# Central-Field Intermolecular Potentials from the Differential

## Transcript Of Central-Field Intermolecular Potentials from the Differential

Central-Field Intermolecular Potentials from the Differential Elastic Scattering of H&) by other Molecules *

BY h O N KUPPERMANRNO,BERJT. GORDONA?ND MICHAEJL. COGGIOLAf Arthur Amos Noyes Laboratory of Chemical Physics,§

California Institute of Technology, Pasadena, California 91109

Received 26th February, 1973

+ Differential elastic scattering cross sections for the systems Hz Oz, SF6, NH3, CO, and CH,

and for D2+02,SF6, and NH3 have been obtained from crossed beam studies. In all cases, rapid quantum oscillations have been resolved which permit the determination of intermolecular potential parameters if a central-field assumption is adopted. These potentials were found to be independent of both the isotopic form of the hydrogen molecule, and the relative collision energy. As a result of this, and the ability of these sphericd potentials quantitativelyto describe the measured scattering, it is concluded that anisotropy effects do not seem to be important in these HZ(Dz)systems.

The determination of interatomic and intermo!ecular potentials from molecular

beam experiments has received considerable attention over the last few years. Early

experiments at high energy with various atomic,l ionicY2and molecular systems

yielded essentially structureless total cross sections. In order to determine the scale

of the potential from such data, it is essential to have absolute cross ~ e c t i o n sw, ~hich

require accurate calibration of beam intensities. It has long been recognized that the

calibration problem can be avoided if the cross section has structural features that

provide an in1Prnal “ calibration.” Recently, rapid quantum oscillations have been

resolved in dif m-rential elastic cross s e c t i o n ~ ,w~ h- ~ich provide the necessary calibra-

tion. The freqaa ncy of such undulations has been related,’* for central-field poten-

tials, to the ranp- of the potential according to the approximate expression

A e =(nrz/pva)=(A/20)

(1)

where A0 is the spacing of the oscillations, p is the reduced mass, v is the relative

collision velocity, a is a range parameter for the potential (e.g., the zero of the poten-

tial), and A is the de Broglie wavelength. As a result, well resolved rapid oscillations

permit the estimation of a independently of the shape and depth of the potential well.

A more quantitative fit to experiment of the differential cross sections calculated from

an assumed potential permits one to determine more quantitatively this potential for

systems subject to central forces. In particular, information about the depth of the

attractive well and the steepness of the repulsive part of the potential can be obtained.

Partly because of the simplicity of interpreting the experiments for central fields,

most of the measurements of quantum oscillations have been for atom-atom scattering.

* Work supported in part by the U.S. Atomic Energy Commission, Report Code No. CALT-

767P4-112.

t present address : Naval Research Laboratory, Washington, D.C.

$ Work performed in partial fulfilment of the requirements for the Ph.D. Degree in Chemistry at

the California Institute of Technology.

$ Contribution No. 4650.

145

146

CENTRAL-FIELD INTERMOLECULAR POTENTIALS

The first molecular system found to have oscillations given by (1) was D2+N2, reported by Winicur et aL6 In the present study, which is a continuation of their work, we have measured the differential elastic cross sections of D2 and H2 scattered by 02,CO, NH3, CH4, and SF,, in order to obtain information about the corresponding intermolecular potentials. The data are discussed from the perspective of a central field approximation. Variation of the relative collision energy and the use of both H2 and D2with the same scattering partner provide a useful test for the validity

of this approximation.

Some of the systems reported here have also been studied in total cross section experiments by Butz et al." and by Aquilante et aZ.I2 Information obtained from total and differential cross section measurements on the same systems are mutually

complementary.

EXPERIMENTAL

The crossed molecular beam apparatus is shown schematicallyin fig. 1 and 2. The main features of the machine are a movable, differentially pumped quadrupole mass spectrometer detector, a differentially pumped supersonic primary beam and a subsonic secondary beam perpendicular to the primary beam, all contained in a bakeable stainless steel 1200I. main vacuum chamber. The beams intersect the axis of the main chamber at the centre of rotation of the detector. The detector chamber is mounted on a semicircular shaped quadrant arm which pivots about the axis of the main chamber, while the detector is free to move along the rim of the quadrant out of the plane of the beams. Thus the detector can scan both colatitudinal and longitudinal angles, although only in-plane measurements were made in the present experiments. Pumping in the main chamber is by means of four 6 in. oil diffusion pumps, each having a nominal trapped speed of 12501./s, and a liquid nitrogen cooled titanium sublimation pump, with a calculated speed of 200001./s for air. The primary beam source chamber and buffer are pumped by a 6 in. oil diffusion pump (1250 IJs) and a 6 in. mercury diffusion pump (150 l./s), respectively.

FIG.1.-Vertical cross section of molecular beam apparatus. N-primary nozzle source, Sskimmer cone, VS-velocity selector, C-primary beam chopper, F-beam flag, CA-secondary beam glass capillary array, IS-electron bombardment ionizer, MF-quadrupole mass filter, EM-electron multiplier, TSP-titanium sublimator pump, OP-Orbion pump, IG-ionization gauge, BV-

bellows operated bakeout valve, VP-Pyrex view port.

All apertures in the apparatus are circular, with the entrance aperture of the detector housing (0.16 cm diam.) located 8.05 cm away from the intersection of the beams. The exit aperture of the primary beam chamber (0.21 cni diam.) is located 7.9cm away from the

A. KUPPERMANN, R . J. GORDON A N D M. J. COGGIOLA

147

scattering centre, and the secondary effusive source (0.16 cm diam.) is 0.5 cm from the centre. The primary beam is formed with the aid of a nozzle-skimmerarrangement giving a measured

- Mach number of 15 and an angular FWHM (full width at half maximum) of 1.4". A

jacket surrounding the nozzle tube permits one to cool the entire nozzle assembly to liquid

. nitrogen temperature

\

PRIMARY 1.4* FWHM

/

0

,5

CM

FIG.2.-Crossed beam geometry.

..2.4' FWHM

I

I {DETECTOR

0 is the measured laboratory scattering angle.

The secondary beam source consists of a glass capillary array attached to the end of a brass tube, which can be tilted out of the plane of the beams by pumping the air out of a stainless steel bellows attached to this source. When the secondary source is tilted, the two beams do not cross, and the background signal intensity can be measured. This procedure is superior to flagging the secondary beam since the latter method tends to modulate the background as well as the signal. The angular width of the secondary beam is 2.4" FWHM. The FWHM cross section of the beam intersection region in the collision plane has the approximate shape of a rectangle0.17 cm along the directionof the primary beam and 0.22 cm along the direction of the secondary beam. The angular resolution of the detector is approximately 2".

