The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies

The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies H . PAULY Instirut fur Angewandte Physik der Universitat Bovn. Bonn. Germany and

J . P . TOENNIES Physikalisrhes lnstitut der Universitat Bonn. Bonn. Germany

I . Intermolecular Potential

.......................................... 201 A . van der Waals Potential ........................................ 203 B. Short Range Repulsive Potential .............................. 210 C . Chemical Potential ............................................ 212 D. Models of the Intermolecular Potential . ........ 213 I1 Molecular Beam Method for the Experimen molecular Forces .................................................. 216 A . Introduction and Definitions .................................. 216 B. Measurement of Integral Total Scattering Cross Sections . . . . . . . . . . 220 C . Measurement of Integral Inelastic and Reactive Cross Sections .... 230 D . Measurement of Differential Cross Sections ...................... 231 E . Summary .................................................... 238 I11. Recent Advances in Experimental Techniques for Molecular Beam Scattering Experiments ............................................ 239 A . Introduction . . ........................... 239 B . Beam Sources ................................................ 240 C . Velocity Selectors ............................................ 249 D. State Selectors . . ...................... 250 E . Detectors ........ ...................... 252 F. Summary .................257 IV Molecular Sca . . . . . . . . . . . . . . . . . . . 257 A . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 B. Classical Scattering by Spherical Symmetric Potentials ............ 258 C . Quantum Mechanical Theory for Calculating Elastic Cross Sections . . 265 D . Inelastic Scattering Theory .................................... 284 V . Atom-Atom Scattering Experiments ................................ 296 A . Introduction ................................................ 296 B. Measurements of Integral Total Cross Sections . . . . . . . . . . . . . . . . . .296 C . Differential Scattering Cross Sections .......................... 305

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.

195

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H . Pauly and J. P . Toennies

VI. Scattering Experiments Involving Molecules. ......................... A. Introduction ................................................ B. Measurements of Nonreactive Integral Total Cross Sections without Rotational State Selection ........................................ C. Measurements of Nonreactive Integral Total Cross Sections with Rotational State Selection ........................................ D. Measurements of Inelastic Cross Sections for Rotational Excitation. . E. Summarizing Remarks on Nonreactive Cross Sections . . . . . . . . . . . . F. Measurements of Chemical Reactions .......................... VII. Concluding Remarks . . . . . . .......... .......... ... List of Symbols .................................................. References ......................................................

31 1 311 312 313 316 321 322 334 335 337

Introduction HISTORICAL SURVEY Early in the history of molecular beam research it was realized that molecular beams were ideally suited for the experimental verification of the kinetic theory of gases. The first experiments having as their aim the measurement of kinetic cross sections were performed by Max Born and Miss E. Bormann in 1920 (Born, 1920). In the following years a large number of fundamental experiments on the properties of atoms and molecules as well as on the basic properties of the beams themselves were reported. These included not only the classical experiments of Stern, e.g., the Stern-Gerlach experiment, but also a number of other experiments on gas kinetics, some of which were too difficult for the available techniques. For example, as early as 1925 Kroger (1925) in Germany unsuccessfully tried to observe a chemical reaction with molecular beams. As in most fields of molecular beam research, however, the first really noteworthy experiments were performed under Stern’s tutelage in Hamburg. In 1933 Knauer (1933a, b) reported differential cross sections for the scattering of the fundamental systems, H,-H, and He-He and these gases on Hg (Zabel, 1933). Other important developments in 1933 were the introduction of the crossed beam technique by Broadway (1933; Fraser and Broadway, 1933) and the first quantum mechanical calculations of atomatom collision cross sections by Massey and Mohr (1933, 1934), all in England. The experimental and theoretical techniques of these papers represent the backbone of modern beam scattering research. Unfortunately, 1933 also marked the forced end of molecular beam research in Hamburg. This event and the subsequent success of beam resonance experiments are probable explanations for the fact that the systematic investigation of intermolecular potentials by molecular beams is only now starting. For instance only ten years ago when Hirschfelder, Curtiss, and Bird published their comprehensive

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nonograph entitled “The Molecular Theory of Gases and Liquids” no mention was made of beam scattering experiments using thermal beams. Since then there has been new interest in the field. As a result of this recent work it appears reasonable to assert that this relatively new method of measuring intermolecular potentials is already as powerful as the older more classical methods of measuring transport properties and the equation of state. Nevertheless, in the past, beam scattering experiments have been actively pursued by only six small groups, presently located at the following institutions: Oak Ridge National Laboratory, University of Bonn, Brown University, University of Wisconsin, General Dynamics in San Diego, and Harvard University. The number of additional groups starting work in the field tempts one to speak of a renaissance of interest in molecular beam scattering. What are the aims of present-day beam scattering experiments ? Generally it is realized that scattering experiments at thermal energies as opposed to scattering experiments in the BeV range will not lead to changes in the fundamental structure of physical theory, the problems of interest merely corresponding to difficult mathematical solutions of the Schrodinger equation. On the other hand, since such solutions are out of the question for most systems, even with the advent of fast computers, there is considerable interest in developing techniques of both an experimental as well as a theoretical nature for learning more about intermolecular potentials and their role in inelastic collision processes and, especially, in the collision dynamics of chemical reactions. The final answers to these problems are long outstanding and, in view of the recent advances of beam scattering techniques, there is some reason to hope that they may be found with the help of such experiments.

SCOPEOF THIS REVIEW The purpose of this review is to survey the achievements of the last five years.’ In this period many of the older concepts and techniques have become obsolete, and for this reason we have attempted a review which, although it covers only recent work, will still be understandable to the uninitiated. By concentrating on the newer developments we also avoid repeating what is already available in a number of excellent monographs or reviews (Ramsey, 1956; Smith, 1955; King and Zacharias, 1956; Schlier, 1957; Estermann, 1959; Frisch, 1959). The scattering work up to 1960 has been reviewed by one of us (Pauly, 1961). The research discussed here is concerned only with thermal beam scattering. A review of the extremely important experiments of Amdur and co-workers using accelerated beams was not possible because of 1 Literature appearing up to the end of 1964 was considered for inclusion in this review.

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lack of space. This work has however been recently surveyed by Mason and Vanderslice (1962). The very interesting experiments on the interactions of beams with surfaces have also been left out. On the other hand we have included work on chemical reactions, which has already been reviewed elsewhere (Datz and Taylor, 1959; Fite and Datz, 1963; Ross and Greene, 1964; Greene et al., 1965; Herschbach, 1965). By including chemical reactions we hope that the connection between these and the elastic scattering experiments will become more apparent. After a brief discussion of the classical methods for investigating intermolecular potentials the present review begins with a survey of recent progress in the theoretical calculation of the intermolecular potential. The method of molecular beam scattering experiments is introduced in Section 11. The problems associated with the interpretation of the experimental results are discussed in this section. A recent reappraisal of the methods used in the absolute calibration of pressure measuring devices shows that the serious discrepancy between theoretical and experimental values of the van der Waals constants is considerably reduced. Although in most recent scattering experiments monochromatic beams are usually used it is still necessary to take account of the velocity averaging introduced by the spread in velocities of the scattering gas. In most experiments this averaging can be corrected for without much loss of information. The available beam intensity and the expected signal-to-noise ratio usually determine which experiments have reasonable chances of success. For this reason we have made an attempt in Section 111 to summarize recent advances in experimental techniques for the construction of beam sources and detectors. Of particular interest at the moment are jet sources, and it now appears that increased beam intensities are attainable with these special sources and also by simply raising the source pressure above that given by the Knudsen criterion. The design problems relating to beam detectors are analyzed, and several recent universal beam detectors are compared with the Langmuir-Taylor detector. The calculation of elastic and inelastic cross sections is discussed in Section IV. The problem is reduced to one of calculating the phase shifts, and, depending on the conditions, several approximate methods have proven to be useful. The high energy approximation, which has been successfully applied to a number of difficult problems involving anisotropic potentials, is derived from the Jeffreys-Wentzel-KramersBrillouin (JWKB) approximation. Because of its universality and relative simplicity, the high energy approximation is emerging as the most useful approximate method for calculating collision processes. The results of atomatom cross-section measurements are surveyed in Section V. The velocity oscillations of the integral (total) cross section provide a precise method for measuring the product w, (the depth of the attractive potential well times the radius at the minimum) independent of the pressure measurement. The recent

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experimental observation of rainbow scattering and interference effects in the differential cross section permit a determination of E and r, separately and even of the shape of the potential in the vicinity of the minimum. Results of scattering experiments in which one of the partners is a molecule are reviewed in Section V1. Recent measurements of inelastic cross sections for rotational excitation indicate that such experiments can provide detailed information on the various terms of the angle-dependent part of the potential between molecules. Finally the scattering results for reactive partners are reviewed. An attempt is made to show how such experiments can provide detailed inforniation on the more complex potential surfaces in terms of which chemical reactions are interpreted.

OTHERMETHODS FOR STUDYING INTERMOLECULAR POTENTIALS In order to assess the relative merits of beam scattering experiments the so-called classical methods are briefly discussed next. For more extensive reviews of these methods see Rowlinson (l960), Cottrell (1956), Buckingham (1961), Dalgarno (1962), and Waldman (1958). Deviations in the equation of state from the perfect gas law are attributed to the intermolecular forces between real molecules. If the pressure is not too high these deviations may be expressed in terms of an expansion in the density (virial equation)

where n is the number of molecules per unit volume, N o is Avogadro’s number, and k is the Boltzmann constant. The temperature-dependent coefficients B(T) and C(T)are designated virial coefficients and take account of deviations due to two-body and three-body collisions, respectively. Classical statistical mechanics gives the following expression for the second virial coefficient for spherical symmetric potentials

An analogous expression may be derived for angle-dependent potentials. In practice the second virial coefficient is only sensitive to the area

(1;

V(r)

.)under the attractive potential curve, and for this reason the

exact shape of the potential cannot be determined by this method. Higher virial coefficients, although tabulated for various realistic potentials, have been of little use because of the difficulties encountered in their measurement.

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H . Pauly and J. P. Toennies

The transport properties of a gas are related by the kinetic theory of gases to a function f, which describes the distribution of velocities and positions of the molecules as they vary with time (Chapman and Cowling, 1952). This distribution function may be expressed in terms of the intermolecular potential by solving the Boltzmann transport equation. A method of successive approximations (Chapman-Enskog method) is used in obtaining a solution of the Boltzmann equation. The so-called second approximation of the Boltzmann equation shows that the phenomenological coefficients of viscosity q, heat conductivity 1, diffusion D,and thermodiffusion can be related to the following general integrals (Chapman and Cowling, 1952; Hirschfelder et al., 1954): Q('*.) = J ~ ~ ~ S ^ e x ~ ( - ~ ~ ) y '-" cosJ9)I(y, '(l 9) sin9 d9 dy, 0

0

where

From the above equation it is seen that the potential comes in only by way of the differential cross section I(?, 9), which is weighted toward large angles by the factor 1 - cod 9 (where I = r = 1 for D and I = r = 2 for q and A). For this reason the variation of transport properties with temperature provides information on the repulsive potential. At very low temperatures it is also possible to study the attractive potential by this technique. Recently Mueller and Brackett (1964) critically compared the sensitivity of viscosity, diffusion, and low-energy scattering experiments to the intermolecular potential. Their analysis shows, that the first two methods are more sensitive to the repulsive potential whereas scattering experiments are more sensitive to the attractive potential. In a somewhat analogous manner the intermolecular potential may be related to observed relaxation times for rotational and vibrational equilibration (Herzfeld and Litovitz, 1959; Cottrell and McCoubrey, 1961). The detailed specific macroscopic rate constant for a transition ij + kl, where ij and kl refer to the quantum states of the two molecules before and after collision, respectively, is given by (Eliason and Hirschfelder, 1959)

k!;

= SSSS/..(%)/,.(..),::(E.

9, cp)g sin 9 d 3 dcp dv, dv,,

where 9 and cp are scattering angles andfi, andf,, are normalized velocity distribution functions for the species A and B. Thef., and& are assumed to be solutions of the appropriate Boltzmann equations. For conditions close to

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201

equilibrium they may be well approximated with the usual Maxwell-Boltzmann distribution functions. The differential inelastic cross section Z:;(E, 9, cp) is in turn related to the intermolecular potential. Unfortunately, a quantum mechanical calculation of Z:!(E, 9, cp) is possible only for grazing collisions (Section IV, D) and under other special circumstances (Takayanagi, 1963 and this volume). Moreover, the available experimental data usually represent an average over distribution functions for the internal energy states. For these reasons an interpretation of the data in terms of an intermolecular potential is a difficult task. The pressure broadening of microwave lines avoids the last difficulty but requires a knowledge of Z:;(E, 9, cp) for all collisions. The above equation also holds for chemical reactions. The remarks made above apply in this case as well. Here, however, Z:,!(E, 9, cp) is the differential cross section for two reactant molecules in the respective quantum states i and j reacting to yield two new molecular species in the states kl. , @:Z 9, cp) is related to complicated potential curves giving the potential for various relative orientations of all the atomic constituents that take part in the reaction. In all these methods the path leading from first principles to the measurable rates is a long and devious one: Schrodinger equation -P potential --f differential cross section -, distribution function averaging + macroscopic rate. As opposed to the classical methods, beam scattering experiments yield directly data on the differential cross sections, thus considerably shortening the chain of calculations from first principles to the measurements.

I. Intermolecular Potential Attractive forces of entirely different orders of magnitude can occur between two neutral atoms or molecules. If the electron spins of two approaching atoms are parallel or if one or both of the partners are rare gas atoms, or molecules for which chemical bonding is not possible, only an extremely weak attractive potential is observed. This potential has a maximum depth of only to lo-' eV and extends out to distances of about > 10 A, which is large compared with the gas kinetic radius of the partners. We shall call this attractive potential the van der Waalspotential. If, on the other hand, the two atoms can form a stable molecule (antiparallel electron spins), an extremely strong attractive potential appears at smaller distances of approach. A strong attractive potential may also appear if an atom or molecule reacts with another molecule, e.g., A BC -,AB C. In both instances the potential may have a depth of several electron volts, corresponding to, in the case of two atoms, the dissociation energy of the stable molecule. We shall call this the chemical potential. If chemical forces exist, the van der

+

+

H. Pauly and J, P. Toennies

202

Waals potential will be observable only at very large separations. At very close distances of approach these attractive potentials are more than compensated for by an extremely strong rapidly rising repulsive potential. These different types of potentials are illustrated in Fig. 1 for the interaction of two hydrogen atoms, which is the only potential for which reliable theoretical results are available. t

140.

t120

-

4100+

80-

+

60-

- + L O -

B

3 s

+ 20:

Y

h

g 5

-.P

E

0-

-

20-

-

LO-

-

60-

-

80-

-100-

-120

0

2

L

6

8

10

12

14

16

Intornucloar distance [atomic units]

FIG.1. Potential curves for the H-H interaction.

The intermolecular forces between nonreacting molecules are generally thought to come about in the following way: At long ranges the mutual perturbations are slight, and it is possible to describe the interactions in terms of the properties of the undisturbed molecules. This is especially true of the

THE STUDY OF INTERMOLECULAR POTENTIALS

203

classical electrostatic and induction forces, which appear only when one of the partners has multipole moments. In the interaction of two atoms the quantum mechanical dispersion force is largely responsible for the attractive potential. This potential comes about due to the mutual perturbation of the electrons in the two atomic systems. The short range repulsive forces arise by way of the overlap of the electron clouds. Thus, the theoretical treatment must take account of exchange, overlap, and possibly correlation effects involving all or most of the electrons of both atoms, a calculation that is still not possible for any but the simplest systems. A.

VAN DER

WAALSPOTENTIAL

1. Atom-Atom Long Range Potential

At long ranges the atom-atom potential can, to a good approximation, be described by the following simple law:

C V(r) = - r6 '

(1-1)

where C is called the interaction or van der Waals constant.2 Typical C values lie in the range between 1 x and 500 x lo-'' erg cm'. A priori values for the interaction constant can be obtained from either a perturbation or a variation calculation. By assuming that the optical polarizabilities can be described by a one-term dispersion formula (either the oscillator strength for one transition predominates or optical transitions are restricted to a narrow energy band) the perturbation result has been reduced by London (1930a) (Margenau, 1939) to the tractable form

where A is the energy term to which transactions occur and cx is the optical p~larizability.~ It is customary to approximate A by using the ionization potential in its place. The combining law c a b = Jcaacb, follows approximately from the London formula, and is useful for estimating interaction constants when better information is not a ~ a i l a b l e . ~ 2 For an Y-S potential it is customary to denote the corresponding interaction constant by C,. In this paper the index will be omitted for the case s = 6. 3 For a discussion of the calculation of and a list of polarizabilities, see Dalgarno (1962). For the optical polarizabilities of the rare gases see Dalgarno and Kingston (1960). Other combining laws are listed in the reviews of Pauly (1961) and Hirschfelder et al. (1954). For a recent discussion of the validity of several combining laws see Thomaes et al. (1962).

204

H . Pauly and J. P . Toennies

A slightly different result was obtained by Slater and Kirkwood (1931), using a variation calculation. By simplifying their treatment to a single closed shell in each atom, they obtained 3 eh ala2 V(r) = - 2r6 m l / , (al/iVl)1/2+ (a,/N,) ‘I2’

(1-3)

where m is the electron mass and Nl and N , are the number of electrons in the outer shells of the two atoms. Recently, Mavroyannis and Stephen (1962) and Salem (1960) have reinterpreted the Slater-Kirkwood formula, and from their result it follows that the total number of electrons in the atom should be used for N , and N,. The London and Slater-Kirkwood formulas agree if A, is interpreted as (Nl /a1)1’2eh/m1/2. Another approximate expression derived by Kirkwood (1932) and Miiller (1933) is

where No is Avogadro’s number and is the diamagnetic susceptibility per gram atom as given by the Langevin formula. Some idea of the accuracy of these formulas follows from Table I taken from the articles by Mavroyannis and Stephen and Salem. As discussed in Section V of this article these theoretical van der Waals constants are of the order of 10-50 % smaller than those measured in molecular beam experiments (Dalgarno and Kingston, 1959). At extremely long ranges (> 300 A) retardation effects resulting from the finite time for photon exchange tend to weaken the dispersion forces’ but appear to have little influence on molecular beam scattering experiments (Fontana, 1963). The perturbing potential used in calculating dispersion forces is an expansion in powers of r - l , of which the first or dipole-dipole term leads to the London formula (Eq. 1-2):

c C8 V(r)=-+-+-+ r6 r8

ClO

ri0

....

From Fontana’s (1961a) calculations of the higher-order terms it follows that at r = 10 8( the dipole-quadrupole term C8 amounts to between 2 and 20 % for light (He) and heavy (Cs) atoms, interacting with like partners, respectively; whereas the quadrupole-quadrupole term, which together with the dipoleoctopole term6 gives C,,, amounts to only between 0.3 % and at most 4 % 5 See the review by Derjaguin (1960) and the recent discussion by Mavroyannis and Stephen (1962). 6 Fontana showed that the previously neglected dipole-octopole term often is larger than the quadrupole-quadrupoleterm, as pointed out previously by Berencz (1960).

TABLE I CALCULATED AND EXPERIMENTAL VAN DER WAALSCONSTANTS M ATOMIC Urna Experimental System

H-H He - He Ne - Ne Ar - Ar Kr - Kr Xe - Xe

Lennard-Jones (12,6) potential

1 a. u. = 0.9571 x

London 7.58 1.31 4.30 52.6 108 -

1.644 10.4 107.7 214 606

erg cm6.

ce

Theory Slater, Kirkwood N fi outer shell 1.74 8.10 67.0 125 259

Slater, Kirkwood N total no. of electrons 7.16 1.75 10.4 116 306 776

Hirschfelder and Lowdin (1959).

KirkwoodMiiller 6.75 1.70 12 135 295 730

c

Accurate calculations 6.499b 1.50c -

Dalgarno and Lynn (1957).

