular Kapitza res istance 3.2 3.3 3.-l 69 High resolutiou tlwnuometers 78 3.2.1 Des ign .. 78 3.2.2 Sensiti vity 81 3.2.3 ::'>Joise . . . 83 3.2.4 Drift rat0s 85 3.2.5 llllplcrn<'ntation and calibration 86 The calorimeter sLage . 91 3.3. 1 Dinwusious 91 3.3.2 \ ·alvt' . . . . 94 3.3.3 Hr lium cavit.~· 95 3.3.4 Location of h<'aters and th0rmomet0rs 99 3.3.5 Heat flow w ithin the calorimeter . 100 Thermal network . . . . . . . . . . . . . 102 3.4.1 T hree stage thermal isolat ion system 102 3.-1.2 Contact r0sistanc0 105 3.4.3 Potential heat leaks . 110 Vll 3.-.1.4 3.5 4 115 Experimcutal coufig ura tiou and interface 117 3.5. 1 T he cryostat aud tlw compu ter interface 117 3.5.2 \ ·ibra liou isolaLiou 123 Exp erimental Methods 124 4.1 125 4.2 4.3 5 T hermal stabili t)' .. . Hrat ca pacity lllNlsnreme>ut techniques 4.1.1 AC calorimetry . 125 4.1.2 Drift calorimetry 126 4.1.3 Discrete pu lse calorimetry 130 4.1.4 f\ lodifit>d discrt>te pulst> calorimetry 133 Cooldowu details . . . . . . . . . . . . . . 137 4.2. 1 Calibrating the HRTs and filliug Lhe cell 137 4.2.2 Taking data 144 \\'arming up 145 Data Analysis 148 5.1 148 Eliminatin g electronic noise and ot her error sources 5. 1.1 SQUID timer reset s . 148 5.1. 2 F lux jumps ... 149 5. 1.3 ~ licroflux jumps. 151 5.1A HRT drift . . . . 154 5.2 F ixing the temperature scale 5.3 Correcting the endplatc· tempf'raturr readings for the singular Kapitza resist <-UICf' 157 . . . . . . . . . . 163 il ff'as urin g the hc•at capacity 164 5.-1.1 Determining thC' size of th<· t C'mJw ra turf' discontinuity 164 5.4.2 Calcul at in g the heat capacit y . 167 5.4.3 Improving t lw noisC' of the data 170 5.5 T he• heat capacity d ata at vario us valu es of Q 174 5.6 Checking the acc uracy of the data analysis . . 174 5.4 \'Ill G.G.l T h e poteutial error in fixing Lite temperature scalP. . . . . . . .).G. 2 T he error d ne to choosi11g, the iu('orr ect point ou l he puls<' a t w hid1 t o <"alculat.<• .:::.T 6 L75 17G Results and Conclus ions 180 G.1 Com paring o m data to t lworrl kal pn•d ietions 180 G.2 Scali n g .. 183 G.3 11<'S('aliug 187 G. 1 Conclusions 189 A A proof for why .:::.cQ I n scales with (Q/Qc) 2 191 B Experimental Details 193 B. l T he heat ers . . . . L9-1 8.2 The g,Nmau iulll n•sisl a H<'P thennomet ers ( G RTs) 193 8.3 Th<' SQCID filt<•rs 198 B. ~l T h<• m agrwt . . . . 200 8.5 T h<' vac uum can lid 20 1 C Catalog of Data Runs 203 D H eat Capacity of a Current Carrying Superconductor 206 Bibliography 210 IX List of Figures L.1 Tlw phasC' diagram of 1 He• '1 1.2 Tlw SJH'cific ltC'al of 111C' -1 1.3 The• SUJH'rfit~id and normal fluid dc•nsitiPs G 1. 1 He• 11 e•xtitat iou SJH'Ct rum 7 l.G Cn•ation of a C'Olllll<'rflow . 9 l.G The• d<'J)I·e•ssim t of p, wi l h I\" caklllat ed wit It ~ 1FT 12 l.l Th<' c·ollnt.c•rflow w lo('ily as a ft~nctiou of' T. 13 1.8 Tlw de' J))'<'ssion of p, with 11· cakllhttPd with HG 20 1.9 Tlworc•tical c•stimatC's for (a) ~ C11 as a funC'tion of r.· = ll '/ ll '1 and (h ) ~Cq a:-. a funC'tion of Q/Q, . . . . . . . . . . . . . . . . . . . . 22 1.10 Thc•or<'l ical c•stimatC's of ~ CQ for n\rious value's of Q as a funC't ion oft 2-l 1.11 Ha ussuta1111's pn•dictio11 of Cq for Q = 12.9 fi\\' jc rn 2.1 1 . . . . . . . . . Comparison of phase• diagrams itt (a) c:om·c•ut ional T-h a11d T-111 span's and (b) s upC'rfluid T-Q and T-{ ·., space's. . . . . . . . . . . . . . . . . 2.:2 23 33 Comparison of the• iso t hC'rntrd <'quat ion of state· (a) in l h<' c·onvc•nt ional li-m plane· and (h) in l hC' su pC'rftu id Q-[ ·, plan<'. 38 :2.3 Sc:akd t<•mpC'ratun• and ordC'r paralltC'tC'r profi lC'S 43 2. 1 T<'IIIJ><'rat un• profil<•s d uriug the en• at ion of all i 111 e•rf~u·e• 46 2.5 Siutt~ l at ion of t h<· Se•lf Organized C ritical 52 2.6 Tlw tc•tnperat ur<' of :-.uperfluid bn•akdown. a:-. a function o f Q -·' - Sample• !teat C'apacity data . . . . . . . . . . . . . . . . . . . 2.8 SC'a ling plot of til<' difl'Prenl iallwat capacit.\' measun•m<'llt s \'S. ( Q jQ, )1 ') (SOC) s t.ate . usi ng t he• t lwor<'t in\ I ntl ue' of Q~ . . . . . . . . . . . . . . . . . . . . . 2.9 59 G1 Scali ng plot of' tiH' diff<'l'<'lll ialiH•at ('apacity measun'HH'IIt s vs. (Q jQ,.)'J. usi up, an <'Ill pi rica! ,·at uc• of Q 0 . . . . . . . . . . . . . . . . . . . . . . G2 X 3.1 Sdt<'llli-1 tic of t he• COlli plet <· c•xprrimrnt al syst <'Ill 66 3.:2 T<>tupcralur<• profilPs for '1llr in a p;rcwil ional firld 67 3.:3 1-i<'at capacil_,. curn·~ in a g ra,·itatioual fic•ld 68 3.'1 T h<' r<'gular and singular 1\:apil za n·sistauce 70 :3.0 Th<• total J'apit za r<'~ista ll<"<' a~ a function of l<'liiJH'ral un•. 3.G Th<• ~ingular "apil:ta rc•sistall<"<' as a fu11ction of r<'duc·<'d nwau houud- ,_ -·) ar.'· tc•mprralllr<' . . . . . . . . . . . . . . . . . . 73 :.3. 7 :-\ lllltll<'rical cak11lal ion of su pNfluid brNtkdmnt IJ 3.8 The• hn•akdown IC'mpNatur<' as a function of Q 78 :.3.9 SdlPmat ic of I he• 1Il1Ts us<•d in our <'xpc•rinwnt . 81 3.10 Sc·usi ti vit.~· vs. tcuq><•rat nn• for the• GdCh and La doprd C:dCh II11Ts 82 3.11 :\ oisc• s prct nun of t IH• La dotwd GdCla t hrrmoiii<'I er 83 3.12 TIH• backgro und noise• d tH' to .) ohnson CUITPII( IlOiS<' in t ltP IIRT capsu 1<• i-\."iS('tll hl.\· . . . . . . . . . . . . . 83 :3.1:3 A.ttachillp; all HRT to a SQCID 88 :3.11 :\ <"alibratiou rnn for IIRTl. thr top plate• HRT 3.1::; The• ndorime•t r r d i nwnsions 93 C<:•ll valvt> sdH·mat.ic 96 3.1() :.3. 17 Exp('rimputal heat capacit.\· of 1Ilr at Q = 0 97 3.18 Th<' c<'ll vol tllll<' 98 . . . . . . . . . . . . . . . . :3.1!) Plct<'<'lll<'ll( of tht> the•nnonwtc•rs and heatNs in the• apparatus 101 3.20 !sot lwnns within I h<• <"<'11 . . . . . 103 3.21 SdH•matic oft lw t lwnnal n<•t \\'Ork lOG 3.22 Schc•ma tic of hPat l<•aks ill I h<· I h<•nHonwl <'T's 107 3.2:~ T he• Cllrrrnt SOllr<'C'S tlsc•d to Sllppl ,\' Q <-\lid Q1-at 109 3.2 I IIrat s inking I h<' JIRT capi llary and ltc•atc•r lc'ad wires 111 3.:20 Pot <'Ill ial brat lrak: llw l\h-Ti H:RT capil lar.v ll..J. 3.2G Tlw tlwrmal s tahilit~· of th<· rxpPriuwntal s.\·slent llG 3.27 SdH•ut al ic of th<' cr~·ostat . . . . . . 118 3.28 LC filtc·rs al tlw lop of the• cryostat 120 XI 3.29 LC' fil t <'rs in t he b n•ako ut box 12 1 3.10 SclH•m at ic o f tlw gas ha ndling syst e m 122 .J. 1 AC calorim et ry sd )(' m atic 127 ·1. 2 D rift calor imt>try sch E'tllatic 129 1.3 Data t ake n wi th t he d rift caloriBH' lr~· m e thod 131 .J.,J 'S pecific heat ' da t a takrn wit h tlJ<• drift calo rimetr.v m <•Lhod 1:32 ·1. 5 S<•ve•ra l p ulsc•s t a kc•JJ w i t.h the d iscr('( <' pulse• c-alor iJtH'Lry w eth od 135 . I.G D isnet<' pu lse• ca lorinwt ry s d H'lii Ht ic 138 -l.l Data take n wi th IIH· disc rC'te pui s<' calorinw t ry 11 1<'1 hod 1;39 -L8 T lw c•x pN ime nt a l confi g uration used t o fill a nd e lllpt y t IH' cell H1 1.9 B ubbl E' hunt : d rt.r rmining t h <' h<•lium fill facto r 1 13 4. 10 T lw uumlw r o f d a t a runs taken as a fun ct io n of Q 14G .).1 Corn •c t i11g fo r r<'s<•t e·rrors 150 .).2 E limi11at ing Hm.: j ulllps . . 152 5.3 E lilllin at ing mi noflu x juwps 153 5.-l Correcting for TTRT drift . . 105 5 ..) Fi nd ing thr p o int locatio n o f thr pulsrs 15G 5.G F inding the mid d le• of th P pulse• lrdgr . ].)8 .). 7 Correcting for IlllT d rift : selected point s 159 5.8 Correc ting fo r 1LHT drift : the d N ivat iV<' m eth od 160 5.9 Fi nding TnAs o n t ht> upward ra mp . . LG1 5. 10 Fi nd in g TnAs o n t lH• d own ward ra111 p 162 5. 11 T h<• si ng ul a r K apit za resistan ce' as a functio n of s u pc•rfluid tem peratu rr 1G5 5.12 T he• he·at cap ac it y at Q = 2 .26 ttV\ )cm 2 us ing Lhn•c di fl'c re ut m easurem e•nts o f t it<:> SliJH'rfluid t.e•mp<'ra t. urc~ . . . . . . . . . . . . . . . . . . . 166 G. L1 O n<' e•xam p k o f tlw po iuts usc•d t o d <'l<•rmin<' t lw l<'lliJ><'raturc drift ou . . . . . . . . . . . . . 168 5. L-1 T h r kin k at ~j a nd TDAS in a drift. d a t a file 170 r iI h N s id <' of a pulse• X ll 5.1G F itting the tE'mJwratun• drift on a pulsr (a) w it h a sm a ll <'tTor rstimat<' a nd (b ) with a large ('rror estimat<• 172 5.1G Pulse• srlcctio n .. 173 5.17 Binning Ill<' data 174 .). l 8 Re•Jn·c·se•nt a tiw h<'al capacity data at various Q 175 5. 19 T l1<' c•xpcriment al s ing ular 1\:a pi tza n•sistance . . 177 G.20 A sdt<•mat ic dnnving t.o illus trate the• <•rror in 6.T dlH' to choosing t he wronp; poiut whrn t.IH' l<'dgcs brforr aud after a pulse• haV<' different s loprs G.l 179 Expe•rimcntalmeasur('ments o f Cq with Q = 3.3 Jt\\" fc m 2 plotted with t I word ical pn•diet ious 18 1 =QfQ,.) G.2 T il e• throrrt.ical scaliug function s, f.d!J G.3 T h<• sca l<'d he•at c·apacit.v data compan•d to the th(•orc• tical scaling func-- 18 1 tions, .f.J, (g) . . . . . . . . . . . . . . .. . . . . . . 183 G..J A plot to <ktNmirw tlw linrarit~· of th<' sealing data l8G u.<>r: T hr heat capac- it~· d:-~.ta scalPel w ith an e•mpirical value• o f Q,. compar<'cl {~ to thP theoretical scaling functions, .f.d {}) 188 G.G The empiric all.v detrrmiuecl temp<'rat urr of s up r rflnid brrakdown 189 B.I T h<' SQUID filtNs 198 B. 2 The• dirncnsious of I h<• m aguc•t 200 B.3 Tlw ,·acuum can lid . . . .. . 201 XIU List of Tables 23 1.1 :2.1 No t a tio n coitVC'Ilt ion s . . . . . . . . 29 2.2 Eq uilbrium and llOtwquilibriutu critical exponents 40 3.1 HRT specific-at ions . . . . . 91 3.2 Cu rrC'nt sourc-e' SJH'cifka t ions 110 B.1 Heater ,·alues . L9-t 13.2 GRT loca.tious . 190 13.3 GRT 1 calibrat ions 196 13. 1 GHT4 calibrations 197 8.G SQUID fill<'r resis tor val ue's 199 B.6 \'acnmn can lid lto lC's and bo lt c ircl<'s 202 C. 1 A catalog of thC' data runs and eommC'n ts on t.lwir noiS('. 2()cJ C.2 A catalog of th<' data nms and th('ir anal ~·sis param<'t c'rs 20.) 1 Chapter 1 Introduction T IH' SIIJH'rfiuid transit ion in 1H<' is an id<•al (<>sting grouud forst udi <'s of' critical phcno nH'liH. It is uniquely s uit<'d fo r this <'llt<'rpris<>: it is a t nmsit.io u in th<' liquid stat<' , so it is not afkcted by strains or <i<>f<'('ts t h at atfrct transit ions in solid s ; it occurs a t low trmJ><' raturC' , so mos t impuriti<'s in t h<' sarnpl<' a rr froz<'n o ut ; a nd finall~', 1)('U UI SE' h elium has a larg<' th r nnal conduct iYity, partic11larly in t lw Sll])('rfluid phasr, its tllPrmal relaxation I itll<'S ar<' <'xtr<' nw ly s h ort , making it possiblr to obtain nwas ur<'BH'ul s al cquili bri um 0 11 rt>aso n ab l<' tim<' scales. The lambda transit ion is , in fact, th<' sharpest coop<'raliw transition disco,·e red so far [1]. As a r<'Sttlt. it is probably t.h<' l><·sl characl<'rizf'd ph as<' trausi t ion o u Parth . And in spac-e: exp<'rim <'nts l o meas un' fini l<' s iz<' <'ff.ects (Co11fill<'d Il<'lium Expcrime nt- C H EX) and t h r heat ('apacity of 1 H<' (Lan thda Point Exp<'rillH' nl LPE ) [2] wen' recently flowu aboard t lw space s hut t l<>, r<'mOYin g tlw OJH' inho tllOg<'Jt <' il _v <'H<:ouut crcd i11 cartlt- bo uud sampl<'s t he grav itationall~' incittC<'O pr<'ssur<' d('p<' nd<'JH"<' of tlH' local transitio11 L<'IIIJ><'rat Ur<'. T lH' rC'sul t s of this <'xpcrin)('nt W<'r<' in r <'ntarkab!C' agr<'<'ffi<'nt with thro r<'l ical pr<'d ict ious. A bow t l)(' s upcrfluid transit io n t.<'m])('rat ur<', T>., 1 H<' is a normal fluid wit It a finitr t h C'rmal coud uctiYi t~·, so t h <' applicat ion of a h<'al flux (T<'at<'s a non<'qu iIi bri 11m stat<' iu which t h<"n' is a temp<'ral ur<' gradi <' nt in t lH' sampl<'. B0low T>., 1H<' is a SU])('rfluid with tH'ar-iufiuite t h erm a l conductivity. Ou t his s icl<" of t hr transit ion it is lhrrrfore possihl<' to a pply a h Pal flux while' m ainLa i11iug a uuiform t.cmp<'ratu r<' throughout tlH' sampl<'. This turns o ut t.o <-r<'a l<' a u uutts ual situation in whic h th<'nuod:vuamics ma.v lw tU-i<'d to d csnil H' a syst em t h a t is not in equilibrium. Tllt' lH'at flux Lhf'rcfore n<'at<'s au <'XcdlC'nt system iu which Lo explo re t h e effects of dynamical coudi t.ions on pitas<' t ransitious. T IH'r<' haw' h<'<'ll a muul><'r o f pr<'dictio w; abo ut the way that a coustaul h <"al flux will a fl.<'ct t h <' s u p0rfluid transit iou . Jt is <'XI><'<"tcd t hat Q will depress the Lrausilion t <'lllJWra tur<' [3 7]. and tha t it w i II s ig n ifinmt 1.\' change t h<' t IH'rmodynaruic- properties 2 iu the vicinity of the transitiou. In particular , it has been predicted that a co11staHt heat fl nx will cause the lwat capacity to diverge at the d epressed t.ransi tion temp er ature [8, 9]. It is expcct.<'d t!Jat it will diverge fa r mor<' s tron g ly than i ts u ear-logrithmic lwhavior in thr absent(' of a beat. cmTeut, and that. this divergcncc occurs whik th<' s uperfluid d<'nsit.y n ' mains fiuit<'. Although there has b een a con sidera ble amount of theore t ical work on t his topic in rcccut. years [4. G, 9- 18], the ph_vsics in thr vicinity of the non-eq uilibrium transition is far from unders tood. Experimental m easurements [19] indicate t hat the temperature at \Yhidt SUJ)('rfluidity vauish<'S is , iHd<'e<L depressed by Q, althougl! t he magnitude of the drpr<'ssion diffe rs sig nificantly from that predic ted by c urreut theories [12, 18]. Tlw first m<'asurem<'nts of t.lw heat capacity compris<' the work of this thes is. One might well ask >Vh.\· such a strong di vergen cc has on).\- now h<'cn obsC'r-v-<'d. After a ll. over the course of t!Jc pas t six ty .\'Cars, there haw' bren nume rou s rxp rrimcnts investigating thC' physics of helium under h eat flo'>v, a nd it would seem t hat if there were a huge anomaly iu the s p ecific heat , one of these experimen ts might well havc stumbled upon it. Th<'re are two main reasons why t his did not happen: 1. Although the h eat capacity diverge nce is expected t o h e quite strong. the t emperature range over which it is predic ted to b e different from t hat of the us ual static trans ition is extremely small. For example , for a heat c urren t o f 10 fJV\' jc m 2 , th<' transition temperature, T'c-(Q) , sl1o uld be depressed by only 1 .1 ,,x, and tlw heat capacity sho uld only be measureably euhaucc•d iu the reg ion wit hin approximatc•ly 1 1tl-( lwlow Tc(Q). Th<'r<'i·o re , in order to con- clusively observe tlw <'ffect o f a hC'at current on t lw lwat capacit~', <'XtrC'mC'ly sensitive thermomC't r:v is <'SS<'ntia l. Thermonwters of this cali h<'r wcr0 only d rvelo ped in recent vrars [20. 21] for LPE. T h C's<' ins trum0nts havc cvolvcd to the po int w he re they a r c IJOW limited o nly hy t h r th<'rmal fluc t uations intrinsic to a 11y finite sample in coutact with a h eat reservoir [2 1, 22] . 2. Und0r llJos t conditions, the heat capaci ty etlha u ccniC'n t is mas ked by van o us otlH'r phcuom eua. v\'hcn high heat c urrents a re applied to 4 He , it becomes d issipa t in·. a nd t herllla l g radiPuts a ppC'ar tha t iutc'rfC'n' wit h a nwasrrn• nrc•nt of t he• IH•at capacity [23. 21]. \\'hm lo\\' IH•at c· urre11 ts an• appli<>d to surwrflu id 1 11 <'. t IH' t<·mpc•ra t un• range• of t he• <'fl'c>ct l><•c·orlles sma ll<'r th an the variat io11 of t Ire· transit io n l<'lllJ ><'ra t Ill'<' t h ro up;ho ut t lw ('(•11 d ue· to p;ra,·ity. T lwrefon•. in ordC'r to !-.C'C' C'dderH·C' of I IH' prc•dictPd ·st rong din•rg('uc:C'·. it \\'as nc•c·ess<-H)' t o t a k<' o ur da t a [23] in a u exJw rinH•nt a l cell t hat was IC'ss tha n 1 nu n high, ow r a ,·c·r.v specific ra nge• of Q. with tlH'nnollr<•lers t !tat h a\'(' a n•sol ut ion l><'t l<•r t !tau w - ro 1-.: [:W]. Befo n • proc·<•<'d i ug to t IH• SJH'C' ifi e t heon•t ira! pr<'d ict ions a nd <'X 1H'ri nwn ts that bot lr mot i\'at<'cl a nd c·omposc•d t Iris p rojc•c·t. I \\'ill fi rst out lin<' a fe\\' of 1IH• essential r<'lation!-. and con<·<•pt s that will he li S<'d in subs('(pH' ll t dNi n lt io ns a nd disc llssions. 1.1 Superfluidity and the two-fluid n1odel Soon aflN lrdiu ru \\'as liq uifi <'d in 1908 h~· Ilc•ikc• Ka nwrl ingh Onnc•s. it lwnwu• c·lc•ar t lra t it had a uunrlw r of st ra nge• pro pN t ies th a t a n • no t shar<'CI h)· ot lr <'r su bst a nC'<'s [27]. For o ne•, IH•litllll rC'flls<•s to fn•c•zc• uuder it s sat nnttc•d vapor JH·c•sstll'<'. In fact, it is t he' on )~· suhst aJH'<' that r<'ma ins a liq uicl a ll t Ir e \\'a.v d o\\'n to absolu t<' zC'ro. It is JH'C'C'Ssary to appl_,. pr<'ssttre!-. i11 <'X<"C'SS of 2,) at mosph N<'!-. to prod II<'<' t he soli d plras<' (Sf'<' F ig. 1.1). C Jl(l('l' a J>t·<•ssun• o f 1 at tllosplrNe, .J lit• Iiq u ifics at 1.21 K. \ \'lu•tJ it is cool<'d fur l h<·r. it uudergoc•s a uotlH'r phase• t ra usformat io n from OIL<' Iiq u id stat<' to a not IH'r. This t ra nsition i!-. accom panic•d h.\' a n<•ar-logrit hrniC' div<'rg<'IIC<' i11 t he• sp <'<"ifi(' hc•at t lr <~t rc·sc•mhl<•s t he• Gn'<'k lc•t IN /\. T lri!-. cun ·c•. s ho\\'n at Yarious n•dtl<·c•d tPm pNat llr<'S in F ig. 1.2. is what gin•s t he• t rans it ion its na llH'. T il<' I rausitio n oc·c·11 rs at a tc'III JH'rat ILJ'C' 1).. = 2. I 7()8 K 1 uudn sat ural <·d vapor pn•ssun• . T lw s ta t,c• Hhow T>-. is know n as Tlc• 1 a ud t he• state• IH'IO\\' T>-. as HP II. \ \' lril<> He• I is a norma l ,· isC'ous fluid. lie• li is a SIIJH'rflnid t hat c•x hi bits a uumbc•r of lllllts tutl <"hararteri st ics. It lras t h<' p <'C UIiar a hili t~· to flo\\' "·i t lrou t dissipat ion . and it 1 T>. = 2 . 176 K 011 tlu• Tr10 l<'lllpPral u re seal<'. i\ lau.'· n•fPr<'HC('s quol!'d in this lhPsis us<' tiH• Tr,t>. valu<• ofT>. = 2. 1720 1\ . H PfPr <'II!'P [30] has <"Oin p il<>d all I IIPasu n••n<'••l~ of th<' equi librium all( ) t r a nsp ort p m p Prt i!'s of 1 1t!' k now u a t th<• ti me• of p nhlicati o n , a nd convPrt <'d ti iPm to til<' T;Hl sndP. 40 Solid He 30 Meltmg § "'a) ..... 20 :l !/) 1/) ~ L1qu1d He I L1quid Hell 0... Cnt1cal p01nt Evaporation 0 1·0 He ga.., 4·0 3·0 2·0 ~ Temperature . ' K Figurf' 1.1 : T IH' phas<' diagram of 1lle. London [29] .) (From Galasic•wicz [:28]. page' 7 aftpr 24 20 I I ~ Oil 16 s "' 0 .c u 12 I::: ·a & en 8 4 0 - I 0 (T-Tl)fK -4 -2 0 2 (T- Tl)ll<r3K 4 6 -20 -10 0 10 20 30 (T-Tl)/10-6 K Figurf' 1.2: Thf' SJWC'i fk lwat of 1TI f' und<'l' t hr sat u rat <'d ,·apor pn•ssurc as a fuucl iou of T - T >... ( Front \\'ilks and 8Ptts [:31 ], pag<' 169 aftc' r 8nC'kinghmn and Fairbank f32].) .) has an <'XI r<•mel.'· larg<' th<'I'mal c·onduct i,·it y. In add it ion. cxperiJn<'uls investigating I ht• ,·iscosit~· of JlC' II giw diffNC'nt rC'stdt s d<'JH'nding on tlw wa.\· ill \\'hi('h they arr pNfornH'(I. .:\[C'asur<'nwnt s indica!<' that it can flo\\' through ('apillaric•s or small ho l<'s \\'it !tout rrsistalf('<' alld thc•rc•f'on• has 110 ,·is('osit.'· at a ll. ll o\\'t'\'t'r. il' au ohj<'cl is dragged throu~h tlw liquid. it <'n('ount <'rs a smalir<'sislcutc·c• that ('hangc•s as a funC'tion ol' t<•mpc•ratun•. impl yi ug that it has a fiuitP , albeit small, Yis('osity. This strange iu('ollsist <'II<'.'' nut I><' cxplaiuC'd b~· t lw two-fluid lllodrl that was first iut roduc·c•d in 1938 h.'' Tisza [1:1]. and <·onsid<'rabl~· llJOclifi<'d SC'\'Nal ~·c•;us later b.\' Landa II [:3 1. :15]. In this ('Ill piriC'a I cks('l'i pt.ion of Sll prrAuicli ty, ll c II is ('()Il l posed or t\\'O c·ontp l etc·l~· intc•rpc•n<'lratiug, insrparahlc· fluids. Om•. the uonnal fluid. possesses au ordiuar.Y ,·isC'osit .'·· and tlw ot ltc•r. tlH• so-<'alled superfluid. has no ,·is('osit .' · at all. TIH• llonnal fluid has dt•nsi l.\' p 11 • vcloC'i ly v, , a nd c nt ropy S. The• SII JH'rfln id has clc•nsit.' · p, and ,·eloC'it.'· v ,. but ('arri<>s no <'nLropy. TIH• total clt'nsit.'· p of I hc• h<'linm is s illl ply IIH• sum of its t \\'O S<' J><U'H I c· d<•11si I i<•s. ( 1.1 ) \\'hilc• the• total dC'n si ty is ntorc• o r less iud<'pcndc•ut of l<'lllpc•ral ure. and is Pq ual to 0.110 g/('Jn ;J [:30]. t ltC' SUJH'rlluid and nonual fluid dcnsiti<•:-. dmng<' a:-. a function of t<'lllj><'ralur<' . Tlw totallllas:-. c·urrent dt•JJsity of tiH' hcliulll. j , C'an l>C' c•xpn•ssed in t C'l'lll:-. of ( he• ('lllT<'ll(S of the I \\'0 fluids j = {JV = fJ 11 V 11 + fJ,,V ,, . ( 1.2) \\'h<'n lwlimn is forC'C'd to flo\\' through Sfll all C'ha nn Pls. th<> nonnal fluid is h('ld to t.lw walls h:v its viscosity, whil<' tlw supC'rfluid flows thro ug h >vvit hon t impc•clc•nce. Au o bj<·<·t drngp,c•d through the· h<•linm . howc•v<'l'. <'nco untNs tlw rc•sistanc<' o ft h P normal fluid . Expc•rinwnts of th<· se('ond kind <'Hil th<'l'<'fOr<' I><• p<'rfomH'd to d<'t<'rlllillC' the dt'JH'nclc•JH'<' of p, and p8 on t C'tH pNa t ur<'. sort. JH'rfonnC'd b~· .\nclronikasln·i li 1\ torsional osci II at o r c•xpC'I' inH•nt of this [3G]. indicate's I hat the norlllal fluid fraetiou. p 11 jp. is c•qual to unity at tlw Sli JH'rfluid transition, but denPasc•s to ,r.c•ro at absolute 6 ~-Point 0.5 1.0 1.5 2.5 2.0 Temperature (°K) Figmc 1.3: T he supcrfiuid and uormal fluid densities as mrasurcd in Audronikashvili's experiment. (From Dounelly [37], page 17.) zero. Thr results of this experiment are shown in Fig. 1.3. Ncar th r lambda point, thr supcrfiuid density vanishes as Ps = Po 6, wlwr<' ( ( 1.3) .3 = 0.671 IS a universal cxpoHent. p0 = 0.38 gjcm·, and t is the reduced temperature, (1.4) A t lworetical basis for thf' two-fluid model was provided in 1941 by Landau [34, 35]. He dc'scrihcd supNfiuid helium as a weakly <'xcit<'d q uant um system , in which a gas of excitations , or quasi particles, movcs within a backgroull([ fluid that lies in an unexcitcd state. At absolute zero, the helium consists C'ntirely of t his perfect background fluid and flows wit hout dissipation. At higlwr temperatures , excitations are creatcd that can collide with each ot lwr and with the walls of thc container, causing t he fluicl to exhibit some of the properties of a norma l fluid , such as \·iscosity. Thc>se cxcit.a- 7 16 ~ 12 ell ~ .... >. 8 0> '- <ll c l.U 4 I I I I I I T=1.1K SVP I 0 10 20 Momentum,p/h (nm- 1 J 30 Fignre 1.4: He II c'xcitatio n spectrum obtained from neu tron scatt ering exp('riuwuts. (From Tilley a nd Tilley [41] , page 44 after Henshaw and Woods [40].) tious an' colkct iv<' <'xcitation:-> of tiH' system. rather thm1 individual excited particles . lu the two fluid wodcl. tJw normal fluid is ickntificcl with thC' gas of quasiparticles. while til<' surH'rfiuid is identified with the' pcrf<'ct background fluid. Landau also pn'srnt<'Cl t hr essrntial fcaturrs of thr quasipart icle energy spectrum. shown in F ig. 1.4. based on phenomenological arguments. He showed that at low energies the dispersion cu rve is a straight Jiue a nd the excitations are phonons. At higher energies. the curve reaches a local maximum before dipping down to a localmiuimum. T he excitations in t he region of the minimum arc called rotous, and tlwy are the ouly other major con tribut ion to the thermodynamic behavior of s uperfiuid hel ium. The thermodynamic fun ctions can Lw calculated very accurately for t his distributiou , except in tlH' Yiciui ty of tlw lambda point , whf'r<' the quasipar ticle density becomes C'XtrC'm dy high [37]. Landau 's spt'etrum was later eonfirmed by neutron-scattering experiments [38 40]. 8 1.2 Heat transport in superfluid 4 He As pr('viousl:v !Il('ntiotH'd , s uperft uid 4 l-Ie has au extremely larg<' thermal cond u ct ivit~·. \\' hen appli('cl heat cmr(' nts an~ small. it is impossibi<' to set up any t('rnperat ur(' gradi ('nt s w hat sor v('r within t h(' hulk liquid . In this limi t, its thrrma l conduct ivity is ('ffcc tiv('l~· infiai t('. H('at flow of this kind must cl ear]~· ))(' attribu trd to a differrnt procrss than standard conduction. It can, in fact, h e cxplain ('d by the two-fluid model. \Nhen heat is applied to helium , the normal fiuid , which carries all the entrap~·, flows frow t he heated end of the cell to the cold end. Because the total density of the liquid rem a ins cons t a nt. t hC' superfluid cornpouent fiows in the opposit<' direction , towards the bea ter. When il reaches t he hot end of the cell, it heats up. conv('rts to normal Auid. and then r<'!,urns one(' agaiu to th<' cold ('nd. The b('at curr<'nt c\c>nsity is Q =PST V n . ( 1.5) \\'heu ther e is no net mass fiow , j = 0, allCI so according to (1.2). p71 V 71 = -p., v 8 • T his implies tha L Q = (Pn + P.~) ST V" = Ps ST (v n - Vs) · (1.6) A hea t current thrrdon' S('ts up a counterflow, W = V n - v B, between thr snpcrftuid a nd the normal fluid velocitirs (srr F ig. 1.5). 1.3 The static superfluid transition \i\' hen it was first discovered , the s uperfluid transtion was compared to t he condensatiou of au ideal Bose gas [29] . It was thought that below T>., t he helium atom s start to p opulate the gro und state of t he system , forming a condensate. It was this compa rison t hat h-ad to T izsa's development of Lhe two-fluid m odel. T he a toms in Q Cu . N ormal Com ponent tilll.illriiiii••••••••Ciiiult Superfluid Component Q Heater F ig ure 1.3: Heat is supplied at the bottom of a cell of H<· II . establis hing a coullLcrflow H' = V 11 - u.. in tlw superfl uid . tl1(' condensate arc in a siuglC' quantum state, so a ll that is necd<'d to describe t.h<•m is a siugk 'cond ensate wm·c·- function ', ( 1. 7) \ Vhen• 1]1 is the Dose field operator and no = 1/·5 gives thC' num ber dcnsit.Y of atoms iu t he condensate. Alt hough this was a compelling s uggestio n , t he lambda transition is clearly not a dirw·t manifestatio n of t h<• coudcnsat.ion of au ideal Bose gas. In helium , t he forces hC'tWC'C'Il the at.oms havC' a s igu ifi caut d f<•ct 0 11 Lh<• behavior of the fluid, and cannot hC' ignorwl. Theory indicate's t hat t hC'SC' iut er actious dC'p lC'tC' the condensate, so t hat C'VC'n a t ahsolut<• zero o nly abont ten percent of t he h<•lium atoms occupy the zero mo mentum stat <' [-12, 43]. Because t he lambda trausitimJ is second ordrr , it c:a11 be dC'scribcd , at least qualitativP I ~', h~· the Landau t heory of phase' t ransitio ns [44] . Its ch ange' iu state u u1 be 10 <ksc-ri IH'd in tNms of a n ord r r paramrt0r that rharactrrizes its distaucc from criticalit~·. For t lw lambd a transi tion . t hr ordrr parameter is identified to b <' t h e condensate wan'- functi on drfined b~· ( I . 7). T lw s u JWrfl uid motion is a respon se to order parame ter phase twists, a nd o n<' h as t IH' rela tion: v .. = - h \ hp, ( l.8) 'II t I whN<' m 1 is tiH' 1H0 atomic mass. The' cha nge in frr0 rnc rgy ~F = &P.- V~ for s m a ll u., then s0rv0s to define t hr r quilibrium p_.. On t hr oth er h and, the conde nsate drnsity n 0 is in grnrral an independent th r n11od.\ ·na mic q ua n t ity, unre lat<'d to fJ., . ~!any o ft he elem ent::, oft he t ra ns i l iou can lw deriwd by t h <' application of Landa u ut<•an- fi<'ld - t IH'ory (~ 1F T) iu the form of a pow<'r-law expansiou oft · a ud it s gradirnt s. T his d<'sc-ription assum<'s that t h <' s.vst<'nt can I><• accurat0Jy d <'S<Tilw d by the• <W<'ragr vaiiH' of t h<' order param et <'r. a ud tH'p,lc•ct s a n.v fluc t uations a bout t hi s m f'all valtH'. ln SIIIH'rfhrid lu, linnr , fluctu ations an' alwa~rs important. and tiH' n' is no t0mprraturr ran g0 for w hich the l\IFT is s trictly valid . As w it h oth er critical point, tran s it ions, t lH'S(' flu c tuations lead to s ingulari tiPs in m a ny properties of the s.vs t e m near t he la mbda p o int , and must be t r<'at<'d t h ro ug h propPr appli cation of critica l scaling laws. r<' uonuali zation group ( RG) t lr<•ory. a ud o t h pr theor£'tica l tools . 1.4 Heat flow and the superfluid transition T h£' a pplication of a lwat flux significant ly a lters the la mbda transition. As was previous ly m£'nti o ned , t he m ost o uts t a ndin g o bservable ch a u ges are predicted to be a d Pprrssio n o f t he t ransitiou lr mperat ure a ud a s trong divergence of the spcci tic heaL. T lw qualitative feat ures of thes<' two e ffrcts can b e derived fro m extremely simple arg uments. 11 1.4.1 The depressed temperature of superfluid breakdown If a heat flux, Q. is a pplied to the bottom of a C('ll of superftuid hrlium, a nd heat is H'movrd from t hr top at an equ al rate , tlH' helium remains isoth ermal, and a cmmtnfiow is crN\te~d lwt.wee'n t he stqwrfl uid and t he normal fluid. For a dosrd cdl thrre is no nrt mass flow , so accordin g to (1.6), Q = Ps STW . vVr can lwncefor t h drop thr vrctor notation ,,·ithout nmhiguit_\·, sincr a ll of tlw motion in th is parti c ular casr is on ly in Oil <' direction (se<' Fig. 1.5). The presence of a couutc>rftow gives the s.vstcm an extr a degree of t hermodynamic freedom, so it is thNefore expected that th e superfluid density, p 8 , sh o uld be a full(·tiou of TV (at least at larger H ') as well as of tcmprraturc and pressure'. There is little' rxperimcntal dat H t.o C'l ucidate the clepeudeucp of Ps on TT '. The only cxp('riment to date' is a measurement b.\' Hess [45] that took place well he! ow T>.. His results ar<' consist<'nt with a calculation of the dependence of Pn (an d h.\' extension Ps) on n· derived from t il e s pectrum of c>lemc>ntary excitat ions [34, 46, 47]. In t h e v icini ty of thP lambda transi1ion, w hNe no Pxperim Pntal meas ure ments exist and t>lemeutary excitatio11 t lH'o ry is expected to fail. w0 must re i.\· on phase transition t heory. Both l\1FT [48] a ud RG theor~r [G] predict 2 that Ps will be sufficieu tly d e pressed by lr that th e quantity q =Ps(lr)W will eventually decrease with iucreasiug H' (sec Figs. l.G and 1.8) . BetaUS(' S and T arc more or less coustanL in the vicinity of the transition , Q <X q according to (1.6). This implies that for a ny g ivrn t('mpC'rature, thrrr is a maximum h('at cu rrent beyond which sup('rflow b rcomes unstable. This lwat cmrcnt occurs at the JWak, when (8Q/Dll")T = 0. Vle may thcreforr d<'fine a set of points in thr T - Q plane' t h at acts as a boundar.\' beyond which s uprrfluidity cannot exist. We label this curve as Tc(Q) , or Q,.(T). The fact that Tc(Q) < T>. follows becausr p -' (ll") > 0 at t ltis point. For experiments in wh ich thr h eat flux , Q, is h eld constant, th e instability occurs w the followin g wa.\'· Far below T>. , t he supcrfluid fraction is re lative ly large, so an appliPd hPat flux res ults in a sma ll counterflow velocity. As the temp erature of 2 Tlw form of p,( !;l ' ) for these t lwories will be ex plicitly di scussed in Sec. 1.6.1. (a) W (arbitrary units) Figur<' l.G: (a) Ps depn'SS<'d by IT". eakulatE?d using m ean-fif'ld thC'or~·; ( h ) tlH' n·sul t iug \·alu €' of q = p,, (Tl')W . l3 inc reasing T J, W (arbilrary units) Figu n' 1.7: F ixed Q, cha ngin g T. T hick solid lines : tlw func tion q = f> 8 (Tl ', T) ll ' for Yario us Yalucs o f T. D aslH'd lin<': T lw Yalu e o f q = Cft>xpt = Q /ST fix<•d in th e <'XJH'rinH' nt . T IH' inters<'ction of a g ive n q(T) curve a nd fJr.x pt g ives t h e coun terflow v(' IO<'it.y indllt<'d by t he a ppli ed ll ('at Hu x Q at t hat t ('m])('rature T. V\' he n T > T(., no valu<' o f \I' satis fies Q = STps( \1 ' ) 11 ', a nd s upc rfiow brea ks d owu . t lw h<'litlm is raised . th<' s,vst<' rn w ill n'act by d err<'asing t he s u perflu id d eusit .v a nd increasi ng t lw coun te rflow w loci t ~·. 13<'<"<UJ S<' Q is fixPcl . i t must do t his in s uch a way t ha t t IH' product of t h(' two n• m ai ns ('Ous t a ut. \ \ ' lwn the t PlllJWrat ure i~ raisc•d t o ~.( Q ), the s~·ste m reaches a point wh en' it will not be able t o in creas e Tl ' w hi le> k<'<' piug Q con s tant. T his h appens wh<>u fJ.~ is s ufficic•ull y d epressed by llw coun t<' rflow t hat a ny fu rt her increase in \I' will c·;uts<' t he product p 8 ( H') n · to deneas<' (s<'<' F ig. 1.7). A t 0 .(Q ) LIH• lwa t flu x c·;w 11 0 lo ng<'r IH' ca rri<•d b,v a c:ou ni<'rflow, a nd s tq H'rflu id i t.v w ill bn•ak clown . 1.4 .2 H e at capacity divergence T he• eli v<'rgeuce of t he heat capaci t.v ('au lw u nders tood i n s imila r t<•nns. . \ s t lw t <'lll pcrat un' is ra ised . t he s upcrfluid d C'ns it ~· decreases. a nd t lH' c·ollnt crflow V<'locit~· must increase in order to carry the ltrat <·urrrnt.. Thr kinetic <'Il erg,v associated with this couutNflow is rqual to 1Ps n ·:~ = (p 5 ll"} 2 /2P.~· Since th<' quantit~· p_, n· is held fix<•d in auy <'XP<'riu Jent in whicb Q is fixrd. an.Y drcrease in fl.~ leads to an incn'as<' i11 t lw ki n<'( ic <'nergy. T IH'rC'fOr<', an increasr in t. hr temperat ure of LiH' systrm is <HTompan ied b~· <UJ inn<' aS<' in t hr kinrtic eurrg_\·. Siuce tlw heaL capacity. C Q· ts si mp!~· th<' amount of <'IH'rg~· that mus t be added Lo raise t iH' tcmperaturr of thr fiuid at fix<>d Q, then' is aJL euhaucewcnL of LlH' heaL capacit.)'. In fact , lwcausc p_, is d<•prcssed b.v I ( ·, the counterflow v<'loci t.y must i JHT<'asc b~· Ill Ore and more to ker•p Q steady aH the tcmperatun' is iun<' llH'ntall)· rais<•d towards Tc(Q). An increasiug amount ofeuerg.v is then.fon' rcqu in•d to r ais<' the t<'mperaturr. and t.h<' hrat capacity ult imat<•ly divrrg<'s at tIt<' iu stab ilit~·. TIH' n<'<"<'Ss ity of tlw div<>rgcncr in thr IH'at capacity cau be put o u a bit finnN ground w ith t.hr hrlp of t.hrnuodynalllics [8]. Th<' prcs<'ll('(' of a CO liJJ(<' rfl ow in supNA.uid IH'li11lll introdu<·<>s a uew pair of coujugat<' variabl<'S to til<' th<'nnodynamic d rsni pt. ion of t he system . T his u<'w pair consists of n· and q. whrrc H · is t h c eountNflow ,·elocity, and <J = p 8 I(' is Lh <' s llJ)('rfiuid •nomcnt.um in t.hc• norm<'\ I fluid frame . The frcr <'IH'rgy at coust ant ckns i t .\' C'<Ul t hen h<' writ ten dF = -srlT + qd ll', (1.9) w hrrr s is t IH' entrop_,. d<•JJsit~·. As wr ha\'(' s<:'eu before, keepi11g Q = STp8 ll ' constant is <'quinllrnt to krcping q constant. At cons tant q, t h<' propN fre<' energy to us<' is <I> (T, q) = F (T, ll ') ~·idding dci? = I( ' q , -sdT - (\ 'dq. Th<' molar heat capacit~· at constant Q is: Cq = - T 1 . (():[)T'l. ! <!?) Q , (1.10) wiH'r<' \ · = 27.38 cm 3 / m ol is thr m o lar Yolurnr. ln ord<•r to s('<' w hy this qu an tit~· nrc·<'ssaril~· clivcrgrs. w<' can e mploy a proof that is identical t 0 t hr Oil (' that rclatrs (",, and C1 ·. show n i 11 a ll standa rd t iJ<•nuodynamics t<'xts [-!9]. \\·<,start \dth thr <'ntrop~· drns ity s (T, \l '). all(] find that 15 Q.<;) ( ():; ) . ds= ( DT 1\ " dT + Dll" T dH . {1.11) Thn<'for<'. C Q = T\. (Ds) = C · T \ ' (~) (DTT') DT + Dll" DT Q T ll Q . {1. 12) From (1.9), W<' obtain t h<' i\ laxwell relatio n Dq) ( D:; ) ( DT 11 " = U\1 " T' (1.13) a nd the <'X Pr<'ssiou for t IH' h0at capaci t:v lwco mcs cQ = C 11 · - T \ · ( Dq ) DT 11 • (on·) . oT " w h<'r<' we h<W<' used t h<' fact that fixin p; Q is cq ui,·alent to fix ing q. We now simply <'mploy th<' idc'ntit~·. ( Dq ) or aw " ( D\oqI" ) T = - 1. ( DT ) 11 . {1.15) t o find ( 1.16) \V<' haw a lr<'ady sceu lh a t s upNfiO'v\' brc•aks d own when (8q/8H '):r = 0. T he fact llt a t this quantity appears in tlw dc'nominator of ( 1.16) implics t hat CQ a lso di verges at this poiut. T hi~ resu lt wi ll I)(' t ru<' so long a s (J 8 is dcpr<'ss<'d s ufficient I.\" by n · t hat lJ = Ps(ll ") l\' <'V<'utuall)· d<'<T<'as<'S with ll ' [sec' F igs . 1.6(h) a nd 1.8(b)J. 16 1.5 The exponents related to the superfiuid transition under heat flow The exponent of the transition temperature depression 1.5.1 For finit<' n1hws of Q, ~. (Q) li e's below T>.. :1 As Q approadt <'S ZC'ro. T,. (Q) approaclt<'s T>.(Q = 0). In orcl<'r to detenniu<' a m on' precis<' relatiouship lw twr('J1 t hesr two t r mprratur<'s , wr call c rnplo.v scaliug argutii<'HLs to d<•t<•mtiuc the s ing ula r bc•h<wior of t ltr heat fJ ux [I]. Th<• idea bchi11d t)l(' s('aling hypot h<•sis is that a ll singularities t hat occur iu t he Yiciuity of t lH· nitical point an· cau s<•d b~, flu c tuat ions that an• corrrlatcd ow•r a kugt h of ordN E,. As thr s~·st<•m approa('hes thr transition. a nd E, ---7 , minoscopi(' 1<-n gth s<·alc•s arc no long<'r important. Tlw o uly le ngth that matt crs is E, . ?\JC';.n T>.. Q = fJ 8 ll' 8T :::::- - {J.- 118 ST. Si nce S all(! Tare constants , the only t<'t'ms t h at <'X hi bit s i ng u Jar lw haY ior an• (.1., and L\. For t he• purpos<' of ('alrula t i ng <•xponents, \\'(' m ay therefor<' write CJc rv (fJ.-c l'.~c ) / l'sr·· T lH• lllllll<'rator is a siug ul ar t<'rm in th<' free <'n rrg,v d011sit,v that is iuwrs<'ly proportional to th<• ('<HT<' lation volunt<\ '2 . p_.,.l'w· E, - d } rv -2- rv . ( 1.17) T h e s('a)ing prop<•rti<•s of t h<' d<'Jtominator can b <' d<'trrmined from the properties of tlH' s tqH'rflu id order paranwter. According to ( 1.8) a nd ( 1. 7), (1.18) T h en•fore. u.,c has tlw ch aractP r of an im ·ersc• lt> ngt h. Si n c·c• the' corr<'lat ion length is t h e onl,v rele vant le ngth ncar the nitical point. o8 , . ""' E,- 1. Putting all tlwse rc•latious tog0l her, we find Q c rv r:. - dE, 1. T h e• corn•lation lc' ngth diwrges as t v, wlwr<' 11 = 0.6705 [50, 51] is the COITC'Iatiolt lc·ngth c•xponcnt and t is t lw redttc<'d t cmperat m e. T herefore. Qc rv t"(cl- tl . Invertiu g this equatio n and s ubs tituting d = 3. we find t h at :J Smnc of t lH• tPxt in this sPction appe<us in Goodstein , C hui , and ll art<'r [7). 17 (1.19) 1.5.2 The exponent of the heat capacity divergence lu t IH' \'icinit :v of tlw diwrg<'nc<'. (1. 16) indicates that CQ ,...., 1/ (DqjfJTV)r· \\'0 can <'xpand q about CJr· t hr \'alut> of lf at t he transition : 2 uq ) . . 1 ( Dq ) 2 q = qr+ ( ~ ( lr - TI ,.)+- o \\ .2 (II . - li .e)+ ... u \ I 11 ., 2 ., 11 ( 1.20) Sinn· (DqjDI \ ') 11 ·, = 0, a nd (8'2 q/Dl 1''2) 11·, < 0, t his implirs l hat Cfc- q,...., ( I l'c- 11')'2. a nd (DqjfJll 'h· ,...., ll'c- 11 ·. T IH'r<'for<'. for m<'asuremrnts at fixPd T < T>., (1.21) Fo r lll<'Ctsun•uH'Ilts at fixc•d Q > 0. wit h T,(Q) dcfinrd b~· Qc(Tc) = Q. one has DQ,. ) CJc(T) = CJr(T,.) + ( UT T=:t ; ( T- T,.) +. · · (1.22) Siuce DQ,.jDT is finite a t T = Tc , CJc - Q,...., Tc - T to h-ading ord c•r, a nd (1.23) when' 0 is LIH.' rr duc:cd t<>mperatur<' wit h r<'SPf'c t to Tc(Q). () = Tc - T - Tc . (1.2--1) T he hc•at capacit .v divNgruce at fixed Q is thc n>forc cousickrably s tronger than it s nC'a r-logri t hm ic bch a\'ior with uo heat ('U rrcu t. T his a JLa l.vsis nrglrcts the flu ct ua ti o ns iu T-1' , which an• prC'diC't<'d to divNp;<' at Tc( Q). The expouput wi ll be infl ue nccd by t hc~w fluctuation s. 18 1.6 Predictions for the magnitude of the heat capacity enhancement Th(' lll ap;nit tl<k of LIH' IH•at capacit~· <'nhanc<'rn<'nt can lw dNived from the expr<•ssion for t.lJ<• free <'ncrgy. 1 Front ( 1. 9), i 11 <'XJ)('ri llt <'nts w here t lw couutf'rflow vdoci t y is held constan t. tlH' change in th<' fr<'<' <'ll<'rgy is fo".p (ll '') II '' c/W '. ~F = F(T. II' ) - F(T, O) = 8 (1.25) Lik<•wis<', w iH'n Q is lwld constant , the dmBge iu tlw r<•kvant fn•c <'nNgy is ~<J> = <J>(T. q) - <l> (T, 0) = - l q 11" dq' = - 0 lq 0 I !!_ d</. Ps (1.26) In ord('f to us<' t hcs<' two <'xpr<'ssions to calc ul at(' t he change in the heat capadt~·, wC' llC<'d to know t IH' dq)('nd a nC<' o f p,. o n 1T•. 1.6.1 The depression of Ps with H · \\'(' wi ll cak ul at<· Ps(T. T\ ') u s in g thrc<' diff<'r<'nt th<'orics: the ~1FT [ 18], which w<' modify h.v us in g <'mpiri cal <'Xpo n<'n ts, th<' tp th eor.Y [52], and RG theor.v. 5 Thes<' t h<'ori<•s a ll start from a nt<'an-fic ld expansion. ( 1.27) which is writt<'u in t h <' fram<' of r<'fNeiH'<' of t h e no rma l fluid. Here n, {J, .J and AI ar<' <'Xpansion ('(H'ffici<'nt s, J/ is ZNO cxc<'pt in Llw 4 • th<•or~·, the tuanoscopic wa,·<' function is giwn by 4 • = 1.f'o e1<1>, a nd v 8 = (t1j m) \1 </> (equ al to th<' laborator.v fram<' n· = u,. - t•,), and m is t he mass of a 1llc atom. T h<· param<'t<'r c1 plays th<' ro l<' of th<' r<'duccd t<>m iH'ratun>, with the equilibrium lll<'<Ul fi('ld transition to SUJ><'rfluidit:v 1 Son &c of th<• tPxt in thE:' u<•xt spction a ppear in C hui. Goodstpin . Hart<'r, and l\Iukhopadhyay [8]. r.wc will cxplkit ly us<' t h<' rPsults of HD [G) in this discussion. l\lor<' r<'ccntl~·. Haussmann h as modifi<>d this t h<•or~· to oht a in a not hPr prPdkt ion for CQ that indudcs the cfrccts of vortic<•s. Th<' r<>sults of Haussmann 's t.lwory will also lw contparcd to om data. ' 19 OC'<"urrinp, whc•n n bc•coiiH' ll<'gativ<'. For a uniforlll 1'., Oil<' obtains (1.28) wh<'r<' k0 = 11/t'Jh. Th<' <'Xpn•ssion for t.•o( ll ' ) is ol>tairH·d 1>.\· minillli7ing P,11 r with r<'SJWct to v 0 . TIH' sup<'rfluid dpnsit.'· i ~ obtain<•d by ~~~h~titutiug v 0 (ll ') hack into F.ur and romputing p, = ( l /t\ )(Uf~ 11 r/Ut\) .\11 tim'<' tlH'oric•s giw p 5 ( 1T ', T) in th<' form f' ,( ll ·. T) = p,( H" - 0. T) j (H). lli<'HII fit' I< I t IH•or~·. for n < 0. OIH' has f. = (1.29) J-.