The heart of the apparatus is an Extranuclear 324-9 quadrupole mass spectrometer l3 mounted in a bakeable double differentially pumped chamber. The operating pressure in the ionization region, measured with an uncalibrated Bendix miniature ionization tube, is typically 2x Torr with the beams on, whereas in the main chamber, it is about 1 x lod6 Torr under these conditions. To obtain such a large pressure differential, we found it necessary to bake the spectrometer housing and Orbion pump for about 8 h at approximately 200°C whenever the machine was pumped down from atmospheric pressure. The mass spectrometer chamber is equipped with a bellows activated valve 6 cm in diameter which is kept open to the main chamber during the bake-out period in order to accelerate the removal of background gas.

Particles entering the mass spectrometer chamber pass successively through a high-efficiency electron impact ionizer, a series of electrostatic focusing lenses, and a 23 cm long Paul l4 quadrupole mass filter. Ions are detected by a 14 stage CuBe electron multiplier whose output is amplifiedby an Extranuclear tuned amplifierfollowed by a PrincetonApplied Research HR-8 phase sensitive detector. The amplified signal is finally converted to digital form by a Raytheon model ADC-24 analog-to-digital converter.

The apparatus is interfaced to an SCC-4700 computer, which serves several functions. First, it tilts the secondary beam in (" on " mode) and out (" off" mode) of the scattering plane. Second, the computer periodically samples and averages the amplified signal and subtracts the background from the total intensity. Third, it calculates the standard deviations for both " on " and " off" modes.

The signal to noise ratio varied from better than 100 at the small scattering angles to a

148

CENTRAL-FIELD INTERMOLECULAR POTENTIALS

minimum of 10 at the largest one. To correct for long term drift in the signal caused by such factors as fluctuations of beam intensities and gradual build-up of background in the mass spectrometer, a fixed scattering angfe (generally between 3.0" and 5.0") was chosen as a reference angle. After the measrtrement of the signal at each scattering angle, the intensity

at the reference angle was remeasured to provide a normalization factor. In this way, individual reIative intensity points were reproducible to within 5 % when remeasured on different days.

RESULTS

+ The differential cross sections for the systems H2 0 2 ,SFt,, CO, NH3, CH4 and + D2 02,SF6, NH3, were all measured using room temperatdre H2 and D2 beams,

+ + with a relative collision

H2 SF6 and H2 NH3

energy systems

of approximately 0.06 eV. were also made using an H2

Measurements beam cooled to

of the liquid

nitrogen temperature, with a relative energy of approximately 0.02eV. In addition,

the SF6 system was studied using a low temperature beam of para-hydrogen. These

experiments scan a wide range in the size, anisotropy and ihitial relative collision

energy of the scattering species, and of the corresponding de Broglie wavelengths.

The measured differential elastic cross sections are shown in fig. 3 to 7 inclusive,

together with the on-line computer determined error bars. The various curves drawn

through the measured points were fitted to the data as described below.

DETERMINATION OF THE INTERMOLECULAR POTENTIAL

In the interpretation of our data we have assumed that the differential elastic cross

sections measured are due to the spherically symmetric part of the intermolecular

potentials. The reason for this assumption and the tests of its validity are described

in the Discussion. In our analysis, a model potential function is assumed and the

potential parameters are varied until a least-squares fit of theory to experiment is

obtained .

In the present analysis we have used a Lennard-Jones (n, 6) potential, where the

repulsive exponent n was either fixed at 12 or 20, or was allowed to vary as a fitted

parameter. In addition, a Morse-cubic spline-van der Waals (MSV) potential l 5

was used in some systems. The MSV potential is defined by

1c(exp[-2p(r -r,)] -2 exp[-p(r -r,,,)]] r

Y ( r ) = cubic spline

- c6r-6

r1

The cubic spline function is a set of five cubic polynomials whose coefficients are

chosen to smoothly join the inner and outer branches of the potential. The end points

were taken such that Y ( r l )= -0.75 E, and r2 = rl +0.2 r,. The fitting parameters

were E, rm, p and C6. The corresponding differential cross sections were accurately

calculated using a partial wave expansion employing both JWKB and high energy

eikonal phase shifts, tested against accurate integration of the radial Schrodinger

equation to assure the validity of this method. In order to compare the computed

cross sections with the data, it is necessary to correct for velocity spread and angular

resolution of the apparatus. In trial calculations, we found that the former effect

tends to dampen the undulations at CM scattering angles > 15"while the latter damp-

ens the small angle scattering to roughly an equal extent. This situation differs from

that of Siska et aZ.15 who found that under their experimental conditions with both

beams supersonic, the effect of angular resolution was dominant at all scattering angles

and that they could lump both corrections into a single effective angular resoIution

function. Consequently, the calculated cross sections were transformed to the

J I

I

I

I

5

10

15

20

25

laboratory angle O/deg

FIG.3

10

15

20

25

laboratory angle O/deg

FIG.4

\

H, + N H 3

H2 + SF6

102-

I 00

I

I

t

I

5

10

15

20

25

IO"1

I

1

I

5

10

15

20

laboratory angle O/deg FIG.5

laboratory angle O/deg FIG6

FIG.3.-Plot of the product of the scattered intensityI times the sine of the angle 0 against 0 in the laboratory system

+ + of referencefor H2 O2and D2 0,collisions. The lower curvehas been shifted downwards by one decade. Points

are experimental, and curves are theoretical fits. The solid curves are the MSV fits, and the corresponding potentials

+ + were used to establish the outer and inner ordinate scales for the Hz O2 and D2 O2 results, respectively. The upper dashed curve is the LJ (12, 6) fit, and the dotted curves are the LJ (11, 6) fits.

+ + FIG.4.-Dif€erential scattering results for (room temperature) H2 SF6 and D2 SF6 collisions. Explanation of the cvrves is the same as fig. 3.

+ FIG.5.-Differential scattering results for (room temperature) Hz+NH3 and Dz NH3collisions. Explanation of

the curves as for fig. 3. The LJ (12, 6) fits were indistinguishable from the LJ (n,6) ones and were not plotted.

+ FIG.6.-Low temperature results for H2 NH, and H2+ SF6collisions. Explanation of curves is given in fig. 3.

The

LJ

(

n

,

6)

curve

+ for Hz+NH3was para-hydrogen

indistinguishable from the SF6were identical to those

MSV curve and was not plotted. shown for normal-hydrogen.

