Muon Capture in the Shell Model Citation Duck, Ian Morley (1961) Muon Capture in the Shell Model. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/VWWY-B590. https://resolver.caltech.edu/CaltechETD:etd-12022004-155343 Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. An effective weak interaction Hamiltonian due to Primakoff has been used in conjunction with the shell model of the nucleus to predict [...] meson capture rates to the bound states of [...], [...] and [...]. All three of these nuclei have a slow inverse [...] decay which should permit detection of a capture by an experiment which is easier than that necessary when the inverse [...] decay is fast. The [...] and [...] capture rates are predictable with sufficient confidence to constitute a test for the presence of the induced pseudoscalar interaction in [...] capture, but are not sufficiently sensitive to the presence of the conserved vector current in the weak interaction to decide on its existence. The [...] predictions are interesting from the point of view of nuclear structure, in that they are an analogue of "allowed but unfavored" transitions in [...] decay theory, and the capture rate is reduced due to the different space symmetries of the states involved. Central values of the predicted capture rates to bound states are, in [...]; [...]; and in [...]. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) ; muons Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Christy, Robert F. (advisor) Mathews, Jon (advisor) Thesis Committee: Unknown, Unknown Defense Date: 1 January 1961 Record Number: CaltechETD:etd-12022004-155343 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-12022004-155343 DOI: 10.7907/VWWY-B590 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 4717 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 02 Dec 2004 Last Modified: 17 Nov 2023 00:03 Thesis Files Preview PDF (Duck_im_1961.pdf) - Final Version See Usage Policy. 3MB Repository Staff Only: item control page

MUON CAPTURE IN THE SHELL MODEL Thesis by lan Morley Duck In Partial Fulfillment of the Requirements For the Degree of Doctor oi Philosophy California Institute of Technology Pasadena, California 1961 ACKNOWLEDGEMENTS Tam grateful to Prolessurs A. ©. Christy and Jon hiathews vho directed my research. I am 4iso pieased to thank Protessor J. F. Elliot! tov a private cormrxmunication of his resulis, Barbara eimmerman and Richard Yolman for their help with caiculations, and Mrs. Lucilie Lozwoya for typing the manuscript. During the course of this work I was supported by a Woodrow Wilson Foundation Summer Fellowship and by a Mand Fellowship. During earlier stages of my graduate studies i held an Imperial Oil of Canada (Lid.) Graduaie esearch Fellowship. Lara very grateful for the essential aupport of these fellowships. P k ABSTRACT An effective weak interaction Hamiltonian due to Primakoff has been used in conjunction with the shell model of the nucleus to . . 16 19 predict , meson capture rates to the bound states of O°, PF ° and He’. All three of these nuclei have a slow inverse » decay which should permit detection of .. capiure by an experirnment which is easier than that necessary when the inverse § decay is fast. The O° and He capture rates are predictable with sufficient confidence to con- stitute a teat for the presence of the induced pseudoscalar interaction in; capture, but are not sufficiently sensitive to the presence of the conserved vector current in the weak interaction to decide on its existence. The F predictions are interesting from the point of view of nuclear structure, in that they are an analogue of “allowed but unfavored”’ transitions in » decay theory, end the capture rate is reduced due to the different space syrnmetries of the states involved. Central values of the predicted capture rates to bound states are, in plé 3 -l , 3 19 Ie = 29 x 10° spec ;inHe Te = 1259 sect: and in F Ye = 1350 sec, Introduction lf NA Preliminary Discussion . Nuclear Physics. . . p Decay of ni . . The 3. Capture Rate. . Phe Individual Transitions Results and Discussion Conclusions . . nid B Preliminary Discussion : The Hyperfine Structure Effect The Capture hate Nuclear Physics . The Rate a . . Numevical Results Comparison with the Single Particle Model The Fate A. : . . . Results ¢ nel Discussion . . Conclusions : . . . . ie? Introduction : ° ° . . Wave Functions and Mairix Elements The Rate 2» . . . . . Conclusions : . . . . Appendices: 1 Matrix Elements for Hole-Pa il Evaluation of the falmi Integrals Reduced Niatrix Eleinents for Numerical Resul Reduced Matrix Elements for « wWapture in Required States ior e.724 Structure of che Nurmerical Resul Numerical Resul ° ° . e . . . « ts for o} 6 ° t oa . te for £19 ts for He 3 ° ° . rticle States je- . . . Wapture in oO 9 Lé my, av eh pee Mr ake wm] tw to Py Ba ws R2 hand ha ke cp oe +e cea Ma ~~] he si oN oS oR we OD <2 € oe INTRODUCTION In the last two years experimental and theoretical advances have stimulated an intense program of research into the problem of the capture of -mesons by nuclei. The production of strong ..-meson beams has made possible increased accuracy and scope of experi- menis, while the establishment of the ¥V-A weak interaction at low momentum transfers has provided a firm foundation for the study of the weak interaction at the high momentum transfers involved in ,- meson capture. As is well known, ;.-mesons stopped in matter are captured into Bohr orbits with n,£~+15 from which they cascade down by X-ray and Auger processes to the 15 state in a time of the order of 19 71° seconds, whichis short compared io the lifetime of the .. meson against capture or decay by the weak interaction. Therefore essentially all .-mesons stopped in matter decay or are captured by nuclear protons froma iS Bohr orbit. Ideally one would study the simplest system, , capture by a proton, to establish the structure of the interaction. For a nurnber of reasons this experiment does not possess the advantages it would appear to. In the first place the cap- ture rate by a single proton is very small and the observation of recoil neutrons is subject to iarge statistical errors. Also, the syster of ;. plus a proton is just not simple. Many complicated aiornic and molecular processes occur. For example, the (.-p) system being neutral can penetrate neighboring hydrogen atoms and form (p-u-p) molecules, changing the capture rate for two reasons. First, the amplitude of the ,. wave function at the proton is different in the u- rnolecule and the yp-atom, and second, the (y-p) spin correlation, which has a dramatic effect on the capture rate, is changed by the formation of the molecule. (For a discussion of the interpretation of » capture rates in liquid hydrogen, see a recent article by Wein- berg (1). } For these and other reasons, it has been the practice to study capture in complex nuclei, usually with Z greater than ¢. The problem of statistics is alleviated, and the (u-nucleus) no longer forms a neutral system free to wander about and undergo complicated molecular processes. Basically two experiments have formed, the total capture rate and the partial capture rate to particular final states. The total capture rate to all final states has been analyzed by Uberall (2) using a Fermi gae model for the nucleus and in a more detailed fashion by Primakoff (3) using a closure approximation to sum the rate over all final nuclear states. This last calculation cor- rectly reproduces the capture rate, the 24 dependence and the exclusion principle inhibition of the capture rate due to the fact that the phase space of the capturing nucleon is reduced compared to that for a free fat nucleon. However, Primakofi's calculetion depends on a nucleon- nucleon correlation length within the nucleus, the everage neutrino en T rgy, and the proton charge dis:iribution within the nucieus, ge that although the overall agreement with experimen: is good the un- certainties in the calculation are such es to mask any detailed struc- ture of the interaction. The experiment of 1 meson capture to a particular final siate Lo wl2 12 . . tye was first done for C -E ", ground etais to ground siate transition, by Godfrey (4) and has since been repeated and analyzed by others (5). In this experiment the © decay of the 5B ground state (7. = 20 milli- 2 seconds) indicates the « meson capture to a bound state of Bo”. There are a nurnber of experimental and theoretical uncertainties involved. The efiiciency of counting decay electrons ig difficult to establish; ecay electrous acé e€asiiy confused with electrons [rum the decay of Ou bound , mesons and with mesonic X-rays. Vheoretically it ia not clear that only the ground sit#te to ground state transition occurs. Y 3 n } . wld e There are five other bound states in Bo which heave uw rather obscure Structure. Itis estimated that approximately ten percent of the transi- tions go to these higher bounc states. The ground state to ground state capture rate can be closely related to the inverse » decay rate, 30 that a lot of the theoretical uncertainty here can be eliminated. Even * so, the experimental uncertainties are lurge euough to preven any € detailed determination of the interaction. Another atternpt to calculate Pes it a partial capture rate has been made by Uberall (6) for the cane of .6 6 ., . Li’ toHe’. “Infortunately this aysterm possesses difficulties pre- venting a good determination 3f the capture rate. The nuclear radius 6. ef Li is so ili-defined as to rnake the calculation of matrix elements er] very uncertain, and the capture rate by a nucleus with % = 3 is small. This system possesses the one advantage that He has only one bound state, and the structure of the & © * system is fairly well understood on the shell model. An examination of the energy levels of light nuclei (7) reveals that there are two other nuclei fcr which the relevant beund states Alé yl? . . are well understood on the ghell snodel, O and ™ °, Experimentally these systems should pessess 2 distinct advantage for measuring a artial capture rate. First, they ere ¥ = 8,9 nucls' so that the capture . ns aw ia) set we €. 490, te inverse.) aece 1 k N= 7,4 sec; 7, for O° = 29 sec) se that a new tyne of experiment becomes possible. Itis no longer necegsary to observe the inverse » decay in the presence of the ,» bearn with the attendant problems of separating 2B decey electrons, |. decay electrons and mesonic ¥-rays, or of determining electron counting efficiencies. Instead, the target (liquid oxygen or fluorine) can be continuously irradiated by the ,. beam, and the induced characteristic radioactivity of the gas boiled off measured to determine the capture rate. On the theoretical side, we shall see that Gamow-Teller type transitions dominate these reactions. The Gamow-TYeller coupling constant in the effective interaction Hamiltonian is sensitive to the nature of any induced pseudoscalar interaction, but not very sensitive to the hypothesis we make concerning a conserved vector current 80 we expect these rates to be able to distinguish various hypotheses concerning the induced pseudoscalar interaction only. A third example, ,. capture by He, wili also be discussed. This case has been discussed previously and essentially correctly. However, in addition to providing a unified discussion of the problem we will point out an error in the previous discussion. In addition a mechanism not previously discussed for a hyperfine structure effect will be pointed out here. The ; Capture Hamiltonian We shall here discuss the interaction Hamiltonian which is con- structed (8) to describe the capture of . mesons by nuclei. The usual weak interaction is expected to be modified by the presence of strong interactions and by the high momentum transfer involved in ,; capture. In the Fermi interaction responsible for » decay prety ty the V-A theory of Feynman and Gell-Mann (9) has become established. The interaction Hamiltonian responsible for j: decay is then ere it,C. is the Wermitian conjuga.e, L+ ¥ § RB and G, the weak interaction coupling constant evaluated from the ». ° Mo » with ht P the proton mass. yi decay rate, is 1,01 x Lo” (2. the hypothesis of a universal Fermi interaction, the same Hamiltonian describes the weal interaction between any four fermions. One has just to replace ., e, v, v by the relevant particles, fhe hypothesis of universality dictates that the coupling constant is still G and the interaction ia still V-4. However, if we adrnit strong inter~ actions (which do not affect the » decay process), the virtual pion effects must be expected to alter the effective interaction. In the particular case of nuclear » decay the interaction is known to be modified to an effective interaction (actually an S matrix element in lowest order in the weak interaction) tH 5 7 SAG, Y,2U,) Apt" Ye 7 By Ye v.14. (2) ~] with g., evaluated from the ;; decay of © # 1.9, ov, on the hypothesis of a conserved vector current in the Fermi interaction Lagrangian, exacily equal to one; and Ba evaluated from the » decay of the neutron equal to -1. 