// 2n. For rom parison with PXJ><'rinH'nt a) datH we• will us<' tlr<' substit utions E, = E,0 (2/) '' , whrn' f.o = l...J.:3 x 10 x Clll [33]. Thr rhara<"l <'ri~ti(' \'(']O('ity n·, ('fl l1 IH' rxpr<'SS('d as II ', I I ·o /'' . wlwr<' ll 'o = 112v/ mf.o = 17!).'-J 111 / S<'C'. 2 We• find that for ~ 1FT. p 8 ( l\" = 0) = '2n 2 1n 2E,"2j!1 rJ, f(t;) thi~ sanH' form for p,(l\" - 0). for v t lH'or.' ·· (1.30) wiH'l'<' I' = - 3Jfn j 3'2 is th<' S<"Hl inp, combi nat ion for JJ whirh \'allislws as n --t 0. indirating that :1! plays no rol<' IH'ar the tram;ition. For TID tiH•ory, .f(t-i} is giwu by <'quatious (3. 12). (C' ll ), and (C3) 111 [G], and is shown in Fig. l.R . . \II thr<'<' t IH•oriPs pn•dict that p, is suf!ieic•utly ci<'prc•ssc·d to c·ausf' su p<•rAow to l>rf'ak down. 1.6.2 The calculation of C 11 • ~<'XI w<' <"Ontpul<' tlw h Nll c·apa.C'it)' r nhaJH'<' IIH' nl using tlH•se <'xpn•ssions for fJ.-(\l ') . \\'<• first tn•at the• <"asc whc·rc th<' cotm tcrflow w locity, II' , i~ lrcld <·ons!Hnt. This 20 (a) 10 0 .. ~ "'.. ~ 0.5 00 ~------~------~------~--------~~ 0.0 0.2 01. K (b) 0.10 •::I"'· -~ ..... 0 OS .... / / / / / / .· / "" 000 00 .: Im fJ 01. 0.2 06 K = Figur<' 1.8: (a) T lw d<'J>I'<'Ssioll o f p, wi t It n II '/ l I '1 , wlH'r<' II '1 is a dtara<·t C'ris t ic v<>locity g i,·cn by 1\ ·1 = h / 111(.. cak ulat('(l using r<'no rmali..mtion group thPory. (Front Ilausstuaun a nd Dohm [G]. Fig. .) .) (h) T lw a ntplitud<' of t iH' s upNA uid <"Hrr<•nt O<'llSi ty, .J.~ fJ 8 /1 8 vPrsus ,., . Solid I iu(' : H<'HOrutal izat io n group t IH•ory. Dash<•d Ii tH': :\l<'an fi<•ld t))('OJ',\'. ( Fro 111 Haussm aun and Dohm [6]. Fig . .J.) =- 21 <'XpC'rillH'lll ha~ IIOt yN he•c•n pC'rformC'd. bu l t ht• ('XJH'('SSion for ('II is e•mJwdded within the• <'Xprc•ssion for l'cJ· thC' quantit .Y ttH'asun•d iu our <'XJH'ri mC'nt. FrotH (1.23 ), t hc· ('hang<• in t he• frc•c• <'Jle•rp;_v due• to the prcsc• ncc• of a c·o111ttNflow is: ~F = fJ.,(O) ll / {" .rf(.r) d.r. (1.31) lo Th<' IH•at ('Clpa(' it~· is r ha11p;e•d by 2 ~C -= -T\ ' (D ~F(T.I\')) DT1 II ( J.:32) II whC'l'e' { · = 27.38 c·m;1/ m ol is t IH' molar \'OluttH' . l 'si ng p_. (O) = Pot<. wlt<'r<' Po - 0.:370 g j cm:l [0 1]. togrliH•r with t lu• scaling rPlation (, = 11 = (2 - n)n. \\'C' obtain ~C'11 1° = - Ccw [3 (31/ - 1) fu".rf(.r) d.r (1.:33) w h<'r<' C'0 = \ ·fJo I I'd / TA = Ll3 .J / lllol l\:. For l\ I FT, this rc•d uc·c•s to For 1. • t heor~· and for liD t IH•or.\'. ( 1.33) is <'nll uatrd using nuBJCrkal dif[en•nt iat ion and int c•gra.tio u. Thc• results ar<' s hown in Fig. 1.9(a). C 11 approadws a finite• !'Oilstant at ,.,., = I!", / ll·, in a ll thrc•c• t he•orie•s. 1.6.3 The calculation of Cq T he• clive•rgc•ncc• of CcJ O('C'IIl'S l><•c·a usc• P.- is ci<'JH·e•ssC'CI by 11 ·. lloWI'\'C'r, thP lwat, capa<'it~· would sti ll be c•nhanl'ed h." t lw prc·~c·nc·c• of a h<'at flux <'''<'ll if p , wen• not d('tm•ssc•d at all. In this limit , p , = f'o t<, and J p 11 ' .2 = - - ] ~<]> = - - 2 ' 2p, ( -Q ST ):l = - - ] (-Q )'.! ,- <. 2p0 8T ( 1.35) 22 ..- 8 ~ 6 (J) ......... 0 s4 (a) - ............. '""";) I:S ...,_,) u lit <I 2 0 0.00 0 . 10 0.20 0.30 0.40 0.50 JC=W/ Wt ..- 100 ~ (J) ...... 0 8 80 60 - 40 0" 20 A (b) A ............. '""";) I:S ...,_,) u <I 0 0 .0 0.2 0 .4 0 .6 0.8 1.0 Q/ QcHD Figure 1.9: T hin line: HD t h cor~r. T hick lin e: l\1FT. T riangles: ~~ t h eory with !II = l. Dashf'd line: no fls deprC'ssion. (a) C ll' (h) CQ. (From Clmi rt al. [8], Fig. 2.) 23 Mean field HD theory H theory t>,. 1/VG 0.397 ., .f(t1,) '2/3 CJo = CJ~/11.'' (W /cm"l.) G082 0. /!)() 73!)3 ? G.>Tl Tab!P 1.1: .\ s utlllllar~· of,.,., . .f(N1 ). and Q,. j t1.". Th<'rrfon•. T1 ' ( 01. ~<1>) :)y"J. v Sill<'<' Q, - Q0 11.'', ( Q = (J2<( (+ l) \ ' , -((nl _ ·J S'"l.T·l -Po >. ( 1.:3G) 11. and (I - '2 - 111, \\' C' can rC'wri t<' this <'XPr<'ssion as t n~c = 11(11+ l)\'Q5 Q :2 Po S'l.T:l >. (Q) (l ..:; ,. 2 ( 1.:37) . which p;i\'C's a finite' <'nhatl<'<'llH'Ill for all Q < Q,.(T) or all T < '/ ~.( Q). \\' he'll p\ is dc' prPSSNI h,Y n ·, th<' IH'H( capacit~· is g iwu by (1.1()), ctlld tlH· hc'al capaci t.Y c•n hau<·<'tll<'ll t is ~C Q = ~C11 + T \ . (Dq/DT):, . . (Dqjulr ) f' ( 1.:38) Su bst i I\) t i ng the' C'X prC'ss ion ddi IH'd ill (]. 29) for P.\ (Tl'). (j t lw hPat capar it~· <'nhancenwnt p , l r = Pol (/I) ll' f''. l><'<·o nH·s: (1.:39) Thc'S(' I'('Sillts ('CIJ) \)(' <'Xprc•ssrd iu (('!'Ill S or the variabl<' !! = Q/CJ,. b.v n•lat ion {} = t.-.{(,.,·)/[11 1 .f(tlc)] oblaill(•d from llw <'xpn'ssiou Q US IIl g t lw p, ( I{ ' )TI' ST. The valuc•s for ,.,-1.. f(i>,.), and Q,.ft 'l." an' lisl<'d in Taui<' 1.1. For the' llH'an- fiPid t heot·~·. 2 II 1 C' (' '1. [(11 + 1) + 5( 1 1 - 1)1> + :2(11- :3),.,-' ] ~ q = c)IIH 2(1 - G1.-'l.) · ( l. 10) 2--1 A t s m a ll (}, this reduc<'s to: (l Al) B y compa ring this expn•ssio n t o (1.36). w<' S<'<' t hat t he fi rst trnn ill t his expansion is th e e nha nn'IIH'n t in CQ t h at wo uld res ult fro111 a cou ut r rflow if p_, we re not de pressed by Tr . Fig ure 1.9(b) s hows tha t all tlm•<• t lH'o ri <•s giv<' a d ivergent C'Q. Agai n , t h e results for t h<' 4' t h eo ry a nd t he l-ID t h ro r:v a r<' o b tained lllllll<'r icall y. Bceaus<' Qc is different fo r t he t hree t heorirs, we h ave usrd Q/Q~.JI) as t lw .r axis. so that a ll t h rN• t h cori <'s can IH' plotted o n t he sam e scale. li Nr Q~. 1 D is t h e cr iti cal heat cur rrnt g ivrn h~· HD . F ig tm' 1.10 s hows th<• di vergcn e<J based ou liD t h eory as a functio n o f t for vaxi o us valu<'S o f Q. (}= 100 Q= 10 (}=I Q=O. I (} = 0.01 Q (, 0 E ~ 4 0 u <l 2 J0 -1< Figurr l . LO: Solid lin rs: ~C'Q, as caknl a t <•d for liD t h eory [G. 8], for various valu es of Q. T h<• nuit s of Q a n • 11 \\' jn n 2. Dott ed-d as hed linE:'s: t he tr m pC'rat ures at w h ich s upe r!! ow was observed to lm·ak d own a t t lwsr Q val ucs b:v D un can . Ah lrrs. and Stcinh0rg [19] [This te mJw rat urr w ill lw call r d fuAs(Q)]. T brs<' liurs iudical<' h ow much o ft h<• e nh a ncPm<• nt o ur exp <•rimr ut <'XJH'<'tC'd to m easun.•. S hadE:'d regio n : t lH• a rea c ut o {l' by the g rav it y width of our O.OG..t em hig h ce ll. 25 Haussmann 's theory Ila ussmann r<'C'<'ntly JWrfornwd anothn RC calcul a tion bas('(! on mod<'l F. His approach permits the order paranwter ('lt';) to rq u a l ZNO both ahovr and lwlow T>., and indud<•s tlw eff<'ct of vortin•s [18]. lie fouud ~ (Q) to be at a slip;ht l ~· lowrr trmprrat un• t ban t h<' value• calcttla t <'d pre,·iously b_ , . Ila ussmaan and Doh 111 [G]. Ile also fouud a st rongc•r <' II haun•nwut of Cq. Ilis fuuctio u n•adH•s a maximum at Tc( Q) instead of showi ng H diwrgc·rH·c•6 (s<'<' Fig. 1.11). 140 120 ,........, ~ -E .. 100 , 0 ---. ......_, Cl .. .. : 80 v 60 40 -6 l -4 -2 0 2 4 !1T T-T.t.. [J.LK] 2 F ig un• 1.11: Cq for Q = 1:2.9 /t\\"jcm . Solid line : theory of Ilaussmann [18]. Dashrd liue: prc•,·io us t hPorif's [8. 9], wiH'r<' vortic<'s \\'<'rr nrglrct<'d. Dottc•d lin r: s p<'cific hC'at w it h Q = 0. T h<' arrow indi catc·s .6.Tc(Q). (Fro m Haussm a un [18], Fig. 7.) 6 SinC'<' tlw div<·rgcncc of CQ is pn•dicted by thNtnody namics. Hanssn mnn's failure to pr<'dict a divergcnn• is due to the fact I hal his calculation tak<'s into account dissipat ivP pfl'Pcts, and I h<'rdor<' smear~ out lh<' I nmsition in n11tch t h<' same way as g raYit .v s uwars out I h<' 7.<'1'0 Q transitio n . In particular !Jb Calculation 1\('C'OUilb [or a finit(' t!'lllj>(>J"atur(' gradient in til<' SII(H'l'fluid phas0. 2G 1. 7 The nature of the superfluid transition under heat flow ThC' criti cal hC'a t flux, Q, .. rd !C'cts a thNnJ odyu a mi c in st abi li t~· in t h(' countc•r fl ow W'loeit.\·. It is t h<• poin t at w hich <·onvC'ct iw heat t rans p o rt can no lo ngN lw s npport<'d b.' · the fluid . a mi s upcrflu idit y nws t b reak d own . T his t ransi t ion is cl c>arl ~· no t th r salllC' as th<' o rdi11ary s utw rfluid tran si t io n , sill<'<' tJw s upe rflu id de ns it.v p 8 is no t eq ual t o Z<'l'O a t 7~.( Q) . It has 1><'<' 11 considered to LH' a na logous to Lh e s pinodal line of a fi rst o rd<· r ph as<' t ra ns it iou . w ll<'r<' a h om ogeJH'Ous IIH' t astable stat<' l><•con lPS u nstabl<• at a fin it(' ,·alu<' o f t h (' o nk r panm w ter (and t h<' SIIJH'rfluid d <•ns it.\') [4. G] . T he s ingula r o bj <•r ts p rodt t<· <>d b:v this instabilit~· a rr vo rl ic('S [14]. T his t ra us itio n a lso i><'ars som e rrs<•mbla u cc t o a d is t i net phase t rans fo rm a tiou , ('V('JI t ho ug h t h erma l grad i('u ts pr<'cl ud <' then' from b<:>ing a n <'quilibrium no rn1 a l ph asr of fini t(' Q o n t lw ot h <•r sick of the t ra ns it io u [8]. \\"h en a s.\'st em is eha rac t erizC'd by a pair of conjuga tc> varia hlrs ( prC'SS II r<'- vo l1une. con ccu t ra t io n-chem ical po t e nti a!. m agnet iza.t io n-magn et ic fi(' ld). a ph as<' tran s iti o n us ua lly O<T IIrs whe n the genera lized s uscep ti bility di verges (gas- liqui d c ri tical p oi nt , bin m·~· tni x ture phas<' sepa ratio n, C uri e poin t ). ln t his pa r t icular ca~w. q and Tr ar<' a n ew conjuga l <' p a ir "·hos<' s uscep tibility, (DII"j uq)T "" (DqjD H' h. • 1 d ivc rg<'s a t T,.( Q). B~· a n a l og~· to o t h N casrs, H ·. i nst<'ad of r>~, m ight b (' t lu• o rci N p a ra uH' t<'r fo r this tra ns it io n . a nd q tlw conjugat <' fi eld . Aeeordiug t o the flu c-l ua t io n-dissipa ti o n rr latio n [55]. t lw m ean squ a r<' flu ct uat io ns iu Tl" a re eq ua l to ( ~I I ·:z) = k 0 T( DH '/ EJq )'f', a nd t h rrrforr di vrrg<' at Tc( Q). It is tiH'S<' flu ct uatio us that mus t sponta u('ous ]_,. generate ,·ort icC's, and lw nc(' te mp <> rat ur<' g ra di <·nt s. b C'for<' ou<' act ua lly readH'S T,.(Q), lradi ng to a brPakd own of tlH' t lH' rnlod .vna mic a pproxima tiou Lo uo n<'q uili bri um s uper flu id i t_v a nd snwari ng th r t ra n s it io n . 1t is y<•l to I><' d <'l <'nnined how d os<' t o T;.(Q) o u<' eau approach b<'forr t !Jis happ<' IIS. a ud heHn' <'X<H" lly h ow s ha rp tll<' crit ical si ng ulariti<•s d iscussf'd in this th rsis r<'all.Y a n•. 27 Chapter 2 The Superfluid Transition in t h e Q- T Plane This chapter presents a rcvi<•w of llli:lii,V of the experiments and theoric•s that inv<•stigate Lll(' ph~rsics of 4 H<' in the pres<'ucc of' a h0at flux. It is comprised o f' the complete t <'XI of' cUI artid<> written for Review8 of Modem Physics by Pct <'r \Veicluuan, David Goodst<'ill, and myself [56]. l! s hou ld providc> the• n<•cessary background all( ! n•!Pnwt cont C'xt for this projec·t. The• art id<' <·one! ud<'s with a bric>f n'vie•w of t he• <'XJH'ri!IH'lll cksnilwd in t lw r<>maining chaptc>rs of this thrsis. Although most oft lw infonuation in th is scct ion w ill br rrpratcd la t<'T' on, it is indudcd hrrr in ordrr to maintain t lw int egrity of tilE' original t<>xt. P<'rhaps it. should b<' considered as a prrvirw of things t 0 ('0111(' 0 Some· of th<' symbols used in this article are different t han those used for the othc•r chapters of this LIH'sis. A f<'\\' of Lll(' difl'erences res ult from t h e fact that this <'XJ><'rinH'Ill is JH'rforme'd c•xt r<'llH'i~· ll<'lll' T>., when' t h<' sup crflu id dc'nsi ty p, all( I t lw no rmal fluid wlocit~· v n ar<' both c•xt r<'lll<'ly s mall. This implic•s that in om rang<' of intNC'St , W - V 11 - V 5 ~ - V.~ . and(/ = t>~ W ~ - J 8 - - p5 V 8 • :-.,lost of the other discrC'panrirs arr s im pi~· due to di ffcrcnccs in not at ion. t·nfort unat<'l.\', this is sym ptomat i(' of t lw fie ld i11 g<'ncra l, \\'her<' t he r<' arc a plrt hora of uolatioua l com·en t ions floatiu g around in t h<' lit<>rature. One of til(' 11101'<' distmbing (and confusing) difE'ercnres is t h<' tNm i nolog~' used for the tempcratme o f s up<'l'fluid breakdown: a number of pap(~rs refer to t he lheor·etical tcmpcratme of' s uperfluid brcakdow11 as T,.(Q) all(l tiH' cxpe•riHteiiLal temperature as TnAs (Q) , while' oLhC'rs refe•r to the e:J:per·imentally measun"d t<'mpcrature of s upcrfluid breakdO'>\'ll as ]~.( Q), and t lw t IH•orc>tieal t<•m pC'rat un· as '1). ( Q). 1 The most s ignificant notation di ffercnees 1 This 0ntirP tlwsis , and all of the• pap0rs puhlish0d out of our r<'SC'arch group, st icks to the first com·cntion. It is our he lief that si ne<' t h<' transition under heat flow is of a very diffpn•nt uat urP. it should he· distinguished in a significant way from the traditional transition at T>,. 28 are summarizcu in Table 2.1. Abstract TltC' bC'havior of liquid '1HC' in a h0at flu x, Q, close to thC' SUJ)('rfluicl transition, can he reprcseut.ed by a ph as<• Jiagram in the Q-T plane aiHl described by means of a llC'W set of critical point exponents, which we introd uce. We review a s triking arra~· of recent expcrim0nts and their t.lwordical intcrpn'tations. Th<' excitement that surrounds this field arises from the fad that advanc<'d thcrmom<'try and the fntur<' availability of a microgrm·i ty experimental platform aboard t he International Space S Latiou will soon open to expcriwental exploration a new frontier of decades of reduceu ternperatures that were previous!~· in acc0ssiblc. 2.1 Introduction Due to its extraord inar~· purit~· and inscnsitivit~· to external perturbations, superftuid 4 Hr has long been the best system for accurate, detailed experimental investigations of phas<· transitious and critical phenomena. The essence of s uperfl uidi ty, however, lies in Llw dynamics of flowing h<:> lium, rather than in equilibri um properties such as the SJH'dfic- heat aucl the superfiuid density. In recent years, cons iu erable experimental attentiou has been focusc~d 011 lbC' behavior of liquid helium , ncar t he lambda transition t.C'mprrature, TJ.. , wh0n a flux of heat, Q, is passC'cl through it [19, 25,57 59]. This artick rrYiC'ws the' wealth of exciting IWW physical phC'nOm<'na uncovered by tlwse experi mC'nts and h~' the parallel tlworctical investigations [4, 11 , 12, 17, 60, 61]. At surticicntl y small Q , su Jwrftuid 4 Hc transports heat <•ssentially without dissipation, by means of superflnid-normal fluid count erfl ow. At hi ghcr Q, or as T approaclws TJ.. , this transport modC' brcaks down. Tn Sec. 2.2 w<· discuss thr phasr diagram of helium in the Q-T plane. In this plane, the supcrfluid statP is houndcd by a transitiou curve, Q c(T) , outside of which dissipativ<:> ftmv takes over. vVe also introduce au d evaluate a new set of critical exponents that arise as a consequence of superflow. 29 This chapter A B c D E F G -f :\fA - u .~ Tor~s( Q) T r(Q) - J ., = - p.~v., Q() w Tv ,\s(Q) Tc(Q) Qo w To ,,s( Q) T c(Q ) = P.,w p = p~ W - U ., Other chapters t 0 Group n t f) - Group ;i q - v .- ( I =W NA ~. (Q) w Qo (/ T>.(Q) - J ,, Qo - u .~ Tab I<' 2. 1: A lis t of tlw diff<'r<'nt no t at io n com •<'ntions liS<'d i11 th is t hc•sis a nd t IH' litNatur<'. Group n: Th<' no tation <·onv<•ntion usc'd in t h<' pap<'rs of D. L. Goodstein, P. B. \\"<'id11ua u . R. \". Duncan. all(l their colla bora tors. Group d: T h<' 11otation COilV<'Htiou used in Lhe pap<'rs of G. AhiE'rs. R. IIaussmann . \ '. Do lun and their col la bora tors. :'\A: not a ppl icabl r. The above table headings: What they refer to: A Tlw red uced t<'lll])('ratun' wit h resp ect. to th<' stat i<' t rausi t io u tcm pera ture: (T>.- T) /T>. B The red uc·ed t <>mperature with respect to the d~·H a m ic tra 11s ition t<>mperature: (Tc- T) /Tr c Tlw count<'rflow wlocit~· D Thr <'XJWrim<'ntally mea.sured ternp<'rat m r or s uperfluid breakdow u [19] E The theoretically predicted tenqH'rature of s uprrfluid breakd own [6, 18] F Tltr stqwrftuid rnomentum cknsit y G T hr n111 pi it ud<' of th e critical heat C'utT<'lll 30 In S<~c. 2.3 we d escribe an inhomogeneous phasC' where t lw heat flows t hrough both normal a nd supcrA uid rf'gions. separated by an interface' region t h at is neither normal nor s uperfiuid. In the normal region t he heat Aux produces a stat ic IC'mperat.UH' gracli<'llt. In the superfluid region , t lH' heat flows aL constan t temperature. Ill the int erface reg ion, a transition bet weeu these types of behavior is mediated by ftuctuat ions i 11 a way t ba t is not :V<'t ac('essiblc cither to experiment or to theory. A recurring theme throughoot tiH' article is the fundamental limitation imposed by t he Earth's acceleration clue to gravity, ge , on th<' resolution of Earth-bas<'d expcri- meuts . Gravity produces a pressure gradient across t he sample, lead ing to a variation in t he local lambda transition temp erature T>.( z) wit h height z according to (2.1) For gcuerality. the possi bilit.v of gravit.v g different from that on Earth is in cluded. For g = .(h LlH' trausit ion temperature t herefore decreases by 1.27 1-"K per cen timeter of columu height. 1f T is t uned so that, say, the center of t he cell is at the local lambda point , t he upper regio n wi ll he s11 J)('rfl uicl w hil<' tlw lm.ver regio n will be normal f-1 uid. The interface betw<'e n , dcfirwd a.s t he n'giou owr which tb0 local prop('rties d iffer significantly from those of a. bulk s~rstem a.t t he sam0 local temperature and pressure , is about 0. 1 mm wide o n Earth [62]. T his figur0 also estimates t he maximum possible critical correlation length iu t he s:vstf>m. T he iuho ruogeuei Ly induced b.v .4e causes t he c ritical sing ulari tics to be round ed o u a scalr s<'t by the heigh t of the c<~ll. Balanciug gravity effects against fini tl' size effects . which 0nt0r if t.lw <·C'll is too small , the' optimal cdl height is about 0.3 mm , leading to round ing ou a scale c =(T - T>.)/T>. 10- Preseut thermometr y [2. 59] is capabl e rv 7 . of resolving red11ced temperatures f 3-4 orders of magnit ude sm aller than this. Iu <'SS0nc:e there is a nc'w frontier, consisting of d ecades of previously unavailable reduced temperature around T>. , that uw o nly I><' t'xplo recl in !llicrogravity. measming cq11ilibrium S]J<'cifi c heat (Lambda Point Exper im ent size effects (Confined H<'lium Exp<'riment Experiments LPE) [2] aud finite CHEX) have rcc<'Iltly flown aboard the 31 s paeP shut tie. Gnwit~· also has a strong effect 011 the iuterfac:c region discussed in See. 2.3. As we haw :-,<'<'11 abo\·<·. the inlNfae<' is cotu pressrd by gravity to a width of about 0.1 nun. I oo s mall I o IH' ~ ~ udied CXJ><'rillH' nl ally. Thr iut erfacc r egion w ill lw st udird in a u <'x t><• rinH'Ill (Critical Dyuami<'s Exp<'rillH'lll DY. A}.IX) presently IH'ing prrpar<>d fo r a low I <'Ill p<'ratur<' microgravi I :v pia t fonu (Low Tem pNat un• }.Iinog ravi t :v Physics Faci lil ~- LT:.lPF) that is to b<' pnrl of tlw IntC'rnational Space• Statiou (ISS). Although gravit~· is cletriliH'Iltal to some mrasurrmcnts, there can al so be interesting uPw physic-s when both gravity aud heat c urrrnt arr prrsrut. In S<•c. 2.4 we desc ribe the so-called self-m~qo.niz('d cTilico.l (SOC) state, in which g and Q conspir<:> to produce a ll<'w . E:'sspntiall.\· homogeneous. uouequilibrium s tate when' tlw t<:>mperat un' T( ::: ) JH"<•cis<'ly paralld~ T,x( .:: ) at a fixed Q-dcpeude11t distanc<'. :\.t larg<•r Q. this state und<'rg<H's a transition from normal to s upcrfluid. with tlH• l<'lllperature g radi<'nt support<'d b.v a st rc•mn of vorl ices in th<' lattrr. Finall.v, in S<'c. 2.5 wr discuss t lw s pc•cifk heats undN heat flow , pn•dict<•cl [8, 9, 12], and subs<'quentl~· confirmed [25], to lw ('rthanred abovr th<' equilibriull l form. with a siugularit.v predict.<'d <lt t.bc• phas<' boundary Qc(T). This <'xperinl<'ll t too ul ti1uatPI.v twcds to b e performed at v<·r.v low Q, and the required microgravi t,v version is <·urr<•utly h<'iug propos<'d. 2.2 N onequilibrium critical phenomena and seal- . 1ng ll ndN conditions when• a unifo rm h<•at flux Q is appli<•cl to s upt'rfluid 1 H<' , a thermal counterflow is cr<'ated in which tlw normal fluid moves along wit.h Q at wlocity U n, a nd t hr s uperfluid mov<'S oppositely with n•locity U 8 . For s uffiC'if'ntl .v smal l IH•a,t cmrent nol too close to T,x [morr prrcisely, onr rcquin's Q < Qc(T). witl1 CJc(T) ddin0cl in Fig. 2.2 all(] (2.2 1) lwlow] t hr temperatur<' T < T>. is c·ssent iall~, unifo rm 32 and t lw hc•at flu x is (2.2) Q = TSrmn· wh<'n' 8 is t !1<• <'Ill rop~ p<'l" unit mas!->. p p, + p 11 is tlw total mass deusit~·. con1pos<'d of SIIJH'rrluid and nonnal fluid part s, and U 11 is th<' normal fluid \'Clocity. Since• tiH'r<' is no tH'I mass flow . Oil<' has j, = j_,. w hc•n• J~ = P.~ u~ and j 11 = p, u, an•, r<'SJH'<'- tin•ly. th<' SUJH'rfluid and normal fluid mass c·utT<'nt dc•usi t i<'S. and u , is llw SIIJ><'rfluid velocity. C lose• toT>-. one> find s <'XJH'rimrntall~· S ~ 8>-. = I .08 .J /g f-.: . p,. ~ r>oicl<. with Po ~ 0.:31 gf cm:1. critical <'X JW11<'111 ( ~ 0.671. and /)11 ~ p ~ 0.1-! gfcma. Thus, in ('XJH'rilll('llt ally IIIUtintll'd Ull it S. u, ~ 1/ 11 ~ - f.9 x 10 ·) -· -:l 1 X 1()- fi Q '/ 1j1\\ ('111 1 ('() - - - r; )<.cm/s (2.3) 1(' 1 Q c m js. ljt\\'fc- m 1 (2.-l) At <'XJH'rinwntall.\· ac·c·c•ssiblc• valuc•s Q - 1 jtV\"jc 1111 aad 1(1 80 Jtm/s and u, is n<'arly four ordc•rs of mag nit udc• smaiiN . 2.2.1 Thermodynamic formalisn1 The• isot h0nnal rondi I ion a !lows an dl"c•c·t i\'<' thermodynamic dc•scri ption of t hr fi 11 it e Q statr [63] . .-\!though nouc•quilibrium sndiu g dc)('s not n·I~· 0 11 this, a more• int uiti Yc dcsni pt io n of a numbc•r of plwnonH•ua SJH'<' ia I to SUJH'rflu idi t~· r<'st!l ts. and so it b worthwhile• pr<'s<•nting tlw th<'or~· in thi s c-ontc•xt. lu tlw fralll(' of refc•relH'<' mo\·ing with tlw normal fluid (iu which th<'r<' is 110 hc•at How) . tlH• difi'PrC'III ial of til(' fr<'<' c•uc•rg_\· cl<'nsi ty F(T. U,) m a~· I><• writ tc•n. dF = - Sr!T -t l s · dU,. , ( •)_,\)~) ill which U ., = u , - u , and (2 .G ) :33 wit h tlH• secoud c•qualit.Y sc•rTrup, as a dcfimlwn of p,(T. C,) when C., is no t smt~ll. 1\c•;u T>-. tire• smallrH•ss ofu 11 implic•s tha t U ., ::= u .. aud J , :::: j,,. ln this fram(' one has tiH' drfiniug n•Jation U ., = (h j 111)\lo. whc•n• 111 is t il(' 1Ilc• atomic mass. aud 117 </>(r ) = r;U8 · r (:2 .7) is the· phase· oft l1<• SliJH•rfluid o rde·r paramrtc•r v = Ide 19 . .vit>ld iug the• us ual hcli('al stnrrt un· of 1. • alip,rH•d a long U ,. TIH• dfc•c·t ivc• tlH•rmod.vnarn i(' dc•script ion (2.3) r<'lic•s on liH' <'xistr11c·c• of at im<'-i!ld<'JH'ndc•llt 1. · in til(' pr<'se•n<·c• of a finite• phas0 g radic•nt . In fa('!. tlwmrall.'· IIII C'IC'atc•d phas<' s lips in tlu• h<')i('al st ructun• (intc•rpn•trd as tuunc•ling bet wee11 di ff'c•n·nt metastable• loC'al rninirn a oft lw fr<'<' rnrrgy) lc•ad to small t<'lllJ>C'ratun• gradic•Jlts a nd clc•c· a~· of SllJH'rfl ow. I Iowe•w•r. t h<' dN·a.\· tinw is e•xtrrmc·l~· largr at low hc•at curT<'n t s a nd t <' lllJWrat m es not loo clos<' to T>-. [G-!]. T lw t hrrmoclynam ic d c•sniptio11 (:2.0) ih ndiu 011 tir nE:' scales sm a ller thau thi s. As U ., in n<'ase•s, both t IH' o rclc•r panwr<'l N ur agu it ud<' It'( C,. T) I a nd t he superfl uid d c• usi t.'· are s u pprt>ssc>d r<•lat in• to t IH•ir c·quilihriull t ,·alrws at C, = 0. Q = 0. \ ·ari a t ions at c·onst ant Q = - TSJ b me• mos t c·on vc•n ic• 11t ly J><'rformrd by dc•fin i 11g t lw L<'P,<' lldrC' t ra11sfomH•d frN' c•rrc•rg_,· <J> (T. J ,) = F - J 8 • U , with clifl'c•rc•llt ial d<T> = - SdT- U s · dJ.,. (2.8) C lose• t oT>-., whc•n• TS' :::= T>-.S>-. is <'ssc•ntially ('Onstaut. \'a ri ations at C'O llstant Q ar<' as~·m ptot ically t IH• sam<' as t hose• M co11st ant J s. For c•xam pic•. t he• SJH'<"ifi<' lr <'at at fix<•d Q ma.v I><' t a ke11 as CQ = T whC'r<' S(T , J -;) -= - (D<T>/ fJT) ( 0De.~,,' ) , (2 .0) ./,.; \\'it h t lw above' obse•ryat ion in lllind , we• sha lllwtH'C'- fort It trc•a t J .. and Q as d i ff'c •ring o nl.Y h~· a constant fact or. 34 2.2.2 Nonequilibrium scaling TlH' fact that U , all( l J , (or Q ) may lw treat<•d a~ tlwrmodynalllirali.Y conjugate ' 'ariablc•s has import a nt <·o nsc•qtH'll<'<'S for tIt<' st ruct Il l'<' of t IH' l he nu od~· n ami c functions IH'ar T>. [Gil]. \Vc pro<·c•<'d by a rralogy witl1 tlw conjug;H<• variablc•s h anclw at <'OUV<'n- tional nit ical poin ts , wlr<'r<' h is t lr c• c•xt<'l'nal rnagnc•tic fiC'Id and 111 til<' nutgrwt izat io rr at a C nri<' point , or h is tlr<' diff<'r<'ncc• from criti('a) prrss ur<' and 111 the• difl'c•n•uc·c• from critical dPnsity at a liq nid-,·apor nitiral poiut. T IH•rc• tlw fn•c• <'TH'I'K\' . \ (T,h ), a na logous to <P. has diff'Pr<'ut ial d. l = -SdT - mdh and it s singular part . 1,, o hC'~·s a n asyllt]Jlo/ t(' s caling fonn [s<'< '. e.g .. [G6]], A , (1~. h) - £ 0 Ic l 2 nA (Doh) lc l.j. . (2. J0) in whic.:h o is lh<• speci fic lwat exponent , ~is t h<' "gap C'x poncnt ... A (.r) is a trrtiYC'rsa l scaling functi on . and £ 0 , Doan· nonuniwrsal scale factors. SJH'C'ifi<>d sa~·. t lw no nu a l izat ions A (0) = uniqu<'l~· A' (0) = 1. T lr en• arC' a('t uall.'· two sealing fund ions. A ± (.r) for ± ( > 0, but we• s ha ll primaril.\ ' \)(' int<•rc•stC'd in the orcl<'l'<'d phase f a nd ('Onsid<'l' o nl.Y A ,·ia. < 0 - A. S imilar! ~·. tlw singular part of tire• s np<'rfluid fn•c• <' lrNgy, <I>, , is <'XJH'<'t.<•d t oo l ><·~· the• scali ng form, (2. 1 J ) in whidr n : : : : - 0.013 [2] is again the us ua l <'q uili brium SJ><'cifi c h<'at <'XJ>Oll<'ll t. ~ Q is t.hc• gap <'X[>O il<' nt for Q, } "(y) is the c < 0 uni v<'rsal scal i.ng function , and .-1 0 • Q 0 ar<' II Oil II II iw·r~al sral<- factors. sp<'ri fi<•d uniq uc•ly ,·ia . say, the normalit at ions }· (0 ) = ) ""(0) = 1 [) '( y) must he an <'\'('ll funetioll of !J du<' to tlw OO\'ious s~·mnwtr~· und<•r sig n 1'('\'C'I'Sa] of Q]. Tlt0 cl c•riYat iw of (2. 10) wit lr n•sp<'c l I o II y idds 111(T, h) = - EoDo l<l A , (Doh lrl~ ) 1 (2.1:2) wlrc•n• the· prim <' d <•no tc•s deri valiv<' with n·spc•<·t t o argumc'nt a11d the o rciN paranwtc•r 35 (a) h m 1st order coex. line 0 (b) Q -- _/_- Tc T 0 Tc T dissipative phase T T F igure 2. 1: Comparison of phase diagram s in (a) conventional T-h and T-m spaces a nd (b ) su pC'rfluid T -Q and T-U.~ space's. HerC', m 0 (T) is thC' spontaneous magul"tization , or t h0 liquid cku sity minus tlH' critical density. T he crit ical lin0s Qc(T) and U.,,r(T), enclose' the region of s ta ble superflow, which corresponds a lso to tbe region of validi ty of t he t hermo cl.v na mic description , and like the thermodynamic dC'scription itself arC' sharp only in th <· a bsm c0 of p hase slips. ThC' d ifferl"nt sha pes of these curves 11 ear T;.. a r<' d C't crmined b:v thr fact t ha t .6..Q > 1, while }1Q < 1 [ser rquations (2. 19)- (2 .22) below] . T h(' nat un• of t h0 cri tical behavior as Q a pproaches Qc from below, and t lw nat ur(' of tlw inho mog0nrous di ssipa ti ve phase for Q > Qc, will be d isc uss('d in later sections. A m ajor difference between (a) all(! (b ) is t he lack of a fi rst order line below T;.. in tlw Ia Uer. T hus, h vanishes t hro ug ho ut the phase co<'xisteucc region in the T -'111 pla ne, whereas Q ex /~ = PsU.~ varif's cont inuously t hro ughout the su p erflu id ph r~sr i11 t he T -Us pl ane . 36 expone nt jj obeys thP scaliu g relation /3 = 2 - n - .6.. T h <.' sccoucl h dcriYativc yields t h <.' orciN panurH't<'l' susceptibilit.v (compressibility, iu the case of a liquid-vap or nitical poillt ) :\ ( T , h ) -_ - EoD0 !f I-'A" (Doh) lEI~ 2 (2. 13) with 1 = n + 2.6. - 2, which thcu ~·i elds the fa tuous Essam-Fisher scaling law o + 2/:} + 1 = 2 [66]. Similarly, Lh<' derivative o f <Ds with respect. to .Is yields Us iu the form (2 .14) when~ B0 = A 0 T~S>../CJ 0 , aud one has the gene rali zed o rder parame ter exp onen t scaliu g relation (2. 15) T he equi librium superftuid d eusity enters the free euergy F v ia a term .6.F.~ = &PsUi for small U8 • T he Legendre trausform yields a term .6.<1? 8 = - J} j 2p 8 fo r small .15 , a nd the inverse of the s uperftuid d e nsity now appears in t he theory as a generalized sw;cepti bil i ty: 1 p,, (T, Q = 0) D:t<I> ) - DJ./ ( T,.J,,=O ) ( au.~ f)J.~ Ro1"'(0) T,./,,=0 (2.16) ki"YQ where R0 = A 0 (T>. S>..fQ 0 ) 2 and t hf' geueralized susceptibilit:v exponent IQ obeys the Essam-Fisher rela ti on [66], /'Q = ( = 2.6. Q + ()' - ()' + 2(iq + /'Q = 2 2 (2. 17) (2. 18) GenPrall y t he gap C'x po nen t is independ en t of n a nd mu st be separately d etermined . Hmwver , in th<' SUJ)('rftuid problem th e J osephson relat ion [sec , e.g., [67] and n'fer- 37 <'n<·rs th<'l'rin ] .Yir lds ( = 2- (\ - (rl - 2)1/. H r re ~ ~ ~o/lrl" wi t II, in diu H•ns io n 2 11 = rl = 1, 11 = ( ~ 0 .611 a nd f.o ~ 1.-J .:\ [68]. d rscrihrs thr d in•rg<'JH'<' of t h e• SIIJWrfl uid ('OlH'r<'ll<'f' lr ng th. ~ = (m 2 kJJT/ h 2 p_,)' l <d 2 >. and the srcond eq u ali t~· fo llows fro m t lw h.\' P('['S('alin g rda tiou 2- n = r/11 [63, 66]. \\'0 th erdore idc11ti f~· ~Q = 2- n - 11 = (d - L) 11. T hi s S('a[iug law implies that Q s('a(c•s with t.h r cross-secti on a l a rN\ f.d vollllll(' e: Q b ecomes sig uificaut wlwu (2.19) 1 o f a cor rdat io n po we r iucideut 0 11 a corrpla ti ou area is or UI (' o rder Q 0 ~g-'. 2 From (2. 15) OIH' obt a ius dQ = /). (2.20) T his relatio n has the iulcrpre tat io n t h at (m/li)U ,,. which has d inH' IISio ns or ill\'Cl'S(' leng l h , S('a(c•s with th<• in\'<'l'S<' <·o rn• la tio n leug t!J ~ - I: the phase• g radic• nt has a signifi('Hlll <•ff<•ct wlwn its " ·avc•lc•Hg l h 27rh / m Us IH'C'Olll<'S co mparabl<' to f. . Not.e t ha t , m on• t~·pi c a ll :v, 0110 b0gin s with t h0 Ia !.I <'I' a ssum pt i011 a nd r ev<•rses t he nb ove a rg tmH' ll t to d rr·ivr the .T osc•phso11 rei a t ion . .Jus t as arbi trari l:v s m a ll h s nwa rs t h0 s ing ular behm·ior JH'ar a Cu ri<• point, O IH' <'X J><'rls a u a rbitrarily sm a ll Q t o Pitlw r s mrar or d rastically a lt 0r tlw la m bda p oint nit iC'al be ltaxio r. T his is cons is t C'nt w ith ~ Q > 0. imply ing t ha t the' scali11g arg ume nt ,1/ di v<•rg<·s as lei ---t 0 a t a ny finit <• Q. spn ·ing Lo d efine Q as a n-levant pe1·t'U1·bation I o t he• la mbda p oint [65]. lu F ig. 2. 1 a sch<'ma t iC' o f t he expec ted ph as<' d iagram is sh own , co ntras ting it with t ha t fo r a <·onv<'lllional cri t ical p oint. T h e litu•s (J 1.(T) aml U8 ,c(T) an• th<' l>ounda ri<•s IH•youd which s upe rfluidily breaks d own : fo r Q > Q 1., due = 2 0rw 111 ay C'stimatf' (J 0 a s fo llows. Two sca lP fart.or univPr sal ity yi<'l ds a forn r C's /..: 0 (/?d~)" /n.kl 2 fo r t iH' s in g ula r p a rt of t h<' <'{ jllilibrium SJWC ific lwat h<'lm,· T>., wlr<'r<' t hC' hyJ>Prllni vC'rsal ratio fl~ :::::: 0.!)0 in d 3 [69]. Tlri s must m a t r h the sraling forru (2. 11) at Q 0 = = = a nd dC't <'rmin<'s .-10 lrl 2 - n = [k llT.x/n(l - n)(2 - n)](RdOd [with tlw c hoice }'(0) 1]. E quatin g t lr <' qua drat ic t C'rrns Aokl 2 0 (Q/Qolci~Q) 2 - Ji!P.< [wi t h the choice } '"(0) 1], o uc obt a in s fi n a lly q 0 = T>.S>.J-A 0 p 0 , a nd Q 0 ~g p0 = 111 kuT>./h ~g 2 ). 2 2 Qo :::::: 30 k\\'jc m 2 . = = = mkiJT'f.S.x R: 12 /hJ-n(l- n)(2- o) (using S 11hst it u t io n o f C'X J>C'ri m f' nt a l numbers yields .4 0 :::::: -21 .Jfnu 3 aud 1 38 (a) (b) m Us U s,c(T ) m 0 (T) ----------+----------h 0 -m o(T) -U s,c(T) Figure 2. 2: Comparison of the isotlH:nnal rq uat ion of statr (a) in tlw convrntional h-111 plan<' and (b) in tlw SliJWrfluid Q-C~ plane. At h = 0 ther<' is a first order transit ion lwt\n'<'n up and down mag nrt izcd states. or bet \\'<'<'11 liquid rtlld \'apor states. :\ s !hi increases. j111(T. h)l iuneasPs as \\'<'11. As IQI increases IU., j increases until th<' s upNfiuid brPaks down at IQI = Q(.(T), at which point ["_,(Q, T) has a squar<' root c·nsp. At this samr point thr s p<'cific hrat at constant ["_, also has a squar<' root c usp [12]. whil<· tlH' sp<'c ific hl'at at constant Q has an inverse squar<' root div<'rgcucc [8] . to s uppn•ss ion of thl' ord<'r paranJ<'I<'r and superfluid density, til(' IH•at cutT<'n t is too larp;<' for the s upcrfluid to support isoth<'l"mal h <'at. tran sport . and a notwquilibrium phase transition to a nrw di ssipat iv<' phas<' occurs. T lw natur<' of this phas<' will lw discussed iu Sr<.:. 2.3 bPlow . In Fig. 2.2 wC' s ketch tlw isotlwrmal rquations of stat<• for liH' com ·eutional and suprrfluid s~·stems. At a fixed subcrit ical temprraturc th(' com·<•ntioual systelll displa.v:-.. th<• us ual first order jump in m at h = 0 b<'(\\'<'<'11 up aud down maguetiz<•d s tat cs. or b('t W<'<'ll liquid aud \'apor states. The superfluid S_YS(('Jll displays a c-ontinuous Y<lriation or C.~ w ith Q, but at IQI = CJc(T ) Cll COUlll<' r S th<• boundary ))('tW<'<'n SliiWrfiuid and di ssipative' phas0s. At this point U8 (Q) has a squarr root. cusp [12], cor-r<•spondi tlg to H st.roug s uppression of f) _, and a di V<'rgeut s uscf'pt ibility (D[r8 /o.l8 )·r· Th is s ing ularity is <'xhibitcd in t lH• s<:al iug form (2.11) as squarr root cusps it1 1"(y) at sonH' fi11i!.<' value y = ±Yc· T his yi<' lds the• pr('dict ions CJc(T) :::::: CJo!Jcki~Q C,,c(T) :::::: Bo1 .,(!Jc)lc!iJQ (2.21) CJc(T) 1 / 6Q , (2.22) 39 in wh ic h f>q = ~ QI (3q = 1 + !ql (Jq. (2.23) The' latter r<'lation g<'ncraliz0s the \\'ido m scaling law m(T,., h) 'X h 116 wit h (5 = ~ l d = 1 +')I tJ which fo llows from (2 .12) in t}l(' limit E --+ 0 [the as~· u1p totic bchavior A' (.r) '""" .rJ!.~ as .r --+ 00 is r<'quin•d for <'OHsistcn cy] [66]. Frolll (2. 17) a n d (2.20 ) Oil<' obtain s cx plicit I~· f>q = d l. As will be disC' ussecl in Sec. 2.5, I he specific lH•ats a t cons t.ant U_. awl Q arc a lso s ing ul a r at Us,c(T) and CJc(T) [8, 9, 12]. The• coh rrcncc lcngt h itsc•lf ob<'YS a sc·aling form (2.21) wit.h ::::(0) = l. Con sidering variations o f E, a t a fixPd value of y = QIQ 0 jfi~ £J, o ne m ay writ<' E.o(QIQo) "Q.t/Q:=:(y ) (2.2.) ) v 1~Q = 11(d - 1), (2.26) serYing I o ckfi1w an <'XJHHt<'llt describing t he characteristiC' nu iatio u of E, wit h Q . Attempts t o \'C' ril'y t he scaling n· lat ions (2.21) a nd (2.26) . with ~Q = 2 11 a nd vq = 112 in d = :3, wi ll he disc ussed in lat e r sectio ns. In Table> 2.2 wc sumnHtrill<' tlH' various critical exponents v\'<' have defined, a lo ng with tlwir values in genera l dimension d and in d = 3. As a fin a l comm en t, we• no tc that wrifications of ccrtai n clynaruical scaling laws in 1 1Ie a r<' com pi icat ed by a f'undanwnta l prohlc m associated wi t. h t h e smalln ess o f' t lH' exponml o. Scaling forms likc (2. 11 ) implicitl y assume t h a t lrl is s uffic icntl.v sm a ll t hat a ll 't n-eiP't mnl scalin g varia bl es may h e ignorC'd . It turns ou1 t ha t n controls tlH' rel<'vaJJ t(' of utw;s and IH'al dij]v sion to thc dy na mic n itica.l behavior [70] through a11 additional scaliug variable' II ' =/okl - o. w berr ')o is a parameter of order uni t~· in t lH' t-. lod<' l F equa tions [70] (d(•sni bing the near-c ri t ical d~rn am ics of' 1l-IC') which cou pl <•s cntrop,v a nd d<'llsit y flu ct. ua l ious to o rder p a ra m eter fluctuati ons. T h c variable w 40 r x p on r n t /) (\ ( ~ (J grnr ral d 1/2 ::; IJ(d) ::; 00 2 - di/ 2 - 0 - 2IJ = (d- 2)IJ 2 - ()' - 1/ = (d - 1)// !iQ /) /q ( l +iq/(3q=d - 1 IJ/ 6.q = 1/(d - J) 6q 1/Q d=3 0. 671 -0.013 0.671 1.342 0.671 0.671 2 J /2 Tab](' 2.2 : Equilbrium and noJwquilibr ium criti cal exponents and t heir values. Note that no rxact rxpressio n fo r the corrr lation k ngth cxpou<'ut. IJ( d) is known , asid<' from the bouudary valu<'s u(2) = oo a nd I;(d :::: 4) = ~' but all oUwr exp onents are eit her giveu exac t!.\· or iu tenns of IJ. There are other independent exponents (e.g., t he cr itical corrr lation f'x ponent TJ :::::= 0.02), but t hey ha pp<'n not to b e involved in a ny of th<' 0xp 0rinw nts W<' di scuss . Exprrssions in t hC' center col umn involving d <'xplicit ly require hyp Nscaling, all(] arc t hereforc valid ouly for d ::; 4. For d :::: 4 the exponents all st ick a t their m<><Ul field values, and may be det ermined by s u bs tituting v - ~ , n = 0 int o t he no n-<'xplicit lY d-dcpcndcnt forms of thr scal in g rrlatious. d oes no t a ppear iu an.\' eq uilibrium scaling function (where s uch flu ctuations m ay lw cm11pl<:'t.ely •·integrated 0 11 t" of t he par tit iou function), bu t does a ppear in those involving dissipa ti ve t ra us po rt , e.g., t hat of the thermal cond uctivity [11]. n < 0 t his variable' vanislH'S as 1<'1 --7 0 bu t extremely slowly: f Since = 10- ?.~ leads only to lei-" :::::= 0.1. T IH' scaling fuuctiou t hen has a very slow parametr ic d c'pcndence on w. leading to "qnasi-scal in g'' [ l l] of thr h<'at conductivity. and hencr to an apparrnt slow variation of tl1 c associatPd rrit ical C'xponc'nt with w. Measured d ynami c critical expone nts affect r d in this way m ay typicall y b e exp ected to d iffer fro m t hr ir t rue asymptotic values by 10-20% [19]. T he scaling phenom ena considered here, however , a re m ainly those associated with pro perties of iso therma l s u perftuids . which t ho ugh o ut of equili brium , are nevertheless a rgm'd to behave as equilibrium thermo d yn a mic system s . To the extent that t his is t rur (i.e., to t he exte ut tha t vortex excitations can be ign ored), t he Model F equat ions again produce a thermod yn amic- ty pe pa r tit ion fu nction wit h entro py/ density fiuctuations iutcgra tcd o ut, and finite Q enforced by a n imposed uniform helical twist in th<' onkr parauH'ter [12]. T h e scaling variable w will again not appear , and one 41 <'XJ)('Cts that quas i-scaling wi ll b<' ahsC'nt in (2. 11) and from al l quaut ities dNin•d from i t , a ud h<'nce that tlH' C'x po ncnts in Tabl<' 2.2 will <'xhibit <'XJW r imcntall~, thei r predicted YaltH'S. On t h <' ot ii N b a nd , t il<' phase' boundari<'s in F ig. 2.l(b) ar<' d d in r d by the on :;e/ of dissipat iw t rans porl. with d i vergcnt fl urtuat ions i 11 the local heat curr<'llt a s tlH'y are a pproached. Such fluctuati on s lead to vo rtex neatiou , dPca.v of s u perf! ow, brPakdown o ft lw dfrct ivc cquili bri u Ill description. a nd the reappParane<' of t lw variabi<- w. In some S<'II S<' w IIlll S( contro l t he "fuzzin<'ss'' oft IH' bo undary Q ,.(T ), and a n <'x prrinwntal t<'s l of Llw rd ations (2.2 1) may Pxhibi t quasi-scal ing <'Wn if, for y s uffici<•nt l.v lrss than y,., tiH' sealing function } '(y) docs not. This issu<' is pot<'nt ially tcstablr by t hr s pecific hrat m rasurem ents d isc ussed in Sec. 2.5 . Cnfortunately, as exhibit(•d i11 Fig. 2.9 be low, prrsrnt data an' constrained to li r s ufficicntl:v far hrlow Q,.(T ) that bo tli asympto tic all() nonasympto t.i c forms fit equal !~, well. 2.3 The nonequilibrium superftuid-normal interface 2.3.1 Interface statics Cons ide r a c.dinclrica l cell w it h <l heat c mTent Q clriYen a long it s axis. labcl<•d b~' co ordinat<' ::, in which til<' u p-stream <'tHlwall , ;; = 0. has T(O) > T>.. In t h<' norm al phaS(' heaL is tra n s ported by th ermal conduc t ion . which at sufficient ly s mall Q is d escri l><•d by the Fourier law (2.27) w h<'r(' t~ (T) ~ Hor 11 is th <· therma l condu c Li vi t.~·, predic ted to d iv<'rg(' at T>. , cousis l<, nt wit h its infinite vain<' at a ll T < 1).. . Expcrilll eutally O ll<' find s [53 . 58, 71] ,., 0 ~ 12 /t\V j cm '2 and Jl ~ O...t-1. ln a wry ta ll ce ll in which T varies s ubstantially a long its len g th , o n<' nw~· Yiew (2.27) as locally valid w ith/\(.:)= ti[T(.:)J so lon g as T(;;) re m a ins s n ffi cient I ~· far a b ove 1).. . "S u ffi ci<'nt ly fa r a boYe T>." may lw quaut ifi<•d by the condition t h a t the tCltlJWrat nr(' drop across a coherence leng th l><' much sm a ll<•r 42 than t h r clrviation fro m T>.: (:2.28) Puttinp, in 1Tlc• paramrt<'rs, this rrquin's r »6x 10 !\ (1 JL\YQ/em· 2 ),.· 1 r :=---~ 0.81. 1 I' + (2.29 ) I/ - As .:; inn<'aS<'S, T( ::: ) will <'V<'nt.ua ll .v viola!