Resonance Broadening of the Sodium D-Lines in Absorption Citation Watanabe, Kenichi (1940) Resonance Broadening of the Sodium D-Lines in Absorption. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/tzfa-n362. https://resolver.caltech.edu/CaltechTHESIS:02142025-214936682 Abstract The resonance broadening of the sodium D-lines in absorption was determined by measuring the contour of the line wings directly. The vapor pressures of sodium ranged from 10 -3 to 75 mm Hg, with negligible amount of foreign gases. Over nearly the entire range, the half-intensity width varied linearly with the density of the absorbing atoms; i.e. Δν 1/2 /N = 0.78 x 10 -7 sec -1 cm 3 --the constant being about twice that predicted by Houston, but much less than the experimental result by Hughes and LLoyd. Above 5 mm pressure, the half-intensity width appeared to increase slightly faster than N. Somewhat in contradiction to theory, the relative width, Δν 1/2 , s /Δν 1/2 ,1' was 1.16. The natural width measured was, within experimental error, consistent with previous values. The inverse square contour for the absorption coefficient was satisfactory up to 5 mm Hg; beyond this pressure an inverse cube contour fitted better. No definite evidence of van der Waal force was observed at high pressures; likewise, no appreciable shifts and asymmetries. MgO windows were used and the temperature determination was made reliable. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Bowen, Ira Sprague Thesis Committee: Unknown, Unknown Defense Date: 1 January 1940 Record Number: CaltechTHESIS:02142025-214936682 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:02142025-214936682 DOI: 10.7907/tzfa-n362 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17004 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 21 Feb 2025 22:47 Last Modified: 21 Feb 2025 22:47 Thesis Files PDF - Final Version See Usage Policy. 25MB Repository Staff Only: item control page

RESONANCE BROADElH NG OF THE SODIUM D-LINES IN AB S ORPrrION Thesis by Kenichi Watanabe In partial fulfilment of the r equirements for the degree of Doct or of Philo s ophy CALIFORNIA INSTITUrrE OF TECHNOLOGY Pasadena 1940 CONTENTS ACKNOWLEDGEMENT I IN'rRODUCT ION 1. 2. :.".i . II Page I Purpose Definjtions Einstein's Coefficient and f-value TH EORY la. lb. 2. 3. 4. 5. /0 Natural Broadening (Classic~l) Natural Broad ening ( Q;uantum Mechanical) Doopler Broadening Lo;enti Collision Broadening Resonance Broadening Broadening due to van der Waal Force III EXPERIMENTAL EVIDENCE 38 IV APPARATUS AND EXPERI J\'fEN TAL PROCEDURE If! 1. 2. 3. 4. 5. 6. V METHOD OF CLLC ULATION 1. 2. 3. 4. VI Source Furnace and Absorption Cell Calibration of Thermocouple Optical System Photographic Photometry General Procedure Microphotometer Vapor Pressure f-values Calculation RESULTS 1. 2. 3. Theoretical Values Experimental Results Oonclus ion APPENDIX LI'r ERATURE 7') I ACKNOWLEDGEMENT The author wishes to express his gratitude to Dr. P. E. Lloyd for introducing him into this work, to Dr. I. S. Bowen for valuable suggestions and guid- ance, to Dr. W. V. Houston for some discussion of the theories, and to :rvr..r. C. Truesdell for taking s ome measurements. ABSTRACT The reso!lance broadenlng of the sodium D-lines in absorption was determ:! ned by measuring the contour of the line wings directly. The vapor pressures of sodium rang ed from 10- 3 to 75 mm Hg, with ne g li g ible amount of foreign gases. Over nee~rly the entire range, the half- intensity width varied linearly ~ith the density of the absorbing atoms; i. e. a))J_/N=0. 7&.lo- 7 sec- 1 cm 3 --the 2 constant being about twice that predicted by Houston, but much less than the experimental result by Hughes and Lloyd. Above 5 mm pressure, the half-intensity width .appeared to increase slightly faster than N. Somewhat in contradiction to theory, the relative width, AY1-_ s//J.Yl 2, was 1.16. , 2, 1 The natural width measured was, within experi- mental error, consistent with previous values. The inverse square contour for the absorption coefficient was satisfactory up to 5 mm Hg ; beyond this pressure an inverse cube contour fitted better. No definite evidence of van der Waal force was observed at high pressures; likewise, no appreciable shifts and asyrmnetries. MgO windows were used and the temperature determination w::>s made reliable. ...3 I 1. INTRODUCTION Purpose The pressure broadening of spectral lines has been the subject of a large number of theoretical and experimental investigations; however, the experimental work has been mainly confined to foreign gas pressure broadening while there has been very little reliable experimental work on resonance broadening -- broadening due to a homogeneous gas. Perhaps the only substantial and quite re- liable work on resonance broading so far was carried out by Hughes and 1loyd1 on potassium resonance lines. They were able to overcome a major experimental difficulty by employing magnesium-oxide windows. However, there was some uncertainty in their temperature determination and in the vapor pressure curve for potassium; also their photometry entailed considerable corrections. From these consider- ations, presumably, sodium would give a more reliaole result. In terms of sodium D-lines in absorption the nature of resonance broadening is investigated -- particularly for the dispersion contour of the lines and the quantitative relationship between the half-intensity width and the number of atoms per unit volume. The experimental values are com- pared with the several theories on resonance Droadening. Other types of broadening a re also taken into consideration where they are appropriate. Asymmetry and shift are tested as well. Aside from a cor~arison with the present theories on resonance broadening , this investi gation may have some value in astrophysics. 2. Definitions In the lit erat u r e on lin e br oad ening , one encoun- ters considerable confusion in the notations and definitions emplo yed by different authors. Here, for the following, some effort is made to systematize these elements. The frequency, y, and the wave-length, A., of a light train is defined in vacuum by A.Y=c, where A is measured in cm. and c is the velocity of li ght in vacuum; while the angular freauency, w, as defined by is frequently used. According to these notations we can write ( 2) The absorption coefficient, ct:...y, for a given frequency is defined in the usual manner by the equation of * Dr. L[inkowski suggests using yfor w/2-,r, in analo gy to tr ~h/21r • logarithmic decrement, ' '7.) (v where di~ is the decrease in the intensity iµ(per unit crosssectionJ when the radiation traverses a path-length dx of unit cross-section (say, one cm. 2 J. If ex~ is constant over the path we can write erg/cm. 2 , (4) where i 0 denotes the incident intensity of radiation and iP the intensity after traversing an absorbing path-length, x. Frequently this equation is written in another form, ( 5) which defines nK, the index of absorption. 11he intensity of absorption, Av, is defined for the frequency interval v and v f dl.J by and can be measured experimentally. For o(x<<-1, one may use the approximation, (7) The intensity distri ·b ution in a broadened spectral line is usually expressed by I(~). For the frequency inter- 6 val )/ and Y-t d)J the quantity I(JJ)dV is proportional to the intensity of radiation absorbed from a radiation of constant spectral distribution as it passes through an infinitely thin layer of absorbing atoms. In general I(y) is not proportional to the intensity of absorption AJJ, unless the latter is measured for a very thin absorb- ing layer. On the other hand, I(v) is identified as being proportional to I(JJ) oz o<.y Ol>' ; that is, ( 8) • The intensity distribution is usually normalized so that /r(~}dY::::. l; however, for an experimenter it is more 0 important to determine the constants in terms of Einstein's coefficients. This is done by taking the area under the absorption curve expressed in terms of A.). o<)I and )) ( or pe.,l and This operation will be exemplified in the following section. The half-intensity width, .aJJ.:2t , in terms of frequency unit, is defined by (9) • We may also use -Y and ML2. in the angular frequency unit and in t h e wave-length unit respectivel y . The ha lf-in- tens ity width is one of t he imp ort ant quantities to be measured in the presen t inve st i gat ion. Sometimes t h e use of half-intensity half-width separately for the l ong wa~e 1 and the short wav e component of a l i ne is convenj_ent for asymmetri c line contours. Two u s eful e quat ions of intens ity distribution are; I())) = const. e-conS t • (~.v) 2 ( 10 ) and ··· - const .- - - - ·· - ( ~ ))) 2 + ( 11) ( A J:i_/ 2 ) 2 ?. 3. Einstein's Coeffici ent and f-value It has been shown 2 that the absorption coefficient, qy, is related to Einstein ' s coefficient of absorption as follows: 1 di,v ci)' h~ , = - iµdx - c ( 12) where Ni and Nj are the numbers of atoms p er unit volume , in the lower a nd the upper quant um states respectively for t he transition i~ j , while Bij and Bji are the Einstein's coefficients of absorption a nd of inducP-d emisr-;ion resp ectively. With certain appropri a te assumptions, na me ly the constB.n cy of B a nd the ap pro x:i ma te const ancv of )I over the line, the Ei nste in 's coe fficient of absorption Bij c an be expres s ed for the tr a nsition i~j by B l• ,l• - C h.))Ni(1 jot,)la (~JJ) e -h))fkT) ( 13) a where the inte gration is over the en tire range of an absorption line. Since at moderate temperature and for visible light the exponential term is negligible, we have cm/gram B·J.J. -- (14) and Ni may be considered equal to N the total number of atoms/volume, provided nearly all the atoms are in the normal state. The oscillator strength or f-value is defined by La.denburg 3 in terms of Bij as follows: .,._ (15) where e is expressed in e. s. u. Hence, by comparing (14) and (15), we obtain • (16) In some cases the f-values have been calculated or determined experimentally. For a summary of the results refer to tables given by Korff and Breit. 