23. This } decay interaction is sufficient to describe the situation in the limit of zero momentum transfer. If we think of the virtual pion effects as being expanded in a power series in K/Mo where k is the momentum transfer in the process and Mi the nucleon mass, then this effective interaction contains renormalization effects of zero order in the momentum transfer, and has an error of the order kKiM,, =~ 1% in a nuclear fs decay with a maximum ( energy of 10 Mev. Ina Fermi interaction involving high momentum transfer, in particular in » meson capture, the momentum transfer is of the order of the u meson mass and kK/My = 10% approximately, and we must seek to keep at least first order corrections in k/MA to the interaction Hamiltonian. (In fact, we will test the hypothesis that these renormalization terms enter with anomalously large coefficients. ) To this end, consider the most general interaction Hamiltonian which a) is Lorentz covariant b) is time reversal invariant c) is G invariant (i.e. charge independent) d) reduces to (Eq. 1) in the absence of strong interactions and e) reduces to (Eq. 2) in the presence of strong interactions at zero momentum transfer. The particular form relevant to the , capture process is (10) 2 2 Y A (k") ¥ )Y + A (x) as re H = G(U,Va U dr { c (k +B (x7) Koy +p (7) K p “ap Up (3) — 5 ————— " m iM p- +31.€ here Ko the momentum iransfer in the interaction, = Pomh Yate in an obvious notation. Requirements (d) and (e), above, on the effec- tive interaction, specify C (0) #1.0, A (0) #1.23. Whe other two b HW terms do not contribute at low momentum transfer. Consider these terms individually. First the term _ 2 O a ——— YY ( Uy Y, U) B i(k ) = s becomes, when we use che Dirac equation (¥ -m)U(P) = 0 to eliminate the lepton momenta, -~ - 2. -( Ua WU) B (k”) ¥ (4) 2 ph 5 1-¥ i with a= —>— . [his is explicity an induced nuclear-pseudoscalar & term. Next rewrite the term C (k”) ¥ . ina@ form separating the 2 spin operator ¢ . (D (k~) will eventually be a renormalization to kK this spin matrix elernent proportional to (ari? for the conserved vector current hypothesis.) Use the identity } -1/2i(% V¥,-Y, ¥ - Tap Kp =-1/2i (V p p a) (P n,) or -2i0 fh ki, Y FP + A7V - FY -Y ap Pp a a a Now using the free particle Dirac equation for the nucleons, with a free particle proton spinor underatood on the right and a neutron spinor on the left, this becomes M,Y%- -~2yY Y 4 2 pir 2 PB, én +7 P + Ay 4M,Y +2k -4p.. (The use of a free particle spinor to describe nucleons in a nucleus introduces no error in first order in k/M,as can be seen from a Foldy- Vouthoyson transformation.) So we have the identity o a Y, = pi/Mp - ki /2M, + soe. (5) oh a 2 Mpe With these we can rewrite the general inveraction Wamiltonian (£q. 3) as #f G{a (x) (TY eu) (yr ) = A (kK a war YY ¥ “ - * via” U Mm Uns G Up - B («*) ( Uy au ? (Ut ‘5 Up ) 5 + As .Cu, ya U) (U,, T py MpUp ) (4) f 7 _ oe ORV RL aU DCG, 7 Up) . a a Be is 2 Mp LC (ke) 2 — —~ =~ k, u,) +e (2 0 gp (HY, au Migr far } 2 - Mp + HOC. in In this form the induced couplings B (k”) and D (k”) appear explicitly 4 i+ as an induced pseudoscalar interaction and as a renormalization to the vector interaction respectively, Now to evaluate D (x4). Notice (11) in (&q. 6) thac ook pam ay teeyyp ge CAPR, ; < ¥ wl ih fe LA COU US HP GU Mp N is the (-1) component of an isovector whose (0) component is similar in structure to the isovector part of the electromagnetic current — re + bese Io gs - i 7 l+f, ) ie Pp TN P oN > eA y@tmdy lis Goo uU Mow? 08 yy, ; N7 2M, Nap Kp (op ta (7) where lp and ry are the anomalous magnetic moments, in units of one nuclear magneton, of the proton and neutron. On the conserved vector current hypothesis, the weak interaction is the self-interaction of an isovector current, the vector part of which is divergence free, and therefore presumably has the same structure as the isovector part of the electromagnetic current which is known to have zero divergence. Then the weak interaction vector current and the isovector part of the electromagnetic current are two components of the same isovector and shoutd suffer the same renormalization by the strong interactions. If this is the case 2 2 C Ok ): D Mk ) as MPN e/2 : e/2 5 il D(x?) = Sc x’). (8) v Bb In this derivaiion we have neglected the difference at Kes m* between it the various electromagnetic form factors. We will compare the pre- dictions of the conserved vector current hypothesis (hereafter referred to as CVC) with the alternative hypothesis that D (k*) = 0, which is essentially the result obtained (10) with no pion current in the weak vector current. To estimate the magnitude of the induced pseudoscalar coupling, consider (12) the simplest diagram which could give rise to such an interaction, the decay of a single virtual pion into a ;. meson anda neutrino, We can evaluate the effective pion-lepton interaction from A v Ny the (7 > ..v) decay rate with an effective t ' w interaction P : N o 7s Py 2 iy ¥ —_— HW TV Bray (ay a * U ) m i (9) B where Bray is an effective coupling constant Py is the pion momentum = ve + Be I is the pion wave function. =m UU this reduces re 4 i“ 5 Using the Dirac equation, wu 20, and ¢U to ae = Bray (a, a Ui) il. (9a) Byuv 12 Then the pion decay rate is toe ge dy dy (am)* o(.4ven) Hye Riv Biusv (any? (any? 2 ™Mar (19) 2 2 a4 = TV ‘m 27 m a m - . T 24 ; . . Here v = = (1 - x) is the neutrino momentum for a pion at rest, ; L ae x 2m /rn and MW" 21/2 so fh 2 2,2 . ea Bay Q =x)" r . (19a) @av (6 7C The total diagram, representing capture of a 1. meson by a nucleon through an effective pseudoscalar interaction is k Bry & Uy 7 Y,u) Z Z2 om (Uv al ). [ Po tm m ot ; r Comparing this with the capture effective Hamilionian, we see that @¢ 2,2, 2 & Say i B°(k° =m™)= 5 4 ow 4 2,2 G m (1 +x’) i 2 2 where we have setk = m . In terms of the pion lifetime (Eq. 10a) te gs =i 2 aad the . lifetime T = G m,.? we get a 3 1207 ; 2 T 5 é 2 Z g. 1 n x ko OS 5 =. 11 BS (x ms) 4% 3% Tr (11) 13 Using r TF 87, x = 9.770 and replacing the unrenormalized pion- rm nucleon coupling constant eg '4n by the renormalized one G*/4n 215, we find B (2 = m* - a ) =9.85. This simple argument can of course tell us nothing about the sign of B (k™). A dispersion theory argument fied (13), in which only the nucleon-antinucleon intermediate state in pion decay is retained, yields essentially the above magnitude and predicts 2 9% a negative sign for B (k”° =m“). Very general arguments (14) which B i- depend on the assumption that the contribution of the one pion pole at P 2 . . ko ® - m dominates the matrix element of the axial vector current > ae) <p Po In > evenat kv =m” ag-ee with the prediction of a negative ue sign for B (c" =m”). In order to determine the sensitivity of the u {3 is capture rates to the assumptions made concerning the induced pseudo- scalar coupling constant, we will test the hypotheses that B A? 2) + 8 ¢) = = ' 9, uf m + 8g, o- So far we have said nothing about the momentum dependence of A (k2), Cc (k*) and D (x°). rhe same dispersion theory arguments b I is mentioned above indicate that these are very siowly varying functions of momentum transfer, and in fact differ by only one or two percent 2 2 . . atk 2m from their values at zero momentum transfer. We will ha ignore this slight momentum dependence and replace them by their 2 fas -. : values atk 20, and reduce the fins] rates iccor tiivly. 14 Before we proceed to consider specific reactions, let us make a non-relativistic approximation on the nucleons in the interaction Hamiltonian. Also, since the ;. meson taking part in the capture re- action is essentially at rest in the lowest Bohr orbit, we can treat the # non-relativistically. For convenience we will put the neutrino wave function in two component form. Then make the substitutions (which are correct to order p/M even for nucleons bound in the nucleus): YP Ue _ (1 - 2M, Wp ~- 4 Un wr; | 2M, ) a uy= 1-s'v) wes wy i 2/2 oi Ux = o(x)w fi |. re 1 / Here yu is a relativistic spinor w is a two component Pauli spinor Me is the nucleon rmass pP,n,v,, designate the particles or their three-momentum whichever is relevant <> is a unit vecior in the direction of the neutrino momentum vv (x) is the ; meson wave function, veduced (5) from its nominal value by the finite nuclear size effect i5 a - = —_ —_ v=k , the momentum transfer, >p-n . Using these substitutions and keeping only firsi order in the various momenta, the nuclear matrix elements occurring in H become ra + « A v a o +e ~ y “y y eee recnesnente, ) Un 7 Yop 4 YT ony », (Px? + T - ON: P w ~ Yn Mp p 7 - + - + _ by Unt MY Uw wy 7 TN (b= n23) c) Uri Um ater )wieis ow N sp 2Mp ° N N p n’ Py Pp p wt Wp e) U4” vw Uo tet oe (w= 12,3) ny Ppp “NT NM” ™p 7 ? f) Ur UuU~ woreow Pp ; P Kp + Ty - ap v - ap A be 8) nT seo UG) ey? Ye Ta: Wp p here enh is the totally 2 cisymmetric tensor, (x PR = 14 3) Oxy is the nucleon spin vo the lepton spin. Now since (L-o,° ¥) = (-¢.° 0) -o,° %) wecan ‘nseri(-o,* 6) L i. L L for later convenience in separai.ng Fermi and Gamow-Teller terms and bring the interaction Ham ltonian inte the fori (&)} ; (i -o,+v) “ive x + L + - Hf = Go (xpe* w) wy, 7 Aww (12) 6 2/2 HP where A = c (x7) (i + vy 2m )-C (x7) co. og fo i fea e 1, Nip Pa v 2 2 a le ~ ee » Ce 2x x too yA oh (oOZrar a} ) (3) P 2 P + A (Ke) 9, "NO OM . v A A - 1 2 A 1 oy c . 2 2 Br (3 Oc) #4 CR") -{o )+ 2D (x yy). i & The various coupling constants we will employ a+ 2 Clk) = ¢ (*%) Fg) =] ro 7 A (x?) z= A (2) -g. 71,23 is 4 A 2 . B i(k) = +8, or 9 t 2 2p (k*) 2 ep oe 8, Ye must aow trey to find ssac-.ions sensitive to the decailed structure of this interaction Hamiltonien. Nsutice in the last term, that Sv! aM = 1, so tha: for « Garnow-Teller type transition, the induced pseudoscalar term can male « contribution of the same order of mag- nitude as the original Gamow-Teller term. We will indeed find that such transitions are sensitive to the value of B (k‘). We now proceed * to a discussion of some specific reactions. ol& Preliminary Discussion 16 All four bound states of N°” are undersiood on the shell model Z (15) as (ip, ,, 3 ,) holes inside the lt core coupled to (id. , 5.2 Zs) 2) t ' : oe / ’ i / particles outside it to give the 1 1, J 267,17, 2°, 37 assignments characterizing these states. 392 - ae Oe - Oo fo) nie t ¥.38 - 112 i- Q=10.4 Mev 6-14 3- i) + oO! oO Shell Mode| Levels for A=16 Nuclei Consider the u capture process from the T 20, J = 9? ole ground~ state to these bound states of nie, For a capturing nucleus with sero angular momentum there is of course no worry about the hyperfine structure effect. Then for the a) T20, Js o* to T #1, J 22° transition, which is the groundstate to groundstate transition, the Gamow-Teller matrix element (¢ ev *) with £21, } #2 is expected io dominate the raie. The Fermi transition is strictly forbidden. The dominant Gamow-Teller matrix element can be obtained from the inverse { decay so this -ate can be determined almost independent of the nuclear model. 1& b) Tec, J*S to Tel, Js transition, again the Fermi matrix element is zero and the Ganiow-Teller matrix element with @ «1, j= 0 ig expected to dominate. his transition should be especially sensitive to assumptions concerning the induced pseudoscalar coupling in the weak interaction. c) T29, J20 toTei, Je#1 transition, we expect a mixed Fermi and Gamow-Teller transition, each withf elandjel. A priori, we would expect these rnaivix elements to be comparable and of the game order of magnitude as the Gamow-Teller matrix elements in the preceding two transitions. + . = 3" ; ‘ d) Te®0, J20 totT2l, J 23° transition, which has matrix elements with £ ® 3 only, we expect a small contribation tc the rate. Even at the large mornentum transfer involved in uw capture,radial matrix ele- ments with f£ >= 3 are very small. The total rate, then, should be dominated by three Gamow- Teller matrix elerments with £ si, j 2 0,1,2. The predominantly Gamow-Teller transitions she it be sensilive to the assumptions raade about the induced pseudcscaiu: teim, und we woula hope that the pre« + - os , gence of an©@ to O transiticn would make the capture rate particu. larly sensitive to the presence of 4 pseudoscalai interaction. 19 Nuclear Physics We require the shell model description of the nié bound states in order to calculate the matrix elements entering the . capture rate. These results are available in the work of Elliott and Flowers (15) but it was decided to repeat the calculations in order to vary the parameters of the nuclear force over a wide range to determine the model sensitiv- ity of the results. The previous results indicate that these states are very close (within ten percent) of the j-j state determined by the hes splitting, so the j-j coupling scheme seems to be the natural descrip- tion of this system and is the one we will use. In addition to describ- ing the bound states of nié we will also test our model against the T 2 0, odd parity states of oh, (A difficulty arises here in the shell model deacription of the T 20, J 3 1” state due to center of mages motion, which must be removed. This amounts to a restriction reduc- ing the number of basis states by one and has been included in the previous work. We will ignore this state in our work.) In the shell model, the degenerate (1 Pi2, 3/2) 0 372, 5/2" 28; 2) configurations of a hole and a particle in a central harmonic oscillator potential are the basis states in which we expand our physical states. The degeneracy is removed and the configuration mixing accomplished by the spin-orbit interaction and the hole-particle interaction. The lowest lying eigenstates of this total Hamiltonian are to be identified with the physical states. 