(' (2.29), and Oll(' ent<'rs a r('gion of nonlinea1· heal tr·a11SfJOJ'/.. In cffc>cl, h' IH'<'OilH'S a stro ng fun ctio n of Q in t his n'gion. f.Ior<'m'<'r, as illustratrd in Fig. 2.:3, at som<' position .::0 , T( .::0 ) = T>., a nd t lw s:vstrm <'ll(('l'S t he s uperrluid phase for .:; > z0 : a nonrqu i li bri um s u JWrfluid no rm a l fluid int<'rfac·<' is getH'rat <•d. Far d owustream frow t his intr rfacP, T( z ) lrw ls o nt at a tcm p<'r at un' T 00 (Q), on<' of the HOnPquilibrium t.hermod.vnanti<" stat es discussNI in S0c. 2.2 abow. Con·<'spondi ngly. the ord<'r parameter m agnit ude JVJ ( z ) J. whiC'h pffecti vel_y \'an is iH's for z < .::0 , grows in tlH' int erfa<"<' n'giou a nd sat urates at a ,·a luc Jv_ (Q)J for z >> .::0 . :1 ThP i nt('l'fac·r is t lw r<·giou over w hich liH' mode of h eat t ram; port converts fro m cond uct ion to s u Jw rfluid c·o tlnl <' rflow. Its C'ha raC' INistic w idth ~(Q), through which T a nd JtJJ ,.,u~· s uhs tantia ll :v (Fig. 2.3). is <'XJ><'c· trd to scak acC'ord ing to (2.25). \\'<' m a.v f'Siimat<' this w id t h S<' llli-quantitatiwl~· h~· dc'fin ing a r<'d U<"<'d lPmpcratur<' S<"a l<' EQ t hroug h eq ua lity iu (2.28) a nd (2.29), a nd d<'fining ~(Q) = ~(cq) ~ 24 ( 1 Q ) - 1'11 w; em')Jl fllll , /'// ~ 0.54. (2.30) Tlw sc·a ling rPiat io n (2.26) pn'dids 1·v = 1/Q (= 1/2 in d = 3) , r<'quiriug llw s<"aliug rf'lation I' = l - ( (~ j i11 d = 3) [70]. T h<' diff'<'rencc IH'IW<'< ' u experimental and theorC'Lical valu e's is pn•sumahl.v du <' to t.IH' quasi-scaling cff<'ct. d <'scr ib ed at tlw cud of Sec. 2.2. Q ua nt itati\'<' r<'normali zation g ro up bas<'d pr('dictiou s for thr full tc•mpera ture profil<' T( : :. . Q), <'xplicitl ~· including t lw quasi-scal ing plH'llOlll<'non [11 , 12), are :3111 faet T (:::) ron tinues to dPnPase slowly wit h ::: in llw supnHuid phaii<' d u<' to rar<' pha~C' slip <'wn ts [17, 18]. soT00 ( ( / ) a nd l ~'oc( CJ)I ar<' not s harp ly d<'fin<'d, b ut we> shall not dw<'ll o n this romp li eat ion lwr<'. 13 6 .0 r 3.0 M I\I' I 4.0 2.0 ~(Q) ·I l\1' ool 2.0 1.0 0.0 0 .0 Normal Phase -2 .0 SF Phase - 1.0 Moo - 4 .0 - 5 .0 Zo 0.0 z 5 .0 -2.0 10.0 Figure 2.:3: Scah-d telll perat un• and ordrr paramrt rr profi lc•s 1H ( Z) a nd \II ( Z), comput<•d withiu tlt r nwa n fir ld approximation [ 1.60]. Hc•re Z = z/f(Q) , 1\IJ I lvl l(Q) 1 aud J/ 'X (T - T 0 )/ {Q). wlH'r0 T0 is tlH• Jllea u fi<'ld transition (('IllJ><'rature a ud / (Q) ex Q I /:! is c•sseut ial l.v tiH• corrdatiou l<·u ~ th {2.2G) iu the· m Pan rl<'ld approx im atiou. \\"ith tiJpsc• scalings the• profilr~ arr indC'pcndc•nt of Q . limited to the normal fluid region z < z0 where t he vanis hing of ·!j; greatly sirnplifirs tlw calculatious. Less qu a nt itat iw predic:tious for tlH' entin' profi](' ar<' limitrd to mean field [4] (which igno res criti cal flu ct uat ions) or largr-N [17, 18] (which replace!) t lw single compl ex ordN paranwtN 'vvith an JV-componrnt vector) approximations. An rxperinwntal m rasurr mrnt of t hr temperat ure profile T( z , Q) within t he interface regio n for a secptence of different Q would all ow a detailed exploration of ncar-crit ical, nonlinc•ar heat transport. It would also provide an experimental LE'st. of the scalin g prwlictiou (2.26). Such a n JCasurcmeut requ irE's a regime iu which ~(Q) is larger than Lhe width ll ' of the tlterm ometer. Present technology places a limit ll' > 50 f.l.lll. From (2.30). heat currents below 1 f..LV\'jcm 2 arc required. Pn'scnt trchnology allows controlled ])('at current s down t o a bout 1 nV\' / nn 2 , leading to~~ 1 mm , so at first sight such an cx pr rimeut a ppears feasibl r. In fact , Ea rth's gravit~r 9e places a fund amenta l lim it on the maximum possible interface wid t h . As described in S<x·. 2.1, gravity produces an equilibriu m (Q = 0) superfluid-norm al flu id interface with width of order ~(.qe) = 100 wn. From (2.30), ~(Q) = ~(g, ) for Q ~ Q 9 = 70 nv\' /cm 2 . For Q of order Q9 , gr avity will begin Lo have a strong effect on the nonequilibrium interface [59], aud for Q « Q9 the heat current will be a small perturbation ou the equilibrium interface. T hus ~(Q) will sat urate' at ~(ge) as Q -7 0 a nd tlw regime E;(Q) » W is unattaina ble ou Earth. For this reascm the critical dy na mics experiment (DYNA~ I X) is curre nt ly being prepared for tlH' microgravity euviroument of tlH' Internatiou al Space Station, with fli ght pla nned for 2004. Heat currC'n ts as low as 5 n\i\)crn 2 will br used. The tC'mperaturc· profile T( :::., Q) will be' measm C'cl t o subnanoKcl vin resolu t ion by slowly moving t h<> in terface pa st a fixed t hermo meter. T lw latter is accompl ished b.v removiu g heat from t he downstream end of t hC' cell s li ghtl~r m ore' slowly thau it enters the u pstream eud , with t he eff<>ct t hat t he normal phase slow ])' invades tlw c<>ll , thus translating t he in terface. 2.3.2 Transition from thermodynamic to interface state \\"0 ha\'C' dis<'ussed thr<'<' possibk dass<'s of s t <•ady HOn <'quilibrium s t at c•s: (<1) <·ouducting normal sta trs with static tc>lllJH'rat ur<' pro fil<' cktrrmi n<'d b~· (2.27); ( b ) isot h<'l'mal t hrr m od .\·na mic stat rs with st <'<HI ~· s upr rHuid counterflow d<'trnnitH'd h~· (2.2); a nd (c) Hl ate>s with a Honequilibrium int r rfacr forming a ·'conversion bo undary'' lwt\\'<'011 s t a L<'s of type (a) and ( b ). TransitionH b etwe>rn (a) a nd (c) occur cont inuously: if st.aL<' (a) is ('Oolrd to Lht• point wh<'r<' Llw d own stream end wa.ll temp0rat.urc• passrs th ro ug h T;., a u iuterfac<' will form o ut o f that <'ndwall and steadily n1ovr upsLn-'am. Cou vc•rs<•ly, as h<:•a t is a dd ed t o s t at<' (c), the int erface will move dow nstrcalll until t h<' iut<'rf<-H'<' disapp<'<Hs iut o tlH' <'ndwall. S imil ar! ~·. tlH' trans ition from (c) t o ( b ) is cont inu o us. Til<' int <•rf;H'<' will disapp<'ar int o t hr 11 pstrram end wall as IH•at is c•xt ract<'<L .Y i<'lcl i ng an isot lH'rma l t h<'n nod.vu;uu ic sta te• at t<'mp r rat un' T 00 (Q). furth r r c•xtrac tio n of hPat w ill causr thr l<'lll(><'ratur<' to dro p be> low T00 ( Q). ThE> nat urP of t hr transition fro m ( b ) t o (c) is lrss d rar. At issur is whrt ll r r thr bulk s upPrfluid. in th<' abs<' n<·r o f a n inLrrfa('<', rccogni zrs T 00 (Q) as a sprcial t.<'mp<'ra1ur<'. \\'ithin tlw mcau fie ld a pproximatio n , tlH' answer is no [4]: Lh<' thermodyn a mi c state• is s tab! <• up to a t r nq)('rat ur<' T;.(Q) . with T;. > Tc(CJ) > T (Q), w hos<' functio nal illv<·rsc• CJc(T) W<' mig ht id<•ntify with th<' b o unda ry in Fig. 2. l (b ), at which t h e• SIIJ><'rflu id d<'Bsity is s uppr<'ssNI to the• point where it is inc:apablc of s uppo r ting the h0at c·tliT<'nt. As s hown in Fig. 2.4, whc•JJ t h<' tlwrmodynamic s tat<' is )l('at<'d abov<' T;.(Q) n compli cat<'d d~·namics rrs ult s, with th<' s~·st(' m fin a l !~· s<'t.tli ng dowu into a stat<• with an int crfacr. S iner T (Q) < Tc(Q) , thr s upcrftui d s ick ac tmtll ~· ('Ools, with a c·omJwHsa tor~· h<'at i ng of t. lw no rma I s idr. Tlw fina l position o f' t h<• i nt rrfacr is d Ptrrlll im~d from en erg~' c·ons<•rva t.i o n . Til<:' ques tion o f whet her or no t a first ordrr transition from ( b ) t o (c) smT iVC's in sonw fo r111 b ryo nd the nl<'an firld approximation is s uhtk, a n d is probably a quc•stiou o f timE-' scale. The sam e thermal nuci N'lt io n of ,·orticrs that kads to a small t<•mp<'ra ture gradien t in t he s upPrfiuid phasr pr<'Stllnably also leads to a continu o us ( b ) to 4G r 110 I I I I I 9 -0.6 \ I \ \ \ \ 8 \ M \ 7 \ \ \ \ \ \ \ - 1.6 6 Q \ \ \ \ \ ~ ~--~~~---~---~---~----------------------------4 - " ::::-::::--- - ,, -----::::. 15 ' ' - ' --- - - - JM 1 -i c _1 20 z 25 30 Figure 2. 1: For bulk scaled t<•mJwrat ur<' (s<'<' captiou to Fig. 2.3) 1\f > !111• t lte unifo rm SUJWrfluid s tate becomes uns table• to a s ta t<' with an iut erface [G1]. C urws 1-G (solid ) s how conS<'CII tiw stablr ( timr i ndrp r ndc•nt ) Sll prrfluid tem prrat u rr profi I<'S obtai ned in t h<' lll<'all fip)c) approximation 11 pon sl ow)~· var~· in g the rig ht- haud wall tPm p<'rat ur<'. C ur v<•s 7-10 s how s uapshots of th<• t inH' c•volntiou, computed within a s imple• OIH'dimensional modeL initiatPd b:v a serirs of phase s lips (not s hown ) after the righthand wall t <'mprrat un• is rai sNI s li g htly a bow that of eurw G. Tlw kc•y fpatur<• is t ltc• Jl<' l temperature drop (front !1/,. t.o fl./00 ) obs(•rwd in the• bnlk SllJH'rflnid . (From \Veidunan and ?diller [Gl].) 47 (c) t.ransi tion, with the int<'rfac0 eut0ri ng continuou sly from the upstream end wall, if the 0xperinwntal heating r ate is infinitesimally slow. Any finit0 heating rate may, howewr. allow a ''sup('rlwating'' of (b), inducing an apparent first order transition to (c) nucleated b~' a random vo rt ex cr0ation event. Such an effect would be analogous to superheating and supercooliug effects at couvent ioual first order transitions, the boundar)' Q,.(T) being analogous t.o a spino dal line al "vhich a local free energy barri<'l" separaliu g the superheated or superwoled state from t he true cqulibrium stale disappears. Exp0ri IW' n I.a.I investigation of tlw (b) to (c) trausi tiou is compl icated by bounclar.v df0cts. T lH' sing ular I\:apitz;a resistance leads to an additional heating of t he upstream Pndwall, which tlH'n acts as a nnclC'at ion point for vort ices ev0n when th0 temperature of the hulk s tqwrfluid is still below T 00 (Q) [25]. C lever cell designs t hat rcducr t lw heat current near the Pnd wall below that in the bulk may event ually allow experimental obscrvatiou of a superheating efl"ecl. 2.3 .3 Interface dynamics Despite the great l)' reduced gravity, the Space Station environment is less than id eal for other reasons. \ 'ibratio nal n oise ( "g-gitter") exists at a level of about 10-:l ge, aud OIH' mi ght worry that such noise might couple strongly to t he interface, and J><'rha ps destabilize• it. U uderslall(liug the cfl"ccls of acceleration noise requires an nndNstanding of (a) the free dynamics of thP iuterfac0 wlwn it is perturbed away from its st<'ady state, <'.g. , whetlH'r or uot the interface is even d ynamically stable, or whrth0r small perturbations might und ngo somr kill(] of "dendritic growth ," and (b) the m culnN in which vihratio u. or other perturbations, couple' to, and pC'rhaps amplif)' t his motion [60]. S uppos(' t hat a slow ,-ariation, z 0 = z0 (r ), wit h thr transvers0 coordimttes r = (.r, u) , is app lied t.o t br interface position a nd is subsequentl y released . Th<' problem is to deriv<' an C'quation of mot ion for z 0 (r , t). In order to motivate thr intercsti11g ph~'s ica l questions. it is HSC'ful to consid er first t h0 a nalogous problem of the motion of 48 an equilibrium intNfac<' b<'tW<'<'n up and clown domains of a n Ising ferrom agnet , with difl:'usive "ModPl A" dynamics [70]. Surface tension acts as a restoring force against p ert urbat ions away from a flat interface, and o11e ma,v derive a diffusion equation for the rPlaxaLion of long waveleng th perturba tions of the interface position: (fJ1 - D\7 2 )z0 (r. t) = 11(r , t). (2 .31) in w hich the value of D depends Oil the rnicroscopic parameters in the m odel, and the driving t crm 11, vanishiug for free relaxation, ~r as been included for complc't.cness. Tlwnnal fluctuations iu tlH' spins lead to a Gaussian whit<' noise form fo r 11 wit h cmwlator (r,(r , t)r,(r', t')) = ( 0 b(r - r' )o(t - t') whcrr ( 0 d r prmls on microsco pic parametN s and on tcmp<'ratnr<'. Using t his form on<' m ay compute the <'qual timr variancr in thr interface positi on ([z0 ( r , t) - z0 ( r' , t )]2 ) C(r - r ') ~ ln 4?TD (lr-r'l) ao ' (2.32) wherr (.Lois an atomic lr ngt.h scalr . Onr sres t h at flu ctuations in the interface p ositi on dive1:qe logarithmica lly with separation: a famo us res ul t kiJ own as interface roughn ess. eneountrred most frequeutlv in disc ussions of crystal facet shapes. T he interface is locall,v stable and flat, but globally wanders arbitraril.v large distances. TIH' result (2 .32) shows that there an-' subtle physical issue's , lying beyond thr much simpler quest ion of stability, arising because t he interface breaks t he contin uous translation inYariau ce of thr s,vstrm . When ever a continuo us symmetry is broken , a Goldston!' mod r is g<'n<'rat<•d, co rresponding in this case to VC'ry slow relaxation of long wavelength prrturbations of the interface position. z 0 ex: ('ik·r· - >.(k) t , wit h (2.31) ,vidding /\ = Dk 2 . Such long wavrlcngth "rnodrs" are highl y susceptible to t hermal or cxtrrnally generat ed noise spectrum ( (k) , and it is the conv<'rgence at sm all k of the integra l 2 (zo(r , t) ) J = (2?T)2 2 d k ((k ) ReA(k)' (2 .33) t ha t dpt enuiu<•s whether or uot t he interfacf' is rough. For t lw case ( ( k) _ (o. (2 .33) divcrge•s logarithm ically, wh icb same d iwrgrm·<' is rr flectf'd ill t lH• corrclator ( 2 .32). Thr ph?sics of t hr li OIH'()llilibrium superfluid-11ornia l in tNf<u·e• is v0ry diffe•n•nl from t hat of llw equilibrium magnE'tic interface. Transport on t hr normal sidr of tlH' inte•rfacc• is d ifl'usi ,.r. but t ransport on t he su p<•rfl uid side is essen! iall.v ballistic. and t he dyn<Ulli<"s of thr int e•rf;u·c• itself iuvolve•s a ve•ry iut ricate <"Oupling of the two. To leading ord e•r o n e finds an <'qna tion of motioll [GO] (D1'L - c'L \1 2 ).::o(r . f) = 11(r , t), (2.31) so t hat the iat<•rface s upports tnwPling wave rxci tat ions, wi t it a w<•ll drfined sperd d Q) of l he· S<U J H' ord <'r as the bulk second so und sp eed. At next-to-lead ing ordf'r one fiuds t hat l he•se• <'xcit a t ions are singalar·ly damped. with A = i('k + Dk 3 12 , so that R0.A k:l/'1. rat lwr tha n k 2 (as for hulk s<•e·orHI sound waY<.'s) at smal l k. Positivity of ReD esta bl ishes int rins ic d~· H am ical stabil it~· of t he int erface. One• may intrrpr<'l t h0 euh auc·<'d damping as arising from t he •.vaves 0 11 t he interface' ·'rub bing" up against tlH' 11onna l pbasc•. :\l orcoVC'r, O I H' finds tha t thC'rma l fl uctuations en ter via a spectrultl ((k) rx k 5 f '1. for IJ , n m ishiug st rongly as k -t 0. T hP physics of this rPstil t is r<'iatc>d to th0 fact that int0rfacr lli Otion is a coopemtive phC'nonw non, invoh·ing <'VaiH's<·e•nt dyna mics o f t he s u perflui d ordrr param <'t<'r t o a dr pt h "'k :l f 'L scali ng as t hr ~ pow<'r of t he• wavd mgth :' T he• microscopic t lwnual noise, whi ch is whit<', must t hen lw a\'('rage•d owr a si milar vo ltt m<' to obtain its 11ct cffc•ct on t he mode, lcadiug ewnl ually to t lw gr<'at ly rNluccd ((k) abcm•. lu cont rast. the• Ising int<•rft:H'<' moves by local spin fli ps a n d IIH' !ll i <"roscopic nois<' is avC'r agC'd o nl .v over a m i C"r osC"opi C' r<•gi o n of w i dt h o 0 and ( r<'m <=tins whi te•. Tlw net n•sult of t lw a t~ al ysis is t hat t he• in tc•gral (2.33) is s t rougly conwrg<~nt at k = 0. a nd th<' snpNflni d- normal int<'rfac·c• is globa lly flat . T his is good rH•ws for DY='J:\.\ lX, wh<'r<' a roug h in trrfar<' wou ld hav<' l0d to substa nt ial smearing of t h<' temp<'rat nrc• pro filr on a scale• varyin g as tlw logari t hm of th<' <·c•ll <Toss-srct ion. 1 1 ' TIH' <'VaJlsn·nt d<.>pth 01 1 t h<' norm a l s id<' is muc h small<'r , scaling as k / '2. For comparison, gnwity wm·<•s ou Huid s urface'S .vi<'ld flu id mot ion to a <l<'pt h ,...., k - 1 proportional to th<.> waY<'I<'n p;t. h . 50 Vib rat iona l accelerat ion noise couples to the in terfac<' i11 t he same way that Ear t h 's gravity docs, t hrough t he ,·ari a Li on iu the local T>.. with pressure. A slight change in T>.. will cause t he interface to t ranslate (for acderation normal to the interface) or tilt (for a<T<'k ra t ion pa ralkl to t he interface) slightly, bu t if t he> chauge is oscillator·y it could r<'sonatc with on<' of t h<' in terfacia l S<'cond sound modes, k ading to a rapicl growth in t he interface mot ion . A cletail r d examination of t he rxpected frequency spectrum of the S pace S ta tion g-gitter , together wit h t he discrete spectrum of s tanding wave modes allowed iu the experimental cell wit hin the planned temperature a nd heat <·urr<'ut raug<'. shows t hat s uch rcsouan('es m ay indPcd occur, bu t t hat t hr siugular dam ping is suffici c> nt ly strong that. t hr interface oscillation amp litude should satu rate' at acce pta bl ~' low levels [73]. T he existence of the iuterfacia l second sound m ode has yet to be tested Pxp erimeutall y. T his migh t. be accomplished b.,. applyiug a sequence of heat pulses to the cell sidrwalluca.r t h<' i11 trrfacP and dPtccti11 g a resp onse at t he opposit<' sid ewall. ScaLt<'rin g of bulk srcond sound pulses off' t hr int<'rfac<', wit h dr trct ion of t hr rdkct<'d pulsrs, might a lso proYidP int N csting information a bout t he c-oupling of bulk and intrrfacc modes. 2.4 The self-organized critical state \Ve have so far disc ussed phenomena i11 which optimal conditions occur in t he abseucc of gravity. ll transpires that there is a very interestiug phenomenon iu which gravity and lH·at currr nt combine to prodnc<' a new type of dynamical state. T he so-called self-organiz ed r.7'itiml (SOC) stat(' occ urs in a ('(•11 which is heat<'d from a h OY(', so t hat g and Q arr parall(•l. 5 Gravity d<'pn•ss<'s th(' lamb da point T>..( z) wit h increasing drpt h accord in g to r' \Vhen Q a nd g an• a utiparalle l (heat from below) t hey cooperat<' t o simp ly prod11c<' a sha r pt?r in t<'rfacf'. T lw na mf' SOC is mo t ivat<'d by non<'qllilibrium states fou nd in '·sandpilt?" models displaying critical power law '·avala nchf'" behavior \vith no appar<'nt t uning parameter [74]. A lt ho ugh the "SO" part of SOC is j ustifiecl for the 4 He state, the "C'' part is uot sine<:> analogous critical power laws have y<'t to b<' demonstrat ed eit her t heoretically or experime ntally. T he name, however, has st uck and we will 110L attem pt. t o al ter couveut iou here. 51 (2 .1 ). while heaL current leads Lo decreasing temperature T( z) wit h d epth according to (2.27). T h<' reduced temperat ure' to( z ) - [T(z)- T..\( z)]/T.~ ,o - '>vhere T..\.o is, say, t he hulk transition t.<'mpcrature if gnwi ty were absc·nL (and hcuce approximate!.\· t h<' transiti on temperat ure~ at t h<' top of th<' ce ll ), contains a competition hetw<'Cll thes(' two df<'cts. On<> might. in fact , imaginc tuning Q in such a way tha t. f( z) is inclependent of z [10]: the sample woulcl a pparently exist in an essentially homogeneous ncar-critical state. In fact , it was argm•d [75, 76] that t he system act ually ''self-organizes" T( z) iu order to enforce a uniform f. Assuming the validity of the Fourier law (2.27), f must he uniquely ddincd by (2.35) whNe fsoc, a fum·tion of the ratio CJ/ g, is t lH' red uced temp erature of the 11ew state. This state has reccntly !wen observed experimentally [58]. T lH' fact t hat h: increas<'s with decreasing T ensures stability of the SOC stat<' to small p<'rturbatious [7.'), 76]. l\Iorc sp<>cifically, an analysis of tlw heat d ift'ttsion equation in the normal phase [61J shows that a pertnrbation f( x ) = fsoc + Jf(x) obe.vs an equation of motiou whose solu t ions are deca:ving plane waves of the form (2.36) with Dsoc = r.:(fsoc)/Cp(Esoc) aud csoc = -IDzT..\ir.:'(tsoc) / TA.OC11 (Esoc), where C11 is tlH' equilibrium SJWcihc lH'at at cous taut pressure. T hus, in addition to the decay controlkd b~· Uw cliffusion constant Dsoc, there is an uuexpected anisotropic; propagation effect where the perturbation moves upstream at speed c50 c;. For the reasonable vahH' Q =50 n\\l'jcm 2 (sec below ), one find s r·soc ~ 2.8 mrn /s, ancl tlw propagation cft"ect shou lcl lw <'Xpcrim entally observable for reasonable cell gcometrics [Gl] . Since r.: divC'rgcs as f -7 0, (2.35) implies that Esoc -7 0 as Q -7 oo. Howcvcr, for large Q t he Fouri<'r law br<'aks down. In particular K-(T>., Q) is fin ite for Q > 0 [11], and the SOC stat<' must th erefore li<' belo·w T..\( z ) for sufficiently large Q > Qsoc· This is consistent with experimental data [58], reproduced in the inset to Fig. 2.5, 52 -2.0 ,-~--~------,-----~----~------~-----,-------------, - - - - - - - - - - - - - - - M=(O) -3.0 M - 4.0 -5.0 ''' Q,G ''' ' ' ''' ' ''' ' G=0.04 I j Fi g ur<:> 2.5: Sim ulat ion of thr SOC state using thr same simplified 1-d model as in Fig. 2.4. For G oc gjQ = 0 the tcmp0rat.urr gradient in the normal phasr (Z < 20) giws way t.o a u asymptotically isot hcn ual s uperfluid phase (Z > 30) with temperature T 00 (Q) < T>.,O · For G > 0 Lhe s upcrfluid phase develops phase slips (vortices in 3-d), and r1 correspondi ng dynamic staircasr struct ure in T( z ), roughly bounded b etween T 00 (z ) a nd Tc(z), to produce thr SOC state. T lw drnsity of phase slips iucrcases with G. Inse t: cxpcrimcutal data replotted from F ig . 4 of [58] showia g t.hc self-organi zation temperature , .6.T(Q) = T(Q, z ) - T>.( z ). On ly for Q < 100 nW jcm 2 (sm all shaded region) is the SOC state in t hr normal phase. (From \ \Teichman and Milkr [61 ].) 53 which shows that tsoc < 0 for Q > 100 n W jcm 2 . Using (2.35), along with (2 .28) defining the validit y of tlw Fourier law. one may show that the previous theory 1s valid for [14, 61] G. 1 Q ( lOOn v\)cm1 ) (l+v)/JL ( )1 -( l+v)/JL !!_ 9e « 1. (2.37) E q11alit:v in (2.37) s<'rves an <'Stimatc for Q soc and yields Qsoc : : : : 60 n W jcm 2 mHkr Eart h's gravity, in vc·ry n·asonahl<' agr<'<'nt<'llt with thf' cxperimNJtal rf'std t. Tlw quf'stiml now r<'mai ns as t o t lw nature' of th<' SOC stat<' below T>... T he st at<' must undergo somf' kind of t ransit ion to superfluidity, but t he fact that it contin ues to support a finit e temperature gradient appears inconsistent with t he isothermal nature of a superfluid . T he resolution of this paradox is shown in Fig. 2.5. Let T 00 (z) = T>..( z ) - 6.T00 (Q) a nd Tc( z ) = T>..( z ) - 6.Tc(Q) define local values of the interface and instabili ty t('lllJH'raturrs discussed in Sec. 2.3.2, wlwrc 6.Tc = T>..,o - Tc(Q) and 6.T00 = T>..,o - T00 (Q) a rr t hf' df'v iations from T>.. in zero gravity. For g = 0 an interface state. represented by t hr uppN cnrw in Fig. 2.5, is formf'd. For g > 0, T( z) first drops below the local trausition at a point z 0 , a nd attempts to asymptote to an iso thermal superfluid at a temperature close Lo T00 (z0 ). However, at a point z 1 > z0 , the descending lin(' Tc( z 1 ) m eets T00 (z0 ) and the s uperfluid becom es um;table. A vortex is generated and crosses the cell , leading to dissipation a nd a finite temperature drop across it. By this mechanism the temperature is a ble to drop below the instability temp erature and Rsymp tote once again to an isothermal superftuid at a temperature close to T 00 (z 1 ). The entire scenario t hen repeats itself approximately periodicall y at a sequ euce of points z,, n = 1,2,3, ... , where Tc( z 11 ) m eets T 00 (Zn- 1 ). T he result is a d ynami c staircase structure, a snapshot of which is represented h:v t he lower curve in Fig. 2.5. T his s tructure fluctua tes in tim e as vortices form a nd a nnihilate, but IS found numerically to siO\v ly move escalator fashio n upstream [G1]. The time resolu t ion of present t hermometry is far too poor to detect fluctua tions m Lhe temperature profile, whose m cau wo uld correspond to a s traight line parallel 5-1 to a nd SOill<'WiH' n ' b<'tw<'<'ll T. (.::) and Tc( z ).6 Exp0riments to d<'t<'<"l Ill<' prwlictrd s t n 'Hlll of \'Ortic<'s . Yia th<' s<'<"OtHI sound nois<' tlwy g<'n<'r<lt<', <H<' in t h<' pla nning s t ag<'S. . \ s i m pi<' ealculat ion s hows t hat the d is t anr<'s b<'t"'<'<'n \·ort ir0s lllUSt seal<' as .:: 11 - ::: 11 1 ~ [~T00 ( Q) - ~Tc(Q)].q1 h· q , incr0as in g with larg<'r Q a nd Slll a ll<' r g. :\It hough tlw SOC sta t<' r0quin•s fin ite g rav it,\·, the regime o f widely separat ed vorl icrs lllay in the futur r pro\·e suffici0ntly interesting that a coutrolled low g raYity <'XperinH'UI will lwconH' dPsirahlc. 2.5 Specific heat at constant heat current 2.5.1 Enhanced specific heat T il(' pr<'S<'Il<"<' of a h t•at <"lllT<'Ilt is pn•dictc•d to <•nhanc<' thr SJH'<'ific heat of SUJ)('rfluid 1 11<• ahow its <'qu ilibrium Yalue . C0 . This ean rasil~· he seen at low Q, s uffici0ntl y fax b<'low T>. , wh<'r<' t IH' fr<'<' <'nNg~· rn h<:Ul<'<'tnrnts ar<' .:::..F(T , U s) = ~ PsU~ and ~<I> (T, J.,) = - J ~j2p8 , in which Ps(T) ~ Poicl( is the equilibrium s upNAuid dcns it:v. Thus, at fi xrd s tqwrfl uid ve locity U ,, , (2.38) whil<' a t fi xed heat c urrent Q. (2.39) C losN to T>.. the s up<'rfJuid d<•usit:v drpeuds s tron gly on heal c urrent all(! LlH' apparcut divergences at T>.. iu (2.38) a nd (2.39) ar c replaced by lH'W siu g ulariti <'s at LIH' phas<' bouudar~· T,. (U8 ) s how n in Fig . 2.1(b). lt. is predicted that ~Cu., will remain fiuit<\ ris iug to a cusp at the phas<' boundary [12]. \vhile ~ Co is pn•di('t<•d t o div<'rg<' [8. 9]. T lw la t ter result fo llows on very general g rounds fro m the thc rmo d .v uamic cou(i A n ' no rmalization g roup calculatio n of this m<'an profil<' a!> a function of Q in t h<' largC'- i\' limit is d('scrihPd i11 [17. 18] . 55 jugacy of U s and J 8 discussed in Sec. 2.2. T he usual thermodynamic m anipula tions imply tlH' rf'lation Cq = Ct,,, + T ( D.Js ) 2 ( 8Us ) . fJT Lf., 8./8 T (2.4.0) Siuc0 tlw susc0pti bility, (8U8 j8J s)r [proportional to th0 slope of the curve in F ig. 2.2(b)] is expPct ed to di vPrge at tlw phase boundary, while (8 J 8 j8T)us = U8 (0Ps/8T)us remains finite. Cq will exhibit a divergent enhancement. There ha ve been no m eas urements of Cu., to da te, but au experimeut of t his kind might lw p0rfornwd in tl)(' prcsrtH'<' of a persist?nt current flowing around a loop, sirnilar t o th0 s uperfluid gyroscop<' cxprriment [77], whrr0 in thr absrncr of vorti<'rs U .- inde0d rr main s fix0d as T is vari ed . A mrasurpmcnt could proVC' difficul t du <' to thr small mag nitude of the enhancement and the challengr of holding U ,. constant whil<> nwasming t he specific heat. However, it has bPen sugges ted th at with VC'r.v fast thermometr.v. a m easurement of Gus might be obtained t hrough a second-sound m easurem ent where the second-sound waves are propagated p erpeudi cular to U s [16). 1\h '<\Sur<'m ent of C'q is m ore straightforward. The predicted divergence of !:::.Cq. togetlJCr with the fact t hat Q cau be ex])('rimentally controlled with great precision, impli es a much more visihl(' experim<'ntal signature. T h0 fi rst experimen tal meas urem<'nts of this quantity wcr0 recently rep ort0d [25) . As will he discussed below. tlwy indicate that t hC' heat capacity is im!C'ed cnhauced, bu t with a magnitude th at is sig nificantly larger than theoretical predict ions. 2.5 .2 The superfluid breakdown temperature Expcrirnrnts pr rformcd at constant Q should find t hat Cq diverges at a temperature Tc(Q) < T>.., d efin ed by inverting (2.21): (2.4 1) wher<' Q6 = Q0 yc and .T = ] / 6.q = l /2v::::::::: 0.746 [5, 7, 72). Bas0d on a renonnalization group analysis of tlw l\Iodcl F <'qnations [70), in au approximation neglecting vorti<"<'S 5G (and h r u<·<· dPca.Y of SUJH'rflow), the predict io n Q 0 ~ 7..! k\\'/C'm 2 was obtainrd [6]. :-. Ion• r<'<"<'nt 1.\ ·, usin g a u c•xt<'nsion of this throry, accounting for dissipation within a larg<·-N approximation , the• value• Q 0 ~ 6.6 k\Y jcm 2 was o btaiurd [18]. Tlwsr tlworrtical r<'s ult s disagrrr with t lH• rrsults of thermal ('Onduclivity <'Xperi m r n ts [19]. T he o nset of thermal n•sist a nee was fouud to 0('(' 11 r a t a t <'IllJ><'l'a turP, whi('h we• ('all TnA s( CJ ). that obe.Ys (2. 11 ), but with .r = 0.813 ± 0.012 a nd Q 0 = 5G8 ± 200 \V / cm 2 . This is a curve iu t he• T -Q plane that fal ls I><'low LlH' tlH'Or<'l icall:v Pst.imal<'<l ~. (Q) for all <·xperimeutall~· c-u·c<•ssible te!llJH'nltur<'S (sec• Fig. 2.6) . .-\ uumber of explanations hav<' IH'<'ll propos<'d for til(' discn•p<UH',Y : (<l) t h e transition at To.-L<;(Q) ltlay h r c:aus<'d b~· a t<'lllJH' ratur<' in stability a t thr crll wall associatrd with t ]H' s ing ul a r Kapitza n•sist.ancr [which raises t h r tcmprrature n rar the bottom (hPat('(l) c•nd platr ahow t hat of the bulk, w hi ch th erefore could serve as a vortPx n uclratio u C'C'nt er]. a nd hr n cr lies below ~· ( Q) [25] ; (b) Tn/\s ma.v br rclat<•d to a graYi t.Y-drpeudent trans it io n . again ly ing below ~- ( Q). and will increase towards Tr (Q) as gra\'it .\' is r ed u<·ed. e.g., by going i11to s pace [18] : (c) s iu<·c the trans ition is onl,v s harpl,\' ddined in LIH• abs<•uce of vorti('<'s , TnAs(Q) wig ht I><• a nalogo us to a spinodal litH' iu a firsl-ord<' r phase traus i t ion [57]: fluet ua tiou- i nd uC'rd vortices nudrate the transition to t. h<· diss ipa tiYe phase. and TDA s( Q) will diffrr from cxp r rim r nt to C'XJH'ri m eut , dc•p<'ndi ng on t IH' lH'a ting rate• usrcl. Since an~· s u p r rfl uid statr above T 00 ( Q) should hr unst able• h~· this m c('han ism , an infini t essima.ll:v s low experiment should find thr trans ition at T 00 (Q). It is poss i blc t hat a ll o f t hrse effects (and perhaps othe rs) an• pr<'sent. The r<'al questio n , wltos<' resolution dcarl.Y req uires m on• rxperimcntal data, is which on<• impos<'s tlw II lOSt se,·en• li tni t a 1 ion 0 11 presC'nt CXJH'rim<•nLs. Tlte a u sw<•r to this quest ion has implica tious for the uwas ure ntent of Cq(T). If effect (c) is dominant, then experime Ht s ha\'(' bas ica ll." a ln•ad .' · n ,·ached tlH• inLriusiC' limit 011 how d ose they can approach th<• divergeuce of CQ(T) (for tht• ra nge of Q explored titus far ). though it m a_Y l>C' possible to desigu an <•xperimcnt with fas t r r hrati ng, rat <'S and fast C'nough thcnnow<>t('rs to n •ach T,.( Q) hcfon • a vortc•x nw nnclcatr. On t he• other h a nd , if proposal (b) is <·m-rcc·t , a spa('<•-based minogravit.y mras nrrtn <>nt of CQ(T) should 57 -6 ~ -6.5 w -7 • -7.5 the DAS data c ? x = 0.8 13 and Q = 568 W/cm- ?. 2 x = 0 .746 and Q 0 = 3400 W/cm 2 x = 0 .746 and Qc() = 7395 W/cm -8~------~------~------~------~~ -0.5 0.5 0 log J 1.5 2 10 [Q( J..LW/cm )] F ig ure 2.6: T hick solid line: Tc(Q), the theoretically predicted temperat ure of superflu id breakdown [6]. T hin solid line: TuAs(Q), wh0rc t he exponent :r and amplitude Q0 a rr chos('n t o best fi t to the observed temperature of superftu icl breakdow n represented by t he data poiuts [19]. Dashed-dotted line: the value of Q 0 thaL, along with the t heoretical valu r J" = l /2 1/, makes t he experim ental heat capac i t~· data m atch the more rcc0nt t hcorctical prediction fo r t he scaling function [18] (sec F ig. 2. 9 b elow). 58 IH' able• to ge•t co nsidrrabl .v dos<'r to Tr(Q) th a n o n <' prrform<'d on thr ground . If proposal (a) is cotTrct , carrfull y d<•s ig nNl g round- basrd rxprrinH' nts mig ht I)(' abk to a pproach Tr(Q) m or<' d os<'l.,·: if TD.\s is due' t o a b oundar.\· <'fff'ct, a C'<'ll cons truc t <•d with a bot t o m plat<' that is 11111 C' h la rgc•r than t hr rross-S<'Ct io n a l arf'a o f t hr bulk hr lium sa mple• could d Pcrras<' th<' s ing ular Ka.pit za rPs is tance s uffici ent!.\· so that the bulk hr l ium can rrach Tr( Q) without a b o undary instability interfering. A C<'ll of this config uration that maintains a reason a ble• g<'omrt.r.\' for heat flow wou ld hav<' to b e fairly ta ll and might thcrcforr lw mon' s uscepti blc to gr av i t..'· df<'cts. 2.5.3 Experimental measurements T hP firs t <'Xpcriment a l m easun•m e nt s o f Cq (T) [23] confirm the pn•dictc•d ('Jlhau<·c•ntell t. but find that it s mag ni t ud<' i~ sig uifieautl.v la rger than c utT(' lll pr<'did io ns [8, 9. 18]. T he d ata wen' takPu owr t h<' ra nge 1 ft\Y j cm 2 2 ::; Q ::; l ft\\'j cm . and a 2 r<'pr<'S<'n lati ve' S<'l at q = 3.5 ft\\'jc tn is s hown in F ig. 2. 7. TIH' ext<•n t of' tlw disagr crmc'n t bc•t wcrn t IH'Ory and c·xprri nwn t cc-1 n h r o bsrrv<'d m orr ck a rly wlwn the• da.ta arf' plot te•d in scakd form. All t.hcori<'s prNlict that. th0 e nha n c·0nH'nt should obf'y a scaling fon u (2.42) w hC'r<' .f.J, (.r) is a uniwrsal scali11p; fu n C't io n . From (2.39) and thC' scaling r<'lations in Tabl0 2.2, for .r « 1 one has f 2 .r2 + O(:r 1 ) [(((+ I )(Q~) 2 /2 poT]SiJ. (2.43) Using th0 the'oretica l <'s timat0s o n<' obtains .f2 = 8.9 .l / mol K for Q(j = 7.4 k\\'jc m 2 a nd f 2 = 7. 0 .J j mol K fo r Q 0 = 6.6 k\V /c m 2 , w lwn• th<> mol a r volumf' 27.38 cm:l / mol has be'<'ll usN I t o obtain familiar uni ts. 1n F ig. 2.8 \\'(' s how th 0 ll<'at capaci ty 0nhancrmC'nt as a function o f t 11<' sca ling 59 110 ...-~ 105 ..- Top_ 1-IRT 0 s ;:::::; 100 '-" Bottom-+ HRT 95 I I 90 T (Q)~I c . I TDAS(Q)-: 80~--~------~--------~--~----~ -3 -2 -1 T- TA (J.1K) 0 Figure 2.7: Sample heat capacity data for Q = 3.5 tt'v\'jcm2 [25]. Thick solid line: equilibrium C 0 obtained from a fit to t he LPE data [2] that is subsequent ly rounded for gravity. T hin solid line: t heoretical prediction [18] that iududcs vortices (rounded for gravity) . Solid circles: data from the average of t he top and bottom thermometers. Open circles: data from the top thcrmometC'r only [b cyond th<' point. markccl {3, wherc th(' bottom (hotter) thermomet<•r was fo und to chang<· its behavior, pcrhaps as a result of a l>oundary heating effect]. T he temperature TnAs marks t he onset of dissipation found in earlier t hermal cond uctivity cxperiments [19], while Tc(Q) is estimat<'cl from a certain th0oretical fit to the data d iscussed later in the text. Inset: sclH•matic diagram of the experimental cell. HRT stauds for high resolution t hermometer. (From Harter , Lee, Ch atto, Wu , C hui , and Goodstein [25], Fig. 1.) GO variable• (Q/Qc) 2 . Since Qc(T) is not actually measured iu the <'xperim('ll(, \\'C' scale tlH' data using t IH' theoretically prc•dictc•d form Qc = Q 0 jfi~Q with Q 0 = 6.6 k\\) nn 2 a nd .:::.Q = 2v = 1.3-12 [18]. As anticipatc•cl. the data for a ll Q 2: :2 ft\\'jcm 2 col lapse on t o a s ingle• line•ar cmn>, \'C'rif.ving that t he• <'XJHHH'nt .:::,.Q is at Jc•ast consistc•nt wit h I )J(' data. H owever. the s lo p e of t IH' exJH'rimental line is f;xpt = 69 ±·I .J / mol K. ap proxin1 atc•J.\· t<'n tim<'s larg<'r than th <' tlworNiral prediction. Au o ptilllis tir <'Xplanation fo r the• di scre pa ucy between th0ory a nd experin1<'nt is that the tlwo ries <H<' producing reasonabl<' estimates for the univC'rsal scaling function f.1 , (.r), but that tlw nonuuiversal a 1uplitud<' Q 0, depending on detailc•d propc•rtiPs of tiH' 111<' s_n;tem and therefore• more' difficult to compute, is PStirnated IC'ss a<-curatc>ly. . \ :--. illu stratc•d in Fig. 2.9(a), tlw dwin•; Q 0 = 3.-1 k\\'jcm 2 in fact pl aces the c•xperinwnt al data on top of the Ill Or<' n·n•nt theoretical curw [18]. \\'i I IJ this cho ice, ~-( Q) li<'s sonH•what abcm' To.\s(Q) (V('rtical dasiH•d litH' in Fig. 2.G). :\. sonH•w l!a t smaller choice• for Qf> would prm·ide an eqnally good fit to the <'arliN ti H'Ol)' [6]. Cufortunatc•ly, all of til<' data lie• at fair!~· s m a ll (Q/Qc) 2 ::::; 0.3 wh<'n' tlH' sc-aling functiou has littl<' st ructnr<' (<'ssenti<llly indis ting uis hable from linear w it hin tiH' sc:at tc•r of tlw clat a,). A tnt<' t<'st \\·ould r<'CJll ire data i 11 t.h<' reg im<' ( Q / Q c) 2 ---+ 1 wiH'r<' t he• se-aling function div<>rges. For completPIH:'SS we also s ho\\' in Fig. :.2.9(1>) an equall.\' good scaling collapsc> based on tlw assumption that Tc(Q) ~ TnAs(Q). Thus, we use Q<.(T) cl<'riwd from (:2.-11) using .r = 0.813 (i .P .. efrectivcly 11 = 0.615) and Q(j = 0.65 k\\' jcm 2 [optimal!,\· ch osen within tlw <'rror ban; quotc•d iu [19]] . Si nce Toi\s places a lower bound o n ~-( Q), the shaqwst condusiou W<' c-an make• a t this stage is thaL t h e data are cousis t<•u t wiLh t· h<' sea ling hypothesis for a fairly broad range o f <'xpcrirnentally aud LIH•on•tically motintt<•cl parameter c:hoic<'s aud that tuorc data closer to Tc(Q) will be r<'quircd for a nit ic-al L<•st of the th<'ory. TIH' e•xperiuwut a l m <'asurc'IIH'Jlts were• t akC'n in a cell Lh aL was only 0.6·1uuu high. a bon t as clos<' to the opt imal h<'ig iJt, as practical eonsicl<'rat ions a llow. AI though Lhe 7 This \'aluC' is in fact within thC' C'Stim <l!C'd margin o f C'rror forth<' amplitude' calculation. Th<' un- eC'rtaint~· of tlu• tlwory is a factor < 2 ( R . Haussmann. private' communication), whiiC' th<• adjustnlC'nt hC'rC' is ~ 1.9. Gl 8 • '1 X * Q'6 -s *0 0 - 0 + ~ '-' 4 0 ~ I> w 1.5 1 J..lW/cm 2.0 1 2.26 2.5 I 2.76 3.0 1 3.26 3.5 1 3.6 1 3.7 1 2 u <I 2 0 • 0 • 0 0.02 0.04 (Q/Q )2 • 0.06 c F igurr 2.8 : Scaling plot of t hf' d iff<'rcntia l heat capacity mrasurem r nts for various values of Q [25]. T h<· cxp r rinwntal da t a are scaled using Q 0 = G.G k\\' jcn/, aud tr nuina tc•d a t t h<• tem perat ure indicated by jJ in Fig. 2.7. T he predicted collapse of the da ta for different Q valuPs onto near ly t he same curve verifies t he basic scaling hy pothesis (2.42). T hin soli d lin<': stra ight linr fit to t hr d at a. Th ick sol id line: t lworf't ica I prf'd ictiou neglec ting dissipa tion [8, 9]. Dashed li1w: t heoret ical prediction includiug dissipatiou [18] . either theoretical curve is ro unded for gravity. (From Har ter. Lee. C hatto, \\"u , C hui , a nd GoodstC'in [25], Fig. 3.) G2 d w 5 u (a) Cl • <l d w 5 u (b) Cl <l 0.1 0.15 0.2 0.25 0.3 (Q/Q )2 c F' ip,urc• :2.!J: :\ltPrnl'ltiv<' sca ling plot of tlH• diff<•r<•nt ial h<'at <·a paci t.v nwasurc•nu•nts for \·arious Yaltws of Q [2:>]. Th<' data s.n ubols ar<' t lw same as those us<'d in Fig. 2. 8. T it<' <'Xt><•rint<•ntal data an• scalc•d usin g Q c(T) dc·ri,·NI front (:2. 11) b_,. (a) usiug I lw 1 t IH•or<'tica l <•xpom•nt valu<• .r - l j 'l.tl - 0.7 1G, but mnplit nd<• Q 0 - :~.I k\\"jc111 c-hos<•n to IH'st match tlH· thcor<•tical scaliug funrt ion: and h.v (b) assuming that 1 ~- (Q);:::::: T 0 1..,· (Q) \\"ith Q(> = O.G3 W j cm and .r = 0.81:~. For th<• Q-rang<' of tlw <'xperiBH'lll al data. t IIC' two anal_,·s<•s an• basic all~· id<'nt ical. with t hr Iines con1 pl<'t <'h· m·c•rlappi11p, to wc·ll wit hiu <'XJ.writu<•ntal n·sol11t ion . Onl~· for high<•r Q clHta would t h<· tm> (its ])('COlli<' disti11p;nishablc•. So lid line: llH'Or<'lind pr<•dict ion tu•p,l<'ctin g dissipat ion [8, D]: Dashc•d lin e•: t hc•on• t ica l pr<'dict ion including dissipatio11 [18]. <'ffpc·t~ of g r;witat io na l roundin p; wN<' t hc•rpforc 111iuimi/c'd . tltc•,· . wc•n• not <'Ill ir<'h·. <'liluiual<'d. Tit<• varia! io n o f T>.. acros~ th<' ('C•ll due• to gra,·it y was 8T>.. ""'8 x 10 x K . . \ reaMHiahl!• nitc•rion for a data point to I><• uuafl'<'c ·t Pd hy gra\'it~· is that o n(' ~ h ou ld lte:wc• IT- T>..l 2: 108T>... Th i~ wou ld rc>st rict tlH' l<'lli(Wraturc• range• of t iH' <'xperiment to mon• than about J ,,K hc•low 1)... Essc•ntially IIOIH' of the• intC'r('sring data showu iu Figs. 2.7 2.9 sat is f\ this nit<'ri o n. ln fa<'t . l><•c·ausc• nwasur<'m<•nts [2:3] hav(:' show n that undN a hc'af flu x Q > J fi\\" j ml'l. , hC' Iinm C'xhi bits a pprc•ciahle d issipation . t h<'r<' is 11 0 ra ng<' of pa ram <' t C'rs fo r wh iC"h this cri t Nion ca n lw sat isfi<>d undc•r Earl h 's gravi ty 111 an isot ltcnllal <'XJ><' rillH'Il f th a t nH'asur<·s C'ci rlos<' to 1~ (Q). T he• small ('('ll h<'ig ht use•d in the• <'xpNinwut predudc•d the u~c· of ~i d C'-wa ll th<'I'nto nt <'lry. The• lu•lium 1<'111 (H'rat u n• was lli N'tsurc>d us ing therntonH't<'rs mo unt c•d on t he• c·<'ll c•nd-plat c·~ and t lu• data wc>r<' correC't <'d for the !:.ing ular l~apit za n•sist anc·c• [78]. T IJC' tc•mt><'rat Ill'<' ran g<' oft he' mc>asurC' nwnt ~ was limit<'d lwC'aus<' t h<' hot tom (hotl!'r) thNmom<'l<'r cha ngc•d it~ lwha,·ior ])('fon• t lH' hulk h<' lium l<'lll JH'ratur<' r<'adH'cl Tv. 1,.,;(Q) [23]. lt was pro pos<•d t hat this change' was cluP to anot hc'r boundar~· dfc'ct rc lat cd to t hc 1\:apit za rC'sist <UJ <'<'. T h<' t <'ll lJH'rat 11 n• at wIt ich t It is plwnonlc>non oc·c·urrc•d is indiC"a t!•d b.\· ,j in Fig. 2. 7. and is the' 111ax imum tc•mpc'rat m e· of the• data s how n 111 P igs. 2.8 and 2.0. As a r<'H ult of t h<' rc•c ht c·<'d ran ge. t he• data in Fig. :2.9 d o not r<'ach hig h e•noug h t<'llll><'rat nrc's t o <'lH'Ollllt Pr much c·un·at ur<' in the> scaling function . :\1c•aHun·nH'llt s that a pproac h cloH<'r to t lt c di V<'rp,cH<'<' ar(' d<'arl.v ll('<'dc•cl. Data up to Tn ~,.,.(Q) s ho uld 1><' c•as il~· obtai nable• u~ing a dc>f'per ('('II c·oHstruc·tc•d with a mid-plauc• t lt<'l'IIIOlll<'lc•r. llowc'\'C'I\ sill('<' ro undi ng du<' to gra\'it~· will be• c•vcn more of a dc•t rinH·nt m·c•r t h<' r<'gion wlwr<' th<' ~('a ling fu11ctiou has ~ignificant c·urn\1 nrc', t he• ill-C'ffc'ds of t h<' clc•<'p <'r cc·ll hc> ig ht will obs('un• som <' oft he' hc•u<'fit gain <'cl h.\' t 11<' ineT<'as<'d t C'lii(H' raturc• range'. The• onl.\' dc'fi nitivC' way to cir<'lllll\'C'IIt the• pro bl c· ms raised in tlw JH·c·,·ious two paragraphs iH to p<'rform n s pac<'-hased IIH'asurelu c•ut of CQ . .\11 c·xp<'rinH'Ilt in the a hs<'nc·c• o f p;rrn·it.'· would ohtaiu data up to Tn,s( Q) without gravitational rounding. pC'rruit t ing an c•xl<'nsion o f tit <' s!'aling d ata into a mo n • rc•,·c•ali ng l<'lll{)('rat urc• rang<'. T hi s would a llow a c·onsidNa bl.' · improwd c•stim a tP of whc•rc' ~.(Q) lie's in r<'lation G-1 to TrMs(Q) [i u part.ir ula r if ~ CJ is s till fin it<' at TnAs(Q) O il <' would c·ondud<' t hat 1~. (Q) > 11l.