Results using

1 50

C E N T R A L -FI E L D I N T E R M O L E C U L A R P O T ENTI A LS

laboratory system and averaged over both the relative collision energy distribution and

the detector angular resolution. The potential parameters were fitted to the data by minimizing the weighted sum

of squares of the differences between the cross sections calculated as just described and the experimental results, treating the vertical scale coefficient as a fitting parameter. For the Lennard-Jones potentials with n fixed, the fitted parameters E and B

were found using a simple Newton’s method. In the case of the MSV (E, r,, p, c6)

and the three parameter Lennard-Jones (E, B, n) potentials, a general method due to Marquardt l 6was used. In the followingsections, the quoted values for the uncertain-

ties of the potential parameters are those corresponding to a 95 % confidence level.

FIG.7.-Differential

10

15

20

25

laboratory angle O/deg

+ + scatteringresults for the H2 CH4 and H2 CO collisions.

were attempted, and they are shown by the solid curves.

Only LJ (12, 6)fits

All the systems were initially fitted with an LJ (12, 6) potential. The optimum

values of E and CT and their 95 % confidence levels are listed in table I together with A,

the de Broglie wavelength for each system, and Q the total cross section as calculated from the partial wave expansion. In addition, the results of the LJ (20, 6) and (n,6) fits are also given in this table. The 02,SF6 and NH3 data were measured with the

+ most accuracy, and hence were chosen for the four parameter MSV fits. The HZ+

CH4 and H2 CO data were of poorer reproducibility quality and for this reason not submitted to such fits. The corresponding parameters are listed in table 2 together with the values for ;1and Q. Various calculated differential cross sections are shown

in fig. 3 to 7 inclusive. In all cases, the Q thus determined was within 10 % of the

value predicted by eqn (1). It is worth emphasizing that while the statistical uncertainties in the fitted potential parameters listed in tables 1 and 2 are often quite small, it does not follow that the “ true ” values of these quantities (e.g., the actual

well depth) must lie within the predicted ranges.

A. KUPPERMANN, R. J. GORDON AND M. J. COGGIOLA

151

In fig. 8, 9 and 10 are shown fitted LJ and MSV potentials for the 0 2 ,SF6 and NH3 data. In each case, the MSV and LJ potential with fitted repulsive parameter are given for the room temperature H, system, while only the MSV fit is given for the

TABLE1.-LENNARD-JONE(Sn, 6) POTENTIAL PARAMETERS AND TOTAL CROSS SECTIONS

II

CIA

e/meV

)./A

Q/A2

12

3.38+ 0.03

7.7f 0.9

0.84

2 0 8 1 15

13.2

3.40f0.04

7.2f 0.9

0.84

182f 12

20

3.46k0.04

7.6+ 1.2

0.84

1691. 15

12

3.510.2

7.3k0.6

0.61

270521

13.6

3.5f0.3

7.0k0.7

0.61

251f:20

20

3.6f0.2

6.7+ 0.9

0.61

222+ 30

12

4.05 10.06

10.4f0.5

0.81

380+41

16.2

4.121.0.04

10.4f0.3

0.81

361 5 30

20

4.151 0.04

10.4k0.8

0.81

335+ 31

12

4.1 5f 0.08

9.6f 0.3

1.52

326f 3 1

16.3

4.18+ 0.05

lO.O* 0.2

1.52

325+ 28

20

4.1410.04

l0.5+ 0.2

1.52

3135 19

12

4.2f 0.4

10.310.6

0.58

3801. 32

19.1

4.2+ 0.2

10.35 0.3

0.58

331+_ 30

20

4.2f 0.2

10.3k 0 . 4

0.58

3345 31

9.1

3.34f0.07

9.6+ 1.2

0.87

2251.15

12

3.45k0.06

9.8k 1.4

0.87

260k 18

9.1

3.34+ 0.09

10.3k0.7

1.56

256k 17

12

3.34f 0.08

10.3f0.8

1.56

2 5 5 1 17

9.2

3.39+ 0.08

9.1k0.8

0.65

2 5 0 5 21

12

3.2610.07

9.1 k 0.7

0.65

245k21

12

3.5k0.1

6.9f 1.5

0.84

2105 18

12

3.7f0.2

9.9+ 1.4

0.87

317+26

corresponding D2systems. Those potentials not shown, were in general, indisting-

uishable from those which were plotted. In the case of the SF6 and NH3 systems, all three potentials are seen to be in very close agreement, while for the O2 systems, the agreement is somewhat poorer. In all cases, however, the potentials overlap throughout the range plotted when the uncertainties in the potential parameters are taken into

TABLE2.-MORSE-SPLINE-VAN

DER WAALS (MSV) POTENTIAL PARAMETERS AND TOTAL

CROSS SECTIONS

system

ls!A

rm/A

E/meV

B

Cs/eVA6

?.JA Q/A2

HZ+02

DZ+02

H2+SF6

DZ+ SFs H2+NH3

DZ+NH3

3.3450.05 3.550.2

4.14f0.02 4.16k0.03

4.2+ 0.2 3.42k0.05 3.23f0.05 3.23k0.05

3.8650.05 4.03 f0.2 4.63 0.02 4.64k0.03 4.62+ 0.2 3.80+0.05 3.88 f 0.05 3.77+ 0.05

7.2+ 0.6 5.2k0.4 64.8k0.7 0.84 213f 19

6.9f 0.9 4.8f0.4 63.1f0.6 0.61 291226

lO.O+ 0.2 6.5+ 0.5 57.2k0.3 0.81 396f 33

10.210.2 6.3k0.6 55.9k0.5 1.52 325)31

10.450.8 6.6f 0.6 54.6k0.5 0.58 389+ 38

9.7k0.5 4.9+ 0.4 58.2k0.6 0.87 288+21

10.2k0.5 4.8 k0.4 59.2+ 0.7 1.56 242f20

9.0k0.8 4.9k0.4 59.810.7 0.65 258-1 18

account. Hence, to within the experimental errors, the potentials for the H2 and D, isotopes are the same for a given scattering partner, and the resulting potential is independent of the mathematical form chosen, and of the de Broglie wavelength. It

1 52

CENTRAL-FIELD INTERMOLECULAR POTENTIALS

should be noted that agreement of the long range regions of the potentials is expected since both the LJ and MSV forms are chosen to have an r6dependence and, in addition, the measured scattering is not very sensitive to this region. The range of intermolecular distances sampled in these experiments, and depicted in fig. 8 to 10 inclusive, was approximately estimated by calculating the classical deflection function from the MSV potentials and considering the range of angles in the CM system covered for each system.

1 H2 + 02

- MSV - LJ( 13.2-61 --

rlA

FIG. 8.-Comparison of the intermolecular potentials over the range of distances sampled for

H 2 + 0 2 ( A = 0.84& and D 2 + 0 2 ( A = 0.61 A), determined from the data in fig. 3. The solid

+ + curve is the H2 O2 MSV potential, while the dashed curve is the Hz O2LJ (n,6) potential. The + dotted curve represents the D2 O2 MSV potential.