20 a) The Nuclear Hamiltonian The spin-orbit interaction in the d shell is determined from 7 7 the ld splitting in o} which is 5.08 Mev (in gr itis 5/27 + 43/2 4.7 Mev but less well established), soV, = 41 f+s with 32 >-2.03 Mev. In the p shell the spin-orbit potential is determined from the 1 Pi,27) P3/2 splitting of 6.33 Mev in nb? (or 6.15 Mev in o}) to be %, = -4.22 Mev (or -4.10 Mev). In the harmonic oscil- lator model we will use, the 2s level would be degenerate with 1/2 the ld level in the absence of a spin-orbit interaction, or 2.03 5/2 Mev above it with the above f+ 6 term. Experimentally the splitting 17 os ar . 17 in©O is 0.871 Mev (in F itis 0.50 Mev), soto fit the CO level structure we require a Id-ls splitting Aas * 1.16 Mev. The simplest assumption to make concerning the two nucleon interaction is that it is a Rosenfeld force with a Yukawa shape and no tensor interaction, em r Tye 5 e ” -3 + 0.7 . _ 4 Ve Vv. 3 (0.3 +0 . 7) mr (14) A strength vo = 40 Mev fits the experimental situation in the A s 16 to 19 region. We will in fact consider a more general exchange mixture (15,16) -Mar Vive (W+MP -HP_+BF ) (14a) Cc mer x T og : ei ‘ s-P where P. 2 (1+ 7°75) Py Poo 5 (1+ 0°55) 89O ~My e + t,¢0, + "7 + . . veve mae (v, vo v5 Vv, TT, V or a "2772 (14b) with V+ wWt+iB-SH-iM Py My ~2 We V5 3 B+ 5 kh vi -;H-i M = A Vig 7 7a M This leaves us four parameters to vary in the nuclear force, obviously an unmanageable task. Elliott and Flowers, from a consideration of the binding energies of the A = 18,19 nuclei (17), have evaluated the triplet-singlet interaction W+M-H-&B = -0.70 and the triplet-triplet interaction WY -M-H+B ® 40.26 with the normalization W+M+H+B 21, leaving the singlet-singlet interaction W -M+H -B =x as one free parameter. The Rosenfeld force corresponds to a value x ® 2, the Visscher-Ferrell force (18) tox 20. We will vary the exchange parameter through x = 0, 1, 2, 3 22 encompassing these two proposed forces and also we will take the strength of the centrai potential to be 30, 40 and 50 Mev, and the radius parameter b = 1.56 and 1.86 fermis, to determine the model sensitivity of the predicted y capture rates. b) The Energy Matrices We require the matrix elements of the nuclear Hamiltonian H 5 2, 4. tea, + D aes +V,, (15) -} -] ; , <1 in the basis (1 p, ,, 2 3,5). (lps), 28 ,,)+ Uris, 1d,,,) 1 1 ~ 1 d,s 2) and (1 P32 1 d, 5) The spin- 1 « 1d,,.) OP), OP). orbit and d-s splitting terms are, of course, diagonal. For the matrix elements of V,, we need the direct and the exchange matrix elements (see AppendixI). Doing the isotopic spin part of the matrix element directly, we are left with for T #1 (16) <V>® wef #2Ve) Crux? , H(- Vit Ve )< v(x)? +( +2Ve2 Co) TY OD, (2 Ve + Von) <o: 25} where E(D) denotes exchange (direct) matrix element and v(x) 2 e /m,r. 23 For T = 0 vos wv fav, vd) HOV g = 3 Ve Cd, $2V_ ute) T,* T+ (-V, -3 V,~) v(x) 7; 75}. (17) The four types of matrix elements occurring cere civen below. Sub- scripts m(r) refer to final (initial) particle states and subscripts n(a) to final (initial) hole states. The reduced matrix elements of the Yukawa potential are evaluated in Appendix II. We need (we use the notation and conventions of Edmonds (19)) 1/2 Gy “4S )+(j en 77 tp GB (v(x) 7. = (-) Gin) ndelrt mé 7) ds, Kis KIM, M, x J; nN mM 1 rs (18) tats +K aa M255, 53M, M, (vx), = ( Oi ent infin ) K Jy 3m K Ig Jn Jn 4s * 4 i. 3 fn f. 2 t. a 3, J J J atl +j 4) L SM “ryt Ae TEA BYE Pd * 5 i} 2 v(x) ) 3(- )* Gos JJ, 22 fn 2 * Jn om Is a Fi daa 5 fot N mf] o<m \ es ee yf, } i824 le any (29) and finally strat; 4 <v(x)o + 275) = 3(-) Gib, 2 PAE) bag, bmn x (-) La tM ‘an J, toe eg ; x 5 a . s"~n ~ rom + Jy J, 3, 4 2 2 4 ze yg ft (21) The lowest energy eigenvalues of these energy matrices, and their eigenfunctions, have been found. In Appendix VU, table l, these eigenvalues and eigenvectors are listed for x 7 2, b =1.56 and vo = 39, 40 and 59 Mev. One parameter remains at our disposal. The strength of the harmonic oscillator potential has been chosen so that the average excitation energy of these five states agrees with experiment for V_=40 Mev. This requires hw213.5 Mev. For comparison, the results of Elliott and Flowers and the j-} limit for these states are also presented. ~arl ihe ,» Decay of N 6 ’ ri 16 : ‘ The ground state of N has a half life of 7.37 seconds against } decay to various states in © . The branching ratio for the ground- state to groundstate transition is 26% and the © value is 10.4 Mev. - + tet a . . . The 2 to® transitionis a unique first forbidden Gamow-Teller decay. In terms of the , decay rate, the ; decay rate is (jem) 7 2,9.5,0\2 2 2) x mE. 2 96 2° (=)? ( =)* (vo) at o71°(s"+ 1°) #(z,0) . (22) r A jo v - r20 Here g, * “1.23 m is the electron mass in units of Q yu is the » meson mass #105 Mev v is the neutrino momentum inp» capture © 35 Mev Q is the nuclear Q value = 10.4 Mev a(t) is the neutrino (electron) momenturn in the » decay in units of ( (es =1- (.7+ m?)') F(z,t) is the coulomb factor for 8 decay. v@ is defined for convenience in the u capture calculation as g vx |i> . (23) Thus v© is just the part of G, (see Eq. 29b and 35) obtained by re- placing j,(v r) by iy r in the radial matrix element. We obtain a value of v © = 9.382, taking the sign from the shell model determination. ‘his is to be compared with a value 0.546 in the j-j Umit, 0.411 using the Eilicit-Flowers result, and 0.325 using our wave functions. We will poscpone a comment on this result until later. Calculation of the Capture Rate In the , capture Hamiltonian, introduce the abbreviations (see Eq. 13) | i c (k’) Q+v/2™M ) rv P 0) a8 A (<7) -vf2%i (Cc (7) +2D (7) ) K Poo i+ @) " emunawanmmeanal {3 (x") +A (x’) -(C (x) +2D (7) ) v . K pe bv eg 2 P Then the matrix element of the interaction becomes HG Yow,’ -7:%) (A+T-B) we. (24) 2 . Here the spatial dependence of the , wave function over the nuclear volume has been ignored, anc LU (x) veplaced iy W (3). The nuclear t ic matrix elements A und B will be defined ina micmen.. ‘The capture rate is 2 ["s j= 27 5 (£.-£,) G* (a) M (27)? 1 a 2 2 2 v G 0 2 Y (0) 2m 27 (25) te car . 2 v is the neutrins momentum, to be taken in the Z direction. @ (3) (z e*)? ‘ is nominally — for a point nucieus. Averaged over initial » states and summed over final neutrino states, che matrix element is 2 (l-o v) ; leo 2M = 5 (A + o° B) (A® + o B*) (oe) = AA*+ BBR, (24) % gum over final and average initial nuclear states is implicit here. . . a + Pseudoscalar terms containing v have been cropped from the rate. Lhe terms A and B can be identified by comparison with df as A A = Gii-Gioveg (27) A A . & “a B *G,e-Givveg- Givxaq-g,vt (28) where the nuclear matrix elements are f rife eT’ [io (29a) N 2 gs <ileye 7 et [> (29) je —= b 2 _ eiver . . q * <f/2,, e ct P |i> (29c) r ° > M . 2 a = ceiver _ op . amare ty Sy e ct bY | i> and t te (29d) S The sum over N is a sum over all the nucleons in the nucleus. r, T , ¢ etc. refer to the nucleon being summed on. 'n terms of these matrix elements 2 2 f AA® = G" -2G° Re (i ¥- gk) (30) BB* 2 “126 og) 7 + - Gag +(G 4 Gy) lhe +2 (G-S,) g, ve (veg t) + 2 Go, Im vig xq* P) 37 4 tpi th ect ge. (31) Oi course, the Hamiltonian is only exact to order k/M go that itis inconsistent to keep the square of the small terms. However, we preter to treat the approximate model exactly so we will carry these terms along. Now to evaluate these matrix elements for the transitions to the bound states of nif using our shell model states. fhe matrix element of + /J2 is common to all these and equal to one, so drop the isotopic spin from further considerations. Then the matrix ele- ment of an operator where Me L £ is a spherical tensor of rank (£,z) operating on space coordinates only and « df ie ig a spherical tensor of rank (K,..) operating on spin variables only, is x jum (™ m M, , dfn <f] HK |i>= Cy i 3/7) i, i 5, 4 L i, x (33) fa a }|m\/6 ula [a Me | a x r fin mle K ihe, f ) <i] HN a> <t i eteli sm. Here C.., J are the expansion coefficients of our shell model states, n’m f subscripts m(n) refer to particle (hole) states. The matrix element summed over magnetic quantum numbers is ; bath ths ~ an Ba oh IRI EC, a ©) Goi, t) x (33') nm f (* Me Me i, im K o£ J, £ f@ gg | <qil Ki la> <e #e tl 1 >. d K This expression suffices for the large matrix elements f{ and g. For the momentum type matrix elements, as shown in Appendix V, we need to replace the orbital reduced matrix element by v ae <t alltel > (’ oi) <t if ult >. The required reduced matrix elements are presented in Appendix III. To evaluate the individual terms in Mu’, it is useful to use the identities for vector products in terms of spherical components (19), 30 Now to the individual terms in the rate. The Fermi matrix element fhasK=z0, <}|l Hf Il 53> 21, and <Q Hele >2 <t_ fle f lla. mm n m™m n With ¥& in the Z direction, Mz © 0 and we can write A i ‘ 2 i f 2 (J,) 6M, 9 65 f Ly, 2,4 22 and f = Je rR 6 Fok : (34) We can identify F, from the general expression above. The Gamow-Teller matrix element, with K =#=1, can be written g {* |e {2 Gy J 1 2 £ and 2.4 2 g = J, 2, G. . (35) - 2 , 2 08 0 o|0 £ Also lye zs J ). [ | ) a G 36 g | f £\, £ 5, ar 1p (36) The momentum type matrix element q can be written 31 (’ | Ms ba,£ Sm, {L Qry + Qgy is f to be identified from (33') with K =Oand<éf |WH,14 >2<@ |2,Ll 2 >, m n m n Then 2.4 2 The crose terms involving gq are 0 07; 0 [+ A iL *® f and ). feet Ct (° ° | °) Im O+gxqee J woe£ty4 wis * ; deid; ReG, Gry. (39) V1 ~ Je / The momentum type matrix element t y can be expressed v4 t (* me Me (” 0 ey 1 £ig/ \i ob (33') with K #1. The contraction is Me\ 3 . J. Tey » a8 we see from £ — £” ce PD G* (Smo 8su Tee £L Then the terms required are ad iw A f+ : J, > (-) dTb Tey 2 (40) rae and the cross term A A f+ ai Reto-gee §. D_ () 53, | (" 0 |) Luc f 1 £' J, (41) * Re Gee Tey . Individual Transitions + - 2 4 ©O toO + the terms entering this rate are g, veg ,t and Ret v-g*, for which two matrix elements are required, namely G, and Tyo O tol - the terms entering here are £“, g’, q’. Refv-q*andImi-gxq*. The individual matrix elements required are Bye G). [0 and 12° + - : + - & to3 - the same terms enter here as didin the O tol transition. The matrix elements required are 0 E r G,. 3 and Qa 32 33 ot to2 - here we require the terms yg qa’. ve g 2 ‘?, Retv: g* and Imv-gxq*. The individual matrix elements entering are G). G,. Q52 and the combination T = “TL, + T,> - T 32° All these matrix elements have been evaluated using the shell model wave functions for a range of parameters. Results are pre- sented in Appendix IV, table 2. The first forbidden Gamow-Teller matrix element G, in the ot to 2” transition can be expressed as G =R(vO), where vO is the £ decay matrix element obtained from the 2° to o* p decay of nie The matrix element is determined in magnitude and its sign is chosen to agree with the sheli model prediction. R is the ratio of the calculated values of G, and v©® whichis very nearly constant and equal to 0.789 for b # 1.56. The total capture rate to the bound states of nié calculated with these matrix elements and with the hypotheses a. z+ 8 Bas 0 with and without CVC, is also presented in Appendix IV, table 3,4. 34 Results and Discussion The nominal capture rate (eqn. 25) . 2,,.2 3 2n 2 (2m) must be reduced for three reasons - to take account of the finite mass of the nucleus (3), the finite size of the nucleus (5), and the momentum dependence of the weak interaction coupling constants (13). The finite mass of the nucleus introduces a factor approximately equal to (2 - 5,./AM, ) so this causes a reduction in the rate of 3.5%. According to the numer-~ ical calculations of Ford and Flamand, the finite size of the nucleus ad a & 2 t <. causes a 6% reduction in OF (3). We reduce the rate 5% to account for the momentum dependence of the coupling constants. Then Pe = 42.6 x i973 (2%) . (25') ha 2 A central value of 2M” #1 gives a capture rate to the bound states of . 3 -] vr. 719.2x10° sec . Now we must try to estimate the precision with which we can predict the matrix elernent 2M. From Appendix IV, table 4, we see that 2n.7 varies by lesa than one percent as we vary Ve through 30, 40 a] and 50 Mev. Additional calculations show that 2é~ is equally independent of x, the exchange parameter in the nuclear force. As x is varied through 0, 1, 2 and 3, ame increases by less than one percent. 