\s(Q)]. PurtltC't'IIIOI'<'. a spar(' <'XJWrillt<'nl wo uld tc•st th<' sup;g<'sl io n t hat To\s(Q) is a gnn·ity a rt il'a r t [18]. a nd permit til<' toust m rtiou of a d <'<'P cell \\'ilh asy ttllll<' l riC' <'ttd plate's to t <.'S I \\'he• I lt<•r TtMs ( Q) is a bound a r.\' <'tf'c•ct. Chapter 3 Thi~ ch apt N Apparatus d i ~c· 11 ~S<'~ mall)' aspN't ~ of t IH' d <•:-.ig n of the e'XI><'riiiH' nl a! apparat u ~ u ~c·d to take· our IH•n t c·apaC'it)· m<'H!->11T'<'llt<'nt s. It \\'ill o11tlitH' t he• rc•cptirC'nH' II( S tH'<'< I<·d to obt.ai u S\I C'c·e•ssf11l lll<'<lSll rr nt<'ll t.s, Lh<' coasLra i 11 t ~ itti JH>S<'d h.v <'XtC'l'n a ! eondi t io11s, and t he• SJ)('C'ifi cations of t h<> fini'll a pparatus . .\ sdH'tnat ic di agram of the <'XJ)('rinu•ut a l SC'tllp is show n in rig. :J.l. 3.1 Limitations on the experimental design 3.1.1 Cell height : gravity and finite s1ze effects Ttt a laborato r.v o n Ea rth . th e d <'JWnd<'nc·c• of 7). o n JH<'Ssttr<' is suc h thnt dT>. / dz = n = 1.:21 jd\./ nu wlu•n• :: is the> \'C'rl ical dist an<·<' in graYit.\·. I3rlow T>,. su p prf]uid ity kc•rp~ h<'l i um i~ot lter111aL but t h<' d is t mu·c· fro nt t hr transi t ion. I =(T>. - T) / 1),, is nonunifo rnt due• to gra,·it .'· (~<'<' Fig. 3.:3). T he' span in rC'd11c·c•d t<'lllJH'rat Ill'<' in a ('d] of h Pight I is r5 1 = ni/T>.. In ordrr to min imil'<' t hr c•ff<'tls o f gravit.v, it is t lwr<'for<' d csimblC' to de•sig n an c·xp<'ri m<'ntal <'<' II to be a~ s hort as p os~ibl<'. Howc•v<'r, if it is too s mall. finitc•-sizc• c•ff<'cts will s ignificantly altc•r the p!J_,·sic-s. ln orciN to <Woid this reg ime'. it is r<'asona blr to n 'quirc• that I 2: 10~. wlH'r<' ~ is lite• C'ohr n•nc·<' l<•ng th. ~ = ~0 I '2/:l ~ 2 x 10 11 I '2f:t c·n1. Th us if we• limit <'XJH'rilll<'nts to th<' rang<' I > lit. W<' find a n abso lu te' lowc•r lilllit for I giV<' Il b.' ' ( 3. 1) <·orre'SJH>IHliug to a n opti mal n'll ltc>ig ht of I ~ 0.3 liUIL llow<•ver, a t<'ll t his slll a ll is ratlwr difhC' ult to C'onstnH·t. Th<' C'alorim<'tC'l' us<•d fo r the <'X JwriuH'ut d0snib<'d in this t lwsis wa:-; <k sig tH•d to i><' approxima l <> I.\' 1 mm t a ll. lu ad uali ly it was mc•asurC'C l to i><' O.G 10 llllll high . c·oiT<'SJ>OIHling to lit= ..J x 10-x a ud au c•xpcrillH'lltal t <' lllp<'ratun' 66 Roughing Pump E :=l d) ::c VJ :.3 0 II) GAS HANDLING/ SYSTEM ::c 0 ~ :=l c. Y ihration Isolation "' 5= ® :2: 0 0 c::: 0.. :2: ~ 0.. ® ~ K i nney Pump QD Bridge Filtered Vacuum can QD SQU ID CRYOSTAT Controllers F ig ure 3. 1: A schematic drawing of the complete cxpcrimcHtal system. 67 z J' Cu / I I I I Super II T~..(z) I I fluid I I I I I Norm al I \ I ' ' .... fluid ... I - Cu 1 gravity ,"" T F igurP 3 .2: T<• mperaturP pro(il('s for 4 He in a gravitio nal field ami a h<'at curn• tlt , Q. A graYitat io na l field causrs th<' transition l<'lliJ)('ratur<' of .J lie to var~· as a function of cell hc•igh t. Solid line: the t rausition tc•mpcratur<' T)l. as a function of c<:>ll height, ::: . Dashed-dot t<•d lin<': when t.IH' teBJJH'ratm<' of til<' sampl<' is adjusted so that t h<' tempPrat ure at the oottom of t he cell is i<'ss than the local transit ion temp<'r<:lt urr, the• trm prrat mr w ill be uniform throughout t. h<' cdl. Dashed line: if t h e tcmperatun• is tllll('d so that t he ('('llt<'r or t !J r crll is at tlw local transition t.rmperaturc, t hcu t h <· top of tlH• c-<•11 is superfluid all( l tit<' hotto111 oft IH• c-d l is normal flu id. S had<'d region: the int<>rfacial region between th<' superfi uid and th<' normal fluid. [Note: not drawn to seal<'. ] rang<' of I 2: 1 x lo -x . G ravi t .v aff<'cts our <'XJH'ri nwnt b.v avNagiu g the h Pat capacity c urve over th0 temperature' rang<' of 5t fo r o ur particu lar cd l he ig h t. This can cause a rather cxtrPllH' rounding effrct. rsprciall~· in t hr vicinity of a divergence. Figure 3.3 illustrates tlw anticipat<'d C'frc•cts of grm·ity on hNtt capac ity tll<'as un•mf'nt s tak<.>u in a 2 llllll aud a 0.64 mm cc•ll . 68 (a) 120 115 ~ 110 :::.:: 0 105 E -- 100 ...., ~ ~ ·u co P- 95 co u Cii 90 ::c 85 Q) ~0 75 -2.5 -2 -1.5 -I -0.5 0 0.5 -0.5 0 0.5 T- T A. (~tK) (b ) 120 115 ~ 110 :::.:: ~ 105 -- 100 ...., ~ ~ u "'P- 95 "'u 90 co u ::c 85 80 75 -2.5 -2 -1.5 -I T- T A. (~K) Figure 3.3: HeaL capac:it.v curves in a gravitational field in (a) a 2 mm high cell (b) a 0.640 mm hig h cell. Tiight thick solid line: R fit to LPE data [2) (at saturated vapor pressure a nd Q = 0). Right Lhin d ash ed line: LPE data rounded for g ravity. Left thick solid liue: Haussmanu 's theoretical prcdit iotJ for CQ [18] (at saturated vapor pressure a nd Q = 3 .5 ft.\\'jcm 2 ). Right thin dashed line: Haussmann 's prediction rounded for gravity. T his c urve is c ut off wheu t h e temperature· at. the bottom of the cell reaches the local valuE' of I:.( Q). 69 3.1.2 Thermometer placement, specific heat error, and temperature instability: the singular Kapitza resistance .\ IH'at flux, Q, fl owing anoss a solid nt etal wall into s up<'rfl uid 1 H<' produc0s a di scontinuity h 0t weeu LlH' t.cmperat un' of the n wtal (1cu) and thr ternpnat nr<' of t h e• s upe rfluid at th<• wall (1'11 · ). This tc•utperat un• julllp is linea r in Q. a no is caus0o by the aco ust ic mismatdt l><•twe<'ll t he• two mal.<' rials. It is known as t lw K a pit:la bo tlllda ry resis L<Utn'. T h<'n' is auot her componrnt t o th0 bo und a ry rC'sistauct• Lhat h as bc·c·n observf'd to b e• wc•ak l ~· s ing ul ar 1war the s up <'rfluid trans itio n [78. 79] (sec• F igs. 3.1, :3.G). This s ing ul a rity comc•s abo ut })('cans<' , i11 a b o undary layer w ithin abo ut a c·orrelation le ng th of the <'JHlplates. the' s upe rfluid / norma l fluid convers iou rate beconH'S too s low t o s upport the t' utirc hc•at flux by countNfiow, n •stilting itt a vN~· small l <'mJwrature g radient n c•ar t IH• eud- platc•s (sec F ig. 3. 1). T h P therma l rc•sis t aJH'<' ac·ross this b o undar~·. Rb = ~n /Q , w here ~T;) = Tw- T,c.,· 1.(with T,c:; p the• te•mJWratur<• o f the bulk he lium ), is thoug ht to div<'rg<' in the• Q--+ 0 limit w h t•Jt TsF approadws T>. . At. the• wall wiH•re the hC'a t fl ows into the cell. Llw l<'lll JWraturc• in the b omHl a r~· la~'E'r is hig her tit an T,c.,·1.• . T he o pposite is true at llw wall w lwre th<' heat <'Xits t lw cell. It is fo und that t h e d a t a for Rb at both walls coll apse outo a s ing le c ur vE' if Rb is expressed a s a functio n of the nH'all ho uucla r.v tc•mpera ture T;) = (T.r.,·p + T 11 ·)/ 2 [79]. T h e p lo t o f Rb vs. Tb is ind<' P<'tHknt of Q. l\ leasurem eut s Lakc u iu th e raugc• 10- 1 > t > 10- 7 iudicat<' that a log- log plo t of t.h e sing ula r bo unda ry resis t.au<·<'. Rb , versu s t.hc• r<' duc<'d bounda ry tempNaturC', tb = 1 - Tb /T>., is nParly a s traigh t liu e [78, 79], so t h at (3.2) ::::} T ,- T . = Q j'J II SF t (1 - (Til + T,c.,·F) ) -~" , 2T>. (3.1) whc•n· {1 a nd ::. /\ a re fittin g pa ra m e ters to t he exp<'riuH•ntal sin gula r 1\.apit za r<•sistanc<'. 70 Top plate ............ I - 4 gravity ! He TsF Tbottom w ~ ! Cu Bollom plate I ~T.bott _j b ... -L Q sr,bott K Temperature Fi gure 3.4: Gr<'.v arrows: th0 t<'mperatun' jump du0 to the r0gular Kapit~a resistance , .6.Tg = Tcu - Til" = QR"". B lack arrows: Llw temperature' jump due Lo the singular Kapitza resistancf', .6.T0 = Tw - T.•ir = Qfl0 . [Note: not drawn to scale.] 71 The mos t r0c0nt dat-a a r0 s hown in F ig. 3.5 [78]. \Vbctl tlH' fit is p C'rfornl<'d over t hr PntirC' tC'mpNa tur<' ra np;c> o f this plot. J = -1.388 x 10-:l K j \Y. and ;;1,· = 0.20:33. The rxis t rncr of t hr regul a r a ll( I s ing ular rompo uPnt s of t IH' I~ apitza r0sist a ncr, s hown i11 Fig. 3.5, have a numlH'r of advrrsc d f<'cts ou t.h<' <'X<'cnt ion of our C'X J)('riment. Thermometer placement I3uth compou<'nts of tiH' Kapit za n's is tanc<' inftu0nc·r th0 tC'mJ><'rature readings of lhcnnompt <'rs mount 0d 011 t h r C<'ll Pn d-plat0s. Th<> fa c t that the' s ingular l<apit za res is tance is a function of temp0ratm e causes SNious complications whrn u sing end plate t h<'rmomr t<'rs for a d~· 1Htn 1 ic m rasur<>ment. This pro blPm can I><' d iminal<'<i b~· using a thc>rmonwt<'r in contact with lhr 1nidplanP of Uw c<'ll s idewall. The t.<'lllJWratnr<' o f t he san1pi<> is t hen r<'<td in a lllclltlle r a ualogu us to a s t andard fou r wire' r<'s istance tllN\Silrc uH' IIL \\"her<' the therm a l boundary rrsi~tanc<'s ar<' <Wo ici c'd iu th0 sam r way t hal coulact resistances an' <'liminated iu tlH' fo ur-w ire' mrasur<' nw nt . v\'e a t t<'Jll pt Pd to hu ild a mid p lane l h<'rm o irH't er int o our expC'rim0nt. C u fort una tcly. it proV<'cl difficult t o nH'<'I bo th the stringent lwip;ht r('quircnH'nts ckscribed in S<'c. 3.1. 1 and to hav<' a functi o nin g mid-plan<' t hermom e t<'r. As wi ll b e described in fu t urc s<'<"l ions. the s ho rt <"<'II height caused t h<' mid pl a nr t hrrmo m el cr to be Lh<'rmally Jiuked in a non-trivial \\'a~· t o 0110 of the cell <'ndplal<'S. Specific h eat error: the specific heat of the boundary layer Brcaus<> t IH' s ing ular l~api t za resis t auce is act ua lly iu LIH' Iluid i Ls<'lf, t he h eat. capaeity of thr upp N a nd !owN b o uuda r.v layers will b e somc'w haL d ifi"<'r<' nt than the' heat capacity of tlH' bulk. A t a red uc<'d t<•mperat III"<' oft = 10 7. about 21< of t lw voltltn<' of t.IH' sampk wo uld be within the::><' boundary layers. Howeve r , bece:ttJS<' the te m[H'ratur<' o f the lop bound ar~· laye r fa lls bc'low T 8 p. whil<' th e l emper at ur<' of t IH' bottom boundary l a~'<'1" falls a b on' Tsp, the t.wo diffc'r<'ut hC'at <·apac i t~· <'ffects would partially cancel , aud t li<' owrall error s ho uld be quit <' s mall. T hi s is fort unat<': si ne<' thr bulk heat capac it~· dos<' to T,.(Q) is unkn own . it wou ld lw impossihlr to accoun t accurate!_,. 72 (a) J.lg Q ..... ..... T A. I ..... 'o... 1.16 ~ 1.14 ..... ..... D..... N ~ I 12 c.:: ~ ~ ..... ..... 0 ..... § ..... . '{) ..... ..... 1.1 ..... 0 0 8 . . . ,o ..... 1.08 0 ,o ~ ..... 1.06 2.1 2. 12 2. 14 T (K) ..... 2. 1g 2.16 (b) I 1.16 (jJ I I 00 1.14 ~ 1.1 2 "'E ~ C§bo <D 0 00 0 0I 0 0 1.1 0 0 c.:: I Q 0 1.08 0 0 0 I I 0 / / 1.06 ------- -- --10- 6 4 I Cf _, I0 - Figurr 3.5: (a) Lin<'a.r plot ofth r total Kapitza resis tanc<' as a function of temperature. Opeu circles: tlw total Kapitza resistance R = R 1,· + R 11 as a fun ction of bulk hdium temperature T. (From Fu, Baddar, Kuehn , and Ahlers (FBKA ) [78], Fig. 2.) (b) Srmi-log plot of thr total Kapitza rcsistanc<' as a function of tPmprrat urr . O pen circles: the total Kapitza r esistance R as a func tion of th<' reduced bulk helium temperature t. Do t t rd liuP: a fit to t he regular componen t of t he Kapitza resistan ce. Rl(. (From FBKA [78], Fig. 4.) 73 0.07 ~ 0.05 ~~ ~ ~ E ~0.03 .D 0::: 0~ (2}. .6 ~ N 0 0 .6 .., 3.9 f..!W/c m- QA' .., ~ 15.8 f.lW/c m') 0 3 1.6 f..!W/cm- + 63. 1 f..!W/cm - X 110.5 f..!W/cm - ') * 22 1 f..!W/cm- ¥.1 (jjf .., 174 f.lW/cm - xo *~ ') 0 X * ') t b F ig ur<' 3.6: Log-log plo t o f thr s ing nl;n boundary resistant<' flb as a functiou o f reduced nwa u bo undary l CHI per at ur(' lb . T his plot was obtai nrd by s u btractiug t h<:> dotted litH' fro m the op<'IL circles 0 11 Fig. 3.5 fo r diffprc nt h<•at currrnts a nd tlw n adj usting th<' x-axis to h r t 11 = 1 - (Tt1 / T>. ) = 1 - [(Tw + T,c;F) / T>..] = t ± [QRb/ 2T>..]· (Fr om FB I...:A [78]. F ig. 5. ) 7--1 for t his rrror. Tt will tlwrrfor<' lJ<' ass umed t h at it is n egligible forth(' analysis of this eX j)('ri ment. Temperature instability at the boundary 1 T ltt> singular l{api lza resist atH'<' might han• a <·o ns id Nably more serio us im pact 011 the rf'su l ls of this cxpcriHH'llt: lwcausc tl H· t<'llljJ<'nttme of t h e helium in contact with tiH' lower wall ( in the h eat from l)('low ('onfi g urat io n ) is highN than Lh r trmperaturc of t h e bulk liq uid . the SUJH'rfluid might IH•<·om e unstabl<' at th<' boundar~· before the hulk fluid r('adH'S Tc(Q). T h <' h,vpot.IH•s is that th<' cxp <'ri m <'nt.a l s upC'rfiuid breakdowu tempcrat m <', ToA s, Ill ip;h t he• d U(' to an cffc•c t at t h r bott nd ar)' was proposed iud<'J>end<•ntly by mys<•lf and RY. Duncan [80]. T lH' arg um rn t goes as follows: AccorcliBg to \\'riduuan and 1\Iil kr, t lw nonn a l-fl uid int Nfac<' probably C' lll<.'rgcs cou tilluousl,v from LIH' wannrr boundary lay<'f [Gl]. Suppose that t h <' interface d c•tach es from thr <' tHlwa ll lltorr or l<•ss when T,, = T,.(Q). w here Tc(Q) is th<• brc•akdowu tc'lllJH'raturc. For lack of an <.'xperimeut.al m easun·mc·n t o f T,. (Q), wc' c·st.intat.c• its val u e u s in g H a ussman n 's pn•d ictiou t, = (Q/6571 \\'jcm 2 ) 0 · 7 11 [18]. L<'l 7~. IH' t h e t<'I11JH'ratnr<' of t lH• b ulk s upcrH uicl at th is point. If om h ypot hesis is c·otTC'c·t , Tx will h<' <'q u al to TDA S . From (3. 2) we haV<' 'T' .L II 0 T 3' -- Q I,.)) t ( Z t, ' - (3.4) whii<' - Til' + Tx T<·-- rr .lb 2 =? 1'11 · = ?T - c - 1'.r· (3.5) Substitu ting thi s relation int o (3 .4) '''<' obtai11 (3.6) =? f .c 1 SomP of tht• t<•xt I r + 2QT>.13 , r- z '' > t c· (3.7) in this s ubsf'ct ion appPars in Harter, LP<'. Chat to, \\'u , Chui. and Goodstein (25]. Oll 0 -7.5 -8 -0.5 0 0.5 1.5 Figure• 3. 7: :\ nunH'rical cakulat io n o f supc•rfluid br<>akdown. obtainc•d h_v sPttin p; tlw bound a r_,. t C'IIIJ H'rat urf' Tb C'qual to t !1<• t h c•orc~ t ical prC'di<-t ion for Tc. Dots: the• DAS dat a. Thill solid lit)(': I hi' lm•akclown tC'mpPrat m f', l.r• dC'tc•nniued num<.'rirall_,. from (3.7). T hic k solid line: lla uss rna rur 's /,.(Q) [18]. T he rc•strlt s o f subst ituting thf' tlwor<'t ical Yai uC' of lc into t lw a bove• eq uation an• s hown in Fig. 1. 7 as I he• I hin solid IinC'. The agrf'C'nlent lwt W<'Cll the• DAS data a nd t he• tlttlll N iC"a l siIll lila t io n is nc·ar l.v prrfC'c t , wit It no adfu8labh· pamm('/ <'7·s. T it <' agr<'C'IIH'nl IH'twe·c'n t IIC' numrri eal rakulatio n a ucl t he• data is quite• c·om t>C'I Iing. In particular, it SC'C'I1lS to <'xpla i n t h C' dihl urbiug d isn<'J><Utc·.v b<'tw<'<'ll tlH' <'x ponent o f t hC' DAS d ata a ud t It<• ex.pon <'Jt t of I he I hC'or.v, which s ho uld IH' qui I e dc•pcnda blr. Tlw formC'r apJ H'ar~ wit It in the• fit (3. 7) via a mmiJination oft lw t hPorC't ical C'Xpo rwnt Lj211 ~ 0.7-l-1 a nd tlw Kapil za c•xponc ul ::;". Th<'S<' r<'strl ts inspirC'd our group to desigtt au <'X)><'riuH•nt to dc•t<'rmiH<' if TnAs(Q) was. i11 fac·t . du C' to a bo undary c•ffect. This <'X J)('rinH'nl is CIIIT<' llt l .v running in o m )aboral o r~·. D uriug the• cou rse• of this im'<'SI igat ion , iI was d iscovc'rc'd that thC'r<' is .vC't auot he r 76 wa.v to n·adt l he• c·otJ(' Ius ion t hat Tn.-\s is d lH' to liH' singul ar I-\ a pi tza n•sistauc<'. 2 .\ppan•nlly tlH'r<' t s <UI inhc•rcnt instahi li t.v built iuto (3.2) that occurs c•xtn•mc•ly dos<' to Tn \S [25]. T IH' proof clO<•s not rely 011 t h<• t h<•on•ti('a] \'a] u<' of 1: (Q). or on an,1; th<•or<'tintl paranret<•rs. and tlH•rpfon' <'liminatc•s possible' Prrors in tire tlrc•oreti('a ] ,·altH's fro111 aff'ecting t h<• c-ondusiou that TD,\s is a bound a r~· d l'Pct. Tlw proof got's as follows: Soh ·ing (:3.:3) for T-,·"' as a function of ~ n, (3.8) o n<' sc<•s tIt at I lwrc is a mzmrnum 18 ,.. for whi ('!t a solution Pxisls. T IH' C'ondit io n Dt 81 /D~7/1 = 0 for this minimum yi<'lds (3.9) Th<• corrcspo ndiug nlhu• of 18 ,.. a t this \'aluC' of ~1 /1 (whi ch we• will call i,) is: (c~o ),. (3.10) rrom (he ('XJH'l'illl<'ll l a ] \ 'Cllti<'S of /J and z 1, on<' o ht aius Q 0 ~ GOO \\' / nn:.! and . l .- (: 1, + 1) 1 ~ 0.83. T hus t iH• i11stabilit.\' t C'nqwrat ure is dose• to TrMs (2. 11). ThP c•xistcuee of/, is n•all.v a n indicatio n that thC' lill<'<tr (in Q) n •lation (3.2) no longC'l' mak<'s scns<' wh<'ll lb --7 0 at fix<'<l finit <· Q, and /, ((J) r<:>ally the n d<'fiii<'S a houuclary of \'a lid i ty of th is IinNtr rc•sist a nc<' approxim al io n. It is vc• r·~· int<'rcst ing, though, t hat t his boundar~· cotT<'sponds V<'l'.\' dmwly to tit<' OIIS<'I of cl issipa t ion ill the DAS c•xpNilll('lll . a nd pr<'s um ahl.v t.o t he· <lc>t.aclull<'lll of t h<' itr tNf<H'<' from th<• boundar~'· To make• a mo r<' acc ura t<' compa rison. we• ll OI<' that th e' logarithmic plot of tlrc• s iug ular bo und a r~· r<'sistanc<' data shown ill F ig. 3.6 has som<' cun·atur<', so it ca nnot h<' fitt<'d ad<'q u at<' l ~· b.v singl<• values o[ z 1.; a nd tJ. Howt"\'C'r, for an.v 111 , Oil<' can do n loC'al lir H'ar fit to find an·ural<• valu<'S fo r t IH'S<' paranH't<'rs. C'ombiniug (3.9) a lld :.! A . Chatto. pri\'al<' communicatioJt ( 1!)99). II (3.10) and soh-ing for Q as a function of lb--= l sp ~n /2T,,. on<' obtains for a gi\·eu lb an insh-lbilit~· at Q = Q i gi\'c u by ) 2T>.o , ~ ,,-tt (~I, b • ~ J, ,-J (3. 11) \\'<' can now pron•<'d din·c-tly frout tlw boundary r<'sistann' data of Fl3K:\ [78] shown in Fig. 3.G to a prediction of su t><'rfluid br<'akdown. \\'<'first find a local fit for d and : 1, at a gi\'<'n 11,. tlH'll con•put <' Q, from (3.1]) and t, from (3 .10). aud finally plot tlwm against each othN on t IH' sanH' graph as th<' DAS data. This is shown in Fig. 3.8. TIH' n•stdts prc•dic-t C'd from tiH' si ngular I\:apit%a n•sistanc<' data arc' <It higll<'r Q valuc•s than t IH' DAS data, but t hP <'Xt. rapo lat<•d agn•c•nwnt with t iH• DAS data is <'xcdl<'nl II C'\'C'tthC'l<'ss . . \fit to our predictt>d points iu the• form I ,= (Q/QuY gin•s .r = 0.8JG3 ± ().0023 and Q 0 - 813 ~ 9 \\'/en /·. which is consist<'n t with th<' 0.\ S fit . From th<' abo\'e <'quatious. OIH' can sho\\' that (T>.- T,)j(T11 - T ,) = (J + : 1,· )/2. Since : 1, < 1 for all//1 this impli<•s that. \\'h<'n the• t<'mt><'ratm<' of tit<' cell bN·omeb unstablc• at 'J ~, 1\ 1• > 1).. On<' \\'Ould <'Xf ><'c-1 that as Ttl' -7 7).., ~T,, will d<'l><'tHI nouliJH•arl~· 011 Q, a nd (:3.2) will no longrr !told tnt<'. Th<' im,tabi lit._y would tiH'r<•for<' c•nt N t IH' c<'ll at a low<'r t <'lllp<'ra t un• than if t lw tcm peraturp drop across the' boundar~' wc'n' to r<'tnain liu C'a r i11 Q. \\.<'would tlt<'rc-fon' cxpc•ct LlH' prNiic l<'d points (o) s hown 011 Fig. 3.8 to I i<' sl ip;ht ly low<'r t han a 11 t'Xl rapolat iou of the' DAS fi l to largc'r r<'d ucC'd t empc•ratur<'s. For comparison. w<· have also plot t c•d th<' r<'<ht<"<'d t<'liiJWratur<' of tlt0 bulk SllfWrfluid wh<'tt T 11 = 1).. Thi:-. shows an <'Wtt dosc•r matdt to th<' D.\S r<'sults. \\'(' condudP frotH the• prcdous analyse's that the' smling of a boundar~·-driv<'n instabilit~·. d<'l<'rrlliiH'd fro111 nlrious nitc•ria [hotlt i11trinsic to th<' singular Kapitza resistattc<' (3.:2), and includi11g asp<'cts of 1~.( Q) \'ia (3./)], all appNH, wit It r0asonabl0 fi tt inp; paramc•t c'rs, to yi<'ld dc•c<'tt l ap,r<'<'m<•nt with T 1M s ( Q). !\lor<' cxpc>rimental data will lw r('quirc•d to understand which crit c•rion is t Iw rnost physically correct Oil<'. 78 -4.---~------~--------~------~~------~ 0 0 -4.5 -5 ......., -5.5 u. ~Vl '-"c -6 01J 0 --6.5 HD prediction forT (Q ) c -7 • o x the DAS data the tempe rature instability tst• when T w=TI.. -8~--~------~--------~------~------~ 0 I 2 3 .., 4 log [Q ( !J.Wicm- )] 10 Figure 3.8: TIH• lm•akdown t <'lllt><'raturc• as a funr t io n of Q. (From llart<•r d a/. [2.)]. F ig. 2.) 3.2 High resolution thermometers On<• oft IH• most illlportant compou<•nt s o f our <'Xt><'riut<'lltal apparat 11 ~ was t h<' t lwr- trH>tnctry, since• all of our data li Ps wit hin s<'n·ral 11K of 1).. Fo rt mtatc-1_,·, iu the• last d<'cad<' . \'C'ry h ip, h r<'solu t iou mag H<'I ic t lwrmom<'t c•rs ( H 11Ts) [8 J]. ha\'C' lw<'ll d<'v<'lo p<'d for st ucl ic•s uc•ar the la 111 bda I ransi t ion. T he IH'st of I h<'s<' t h<'rntollwtNs Pxh i bit low fn·quency noise• o n t h <' order of 10 !l to 10 3.2.1 11 /\' / .,JTi'Z. :l Design In I hPnn od:vn a mi c t<'rms. a lhennomel(' 7' is uwdc• lc·d as a su hs.vs tr m in cont act wit h a hNtt rrsNvoir. T he• reser·uoir·, I h<' obj c•c·t whos<' l<'tll(><'rat un• is to 1><' nwasurc•d. is ;1SomP of IIH' I Pxt in t Itt> n<•xl f<•w s<'<"l ions app('ar s iu L<'<'. Ha rt <'l'. and C hui (2GJ a nd Day, llahn. C hui, Hart<•r, R ow<' and Lipa (:.2:.2]. 79 ass urn rcl to br isotlwrmal and to havr infinitr h rat capacit~·. Til E' therlllOJII<>t<' r inc-orporatrs an 0asi!~· nu'asttr<'cl quautit~·, like• tl10 mag netization o f a salt. wltos<' IIH'all ntiLH' is a functi on of t em p0rat 11 r<' a nd ca11 thus I><' used to iufN t lu• r<'S('r voir t <'Ill pe r- at un'. From s t a tis tical mechauics. it is well known that thennod~· namic propc>rtiPs und<•rgo c-outiuual fiuct uatious a bout t h<'ir ll}('an valu0s. In t h<' c-a s<' o r a m agn<'t ic t IH'l'lllOIIH'tC'r, this pheuolllCllOll !<>ads t o rn agn0tization fiuct uaticms ovc>r a fr<'qurncy band from de to t lt<' inn·rsc or i1 chan\ c-t.<•ris t.ic tinlf' constant. If tl}(' mag nit ud r of t IH'S<' flurt.uat ions arc com parabl e to tlw r<'soluti o n of thr thrrmom0t.rr, Uwv appear as n o is<' in th0 t0mperatur0 nwastll'<'lll01lt. Anotlwr source of flu ct ual iou n o ise in a tlt01'1110ill<'t<'r is the temp<'ratur<' variat io n caused bv en erg_,· t ra usfN throug h th0 t h<'J'ma l link t o a rcscn ·oir . HRTs a r<' c urrcHtly limited b.' ' l his latter form of no is<'. Concept l11 it s bas ic form. au HRT <'OHsists of a paramag n 0t ic salt pill ti ght! ~· <·oupl N I t.o a SUJH' n ·o nduct ing pick-ttp coil. ,\ SUJH'rco ncluct ing flu x tul){', s urroundin g the' IITIT, is us<'d t.o trap a V<'I'." s tahl<' DC fic•ld ("" 10 mTrs la). The salt is tlwrma, ll~· cotrpkd to th<' h0lium samplr through a g rid of hi g h pmit~r (99 .999%) copprr heat fin s. Ilig h puri t~· coppN is u sC'd to max im iz<' the t hrrm a l cond ueti \'it~·. T he pic·k- up coil is coun<'<'l<'d t o a SUJWITonduc Ling quant Ulll int Prfrrcncr d evice (SQUID). which llH'asu r<'s ch a ng<'S i11 t h<• mag netization of the salt as a fuuctiou of tpm JH'rat Ill'<'. Idc•ally. t h <'S<' t }l('l'llJOill et <'rs s hould be o p<'rat <'d ill a t <'IIIJ><'rat llr<' raug0 wlH'r{' t lH' salt pi II is in th<' paramag netic phasr and is c los<' to its C urie temperaturr , T,., whrre it s mag nrt.ization is hig h!.\' trmprra tm<' d Pp r ndc•nt . In l h0 f0rrom agnf'ti c ph asr , discontinuo us dntng<'s in tlw Aux d0n sit~· dur t.o switchin g of m ;-tgJwtic dom ains c•ff<·ct [82] thc Barkhauscn can appPar as a s ig nificant noise• contributio n in t h c thNmomN<'r m 0a- s tm'nH' nts. !\los t HRTs us <' salt pills rnacl<> o f cr:vstallin0 coppN am m o ni11m brornid r [C'u(:\'II 1):2D r 1• 2R 2 0 ] (CAB) whi('h h as a Tc o f a pproximat<'h' 1.8 K I1 rc0 nt dc•sig ns us0 GdCI:!. which is dwmicall.v l0ss r0ac t iw• than CAB a nd can h<' g rown dirC'ct l~r o nt o tl10 copprr fins. In addition. CdCia is approximatC'l ~· five• timrs as sc•n s itin• as C AB a t th r SIIJWrfluid tra n s itio n t r mp Nat m 0. T>... 80 Despil e thes<' favorable properties, Tc for G d Cl:~ liC's j11sl above T>-. at 2.2 K, causing t he salt to operate in tlw ferromagnC'tic ph ase. In an attC'mpt to overcomC' t his problem , w<' constructed several thcnnomcters using GdCl:i dopC'd with La3+, a non- magnetic ion. This was done in order to deprrss the C urie temperature of t he salt. GdC l:~ and LaC~:! arr isomorphous salts. In a solid solu tion, the diamagnetic La3+ ions subst itu te for the paramagurtic Gd3+ ions to form crystals in which both ions are prc1:>eut [83] . For a uou-m agnetic iou such as La3+, t he depressiou in Tc is expected to be approximately linear with eloping coucent ration [84 J. Construction Thr HRTs in our C'Xprrim r nt WNC' constructC'd and assC'mhlC'cl at t he .kt Propul sion Laboratory (.JPL ) and a t C<tltC'ch h:v Talso C hui , Ri cha rd Le(', and m~rsclf. 4 Thr salt was prepared from 9-1.5 g of GdC1:3·6Il:z0 a nd 2.7 g of LaCh, producing 63.74 g of Gd 0 . 958 :;Lao.o.J 1 sC I :~. The hydrated GdCh was firsL mixrd with LaCl 3 and then dehy- drated , us ing a reflux COJHJeus ing tcchuique wit h t hionyl chloride [86]. The mixture was heated iu a dry nitrogC'n atm osphere until the sample melted. The m olten salt was p oured into a BcCu capsule, fl owing in-between tlw enclosed c:opper heat fins, and kft to solidi{v. TlH· assembly was sanded to removC' excess salt and sealed, using a BeC u cap a nd cad mium sold<•r. BeCu was chosen in order to reducC' background noise caused by eurrC'nt f-luc t uations in thC' capsuk wall, while maximizing thermal conduction to t h<' salt pill. Cadmium solckr was used lwcause it is not superconcluct ing at 2 K , and would therefore not distort t he field trapped by the flux t ube. A supcrconducting pick-up coil made of 3 m il niobium wire fit t ightly around the capsule. ILs diuwusions wen· o ptimized to m atch the iuput impedance of t he SQUID and Lo cover the salt s ufficiently to ensure maximum energy transfer. T he twisted pair leads of the pick-up coil were threaded into a 9 mil i. d. Teflon t ube . a ud then into a 15 mil i.d. uiobium-titauium capillary shield. T he Teflon t ubing was used to pn·vcnt shorts between the leads and the 1'\b-Ti capillary. After the twisted pair 4 For a d etailed description of t he HRT constru ction technique, sec the .JPL HRT Fabrication and Assembly R eport [85). 81 reservoir Hi gh purity copper rod Nb flux tube magnetome ter Twisted pair inside Nb-Ti capi llary Hi gh purity copper heat fins Salt pill Fip;ur<' 3. 9: Sth<'rllatir of th<' IJI1Ts us<'d iu our ex peri mcuL. (From Lee , Ilarf <'l", and C llu i [26], Fip;. 1.) a ud it s casiug were inserted , U1e capillary was injected with grease to pr<'wllt noise' du<' to m icrophonic Yibraliou. Pur<' uiobi um can also be usc'd for t IH' capi 1\ ar_v, but niobiun1- t itaui um has a highN tlH•rmal r<'sislanc<'. and is th<'r<'forc' pn'f<'rred in or<kr to 111inimize the' inAuC'n<·c• of heat lc'aks. Thr capsu l<' was cadmium solci<'r<'d to a n anueaiC'd. hig h pmity (99.999o/c.) coppN rod that \\"as put into dos<' thNmal contact to the• objrct wllosc tc•mpNatur<' was to I)(' rn<'aSmC'd. (SrC' Fip;. 3.9 and thr inset o f Fig. 3.11.) 3.2.2 Sensitivity \Vc• cons tructed thr<'<' l-InTs Ollt of GdC1:1 doped with lanthanum, all(! orH' Ollt of p u r<' GdC \:1. T IH' thn•e dop<'cl salts wc'r<' mollntrd dir<'ctl:v on tlw calorim<'f<'r, and th<' pure' GdCI 3 salt \\"as mount<'d on a radiation s hi<'ld that surround<'d tlw <'X JH'ri nwnt. Srn s itiYit.v \·crs11s tC'mp<'ratllr<' m<'as ur<'nl<'nts W<'r<' mad<' for both t h<' pun• a.ucl dop<'d GdC1:1 I-TI1Ts by slow ) ~· cooling th<' lwlium sam pi<' and r<'rordiug t lw out put o f the Figm<' 3. 10: S0nsitivity vs. temp0rat urC' for GdCI :1 (das h ed lim~) and La doped G d Ch (solid lin<'s) HRTs, with tC'm])('raturC' d<'t<'rmincd by tlw calibrated GRTs. (Fro m Lee, 1-larl<' r , a nd C hui [26], Fig. 2.) ITRTs alo n g with the m C'asure nH' nts o f calibrat 0d germanium resis t a HCP thermometers ( G RTs) m o u nt 0d dose to t lH' IIRTs. T lw S<-'Hs i t i v i L.\' of each t hNmo m e t c r is plott <'d w rs us t <'HI per at ur<> in Fig. 3. 10. T h e se11s it i \"it ·" is d cfi11cd as tlw <·h a Hg <' iu t h <• lllllllb<•r of m ag tH'tic flu x quanta. ¢ 0 =h j "2e ~ 2.07 x 10- J!i Weber , n •cord <•d by th<• SQUID m agn<•t o ut C't<'r over a g iveu c-hang<' in L<'tll JH'ra t un•. as m casun•cl by t h<• a djac<' nt G RT. T h<' pun• G d Ch HRT h ad a max illllllll sensitivity of 28.9 cf>o/ ftl'. at 2.200 K , w hil<• th<• dop('d HRT had a nraxin1UJJJ s<'ns itivit.v of 23.6 cf>o/!t,l'. at 2. 185 K T his cl<-arl.v d<•mons trat<'S t hat tbc C uri<' t<'lllp<•ratm<' had h<'<'ll d<'pr<•ss<•d. ITowev<'r, th <' d <' pr<•ssion in Tc by 0.7% was s mall<•r than <'Xp<'<:t<'d for a 4% La conc<•ntration [84]. and this IowN Tc is s till a bO\'<' 1h<' l<' HIJH' ntt 11 r<' raug<' for exp<•rim<'n ts conducted ,-<'r_v IH'<H to T >-.."' T his m ay lw du<• to inho mog <'IH'ous doping of the G d C I:1 salt due to r'Th<'S<' t ll<'rmonl<'t('rs \\"<'!"<', IIOI)('IIwl<'SS, u s<'d for our lwat capacity <'XJ><'rilll<'lll s. Tlw 83 0.25,..-------.-------,---...,.....-----,.---.. --.. N Pure coppe r :r: (';- 0.2 T o thermal reservo ir ~ Nb-tube 0 <', I - 0 0.15 0 ' --' SQUID pick-up coil E 2 u0 0.1 0.. Thermal posts r:/) 1-. 0 ... 6 0.05 Q.. •• .. -.. .. -·-- .. 0 2 3 Frequency (Hz) 4 5 F ig ure 3. 11 : C ircl<'s: tlOisc s p <'d rtllll of the• La d o p ed G d C h tlwnuo nH•t e•r. Solid lilH': fit t o t h e• ftu c tu atio n-dissipa tiou t h ro rem (I?D T). (From Lee, H a rte r , a nd C hui [2G], F ig. 3.) Insrt : a sclH•m a ti<' drawing o f a n HRT. (From D a~· et al. [22], F ig. 4.) cl us t<•ring o f th e LaCl 3 while• t h e• salt solidifi <•d . LaCl3 h as a m d ting l<'lllJH'r at Ill'<' of 8G0°C c·om par<•d to G09°C for G d C 1:1 [87]. T he• reduc tio n in sensiti v it.Y. by l 8Yc, is t bo ug ht to lw d u<· t o t lw a ni sot ro pi e s use·e•pt ihi! i ty o f t h r G d C l 3 n~·s t a ls t hat \\'e'r<' fornw d durin g solidificat ion [88]. 3.2.3 Noise To tlH'as un• the HRT the rma l uo ise d r us i ty, th<· shield tc•nqwrat ure was adjusted so t.ha t t h e• cooliug r a t.<' of th <' calo rinwt c•r w a s lc•ss t.ha n 0 .01 ul\:js. i\ lcasu n'lll<' tlt s W<'H' take n ,) I' K below T>.. T h<• o u t put o f tlw HRT was filt.<'r('(l at 10Hz. contl<'ctc•d t o a SJ><'<" trut n <Utal yz<'r, ami t (l(' p ow<'l' SJH'<" tra l d <' n s it~· (PSD ) d Ptc•rm itwd (srr F ig. 3.11). T he• m a in source o f 110ise is caus('(( h~· t r mpe•ra tu rr Yariat io n dn r to c• n <· rg~· fl uc tuDarkh a iiSS<'II df<'ct was far from our lim it ing HOiS<' sour\<' . 84 a tions through t lw thermal link to the reservoir. The fluctuat ion-dissipation t heorem (FDT) predicts a specific relation for t his therma l no ise density [21. 89] : (3.12) wherC' (T'2) J+ is thC' j)O\VN spC'ctral density of temperat ure fl uct uations, defined for p ositive frequencies. Here T aud C represent the t her mal relaxation time and heaL capaci t~·, resp ectively. of the salt pill. T hey are rela ted by T = RC, where R is the therma l impedance between the saiL pill a nd the helium sample. Iu t he low freque ncy limit , (3 .1 2) reduces to J4Rk 8 T 2 , indicatiug t ha t. R is thr only relevant par am etf'r t hat cau be adjusted in order to rrd ucC' noisr originating fro m t herm al flu ctu ations [22]. Agn'cm f'nt with thr FDT can lw verifiC'd by fi tting (3. 12) to thr noise spectrum. T h is rcquirrs knowing t hC' valur of T and C for t he salt pill. T he heat capacity was m easured direct!)· by a heat pulse technique, with the helium r eservoir em pty. After taking into account the addi t ion al heat capacity of t he co pper reservoir, each salt pill was determined to have C = 82 ± 2 m .J /K. T his agrccd ·w ith a n estimate of C calcul ated using tl w volume of t he salt (0.24 cnr3 ), the densit.\' of GdC h ( 4.52 gjcm 3 ) [87], ami t lw zN o fidd heat capacity da ta of Hovi et al. [90]. T he thermal rC'laxat ion t im(' was detC'rmiu cd b)' o bse rving t hr re>sp onsc of thr HRT to a pulse of heat appli r d to tlH' hr lium rcsrrvoir, whr n full. T his _vir ldr d T ~ 1.0 s which is c-onsist<'nt with the valt10 c!C'te rmitwd from thC' 6t to thr no isC' sp rctru m, T = 0.7 ± 0. 1 s (sc<' F ig. 3. 11 ). T his gives an estim ate R = 8.5 ± 1.5 K/ Vl for t he therm al impedance. At low fr<'q ueucies, thE' uoise density wa:-; 5 x 10- 11 K / Vlli. which is eq ual to t he noisC' d C'nsi ty of I litTs made wit h salt pills of p ure G dC 1:3 [22]. T he PSD was fo und to contain a sm a ll but obser vable backgro und noise clue to .J ohnson current noisE' in the capsule assem bly. T his was confirmed from meas urem ents of a 'dumlll)'' caps ul e, where the salt pill was absent. By disassembliug the caps ule and taking 110iSE' m eas urem euts of cad1 compoueuL, it was show n that m os t of t he current noise originated from the copper heat fills, and not from the BcCu caps ulP or t he cop pt'r therm al 85 0.025 r---------r---------~--------r---------r---------, 0.02 BcCu capsule+ heat fin s + thermal link BcCu capsule+ heat fin s BcCu capsule 0.015 E ::l b ~ 0.. 0.0 1 {/') '~ ~ 0 0.. 0.005 2 3 5 4 Frequency (Hz) Fip,ur<' 3. 12: Tlw backg ro und no isr dur to .Johnson ctnT<'nl noise• in t lt r c-ornpone•nt s of t he• caps ul<• a~-;sc· m bl.\·. Iink . T his b ackg round noise• h a d a m agn it udc• o f a pproxi m a t e·l ~· 1.8 x 10 1 <Po/ Jfu. ro t-r<'sp onding to a P SD of ;) x 10 - :.~: 1 J\::.1 / liz. 3.2.4 Drift rates All HnTs rx hihit s m a ll drift ra t rs I hat mus t lw ta ke•n into <H-c-oun t w hPn tn a king pr<•cis<' nu:>asun•nH•nts ove r lo ng prri ods o f t i 11H'. T he' ori g i11 of tIt is d r ift is t h o ug ht I o I><' du C' <'ithC'r to flux n e'<'P in the• nio bium flu x I ubr, or th r nn a l rc•Jaxatio n oft hr salt pill th at causC>s c ha ng<•:-. in it s m agn<' l izatio u. It ha!-i be•rn obs<' tT<'CI [91 ] t ha t , fo llowing t h<' iuitia l coo l d owu o f th<' salt pill. the drift ra tr d rc-a.vs to a constant nduc• owr a p<'riod o f se•ve•ra l d ays. En·n S ill all t <'lll()('raturC' cha nge's ·rc·-rtw rg izC'· I hC' drift , whi e·h thC'n d e•ca.Ys with a fasl t>r time c-onstant hack to a s t C'ad~· n\ lu c. T he• drift rate•:-. o f the d o p<'d llRTs wprr m rasurr d fo r <'YPr~· cla t a ru n (spe• a d e•- 86 script io n o ft It<• ana l.vsis t<•chniquc• iu S<•c . .J. l.l ). T lw.\' drifted bo th up a nd down in t <'lliJH'ratun', a t rat <•s that n w g<·d from ;t,('l'O to 10 "<Po/S('<' = l.G x 10 - l :.! 1\:j sec. T his c·o n Pspond s to a L<'lllp('t'at un• drift of aro uud ~ nK / Itour . a ud cau be account <•d for during d ;tt a analysis. Fo r <·ontparison, t h<' drift rat c•s of p un• G dC I:1 awl C.\I3 liHTs ar<' ~ 10 I'.! K /s and ~ 10 3.2.5 Implementation and calibration I :l K /s, r<'SJ><'di w ly [9 1]. SQUIDs, filters, and computer interface Each of t IH· IJRTs in 0111' ('X (H'l'inu·nt was a t tach<>d to a 220 ~ll lz nF SQCID sys- t C' lll that was ma nufact ur('d by Q uant 11111 D c•sigu , Inc. (QD). EaclJ SQL"ID ch ip was mo1mtNI o n a sa pphin' suhst rat<• fo r IH'tt<' r thc•ruta l ('OlJ(Iucti vit.\' a nd stabilit y. T h r SQCIDs th<•msr}yc•s W<'r<' s hi <'ldc•d by ni o hi11111 fiux I ul><·s a nd IllOIIIl L<'d 011 a bracket. t ha t was S<'para t <' from a ll ot h<'l' t h<'rma I st agc•s. D u ri ug t IH• <'XJ><•ri Il l<' II t , t IH• brack<•t was a t a.p proxima t<'ly -l K a nd was not I h<'rm a ll.\' C'Oill roll<'d. f\Jpasllr<' II H' nts by ot hc•r gro ups iudicat P tha t I h<• JInT no is<' C'an h<' sonwwhat r<'d IIC'<'d b~· sc•n·oi ng t h<• SQ l ' ID 111011111iug brack<'l at low I<'IIIJH'ra tu rc•s [91]. Eal'h SQU ID was in t<•rfac<'<l to a 186 <·o mput<•r t hro ugh a fl uxcountc•r hoard 6 ci<'s ign<'d b.v P <'t<'l' D a.\·. a pos tdor in the lab a t th<' t inw of m~· a rr iYal in t h<' g ro u p. T IH• fin xl'ount <'l' hoard n •;ul the· unfilt <•rcd ana log voltagE> sig na l fro111 t lH• SQCID c·ont ro i!Pr. [f I II<' sig na l was gn •at<•r th an t lw \'olt ag<' eq ui\'a lc> ut to ,....., 0.8 c/>0 , it S<'lll o ut a n's<'l s igna l to the cout ro ller, and tlH' analog s igna l dro pp<'d qu ick!,\• to Z<'ro. At t lw sam<' tim<'. th <• flu x c·ount<' r was in<·rc' nl<'ll l<'d hy om', a nd a t itner s ig na l was s ta rt ed tha t m<•asm NI tlw tim<' t ha t had <'la psc•d s in<"<' t he· r<'s<'l sig na l had been sc•nt to the <·ont roller. T il<• ,·o lt ag<' signa ls W<'r<' s<'JH'rat<'l,Y fi ii N<'<I in t he• SQC'I D <·out roll<'l's and n•ad din•d l.v iut o a Da t a :\quis itio n hoard (D.\ Q) manufa('t urc•d b~· :\at ional Inst ruHH'nts. T h<• sig na l was fi lt Pr<'d to ma tch th <' , ~·qni s t nitc•rio u for om data aquisition rat<' of 1 lL~. T his was ;H·co mplishc>d b.\· modify in g t he• filt<'T's. shown in Fig. B.l. t ha t \ \ '('r<' "Fori\ diag ratu of t ilt• flux<·ouui Pr hoar d , S<'<' tit<' .JPL lli1T Fa bril'ation a nd Assem b ly 11 <•port. [85] . 87 part of the• comine•rcial package to haw a cutoff o f 0. I H z. 7 D iagrams of t!JP m od ifie•d filte•rs a r<' in('ludc•d in .\pp<>ndix B. :\ Lah\'IEW prog ram rmd in the filte'r ed analog voltag<' ntluC's fro m thr DAQ. and adde'd the•m t o t h<' dig ital e·ountiug s ig n a l n'ad iu from tiH• flu xcount c'r board. T his gaw til<' t ota l tc'lllpC'ratun•. in tlllit ~ o f ¢ 0 • n ·ad by t he HHTs. Bc•caww there' is som <' eh·, la.' · lw t wc'e' n the' t i 11 H' t hilt t IH' n•s<'t s ig nal is rc•cei VC'd b~· the SQ U lD cont rollrr a nd th<' timr that thr flt1 xcountc·r inc rrmr nts it s vahH>, a spike a ppear~ in t he• d ata at tlw time of the rc·sC:'t . T his ca n c'asil~· br rrmov<'cl during d ata analy~i ~ b.v throwing away auy data po int s t akrn in so m e' sma ll window ( us ua l I ~· about half a s<'<'OIHI ) a ft c•r t lw t im<'r ~ i gn al h ad lwgun countin g (se'<' S<:'c ..). ] .l). Thi~ flu xc·o uut e'r syste•m C'ould not count fast<'r than approximate'!~·;)() 6 0 / sc'e·. For tlw sc•ns iti v it .v le'v<'ls of our c•xp<'ritncut , th a t nt(;'ant th a t if the tr mp rratnn' c h angc•d a t a rat<' gn'atC'r t !tan '""' 8 I' h.)s<'<·. t lw t IH'I'IUOllleters would n o longer g ive' a n acc nra t <' reading .. \!tho ug h this \\'<l.S f'airl.v (O K . vc'r~·) inc·om·<>nicnt . and put a s light n'st r iction o n t lw lli<llllWr in which <'X I><'rinH' nt s \\'e'n' ('Ouductc•d. it was not diffic ult to avoid passing this thn,~hold. T lw cou nting limit atio n was due to a s pik<' at the o utput of tlw QD SQUID. Fut u rr it<'rat i on~ of t lw <'XJ><'rinwut should co n ~idc•r u~iu g auot lH•r s~·s t <'Ill- t b e Low Tcm prrat u r<' G ro up a t .TPL It as bc•gu n t <'s ting a S t a r C'r.v o <'l<'c t ronic DC SQC1D to ut roller. a nd t h r~· have fo und t.hat it c·otmts up t o 1500 </Jo/se'<' [92). Heat switches T h<'r<' wa:-. otH' ot h<'r Iinti t at io n o u the OJH'ratio n of t h<' HJ1Ts. If t h<' magH<'t izat io n of t.lw III1T salt pills ilHT<'ast•d siguificau t l.' · during thP cours<' of a tC'mJWratmr s w<'e'p. t h<' l<'ad s of it s nio bium pi('k- up <'oil s tarted t o approach nitical C'I IIT<' nt , a ucl its t<'mp<>ratur<' rNtdi n g l><'gan t o l><'h avc• inC'ons is t e ntl y a ll([ in ac·c· ura tC'l.'·· To <Woid this prohlc' nt , a lwat switch llliHk o ut o f m a nganin wire' w ith copper-dad ni o bium- titanium wirr lrad s was wrapp<'d aro und and CE- vamishc'd to the• pickup coil 's nio bium- t it a nium T iw <IP~i~n of 1h<• modifiNI filt ers wa~ giv<'ll to us hy QD , a nd W<' c·xperiiiH'Ill<llly confinuPcl tl l(' cutoff aflc•t t h<' dtau g<'s W<'r<' mad!'. 7 88 leads: made of copper-clad Nb-Ti wire I I ------- Jf--- ' Nb capi llar SQUID ----. ---~ heat switch: made of manganin wire and attached wi th GE varni sh Nb-Ti lead wire shunt: made of manganin wire superconducting contact screw non-superconducting washer F ig ur<' 3.1:3: Attaching an II RT to a SQU ID. T h<' s hunt ac-ts as lh<' R in an RC low pass filt<'J' fornH•d with I h<' indudarH·c• of t h<' Slf()(' n ·mH iuc-t ing c-oi l. T IH' lwat swi tch is us<'d to dri,·<· I h<' :\"b kad uonnal. dumpiug tlw c urreut \\' IH' n it lwconws near nitical. IC'ad wirc•s (s<'<' Fig. 3. 13). 11 \ \"IH'll a ,·olt ag<' was a pplic•d nnoss tlH• l1<•at s \\'il rh, it ll<'atc•d up , dri,·ing tlw l0ad \\'ir<'s norm al, th<'r<'hy c•liminating th !'ir <" lllT<'ut. Tu ordc•r to <'usur<' tha t tlw tlwrn• om<'t<'l"s h a d the• s<HliC' calibration fro m on<' run to t lw JH'XI , the h<'at switch for <'ach TIRT was firNI \\'it hin a ttl..: ofT>. at t h<' hrginning of <'YNY <'Xpcrimeutal run. 13<'raus<' our hNlt capacit~· data SW<'<'PS CO\'N<'d a t<'mpNaturc' rang<' o f ouly s<'V<'ra.l ttK. tit<• lend CU IT<' lll s n<'V<'r a pproadt<•d t h<' ir nitical ,·a lu<'s during standard op<'r at ion. ~N iobiu 11 1 (('ad win•s can also IH' mwd , althoup,h this s hould lw avoid<'d sine<' pun• niobium tc•ncls to tum brittlt> a nd hn•ak a l low l<'lllf>Prat urt':-.. 