4 In genera.1 5 . _ (17) where n =-1 for principal series of the alkalis and :f ji = ~(gj/g1 )fij in general. If we assume that the intensity distri bution is represented by only eq. (11) we may determine the consta nt by u s ing eq. (16). (8) we have By K(LlJJ ::-/21T ) d : =- 2 {18) )J Subs tituting olµ i n to (16) we obt a i n rea dily _mcK f . . == J l. rre2N Tf j ( AJJ½/ 2 ) d (L\ JJ ) (4Y) 2 -t ( ~½}2) 2 - mcK 2 7fe N (19) 2 2 K = ,re N f ( me ) -1 sec -1 cm -1 • (20) This value will be used in the calculation of resonance broadening. II THEORY Although this thesis is primarily concerned with the resonance broadening of the D-lines of sodium, a ·b rief review of the various other causes for broadening of sp ectral lines is given. Since, after all, the effect s occur simultaneously, more or less, and it is particularly important for an experimenter to know wha t effect s are most prominent under cert3.in experimental conditions and how to differentiate the several effects. He must also h3.Ve some method of correlating his experimental data with theoretically derived expressions. Moreover, the theories them- selves entail some assumptions and approximations. For detail treatment of the theories, the reader must refer to original papers or to more comprehensive expositions. 6 , 7 , 8 The different types of broadening of a spectral line are classified as: 1. effects relating to individual atoms: a. natural broadening b. Doppler broadening 2. effects relating to interacting atoms: a. resonance broadening (homogeneous gas) b. foreign-gas pressure broadening c. broadening due to fields of charged particles II Broadening due to van der Vifaal type of force is included under 2b. This classification is merely for conven- ience; for the various effects are not all independent and any method of differentiating the de gree and. kind of interacting force would be rath er arbitrary. Moreover, in some cases, there are several different theoretical approaches to the same problem -- not without some confusion. In the following review, where it is appropriate both the classical and the quantum mechanical treatment will be considered and emphasis will oe laid on theories which seem to be reliable or which are significant to the present investigation. la. Natural broadening (Classically) T'ne classical theory of natural line ·broadening is based on Maxwell-Lorentz theory of electromagnetic field. By means of retarded potentials, very adequate theories have been developed for treating the field produced by a moving point charge and the emission of radiation. 9, 10 ,ll Here consider as a source of radiation a number of moving point charges, the sum of whose moments is given by (21) where ei represents the charge for each particle and xi the small displacement from a specified origin, O. If we assume the particles to De moving slowly and make some proper approximations, the Poynting vector for such a system at a large distance R from O (R >> IXil ) is found to -b e (in Gaussian units} • (22) The bracket in this expression indicates the retarded time, t-R/c, at the origin and Qis the polar angle between the axis of p and the direction of observation. When Sis integrated over an entire sphere of lar ge radius R, since S is radial we o·btain the total energy radiated per unit time, dw dt (23) However, a moving charge which is radiating must satisfy the equations for conservation of energy and of momentum; in other words, the reaction force must be considered. Since this rea ction force is i n general small, in th e first approximation it canoe neglected compared to the externa, l force s ; i. e. the equB,t i on of mot ion i s s i np l y , F ma, wh ere Fi s the total externa l force. ( 24 ) I n the second 13 approxi:c,:J,tion, to conserve ener gy we introduce a reaction force, I. Thus the equation of motion becomes -F -t -f = ma . (25) According to the energy equation for two times t 1 and t 2 f is obtained by using eq. (23) and ti::,r integration; i.e. 2 e2 f::: - - 3 - - a. 3 c (26) • As a simple model of radiating particle we take a linear harmonic oscillator whose moment is (27) p == ex where x=x 0 exp(iW0 t). From the preceeding considerations the equation of motion for this oscillator becomes m:x:-= -m~x +- ( 2e 2/3c 3 )x (28) • Since xis small, as a first approximation it is replaced by w 5:x:. If we let Y ~ ( 2e 2w5/3c 3m J the solution of ( 28) is approximately (29) • The average energy over one period is = ·2 2 2 W::::: m(x + l(.) 0 x }/2.:::. W exp(-rt} 0 • (30) I'/- The amplitude of radiation from the oscillator is therefore A( t } = A0 exp ( - rt/ 2 -+ i ~ t ) ( 31} • If we treat this by Fourier analysis (refer to Appendix 1) we obtain the familiar intensity distribution, ' ( 32) Where I(Lv) is normalized to unity. The half-intensity width is found to be in wave length unit ~{1 =- 2-rrc £).W/w~-= 21fCY /t<J~ =- 411e 2 /3mc 2 cm. 1.17 x 10-4 i which is a constant. (33) In the general case this is contra- dicted by the quantum mechanical solution; however, for the alkalis the final results are identical. lb. Natural Broadenin& ( Quantum Mechanically) A very satisfactory treatment of natural broaden- ing has been given quantum mechanically by Weisskop:f and Wigner 12 • An outline of their treatment is given here. The quantum mechanical treatment of interaction of radiation with matter provides the basis for natural broadening. Since the interaction energy is considered small and the method of solution is approximate, the result IS is correct to the first order in the interaction energy. According to the method of perturbation, the Hamiltonian His assumed to be expressible in the form, H ::: H 0 t- H' The characteristic values and functions ( 35 ) "Yl,._ of Ho are known ,and H' is the perturbing term which is small compared to H0 • In the classical treatment one uses a reaction force to conserve energy; while the perturbation theory also provides for this but without the hi gher approximations entailed in the case of self force. By using Schroedinger'.s equation, ( 36) the solution is obtained in a series of the characteristic functions of H0 (37) where 1/1n is a function of the coordinates of the unperturbed system including spin variables. In the usual manner we obtain the differential e quations of bn(t). [Refer to some text in quantum mechanics. ing, reference is made to Heitler 13]. In the follow- where (39) • Weisskop:f and Wigner, according to Dirac's theory of light, succeeded in solving eq. tion holds for t comparable with the reciprocal of the transition probability. only two states at the time (38} so that the solu- First the case of an atom with m, n (Ero- E:n_::: llw) is considered; and t::: 0 the atom is in the excited state m and there are no initial li eht quanta in the radiation field. This simplifies the analysis. 11he differential equation (38) becomes (40a) (40b) where bmo and bnlk are the probability amplitudes, while t he initial conditions are ½n 0 (0) =-1 and bnlk (O)= o. To solve the differential equation, let bmo (t) = exp (- rt/2) vn1ich s ati s fie s the i n itial conditions. (41) The va l u e of i s solved to give t h e n earest approximation. r Eg . (4 0b) become s • (42) Since~- En=1iw0 , the s olution i s (4 3) Equation (40a) mus t a l s o be sati s f i e d . ~s i n g (43 ) we have • (44) The sunmation can be replaced by an integration over all the frequencies w. It can be shown that by integrating over all directions, eq. (44) reduces to (45) where fk is the number of states per cm 3 and per energy interval d.E, and fd.fL is the integration over all direc- tions, and Amn is Einstein's coefficient of spontaneous transition probability (per unit time for emission m _,.. n) After a time probability that ht-Vic t very large compared to 1/r the has been emitted is • ( 46) From this, after integrating and using eq. (45) one obtains I {w) where I O -= 1-fw. {47) :I.1his equation is similar to the one 1 classically derived. In accordance to the uncertainty principle, the uncertainty in the energy of the excited state is AE = -1i/At =1ia"'; since the life time of the excited state is 1/r. Equation (45) may be transformed by using (48) where the g's are the quantum weights and by using eq.(15) which defines f-value; one obtains readily , ( 49) the half-intensity width in frequency unit for a resonance line. 2 2 It is seen t hat, for the resonance lines ( Si+.P1 i 2 2, 2 Si +-2 Pi ) of the alkalis, the half-intensity widths are 2 2 the same for the doublet, and moreover, the values are identical to the classical ones. Weisskopf and Wigner (loc. cit.) also treated the ca se of atoms with several states, but only a few remarks will be given here. The dispersion equation is again s imi- lar to eq. (47) and the half-intensity width is given for a line an-.am by the sum of t h e half-widths of the two levels (50) or · ~~ = 4ne 2 (mc 3 }-l m-..fi + i:I.1he above considerations were in terms of emission; however, the result is essentially the same for absorption. ,zo li'or this case refer to Appendix II. 2. Doppler p roadening The thermal motion of a radiating or a ·b sorbing atom causes line broadening ; for, according to Doppler's principle, there is a shift in the normal frequency )) 0 • This shift depends on the linear velocity Vx as follows: (53) • If we assume a Maxwell-Boltzmann distribution of velocities for the atoms -Ceach mas~ m) in a gas at a temperature, T, namely, (54) where _J3 = m/2kT and in a velocity range dn/n is the fraction of a toms withVx and vx;- dvx, we o·btain the inten- sity distribution for Doppler broadening , (55} which is a Gaussian distribution. The half-intensity width is found to be cm. ( 56) For sodium D-lines at which is much larger than the natural width. It seems that it would be suitable to measure natural width in the X-ray region; however, Ehrenberg, Mark, and Susich 14 were unsuccessful in their attempts due to the interference of fine structure lines. Nevertheless since the wings of a Doppler distribution fall off exponentially while those of natural broadening decrease with ( Al.J )- 2 , measurements can be made even in the visible region1 5 . A brief consideration is given for the combined Doppler and natural ·broadening. Tu.e to the motion o:f the atom the center of the damping distribution is displaced from ).