35 Also, as the nuclear radius parameter is increased from b = 1.56 fermis to b 21.86 fermis, the rate decreases by about four percent. The reagon for the lack of dependence of the rate upon the nuclear force parameters is the fortuitous circumstance that the capture transition is predominantly (~89%) to the groundstate of nie and the dominant matrix element G, can be determined experimentally from the inverse p decay rate. Itis clear then that the uncertainty in the choice of nuclear parameters {s not the significant source of error in the pre- dicted rate. Here itis necessary to discuss the discrepancy which persists between our description of the A #16 sheli model states and that of Elliott and Flowers. The difference is seen to be small but it has a significant effect upon the ;. capture rate. On the basie of comparison to experiment there appears to be little to choose between them. Our Pf 26 levels are in the correct order. The T * 1 levels are grouped within 3 Mev, compared to a grouping within 1 Mev for EF and 392 kev experimentally. Our predicted value of v© agrees with experiment atx 8 2, Vo 230 Mev. The EF result is 5% high atx ® 2, vo = 40 Mev. Both of these are satisfactory results, since v © is particularly sen- sitive to a cancellation between the contributions from the (i Prd d 5/2) 1 /2 1d 5/2) components which dominate the J = 2” T = 1 state. d (lp, and { P, Actually, the really significant difference between the two results as far as predicting the ;. capture rate is concerned is just one, namely the structure of theJ = 0 ‘{ =i state. Both results are within ten 36 percent of the jj Hmii but the admixture of (1 Pye 1d 3/2) is of opposite sign in the two cases. The matrix element of GAT) to the (1 P37 1d 3/2) state is roughly cwice as large as to the (1 Pye i diy 2) state so that this difference in sign causes a considerable enhancement (above the jj limit) in these two matrix elernents for EF, whereas they are consider- ably reduced in our case. This single feature is primarily responsible for the different predictions of the two sets of wave functions. A careful examination of the energy matrices supports our conclusions. Weare forced to accept our results over those of EF. Since roughly eighty percent of the rate is accounted for by the experimentally determined matrix element G). we can safely make a liberal assumption about the uncertainty in the ' predicted’ part of the rate. We will consider the possibility of a 30% uncertainty in this part, {2 (o st oe or Conclusions The predicted rates for the various hypotheses including a six percent uncertainty are Case 1 2m = 1.00 - 0.88 2 1.70 - 1.50 3 1.29 - 4.13 4 -8i- 71 5 1.55 - 1,38 b 1.09 - Fé It is seen that it will not be possible to tell, even within these predictions, whether or not the conserved vector current is present. In fact, within these uncertainties, there is a slight overlap between cases | and 6 which have 8, =+8 Ba and zero. pl? Preliminary Discussion gl? and ol? provide another example of nuclei all of whose known normal (positive) parity bound states are understood on the basis of the intermediate coupling shell model (17). In this model the nucleus is treated as a closed olf shell plus three extra core nucleons in ld or 2s states. The contiguration mixing is accomplished by extra core nucleon-nucleon interactions, plus d-s splitting and i-e splitting for each nucleon. The structure of the relevant nuclear states ob- tained by diagonalizing this interaction is presented in appendix VI (17 and 20). The notation used is that of Jahn (21) and we use his scheme to classify states of three particles in the d-shell. a“s states are classified by their a? parents, de? states by their a” parents, and the 8° state is unique. 1.43 Y2t [e) o' %* 1.56 Q= 4.74 Mev 2" 0.1%8 5,7 ° “+ Shell Model S fates for A =19 Nuclei + Let us examine the ; capture process from the T =1/2, J 31/2 19 + i+ + ground state of F to the fT 23/2, Js 5/2, 3/2, 1/2 ground and excited states of O . First of ail, we notice that the inverse 6 decay -~yat apot. . . from J*5/2 toJ s1/2 is second-forbidden; the $ decay proceeds 19 t,o . to excited states of F soit will not be possible te replace any calcu- lated matrix element by an experimentally determined one. Consider the capture reactions intuitively, on the basis of the large Fermi (e”" ") and Gamow-Teller (ce ’'*) terms in the interaction Hamil- tonian. + + as . a) 1/21/2 -3/21/2 transition: in 3 decay, we would say that a Fermi transition between such states is forbidden by the change in isotopic spin, since we can factor out a coordinate type matrix element <1? leaving the matrix element < rv? which vanishes between states of different isotopic spin. In » capture, a similar situation exists. To the extent that the matrix element Ci,(vr)? is independent of the basis states, it can be factored too, leaving the matrix element re T > which vanishes asin 6 decay. In fact, & i(vr) > is not exactly independent of the basis states, s0 we would expect the Fermi matrix element to be a small difference between the contributions from various basis states. No such factorization is possible in the Gamow-Teller matrix element. Off-hand, one would expect this term to dominate the whole transition rate. (In fact, we shall see that this is not the case, due to the different dominant space symmetries of these states.) 39 b) 1/2 1/2" - 3/2 3/2°: the Fermi matrix element has L # 2 only. With the large momentum transfer, this matrix element could be large. However, experience shows that large changes in L are not favored and we expect this transition, too, to be dominated by the Gamow-Teller matrix element with L = 0. c) v2 1/2" - 3/2 5/2": a priori, itis not possible to say any- thing very specific regarding this transition. We do expect this rate to be small compared to the rates for the transitions with smaller changes in angular momentum though. It is worth remarking here about the effect of possible higher excited bound states in ol, The work of Zimmerman (22) indicates that there are no other bound states at excitation energies less than 2.4% Mev. Above this energy there is no experimental information. The shell model predicts the next positive parity states to be J # 7/2", 9/2" and sa’. States of such high angular momentum would make a very small contribution to the capture rate, and could be neglected or corrected for using just their dominant shell model states for an approximate calculation. Another possibility is that there might be negative parity atates of low angular momentum. A description of such states in the shell model is not available, and they are usually interpreted in the cluster model (23) as being, for example, states of an alpha particle bound to a cb negative parity core. Intuitively, we expect the weak interaction to be very inefficient in causing such a large rearrangement 40 of nucleons. The expectation is borne out, for example, by the ab- sence of a first forbidden Gamow-Teller transition from the 5/2* groundstate of 0)? to the 1/2” first excited state in rl? in competition with the slow allowed but unfavored transitions that do proceed. Fin- ally, if positive parity bound states of low angular momentum are found in }, this work will have to be supplemented by their inclusion using their shell model descriptions. On the basis of these considerations, we expect the total capture rate to the bound states of 0)? to be dominated by the Gamow-Teller terms in the Hamiltonian. As mentioned in the introduction, such transitions are expected to be sengitive to the presence of an induced peeudoscalar term in the interaction Hamiltonian. The Hyperfine Structure Effect pp Capture by r)? is complicated by the fact that the nucleus has non-zero spin, in this case spini/2. The mesonic atom will exist in hyperfine states with total angular momentum F 21 or sero and in these states there will be a different correlation between the » meson spin and the nuclear angular momentum J. Since the spin of the extra core proton will be correlated with the angular momentum J, then there will be a different correlation between the ,. spin and the spin of the extra core proton in the two hyperfine states. This correlation will not way 16 affect the » capture by the protons within the © core but will for cap- ture by the outside proton. An examination of the capture Hamiltonian reveals two terms dependent on the , -proton spin correlation, namely A T,. On, and 7 Try (24). A dixect calculation for : capture from F = 0 and F * 1 hyperfine states by a single proton shows that with the Primakoff Hamiltonian Treo Peet This large dependence of the capture rate on the .-proton spin cor- relation makes it necessary to treat the two hyperfine levels inde- pendently, and to correct for any transfer between these levels. Even in a complex nucleus we expect this hyperfine effect to necessitate a separate treatment of the individual hyperfine levels. the hyperfine splitting in a .-mesic atom i6 m Aves — ~ m € z° AY, a 6x10'3 2? sec / (where 4 ae s1.4x10° sec ig the HFS splitting in hydrogen). The finite width of these states arises from (3). a) « capture and decay for which fi ~ 10" sec” and conversion between these siales by 42 b) collisional de-excitation (This ig emall except for the highly mobile neutral hydrogen ..-mesic atom.), c) magnetic dipole radiation with a rate as) 13 m 3 [ ‘= 19 m (e*) Use | = 103 aec™, for Fi? and is dg) Auger electron ejection with a rate m 3 ia ~ - [= 10m (e*)' Z (se = 10° sec } . 3+ Thus, in estimating the effect of the HFS effect, we may neglect the overlap of these two states and assume thai they decay independently, starting with an initially statistical population. The triplet hyperfine level lies highest since the magnetic moment of * is positive. Now let by the total ; disappearance rate from the hyperfine w 1+ level with F s J +1/2 A= be the » capture rate to bound states from the HFS level with Fa J+1/2 A be the conversion rate from F z=J+l/2toF 2#J-1/2 N + be the probability that a , is in the HFS level F» J +1/2 at time t after its arrival in the as 1/2 level of the } atom, with the initial values N+o = 3/4 N-o 81/4 43 N= be the probability that a). has been captured to a bound state from the HFS level F 2 J + 1/2, with the initial + values N= * 0. From the solutions of the differential equations governing this process we find, as t oo, + - a r : A + c 3 - c i. 3 T c A, + Ag 4 c r_ 4 4 A, TAL Let us treat these corrections - the HFS effect and the conversion effect - to first order only. Introduce the average disappearance rate + - Ss AA r + = 8 -.7 3 ; . A zatit+ z) and X FA(1- 54). To first order in 3 and Ns 3/42°41/42" and define so that A/a we get from the above that the total probability of capture to bound states is rx + - 1/3 + 1 .- fa ¢ 3 ee oe > 2e-{=>rR ¢+—r + (= 2d = - XR =A, Net NTN, 147% t7GZ* (Te + 4 x / Re ro). (43) 3 \+ lve , Here A= Z ro + z a. is the capture rate for no correlation between .. and proton spins, or for a statistical population of these levels. For a crude order of mag: itade estimate, take } 2 0.10 r/R 20.10 anda” -at yt Cc (a R +A. The correction due to HFS and conversion effects is then of the order of ten percent. We can make a more detailed estimate of the HTS effect using a result presented by Primakoff (3). From this work, 44 2b L gp 24+) (44) a 22 Pp J (7 +3) where J is the nuclear angular momentum, Z the nuclear charge (for an odd Z, odd A nucleus) and C773 » is the ground state expectation value of the scalar product of J with the spin of the odd proton. b/a isa ratio of coupling constants equal to -0.95. This result rests on the closure approximation and the approximation that the exclusion principle inhibition factor is the same for the core protons as for the extra core proton. It also includes the assumption that the j;. decay rate is the same as the p capture rate. For the case of gl? then, 3 ® -.14 <o JID . With the assumption of a single particle model for F'”, Gy 97 = 3/2 and § ® -. 21, The single particle model for gl? in which the ground state is taken to be a 2 Sa/2 proton, has the justification that the magnetic moment of this state is almost exactly equal to the single particle value (2. 63 compared to 2.79). This is really no guarantee that other properties coincide with their single particle model predictions. For example, consider the orbital angular momentum of the extra core . 2 - proton. On the single particle model <t p » = zero. Using the complete shell model description of the ground state (see Appendix VI, use the V # 40 eigenvector), we find a 48% probability that the proton is in a d-state and 52% probability that it is in an s-state, so 45 Xs . > = 2.87. One might thus expect (TI? to be considerably less than 3/2. The matrix element <2, 5 (1 + Ce ) g r> has been evaluated using the methods to be discussed later, and the shell model states in Appendix VI. In fact, these detailed results differ but little from the single particle model. We obtain Vz30 V=40 V#50 <7 +3) 1.34 1.39 1.43 which reduces our estimaie of 3 to -.29, More detailed estimates of Are the transfer rate between hyperfine states, and of 5 have recently been made by Uberall (25). His prediction for Ae is essentially the one above, but he finds, using closure and the complete shell model wave function for pi? that Primakoff's approximations are not very good. In particular, he finds that the exclusion principle inhibition factor is considerably larger for core protons than for the extra core proton so that in Primakoff's estimate Z should be reduced and 4. thus increased to 5 # -0.60. Calculation of the Capture Rate From the preceding discussion for the case of of, the capture rate is (see Eq. 25) 46 2,, .43 3.2 [' : G (Ze) Vv 2m‘ (45) 27 where M is the matrix element between » and initial nuclear states and neutrino and final nuclear states of the interaction Hamiltonian (with Up “(0) removed), which we will writemM «<1 df li?. In the case of pl? it ia necessary to sum Vis over final states (neutrino and nuclear states independently) and average over initial states, but in this average the HFS states F » J + 1/2 must be treated separately. Denote these as a = me (averaged over HFS states with F * 1) and d. ® M2 (averaged over HFS states with F = 0). In order to carry out a sum over , and initial nuclear states independently rather than coupled, insert projection operators P+ and P- with the properties pP+(F se i>: |Fe1> p- |F20>*#|Fe0> P+|F 2 0p 2* 0 p-|Pel?es 0. By analogy with the two particle case, the required operators are 1 1 P- 2 7 (1-25 -UL) Pees (3+25-°U) where (QJ, is the lepton spin and J the nuclear angular momentum. Then AL y <ele pel i> lel e> and d. z ele p.