-

-

206

H. Pauly and J. P. Toennies

(for Cs). The third-order perturbation terms, varying as Y - ~ ,do not contribute to the dispersion potential. A further modification of the long range potential is introduced by the adiabatic coupling of the electronic and nuclear motions (the well-known Born-Oppenheimer approximation consists of neglecting this coupling). If one of the partners has either orbital or spin angular momentum, as is the case in scattering experiments with the alkali atoms, the coupling is of the order of magnitude of the van der Waals attraction and is supposed to vary with r - 2 (Dalgarno and McCarroll, 1956, 1957). On the basis of the same coupling Wu (1956) (Wu and Bhatia, 1956) has shown that two He3 atoms have a weaker binding tLan two He4 atoms. The interactions of excited alkali atoms including resonance and spin orbit coupling have also been reconsidered recently (Fontana, 1961b, 1962; Mulliken, 1960). Finally, it should be mentioned that Schlier has calculated the anisotropy in the van der Waals potential between an excited atom (Ga 2P3/2,1/2 and 2P3/2,3/2)and a rare gas atom (Berkling et al., 1962a). 2. Atom-Molecule Long Range Potential

The dispersion potential' for this case is complicated by the fact that the anisotropy in the molecular polarizability must be accounted for. Using a simplified theoretical model London (1942) calculated the interaction constant between two molecules:

+ C){sin Y, sin Y 2cos(E2 - El) cos + 3(B - C) C O S ~ Y

C(Yl,El, Y2,E2) = ( A - B - B'

- 2 cos Y 1

Y2}'

(1-5)

+ 3(B' - C) C O S ~+Y( B~ + B' + 4C), where A , B, B', and Care defined in terms of a l l , al I!, a2', a211 (the polarizabilities perpendicular to and parallel to the molecular axis) and characteristic energies ASl', hV,ll, etc., and the angles are those defined in Fig. 2. If one of the molecules is replaced by an atom (denoted by 1) the potential can be well approximated by

(1-6)

7

For a review see Pitzer (1959).

THE STUDY OF INTERMOLECULAR POTENTIALS

207

molecule 2

molecule I

FIG.2. Coordinate system for angle-dependent interaction potentials.

where 111, c = 3a1az -

1,

+12

- al

qdisp

a2

= a''

+ 2u1

= +(a"

+ 2al)

( I are ionization potentials), (qdispis the anisotropy factor

for the dispersion potential), is the average poiarizability of the molecule),

(az

P,(cos Y) = Legendre polynomial. The factor of interest in the above formula is the anisotropy factor q d i s p for the dispersion interaction. The interaction constant C in the above equation is the same as obtained previously from the London formula (in deriving Eq. (1-6) it was assumed that Vl = $I); qdispmay be estimated from the anisotropy of the molecular polarizability, for which values are available for simple systems (Hirschfelder ef al., 1954). Recently an attempt has been made to calculate quantum mechanically the anisotropy of the polarizability of simple molecules (Kolker and Karplus, 1963). Equation (1-6) will also apply to a good approximation if one of two molecules rotates much faster than the other molecule and many times in the course of a collision. London (1942) and Sparnaay (1959) [see also Longuet-Higgins and Salem (1961)l have pointed out that it is not valid to assume additivity in treating the dispersion interaction between complicated molecules. Deviations resulting from such an assumption may amount to 10-30 % depending on the symmetry of the molecules. In addition to the already mentioned dispersion forces, the induction forces arising from the interaction of the polarizable atom with the field produced by the multipole moments in the molecule must be considered. Practically all

H . Pauly and J . P . Toennies

208

molecules have quadrupole moments,' whereas heteronuclear molecules will, in addition, have dipole moments. At the present time nothing is known about the octupole moments of molecules. Of the interactions involving known multipoles the interaction of the induced dipole moment in the atom with the molecular quadrupole moment' Q is sometimes largest (if the ratio 24Q2/p2r> 1 is satisfied) and is given by 12 V(rl Y) = - - ulpZQzcos3Y2 I'

12

=

-r' u1~2Q2[3pi(cos"2)

+ 3Pdcos y 2 ) 1 9

(1-7)

where Q = 4 J (32' - r2)p dz, p ds is the charge in the volume element ds at r. Another term" is introduced by the interaction of the induced dipole moment in the atom with the dipole moment of the molecule 2

V ( r , Y)

=

-u1P2 (1 + 3 cos2Y2) 2r6

- -u1uz2 [l

r6

+ Y,(COS

(1-8) Y2)].

In the case of molecules with small dipole moments ( < 1D) these additional terms are small compared to the spherically symmetric or anisotropic part of the dispersion term. In the case of large dipole moments, on the other hand, the induction terms can be comparable with the dispersion terms. Additional terms with cos2Y and cos4Y, which fall off as r-', are known to exist (Bennewitz et al., 1964). 3. Molecule-Molecule Long Range Potential

In addition to the already mentioned dispersion and induction forces, classical electrostatic interactions, denoted by @, must be accounted for in describing the potential between two nonspherical molecules : The number of induction terms is doubled to take account of the multipole field of the second molecule.''

* Spherical tops are an exception.

For a compilation of known quadrupole moments see Buckingham (1959-1960). cos Y2 induction term is not expected if the particles do not carry permanent charges. l1 For a comprehensive review of molecule-molecule interactions see Margenau (1939). loA

209

THE STUDY OF INTERMOLECULAR POTENTIALS

The general electrostatic potential is described by a double series expansion in r - ’ (Hirschfelder et al., 1954). When the distance between the molecules carrying the multipoles is large compared with the dimensions of the molecules, which is the usual situation in long range molecular interactions, then the series can be simplified to

where the individual terms account for the following interactions : dipoledipole, dipole-quadrupole (2 terms), quadrupole-quadrupole, and dipoleoctupole (2 terms). Theffunctions are functions only of the angles shown in Fig. 2. The first two f functions are given below, the others may be readily derived from the general expansion for the electrostatic potential” : jJl, 2) = - 2 cos Y, cos Y z =

+ sin Ylsin Y,

cos(Z:, - El)

- {2~0’(1)C0’(2)+ C51(1)C!+1(2)+ Cy1(1)C!1(2)}y

(I-1Oa)

fPQ(l,2) = ~ { C OY1(3 S cos Y, - 1) - 2 sin Yi sin Y2 cos Yz cos Elcos S2 - 2 sin Ylsin Yz cos Y, sin Zlsin Z z } = 3( C,1(1)C,2(2)

1 +[C!

J5

l(l)C: ,(2)

+ C:,(l)C?

,(2)]).

(I-lob)

So far mention has been made only of the potential involving stationary molecules. In low velocity molecular beam experiments typical molecules with a Maxwell-Boltzmann distribution of rotational energy corresponding to room temperature usually undergo several rotations during a collision. To a first approximation it may then be possible to use the expectation value of the potential in calculations of scattering cross sections. If such a first-order term disappears then a second-order perturbation calculation has to be performed just as is done in calculations of the electronic dispersion potential. For rapidly rotating molecules interacting with atoms this means that the cos3Y potential term disappears entirely, and only terms of even parity, e.g., cosz’l”, cos4Y contribute to the average potential. 12

Cg* is Racah’s notation for 2k

+1

Y k d y l ‘ , 9).

This is a convenient notation for describing angle-dependent potentials. Yk,(’fr,E) is a normalized spherical harmonic.

210

H . Pauly and J. P. Toennies

If one of two molecules rotates rapidly then the electrostatic dipole-dipole interaction vanishes. The interaction with the quadrupole moment of the rotating molecule remains, however, because of its even parity. The quadrupole-dipole electrostatic potential is expected to be dominant in such cases. The average quadrupole moment of a rotating molecule is about 4 the value calculated in a molecule-fixed coordinate system. The second-order dipole-dipole potential leads to an additional relatively weak potential term, the sign of which depends on the m state of the rotating molecule, provided it has time to orient itself in the field of the dipole during a collision (Toennies, 1965). If both molecules rotate many times in the course of a collision then the potential can no longer be treated classically, and the dipole-dipole potential disappears in first order.I3 A second-order interaction, which can be interpreted as resulting from the mutual interaction of induced dipoles (Keesom alignment), leads to either an attractive or repulsive potential, depending on the rotational states of the dipoles (London, 1930b). In collisions the results of such potential calculations may be modified by the coupling of the angular momenta of the molecules. For example, Hornig and Hirschfelder (1956) point out that the dipole-dipole potential commutes with the component of the total molecular angular momentum along the line connecting the centers of mass. The potential calculated according to such a coupling scheme will be different from the potential calculated if both molecules are assumed to be coupled to the electric field. More theoretical and experimental work is necessary before the conditions for which coupling is important in specifying the potential become clear. Finally, it should be pointed out that when the rotational period approaches the time of a collision the concept of an averaged potential is meaningless because of the occurrence of rotational transitions during the course of the collision. A discussion of the calculation of such inelastic cross sections is deferred to Section IV.

B. SHORTRANGEREPULSIVE POTENTIAL The a priori calculation of the potential at short ranges is difficult since the perturbation method is no longer applicable, and the variation method, requiring a knowledge of the molecular or united atom wave functions, must be used. The results of such calculations are best described by an exponential l3 The interaction between two symmetric tops is an exception since a component of the averaged dipole moment remains in the z direction (first-order Stark effect). The quantum mechanical calculations were performed by Margenau and Warren (1937). See also Carroll (1938).

THE STUDY OF INTERMOLECULAR POTENTIALS

21 1

term, but, depending on the extent of the calculations, the sum of two exponential terms may be necessary. Since only a relatively small number of systems, most of which have as yet not been studied by beam techniques, have been carefully investigated theoretically, a review of this work is beyond the realm of the present paper. A partial list of systems for which calculations are available include: H H, H He, H + Li, H Be, H + H,, H, + H,, He + He, Li + Li, Ne + Ne, Ar + Ar, as well as H + H*, H + Be*, He + He*, Be + Be* ; where * denotes an excited state. For references concerning this work a number of recent reviews may be referred to (Abrahamson, 1963; Buckingham, 1961 ; Cottrell, 1956; Hirschfelder et al., 1954). Recently the He-He interaction at short ranges (0-1.0 A) has been reinvestigated (Phillipson, 1962; Kim, 1962). The results are of a fairly high accuracy, and the error is estimated to be well under 1 eV. On the basis of these calculations there now appears to be a definite discrepancy between these theoretical results and experimental results from high velocity atomic beam experiment^,'^ which amounts to a factor of 1.2 to 2.8, dependingon the internuclear distance. One possible explanation is that inelastic processes involving excitation of higher electronic states (Thorson, 1963) may be occurring at the high relative velocities (> 3 eV) encountered in these experiments. If this explanation is correct it would appear that the concept of an intermolecular potential is limited to energies less than a few electron volts. Of special interest are the repulsive interactions between excited state atoms and ground state atoms, since the former are easily detected experimentally (see Section 111) and since there is some theoretical evidence for the presence of a hump with a double minimum in the case of He('S) and He(3S) (Brigman et af., 1961). Very little is known about the anisotropy of the repulsive part of the molecule atom potential. In most quantum mechanical calculations of simple systems' the potential is calculated for a few different relative orientations. In one instance the calculated potential (Roberts, 1963) for He-H, was fitted to an expression of the type

+

+

V ( r , Y ) = Ae-"'[ I

+

+ GP,(COSY ) ] ,

(1-11)

where, for this system, 6 = 0.375. Unfortunately, only 1s atomic wave functions were used in constructing the over-all wave function in this calculation. More accurate calculations of a similar type, which are fitted to series expansions in Legendre functions, are needed for calculations of inelastic cross sections. Since such a priori calculations are lacking in most cases, it has been necessary to assume additivity of the repulsive forces between the individual For an excellent review see Mason and Vanderslice (1962). See, for example, Mason and Hirschfelder (1957): Hz-Hz, Hz-H*. See also the references in Section I, C. l4

212

H . Pauiy and J. P . Toennies

atoms making up the molecules (two-center models). This is the potential usually used in calculations of rotational excitation (see article by K. Takayanagi). The conditions under which this procedure is justified have been investigated theoretically (Salem, 1961). Roberts (1963) compared his quantum mechanical potential with one based on a two-center model and found the latter to be in error by a maximum of 50 % for end-on approach (Y = 0). For studying the scattering of homonuclear molecules it is sometimes advantageous to convert to prolate spheroidal coordinates (Craggs and Massey, 1959). C. CHEMICAL POTENTIAL Chemical reactions of the simple type A + B C + AB + C can also be interpreted in terms of an intermolecular potential. In this case the potential is a function of the three internuclear distances TAB, rBc,and rAC and is usually denoted by v ( r A B , rBC,TAC). A multidimensional potential of this type, when incorporated in a suitable theory of scattering, should provide a complete description of the kinematics of chemical reactions. A necessary condition for the applicability of the potential concept is that the displacement of the three atoms relative to one another proceeds so slowly that excitation of higher electronic states does not occur. It appears that this condition is fulfilled for a large number of chemical reactions (London, 1929). The a priori theoretical calculation of such potential energy surfaces is an extremely formidable task. Even in the case of H H, no attempt has been made to carry out as extensive a treatment as James and Coolidge did for the H, molecule. One of the best recent variational calculations by Boys and Shavitt (1959), (Shavitt, 1959), for the H,-H system yielded an activation energy of 14.8 kcal/mole for the reaction which is to be compared to the experimental value of k8.5 kcal/mole. Because of such difficulties, other approximate methods have been developed. Since the validity of the approximations has been ascertained in the past by comparing calculated and experimental activation energies, only a linear configuration of the three atoms had to be considered.16 This simplification not only makes the approximate quantum mechanical formulas first proposed by London more tractable, but also permits the use of known intramolecular potential constants for the individual molecules that participate in the reaction. This semiempirical method was first developed by

+

l8 Approximate calculations show that this direction of approach is the most favorable for chemical reactions. See Porter and Karplus (1964).

THE STUDY OF INTERMOLECULAR POTENTIALS

213

Eyring and Polanyi (Glasstone et al., 1941) and has recently been modified by Sat0 (1955). In modified form the approximation appears to give reasonable values of activation energies for some systems. Furthermore, the modification by Sat0 no longer leads to predictions of stable species of the type ABC, which had been predicted by Eyring and Polanyi and for which definitive experimental evidence is lacking. Recently, Porter and Karplus (1964) have extended the London-EyringPolanyi-Sat0 semiempirical method by including the previously neglected overlap and three-center integrals. Thus it was possible to carry out meaningful calculations for nonlinear configurations. Their results are shown in Fig. 3 for two different configurations of the three atoms. It is hoped that molecular beam scattering experiments will eventually provide sufficiently detailed information on the angular distribution and excitation of products to permit a choice between the potentials obtained using various approximations. Since the Sat0 modification relies on theoretical calculations of the repulsive potential, which are only available for H,, it cannot be expected that these approximate methods will be as successful when applied to other systems.

D. MODELS OF THE INTERMOLECULAR

POTENTIAL

On account of the difficulties encountered in calculating the intermolecular potential, particularly at small distances of approach, it has been necessary to assume mathematical models for the potential. These models must not only give a reasonable description of the general shape of the expected potential, but must also be sufficiently simple to be used in actual calculations. Three formulas, which have found wide application in explaining equation-of-state data and transport property measurements, are frequently used in analyzing molecular beam scattering data." These formulas are briefly discussed below. ( 1 ) The Lennard-Jones (n,6 ) potential:

(1-12) The meaning of the symbols can be derived from Fig. 4. Quite frequently n is set equal to 12, in which case the experimental data are used to find rm or ro and E . For tables of values of ro and E for various interaction partners from viscosity measurements and second virial data see Hirschfelder et al. (1954) and Rowlinson (1960). 17 Cohen and Blanchard (1962) have recently suggested a realistic potential function, which is mathematically related to a ground state wave function.

H . Pauly and J. P . Toennies

214

Vkr)

!!

1t

1.0

3.0

4.0

3.0

u)

20 1.0 0.5

03 VkV)

2.0

1.0

3.0

4.0

3.0

FIG.3. Multidimensional chemical potential (Porter and Karplus, 1964). Lines of constant potential are plotted as a function of the two internuclear distances.

THE STUDY OF INTERMOLECULAR POTENTIALS

+

215

12 10

0.8 06

& s

O4

2,

P 0.2

t

.-a c

d

0

a2

U

9

g

a

0.4

0.c 0.f

- 1s I

0.5

10

1.5

Reduced intornuclear distance

2.0

2.5

I rm

FIG.4. The intermolecular potential between two atoms.

( 2 ) The Kihara (1953)potential:

=

co;

r

OI-
rm

By introducing the additional parameter a’ Kihara has made the LennardJones ( 1 2 , 6 ) potential more flexible by shifting the point for V(r) = 00 from 0 to d r , .

(3) The Buckingham potential: (1-14)

216

H . Pauly and J. P. Toennies

The Buckingham potential has the advantage over the Kihara potential in that the exponential repulsive potential seems more realistic than a simple power law. Only for small a is the maximum of the repulsive part (the potential goes to - 00 for r -,0) so low that it will have an influence on results calculated for molecular beam experiments. The repulsive part of the Buckingham potential can be fitted to the Lennard-Jones (12, 6) potential in the vicinity of the minimum, with the result that a = 13.772. A comparison between the values for r,,/rm,(dV/dr),,, and (d2V/drZ),,,as well as the asymptotic value for the interaction constant C, for the three potentials is tabulated in Table 11. As more refined experimental data become available, the above mentioned potential models will undoubtedly become inadequate. Two possibilities present themselves. In investigations of the solid state of rare gases (Pollak, 1964) different regions of the potential are described using different model potentials and these are joined together. This may be avoided by using a many parameter function, such as one suggested by Boys and Shavitt (1959), which permits the introduction of correction terms in higher order. (4) The two-body chemical potential: Since in most scattering experiments on chemical reactions the results are averages over the orientations of the molecules a simple spherically symmetric model of the potential is usually used to facilitate the interpretation of the results. One potential model that has been proposed to explain the anomalous behavior of the elastic scattering of K on HBr is shown in Fig. 5 (Herschbach and Kwei, 1963). This onedimensional potential represents approximately a cross section through a multidimensional potential of the type shown in Fig. 4. The outer minimum is the usual van:der Waals attraction (the dotted line corresponds to one of the potential models described above), while the inner minimum is characteristic for chemical reactions of the type K + HBr. The repulsive core takes account of the fact that a reaction does not occur except for a very small fraction of collisions with small impact parameters, which are neglected here. The potential curve shown in Fig. 5 has not as yet been checked by experiment, but may prove useful in analyzing elastic scattering cross sections measured with partners that can react with each other.

11. Molecular Beam Method for the Experimental Determination of Intermolecular Forces A. INTRODUCTION AND DEFINITIONS In this section the methods by which cross sections are experimentally measured are discussed. Special attention is given to the problem of correcting for deviations of the actual apparatus from the ideal behavior, and to sources

217

THE STUDY OF INTERMOLECULAR POTENTIALS

-5t I

l

l

l

l

0.5

l

l

l

l

Reduced internuclear

l

1.0

l

l

distance

l

l

l

15

l

l

Irm

FIG.5. Potential model for reactive collisions (Herschbach and Kwei, 1963). The dotted line shows the Lennard-Jones potential.

of error that may be present in such measurements. Details concerning the actual construction of the elements of the apparatus are postponed to the next section. A molecular beam is defined as a directed ray of neutral atoms or molecules in a high vacuum. The particle density is made so small that collisions among particles in the beam can be neglected." In addition, the vacuum in the apparatus must be so high ( E Torr) that collisions between the directed beam particles and the molecules in the apparatus are improbable. Such a beam may be simply produced by allowing a gas or a vapor to effuse through a hole from a chamber, which is denoted as a molecular beam source or 18 The mean free path of molecules in a beam for collisions with other molecules in the beam has been calculated by Troitskii (1962).

