Intermolecular Forces from Atomic Line-Shape Experiments

1368 HARRY C. JACOBSON

ACKNOWLEDGMENTS

I am very grateful to Professor S. Y. Chen forfurnishing the line-shape data which made this anal-ysis possible. I would also like to thank Dr. C. L.

Chen and Dr. A. V. Phelps for communicatingtheir results to me. Finally, I would like to ac-knowledge the assistance of A. Atakan who carried outthe calculations using the general pressures ap-proach.

*Work supported by the National Aeronautics and SpaceAdministration under Research Grant No. NGL-43-001-006.

~M. Baranger, Phys. Rev. 112, 855 (1958).H. Margenau and M. Lewis, Rev. Mod. Phys. 31,

569 (1959).The extensive formal developments and the variety of

applications of moment theory preclude a complete ac-counting. A few references pertinent to the present paperare listed below. See also, R. G. Gordon, in StochasticProcesses in Chemical Physics, edited by K. E. Shuler(lnterscience, New York, 1969), p. 79.

J. P. Vinti, Phys. Rev. 41, 432 (1932).R. G. Gordon, J. Chem. Phys. 41, 1819 (1964).R. G. Gordon, J. Chem. Phys. 54, 663 (1971).'J. D. Poll and J. Van Kranendonk, Can. J. Phys.

39, 189 (1961); R. P. Futrelle, Phys. Rev. Letters 19,479 (1967); and H. B. Levine, ibid. 21, 1512 (1968).

R. Kubo, in Stochastic Processes in Chemical Physics,edited by K. E. Shuler enterscience, New York, 1969),p. 101; R. P. Futrelle (unpublished).

9H. Margenau and H. C. Jacobson, J. Quant. Spectry.Radiative Transfer 3, 35 (1963).

R. M. Herman, Phys. Rev. 132, 262 (1963); H. Mar-genau and G. M. Murphy, The Mathematics of Physicsand Chemistry (Van Nostrand, New York, 1964), Vol. II,p. 413.

I~R. L. Fox and H. C. Jacobson, Phys. Rev. 188, 232(1969).

R. O. Garrett and S. Y. Chdn, Phys. Rev. 144, 66(1966); S. Y. Chdn and R. O. Garrett, ibid. 144, 59(1966); S. Y. Chdn, D. E. Gilbert, and D. K. Tan, ibid.184, 51 (1969).

~3C. L. Chen and A. V. Phelps, Phys. Rev. 173, 62(1968); and C. L. Chen and A. V. Phelps (private com-munication) .

~4H. Margenau, J. Quant. Spectry. Radiative Transfer3, 445 (1963).

' R. B. Bernstein and J. T. Muckerman, in Intermo-lecular Eorces, edited by J. O. Hirschfelder onterscience,New York, 1967), Chap. 8.

A. D. Buckingham and J. A. Pople, Trans. FaradaySoc. 51, 1173 (1955).

~'The data of Phelps and Chen are relative to the linemaximum. The figures in the table are relative to theunperturbed line position. According to Chdn and Garrett[Phys. Rev. 144, 59 (1966)] the shift for rd =0.81 is 0. 19cm ~. In order to infer moments the area near line centerhad to be estimated. This was done by fitting a Lorentzline shape to the available data. The inferred half-widthat half-height was 0.35 cm ~, a value consistent with theresults of Ref. 12. The moments were calculated as-suming a cell temperature of 410 K.

~ W. E. Baylis, J. Chem. Phys. 51, 2665 (1969).~9M. Takeo, Phys. Rev. A 1, 1143 (1970); J. M. Farr

and W. R. Hindmarsh, Phys. Letters 27A, 512 (1968);R. P. Futrelle (unpublished); R. G. Breene, Jr. , Phys.Rev. A 2, 1164 (1970).