89 Calibration TlH· first st cp in calibrating t lH• IIRTs was to trap a mag net ie fi(' ld in t IJ<• Hux t u bcs. Tl1is was d o11<' b.,. rais ing t lH• l<' lllJ><'ra t ur<' o f t h (• cxpcrimeu t above 9 K , tlH' supen:oud uct i ug t <'Ill perat ur<' oft he' :'\I> !lux tulH's. a pplying a c urrcu l to a m agu e t surrouuding the Yannlln can to g ive· the des ired field ntl ue, and tlwn cooling t IH' <'XJH' rimcu t below 9 l\ . B<'causc t IH' t h ermomC'l <'r s<•ns i t i vity is proportional to the magm't i'l.:at io u of the salt s, t lw hig her the mag rwt.ic field , thr higllc•r the sPns itivit.v (¢0 / K ) of t he HRT. \Ve IISE'd fiPlds of appr oxi m a t.r ly I I00 Gau ss for various runs. For th<' final hrat capaeit~· run '''<'applied 0 .25 A mps to t h<' mag nct leads. which corrrsponds to a ficld of a pprox ima t<•ly 30 Gauss. T he pr('c is<' valu<' of thc field uscd was n ot mrasurcd, s ince it s ah~olu te Yaluc was no t i111po rtant only its rrlatiw valu<' as it affrct<'d JIRT S<' rl si ti vi t.v. T h<• ll <'xt st <'P in calihrat ing c•aeh HRT was to dctermiH<' its \'olt /<Po ralio. T his nnmbcr is rclat c•d t o thc tank c in· uit o f <'ach HHT's SQUID. a llCI is t hc r('fOr<' sligh t ly diffNr nt from o nc HRT to tll<' JH'XI. T il<' <'asies t. way to find t hi s rat io is to servo t he s hicld s t.agc until th<' t<'m Jwratur<' of the• calorimet er has an cxtrenwly s low drift rate. T il<' drift ratc s hould bc s ufficirnt I~· s low t.hM it take's 1-5 miuut <'s fo r tlw I hc•mJom<'ter t o go throu g h the tempcraturc rangc cquival('nt to o nc flux quantum. This s low ratc <·u s urPs t h at there are man.'· d a ta poin ts tak<'ll p<'r 4>0 , a nd t h at thC' nnfilt C'rcd analog signa l from I h e SQGID coulrolkr will look m ore o r less like a saw-tooth wave. Th(' amp] it ud<' of the' t riang lc wa\'C' is t IH• \ ) ¢ 0 ratio. The app roxima t<• val U<' tall he d <'termin<'d hy fitting lin<'S to th<' edg<'s of the wave a nd then <'xtrapola liug to get t lw hc•ip;ht. Sewral m ore• d<'cilllal place's can IH' obtain ed for the fil by plugging in t il<' t<' utatin' value to a softwart' s iwulatc•d flux conllt<'r aud aLLCtll ptiug to lllak<' a cont inuou s t<'Hq><'raturc• s ig nal frorn t IH' da,La. By 'l.:Ooming iu 0 11 tlH• regious where tiH' flu x count ch a n ges. th<· V/1Jo vain<' cau h <' tweaked Ulltil tlw curve umtdH•s ou C'ach s id C' of tllC' di scontinuity. O n e<' t he final \ )¢ 0 valu(' was ohtai n<•d, it was <'llt<•red into a Lab\'IE\\' program that antom a tieall.v add<'d th<' a ual og signal and the flux count t o o btain a temp<'ratun' rcading in units of </>0 . 90 Thr nrxt s trp in tlu• e·alibration process was to dete rmine how many ¢>0 the flu xcountrr swrc•ps throug h when the HRT salt change's b.v one• Keh·i u. This value va ri<'!'> drpcndin g on thr compos ition and thE' amount of the sal t use'd in t he• HRT, aud o n tlw mag nit udc of thr mag nr tic fiC'ld that was froz<'n into t lw flu x tub<'. The calibration was prrfo rmrd twice fo r rach run, oncr over a largr tcmprrat ur<' raugr from a pproxima t e ly 1.8 K t o approximately 2.5 K (as s hown in Fig. 3.10), and ouce withiu only ±2 ml\: of T)... T lH' cali bration runs we're antomatrd by a Lab VIE\V program that s<'r voed an upper stagP of the expf'rinwnt, incremeutiug t lH' servo point <'V<'r.'· frw s<'COJI(ls. until the• HRT traV<'rsed the e ntire• l<'mperature range' of t hr cal ibrat io n . Fo r the large' ra ng<' cali brat ion. t he• Lab \'IE\\' prog ram a lso s uppli <>s a voltag<' pul se from t lw DAQ to thr heal swit e·h o f each HRT <'V<'ry 20 miuule's, so that thr le•ads nrver reach criti cal c urr<'nt. A Itho ug h this appc•ars as a la r ge spi k<' i 11 tlH' calibratio n d a t a, iI can rasi 1~, be rrmovrd dmi ng data a n a l:vsis. T h e wide' range' calibrat io n was oul.'· used t o map out t hr widr rang<' sc nsit ivi t i~' of th<' HRT. T lw hea t capacil.\' data was anal.YzC'd with th<' na rrow ra nge calculation, which did n o t requin• heat switche•s to IH' fin•d. For hot h calibrations, the• total flux rcadiug of the IIRT is plot ted YS. t he le rnp<'ra t u r<' r<'Hding o f a cal i brat c•d G RT loca t N l i rr close proximity, whe re b oth axes arc' n ormali zc'd so l hat T).. is e'qual to z<•ro. For the' fiuN calibration, which is t h e one used to ana l.vze' t h<' h<'at capaC'i ty data, tire' n•s ul t i ug c urve is then fi l ted to the form: (3.13) whrrr </> = n¢>0 is t h r numb<•r of flux quanta. T ir e values o f I he• coc>ffici Pnts for t h<' HRTs us<'d dmin g o m heat c·apacity data ruu at'<' lis l.<'d iu Tahl<• 3. 1. An example calibration mn is s hown in Fig. 3. 14. As can hr S<'<'n fro m the• values of th<' co<'fficicrr ts, the' fit iu (3.13) is ll<'arly linear O\ '<'l' th <' lirnit<'d t<'lll JH'raturr rang<' of our JJH'asure!IH'IIl ~. Tlwr<'fo rc, for most purpos<'s, o nl.v coefficien t ' h' is n rrd<'d for data anal ~'sis, as i I gives a vC'Q' acc ura t e co nvtc' rs ion bctw<'en thr nnmhrr of ¢>0 in o nr fi,K . 91 Locat ion Salt V /c/>o H b (' d cr II GdCI:1 doprd G.309 -2. 7-!37 0.1618 - 1.3811 2. 11 G8 hot. tom w/ La ('(' 11 top GclCla clop<•d w/ La G.l7 HRT2 :-.hield pun· GdCh '? ., HRT3 cell m icl p lan<' GdCI: 1 dop<•d 6.9-1 -7.9392 HRTO HRTl -7 ..)073 0.178.) x w-i xlO 12 - 3.9479 3.3076 X 1() -i x lO 1:.! •) ., ., 0. 128:3 -5. 297[) 2.6·118 X x lO II wj La x 10 Tal> I<• :3. 1: TinT SJW<"i fica t ions. Data is not <H'ai labl<· for H HT2 l><•causC' t his stag<' has littl<• hPat <"apacity. making t IH'rmal c·oll trol difficult and cal ibration cballeuging . . \pproxitnal<' ,·altH·~ dPri\'<'d from the otiH'r stagr:-. ar<' s uffiC"i<•nt for all <'Xp<'rinwntal purpos<•s. 3.3 The calorimeter stage 3.3.1 Ditne nsions TlH• calorim<'l<'r th at \\' <' usrd in our <'XJ><'rinH'nt was d<•sign<'d and built by Prt<'r Day :-.oott aftpr m.v arri,·a l in th<• g roup . T h<' top and hot 10111 <'IHiplat<'S w<'r<' built of ox~·g<'n fr<'<' hig h co udu ct i,·it~· (OFTIC) copprr. and WN<' amt<'al<•d to furthrr iucn•asc• thc•ir tlwnnal <"OBductaucp (se<' Fig. :3.1.)). Tlw <•ndpl<lt<'s wrn' hraz<•d to a 0.025" thick stai nl<•ss-st <'<'I sid<'wall. Tlw fact t ha l th<• eudplat <'S \\'<'!'<' hraz<'CI to t hr sid<>wall tu rnrd out l o I><' q 11 it<' fort ui t ous. .\ It hough it was uot kuown a t tIt<' t itn<' of const ruction . an.'· joining t <·chu iq tH' that I<'H\'<'S a s nwl l gap h<·t W<'<'H Lht' sid<•wall and tlw <·opp<>r lrdp,r of t h<' <' tHiplat.<' p,in•s a larp,<' Q-d<'peud aul sign a l ou L)J(' eudplal<' t.h<' nno llletrrs at t rmp<'rat un•s II<'HI' T>. [93]. This sig nal vari<·s from Ol H' c:<'l l to auoth<'r. aud is knowJJ as th<' anon talous I~apit ;.a r<'sis latH'<'. B<><·aus<• W<' w<'r<' forc<·d to us<' o ur top and hot t 0111 t IH'I'IIIOllt<'l rr:-. to t ak<' o ur data. t It is <'m'ct would ltm·£> swam p<•d o ur :-.igual and pn•,·rntrd us from obtaining nwan i ngful IIH'asu r<'IIH'IIt :-.. 92 1.5 :::.::: E ;:-< 0.5 '- ~ a::: 0 I -0.5 0 "7 E- a::: 0 - 1 - 1.5 -2~--------~----------~----------~--------__J - I -0.5 0 0.5 IIRTI - HRTI (TA) (<J> ) 0 Figm<' ;~.1 ,1 : .\ calibration ruu for IIHT1, taken in fil(' 090G98.01. . \ s JH'<''· iollsi.Y llH'lll iotwd . t IH' hc•ig ht of t he• sidc•wall was II Htde· as sma ll as prarl ically poss ibl0 to m·oid th<' insidio us 0A'c•cts of gr<n-ity. T he• dicUII<'tC'r oft he' <"<'I I wa~ dlOS<' ll to be• 2.75". T hi s gav<' t lw sampl0 a drc·c•nt amount of total h<·at capacity (ap- proximalrl.v 10 .1 / K ), and t hPrc• for0 a la rp,<' <'qnilibrium ti m<' constant to t.lw s hic•ld ( approxi11tat <'I.' · 2. 1 hours). TIH' dimc•nsious oft ht> c·c•ll an· !-i how11 iu Fig. 3.15. The total volun H' of tlw c·op1wr part s shown in t IIi ~ fi g m <' is approximat Ply 12 1.33 r m:1. At 2 K , t hr spPci fie IH•at of cop prr is around ().()3 m.l /g 1\: [91 ]. whi<"h impli<'s that thr hC'at <"apaC'ity of tiH' ('Opprr port ion of o m cc•ll should he• aronncl 3:3.;)67 m.l/ K . At the• <'lid of t IH· <'XI><'rinwnl. lh<' hc•at rapacit:v of tlw <'lllpty c·c•ll was lll <'HStll'<'d by injc•ct. ing a pnls<' of hc•al aud uwasmiug tlw hc•ight of t lw rrsu lti11g t<'lllp<'ralmr di scoutinuit~·. T hi s nwasun•me'nl was takr n sc•,·rral limc•s a t 1.89 K , a nd thr a\'<'rag<' valur wa s dc•trnni urd to I><' O.lR .J j K . The• n' tmtining hC'at capacit y is dur to thC' hrat capacit~· of tlw tim'<' HRT s;llt s in do~e thermal <"Outa<"t \\'it h t hC' calo ri lll<'ter s t ap,C' [2G] Y HTJw carpful l'('ad<•r will no ticl' t h at til<' h t>al capacity o f th<' c·opJ><'r plu~ thtPf' tiutP~ t ltP lwat 03 2.75 1.415 0.125 0 . 125 0.375 ! 0.125 t OFHC Annea led 2.0 Copper llcatcr oW Heater 5 W l .o25 ~ Figurr 3. L.): Thr c-alo rirn<'l <'r dinH'II sions. JH"<'<·i!-.<'1~· to seal<'.] All 11llllllw r s ar<' in indH'!-.. [:-Jot drawu 9-1 3.3.2 Valve Low t <'mlw ra t urP \'alv<'s ar<' n otori ou s )~- tricky. cons is tently wit bo ut leaking. It is diffi c ult to g0.t t h<'lll to wo rk Bee a us<' tlw thermal st.abili t~· of o ur sam pi e \\"as so <'SS<'ntia l for tlH' s uccess o f o ur <'XI><'rimc•nt , it was u ecessar_,. that ou r \'alw• han' an absol u t <'l.v n cglig i ole leak rate>, s iw·c• a u.v helium in t h e fill line would give a devastating IH•at kak to t he• bath. For this reason , we• did n ot u se a pressur<' activat cd val ve t h at n•quin•d pr<'SS UI'<' t o dose t.)w va lve• (d efault ope n). vV<' f<'an•cl t h a t tltc> lt e•Jiuut gas usc>cl t o kN•p the Yah:e s hut dnring the• data run might h ave' acted as a s hort to th<' ha t h . 'We• suhsc•qu <'ntl~- lrarn<'d that oth<'r g roups haY<' u sed this valv<' SIH'C<'ssfull .v, sin cE> tiH' prcssun.• d c pr<'SSE'd th e ,·a lu <' of TA in tlw fill lin<' [92]. Our cell used a simple mech a nical valv<' (sE>E> Fig. 3. 1G). :\ mE>chanical pl unger fro111 room tc m perat ure. which was attached to a n a llen head 0 11 t lw cold e nd and a fi tting for a t o rque wr<'Itclt o n t.ltc• warm c•nd. was fed into tlH' ,·acuum cau t hrough an o-ring fit ti ng that a llmwd t he• plungc•r t.o ro t a t e. 10 T IH' plun ger was usN) to turn a 10-..J.O scT<'W, which clc•fi<'cted a t hin copp<'l' d ia phrag rn ont o a kn i f<'-<'d gc' scat m ac hi tH'd o nt o t he fi ll line· on th<' top <'ndplatC'. 11 A l<'ak tig ht S<'al was caus<'d by Lh<' p lastic dc-fo rmatiou of t he dia phragm. 8c•for<' rv<'r~- cool-down a s m al l amonnt of indium was llle lt cd o tt t he botto m o f t h<' diaphragnt in t h<' location ·w hen• it makc•s coJt tac t w ith t he• k11ife-cdge seat. T his a llow<•d the• valw to be shut multipll' times w it ho ut leaki ng. It took rv23 lh- iu of t orque' to s hut the diaphra m . If the valve had to be shu t tuu)Liple t iuws on a cooldow11, s li g ht ly m o n ' t o rqu c• was used ea ch t in w. Aft c'r the \'alve• was shut , tlw fill line was hookc•d up to a c·harcoal cryopump to rc•m ovc' any hel inlll t ha t mig ht IC'ak o n t. 1:2 capacity valu f' quotPd for thf' IIRT sa lt s in SC'C·. 3.2.3 doC's not add up to ().18 .J j K . This is not an OV<'rsight. o r a mist.akf'. During thf' run !.ha t we· usf'd to characterize tlw HRTs, a higher magn<'t.ic fidel was used t han duriug the run wlwn• wc took ou r heat capacity data, all(! t.lw heat. capacity of t lw salt s is a fun ction of mag nf'tic fie ld. T he' total heat capacit:v of t he c<'ll d uriup, t.h<• tlwrmon tC't er chanH·t !'rizatiou run was a pproximately 0.28 .J / K . 10 \Vhen tuming the plunger. car<' had to b(• t akPn to a\·oid lateral lllotion , or t lw o- ring would becont<' dis placPd . and the vacuum can would qui ckly fill with eno ugh lwlium to ruin a run . 11 Tlw top- plate o f our cell was annealed, a nd so after o ne or two usPs the 'kni fe-edge' lwcarne quite bl unt . It functioned anyway, hut futun· d<'sig ns of this vaiY<' sho uld <'it her not an neal t lw copper or us<· a st ai n less st e<'l kniff' f'dgf' soldf'rf'<l to t lw top plate. 1 :! As wi ll he desnihed in future chapt<•rs, wf' n was ured the a mount of h!'lium in t IH' ecll al t he The• adnulta~c·s of this vain• dc•sign is its simpli<"ity aud dl'c•cti,·c•rH•ss. !Ls disadYant agr is t hnt t IH' d i nwnsions of t IH' rnc•chaniral d iaphragn 1 p11 t s cons! rain Is 011 t lw aspc•ct ratio of t hr cc•ll i11 a wa.Y that is not idc•al for t h<'rmal h<'at fl ow. For rut 1m• <'XJ>C'rirucnts it would be advantag<'ous to chang<' to a II lOr<' Y<'rsat il<' vaiY<' d0sign. s uc·il as t h<' Oil<' c·orlstrudc•d by Hi<.:hard Packard 's ~roup at Brrk<'l<'~' [95] or Yuanming Liu's at t he• .Jc•t Propulsion Laboratory [9G]. 3.3.3 Helium cavity Th<' pr<'cis<' ,·olume of t hr lwlium rompartnH•nt of t h<' c·<'ll was dN<'rmirwd during the• <'XJH'riliH'Ill. .\(though t}w dialll('t(-'1' of tll(' ('('II C'OII)d <'asil.Y h<' IIIE'HSilf<'d din•<·tl~· whc•n t he• c·e·ll was warm. the· J>I'c•dsc• depth was somewhat morr difficult to d<'t<'nnine. \\' he•n it was originall.\ · built. the• dis tarH·c· IH'tW<•en tlH• two c rH.lplat<'s was int<'IHI<'d to ))(' I ntlll. Hmn•,·c•r. l>C'twc•e•n small <'ITors in machiniu~ tlw d<'plh of the sid(•wallledge and tlw sid<'wall itse•lf. and t IH• side•wall uot IH'ing flu s h against tll(' end plate during tlw brazC'. tlw <wc·unu·.v of this dc•pth was uot guarante<'d. \\'<' rue•asur<'d the artual value· of LIH· cdl hc•ig ht to g n•at pn•cisiou b.v using thr<'<' te•chniqurs : 1. \\'<' fillc•d the• c<'ll wit It lwlium and thrn ntN\s urc•d its h<'at capadty as <1 fund ion uf t<'lllJH'rature (sE><' Fig. 3.17). Since the• ll<'at capacity ofhc•lium in tlw ahse' ll<'<' of a heat current is known to gr<'<tt precis ion .. '''<' cou ld subtract off' t.he• hc•at c·a pal'i t.\' of 1 IH• <'Ill pt~· ce•ll from our data, all( I tl H'll match it to t he• known c·11rvc• h.v using t lt r IIIIJIII><'r of mol<•s in the• sample as au adjust.ablr pararnrtrr. B<'e·ausc• the• width oft h<• cdl is known, t IH• deusit~· of liquid IH?Iium could then IH' use•d to dc·rivc· the· heig ht of the· c·e•ll. :2 . •\t the• c•nd of om c•xpC'riutent, we• cxt.racL<·d all of the he lium ill ou r cell into a nllihratC'd ,·olunH>, and mcasur('(l the' n umbc•r of moles in tlH• samplr. \\'<> vc•rific•d the• llllllll><'r dc•tc•rmin<'d h.v lll<'tlwd {1), eonfirmiug the cslimatNI valll<' of th<' c<'l l hc·ig ht . IH•ginning of tlw run and s<>vc•ral months lat c•r right hc.fore WI' warnwd up. \\'p found l lw <'xact amount of lwliuJn a;; originall.' · present , indieatiug thai our w d\'P did not lrak, a nd llw cryopump wa.'i an unn<'<'<'S::.ar.'· pn•<·aulion. 9G To the top o f the cryostat V"c"hcad ------. ~.,. Turn rod Shield stage Valve Valve kn ife HRTI post 'Dead vo lu me ' Main cylinder of Helium H eater 6W Nb flux tube Heater SW ...... ___ ... - Figun• 3.1G: A schemaLic drawiug o f Lhc> valve that seals tlw h0lium in th<' cell. A tll<'C ha nical tnrnrod frolll th<' t·op of tlH' cryostat turns a ] 0-clO sn<'w, t.hat prc•ssc•s down ou a thin copper diaphragm, which seals agaiust a knife edge s0aL. [No t drawn pr<•cisc•l_\· to scale.] 97 100 ........ ::.::: - 95 u 90 0 E ......, "--' ~ C"j 0.. ell u ~ ~ ::r:: 85 80~----~----~----~------~----~----~ -3 -2.5 -2 -1.5 -1 -0.5 0 T- TA. (!lK) FigurC' :u 7: I TC'at capac-i t~· c un·c· at z<.'ro heat fiu x. Solid litH': fit to I h<• LPE data [2] , routHic•d fort IH· g ravit .v width or a O.G..J. uun <:<•II. Opeu circles: our pub<' data tak<•u at ...:ero Q, usi ng tit<• nu111ber of m o lc•s or llE'Iium in t.IH• sample• as an adjttst.ablC' para11tNC'r. The• fit indicates that thC' h<'ight oft lw c<>ll ish = O.G I n1111. TIH' <'X<"<•llent agrC'<'IIH'IIt bC't WC'<'JJ t ltC' gnn-ity rounding of tIt<' data aud t IH' t hc•oret ical cu rw is a lso <'vid<•nt. :3. As dPsnilwd in Sfc. 3.1.1. gnn- it~· CT<'atC's a h~·drostatic pn•ssun• gradieut that caus<•s the trausition t<•mpcratun• to Yar.\· as a function of hc•ight "·ithin thC' edt. This It as the t'll'eet of rounding t hC' spC'cific hC'a t ovC'r a tC'mpC'rat tJrC' rang<' d<•t c•nu in<'d by the cc•ll h<•ight. Tltc• d<'<'P<'r the c·C'll, t hC' more t h<' rounding that o<·<·ms. Tit<• d<•grC'<' of roundiug call th<'rcfor<' bC' usC'd to giw a prC'cisC' cC'Il c!C'pt h (s<'<' Figs. 3. 17 and :3.3). All of t hC'S<' nwthods indintt<•d that the• c<'ll coni ainC'd 0.089G moles of lteliuHJ, had a ,·olttmC' of 2.-1.)2 cm=1 • and t lwrC'fon' had a h<'ig ht of O.G.JO IIlii I. 98 'Dcall' Volume 0.0-1 mm (0.064 em) 0.05 "(0. 127cm ) 2.750"(o.985cm) Figur<' 3.18: SdH•nwtic of tlw r<•ll ,·oluw<'. T il(' ·dead H>l uuH•' ref<•rs to th<• lill litH' aud Yaln• s<•a t that are o ut s id<• tlH· main c~· lind<>r of helium . ldPally this ,·olunH' is uot fill<'d with liquid ltclittllt, but with H bubbl<' of hc•limn vapor. ' D ead volume' B<•rau1-><' of th<' d<•sign of our fill lin<' and YaiY<', o ur n•ll COIIIHitH'd a n•rtain autouut of "dead ,·ohuue · that la.Y outsid<> of th<• m ain cy lind<'r of h<'liunL If this YolnnH' was hllc•d with SUI><'rfluid helium . thPlt it would haw• di1-.tort<'d th<' uniform h<'at flow in til<' sampl<•. lu additiou. l><•<·attsc the hrat flow t hro ug h tlw thin fill lin<' would h<' larg<'l" t ha u for t he• n•s t oft IJ<• sample. t he h<•at capacity CQ p<'r mole o f tit<• h<'lium in this Yolum <' would h e• <•nh a ttt<•d owr tha t o f Lhe bulk hel ium. fd<'all .v, thi s 'd<•ad volunH• ' s ho uld I><• made as s mall as possi bl<• wheu t lw ealorimctrr is drsigu<•d . From Fig. 3. 18, it u w I><• cal<-ul at<•d that , for our cell. it s YoluHH' was 0. 121) \ t)\ = 7f ( - 2 2 X 1 (0.:~8 1 ) 1.2!ii + -7r - 3 2 2 X 0.203 = 0.023G t m '.1. (:3.1-!) T h<•rc•fo r<•. tlH· p<'lT<'ll!ag<• o f tlH• C<'ll oer upi t•d b~· the "dead ' ,·olume was approximat <' 1 ~· \ ·1>, \ 1wr X I()() ~ 0.02:3G X 10() = 0.9.:J %. 2 A 7G (:3.13} fn O llr ('X p<'rii1H'111 . \\'(' il( ((' IIIJ>t('( ( to limit tlH• infltH' II ('(' of t it<' d<•ad VOIUIIH' b.\' 0 111.\ ' filling t It <• <"<'II with enough he•lium to o<·c·upy thr ntain c,vlindrr. Thr idra was that t liP 'de·ad volu nt<' would th<•n IH' filled with a ·bubhl<' of IH'Ii um vapor. Unfort un at<'l.v, it is mor<' or 1<-ss intpossihiP to \'('rify the• !oration of the bubble without going to rat IH'r drastiC' IIH'asurc·s (w hic·h we• did no t ). In au~· c·as0. th<• exislaut<' of a 'bubbl<•'. rrgardl<'ss of its locat io11. assur<'d that o ur <'XJWrinH'lll \\'as JWrforuwd at const aut sat urat<'d ,·apor pr0ssurc• (S\'P). and ('Ould t hc•n•forc• I><• <·ontpan•d to tlt<• abundaH<'<' of theoretical pn•dictions and <'xprrinwuta l data dc•riwd tllHIN tltrsc• ('Ouditions. 3.3.4 Location of heaters and thermometers T hrr<' '''<'n' t,,.o hNltNs lo('at<'d din•<· I I~· on t.ltc• calorinH•t N. Thr.\' w<·n• both four wir<' IH•at Ns. coustntc·tNI o ut of <'wtuhout \\'ire•. c·ount<•nvoutHI to IH' uou-i nducti,·<· a nd G E ntmishe•d dir<'ct I~· to t h<' ('OppN. Thc•y WN<' bot It lo('at c•d IH'Iow t It<• IH•liulll sample•. IJ e•HtN 6\\' , which J>rm·idc•d the• 111ai11 lwat ('lll'l'(' llt (J , \\'aS 31:2.31 n aud lo<·at('d approximat <'I." L.-1:2'' l)('lcl\\· t lw hPI i um sam pic•. He•atN ::> \ \'. whi('h pro\'i d c•d the ('alo rillt('ll'_\' he•at ('IIIT<'nt . CJcal· \\'ilS 36.96 n. and \\'HS loC'a( (•d approxillla( C' l ~· 2.2.) .. l>e•lo\\' t he• ltrlium sampl<' (s<'<' Pig. 3.13). Botlt IH'atNs W<'r<' crimr)('d at UH' ('alori m <'tc•r to Sllf H'H'OtHltH·t ing niobi un1-Litanium lead win•s wh iC'h W<'l'<' h<'at-sllnk at <'<H'h progn•ssin•l .Y ('Oidc•r (highc•r) stage• until rc•athing tlw t o p of t.h<' \'acuu111 can (se<' Fig. 1.16). ThC'rc W<'l'<' four thNIIIOill<'t<•rs locat('d din•<·tly 0 11 tlw calori111etPr: 1. IIRTO t:l was c·- be•a ut wel<l<-cl l o tlH• bot t o nt of t he• calo ri tn<'t C'l' 2·· h<'low t h<' lt<'litllll s<unplP. Th<' post. of HRTO was a pproxint al<·l.\· T long and h11ng b(']ow t lw C'alori nt<'t<'l'. The• IIRT salt . whi<'h was 0.9.j" long , was lo('at<'d at tIt <' C'nd of' t he' post . 2. TlRTI was c·-lwam '"c•ld<•d to a tab ]o('at<•d 011 th<' top endplatc• of tlw calorinwtc•r. l t \\'as locH t ('(I 0.:313" a lww the• lwl i um sam pit' . Tit<' post of IIRT 1 was approximat<'ly 10'' long. It was h<'ut a ro und tit<' ndorinu• ter, so t hat tl w <'lid l:l A uotC' for those• who mig ht a<'tually look at t h(• lab hooks for this <·xp<'rinH'nt: for son I<' r('ason that I cannot ('Xplaiu. liRTO was abo rcf<'IT<'d to as IIRT.t at tiuws during the• ('Xp<'rinwut. ~ly apologi<'s for any <·onfu~iou that this 111ay <·ausP. 100 of t he• IIRT post and t he• salt Ia.'· parallel to t ho:-.c• of IlHTO uudcmcath the ('alorinH't N (sc•c• Fig. 3. 19). :3. GTIT..J was also locatC'<l 0.313" ahoY<' t hr Jwliun1 sampl<'. so that it would gi\'(' t hc> sanw lC'IllJWraturC' rC'ading as ITRT I. It was c·oatC'd \Yith siliC'onc• lwat-siuking c·ompouud and tighth· fittc•d into a copprr foil that \\·as indium soldC'r<'Cl to the• top- plat<' of th<' calo ri llt<'l <'r. I. 1IRT3 \\'as c•- IH'alll wc•ldc•d to a knifc•-<'dg<' ring I hat wa:-. in pr<':-.s<'d c:out act with t h<' st airdl'ss-st <'<'I sidc•wal l. It was int c•ndc•d 1o a('( as a rnid-plaue t h<'l'lllOilH't t•r. Figmc• 3. 20 s hows wh.\' tl1 is dc•sigu was not Sll<·<·c•ssfll l. 3.3.5 Heat flow within the calorimeter B<•<·ausc• thc•n• is signi ficant hc>at flow in this rxprrinH'nt , t lw C'<'ll should id l'ally haw ()('<'II dC'sigtH'd so that t lw isot IH'l'lllS \\'it hiu t hr ('Opp<'l' <H<' flat in t IH• rq~ion IH't \\'C'<'n IIH' sample• and I he thcnHOill<'t<•rs. Duncan and Ahl<'rs [79] prrfonnC'd a uunwrical simulatio n using the tlwrrual difl'usion <'quatiou discrc•tel.\· in c~· liudrical coordinate's. to dc·t<'l'lllillc' what c·onstraiut s this n•quir<'lllCHt puts on tlH' dinH'nsions of a cC'll. T h<'.\' discown•d that iu ordc•r to c•usun• fl a t isot herr us at th<• hclilllll surface•. it is tH•c·c•ssar.Y to us<' au rndplatc• with an aspc•c·t ratio (dicunc•t er/ le•ug th ) of one or 1<-ss in ordN to sC' paratc• thr IH•a tc·r fro nt the• sa mple>. lr1 ot.IH•r words, i11 order to e11sun• that tlw hC'at <'nt<'l'ing o ur ('<'II was 11niform anoss tiH• width of tiH• ('<'II. the• h<•atcr (G \\' ) should han• h<'rn lora tc>d at le-ast 2.70" bC'low the• sa111plr.. Likc'\\'iS<'. tlH' tlwrrno lne'IC'l' ( H nT 1) s hould han· bC'C'Il loc·a t<•d 2. 75" abO\'(' t he• Salll pic•. l ' u fort uu at<•l.\', the dc•sign oft he val\'C' llladc• th<' rc•alizatiou of this SC'c·ond nitc•rion impossible•. To dc•t <'rill ill(' t hC' C'ffC'ct that t hC' locatio ns or O il[' lwat c•rs and t hNillOill<'l<'l'S lllight hav<' ou o ur C'XI><'riuwnt. \\'C' pNforlllC'cl a finitc•-c•l<'nH•nt ana l.vsis on our cc•ll. \\'<' dis('O\"C'r<'d th a t a lth o ugh our dC'sign was no t id<'al. t hC' t h<'rmal problc•rns W<'r<' not too sewn•. The res ults of this anal:vsis an• shown in Fig. 3.20. T hr isot lwnus i11 t lw <·op)H'r near t he ~ampl e ~urf~H'<' and th<' t.IH' rnlOillei.NS arr ll <'ctrly flat. On thr JOL HRT: High Resolution Thermometer 0 GRT: Germanium Resistance Thermometer Heater ~ Vacuum Can IK pot Heater 7W HRTI (top) 0.640 mm i Helium HRT3 Sample Shield Heater 6W (shield) Magnet 0 0 0 0 0 0 0 (boltom) ______.~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 or Figur<' 3.19: Sdl<'ll!al ic of the apparatus. illust rat iup, t hC' locatio n al l t hr lH'at<'rS aud t iH'rnloHwt er~ us<•d i t1 the ex peri nH'nt. Tlw lwat <"IIIT<'tlt Q is applied with h(•at<'r G\\' and tlw ralorimPtr~· lwal curTPnt CJra1 with hP<tl<'r ;)\\' . (:\'ot<': not drawu to scalE>.] 102 oth<'r ha nd, t h<' isot h erms in t he• Yicinit.\' of t he• 111 id plan<' thc•rnJOlll<'l<·r arc clearly not flat, i nd icat i ng t ha t , as SII SJH'ct r d , HRT3 d id no t giv<' a t m st worthy indication of t h<' helium tcmJ H'r a t ur<'. 3.4 Thermal network Becaww o f t hr rr lat ivel y largr hrat C:UITI"nt s a nd na rrow l<' nl JH'rat un• rang<' of o ur experinw n L t hr t lw rm a I ne•twork was a n i rnportaut asJwct of tlw d<'sign of t hc• a.pparat us. It was essenti a l t ha t a ll t h er mal pat hs w0r0 ac(·oun tc•d for. and that t h<' dis trib ution iu t herm al r<'sist a m·<' a nd heat capacit i<'s W<'rl" d0signrd so that wr could bot h h av<' s ufficien t t h<'rllla l contro l and a p ply t h e necessary hPat curr<'nts and still havl" t h<' sam pk b r s ta bl e• w he n at the la mbda tem per ature. 3.4.1 Three stage the rmal is olation syst e m T hc• calori llH' I c•r was mo u11 t C'd 0 11 a t hree s tage t h crma l isola t ion s.vst em. In its basic form . <'<H'h s t np;0 c·on sis t<'< l o f H 4" coppe r ring plated iu gold , a GilT, and a h t>ater mad<' of e•Yano h m win• non-ind uct iw ly wouud aro und a copper post also p lated iu gold. 1 1 T IH' h eat0rs and C TITs w0re sc r<'w<'CI t ig htly to t he• st age• ring. The· gold plating was to improve tlw t h0rma l contact lwtween t hem. T ht> top s tage was coun <'<"t<•d to a 1 K p ot, w hose h d ium fe<'CI capi ll ar.v 's irn p l"d e'll ('l" was mc•as tm•dt!i to be Z = 0.2 x 10 12 <·m :1. Accord ing to Richardson and McCl intock [98]. t h is slw uld h ave' g iven a base tcm peratmc of 1.55 I\' and a cool ing pow<'r of 20 m\\'. T he· hasr temp <'rat u r<' of our 1 K p ot wa~ mrasured to lw 1.68 h.. Its behavior was a hit fl a ky. T il(' ptun p ing power ucc<'ssar~' to provid<· a s table , non-oscillat o r,v, tcmpr ra tttr<' base vari<•d as a fun ction of ll t>lium bath level. T h<• t.em per atnrc wonld re m a in stable• for approximat d y 15 h o urs aL a g iven p u mping power , after w hich it wou ld h<•gi n to oscillat<' wi ld ly. If t h e valve• to th<• p ump was t h e n shu t a lmost alllhr wa~· 14 15 down , t h e rl"h~· decrrasi ng t hr p umping (H>WC'r. the ostillat ion s would stop aud 8<'<' A ppC'11dix I3 for hC'<'l.tC'r valu <'s aud G RT eal ibrations. T lw im pC'dC'rJC'<' was m C'asuJ'('(] using t lw tech n iq ue <ksnibed iu Rirhardson [97], p. 55. 103 Copper top end plate Color: T Vector field: -grad(T) J (Cill) Mid-plane thermometer: pressed contact ~T Kaput a ~TK""'''" Copper bott om end plate Stainless steel sidewall * ~W/cm Q= I 1 2 Figur<' :3.20: A fiuite elcmrnl a ua l.vsis of thr thrrmal profik of thr crll. Isotherms an' at 10 uK separation. The dis tor ted isot hrnns IH'ar the stainless sl<'d sidewall iudi cal<' wh:v th<' midpl an<' tlwrmonH'kr did uot g ive accurate tempcratuH' readings. 10 l t lu• I<'IIIJ)('ra lure wo uld s ta bilize• fo r a uolhN 15 ho urs. It was this JWCttl iaril .v that put a t inH' limit o n t IH' le• ng tl! o f our ex perimenta l runs. The t o p st age was :wrvoecl us ing t.lw G HT and a QD He•sis ta lle<' Bridge• run throug h Lab\ 'IE \\' . During til e JH'riods th a t the• 1 [( pot was be• having, it was cout rolle•d to within ± 10 11 K . Th e• t hird s tag<' co nsist C'C I of a rad ia tiou sh ic•ld tha t cotn pkt Pl:v SlliTOIII I<i<•d t he• calo rin H'I<'r a nd th0 ll!1Ts. It was coHstnKt <'d of coppc·r a nd pl a ted itl gold . . \ c-opper co il t ha t was pr<'ssuriz<'d with he•lium gas whil e• a t 77 K . seal<'d . and soldNC'd shut was at tached to t he• I> o tt 0 111 of t lw s h ie•ld. T his \H l S to itHT<'aS<' t lw heat capaci t~· of this s tage. tlu•r<> by imprm·ing its tiH•rm a l control. T hro ug hout all h<'at capaci t~· ruus it was sN nH'd using a n liHT t o within a bo ut ± 0.2 JLK . All s t ag<'s we•n• s t ruet mall~· rotlllPCt Nl h.v tbiu wa ll s t a inl<'ss-st <'<'I p osts wit h h igh t lwnnal n•s is t ivi t ~·. braze•d to t IH· coppc• r riu gs. The.v were• a lso liukc•d hy a thi 11 wall CUJH'rll icke•l c irc ul at or, thaI was t IH'nua ll.v audwrc•d t o all of th e s( agc•s. T he eire ulat or was us0d to cool I h<' e•xp<'rilll etll from 77 K to J K, b.v appl ., ·ing a pproximat ely 3 psi of hc•lium gas at o ne <'nd of t h<' circulat o r . and pumping it o ut t he• ot ll<'r c• nd. lt was cvac:ua tc•d during d a t a r uns. Due• to the• lH'at CIIITe•nt s usNI in t il<' <'X p<'rinw nt, t h<'r<' had t o I><' a fair! ~· s t ro ng thC'rma llink i><•tw<'en all o f thE> stag<'s t h<' s ta inle•ss-st<'<'i pos ts wC'r<' not ne•arly suffi- ci<'nt . T he• n trio tt s stag<•s W<' r<' ther<'fon • conn Pc ted using brass rods solclNed I o copp<'r fo ils which WN<' t IH'II sc n•wc•d d own to t he• s lag<· rings. Due• to th<' ad vrrst' e• ff<'rl th at this had 0 11 t lH'rrllal control. t IH' linking n•s is ta nces wen• chosen to IH' as large as possible• givc•n the• lwa t nuT('III s that we• iutc•nd<'d to usc• for o ur Pxperiuw 11t. .~ Iea­ sur<'m<•nts of the• the•nn a l rc•s is tiYit ,\· of 1He• indica te• tha t llH• flu id begi ns to c·x!Jibit dissipa tio n for lwat fltt xe·:-; la rg<'r th a n J f t\\' jc nl'l [23]. so we• chose this as o ur upp<'r heat cut-re•nt limit. For o ur cC'Il. this co rTe'spo nd<'d to a t o ta l pow<'r of 1:)3.3 p \\' . For d C'c<' nt t hN tna l control. it was ne·ce•ssar.\' t o se•n ·o tlH' top stage• l<'lllJH'ra t urc• slightl.v a bove t 1)(' bas<' t <' Ill p<'ratm<' of t he• 1 K po t , a nd w<' t hC'rc•for<' e•st ima t <'d l hat o pti ma l d a ta could lw o bt a in<'d if th <' t o tal n•sis t anc<' i)('lW<'<'II stag<' o ne• a nd t he• ca lori nH'te•r l05 was: 1? _ ~T ::::::: 2.1768 K 1.90 K ::::::: :- . . - Q 153.3 fi\\' 180o h. 1\\ . (3.16) lu add itiou, it wm. ad\'antap;c•ous to put thr la rp;rst portion oft his r<'s ist arH't" IH•t w<'<'ll t h<' ~hi<'ld all(! the• calorinwtN, so that a ny thrrrnal no ise• on t hr shi r ld stag<' would lwn• a minimal infiu<'llC<' 0 11 tlH' sampl <'. Thr t hrnnal r<'sistanc·c·s brt Wf'f'n tlw stap,C's w<'r<' rtH'as u r<'<l. and their ,·a] u <·~ an• show n in Fig. 3.:21. TIH' t ota l rc•sist a nc·<' t n n l<'d o nt, to lw l 6G8.1G K / W . clos(' to t.lH' desi red value. T il(' the rma l link lwtw<'<'n t he sh i<'ld and t he• ralori rnc•t <'I' was c·om JH>S<'d of thr(;'e t hi11 brass rods, which w<•rp at t ach<'d nt lo<"at io ns sPparatc•d b~· I 20° ll<'<H t IH' c•dg<' of lh<' ralorin H'l<'r. T his desigu was to distribute• t he• h('at fl ow as <'\'<'Ill.' · as possible o ut oft h<' top oft he· rc>ll. Sidewall thermal resistance T lw t hPrm a l nrt work bC't\\'C'<'n t h<• sid<'wall and t he• JH~Ts [shown i11 Pig. 3. 22( I>)] was also in,·es tip;at<'d by injc•rting a hra t c·urr<'tlt of 32.08 n\\' into tllC' c•mpt:v c<•ll. and rrH'asuring tlH• l<'lllJ)('rat un• dis pl ac<' nr<'nt of t hr linTs. T his addc•cl q chang('(! t he• t c•uqwmtm<' of linT 1 by 30.8 Jll\, of IlnT3 by :33.3 ,11\ . a nd of I InTO by 36.-l flh:. T his impli <'s that t.Jw n•sist aJH'<' hct W('<'n lll1T l all(! IIRT3 was R sw 1 ::::::: 75.66 K j \\'. and the• n•sistarH·c• h<•tw<'<'ll HRT:3 a nd llRTO wa!> R s\\·2 ::::::: 99.03 K j \\'. In This confirms that o nr nr id plari<' t lu•rmolll<'t<'r. HHT:3. was hiasPd towards t ht• top plate of t h<• <·<'II. 3.4.2 Contact resistance Thr fart t Itat o m <'XJH'rillH'lll was sc•Jisiti v<' to <•xtreuH•l,v s 111a ll dta ug<'s i11 l<'m]wraturr mPant that \\'(' also had t o I><• c·ou<·c•ru<•d a bo ut the <•([eels of contact n•sistan<·<>. At d iff<'r<'ll t t i nws we• w<'r<' plagu<' d h_,. hotlr t IH' t hNrna l and electrica l ,·ari(-'t i<'~. r;This is of tlw sanw ordc•r of magnitud<' to tlw anli<'ipatt•d \'alu<•. Giv<'ll that tlw roudurth·it~ of stai n lpss-st<'<'l is about 0.002 \\) K Clll, a 0.025" t hick SS sid<•wall th at was 2.75" iu di aruel<>r a ud l 111111 high wou ld hm·<' a u m·prall n •sis tanc<• of a pproxitu a l<'ly 9 1 K f \\' . 1 106 1 Kpot R "" 8 K/Watt Stage 1 R 1F 407.38 K/Watt Stage 2 R 2,= 395.39 K/Watt Stage 3: Shield R \4= 865.99 KJWatt Stage 4: Calorimeter Figme 3.21: Schema tic of the thermal network of th<' apparatus . .6.Ti repres<'nts the thermal nois<' on stag<' 'i whcu stages 1 a nd 3 arc under servo with q applied . 107 (a) HRT3 post ' -- HRTO post M echanical s upport ~ Pick up coil (b) (c) F igur<' 3.22: (a) Sc hematic of the cell with HRTs. (b ) ThE' t IH?nn al net work of Llw stainless-steel sidewall. (c) The therm al network a nd possi blc heat leak of the b flux tube to tlw top-plat<'. 108 The rmal contact r e sistance proble m s Dmin g one• of om early r uns. we noticed t ha t t here was a myst0rious var iable thermal resista nce t hat would occasiona lly appear lwtwccn the top plate and the valv<' d ia phragm of t lw c<' ll (s<'<' Fig. 3.16) . It wasn 't larg<', hu t it was enough to ruin a dl-\ta run if it changed part way t hrough. We bad attached t hrs0 t wo j oints (and a ll other j oints) wit h Apiezon greasr in order to increas<' t lw n nal contact. Apparently this was uot a wise choice. It seems t hat Apiezon grease, and most other forms of vacuum grease, fract ure at low tem peratures. aud arc not good for therma l joints [99, 100]. D uring our ucxt warm up. we d eaued off the grease a nd replaced it with a thin layer of sili tOJH' heat sink com pound with Zu0 2 (Dow Corning 340) . WI:' wwd this com p ound when conn ec ting all G RTs. IH•aters, heat sinking posts, a nd circul ator anchors to t lw t herm al stagrs, as wdl as wlwn assembling t lw various com ponents of t he shi eld and t hr calorim eter . After t hat , w<' n<'v<'r encounter ed a ny problem with t hermal contact resista nce. Electric al cont a ct r esistance proble m s D uring another r un , t he temperat ure of our calorimeter stage would occasionally start to drift , <->veu wheu t he shield temp era ture rem ained extrem ely constant. This oul.v happrnNl clming r uns when Q wa~ applied to t he cell. Aft.er some prelimin ary testing, it !)('came clrar t ha t t lw vo ltage th at was br ing applied to heater 6~ to ge ner ate Q 7 was changing over t ime. After a few fa iS<' starts, wr discov<'rrd t hat it vvas d uc to problems wit h contact r<'sistancc on t he aluminum fo ur pin connectors located on our breakout box. Appa rent ly t hey had built up a fair am ount of oxide, whi ch ca.used a n in termi ttent contact resistance t o be added to t he lead resistance. Vve did two t hings to cure t his pro blem : We first replaced all of the aluminum four pin conn ectors vv it h Delerouix ' military' connec tors (Ampheuol P T 06A-8) . T hese not only have gold plated pius for optima l contact , but they also lock into place, preven ting gra\·ity from looseni ng the connection. T his climiuatC'd the problem . 109 As an <'xtra safPt y nt<•as un', w<' also a ttadH•d h<'aters G\\ ' a ud 5\\. to t urr<•Ht sourc<'s that w<'r<' s uppliPd h.\' th<' two DAQ volt.ag<' channe ls. Th<• p ower g<'IL<'ntt<•d in th<' h<'al <'l'S would t lwn lw i ns<•n s i tiw to t. h<· rC'sis t.ancPs of' Ill(' leads, cout act. aud otherwis<'. T hr current som<·C's that we install ed arc s hown itt Fig. 3.23. For constant current o p<•ratiou. R:d R 1 is s<'t equal to R 1/ R 2 , so that (3.17) f (·onstant - R Y," R, -J\/V'--r------, I, ~ Y oul R, Figure 3.23: Th e• curr<'nt source us<·d to s uppl y Q a nd Q cai· \ in <·om<'S from the o utput of a DAQ <·ontro lkd b.v a La h \ 'JEvV \'irt ual los t rumeut. \\'<' us C'd only precision (±0 .01 %) resis tors in the cons I ruc tion of the c utT<•u t so mT<'S. T hC' r<'s is tam·<· values we ins talkd iu our t wo current sources o ur lis t<•d in Tabk 3.2. T hC',V wen• dws<' n t o g ive optimal power for our hea t c urrent ra nges, us ing tlw fact tha t t.lw maximtmt volt ag<' put out by the DAQ was 10 Y . Tlw dC's ig n o f' t hC'S<' <·tnT<'nt sources W<' re chosen so that if C'ither sid e of Llw h eater IH'cam C' s hortNI t o grouud , tit<• c11rrC'nt s uppli<'d t.o th<• sampk wou ld be minima l. T his was t o pr<'V<'llt ruin<'Cl mn s a nd cxp loclc•d cel ls. F'or t.he resist.au<·<•s lis t.<->d above, t.h<• 110 Heater Rt R'2 R :l Rt Max. Current Max. Q 5W 100 kS1 100 kS1 10 kS1 10 kS1 100 fl..\ 9.65 n \Yjcm 2 6W I 0 kS1 10 kO 1 kO 1 kS1 l mA 8. 15 fi\\ )cm 2 Table 3.2: C urrent :;o urc<> specifications. rnax inlltlll currrut suppli<>d in Lit<' cv<'Jlt of a s hort would br 4.7 mA, whidt corr<'spouds to a powc•r of 6!)0 ,,vv. Giv<'Jt om thc•nnal JH'lwork, this would IPad to a chaugc• iu t lw tPm1wraturr of thr calorimrt<'r st age• of 1.145 K, whidt would not lc·ad to auy problrms that could 110t rasil~· br r<'m<'<liPcl. Tlw c urrc>nt sources wrrr tcstcd for a s imulatc>d changr of 10 S1 in kad resistancc. Thr ,·oltage across t he load (the s imulatcd hPater) was stable to less than 0.001 mY, giving a p<'rceulage ,·ollagc c•rror o f less than 0.0005o/t., and a J)('tTcntag<' hcat cu rrent error of lc·ss than 0.0019/(. Bc•causc• 10 n is far larger than au.v r easonable• variabilit :v i11 the· IC'ad r<'sist ancc, ac-tual c•rrors s hould be> far below thcsr valuc·s. 3.4.3 Potential heat leaks For t IH• sample• to be thermally stable• to the tweded tolerance, it was c•ssc•ntial to lllakc• s un• that the thermal rc•sist<trH·e•s hrt W<'<'ll our stages were couslanl at a ll lilllcs. This nt<'<lllt that a ll lcad wirc•s. niobium capi ll aric•s. circulators. and brass n•sistance bars had to lw S<'curc>cl firm ! ~· in pl;.u·c•, so that tlw.v would not change• posit io n and intE'nllittcntly conduct hrat t o unwantc•d locations. The l0ad \\'irc•s wrrr all hrat sunk to PVE>r~· colder (higher) stag<' by GE-,·arnishing its twist0d p airs a round a gold-piM<'d pos t that was tlwrwally anchorC'd to tlla.t stag<' (s<'<' Fig. 3.24). Tlt<'.Y we' re' LhNeforc• imnwbil<', aud had coustant thermal rcsista ncrs. Tlte IIIlT niobium capillari<'s wrn• sccurrd in a. s imilar maml<'r. Trn to fift<•c'n sLrauds of tOJ>JWr wire wert' wrappcd around thr capillary at thr locat ion oft lw hrat si uking. a ud attached wilh GE-,·amish. Tlw other ends of thr copp<'r strands W<'r<' 11 1 To stage 2 Copper T ab [ • heal sink compo und - - - Heal si nki ng post: C u plated in Au \ Shield stage: C u plated in A u Nb-Ti lead w ires GE varni sh C uNi crimps C igarelte paper Ni obium-Titanium ca pillary Heater 5W : evanhom w ire GE varni sh N iobium w ire Picku p coil ~ -1 1- Post d iameter = 0. 125 .. F ig ur<' 3.2..t: H<'aJ sinking tlw HRT capillary a nd h <'alN kad wires. 112 solden•d to a ropprr tab, whi('h was paiutrd with h eat siuk compound. and firmly srru rrd to t hr s t agr. Niobium HRT capillary AI thoug h this m<'t hod of Sf'(' uri ng t h r niohi u Ill capillarirs r ns urc•d that t lwy would not givr an int0rmitt.cnt t lwrmal rrsistaucr hNwe'<'IJ s tagrs. it was not dear t hat the• uncha ngi ng the rma l rcsisUwcr that th<'y did gi vr would not b r " probif'm. Aftrr all , the• N b-Ti capillaries wNc essent ia lly running from thr bottom, middlr, and top of t IH• u•IJ to coldc·r stages. T h is would seem to be a perfect path for h eat to r scaJH' from 6\ \" a nd ,)\\' a nd avoid goiug t hro ug h the cell of he lium. Si nce the effpct I hat wr wc•n• I r.ving to measun• was stwngly dc pe•Hclent ou t he magu itude of the lH•at curr<'nt , it was important to kuow if this wo uld be• a large <'frecl. To make sure t h at it was not, an anab·sis was [H'rfo nne•d 011 t he• crllnC't work. An arbit rar~· heat c urrent of Q = 3.8 /t\\'jcm 2 was ch os<'n for the• analysis. The• nc•twork use•d is shown in Fig. 3.25. From Fig. 3.20, we S<'<' that: Qj - Q, - Q:\ = () (3 .18) Q ;; - Q:l - Q'l = 0 (3.19) Q-1 - Q 7 - Qn = 0 (3.20) Q s - Qc; - Q r, = 0 (3.21) Substituting .:::.T = QR (for rxamplr. 7; ('" - T~'t> = Q5 Rjoin 1 ), we• wwcl thr following c·st imat <'d values for tcnqwrat ure a11d resis t a n('C n1l ues: 113 'T~hicld 2.0--19 K TJJp 2.1768 K njoint 10 K/Vv flsw 10 1\. j \\' 20.5 pJ\.j\\' RJ-(apit/.i\ 866 Kj \\' R;ll fluh(1) - 9.56 X 106 K / V.' flub(2 ) 1.] 9 X 106 I\./ W Rub(3) 3.96 X lO(i K j \V Rl ink 33-l K /W To obtai n t iH'S<' rstimat<•s, w<' usrd t lH' fact t hat. the thermal resistauc<> of Lh<' . bTi capi ll ary is 1 em Kj\\', and t hat it has an outc•r diarnrtrr of 0. 660 rnm, and <lll inllf'r cliamrtrr of approximatrly 0.-106 mm. Furthrrmorr, R nb(l) was approximatdy 8 inch es lou g. Rub(2) was approximate!~· 1 inch long. a ud Rnb(3) was approxinHltrl~· half au inch lo11g. S ubst itut iu g a ll of t. h rse values iut o the eq uaLiou s and sol ving for T 1 , T 2 , T;('P· aud Tcp. \\' C' found: T1 2.17679957 K T2 2.17527228 K Ttcp 2.17679099 K Ttp 2. 1753-1244 K From thcs<' Yalucs. we can determine t hat the heal flux out of thr top of the cell would be 113 Ji\ \ ' , or "' 3.8 p\\'jcm 2 . a nd t h at t he heat fl ow out. of thr sidrwall through the nio biulll capi llar_,. is 4.5 x 10- ll 11\\'. C learly Lhcre is n o problelll w it h stray brat Je-Rks through I h<' l\'b- Ti capillari<'s. N iobium flux tube Anot.h<'r p ot cu tial so ur<:c of h<'al leaks was h<'l wren t h e HRTs a nd thr bottom of t hr cell, t.ltro ugiJ tl1c niobiuw flux t ubes. T he flu x tub<·s WE're both fir111l_v mountrcl on t he' 114 links (a) Shield stage ~ · Bottom endpl atc HRT3 post ..-···· ·············· .......··· ····························... [C (b) QM t Tt;bield R34 Q2 t t Rnb (3) t Q7 Rn b(2) t Q4 t Q. Rlin~ Tz T,P Qs ··········································· Q6 Rjoint R link T, Ttcp Q3 RKapitza Rnb (1) Tac Figm <' 3.25: Pot<'ntia l heat lrak: t h<' Nh HRT capillary. 115 I >OI tom of t IH' c·c·ll and a lso in solid contac·t wit II t lw nwdwn ical s11 pport ho lckrs of t h<' LIRT coppC'r pos ts [sN' Fip,. :3.22 (a) and(<") ]. If tlw c·ondu <"t i,·it .v of the• flux tulw was too hig h, t.hc• 1op- plat e• and 111 icl plaJH' t h<'l'lliOIII<'t<'l's wo uld h<' ad n•rs<'l.v in Hu<'IIC'<'d h.v t he• hot t N t <'Ill per at un• of t hP bottom plate . . \ quiC'k <"akulation shows that tlw tc•mp<'l'alun• of llRTl will bP equa l to: ]' ) Rc·u T IIRI 1 = T.tp + (rr .l h p lp R n (3.:22) Cu+ Ll FJ ' whPrc• T1 " is the• l<'IIIJ><'rature• of the• lop plat<', 71>P is t.hc• L<'lllpe•rat.ure• of th<' hotlon1 pia Ie>, R cu i ~ t he• !'Psi~ I <Utc·e• o!' t h<' IIRT c·opJWr post. and R FT i~ the n·~ is t an('(' of t h<' Aux t uiH' . . \t 2 )( , alln<'alPd <"Oppe•r ha~ a t hNmal c·o11dudivit .v A ~ lG \\'jcm 1(, and bot h stainle•ss-stc•c• l a nd nio bium (the• two c·olltpOIH'n(s of tlw flux tuh<'s) have• a th<'nnal c·ondu<"t i\· it~· A~ 10 :l \\'jC' m !