TABLE I1 COMPARISON BETWEEN CHARACTERISTIC INTERCEPTS AND DERIVATIVES AND ASYMPTOTIC BEHAVIOR FOR LENNARD-JONES (n,6), KIHARA, AND BUCKING- PO~ENTIALS Lennard-Jones (n,6 )

Kihara

THE

Buckingham

3

ro rm

24.2lI6

Y

rm(1 -a')

31 31

728

rVn2(1-a')2 C(r+

CO)

2&rme(l -a')6

8

3

THE STUDY OF INTERMOLECULAR POTENTIALS

219

molecular beam “oven.” Slits or apertures are used to collimate the beam. If a chamber of length L containing a gas or vapor with density n is introduced into the path of such a molecular beam, then collisions will take place between the gas molecules and the beam molecules, and the latter will be deflected from their original direction. The types of collision that occur in the thermal energy range may be classified as follows: elastic, inelastic, and reactive. An elastic collision is one in which neither the internal energy state nor the molecular composition of the partners is changed (e.g. atom-atom collisions at thermal energies). An inelastic collision is one in which the molecules experience a change in their energy states only (e.g., rotational excitation), and, finally, a reactive collision is defined as one in which the partners have a different composition after the collision. As a consequence of the collisions in the scattering chamber the beam intensity 4ois attenuated to 4 according to Beer’s law,

9 = 9oexp( - nak,(g)L).

(11-1)

The total integral collision cross section ok,(g) is, in general, the sum of the elastic scattering cross section and the sum over all inelastic or reactive cross sections.’’ In this article the integral over scattering angles of the differential cross section is designated as the integral cross section. It has been necessary to introduce this term in order to permit us to distinguish between, for example, an integral elastic cross section and an integral total cross section. In the terminology usually found in the literature both of these cross sections are referred to as total cross sections. afot(g)depends on the intermolecular potential, on the quantum states of the colliding molecules, denoted by p , and on the relative velocity g of the colliding partners. In general, the empirically determined value of o,,,(g) is a mean value, averaged over the distributions for the internal energy states and the relative velocities of the colliding particles. The scattering chamber can also be replaced by a second molecular beam (method of crossed beams). In the case of crossed beams the beam for which the scattering is observed is called the primary beam, index 1, and the beam that produces the scattering is called the secondary beam, index 2. The method of crossed beams has the advantage that the relative velocity and the scattering region are better defined, but the inherent disadvantage that the integral of the density over the beam path is more difficult to determine absolutely. A measurement of the intensity of the scattered primary beam molecules as a function of the scattering angles 0 and @ yields the differential scattering l Q Depolarization cross sections may not be associated with an observable angular deflection and for this reason are a possible exception.

H , Pauly and J. P . Toennies

220

cross section Zvp(O, @, g) where the index v denotes the particular type of collision being measured. The differential cross section is related to the integral cross section by avl'=

s

IvP(@,

cp, g) dQ,

(11-2)

where d!2 = sin 0 d 0 d@. The differential cross section and the integral cross section are the two quantities measured in a scattering experiment. If only elastic collisions can occur then the measurement of these quantities is straightforward. For measuring inelastic integral and differential cross sections aj,',{(g) and Z:;{(@, @, g) it is necessary to specify the internal energy state of the molecules before the collision, denoted by i, and analyze the molecules for their state f after the collision. Of course indirect information on inelastic collision cross sections may also be obtained by measuring the change of velocity of the scattered primary beam. If a chemical reaction i -+ f (where, in this case, i refers to the reactants and f to the products), between the beam molecules and the target molecules is possible, not only the angular distribution of the elastically scattered primary beam molecules, but also the angular distribution of the reaction products can be observed. By integrating the measured angular distribution of the reaction products over the whole solid angle, the integral reaction cross section (g) may be obtained. Table I11 summarizes the different cross sections that are measured. The notation used in Table 111will be used throughout the remainder of this article.

B. MEASUREMENT OF INTEGRAL TOTAL SCATTERING CROSS SECTIONS Figure 6 shows a schematic diagram of a simple apparatus used for measuring integral total cross sections. The beam leaves the oven (0),is collimated by the slits (S,) and (SJ, and after passing through the scattering chamber (C) is finally detected at (D).

I

I

I

I

I

I

FIG. 6. Schematic diagram of a simple apparatus for measuring integral total cross sections (Esterrnann et al. 1947b).

TABLE 111

SUMMARYOF NOTATION FOR MOLECULAR BEAMCROSSSECTIONS Integral cross section Type of process being measured

Elastic Inelastic i =initial rotational state f =final rotational state Total Reaction i =reactants f = products

u#(g)

For specified initial and final states

Sum over final states

Differential cross section I#(@. @, 9) For specified initial and h a 1 states

Sum over h a 1 states

222

H. Pauly and J. P. Toennies

The method consists of measuring the attenuated intensity at different scattering chamber pressures. The integral total cross section is obtained from the slope of the plot of l o g 9 versus n, the density in the scattering chamber. Figure 7 shows several examples of such intensity versus pressure diagrams. In evaluating such results the following factors must be considered.

1. The Effect of the Velocity Distributions of the Scattering Gas and of the Primary Beam on the Measurement of Inteyral Total Cross Sections As already mentioned, Eq. (11-1) applies only to the ideal case in which the scattering gas is stationary before collision and the primary beam molecules

I

I

1

2

3

I

1

4

5

6

7

Pressure (arbitrary units)

FIG.7. Attenuated intensity versus pressure diagrams. The different curves are for different primary beam velocities.

THE STUDY OF INTERMOLECULAR POTENTIALS

223

have the same velocity. Whereas it is now common practice to use monoenergetic primary beams with a velocity u1 it is basically impossible to velocity select the gas in the scattering chamber, and only by cooling the scattering chamber or by using a secondary beam can the thermal motion of the scattering gas be reduced. Thus it is necessary to replace Eq. (11-1) by

9 = $0 exP(-

n~fblcrr(u1)L).

(11-3)

The effective scattering cross section ~fb~,,,(u,) can be calculated by averaging over the velocity distribution in the scattering gas provided that the velocity dependence of the total cross section is known. For this purpose it is convenient to define a new function F(x) by G3t,,,(u1> = atl(ul)F(x>,

(11-4)

where x = ul/c2, and c2 is the most probable velocity in the scattering gas, defined by c2 = (2kT2/m2)1/2.The correction function F(x) depends not only on the velocity dependence of the integral total cross section but also on the type of scattering arrangement used (scattering chamber or secondary beam). Correction functions have been calculated (Berkling et al. 1962b) for different experimental situations for a velocity-dependent cross section of the type (see Section IV) aPtot (9)

-Z/(s-

1).

(11-5)

Table IV provides a survey of the available correction functions. These functions have been tabulated for s = 6 (elastic long range van der Waals interaction) and s = 00 (elastic hard spheres) in the paper by Berkling et al. (1962b). In this same paper the correction functions for a nonspherical symmetric cross section have also been derived. Figure 8 shows the behavior of the correction function Fao(s, x ) for different values of s." If the primary beam is not velocity-selected then the correction function must take account of the distribution of velocities in the primary beam as In this case al0,,,,(ul)in Eq. (11-4) is replaced by ~tbl,,,(ul)

= 4dc,)G(y),

(11-6)

where y = cl/c2. c1 is the most probable velocity in the primary beam source. The function G(y) will also depend on the experimental arrangement and the velocity dependence of the integral scattering cross section. Table IV also summarizes the different G(y) functions. Copies of Fa& x ) tables may be obtained by writing to one of the authors (JPT). Although the correct G ( y ) had already been calculated by Wood and Torrey (Wood, 1943; Foner, 1945) the approximate correction function of Rosin and Rabi (1935) was used in interpreting the results of most scattering experiments up to 1962. 20

21

224

H . PauIy and J. P . Toennies TABLE IV SUMMARY OF

NOTATION FOR CORRECTION FUNCTIONS FOR SPHERICAL SYMMETRIC CROSSSECTIONS” u;ot(g) = u,”,,(uo). (uo/d2’(*--1)

Experimental arrangement

Type of detector n(1) dl

Scattering of a molecular beam in a scattering chamber with a Maxwell-Boltzmann velocity distribution of the scattering gas

Surface detector (e.g., LangmuirTaylor detector)

Scattering of a molecular beam on a secondary beam with a Maxwell-Boltzmann velocity distribution

Surface detector

Scattering of a molecular beam on a secondary beam from an orifice source with cosine angular distribution law

Densit:, detector (e.g., electron bombardment detector)

F(x)

G(Y)

nL

nL

Density detector Surface detector

-

nof 2n D Density detector

0 Notation: no is the density in the secondary beam oven, f the area of the secondary beam opening, D the distance of scattering center from the secondary beam source. The functions F(x) and G(y) are related to the hypergeometric functions 1F1 and 2 s . See K. Berkling et a(. (1962b). The functions Fuo, Fbo, Gao, Gbo, Gao’, and Gbo’ for s = 6 (elastic v& der Waals interaction) and s = co (elastic hard spheres) are also tabulated in this paper.

Since the velocity dependence of the integral elastic scattering cross section for collisions between heavy atoms and molecules is well approximated by Eq. (11-5) [see Sections IV and V], these averaging processes are no longer a serious source of error in measurements of integral elastic cross sections. The elastic scattering of light atoms is an exception, since the velocity dependence of the cross section can no longer be expressed in a simple analytic form, and a numerical integration is therefore necessary for each special case. Figure 9 shows the effect of the averaging in the scattering of a monochromatic primary beam on a secondary beam with a thermal velocity distribution. The calculations correspond to the example Li-Kr. Figure 9 also illustrates how the velocity dependence of the effective integral elastic cross section relative to the integral elastic cross section for a stationary scattering atom (T2 = 0) changes with the temperature of the secondary beam.

225

THE STUDY OF INTERMOLECULAR POTENTIALS

FIG.8. The correction function FUO(S, x ) as a function of x = v ~ / c afor different s.

2. The Efect of the Finite Angular Resolution on the Measurement of Integral Total Cross Sections According to Eq. (11-2) all scattered particles contribute to the integral total cross section. Because of the finite dimensions of the beam and the detector this is not entirely true in practice. As a result the measured effective cross section is too small by an amount that depends on the solid angle An within which scattering is not observed. The error in 0 (indices have been left off for the sake of clarity) is

Ac

=

@)

do,

(11-7)

where the integral extends over the solid angle An which characterizes the resolving power of the apparatus. An exact calculation of An requires a knowledge of the differential cross section I(@, @) in the laboratory system. The

H. Pauly and J. P. Toennies

226

2oc

0

m Primary

Zoo0

3Ooo

beam velocity V,

FIG.9. Calculated effective integral elastic cross sections for scattering of a monochromatic lithium beam with velocity UI on a krypton secondary beam with source temperature T.The curve for T = 0 was calculated for a Lennard-Jones (12,6) potential. The other curves have been successively shifted downward by 40 Az.

227

THE STUDY OF INTERMOLECULAR POTENTIALS

long range attractive forces lead to a concentration of the scattering in the forward direction, so that the elastic differential cross section rises rapidly with decreasing angle. The differential cross section becomes nearly independent of the angle at angles smaller than a characteristic limiting angle ao. This limiting angle corresponds to the quantum mechanical uncertainty in the position of the scattering particle during the collision event and, in the center-of-mass system, is approximately given by (11-8) If the resolving power of the apparatus is of the order of magnitude of this limiting angle (transformed into the laboratory system) the error in the measured total cross section is 5-10 %, depending on the colliding particles and the special experimental setup. The laboratory system angle corresponding to a, is denoted by A,,. Table V gives values of the limiting angle A. for commonly used scattering partners. In order to measure the total cross section with an error smaller than 1 % it is necessary to make the angular resolution of the apparatus about of the critical angle A,. TABLE V THELIMITING ANGLEIN THE LABORATORY SYSTEM AO (minutes of arc) FOR DIFFERENT SCATTERING PARTNERS Incident atom

Li Na K

Target atom Ar

Kr

Xe

Hg

12.6 5.6

11.5 5.0 3.3

10.5 4.6 3.0

1.4 2.5 2.1

3.1

For reasons of intensity, beams of rectangular cross section, with beam height much greater than beam width, are almost universally used. With such an arrangement the smallest scattering angle leading to observable scattering depends on the angle CD, so that it is difficult to assign an angle of resolution to the scattering apparatus. For this reason Kusch (1964) has recently suggested a criterium for calculating the resolving power for a particular'apparatus using rectangular beams. In Kusch's calculations the efficiency of the detector is defined by q = 9'19 where 9'is the intensity, measured at the detector, of beam molecules that have been scattered through an angle 0, while 9 is the detected intensity of these same molecules without scattering. For example, ~ ( 0=)1 implies that none of the molecules that have been

H. Pauly and J. P . Toennies

228

scattered through an angle 0 are counted as having been scattered. ~ ( 0 ) can be calculated for any particular geometry. The resolving power O0is then defined as the smallest scattering angle for which ~ ( 0 ,=) 4. This definition has the advantage that it allows a comparison of results obtained with different experimental arrangements. For the same resolving power O0,Ao will be different for different scattering partners. Using measurements of elastic differential cross sections at small angles it has been possible to determine the integral cross section as a function of the resolving power for a series of scattering partners (Pauly, 1959; Helbing and Pauly, 1964). Figure 10 shows several examples. Here, the integral cross section relative to the cross section for ideal resolving power has been plotted as a function of the resolving power in the laboratory system. The geometry is of the usual rectangular slit type. The order of magnitude given above for the resolving power necessary to measure the total cross section with a given error has been taken from these measurements.

3. The Determination of the Density in the Scattering Chamber. For the accurate determination of the absolute value of the total cross section, the density in the scattering chamber or more correctly the quantity j n(l) dl over the path 1 of the primary beam must be known. The density in x

5'

m'

20'

30'

40'

50'

Angle 64 mdutbn A, (Iah sysl.)

FIG.10. Measured integral cross sections as a function of the resolving power in the laboratory system. A rectangular oven slit and detector were used.

THE STUDY OF INTERMOLECULAR POTENTIALS

229

the scattering chamber is usually measured by a pressure-measuring device appropriate to the pressure region : McLeod manometer, ionization gauge, or a Knudsen manometer. Of these the McLeod manometer has been preferred. Recently, it has been demonstrated, however, that pressures measured with this instrument may be consistently too small (Meinke and Reich, 1962; Gaede, 1915). For this reason a dynamic method has been suggested for producing a known pressure in the scattering chamber (Bennewitz and Dohmann, 1965a). A similar device is commercially available (Varian). Figure 11 shows the apparatus with which it is possible to produce a gas density for which the pressure is known absolutely to 1 %. The scattering chamber (1) is evacuated through a standard leak (2), the conductance of which can be calculated from the geometry (Bureau et al., 1952). The high pressure on the inlet side is measured with a U-tube. After closing the valves (4) and (5) the gas is admitted to the scattering chamber through the needle valve (6), and the mercury level in part A of the apparatus rises gradually with time. From the resulting flow rate and the known leak rate the pressure is determined. The extreme accuracy is achieved by measuring the time rate of change in capacity between a concentric cylinder and the mercury level in part A.

scattaring charnbar (1 1

steel maasuringcall

FIG.1 1 . Schematic diagram of an apparatus for producing absolute pressures to within

1 %.

A further source of error in the density determination is the gas that flows out of the entrance and exit slits of the scattering chamber. The gas flow can be calculated from the geometry of the slits, and a correction can be obtained. The magnitude of the required correction term increases with increasing

230

H . Pauly and J. P. Toennies

pressure in the scattering chamber. On the other hand the pressure measurement is always more precise at higher pressures. Consequently, the length of the scattering chamber is dictated by a compromise. The error due to gas losses through the slits can for example be avoided in a two-beam experiment (common oven and detector). Both beams pass through the same scattering chamber, but the paths are of different lengths, as shown in Fig. 12. From the difference in path lengths and the attenuation of the two beams, an absolute value for the cross section may be determined. Another method for checking the density measurement is to measure the cross section for partners A and B with A as primary beam and B as scattering gas and vice versa (Buck, 1964). Once one cross section has been established absolutely other cross sections will only need to be measured relative to this one cross section.

FIG.12. Schematic diagram of a two-beam experiment for determining absolute integral cross sections.

The K-Ar cross section has been suggested as a standard cross section, since it has been measured independently at a large number of laboratories (Toschek, 1962; Bennewitz and Dohman, 1965b). C . MEASUREMENT OF INTEGRAL INELASTIC AND REACTIVE CROSSSECTIONS In experiments on rotational excitation it is found that most of the inelastically scattered molecules are scattered into the forward direction (see Section VI D). In these experiments it is necessary to take account of the attenuation due to integral total scattering. Similar conditions will prevail for a chemical reaction between a very fast primary reactant beam and a light slow scattering gas, and the results obtained here may be adapted to this situation. To derive the pressure dependence of this type of scattering the example of rotational excitation involving the transition i -,f is used. In the experiment molecules in the state i enter the scattering chamber, and a detection arrangement with a limited acceptance angle a (a > a,,), which is sensitive only to molecules in the state f, is situated at the other

THE STUDY OF INTERMOLECULAR POTENTIALS

23 1

end of the scattering chamber. Per unit scattering length, dL, the increase in intensity of f molecules is proportional to o!z{(c1). This increase is partly compensated by events in which f molecules are either lost by scattering through an angle larger than c1 or are scattered with an angle smaller than c1 but undergo a transition to some final state s # f. This last cross section is well approximated by of,,. If inelastic processes are improbable then o:,, will be roughly equal to os,(a), which is a function of the resolving power, whereas if inelastic processes predominate then a total cross section independent of the acceptance angle will be more appropriate. ofol is easily obtained by measuring the total scattering of molecules in the state f with the same apparatus. The change in intensity of molecules in the f state is given by

d 9 f = 9'oiz;(a)n d L - S'af,,n dL.

(11-9)

If 0;;; = 0 then Eq. (11-9) reduces to the differential form of Eq. (11-1). The change in the intensity of molecules in the i state is given by d Y i = 9'ofz;(a)n d L - 9io:olnd L .

(11-10)

In general to a good approximation (< 10 %), oiotz o:ot and Yfor$(u) < 9io~ot(c1). Neglecting the first term in Eq. (11-10) it is possible to solve the above equations to give

Y f / f o i= oj,',f(a)nL exp( - a:o,nL),

(11-11)

where 9oi is the unattenuated intensity of molecules in the state i. Equation (11-1 1) has a maximum when nmax= (ofoIL)-l, and therefore Eq. (11-11) may alternatively be written as (11-12) Figure 13 shows the form of Eq. (IJ-12). The ratio o~,'d;(cl)/o~o, is determined by fitting measured curves to a calculated curve similar to the one in Fig. 13. From an independent measurement of oiot,oi,;!(ct) may then be obtained.

D. MEASUREMENT OF DIFFERENTIAL CROSS SECTIONS The problems associated with the measurement of differential cross sections for elastic and reactive scattering are basically the same and are therefore treated together here. An essential requirement for the measurement of differential cross sections is a well-defined scattering center. Consequently, a crossed beam is often used in place of a scattering chamber. This has the disadvantage that the particle density in the scattering region cannot be accurately known, and therefore, absolute measurements are difficult to make.

H. Pauly and J. P. Toennies

232

0

1

2

3

n -

4

5

nmox

Fro. 13. Calculated reduced intensity of inelastically scattered molecules as a function of relative scattering chamber gas density.

A calculation of the secondary beam intensity at the scattering center presupposes a knowledge of the angular distribution of the secondary beam and the distance of the secondary beam source from the scattering region. In the case of an ideal orifice in which the secondary beam has a cosine distribution, a reasonably accurate result may be calculated (Pauly, 1957). Many-channel arrays (Zacharias ovens), however, provide better peaking in the forward direction and higher intensities, but the angular distribution of this type of beam source is not accurately known (see Section 111). It is, however, possible to calibrate the particle density in the secondary beam by using a collision pair for which the total cross section is available (for instance, from measurements in a gas-filled scattering chamber). Because of the concomitant intensity loss it is usually not feasible to use a velocity-selected secondary beam. Instead a fast monochromatic primary beam of light particles is scattered on a slow secondary beam of heavy particles. In this arrangement the velocity distribution of the secondary beam is of little importance. An improvement in the velocity resolution is also possible by using a Lava1 nozzle as a secondary beam source. Such a beam has a much narrower velocity distribution without loss in intensity.