PHYSICAL REVIEW A VOLUME 4, NUMBER 4 OCTOBER 1971

Intermolecular Forces from Atomic Line-Shape Experiments*

Harry C. JacobsonDepartment of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37916

(Received 19 April 1971)

A systematic convergent method of studying intermolecular forces in excited states is de-tailed. Extensive comparison between theory and experiment indicates that high-resolutionline-shape experiments over a wide range of frequencies will permit the inference of param-eters in realistic forms of the potential energy. The analysis focuses on the P&&2- S&~2 re-sonance line of cesium pressurized by helium, argon, and xenon. When the model is usedwith ground-state potentials inferred from atomic-beam experiments, and when the excitedstate is described by V=G/R —D/Re, the following reliable parameters were determined:helium, G = 0.459 + 0.060 and D = —0.614 +0.061; argon, G = 6.48 + 0. 20 and D = 5. 23 + 0.05;xenon, G = 53.6 + 5. 8 and D = 19.1 +0. 57, where the units of G are 10 erg cm and of D are10 5 ergcme.

I. INTRODUCTION

Information about intermolecular forces betweensystems in the ground state is available from a

variety of experiments'; however, line-shape ex-periments afford the only generally useful probeof excited-state interactions. Extensive efforthas been devoted to the study of interactions in the

4 INTERMOLECULAR FORCES FROM ATOMIC LINE- SHAPE EXPERIMENTS

microwave and far-infrared regions of the spec-trum""; however, only sporadic attention hasbeen directed to other interesting spectral re-gions. '" Also, generalization is difficult becausemany of the results appear as tables of data for aspecific system. Individual features of a spectralline, such as the width, shift and asymmetry, orsatellite position' have been used to establish pa-rameters in an assumed form of the interactionenergy. No general use has been made of thewealth of information available in the line shapeitself. This is certainly understandable since theliterature rarely includes data on line shapes.

It seems clear, first, that the entire shape ora, large number of moments of the shape must beused in the relevant computations if a maximumamount of information is to be recovered. Second,if the information is to have a general utility, itmust correlate many experiments done under dif-ferent conditions and not simply reproduce the ex-perimental facts for the experiment from whichthe information was inferred. Also, it is possibleto lapse into curve fitting of little significanceunless the model is well defined and a small numberof parameters is used initially. And third, theanalysis must be systematic and it should yield adefinite determination about the significance of anyparticular parameter.

In the present paper a well-defined realisticmodel is used to study the line shapes of the 'P»,resonance line of cesium pressurized by helium,argon, and xenon. A systematic convergent meth-od of studying atomic line shapes is proposed inSec. II, a method which can be readily applied inas much detail as is appropriate to any line-shapeproblem. The usefulness of the technique is es-tablished in Sec. III which compares the highlightsof a series of numerical calculations with experi-ment.

II. METHOD

The line shape can be written in the form

E(~)= — C(t) e'"'dt27r

in which C(t) is the correlation function. C(t) canbe approximated in a variety of ways, but in everyevent depends on the form of the interaction energy.If the potentials are considered as functions of Pparameters k~, then a multiple regression analysiscould be used to determine which of the k, aresignificant and what their values are. It would bedesirabLe but impractical to minimize

J= dw F u —— C k~, kz, . . . , k~, t e™tdt.17T 00

An entirely equivalent and feasible requirement isthe minimization of

J= — dt f t —C, t

where f(t) is the Fourier transform of E(u). Next,if C(k, t) is expanded in a multiple Taylor seriesabout an initial set of parameters k0, then theequations for k —k0= 5k are

6k=A y y

with

y; = dt Re —' [f C(ko, -t)]*~C(k, t

0 &k;

and

9C(k, t) dC(k, t)ak,.

C(t)=exp[-4m f R dRe "" (l —e' )],where p= (Vf —V;)tfh, V;(f) is the potential in theinitial (final) state, n is the number density ofperturbers, T is the temperature, and R is theintermolecular distance. This in turn requiresradial integrations of C and its derivatives at eachiteration.

This approach, which is both systematic andconvergent, satisfies two of the requirements in-dicated in the Introduction: It uses all of the in-formation, and it can be used to establish whetheror not a particular parameter has any statisticalsignificance in the explanation of the experiment.To determine whether the procedure has any gen-eral utility it was necessa, ry to study a large bodyof data for which line shapes were available.