30.0

20.0 -

-% 10.0-

E

n

0.0-

- H, + sF6

MSV LJ (16.2-6)---

Dz SFG +

MSV

........s

-10.0-

' -20.03,0

1

t

I

1

I

1

4.0

5.0

6.0

7

FIG.9.-Comparison of the intermolecular potentials for H2+SF6 ( A = 0.81 A) and D2+SF6 (A = 0.58 A). Explanation of the curves is given in fig. 8. The correspondingcurves for H2+SF6

at h = 1.52 A are indistinguishablefrom those at A = 0.87 A, within plotting accuracy.

A . KUPPERMANN, R. J . G O R D O N A N D M. J . C O G G I O L A

153

20.0 -

5 10.0 -

1 A

2 0.0-

-10.0 -

- H2+ N H 3

MSV

LJ (9.1-6)---

D M

2 S

+ V

NH3

........

-20.01

1

I

I

I

I

I

1

3.0

4.0

5.0

6.0

I 3

rlA

+ FIG. 10.-Comparison of the intermolecular potentials for H Z + N H 3( A = 0.87A) and D Z + N H 3

( A = 0.65 A). Explanation of the curves is given in fig. 8. The correspondingcurves for H2 NH3 at X = 1.56 A are indistinguishablefrom those at h = 0.87 A, within plotting accuracy.

DISCUSSION

Ford and Wheeler l1 have shown by semi-classical techniques and for a centralfield potential having an overall shape analogous to that of a n LJ (12,6) potential that when the deflection function has a relative extremum, interference between the attractive and repulsive branches leads to rapid oscillations superimposed on the broader supranumerary rainbow undulations. In the past, oscillations of the sort reported here have been described qualitatively as resulting from such an interference effect. This description is incorrect for our systems because in the quantum limit, where the de Broglie wavelength becomes comparable to the potential range, the Ford and Wheeler analysis is inapplicable. The breakdown of the semi-classicaldescription is seen in at least two ways. First, we have observed strong undulations at angles considerably larger than the rainbow angle, whereas the semi-classical description

+ predicts that oscillations die out rapidly on the dark side of the rainbow. For

example, the LJ (12, 6) fit for the H2 O2 system predicts a classical rainbow at 15" in the CM, whereas we see strong oscillations out to 25". Indeed, the absence of rainbows both in theory and experiment for these systems shows that the semiclassical approach cannot be used here. Second, accurate quantum mechanical theoretical calculations predict oscillations with a spacing given by eqn (1) for purely repulsive potentials with monotonic deflection functions. The Ford and Wheeler analysis, however, reduces to the classical result whenever the deflection function is single-branched, and no undulations are possible. The oscillations in our systems are more accurately described as a diffraction effect produced at the steep repulsive wall of the p0tentia1.l~ The presence of an attractive well intensifies the diffraction oscillations and can increase their frequency since in this case the appropriate range parameter to use in eqn (1) is r, rather than 0. However, since in most cases the van der Waals minimum occurs at a distance ,Y only slightly larger than the zero of the potential, the frequency of the undulations is only slightly affected by the presence of the well.

The intermolecular potentials of the systems we have studied are anisotropic ;

1 54

CENTRAL-FIELD INTERMOLECULAR POTENTIALS

consequently, the interpretation of our data is more complicated than for atom-atom

scattering. One approximate way of coping with this difficulty is to separate the

potential into a spherical and an anisotropic part. We then assume that the effect

of the latter is unimportant due to a combination of rotational averaging and the

likelihood that the decrease of the magnitude of this anisotropy with the intermolecu-

lar distance, Y, is more rapid than that of the spherically symmetric part, making it

already sufficientlysmall for the distance range sampled by the present experiments.

A partial wave expansion can then be used to determine the isotropic part of the

potential, as was done in the previous section. Such an analysis, however, is not

necessarily correct since the anisotropy may dampen or “ quench ” the oscillations and

possibly shift their locations. Rothe and Helbing 2o and Kramer and LeBreton 21

report quenching of the glory undulations in the total scattering cross section of

alkali atoms by various large asymmetric molecules. On the other hand, Aquilante

et aZ.I2find no evidence for quenching in the glory scattering of D2 by N2 and several

hydrocarbons. Also, Butz et a l l 1 were able to fit the glory undulations in the total

cross sections of He, HD and D, scattered by CH4, N2, O,, NO and CO using a

spherical Lennard-Jones (12, 6) potential. Only the CO, glories appeared slightly

dampened, as compared with their theoretical calculations. Turning to the rainbow

maximum, Anlauf et aL2, found that for Ar+N, it was weaker than expected from a

Lennard-Jones (n, 6) potential (with best fit obtained for n = 20), and attribute this

+ difference to

Ar N2 with

+ quenching. Similarly, Cavallini et ~ 1 com. pa~red ~the rainbow

that of Ar Ar and attributed the dampening of its intensity and

of the

shift of its position to higher angles to anisotropy effects. Tully and Lee,24 after

studying the same Ar+N, system, assume that the shift in the rainbow position to

larger angles is negligible, but that the quenching is not, and get a slightly deeper well

than Anlauf et al. Stolte 25 measured the total cross section of Ar+NO with the

rotational quantum numbers of NO selected to be J = MJ = 3a n d J = MJ = 4,and

found that the anisotropic contribution to the total cross section is less than 1 %.

Farrar and Lee 26 have seen rapid quantum oscillations in the differential elastic

scattering cross section for the p-H, +p-€3, system, and were able to interpret their

data using a central-field assumption. We now consider the theoretical calculations

on anisotropy effects on differential elastic cross sections done so far.

Cross 27 found in an approximate semi-classical calculation, using a potential

with an isotropic part similar to that of K + K r that anisotropy can significantly

quench glory, rainbow and “ rapid ” oscillations. However, Cross’ theory, which is

based on the Ford and Wheeler treatment of interference between different branches

of the deflection function, is inapplicable to our systems where the undulations are

produced to a large extent by diffraction at the steep repulsive wall of the potential.

Furthermore, he assumes that the dependence of the isotropic and anisotropic parts

of the potential is identical, an assumption subject to question. Finally, the systems

treated in the present paper are more highly quantum than that considered by Cross,

and the anisotropic effects are expected to be quantitatively different. Wagner and

+ M c K o ~ i,n~a~n exact solution of the Schrodinger equation for the scattering of

Ar H,, found no significant quenching or shifting of the rapid quantum undulations.

However, their results provide only a lower estimate on these effects since H2 is more

isotropic than other diatomic molecules, and rotational transitions, which play an

important role in quenching, are less likely for low energy collisions with H,.