(. Thi s impliC's that !'or t IH' diiiH'IIs ioll s oft h<' llHT post and the• f-lux tube>, RF·1/R<'u ~ 14x lOa. Tlw tC'ntprratmc•e•JTOr is thc•r<'forc• on th(' o rd er of Oil<' pa rt in l<'n th ousand , and c•xtrc•mc• l,v ne•gligihl<'. 3.4.4 Thennal stability In o rd er to l c•st how wC'll W<' had manappd to C'OIIt rot t he• t ime• d<'JH'IHie•nc·e• of' t he• thc•nual JH•twork. a ud the c·onsist <' II <',\' of ot lwr thc•rmal pa mm<'t c·rs. we• did a t <'St of tlw thC'rtnal stability of our syst<>m . ln tiH' ai>~<> IIC'<' of a Jwa t CIIITC'nt , wf' S<'rYo<'d th<' shic•ld stag<' so that t he• helium sample was in t lw vici nit~· ofT>.. with approximatd.v ZNO t<'mperat urc• drift rate•. \\"c• kft the s hi<'ld stage at a ('Ons ta nt lf' lllJH' raturf' for aho u t 18 ho m s. whic·h was ahou t as long as t h<' 1 K po t wou ld allow without acting up . .\ s C"an lw S<'<'n in Fig. :3.26. t IH'r<' is no m·('rall t in H•-d<'J)(' Ilde•nt n •sis tanc·e•. and owrall s tabilit~· is wit hitt a fraC't io n ol' a 11K. llG ftle 032999.01 10 "0 iii E "'I 0 ~ 0:: J: 5 -10 0 2 4 6 8 10 12 14 16 18 2 4 6 8 10 lime (hrs) 12 14 16 18 02 0 1 ~ f= 0 0:: J: -0 1 -02 0 Figm(' 1.26: TIH' t.lwrmal s tabilit y of our experimental sys tem. The un it s of t ltr shiC'ld (HHT:2) are in c/Yo. lll 3.5 Experimental configuration and interface 3.5.1 The cryostat and the computer interface T IH' C'alorinwte•r. shi<· ld . aud tlJCrmal isolation stages were• mounted insid<' of an a lulltiuunt nt<'lllllll ran . . \lumiuutll was used b<.'eausc> it is non-magnc•t ic. and the•r<'for<' had a minimal intpact 011 t h<' fu11rtiouiug of th<' SQUIDs. A magn<'t was wound around tlw bottom oft IH• ,·,u·uum can, dirc•c-t ly outsid(• tIt<' location of tlw HHT :;all pills (s<'C Fig. 3.10). The• IIHlgne•t was wound of A\\.G #lG roppN wire• with pol.vllt<'mtal~·z<' insulation . It was basrd on a dc•sign b~· John Lipa·s group at Stanford. lt was coust ructc·d to localize• t hr fi 0ld in the• <"<'nt N oft h<' ntag ne•t, and t h<'l·<'h.\' prc•vc•nt tiH' fi<•ld front alr<•cting an~· t,hing but th<.' llRT flux tniH's. 17 l3C'catts<' the rnagu<'l is only t unr<'d ou when the• fic>ld is trappc>d in the• flux t ulws. and not for t lu• dmaJion of I lw <'Ill ir<' <'Xt><'riHH'Ht. it is not important that t IH' <'OPP<'r wirrs g<'rH'rate• a fair amount of h<'al. Tlu• pron•ss on!_,. takes t<'ll to twent~· minute's. and th<'r<'forc> doC's not boil off a significant antount of helium . The• resislaru·<' of tlH' magnc•t win' is 12.ii n at roolll 1('lll]H'rat Ur<'. 2.0 n at II 1\: , and 0.0 n at 1 1\:. ~leasuring this rcsistaJH'(' was I hNc>forc• au e•xcc•llc•nt way to d<'l<'rmirw the• trnq.wra.ture oft he outside• of t h<' <'Xp<'ri 111<'111 d 11 ri np; cool-dowu. The• vrtntum <"an was motmt<•d 0 11 a probe• that has s t aiukss-st<'d s upport I uh<'s runuing to a top brass pla.tr (srC' Pip;. 3.27) .. \lso rnnning IH'twc•c•n the• vacuum can and the• top plat<• were•: the• pumpinp, lin<' for the· 1 K pot , tlw lll<'Chanieal rod to shut tiH' C'e•ll Yal\'('. four HF lines for t [)(' SQriDs. tIt<' <'<'II fill lin<'. thC' circulator capillari<•s, a hPlium le•vc•l llt<'l<'r. a charcoal cr.Y opuntp for tlw fill lin<'. aud approximatc•ly .)0 twistc•d pairs of win·~ use•d to run t)J(' !waiNS. cnTs, a nd hrat switch<'S. TII<'S(' win•s c•nt N<'d t II<' ntntum e<IJJ through a IH'rm et ic BPndix <·on nc>ctor . . \11 of I lt<'S<' it c•ms w<'r<' t ltrrad<'d through holes cut inl o a S<'ric•s of e·opp<'r radiation haffl<'s. wIt iclt pr<' \'<'111 Nl room t <'tllJH'raLun• rad iaLiou front corning down from I h<' top platr of tit<• cryostat. Tinfoil was e•poxied around what was krt of th<• OJH'nings iu tltc•s<' ltol<'s to e•ye•u furth<•r protc•ct. against radiatiou aud furtlwr prolo np; thr hC'lium 17 ThP ma~;n<'t s pc•c-ifkat ions ar<' gi\'C'n in .\pJWndix 0. 11 8 r-------~)1~ To Roughi ng Pump Fro m gas He bottl e j::y_ I h To K inney Pump Filter box ~~ j >. Bu bble level ', IIII . T or p late I > He lium ha th I I < Circ ulator I . I ., LN2 oute r j acket ........; · - v, I ~---' > Vac uum .jacket ·,• I I < 1-1- mc ta l sh ie ld-----+ > I Pumping line '. . > Mangan in tw isted pai rs ! I .. ~·- -~·- :~Y: ., I I ~') l I I I Kp ot I ,.,.,..,_ ~ ! He feed capi ll a ry Rad iatio n ba ffl es ' < l. < > 52-pi n connector J A llen -key sere wdrivcr tfl:J , 1r > Ia ll a t.l. ll a t.l. Ce II valve ~IL .IIII , ~ L...:"" Mag ne t ~~r Vacuum can .. > ., . ' I, !:>:1 S uppo rt base I~ I < < !:>:1 ~ I S uppo rt screws Figure 3.27: A schema tic drawiug of Lhe cryostat. 119 bath lifC'. T hC' magnet lc>ads wC'r<' at tach ed t o banana plugs conuC'<"t<'d to on<' of the lowN rad iat io n b affies. LC'ad wirc•s wrrC' o nly insrrtrd to turn o n thr magne'tic firld a ud w<•rc remO\·ed for th<' remaind<'r of t hr r un . which prevent<'cl a s tron g h<'at l<'ak to roou1 trmpcraturc. During t.IH' <'XJ)('rinwnt . tl w pro h<' was iusert <•d into a hcliu111 d ewar t ha t had both a n o utN liquid nitrogciL (LN2) cav it.~r a nd au outer vac uum j acket to Ll w nuall y shield t hr hrlium bat h from roon1 t rm pera tnr<'. Surrounding thr <'nti rr d c•war was a doublr walled Jl-lliPt a l s hie ld to prrvrnt m agnetic· fiC"ld s frorn a ffrc ti n g t lH' SQUIDs. T he lOU lead wires were at tad1<>d to a n LC fil ter as t h e~' ent<'r<>d and rxit<•cl t }J(' cryostat in o rde r to prPveut RF pickup. T h e reson a nce of t his <.:irc·ui t was 3.3 l\lllz. and it s con figuration is s hown in Fig. 3. 28. After leaving lhc fil t erbox, t hP w ires ran t o a brPakou t h ox. From thrrr th r wirC's for t h r GRTs and for the' ht'atrr on s t agr 1 (Ill ) wrrr attarhrd to thr QD R <'sis t a nr<' Bridgr. T hr b ridge' was a tt ached to a ~ation al Instruments G PIB hoard installrd o n a 486 rompu t rr and cont rollrd through a Lah\ .IE \\' program . T h e oUter heater wires werc attachrd clirrctl.v to two diff<'r<'nt DAQ hoards th a t wne run frolll the· sam e com puter. At t h r breakout l>ox, <'ve'r)' wire was fil t<'r<'d by a not hc•r L(' fil t c•r constr ucted by wrappiug t h em around a loss)· fNridC' torus a nd connrrting t ll<' w ire's of Pach t wisted pair with a capacitor (s<'e' F ig. 3.29). T hr RF lines W<'r<' hooke·d to SQUID contolkrs that wrr<' attachrcl both to tlw DAQ boards a nd to the ftuxconnt er hoard s mentiorwd in Src. 3.2.5. T he• pu111piug line for Lh<' 1 K pot wcnl loa Kinney pump located in an adjacent roo111. This hc'l t><'d to cut down 0 11 t IH' vi brat ion presen t ir1 t lH' room. All othe r lines lraving t hr c r~·ostat wNe' att achrcl thro ugh a sNirs of valves (shown i 11 F ig. 3.1), aud run t hrough f" copper pipe t o a gas haudling sys tem (sh own iu F ig. 3.30) Lbat was usc'Cl to s uppl y tlle rrgular hC'Ii111n gas, ultra- pure• helium gas, a ud roughing pump I i ti C'S. 120 FILTERBOX From brea ko ut box VACUUM CAN L= I mH To cryostat C= 2000 pF 1 L= 1 mH To breakout box C= 2000 pF From cryosta t o( 1 F ig ure 3.28: The F ilter-box at the top of t h<' cryostat.. T h ese LC filters a rc on cv<'ry wir(• entering and Pxit iug the cryostat to prcv<~ut RF pickup. 121 To the computer t ~ From the computer BREAKOUT BOX Twisted pair lead wires / Ferride core torus Capacitor: lOOpF From the filter box/cryostat t I To the filter 't box/cryostat Figure 3 .29: T b<' breakout box filters. Thes<' L C filters 'vV<'r<' on <'V<'ry twist<'d pair of l0ad w ires in t lH' oxp crimont . 122 Q COLD COLD TRAP TRAP UP! I \4'' copper pipe to to ultra-pure Gas He holllc Ml' I'M •;.." copper pipe Gas He bou le o;..-· copper pipe ® 0 I " i.d. s.s. nexy tube to roughing pump B: Valve Pressure gauge ± 30 psi Pressure blow off val ve 30 psi (i) Thermocouple gauge I-I G: H astings vacuum gauge UPII : Ultra-pure G l-le Circulator Mini -Fridge Pumping line Quic k release clamps C: Mr: PM : Q: Figure 3.30: A schem atic drawing of the gas haw:lling syst em. 123 3.5.2 Vibration isolation The t emperature 111easurements of t he HRTs were wry sensitive to vibration. The crvostat itself was mounted 011 a piece of concrete in the sub-s ub-basem ent of the Sloan Laboratorv. on a piece of concrete that was sepa rated from the resL of the buildiug. Gcucra l builc.liug vibrations therefore did uot affect the experiment. In-betwrcn Lhc cryostat. aud the gas handling system, the ~" copper pipes wen• attached to ~,, flc•x- tubes, and run through a la rge bucket of sand. This damped a ny vibrati on from thr pumps or from proplr walking around on thr floor n ext to tlw cryostat. As an added prrcau t ion, the area around the eryostat was roped off during any data runs to ent down on flo or Yibration. 111 T he fina l source of vibration came from an unexpected place. T he outer LN2 jacket for our c:r~'ostaL was plugged at tht• top and had a 2 psi blowoff valve to let out the nitrogeu gas boil-off. T he slight. hissiug of this blow-off val vc was e11ough to cause nolic:c'able 11oisc in our data. To cure t he problem we took off the cap to the inlet and replac(•d it wit h aU-shaped open copper pipe. So long as thrr<• was LN2 in the jacket , there was a lways positiw pressure , and the inlet did not icc over all(! t he noisr stopped. 1 8WP could act ua ll y ~ee t hP noise iu our data from people walki ng in Deliveries of LN2 (kwars wnP pa rt icular ly devastatin g. t lw vicini ty of t lw cryostat. 12-l Chapter 4 Experimental Methods T im'<' d i fkn•ut mNhods W<'I'<' us<'d to I ake• heat capaci t~· data d uriug I h<' c·o u rse' oft his projc•c·t: :-\C C'alorimctry, drift ca lorim<'t r.Y, and discr<'t<' pulse' calorim<'tr~·. A full S<'t of l ,a h\ 'IE\V programs was wri tt.c•n to control the• <'Xp<'rim<'nl using <'HC'h rn <'thod . Por all data. nrns, r<'gardkss of nl<'thod us<'d, th<' hrat current Q was suppl ird to h<'aLN 6\<\' (s<'<' Fig. 3.19) h~· a current so un·e pow<'rf'd by a voltage output from OIIC' of Lhe ('han rw ls of a :"Jational Ius trunl('nts AT-~IIO-lGX 16-bit DAQ board. The heat usf'd for the' caloriuwlr~· meas urement. Cdcal· was suppli<'d to ]wat er 5\\' by auot h e'r c·urrc•nt sour<·c• J><)\\'C'rc·d b.\' a ,·ol t age output fro11r a sc•c·mHI chann<'l on Lh<' satn<' DAQ. T il(' top s tag<' was S('rVof'd to approximal<'l.v ± 10 11K with a QD bridge' controllc•d h.\' Lah\'IE\\' through a l'\at ional Ins tnJmc'ul s CPIB board. Th<' t<'mp<'ratur<' lll<'as un'nH'nt for this se•n·o was provid<'d by a Lake'slror<' C'<l I i hraLc'd g<'rman i u m resistance' t hNmomrt<'r. The' s hi <' ld was S<'l'VO<'d to approxinlat.<'ly ± 0. 2 JtK ·w ith a PID program writteu in Lah\ ' TE\\'. Ill1T2 provided t h r LrmpNat Ill'<' m easure ment for this ser vo, and the h eat was s uppli c•d b.' ' a ,-ollage o utput b.v a chaunel 011 a J\ational Ins trument s AT-AO-G 12-bit DAQ across h eater H3. Tlw te nlJH'ra t urc of Lhe helium sam pi<' was nlC'as ured by t hNIIIOllH' tcrs HRT3 , HRTl. aud llHTO. Each cxpcrinH'nt was run by a mas ter Lab\ ' IE\\' program that coordiuat <'d l h<' \'arious S<'rvos. t urued off and 0 11 the' volt a g<'S to th<' h ratc'rs, rc•ad in tlH' tenqwratun's of al l of the stag<'s, and s av<'d th <' data. to dis k . ]·r . -0 4.1 Heat capacity measurement techniques 4.1.1 AC calorimetry The• fir~t t<'dlllique that '''c•1 tl~('d was .-\.C' calorim(•tr.' ·· Thi~ mrthod has a numbrr of ad\'antagc•s o\'<'r ot h<-r t<•thniqtu•s: it c•limiuatc•s the adv<'rS<' <'ffc•c·ts o f ~tra.' · lwat IPaks, and it c•asil~· pc•nuits C'adt data poiut to IH' avNagc•d O\·er lo ng pNiods oft inH'. impro,·ing t lw s ig na l to tJOis<' ratio. In this m <'t hod , I he• sam pi<' is at t adl('d through a known t h<'rnt;-tl n•sbt a JH"<' to a hat h at constant tPIIIJ)('ratmc [ 101]. For om c•xpc'rillH'nl , t he• hath is the• ~h i<' ld . a nd llw thC'l'utal rrsis ta n cc• is /?;11 . Jl c•at is app liN i lo t h<' sam p le• t hroug h a lu•at<'l' (.) \\') at Q 0 c·os ( ~w1). all( I t h<' t r m prratm·c• is ntras ll r<'d h~· a l hc•rn tonwt c•r tuount C'd 011 t lH• <"alori m<'l r r· (s<'<' Figs. :3. J 9 and -1.1). T hi s t hrnno nwt <'r 11l<'aS1ll'<'S (-t.l) whrrr T .,hi«'ld is t he• t <'m pc•rat Ill'<' of t he• s hic•ld . C is t he IH•at ('apac.:i l,\· of the sam plc . a nd n is a phase• offsc•t of litt le• import a 11c·c• so loug as I he• fo llowing <'XJH' rillt<'ntal <"Oildi t ions are> me•t: l. T lw h('at C'apaC'it y of the sample• is muclr la rp,N tha11 thr lwat C'apadtiPs of t iH• t hc•nuouJc>t c•r a11d t IH• h eate•r; 2. T lw t hr rma l n •sistanc·<'s hc•t.'·"<'<'ll tiH' sautpl <' a nd tit <' IH•atC'l'. l hPrm omctC'r. and hat it do no t ch ange> as a fun('t ion oft imc•; :3. wr., >> l. wiH•rc• r,, is t lw rrlaxa ti o u tim r o f t lw santpi P: '1. T IH• rc laxa t io11 l ittr<'S of t he• hcal<'r a ud the> lit(•nuonwtcr. r 11 and r 0 • M<' rxt rc•nwl~· s hort <·omparNI lo t IH• frc>quell<',\' w. 1 Allhou~h I wa~ tlw only on<• who took discr<'t<• pube calorim<•try data , at variow., tinws duriuf!, this p rojN·t, Talso Chui a nd Ri('hard L e<• were involwd in llw <'X<'<'ulion of t h<• <'xp<•rinwnt nl p ro<"<'d urc•s, and tlw whole p,roup was invoh·<'d in makiup, d<'('bions ron c<'rn ing th<' <'XJH'rinu•nl . I <llll tlwrpfor<· u s inp, t lw plura l for all ac tiv<• a nd pass iv<' <'X I><' riru Pnt at a<"t ions. 12G T lw magnitude' of the' sin usoidHI c-ompotwnt of tl10 t<'mprratur<' oscillation is T 0 = Q 0 /(2...,)C'). so the' hC'at capt-wit~· oft lw sampl0 can lw cl<>terminC'd b~· nwasuring T 0 : (' = CJo . 2wT0 (-L2) T hr heat capacit~·, C. is an•raged ovN tiH' t emp<•rat ure range of t.hP osc- ill ation. Til<' initi a l results of o m a tt e mpt at AC calorim etry gaYP rathN baffling n•sttlts. T h <• amplitude' of the AC t0mp<•rat.un• s ignal seemed to indicate that th<• lwat capac- ity of our cc•ll was significan t ly difrN<'nl than we expected. It app0ared, in fact , to lw <'VCtt smaller than the well-kuow11 heat capacity of 1H e in th<> ahsrtH"<' of a heat flux. As it would turn out. this sl range res ult was due to the fact that our midplane t h<•mtOIIH'l N was sh ort <>d I o the lop plat<' of the cell. and was tlH'n>for<' lwi11g iuflu<' nc<•d b.v t h<' s ingular Kapitza n•sistauce at t h e top boundary. Sim·<' the m ag11it udc o f tIt is r<•sist aucc changes as a fnnrt ion o f I Pmpcrature. con eli Liou (2) of the above' rxpcrinw nta l n•quirrm<·nts was ,·iolHt.<•d, and (4.2) was uot an ;.u·cuntl<' formula for th<' brat capacity. V\fC' wou ld n ot discov<'r that this was our probl<•m until W<' tri0d oth<'r calori mrtry mrthocls that wN<' somrwhat morr straightforward to inl <'rpr<'t. 4.1.2 Drift calorimetry \\"r nrx t tunwd to t lw drift calori nwtr_y tedmiqur. Likr th0 AC mC'thocl , tlw drift mrt hod also eaijily allows t hr data to be aYerag<'d owr many c~-c l <•s. The standard technique For this t.C'chuique, the s hi<'ld tc>rnp<' r at. urC' is controlled so that the' t<'lllJH' rat ur0 of tIt<' hcli tllll sampl<• drifts dovvuwards a t S('Veral uK/sec. The tem peratu re' rauge of the <'xperimcut is d<'fitrc•d h~ a minimu111 temperatur<', T,11 ; 11 , and a maximum tc tll)H-'rature, Trnax· Th<' I <'Ill JH'rature is JWrmi t I C'd I o d rift down u uti! it readws T,,.;,.. \Nhcu it falls h r low Tmin· anotlrrr hC'at Cll tTC'IIt, CJcal· is s upplird to the sampk. T IH• magnitude of CJcal is chosrn so that wlwn it is applied , the tC'lllJ)('rature of the h<'lium drifts Computer Stagl! 1: Shidd Servo Loop: Lahview PID R14 := 850 K/W I HRT I I OAQ I. ch I Q --+-------:~~ I Hcawr 6W I DAQ I , ch 2 (1) 2 Stagl! 4: Calorimeter Fip,ur0 4. l : Sdt<'nt ati<' o f Lit<• AC llt<•Lh od o f lH•at. capacit y Jll<'asun•m <•u(. A IH•at flu x, Q. i~ ~uppli<•d to tliC bot tom oft h<• c0ll. Th<' t<•mpC'rat ur<' o f tlw ~ hi C'ld is <·ontrollC'd to ± 0.2ftK. ,\ s innsoirlal IH•at <· mT<'nt , Q.:al = Q 0 cos is ~mppli <•d to th<• bot tow of tlw <'<'II . <·ausing tlw sampl<• tc•mt><'ratun• to osdllat<• with fn•qu<'ll<'." oJ.J. Thf' a m p li tnd<' of tit<• os<"i llat ion , T0 . is im·Nscl.'· proporti ona l to th<' h<'at <"apacity C. Cwt) . 128 upwards at srwral nK /s<•c·. It is allowC'd to drift upwards until it n•adH'S Tmax· At that point. Qcal is turnrd o A', a nd tlH' trmpcratlii'C' oncr again drifts down to 7;11 ; 0 . T his procrss is rr])('atC'd 5- 10 t im<'s so t hat the data can b C' hi nn<'d and awragrd. Thr sh irlcl l E'IHperat ure rem a in s uuc ha ng<'d throughout the m easuremen t. (S<'r Figs. 4.2 a nd 4.3.) T he lu•a t capacity can IH' <'asily d c•rivc•d. On lhC' dowrnvard r a mp , w hC'n ouly q is ap plic•d to t h<' b ott o m of t lw c<'ll , Q = T- 7~hi('ld + C(T) R (dT) dt ( 4.3) dowu w hrn· T is t IH' t <'111 p Prat un• of t h <' sample>, Tshi<·ld is llw tem pcratun' of t!H' s hic,ld, R is t h r t h rrmal r<'sist a ncr h <'twc·rn t h<' t o p of t h<' caloriHH't<'r a nd t IH' s h i<'ld ( i u our casr fl = R:1 1). a nd C(T) is t h r hrat cap acity of t h r samplr. On thr upward ramp, c wh<'u Qcal is also t urned on. Q + CJcal = T - ;~hield + C(T) ((r:;) up Sill('<' th<' s hiPicl t<'lllJ>C'ratur(' is k<•pt constant throughout tlH' lll<'asur<'nH'Ul , lhe lwa t capacit~· is found b~· s ubtractin g ("1.3) l'ro m (4.4.), CJcal C ( T) = ~:-:---:-----:---=-::--:-(dT jc/1) 11 1> - (dT / df )down. (-1.5) A variation on the standard technique For o ur C'X)J<'rimrut, Tmin and T.nax W<'l'<' chos<.>n 111 relation to TnAs(Q). 7; 11 i 11 was c·hos<'n to b<' 3-5 jiJ-\. bc•low TnAs(Q) , and 7~uax \>Vas dwseu Lobe just aboV<' TnAs(Q). A hra t cu rrr nt Q was appl i<'d Lo IH'at<' r GV\' at t h e vPry IH'giuuiug of t.IH' <'XIJC'rilll<'UL. On<' up-d own ramp SC'<JU C'lH'<' was t a k<'ll wit l1 Q applied. In ordrr to compare the• s pc•ci fic IH'a L a t co nstant h eat flu x wi th th(' well-known static sprei fi c brat. Q was s low l.v t lll'll<'d off' at t hC' C'nd of t he first ralllp, all() anothrr ramp S<'qnrnc<' was takr n iu t h <' ahs<'ll('(' o f a h eat flux. Q was turu<•d of[ and 011 in h<'t wr<'n r,·rr~· np-dowu ramp srqu r nc<' (s<'<' Fig. 4.3). It was t his particular t <'rhniqur 129 Computer DAQ2 Heater H4 Stage 3: Shield Ser vo Loop: Labview PID RF SQU1D R34=: 850 KIW RF SQU1D ~ He Stage 4: Calorimeter DAQ I. chI Q DAQ I, ch 2 Figure 4.2: Schem atic of the drift method of heat capacity m easurement. A heat Hux, Q, is supplied to th<' bottom of th<' cell . T h0 t<'mpt'ratur<' of the shield is controlled to ± 0.2 p,l-\ at a t empcrat.nr<' chosen so that , wiLh Q applied , t.lw temperat ure of tlw sample is within several pK of TrMs(Q) a ud drifts dowuwards at rv 5 uK/scc. Oucc th(' t<'mperature reaclws a fix<'d t<'mpf'raturc T;llin , locat('d rv 5 JtK !)('}ow TnAs( Q) , a S<'cond heat cnrr<'nt, Qcah is supplied to t he bottom of t he C<'ll. T he magn itude of Qcai is choseu so t ha t tlH' temperature of the sample drifts upwards at ,....., 5 nK/sec. vVhen the sample temperat ure reaches a fixed temperature, Trnax "' 1 JlK above TnAs(Q). Q,a1 is turnf'd off. T lw tcrnpcrature of t he sample then oncf' mor<' drifts downwards, and tlH' process is repeat<•d ind cfini tcly. 130 that tt ncovcrrd the fact that thr mid- plau r t he rmo m <'trr was s hort<'d t o the top plat e. T lw specific lH•at m casur<' m c nts t a kcn at ZC'l'O hrat flu x wrrr in rxec llcnt agreemen t with llH'asurc nt<' nts in t lw literal m<' [2]. IJowrvcr, whrn a h rat flux was app li<' d , the tcm prrat nr<' s('al<' a pp<'a.rrd to s hift rr la t i vr to tlw t em p er at urr seal<' for t IH' q = 0 meas ur<'mPnt s (srr Fig. 4.4). lt was. in ft:tcl , s hift<•cl as a dir<'C'I result o f thr K apitza rcsistancr brtwrrn IIRT3 and the sampk. If t h<' t<'lll!Wratun-' sit ift wen' due onl y t.o tlw uniform of[set of thr regul a r comp on Pnt of t he K a pit za r<'sistane·c'. \W' could have' c·ontimt<'<l to u sr thr drift m et hod to pursue hc'al capacity rrsults. In that C'asr, thr b rat eapacit.y data wo uld have b r<'ll ;.u·<·urat<'. o n!:-.· s it i f'tc'd in tem p r raturr. The s ing ular compon ent of Llw Kapi tza rrsis t an<·e>, lt owc'v<'r , is a fnnetion o f t em pNat ure. T h erefor<': (4.6) when' T 0 is tit <' l<'lllJH'ratur(' nH'asurrd h~· HRT3. T 11 (' is tiH' he lium Lcmperatur<'. ,j.n is the' (<'ll1I)('nll urr drop dur to til<' s ing ular Kapitza rc>sistauc<-', a nd T 0 is the boundary tcmp<' rat urr dC's('ribrd in Src. 3.1.2 . Extractiug dT11 ,.jdt fro lll the llH'as ur<' nH'Ill of T0 to drt<'l'mine' t.he heat capaci t:-.· of the sam pie from t h e d ata i!-> <'xtraord i uarily d iffiC' ult , if not i 111 possi bl<' . 4.1.3 Discrete pulse calorimetry \\'itb this discoYery, \\'<'fi na lly W<' abaudo n<'<l o ur so-called ' mid-planr' thermometer all(l t.umC'd t o tlw di sne't.C' pnl s<' ('a]oriuwt r,v m eth od . T hi s m Pt ho d is extremely simpl<•. .:-\ s ing le• he'at pulse , ...:l CJcal • is injc•C"trd int o tlw samp le>, a nd Lhe r<'suiLing discontinuity in th e t.<'UJP<'ratur<', ~T , g iv<'s th<' lwat. capa<"il y: C = _j. Qcal . .:lT St ra~· lwat lr aks can causr sonw rrror. but this efi'<'ct can be minimizt>d by extrap- o lat in g thC' temperatur<' traj rc t o ries b oth befo re aud a fte r the discoutinuit.y to the 132 1 4Ur-----------~--------~--~-- (a) ,-._ ~ 120 0 E ;::::; 100 '-' u -4.5 -3.5 -2.5 -1.5 -0.5 0.5 ~T = T - TA(J.!K) J 30r-------------------+-~~--- (b) 11 0 u 90 70 ~----------------~------~~ -2.5 -1.5 ~T -0.5 0.5 = T - TA(J.LK) F igure 4.4: Sp ecific heat d a ta t a ken wi th tlH• d r ift m ethod . T he d ata a r<' t a k<' n with t iH' mid-pla n<' t lw rnlOill<'t<'r. F iles a t no n-zero Q are a ffec t <'d by a Kap i t za n•sist a n<·<• a nd do not g i \'(' va Iid r<'adings. Solid c ircl es: n o hrat fi ux. Open cirdes: 2 Q = 0.2 13 f t\Y j cm "2. Tria ngles: Q = 1.12 f t\ A. 'jc m . Squan•s: Q = 3.58 f t\\)cm"L. (a) Cnshi ftrd data. Arro""s indicat e wh en• dissipative fluid <'nt<•rs th e cell. (b ) T he• d a.t a s hift <'d in t<'mprra t ur<' t.o a dj ns t fo r b o unda ry rrsistan c<'. (A v<'rs ion of t h is figu r<' was pu blis hed in lla rt.<'r, Lee, C h ui , a nd Good stein [102], F ig. 2.) 133 middle of the pulse. Thf' error will be negligible if: 1. The tempe rature of the sample has minimal drift before the pulse; 2. The tinw constant of the system is extwmcly long, so that the temperature drift after the pulse is also small. Both of these cond itions wen' c'asily m et for our cxperimeut. We not. only had excellent thermal control to sati sfy roudition (1), but tbc time· constant of om sample was r = nc ~ 8,640 S('C = 2.4lus, which clearly satisfies condition (2) . As in the AC tC'chnique , the heat capacity value' expressed in ( 4. 7) is averaged over the temperature range of .6.T. The magnitude of fl.Q should therefor<:> be chosen so t ha t the resulting .6.T is much smaller thau the temp<:>rature range of any exp ected features iu the data . llowever, for a decent signal to noise ratio, fl.T also needs to be significantly larger t ba11 the noise of LlH' thermometers. These problem s were not significant for ou r exp erimental system: the uoisn of our thermometers was extremely low, and we were trying to observe a strong divergeuce not subtle details in t h<> data. V•/r applird a pulsr of approximatrly 20 pAmps across a 36.963 D rf'sistor (hratcr 5\iV) for 500 rnsrc. This inj<>cted "' 0. 7 JIJ into the samplf', resulting in a temperature s tep of "' 100 nK. In retrosp ec- t. this was slightly larger thau ideal. Since it was p erformed on earth . our cxperimeut had au intrinsic rounding due to gravity. For our celL Lhis averaging was over the temperature range fJT rv 1.27 ttK/cm x 0.064 em = 81 nK. If wr had matcll('cl tlw rounding due' t o fl.Qcal to the rouuding dU(' to gravit:v, then minimal information would have bec11 los t. 4.1.4 Modified discrete pulse calorimetry BecausP of the large time constant of our s:vst em , the difference iu the temperature drift rates on either side of a single heat pulse was ex tremely sma ll [sf'e F ig . 5.6(a)]. How<·ver , a fter muHiplc pulses, the systC'm lwcame noti c-rably out of equilibrium, aud LIH' drift rates before and after the pulses increased accordiugly. To comba t 13..J this prohklll, we did no t keep the s hie ld temperature co11s ta nt throughout tlw C'ntirr t emper at ure range of our nH'asun·mcut. The iuitia l shield S('L point was cboscll so that , vvit h Q appli<•d, t.hr tC'mperattll'C' o f the samplP was r v 3-5 11K IH'IO\\' ToAs(Q). and had a drift rat r lc•ss th a n 0.1 nK /s<'C. After thi s drift rate was c•stahlish<'d. the n•ll t r mprrat ure was m ras urcd for o u e to two minutes. T il<' first hC'at puis<' was t hC'n appl i<'< l, a nd the t<'m JWrat ur<' was meas ured fo r anoth<'r <HtC' to two minul rs without cha ng ing th e s hield tE'mJ><'ratun•. At the <'ltd of thr two minutes, t lH• shirlcl lf'Htpcrat. urc' was adjusted to uull the drift r a te• o f I he• samplr [s<'<' Fig. 5.6(b )]. \ \ 'hm the drift rat<' was meas ured I o I><' anomalow;ly large• as comparc•d t o a pre-d e fined ('Utoff valu<', thP s hiPld c:orrC'<"tion was not <'X<'<' tlt C'd. This prewuted the control p rogram from dlunpiug a lot of hC'al iuto thC' s:vstem aJtC'I' reading a flu x j ump2 (or j u n1ps) . <'ausc•d by c•lc•ctronic noisC', and avC'l'ted the rmal disast<·r . Th<' h<'al flow <'quat ions imp!~, that thC' c·otT<'ct a m o unt to raise the s hield tcmpC'ratur<' t o null the drift ra.t <' is: " ·hen' ~T11 " is tl1C' trmpC'ratnrC' stC'p heig ht due• to the pulse. R is the tlwrmal n'sistan cr hrtwN' n th r sam piC' and the s hif'ld ( R :1 J), a nd C is t he heat capacity o r t 11(' sample. JlOW('V('J' , bccausP T = nc is so large for our system, t ht' firs t Lf'I'lll tau br ig no red . T IH' rPfore. the s hie ld tcmpcratun~ s h o uld be r aised by (..J.9) whe n• d<P 11 c/dl was thC' <1.\'('ntp;C' drift ratr ))pfo r C' adjustrrwnt , mC'as ured in ¢/sec. There fore. f ::::= R :3 1C · ( K shield / ,;II"), wher<' Kshicld is t hP convers io n rate of the shi f' ld tlH•rmomc•I<'J', HRT2. in cb0 / K , and n; 11 c is thC' conwrs io n ratf' of whichPYPr HRT \\'aS 2 F lux jumps will h<' discussc>d i11 m on? depth i11 thP data analysis cha pter . They are esseutially unitary jumps in til(' count o f t hf' flu x counter board that are caused by R F noise. T lwy can be easily rc' llJ OV<'d in data a na lys is, hut ar<' more diAicult t o account for wiiPn rorr<'rtions arf' done> in r<'al tim<'. 135 - 0.65 Q .3 - 0.7 .. .. ~-< ~ -0.75 0:::: :r: - 0.8 t- 0:::: :c (a) - 0. 85 - 0.9 r 1 6 14 61 6 6 18 620 622 time (minutes) - 0.7 14 - 0.7 15 ~ ~ .3 - 0.7 J 6 ~~ ::::--0.7 17 10:::: :r: - 0.7 18 I ~ - 0.7 19 :r: (b) -0.72 - 0.72 1 61 8 6 18.5 6 19 6 19.5 620 620.5 time (minutes) F ig ure 4 .5: Data taken wit h the discre te pulse calor im etry method at Q = 3.7 1;,Wjcm'l. (from file 0 22399.01). (a) A sequence o f t hree pul ses. (b) One pulse. T his plot is the a rea enclosed by t he box in (a) . T hi s pulse had a n unusua lly large drift ra t e a fter t h0 p ul se. and t herefore illustrates how changing the shield tem perat ure aft er the p ulse is ver.v effective iu uulliug t he sam ple drift rate. I t sho uld also be noted that altho ugh t lw after pulse drift rate looks C] ui tc st<'cp on t he scak of pl ot (h ), it is still very small compa red to t he pulse heig ht , as cau be sccu on the scale of plot (a). L:3G ust'd to nH'asurt' lhc drift rate of til<' helium l('mperature (for most runs TinT3 was us<•d) . AI tho ug h all of the HRT conve'rsion factors remaiu mor<' of IC'ss eonst ant O\"<'r t.he range of several til~. th<'.v vary s i gnificant!~' ovN a rang(' of OJH' or two mK. Tlw calorinwt('r stag<' HnT conversion factor. 11: 11 " , was tlH'refore liH' sanw for all runs. TT owever, sine<' the s hi0ld set point t 0111 p<'rature was a l tc·red in proportiou to Q , th<' s hi<•ld conV<'rsion factor, h:s hi<·Jd, chang<'d s ig nificantl y from one run to tlH' rwxt. Thcr<'fore , f was differ<'nt for a ll run s taken at a disti nct values of Q, aud was empirically determined for each data ruu. After tll<' s hield was adjust0d to null t.he• drift rate. tlw telllJ><'raturc' was me•as u•wl for auot IH'r two minutes .. ami th<' JH·on•ss began again. TlH' entire SC'< PH'nc<>, illust.rate•d in Figs. ,1.7 and 5.G was: 1. Take• t<'mperature data fo r two minutes: 2. Inj ect a heat puls<' into t lH' salllpk; 3. Take data for anotber two m iw rt.c•s, holding tire shield t<'m JWratun' at t lr<' sam<' ~wt.point as it was b0for<' the• puis<' ; 4. Adjust t lr <' s hield srt point to null t h<' drift rat<': G. Hetunr to step (1) and r<>J><'at until th<' cdl tc•mperature is ahoVC' ToAs(Q). On<' of the disadvantag<'s of tlw discn•t<' pu is<' n1<'thod is that th<' t<'chn iqur doC's not easily permit av<'raging. In ord<'r to combat this problrm , W<' n'p<'atNI tlw scqu<'IIC<' li st<'cl abo\'C' 5-10 tim<'s per run. Aft<'r tlw L<'mperature of the sample, T 11 (., weut above T 1Ms(Q) , the shi (' ld tempNat ur<' w;-ts low<>r('d so that. Til<' l>q?,ctl l to coo l al a rat.<' of 0.1-1 nK /sec. This slow rat<• e ns u rc•d that t he t. hermom et <'l's wou ld stay wit hin calibratio n . l3C'eausr wr wcr<' also limi ted to 15 ho ur data runs. a ud we wa11tcd to averag(' as many times as possibl e , we did not switch Q off and on during the discrete• pulse data runs. lnstcad. we m easured the Q = 0 h eat capacity data 0 11 S<' paral<' ruus S('V<'ral limes throug hout tlw cooldown (s('(' Appendix C). All Q = 0 fil<'s w<'n' 1:37 111 0XC0l10nt agrC'<>nwnt wit h one a not h<'r (and with pre,·iousl.v p u blisltcd lllNtSIIr<'mC'nt s [2]), indicating that nou<' of Lh<> relevant experimental param<'t <'rs chang<'d s i gnificant!~· O\'<'r th<' course of tlH' run . 4.2 Cooldown details T lw first st<'p for <'V<'ry cooldown was t,lw sant<' as for E'VE'ry other low-tcmpcratun' physics <'XP<'rinl<'n t: g<'t th<' sam pie t o .J h: without vacuum can leaks, electical shorts, or plugg<'d capi ll aries. \VhPn t his was ult imately accomplishPd. tlt<' next st<'P was to charact N iz<' thP <'XJWrimcntal syslE>m . \'\'e: l. Del ermi n<'d the t iH'n nal JH'l work of I h<' s~·st0 m. Tit is was don<' b~· applying wtrious IH'at ClllT<'nts to t h<' s~·stc' rn with h<'at<'r 6\\ · and mf'as uri ng t IH' r0sul t ing chang<'s in t<'mperaturC's of I It(' various stag<'s; 2. J\T0asnn•cl the attributps of tiH' 1 I\ pot. \\'E' dE'tcrmirlE'd both it s base t<'lliJH'ratttrC' and total cool ing power. T his indicaL<'d t he m aximum lH'aL curT<'nt. that cou ld be passed through tbe s.vsL<'lll whil<' til<' cell tcmperat un' was nNtr T>.; 3. l\leasured the heat capacit y of I h<' <'mpty c<'ll. T his was don<' using the staudard disnet<' puis<' tPdwiqu<'. 4.2.1 Calibrating the HRTs and filling the cell One<' tiH' g<'JH' ral charactrristi<·s oft h<' s~·s t<'m wf'r0 det<'rmin<'d , t h<' next st<'p was to p11 t in tlw magn<'tic fidd and t lt C'n cali bra tf' t 11<' t hf'rmom<'ters, as was clf'scri b<'d in Sec ..).2.5. T his h ad to h<' don<' IH'fore t. h<' cC'll was fillc•d , since fr<'C' Zing th<' magn<'tic fit•ld into t he flux tulws required ra isi11g t.hc• tC'mpC'rature of t hf' system abov<' 9 K, t. h<' transitio11 t<'nlJwrature of niobimn. A closNI ce ll of helium t0uds to react poo rly (or morc pr('cis<'l.v, explosiYC'ly) to th<'S<' sorts of tC'mperatures. 138 Computer DAQ 2 Heater 1-14 Stage 3: Shield Servo Loop: Lahview PID R 34 := 850 K/W RFSQUID I HRT I I I OAQ I. ch I Shield Set point Q JL He Stage 4: Calorimetl:r I Heater 6W I DAQ I. ch 2 L1Q F ig un• --!.6: Sch<'matic of th<' discr<'t<' pulse lwat capacit_y measuremeul us<'d iu our <'xpNimcut. A hc'at fiux , Q , is s upplic'd to hNttN GvV. T lw t<•mpcrature oft he shield is controlled to withiu ± 0.2 pK of a tPmpcrat ur<' T 1, a t<'ml><'r<lturc c hosen so that the tE'mpera t ur<' of the sam piP is approximately 3 - 4 JtK be' low To As (Q) with a drift rat<' l<'ss thau 10 11 K /s<'c ..-\puis<' train is suppli<•d lo heal<' r 3\\'. Oucc the t<•mpcralur<' reaches a fixed tentperatur<' 1~ 11 ax locat<'d approximat<'ly 1 ttK abov<' TnAs(Q) , Lh<' pulse train is disco nlillu<'d , all() Lhc shield L<'HlpNatur<' sC'tpoint is low<'r<'d to a l<'mp<'ratur<' T'2, chos<'n so that tlw I1C'Iium cools at a rate of G- 10 uh:js<'e. \\.'h<'Il lh<' t<'lllp<'rat urc' n';H.:h<'s t IH' initia l t <'lllp<'ratur<' of th<' nwas ur<'m<'nt. th<' s hield setpoint is retumcd to its original value, and the pulse· train is r<'s um<'d. This prOC<'SS is r<'JWat<'d a numb<'r of time's for every run. 139 0.5.---------~--------~--------~----~ Q-0.5 ::1. ........., ..---. E--.< - 1 ......... E-- - 1.5 0::: ::r.: -2 E-- -2.5 0::: :r: -3 -3.5 0 5 10 time (hrs) 0 ..... 15 (b) (l.) N c e 1.5 ...... ..D ~ I I I " Q) :.a 0.5 Vl Q L---------~------~--~~----~--~--~~ 0 5 10 time (hrs) 15 Figure --1.7: Dat a t.ak<' n with t h <• discrrtr puis<' calorimetry method a t Q = 3.7 p,v\'jcu/ (from fi!P 0:22399.01). (a) T he <'utir<' run UH'as urrd by t.ll<' top t hrrmomrt<'r , Ti nT J . (b) T h<' s hi eld l<'lllpNat.ure throughout tbc ruu. T ll<' utu•vcHcss of tlH' shield adj us t nwnt 01 1 t Iw hrs t t wo ramps occu rrrcl w h<'n <'l<'dronir no is<' caus<'d flux jumps in lh<' lh<'rlliOilleters. T h is problc111 wi II br clisntss('(l furth<'r i11 t h<' nrxt chapter. 140 Filling the cell .\ftN I h<• HRTs \\'<'r<' satisfactoril.v <"a librated. tiH' <"alorirn<'t<'r was fillc•d with iso- topi<"ally pun• 111 <', with a ;111<' COJtt'e ut rat iou of oui.Y ().01 parts p<'r billion (.\' = 1 11 ). Although tlw <·fl'c·ct of ·1 1Ie• ('OiltaJnination 011 T((Q) is not 1 [: Tlr]/[ 1Jc•] = I x 10 known. its dfc•c·t on T>.. has IH•c•n ILH'asurcd [ L0:3] to I><• r/T>.. / ri.Y = - 1.4 I\: . If th<' c•fl'<•cL on Tr( Q) is of e·omparablc map,uilud<•. a <"OJH'C'Ill rat ion of .\' = 0.07 pph would not aff<'cl om r<'stdts. lu fad . it ('aulw argtwd that , in our range• ofQ. t and.\ , the• pff<•ct of the• hc•at flu x would l><' to S\\'<'<'P t IH• 3 He• o ut of th<' c<'IL producing a film of le•ss than a siup,le Jnonola.wr at t h<' <"oolc•r <'lldplnt<'. T l1c•rrfon•. wilh r<'SP<'<"I to :1H<' impurit ic•s, t hc• heat flux rna.'· actually illlJ)l'O\'<' tiH• pr<'<"ision of the uH•as ur<'llH'lll. lllaking i:-.ot ropintlly pun• he•l ium lln11<'C'<'SSH I'y. liow<'\·e•r, it liC'n'r hurts to I><• 0\'<'r-caut ious , so wc• us<'d it anyway. Be•forc• filling th <' e·<'lL \\'<' flu shrd the• room l<•mpPralun• pip<• work tlm•e• tim<'s wit h nltra-purc• 1Tic• to make• s un• tlH•rc wc•n• uo lillg<•rin g c·out.arninants. \\'c• th<'n Op<'n<'cl t lw r<'gulator (\·aln• I in Fig. 1.8:1) to ;3 psi. and t1H'I1 OJH'JH'cl t h<' valw at the top of tiH' n,vostat ( ,·ah·<· 1). \\'c• waulc•d to Hntk<' S ill'<' l hat om m<'Hsnrc• mcnt s WE'l'<' at saturat Pd vapor pre~sur<' (S\'P). :-;o we• had to mak<' sur<' to fill til<' c<'ll with just C'noup;h lwlium so that it would he ue•;uly, but not <·ornpl<·t<'l~·. fill<>d at t lw lambda t<>mpcralure. L3ecau:-.c• t h<' molar vohtuH' of h<'lilllll clc•<TNlS<'S with itHTC'asiug temJwrature in the supNfl uid regiou. tlti~ r<'quin•nwut would be• mC't so long as \\'(' filled tlw <:<•II at a l<'lllp<'l'at ur<' ab<)\'(' T>., . Sin('(' a larp;<' bubl>l<> would distort LlH' hpat fl ow, '''<' also wanted to malw sun• that w<• didn ' t put in too little• helium. Ide•ally. W<' \\'antc•d to fill tlw ('<'II wit.h just c•nough hPiium so that lit<' bubbl<> would c·o mpl<'t<'l~· oc·rnp,v the 'dc•ad ,·olunl<'' oft h<' c·<'ll. From t h<· <· urv<• of molar ,·oluuw vc•rsus t<'tnJWrat un• [10 1]. W<' found that if W<' fill<'cl tit<• <"alorim<'ler at 2.G7 K, w<• would ltaV<' a bubhl<' filling 0.1 7< of the• c·c•ll YOI11me at 7).. \\'e• aimed below tltc• ·de•ad Yolullt<' JH' r<·e•ntag<' of 0.957< si n<"<' W<' kn<'w that t h<> action of t IH• nwdtani<'a l plung<•r would h<>at up t lw C'<'li. forc·ing SO III(' oft h€' helium into I h<' fiJI lin<•. a nd c]('("J'('HSillp; t IH' hrJium fill faetor :1 For 11H• r<>utaindPr of this dwpt<•r, va}q• numb<'rs will n.f<>r to thC' YHln• muuhC'rs lallC'IPd in Fig. -1.~ . 141 Roughing Pump Q:D 0~ D c ..... - ~(D ~ T / Pressure gauge Thermocouple gauge cp Cf 1/1 cp ,0 Calibraled volume .A l G)-.< Extra volume - ~ ~ "0 ...., = Top plate ('l> :r: ('l> '---- I Fill line --" , CD 0~ ~ Cryostat I cell I Fi g m <' 4 .8: T lw cr,vostat confi g ura tion used to fill the cell IJ<'fon· til<' run a nd <'Xtrac·t the h<'liuul aft<'I' the run . T IH' ·calibrat<'d volume' and '0xtra volume ' \\'Cl'<' o uly present during h<'li um extract io n . below t IH· · t a rg<'t · . W hile th<• fill liu<' was OJ)(' JI t.o the ultra-pur<' b o t tic of h e liuu1 , we• scrvocd LlH• cdl t<'mpNatur<' t o 2.6 7 K. VVh<'n W<' had coufirmecl that the cell was fillNI by dH•cking t hat the prcssun• of tlw fi II line no lo nger d ropJ><'d when t h<' ,·al \'(' to t l~e• hdi um bo tt le• (,·alv<' 7) \\·as closed o H', w<· s hut the nllw at the• top o f the CT~·ostat (va lve 2) to pr<'v<'nt helium fro111 <'scaping wh0u we' closed the t<'ll valve (valvP 1). \Ye LIH'n s hut the· ntlv<· to t he' c<'ll by ro tating LhP tururod with 25 lb- iu of t orque. This In<•chanical act io n rais<'d the' te•mpPraturc of tlH' cell to 4.3-1 K , fo rcing som<' fracti o n o f' the' ll<'lium in th<' C"cl! into t h<' fill- liue. \\'c pumped out the fill-lin e unt il it no longN gaw a cl<'tc·ctahlc reading 0 11 t he leak-de t ector. \\'c· thc u opc·ncd it up to a charC"oal n~ropump for th<' dura tio n of Lh<' ruu , wll cr<' it m aiutaiued a pn'ss un~ less than 10 :J torr . 1-!2 The bubble hunt In orci N t o ckt<'rmirw c•xact ly how much helium we• ha d iu our C'f'll. w<' follm\·c•d a p roC'rdurc· v<'r.Y s imil a r I o t.h at used in t he H RT ('alibrat.iou . which we a fl'c•ctio na t ly t. r nn <'d 'tlw bubble hunt'. \1\'e used a La bV I E \ \' prog ra m t o s lowly ramp th r tc•mp e ral ure of tlre cell fro n t 2.7 1\: to 1.8 K Ov<'r a 20 ho ur JH'riod (an awrngr ra t r o f 12.5 pK/ sec m 1.08 K / d ay). \\"e hookc•d up th r HRT h0at switchrs (drsc ribed iu S<•c. 3.2.5) t o~' DAQ, and a ut om at ically firrd t h r tn r vrry 20 minutes. T he Ht ol a r ,·ohtllH' o f he lium has <\ mini m um a t a pproxim at<'l.v 6 mK a bon~ T>. . caus ing t lw Yolu rnr oc·c·upic•cl b.v th e h r liu m to c·h a nge as a fun c tion of tc•m pcraturc>. T hNrfon >, as the liquid is IH•at ecl a bov0 t he tra ns itio n , if t he bubble is s m a ll f' no ug h , it will rw ntu al l~· disRpprar. fi lling t h e• ('n t ire ce ll with h e lium . A t th is p o int. t iH' h 0at capaci ty of t he sam piP cha nges from Cs\·J> to C1 · • a nd t lw heat ing ri'lt,<' of t h<' !te l i11 nt a lso c ha rtges. T h f' r<' for e, the tempera ture drift rate w ill cxh i bit a s lo pe r h a ngr w hc•n t he bubbl e' disappears (see F ig. 1.!)). \\' hc•n t he• samplc• is cooi<'d as o p posrd to heated. in addition to the slop e cha n g<', t here w ill a lso b r a n act ua l j um p in th <' tcmpcra tu r<' due to tlH' heat r <'l<•as<•d w hen t.he lmbble is f'o rmrd. B~· o bsNving a l w ha t l <'IliJH'ra t un' this di scont inuit y o<Tu rs. a nd m a t ching it wi th t h e app ropria te m olar voluiiH', il is possibi<o to d r t c•rmine hm\· much lw lium is in l he cell , a nd therefore how la rge a hu hhl(• will <•xist a t T>. d 11 ring t lw lllNts urcm cHt s. T h e t<'lllJ ><'r Ht ure disco ntinuit~· during o ur hubbl r hunt O('C' Urred a t 2.371 K . T h <• m ola r YOhll1H' of lw liu m. 1 ;11 , at t hi s tc• rn pera t ure is a pproximately 27.530 cm 3 / mol. i nd icati ug th a t du r ing o ur rxp erimcut . the bu bbl<• size ·wo uld he: 1 i>Uhbi" 1 ;.,.11 - - - ::::= 1 - 1 ~n (2. 1768 K ) = 1 - 27.530 ::::= () . ()O-o. 1 ;" (2.371 K ) 27.287 (-1. 10) T h<·rc.>fore . o m bubble• wo uld occ upy a pprox ima t d~· 0 .5~ of I hc• C'cll. or s lig ht I ~' IC'ss t ha n t he ·dead volm1w'. 113 0.8 0.7 (a) 0 :... ~ 0.6 c (';3 -~ 0.5 ..c .... (';3 I ..-... 0.4 ~ E 0.3 ~ ~ 0.2 :c 0.1 0 0 20 40 60 100 80 120 140 time (seconds) -3 -3.5 ..-... -4 u Cl) ;2 -4.5 a -o -5 ~ ~ -5.5 :c -o -- - -6 (b ) -6.5 -7 2.355 2.36 2.365 2.37 2.375 2.38 2.385 GRT4 (K) FigurP -1.9: Thr bltbbl<• hunt : drtNJuiuinp, th<' h<•liun1 fill factor (from Fik 000298 .02). (n) TIH' discontin11ity in tc•mtwri'lt urc• that rc•sttlts from b~tbble fonuatiou (b) Tlu• derivatiw or (a), plot tc•d V(' I"S U S GRT trmprrat 111"(' . Tlw dNivat iw ckarl:v iudicatc•s the discoutinuit~· in ~;!opt• thal n•sults as th<' run·c· switdiC's from C 1 to C's\'1'· The• otlwr spik<'S arr a rrs11lt oft IH' IH•a t switch firing at regu lar intervals. 