| i> «i le le> where we sum over uncoupled initial states. Two rates are required: Awe att 4 a 3 ielXlip? (46a) wm [bo since P+ + P- 21, and ay rare =. 2 (Eel Ms) OC) Fle] (460) oy P+ 8 Urs with Pe - 3 - 3 ri . Calculation of AD will be postponed until after the discussion of X. From the previous work 2A = AA*+BB*® where (see Eq. 30 and 31) AA® 3 a 22 c Re (f q*) (47) 2 2 2 A 2 A * - ye = f : . P BBs Gig +(G) 2G,G,) | Fg | + 2(G. Gg Re(v'g o*) (48) e * 4+2G.G Im(vegxqt) +g t +Giq . As in the previous work, the approximate model is being treated exactly. The notation is the same as that used previously. f, g, q and t are the nuclear matrix elements defined in equation (29a -d} , Nuclear Physics There follows a discussion of the evaluation of the various nuclear rnatrix elements using the complete shell model specification of the nuclear states. Ina later section the single particle model predictions will be compared with those of the shell model. 48 The antisymmetrical shell model wave functions for three particles in the (id, 2s) shells are required. The phase conventions used are those of Elliott and Flowers (17) and Jahn (21). The notation is subscript 123: - label the three extra core nucleons TLS d - completely specify the three particle state ha K - completely specify the two particle state Casita x ft a; TLS 8 > - the fractional percentage coefficient (FPC) for expanding a three particle state in terms of its two particle parents la’; wan? - antisymmetric state of two particles (with magnetic quantum numbers sup- pressed) . The wave functions, with summation over all magnetic quantum numbers except Ta and Jz understood, are, in LS coupling: la? a; tusss> = <a’, wax lba’; rus3> Jas wax 2, laD 3 ( ty °) ( 4, | Lz ( be % ¢t IT/\2 @#@ iu] \}y 2 ") J (49a) " [ La S Ss L 49 (1a)* (2s) | a*s; TLSIK ) = nll a’; wax? ls 7, + (-)P P | 2° It 9 @ IL/ \y 4 (where (-)P IP isa sum over permutations of the three particles to a S$ t (49b) antisymmetrize the state) (1a) (28)° [as*; TLSJKY = les oan, la ?, + (-)? P} 3 3 (20)? \.3, ruzossd> = <s% @ s 0 aul} ss tLe0s 3? 16%; eo Ae? Is >, 2 tir] \3 4 —w b nu Ja (49d) 59 The (28) state is unique. However, the phase of the FPC is not given in either of the previous works. The phase used here was chosen by analogy to the FPC of the similar configuration for (1a)? Thus, since ¢13p)} ae [3] 225) = - and <3'p I a [3] 22, s +E ’ 2 2 we have taken <133i\ 3? (3) 22. =. 5 anaes 5° [3] 225) 2 aA . In the matrix elements, the particles within the closed shells can be ignored. The interaction Hamiltonian is then 24 = Da Hi for the three extra core nucleons only. A matrix element of this operator between a state in of and a state in gl is <I] Y [29> x 2, 3)? |a> Ce | a> Cal x, > . (5¢) The basis states la), | s> and the expansion coefiicients are given in Appendix VI. The following types of matrix elements of the single particle operator #, occur: a) <3/ i, 1a? = (rpcyrpc) <7 la®, {sal Hlar, vy Pal Haye A (ewcy <a? lat? Gi dlar, ) <a] Yi la2sre 2 (recy) 71a? 1, lH, le>, a) <a?ol | a2e)= Hel sldgl> 5 + 13 S01 fle? 1, pS els? , 13 2 5i e) Las*| Hf, las) 2 = on a, | H, la, 5,7" f) <as4| x, | 23> =F (FPC) 12 <6? | 5%> - ,<¢ | H,| >, . In order to evaluate sums on magnetic quantum numbers in these matrix Ms ia ~ elements, expand x, - aad s dx Tt where ™ . _dfz* operates only on space coordinates and is a spherical tensor of order (L, M,), s He operates only on spin coordinates and is a spherical tensor of order (K,y). Also introduce reduced matrix elements (see Appendix V) of these operators, for example the reduced matrix element Cf, I He 2,7 is defined by mi Me <2, melo detlt m.> -( tok I< [ale . (sy Mf Le Following are the various types of matrix elements occurring in the », capture. Primed quantities refer to final states. Certain summations are implicit. (In all these a common term wo? FLT mM ' Me | i, K £ Jj, J J' t will later go out in a sum over nuclear states.) wie (3 5") has been omitted. In the previous order 52 a>| # la? z ca? pada? ris 37a; salar, rL's's> x bo, teT+A+e +L 4S '41 (60 FFL S$'L'y? (-) 7 x Ss Lb J (" el |a{\ ff oi oT ° ty | ( ol. 7 re daa}. s' L's! K £ jy <5 ee, ly? Sal He lla> (52a) <a? | Hf, ia? >= 5 <a*; tan) a}, rus s> x arn ad +T4h +S '4h4L (2 Ef 82 (-)' P+ +5 '+£4L41 T -lijnmi\/ro oi a\f/s Kk Ss! S Ll J z a Sez dee x voaqryls oe gy a 3 subi K £ j, Cy Wet b> <s Ite lad (52b) 1 <a¥] He 1a?s? 3 a, etanl}d ; Tis 2’ > x aaa ck Tv ' (2 Gh Sy (yt Ttass +1 x T, -) Til /T 1 or) {Ss Kk Ss! Sou fF : . £2 Set ToL Ty\e t g]/ le 4 8 S' ou’ J) x K £ j, <3 ae Ja><s Il fe lar (52c) 53 In this case, there are two types of terms, s~-s and d-d: CaPal ify ae? 5» , +T+4 454K45! ' a iad Se fy (.) #5'$4L4J'+1 x 3 4 Pert Opn < allay Cuil Hells? > > ga} Kasi Hla? 5). ane (60 6otts 8 Sst LL (yt tT tesae sete [* “1 x T 1 Satire ar [s L I\ Ay if oy iS Za lI | oll, > os a NGI Ay ll 27 NOU TVA A, s! L! J! N K L£ an (52a') 2 A A A "i 2B é L i Cas"| Ys la’s/? 2 (2 tt! T S! L)? x (.yTHOtL ss +L '4£ +1 (ae -1 "| bau brie bre x ay 2 2 2 : K zt i S| aH a? Gll Hel D t ¢£ T/\& S (52e) 54 and finally Las?! 42? 2 PGP jt oanll 3; Toss” x Sn Anand (= TS S'L)? (-) Tz «l tt+T+A45'41 ( l T x Tv ' (" 1 ae K Yt SiS s Oo J db 3 24 3/\4 s'utg'| x K £ iy Call ba> Cell Hel ar. (52£) The Rate aA Before we evaluate m2 averaged over initial nuclear states 3 and summed over final ones remove a factor = from each matrix Jz element. Thus iver =Sctle Glirp~- ae 2 v2 Take this factor 9/2 right out of mM’. The » capture Hamiltonian is now in the form of the operator iH, For the Fermi matrix element K # 0 and 55 “J, J; jg! nag Jed, Me f= (J J')* (-) t/ % «+ Averaged over Jz and summed over J} (with Ms=O fixed) 2.2 wae fo = fi Zs a (53) Fr is to be evaluated from the above general matrix elements using Cah Har er and CL I Hele? = Cole Se | LD. The Gamow-Teller matrix element has K =], AA Ps - - ' 1s gc (940% (27% (Te ve ™)(' m "| Gy. J ' | 57M &£ iy , j and ge: J' > — a", . (54) gd £ For ¥-g, take @ in the z direction so that , = M,2 m, = 9. Then | ‘ 0 L£ jf Mi 2 Jeg)? ede? ), (i rly uc} , (55) 1 Gge (FHN7(-) 7 [- 3, J J! The momentum type matrix element with K #1 can be written (see Appendix V) 56 aad J-J -J J' |[m v Weim » O |My t = (F99F%(-) # 2% "| ")/ ) Te py J dg i /\2 L i 1 Lit and the contraction J-J oJ J' 10 ~ 4 4, 4 z ( z Zz L+L ya p) Ty Lj, involves the reduced matrix element <b, Nf ull L? , but aside from this difference has the same matrix elements as G; L 1 The terms in ) involving t are 2.4, aaa 2 and Ret-gthe J' Re Q_. (-)" ”) Lik [2 my f For the momentum type matrix element with K 2 0 we can write " ° Me i) (- 0 j,/ \o Lt j/ Mook involves the reduced matrix element <4 ful t ? m “J ot ; £ nL," vag I, [F, Qe Li, but otherwise has the same matrix elements as ie . The three terme required are 57 L 2 q feu E aie Wry (58) * A bl 0 : -Qg%@ s J! Refveg J Re/ Qe; She (59) 0 0 F fuj, 7 (' L and finally Im S-gxqeed' Red, Ja F(A Se x (60) Lit'j, 1 1 (* L ‘\(s o lo * 11 4 L£1l1/ G,1Q ie £1) Numerical Results a) 4/2 1/2" to 3/2 1/2* transition: The matrix elements required here are Fo (and Q which differa only in the reduced matrix ele- arn VLU T,.. and Q,,,. These have been ment), G, 0 (and T au)? ill lid ou’ Gi, (and T evaluated using the states in Appendix VI. Numerical results are pre- sented in Appendix VII. b) 1/2 v2" to 3/2 3/2* transition: Here we need the matrix elements F, (and Qo12" Q 532) Gio (and Toy) Gy, (and Toys G5 T,,, and Qy1° Results are presented in Appendix VII. lil c) 1/2 f2* to 3/2 5/2* transition: The matrix elements that enter are F, (and Q512 G,, (and T_..), Gs, (and T 433)" T4333 and 333° These results too are presented in Appendix VII. » Q232)) Go» 233 58 In addition to the individual matrix elements, Appendix VII contains also the separate terms in the rate 2 summed over the three bound state transitions. The coupling constants for the various hypo- theses are in Appendix VII, table 2, and the rate D calculated with these is listed in Appendix VII, table 3. Comparison with the Single Particle Model {tis interesting to know just how closely the single particle model corresponds to the shell model for the A219 nuclei. In the single particle mode], the A 219 nuclei are considered as a single particle coupled toan(L #£0,S 20,T 21, Tz -l) A 2#l8core. The J2 iat states are supposed to bea Zs single particle and the J = 3/2", 5/ 2* 1/2 ;> Single particle states. We will calculate the e& eo states are 1 43/2, uw Capture matrix elements on this simple model. The matrix element Me aw CT of a single particle operator 4 > tds He ir between these single particle states (again omitting a factor " [ Me | i, K L i, ) 1s Ig’ j K+L4j <t| Yam « ba 3 31/2 LL, [ K 4] (-) , x (61) -J_ J V2 as Coll Hell Lerdava lly, a/2>. 59 The matrix elements for » capture in this model are listed in Appendix VII. A factor (3, Jax 107* has been taken out as in the previous work. There are a number of major differences in the predictions of these two models. a) In the 1/2 1/27 -3/2 1/2" transition, the single particle model does not (in fact, cannot, as itis constructed) reflect the isotopic spin inhibition of the Fermi matrix element Fy which the full shell model does quite explicitly. Neither does the single particle model repro- duce the cancellations which in the shell model results in a much reduced Gamow-Teller matrix element Gigs so even if we were to arbitrarily drop the Fermi term, the single particle model results would in no way correspond to those of the shell model for this transition. b) In the 1/2 1/2°-3/2 3/2* transitions, the matrix elements F,, G,, andG correspond fairly well (except for a common sign) 2° "12 22 in the two models. Gigs the deminant matrix element in the shell model, does not occur in the single particle model. c) For the 1/2 1/2*-3/2 5/2* transitions, the predictions of the two modele differ almost by an order of magnitude. We have to conclude a posteriori that the single particle model is not adequate and the shell model is required. Itis true, of course, that the amount of work required in the shell model calculation could be greatly reduced by keeping only the dominant states. £0 The Rate y\ D In order to correct for the HFS effect and for transfer between the HFS states, we need to know Ape Ae A= 8 (<A LH is?) (46b) In the previous notation (Eq. 26), this rate is ~2(2 Ay) = > tr (1- OL v (A+ O2-B) Or J(ae + To BA) =2Z2ReAJ-B* +i BxJ-B* (62) where nuclear matrix elements are implicit and pseudoscalar terms have been dropped. The terms that enter 2 Re AJ:B*, in terms of the nuclear matrix elements previously defined, are: Ti4Ititj) 2ReG, G,fJ-g* = 2Re GyG, (-) tj x 3 [4 i | L ‘ ? * 6 2 33 5} dee’ Fy GH, (63a) J+s'tj) -2ReGy Gp £J'0(v-g)*¥ = 2Re Gy G-) JS\5 iv x (° 0 tt 0 ale j' , 1 £457 \) e'lat/\rI 3 J') F Ge 63b J, £15) Lie (63b) ' A 2Re Gee sei (Sxqe = 2Re G2 (aT 3g rf § je x (2 j' ) ( Lt } (°° jou w\ 23 4 bo ou! ) FO (6 f/ 2 rey! 3c) 61 1 ‘ow -2Re Gyg, f{J-vte=2Re G, a, (rete J! x | 32 [¢ Li ae a 2 yoy gy) OWL" E TH, (63d) a -2ReG, G, ¢-qJ-gh=2ReEG, G, (-) pS Fh x (3 j' are a Jos Nr uly d je je Qerj, GH (63e) . J+S'4145) rg 2Re Gy Gp vq J-v (%-g)#=2ReG, G, (-) v5 jf, x (+ joo (° 0 " ° 0) (2 ol). JJ si \l Li j 1 £ ju ME §, MY! Bit Aer, Gh, (63f) ~2Re o? Veg Ji (¥xq)* = 2Re a’ (-)779 +h 33" ji x (“ 0 "VC 0 {0 [3 i i i, it 1 Lig/ \. “gvAs gs osv\ao1i L)* Dye Ape Aer, Or}: (63g) . ee’ (3% 2ReGy ga VqsGthe2ReG,g, (-)* J! ra x ( 0 °) (° 0 ifs j! , ro Lig/\ Ly, I Bye OL Op, THpry: (63h) 62 IniB x J:-B* the following terms are required: '} +£ na ;' a] iGe gxdegeeG? (7PM 32g [3 ji ') (3 i! | ——— J \s gosh ki 1 £ va Sff00 Git GHiy: (64a) 2 2, THT HltLeye iG” (xq) x J+ (%xq)* = GY (-) 1B, RIK dpe sje x (3 j' at 0 J J ji L L'! er ee a ee ae oe i K L/\1 1 et (64) nn ERT! 2 Re{ i Ga Gp (gx): 4(<- gp} = 2ReG, Gp 3 J'j(-) x fo ei’ 2) i, yp j" ‘). tcl elills ag 114 G je G¥ie! (64c) T+T'414j1 -2Re Gy G,(gxJ)-(vxq)* = 2Re Gy Ga (-) 3j jis’ x 3§ (° 0 °) (3 j! ‘(3 j! )(' 1 ‘| z 1 “ulé/\y go oY Nr 1 F/ AG Lie ies Gig Mery (644) 63 t ALAN 2Re {i Gp g_lexJ)% | = 2Re Ga ga (-)) 14 35x [> 3 VO j' ‘HE of) Bis x Jogos 1 £/\1 wile Gje Terug (646) °| K ji $ t 2Re {c, G(%-g $03) (x9) = 2ReG, G, we JJ i 0 ofo\fo ojo\fi 3 1 ; byt! x 1 jiK/ (i. L'|K/ fi gf ij, iL’ K Gye Qeuy (64f) and finally a A J++ 2! -2Re {Gy g, (vxq)x J-v t} =2ReGyg, (-) x AA ‘ ? 357, Jeg! Bie (2 . / x JI J J' (* o|]o ° 0 | 1 1 3 » ule/(r gtie/fa gt a, |” 1K L Qe, Tey. (64g) In all these, primed quantifies refer to the final state, and unprimed to the initial state and summation over repeated indices is implicit. 