> 0 to Y0 tA where 4=Vav/c. Now the proba- bili ty of displacement for the interval .6. to D-t- d.c,. is e-A;b 0A and the resulting damping distribution is (57) - 0- where b : : . .$d ( ln2 )-±-.1 and ~ and Jd are the half-intensity widths of natural and Doppler broadening respectively. The integral can be solved approximately for the extreme cases. For approximate expressions and values refer to Mitchell and Zemanskyl6. It is to be noted that the deviation from the dispersion distribution is less than one percent even when .t.)1=12-b. 3. Lorent~ Golli s ion Broadening , '.l1he simple s t form of collision broadening was treated by Lorentz 17 following the suggestion by L~ich el son 18 . 'l'he physical situation a nd the analysis is a s follows. .An oscilla tor with normal frequenc y w 0 is subj ected to collision~at time interval 'T. assumptions are: The (1) the mean time between collisions is large compared to the time of collision, 12) each collision stops the oscillation completely - the energy of oscillation being converted into kinetic energy -or the oscillation may be stopped during the collision, only to resume the same frequency with a possible change in the phase and amplitude after t he collision, (3) the \ ti me interval is distr.i buted according to the stati s tical l aw f or the interval ,rand 'Tr d?"' , (60) where the mean value of 7' is given by I 60a) If we take the first alternative of assumption (2) above; in other words, if the amplitude of oscillation is for and after t h e above intervalj A(t)-=-0 a Fourier analysis gives a non-monochromatic li ght. We write A ( t ) = ( 1/211 ) / ~ (cu) eiwt dw ( 61) - """' and by Fourier analysis, we obtain ·1"" A(w) ~ (i/2r,) /•i(Wo-W)tdt • Now the intensity, of r (62) I A(w)j 2 , is averaged over all values using eq. ( 60); the result is I (w) ( 63) where , - i may be c@lled the life time due to collision. Furt hermore, if the radia tion s tarts aga in aft er an impact with a changed amplitude and phase, the resulting intensity distribution is iden t ical to the one just obtained provided t h e r adiation during the impact is neglected. The above intens it y distribution is the same as that for natural broadening and in fact it can be shown that when the two types are treated together the half-intensity widths are additive in the resulting dispersion equation: I (w) - const. (64) To :find the magnitude of the broadening by collision, we must find the value of r . Let ',; -l be the number of collisions per second according to kinetic theory; then we have by (6 0a) ( 65) Furthermore, from kinetic theory , where; is here called the 11 ( 66) o_ptical collision diametern, --v is the root mean square velocity of the atoms, and N the number of atoms per unit volume. Combining (65) and (66) one obtains ( 67) and by further substituting the kinetic theory value of . v, one obtains ( 68) ' where N is the number of foreign gas atoms per unit volume when the latter is large compared to the number of radiating atom, and m and m are their respective 1 2 ma s ses. This formula is applicable to broadening to f orei g.a gases or to a toms of the same kind. If the kinetic theory value o'f J is substituted, as Lorentz did, the half-intensity width is too small; in fact the line broadening experiment give comsistently larger value off than the kinetic theory value. Not much meaning ca n be attached to this "optical collision diameter" in its cla ss ical form. ¥?eisskopf 19 has shown that a phase change of emitted wave is caused even by the passage of a disturbing particle at relatively large distances. In treating Lorentz broadening, the radiation during collision was ne glected. This would lead to what is called "quenching" of radiation; however, experiments do not seem to support this mechanism in general. In the absence of a quenching gas, the mechanism of collision damping is treated with the inclusion of the radiation during the collision. Lenz 20 and Kallrnann and London 21 have pointed out that a sufficiently large phase change at collision is equivalent to a sudden termination of radiatio n . They defined J as the distance between t wo intera cting atoms when t:he phase change is about 1r radians. Ja.blonski 22 considered t 1.1 e temp orary formation of a pseudo mo lecule consisting of an a.ts or.b ing a tom and a foreign ga s molecule. In terms of Franck-Condon potential curves of initial and final states, he explained as ymm etry and shift; however, he did not obtain an expression for the intensity distribution. Weisskopf 19 more recently has obtained a method of calculating fas defined above. The phase change intro- duced by a passage of a particle is A /> :=f:LlCJ~r) d t ( 69) AT where the integration is over the time of the collision and w0 (r) is a function of the distance, r, between the colliding members and is, therefore, a function oft. The size of J' depends on the change of energy level, A E:::. ( 70) ----h 4W and consequently also on the potential of the interacting forces between the two particles. The perturbing particle is considered to be moving in a straight line (distance d from the radiating atom) with a uniform speed v so that we have Thus according to this mechanism, the magnitude of ..f depends on the distance ·between the interacting a toms when one of these, the radiating atom, is suffering a considerable phase change. Weisskopf defines ..J so that the chosen value of d makes Ap ,....,,1, ( 72) ' where the intergration is taken over infinite limits since the contribution beyond Ar, is small. The above consider- ation applies to both foreign gas pressure broadening and resonance broadening but here the former will be taken and the latter will be discussed in the next section. When t h e p ertur bi ng pa rticle is a f orei en a tom, A.W i s given approximate l y by (7 3 ) ' which is a form of London 1 s 23 result. bis in general positive and of the order 10- 32 or 10- 31 cm 6 sec- 1 . Using eq. (73) and (71) in (72) one obtains c,a -2.nb/ A/,cc (V2t 2 + d2 J-:3 • dt=:3ir2 b/4Vd5. (74) -oP I:f one chooses d such that 4/,......,1, one obtains (75) • W h.en this is inserted in eq. ( 6 7), ( 76) • H. Kuhn 24 uses Ll-/,r...,7{ instead A/,,_ l; which makes the result different by only this factor. The value of ..J' obtained from (75} approximately checks with the value obtained -experimentally from the half- width according to eg. (67). Lenz 25 has worked out on the basis of a more general expression for AW , namely, ,2.J (77) • He defined f to be the distance of interacting atoms when the phase change is 2~. J = 1 .5o ( c/v) 1 / (p-l) The result is: (78) • Refer to Michell and Zemansky (loc. cit.) for methods of determining p. 4. Resonance Broadening By resonance broadening we mean the broadening of ... a line which results from the mutual interaction of atoms of the same kind under moderate pressures -- pressures for which the mean free time for collisions is large compared to the time for a single collision. We limit this type of broadening to perturbation energy of the form, , (80) which may be obtained by considering the coupling force between two dipoles. Al though van der Waal type of perturbation (....... r- 6 ) exi s ts simultaneously, it is ne gligible compared to resonance perturbation energy up to a moderate pressure and therefore will be ne glected in this section. Holtzmark 27 early treated resonance broadening classically in terms of coupled oscillators but his result was unsatisfactory. Later Schtitz and Mensing 28 carried out Holtzmark's method more properly and concluded that the line breadth varied linea rly with Nf, without giving the dispersion formula. According to Weis s kopf's modified Lorentz t heory , the half-intensity width canoe calculated for resonance broadening ·by combining equations ( 72), ( 71), and ( 80) as follows; • ( 81} Since Weisskopf puts J~ ~ 1 for d = f ; we have by using B =e 2 :r 12 /16 h~, f s 2 (82) (e f12/41mvov }½ Hence, from eq. (67) sec -1 , ( 83) While the dispersion equation was given in the preceding section. Margenau 26 has treated the pressure effect by * f 12 seems to mean see reference 7 fmn ( gnf Ism) as in eq. ( 4_9 ). 0/ statistical mechanics. Assuming the dispersion equation (11), for the half-intensity width of resonance broaden- ing, he has given sec-l (84) t Which is a little larger than Weisskopf's value and has emphasized that J did not enter into his treatment. Furssow and Wlassow 29 have refined Weisskopf's theory and have treated resonance broadening both classically and quantum mechanically. According to them, ·during the distur·bance of the excited atom by an unexcited atom of the same kind, there is a finite probability of energy transfer from the first to the second and consequently there is a decrease in the lifetime of the excited a tom depending on the ener gy exchange. Instead of a phase change, according to eg. ( 69), as assumed by Weisskopf and others, they assumed an amplitude change to take place in the radiating wave during collision and hence the broadening of a line. '.l'his amplitude change., which takes place suddenly when the interacting atoms are relatively far apart 1 may even have a reversal in sign. In the case of natural ·broadening the unsharpness of the energy level is due to the reduction of the finite lifetime of the excited state occasioned by the interaction of an isolated atom with the radiation field. In the case of pressure broadening the reduction of the lifetime is due predominantly to the interaction of the excited a tom with its neigh.b oring a toms. Hence the physical picture of the interaction is much more complex. Collision theories as a rule treat the interaction of only two atoms at first and later the approximate effect of the rest of the (more distant) atoms is added. The quantum mechanical method of solving collision problem is to assume an appropriate wave function representing the energy states of the two atoms and the mutual interaction between the atoms and the radiation field. 