4.2.2 Taking data Leveling the dewar At til(• h<'ginning of f'vf'r~· data rm1. Wf' lf'Y<'lf'd t h0 experimental cd l to minimiz<' tht• efff'cts of gnwit~· 011 our data. \Yf' used a fa irly nude tC'chniquc to d o this. B<>for<• cooling down, wf' l<'wi<'d th<' c<· ll with r Ps pect to LhP top-plate of our cryostat. \Y<' built a platJornt that was attached to thE:' cell and put a 2-d bubhl<• lrvrl on both thr platform and th<' top-plate, and adjusted Lhr vacuum cau supports tm til thr bubblrlevcls both indicaU•d that tlH• two surfac<'s wrr<' horizontal and parall<•l. Oncr cold. w<· t IH•n adjusted the s upport snrws oft lw d<•war until a 2-d bubble-level indicat<•d t hat t ltr t o p-plat<• was h orizon tal (srr Fig. 3.27). Usillg slt <'<'ts of pap<'r. a micromrt<'r, a nd a sin0-bar, Wf' pC'rfonn0d a test to s<.'<' what sort of r0solutiou W<' had during this l<•ve liug process. \\'(' dct<•rmitH'd that by lC'v<'ling tlw 2-d bubhl<'-lf'n• l h.'' <'ye. we cou ld rcsolYc up to ± 0.03° . If W<' assunH' maxiHHtllt <'ITor. this corTPspouds to a tilt of approximat<'iy 36.0 fllll to our 6.985 <·m diawcter <·Pll. AlUwug h s mall in absolut e terms, our t·ell was oul.v 0.06-! <·m tall, so this <'OlT<'spouds to a height <'rror of nearly 5%. Fortunately our data <U<' not all that SC'Hsitiv<' to gradt~·. A lthough a 5o/c l<'\'f'ling c•rror would slight!.\' in crras<' t ll<' heat capacity <•nhanc<.'ment, it wou ld certainly not c·!t ang<· it qua li tatiY<'l.Y. For <Ul~' <'XJWrim <'nt in which it. is 0ssential that the c<>ll I><' ]('V('I , sodt as thos0 perfomt<'d b.v Da~· et al. [58, 09], more care sltould be taken. Au exc<> lknt tC'cllllique was dC'v<'lOp<'d b." Il.\'. Duncan 's group, and is described in d<•tail in [39]. The data sets After kvcliug the cell. we began takiug data. \'\'<' first appli<•d Q to hNtt<'l' 6\V, brought S tage l into scn·o us ing GRTl all(l tlH• QD bridge•. lo('(t(c•d ToAs(Q), aud Lhcu chose the sC'Lpoint of LIH• s hi <•ld scrn> so that the• l<'lllJH'ratun· of th<' calorinH't<'l' was several pi\. below Tn,\s(Q) wit h a low drift rat<•. \\'p th<•n wait<'d forth<' s:vst<•m to come s ufficieuLly into cquilibrituu that Lh<• drift rate or LlH' sample was l0ss than 145 G X w - l <Polsce~ 0. 1 nh:jsec. \\'(' thm s t a r t<•d the• La b\'IE \Y prog ra m th a t <'X<'ClltC'd 1he• modified discrpte• iu F ig . l. 7. pulse• calori nwt r.r nwt ho d d <'scribed in SC'c . .J.l ..J . a nd s hown \ Yc· t hf•n lrft e•wr.rt It i ng a lone for t he' n<'xt 15-20 hours. 1 \ \'hC'n th e r un was on'r. W<' m easnr<'d t lw curre•nt s W<' ha d applif'd t o both lwat<'rs ;:)\\. and 6\\. b.v nwas uring t h<' Yoltag<' drop across prrcision res istors that WN<' iu S<'ries with t he' he'ate'rs. \\.e• t hen a dju s tc·d th e• J [\" p ot to diminat<' an~· osci ll a tion s. t ransfc •rrcd hc, limu if JH'('(I<'d , r<'-levclcd t hr cC'll , and h<'gan again . Tt be'camc ckar C'arl ~· o n in tlw cooldown that we• were u o t g e•tling mueh of a disl i ng uis ha bl<' s ig n a l for lwat c uncut s le'ss t ha n 1 I'\Y I trn . \ Ve t h en• for<' <'OJH'e'n trate•d 2 o u r e· f!'orts iu thP hea t c urre nt rauge 1 fL\Y i c n/· ::; Q ::; ~1 JL\\"jc- 111 2. \\'c• t ook 20 SII<T<'ssful dal a ~C'ts iu all. T h<'i r d is t ri hut ion as a functio n of q is shown i 11 Fig . ...t .10. and t he•i r fi I<' na uws and d<'t ai b an· g i \'C' Il i 11 :\p p<'nd ix C. 4.3 Warming up A l'l <'r all tiH' daLa runs were fiuis hcd . t hen• we re a f<'w things l<' ft Lo do h c•f'orc• warlllin g up the• cell. \Ve re uwas urPd the therllla lne t work t o make• s ur(' that it lt ad n ' t c han ge'd d uriug o ur ruu d uc to bad thermal C'O nt ads. he• at k a ks, gas in Llw vacuum can , <'t c. Fortt111 a tdy. it hachd. \\'<'also pc·rform e•d ano thN ' bubbl<' hunt ' in o rdN to drt<'rminc \\'IH't he•r we' had lo s t a nd lH'lium durin g t ll<' c·ours<' of t h e run dur to a leak.\' Ya h ·c. T IH' t<'mp Nature jump O IH'<' again o<·<·mr<'d at 2.37 K . indicating that the a m o unt o f lwliu m in our cdl a lso had no t c·hang<'d during t h e cooldowu :' \\'c thc u usc•d a P hilips PP. f 6G72 hig h r<'solu t ion tilll<'rlco unt cr ( 1 GHz) to m casun' t h e ('Ons is t(')lcy of tiH' pulsc>s o utpu t by o ur DAQ b oard. \\'(' found that th<' erro r iu the' puis<' w idth was ± 0.1 JLS<>c, o r 2 x lo-r, %. B<'fon• warming up , t h e fina l s l<'J.> was to t'xtra<.: l t h e• helituu fro lll the c·c•ll and dir<'c t 1.\' IIH'as ur<' the uuJnbe r of mole's that we had us('d iu o ur c•xpNinH'n t. V\"c• s<'t 1T h<• 111or<' alou<' we left it , t he better th<• da t a. The qua lity of the data tak<'n m·<'rnig ht was bNt<'r t ha n th<' data tak<·n during th<· day. Thl:' only drawback wm; a large :spi k<' that app<•an•d in t lw d ata a t a pproximat<'l.V 6 am wlwn thP {'ampus tame t o li fe. T his could a l wa~·s be r<'lllOY<'d lat<'r during data a nalysis. '' All of tlw puis<' dflta r uns W<'l'<' tak<'n in a sing l<' cooldown. s ip, u i li {'a n tl~· 14G 3 2.5 cr. c: ::J 2 :.... "0 ~ 1.5 .0 E ::J c: 0 .5 0 0 1 3 2 4 2 Q (j..LW/cm ) F igurC' 4. 10: T I1C' nu m ber of d ata runs taken as a function of Q. 'vVf:> concf:>ntratf>d o n t he larger values of Q since this was t he range wlw rc th C' largrst heat capacit:v· enha ncement was o bser ved. up t he cryostat in the configura tion shown in F ig. 4.8. Vvr used a P aroscicntific, Inc. D igiqua rtz quar tz-crystal press ure gaug<" t ha t had an accuracy of 0.01 % and a rrsolution of 0.0001 %. It could wit hstancl a maximum JWC'ssurr of approxirnatrly 1.2 ba r. O ur calibratf'd volum C' was 398.329 cm 3 . If a ll of t hf' liqu id helium in our cell were to expand into t he gas phase, it would occupy approximately 2.3 liters at RTP. ~·c t hcrcfo re acldr d an extra volume to t he system so t hat t he maximum possiblC' pressu re• t ha t t he gaugr would srr during t hr rxtraction was around 1 bar. vVitb thC' valvC' to tlw cryostat (valve 2) closC'd, we pu t hC'liu rn gas into thr room tempr rat un' systrm a nd ca librated the volume• of thr piprwor k and of thC' 'C'xtra volume·. W <• found t hat, together , thry occupied 2.447 li t0rs. 'vV<· t hen extract<·d thr helium in fi ve s teps, t aking o ut successively sm aller am oun ts , until we were down to t he resolu t ion of our prcssurC' gauge. In tota l, we measured t h at t here werC' 0.08918 molC's of hC'lium in our cell. D uring a prC'vio us run (onC' of t he drift calorimet ry runs), we had ovcrfi lkd thC' 147 crll.G B<'<'<UJS<' the comprrssi blility of liquid helium is c•xtn•mcly s m a lL we• would expec t that tlH· numbe r of m o k s in the cell during that ruu would br d ose• to t ll<' total uuml><'r of moles tha t cou ld fit in Lh<' cell whi le• at S\'P. At tlw end of that run , we• <'xt ract c•d 0.08962 mol0s. Th0 percent chan g<' was then•for0: 0 .08962 - 0.08918 O() g C/ 0.08962 X l = ().4 l I <' (·l.lJ) which is in <'x<·<•ll<'nt agn '<' IIH'Ilt with our Ill<'HSIII'<'d bubbl<' size•. Aft<'r the• lwlinm was c•xtractc•d, W<' one<' again nwasured th <' hea t capacity of th <' em pty c0lL a nd then " ·;-unH'd up the experim ent. It was timr to b egin a nahrziug the data ... <i lll oL IH•r words, fill Pd it to a press u re a bove lhc SVP. 148 Chapter 5 Data Analysis T his c hapter will d('scribe th e I1H'tl10ds us<•d to a n a lyze the fiual S[Wrific hc•at data. ln gcu<'ra l, llH' discret<• puis<' lll<'l hod has a rat IH'r s im pie d ata a ua lysis pron•du re. T h<' m ai n contplieat.ious S[Wr ifi r to o m exp rrim r n t wC're: 1. E liminating electronic no ise aud IIHT drift ; 2. Drtcrm iuiug thr absol ute t<'lllp<'ratnrr sc-alr: 3. Cor rc;>cting t he end plate t em perature m easurements for t h r siugular Kapi t za r<'s is tanc<'. :\ft N t.h0sc things W<'r<' acro mplishrd, find ing t b<' heal capacity was llt<'ll o nly a m attrr of fittiu g lines to th e slop<' in tc•mperature o n the sidrs of each pulse, and th<'11 <'XL rapolati ug these lines t o Jiud t he diffc rcnc<' in h ciglt t in t.lH' midcll<' of e'ac-l t pulse . 5.1 Eliminating electronic noise and other error sources Ther<' were a number of t hings wr had to do to dean up our data h<>fon' ns iug it to clet<•nniuP tlH' heat capaci ty. Brcause' tlH' efr<•ct that wr wrre looking for was s upposrd to I><• fairly s ma II , dimiua tiug sp uri o us s ig na 1s JJl<'<llJt t lw difference bet w<'<' ll haYi ng t he h eat capaei t y <'Uha nremen t los t in tl1<' n o is<' a nd having it lw clrarl.v d isting n ishablr. 5 .1.1 SQUID titner resets As described in Sf'c. 3.2.5. if t he a na log part of tlH' sigu al is tak<'ll imm<'diat<'ly afte'r the• Jinx coun ter is iucrem PHt<'d (and a r<'set is S('JJ( to t he controlle'r ), it s value' is not ,·alid , because it has not yPt ha d time' to s<'tt !P. T he s<'tt ling tim<' is d<'t<'rmin('(] h~· 1-19 Lh<• analog filter used to eondit iou t h<• sig na l (s<'<' :\pJ)('ndix B.3 ). As a r esult , a s pik<> a ppears iu tlH' data a t t IH' t i Ill<' of t lw r<'sC't . T h<•sc s pi k<'s eau b <' sren iu Fig. 3.1 cv<•ry LimC' tiH• data counts a multiplr of ¢ 0. ThNw s pik<'s eau <'asily I)(' r<'m ovcd during d ata anal.vsis. \\'h ell the fluxco unt<' r sends a r<'S<'t s ig nal t o t h r SQ C ID contro lle r . it also starts a tinH'r that is rc•<·ordC'd b~· the compulN. T his tiwcr s ig n a l can b e us<'<l to <•limiuatc• ait~· data p o ints t a ken in th<• vici11i ty of Lh<• r<'SC' t. sig ual. Tb<• s ize of tlw window ll<'< 'd<'CI to rom pletcl~r dim i nat e the s pik<• \'cHi<•s d<'JWndillg ott the t<•mperatur<' drift ratr and tlw filter settings o f the SQUID controll<•rs. TlH' drift dat a sh own in F ig. 5.1 were filterc•d by a filt er' with a cutoff of 0. I Tlz, and had a fair!~· la rge drift rat<' o f 5 nK /src. \\'<· h a d to throw out data in a window of 1 second after the rt'S<'t t o <'limin at<' tiH· spik<•s. For thC' pulse• da ta. where th<• drift rat<• o tt t. h<' pulse l<'dge ( the• data t a k<'ll in bC'twr('n s ubsrquent puls<•s) was <'xtr<•mc>l.v minirnal , a 100 m src window was suffirirut. 5 . 1.2 Flux jumps Radio F r('(Jll <'ncy (RF ) nois<' intrrf0rcd with th <' <'lcrtro nics o u tlH' fluxco unter boa rd. all(l eaus<'d thr flu x count. t o jump diseont iuu ou sl~· b~· au int egra l number of flu x quanta. \ \"hr n we first set up th e cxp Priuwnt , we <'Xp 0riell(:<•d aro und OJtC' flux jump pe r minut<'. W<• r limina t ed some of tlw probl<' lll by putting low pass filt0rs on a ll inputs and outputs of Lh<' fluxco untcr board . Altlrongb thi s llt <Hk t.hr experimrnt ro bus t agni us t t umiag 011 and o ft" light sw i t rites i11 adjacent rooms, it was still srnsi t i vc to sold0ring iro n switr hrs. flickNin g flouresc<'nt lig hts, dri lls, a nd othrr unus ua ll.v high RF no ise• sources . During a norlll a l run, wP go t an ave rage of 1-3 flu x jumps p e•r ho ur. Occasionall:v, when contractors wrre workiu g ncarb:v, or an overhead llourcsceut bulb startrd to dir o ul , w<> wou ld P,<'L co ns iderably m o re . One of tlH' mor<' afflict<'d ru11s is s hown in Fig . .J .2. AJtltottgh a nuisanc<>, t lt es(' flux jumps could <'asily be a<Tount <'d for during d a t H analysis. It was simp ly a rnat tcr of fiudiug Llw location of the jumps. and t he'll 1 ThP lil U•r s pPcifieatio Hs an• g iw n in Appendix D. 150 -42 -44 ...-... 0 -e- ;:;:: -46 r::r: 0::: -48 -50 2700 2710 2720 2730 2740 point number F igure 5.1: Correcting for flux-counting reset errors. A drift calorimetry fil<' at Q = 2 0.243 1iW /cm (file 011498.01). Top line: The data before correction fo r SQUID timer resets. Bottom line: the data after SQUID timer correction, shifted dowu by 3 ¢ 0 . Data points wen' taken a pproxim ately once a second, filtered at 0.4 Hz , and cleaned by elimiuaLing t he data taken within oue second after the reset signal was received. subtracting t he a.ppropriat<' number of quanta from the flux count for all subsequent data points. This prorrss was automated by a program 2 that took the difference lwtween t he HnT data of adjacent points, ~ = HRT (n + 1) - HRT(u), ( 5. 1) a nd tlH' n plotted them as a functi on of p oint number. n. Tbr program t hen scanued ~ to sec if it was within certaiu cutoff values. T hese cu toff values were d<'t<'rmin<'d by tb0 pnls<' height and tlw drift rate down for the discrete puls0 calorimetry data. Typically, the puis<• height was around 0.6 ¢ 0 , all(! th<' drift raU· on the downward ramps was -5-6 x 1o- 3 ¢ 0 /sec. Therefore we usu a ll y S<'t t h<' cutoff valu<'s so that we 2 T h<' program is written in 1\ latla,b a nd is rall0d ·rieanJl ux2 .m ' . 101 would c.:ouut: ~ ~ O.G <Po + 1 </>o - - 1 </>o 1 rH 10 L :20 { ~ > 0.2 ¢ 0 aud - j II dHRT 3 dt (n ) :=:; - 5 x I 0 ¢ 0 /sPr 10 { ~ :=:; - 0. -1 <Po wiH'rP t lw firs t +<Po coudit ion is to account for positiv<' flu x jumps Oil th <' upward ralllps, the S(-'<"Oud +<Po condition is Lo account for posi tiv<' flu x jumps o n t h<' downward ralllps, aud Ill<· - dJ 0 condition tak<'s nm• ofhoth upwards and downward ramps. This t<•chuiqu<' was <'xt r<'lll<'l~· pff<'ctiw, and allmY<'d u s to adjust for runs that had as many a s S<'Vcral hundn•d flux jnmps, s uch a s th<' OIH' sh own iu Fig. 5 .:2. Tlw on<' plac<' th<lf this tPchniquc faikd was when a flux jump o<.:cun·<'d on HRTO rig ht on a puis<' on t lw nonnal fluid s id<•. A s ran h<• observNI in Fig..).G(b) , HRTO puis <' heig hts iu this r<'gion were cousidt•rabl y larg<'r than o nr ¢ 0 . T hr prog ram was therefo r<' not ruu o n tiH' HRTO puls<' dnta ahovr th<' transition. lns t<•ad. fiux jumps W<'r<' found by plo t t iu g th <' IIRTO data wrs us the IIRT 1 data and lookiug for discontiuuous of[s<•ts of lllultipl<'s of ¢ 0 lwtwf'<'n th <' different ramps. 5.1.3 Microfl ux j urn ps T IH' final typ<' o f <'kct.ronic noisr vvP ha d to e li111inatf' wen• the micro llux jumps. 'I'd icroftux jumps' is a bit o f a misnomer. T h e minoflux jumps wer<' neitlH'r 10- 6 ¢ 0 iu mag nitud e', nor a tru r jurup. They w<'rC' m ort' of a blip [s<'C Fig. 3.3(a)] tlr<tl r<'S<'ll Jbkd tl 1P timer rrsPts discussN I in Se<.:. 5. 1.1. The o rigiu of the u anH' is a compktc mys t<'ry. B ut t lw name' oncP gi VC'll , stuck. Sin er:' it appears throug hout t h<· la b books, I will not c ha ng<' it h<'r<'. Lik(' it s name, t lw precise orig in o f th(' minoflux jumps th<'lllS<'lvPs is not. known . T h e~' clearl.v do 110t n •prcscnl Lh<'rmal prop<•rt.i<•s of t.bc systC'm , and appear to hC' due to souH' sort of ei<'ctrouic uois<'. A s with th<• flux jumps, it was <'asy to elimiu a l<' tlw microflux jumps during data 152 0 1-o (!) N cro 150 (a) 1-o ....... ..0 1-o ro I I 100 ,....... 0 ~ E0::: ::r: 50 ~ 0 E0::: :r: 5 10 15 20 25 20 25 time (hours) 8 100 (b ) (!) N ;>-. ..... ro 1-o .'::! ..0 80 a ,....... 60 0 ~ _...... E0::: ::r: 40 ~ g 20 0 ::r: 5 10 15 time (ho urs) Figure 5.2: Correcting for flux j umps. A discrrtc pulse calorimetry file at Q = 3.2G J.LV'..' jcm 'l. (file 030699.01). (a) The data before it is corrected. This file has mauy morr flux jumps t han us ual. The top curw is H.RTO and the bottom curve is HRTl. Thn'<' of thr la.rger flux jumps arc indicated b~, arrows. (b) The data after the flux jumps ar<' removed. 153 -147 -148 (a) -149 .---._ - 150 0 ~ -151 ("') r 0::: -152 ::r: -153 - 154 -155 -156 3510 3520 3530 3540 3550 point number -5 -6 ,..--.,. u Cl) C/l ;;;:: -7 t: '-' -- -8 0::: -9 ~ "0 ("') r ::r: "0 -10 (b) - 11 2500 3000 3500 4000 point number F ig urr 5.3: E limin a tin g micro Aux jumps. A drift calorimrtr:v fil<' at Q = 0.243 f-L\11.' /en/· (file 011498. 01 ). (a) A microflux jump o n HllT3. T he top curve is bcforr eorrection. T he bot tom curve is a fter correction. and is shifted down b~· 3 ¢ 0 . (h) T h<' <krivati w of HRT3. The' top curve is lwfore correct io n. T here ar<' four mic rofiu x jumps iu this range. T he bo ttom curve is a fter correction, and is sh ifted down by 3 nK/ sec. 154 analysis. This procedure was automated by a program:3 that also took tbc 2-point difference d efined in (5.1), and scanned .6. for a negative spik(' closely coupled to a positiv<' spike. WIH'n it found this signaturC' at point 11 , it replaced HRT(n) with [HRT (n + I ) + HRT(n- 1)]/2. This C'ffrctivdy C'liminated tlw microflux jumps [sC'C' Fig. 5.3(h)]. 5 .1.4 HRT drift As was (k scril)('d in SC'r. 3.2.4, all HRTs C'xhihit small drift rates. These drift ratC's mu st be taken into account wlwn making prC'cise measuremC'nts over long periods of time. The IIRT drift ra tes are not the same from run to run, a nd must be measured empirically for c'ver.v data file. The drift rates were usual!~' more or less constant over the length of one fil e. although occasionally they would exhibit small changes from one part of a fil<- to aaothcr. vVC' found that the drift rates of HRT1 (top t hermom eter) wNc usually slightly higher than thosr of HRTO (bottom thcnuornct.cr) . and both WC'H' on thC' orckr of s<•wral fi ¢ 0 /scc, or approximately 15 nK/ day. The drift rat.C' of the t lwrmometers can lw d etermined hy plotting the tcmpC'rflture reading of IIRT 1 vs. th e temprrature reading of HRT0 4 for each file. Because HRT 1 and HRTO are thermometers adjusted to read the ternperature of t.lw same ohjcct, this plot should be a straight line. whose s lo pe is the ratio of ,.,; 11 Rn to ,.,; II RTo , where ,.,; is the number of ¢ 0 / K for each thermometer. However, if HRT1 drifts relative to HRTO. this straight liuc will move for each subsequent ramp (see F igs. 5.4 and 5.7) . vVhrn t hC' drift rat.('S are constant , thP~' can lH' m ore or less tkl<'rmincd by fi nding out what valuC's of f 1 and fu usrd iu a plot of [HRT l + j, · t] vs. [HRTO + fo · t], where t is time , c-ausr all data points to collapse onto a siuglr curve'. It is easiest to observe t h0 drift rates on thr downwa rd r amps, b<'causc t hey contain data poi11ts that are continuous in tc>mpC'raturP (se(' Fig. 5.4). How<'V<'r, they also have considerably more' noise than the data on t hC' puis<' kclges. 1 : T his program was a lso written in Mat lab a nd is cal10d 'ckan_microflux.m'. 4 Both UHTl a nd I-InTO should first be adjust0cl for t.lw singular l\:apitza n'sistancC' by t he m0t hod described in S0c. 5 .3 so that thPy bot h read the temperatur e of t he h elium sample. 155 -17 (a) • ,.-.. 0 ~1 7.2 _....._ C/") <( 0 r-< - 174 ~ . • .. • • 0:::: ::r: -17.6 I g 0:::: -17.8 ::r: - 18 - 14.3 - 14.2 - 14. 1 - 14 - 13.9 -13.8 - 13.7 -13.6 HRTJ - HRTI(TDAS ) (<j> 0 ) - 17 (b) .. ~ (/) <( 0 c - 17.4 g 0:::: ::r: - 17.6 I 0 ~ - 17.8 ::r: - 14.3 - 14 .2 - 14. 1 - 14 - 13.9 - 13.8 -13.7 - 13.6 HRTI - HRTL(TDAS) (¢0) F igure 5A: Correcting for HRT drift o n a Q = 3.6 fiVVjcm 2 file (030999.01). IIRT1 vs. HRTO tlw wholf' fik. The small clumps of points [lik<> th<' ones at (-13.7, -17.2) and (-13. 75,-17.3)] arc the upward p ulse' ledges. The continuous lines are the do\<vuwarcl ramps . T he scattered poin ts th at do no t lie on the lines arc due to t he spikes immediately following a pu lse (an f'xamp l<> of which can hC' obscrVPd in Fig. 5.13) that ar<' caus<'d h~r SQUID timer resets (sec Sec. 5.1.1). (a) Before' drift corn'ction . (b) After most of the drift is corrected. Ou t his file, one of the ramps has a slightly different drift rate thau t he ot hers. 11)6 Using the pulse ledges ln ordC:'r to fine-tune j 1 and .f0 , w<' th<'r<'fOr<' us<'Cl t h c data from the puis<' l<'d p;<'S. 'Ye fouud t h r trmperaturcs llnTllt•dg(· and IlRT0 1PdgP in the middle of rach pulsr IN lge, and plott ed t h em against o nC:' a not hN (S<'<' Fig. 5.4). This was a lso an aut o matC:'cl prO("('SS.') Once again. th<' program first round lhe 2- p o int tlif[N('!l("(', ...."l. and t IH'Il locat<'d tlH· pulses by sorting for all points where ...."l 2: 0.15 ¢ 0 (sec Fig. 5.5). lt sawd t.he p oint 1111111lwrs of t.hc• puls<'S as Llw S<'l { P}. Because LlH•rc• is Oil<' data poi11t on 0ach pnls0's rising 0dg<', Llw lltlltJIH'r 0.1 5 (/>o was chosen b<•caus<' it was a lit tl<• l0ss than ha If of t lw small0st pulse's tot at ltc•igh t. It t h0n recordrd t h<• H RT vain <' hal fwa~· in bctw<>rn <'ach pui s<' (s('<' F ip,. 5.6) as HRTI ..dg<'· B~· adjustiug / 1 and fo on th0 plot of llRT l 1pdg<> + .f1 • t vs. HRT0 1PdgP + .fo · I, t lw drift rates could b<' mon• <Kc urat<'l~· det<'nuined (sPP Fig. 5. 7). 5.-----.-----.-----.-----~----~----~-. 4 ~2 f:;' ~ :r: II <l 0 I 000 2000 3000 4000 5000 6000 point number, n Figure 5.5: F inding tlw pul s<'s. Thr .v-axis is thr 2-point diff<'r<'nC(' of TinTO on a Q = :3.6 p\\'jcm 2 file (030999.0 1): ...."l = HRTO (n+l) - HRTO(n). Th<· program finds all tlw poiuts with ...."l 2: O.lfi ¢ 0 a11d saV<'s tlt<'lll iu the variable P. Th<>s<" an-' indic-ated b_,. circles ( o). ;\ otc that the uoisc at ...."l = 0 ( the data on t hr pulsr kd g<'s) i ll<T<'as<'s whm dissipatiw fluid is in th<' c-rll for 11 > 6000 and n < 500. 5Tbb process wa~ writtPn into a t\ latlah program , 'tC'mp!LpulsP_timP.m· 157 Using the d e rivative of the pulse ledges For some fil es, it was d iffi cult to independent ly gncss ft and fo to makr t he vario us r amps collapse. For thrse casrs, wc came up with a method that examines the drift of rach thermom<:t<:r S('parate l~'· This met hod works best on drift calorimetry files, but could also be used o n discrete pulse fiks. ~We first took the time de rivative of HRTtedgE.', an d then plotted HRTt,·dgl' ,·s. dHHTtedge/dt (see Fig. 5.8). Because t here were a Humber of ramps, for no u-driftiug t hermometers, this curve should retracc itself rnauy times. A cons tant drift rate, 011 the• other ha nd , would change dHRTtedge/dt by a constant of!"sc>t. aucl cause' HRT 1Ntg"' to move over time. By measuring the total distance' hf'twef'n HRTtC'dg<: 0 11 the first ramp all(] t h r last ramp, and thc11 divicli11g this nnrnlw r b.Y the l otal time of the file•. an a pproximate drift ratr can I)(' detcrmirwd for that particular thermometer. For most files, we uscd a ll of thes<' methods. Wr first separatrl~, found the d rift ra tes for HRTl an<.l HRTO using the derivatiVP m ethod; we then confirm ed that t hese were correct by plotting HilT 1 vs. HRTO and looking at the downward ramps. \Ve then fiuc•-t uued the rates b.v plotting HRTlt.,dge vs. HRTOt<:dgp · For sonw files, the drift ra te> d id uot remain constant th rougho ut the run. For example, t.lw file p lot ted in F igs. 5.4 aud 5.7 had a different drift rate betwee11 t he first ramp a ncl the rest of t IH' fil e. On fil es like> this one, we had two opLious. If t he differing data was only on one ramp, we' often would throw away t he anomalous ramp, a nd only analyze t he remaining da ta. If, however , there were' several ramps that did not bc•havc, we found t hat we co uld make the d ata eoll a psr hy adjust ing C'ach deviating r amp hy a fixed amount. T his analysis showed that most of the t hermomctC'r drift t ook p lace during t he downward po rtion s of the data filrs. 5.2 Fixing the temperature scale HRTs arc sccoud ar~' thermometers. and d o not give a reading of t he absolute tc mlH'raLurc. It. was therefore uecessary to fix the temperature scale for each run by 158 0 - ~ 6-0. 1 - Vl <( c_CL0.2 E-< ex::: -0.3 ::r:: I --0.4 E-< ex::: :r: -0.5 (a) . -0.6 5400 5600 5800 6000 6200 6400 6600 6200 6400 6600 time (sec) 3 Q 2.5 6 2 5400 5600 5800 6000 time (sec) F inding tlw midd l<' of th<' pui s<' kd gC' on a Q = 3.6 ft\\' j cm 2 file' (030999.01 ). ~ Iid \nl~' lwtwc'<'ll the' spik<'s found 0 11 Fig. 5.5, W<' lontl<' the puis(' )('dgPs. The arrows iud icat<' Llw points plotted in Fig. 5.9. Dissipative' fluid enters th<' c<'ll aro und poin t 6300. )Jote that t hC' HRTO pu is<' h<'ig hts b egin to incrNtS<' at t his point , as t h <' bottom l ay0r of fluid l><'gi ns to hrat 11 p. Soou after t his poi 11 t. tlH' HRTl pulse' he ig hts lH'gi n to d<'crC'aS<', s iJH"(' I h P h eat s tart s goiug iuto h eati ng up Ll1P bottom Jaypr of uormal fluid. FigurP 5.6: 159 - 11 ~-12 0 • -e..._.. ---u,...-]3 <( E--0 25 -14 ~ -15 -17 - 18 • -·· . -~ I ::r: ......... . ••• • E-- E= - 16 0:: .,. . ,. .... ........,... .... ~· • .• : •• ~0 ~ - 13 0 -12 -11 -10 HRTl- HRT1(TDAS) (<j> ) -9 0 Fig m <' G.7: Con<'cling for HRT drift on a C2 = 3.6 pW /cm 'l. file- (030999.0 1). HRTl vs. IIRTO ·middle of t he ledge' points. Solid dots: before drift correction. Open circles: after most of the drift is corrected. On this file, one of th<' ramps has a slight ly diff<'r<'n t drift rat(' than thC' rC'st of the file' . Tlw points in each d uster of 10 or so p oiuts an' from \·arious ramps. matching a fcat.me in our data with a pr<'Yiou sl:v mC'asun •d q uan tity. Vve did t his by corrclat.iug the trnq )('ratllr<' at which Sll prrflllid tnrrH'cl to dissipati vr fluid in o ur C<'ll with To ,.\s(Q) [19]. This point was dct<'l'mined by first obsrrving on which temp C'rature pulse led ge t he nois<' of th<' bottom thennonteter suddenly incrC'ased (see Fig. 5.9) during the upward ra mp. T his set au upper limit (th c• temperature of th<.' fi n;t u oby led ge), as well as a lower limi t (the t<'mperature of the previo us led g<') for TDAs(Q). To g<'t a mo re precise location . we then looked at the dowuward ramp (see Fig. 5.10) and picked out the tem perature where Lhe bottom thermo meter became less noisy and changed the most rapid l.v. vVe theu fixed the temperature scale of the Lop and bottom thermometers by scLLiug their temperatures at this point equal to To As( Q) ± Q · R 6, 160 4.2 4 ~3.8 --E 3.6 rr. 0 -e- '--' ...... "0 ::::. 3.4 F-< 0::: ::c "0 3.2 3 2.8 -4 -3 - ] -2 0 HRTl - HRTl (T ) (<\> ) 1 2 0 Figure 5.8: Correcting for HRT drifL on a Q = 0 file (021699.01 ). HRT1 vs. dHRT 1/ dt , when· HRT 1 is comp osed of the 'middle of the ledge' points. Because then' are many ra m ps, t his cmw should rc·tracc itself many times. Any HRT drift will be observed as a width in t he x-axis spread of t he various curvc•s. O n this fi le, the spread iu HRT1 is approxima tely 0.5 ¢ 0 . Since the file is approximately 25.5 hours long, t his corresponds to a drift ra te o f ~ 5.4 J1¢o/sec. when· Rb was th<' singular bounda ry rcsistaun• discussed in Sec. 3.1.2 . Iu other words. if ll oAs was t lw point numlwr at which 71)As(Q) was o bserved, we set T l = HRTl- HRT 1(noAs) + ToAs(Q) - .0.Tc[ToAs(Q)], (5.2) and where D.T,. = Q · Rt and 6T" = Q · R~ arc equ a l to t he tempE'ratu re dro!J across the hotter and colder boundary layers. respectively; Rt all([ Rb arc the singula r bo undary resistances of the hotter a mi colder boundary layer as a functiou of super:ftuid temper-ature iust cad of bo'undar·y layer· tempemtur·e; and 80 is a constant offsrL t hat lGl 8 ~ (a) a o6 r- 0.:::: ::r: ... - ( 4 5 (b) "'0 0 r0::: () ::c "'0 -5 5200 5300 5400 5500 5600 5700 5800 5900 point number Fip,m<' 3.9: Finding TI>,\S on a Q = :3.G fi\\'jc ur'l. file• (030999.01) ou t lw upward nuup. (a) III1TO iu fiT-\. arbitrar.'· zc•ro. (b ) dHI1TO /dt (u K / sC'('). B~· ohs<'rYing \\'h('ll tlH· dc•riYatin· or the• bot tolll I IH'l'IIIO!ll<'I<'I' ()('('()JII('S nois,\·. ())I(' can dC'IC'rminf' whPu dissipat in• fluid <' ltl Prs t lw ('Pll. .:\s soon as this happ<'us , tlw temJ><' rat ur<' or tlH' bottom boundary la.\'('1' lwgins to hc•at up, absorbing au ('\'('1' greater frattion or I lw lwat. as c·om par<•d Io t h<' bulk . and t hC' pulse' hc•ight nwasurC'd b.Y I he• hot tom t}H'rtllOilH'tC' J' s tarts to itHTC'ilSe'. will be explained furtlrc•r bPlow. T 1 aud Tr.J are S<'l to lw <·qual to the lc•rnJ><'rat un•s at the wall (711') at the• lop and hottoru of tiH' cdl. resp<'diwl.v. lu the• superfluid phase the·.'· t hc•n{on· difl'c•r b.v t IH• total te' IIIJ><'rat urc• drop anoss tiH' top and bottom bonudaric•s. (5.4) 11, and Rt, ar<' th<' boundar.' · tc•mpNaturc• and rc•sistanc·c• mc>asurc•d b.\· FBI-\.\ [/8], Q is I h<' hC'al c·trrTC'nt use•d in LIH' nwasur<'tn<'lll bc•ing C'on·c•c l<'d, and ~~~~ is !.II(• inkrrl'd n•d uced he•li 11111 te111 p<'rat ure at I lw top wall ( with resp ect to t he• lambda t rausi liou tc•ttrperaturc• at the top of tlw n•ll , Tl ~'). \\'f' th<'ll iuterpolatpd from this graph to 0 JG2 15 ~----~------~------~------~------~ (a) 0 .:0 0-10 E-< ~ -20 -30 ( b) "'0 800 1000 1200 1400 1600 point number Figt tr<' 5.10: Fi nding TnAs 0 11 a Q = 3.6 Jt\Yjcm'2 filr (030999.01 ) o n t lw dow nward ramp . (a) HTITO in Jt l\:, arbitrary zc•ro. (b) dHRTO/ cl t (nl\: j sc•c) . B.v obsc' rv iug wiH•n t he• dcriYat in~ of tlw bot to m tlH' nHo m etcr becomes q uiet , o11c can dc•t enni m• w hc•u dissipa ti ve fluid leaves t h e crll. T h e temperature a t which t his occurs on the d ow nward ra mp s h ou ld lw in-l)('tWN'n the• last ctnict temperature' pulse lc•dgc <t])(l t h e• fi rst no isy tc•mperaturc pulsr kd g c• 0 11 t.IH' upward ramp. fiud t h e value of ~Tc = Q · Rb a t TnAs (Q). Simila rly, we found ~Th by plotting Ru wrs us th<' reduced h elium temperature a t t h C' bottom wall (wit h rrsprct to the lam bda transit ion temperatun• at t he' b o tt o m o f the cell. T~ottom ) : (5.5) whc'n' n = 1.274-1248 x 10 c; K /cm is t il <' s lopr of th<' tran s it ion t<'lllp<'r at ttrc iu g ravity, a nd I = O.OG-1 em is th e brig h t of o ur cC'll. The' two c urv<'s d esni h rd b.v (5.4) a mi (5.5) a rc s hown in Fig. 5. 1J . It was u ot n ccf'ssary to a djus t th e t r mpc rat urc seal<' b~· the' rrgular cotnpom•rJt o f t hl' K apitza n•sistane<' because it is n o t a strong fun ction of t<'lllpcrat urc O\'N t hr ra11gc• of our data. Ther efore. o ur t hcrm omctcrs were a djus ted to read t h<' tern JH'ratur<' oft h e• SUJ><'rfluid rig ht next to t h e wall, T 11 · • 163 5.3 Correcting the endplate temperature readings for the singular Kapitza resistance T lw n <>xt data analysis stt'p was to cmTt'ct t he rt'adings of IIRT l and H RTO for t lw singular 1\:apitza rt'sistan c<>, in order to obtain readings of th<' tempC'raturr of th <' h elium sample. We used the data of FBKA [78] to plot R 0 vs. t he eudplatc temperat urcs 6 t 10 p and tbott. and theu iuterpolated t h ese cur ves to fiud the correction for each p oint of TO a nd Tl defined iu (5.2) a nd (5.3). Thrrr w<'r<' a numbrr o f points t hat we could not dirrct\)' adjust in t his wa~·. since our data canw cou sicl<'rably dosrr to thr lam bd a transition than t h 0 FBKA data. In order to corrrct thrsr points, wr <'xtrapolatrd thr FBKA data to lower reducrd tempf'ratures. \\"e a ltered t h e way this extr apolatio n was p0rformrd until we could verify t h a t it was correct. \Vc did t his by first clcfiniug three different meas ur<>mcnts of tlH' supcdl uicl t<>tll pcrat ur<': Tl + TO 2 ---- - f 2 "3 T:sF + :r:~F (5.G) 2 Tl + !::..Tc TO- !::..T" Tl + TO 2 = ---- (5.7) ' (5.8) w h crr !::..T" a nd ~yc arr t lw t r rnprratm <' drops across th<' hottrr and coldC'r houn daries over thr t<'mpNatme range of both t h<' FBKA data and its C'Xtension. If we com p ute t hf' lwat capaci ty from t h<' data usiug thcs<' thr0e C'xpr<'ssiou s for t h<' superflu id tcmp0rat 11 rr , WC' 0xpcct to get t hrc'<' diff"cr<'nt curv<.'S t hat will agrr<' only if 1. T h r siugular Kapitza resistauct' of o ur cell agr ees with the data of FBK A. aud lGJ 2. Th<' s ing ular I~apitza rC's is tancr of om cc• ll at lower rw lt•c·C'd t C'lllp<'ratur<>s is tlH' sanw a s o ur extension o f t h<' FBKA data. vV<' found that far awa~· frolll 1). the three heat c:a p acit~· c urves a lmost always agrred , indi cating that (1) was g<'IH'rall y ~atisficd. For lhe hig h C'st temperatures (lowest rcduc<'d t<'lrlJlC'ratm<'s) oft IH' FBKA data, the three h eat capacity c urves did n ot quit<' ov<'rlap (s<'<' F ig. :5.12), indic·at in g that tlwr<' was a small rang<' wlH're the sing ular Kapitza rrsi s t <:U1C<' of our ('('II did n ot agrN' wi th that o f FBKA. In order to make Lhe c urves agn'<~ in thC' tempera Lurr rang<' closc•st t o the transition, w<' had to a djust the wa~· in which we did the exte ns ion . \\'e ultimatrly found that if\\'<' <'Xt<'nd<'d th<' data by performing a first o rd<'r fit t o the curve [logLO{l&). log 10 (Ji\)] to g<'t (5.9) \W' could makt> t h e l h n'<' IH'at capacity cmvrs agree quite wel l. This extrap olation is s hown a s triang!Ps (6) all(l circles (o) in Fig. 5. 11. l l is. in fact, a n indirect m<'as ur<' lll <'lll oft h<' sin g ul a r 1-\:apitza resistan ce at. lower reduc<'d t emper atures. 5.4 Measuring the heat capacity 5.4.1 Determining the s1ze of the temperature discontinuity After tlw Kapitza correct ion was made. it wa s time to calc ula t <' th e h eat capacity us ing t h<' t f'lll pcratur<' <'xprrssio ns I:~F. I:~F, a nd I:~F. F or rach puis<>, the }l('ig ht of t h ~· t <:'IUJ)('rat un' discont inuit .v . ....),T~F· was m <'a s ured by l. FiLtiug a lin<' to tlw Lempt>rature data p o iut.s composing ha lf o f th<' ledg<' lwfon' the pulse, and a litH' to the tcmprraturr dat a points composing half of t IH' kdg(' aft<•r t h C' puis<' [s<>e Fig. 5. 13{a)]. In ord<'r t o makr sure that thr fit did no t <·on tai n a ny of the points o n the pulsr itself o r too dos<' to th<' s hi<'ld t<•mp<' r a ture adj us tm<'nt , the fit o n tlw l<'dg<', befor<' the kt h pulse at p oint P (k), 165 0. 4 .---~-------r------~--------r-------,---~ 0.35 (a) sz 0.3 3 • *• * f<l . .•• .,. :t '.. 0.25 ~·.·· 0.2 2 0 .. .., ... ""*··· 4 3 6 X 10- 1 sF 0 .4 0 0 o oo ooo 0.35 sz 0.3 •* 3 • *· :.t ~ ~ ••~ 0.25 (b) 0.2 "'- ' •• ~ F igure 5.11 : T he singular Kapilza resistance as a fun ction of s u perfl uid temperature, for Q = 3.6 ft'vV jcm"l.. Dots (• ): tlH' temp erature d rop across the coo ler hounclar:v (FBKA [78] d ata,). S ta rs ( *): the tem per at ure d rop across t he hotter boundar)' (FBKA). O peu circles (o ): the tem perature dro p across the cooler boundary (ext ens ion of t he FBI\:A data). Triangk s (6): t h0 tem perature d rop across t h<' hotter bonn<iar)' (<'xteusiou of tlH' FBKA data). (a) Linear plot. (b) Log-log plot. The curvat ure is due to using tsr- vs. tb· 166 Figur<' ::i. 12: Biuned heat capacit.\' data at Q = 2.26 Jt\\' j cm 1 (file 030499.02) using three> different mpasurc'lll<' n ts of t lw Sl!J)('rfluid t0mp0rat urc. Solid l in<': Q = 0 data (fit to the LPE data , rou rHlrd for g ravity). Opcu c-ircles: using th e average' of t lH' t.op a11d bottom tlH'rr IIOIIH'L<'rs, T~F. Sq uares: using the top Lhermom etrr, T4F. Triangi<'s: us iug the bottom thf'rmom<'tcr, T~1F. Dotted line: Tr>As(Q) [19]. Dashrd line: Ilaussmann's Tr(Q) [18]. Not<' that thN<' is a small allrOIIIIt of disagrcrnH'nt at T - T>-. ~ - 1.0 /II\: :::} 1sF ~ 0.689 x 10 6 . As c·<ut he ohs<'r\·c•d 011 Fig. 0.11, tlti:-; disagr<'<'HI<'nt occurs at t h e• very lowest rc•du<·c•d temperatures of the FBKA data, indicating t hat for this rauge Lhf' singula r Kapit za rcsistanc<' o f our cc'll did not agrrr with thei r n'su lt s. is JH'rfomu•d frotll point miul = P(k- 1) + l<'llgthl 2 to point maxl = P(k) - 7, +7 167 a nd tlw fit on t lH' lcdg<' aft <'r t he pulse• is p <'rformed fro n1 poiut min2 = P( k) + 7 to point max2 = P(k) - l0ngt h2 - 51 2 whN<' P (k) is the• sc• t o f p oint nullii H'rs of t.lw p ulsc•s ill ustrated in Fig. 5.5~ l0np;th l = P(k) - P(k - 1) is t iH' m mii >N of p oi nts 011 l h<' first l<'dg<·: and l0n p,t h2 = P (k + 1) - P (k) is t h<• lllllllhN of p oints 0 11 tliC' S<'cond ledg<' (s<'<' Fig. 5. 13 b): 2. Extra p ola ting t hese• I wo liucs to Uw p ulse poin t, P(k)~ 3. C'aku lati11g t lH· h<'ight di ff<'r<'HCC IH't w<'<'ll t hP Pxtrapolated liues at th e poiut P(k). T his ct iffcr<'IH"<' is <'q ua l t o ~T~ F· 5 .4.2 Calculating the heat capacity T h e speci fie heat ( p <'r mo l<') of t h<' lw l i mn sam plr was t lwn caku lat rd I hrc<' di fl"erent ways , using the vari o us t<'m peratu r<' mNl.Silr<'llH'nt s ~T4 P: .6.Qcal SllC,,., = ( ~ T~F - liCrl'll ) , j"t\. wh<'f<' HC'c(•ll = 0.20675 .J / K is tlH' heat capacity of the em pty cell. and (5. 10) . = 0.089 18 is t iH' numh<'r of mo le's o f h<•l ium iu the sample'. The t e mpe rature range o f the h eat capa city m e a sure m e nts \ Yr would <'XJWCt t hat <l.ll of t h<' d ata taken at temperatu res ]pss than TnAs(Q) shou ld g ive good hc•at capacity n ·sult s. At ltig hc•r t<'lU pcrat urcs. th <' fl uid iu th C' bottom of th<' cc•ll h<'COllH'S dissipa ti ve>, a nd some o f t he heat from Q a nd CJral begins to heat 1G8 - 0.42 - 0.44 (a) Q -0.46 a (~ 0 ~ - 0.48 :I: -0.5 - 0.52 1---- -----._..l 1750 1800 1850 1900 1950 Lime (~) -0.426 -0.4262 (b) -0.4264 ,....__ ::.::: -0.4266 a ~ -0.4268 0:: :r: - 0.427 -0.4272 -0.4274 -0.4276 u.....__.___ 1860 ___._ __ . __ _..___ 1880 1900 1920 _ . ._ __.___ 1940 1960 ___._ __, 1980 time (s) FigmP 5.13: Thr points usrd fo r a tr mJWra tm<' fi t ou a Q = 3.7 Ji\\' j cn / file (022399.01 ). (a.) An <'lllir<• pulse. T IH· open circk is the poiut, ch osen as tlr P ·puls<' location,' P (k), found iu Fig. 5.5. T h<> arrows indica.tr thr poi11ts usrcl to fit thr trmpPrature clrifl bcforr and a.ft<' r th<' pulse. D ott<' d lirws: the <'xtrapolatiou of the fitted linrs. Das h-d o tted lirw: t lr <• h eig h t. ~T. (b) Th<• fit to the upper ledge of LIH' puis<> s lrowu iu (a). The thick liu c is t h e fit , audit is only drawn OY<'r the rangr of t he fit. The erro r c·s timate for t.his fit was 9.G5 x 10- 5 Jd( [sc•e (5.11) to <'xplaill t his quantity]. For bot h plots, t lw .' ·-axis h as an arbitrary zer o. The minimum at t = 1930 s is wh er<' t hr shield t <' IIIJH'rat ur<' adjustnH:' nt was m ade. 169 up this layer. As can be obsern'd in Fig. 3.G. this caus<'s tlw bottom t h<'rmomPtN to s hoo t up, a nd the appar cul ' heat capac it y' [t h e quantity calculatf'd in (5.10)] to plumHl<'t. On Ut<• other ha nd , lPss h eal uow goes through t h<' to p of the cell , so t.lw puis<' siz<•s IIH'asttr<'d by tlH· top thcnnonwt<>r begiu to decreasP, and the appar<'nt 'h<'at cap<lc it ~·· be•gins t o ris<• rapidly. Although t hi s !)('hax ior is just w hat w<• obsNv<' for Lit<' top t hermonwl<'r (sec point n in Fig. 5. 12), W<' found tha t th<' te'lllf)('ral.ttr<' st<'p s nt<•asurcd by tlw bott.Oill t lwrmom<'t <'r s udcknl y lwcanH' largc•r t ha n those· lll('asured b~· the top Uwrmonwt.e•r at a temp<•ratur<' brio"· T 1Ms (Q) (s<'<' point rJ in F ig. 5.12). A c·a r<'ful e•xaminatio11 of tlw drift data indi catrs t h at a distinct kink in the• drift rat<' of th<' bottom th<'rmonwter can be clearly observed at this t <'mperatur<' (s<'<' Fig. 5. 14). T his ph<'nomcnon occurrrd at rongldy th<' same value• of T 11 · at the bottom surfacE' for all val u<>s of Q us<'CI in tlw c•xp<'ri nwut. This value• of Til' corresponds to a local colwr<'nce length, E., of a few m inonwt C'rs, w hi<·h we• beli<'vc to be of the samP order as the surface ro ughness of th<' wall. \\"hen E. l><•cotJH'S larger th;w th <• surface roughness. the cffectiYc area throu g h w hi ch hC'at passc•s from Lite iuLerfac<' into t h e bulk hcliuu1 is reduced, thereby increasin g the apparent Rb. A simple dwck indicat<'s that the coh<'rcn ce length at LIH• top surfac<' s hould never ])('cOnH' as larg<' as t he• surfac<' roughness b<'fore dissipatiou enters t IH• cell. T h e fact that the bottom tlwrntOmC'ter lwgins to rnisbehav<' at T13 (Q) < TDAs (Q) h as S<'veral implication s for o ur experiment . For o ne, we 1wed to take this b <'havior into account when fixiug t h<' absolute t<'mperature of th<' JTTITO , the bottom th<'rmom<'tc'r. To adjust for the apparent increasE' in ..).T11 near the transition. W<' add<'d an offs<'l 60 to tlw temp<'l"at m <' T0 defined at T 1Ms (Q) in (5.3). T his phenomenon also means that we' cannot reall y trust o ur h <•at rapacity data for t<•mpcratures grNtt.er t.han TIJ . \!\'<' th<'rcfor<' w<•re not able to get as close to t he Lrausition as w<' had orig inally hopC'd. 170 0 . 07~----~----~----~----~------~----~----~~ HRT I : top thermometer HRTO: bottom thermometer 0.06 -0.05 u (l) r./) ~0.04 a ....; "'0 ~ 0.03 et:: ::r: "'0 0.02 0.01 r====:=:::======--~~....:..:..:...