(4)

III. COMPARISON WITH EXPERIMENT

The principal series of cesium perturbed byrare gases has been the subject of extensive ex-perimental effort by Chen and his collaborators. 'Their results for the 'P»2 —'S»2 resonance linepressurized by helium, argon, and xenon wereselected for detailed study.

(This assumes that linear terms are retained inthe expansion. )

Since it is C and not F that theory provides,Eg. (2) is a reasonable form to study. The equa-tion also emphasizes the potential importance ofhigh-resolution experiments which cover a widerange of frequencies, since the former determinesthe range available in the t integration and the lat-ter fixes the resolution in t.

Even in the simplest cases, the calculation isnot as straightforward as the usual least-squaresmethod because the functional form of C(t) is notknown. For example, when the criteria for validapplication of the quasistatic theory are satisfied,C has the form

1370 HARRY C. JACOB SON

I.Q

Q,7

KCC

COK

v) Q4Z'

FIG. 1. Line shapes of the longerwavelength component of the cesium res-onance lines in argon at, a relative den-sity of 60. 8: (circles) experimental datafrom Chen; (solid line) according toquasistatic theory. (Frequency here -imeasured in units of wave number, cm .)

O, l

-70I

—60I

-50I

-40I

-30I

-20I

-IO

FREQUENCY ( cm ')

A. Cesium-Argon

A line at a relative density ~ of 60. 8 was used toestablish the two final-state parameters in a Len-nard- Jones 6-12 potential Vf = G/R —D/R, whenthe initial state was described by a potential ofthe same form with constants A and B taken fromatomic-beam experiments. ""' Because the re-quirements for valid application of the quasistatictheory are well met for the experimental condi-tions, Eq. (4) was used for the correlation func-tion. The analysis, using A = 3. 68 & 10 '0 and8 = 3. 30&10 ' cgs units, leads to values ofG = (6. 46+ 0. 20) x 10 ' and D = (5. 23 y 0 05) x10cgs units. (Here the uncertainties are +Ss, whereS= [pF (p, n —p)]", and s;= [JA;/(n —p)] ';P is the number of parameters, n the number ofdata points, and 57~ values were used for I . Thequantities J and A are defined in Eq. 3. )

In order to determine whether or not such a cal-culation has any utility, these constants, with Eq.(4) for C, were used in (1) to determine the line

TABLE I. Observed and calculated shifts and widths forargon using Eq. (4) (in cm ).

Relativedensity r

ShiftObserved Calculated

WidthObserved Calculated

13.230.060. 8

100.0

—5.4—15.0—32. 2—50. 0

—14.0—32. 5—54. 0

10.220. 436.557.0

26. 540. 055.0

shape at a number of other relative densities.Figure 1 indicates that the agreement betweencalculated and experimental line shapes at x= 60. 8is good. Table I compares some widths and shiftsat other densities. The data are reproduced within30% in the range where the approximations shouldbe good. (The results are within 10% if the widthat r = 30 is excluded. ) In order to give the methoda more severe test by going to lower densities,the general pressures theor~~ was used to computethe )ine shapes. The results, shown in Table II,again indicate that widths and shifts can be well

4 INT ERMOLE CU LAR FORCE S FROM ATOMIC LINE- SHAPE EXPERIMENT S 1371

TABLE II. Observed and calculated shifts and widths forargon using general pressures method (in cm i).

TABLE IV. Observed and calculated shifts and widthsfor xenon using general pressures method (in cm ).