The range of intermolecular distances sampled in the present experiments, esti-

mated by a semi-classical analysis as described in the previous section, and depicted

in fig. 8 to 10 inclusive, includes part of the repulsive wall and the minimum in the

attractive well. We conclude from the present experiments that in this range, and

BY h O N KUPPERMANRNO,BERJT. GORDONA?ND MICHAEJL. COGGIOLAf Arthur Amos Noyes Laboratory of Chemical Physics,§

California Institute of Technology, Pasadena, California 91109

Received 26th February, 1973

+ Differential elastic scattering cross sections for the systems Hz Oz, SF6, NH3, CO, and CH,

and for D2+02,SF6, and NH3 have been obtained from crossed beam studies. In all cases, rapid quantum oscillations have been resolved which permit the determination of intermolecular potential parameters if a central-field assumption is adopted. These potentials were found to be independent of both the isotopic form of the hydrogen molecule, and the relative collision energy. As a result of this, and the ability of these sphericd potentials quantitativelyto describe the measured scattering, it is concluded that anisotropy effects do not seem to be important in these HZ(Dz)systems.

The determination of interatomic and intermo!ecular potentials from molecular

beam experiments has received considerable attention over the last few years. Early

experiments at high energy with various atomic,l ionicY2and molecular systems

yielded essentially structureless total cross sections. In order to determine the scale

of the potential from such data, it is essential to have absolute cross ~ e c t i o n sw, ~hich

require accurate calibration of beam intensities. It has long been recognized that the

calibration problem can be avoided if the cross section has structural features that

provide an in1Prnal “ calibration.” Recently, rapid quantum oscillations have been

resolved in dif m-rential elastic cross s e c t i o n ~ ,w~ h- ~ich provide the necessary calibra-

tion. The freqaa ncy of such undulations has been related,’* for central-field poten-

tials, to the ranp- of the potential according to the approximate expression

A e =(nrz/pva)=(A/20)

(1)

where A0 is the spacing of the oscillations, p is the reduced mass, v is the relative

collision velocity, a is a range parameter for the potential (e.g., the zero of the poten-

tial), and A is the de Broglie wavelength. As a result, well resolved rapid oscillations

permit the estimation of a independently of the shape and depth of the potential well.

A more quantitative fit to experiment of the differential cross sections calculated from

an assumed potential permits one to determine more quantitatively this potential for

systems subject to central forces. In particular, information about the depth of the

attractive well and the steepness of the repulsive part of the potential can be obtained.

Partly because of the simplicity of interpreting the experiments for central fields,

most of the measurements of quantum oscillations have been for atom-atom scattering.

* Work supported in part by the U.S. Atomic Energy Commission, Report Code No. CALT-

767P4-112.

t present address : Naval Research Laboratory, Washington, D.C.

$ Work performed in partial fulfilment of the requirements for the Ph.D. Degree in Chemistry at

the California Institute of Technology.

$ Contribution No. 4650.

145

146

CENTRAL-FIELD INTERMOLECULAR POTENTIALS

The first molecular system found to have oscillations given by (1) was D2+N2, reported by Winicur et aL6 In the present study, which is a continuation of their work, we have measured the differential elastic cross sections of D2 and H2 scattered by 02,CO, NH3, CH4, and SF,, in order to obtain information about the corresponding intermolecular potentials. The data are discussed from the perspective of a central field approximation. Variation of the relative collision energy and the use of both H2 and D2with the same scattering partner provide a useful test for the validity

of this approximation.

Some of the systems reported here have also been studied in total cross section experiments by Butz et al." and by Aquilante et aZ.I2 Information obtained from total and differential cross section measurements on the same systems are mutually

complementary.

EXPERIMENTAL

The crossed molecular beam apparatus is shown schematicallyin fig. 1 and 2. The main features of the machine are a movable, differentially pumped quadrupole mass spectrometer detector, a differentially pumped supersonic primary beam and a subsonic secondary beam perpendicular to the primary beam, all contained in a bakeable stainless steel 1200I. main vacuum chamber. The beams intersect the axis of the main chamber at the centre of rotation of the detector. The detector chamber is mounted on a semicircular shaped quadrant arm which pivots about the axis of the main chamber, while the detector is free to move along the rim of the quadrant out of the plane of the beams. Thus the detector can scan both colatitudinal and longitudinal angles, although only in-plane measurements were made in the present experiments. Pumping in the main chamber is by means of four 6 in. oil diffusion pumps, each having a nominal trapped speed of 12501./s, and a liquid nitrogen cooled titanium sublimation pump, with a calculated speed of 200001./s for air. The primary beam source chamber and buffer are pumped by a 6 in. oil diffusion pump (1250 IJs) and a 6 in. mercury diffusion pump (150 l./s), respectively.

FIG.1.-Vertical cross section of molecular beam apparatus. N-primary nozzle source, Sskimmer cone, VS-velocity selector, C-primary beam chopper, F-beam flag, CA-secondary beam glass capillary array, IS-electron bombardment ionizer, MF-quadrupole mass filter, EM-electron multiplier, TSP-titanium sublimator pump, OP-Orbion pump, IG-ionization gauge, BV-

bellows operated bakeout valve, VP-Pyrex view port.

All apertures in the apparatus are circular, with the entrance aperture of the detector housing (0.16 cm diam.) located 8.05 cm away from the intersection of the beams. The exit aperture of the primary beam chamber (0.21 cni diam.) is located 7.9cm away from the

A. KUPPERMANN, R . J. GORDON A N D M. J. COGGIOLA

147

scattering centre, and the secondary effusive source (0.16 cm diam.) is 0.5 cm from the centre. The primary beam is formed with the aid of a nozzle-skimmerarrangement giving a measured

- Mach number of 15 and an angular FWHM (full width at half maximum) of 1.4". A

jacket surrounding the nozzle tube permits one to cool the entire nozzle assembly to liquid

. nitrogen temperature

\

PRIMARY 1.4* FWHM

/

0

,5

CM

FIG.2.-Crossed beam geometry.

..2.4' FWHM

I

I {DETECTOR

0 is the measured laboratory scattering angle.

The secondary beam source consists of a glass capillary array attached to the end of a brass tube, which can be tilted out of the plane of the beams by pumping the air out of a stainless steel bellows attached to this source. When the secondary source is tilted, the two beams do not cross, and the background signal intensity can be measured. This procedure is superior to flagging the secondary beam since the latter method tends to modulate the background as well as the signal. The angular width of the secondary beam is 2.4" FWHM. The FWHM cross section of the beam intersection region in the collision plane has the approximate shape of a rectangle0.17 cm along the directionof the primary beam and 0.22 cm along the direction of the secondary beam. The angular resolution of the detector is approximately 2".