'l'he wave function sare expressed as a sum of terms with coefficients which are functions of time. r 'l 1his is sub- stituted into Schroedingersequation of the form, (0) -t H(l)-t H(2 )-t Vd(l,2, t H- V(0,1,2, t )- ili.; .t]lf1 (0,1,2, t) =0 where ( 85) 1f represents the wave function; H(O),H(l), and H(2) represents the energy of the radiation field and of the atoms respectively; Vd(l,2,tJ is the important dipole interaction energy , and V(0,1,2,t) the interaction ener gy between the atoms and the radiation field. On substituting \ .33 the assumed wave function into Schroedinger's equation in the usual manner, one obtains the differential equations of the coefficients. These differential equations are solved more or less approximately --for instance, first neglecting the radiation field -- and one obtains solutions of the coefficients in terms of integral of the matrix elements of the form lliVa\2) where Va is the interaction energy of two dipoles which has the form represented in eq. (80j A knowledge of these coefficients provide the required information more or less, and one may obtain the dispersion equation by assuming equation ( 60) for exponential decay . Furs s ow and Wla ss ow have o-btained an ~xpre ss ion for the half-intensit y widt h s i milar to Wei s s kop f 1 s but larger by a factor of 8/3: sec- 1 • ( 86) Jablonski 30 has attacked the problem of line broa dening by a r a t h er general meth od {Franck-Condon principle); however, he ha s not given a ca lcula ted value for resonance broadening . Recently Dr. Houston 31 has critically analyzed the previous methods of treating collision broadening and has concluded that the three different approaches are not independent but are quantum mechanical approximations to the same problem: therefore, the results cannot be added. Furthermore, in the same pa per, the problem of fine structure is considered and his analysis shows that the broadening is the same for the mem bers of a RussellSaunders multiplet. For the half-intensit y width, he gives (87) where S(J,J'} is the strength of the line as defined in + Condon and Shortly, and J and J' are the inner quantum numbers for the ground and excited states respectively. For the resonance lines of the alkali -- using S( ~,½) e2b./4,,2mH, -- the half-intensity width is sec- 1 • (88j The 'work of Hughes and Lloyd supports the e qua lity of the broadenings of t he doublet. The t heoretical results o·btained in t his s ec t ion will ·be later compared with experimental va lues. How- ever, the f-value s occurring in e q . (83), (84), and (86) 'l- Condon and Shortly, "The Theory of Atomic Spectra 11 p. 98. a.re rather ambiguous when considered in terms of fine structure, It seems that the quantum weights should appear with the f-values as in eq. (49) or (50a) and (87}. Preference is given to the form represented by equation (87). 5. Broadening due to ~ :: der Waa.l Type of Force In the present investigation, the broadening due to foreign gases is ma.de negligible; however, at relatively high pressures (above 5 mm of sodium vapor pressure) the vapor pressure of Na. 2 molecules becomes appreciable and therefore may be treated as being due to a for·eign gas. Hence, a brief consideration is given to the broad- ening due to perturbation energy of the van der Wa.a.l type. London 23 has considered this type of force and has given methods for calculating it . .M.a.rgenau32 has derived an equation of van der Waal's potential applicable to spectroscopic purposes. The perturbation energy is given as follows: (89) where h.b is the coefficient and b is usually positive. This expression was used in connection with Lorentz collision theory for foreign gas pressure, namely in eq. (76); and the result is considered applicable J(, where it can be assumed that the mean free time is large · compared to the duration of collision. However, for very high foreign gas pressure, this assumption can no longer ·be used. The analysis shows there is ·both shift and asymmetry of the line under hi gh pressures. Kuhn 24 , 34 and London 33 treats the validit y of We i s skopf's modified Lorentz theory . Kuhn 24 conclu des t hat the small frequency changes are due to the simultaneous action of many nei ghb oring atoms and the larger frequency changes are caused ·by the action of a single atom. He gives an intensity distribution of the form, , (90} which is valid at the far wings of a line. For the shift (in the red for positive b) he gives ,J ,o 2 max =- cons t • N ~ (91) • lvfargenau 32 likewise treats the problem by sta tistical mechanics and gives for the intensity distribution for very high gas pressures, I())) , ( 92) where s , (92') J7 where bis the constant in eq. (89). to the form ( 90) for (J) 0 Eq. (92) reduces .» )>> 10 bN 2 • The half-inten- si ty width for (92) is sec- 1 ' (93) whereas the shift of the line maximum is »o - Ymax == (2o.-; 3) 3 bN 2 • (94) Although presumably in the present investigation the shift and the asymmetry due to foreign gas pressure is small, these will ·be checked a.s well as the shape of the line contour. d8 III PREVIOlJS EXPERIMEN'rAL EVIDENCE No Bttempt wlll be m8 n e to give a com1· lete review of the ex9 erimental work ori the entirA field of linP brnadening; only works pertinent to the present investig ation will be em~1hasized. For :more com- plete compilation refer to a comprehensive review6,7. a. Natural Broadening iVIinkowski 35 obtained the natural line width for Na in very close a greement to theory by a method of mag netic rotation and line contour. veri long path length the method of totR l line contour. J:3y using R (1 meter) he has also verified Bb s orption and the sha9 e of the He found that beyond 10- 2 mm of sodium van or pressure, resonance broadening became a9~~•r e ciable . S1nce this ia just within the lower limit of the present invest i ; at i on, a comp arison wi ll be made with his results. ~9 Weing eroff '~8 Likewise Schiltz._.,, --' · , snd Korrr3 7 verified the line contour and the constant natural width (mainly for sodium) which s tarts to deviate from about 10- 2 to 10- 2 mm upwards. As a whole the results on natural broadening seem to be satisfactory. b. Resona~c~ Broadening ~he experiments in the region of resonance broadening (va9 or pressure of the alkalis ab ove 10-2 mm Hg.) are meager and mostly unreli a ble. Harrison and Slater 40 verified the lin e c ontour of sodium princi p al series but they obtained a de p ende nce of the halfJ_ intensity width with N~ . Similar conclus io n s on the half-intensity width were obt2.ined by 'l'ru.mpy 42 oni Rg 2537 and by 1!'J aibel 41 on Cs p rinci pal s eries. 'l'hese ex- p erimenter either used some forei g n gas or very limited pressure rang e; hence their results have little v a lue. On the othe r hand Orthmann and Pring sheim 36 working with Hg 2537 seem to conclude a linear increase of the width with N. Thus the only substantial and quit e r e liable work is b y Hughes and Lloyd 1 on K r e sonance li nes. ' 1!i th sup erior experimental technique, they were able to cover a pressure range 10- 3 to 2xl0 1 mm Hg. 'I1hey found: ( 1) the dis p ersion equation is for the most p art satisfactory, (2) the line width varied linearly with N, (3) the h a lf- intensity width were e qual for the two members of the doublet, in a greement with the theory by Houston 31 , but the absolute magn i tude of' the broadeni ng were several times larger than p redicted by theory, in fact five ti~ e s larger than the value of eq. (88). Although not quite within the region of' reso·;;4 nance broadening , the vwrk by Kuhn '--' on homo g emous Hg vapor pressure may be mentioned. ( .6))) -3/2 He has verified the law for the long wave len&;th wing of Hg 2 537 under hi gh pressures. 41 IV AP PP.RA'l1 US AND EXFERIMEN'11AL PROCEDUh_b; ~ince the a pp aratus and the procedure emp loyed were essentially the same as those described in detail by P. E. Lloyd (in his thesis 1937, C. I. t. ), the descri p tion here will be for the most part brief and only the modifications will be described thoroughly. 1. 0ource The la.mp is the same used by Lloyd exce p t· the sup- port for the elJctrodes was rebuilt. A re p laceable tung s- ten filament (about 1 mm. in diameter and 2.5 cm. long ) is rigidly sup :J orted by two nicl-rnl electrodes. 'l'he fila- ment is operated in vacuum at about 70 to 73 amperes corresponding to a filament temperature of about 32000 K measured with an optical pyrometer. The electrodes are cooled by air to prevent the seal from melting. The current can be maintained constant to within one ampere by a rheostat. Microphotometer traces showed that the variation of sensitivity of the photographic plates plus th~ variation of the intensity of the continuous source over the spectral region investigated was ap proximately linear and nearly constant. 'l'his simp lified the subsequent photometry. The -image of the filament at the slit of the spectrograph was large enough and fairly uniform; moreover, it was found easy to align it with the slit as soon BS the lamp was evacuated. 