THE STUDY OF INTERMOLECULAR POTENTIALS

233

Because of the low intensity encountered in measurements of differential cross sections (105-106less intensity than in the primary beam) the resolving power in these experiments is usually not very large. Sometimes it is necessary to take account of the decreased resolution in an analysis of the data (Beck, 1962). Finally, one other precaution must be considered in setting up a differential cross section experiment. If the intense primary beam strikes the walls of the apparatus a part of it will be reflected and lead to a serious background. To avoid this it is advisable to permit the unscattered primary beam to enter an auxiliary vacuum chamber, which is connected to the main chamber only by a small hole. A comparison with theoretical differential cross sections requires a transformation of the measurements into the center-of-mass system. This transformation is particularly simple if the secondary beam particles are stationary before the collision and the primary beam particles all have the same velocity. In general, however, the secondary beam particles are in motion and do not have a uniform velocity, making the transformation more complicated. As a result, the measured differential cross section at a given angle consists of contributions from a region of angles in the center-of-mass system. The transformation is particularly simple when both beams have a uniform velocity and cross each other at fixed angle. This case has been treated by Helbing and others (Helbing, 1963; Herschbach, 1960a; Russek, 1960; Datz et al., 1961a; Morse and Bernstein, 1962) and will be discussed here. The notation is defined in Table VI. The relative velocity g of the colliding partners before and after collision is given by g' = V l ' - v2i; gf = y l f - v2f. In the center-of-mass system the collision leads to a change in direction and, if energy transfer occurs, to a change in magnitude of g. The scattering angle 9 in the center-of-mass system is defined as the angle between g' and gf. To relate this angle to the corresponding angles in the laboratory system it is necessary to transform the laboratory velocities to the center-of-mass system. The velocity of the center-of-mass is always the same before and after the collision and is given by

v, =

m1

Vli

m 1 + m2

+ m 1 + m2 m2

V2i.

The equations for transforming the velocities in the laboratory system to the center-of-mass system are (see Fig. 14): . . V l l = v;, + v,, V l f = v:, + vc,

.

v2'

.

= v;,+

v,,

v 2 f = v2,

+ vc.

TABLE VI NOTATION OF MECHANICAL VARIABLES FOR COLLISIONS After collision

Before collision

Elastic and inelastic

Masses Internal energy E'

Relative kinetic energy Center-ofmass system Velocities

Laboratory system

= E'

+

+

Reactive

ml

m3

m2

m4

Elf

E3f E4f

E2f

0= E'

( E I ~- EI') (E21-

Center-ofmass system

+ [(El' + + + D4O) + + D1° + Da0)l E2'

- (E3f

E2f

D3O

E4f

Laboratory system

Center-ofmass system

Vlf,

01'

v3fc

031

vzf,

v2f

v4f,

v4f

Laboratory system

THE STUDY OF INTERMOLECULAR POTENTIALS

235

FIG.14. Velocity diagram for cross beam kinematics. The example shown is for elastic scattering. The solid lines represent velocity vectors before collision. The dotted lines represent the velocity vectors after collision.

The kinematics of collisions may be visualized with the aid of a velocity diagram of the type shown in Fig. 14. Here velocity vectors of the molecules before and after collision are shown for the example of elastic scattering. This type of diagram has the advantage that the ends of the vectors correspond to the laboratory positions of the particles at various times. Thus at time t = - z the molecules are at the lower left-hand part of the diagram; at t = 0 they collide, and at t = + z they will be located at the tips of the vectors at the upper right. If inelastic processes occur then the same diagram may be used with the exception that the relative velocity will not only be rotated as a result of the collision but will also be either lengthened (exothermic process) or shortened (endothermic process). From the kinematics of the collision it is possible to derive expressions for relating scattering angles and solid angles in the center-of-mass and laboratory systems. The results for the transformation of angles and solid angles from the center-of-mass system to the laboratory system are presented in Table VII. The following conventions in addition to those used in the previous table have been used. 9 and rp are the scattering angles in the center-of-mass system. 9 is

H . Pauly and J. P . Toennies

236

TABLE VII FORMULAS FOR THE TRANSFORMATION OF ANGLES AND SOLID ANGLESFROM THE CENTER-OF-MASS SYST~M TO THE LABORATORY SYSTEM

dR -=

+ &c + h2kbl

k2 hl cos 6 -ha

dw

where k =

f ( m )=

Vli

=

(z)

A3

(1

va2 +- 2 vz -cosy Dl v12

y is the angle between vli and vai 1 for elastic and inelastic collisions

m1m4 mam3

fh)= -for reactive collisions A E = Er - Ei M=mi+ma

a = sin 9 sin 6 cos p - cos 9 cos 6

/)=I+-

1 A2

c = cos 9-

tan 6 =

2

- -cos6

h

1 -

A

vai sin y Vli

6 + cos A

- u2'cos y

OIS57l

the deflection angle and cp is the azimuthal angle. cp may be deiined with respect to an axis perpendicular to vli and vZi. @ and CD are the corresponding angles in the laboratory system. 0 is defined for the scattered particles as the angle between v1 and vlf, the same applying to particle 2. CD must, of course, be defined with respect to the same axis used in defining cp. In the formulas in

THE STUDY OF INTERMOLECULAR POTENTIALS

237

Table VII the index 2 may be exchanged with the index 1 so that the equations are the same for both particles. The formulas are valid for all types of collisions including reactive collisions. In the latter case, however, it is necessary f to make the following substitutions: u:, -,u3,, u2, + u4,. Furthermore, in the case of chemical reactions the functionf(m) defined in Table VII is no longer equal to 1. The reverse transformation, i.e., from the laboratory system to the centerof-mass system, leads to similar equations. They are not included here and may be found in the literature (Helbing, 1963; Morse and Bernstein, 1962). It must be noted however that this transformation is not always unique and that one laboratory system angle may correspond to two center-of-mass system angles. The condition for a one-to-one correspondence between angles in both systems is

The quantities appearing in Eq. (11-13) are defined in Table VII. In transforming experimental differential cross sections, additional difficulties are encountered because of the unavoidable distribution in the direction and magnitude of u2. A unique transformation of the effective laboratory system differential cross section Zterf(O,0,vl) to the center-of-mass system is not possible under these circumstances, even if the previous condition, Eq. (11-13), is satisfied. The reverse process of calculating an effective differential cross section from an assumed differential cross section in the center-ofmass system is however possible and is best performed in two steps: (1) Using Helbings results, the center-of-mass differential cross section Z(9, cp, g) dw is transformed to the laboratory system to give Z(0, CP, ul, v2) dn. (2) This quantity is averaged over the velocity distribution in the secondary beam in the following way

E(v) is the efficiency of the detector, which may be a function of the beam velocity. A surface detector such as the Langmuir-Taylor detector is essentially independent of v, whereas for a density detector, such as an electron bombardement detector, E(u) is inversely proportional to v. For large scattering angles vlf will be considerably different from vli. In Eq. (11-14) it is assumed that the distribution function f ( u 2 ) is independent of the position at which the collision occurs. This is not true in the case of an ideal orifice.

H . Pauly and J. P . Toennies

238

More generally (11-14) must be replaced by the following expression

I,,,(@,

@, v1) =

E(U1')

0

E(ul') 1 - exp( - z o )

where

s

d z = n2f,(v,, and

20 =

exp( -2) I(@, @, u l , u2) dR dz, (11-15)

I)

9

dv, dl

"1

joLyv,d..

If a monochromatic primary beam is not used, then the above result has to be averaged over the distribution fl(uli) (Helbing, 1963). Equations (11-14) and (11-15) have been derived for the usual case of small densities in the scattering chamber or secondary beam. With increasing scattering gas pressure the observed intensity will deviate from a linear behavior with pressure (Knauer, 1933a) in much the same way as with inelastic rotational scattering. Other experimental difficulties, such as an excessive pressure rise in the main chamber, make measurements of differential cross sections under these conditions undesirable. In discussing measurements of differential cross sections it is often convenient to classify collisions according to the ratio of the speed of the molecules to be detected, e.g., ulf or u3 (in the case of a chemical reaction), to the speed of the center-of-mass system (Herschbach, 1962). This ratio largely determines the kinematics in the laboratory system and, therefore, the cross sections that can best be measured. Three representative cases are summarized in Table VIII. E. SUMMARY The problems involved in the measurement of integral and differential cross sections for elastic, inelastic, and chemically reactive collisions have been discussed in this section. The most difficult aspect of such measurements is the absolute determination of the cross sections. Recent progress along these lines indicates that it is now possible to make absolute measurements with an accuracy of better than 5 %. Correction formulas for the averaging over relative velocities resulting from the velocity distribution of the secondary beam are discussed. Only in experiments designed to measure the oscillations in the

239

THE STUDY OF INTERMOLECULAR POTENTIALS

integral cross section as a function of primary beam velocity or in the differential cross section as a function of angle is information lost because of velocity averaging. TABLE VIII TYPESOF KINEMATICS ENCOUNTERED IN MOLECULAR COLLISIONS

Easy to measure Case A

Dc

u3

uine+”g),

e.g., rotational excitation

Difficult to measure

utot(d,

I”& ( 0 , @, 9)

Ureact’Q)

e.g., the product KBr of the reaction K + HBr + KBr -t H

+

Case B

u3

Case C

u3 N , u c

uc

Iv” (0, @, 9). U t o t k ) , e.g., differential elastic cross section

Oi.e1”(9),

uresct’(9)

av”(d, I v ” ( @ , @, 9)

111. Recent Advances in Experimental Techniques for Molecular Beam Scattering Experiments A. INTRODUCTION

In practice a scattering apparatus is usually more sophisticated than the one shown in Fig. 6 . Ultimately the aim is to have an apparatus with which it is possible to specify completely the translational energy and the internal energy of the molecules before and after scattering (Fig. 15). Experiments of this type are, however, generally not possible because of the extremely low intensities available with conventional molecular beam sources. Nevertheless, recent experiments with special high intensity sources and improvements in detectors are encouraging. These developments in the design and construction of beam sources and detectors as well as velocity and state selectors are reviewed in this section. Emphasis is placed on the importance of various parameters, whereas less attention isgiven to the experimental details which can be found in the cited literature.

H . Pauly and J. P. Toennies

240

/

, , /

/

/

.

,

/

-

-

.. \

/

\

\

\ \

/

/

I

I

I

/

. _.-.

V

- . - .-

. L ._I

I

I

\\ ‘ d \

/

/

/

I

I

FIG. 15. Schematic diagram of an apparatus for crossed beam scattering experiments with state or velocity selectors.

B. BEAMSOURCES I . Thin- Walled Orifice The commonly used oven with a thin-walled orifice as a source of molecular beams has the advantage of being simple and compact and being useable with practically all molecules and most atoms. The orifice may be circular or have the form of a long slit. Whenever possible the latter should be used since it provides more intensity with only a small loss in angular resolving power. It is customary to operate these ovens at pressures such that the mean free path A in the oven is of the order of the diameter d of the hole (Knudsen condition). For Knudsen flow (A > d ) the intensity in the forward direction calculated for an ideal orifice of area F, cm2 and source pressure Po Torr at a temperature of T degrees Kelvin for a gas or a vapor with molecular weight M is PoFo molecules f ( 0 ) = 1.12 x 1022(111-1) JE sr sec

[

1.

24 1

THE STUDY OF INTERMOLECULAR POTENTIALS

All intensities referred to in this section are understood to apply to the forward direction. For a slit with a width equal to the mean free path in the oven and a height equal to 1 mm the intensity at a distance of 1 m from the source is, within an order of magnitude, the same for all gases and is equal toz2

$(O)

= 5 x 1o’O

or Y(0) = 5 x 10l6

1

molecules sec mm2 (detector area) molecules sr sec

[

]

Following a suggestion of Bennewitz (Bennewitz and Wedemeyer, 1963) we call such a beam a “ standard beam.” Conventional ovens have been adequately described in the literature (King and Zacharias, 1956; Ramsey, 1956). By passing a large current through a thin-walled tungsten tube with a narrow slit in one side, extreme source temperatures have been achieved (Lew, 1949). With such an oven it has been possible to produce thermally dissociated atomic beams with 90 % H atoms and as much as 20 % N atoms (Hendrie. 1954; Kleinpoppen, 1961;Everhardt, 1962). Ground state and excited atoms can also be produced by using a Wood’s discharge, microwave arc, high frequency (20 Mc) discharges or by optical or electron impact e~citation.’~ The presence of metastables in a beam that is to consist of ground state atoms only may require the use of special selection (e.g., magnetic field) to suppress the metastables. In general the alkali halide molecules have a tendency to form dimers (Miller and Kusch, 1956) at the temperatures required to vaporize them sufficiently. This may be overcome by using a two-chamber oven (Berkowitz et al., 1962) in which the exit chamber is operated at a higher temperature, at which the dimers tend to dissociate. In this way it is possible to obtain slightly higher translational velocities as well. “ Cloud ” formation in front of the oven orifice has been made responsible for the observed tapering off in intensity and broadening of the beam at oven pressures above that corresponding to the Knudsen condition A d (Kratzenstein, 1935). Recently considerable attention has been given to the complicated problem of rarefied flow through an orifice at pressures at which a transition from molecular to hydrodynamic flow is to be expected (Howard, 1961 ; Gustafsen and Kiel, 1964). Experiments with adequate pumping speed

-

22 Calculated for a kinetic cross section of 50 x 10-10 cmz and a beam velocity of 5 x lo4 cmlsec. 23See the review by King and Zacharias (1956) for an excellent description of such sources. Recently sources for the following beams have been described in the literature: H atoms (Fleischmann, 1961), 0 atoms (Nutt and Biddlestone, 1962), He metastables (Lichten, 1962).

H. Pauly and J. P. Toennies

242

between orifice and collimator indicate that a considerable increase in intensity can be obtained by raising the source pressure above the Knudsen condition. For example, Zapata et al. (1960) report a 30-fold N2 beam intensity increase on raising the source pressure from d/A 1 to d / A 70. Similar results have been reported by Becker and Henkes (1956), who obtained & of the intensity obtained with a Laval jet with the same cross section (for a discussion of Laval jets, see Section 111, A, 3) at the same total flow rate. From these results it would appear that an order of magnitude increase of beam intensity can be obtained by raising the source pressure above the Knudsen condition, provided that sufficient pumping speed is available (Cassignol, 1963). With ovens it is not possible to reach energies above 3500" K for atoms, and, usually, because of thermal dissociation, the optimum temperatures for molecules will be less.

-

-

2. Many-Channel Array

-

A single channel that is much longer than wide, 1% d when operated at a pressure A 1, has an angular spread at half-intensity that is 3 times smaller than that of a hole (Clawing, 1929; Giinther, 1957). The directivity decreases however with increasing source pressure and beam intensity. By using a large number of channels in cases in which the size of the emitting source is unimportant it is possible to obtain large beam intensities with improved directivity for the same total flow through the source (King and Zachararias, 1956). The flow properties of a many-channel source are compared with those of a thin-walled orifice in Fig. 16. Giordmaine and Wang (1960) have recently completed a theoretical investigation of a many-channel oven. For the forward intensity 9(0) they obtained (111-2)

where F is the total source area, z the transparency, m the number of holes, N the total flow rate in particles per second, CJ the scattering cross section, and fi the average velocity of the molecules in the oven. The half-width of the beam is (111-3)

-,x/3, and therefore 9(0)array -+ Y(0)orifice in For large flow rates accordance with the experimental results shown in Fig. 16. The directivity of a

243

THE STUDY OF INTERMOLECULAR POTENTIALS

2.5 1015

3(0)

2.5 tot3

2.5x 10’’

2.5x 10” 1015

d6

1017

10l8

Molecules

1019

FIG.16. Measured forward intensity 4(0) as a function of total flow N for a manychannel array (1) and an orifice source (2) (Becker, 1961).

many-channel array is a definite advantage when operating with noncondensable gases, because of limited pumping speeds, or when working with beams of expensive materials, such as isotopes. From Eq. (111-2) it follows that the advantage gained by using a particular many-channel array is given by FTm. In comparing different methods for constructing these beam sources we will use Tm/Fas a characteristic parameter, since it is usually desirable to keep the source area, which is related to the angular resolution of the apparatus, as small as possible. Table IX gives values of ( T ~ / F ) ’obtained /~ by different workers using various construction techniques. From Table IX it would appear that there is little room for further development of these ovens. The most efficient technique, described by Hanes (1960), uses a plastic matrix in which tightly packed copper wires are first imbedded and later leached out, leaving channels of the same diameter (20 p) as the wire. For high temperature operations the original crinkly foil technique shown in Fig. 17 is still the best. Photoetching of thin metal foils would seem to be a suitable method for obtaining larger Tm/F values. Using the crinkly foil technique, Becker (1961) achieved g(0)array/g(O)orifice 7.5 at a total flow rate of 10’’ molecules/sr/sec. His experimental results are in rough N

H . Pauly and J. P. Toennies

244

TABLE IX

VALUESOF

THE

CHARACTERISTICPARAMETER ( ~ r n / F ) ' FOR /~ MANYCHANNEL ARRAYS&

~~

Method of construction

Hole radius (mm)

Number of holes

Area F (mm2)

(<)I14

Crinkly foil

0.07

1.30 x 103

20

2.4

Klystron grid

0.17

224

20

1.3

Holey foil stack

0.024

1.28 x

lo4

133

2.0

Crinkly foil

0.027

1.80 x 104

95

3.1

Plastic matrix Crinkly foil

0.020 0.028

530 7 x 103

1 26

4.3 3.7

a

References Minten and Osberghaus (1958) Giordmaine and Wang (1960), Source A Giordmaine and Wang (1960), Source B Giordmaine and Wang (1960),Source C Hanes (1960) Becker (1961),Source K5

See text for meaning of symbols.

agreement with the predicted behavior, with the exception that he observes a N-'I3 intensity dependence with C 0 2 and NH, instead of the predicted N - ' I 2 dependence. Furthermore, he concludes that there is no mutual interference between neighboring channels, as suggested by Helmer et al. (1960). For certain applications Becker recommends a stack of slit-shaped channels as being nearly as efficient as a many-channel oven but easier to produce. Beams coming from long channels have velocity distributions slightly shifted to higher velocities

7

4

2

5m m / L

FIG.17. Detail of crinkly foil construction of many-channel array.

THE STUDY OF INTERMOLECULAR POTENTIALS

245

since low velocity molecules are preferentially scattered as the beam passes through the channel (Hostettler and Bernstein, 1960). The distortion of the velocity distribution is a definite disadvantage in evaluating certain types of scattering results (see Section VI, E, 3). A reliable method for predicting the actual velocity distribution, which probably will depend on the operating pressures, is not available. Photoetched window screens24with small rectangular holes, 7.5 p on one side, have been suggested for obtaining higher intensities without a distortion of the velocity distribution (Datz et al., 1961b). 3. Laval Jet By using a Laval jet, on the other hand, it is possible to achieve supersonic flow, and this appears more desirable if extremely high intensities are desired (Kantrowitz and Grey, 1951; Kistiakowsky and Slichter, 1951). For an excellent detailed discussion of Laval jets as well as other beam sources, see the article by Anderson et al. (p. 345, this volume). Figure 18 shows the experimental setup as it is customarily used. In addition to the Laval jet, a simple converging nozzle has also been used (Deckers and Fenn, 1963; Scott and Drewry, 1963). This has the advantage that hydrodynamic calculations for this case have been carried out. The skimmer construction is critical since, if the angle 6 in Fig. 18 is too large, the shock wave at the skimmer will be detached and spread itself over the mouth of the skimmer, resulting in sonic flow at the downstream side of the skimmer. Under ideal conditions the beam intensity attainable with a Laval nozzle is given by

where S, is the skimmer area, no the particle density in the settling chamber, a, the sound speed in the settling chamber; y is the ratio of specific heats, and M is the Mach number at the skimmer throat. The Laval nozzle source is compared with several other sources in Table X. From the table it is obvious that inherent in this type of source is the need of much larger pumping speeds in the source chamber. Deckers and Fenn had a pumping speed of 5000 liters/sec in the nozzle chamber and 40,000 liters/sec in the experimental chamber. Even with these pumping speeds they were not able to obtain a vacuum in the chamber below a few times Torr. Campargue (1964) has recently described an improved design requiring less pumping speed. This disadvantage would not be encountered in experiments using easily condensable vapors and gases. This has been tried in only one known 24

Obtainable from Buckbee Mears and Company, St. Paul, Minnesota.