Relativedensity

Shift WidthObserved Calculated Observed Calculated

0. 52. 6

13.2

60. 8

—0.12—0.65—5.432 ~ 2

—0. 10—0.38—3.0

—30.6

0.401.8

10.2

36.5

0.401.88.6

46. 7

5.09.8

26. 741.3

—1.7—4 7

—47—68

—1.6—5.6

—51—78

4. 730.966. 081.3

4. 312.979. 2

99.4

G (f )t(A+ih)-

was compared with

f(f) e ((Ul+(d)

(5)

(6)

where u and d are the observed width (one-haU thefull width at half-maximum intensity) and shift,respectively, of a line which was assumed to havea dispersion profile. The theoretical width and

shift (k and h, respectively) were calculated fromthe real and imaginary parts of

g(t)= 2)(vnt f e (~" (1 —e' )rdr, (7)

where v is the mean velocity, n is the number den-sity, and

6= h f (V~ —V)dt

In (8) the Lennard-Jones forms were assumed asbefore and R'=r +v'f~. In this case J [Eq. (2)j can

calculated over a large range using a simple mod-el for the potential. (In this connection it is wellto recall that the constants, with their quoted un-

certainties, refer to the model which uses thequasistatic approximation. Other, slightly dif-ferent, constants lead to improved agreement be-tween experiment and the general pressurestheory. ) The results and a consideration of Fig.1 suggest that the only way our knowledge of thedetails of the potential can be improved is by the

analysis of the entire line, including accurate mea-surements well away from line center.

To provide an additional test of the systematicmethod, a correlation function corresponding tothe phase-shift theory, which is appropriate atlow densities for the systems under study, wasused in Eqs. (2) and (3). Specifically,

be evaluated analytically. When the integrals overimpact parameter are performed numerically, the

iterative calculation leads to converged values forthe parameters in the final-state potentials. Hereof course only two data are used; in a sense, allof the observations which lead to the conclusion that

f(f) is given by (6) are employed, but the resultsare very sensitive to the two inputs. For the ex-periments in question the pressure broadenedwidth at low densities has to be inferred in the

presence of hyperfine and instrumental effects.Also, the lines are not Lorentz shaped well away

from line center. As a consequence, the parame-ters obtained in this way are not as reliable asthose given above and there seems to be little pointin recording them.

It is interesting, however, to compare the ob-served values with phase-shift results which usethe constants obtained at r= 60. 8. The source ofconstants for helium and xenon is described later.The comparison in Table III indicates a lack ofgood agreement, which could originate in severalplaces, e. g. , in the choice of a velocity parame-ter for Eq. (7), or simply in the fact that there isno a Priori reason that they should agree. Theimportant feature of the calculation is that the re-sults depend strongly on the values used for theexcited-state parameters. For example, the pre-ferred values for argon correspond to a well ofdepth 1.1 &10 ' ergs at 5. 4 A; the values in TableIII would agree within 10%%ug if the well is 2. 5~10 'ergs deep and positioned at 5. 75 A. This, in turn,implies that the low-density results will be im-portant in establishing the form of more elaboratepotentials.

B. Cesium-Xenon

Relative ShiftPerturber density r Observed Calculated

Full WidthObserv ed Calculated

HeliumArgonXenon

40. 160.826. 7

+0.13—0. 24—0. 25

+0.23—0. 12—0. 20

0.700.800.62

0.370 ~ 280. 54

TABLE III. Comparison of experiments at r =1 withphase-shift results which employ parameters obtained athigh relative densities (in cm ).

Again the analysis using the quasistatic formfor C was used to analyze the I'„2—S„,resonance2 2

line pressurized by xenon at a relative density of26. 7. The calculation leads to G = (53. 6 ~5. 8)x10 ' and D= (19.1+0. 57)x10 ' cgs units whenA = 12. 3&10 ' and B= '7. 9&&10"cgs units. Thesevalues were used in a general pressures calcula-tion, the results of which are compared with ex-periment in Table IV and Fig. 2. The experimentalline at r = 9. 8 shows two intense peaks and the one


Ieo—

0— EXPETHEO

0,6—

g) 0,4—

FIG. 2. Line shapes of the longerwavelength component of the cesiumresonance lines in xenon at a relativedensit of 26'

y .7: (circles) experimen-tal data from Chen; (solid line) accord-ing to general pressures theory.(Frequency here measured in 'ts ofwave number, cm '. )

0- 250I

-200 - 150 -50

FREQUENCY (cm ')

at ~= 5 shohows a small satellite. These constawhich were obtained atthe shapes well at the 1

' ' s aa r= 6. 7, do not ree ower densities. This fa

and consideration of T bla eIVand F' . 2is act

more indicate that a moig. 2 once

a a more complicated form for& can be profitably employed. Even so, the

widths and shifts are re rwi in 25iq if the width datum at r = 9. 8 is o

s i analysis are given in Table III.