'l'he life of the filament was on the average over ten hours provided the vacuum was good and the current regulated. r ·c is fe.irly easy to replace the filament by removing the ground-glass joint; however, the joint should be removed whenever the lamp is not being used for a long period. The intensity of the lamp was high enough so that exposures were mostly less than one hour for the plates used. 2. Furnace and Absor-otion Cell 'l'he furnace used was designed and built by Hughes and Lloyd. For details refer to the thesis by Lloyd, particularly Figure 4. 'l'he vacuum furnace consists of ( a) a :;:iorcelain cylinder (30 cm. long and 2.6 cm. internal diameter) wound externally with molybdenum wire in such a way that the central section has less winding than the ends, (b)·a pair of concentric, vacuum-tight, water-cooled steel cylinders which surround the heating element, ( c) a pair of removable "plugs" which are fitted to the ends of the cylinder by rubber gaskets and which are provided with glass windows allowing light to enter and to exit, and (d) outlets for the electrodes, the thermocouple, and the evacuating system. It was found difficult to insert and to remove the thermocou p le according to the original design, so later a ground-glass joint was added to facilitate this operation. i .. .. .. .. .. .. .. . . . . ...... . . . .. - ~c==::::::::~r~r~==~~====::=Ll~=n=~~::::~===:::J6 --~~-F____ :1);6 ·~ d ~ ~ - - G_ _ c=::#'" .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . .. .. .. .. .. .. .. .. ~ Fi p: . 1 C Fi g . 2 AB50R'PTION CELL 3 s Fi g . 3 'l'he absorp tion cell (A) in .t' ig. 1 sli de s snug- ly into the por celain cylinder (13) and the thermoc ouple (-i·) makes a metallic contact (C) with the cell. The MgO win- dows (M,N) for the cell are sup ported loosely by cop 9er tubes ( D, E) which also fit the cylinder. .t'Urthermore, soaps tone cylinders (F,G) heat insulate the cell from the plugs . The ends of "the absorption cell are provi- ded with rings (R,S). One of the plug s has a sylphbn and can be screwed into the cylinder. bY turni ng t his p lug , one can exert sufficient pressure on the windows to obtain a seal due to the collap se 01' the rings (R, 0). 'l'his operation can be carried out with the s ys tem unde r vacuum. J:t·or a short path length ( about; O. 2 cm.), the design of' the steel absorption cell is shown in .t''ig. 2, (A) and (B) are the MgO windows. (C) is the cup where the thermocoup le makes contact. (D) is a co p~ er gasket which fits very snugly into the cell. When the steel p iece(}) is pressed upon (D), the co p~er gasket makes a seal for the ~indow (A); likewise for window (.I:\). Sodiu.~ is inserted into (K) which opens into the cell and a copi:, er screw (.lG) seals it fro m the outside. A 110 volt direct current from the generator in We st Bridge Laboratory was used with a voltag e regulator. Throughout the inv e sti gation the wattage of the furnace was measured; this gave a rough estimate of the t emp erature. It was possible to maintai n the temperature con- stant to within one degree. A pressure of 10- 5 mm . Hg measured bys. McLeod gauge was usually obtained for the furnace by means of a Hyvac pu~'11p and a two-sta ge merc_ury diffusion pump. R. 'i'. Brice43 h8.S made a study of the propertj_es of MgO crystals, and has found that MgO window is cs.pa - ble of resisting the attacks of alkali vapor s t o a very high temp erature. 'l'he op tical ~)ro p erties of .MgO crystals are also suitable for the study of the alkalis. Lloyd successfully used the MgO windows to study the resonance lines of _potassium. . The pre sent experiment shows that the windows (2.5 mm. thick) were not aVi: reciably affected by sodium vapor up to 700°\.J and h 2v e prov ed entirely satisfactory. 3. Calibrati on of Thermocou p le ~here is some doubt as to th e s ccuracy of the temperatur e deter rn.i nation in the work by ttughe s and Lloyd (loc. cit.) in addi ti on to t he unc er tainty in the v e.por pr essur e curve for p otcssiurr.. calibrated his thermo c ou~: -, le 11 Al t hough Lloyrl i n s i tu 11 , he fo u::---.c. 8. rela- tively lar ge temperature diff er ence (12.4° per 1000 ) betwe en the vplues cal ibrated brated outside the furna.ce. 11 in situ" ~md tr 0s e cali - 'i'his ~ndicate s a l ar ge t emperature c:rad ir-mt between the th er mo cou~le wires ...,n-'3_ the absor; t io n cell; in fRct th1 ~ wa p expe cted since the junctio n of t he thermocou; le Sinc e 1 "' Rn WAS covered with insul a te. a ccur ~t e de t errn~nati o~ cf t e- ~err t ur e hi glrnly i 'ri·-ort ant in -c;he C["_Jcul r ti on, much r- ,,.,., :rr-1sis 0 wa 2 considered justified. A n et a llic c ont a ct w2s made be t ween the th er Mocouple wires and the abs orption cell. A small steel conical cup was soldered onto the a osorption cell. Into this cup some pure tin or some s~ecially prepR~ ea alloy (52.5 ;, arts J: H , 32 par ts Pb, 15.5 parts ;:;,n) melting ::it qo 0 c w".ls ;:- l a ced. When the metal melted, the t hermocou- p le junction fell jnto the cup and ~::ive a met a llic cont ac t. Mo st of the time the alloy wa s used. ~reliminary tests were made to see whether ..,he thermocou~J le vdres were attacked by the ,:i_Jloy. It was round that; the chro- mel and alumel wires were not apprec iab ly atb, cked by the al loy*. Both tin and alloy g ave ap preciable vapor pres sure above 550° C; ho wev er, wi-ch the v acuum system running the pressure in "the furnace was a.bout; 10-4mn1 * Constantine wire wR s readily dis s olved in the alloy heated to about 300°c i or sever a l days; wherea s j_ron, tungs ten, molyoednum, :nichrome v\r ere no t afi ec ted. 4-7 of Hg at 550° inste 2.d of 10- 5 mm. Thus ev en for the few plates above this temperature, this vapor :presf'1-1re was not considered serious. 'l'he calibration of the thermoco LL"'.=) le w2. s carried out "in situ" using stri '.-) S of the above menti oned alloy, tin, l ead , zinc, and alumi num. br a ted i n air. It wP.,s also cali- The electromotive forc e ( about 10 mil- livolt per 25o 0 c) measur ed by means of a ; o t~ nt l o~e ter wa s plotted ag ains t temp er 2ture. The two calibrat ion curves diff ered about :2 .::; 0 2er 1 000 . 'l'he c a l ibrat j_on wa s r epeat ed several times and the results were found to be reproducible, t herefore, it was i nferred that t he contact of the thermocoup le i tse lf was qui te repr oducibl e . Subse quently t he calibration curv e determined 11 in situ" was used to deterrnine the vapor pres sure oi s odium. . 4. Opt i c a l Svstem Besides the two MgO windo ws and th e two gl a ss windov,s of the furrn.1.ce, three th in len ses (l, ? , 2 ) 2nd a prism (P) were inter ~os ed b e twe en t he source (L) and the slit (S) of the s pe ctrograph (Se e Fi gure ~ ). Before takin[ each pi c ture th e geometry was adj us ted to g ive a u n iform imar, e of the filan-ient on the slit. T'he Rowland s p ectrozra: : ; h-:i- us ed has a conc~rve gratinf of 21-foot focal l ength. In the first or der -:c Loc?.ted in Room 1 Bri o ' e Labor a t ory the dispersion is 2.64: 0 A per mm. determcned by :r,,_e8:'.1S of iron arc lines and the resolvinc ~iower :~s estir-1:ctted to be about 70,000. The spectrorra0h was use~ entirPly in the first order, hence a yellow ( ~ratten Kn) fil t8 r was ..:; p laced befor e the slit. 'lhe ::tlignment of the [ratinc wr-.s tested first visually for tilt and ill ~~nqtion and se con~ ~hotorra 1 ni1ical ly vii t11 the aid of Fe arc s :·,ectra unt-;I "" r e latively shar ~~' syrr~:i;etric imaf r'. v. as obta:1.ned . 1 Very little re adjus tment was necessary thereafter. ·-1.'he slit vrn s cle sned and lrn_:;-: t c ':l v ered by o:re of the lenses. ~he micro~eter screw for the sl~t wss found t'.) have some l a c and t·h~e zero of th e scale: v. 10 const a nt. c:o t It was f ound , however, t hat by turning the screw tn Mimis one l ong . , en~ then re t urnin~ t o zero the c 0 ale For th e narrow 11res thP slit w~dt h was set at about 14 to 15 microns and for the wide lines to 70 microns. Spectrographic -lates werP selected 0n the b8 s is of high sensitivity and uniformity of se ns-i tiv:lty over a wide spectrRl range to minimize corrections. 3 man I-D plates (14 kostly East- II x 10 11 ) vrere used; and tor thG widest lines Ea stman I-D :;-·lates (7/4tt x 18i') were satis1actory , while for the narrowest lines, in addition to I-D ~Jlates, Ae::,f e. Super)an Pr e ss f'ilr:1s were tried. 'l'he I-D r l 2t e is rel ati vely f r.st but its sensitivity fa lls off rap i d ly r t about 6100 i; ·while the l-b is onl y about hr.lf 2.S f2.st and us able much beyond 6100 ~- 'l'he fi lm W2S used for the narrowest line r fo r its fine- gra ined emulsions, but very little advantage was appreciated. was as 2. ~he exp osure ti ~ e rule l ess than an hour vvith no hype rsensitiu tion. 1 Althou gh this process increases the speed of t h e ~lat es by a factor of sbout four, the error i nt roduced cert2.:i.nly would offset the economy in time. The pJ.2. tes v1ere develo.r, ed in a 25% solution of Eas tman formula D-19 for four minutes. \!1.Thi l e in the de- veloper the plates were brushed v:i th c ame l's hair brush. The Dl a tes we re fixed in Eastman F-1 solution for 10 to 15 minutes, was h ed in running wat8r for h&lf 8.n hour, rinsed c arefully with distilled water and dried. The intensity distributions of the r e sonance lines were determined by means of a pa ir of step-weakeners made of lantern slid e p l a tes. ~he step-weakeners were placed in the p l a te h older 1.5 mm before the spectro graphic pl ates in such pos itions that they would be on both sides of the sodium D-lines. 'i'hus the absorp- tion D-lines and the step-we akeners were pho tographed simultaneously for e a ch :J late. ty comrrnrin[ the d ens i- JO ties of the lines and those of the steps, the transmission contour of the line could be measured. 'l'his comparison is carried out with the aid of a microphoto met er. 