64 Numerical results for these terms are given in Appendix VII, Table 1. The rate A D according to the various hypotheses appears in Appendix VII, Table 3. An examination of these results indicates that A D {a dominated by two terms. The firstis the Gamow-Teller term Gig leading to + capture to the 3/2 3/2 state from the triplet HFS state. This term is compensated by a large cross term between G, and Q The 0 111 rate A D is never larger than one half y , so the HFS and transfer correction to the rate, using the estimates for 3 and i. in the preliminary discuasion, is going to be in the neighborhood of ten per- cent. Results and Discussion The capture rate in units of the decay rate of the 1; is ["c aa = 96 w(x) (Ze)? (2A). (65) 2 According to the numerical results of Ford and Fiamand (5), Yi. (0) should be reduced from its nominal value by 8% due to the effect of finite nuclear size. We also reduce the rate another 5% to account for the momentum dependence of the coupling constants. Then Te r. = 0.306 «1077 (2A) . 65 For example, for 2 y = 100 this gives a capture rate to the bound siates of 0? of [ie =1. 35x 193 sec”), We are now faced with the difficult task of estimating the un- certainty in our result. One source, the HFS effect and the transfer effect, can be corrected for. The dependence on the nuclear radius is slight since the dominant matrix element is G The troublesome 10° source of uncertainty is the shell model calculation itself. There ia the ground state magnetic moment of Fl? to check against experiment, and as well there are three B decay rates in the A = 19 system which we can compare to determine the effectiveness of the shell model description of these states. The magnetic moment predicted is 2.80 - 2.87 nuclear mag- -S3. This is an error : netone compared :o the experimental value o of 7% in a matrix element, or 14% ina rate. A serious feature of this result is that the calculated value equals the experimental value for no values of the nuclear force parameter Ve: yl? p decay, a superallowed transition, agrees well The Ne with its shell model value. The shell model result is slowly varying and never differs by more than 20% from its experimental value for Vv. ® 0 to 50 Mev. At Vv. 2 30 Mev the error is -12%. 19 19 . The O -F " (3/2) 8 decay rate predicted varies from log ft ® 3.4 at V. = 0 to log ft 2 4.7 at Vv. = 50 Mev. The predicted rate agrees 66 with experiment ai Vv. = 25 Mev and is 15% high at ve 2 30 Mev. The 0}? pl? (5/2) B decay rate predicted varies from log ft = 4.2 to log ft = 8 as Ve varies from 0 to 50 Mev. The shell model prediction agrees with experiment at Vv. = 30 Mev. There are some encouraging features about these results. In the first place the predicted resulis are within 15% of the expert- mental results for all rates at Ve = 30 Mev. ‘Chis gives us a feeling that an error estimate of 15-29% in our results at vo t 30 Mev is reasonable. The pcapture raie A , however, is remarkably inde- pendent of Vo and since we expect the theoretical value to intersect the experimental one for some value of Vor we could be even more conservative in our error estimates. However, we will consider a possible error of 20% in the rate as a liberal estimate. A 20% uncertainty in our results completely obscures the dif- ference between the assumptions of a CVC or not. However, the dif- ference between the various hypotheses concerning the induced pseudo- scalar still remain. If we include a 19% experimental error, or a total uncertainty of 30%, then the predictions are for g,* +8 a, 2A 2 65-140 -8 g, 200-350 0 1 20-230 67 Concerning the individual matrix elements involved, there are some points worth closer examination. First, the Fermi matrix ele- ment Fo in the 1/2 1/ 2° - 3/2 ae transition is greatly reduced over its single particle value due to the isotopic spin change inhibition as we expect. A surprising result which can be understood in retrospect is the drastic cancellation in the Gamow-Teller matrix element Go in this transition. An examination of the dominant space symmetries in the shell model description of the A = 19 states (17) reveals the ex- planation for this reduction of G.). The dominant (90%) space sym- metry of the T «1/2 states of A «19 is the symmetry (3] (i.e., a space wave function totally symmetric under interchange of any two particles), whereas the dominant (again close to 90%) space symmetry of the T # 3/2 states is the space symmetry (21] (i.e., a space wave function symmetric under interchange of particles one and two, but antisymmetric under interchange of one and three or two and three). States of different space symmetry are of course orthogonal, so that the matrix element (1) between such siates vanishes. This gives us a more detailed understanding of the mechanism of the isotopic spin change inhibition of the matrix element Fo But now we can understand algo the reduction of Gyo? which is essentially the mairix element x v Je (vr) > , or, to the extent that x jo(vr)? ia factorable, itis just the matrix element g T> which vanishes between states of different space symmetry. 68 Vhe T * 3/2, Js 3/2", 5/ 2" states also belong predominantly to the space symmetry type {21] . This gives us an explanation of the reduction from the single particle value of the matrix element Gio in the 1/2". 3/2" transition. The small value of other matrix elements can also be explained by this inability fo span different space symmetry types. Even for matrix elements with ¢ = 2, < j.(ve? is roughly independent of basis states and can in a crude sense be factored leaving a vanishing matrix element Li ». (This same "unfavored' character of the matrix elements between T * 3/2 and T 21/2 states is responsible for the slowness of the allowed 3 decays from 0? to ri?) This reduction in the », capture matrix elemenis due to their unfavored nature is fortunately not sensitive to the assumptions con- cerning the nuclear force strength parameter Vie so does not cause a wide variation in the predicted capture rate for ve = 30, 40 and 50 Mev. Itis true that the symmetry of these states, and hence the can- cellations in the matrix elements, might be sensitive to the exchange mixture used in the nuclear force. The only reassurance we have on this score is the euccesa of the Elliott-Flowers wave functions (which use a Rosenfeld mixture) in predicting the unfavored 6 decay rates. There are two features of A» that deserve comment, although there is no clear explanation for them. The first feature is the sensitive dependence of Ap on the nuclear force parameter Ves Fortunately, é9 the rate A does not have such a wide variation. The second feature is the fact that A D does not dernonstrate the dominance of the cap- ture by the F ® 0 state which one would expect for capture by a single proton {and which is demonstrated by the total capture rate to all final atates (25) ). The reason for this must be in a secondary effect overwhelming the primary HFS effect. The mechanism that must be responsible is an exclusion principle effect inhibiting creation of a neutron in a bound state from ,. capture in the F = 0 state. Conclusion If we can believe the shell model within twenty percent, and the B decay rates indicate that we can, then the capture rate to the bound states of o!? provides a distinction between the hypotheses concerning the induced pseudoscalar interaction. It is clear of course that a work such as this which is so dependent upon a model cannot provide a definitive test of the coupling involved. Instead, the interest in this work must be from the point of view of nuclear physics rather than particle physics, so that eventually with a well established ; capture interaction, these results will provide a test of the shell model and in fact supplement the study begun by Elliott and Flowers. 70 He? Introduction An example of » capture which has gained increased interest recently is ;, capture in He? to the groundstate of 113(26). The wave functions of the A = 3 groundstates are known with some certainty (27), so that the capture rate can be predicted with considerable confidence. This problem has in fact been treated before (28). Our results are in close agreement with the previous treatment. The purpose of this discussion will be to complete the previous treatment by including momentum type matrix elements and L = 2 matrix elements previously omitted. It turns out that these terms, which we might expect to make approximately ten percent contributions, in fact only amount to about two percent of the rate. In addition to these slight improvements we discuss the HFS effect (3) in » capture by He, aneven Z nucleus. Experimentally this case poses a difficult problem, that of determining the counting efficiency with which the inverse B decay electrons are detected. The maximum electron energy in the H-He? transition is only 53 kev. The long half life (12.4 years) makes it possible to detect the inverse p decay away from the , beam back- ground, but also serves to reduce the counting rate. On the theoretical side there are a number of uncertainties in the predicted capture rate, none of which appear to preclude about 7 five percent accuracy of our results however. These are the mean square radius (and to a lesser extent the shape) of the charge distri- bution of the A * 3 bound states, the wave func:ions of the A * 3 ground~ states (which may have D state admixtures due to the tensor force as in the case of the deuteron), meson exchange effects and the HFS effect. Information on the charge radius of the A ® 3 system is avail- able from the coulomb energy difference between Hw? and He? which is 0.76 Mev and can be completely ascribed to the coulomb energy. This gives R = 1.6/4, «© 2.24 fermis for a uniform spherical distribution. Werntz (28) gives an extensive discugsion on the effect of various shapes and sizes of charge distribution. Here we will content ourselves with using the above as a central value for the root mean equare radius to determine the length parameter in harmonic oscillator wave func- tione with which we will evaluate radial matrix elements. The He’, H? groundatates are T=1/2, J # 1/2" mirror states 22 24 for which the only compatible assignments are 22, Py 12° P and 24) 1/2’ 1/2 1/2" Verde (27) summarizes arguments which conclude that 2 tg a groundstate wavefunction which is 96% (symmetric in space S112 coordinates, totally antisymmetric in spin and isotopic spin coordin- ates) plus 4% 245, can explain the groundstate magnetic moments 1/2 and (by the exclusion of the P states) the anomalously small deuteron neutron and proton capture cross sections to these states. We will use this assignment. A difference between our treatment and that of 72 Werntsz will be the fact that cross terms between these S and D com- ponents do occur in the capture rate, so that the unknown phase of the D state admixture introduces an uncertainty in the capture rate (which amounts to only one percent however). Meson exchange effects are presumed to explain three features of the A = 3 system; the discrepancy of 0.27 which still remains in the ; . 24 . . magnetic moments even with a 4% D admixture (which corrects 1/2 the sum of the magnetic moments of H? and He? but not the individual moments), the neutron capture cross section (which would vanish for a pure ae (space symmetric) state, and cannot be explained by an electric quadrupole matrix element to the a ee component of the groundstate), and also the ft value for the superallowed HO to He? decay (29) which deviates by 2% from that for neutron decay. Primakoff (8) argues that meson exchange currents will make contri- butions of the order of 1% to the capture rate, but this remains to be convincingly demonstrated. We will ignore these effects here. The HFS effect exists for ,. capture by He? due to a mechaniam indirectly related to that discussed by Primakoff (3). Consider a simple model. Let t ( 4 4 ) be a neutron (proton, » meson) with spin up. Then for capture in the F = 0 state the configuration which is antisymmetric for like particles is (J f t + ) and for capture in the F = 1 configurationitis (J $??). Fora V-A inieraction neglect » capture by 4 proton with parallel spin. Then, if A ia the apin-flip amplitude for , capture by a proton with opposite spin, and B is the non-spin-flip amplitude we have symbolically (Tt dere Ald dev (F*h¢)= BCE tLe v . and there results a hyperfine effect if B? Es A” . For a pure V-A interaction the spin-flip and non-spin-flip rates are equal, and there would be no HFS effect. For a renormal- ized V-A interaction we expect no more than, say, twenty percent difference between the F * 0 and F =1 rates. In addition, for He? there does not exist a mechaniam to disturb the statistical population of the HFS levels in a time comparable to the lifetime of the , meson exchange in close collisions between neutral (j;- p ) atoms and protons. This mechanism is not important here because the coulomb barrier 3 . ; ; 3 1 prevente (,.-He ) atoms making close collisions with He” nuclei, the overlap of the nuclear wave functions is smali and the exchange ampli- tude reduced. Wave Functions and Matrix Elements Here we can borrow the notation and techniques used in the shell model discussion of gl? From equation (49d) we have the wave 74 tunction for a state 75 [3] (i.e., a state symmetric in space, antisymmetric in spin and isotopic spin), 22 Si/2 Ie’; TLz0ssJs>¢ <7; ta; «| a; TS; 38> x [* tz *\ [ Aa *) [5 0 *) x tir/ 42 Is/\s ol 3 ls? ta x 12 18); (66) wi~ where T#S2#J 21/2 . 