H. Pauly and J. P. Toennies

246

sklmmer

lava1 nozzle

-

A

experimental chamber

settling chamber

to

to pumps

FIG.18. Schematic diagram of a nozzle source. TABLE X CoMPARISON OF

Source

Typical value for ratio of theoretical forward intensity to total flow

DIFFERENT TYPES OF BEAMSOURCES

Source area

Measured forward intensity molecules

References

Single orifice,

0.3

0.77 mm

2.5 x 1017(Hz)

Becker and Bier (1954)

Single orifice,

1.4

0.77 mm

2.5 x 10l8(Hz)

Becker and Bier (1954)

h>d

h
Many-channel array Laval nozzle

2.4

11

mm

-12 (M= 4) 0.77 m 0.2 ( M = 10)

5

x IO"J(C0a) Giordmaine and Wang

1.8 x lOle (Hz)

(1960)

Becker and Bier (1954)

instance (Hundhausen and Pauly, 1964a) and appears to be very promising. Some acceleration of the beam molecules amounting to about a factor 2 in the velocity can be achieved by adding He or H2 to the source gas (Becker and Bier, 1954; Becker et al., 1955; Waterman and Stern, 1959). If sufficient amounts of the light gas are added, the beam velocity can be raised to that of the light gas, at a sacrifice of the beam intensity of the heavier component. For many applications the Laval nozzle has the advantage that the expanded gases leaving the nozzle are at a low temperature, corresponding to 50°K. As a result the velocity spread is about a factor two narrower than with an

-

THE STUDY OF INTERMOLECULAR POTENTIALS

247

orifice (Becker and Henkes, 1956; Valleau and Deckers, 1963; Phipps e t al., 1963), and the lower internal energy states are preferentially populated. The Lava1 nozzle has also been combined with a high frequency discharge to obtain a high intensity H atom beam. Recombination is avoided by allowing the discharge to occur in the skimmer (Fleischmann, 1961), which may, however, lead to serious flow disturbances.

4. Fast Beam Sources The previously mentioned sources can be used to produce beams with measurable intensities up to energies of about 0.5 eV. Several suggestions have been made and tried for obtaining neutral beams of higher energy. a. Shock Tube and Supersonic Nozzle. By using the high temperature region behind a reflected shock wave it is possible to raise the source temperature to several thousand degrees and the beam energy to several electron volts. This method has several disadvantages : Since the beam samples the equilibrated gas behind the reflected shock it consists mostly of atoms with a small fraction of molecules and, under certain conditions, ions. Since shock wave repetition rates are of the order of 5 min per run the available time for adjusting and optimizing the apparatus will be severely restricted. Nevertheless, the method has been shown to work (Skinner, 1961). The apparatus used is shown schematically in Fig. 19.

I

shock front

---

1 todurnp tanks

'

I

1

experimental chamber

to dump

tanks

FIG.19. Shock wave with nozzle.

b. Neutralization of Zons. For attaining very high energies the ionization of atoms and subsequent electric acceleration of the ions followed by neutralization is an extremely reliable and well-developed technique (Beek, 1934; Amdur and Harkness, 1954). Surface and space charges make the extension of this method to lower energies difficult. Nevertheless, it has recently been

H . Pauly and J . P . Toennies

248

possible to operate down to energies of the order of 10 eV (Utterback and Miller, 1961; Hollstein, 1962; Devienne, 1962; Devienne and Souquet, 1961) with intensities of the order of lo9 molecules/sec. The method can be applied equally well to atoms and molecules. An inherent disadvantage of this process is that a portion of the beam may be in excited states, which must be removed from the beam by a magnetic field. Figure 20 shows the apparatus customarily used.

electron gun

I

monochomata

neutralization chamber

I sweeping f i d

FIG.20. Neutralized ion molecular beam source.

c. Rotating Impeller. Moon and co-workers (Moon, 1953; Bull and Moon, 1954) have developed yet another method for accelerating beams. A mechanical impeller is swept through the source chamber at a high speed, corresponding to a linear velocity of 10’ cmlsec, producing a pulsating beam through the oven hole. This method has the advantage over the shock tube high temperature source in that all easily vaporisable molecules and atoms may be accelerated and that the repetition rate is much higher. The attainable energies are, however, limited. Here as with the other methods more evaluation work is necessary. d. Other Methods. Three other methods for producing beams of intermediate energy (1-10 eV) have been proposed, but none has been tried out, to the authors’ knowledge, as a molecular beam source. Sputtered target atoms arising during high energy ionic bombardment (50 keV) have energies in the range between 0 and 250 eV and could be used for beam experiments (Wehner, 1959). An estimate of the sputtered beam intensity at energies between 1 and 4eV with an incident ion current of 100 pA gives atoms 9 ( 0 ) N 2 x 10l6 Since this intensity is evenly distributed over a large velocity range, velocity selection will reduce the intensity by a factor of about 100 (Beuscher and Kopitzki, 1964). Only solid targets (mostly metals with atomic numbers greater than 20) have been investigated to date (Almkn and Bruce, 1961). In one case it was possible to sputter potassium (Stein and Hurlbut, 1961). At

THE STUDY OF INTERMOLECULAR POTENTIALS

249

high energies with single-crystal targets the sputtered atoms are preferentially emitted along crystal directions with the largest atom density, but this natural collimation is offset by a reduced sputtering rate (Southern et al., 1963). The method has the disadvantage that a velocity selector operating at extremely high speeds is necessary. Polar molecules may be accelerated by pulsed inhomogeneous electric fields in a manner somewhat analogous to a linear accelerator. The method has been applied to produce slow beams (King, 1964). For beam acceleration the method has the disadvantage that the energy gain per stage is small or only of the order of several degrees Kelvin. Finally, one other method has been suggested for obtaining neutral molecule beams in the intermediate energy range. The method consists of directing the positive and negative ions of a continuous discharge into a tube in which the ions transfer their linear momentum to neutral molecules (MoncrieffYeates, 1960).

C. VELOCITY SELECTORS Two basic types of velocity selectors seem particularly suitable for molecular beam scattering experiments. The simple Fizeau type, consisting of staggered slotted disks on a common rotating shaft, was first applied to molecular beams by Eldridge (1927) and Lammert (1929) and was further developed by Bennewitz (1956) who used an induction motor. Recently several designs have been described in the literature using commercially obtainable commutator motors (Hostettler and Bernstein, 1960; Trujillo et al., 1962). The motors were obtained from Syntorque Inc., West Hurley, New York. The induction motor drive may, however, have certain advantages when working with bakeoutable metal systems, since the stator field may be mounted externally and removed during bakeout. Such velocity selectors have the property that the product of maximum transmission times the resolving power is given by (111-4) where T is the transmission in the maximum ( T = u/c, where a is the slit width and c is the sum of slit width plus the width of the tooth). R is the velocity resolving power [ R = o/Av = d/a, where dl(2nr) is the angle between the slit in the first disk and the corresponding slit in the last disk], r is the radius of the disks. L is the distance between first and last disk, and o is the radial velocity of the rotor. P,, is the velocity of the transmitted beam. The highest reported rotor speeds are of the order of 40,000 rpm (670 rps). For obtaining

250

H. Pauly and J. P. Toennies

much higher speeds a magnetic suspension would probably be necessary. Another method for velocity selection at high velocities is to use two independently driven slotted disks having a large separation (Marcus and Mcfee, 1959). The velocity of the transmitted beam is determined by measuring the phase shift between the two disks. A second type of rotor often used for neutron velocity selection and first applied to molecular beams by Margrave and colleagues (Grosser et al., 1963) consists of a single disk with slightly curved radial slots. The arrangement is shown in Fig. 21.

FIG.21. Radial slot velocity selector.

This velocity selector has the advantage that two beams can be selected simultaneously with the same velocity. The scattering chamber and detectors have to be placed in the central hole. In an ingenious application of this rotor, Colgate (Muschlitz, 1963) intends to place the bottom side of the rotating disk in close proximity with a stationary surface. In this way the selector serves simultaneously as a pump for the scattering chamber. Both types of velocity selectors have a maximum transmission for the desired velocity of about 50 %. D. STATESELECTORS

The state selection of atomic and molecular beams in inhomogeneous electric and magnetic fields has been developed in connection with molecular beam high frequency resonance measurements of the hyperfine structure (see Ramsey, 1956). Since atoms are not known to possess electric moments they are separable only in a magnetic field. Using a relatively short (12 an long) magnetic six-pole field it is possible to separate out and focus one of the electron spin states of H (Clausnitzer, 1959; Christensen and Hamilton, 1959). By using longer fields and greater field strengths heavier paramagnetic atoms can be focused in a similar way. Practically all molecules on the other

THE STUDY OF INTERMOLECULAR POTENTIALS

251

hand are in ‘Z states and therefore can be deflected in magnetic fields” only by way of relatively weak higher-order interactions. Diatomic polar molecules can be focused and separated according to rotational states in an electric four-pole field. Under suitable conditions the molecules can be shown to have simple sinusoidal paths in the homogeneous field. The focusing action is strongest for the sparsely populated angular momentum states ( J , M ) = (1, 0), (2,0), and ( 3 , O ) and results in an increase in intensity by a factor of approximately 100 over that which would be obtained without focusing (Bennewitz et al., 1955, 1964). In the same field the upper inversion state of NH, can be selected (Gordon et al., 1955;Vonbun, 1958). Symmetric top molecules with J N K and M N - J , i.e., those with symmetry axis along the field direction, are focusable in a six-pole field (Kramer and Bernstein, 1964), whereas molecules for which the product K M = 0 are focusable in four-pole fields. All molecules with J = K = M or J = M a r e difficult to focus since they are deflected outwards in four-pole and six-pole fields. For these molecules three types of two-pole fields are available and are shown schematically in Fig. 22. All of these fields have the disadvantage that the molecules do not follow simple trajectories and that only a small focusing effect can be achieved for small apertures. For a field of the type shown in Fig. 22(c), the latter difficulty can be compensated by suitable pole-piece shaping (Friedmann, 1959). A cylindrically symmetric field formed by mounting a wire in the center of a cylinder has also been used for focusing NH molecules with a similar Stark effect as the J = M molecules [see Fig. 22(a)] (Helmer et al., 1960). A state selector for vibrational states worlung on similar principles does not appear to be feasible at the present time.

FIG.22. Schematic cross sections through three types of two-pole deflection fields. 26 An exception to this rule are salts of the type MXFeC12, where MX is an alkali halide, which have magnetic moments of the order of 70 nm (Kusch, 1960).

H . Pauly and J. P . Toennies

252 E. DETECTORS

The detector is almost as important as the source for carrying out a scattering experiment and quite often more difficult to construct. Partly because of the greater demands placed on the detector and partly because of recent advances in the design of the electron bombardment detector, here abbreviated by EB (Wessel and Lew, 1953; Fricke, 1955; Bernhard, 1957; Quinn el al., 1958; Bennewitz and Wedemeyer, 1963; Weiss, 1961; Aberth, 1961), this detector together with the Langmuir-Taylor detector (LT) have virtually displaced all other types of detectors, such as the Pirani detector. For this reason the present discussion is restricted largely to these two detectors, and for information on the other types the reader is referred to the literature (King and Zacharias, 1956; Ramsey, 1956). A brief account of the working principles of these detectors follows: With an EB detector, beam molecules are first ionized by an intense electron beam. Unfortunately, the background gas will be ionized as well (a standard beam corresponds to a pressure of only 3x Torr, whereas an easily achieved vacuum is about 3 x lo-’ Torr) and a mass spectrometer is therefore necessary to pick out the ions coming from the molecular beam. Figure 23 shows a schematic diagram of an EB detector that uses a Paul-Steinwedel (Paul et al., 1958) mass filter. The LT detector on the other hand takes advantage of the fact that beam atoms are ionized on contact with surfaces having a work function Q, greater than the ionization potential I of the atom (the atom may also be part of a beam molecule). By using suitable surfaces such as oxidized tungsten, @ 6.0 eV, it is possible to ionize the alkali metals and other elements withloosely attached outer electrons, e.g., Ga, In, T1, etc. The desirable property of this detector, that most of the background gas in the apparatus is not ionized, is also a disadvantage, since only certain atoms are detectable.

-

cold t r a p ( N 2 ) f l

ion source

__

LT-detector

mass filter I,

molecular beam

I

,

I

,,

1

FIG.23. Schematic diagram of an electron bombardment detector using a Paul-Steinwedel mass filter.

THE STUDY OF INTERMOLECULAR POTENTIALS

253

Before discussing these detectors in detail it is important to bear in mind the sources of noise, denoted by N , that may be encountered. In order of importance they are : (1) Random fluctuations in the background intensity,

where e is the electron charge, Av the bandwidth, and 9,, the background intensity in amperes. (2) Multiplier noise resulting from the statistical distribution in the amplification factor for the individual stages of the multiplier, Nlast dynode

- -J” v-

Nfirstdynode,

where Vis the average gain per dynode stage. (3) Vacuum fluctuations in the molecular beam chamber. The pressure fluctuations of oil diffusion pumps can be reduced to about several times lo-’ Torr. The resulting beam fluctuations depend in turn on the length of the beam and the angular resolving power. (4) Natural statistical fluctuation of the signal, N

= J2e

AV yS,

where 9,is the signal intensity in amperes. The use of a mass spectrometer almost always reduces the background intensity and the noise coming from the first source. The second source of noise can be eliminated by either avoiding the use of a multiplier or, in the case of very feeble signals, by counting the individual pulses. The third source of noise can be reduced only by shortening the beam path or by improving the vacuum. It can also be reduced by using a beam chopper set at a frequency different from that of the pressure fluctuations and by rectifying the chopped signal in phase (lock-in amplifier). Finally, it is possible to improve on the statistical noise by either increasing the beam intensity or the measuring time. The signal-to-noise ratio from the first source for an ideal detector26 with a mass spectrometer can be shown to be equal to (Schlier, 1957; Bennewitz and Wedemeyer, 1963) (111-5) 26 An ideal EB detector is one in which the ionization volumes for beam and background are the same.

H . Pauly and J . P . Toennies

254

where fi is the average velocity of the beam molecules (the way in which the average velocity is calculated depends slightly on the type of detector used) and where nb is the particle density of all background molecules in the detector. E is the relative ionization efficiency for the background and is defined as the ratio of the number of ions of all masses produced per background molecule (air) to the number of ions of the desired mass produced per beam molecule.27 g describes the reduction in the background intensity at the observed mass obtained by using the mass spectrometer. This reduction factor depends in a complicated way on the type of background, the beam molecule, and the resolving power of the mass spectrometer. k is the fraction of beam molecules that passes through the ionizer and is ionized, andf'is the fraction of ionized beam molecules that enters the mass spectrometer, while T is the transmission of the mass spectrometer. With this expression it is possible to discuss the theoretical performance of both the EB and the LT detectors. Our experience at the Bonn Institute with the two types of detectors using the same type of mass spectrometer (Paul-Steinwedel four-pole field) in similar oil diffusion pumped brass apparatuses can be summarized in the following tabulation EB detector (Kr) k 9 &

10-2 10-3

51

LT detector (TIF)

51

10-3 10-3

From this comparison a theoretical ratio of the signal-to-noise ratios for the two detectors can be arrived at with the help of Eq. (111-5). The result is NEB/NLT w 3 x 10' and is in good agreement with the measured sensitivities. Since E and g are large for oil vapor this is a particularly undesirable component in the background gas in the detector chamber. By surrounding the detector with a liquid nitrogen cold trap it is possible to reduce the partial pressure of the hydrocarbon vapors by a factor of lo2-lo3. In addition to the alkali atoms (Schroen, 1963) and their salts, thallium and the thallium salts also appear to have a high ionization efficiency28 on an oxidized tungsten wire. Other atoms and their salts that can be detected on oxidized tungsten with only slightly lower efficiencies are Ga, In, and Ba. For K, Rb, and Cs, efficiencies close to 100 % can also be obtained on tungsten 27 E may be approximated by using ionization gauge calibration factors defined relative to air. 28 The ionization efficiency for TlF has recently been measured to be 90% (Schiirle, 1964).

THE STUDY OF INTERMOLECULAR POTENTIALS

255

wires without oxidation. Datz and Taylor (1956) have shown that Pt and Pt (with 8 % W) wires have low ionization efficiencies (< 1 %) for the alkali halides but a reasonably high efficiency (10-40 %) for the metals. A detector consisting of a Pt or Pt (8 % W) wire together with a W wire can, therefore, be used to detect beams of the alkali halides in the presence of atom beams of the same alkali. Recently it has been shown that the selective behavior of Pt (8 ”/, W) wires depends on a special state of surface contamination (Touw and Trischka, 1963). For this reason extreme care must be exercised with these detectors if precise quantitative measurements are required. They can be avoided altogether if a magnetic deflecting field [see Fig. 22(b)] is used to prevent the paramagnetic alkali atoms from striking the detector (Herm et al., 1964). One important disadvantage with the LT detector is that even small amounts of alkali impurities in the tungsten wires will lead to a serious background. Thallium apparently does not appear as a background, and this is one reason why thallium salts are used for molecular beam experiments. Methods have recently been developed for improving the purity of these wires, and the alkali background coming from such ultra-pure wires can be a factor 10’ smaller than with previously available wires. It is also possible to prepare ultra-pure wires in a simple apparatus (Greene, 1961). Detectors using spectroscopically pure platinum or Pt (8 % W) (@ 5.5 eV) generally have a much lower alkali background and should be used whenever possible. Hydrocarbons from oil pumps are readily cracked to free radicals which in turn are easily ionized on hot oxidized tungsten wires and contribute an additional background, which is distributed more or less uniformly over the mass range 50-225 (Bennewitz et al., 1964; Schurle, 1964). Thus, despite cold traps and a when such hydrocarbons are present. high resolution, g is limited to Excited atoms in metastable states may also be detected with a LT detector since the ionization energy of such atoms is usually considerably reduced (Faust and McDermott, 1963). With most metastables the conversion efficiency is of the order of a few percent. Molecules containing atoms with strong electron affinities can also be detected by negative ion formation with a detector similar to the LT detector (Trischka et al., 1952). The efficiency of this process, however, is low or about

-

0.1 %.

For other atoms and molecular beams an electron bombardment ionizer is necessary. The design and construction of such ionizers depends essentially on the type of mass spectrometer with which they are used. In general it is difficult to ionize much more than 1 % of the beam, and this is the essential limit on the sensitivity of these detectors (see Table XI). In order to reduce the background intensity of molecules with the same mass as the beam molecules, ultrahigh vacua ( Torr) are desirable in the ionizer region. N

H . Pauly and J. P . Toennies

256

TABLE XI COMPARISON OF SEVERAL RECENTMOLECULAR BEAMDETECTORS

Detector

Transmission Ionization Background Signal of mass efficiency pressure registra(Torr) tion spectrometer a = Kf

EB without mass spectrometer

-

9x

EB with a 4-pole mass filter

0.5

2.5 x

EB with magnetic mass spectrometer

0.25

4x

LT without mass spectrometer

(1.0)

(1.0)

N

LT with 4-pole mass filter

0.5

1.0

N

a

2 x lo-*

S T ~ N D A R DReferences BEAM

Lock in lo3 (K) 64 Hz T = 10 sec

Friedmann (1959)

Lock in 2 x lo4 (Kr) Bennewitz 62 Hz and T = 2.4 sec Wedemeyer (1963)

N

a

($)

3 x

10-6

Lock in 23 Hz T= ?

loe (Kr)

Dc amplified T < 1 sec

3 x lo4(K)

Pulse counting T 3 sec

3 x lo6(TI) Bennewitz et al. (1964)

Weiss (1961)

-

N

Includes 50 % transparency of many-channel aperture by which beam enters detector.

In choosing a mass spectrometer for the detector, the ratio Tflg has to be considered carefully. The Paul-Steinwedel mass filter (Paul et al., 1958) has the advantage of a high transmission combined with a large resolving power, e.g., T = 0.50 and M / A M = 85, and has a resolving power that is independent of the mass and injection velocity of the ions. So far, however, it has been difficult to achieve a large value for the product kf, especially when slitshaped beams are to be detected. Values of the product kf for various detectors are listed in Table XI. Both systems have advantages, depending on the application. By using a “lock-in’’ amplifier a certain improvement in the signal-tonoise ratio from sources of noise of the type listed under (3) in Section 111, E, can be achieved as already mentioned. Harrison et al. (1964) have calculated the velocity dispersion of modulated beams. Bennewitz and Wedemeyer (1963) have shown that an improvement of a factor 6 in SIN can be obtained by using only a narrow band width amplifier (e.g., A V ~=, 0.8 ~ cps at 62 cps), whereas the phase-sensitive detection brings only an additional 60 %.