'l'hus it is highly important to have an accur- ate calibration of the step-weakeners. Each step-weakener has fourteen steps, each about 1½ mm wide and ap~roximately of uniform density. Since the two step-weal-r nners were made from the same exposures the corres p onding steps have about the se..me densities--the calibration checks this point. ~he transmission of the step-weakeners were calibrated by a densitometer using a K2 'Nratten filter, by a Krfrss microphotometer and above all photographically using the same optical system as in the experiment. 'l'he results of the first two methods were not used, since these methods do not measure most of the diffused lights scattered by the step-weakener plates. Since the las~ method best reproduces the optic2l system, emphasis was laid on this method of calibration. A series of rotating sectors 45 was used to calibrate photographically. The fraction of incident light allowed to pass through the several sectors were 1, 5/6, 3/4, 2/3, 7/9, 1/2, 1/4, and 1/12. calibration of the higher transmissions was emphasized because these determine the wings of the absorption lines. 'l'he sectors were rotated at a speed of 1725 revolutions per minui:;e to avoid intermittent effect 46 , and the current in the cource lamp was kept at 70 amperes constant to one ampere. Each step of the phtographic plate corresponding to each sector was exposed for 3 or 4 minutes (accurate to within one sec.) with the slit at 50 microns. The two step-weakeners were lrnpt before the plate as usual. By means of a microphotometer the densi-cies were calculated and the result was plotted on a D-logE curve, \'.' hich was approximately linear. Care was taken in selecting the appropriate region on the plate for the microphotometer, since the density is not constant over a stri p and the sensi tj_vi ty of the ~J la tes varies slightly. Finally the transmission of the steps of the step-weakeners vrnre interpolated from the graph. 1''ive calibrations were made and the weighed average of -che transmissions is given in the following table together with the values determined by the first two methods: Table I Step 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Photographic method 96 86 81½78 72 66 56 47 38 30 18 10 6 4 Microphotometer 97 82 74 68 61 53 42 32 22 15 8 4 2 1 Densitometer 86 71 61 54 45 38 29 22 15 10 5-} 3 2 1 ( 'l'he values are in /; tri=msmi s sion) Subsequently the photographic c2libration was adopted as the most ap propriate. Its use later showed that the points fall on a fairly smooth consistent curve. Nevertheless, -che error in c,hotometry must be quite large.,.easily twenty or thirty percent in the final result of the magnitude of half-intensity width. 6. General Procedure Before each series of experiment, the absorption cell and the MgO windows are heated to about 600° c; within the furnace. AJter cooling, chemically pure sodium freshly cut under xylene is ~laced within the cell filled with CO2 gas. The MgO windows, the cop p er tubes, and the soapstone tubes are ali e;ned. Then the furnace is closed and evacuated to remove the xylene, CO 2 and air. Before the cell is se a led the furnace is heated over night to slightly above the melting r oint of sodiw71 to remove the occluded gas in the sodium. Finally the 11 p lug 11 is screwed in tightly to form a good seal for the absor p tion cell. The temperature in the furnace reaches equilibriu..rn after several hours. It is possible to take two p ictures per day, since the exposure ti:ne vms u s ually less than one hour. 'l'he electromotive rorce of the thermocouple is read a number of times during eacn exposure, a s well the pressure. th e absorption tube lengths used were 7.49 cm 8 .S a.nd 0.2 ? cm. Some attempts were made to avoid sodium dro p lets 1·rom forming on the Mg O vJindows. Circular p ieces of steel wire near the circumference of the windows ap parently hel p ed this difficulty. Anyway the pre- sence of t he droplets was an indication of equiJ.ibrium of the vap or and li quid pha se. .tturthermore, after e a c h run, when the cell was cooled, addit ional preci pi t ation we_s ob served. V 1. ME'l'HOD 01, CALCULA'l'ION Micro p hotometer A Krtiss miclu'ophotometer was used to d et er- mine the variation of transmiss i on of the lihes with wave-leng th. The micfiro p hotometer is capable of mag- nifying the leng th (i.e. the wave-leng th scal e ) o f the p late in the rattos 1:1, 2:1, 6:1, and 40:1. These s p ecified ratios wer P found to be correct e x ce p t last should read 39.7:1. the The maximum lag of thP quartz fibr e was found to be about l.57a ·for both a 4 cm. 2-nd an 8cm. deflection. 'l'he defl ec tion was not quite linear on the scale as exp ected from the g eometry of the instrument; but this is not serious, since the trace s of the l i nes and the step-weakener wer e mea s ured to- g ether. 'i'he effective slit widths used were 10 mi cro ris and 20 microns and the effectiv e leng th e_bout 1 :mm. 'Ihe mi cro- p hotometer p lates (9x24 crn. 2 ) were run by motor; and about 20 minutes was re c1u ired for eech p late. The drift of the zero was almost negli g ible after the in s trument had been left standing in op eration for a few hours. In spite of· the several factors, the corrections to be made were as a rule relati'vely small -- in fact, p robably smaller than the uncertainties in the calibration of the ste p -weakeners. The coordinates of the micro p hotometer traces were measured accurately enoug h by a com~ arator in the usual manner. These coordinates with some corrections made possible to p lot the contour of the lin e s in terms of wave-leng th and transmission. ~o obtain the trans- mission, it wa s necessary to interp olate from the two step -we2kener traces; however, since the curv e s were close enou gh that a linear interpolation was taken as sufficiently accurate. Usually the wings of the lines corr e s p onding to transmiss i on from .40 to . 90 were measured; while for very narrow lines incom:-letely res o lved, the entire contour was taken. There vms alwa.ys the in- herent difficulty of determining the extreme wings of the lines where the transmission can be assumed to be 1.00 . Symmetry was tested for each line and for the very wide lines, the shift was exar.1ined in terY.!1.s of iron-arc snectra. 2. Vapor Pressure Fortunately, the available data for sodium vap or -ores sure seem to be quite sufficient for the present purpose. vapor Since the exp erimental p oints on p res sure are extensive and since the em!"'. irical curve quite adequately rer;r-esents them, no extra L: olation· is necessary over the entire rang· e investi g ated (from lxlo- 3 mm. to 8xl0 m..rn.). The f ollov1i nc; equation by E. 'l'hiele 48 was adopted: log10PNa== ~(2624:4)/(4. 573T) -1.178 lo gT +11.402 The ::)ressure i s give n i n mm. of Hg. (100) This e r::uation is probably accurate to within a few o erc ent for the rang e covering from 200° to 600° C. It was obtained by 'l'hiel e from his own experiment and from that of Edmondson and Egerton49 ; also a rather complete su...~mary of exp erimental results for sodium is ,s iven. F'or relatively high terr..J er B.ture the V8t or pr es sure of sodium molecule becomes ap··r eciabl e a nd mu st be taken i nto a ccount. Thiel e give s : (101) According to thes e t wo e quation s a t 701° K, PN a .:=:. 0. 731 m.rn. and pNa-:::::. 1. 37xl0 -2 , at 771 0 K Pw~ 3. 6'27 2 2 and pN 8 - l.llxlO-l a~d so on. Knowing the v ap or pressure and a ssumi n c p erfect g as lB w t o hold, one ea s tly calculate s t h e number of atoms p er cm 3 to be N =9. 7lxlo 18 p/J: where n is expre ss ed i n nnn. of Hg. (102) 3. f-values The f-values have been determined experimentally by a number of methods. and Breit 50 . For a summary refer to Korff For the sodium D-lines the f-values have ·been measured by ( 1) La.den burg and Minkowski 51 :.according to the method of magnetic rotation, and (2) by Minkowski 52 in measuring natural width. Table Method II ( 1) ( 2) f1 for 5896 ( D1) 0.36 0 .32 f2 for 5890 ( D2 ) 0. '71 0.64 2 .o ± .1 2.0 ±..(0.05) ratio ( f 2/ f1) Fermi 53 theoretically estimated the ratio, f 2/f1, to ·be approximately two. For the present _purpose f 2 :::- 2/3 were used for the f 1 =1/3 and Di and D2 lines respectively. It is to be noted that probably measurements of :f-values by using lines ·oroadened ·by collision are not reliable. In fact there is some doubt in using the f-values in the case of lines strongly broadened by resonance forces or by collision. Moreover, in most of the theories on resonance broadening, there is much ambiguity in the mean- ing of the f-values involved. This must be clarified be- fore a satisfactory comparison of the theoretical and the experimental values could be made. 4. Calculation (a) In most cases the calculation is rather direct without much corrections. Since the lines are as a rule symmetrical and when there is no over-lapping of the Dlines, the two wings of each line are averaged to-gether. If the dispersion equation is assumed to be cx:x =- -log(I/I 0 )-==t VVhere (103) 2 K = (lfe /mc }Nf sec- 1 which was o·b tained in I sec. 3, xis the path length, and e is expressed in e. s. u. ; we can determine whether the experimental points fit this contour. Since within sufficient accuracy (AP) 2 =- c 2 (4i) 2 ( ,.\.. 4 )-l we may write 0 ' 2 cm . (104) Or more briefly, y - Cz C' cm 2 (105) and where C - y = ( A..l }2 2 ;\_5e N:fx AV½ 2mc 31ogel0 For the D-lines of sodium ( f 1 :::.1/3, cm 2 and • (106) f 2 ::=. 2/3) (107) i = 1 or 2 corresponds to Di and D2 respectively, and k1 = . 812 x 10-41 and k2 :: l .634 x 10-41 • In other words by measuring the slope C, of where the subscript the straight line portion of the curve obtained in the z-y plane, one can determine the half-intensity width, c.J. · (k.Nx)-l J. sec.-l (108) In general C' is negligible compared to the other terms, so that the straight line passes approximately through the origin. For vapor pressure of sodium above 5xio- 2mm. -of Hg. the lines were considered wide enough and therefore sufficiently resolved by the slit width used. quently, no corrections were made for it. Conse- On the other hand, the sensitivity of plate varied sli ghtly, so a small correction was made. A small correction was made also for the lag in the michrophotometer fi -b re but this did not cause any appreciable difference. Thus for vapor pressure above 5xio- 2 mm. the result should be rather reliable, provided overlapping is taken in account for pres• sures above 0.4 mm. (b) The lines for pressures below 5xio- 2 mm. were incompletely resolved. 