3 3 T, 8 +1/2 for He’, -1/2 for H and the FPC are - > fort20, 4 ®1 and + p> fortsl, A420, Ag itis written this wave function implies a sapara function but in fact such a restriction is not necessary. For the Diya state, since it enters only with a weight of 4%, ii suffices to take a simple wave function which is separable. From equation(49c) we have for the (de ) configuration, which is the simplest one for this assign- ment, 24 Die lds’; TLS IK> 3 [s*; tox zla?, + (+) pi (: te i 0 iN b, oly a $s ¢t 2 0 4 418/\8 Lig qT, T Ly L (67) 75 where T 21/2, 5823/2, Ls2, J 21/2 and 82] for S = 3/2 t = 9 for | 5°; to su, to be antisymmetric. Now we want matrix elements of an operator # = D. H: Me 19h ,- where H: = ok dt, ste ©, in our former notation. These are 2 Kez, [yf] 22g s 1/2 1/2 <8"; axl} 8; T's! Dy 52, ta « |} 53; TS 3y 54 To *" (s 1 pe K 4 ¢ 2/\r afr ds3 dsex by. (* K 4 CG lH, lls) <s lle Ile? - boo Ty Tv tp (68) J K £ Ij, which will go out in the sum over final states and average over initial Here we've omitted a term (-) 7-42 (J 513 (-* Je J og' states. Primed quantities refer to the final state, unprimed to the initial state. Again, the reduced matrix element Ls t) He | > does not require a separable space wave function although we will use one to evaluate it. 24 Dip2 figurations (where now we do need to use separable space wave func- For matrix elements between the “S19 and the con- tions) 76 C44p_ |H P28, 27» X02; 2a/}e3; 10; 3 (2 $s Sis)? (-yttT+445'41 [ “1 me 1 q (s K 4 so J T Lity\3 t 3/\3 a2 §// S's K £ j, Call Hullasz> <o ll He lla? (69) : 22 24 (and a similar expression for < Siy2 [Hf Diy) also we need 24 24 hp a ma redag atte < Dy, _|4 Di )9? . Go tS Lety? (-)? $h+s 414 ° \K £ j/ <ya2lHellirer Call Heh a? . (70) The matrix elements for s-s transitions between the Dr configura tions vanish because this would be at 0 tot = 0 transition. The Rate . at From our results for , capture in F we have the capture rate (Eq. 45) for a statistical population of the HFS levels r « G? (Ze)? a ve (2A) (71) 2 1* 77 where (see Eq. 46a et seg.) 2 A ® AA®+BB* , 2 ,2 2 - with AA*® = GUf - 2G) Re(f ¥-q*) and = BB* = G* g“+ (G7? -26,G,) Iv-g|7 + 2(G, -G, )g, Re(O-g tm) A 22 2 2 +2G,G Im (v-gxq*) +g. t +Giq In the 1/2" to 1/2" transition for » capture in He? to the groundstate of w? the nonvanishing matrix elements are (Eq.53-40) Fo G G 10’ ~12’ T u Q5 .T We will evaluate the large matrix Ol 1 T and Q 10° “ll 2il il’ elements (f and g) between states including a D state admixture a2 22 24 1-€ Si/2 +E Dire where Pag = 0.04 and i€| = 0.20. We will just include the contri- bution to the momentum type matrix elements (t and q) from the domin« ant configuration ee so that to this approximation we only keep Ton and Qoio° Using equations 58, 69 and 70 we obtain Foe c QQ - 6%) <ellet" 4. |] 3s) + é* La lle t? , || a? (72a) Gio? 5 (l-<%) Ks len. a> + e* Kalle**’E-0 ll a? (7 2b) 3 Gs» = if € l1-€? Cellet" >, | ay - ia calle 5, ] a> (72c) Quo Fy <ell Lz 0 Leill s> (72d) and 78 Ton 2 fp <8 £20 Lezille>. (72e) The terms entering the rate (2)) are (Eq. 53 et seg.) fz ar? (73a) ce] Pa = 2(3 Ge + 3 Ge") (73b) Re(f v-q*) ® 2 Re(F. Qs) (73c) rir, (734) a ; O10 (73e) Rev-g t® = - Zz Re (Gig - {2 G,>) Ty (73f) lel? = 2(G,9-/% G,,)° (738) We include the square of small terms (Eq. 73d,e) in spite of the in- consistency of doing so merely for the sake of completeness. Terms of first order in the nucleon momentum (Eq. 73c,f) which were omitted previously are also included. The term [¥-g| 4 (Eq. 73g) was treated incorrectly by Wernts (28) who averaged over neutrino direc- tions assuming only £ 20 matrix elements and lost the cross term between Gio and G, This term is linear in the D state amplitude 2 whose sign is unknown and this constitutes an extra source of uncer- tainty in the calculation of the rate. In Appendix VUI we evaluate the reduced matrix elements required using the simplest space wave functions with a length 79 parameter b= 0.80, 1.0 and 1.2 fermis (compared to the usual cri- terion b = 0.468 R = 1.05 derived from the coulomb energy). In addition we present the rates (2A ) and (2A p) (see Eq. 62) calculated with the various hypotheses on the coupling constants. Conclusions The capture rate (Eq. 45), including effects of finite nuclear mass (3), and a 5% reduction due to coupling constant form factors (10) is Fie 250(2a) eee”! (74) With a central value of 2A = 5, we get a capture rate 1250 sec”, An examination of the terms entering the rate indicates that momentum type matrix elements reduce the rate by about 2%. Inclu- sion of the D state reduces the rate (due to the reduced Gamow-Teller matrix element between S = 3/2 states) by about 6% from that calculated with a pure Sh configuration. The term linear in the D state amp- litude amounts to about 1% of the total rate, and must constitute a source of error. We can assign a 5% uncertainty to the results due to the uncertainty in the nuclear radius. If the nuclear radius can be determined more precisely, and if the experiment can be done with one or two percent accuracy, it might then be interesting to do the radiative corrections to the capture rate and to make a more careful study of meson exchange effects. 80 APPENDICES $1 APPENDIX I. Matrix Elements for Hole-Particle States In che shell model, the groundstate of ol ia a double closed shell and the odd parity states of ol and nié are considered to be states of one hole in the closed shell plus one particle outside the closed shell. Here we derive the required matrix elements for trans- itions between these hole-particle states and between a hole-particle state and the groundstate. We also present the matrix elements of two particle operators between hole-particle states. Denote a state of n fermions by + fala, &, (1.1) where g, is the vacuum state, and the creation-destruction operators ( yes § satiaf 16 anticommutation relations (a, ,a + Jur » The a UA a / aoe. product runs over the states A occupied by the n fermions. Denote a closed shell by the state Y= TT ay &, (1. 2) AEK where A runs over all the states within the closed shell, which are numbered up to K. Then a state with one hole in a closed shell is Yy 2 ai ¥, where m< K (I. 2a) m and a state with one particle outside a closed shell is BZ Y = ar Y, where m>K. (i. 2b) da Now introduce creation-destruction operators Bos and b. for aoles & 38 + + oe band by the correspondence a ->b | and a‘—b for m< &. States Tt ¥e2 m wad with one particle outside a closed shell and one hole within are + + : ~ = 3 of eo y, v a bo Y, with n VHX Pm. {X, 2c) Let us evaluate matrix elements for various physical operators in the second quantized representation. A single particle operator, with creation-destruction operators antisymametrized, is QO = 2. <x) 10x) | Usta> a/2 (ata, - sat) (1. 3) r where r,s run over all states, Ii we rnake the artificial separation into holes and particlee, the operater becomes O = 0, + O,+ 0, wherg BS = 2 SP ter Ota 1c? 1/2 (aba, -a,aty (1. 3a) S37 O 2 = Ae eceS4, 2) Ole) [Ge 1/2 (v, bE = v2 o,) = 2 CW er (xx) 18 (x) IP, (2)> * (arbe + bias) s<« 2 which are, respectively, particle-particle, hole-hole and particle+ hole or hole-particle transitions. ‘’e will want the roatrix element of such a single particle operator between states of one particle and one hole. The matrix elernent of 0, vanishes between suck states. The 3 remaining two result in matrix elements for particle-particle and hole-hole transitions, so that, if a,c denote particle states and b,d hole states, the matrix element is CY ab Olea? . Oba CY(0)| OC) [Rx (I. 4) Sag Lgl} 0)] 4.60) - Had we not antisymmetrized the creation-destruction operators in O there would have been diagonal matrix elements of the form Sac d ba - Cu (al O (x) | O.¢x) > which would have contributed to the energy of the state Y ob and which we could have removed by lumping into the central "single particle-closed shell" interaction. We will employ this shortcut to eliminate similar diagonal matrix elements of two particle operators, without doing all the extra work of antisymmetrizing the operator. Then, using this shortcut, the two particle operator is H z 1/2 2 Hf ay a; aye ijk ijkl (I. 5) a where HE sree 2 SH Yiceyl soo! atay Uy cn? 2 + since ef 2 H jitk 12 Here ijké@ run 21° over all states, and we have yet to transfer to hole-particle notation. A matrix element taken between states of cue hole and one particle is CY gom| J | Vear?l5 W esnce Stl Pata laf af aay) abet \¥.> e where (m, r) are particle states, (n, s}) hole states, and ijk 2 refer to both. There are sixteen possible combinations, ijkl greater or lesa than %, of which the only ones with non-vanishing matrix ele- ments are + + a, bya, by , ar bibs a, b 133 7a, bp > by a, 7 Oey: and b, i>; PL All told, the matrix element of these operators is CV ma | H1Y 28? = + nes va 7s; ar * Sas Amr * bar Nae ‘ The single particle (single hole) parts of the matrix element Sans \mr 'Smr/ ne Come have been eliminated by using ordered operators in d€ or can now be lumped into the central potential. In any case, we will drop these terms from the two particle matrix element and retain cy ran | | Y rs = + mejen 7H mejar (I. 6) which is just an exchange minus a direct matrix element. One other type of matrix element arises, that of a one particle operator between the closed shell and a one hole, one particle state. Now only the © 3 Part of O has a non-vanishing matrix element, and the groundetate transition matrix element is 85 CH OLY par ECLIKY (1G 001 U0? va ah ot | 407 = CY (x)|O(x)|U,()? - (1.7) There remains only the problem of coupling hole and particle states -eigenstates of I," ws Joy Jug (as well as of isotopic spin) - to form eigenstates of the total angular momentum y¢ and J, where J 2 J, - Ju This problem can be treated by using the time reversed state (19), for which J->-J, for the hole state. The time reversed, or now the hole state, is (jem) so, = (7 uli. -m). (I. 8) t+t x The isotopic spin of the hole state also introduces a factor (-) ~. Finally, let us write down the complete matrix elements, between hole-particle states which are eigenstates of J, Jo T and TS We use jj coupling and consistently denote particle states by the sub- scripts (m,r) and hole states by the subscripts (n,s). Subscripts f and i refer to final and initial states. a) For one particle operators 86 Cin =ma)H(js =m )4( tate )#(ty te) CY sanlOlY 42 * () x {Dna Y 29109 [Beta = 3 UP (9 960) [ato >| (1.9) b) For two particle operators (Jj, -ma)+(js ~m,)+(t. ~te )+(t, +ta,) Ky ltle et &) . +P 0) 4(2| | U(2y Gaay> oe My, Jn jm fi\ sO f¢ V4? Us > i at f pew N \ 2)| Hy» 1A k2) 0)> | (1,10) and c) For matrix elements in groundstate transitions Gs, bad m.)+(t., ~ti,) Ce al Ol¥e> = () x 1 ¢alOealdlaeo (1.12) ( -m, m. Jn jm 87 APPENDIX Il. Evaluation of the Talmi Integrals. The reduced matrix elements of the two body interactions are readily evaluated in the model where we use harmonic oscillator wave functions and a Yukawa shaped potential. In harmonic oscillator units “1/2 (where the unit of length is b = (2Mw) = 1) with h #21, the nor- malized radial functions Rg are 2 -x c72 (x"- 3/2) e 1/2 ox 7/2 R, * (8/3/R ) (x) @ (11. 1) 2 -x /2 Ry ? (8/3Jr ile R, = ( $ a3 )/? (x) and the Yukawa potential for the two body interaction is “HY Ln . Viredee fen, t 2. vba ve (2 xe GD, with Yi2 =x-y. (II. 2) The reduced matrix elements of the two body interaction defined by CY (mgt) Fa(may2) |v 2) [Pmt Po(mizy> = m @ £ K e Ma fs Mp 4 K fs i. Keen (2) I vylegay 4 (2) (11. 3) 88 are KL) 4,(2) lve Jey Yas 2 /E Be A B « | c D » ax f (: 0 alk 0 a x k. K fa fs K Lp z0k JT (i.4) where A ze kre fxtax y2ay R(x) Rolo) vq (oy) Rev) Bylo) ax dy Ryle) RX) Rely) Roly) Vera) PCED » (11. 5) The change of variables x eR 41/27 7 *R-1/27 with the substitutions dx d*y = ap ay <e zp 4 z 1? *Rrt with t * cos ZR, y” = r+ r «-Rrt x ¥ s pn? 1/4 2" xy allows us to write (30) + fr? an var at (2,608 60} (RO) Ryo} Ved BGA (11. 5") 89 The use of harmonic oscillator radial functions makes it possible to do the integrals over R and t directly leaving integrals over r of the type oo 2 2 -(r 7/2 + yr) 1/2 ,. | dre won e I, where ° a “x2 2ntl the integrals 1 s / dx e (x-y) must be ta done numerically. The required integrals for the two values of the nuclear radius used are Th be 1.56f Mebantl10 b2l.86f mab= 1,30 1) 0.17274 0.11388 1, 0. 17809 0. 10583 t 0.42989 0. 23361 1, 1. 6889 0.85055 and the reduced matrix elements required are = 08, = 1K = olf, 224, 21K = ot, =24,= 1K = alt, O04, 21|[K = le, 04,2 1(K = I(t, 204, =1|/K = I]t, 204, 21K = 1fe = 24, =1||K = 3]f, 90 b=1.56 fermis b=1. 86 fermis +0. +0. +0 -0. +0. 1011 09751 - 3301 2899 3741 .2124 . 1003 3151 +0, +0, +0. -0 +0. +0. -0 +0. 06256 06597 2505 - 2158 3307 1545 .907014 2450 91 APPENDIX Ill. Reduced Matrix Elements for |: Capture in olf There are two types of radial matrix elements arising in the p Capture calculation in of, jee) a) I,, = [ R, (x) Jp (vr) R,(r) r? dr and b) $ ito 2 ss a Jy (vr) Re (r) (2 + 2] R(x) rar bM o dr co Pie Le - a f by (vr) R* (r) (s_ . 4] Rr) rar. fe) dr r Here b = 1.56 fermis and 1. 86 fermis nucleon mass. bM # 7. 24 and 8. 64 At] M v ® neutrino momentum in units of b 0.700 for b = 1.56 fermis 0.835 for b = 1. 86 fermis These integrals are easily evaluated using a series expansion for the spherical Bessel functions. Even with the large values of v , adequate results can be got with two or three terms in such an expansion. Following is a table of these integrals. 