THE STUDY OF INTERMOLECULAR POTENTIALS

257

F. SUMMARY In this section an attempt has been made to summarize briefly some of the recent advances in experimental techniques that are of crucial importance in carrying out the more sophisticated scattering experiments. I n planning a scattering experiment, Eq. (111-5) and the other sources of noise listed in Section 111, D have to be considered with great care. To help in estimating the intensity at the detector some characteristic values of beam attenuation for the components in a typical apparatus are summarized in the accompanying tabulation. Component Velocity selector

Beam attenuation 5 x 10-0

Rotational state selector including focusing

10-1 to 10-2

Fraction of intensity at small angles

10-1 to 10-8

Fraction of intensity at large angles

10-4 to 10-7

Over-all LT detector efficiency

1 (alkali metals and salts)

Over-all EB detector efficiency

10-2 to 10-8 (other beams)

Finally, from Eq. (111-5) it follows that an improvement in beam intensity leads to a proportional increase in the signal-to-noise ratio, whereas a corresponding improvement in one property of the detector enters only as the square root.

IV. Molecular Scattering Theory A. INTRODUCTION

The calculation of the scattering cross sections measured in molecular beam experiments usually requires the consideration of an extremely large number of partial waves. In an extreme case, for example, involving the strong dispersion interaction between alkali atoms, kb,,, x l,,, is of the order of 1O00, For this reason the direct application of the exact quantum mechanical method, in which the phase shift is determined for each partial wave, is prohibitively time consuming, and has been carried out only for scattering partners for which i,,, is not large (I,,, = 50). Fortunately, the large angle scattering can often be treated classically, and only at small angles is it necessary to take

H. Pauly and J. P. Toennies

258

account of the wave nature of the colliding particles. Recently, several approximate methods have been developed and successfully applied to the calculation of scattering at small angles. The discussion of this section starts with the classical description of elastic scattering by an intermolecular potential of spherical symmetry. The classical scattering by an angle-dependent potential is not discussed, since transitions between energy states cannot be dealt with classically. The classical description of scattering by a multidimensional potential is discussed in Section VI. The quantum mechanical treatment and the semiclassical approximation are then described. In the second part of the section a theory of inelastic scattering is outlined. Emphasis will be placed on the assumptions and applicability of the theories and not so much on a rigorous mathematical development. In some of the sections it will prove to be more convenient to introduce reduced parameters. This is advantageous only if special potential models, such as those described in Section I, D, are used. By using reduced quantities it is possible to readily adapt the formulas and numerical calculations to various specific collision partners. The following reduced parameters appear in Sections IV, B and IV, C, 3:

B = b/r,

(reduced impact parameter)

P = rlr,

(reduced distance)

K

(reduced energy)

= E/E

U ( p ) = V(p)/E

In addition, the abbreviation A

(reduced potential). = kr,

is used in these sections.

B. CLASSICAL SCATTERING BY SPHERICAL SYMMETRIC POTENTIALS The two-particle collision problem is formally reduced to a one-particle problem by a transformation to the center-of-mass system. Thus one needs to consider only the scattering of a particle [of reduced mass p = rnlrn2/(rnl + mz)] which is moving with the relative velocity g with respect to an infinitely heavy stationary particle. The angle of deflection 9, as a function of the impact parameter byis obtained directly from the conservation of energy and angular momentum : (IV-1) pminis the reduced distance of closest approach and corresponds to the largest value of p for which the radical is zero. Equation (IV-1) relates an angle of deflection to every impact parameter. In the case of a typical intermolecular

THE STUDY OF INTERMOLECULAR POTENTIALS

259

potential this relation cannot be uniquely inverted, with the result that there can be several impact parameters that lead to the same angle of deflection. In calculating the differential cross section, therefore, it is necessary to sum the contributions from all impact parameters leading to the same scattering angle:

(IV-2) For realistic potentials (see Section I, D) the classical deflection function 9(p) cannot be calculated in closed form. Table XI1 gives a summary of known numerical calculations. Figure 24 shows the typical behavior of the deflection function for a realistic potential (Lennard-Jones potential). Collisions with positive scattering angle can be attributed largely to the repulsive portion of the potential at small impact parameters. The scattering angle decreases with increasing impact parameter, until the average repulsive and attractive forces just cancel, and no deflection is observed (impact parameter Do). For still larger impact parameters the long range attractive forces lead to a change in the direction of deflection to negative angles. After going to a maximum at pr the deflection function converges to zero as p tends to infinity. For small values of the reduced energy K (the exact value depends on the potential) the maximum negative angle of deflection corresponding to the impact

FIG.24. Calculated classical deflection function8@) at different reduced energies K for a Lennard-Jones (10, 6) potential. The diagram in the upper right-hand comer gives the definition of various reduced impact parameters.

h,

G l

0

TABLE XI1 SURVEY OF CLASSICALSCATTERING CALCULATIONS

Classical deflection function

Differential cross section

Tables of s(& K )

/%(a, at(K),&"(K)

Tables O f Z ( 8 )

Hirschfelder ef ul.

Potential Lennard-Jones (12, 6) Lennard-Jones (n,6) Kihara (12, 6) Buckingham (a,6)

(1954)

Diiren and Pauly (1964)

Total cross section

%

e

kl

iz

-

flo(m,ao'(K), ao"(K) -

Diiren and Pauly

-

-

-

Diiren and Pauly

(1964)

-

3

-

9 s. 8

-

Schlier (1963)

-

-

Diiren and Pauly (1963)

-

(1964)

-

Mason (1957)

S(a)

Mason (1957)

a 3

rl

4

3

THE STUDY OF INTERMOLECULAR POTENTIALS

26 1

parameter p, goes to infinity (orbiting collisions). In general these orbiting collisions contribute only little to the cross sections, since the derivative dp/d9 [compare with Eq. (IV-2)] is very small in the vicinity of P, (see Fig. 24). For angles 9 < 9, three different impact parameters PI, Pz, and p3 contribute to the differential cross section, whereas for 9 > 9, only one impact parameter p1makes a contribution. Since the classical cross section contains the factor (dS/dj)-',it will have a singularity where d9/dPvanishes (see Fig. 24). Since the optica1,analog of this phenomenon is responsible for rainbows, it is customary to refer to the scattering in the neighborhood of the angle 9, as rainbow scattering. Near the rainbow angle 9, the deflection function may be expanded in the form with

9 = 9,

+ q(P -

Pr)',

(IV-3)

From this expansion, the behavior of the rainbow part of the differential cross section (only considering the contributions pzld/?,/d91and /I3 IdP3/d9Iin Eq. (IV-2) and neglecting the additional contribution Qlld&/d91,see Fig. 24) can be easily derived. On the bright side of the rainbow (9 < 9,), the rainbow cross section will be

(IV-4) whereas for the dark side (9 > 9,) the rainbow cross section will be zero. The additional branch of the deflection function, which describes the scattering at small impact parameters (& IdP,/d91),gives a nearly angle-independent contribution to the rainbow cross section calculated in Eq. (IV-4). Figure 25 shows the characteristic behavior of the classically calculated differential scattering cross section as a function of the angle with the reduced energy K as parameter. The large increase in the differential cross section at small angles may be attributed to the scattering at large impact parameters for which dp/d9-+ co. For these large impact parameters the integral (IV-I) can be expanded, and only the attractive term in the potential V(r) = -C/r" need be considered, with the result that

(3b'

9 = (s - l)f(s) - 7 ,

where

(IV-5)

262

H. Pauly and J. P. Toennies

0

FIG.25. Differential scattering cross section (weighted with sin@ as a function of the angle of deflection in the center-of-mass system for different reduced energies K. Classically calculated for a Lennard-Jones (10,6) potential.

Mott-Smith (1960) has recently derived an expression for the classical deflection function for arbitrary angles and power law potentials. From Eq. (IV-5) the differential scattering cross section can be calculated easily :

(IV-6) Superimposed on the singularity at 9 = 0 resulting from the long range attractive forces [see Eq. (IV-6)], there is a second singularity coming from collisions with an impact parameter Po. This singularity as well as all others which are entirely a result of sin 9 = 0 are designated “ glory effect” in analogy to optics (Ford and Wheeler, 1959). Additional glory singularities may occur, if the deflection function passes through n, 2n, etc., at impact parameters P # 0 (for example in the case of orbiting). The glory effect for elastic head-on collisions corresponding to 9 = n is suppressed since P goes to zero in Eq. (IV-2). This, however, may not be the case if inelastic processes such as chemical reactions occur (Herschbach, 1962). A general criterium for the occurrence of a glory for head-on collisions is

THE STUDY OF INTERMOLECULAR POTENTIALS

263

where Preact(/?) is the probability of reaction and 9, is the deflection angle for the product molecule (glf for inelastic scattering). In the case of elastic scattering the forward glory at 9 = 0 is swamped by the unattenuated primary beam intensity; however, in the case of inelastic processes it may be removed sufficiently from the unattenuated primary beam to make it observable. Of considerable interest in connection with chemical reactions is the backward glory 9 = 71, since the associated increase in intensity will make the observation of this otherwise weak scattering possible. The dependence of the rainbow singularity on the reduced energy permits a determination of the potential parameter E from measurements of the differential cross section at different energies. Furthermore, the differential cross section at small angles provides information on the attractive part of the interaction potential (parameters C and s), whereas measurements of the differential cross section at large angles (large compared with the rainbow angle) may be used to give information on the repulsive part of the potential. In the case of small angle scattering as well as in the case of rainbow and glory scattering, quantum mechanical modifications of the above results must be taken into account. The classical scattering by potentials proposed for the interaction of reactive molecules (see Section I, D, 4) can be treated in the same way as described above. The behavior of the classical deflection function 9(p) is shown in Fig. 26. In general, there may be five different impact parameters leading to the same angle of deflection. The existence of at most three impact parameters for which d9ldP = 0 gives rise to two or possibly three rainbow singularities if orbiting does not occur (Herschbach and Kwei, 1963). The experimental observation of these rainbows should provide a good method to check the usefulness of this potential model. From Eq. (IV-6) it follows that the classically calculated total scattering cross section diverges for small angles. With the help of the relationship Z(9) sin 9 d 9 = 2mm2

(IV-7)

it is possible to calculate a scattering cross section that accounts for all scattering with angles larger than a certain angle CI.If pl, p2, and p3 are the impact parameters corresponding to CI then (see Fig. 24) -"a: - j332 nrnl

For small values of goes to

CI,

+ Pl2 - /I2'

M

P32 + 2P0 A&

(IV-8)

p1 approaches p2, and the classical cross section

264

H . Pauly and J. P. Toennies (IV-9a)

P3 is almost entirely determined by the attractive part of the potential, for which the integration of Eq. (IV-6) leads to

(IV-9b) Once more the divergence of the total cross section for u + O becomes apparent. From the uncertainty principle it follows that the uncertainty in the scattering angle will be largest for grazing collisions with large impact parameters. Since the magnitude of the scattering angle for such collisions is also small the uncertainty in the angle will become larger than the scattering angle, and it is no longer possible to use the classical deflection function and quantum mechanical calculations are necessary.

Reduced impact parameter 13

FIG.26. Classical deflection function reduced energies K .

for a potential with two minima for different

265

THE STUDY OF INTERMOLECULAR POTENTIALS

c. QUANTUMMECHANICAL THEORY FOR CALCULATING ELMTIC CROSS SECTIONS

1. Basic Formulas

In the quantum mechanical theory of elastic scattering the problem is to solve the Schrodinger equation for motion in a spherical symmetric potential V(r).Here as in the classical case it is necessary only to consider the scattering of a particle of reduced mass p on a stationary scattering center. The Schrodinger equation may be written as (IV-10) ) ' the / ~ wave number of the incoming wave. The desired where k = ( 2 ~ E / h ~ is solutions must have an asymptotic form corresponding to an incident plane wave and an outgoing spherical wave with amplitudef(9) : +(r) = exp ikz

+f(9) -exp ikr. r

(IV-11)

If the density of the incident wave is unity, then the incident flux is g particles/ cm2 sec where g is the relative velocity. The flux scattered through a solid angle do is then 9 ( 9 ) = lf(9>I2Q do.

(IV- 12)

The differential cross section Z(9) is the fraction of the incident flux scattered into an angle do and is given by = lf(9)I2.

(IV-13)

The integral cross section, defined as the integral of the differential cross section over the entire unit sphere, is 0

= J(m;(

9) do.

(IV-14)

In solving Eq. (IV-10) the incident wave is expanded in a sum of partial waves, each with orbital angular momentum 1. The result forf(9) obtained by this method of partial waves is i "

f(9) = - - C (21 2k I = O

+ l)(exp 2iq1- l)PI(cos $),

(IV-15)

where PI(cos 9) are the Legendre functions. The phase shifts ql for each of the partial waves are required in order to calculate the scattering amplitude from

266

H. Pauly and J. P . Toennies

Eq. (IV-15). This problem is discussed below. Inserting Eq. (IV-15) into Eq. (IV-14) and using the orthogonality properties of the Legendre functions the integral cross section reduces to (IV-16) 2. Approximate Methods for Calculating Phase Shifts In the quantum mechanical method, the scattering phases are determined by the asymptotic behavior of the radial eigenfunctions R,(r) for each partial wave. These in turn are solutions of the radial wave equation

In general, the solution of Eq. (IV-17) must be carried out using numerical methods (Collatz, 1959). As the number of partial waves (I,,,,, x ka = 2na/l) increases the amount of numerical work required becomes extremely large. For this reason the exact method is pretty much restricted to problems with l,,,,, x 50 or, in other words, to such systems, in which one of the partners is very light, e.g., H, H,, and He. Table XI11 summarizes available quantum mechanical calculations of scattering cross sections. Figure 27 shows an example of a quantum mechanical calculation of the differential cross section for an assumed H,-Hg potential (Bernstein, 1960). Quantum mechanical calculations are also useful for assessing the validity of the approximate methods to be discussed later (Marchi and Mueller, 1962, 1963). Since quantum mechanical calculations are necessary for only a limited number of collision partners or, in the case of other systems, for only a negligible region of small impact parameters, approximate methods are generally preferred to these tedious calculations. Two criteria are of importance in selecting various approximate methods for the calculation of phases. These are the ratio all of the dimension of the potential denoted by a, to the wavelength of the incoming wave 1,and the ratio a/6 of the dimension of the potential to the quantum mechanical uncertainty in the position of the particle 6.” If both ratios are large, then the scattering particle will follow very closely a classical trajectory, and the classical scattering theory of Section IV, C, 1 may be used :

a’n 9 a16 $= 1 29

classical trajectory.

The ratio a16 is of the order of ermlfig.

TABLE XI11 S ~ W OF Y EXACTQ U A N T U MECHANICAL M CALCULATIONS ~~

References Halpern and Buckingham (1955) Cohen et a!. (1956) Buckingham et af. (1958) Bernstein (1960) Bernstein (1961) Bernstein and Morse (1964) Diiren and Pauly (1964) Mueller and Brackett ( 1 964)

Calculated quantities

Potential

Collision partners (parameter range)

Buckingham

He-He K < 1, A < 3

T < 5°K

Differential cross section Integral cross section UW) Phase shifts differential cross section Differential and integral cross sections Differential and integral cross sections

Lennard-Jones (12, 6 ) Modified Buckingham Lennard-Jones (12, 6 )

Hz-HZ K=5,Aw8 Hz-HZ K < 3, A < 7 K b 7, A b 30 Hz-Hg

TwWK

Lennard-Jones (12, 6)

K b 7. A b 30

Integral cross sections

Lennard-Jones (12, 6)

Integral cross section

000

o(g)

Integral cross sections dK)

Modified

Lennard-Jones (12, 6)

He-He He-Ar He-Xe He-Hg Ne-Ne CH4-CH4

~~

Temperature range

T < 70°K

T w 300°K

T w 300°K

268

H. Pauly and J. P. Toennies

- quantum calculation

--I

classical calculation

-

K 15

I I I

A 14)

I I

I

I I

, 1

\ i \

\

\

n

I

/

Deflection angle in the center of

~ Q S Ssystem

[deg]

FIG.27. Differential cross section as a function of the angle calculated by the exact method of partial waves (Bernstein, 1960) for the Ha-Hg scattering.

If both ratios approach one then a classical trajectory loses its meaning, and the wave nature of the particles must be taken into account: a/n 5 quantum effects, 4 6 51 In the very extreme case (u/A 4 1, a/S 4 1) it is necessary to proceed, as indicated in Section IV, C, 1 and solve the radial wave equation for each partial

269

THE STUDY OF INTERMOLECULAR POTENTIALS

wave. Between these two extremes it is possible to calculate the phases using approximate methods such as the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) method and use these phases in the quantum mechanical formulas derived above. a. The Born Approximation (a/6 4 1). The basic assumption in the Born approximation is that the perturbation by the interaction potential is small. In connection with molecular beam experiments, however, it is convenient to specialize the Born approximation to cases in which a/A > 1, and this is done in the next section. Then, a/6 4 1 is a sufficient condition for the Born approximation. The other extreme case, a/L 4 1, is of only little interest for molecular beam scattering experiments because of the restricted region in which it is useful. The general Born expression for the scattering phases is (IV-18) where JI+1,2(kr) are Bessel functions. b. Semiclassical Region (a/A % 1;a/6 arbitrary).This is the region of greatest interest for molecular beam scattering experiments. The JWKB expression for the phases given below is valid in the entire region a/L % 1,

(:u

qJWK"(P)= A

1-

p'

112

7)

dp

-

sp:(

1-

-

5)"'

dp] , (IV-19)

where I + 4 has been replaced by kb = Ap. The parameter A = k r , appears as a multiplicative factor in Eq. (IV-19), implying that the JWKB phases are dependent on rmonly through the factor A and the reduced impact parameter p = b/r,,,.Thus the JWKB phases can easily be adapted to different systems, which is not possible in the case of the rigorous quantum mechanical phase shifts. Figure 28 shows a typical plot of the reduced phase q* = ? / A as a function of the reduced impact parameter 8. A comparison of JWKB phases calculated for a Lennard-Jones potential at a/A + 1 with quantum mechanical phases indicates that the JWKB phases give correct results to within a few percent (Marchi and Mueller, 1963). A comparison of Eq. (IV-19) with the classical deflection function shows that the JWKB phases satisfy the following relation : (IV-20) This is sometimes referred to as the semiclassical equivalence relationship. Equation (IV-20) may be used for obtaining phases from the classical deflection function (Bernstein, 1962a; Smith, 1964).

270

H. Pauly and J. P . Toennies

+0.5

1

I

I

I

0.5

1.0

1.5

I

I

2.0

2.5

0.4 0.3

-

0.2 0.1

*

$7

-

0 . 0.1

-

0.2

-

0.3 0.4 -

0.6 -

0.5

0.a

Reduced Impact parameter

p

FIG.28. Reduced phase shift as a function of the reduced impact parameter ~ ( p ) ,

c. High Energy Phases. At the large impact parameters generally implied by a/A $ 1 the intermolecular potential averaged over the collision path is usually considerably smaller than the relative kinetic energy. As a consequence, it is reasonable to assume that the angular deflection will be small and that the trajectory may be approximated by a straight line:

r2 = b2 + z2.