The heart of the apparatus is an Extranuclear 324-9 quadrupole mass spectrometer l3 mounted in a bakeable double differentially pumped chamber. The operating pressure in the ionization region, measured with an uncalibrated Bendix miniature ionization tube, is typically 2x Torr with the beams on, whereas in the main chamber, it is about 1 x lod6 Torr under these conditions. To obtain such a large pressure differential, we found it necessary to bake the spectrometer housing and Orbion pump for about 8 h at approximately 200°C whenever the machine was pumped down from atmospheric pressure. The mass spectrometer chamber is equipped with a bellows activated valve 6 cm in diameter which is kept open to the main chamber during the bake-out period in order to accelerate the removal of background gas.

Particles entering the mass spectrometer chamber pass successively through a high-efficiency electron impact ionizer, a series of electrostatic focusing lenses, and a 23 cm long Paul l4 quadrupole mass filter. Ions are detected by a 14 stage CuBe electron multiplier whose output is amplifiedby an Extranuclear tuned amplifierfollowed by a PrincetonApplied Research HR-8 phase sensitive detector. The amplified signal is finally converted to digital form by a Raytheon model ADC-24 analog-to-digital converter.

The apparatus is interfaced to an SCC-4700 computer, which serves several functions. First, it tilts the secondary beam in (" on " mode) and out (" off" mode) of the scattering plane. Second, the computer periodically samples and averages the amplified signal and subtracts the background from the total intensity. Third, it calculates the standard deviations for both " on " and " off" modes.

The signal to noise ratio varied from better than 100 at the small scattering angles to a

148

CENTRAL-FIELD INTERMOLECULAR POTENTIALS

minimum of 10 at the largest one. To correct for long term drift in the signal caused by such factors as fluctuations of beam intensities and gradual build-up of background in the mass spectrometer, a fixed scattering angfe (generally between 3.0" and 5.0") was chosen as a reference angle. After the measrtrement of the signal at each scattering angle, the intensity

at the reference angle was remeasured to provide a normalization factor. In this way, individual reIative intensity points were reproducible to within 5 % when remeasured on different days.

RESULTS

+ The differential cross sections for the systems H2 0 2 ,SFt,, CO, NH3, CH4 and + D2 02,SF6, NH3, were all measured using room temperatdre H2 and D2 beams,

+ + with a relative collision

H2 SF6 and H2 NH3

energy systems

of approximately 0.06 eV. were also made using an H2

Measurements beam cooled to

of the liquid

nitrogen temperature, with a relative energy of approximately 0.02eV. In addition,

the SF6 system was studied using a low temperature beam of para-hydrogen. These

experiments scan a wide range in the size, anisotropy and ihitial relative collision

energy of the scattering species, and of the corresponding de Broglie wavelengths.

The measured differential elastic cross sections are shown in fig. 3 to 7 inclusive,

together with the on-line computer determined error bars. The various curves drawn

through the measured points were fitted to the data as described below.

DETERMINATION OF THE INTERMOLECULAR POTENTIAL

In the interpretation of our data we have assumed that the differential elastic cross

sections measured are due to the spherically symmetric part of the intermolecular

potentials. The reason for this assumption and the tests of its validity are described

in the Discussion. In our analysis, a model potential function is assumed and the

potential parameters are varied until a least-squares fit of theory to experiment is

obtained .

In the present analysis we have used a Lennard-Jones (n, 6) potential, where the

repulsive exponent n was either fixed at 12 or 20, or was allowed to vary as a fitted

parameter. In addition, a Morse-cubic spline-van der Waals (MSV) potential l 5

was used in some systems. The MSV potential is defined by

1c(exp[-2p(r -r,)] -2 exp[-p(r -r,,,)]] r

Y ( r ) = cubic spline

- c6r-6

r1

The cubic spline function is a set of five cubic polynomials whose coefficients are

chosen to smoothly join the inner and outer branches of the potential. The end points

were taken such that Y ( r l )= -0.75 E, and r2 = rl +0.2 r,. The fitting parameters

were E, rm, p and C6. The corresponding differential cross sections were accurately

calculated using a partial wave expansion employing both JWKB and high energy

eikonal phase shifts, tested against accurate integration of the radial Schrodinger

equation to assure the validity of this method. In order to compare the computed

cross sections with the data, it is necessary to correct for velocity spread and angular

resolution of the apparatus. In trial calculations, we found that the former effect

tends to dampen the undulations at CM scattering angles > 15"while the latter damp-

ens the small angle scattering to roughly an equal extent. This situation differs from

that of Siska et aZ.15 who found that under their experimental conditions with both

beams supersonic, the effect of angular resolution was dominant at all scattering angles

and that they could lump both corrections into a single effective angular resoIution

function. Consequently, the calculated cross sections were transformed to the

J I

I

I

I

5

10

15

20

25

laboratory angle O/deg

FIG.3

10

15

20

25

laboratory angle O/deg

FIG.4

\

H, + N H 3

H2 + SF6

102-

I 00

I

I

t

I

5

10

15

20

25

IO"1

I

1

I

5

10

15

20

laboratory angle O/deg FIG.5

laboratory angle O/deg FIG6

FIG.3.-Plot of the product of the scattered intensityI times the sine of the angle 0 against 0 in the laboratory system

+ + of referencefor H2 O2and D2 0,collisions. The lower curvehas been shifted downwards by one decade. Points

are experimental, and curves are theoretical fits. The solid curves are the MSV fits, and the corresponding potentials

+ + were used to establish the outer and inner ordinate scales for the Hz O2 and D2 O2 results, respectively. The upper dashed curve is the LJ (12, 6) fit, and the dotted curves are the LJ (11, 6) fits.

+ + FIG.4.-Dif€erential scattering results for (room temperature) H2 SF6 and D2 SF6 collisions. Explanation of the cvrves is the same as fig. 3.

+ FIG.5.-Differential scattering results for (room temperature) Hz+NH3 and Dz NH3collisions. Explanation of

the curves as for fig. 3. The LJ (12, 6) fits were indistinguishable from the LJ (n,6) ones and were not plotted.

+ FIG.6.-Low temperature results for H2 NH, and H2+ SF6collisions. Explanation of curves is given in fig. 3.

The

LJ

(

n

,

6)

curve

+ for Hz+NH3was para-hydrogen

indistinguishable from the SF6were identical to those

MSV curve and was not plotted. shown for normal-hydrogen.

Results using

1 50

C E N T R A L -FI E L D I N T E R M O L E C U L A R P O T ENTI A LS

laboratory system and averaged over both the relative collision energy distribution and

the detector angular resolution. The potential parameters were fitted to the data by minimizing the weighted sum

of squares of the differences between the cross sections calculated as just described and the experimental results, treating the vertical scale coefficient as a fitting parameter. For the Lennard-Jones potentials with n fixed, the fitted parameters E and B

were found using a simple Newton’s method. In the case of the MSV (E, r,, p, c6)

and the three parameter Lennard-Jones (E, B, n) potentials, a general method due to Marquardt l 6was used. In the followingsections, the quoted values for the uncertain-

ties of the potential parameters are those corresponding to a 95 % confidence level.