'I1he slit width of the spectro- graph was 14 or 15 microns and that of the microphotometer 10 microns. The resolving power of the photo graphic plate is a few times smalleT. These factors would tend to flatten the contour of the lines. Consequently , one cannot com- pare their contours with the dispersion e quation . We may trea t t h i s cas e by the meth od of total a bs orption 54 • In principle although the lin es a re broa dened and ·fl a ttene d by the fi n ite r eso lvin g p ower of th e i nst r umen t, t h e a rea un der t h e transmission curve re mai ns constant. That i s , I - )d( D-A ) ;::::; c anst. (10 9 ) Io Le t tin g (I/I 0 l A.t\. Jc 2 = exp (-c<x l "" exp f r~t~~~~/2 i2J where i s the half-int ens it y wi dth i n wave-l ent, t h uni ts and C is defined in eq. (106), one obtains This inte gral can be solved in terms of the integral representation of Bes sel function (111) so that (112) where r=4Clog8 10( .6.4)- 2 t which for the present purpose is ,-..,..r 105 • For r>> 1 the solution is shown54 to be , A = (4 ClogelO f 2 (113) or This method was applied to narrow li nes incompletely resolved, although not too much reliance can be put on * the results. '!'- Particularly the extreme wings are difficult i.Jr. 1viinkowski in a discussion suggested it should be applicable even for rather short tube lengths {several cm.) to measure. The area was measured by planimeter with about one per cent accuracy. Near the region of natural broadening the half-intensity widths of resonance broadening and of natural broadening were considered simply additive on the basis of the similarity of the dispersion equations. Hence the natural width was subtracted from the total measured half-intensity width. On the basis of II sec. 2 the Doppler ·broadening was neglected. ( c). For vapor pressure above O.5 mm of Hg, the D-lines overlap partially or completely. For partial overlapping some correction or a change in the method of calculation is necessary. The absorption coefficients for the overlapped region are considered simply additive, c2 logel0/ ( M 2 }2 ] (114) One must make correction more or less according to the degree of overlapping. Lloyd treats in his thesis a .few cases of overlapping lines. For sodium, the D-lines were close enough ( A,{ 12 6i) that in several cases the lines were trea ted as a single line after taking care of .1A 12 • A few plates were not useable since the members of the Na 2 bands were 63 so numerous and prominent that the background was unreliable. Asymmetry was tested for most of the lines, and for the wider lines any shift was investigated by means of iron arc spectra. For very high vapor pressure, the contour was compared with the general form, oCx =-= (AV )-p (115) to determine Whether other form of contour representation would be more satisfactory. V .L 1. RE.::iuL'l'0 ~heoret i cal Values For convenience the t h e oretical values of res o n- ance h al f-intensj_ty width obts.i ned previously are summarized: (A) 2 ~))½ == le f 12/ 4nm .).6) 1\j sec- 1 \ b) 2 6- J.,7½::: l e f 12/6m)J0 JN sec (C) A.V l ::: ( 2e 2 f 12/3,ym.>lO ) N sec- 1 (D) Ll.)) _1_ =- -1 2 --Wei s skopf (116) --Marg ena u (117) - - Fur sso'-:r & 1'.' l a ss ow (118) 1 2 622S(J,J 1 )N/h(2J+l)( 2 J. 1 -t- l ) sec-1--Houston (119 ) The mean num erical valu e s ac cording to t h ese theories for t he r e sona nc e l i n e 2 of so d i um , pot s s s i um, a nd r ubi di u:1 n.r e l is t ed i n the fo l lo':.' ing t a bl e : Tabl e Theory Element (A) 7 -1 6))_1"xlO ( sec ) 2 III (B) M 1_x10 7 8 (C) (D) Ll.Vixl0 7 Li.1,\xl0 7 2 2 Na O. 39fN O. 8 3fN 1. 05fN 0 .49N K 0. 5lf'N l.08f N 1 . 37fN 0 .63N Rb O. 5 cfN l.llfN l.4lf N 0 .65N As me ntio ned b efo r e , the f-valu e s o c cur ri ng in eq. ( 116 ) , t5 (117), and(ll8) are ambiguous so that the relative broadening of the members of a multiplet cannot be specified. However, since the quantum mechanical treatment for res onance broadening is similar to that for naturEtl broadening, one may j_nfer that the quantum v.r eie;hts should enter as in the latt er ~ In other words, the half-intensity width is ,, inferred to be equal for the members of the alkali resonance lines. On the other hand, theory (D) specifically gives a one to one ratio for the same lines --neglecting the slight difference in the wave-leng ths. For resonance broad ening the dis persio n equation (103) is taken to be correct and the contour is predicted to be quite symmetrical. However, at relatively high pres- sures the theory on van der Wael force predicts asymmetry, 2 shift, and a variation of t t.e half-intensity width vrith N r ather than with N. Moreover the contour would not be represented by equation (103) even at the wings. The theoretical value of natural width {confirmed by experiments) is constant and equal for the D-lines, and the calculated value is LlV -~- = 1.01 x 107 {120) For potassium and rubidium the corresponding values for the natural width are 0.60xl0 7 and 0.75xl0 7 respectively. 6b 2. Experimental Results The experimental results are summarized i n 'l'able IV and Figure 4 . Two different tube lengths were used end several runs were ma.de. Nearly half of the trials wer e ' unsatisfactory due to leakage in the seal of the absorpt- ion cell or to formation of dro p lets on the windows , and, therefore, only plates which represented reliable vap or pressures were utilized. The range of transmission nea s- ured for most of the plates was fro m about 0.35 to 0 .95. In Table IV the pressures were determi ne by eq. ( 100) e.nd N by eq.· ( 102). slo p e of eq. C1 and C2 were measured by the ( 105) and A-Yi by eq . 2 ( 10E3). 'l'hes e d.JJ1 1 s ~ are the measured half-intensity widths minus the natural width which is given by e q . (120). The subscript f aft er plat e s 22, 25, and 27 indicat e s the use of film. In Fi s ure 4 the dashed line indicate the positi on of the natural width for sodi1. .:-ii. D-lines and the solid li ne s 1 the mean JOsitions f o r the two components. Plat e s 22f, 24, 25f , and 26 v,ere det 8r min ed by the area method, since the line s vrere inc omp letely resolv ed . The r esults for these are a bout the sene order of me-[nitud e a s the value for natural width . 'l'he narrov;es t li ne s are r.ot very reliabl e and gi v e ,v8lues for re sonance wj_dth somewhat to o hi[h. The total wic.th for the l ines on pl8.te t7 Table IV Tube l e ngth 7. 39 cm. Pl a t e T p ( :n1Y1 ) N 22f 310 9. 34 X 10- 4 1.78 X 1 0 13 24 550 5.50 10- 3 9 .71 X 1 0 1 3 25f . 569 1.18 X 10 - 2 2.02 X 1014 26 57 0 l. ?3 2 .10 27f 601 3. 8 1 6.16 28 601 3 . 81 6.16 29 633 1.01 x 1 0 - 1 1.54 32 783 4 .6 2 Tube len[th X X l OO X 10 15 5. 7 3 X 1 0 16 0. 22 cm. 46 64 5 0 .157 2 . 37 X 1 0 1 5 47 669 0 . 314 4 . 56 48 701 0.729 1. 0 1 X 1016 53 64 3ft 0 .150 2 .27 X 1015 54 719 1.13 1.53 X 1016 55 796 5.96 7.28 56 84 0 13. 3 17 1. 54 X 10 57 878 2 5.0 2. 27 58 951 7 '2 . 3 7. 38 ( c 0 ~+- -l_nued on next pace ) 68 Tube l c::ig th Plate '7.39 c111 . Cl C2 0 d>'.J~ s 2, ti')!} ' 1 22f 7.6 xl0 ·- 0 l.53xlo- 19 -0.30xl0 7 -0.29x10 7 ?4 l.'.?5xlo- 19 2.60 1.11 1.21 ?:Sf :.36 7 . 45 1.95 2.04 26 3.21 6. 54 1. 5::S 1. ;)8 '.?'7f ?c04xlo- 18 4 . 25xlo- 12 4.51 4. '71_ '.28 1.66 3.8 5 :: .49 4 .16 29 6. c~ l.6Sxlo- 17 5.84 8.04 3.4 2 xlo- 14 32 2.c6xl0 9 rrub e lencth 46 8.45x10-l<J '"' .:_6xlo- 18 l.90xl0 8 :; • 11.4xl0 47 3.43xlo- 18 8.85 4.12 5.30 48 1.74xlo- 17 3.60xl0- 9.4:: 9.73 53 7. 82xlo- 19 l.88xl0- 18 54 2. 59xlo- 1 7 6.15xlo- 17 17 l.8cx10 8 9.40 2.2ox10 8 l.llxl0 -15 5.7lxl0 9 56 l.35xlo- 14 l.63xlOlO 57 4.85 3.24 58 3.1 xio- 13 7.8 2.24xl0 8 ( continu ed on next pr.ge) 9 Tube le n z;th 7.39 cm. Pl at e ~))Jc.. 1/N :_;, ~ t\ )J½, s/N 22f ? ? 24 1.14 X 10- 7 1.25 X 1 0- 7 1.09 25f 0 .97 1.02 1.05 26 0.73 0.75 1.0 3 27f 0.73 0.77 1.05 28 0 .57 0.68 1.19 29 0.38 0.52 1. 38 "' ~ 0.97 0.59 32 Tube length A~ , s/ LlJJ .1: , , 0.22 Crl . 46 0. 80 1.03 1.28 47 0.90 1.14 1.29 48 0 .93 0.96 1.04 53 0.80 0.97 1. 20 54 0.6 3 0.74 1.18 55 '.). 79 56 1. 06 57 1.17 58 1.1 Mean o.71 2 (27--- 55 inclusiv e ) 0.839 Me a n ( 24 - - -54) 1.16 - ,70 ' I:' .. _t:t: ' rrJ ...., I :tt•·• .C. +· I:. j jj .. -1-4 + . ! ~, ·- •-::. + ,. -1.' _,. F:w,re :I I-·- i', +1 f; .,, ~ g I"' . . t :J i'q ~ '_._. ___. ; ·- ~, ,__ , l + I ~~ I ~ ~r.:., .. ;!j ., ~ j j , ~1 ,-1. 't t' i '"' :;:i_: I+ I 11£ .. .... lli· - d ,_ ,--·,µ J'"'." 1: •H I; , i:.. ·-• +<;.;. t:!:t •· ,_ ..;:: L., . - j. f •·· ,. . . . :1 - _:,1-, ~- ., :1 rt; ::'"i:,= ~1 ;.. _ti ~ l'l: t ~ t;j r I~ ....; tr: . - . ~ -· ii ~ -r~ ,n ~ , ... I! ._;_~• r , r.mf S:t:1 ..:..,.;:, ll ~:,:, - l;./ t'·•tj ; + ,. ~.::.-1·; w □ ~ -~ : :i -1. ; -~ I->·+ J._:;.i:J½ - .:• '.. '· - ~ It t . ~ L;.: ., 'i. ,::-4 t· !t, ,-....;:.:;-i·~•:.tc,i;.i±r fh-1L,.,.,,.1_E.:;:.L m~~ l1 ...,J1!1: - - - - - ~--~--..._..,.-J:IJ ' ,,.., ·<,:n · ·-1 rrH'1"·....__-""--!I'-h..l'.....;.....--- - - - - - ~ - - - - - - . l . 1 ! _ ~ ~;.J 71 22f is less than the natural width -- hence the negat ive re sult in Table IV. In the same table the average value 7 o:f the two components for .dY :/N is equal to O. 776xl0-2 3 sec -1 cm. • t ely measure Using this value one may approxima the natural width from the total measured widths (l.Olx 10 7+th e value in Table IV) for plates 22:f, 24, 25f, and 26. The results are in terms of the means of the doublet: Plate 22:f ( 0 . 72 - 0 .14 )xl O7 = 0 . 5 8xl O7 24 ( 2 •1 7 - 25:f ( 3.01 - 1 . 5 7 ) " 26 (2.57 - 1.63) Q • 76 ) 1.41 II II ~ sec- 1 Tl fl 1 . 