If a straight line trajectory is assumed in Eq. (IV-19) for the JWKB phase an expansion in powers of VIE can be obtained, the first term of which is (IV-21) The phase shift obtained in this way is identical to the phase shift appearing in an approximation developed for treating nuclear interactions at extremely high energies (Moliere, 1947). For this reason it is designated " high energy approximation " in the subsequent discussion. The phase shift expression

THE STUDY OF INTERMOLECULAR POTENTIALS

27 1

(IV-21) is also encountered in optics where the corresponding region is called anomalous diffraction. Glauber (1959) was able to derive the same result using quantum mechanical scattering theory. He had to assume that the back scattering was negligible and that the wave function varied slowly over a wavelength. His derivation leads to the following regions of scattering angles in which the approximation is valid:

Schiff (1956) has obtained similar results by approximating the individual terms in the Born series by the method of stationary phases. In view of the fact that the scatterer pierces the potential field along a straight line without being radially scattered, it is not surprising that this approximation is also valid for anisotropic potentials of the general form V(x,y , z). Consequently, the approximation has proven to be very useful in treating the total and inelastic scattering of atoms and molecules on molecules. d. Validity Diagram. The regions in which the above approximations are best suited for calculating phases may be summarized in the form of a diagram. For this purpose a/A has been replaced by the angular momentum quantum number I of the characteristic partial wave. Since it may be shown that to a good approximation a/S is equivalent to 2qhe,the latter is used in the diagram in place of the former. Finally it is noted that, if I PI, the absolute value of the average potential is defined as

I VI -

:J_:,

=-

V ( X ,y , Z ) dz,

(IV-22)

the following relationship holds :

IVI

- I = 2qy.

E

In the validity diagram, Fig. 29, the logarithm of lVl/E has been plotted against the logarithm of 1. IYI/E extends from 1 to and I from 1 to lo3, 2qheis constant along the diagonals and runs from to lo3. Finally, it is pointed out that the scattering angle defined in Eq. (IV-5) for small values of (VI/Eisproportional to I VIIE,

171

9 = (s - I)-.

E

H . Pauly and J. P . Toennies

272

quantum mochanicsl

1

10

somiclassical

io2

10'

FIG.29. Validity diagram for the calculation of scattering phase shifts.

The IF]/,!?,1 plane is divided into two halves, the dividing line being arbitrarily set at 1 = 30. To the left of the boundary 1 < 30, corresponding to small values of a/A, a rigorous quantum mechanical calculation is expected to be required. Only in the case that 2qheis small, a certain simplification is possible by using the Born approximation. The more important semiclassical region for I > 30 is described by the JWKB phases. For large 2qh"the crosssection results go over into those obtained from classical theories, while for IVI/E < 1 the phase expression simplifies to the high energy expression. If a straight line trajectory is inserted in the Born approximation expression for the phase shift [Eq. (IV-18)] a result identical with the high energy phase is obtained (Massey and Smith, 1933; Wu and Ohmura, 1962). For this reason

THE STUDY OF INTERMOLECULAR POTENTIALS

273

the high energy phases are sometimes referred to as Jeffreys-Born phases. In reduced notation I V I/E and I are replaced by 1/K and A , respectively. The boundaries in Fig. 29 are quite arbitrary, and it may well be that some of the theories have larger realms of validity. Nevertheless, it is hoped that the diagram is of use in deciding which of the approximate methods is most appropriate for treating the scattering from a particular region of impact parameters. In the next section further approximations implied by the semiclassical method are discussed and applied to the calculation of differential cross sections.

3. Semiclassical Calculation of Differential Cross Sections The semiclassical calculation of the differential cross section may be defined by a set of mathematical approximations first introduced into the exact quantum mechanical expression (IV-15) by Ford and Wheeler (1959) in order to simplify the numerical calculations. For general methods see Massey (1956) and for a discussion similar to the one presented here see Bernstein (1965). (1) The phase shift q1 is replaced by its JWKB approximate value [Eq. (IV- 19)]. (2) The Legendre polynomials are replaced by their asymptotic expressions, which are valid for large angular momentum I : (a) For sin 9 5 1//,

+ 319).

P,(cos 9) E (cos 9)'JO([I

(IV-23)

(b) For sin 9 ,> 1//, P,(cos 9) 2 [+(l

+ 3)n sin

sin [ l

+ 339 + 4n).

(IV-24)

(3) The summation in Eq. (IV-15) is replaced by an integration. Even with these simplifications it is not possible to calculate the differential cross section in closed form for realistic potentials. Since an analytic expression for the differential cross section is desirable in order to discuss its general behavior, a fourth approximation is necessary: The integral for the scattering amplitude is evaluated using the method of stationary phases. In applying these methods to evaluate the scattering amplitude integral it is convenient to distinguish between two angular regions depending on whether the approximations (2a) or (2b) for the Legendre polynomials may be employed.

H. Pauly and J. P . Toennies

214

a. Small Angle Scattering [Approximation( t a )for the Legendre Polynomials]. In order to use this approximation, the following condition must be satisfied for all angular momenta 1: 1

sin 9 5 -,

1max

where I,,, is the largest value of I that makes a significant contribution to the scattering. Since I,,, !z ka, where a is the size of the potential, the above condition may be written as sin 9 5 -. ka 1

Thus the scattering amplitude reduces to ra,

do + :

f(9) = -

(21

l)(exp 2iqI - l)Jo([l

+ $39)d l .

(IV-25)

As already mentioned, Eq. (IV-25) can only be integrated numerically. In order to extract the physical content of Eq. (IV-25), the integration over I is divided into the following four intervals (for illustration see Fig. 30, which shows a typical plot of the JWKB phases as a function of the angular momentum 1): 1.

0 I1 < 10 - Al12,

+ All2

2.

1, - A112 I 1 I lo

3.

1, -k Allz < 11 m,

4.

m
In the first and third intervals the phases are large and change rapidly with changing 1. As a result exp 2 4 , is an oscillatory function depending on I , and the integral in these regions averages to zero (random phase approximation). The boundary m is chosen so that the phases are sufficiently small for I > m to permit an expansion of the exponential, exp 2iqI - 1 = 2iql - 4q12. Furthermore the integral (IV-19) for the JWKB phases may be expanded for large angular momenta I > m : (IV-26)

275

THE STUDY OF INTERMOLECULAR POTENTIALS

P

i

. \

Q-

FIG.30. Typical behavior of the classical deflection function

~(m,and the functions tp f@).

a(/$, phase shift function

Since the phases for large I are nearly completely determined by the attractive part of the potential, which may be written in the form -C/rs, Eq. (IV-26) leads finally to the result, valid for large I,

(IV-27) with

H . Pauly and J. P . Toennies

216

The second interval centered about lo is characterized by

(2) = o l=lo

(region of stationary phase). In the neighborhood of 1, there will be a large contribution to the integral (IV-25) since exp 2ir], is not oscillating. In this region, the phase function may be replaced by a parabola: = 1(20)

+ 4 2 - lo?

(IV-28)

Using relation (IV-20) between the phase function and the classical deflection function, the parameter K can be calculated from the derivative of the classical deflection function at the point Po: (IV-29) The boundary m is, to a certain extent, arbitrary. Following Massey and Mohr (1933), m is defined by (IV-30) with the result that

where C

co=-y.

Errn The final result for the scattering amplitude [Eq. (IV-25)] at small angles is

(IV-31) where

n

L

Al(X) = - Jl(X), X

and Jl(x) is a Bessel function of first order.

277

THE STUDY OF INTERMOLECULAR POTENTIALS

The functions GZs-4(x) and G s - 3 ( ~are ) special cases of the functions Gn(x)= nxnI:J,((t)t-n-1

dt

that may be expanded in the following forms: For n = 2v,

where C , is Eulers constant. For n = 2v + 1 , G,(x) = (- 1)'"

22vy !XZV+l - (2v (2v 1)(2v!)2

+

+ 1) 5);( j=O

zi

1 (2j - 2v - 1)0'!)2'

The above result together with the expansion of the Bessel function permits a calculation of the scattering amplitude f(9) and, of course, the differential cross section Z(9) = If(9)I2for small angles. The contribution due to the glory effect (from scattering in the vicinity of Do) is, depending on A--(s+3)'(Zs-2), small and is usually important only for light atoms. The differential cross section calculated by Eq. (IV-3 I) shows oscillatory behavior. The main contribution results from the first term involving the Bessel function A,(x). Consequently, the minima of the differential cross section are approximately given by the zeros of AI(m8). Thus one obtains for the first minimum in Z(9) for a Lennard-Jones (n, 6) potential: 3.832

=

A

(

3nnA ) - ' I 5 16(n - 6)K

The potential parameters n and E being of little influence on the oscillations of the differential cross section at small angles, the potential parameter rmshould be determined with fair accuracy from the experimental observation of these interference patterns. For very small angles requiring only the first terms in the series off(9) the result for the differential cross section may be reduced in good approximation to [Helbing and Pauly (1964); Mason et al. (1964)l

(IV-32)

with gi(6) = 0.4275 and gZ(6) = 0.6091. In contrast to the classical result, Z(9) remains finite for 9 = 0.

278

H. Pauly and J. P. Toennies

b. Large Angle Scattering [Approximation (2b) for the Legendre Polynomials]. The substitution of approximation (2b) into Eq. (IV-15) yields the following expression for the semiclassical scattering amplitude :

with V* = 2 ~ k 1 (2

+ +)9 k 1114.

According to the approximation of stationary phase only the regions with drp,/dl= 0 need to be considered in the integration of Eq. (IV-33). Because of the relationship between the JWKB phases and the classical deflection function [Eq. (IV-20)] these regions are identical with the impact parameters that contribute to the classical differential cross section. This relationship is shown in Fig. 30. For 9 c 9, 'p+ has two regions of stationary phase and cponly one. For 9 > 9, only cp- has one region of stationary phase. For calculating the scattering amplitude, 'p+ and cp- are approximated by parabolas in the region of stationary phase. Provided that these regions are sufficiently separated, the scattering amplitude may be calculated from Eq. (IV-33) with the result (IV-34) where Zj(9) are the classical differential cross sections [Eq. (IV-2)] and wherej goes from one to three for a potential with a single minimum. The phase angles ctj are related to the phase shifts by 2q-2(1+3)1'-

(IV-35)

where the primes (' and ") indicate first and second derivatives of q with I . The differential cross section calculated from this expression is similar to the classical differential cross section, but with oscillating interference terms. In discussing this interference it is necessary to distinguish not only between positive and negative scattering angles, but also between positive and negative impact parameters. Figure 31 shows the deflection function for positive and negative impact parameters. The three impact parameters that contribute to the interference at a certain angle are denoted in this figure by PI, pz, and p3. The angular separation of any two adjacent interference maxima centered about the angle 9 corresponding to the impact parameters pi and Pk is given to a good approximation by (IV-36)

$

Z i

I

I

I

I I

n

THE STUDY OF INTERMOLECULAR POTENTIALS

I

z

N

+

?

/f

t I I

I I

%

279

FIG.31. Classical deflection function for positive and negative impact parameters. The deflection function parameters have been shifted by 360" in order to illustrate interference.

H . Pauly and J. P . Toennies

280

From Eq. (IV-36) it follows that the interference maxima will be closely spaced for large ISj - pkl and widely separated only for small Ipj - pkl. Only in the case of light atoms will the interference extrema resulting from the impact parameters p1 and p2 or p1 and p3, respectively, be sufficiently separated (A9 = 2n/[AIp1 p2,31],see Fig. 31) to permit their observation. For heavier atoms only interference between p2 and p3 will be observable [A9 = 2n/(A(p3- pzl)] and the corresponding interference maxima are referred to as supernumerary rainbows (Mason and Monchick, 1964). The observability of these extrema presupposes a narrow distribution of relative velocities and a good angular resolving power. When these conditions are not fulfilled the oscillations will average out to give the classical result. The region of rainbow scattering cannot be described by Eq. (IV-34) but must be treated differently since two regions of stationary phase overlap in this case. The contribution to the scattering amplitude from the vicinity of the rainbow angle can, however, be calculated simply using the approximation, Eq. (IV-3), for the deflection function, to give:

+

(IV-37) with

x = (j- 1 / 3 A 2 / 3 ( 9 - $r).

Ai(x) is the Airy integral defined by 1 Ai(x) = 2n

-m

exp( ixt

+

f3)

dt.

An additional contribution to the scattering amplitude comes from the “ stationary” region for cp- (analogous to the classical calculation). This term interferes with (IV-37) as discussed above, but the interference will generally not be observable because of the closely spaced oscillations. Neglecting this only slightly angle-dependent contribution and the interference effect (in complete analogy with the classical treatment) the differential scattering cross section in the vicinity of the rainbow angle is given by (IV-38) For angles 9 < 9, (negative x) I,($) is an oscillating function. The first maximum for large angles 9 is called the rainbow, and the successive maxima for smaller 9 are called supernumary rainbows. Since the deflection function for realistic potentials differs from the behavior given by the approximation (IV-3), higher order terms are required and lead to a shift in the positions of the supernumary rainbows. It is interesting to note that the maximum of the

28 1

THE STUDY OF INTERMOLECULAR POTENTIALS

differential cross section does not coincide with the position of the classical rainbow singularity, but is shifted to smaller angles ,,9,

= 9, -

(g2)1’3

1.0188 -

.

(1V-39)

As is to be expected this shift disappears for large A . From the semiclassical treatment described above it follows that for large values of A (corresponding to heavy particles) the differential scattering cross section is changed only to a small extent by quantum effects except at very small angles. In cases for which A I the semiclassical method breaks down, and an exact quantum mechanical calculation has to be performed.

-

4. Semiclassical Calculation of Integral Cross Sections

With the optical theorem (IV-40) and the approximation (IV-31) for the scattering amplitude at small angles it follows that (Al(0) = 1 and G,(O) = 1) - 4ooJl‘i

or CT

= oMM

-

9o‘A cos(2q0 -

i)

(IV-41)

+ Ad.

The integral total cross section o contains thus a contribution oMM from the attractive long range part of the potential. The presence of a short range repulsive potential introduces an additional contribution Ad [see Eq. (IV-8) for the classical analog] that comes from the glory effect [interval 2 in connection with Eq. (IV-25)]. dMMis a monotonic function of the velocity, whereas the correction term Ad oscillates with the velocity. The above result, Eq. (IV-41), can also be derived directly from Eq. (IV-16) for the integral elastic cross section, making the same approximations as in the case of the small angle scattering amplitude (Bernstein, 1962b, 1963b; Diiren and Pauly, 1963). Since it is possible to integrate C,f(s)k”r,” dl 2K1S-l

(IV-42)

in closed form [not possible for f(9) in the case of 9 # 01, the somewhat arbitrary assumption in the choice of the boundary q, = 4 can be overcome.

H . Pauly and J. P. Toennies

282

This method (Landau and Lifschitz, 1959) gives, in Eq. (IV-41), instead of QMM a value uLL, which is given by uLL

2 4 s - 2) - l])! sin(n/[s - 11) ~

= (s - 1)(2s - 3)(2/[s

M M .

For s = 6 the difference between uLL and CTMM is 7 %. Exact quantummechanical calculations of integral cross sections for a monotonic potential function show that the constant obtained with the Landau-Lifschitz approximation is correct to within 1.5 % (Bernstein and Kramer, 1963). An improvement in the accuracy of Eq. (IV-41) is thus possible by using cLL in place of bMM. Finally it follows, from Eq. (IV-41), that the amplitude of the oscillations in the velocity dependence of the integral cross section are given by (IV-43) Numerical calculations for a Kihara potential show that Po is nearly independent of K and a’ (Diiren and Pauly, 1963). This conclusion applies to other potential models as well. The same result holds approximately for 9,’ for reduced energies in the range 0.5 < K < 5. Thus the relative amplitude depends on the potential parameters E and rm and the reduced mass p in the following way : (IV-44) Equation (IV-44) shows that the oscillations are expected to be largest for light atoms. With increasing K , however, 9,’ depends more strongly on K and finally, in the high energy limit, 9,’ 1/K. In this region, approximately characterized by K > 5, the relative amplitude becomes independent of the reduced particle mass. The oscillatory behavior of the integral cross section in the high energy limit is discussed in Section V. The calculation of the differential cross section at small angles as described in Section IV, C, 3 and the results for the integral cross section discussed above are valid only when the maximum phases are sufficiently large, vo 4 1. For large values of the reduced energy K the approximations leading to (IV-31) are no longer valid. In this region, the high energy approximation, as already mentioned above, can be used to calculate the integral cross section as a function of the relative velocity. The regions of the semiclassical approximation and the high energy approximation overlap, so that the whole velocity range may be covered by these approximations. The velocity dependence of the integral cross section is shown in Fig. 32, calculated for a

-

283

THE STUDY OF INTERMOLECULAR POTENTIALS

Lennard-Jones (12, 6) potential with parameters E and r, corresponding to the system K-Kr. For high energies, CJ is a monotonic function of g, since the collision is governed mainly by the repulsive part of the potential. As the attractive part of the potential becomes more important, oscillations in the integral cross section are observed. The validity regions of the high energy approximation and of the semiclassical approximation are indicated in Fig. 32. I

I

I

,

,

,

I

,

,

I

,

,

,

,

.

,

,

region d the semi-closwrrl m m d m region of the hlgh m n g y approrlmolion

05

-

1

Lmmrd-h)oms(l2.6) wtenlml

OL-

0302

-

01

-

I

power law repllrion V(r)

l

3

l

4

I

6

I

I

B10

20

1

,

30 CO

I

1

1

60 80 100

-4

- ;ii

1

I

200

1

1

I

l

300 400 alO' c"/+cc

l

I

FIG.32. Calculated integral cross sections for the system K-Kr as a function of the relative velocity. The high energy approximation was used for the region of velocities greater than the last maximum.

The measurement of the velocity dependence of the integral cross section is a powerful method for obtaining information on the intermolecular potential. For heavy partners (large reduced mass), where ACJ< CJLL, absolute measurements of integral cross sections can be used to determine the potential constant C . Furthermore, measurements of the velocity dependence of the integral cross section permit an investigation of the distance dependence of the attractive part of the potential given by the exponent s. In the case of light atoms (small reduced mass), it is even possible to determine the product er, directly by observing the positions of the extrema, which in turn depend on the maximum phases 'lo. These maximum phase shifts ylo are given by the JWKB approximation (IV-45) depending mainly on the product er,. Bernstein (1963b) has shown that the number of maxima in the velocity dependence of the integral cross section is related to the number of quantum mechanical bound states of the potential well.

284

H . Pauly and J. P . Toennies

The parameters jo and 9,' required for a calculation of Ao have been tabulated for a Kihara potential (see Table XII). In addition the maximum phases have also been tabulated for this potential (Duren and Pauly, 1964). Numerical calculations of integral cross sections based on the high energy approximation have been performed by Bernstein (1963a) for a LennardJones (n, 6) potential and a Buckingham potential.

D. INELASTIC SCATTERING THEORY 1 . Introduction

In discussing the scattering of molecules it is necessary to consider inelastic processes involving transitions between different rotational levels. Vibrational transitions are in general very much less probable at the energies available in a typical beam experiment and will not be dealt with here. Again it is possible to divide the quantum mechanical methods for the cab culation of inelastic collision cross sections into two regions. In the first region, characterized by large values of a/S, usually associated with large angles, it is possible to use classical deflection functions to obtain the trajectory. The transition probability can then be calculated using time-dependent perturbation theory (Takayanagi, 1963). The change in velocity accompanying the inelastic collisions, which is always a problem in such calculations, may be accounted for approximately by using the geometric mean of the velocities before and after the collision (Breit and Daitch, 1955; Lawley and Ross, 1964). The calculation of the classical trajectories in an angle-dependent potential is a difficult task (Cross and Herschbach, 1964). For only slightly asymmetric molecules, however, inelastic scattering cross sections may be approximated by using the classical deflection function for a spherically symmetric potential. In the region of small angles two methods have proven to be useful. The first method involves the use of the high energy approximation in a time-dependent perturbation formulation. The results of this approximation are identical to those obtained using a straight line classical trajectory. The interpretation of the results of this theory is the subject of the first part of this subsection. The second method, an exact quantum mechanical treatment, starts with a partial wave analysis involving the total angular momentum J(J = 1 + j, i-j,) in place of 1. Then for each of these new partial waves it is necessary to solve a system of coupled equations. In the case of a weak interaction and a small transition probability, neglect of most of the coupling terms (matrix elements) is permissible, and an approximate solution can then be obtained. This approximation is called the method of distorted waves and is discussed elsewhere in this volume by K. Takayanagi. The distorted wave approximation has the advantage that it takes account of the angular deflection of the scattered

THE STUDY OF INTERMOLECULAR POTENTIALS

285

particle, which is not accounted for in the high energy approximation. It has the same disadvantages as the exact quantum mechanical method in that phase shifts have to be calculated for each partial wave. For strong inelastic interactions the distorted wave treatment gives values for the transition probability that are usually too large. Because of these limitations the distorted wave method is probably most suitable for calculating the scattering at low temperatures from a slightly anisotropic potential and for calculating vibrational excitation.