FIG.7.-Differential

10

15

20

25

laboratory angle O/deg

+ + scatteringresults for the H2 CH4 and H2 CO collisions.

were attempted, and they are shown by the solid curves.

Only LJ (12, 6)fits

All the systems were initially fitted with an LJ (12, 6) potential. The optimum

values of E and CT and their 95 % confidence levels are listed in table I together with A,

the de Broglie wavelength for each system, and Q the total cross section as calculated from the partial wave expansion. In addition, the results of the LJ (20, 6) and (n,6) fits are also given in this table. The 02,SF6 and NH3 data were measured with the

+ most accuracy, and hence were chosen for the four parameter MSV fits. The HZ+

CH4 and H2 CO data were of poorer reproducibility quality and for this reason not submitted to such fits. The corresponding parameters are listed in table 2 together with the values for ;1and Q. Various calculated differential cross sections are shown

in fig. 3 to 7 inclusive. In all cases, the Q thus determined was within 10 % of the

value predicted by eqn (1). It is worth emphasizing that while the statistical uncertainties in the fitted potential parameters listed in tables 1 and 2 are often quite small, it does not follow that the “ true ” values of these quantities (e.g., the actual

well depth) must lie within the predicted ranges.

A. KUPPERMANN, R. J. GORDON AND M. J. COGGIOLA

151

In fig. 8, 9 and 10 are shown fitted LJ and MSV potentials for the 0 2 ,SF6 and NH3 data. In each case, the MSV and LJ potential with fitted repulsive parameter are given for the room temperature H, system, while only the MSV fit is given for the

TABLE1.-LENNARD-JONE(Sn, 6) POTENTIAL PARAMETERS AND TOTAL CROSS SECTIONS

II

CIA

e/meV

)./A

Q/A2

12

3.38+ 0.03

7.7f 0.9

0.84

2 0 8 1 15

13.2

3.40f0.04

7.2f 0.9

0.84

182f 12

20

3.46k0.04

7.6+ 1.2

0.84

1691. 15

12

3.510.2

7.3k0.6

0.61

270521

13.6

3.5f0.3

7.0k0.7

0.61

251f:20

20

3.6f0.2

6.7+ 0.9

0.61

222+ 30

12

4.05 10.06

10.4f0.5

0.81

380+41

16.2

4.121.0.04

10.4f0.3

0.81

361 5 30

20

4.151 0.04

10.4k0.8

0.81

335+ 31

12

4.1 5f 0.08

9.6f 0.3

1.52

326f 3 1

16.3

4.18+ 0.05

lO.O* 0.2

1.52

325+ 28

20

4.1410.04

l0.5+ 0.2

1.52

3135 19

12

4.2f 0.4

10.310.6

0.58

3801. 32

19.1

4.2+ 0.2

10.35 0.3

0.58

331+_ 30

20

4.2f 0.2

10.3k 0 . 4

0.58

3345 31

9.1

3.34f0.07

9.6+ 1.2

0.87

2251.15

12

3.45k0.06

9.8k 1.4

0.87

260k 18

9.1

3.34+ 0.09

10.3k0.7

1.56

256k 17

12

3.34f 0.08

10.3f0.8

1.56

2 5 5 1 17

9.2

3.39+ 0.08

9.1k0.8

0.65

2 5 0 5 21

12

3.2610.07

9.1 k 0.7

0.65

245k21

12

3.5k0.1

6.9f 1.5

0.84

2105 18

12

3.7f0.2

9.9+ 1.4

0.87

317+26

corresponding D2systems. Those potentials not shown, were in general, indisting-

uishable from those which were plotted. In the case of the SF6 and NH3 systems, all three potentials are seen to be in very close agreement, while for the O2 systems, the agreement is somewhat poorer. In all cases, however, the potentials overlap throughout the range plotted when the uncertainties in the potential parameters are taken into

TABLE2.-MORSE-SPLINE-VAN

DER WAALS (MSV) POTENTIAL PARAMETERS AND TOTAL

CROSS SECTIONS

system

ls!A

rm/A

E/meV

B

Cs/eVA6

?.JA Q/A2

HZ+02

DZ+02

H2+SF6

DZ+ SFs H2+NH3

DZ+NH3

3.3450.05 3.550.2

4.14f0.02 4.16k0.03

4.2+ 0.2 3.42k0.05 3.23f0.05 3.23k0.05

3.8650.05 4.03 f0.2 4.63 0.02 4.64k0.03 4.62+ 0.2 3.80+0.05 3.88 f 0.05 3.77+ 0.05

7.2+ 0.6 5.2k0.4 64.8k0.7 0.84 213f 19

6.9f 0.9 4.8f0.4 63.1f0.6 0.61 291226

lO.O+ 0.2 6.5+ 0.5 57.2k0.3 0.81 396f 33

10.210.2 6.3k0.6 55.9k0.5 1.52 325)31

10.450.8 6.6f 0.6 54.6k0.5 0.58 389+ 38

9.7k0.5 4.9+ 0.4 58.2k0.6 0.87 288+21

10.2k0.5 4.8 k0.4 59.2+ 0.7 1.56 242f20

9.0k0.8 4.9k0.4 59.810.7 0.65 258-1 18

account. Hence, to within the experimental errors, the potentials for the H2 and D, isotopes are the same for a given scattering partner, and the resulting potential is independent of the mathematical form chosen, and of the de Broglie wavelength. It

1 52

CENTRAL-FIELD INTERMOLECULAR POTENTIALS

should be noted that agreement of the long range regions of the potentials is expected since both the LJ and MSV forms are chosen to have an r6dependence and, in addition, the measured scattering is not very sensitive to this region. The range of intermolecular distances sampled in these experiments, and depicted in fig. 8 to 10 inclusive, was approximately estimated by calculating the classical deflection function from the MSV potentials and considering the range of angles in the CM system covered for each system.

1 H2 + 02

- MSV - LJ( 13.2-61 --

rlA

FIG. 8.-Comparison of the intermolecular potentials over the range of distances sampled for

H 2 + 0 2 ( A = 0.84& and D 2 + 0 2 ( A = 0.61 A), determined from the data in fig. 3. The solid

+ + curve is the H2 O2 MSV potential, while the dashed curve is the Hz O2LJ (n,6) potential. The + dotted curve represents the D2 O2 MSV potential.