C. Cesium-Helium

The experimental result40 1 was used with t o

suit at a relative densit fthe quasistatic form of C t

infer the parameters for helium. H

known. A chin e ground- state otep ntial is not wellc oice that is consistent w' p

used to 'nfaws and other caalculations was

o i er the results that arethat 11 bwi e called case 1. T '

a are preferred and

x10-'»his set uses A = 0. 891

= 0. 426 x 10 G = 0= ( . 459 +0. 060)x10- 0

an = —0. 614+0.061)x10 c s un

comparison betwcgs units. The

'on e een theory and ex er'

given in Table V fperiment is

lation and in Tabla e for the general resp ssures calcu-in a le III for the phase- hift- s computa-

tion. Figure 3 corn areswith ex er'

p s the line shape at ~= 40. 1wi experiment and indicates that threliable. Th

a e results aree negative value of D

o en ia is repulsive for all R desome comment. Fir t

eservesirs, the model was used

a number of ground- st twith

n -s a e potentials and in ecase the method led t

in every

Second the 'e o negative values fo Dr

e ' a e van dere implication is not th t thinteraction is absent but ra ther that re-

in eractions are important at lar eof' te ole 1 d t

'llpotent' 1.

cura e ma inpp g of a more realistic

As an example of the kind of calculatiout to support the

in o c culation carriede parameters of case 1, a com-

putation which employs anothercase 2,

'ano er set of constants,

The parametere, is compared with experime t ' F'n in ig. 3.

B = 0. 426 x10p eters for case 2 are A = 0 425x10 ",0, G = (0. 151+ 0. 051)x10 and

D= ( 0. 884+0. 095)x10"state potential wh

D= —. . . Case 2hasa rg ound-

ial which is deeper (ez= 0. 107x10 '4

ergs) and closer (R2= 5. 2 A( 2= . A) than case 1 (e, =0.0513ergs, R&= 5. 9 A . Thee analysis of data at

. 1 yields the excited-state aragain, is negative.

TABLE V. Obsserved and calculatedhelium using the ene

shifts and widths fore general pressures m th de o 'in cm ~).

D. Summary of Results

Relativedensity

7.121.240. 1

+ 2.47+ 7.21+13.7


+ 1.3+ 5.5+13.0

5.518.035.6

7.820. 836.8


The result s of the present anal sis amarized in Tabl VI.

ysis are sum-a e ~ This table

e van der Waals const p

of the moments of the e e 'o e experimental line shapes.

4 INTERMOLECULAR FORCE S FROM ATOMIC LINE- SHAPE EXPERIMENTS 1373

-102TABLE Vl. Summary of potential parameters for cesium —rare-gas systems. Units: short-range constants, A!G(10

ergcm ); long-range constants, B/D (10 ' ergcm ); well depth, e(10 erg); and position of well, R, (A).

Perturberstate

This paperA/GB/D

R~

Mahan (Ref. 9)B/D

Baylis (Ref. 8)

R~

Moments (Ref. 4)

R~

Initial

0.8910.4260.0515.90

0.407

0.0506.34

0.0515.90

HeliumFinal

0.459—0.614

0.775

0.0119.1

Initial

3.683.300.745. 29

3.19

0.9025. 24

0.745. 29

ArgonFinal

6.485. 231.055.40

6. 13

0. 214(1.55)7.96 (3.86)

1.185. 24

Initial

12.37.91.275. 60

7. 71

2. 184. 91

1.275.60

XenonFinal

53.619.11.706.18

14.9

0.429 (10.7)8.25(3. 86)

3.045.52

Comparisons must be made with caution. In thepresent paper the coefficient of the R term in thepotential represents all of the "long-range" effects,

not just the C8 calculated by Mahan. gn fact, theresults for argon and xenon agree within about20%. ) The potentials computed by Baylis contain