92 b = 1.56 fermis b = 1.86 fermis om 181 194 1,, 310 347 1,3 0159 0254 £ n00 .116 . 9888 £ io . 0306 0278 £ 36 . 00224 00202 £ 592 .0210 . 0260 $ 942 . 000872 00149 £ 502 163 120 - . 0184 . 0208 922 APPENDIX III. 93 The reduced matrix elements required, Ct fe a> and <e_l AL le > are (all these are times a factor -i) <olili> <afilid <alsli> <oli,olli> Coli.2l> Kalli,oli> Calli,2ll> <allz,2lji> Call3,2[> Kall 3.4] > b 23.56 fermis 544 - 588 . 0565 ~116 0582 2193 » 9232 0243 »9126 . 00356 . 86 fermis - 582 . 658 - 0900 . 0888 . 0659 0758 . 0280 0310 .0145 . 00619 a) whe APPENDIX IV TABLE 1 Structure of the Atz1l6 States on the Shell Model. The results presented are for x = 2, b = 1.56 fermis and Vo = 30, 40, 50 Mev. For comparison, the results of Elliott and Flowers (EF) as well as the (jj) limit are presented. The experi- mental (Ex) and calculated (E,) energies are given fitted at vi 40 Mev with Rw 213.5 Mev. “f fe Le*+ Lg'+ €f°- Aq 6b 625 °+ Ott ‘+ L9z°- as $1 9=Ka Pel 16 "+ 7°9"+ bbI °° oF _€=£ 0=L bE 'e 9€ + C68 "+ €tl*- 0€ =9A T cf OF + 28 °+ €1°- Late Zot Aq ga 9 96% "+ ean t+ TTE “+ 68t°- EPO *- os 88 "8=Xq $O°L SSE ‘+ GLG "+ €Lz + 1gtt- Sec" OF _@=£ OFL 02°8 122 "+ 626° O22 "+ 2LI- LoL? Of =A . ff gt t- 56 t+ QC "+ aa o°2! 96S °- vant gz *- 0S L°OLExXg g°t LOv"- e738 + 8Sz°- OF _€=f£ 1=1 put OIR'- Clg "+ oz? *- CE=OA = rt SO! [LE ‘+ VoL t+ eet CEP t+ 602 *- og BOT=KY 57a! P82 “+ S$ g'+ LLO*~ 1980+ Suits OF 27°F ti gol OZ *+ $6 "+ oz! *- eZ '+ QE *- GE =A "1 et ga "+ Zt "+ Tet: 910s 86" Ag €°Z1 $92 “+ P21 + QTE ‘+ Let + 068° 0S g*OI=*F G°2Z1 12z "+ OBC "+ Zee "+ 890 °+ S68 OF _U=f 121 Lvl 861 “+ CHO + SSI *+ S10 ‘+ 996° O£=9A 7 ff 0S0 "+ S66" ec + FI ze1‘- 266° os S‘OT=*a PE] gli*- £66" OF Of T= $21 001 *- S66" CE =A Ls ope Ce Oe Oe hey Ba OM pr Oat Mog 2 Ray eee Fay "LT 21Qel. *l XIGNEddV The matrix elements for 96 . capture in ote below, for nuclear parameters b = 1.56 fermis, x = 2 and Vc are presented 30, 40, 50 Mev. For comparison results obtained using the wave func- P g tions of Elliott and Flowers (EF) are presented, as well as those obtained in the (jj) limit. Table 2. Vc=30 40 50 EF jj of .07 G, -. i192 -. 182 -. 172 -. 286 ~. 256 Ti, +130 4.125 +.120 +. 196 +. 164 of F -.0388 -.0302 -.0474 +.080 +, 148 G, 4.117 +.115 +.105 +. 199 +, 209 Qi 4.00392 +.00116 +.90705 -.0319 -. 0546 Qi> -.69740 +-.00720 -.00674 -.00916 -.0122 of .3” F, «0111 -.00971 -.00865 -.011} -.0147 G, ~.00938 -.00737 -.00587 -.0172 -.0170 Q, +#.00296 +.090259 +.00232 +.00294 +.00390 Q,, +.90046 +.00040 +.00036 +.00046 +.00061 ot.2” G, 4.392 +. 257 +.197 +.324 +.430 G, #.900i2 +.90100 ~-.00203 +.90629 +.00450 Q,, -.00950 -.9108 ~.0:21 -.99557 -.00531 T -.00316 -.00398 -.00399 -.0012 +.00929 vO +. 383 +. 325 +. 246 +.411 +. 546 R .789 .798 899 +. 788 +.788 APPENDIX 4. Table 3. A * Reiveg A lin 9+ axg* Re veg t® V .=30 90537 2°93: »2i8 ~91693 ~ 090067 -. 961909 =.0ii6 -. 9280 7 49 . 90349 ~529 227 973% 00074 -. 00053 #5 3e ~. 0265 The individual terms in the rate for 50 . 30727 533 223 0145 -O010i ~,. 00144 ~. 9136 ~. 0245 I capture inO . . 00565 » 00278 Table 4. The rate am“ on the various hypotheses. Tase } V =39 c » 94} 1. 69 l.al 06) i. 47 1.03 0241 057} i. 28 2. i ro i2 63 05 94 - 40 ij -0672 651 ~ 24) ~ 0270 . 00917 09973 » 0384 .0392 98 1g APPENDIX V. Reduced Matrix Flements for u Capture in F i Fy We refer te the general formulze for matrix slermenta in the A= 19 system. The reduced matrix elements for the spin variables ave a) K=0 a ae Kif2lle = 9 || a/2> = 2 b) K=1 sheet. Casals st asade Se. To calculate the reduced matrix elements in the space variables we use the harrnonic oscillator wave functions o* —s~ Y, = e/a yl? (x? ~ syaye * ve ox” (vet) s & 1/2 (2 2 a y, = (2/5 Sik ) x @ Y5 and also the expansion fa an me eee a (-Y 4m ftwey VAR) OY. Ore 2) Here we understand summation over reneated indices, and we have 2 £/2 x = efb verhs b= 1.7 fermis 8: 109 ev w= 9,850, Then, for example, Lale te: | a> = (* MIT) Calle Mg lle? ty = 3) 99 : eis: defines reduced matrix elements of e ® with £ =0, 2and 4, There are five such reduced matrix elements required, which we evaluate for § in the & direction. They are a) <dfle"** ceolla> = 0.642 v) <a let?’ cezlla> = 0.347 ce) <d le"** gaa lla> = 0.034 a) <e let ceo lls > = 0.652 e) de |le"®* cee a> = -.580. We also need reduced matrix elements of rnomentum type matrix elements <r fet® Berm |e > and to apply the general expressions derived for these matrix elements we need to write this as a reduced matrix element depending on (1,m,) m: Mz hie £ Me de and (i,m) through ( ] . In order to apply the gradient formula write [tame P = Py Ya, then wig: u l/2, ee "| £m,> ea (anny! ju (vx) Fo, x [ =) Po) 8) (oe an R Lo t,|K/\L fy “) yw K (Vv -4) where we gum over L, K, Ww and take § in the 2 direction. The _ matrix element then is f. 7 \ 1/2 wiser . i Re va feeb <tlet? ” Peamli> = bon (ci) (4) x (" 9) [0 ms L £iK/ L {. [ads (5, x) Zu VC} OMe Be} eves) and using the gradient formula (19) this becomes na Lat ye ( Et 0 = oi 47 )°' bM + 4 & Wed, °) (om K/ \l & f 24.43, 24h) f i i ‘ (v-5! Here we have introduced the notation (+) po 2 f. & = x dx j, (vx) ¥ [g i a. and t,t f f * dx x ‘ 0 f£ i fo’ dx x ‘ To fit the pattern of our general formulae we need i f 1 £4) 1/2_ (+) f V2 . (Sxan/ Put PK sat ( bh) , Atk +) (° cn Pele) aE fio te « iL 4, K A. lik f @oK 4j (V-6) [™ My | a) 4. va f, 1 LILs. Then we can define reduced mornentum type matrix elements by oe, m. 1¥4 ™, KO Celertt penal = (BM IP) MOU) Ge old f 4 t (v-7) There are ten such elements, not all of which are significantly large. The integrals required are E ae =. ae ¢ aig 7 +374 3107 ~. 307 a2 = 7 624 $ a2 27012 a2 77-074 $ 552 = 4-022 ae = -.0031 and the reduced matrix elements are C2 | Ka <2 | £2 C2 || C2 <2 | y.| Qo | Ko | Co | L =9 fi L 22 L =2 £33 L =4 Ll =4 L =0 £ =2 L x2 b=: |2> L=i 2? L=3 |2> La3llz> L-s|2> Las [2 L =i flop L =iljz> L=3lle> i92 2 -.9620 . 143 . 0074 . 0160 . 0144 .90025 ~60153 20665 0037 ~ 9069 mA bu “DRPENDIX Vi. Structure of the Required States for A = 19 In the following table we give the structure of the states in F 19 . . and QO © which enter our work. The states are given for three values of the central two particle (Yukawa with a Rosenfeld exchange mixture) interaction (30, 40 and 50 Mev), and refer to a standard convention (17) for relative phases. The notation requires a bit of explanation. For example, a) a? (3](10)*p means 3(d) particles ina state totally sym- metric in space, belonging to the representation (10) of the subgroup (R 5) of the permutation group, with total spin 1/2 and orbital angular momentum 2. 2 2 . ' b) d's 13 “S means a parent state of two (d) particles ina singlet isotopic spin state, triplet spin state, with zero orbital angular momentum plus an (s) particle LS coupled to thie parent state to give the relevant final state. The ds” and Pi states have a similar designation. 104 PPS* OFS" TIS’ S. {e] ElEe- Ooze ° LS6Z° 6£0°- OfO°- 220°- MO iIE TO0°- OGCc’ 2uo°rr re ey Gq €! SP £00° 000" 900° Sé0° col’ F60° vin $S0° GSO * SPO" “ly EE ay ott LIt0° 020° O9Z0° 620° O80°- E80" LEZ*- OVE HSE" GSI- OBI LIz- cl +, S9P° CEFF Lee ELE OBE BBE - O80°- COl’- Q92I°- dy €F 460° O2@T* FFP - dz oii De £! 9S0°- 090°- F90°- SOR = O8E°- PHEE'- Mz {€ G20°- O£0°- b6PO'- Ge ¢t 9L8° 098° S$8 ° £0g° GIS" BIS’ Sz l¢ S4 GPS °- OfS*- BER '- Sz €!1 SP 89T°- O61 °- BOZ*- Tli’= O@l°- SET*- an x Ce2@*- OG2"- LBZ *- E€6T° O22° GS2° EHIO*~ OFTO'- 9EIO’- dn 8io°- CEO" 6£0"- dz (i!) [i] On at S8E° GOLF’ SzP- Jz G10°*- C&G" 6€0°- Ch 681° O02 *- 907°- Z9H'- OB - O1G"- dy, €60°> Olt 927° da OBLO°" OLO°- CPSC 610° oot: 92T° dz (12) a, 892° OL" O9L° 660° OFO* $20" Cy (01) [12] By Zi€r- GE*- BIE°- S_ (og) Gy (01) [8] ¢P 0S OF of 0S OF Of OS OF of OS OP of 2/S=f 2/1 z/e=f 2 /ERL Z/t=L 0 2/€FL @it=fL 2 /tHL 9383S 105 APPENDIX VII. Numerical Results for F 19 In the following table are presented the matrix elements for » capture by rl, These are evaluated with the shell model wave functions of Elliott and Flowers (17) given in Appendix VI, as a func- tion of the two nucleon interaction V. Also, for comparison, matrix elements evaluated on the single particle (SP) model are presented. Recall that a factor (3/ J2) x 107% has been removed from all these. 106 TABLE I Term Vv =30 v=40 V=50 SP 1/21/27-3; 21/2" Fl 1133 =. 145 ~ ids “17.7 G7 444 -. 802 ~ i. 086 $17.7 G, +.198 +. 146 #. 125 Qoyy 10899 +.0054 4.0634 + 1.81 Tp}, +-00512 4.0378 +. 0648 - 1.81 T,,, +:00447 +.00338 +. 00284 Ty) 7-178 -. 198 -. 243 OQ, 716 -.741 -.708 1/21/2*=3/23/2" F, +.609 +. 601 +. 602 - 3.20 Gy -2-687 -2. 661 2.577 Gj, 41-505 +1.532 -#1,.576 ~ 5.05 G,, +.805 +.877 +.949 - 3.9) Q,,, +:00536 +.00516 = -+.00521 + 0199 Q55, ~ 0114 =. 0101 -.0094i + .0376 Ty), #261 +. 260 +. 253 T,), #:0329 9 +.0327 +, 0331 + .9316 T 4, +618 +. 639 +. 669 Q4, 7950 -.997 - 1.038 1/21/2°-3/25/2* F, +190 +. 253 +. 289 + 3.17 G, =. 113 -. 256 =. 346 - 2.60 G,, -. 649 -. 820 -.932 ~ 4.12 O34 . - - Q,,, +-00353 +.90379 +.00404 =. 0199 Q,4. 7100427 = -.90492 -,00549 =. 0374 T #0115 #.01295 -+..01357 233 ioe) ~ Table [ (continued) Term V=39 V=49 V=50 oP 1/21'2-3/25/ 2* (continued) Ya3300 7 " " 333 . 60429 -.900972 -.000757 2533 +. 599027 +,000113 +.100208 Total f 1. 7/35 1.870 1.992 g° (99. 61 103.6 104.8 e- gl" 54. 66 56.05 55.98 ° £793 .901 i. 229 q’ 4,643 5.077 5.317 Re fv" q* +.0449 . 0468 . 0474 Im +: gxq* 12.56 i3. 84 14.21 Rev-g t* -5.97 -6.14 -6.38 i(gxJ)- g* 33.93 28.50 2:.81 i(¥xq)xJ+ (vxq)* £339 35 45 ~i(gxJ)-o (¥: g)* -7.65 -5. 60 ~2.95 -(gxJ)- (vxq)* 13.32 13.43 13. 64 ~i(gxJ): v t* .552 .756 1.04 (v+ g vxJ)-(%xq)* -4.3) -4.17 ~4.05 -(¥xq) xJ-0 -.76 ~. 86 -.98 f J: g* 3.48 3.97 4.46 -f£ I'v v3 g* -3.76 -3.92 ~4.04 +i f J+ (¥xq)* -.93 -.98 -. 100 -f£J°¥ +.58 +. 60 +.62 -v-q J°g* -.118 -.979 -. 066 veg dv (v+g)* +. 068 +,056 +, 046 “i ¥'q J+ (vxq)* +. 041 +. 046 +. 043 vq J-¥ tm ~.017 -.019 ~.02i 108 TABLE 2 The coupling constants in the yp capture Hamiltonian are listed. Cases 1, 2, 3 assume g = +8 By » -8 Ba 0 and a conserved vector current. * ok Cases 4, 5, 6 assume Sp = +8 Bar > 8 Bae 0 and no conserved vector current. l 2 3 4 5 6 Gy 1.9053 1.953 1.053 1.053 1.953 1.053 G, ~i.480 -1.489 -1.489 ~1. 283 -1, 283 ~1. 283 ~ .795 + .340 - . 183 - .510 + .586 + .012 i093 TABLE 3 The rates 2» and AD for the various hypotheses | to 6 calculated with the matrix elements of Table | are given below: V..230 V=40 V=50 2r Case | 111.6 115.2 118.2 2 275.6 285.5 286.9 3 178.6 184.90 187.2 4 91.4 94.1 96.5 5 266. 1 278.8 276. } 6 158.2 162.6 165.3 ry Case | -46. 14 -27.92 -19. 64 Fa -62. 24 -26.71 +32. 64 3 -45. Zi ~18.08 418.77 4 -27. 98 - 6.74 415.33 5 30.58 + .566 +42. 48 6 -2€.i1 - 3.29 +28. 30 119 APPENDIX VIII. The reduced matrix elements required in ,» capture by He”, evaluated with the harmonic oscillator wave functions ~ 2/2 Ye = 4 e if Yo (tw - ~ ele Yo = fa ye UI? yt 3Jn “ where zw=2r/b ; b=z9.8, !.0 and !1.2 fermis are, for s = 100 Mev, Table 1 b=0.8 1.0 i.2 <slle'®'* c=o |s> 960 938 920 <afle® * c=o lla> 908 800 . 804 <sfjie° © L=2 a> ~. 102 -. 154 -.2i7 <dlle* * £=2 la 097 wer 201 <a £20 L=1 |s> -.043 -. 042 -. 041 The matrix elements entering the rate, for a D state amplitude E= #0. 20 are Table 2 b=0.8 1.0 1.2 2 F, .958 .934 2915 2G) .934 911 . 894 2 A154 -.943 -.042 ~. 041 2 Th), -.043 ~. 042 -.041 G> (é = +. 20) -.012 -.018 -.025 +.009 +,014 +.020 G,(é= -. 20) ‘11 Here we present the individual terms entering the rates (2A) and (2a,,). For the terms in (2A) we also present the results for € =0, b=1.0fermis. In calculating (2a) we have taken € to be positive. Table 3 b=0. 8 1.0 1.2 € =0 b=1.0 (2A) 2 f 912 . 872 . 837 . 880 ge? 2.616 2.490 2.397 2.639 * 0006 . 0006 . 0006 . 0006 q° . 9006 . 9006 . 0006 . 9006 Re { %- q* .024 .023 . 022 .023 Re veg . 023 .022 02! .022 log)? €=t.20 .892 859 839 . 880 €z-.210 . 857 . 807 767 (24) i(gxJ)- o* ~2.616 -2.490 -2.397 -2. 639 ~i(gxJ)-v(v> g)* +.872 +. 839 +.799 +. 880 ~i(gxJ)*v t® .023 .922 .021 .923 {J+ g* -1. 342 -1.276 ~i.227 -1, 320 -f J: ¥(v> g)* 2775 . 737 . 708 »Th2 -f J-¥ t®* -. 921 -.020 -.019 ~.020 -v'q J’ g* +.034 +.033 +,932 +.034 vq dev (t+ g)* -.912 -.011 -. 011 -.O11 vig J+> t# -.0093 -.9003 ~.90093 -.0003 Li2 The rates (2X) and (2a pb? calculated with the coupling constants of Appendix VII are presented in table 4. We take € = +. 20, ‘Sable 4 b=0.8 i.0 tr2 bs1.0, € 20 aa Case | 5. 23 4.96 4.¢5 5.26 2 7.60 7. 24 o. 98 7.60 3 6.17 5.86 5.63 6.19 4 4.27 4.05 3.83 4.29 5 6.78 5, 49 6.26 &. 88 6 5.21 4.95 4.76 5.21 PA ry Case | 2.23 2.12 2.04 2.53 2 4.92 4.69 4.53 5.29 4 . 890 . 768 . 739 0997 5 2.65 2.54 2.46 2.99 Co Pan m” co _ Oo aw ew Ly > oo w 10, REPERENCES ainheas vairsa >) ater atitawvg fi y* S. Weinberg, Physical Review Letters (i960) , 575-578. [> H. Uberall, Nuovo Cimento (1957), 6, 533-540. H. Primakoff, Reviews of Modern Physics (1959), 31, 802-821, T. Godfrey, Ph. D. Thesis, Princeton University (1954), and Physical Review (.753), 92, 512-513. K. Ford and G. Flamand, Physical Review (i960), 116, i59i- 1596. H. Uberall, Physical Review (1959), 116, 282-294. F. Ajzenberg and T. Lauritsen, Nuclear Physics (1959), Lt, 250-258. A. Fujii and H. Primakoff, Nuovo Cimento (1959), 12, 326-355. R. Feynman and M. Gell-Mann, Physical Review (1958), 109, 193-198. M. Goldberger and S. 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