30.0

20.0 -

-% 10.0-

E

n

0.0-

- H, + sF6

MSV LJ (16.2-6)---

Dz SFG +

MSV

........s

-10.0-

' -20.03,0

1

t

I

1

I

1

4.0

5.0

6.0

7

FIG.9.-Comparison of the intermolecular potentials for H2+SF6 ( A = 0.81 A) and D2+SF6 (A = 0.58 A). Explanation of the curves is given in fig. 8. The correspondingcurves for H2+SF6

at h = 1.52 A are indistinguishablefrom those at A = 0.87 A, within plotting accuracy.

A . KUPPERMANN, R. J . G O R D O N A N D M. J . C O G G I O L A

153

20.0 -

5 10.0 -

1 A

2 0.0-

-10.0 -

- H2+ N H 3

MSV

LJ (9.1-6)---

D M

2 S

+ V

NH3

........

-20.01

1

I

I

I

I

I

1

3.0

4.0

5.0

6.0

I 3

rlA

+ FIG. 10.-Comparison of the intermolecular potentials for H Z + N H 3( A = 0.87A) and D Z + N H 3

( A = 0.65 A). Explanation of the curves is given in fig. 8. The correspondingcurves for H2 NH3 at X = 1.56 A are indistinguishablefrom those at h = 0.87 A, within plotting accuracy.

DISCUSSION

Ford and Wheeler l1 have shown by semi-classical techniques and for a centralfield potential having an overall shape analogous to that of a n LJ (12,6) potential that when the deflection function has a relative extremum, interference between the attractive and repulsive branches leads to rapid oscillations superimposed on the broader supranumerary rainbow undulations. In the past, oscillations of the sort reported here have been described qualitatively as resulting from such an interference effect. This description is incorrect for our systems because in the quantum limit, where the de Broglie wavelength becomes comparable to the potential range, the Ford and Wheeler analysis is inapplicable. The breakdown of the semi-classicaldescription is seen in at least two ways. First, we have observed strong undulations at angles considerably larger than the rainbow angle, whereas the semi-classical description

+ predicts that oscillations die out rapidly on the dark side of the rainbow. For

example, the LJ (12, 6) fit for the H2 O2 system predicts a classical rainbow at 15" in the CM, whereas we see strong oscillations out to 25". Indeed, the absence of rainbows both in theory and experiment for these systems shows that the semiclassical approach cannot be used here. Second, accurate quantum mechanical theoretical calculations predict oscillations with a spacing given by eqn (1) for purely repulsive potentials with monotonic deflection functions. The Ford and Wheeler analysis, however, reduces to the classical result whenever the deflection function is single-branched, and no undulations are possible. The oscillations in our systems are more accurately described as a diffraction effect produced at the steep repulsive wall of the p0tentia1.l~ The presence of an attractive well intensifies the diffraction oscillations and can increase their frequency since in this case the appropriate range parameter to use in eqn (1) is r, rather than 0. However, since in most cases the van der Waals minimum occurs at a distance ,Y only slightly larger than the zero of the potential, the frequency of the undulations is only slightly affected by the presence of the well.

The intermolecular potentials of the systems we have studied are anisotropic ;

1 54

CENTRAL-FIELD INTERMOLECULAR POTENTIALS

consequently, the interpretation of our data is more complicated than for atom-atom

scattering. One approximate way of coping with this difficulty is to separate the

potential into a spherical and an anisotropic part. We then assume that the effect

of the latter is unimportant due to a combination of rotational averaging and the

likelihood that the decrease of the magnitude of this anisotropy with the intermolecu-

lar distance, Y, is more rapid than that of the spherically symmetric part, making it

already sufficientlysmall for the distance range sampled by the present experiments.

A partial wave expansion can then be used to determine the isotropic part of the

potential, as was done in the previous section. Such an analysis, however, is not

necessarily correct since the anisotropy may dampen or “ quench ” the oscillations and

possibly shift their locations. Rothe and Helbing 2o and Kramer and LeBreton 21

report quenching of the glory undulations in the total scattering cross section of

alkali atoms by various large asymmetric molecules. On the other hand, Aquilante

et aZ.I2find no evidence for quenching in the glory scattering of D2 by N2 and several

hydrocarbons. Also, Butz et a l l 1 were able to fit the glory undulations in the total

cross sections of He, HD and D, scattered by CH4, N2, O,, NO and CO using a

spherical Lennard-Jones (12, 6) potential. Only the CO, glories appeared slightly

dampened, as compared with their theoretical calculations. Turning to the rainbow

maximum, Anlauf et aL2, found that for Ar+N, it was weaker than expected from a

Lennard-Jones (n, 6) potential (with best fit obtained for n = 20), and attribute this

+ difference to

Ar N2 with

+ quenching. Similarly, Cavallini et ~ 1 com. pa~red ~the rainbow

that of Ar Ar and attributed the dampening of its intensity and

of the

shift of its position to higher angles to anisotropy effects. Tully and Lee,24 after

studying the same Ar+N, system, assume that the shift in the rainbow position to

larger angles is negligible, but that the quenching is not, and get a slightly deeper well

than Anlauf et al. Stolte 25 measured the total cross section of Ar+NO with the

rotational quantum numbers of NO selected to be J = MJ = 3a n d J = MJ = 4,and

found that the anisotropic contribution to the total cross section is less than 1 %.

Farrar and Lee 26 have seen rapid quantum oscillations in the differential elastic

scattering cross section for the p-H, +p-€3, system, and were able to interpret their

data using a central-field assumption. We now consider the theoretical calculations

on anisotropy effects on differential elastic cross sections done so far.

Cross 27 found in an approximate semi-classical calculation, using a potential

with an isotropic part similar to that of K + K r that anisotropy can significantly

quench glory, rainbow and “ rapid ” oscillations. However, Cross’ theory, which is

based on the Ford and Wheeler treatment of interference between different branches

of the deflection function, is inapplicable to our systems where the undulations are

produced to a large extent by diffraction at the steep repulsive wall of the potential.

Furthermore, he assumes that the dependence of the isotropic and anisotropic parts

of the potential is identical, an assumption subject to question. Finally, the systems

treated in the present paper are more highly quantum than that considered by Cross,

and the anisotropic effects are expected to be quantitatively different. Wagner and

+ M c K o ~ i,n~a~n exact solution of the Schrodinger equation for the scattering of

Ar H,, found no significant quenching or shifting of the rapid quantum undulations.

However, their results provide only a lower estimate on these effects since H2 is more

isotropic than other diatomic molecules, and rotational transitions, which play an

important role in quenching, are less likely for low energy collisions with H,.

The range of intermolecular distances sampled in the present experiments, esti-

mated by a semi-classical analysis as described in the previous section, and depicted

in fig. 8 to 10 inclusive, includes part of the repulsive wall and the minimum in the

attractive well. We conclude from the present experiments that in this range, and