44 " II 0.94 fl II These give a mean value for natural width equal to 1.09 x10 7 sec- 1 which is within experimental error in agreement with the results on natural width for sodium. and Korff 37 Both Sch~tz 39 ~ observed that the resonance broadening sets in when N is about 5x10 13 , while 1'!i inkowski 35 , a ·b out 1.5xlOM:. According to the present investigation, resonance width is about 23% of the natura l width a t about 3xl0 13 . measurements are very a ccurate, this woul d mean t hat pr obably the measured resona.nci.e width in the present case is too high by 30 or 50%~ This is possible. However, the result s here a re much more consistent than those of Hu ghes and Lloyd1 when compa red to results on natura l width. This fact is reassuring. '.l'he lines in plates 48 and 54 _are partially. overlapp ed, while in 32, 55, 56, 57, and 58, ..they are completely overlapped and were treated as single lines. This treatment was considered justified since the A~ 1 s measured for the wings were much larger than half the separao tion (6A/2) of the two lines; i. e. the AV 1 s measured were lar g er than 50 R. In a log scale the resonance half'-intens:ity width are pl0tted ag ainst the corres ponding N's in Fig. 6. The result shows the linear depend ence of these as predicted by theory and as confirmed by Hughes and Lloyd. However, the absolute magnitudes are greatly different. Hughes and Lloyd 1 obtained AV ½/N::: 3. 2xlo- 7 for the two resonance lines of potassium --five times the theor e tical value (0.63xlo- 7 ) by Houston 31 . A)J½/N-= O. 712xlo- 7 and pectively. For the present case of sodium = 0.-8'39:xlo- 7 for Di and D2 res- The mean va.lue ( 0 .77 0x10- 7 ) is thus nesrly two times the theoretical value (0.49xlo- 7 ) by Houston. Although the error in the pres ent invest:i.fat:ton is larg e it is unlikely that the true value is less than 0.7xlo- 7 . Nevertheless, looking at the experimental values one may conclude that the true value is about 0.7xlo- 7 since nearly all the experimental errors would tend to me.ke the result too high. 73 I ~ 2 -3 4- s- 6 t ~(¼l l" 2 z 71- 75 A more serious discrepency is the ratio of the half-intensity widths. Hughes e.nd Lloyd's r esult is in agreement with the theory b y Houston. for ~>-i2, 8 / AY J,.1 . ~ They obta j_ned 1.05 For the present c s.se of so dium the mean value is 1.16 and the individua l ratios are consis tently lar g er than 1.00*. For the narrow lines the ratio seem to be nearer 1:1. The error in measuring the rat io is considerab ly less than i n measuring the half wid ths and this discrepancy canno t be v1holly attributed to experimental errors. Nearly all the l ines are well repre2ented by the inv erse square contour. For e x run:ple, refer to Fig. 5 for the case of plate 47. Thi s is al s o in agreement wi th the theory and the results of Eu ghes and Lloyd. Howev er , the incompletely resolved lines were not adequately re presented by this contour. Except ions are also the lines on p lates 32, 56, 57, and probably 58. Thes e did not fit a t all, hence when compared to lo g ( Io/I) ={( 6.l )-p, the values were found to be as follows: {!During a seminar Dr. Bowen expressed a belief that Dr. Lloy d's result should be l a rger than 1.05 -- a bout 1.3 or 1.4 (according to his rough measurem e nts). Dr. Lloyd h ad to make c ons iderable corre ct i on s for varying sensitivity of the IR plates used. Also some unpublished results of ~ilr. Shang -Yi Chen g iv e s 1.6 for the ratio of the half ,vidths of the re s onance li nes of rubidium. 76 Plate short W8.Ve wi ng Ps 32 3 .3 56 3.0 l ong ' 'iBV e wing Pl 3 .0 5'7 2 . '7 not a ccur a t ely measurable 58 These resu lts are not very conclu s ive, si~ce t he v a riation of the sensitivity is considerable for these very wide lines. However, they ar e defi nite en0u ch to show t hat the absorption coefficient fall s off much faster than the i nvers e s qu are representation, for pressures 2.bov e 5 or 10 m..-rn . a nd Lloyd 1 giv e s Ps 20 mm pressure. Hughes = 2 .5 and p 1 = 2.0 for pot a ssiun a t In t erms of h omoe:'3neous Hg va:J or :1-\uri..n 34 found Ps e qual to about 3 , whil e he measured Pl=- 3/2 in agreement with the theo r y for vrn der ''Jra l t yp e of force. Thus in the pre sent C8.se there is v ery little evidence f or van der Waal i nteract i on . Acc ord i ng to F'i sure 4, t he half- intensity width seems to increase sl ightly f as t er than N at pressures ab ove 5m111 but not 2.s fast as N2 ; howev 0r, these valu e s are only appr oximate, s i nce t hey were determined from the i nverse square co ntour . The pos si bility of me.king me2surerri.ent s fo r sodium above 50mm is l ir'lited by the conspicuous 2.p~J e2,r ance of the members of Na 2 b ands ab0v e a. mill i= c t er vapor pr essur e 77 of Na. These are observable even on plate 29 at 0.1 i:nm. They start strongly from the red and cover the entire region of the photo f raphic pl a te. A number of pl'."'tes for high pressures were useless, for there wa s llardly '2.ny :ne 9surable background due to the absorption of the bands. It mi ['ht be possible to meP.sur e somev.'hat b ey0nd 70 P.ltr: if the sl it width of th e spectro ~r 0,~ is kept narrow and the absorption tub 0 leng th is b elow O. 2 cm --2, r Bther d1ff'icult task. The lfa2 b ands were early studj_ ed by i'for:,rJ 55 •,md lat er more e::actly i nves t iga:ted by Loomis and 'food 56 , also by Fredrickson and Watson 57 . The bf' T'.d systems ( a r ed and a c reen) wer ~ ~~ c ntified r oughly in the ~re se nt wo~~ , but ~o s tudi e s 1v2, s made of them. T~c ~~c~oph otometer t r a ces of thP D-l~nes ~h o~~d rir::: ry f .J r sor.1e oi t}:1e cont 01.1.rs wer·e of the ord er due to shift was observe~. The Lorentz collision diar.. eter is giv en bye ~. ( 67) f= for T ~ 550°, v =- l.lxl0 5 cm/sec; this value f ives for 78 0 7 r- ~7 ~8 ~9 about 100 A a little lower t h rm the v s lues by other•_,,) ' ..., ' V for s odium. 3. Conclusi on The ex:;::i erimen t a l error in det er m:i_n inc b-YJ_/N i s /~ estimated to be about thirty ~ c rcent --mostly due t o Grrors in pho tome t~7 and r "rr.l~r to er r or s jc1 ation. possibly s o·r>J eV'~ '.,tt too The experiment r l r s sult :c_ ~ t ern:'.) '?r a tur e d et '?rm:i_n- high -- p erha r s .1Y"}/N =- O. 7xl 0 -'7 r nther t hr ::_ 2 .S x lC: - 7 . It is su gg ested thAt t he eYp er ine n t o ~ , ot r s Al u m b o repeated v1 i th i mpr oved t e crn1i que in t e!"l) e r a ture c'l e ter::nina tion. Sinc e the bo::,.d2 nre rr:o r e troul'le s or:~e in th 0 ca se of sodium tha n of :::, otassi nm t he latter .would b e mo re suitabl e for pressures e.bove 10 mm . More c onsid er a tio n should b e c iven to t he rel a tive half-intensity widths of multiplets both theoretically and e x p erimentally; in othe r word s , a further analysis n e c e ssary in the f-values. A more accurate d etermination of the line con t our would be pos~ible with i mproved photometry. This would be helpful in study i n g the c ont our at hi :::h pressur<:s where the inverse squar e contour fails. Prob a bly it i s im:;;o s s- i ble to study v nn d er 'Fiaal force with Na due to the interference of the Na2 bands. ' <..l T:J APPEN1)IX I* Fou rier ir.ter gral can be writ t en in the form 1=f(w) e iwt dw A(t) -(>o where f (w) = (l/2n) f 00 A(t) e-iwt dt 0 - Aoe -~t/2 eiL--lot Let A( t) then f (iv) - (Ao/z-n-) f =et½ ,--+ i ( wo- w ~t' d.t 0 Ao Since I(w) ~ const. I(w) * 2 jf(w) 1 , we obt ain c ons t. • Courant and Hilb e rt, "Me thoden der Mathematisc he Physik" Sprincer, Berlin , 1924 Chap . ii 6 30 APPENDIX II THE CASE FOR ABSORPTION Consider a number of similar os cillator whose density, N oscillators per unit volume, is such that the mutual interaction of the oscillators ma y be ne glected. An incident light beam of constant intensity in the nei ghborhood of IJ..6 (the natural frequency of the os cillator) will cause force vibration of the oscillators. ~ne equa- tion of motion of the k-th oscillator will be given by .where }""2e 2w§/3mc 3 and E (w)-=- e 1 (wt -+ .Jk) is the field strength in the region of the k-th oscillator. After a sufficiently long time the solution is eE ( for w near (1.) 0 ) The ~verage work performed by the light beam per sec, in othe:i;.- words, the average energy absorbed per sec. by an oscillator is given by 8/ • • L1 t This will g ive for N oscillators a distribution I (lJ) const. , and the half-intensity width is again e_s bef ore . 3~ LITERA:I.1 URE l Hughes & Lloyd, 2 Tolman, Phys. Rev. Tolman, nstatistical Mechanics with Applications Phys. Rev. 52, 1215(1937) 23, 609(1924) to Physics and Chemistry," Chemical Catalogue Co., 3 Ladenburg, Zeit. Phys. La.denburg & Reiche, 4 Korff & Breit, 5 W. Kuhn, 11, 596(1923) Rev. Mod. Phys. Reiche & Thomas, 6 Weisskopf, 7 lfu.rgenau & Watson, 8 Finkelnburg, N.Y. (1927) 4, 451(1921) Natur. Zeit. Phys. Chap. 16 4, 471(1932) 33, 408(1925) Zeit. Phys. Phys. Zeit. 34, 510(1925) 34, 1(1933) Rev. Mod. Phys. 8, 22(1936) "Kontinuierliche Spektren" XII, Julius Springer, Berlin(l938) .,_ 9 Lorentz, "Theory of the Electronsn 10 Hei tler, "Quantum Theory of Radiation"', 0xford(l936) 11 Joos, "Theoretical Physics'' Leipzig(l916) p322-328, G.E.Stechert, N. Y. (1934} 12 Weisskopf & Wigner, Weisskopf & Wigner, Also refer to Hoyt, Zeit. _Phys. 11 " Phys. Rev. 63, 54(1930) 65, 18(1931) 36, 860(1931) J3 13. Houston, 11 Priciples of Quantu m Mec h anic s II mimeo- g Paph note s (unpublished) 14 mrenberg & Mark, Zeit. Phys. II Ehrenberg & Susich, 15 Minkowski, 16 Mitchell & Zemansky, Zeit. Phys. R-toms 11 , 17 II Lorentz, 11 42, 823(1927) 36, 839(1926) ' 1Resonance Radiat ion and Excited MacMillan Co. N. Y. (1934) Proc. Arnst, Acad. II 42, 807(19 27) 8 , 591(1900) Theory of Electrons" Leipzig (1916) 18 Michelson, Astro. Jour. 2, 251(1895) 19 Weisskopf, Zeit. Phys. 75, 287 (1932) II II 20 Len;, Zeit. Phys. 21 Kallmann &. London, 22 Jablonski, 23 London, 2G, 299(1924) Zeit. Phy s. Chem. Zeit. 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Rev. 33, 429(1927) Fip:ure 7 l!.\ \..: "' ~ Fi gure 8 -- Fi p;ure 9