~-· O L---~----~----~----~--~----~----~~ 1370 1375 1380 1385 1390 1395 1400 1405 time (s) F igure 5.11: A drift data file (101098.01) takrn at Q = 2.901 t'W/ ern 2 . T lrr two t IH•rmom<'l crs are not corrrelcd for t hr Kapitza rrsistancr. Si nee the si ngular Kapi t za rc•sistanc<' is a function of t<•mp cratm r. t hr ir dNivativrs ar<' not idrnt iea l, <'V<'ll in tlr<• supcrfluid. Poin t f) p oints to a kink in thr cl<>rivativ<' iu HRTO, whor<' Wt' beliPV(' t lw coherencr lrngt h f. IH'comcs large•r tha n the s urface• rougluH'SS. ~ ot<• tlr at not hiug happens in HTITl at t his tc•m peraturc•. Point o iudicates a uot h<'l" kink in the drrivatiw ofiiRTO. T his is til<' p oint when• d issi pa tion cnt (•rs Lhe cdl. a nd corresponds to TDAs · A d<'cr<'as<' in th<' drift ntk of HRTl a l::-;o occurs at n. 5.4.3 Improving the noise of the data Throwing out noisy pulses ln orckr t o i111p ron~ tlH' uo ist' o f o ur h eat capac it y llH'ilSilrt'llH'IIt. w<' on l_\' usNI thC' pulsrs that had low <'rror <>sLimatC's to thC' line's fit to t lw d ata on both th<' trailing a nd foll owing lc•clgrs. l\ los L of this uoisc was du r to tlw instabili ty of the• 1 K pot as a function or bath l<'v<·l. Th<' fit to the data poiuts on <'ach ledge was pcrforme•d using th<' lC'ast -sq uarc•s fit function (" polyfi t.m') in l\ latlab. T he error rst im ate for C'ach point, a 11 , was <'valtJ nt<•d by t his !"unction. H, for a giwn line fitto N data point s 171 on a pulse• led ge, 1 N "\' 2 ~ ayi ~ tv 1= 1 2 X 10 .4 tLK, (5.11) wr wo uld <"Ount t hr pui s<' t owards thr hc•at <"apacit~· m easun•mc·nt. Oth er wise, wr would thro\\' o ut t lw data dNivrd from that puiS<'. In Fig. 5.1.), two fits arr s hown t.hr first o n<' has au an:rpLa bl<' fi t. Prror, while the second OlH' lirs just o ut s iclr the accept a hi<• raug<' , and was t hcn>f'o re not c·otlllt Pd for the h eat ('apati ty measureweut. Figure .J.l6(a) s hows a run whrre thr first coupl r ramps h ad a high rrjrction rat<• du<' to thr os<"ill ating 1 I< p o t . F ig ure 5.16(b) s hows t h e sr<"ond of these upward ramps, '>vhen• approximatrl.v o n r third of tlw pulsrs WN<' n•j<•cted du <' to high <'ITOr e>st,im at<•s. T his t <•dmiqm• c•HsurPd tha t only data taken wi th low thermal noise wo uld couut t 0\Yards tiH' h<•at <"Hpa<"ity mcasurcm<•ut.. Binning the data Thr oth<•r thing wr did t o lowN tbe n o is<' ol' t.h<• heat capacity llH'as urem ent was to ave rag<' th r data. As d esni lwd in S<'c. -1.1.4, rach rampin g srqurnce was rrpra.trd 5-10 tinws JH'r run. lu o rd N to p erfo rm the• data aYeraging, w<' d efin ed a number of tern prrat urr bins. a nd t h r n avrragrd bot h t lw trm pcratur<' and I h<• hPat capart i t.v over t h <' p oints t h at fell w ithiu <'etch bin . Far Rway from t h r transit ion , the heat <"apaci ty data are fair!.'· f-lat , a nd t h e bin sizes could be ch osen to hr qui t0 la rgr without losing a n.v in format ion. H owrwr, as tlw t <'lllJH'rat ur<' a pproadr<'d 7~.( Q). t h e s lo p<> of the h eat capacity data began to rise st<•<'pl :v. ln this range, a h ig h<'r <l<'ns i ty of bins was t herefo r<' rcq uired . For L('lll p<'rat ures l <·s~ t Itall 2 ttl( bf'IO\\' T>,, W<' <lf'fin C'd I 0 bins/ftl( , a nd for !Jig b <• r (('rnperat llr<'s, W(' clrfinrd 25 bins/ tth: . A n <'xampk of th<• rurrulH'r of points pN bi11 is s hown in Fig. 3.17. 172 (a) -3.3735 -.. -3.374 ~ ::1. '-' 0 ~ -3 .3745 ::r: -3.375 -3.3755 210 220 230 240 250 260 270 280 time (s) (h) -0.101 -0.102 -.. -0.103 ~ ::1. '-"" 0 ~ -0.104 -0.105 -0.106 2610 2620 2630 2640 2650 2660 time (s) F ig urC' 5. 15: F itting thC' t.(•Juprra.turC' drift. 0 11 a Q = 3.7 /i\Vjcn/· file (022399.01). (a) A puis<' w hosC' fit h a c! an nror C'stimate <·qual t o G.42 x 10-" JtK. (b) A puh>c wh ost' fit h ad Hll <'rror <'stimat <' <·qual to 2.54 x 10- 1 JtK. T his puls<' wo uld 110t ha.vC' brrn scl{'(.:ted. For bolh p lots, thC' y-axis h as a u a.r bitnu~· zero. (a) - I - 1.5 ,-.... ~ 2--2.5 f--. 0::: :::r: -3 -3.5 -4.5 0 15 10 5 time (hrs) (h) f--. 0::: ~ -2.5 f--. 0::: :::r: -3 3.8 3.9 4 4. I 4.2 4.3 4.4 4.5 time (hrs) Figurr 3.1 G: Pu lsr se•le•e· t ion 011 a Q = 3. 7 I' W / <"111 file• ( 0:22:399.01). T h<' pulse·~ that \\'e•rc• f>de•c·t<'d to usc• for lh<' ll<'at capa<'it ,\ · data arc• mnrk<•d with a dot on the• upward t<•mpcrat un• rise. T IH• ot Iter puls<•s wen• n•jc•cted l><'cause• t ll<'ir c·rror est itnal<·s W<'r<' abovr the t' llt ofL (a) T bc• wlt o lC' fil<-. T his filr was takc•n imuH•diat<·l.v aft<•r a helium transfPr. and so the• noise• at t lw IH'ginniug of the• lile· was particularly high. As a n•sult. mor<• point~ wc•n• t hrowu awa_,. duriug tlw first two ramp~. The point~ thrown away o u t lw latN ra mps W<'I'<' n•.i<'<'l<'d bpcause of bad srrYO n•spousc. (b) .\ portion of thr srcorHI ra11rp . whNr t hr LJ\ pot was ))('gi nnin g to srtt lc• down. 1 8r-----~------~-----,~------,-----~ 7 QL---~UU~~~~~~~ -5 - I -4 -3 -2 T- T/.. (JlK)- average bin tempe rature 0 2 F igurr 3. 17: T he distribution of point s within the bins of a Q = 3.71 tt\V jcm fil<- (022399.01) for th<' average or the two t hr rmorrwtrrs. On this fil<' , t hN<' W(')'(' 4 0-l ac·c·c•pt <'d pu Jses. T h rr<' wc'n' 18 bins for T - T>. < - 2 tLK, aud 43 bins for - 2 tth: < T - T>. < ToAs(Q) - T>.. 5.5 The heat capacity data at various values of Q :\s ohsc•n·N I in Fig. 5.18 , our various m <'asurem cu ts of CQ agree well with the heat capa.c·i t~· at Q = 0 for all temprratun•s !owN than abo ut 1 ttl\. below T>.. Bet Wf'<' ll 1 ttK and 0.5 tLK below T>. , a s ig nificant increase of thr h eat capacity is ohsNv<'d. This is the first direct cxperilllen tal rvid<'n C<' for the increase in Cq nrar T(.(Q). The m agnitude• of' t ll<' lH•at capacity r nha iH'<'IllCHL increases wit h Q. T h ese n •s ults will be co rn pared with <' lliT<'nt tlJCo retical pn•dicl ions iu C hapte r G. 5.6 Checking the accuracy of the data analysis I3c>for<' compariug how our r<'su lts nwasur<' up to t h<'ory. it is wor th taking a lit t I<' tinw to c•n surr that t lw possibl<' Prrors du<' to our nwthod o f d ata ana lysis an' n·latively s mall. 115 98 - 96 • Q = 0 (fit to LPE data) Q = 0 (our pulse data) Q = 1.51 jlW/cmQ = 2.26 Q=3.0 1 Q = 3.7 1 ") ~ 94 0 _§ 92 -.. ........ c ·-u 90 C;:j c. u.... 88 C;:j ~ 86 84 82 -4 -3.5 -3 -2.5 -2 -1.5 -I -0.5 T - T A. (Jl K ) Figun• .). 18: H<'pr<'s<•n tativc• lu•a t rapacity data at ntrious Q. T h<' C'Q cun·c•s ar<' tc•ruJiuatc•d a t ~J(Q), tlH• triupc•ratur<' m arked by ;J iu Fig . .: > .12. 5.6.1 The potential error in fixing the temperature scale T IH' acTurac.\· of our d a ta anal.,·s is l<'r hniqtl<' is st ron gl~· d<'p<'ndc'nt on whC't hN or not the• point lllllllh <'r of TI>As (Q) was chosen corrc•c th·. If thr wronp.; t<'lllJH'rat ur<' was choseu. tlwn t lw singular Kapi t z.a adj ustmeu t wou ld be don<' at t he• wrong t em pNat un• values. and th<• hrat rapac ity calcu lation wou ld not IH' cotT<'d. Fort unat<'ly. t lu•n • an• s<'\"<'ra l dwrb that c·a n IH' JH'rforuwd to \"Nif_,. that th<' right valtJ<' of ToAs(Q) was 1. TIH' IH•at capadt~· at a ll u o n-zt>ro Q valu<'S shou kl ap,r<'<' wiLh tllC' t h<• heat c:apacit.\" c·uiT<.'s tak<'n at Q = 0. fo r l<' lnpcrat urcs lower than about 1 jd\. h<'low 2. TIH• IH•at capadt.\" m<'as tm•d with ~~ 1 .. . T;F. and 1~F should p,iw ickntical n•sult s. ~3. Far fro111 t l)(' tran sit ion t <'IIIJ H'ral ur<>, (1(l T 1 ) /2 s hould IH' approxi mat c• l.' · c•qual 176 to the FBKA uwasureme ut of !:1T = Q · R&(CJ). In this temperatm e range, t lH•n• sho ul d I>C' little differeucc bHw<-'<'11 !:1T11 and !:1Tc. In addition , closr to thr transition t ernprrature, (To- Tl) /2 should lie halfway in- l)('twC'en t::,Tc and f:1Th (srr Fig. 5. 19) . A ll of tlwse dwcks confirm that o ur method of choosiug ToAs(Q) was quitr accuratr. T\'onrthrlrss, in futnre exprrimcnts, it would lw worth attempting to ramp Q off a nd on for rvNy run , so t hat t he temperature scale can be definitively determined by the location of T>. at Q = 0. As previo usly mentioned, we were unable to do t his duriug our cooldown because of Lime limi ts se t by a bath level dependant oscillatiug 1 K pot aud a fluxcountcr that would lmw <"Ount for fast temperatnrr scans. However, hewing T>. as a ma rker wou ld haYc brrn extremely usrful and saved many headaches during data anal~·sis. 5.6.2 The error due to choosing the incorrect point on the pulse at which to calculate 6T For t he pulse m ethod , t he heat capacity C sho uld be calculated along the temperature traj ector~r in such a way thaL: 6.Qcal = 1 dT C - dt. puiS(' cit (5 .12) If we simplify the process by calculatiug C = !1Qcal/ !:1T, we will o btain the exact same H'sul t so long as 6.T, t he height diffrrence between t he two extrapolated li1H' fits , is m0asmw l at tlw data poi nt at wh ich Area. 1 a.nd ArN\ 2, shown in F ig. 5.20, arr eq ual. T he amoun t of error in th<' heat capacity mcasur<'m ent cansrd if t he point at which !:1T is calculated is chosf'n incotTf'ct ly, wi II dcp0nd on t hf' d iffNrnc:e l)('tWf'NJ the sloprs of t hf' tf'mpera.t m r lrcig<'s prcced ing a nd followin g tlw heat pulsf'. If t here is no diff'Ncncc lw tw<'en t hf' two slopf's, t hen 6T wi ll l>C' indf'pf'ndrnt of what p oint is chosen [sec F ig. 5.20(a.) ]. Howrver, if t here is a. large diffrr('l1ce betw<'rn t h<' slopes, 177 0.36 0.34 (a) __....,0.32 ~ 3 0.3 f-- <l 0.28 0.26 0.24 0.22 '---~---'------'------'------' 0.5 1.5 0 0.4 .---------.---,---.------,----...,..------, 0.35 (b) ~ 0.3 3 £-<l 0.25 0.2 L----~--~-~-~---~--~ 0.4 0.6 0.8 2 X 10- 6 Figure 5.19: The experi meutal singular Kapitza resistauc<' on the last six ramps of 2 a Q = 3.6 tt'vV /cm file (030999.01). Op<'n circle's (o): .6.Tc, th<' t<'mpcraturc drop across the cool<'r boundary (both tlw FBKA data a nd i ts <'xtcusiou); triangles ( ~): .6.T" , t h e temperature drop across the hotter boundary (both the FBKA data and its <>xt<'nsion); solid dots (• ): th<' mC'as urcd sin gular Kapitza rC'sistance, found by plottin g (HRTO,cctgc + HRT1,Pctge)/2 vs. (HRTO,('ctgl' - HRTl,ectgc)/2; /3: th<' tcmperatm<' at which the bottom thermometer begins to misbehave; / top: t h e lowest temperature of the FBKA data for .6.Tr; / bon: tJH• low<'st temperat11n' of the FBKA data for .6.Th. (a) Lin<'ar plot. (b) Log-log plot. The pnls<' data should agrc<' with th<' FBKA data far awa.v from T;.., and be halfway in between .6.Tc and .6.T" dose to the transition. 178 ~Twill IH' a s trong fun c tion of whi('h po int is chosen [see Fig. G.20 (b)]. \\·e can make som e estimate of what th<' possible error in o ur hea t capacity tll<'as un' mrnt du <' t o this effect mig ht IH'. TIH' pulsrs on a typical file 7 had an avrragr d i ff<'r<'tH"C' of 3 x 10 5 JLK /s<'c brt wrrn t hr s loprs oft hr preceding and fo ll owing lrdges. TlH' <'tTor in choosing at what p oint ~T s hould br calcu la ted is a t most two point s. 'vvhich corrrs ponds to approx im at('l:v t.wo srconds. Therefore, llw erTor ir1 calcul at ing .6T is aro und 8( .6T) = 6 x 10-'' ftK. S iuc<', for o ur data, ~T ~ 0.1 p,K , this r<'pr<'sents au <'lTor of a pproxim at<:>l.v: b ( ~T ) - - X 1()() ~ ~T 6 x 10 5 X 100 = 0.06~, 0.1 wlri<·h is around ten tilll<'s s maliPr t Iran th<' noisr of o ur data. 7 \\"<' us<'d fil<' 022399.01 for t his cakul at io n . (5. 13) 179 (a) Area 2 ---------------~x~~ Area I Area I Figur<' 5.20: A schematic drawin g of the Nror due to choosing t he wrong point when th(' lcdg<'S before and after a pulse have different slopes. Opru circles: data points. Dashed linr: the fit to t he data points. The correct point to calculatP !::.T is chosm w hen Area 1 = An'a 2. (a) No differen ce between the slop es of the preced ing and fo llovving lrdgC's. Sll acled triangles: t.lw arf'a between t hC' ac t ual <iata ( tlw sloped line) a nd the ideal line used Lo calculate the heat capacity (ver tical line) (b) A large difference between the slopes of t he preceding and following led ges. 180 Chapter 6 Results and Conclusions Iu C ha pi Pr 1, we d emous t rated th a t the h eat capacity of s uperfluid 1Il<' a t C"Onst a ut lwat flux, CQ . is predictC'd to diverge at a Q-dep<'ndcut transit iou L<·mpc·ra t 11 rc·. lu C hapt.c•r 3, we' s howed thai tlH' firs t uwasur<' UH' nl.s of thr beat capacit y iudicate• that CcJ is, inde•c•d , euhancrd a s a fuuct io11 of Q. Tltr qu <•stio n rc•ma ins. howrvN , whrt IH•r t.lw t iH•orr ti cal prrd iction for CQ and t hr rx prri menta l rcs u Its share' m o n • l ha n gen er a l qua l i I at i \ 'C' ag reem e nt. 6.1 Comparing our data to theoretical predictions Our hrat capacity data s how a considc•rably largrr enh;:mc<'nwnt th a n prrcli clPd by c tuTc'nt tlworirs. This is d ea rly illus tratr d in F ig. G.l. where a heat capacit~· curve taken a l Q = 3.5 f.LW feu/ is s howu a lo ng with the theories of HaussnH:Ullt [18] aud liD [G, 8]. To d c tenniuc the cx tc•ut of l.lH• di sagn~cment , we 11ce•d to c•x;-uniit<' how our dat a com parPs I o the· t IH•ory a l various Iimi ts. All thcorirs pn•clict 1 t bat t he• c•nlta ncC'HH'Itt s ho uld o he~· scaling oft ltc• for111 (G. l ) when• !.1. (.r) is a uni ,·crsal scaliug func t ion a nd {2 = Q / Q c· Siu cc we' know that ::::.cQ is ana l.vl ic a t s m a ll Q for I -=f. 0. this impli <'S that (G. 2) T he fi rs t INm of (6 .2) would be> thf' exact rrs ult fo r ::::.cQ tn if p, were uot d 0pressccl b y \{ " in Se•c. 1.6.3 wf' saw th a i this was true iu the ~1FT limit. It is eas.v to sh ow 1 A l:ihort proof of why t::. CQ L" lllU S l srale in t h is way is g iven in Appendix A . 182 Thcn'fore, C,1 = Co + A I q 2 + ... (6.6) 2 A'= T 1' ~-) ( DT 2 2p.s q~ =O ' . (6.7) whnc Lhe codfici('llt of q'L is: 1 whcr<' A= A'(Q,.jST)'l.. Siner A' (1<-pcnds o nly on thc cquilibrium p8 (q = 0, T), it is complctC'I:v dct C'nninC'd h.v t.ltC' two fluid mod<'l. From (6.2), wc ran thus dcfine fom different limits: 1. \?\' hen {! 2 is sufficiently small. t::.CQ ~ 0. and CQ = C 0 , the heaL capacity at zero Q. 2. \Vlwu on ly tlH' kading order t<'rm mattcrs , thc mag nit ude of t::.CQ docs not dep<'nd on tlw wa~' in which Ps is depressed by Tr. In this n'gion, the physics of thr heat capacity E:'llhancemcut is entirely due Lo the two-fluid model. 3. \:\' hen rl :S 1. ::::.cQ shows a div<'rg(•ncc that depends on t he depression of Ps with H '. 4. \,Yhen {!'1. > L the helium is dissipative fluid , and CQ is not defined. As we llavc alread y noted , tbc data show excellent agreement with t heory in limit (1), wh('H' no cnhanccnwnt. is cxpcc:tcd. Very dose to thr transition, in limi t (3), the prediction for CQ is not. vcr~· rcli abk, sin('C' it is hcaYil y dcpcndC'nt 011 t he form of p 8 (TT'), of which li ttk is known. HowC'ver , WC' have not y<'t examined the int.ermecliat.e region (2) , {!2 < 1, •vhcrC' t he two-fluid model should clominat.f' t he physics. We woulcl expert the agr<'f'ment brtwrC'n om data and th<' prc·dirtion in this r<'gion to h<' fai rly good , sincC' t lw two-fluid model is quite· dC'pcndablc. To cktNmin(' what part of our data falls within this limit. W<' r<'rall t hat: (6.8) 183 whrn• :1: = 1/2v, all(! (6.9) This expression allows us to express (}as a fuuction oft, whc•u the data is tak<·u at fixed Q: (6. 10) For th0. data shown in Fig. 6.1, takrn at Q = 3.6 /tvV/rm 2 , this givrs (} = 5.48 x w - ro t u, whrrr w<• have used the most rPceut valu e for Q 0 [18] . If we set the in termediate g 2 limit (2) to be defiucd as those temperatures for whicll 0.01 :::; rl :::; 0.1, we fiud that we should expect thr I wo-fluid model to domiuate tlw physics when 1.82 x 10- 7 :::; t :::; -1.41 x 10- 7 , or wheu - 0.96 t.tK :S T - T;.. :::; - 0.396 ,_~,K . Therefor<', most of the data below l {j where au cuhanceuwnt is observed is iu the intrn nrdiatr g2 limit. Thus , if one lw lirvrs that the thcorrtiral wtlnc of Q 0 is rorr<:'rt, thr disagreement hrtwrrn our data and t lw throrrtiral rurv<' is rather disturbing. 6.2 Scaling T lw cxt<•nt of the discrepancy brtwreu theory aud exp<'rimcnt can b<' further elucidated by act u al)~· plotting thr rl.ata in scaled form. Fignrr 6.2 shows the scaling functions J.1., (g 2 ) for Haussmann's t heory [18], for HD throry [6], and for t hr casr when p8 is not depressed by a counterflow. Pigun• 6.3 shows our heat capacity data plot.tC'd in scakd form. The data are takrn at constant Q. changing Q, by changing T, as was shown in (6.9). They are meas ured with the averagC' of thP top aud bottom thrrmometcrs (T~r-), and are therefore terminated at point f3 marked in Fig. 5. 12. The most recent theoretical value of Q 0 [18] is used Lo scale the data. T he data agree well vvith the scaling prediction for all but Lhe lowest heat cur- 184 70 .-----~------~------~------.-----~ 60 50 No p depression s Theory of HD ( 1996) Theory of Haussmann ( J 999) / I I I I 20 0.2 0.4 0.6 0.8 (Q/Q )2 c Figur0 6.2: Thr t lworrtical scaling functiOJJS , f.J•. T hr theory of HD was calc ulated in R<'fs. [8, 9]. T lw t lwory of Haussrnann was calculated in R<'f. [18]. T lws<' two th0ories a rc scaled by their own values of Cdr·· t hry fall on a sing!<' curw, as a nticipated. They end when (Q/Qc) 2 ~ 0.1 , or (Q/Qc) ~ 0.3 , and t herefor<' Ji0 within the intcrmcdiat0 r./ limit. Vic would thus exp ec t .fJ.. to be proportional to {2 2 . As shown in Fig. 6.3. neit hrr of the tlwori<'s depart much from linearity in t h is region. vVe do fill d t hat our data for 2 2 2 /LW /em ~ Q ~ 4 pW /em can be represeuted in the form (G.ll) whcr0 A = 69 ± -! .J / mol K. Howcwr , t h0 t h0orctical rxprcssion for the firs t order term givrn in (6.7) gives a value of A = A'(Q,.j ST) 2 almost exactly ten t imes Jess than :~o ne possible explanation for why t he d a ta of the lowest heat current files do 110t conform is that the ir e nhancements are expecte d to occur at lower reduced tempe ratures. T hey therefo r<" re ly the most on thf> sing ula r Ka pitza resistance correction . If therf> is any E'rror with o ur extension of the FBKA data , these filf>s will be infiue nc<'d tlw most.. !\lore Pxpcrimcntal data, prderably with a lllidplanc thermonwtcr, is needed befor<' the fi na l verd ic t can b<' issued. 185 • 1.51 ).lW/cm 2.01 2 .26 2.51 2 .76 3 .01 3 .26 3.5 1 3.61 3.71 \1 ~8 ~ - * 0 E ;::::.;6 '--" 4-<-, II X *0 0 + 0 4 I> <::l....... 2 0 0 0 u <]2 • ---- -- 0 • 0 0.02 0 0.04 • • 0.06 (Q/Q )2 0.08 0.1 c F ig ure 6.3: T he scaled h eat capaci ty d a t a compared Lo the t h eoretical scaliug full(·tion s , .f.1s . Each theory is scaled by its own value o f Q(., a n d t he experimen tal data is sealed by Ha ussmann 's Qr. T hin solid lin e: t h e fit to t h C' data expressed in (6.11). T hick solid lin<': t h eory o f C hui et al. [8] b ased on H D t heory [6]. Dashed lin e: theory of Ha ussm a nn [18]. t his exp erimenta lly rneasur<'d value. T hu s, if the t h 0or rt ical va h10 of Q0 is corrrct, t hr two- Anicl m od 0l (wit h Ps indC'p cnd<'nt of o) cannot explain o u r result. O n e wa:v to test w h C'thcr the t lworetical val u c of Q 0 is correct is to sec if we t rul y a rc in the intermcdi at0 rl ra ng<', w here only t he first order t erm in (6.2) m atters. To do t his, we first di v ide b oth sid es of (6.2) by Q 2 t- 2 v to obtain : '' CQ t' +2 1 .:....l. Q 2 = _;.\ 0 + B 0 (Q"l t -4v) + C0 (Q"l t -4v) + ... ' 2 (6.12) 186 where Ao - A - QB - I/ ( /! + 1) { . - -7 r;- 2p0 S 2 T; - L607o x 10 (6 .13) . Ha ussm a nu and Dohm 's theory [6] calculated A 0 = 1.14 x 10- 7 , a nd Ha ussmann 's t o sec whether the data have a n)' cnrvnt nrP at all (implying that they are influenced b~r higher order terms), or w hether the.v fall o n a straight. horizontal line ( imp l~·ing that they arc dominated by the leadiug order tern1). P lott ing it in t his wa.v a lso eliminates any dependance ou the theore tical value o f Q 0 . As cau be seen in Fig. 6.4 , th0 d ata arc scattered about a constant value that is exactly ten times larger thau the' val uc' expected if Ps were no t deprcss(•d by Tl ' . 6 X JQ2.5~~----~----~~------~------~----~ 2 o._ N 1.5 Cl N + 1 • • • 2 • 3 4 F ig mc 6.4: A plot to (kt ermiue the linearity of the scaling data. The data symbols a r e the same as t hose usc'cl in Fig. 6.3. Q is in uni ts of W jcm"2. The units of the u-axis arc' (.J j mol K) /( vYj cm 2 ) 2 . and the uni ts of the> :r-axis a rc (W /crn 2 ) 2 . Solid line: y = 0. Dotted-dash<-'d line: y = 1.6075 x 10- 7 . Dashed line: .lJ = 1.6075 x 10- o. 187 6. 3 Rescaling One possible (and optimistic) expla uatio11 for the' discrepaucy bctweeu the theor<'tical curves and t he experimental measurements is that our data a rc not, iu fact , in t he low rl limit. Alt hough it certainl y appears that. o ur data seal<' to a linear function, it is possible that t lH' scatt<'l" in o ur d ata simply obscure's tlw undc,rlying curvature. Iu oth<'r words, perhaps t he t hcori<'s a rC' producin g reasonable C'Stim atc'S for t lw universal scaling fnuction .!J, (r/), but that th<' nonnni versal amplit ude Q 0 , definC'd by (6.8), is <'Sti mat.ed h'ss accurately. T hi s would not lH' all that Sll rprising, si nct> the calcu lation of Q 0 dep<'nds on detailed properties of t he 4 H<' systcm and is considerably more difficult to comp ute~ than f. 1., ((/). As illustratt>d in Fig. 6.5 (a), t he choice Q 0 = 3460 V\' /cm 2 plact>s tht> t>x per imental data o n top of Haussman11's t heoretical curve [18]." \Vith this choice, '1~. (Q) lies just a bove ToAs(Q) iu the temperature range of t he cxpcrimcutal data (sec' the clashed-dottC'd liue in F ig. G.6). A somewhat smaller choice fo r Q 0 would provide au t>q ua lly good fit to thr earlier theory [G]. U ufortuu at<'l.v. all of the data, ew~u resc:alcd iu t his manner, lie• at r/ :S 0.3 (or Q/CJr :S O.G) wbere t h<' sc:ali ug fuuctiou h as little s tructure. In this range, the scaling fuu<:tiotJ like o nr d ata is <'ssentially iudistiug ui shabl<' from linear within the scatter of t h<' data. A t rue t<•st of the resealing hypothesis woukl require data in the rrgimc (Q/Qr) 2 ---+ 1 where' the scaling function diverges. For complt>tenC'ss, we a lso show in F ig. 6.5( b) a n t>quall y good scaling collapse based ou thr assumpti on that Tc(Q) ~ TDAs(Q). For this plot , we have used Q<.(T) derived from (6.8) wit h .r = 0.813 (i.e. , effectively v = 0.615) and Q 0 = G50 W / cm 2 (optima lly chosen wit hin the error bars quoted in [19]). Since TnAs(Q) places a lower bound on Tc( (J), the sharpest conclusion we cau m ake at Lhis stage is that tlw data a rc consisteul with t he scaling hy p othesis for a fairl~· broad range of experimentally ami theoret icall ~, m oti vatc'd parameter choices , a ud that more data doser to Tc(q) will be required for a critical test of the theory. ~ P. B. \Ve ichman. private commun ication (2000). 5 T his value is in fact wit!tin t h<' Pstima t cd m argiu of error for tlw ampli tud<' calculatiou. The unccrtaiu t.v of t!te theory is a factor < 2 [R. Haussmann, private COHl ll1Ullication (1999)], while the a djustment, here is ~ 1.9. 188 0 / ~6 ~ <> (a) 0 E ..._ ::::!. ~:S_ u <> 4 0 <l 0 0. 1 0.2 0 .3 0.4 0 / ~6 ~ (b) 0 E ..._ ::::!. ~:S_ u 4 0 <J .._~ 0. 1 0.2 (Q/Q )2 0.3 0.4 c F igu n• 6.5: A lte•rna ti w scaliup; plo t of th C' IH•at capacity m easure ment s for vario us value's o f Q [23]. T h e d a ta symbols ar<' t h<' sam e• as t hose usrd in F ig. 6.3. T h r c•xpr rint c ntal d a t a a re scaled us ing Q(.(T) ekri w d fro lll (G.8) by (a) us ing t be• tlH'or<• t ica l 2 rxp on r nt valu e .r = 1/2 1/ = 0 .74G , but amplit.udf' Q 0 = 34GO \\' /c m d10scu to best m a tch tlw t lwowtic:al scaling functio n ; a nd by (h ) assuming t ha t Tc(Q) ~ Tn .1s(Q) 2 wit.h Q 0 = 650 \\' /cm a nd .r = 0.813. For t he• Q-ra u p;r o f t he e•x pNinwn tal d a ta , the• t wo anal ~·st>s a r e b asicall.v identi cal, with Lhe lines completd y overla pping to we'll w it hiu C'X pNinwnta l r<'so lu t io n . O ul ~· for hi p;h N Q d a ta wo uld t h<.' two fit s lwc·oute' d ist ing uis ha bl c. Solid Ii tl<': I heorr t ical predict ion nrglrct ing d issip M io n [8, 9]. Dash<'d litw: theor t>tical p red icti on including dissipation [18]. Ar rows: the int N Ill ('(]i atC' r/ = (Q/ Qc) "l rC'p;io n . w lwrC' thC' first coeffi cient o f thC' rx pa ns ion in (G.7) clout i ua t es. 189 -6 -6.5 ,_. 0 OCJ 0 -7 the DAS data x = 0.8 13 and x = 0.746 and x = 0.746 and -7.5 -8 ~--------~------~~------~~------~--~ -0.5 0 0.5 1.5 Figun• G.G: Thick solid Iin<': ~ ( Q). t IH• l h<•orc•t icall_,. pr('(l ict<•cl t<'m p<>ra tur0 of s u p<'rHuicl br0a kdown [GJ: thin solid liup: TrH<>( CJ) , wlwn• the• <'X J><Hwnt .r and amplitud0 Q 0 an• dros<'n 1o lwst fit tot h<' ohsnvc•d l<'mpcrat ur<' of sup<•rfi uid br<>akdown rPpr<'s<>nl<'d h~· llw data po int s [H>]: clashrcl-dot tc•d liur: thr value• of Q 0 that. along with the t IH'orr t ical vahrc• .r l /"2JJ. makc•s t he• <'Xtwrint<'tl tat lH'at ntpaci t.v data m a t c-It tlw more• rc•n•nt th<'orc•tical pr<'dic-ti o u for tlw scaling functio n [18] (s<'<' Fig. G.5 ). 6.4 Conclusions Our data p;iv<' tlw firs t nmelusiv<' c•vid<'tH"<' I hat Cq, I he brat capacity of '11I<' at fixC'd Q. is c•uhaut<'d as a function of Q. In add iti on , l<J scalf's with {/- = (CJ/ Q,)"l·. as pr<'dic-tC'd. How<'V<'r. eith (•1·: 1. ~Cq is <"OJtsickra hi~· la rgc•r I han t hc•oret icall.v pr<>d ict c•d m· 2. Tc(Q) is much closrr to T 1)\s th an curr<'ut theory pn·diets, and our data arc• t lH'rrfor<' st ro u p;l~· inHu<·n<·<•d by tlw brginuiug of th<• cli\"C't"g('lln' at 7'r.(Q). Data lying doser to '£ (Q) Hn' JH'<'<kd to d<·t<•rtniuc• wlric·IJ of t lt c•s<' two options is corr<'cl. or i r t h<'n' is .vet anot h<•r unexplon•d possi hili t~· tit at <'xplai us onr n•sults. 190 T h<' a nswrr shonld illuminat(• t h r Wl\Y t h at p,, is clr p r<'ssed b~· Q as wrll as t hr act ua l ,·alu<' o f J:.(Q). 10 1 Appendix A A proof for why 6CQ t -(lt scales with (Q/ Qr-)2 Accord ing to ( 1. 26). t lw cha ng<' in t h<' fr<'<' <'ltNgy d u<' to a c-onstant heat flu x. Q. is t' ~ <I> = - lo rr dq , (A.l ) \\'h<'r<' q = Ps n ·. \\'<• now defin<> t h<' scalin g variable' {! h_,. t lw <'qu ati o n (c\.2 ) wh<'r<' q( = CJr/ST = (Q 0 /ST) (l." . \ \'('now s ho\\' t hat if Ps(T. Q) sca lrs iu t h r form .) ) = p,(T, O) = Po 6 [J , (T , (.~ . g(u) g(Q) • thrn t hr s pc•cific heat Cq \\· il l scl'lk as we ll . HC'rC' ( (A.3 ) = 11. and g(g) = 1/ j(H) [s<'<' (1.:29)] is a smooth uni w'rsal sc·aling function, w it h g (O) = 1. t ha t su pprrssC's p8 as (} is iucrcas<'d. Substit u t in g t his C'xprrssion int o (A. l ), wr find ~<J) = (j 1 _ ____s:__<G(g) . G(g) {Jot =1g.T,r; (.r) d.?'. o (.\.-1 ) with uni w•rsa1 sca lin g fun ction G(Q) ~ g1 /2 fo r sma ll g, <'quival<'nt to t hr two fluid resul t ..6.<1> ~ q1 j 2p,(T, 0). Fro nt (A.2), {} = (Q/Q 0 ) I 1 ", a nd r! ) 211 ( Q) (D Dt q = T>. Q o (. \ .5) 192 From (A.4), Q2 ~<P = - S2T~ t2-l~ G(g), >.Po whN(' tlw sprcifie iH'at rxponrnt is n (A.6) = 2 - 3v (the usual hy pNscaling rrlati on in d = 3 dimensions), and it. is pNmissibk t.o replace T hy T>. in tlw prrfactor in th0 criti cal region. This df'mon strat('S C'xpli cit ly that t h r frc0 f'nf'rg.v incr<'mcn t scales with the expected rxpoucut. Proceeding uow to the specific heat, we have (A.7) and (A.8) T he specific hrat Puhaucemeul obeys Llw requisite scaling form: (A.9) where the amplitude is g iven C 0 = 1"Q5/ p0 S 2 T] . and th0 uniwrsal scaling func tion o beys f .Js (g) ::::::: v(v + l )g2 /2 for sma ll {), equivalent t o (1.36). 193 Appendix B Experimental Details This App(l rHiix contain s vf'r.\' SIW<"ifi(' dNails about tlw <'<p ripnH' nt us<'d during t hi~ ex,)('rim c nt. It \\'ill probabl.v be of little int eres t to a ny b od_\' <'X{'Cpt for Xiukai \\'u, Audrcw C ha t t o, and ot hr r fn ture iniH'ritors o f' m,v <'xp crirnrnt. It is iul <'n<h'd to limit tiH' amo unt of time that the." are for{'rd to s pend digging t hrough m y IHIIIH'rous lab books, sorting through e nd less d0sn ipt ious o f leak checking a nd cooldown proc(ld Ur<'S in o rd <'r to f·iud <'sscut.ial descriptions of lab <'q ui pmcnt and ('ali brat ion s. T havf' tried to include all of' t he iufornratiou th a t 1 h ad to lruut for at one' timP or a not h er during m ~· mn. 19 1 B.l The heaters Heater location resistance #of wires purpose Hl S tag<' 1 235 .3 n 2 St'' rvo S l H2 S tag<' 2 2 70.7 r2 2 ser vo S2 H3 S t a p;e :3 (shic•ld ) 2-10.1 n 2 sc•rvo shid d H4 Calo ri ttl<' I <'r (top) 92A.J39 n 4 h old/ sw itch Q t o t lw top 7W CalorinrPt er (t o p) 49. 5.JG n 4 ho ld / switch Q t o th e top 6W Calo rinwt <'r (bott o m ) 312.51-t n 4 a p ply Q 5W Calori rnc Ler ( b o tt o n1 ) 36.963 n 4 a pply Q cnl T a bk D. l: llc•a t Pr va l u<'S . T he lis ted 2-wire resist a nee YahH'S inc-1 ud <' t h e k ad r<'s ist a nc·cs. \\'it bo ut the lea d wire's , these hcatC'rS wen' approxiulal<•l_v 50 n <'etch. 195 B.2 The germanium (GRTs) resistance thermometers location serial number R vs. T calibration curves taken b y: GRTI St.ag<' 1 25770 Lak(•slwre GRT2 SLag<' 2 26294 P.K. Day GRT3 Stage 3 26293 P.K. Da.Y GRT4 Calorimeter 25771 Lakeshore GRT5 SQUID holdiug bracket 26295 P . K . Day Table B. 2: C RT locati ons. 196 T IH' GRT calibrations ar<' fit loan equation of t lw form: 2 + ... (B. J) 2 + ... (B.2) l11 T = ao + o,luR + a 2 (laR) aud In fl = b0 + b1 In T + b2 (In T ) r 1.·1 < T < 2.7 2.7 < T < 1..3 1.·1 < T < 0.0 oo 6. 209144.9 2.0064398 4. 5·129196 (/ 1 -0.9014 7248 0.5986;)212 -0. 35c187907 (1 2 0.011739325 -0.13665641 -0.017659023 U :J -0.00076388126 0.0063042152 0.0013741778 bo 12.:339725 12.6-JJ 712 12.350092 bl --1.-1024999 -5 .2370130 --1.4608890 bz (). 9;) 767928 1.7394952 1.0596051 b:l -0.] 2858632 -0.37923448 -0.1837,1704 Table B.3: GHT 1 (GnT25770) cal ibration s for t.lm'e difkn'nt temp<'raLun· raug('S. 197 1.4 < T < '2.7 2.7 < T < 4.5 1.4 < T < 3.0 5 < T < 20 ao G.-1799551 2.4105263 4.9350023 5.7171536 (II -0.99398 763 0.-1-1799897 -0.-19672422 -1.3259221 (l'l 0.05185 1284 -O. J 19 10656 -0.0016523896 0.18678894 n:l -0.00111-13260 0. 00563804-11 0.00078669171 -0.011 15571 1 bo L2.292540 12.613498 12.303673 ? bt -4.3835-191 -5.2727602 --1. 1-177331 ? b'l 0.91638015 1. 7574867 1.0308806 ? ba -0.10776608 -0.38211452 -0.17106898 ? Table B..+: CRT-! (G RT25771) calibraLiom; for four diffcrcut tempcraLun' ranges. .... -rj c- --; '? ;:::!> ~ '"' ~ '? ..,0 '1l v _, ..-; :r. :l.: -:: ,.... --; 0 M" :r. :fJ ~ t;J CJ• ..-; '-' ~ ro (t (f; :r. R52= IO.OK- 0.1 "l: I RS-': I O.OK - O.J t?c C36 = 1 J.IF C39 =I J.IF R91= I O.OK-0. 1"'< M- C/J ('t) ::n RI O- - -.., u ~ cj D r.n ('t) ~ ::r ~ .to Rll-- R9--- R8 -- - 2:.: ..-; ..., r. ....... "-' r- R7 - - - Rl--R2--- ...__... tJ I I R6 -- - T C37 = I J.IF R.l--R4 - - R5 - - - T C3 ~ = I J.IF u. _, ~ 0 "?: w cs· ,. . . 5:; cC..::jq" ~ >(X) 1!)9 SQUID filter r esistor numb ers Original resistor value Modified resistor value Rl 5.G2Kn 5.G2 Kn R2 3.G2 Kn G.G2KO R3 22.GK0 22.GKO R4 3.92Kn 510Kn R5 2·1.3Kn 24.3Kn R6 3.92Kn 5/0 t~n R7 3.nKn 249KS2 R8 ·UJ9Kr2 .J9.9Kn R9 :3.9Kn 249Kn RlO 13.oKn 13.oKn Rll 2A3Kn 2A3KS2 Tab le' B.5: SqU ID fi ltc'r r<'sist or val ues, bcfon' a ud after u w dificat iou. 200 B.4 The magnet 0.53" 0.53" 3" 12.25" I 12 layers of wire F igure 13.2: T he dinH'usions of thr magn(•t so lrnoid. \\' ire: A\ \ "G # 16 coppC'r wi r<'. pol~·t }J('rtlla l<'ZC' im;u lat.c•cl. 2671 ft. long. \Vcight: 21 pounds (30.6 pounds with thr vacuum can) R<'sistanc<': 12.5 n at room tcmperat ure 2.0 nat 77 K 0.5 n at .f T\ dH/di: 110.71 Gj.\ :201 B.5 The vacuum can lid '· 7~~1 hoi<< i- tJ F ig nr<• 13.3: T lw vaC' unm can lid . :202 Hole type # of holes Hole diameter Radius from plate cntr Top Bolt Circle (BC) Bottom BC A L 1.000" 0.0 1.350'" 1.33()"' B 2 0.500" 1.500'' 0.850" 0.975" c 3 0.500'" 1.500" . 0.830'' - D J 0.375" 1. 750"' 0.725"' - E G 0.625" 2.750" 0.975'"' - F 2 1.000'" :2.600" 1.350" - H 3 0.375" 2.600'" - 0.725"' J 12 6-32 - - 6.500" BC'O K 3 10-24 - 7.200" - BCO L 16 10-2-1 - 7...100" - BC'O M ·1 9 /:32" 3.7-to·· - - BC'O N 6 6-32 3.875" - - BC'O comme nts Table• B.G: \ ·acu mn can lid h ol(•s ami b o lt circles. BCO: Bolt circle only. Not<•s: 1) Type ~ b o les arc ra di a l ho l0s around th <' rim of tlw plat<' 2) A ll bo lt holc•s an' #G-32 D +T, G plan's at 60°, 0.30" det>p blind h o l<>s except typ<' K a nd L , w hich <H<' # 10-2-l. Typ<' L is a t hru- hoiC', but t~·1w 1...: is blind. 203 Appendix C Catalog of Data Runs T his A ppPndix lis ts the fil<•s lak<•u with th <' discrete pulse calorimetry mrthod that gave s at i s fac tor~· d a ta. ThP first I abl<' <'Oiltmcnl s 0 11 t!Jc u ois<' of the fil<•s, aucl t h<' second tal>l<' giY<'S t lw paranwtc•rs us<'d during data ;-w alysis . File name Q Comments 021099.01 3 \'pr~· quie•t data. Flux j umps iu Ill<' s h i<'ld m a ke• tlw down ward drift rat<' change. 021199.01 021190.02 2.76 Combine• the 2 ramps fro m 021199.01 a nd the first 3 from 021199.02 for Yery good n ois<'. T he last 2 ramps of 021199.02 ruiuccl b.r 1 K p o t oscillation s. 02 1399.01 2 The last 2 ramps of 021199.02 ruined h:v 1 K pot osc-illat ions. 021499. 01 1 Good noisr o n a ll n unps. Can ' t rrsolv<' <'nh att<'PJm•nl o f CQ· 021699.01 () Good noise' o n a ll ramps. 021899. 01 0 fairly good noise•. 022199. 01 3 Noise is fa ir. Repeats th<• res ult s of 02109£).01 well. 022299. 01 3.f> \'Ny few flux jumps. Hamps 2-5 h ave v<·r~· low n oise'. 022399.01 3.7 \ 'C'ry good noi se. Very few flux jumps uu Li l the <·o11traC'I ors a rrive in t h e m o rning. 030199.01 •) -· 0 F lux jumps o n the s hirk! caus<' s onw problc•ms, b ut wit h a t ig ht sortin g c-utoff, noise is dc•cc•nt. 030399.01 1.5 Excdl <'ut n oise' t hro ug ho ut 25 hr file. 030499.02 2.2G Ex<"<'ll<'nL 11 oise througho ut 22 hr fi k. - T his tab le• is continu<'d o u the next, pag<' 20..! File name Q Comments 030599.01 2.G \ 'rr." good noise' throu ghout 20 hr filc-. 030699.01 0.26 Lit<'rall:v hundre ds of flu x jumps. After t h <'~' an• ckan<'d up, the twisP is d ecent. 030799.01 3.5 .\"ot many flu x jumps. Fairly good noise. First ramp has a cliire reuL drift r ate. 030999.01 3.6 Noise is good for the whol0 file. First ramp h as a differpn t drift rat<'. 031199.01 031599.01 3.8 A large n w g<' file that goes well into tlH' normal fluid sic!<' so t h a t W<' can plo t H RTO ,·s. IIHTO. 31899.01 0 Test to see• if B:-JC measuring voltage? applied to CJcal lllakes a d iffe rPuce to t he d ata. Fi rs t 4 ramps t h0 BNC is plugged iu . Last 3 tlw BN C is n ot plugged in. (It didn ' t 111 ake• a differ<' n ce.) Tab!<' C. l: A catalog o f t he data runs , their a pproximate hrat c urrPnts, a nd comm<'nts on their n ois<'. For JYlOr<' precisr valu es of Q, se•e• Table C.2. T lw n ext ta ble gives tlH' paranwt.rrs used durin g the data a,na lysis procrss descrilwd in C hapter 3. The column h eadin gs (t hat aren ' t complrtely obv ious) arc: l eal: T h e• mag nit ud<' of the c utT<'llt a pplied to h eater 5\\' during tlw h eat pulsE>. T hi s quantit y is m easured b~· measuring the Yolt.age dro p across a 100 kn precision rc•sistor. 1 Th<· pmver a ppli rd to tl1<' cell is P = I/a1Rsw = I/a 1 x 36.063 n. Drift Rate: T h0 t0mperature drift rate oft h e' HRTs as a function of t inH:'. T his drift rat<' is thoug h t to Lw due to e ither flux <T<'<'P in th<' niobium flux tubc•s or t h erma l r0laxat.io n oft lw salt pills. For tlw drift nH' thod it is cl et.crm ined by tracking t h<' rnow•nH' tlt of T>. or TDAS for s ubsequent r a mps. For Lhe pulse llH?lhod, it is d <'t<'nnitH'd by both 1hat a ud by plot Ling HRT 1 ' 's. IIRTO. Offset in HRTO: T his is an empiricall y df't("'rtni tH·d u11mbcr that sh o uld be• e'q ual to t lw t<'lllp<'rat nr<' s hirt in HRTO du<' to the offset at the bottom at poi nt d (as desni lw d in SPc. 5.·1. 2). 1This n>lt age> \ ' 'calcJ•c2. 111 '. = 100 kf2 · l eal is the voltage requestc>d by the i\ Iatlab a ualysis progra111 205 File name #of ramps Offset in HRTO (JtK) Point # of TnAs( Q ) (J lA) Drift rate of HRTl (wl>o/sec) Drift rate of HRTO (wl>o/sec) 3.00853 0:21099.01 v 0.132 J7,660 20.082 -L.2 -8 2. 758272 021199.01 02119902 2 5 0.129 18,600 20.08:3 5 2 0 0 2.00707 021:399.01 7 0.1 37 lG,llO 20.083 0 -1.0 L.00488 021499.0 I 7 0.01 18 ,625 20.080 3 -0.5 0 021699.01 7 0 9,102 20.079 -5 -2.7 0 021 899.01 7 0 6.828 20.076 0 () 3.010287 0:22 199.01 G 0.085 20 ,620 20.084 I -2 3.31203 022299.01 6 0.215 18 .950 20.092 5 -0.3 3. 71249 022399.01 6 0.28 17,850 20.091 8 0 2.308046 030199.01 5 0.11 57,974 - 0 () 1.3072 030399.01 9.5 0.05 8 .600 20.090 -0.5 () 2.259107 030499.02 9 0.07 69,080 20.088 1.3 -1.4 2.5097 030599.01 I 0.056 16.930 20.090 5.-l -3 3.2612-t 030699.01 8 0.027 8,850 20.089 2 -2 3.5118-+5 030799.01 7 0.08 26 .310 20.091 0.7 -2 3.61212 030999. 01 8 0.16 18, 700 20.090 2.5 -l 3.81278 031199. () 1 4 2,282 20.094 2 -0.3 0 031899. 01 7 5,5 JA 20.092 10 8 Q 0 l eal Tahi<' C.2: A nttalog of thr data runs and th<'ir a na lysis param<' (Ns . 20() Appendix D Heat Capacity of a Current Carrying Superconductor T h e p!Jysics o f 1He under a hea t flow can lw a pplied t o t.IH' casr of a su prrcond ucto r can y iug a coustant c k ctric c urren t. T his was desc rilwd iu an article by David Goodst<•in, Talso Chui, a nd m:vself t h at a p p<•ar t>d in Physics LettfTs A [106]. T his App m dix consists of t lH' text of t hat arti<'h •. Abs trac t \Ye prrsr nt an a nalysis o f t hr hrat cap ac il )' of a s uprrconductor car rying a constant a ppl ic•cl <'l<•c·t ri c <· urr<'n L \\'e fi ud t ha t t iH' heat capaci ty d iverges with an expon rnt of 0.5 at a dc•p ressed t ra n sitio n l <'mperat un'. T h is r0sult is s im ilar to a r<>crnt calc11lation o f tlw ltrat cap aci ty of s upN fluid hr lium unck r a n a ppli rd IH•Ht c·urrc•u t. T h<• G in zbu rg- Landa u ( G L) <>quation s [107] gin'' a rcm arkabl.v accurate dcscri pt ion of s np N condu ctors in t he Yici ni ty of the s up<'ITOn d uc tiug lrau si liou . It is well known th at t h0 G L 0qu a tions pn•d ict. that s u pC'l'co nd uc Li vi ty will lm•ak dow u at a cr itical currr nt th at occurs wh C'n t lw vC'Iocity of t lH' Coop er pairs. I/8 , is [108] (D. l ) w h ew ~ is the eucrgy gap a nd fJ F t he F<•rmi m om C'ntum . It dors not scC'm to hav<' b<'<'n u ot iced , lwweYer , th a t in a c urre nl- biasrd sam p le, t h<' sanw p hysics leads to a d i vcrgt•Jit IH•at capacity. T he arg ume nt is as fo llows: T h<• l'rec e ucrgy d cns i t.\' of a uniform sam pic is g i vcn b~· G L as (D.2) 207 wher<' .fo pertains to the normal state. If tlw sample carries a uuiform curT<'Il(. tlH' order parameter is giv<'n h.Y (D.3 ) when' 4'o ami k ar{' constant. Then. minimizing f with respect to 1,•* ~·ields (DA ) wlH'r<'. atcordiug to the usual GL arguments, rJ is a constant , n = -no(Tr- T ) (D.5 ) "(k :!. = -l 111. II :!. 2 .~ (D.6) \-vhcrc a 0 is a positive constant , when' 111' is the mass of a Cooprr pair. a nd the number deusit~· of Cooper pairs is given by (D.7) Thr SII J><'r ClltT<'nt is tlH'tt gin'u by (D.8) wlwr<' ('' is t IH' charg<' on a Cooper pair. It follows that. · "'- (n + 3!' 1.: :~. ). (-DJ,) 8118 (D .9) T At small k [or v,. . which is equivaleut to k according to (D.6)] tltis dcrivat iv<' is positi\'(' a nd nearly constant. but because n., is r{'ducecl by itHT<'asing k according to (D.7). th<• dNi,·atiw is drin•n to zero at a critical val ue of k, and thr super current 208 becom es uns ta ble along a c urw in t h e T - k p la ne giw n b.v 2 et + 3r k = o. (0.10) Eq ua t ion 0.10 1wluces to (D. l ) if w0 r<'cogn izc· that (-n/r)~ = ~/f!IIp w h<'r<' 11"' is t h e F<•rmi w locit.v. l t foll ows fro m (0 .-1) tha t t he <'ntropy dcnsit:v in the T - k planc is giwu hy (D.ll) At a uy coustaut ,·a lue of k. this gi,·cs risP to l h<• us ua l fi nite sl<'p in th<' heat capacity, (D. 12) w hen• C'.- ami C'11 a re t hc s n percond uct ing a nd norm a l hcat capaci t ics. How<'vC'r, if inst<•ad of holdiu g k constant we hold J , constant (as in a c·ntTcn t- bias<'d samplc), til<' II C _ C .1.. - a6T " + .13 + 2a 0 rk ( Dk) (1 DT (0. 13) .1,, B nt. using (0.7) a n d (D.8), uk ) (8T (D. l -1) .!, It follows th a t CJ, di wrgcs a long th c nitical cmT<'n t curve• given by (0.10). If wr wri te t h<' solu tiou o f (0. 10) as a curw in th r T- k pla ne. Tc( k), t lw u it is easily s lt0\\'11 th a t [8] t lH• s in gular con t ri b u t io n is prop or t ion a l t.o (Tc(k) - T) l / '2. B chav io r a u a logous to t. his h as lw<'n prcdic t Nl to occur in s n pcrfiuid hdium in a coust.ant ileal fiu x [8]. IIow<'vcr , in th <' cas<' o f s u pcrflnidi t.y, t his kind of nH'<Ul fic•ld t h0ory in uo t dependa bl<•. T il <' p hc nom c no n is partic ul arly int<'r<'Sting in t lH• case of s n p<'r<.:ond uct ivi ty. lwcaus<' in this cas<' t hr G L <'quat ion s arr t bought to be val id wit ho ut correct ion for t hc rm al fln c t ua tio ns, a nd b0cause th<' lH•at cap acity at constan t 209 ( ' UIT('Ill is r<' l<'n Ult t o d <•v ic<' a pplicatio us s uch as Lra us it io u-<'dge bolom e ters [109]. \\·e know o f no ot hN ins t a nc<' iu which a lll<' aJl fie ld t h eory pn•dicts a PO\·V<'r- law cli vrrg('n c<' in t. h(' lwa t capa cit y. It s ho uld al so IH' n o ted th a t t IH' s<:Un€' a na lys is lea d s on(' to ('Xp('cl tha t t.h (' m <•an-squa n · w loci t ~· flu c tuatio ns, ( 1/.~), s ho uld a lso d i v<•r g<' a lo ng T( (k). 2LO Bibliography [1] .J. .\ . 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