I.O

0.9

K

5

Q4

X

FIG. 3. Line shapes of the longerwavelength component of the cesiumresonance lines in helium at a rela-tive density of 40. 1: (circles) experi-mental data from Chen; (solid line)case 1; (dashed line) case 2. Thelines were computed with the generalpressures theory using constants forcases 1 and 2 which are given in thetext. (Frequency here measured inunits of wave number, cm '. )

0.2

0 I

-20I

—IO

I

IOI

20I

50I

40I

50I

60 70

FREQUENCY (cm ')


much more detail than a Lennard- Jones potentialcan accommodate. For example, the excited-state potentials for argon and xenon contain twominima, a shallow one at large R and a deeperone at small R, as Table VI shows. Finally, theconstants of the present paper are more reliablethan those obtained by analyzing the first two mo-ments of the lines, because more data are employeddirectly in the determination and because the mo-ments are very sensitive to the line shape far fromthe line center. (The constants obtained from mo-ments of the cesium-xenon line did not yield agree-ment between experimental and theoretical lineshapes. )

The present result for the cesium-helium ex-cited-state potential shows good qualitative agree-ment with the calculated one. ' The calculated po-tential contains a very weak attractive region at

0

an intermolecular distance of about 9 A; the po-tential is repulsive over most of the region of im-portance for the line-shape calculation, becomingstrongly repulsive at about 3 A. The potential in-ferred in the present paper is repulsive every-where, becoming strongly repulsive at about 3. 5 A.

IV. CONCLUSIONS

When the analysis described in this paper wasinitiated, it seemed unreasonable to expect thattwo parameters in an excited-state potential wouldexplain one line shape. Preliminary calculationsindicated that, in fact, the model had a wide rangeof validity. The method described in Sec. II fur-nished a systematic technique for the study de-scribed in Sec. III. The conclusions are that thegross features of atomic spectral lines can be pre-dicted in a reliable way using the model, thatrealistic excited-state potentials can be inferredfrom line shapes for comparison with theoreticalcalculations, and that high-resolution data over awide range of frequencies must be available, pref-erably in tabular form in order to achieve a morecomplete understanding of neutral atom interactions.

ACKNOWLEDGMENTS

I am very grateful to Professor S. Y. Chen forproviding the line-shape data which made thisanalysis possible. I would also like to thankAhmet Atakan for his careful computations usingthe general pressures method.

Work supported by the National Aeronautics and SpaceAdministration under Research Grant No. NGL-43-001-006.

~(a) E. A. Mason and L. Monchick, in IntermolecularForces, Vol. 12 of Advances in Chemical Physics, editedby J. O. Hirschfelder (Interscience, New York, 1967),Chap. 7; (b) R. B. Bernstein and J. T. Muckerman, ibid.Chap. 8; (c) G. Birnbaum, ibid. Chap. 9.

W. Behmenberg, Z. Astrophys. 69, 368 (1968); W. R.Hindmarsh et al. , Proc. Roy. Soc. (London) A297, 296(1967); S. Y. Chdn, in Proceedings of the InternationalConference on Optical Pumping and Atomic Line Shape,Warsaw, Poland, 1968 (unpublished).

M. Takeo, Phys. Rev. A 1, 1143 (1970).H. C. Jacobson, preceding paper, Phys. Rev. A 4

1363 (1971).R. O. Garrett and S. Y. Ch6n, Phys. Rev. 144, 66

(1966); S. Y. Chbn and R. O. Garrett, ibid. 144, 59(1966); S. Y. Chdn, D. E. Gilbert, and D. K. Tan, ibid.184, 51 (1969).

R. H. Ritchie and V. E. Anderson, Nucl. Instr.Methods 45, 277 (1966).

'R. L. Fox and H. C. Jacobson, Phys. Rev. 188, 232(1969).

W. E. Baylis, J. Chem. Phys. 51, 2665 (1969).G. D. Mahan, J. Chem. Phys. 50, 2755 (1969).

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Intermolecular Forces from Atomic Line-Shape Experiments