2. The High Energy Approximation for Inelastic Scattering a. General Theory. In the following the high energy approximation for inelastic collisions is discussed in terms of the general quantum mechanical theory of inelastic scattering (Burhop, 1961; Massey, 1956). After listing the assumptions of the theory, the imaginary part of the phase shift, which is directly related to the extent to which the collision is inelastic, is calculated using the high energy approximation. By way of the imaginary phase shift, simplified formulas for the general behavior of inelastic and total cross sections are derived. The basic assumption implied in the application of the high energy approximation for calculating rotational excitation cross sections is that the coupling between the orbital angular momentum I and the rotational angular momentum j =jl+ j , of the molecules may be neglected. This assumption would appear to be best justified for collisions in which 1 B j,, 1 S j2, and for weak interactions for which, to a good approximation, the total angular momentum may be written as J w 1. This assumption implies that the change in I accompanying a rotational transition may be ignored (see below). Fortunately, the above inequalities correspond to just the situation encountered in small angle scattering experiments with molecules in selected rotational states. With the above assumption the wave function for a scattered wave in some particular channel given by the quantum numbers J, M , 1,j , , j , , where J is the total angular momentum quantum number and M its projection along the field direction, reduces to a simple product of wave functions given by JM $lj1j2

+

4i~mt4j2rn~~~mr,j~rnl,izm2,

where the 4’s denote the unperturbed wave functions of the molecules and 9t,n81,j,rnl,j2m2 is the radial wave function for a partial wave Im, and for the rotational states j l , rn, and j 2 , rn,. Using the method of the high energy approximation in a time-dependent perturbation formulation the following result is obtained for the scattering matrix, which is equal to the amplitude of the scattered outgoing wave in the forward direction carrying particles in

H . Pauly and J. P . Toennies

286

the final state (Glauber, 1959). This result applies only to the situation in which the molecules in the incoming wave are all in one rotational state:

The molecular wave functions have been abbreviated by

where q is a vector giving the position of the molecule axis with respect to a laboratory coordinate system and i and f indicate initial and final states. Two important additional assumptions were made in deriving the above result from the high energy approximation. The first assumption concerns the change in linear momentum, which is only properly accounted for when AEIE 1, where A& is the change in energy accompanying the rotational transition. Secondly, it has been assumed that the rotational motion of the target molecule with respect to the passing scatterer can be negle~ted.~’ If this is not justified the time dependence of the potential must be accounted for in the expansion of the exponential (Dirac, 1947; Alder and Winther, 1960). Under these circumstances, given by ‘rrOt/tcoll< 1 (where ‘rrot is the rotation period and tcollis the time of a collision), the transition probability will generally be small. In high velocity collisions, on the other hand, ‘rrot/ tcoll2 1, and the transition probability is expected to be large (adiabatic or Massey criterium). In this limit which isdiscussed here the transition probability no longer depends on trot/tcoll,but depends on the high energy phase shift. In the case of angle-dependent potentials the computation of the phase shift is somewhat more complicated than for spherically symmetric potentials. The potential between the scatterer (for simplicity an atom) and the perturbed molecule is a function of the angle between the internuclear axis q of the molecule and the vector r connecting the centers of mass of the two particles (see Fig. 33). For the calculation of the phase in the high energy approximation it is only necessary to integrate over a straight line trajectory given by r2 = 6’ + z2. For any fixed direction of q the integration will depend on the x , y coordinates of the collision trajectory in a plane perpendicular to z = gt and upon the relative orientation of this plane to the coordinate system X , Y , Z = E (E is the electric field direction) in which q is defined. The inelastic, elastic and total cross sections, however, depend only on the angle between g and E.

-=

3 0 If the interaction time is of the order of the rotation time a type of spin orbit coupling can occur. For a collision with I Iijl different differential scattering cross sections at the same angle are expected for the two different relative orientations 1 j l = [I jll and 1 jl = ( I - hl.

+

+

+

THE STUDY OF INTERMOLECULAR POTENTIALS

287

FIG.33. Coordinate system for calculating the phase shift for an angular-dependent potential.

The inelastic cross section may be calculated directly from the scattering matrix. The result for the inelastic and elastic cross sections for specific quantum states is (Glauber, 1959) 0;;:

= 2n

05;”

= 2n

and

Iamb Jamb

db I(i[exp( - i2qbe)- 1]f)12

(IV-47)

db I(i[exp( - i21lhe)- 1]i)I2,

(IV-48)

where qhe is the high energy phase lthe =

1 1 ZGJ-,

+m

Vq,, q 2 , x, Y , z ) dz.

The total cross section is defined by O.fot=

c

f#i

0;;:

+ .I,.

(IV-49)

From the closure property of the rotator eigenfunctions it may be shown that

c I(ilHlf)12 f

=

(ilH21i),

(IV-50)

and the following result for clotis obtained31 : 31 The contribution from transitions for which the high energy approximation does not hold is negligible because, first of all, the transition probabilities of such collisions are relatively small and, secondly, because these collisions occur at small impact parameters and contribute therefore only little to the integral cross sections.

288

H. Pauly and J. P. Toennies uio,= 272

Jamb

db[(ilexp( - i2qhe)- 1I'i)].

(IV-51)

Subtracting ueIfrom aIoIgives a result for the sum over all inelastic cross sections denoted by ainel u.ine1 =

C

ffi

= 2 n j y b db [l - I(ilexp(-i2qhe)li)1'].

(IV-52)

These results may be interpreted by separating the phase q into an angledependent part denoted by 4 and a spherically symmetric part 4 associated with the corresponding terms in the potential

v = 8 + c1 c,cos"Y. W

"=

This is always possible for the high energy phases. Furthermore, a new quantity called the reduced partial cross section ubis introduced. ub is defined for a specific b corresponding to an 1 by u = 272 j b db ab; thus u!izf may be interpreted as the transition probability for a collision with impact parameter b. After some computation the following expressions are obtained for the reduced partial cross sections: ,,h,i-+f inel

-

(IV-53)

I(iIexp(-2il?)If)12,

11

- exp(-2i$)(i)exp(-2i~)Ii))', ab.i inel - 1 - (ilexp(-2iq)li)(iIexp(+2iq)i), 0 ;;

=

up;: = 2 - [exp( +2irj)(ilexp(

+ 2i9)li)

+ exp( -2irj)(i lexp( - 2i#[ i)].

(IV-54) (IV-55) (IV-56)

Equation (IV-53) implies that only the angle-dependent part of the potential is responsible for a quantum jump between two rotational states. From the remaining three equations it follows that in the extreme case as 9 + 0, 0 and 08, --t upot.On the other hand as 4 + 0, u:l does not disappear. The resulting scattering is called " diffraction " or " shadow " scattering to distinguish it from " potential " scattering coming from the fi term only. This result means simply that not all molecules that undergo collisions in an angledependent potential make a quantum jump. Of primary interest in connection with inelastic scattering is the scattering matrix. From Eq. (IV-46) it follows that for the high energy approximation an element of the scattering matrix is given by

sff= exp(-2irj)(ilexp(-2i@)lf).

(IV-57)

THE STUDY OF INTERMOLECULAR POTENTIALS

289

and o!;( may be derived by introducing The relative magnitudes of o:,, real and imaginary phase components t1and Cl, respectively. The imaginary phase component is related to the absorption of particles in one quantum state (e.g., by transitions to some other state). The elastic scattering matrix element follows from Eq. (IV-57) and is used to interpret 5, and h:

sfi= exp( -2i(t1

- ill))= exp(-2tji)(ilexp(-2ifi)li),

and setting (ilexp(-2iQ)li)

= exp(-2i(Q’

- iff)),

the real and imaginary phase components are

t, = tj + 4‘; c1 = 4’’. With the substitution M = exp( - 25,) the reduced partial cross sections may be written as opii = 1 - 2M cos 2ti M 2 ,

+

&el ob,i

to,

=

1 - M2,

-2

(IV-5 8)

- 2M cos 25‘1.

Since M is related to the imaginary component of the phase shift it is a measure of the inelasticity of the interaction. For a pure elastic collision M is 1 and = 0. As the imaginary component of the phase shift increases M deapproaches 1. Figure 34 shows plots of the reduced partial creases and cross sections as a function of M and cos 25‘,. For rj = 0 the behavior of the reduced partial cross sections depends only on the cos2t term, this term leading to the oscillations in the reduced partial elastic cross section between 0 and 2. For 4 # 0 the behavior of the reduced partial cross sections is given by similar oscillations between the limiting curves cos 2 t = 0 and cos 25 = 1 with the abscissa moving in toward smaller values of M as 4 increases. It is interesting to note that the reduced sum over all inelastic cross sections o!ie, does not undergo similar oscillations. This may be attributed to constructive interference between the individual inelastic cross sections, which do have oscillations. In the random phase approximation the elastic cross section is always larger on the average than the inelastic cross section. In using the random phase approximation for calculating total cross sections it appears that a value larger than 1 should be used for the average reduced partial cross section for the small impact parameter region. In interpreting the curves in Fig. 34 it must be remembered that for large impact parameters tj and 4 are monotonic functions of b-’ taken to some power. At small impact parameters repulsive forces become important, and i and 4 will have maxima if the repulsive forces are included. The usefulness of the high energy approximation when repulsive forces are important has yet to be carefully examined.

H. Pauly and J. P. Toennies

290

cos2g = 0

2.0 b

0.1

cos25 =025

cos2g '0.5

1.0

cos25 r0.75

0

0.5

1.0

20

c o s q =la COS25

6Lt

=0

cos2E =0.25

1.0 .

~ 0 ~ =0.5 2 s

cos2E 10.75

0

extreme inelastic SCdtWlng

M

0.5

cos2E -1.0 -M

elastic scattering

FIG.34. Reduced partial cross sections as a function of the inelasticity parameter = exp(- 2v).The curves were obtained from the high energy approximation.

THE STUDY OF INTERMOLECULAR POTENTIALS

29 1

Recently Bernstein et al. (1963), using a statistical approximation in conjunction with distorted wave calculations, showed that the ratio of the inelastic to total cross sections is always less than +.This result agrees with the high energy approximation. From Fig. 34 it is apparent that the ratio t. corresponds to the extreme inelastic case given by M = 0. In concluding this general discussion of highenergy, it is pointed out that the approximation does not take proper account of large angle scattering produced by the repulsive part of the potential nor does it allow for coupling between angular momentum. b. Inelastic Transition Cross Sections. Of particular interest are the inelastic cross sections for specified quantum transitions and the total cross section since these are most accessible to measurement. The reduced inelastic cross section for a given rotational transitionjimi -+ jfmf and for a given relative orientation of scattering trajectory with respect to the orienting electric field direction may be calculated from Eq. (IV-53). Equation (IV-53) may be interpreted as the transition probability for a collision with a certain impact parameter 6. The exact evaluation of Eq. (IV-53) is extremely difficult, especially if proper account of the orientation averaging always encountered in beam experiments is to be included. The transition probability for an angle-dependent potential of the general type

C V(r, Y ) = - - [l r6

+ qP,(cos Y)]

(IV-59)

(see Section I, A, 2) and for one given direction of approach have been calculated (Kramer and Bernstein, 1964). The curves in Fig. 35 show the typical behavior for the inelastic and total reduced partial cross sections which have been weighted with the impact parameter to give the partial cross section (= bob) and are plotted versus the impact parameter. The area under the lower curve represents the entire inelastic transition cross section 0:;;. From these results it is evident that the largest part of the cross sections comes from the region of large impact parameterswhere the conditions for thevalidity of the theory are best satisfied. Fortunately, recently measured inelastic cross sections can be accounted for in terms of small angle scattering, and in this limit (4 < 1) the high energy approximation reduces to the Born approximation result. The experimentally measured inelastic cross section is defined in terms of the acceptance angle Oo within which the inelastic scattered molecules are detected by theapparatus. The experimental inelastic cross section is related to the theoretical cross section by ginel*,, i+f

- n:J -

,@(;A:)]

0) sin 0 dO d 0

(IV-60) = 2n I I b d b l(i12i91f)12.

292

H . Pauly and J. P . Toennies

b-

b0

b-

FIG.35. Typical behavior of the partial total and the partial inelastic transiton cross sections. The dashed diagonal line corresponds to a = 2 in both curves. The dashed curve ' ~ ' )shows b the behavior in the Born approximation. The shaded area is in the c ~ ~ , $ ~ - f plot proportional to the experimental inelastic cross section.

It is to be noted that b, is the impact parameter corresponding to 0,. The evaluation of Eq. (IV-60) in terms of potential parameters is relatively straightforward for a molecule-atom anisotropic potential3* of the type (IV-61) 32 For simplicity, only the molecule-atom potential is considered here. The extension to two molecules has been carried out elsewhere (Toennies, 1965).

293

THE STUDY OF INTERMOLECULAR POTENTIALS

where DA,iis a coefficient that depends on the molecular and atomic properties (see Section I). As mentioned previously the integral over the straight line path depends on the relative orientation of g (i.e., z ) with respect to E (i.e., Z ) . For the sake of brevity the orientation dependence of the phase will not be considered explicitly here. For one given orientation, say g 11 E the phase may be written as

where 9, cp give the orientation of the molecule axis with respect to 2, and the cltm are constants which, in general, depend on the relative orientation of g with respect to E. Inserting Eq. (IV-62) into Eq. (IV-53) the following result is obtained for the inelastic cross section for the case that the specified transition is produced by one term corresponding to some specified A and t in Eq. (IV-61):

In order to calculate the matrix element in (IV-63) the rotator eigenfunctions i and f must of course be defined with respect to the same coordinate system as 9 and cp. bo is related to the total integrated cross section (if oine,e ol,,) measured with the same apparatus by

(2)

112

= b,.

Alternatively b, may also be estimated from the classical deflection function [Eq. (IV-9)] if the angular acceptance angle 0,of the apparatus is known. For this calculation, the interaction constant derived from a measurement of the total cross section with an apparatus with a resolving power greater than the critical resolving power (see Section 11, B, 2) is also required. b, may be calculated in this way only if the acceptance angle is greater than the limiting angle a,,, defined in Eq. (11-8), for which it is possible to use a classical trajectory (a/6 > 1). The situation implied by the simultaneous validity of the Born approximation and the classical deflection function is illustrated in Fig. 35. It is to be noted that b, is smaller than the b for which the partial total cross section has its first maximum (corresponding to a/6 1). Before Eq. (IV-63) can be compared with the experimental results it has to be averaged to take account of (1) the different possible directions of the trajectories with respect to the electric field direction, (2) the geometry of the apparatus, and (3) the velocity distribution of scattering gas molecules, which will also enter into (1) and (2).

-

H . Pauly and J. P . Toennies

294

The properties of the matrix elements of the phase lead to the following selection rules for the small angle scattering (Toennies, 1963, 1965). (1) The Am selection rules depend upon the direction of approach of the trajectory of the perturbing atom with respect to the electric field direction. This direction dependence is a function of the angular dependence of the potential. (2) The Aj rules for the angle-dependent potential given in Eq. (IV-61) may be summarized by Aj = I , I - 2,

..., - 2

or for a potential of the type

V ( r , vt, 2 1 , by

y2, 2 2 )

=

1

lllzmirnz

Aj,

= lt,

D~~~2rnlrn2~-'Yllr~ n1 ~ () ~Yt I~ 2

rn,(~2,~2)

1, - 2, ..., -It.

For example, for electrostatic interactions the equivalent 1, is 1 for an interaction by way of the dipole in molecule 1 or 2 for an interaction by way of the quadrupole, etc. Thus the inelastic cross section for each specific Aj, Am transition is a sensitive measure of the magnitude of the corresponding term, or terms, in the expansion of the angle-dependent potential. These results may be given a simple physical interpretation: The above theory applies to the special case in which the scattering molecule sweeps past the primary beam molecule in a time short compared to the rotation time of the primary beam molecule, qot< tco,,.Consequently, components in the electric field parallel to the trajectory average to zero, and the dipole or quadrupole of the primary beam molecules " sees " a strong alternating inhomogeneous electric field directed perpendicular to the classical path. This field induces the rotational quantum jump. Since the field contains higher-order derivatives quadrupole and higher-order transitions as well as dipole transitions are expected to occur. This simple picture also provides an explanation for the postulated orientation dependence of Am transitions, briefly mentioned above. c. Total Cross Sections for an Angle-Dependent Potential. Equation (IV-51) has been evaluated for the molecule-atom potential given by Eq. (IV-59) and for the molecular states ( j , m ) = (1, 0), (1, l), (2,0), ( 2 , 2) (Bennewitz et al., 1964). More recently Reuss (1964) has evaluated the results of distorted wave calculations of total cross sections (Arthurs and Dalgarno, 1960) by using the random phase approximation. For small values of the anisotropy

THE STUDY OF INTERMOLECULAR POTENTIALS

295

factor there is good agreement with the above-mentioned results based on the high energy approximation. In general, it may be shown that odd terms [P,(cos Y), P,(cos Y), etc.] in the angle-dependent potential contribute little to the total cross section, so that a measurement of the ratio of the cross sections for (j , m) = (1,O) and (1, 1) provides information on the even cos'Y term in the potential. In the case of two molecules the calculations are more difficult. The scattering of a dipole in a low rotational state [ ( j ,m) = (1,0), (2,0), and (3, O)] on a second molecule carrying multipoles but at room temperature was estimated in the following way (Toennies, 1965): The Born approximation was used in the region of large impact parameters and the approximation of random phases in the remaining region. According to Eqs. (IV-58) and as shown in Fig. 35 the random phase approximation is a lower limit. Orientation averaging was properly accounted for in the large impact parameter region. In this approximation the potential was averaged over the eigenfunctions of the scattering molecule, which is justified on the basis of the adiabatic criterium ( ~ r o t / f c o l l< 1, corresponding to a small transition probability). As a result of this averaging the interaction with a rapidly rotating dipole (relative to the collision time) in the scattering molecule was found to disappear in first order (see Section I, A, 3). For the case of symmetric tops and molecules with quadrupole moments, however, the multipoles remain despite the rotations. The results for these two cases for the effective interaction constants Cse,,, obtained by comparing the calculated cross sections with the Massey and Mohr formula for the same s, are listed in Table XIV. y is defined by = PlPzY,

where pl and p 2 are some set of multipoles in the molecules 1 and 2. TABLE XIV EFFECTIVE INTERACTION CONSTANTS FOR ANGLE-DEPENDENT POTENTIALS

Partners

Dipole-sym. top Dipole-quadrupole

Interacting multipoles ( p ) PI

PZ

P1 PI

Pa Qa

Y

0.225 0.089

296

H . Pauly and J . P . Toennies

V. Atom-Atom Scattering Experiments A. INTRODUCTION

The early atom-atom scattering experiments were concerned almost exclusively with the determination of integral elastic cross sections. For these measurements, beams with a Maxwell velocity distribution were used. The first direct measurements of elastic cross sections as a function of primary beam velocity made use of the velocity dependence of the deflection by gravity (Estermann ef al., 1947a) and showed that slow atoms were scattered more readily than fast atoms (Estermann et al., 1947b), in agreement with the predictions based on the kinetic gas theory. It is only in the last few years, however, that the use of a velocity selector in the primary beam has made it possible to obtain detailed information on intermolecular potentials. Only scattering experiments with velocity selected beams are discussed here. The earlier experiments and the results derived from them may be found elsewhere (Pauly, 1961). Recent measurements of integral elastic cross sections with monoenergetic primary beams have made it possible not only to investigate the distance dependence of the van der Waals forces, but also to observe the small effect of the repulsive forces at thermal energies of the collision partners (glory effect). By state selecting beams of atoms according to the Zeeman levels it has, furthermore, been possible to measure the asymmetry of the van der Waals forces in atom-atom collisions. Progress has also been made in the measurement of differential cross sections. In particular the experimental observation of rainbow scattering has greatly improved the accuracy with which the intermolecular potential can be determined. Interference patterns resulting from the wave nature of the particles have also been observed in recent experiments. Most of the experiments reported in this section have been carried out with alkali atom primary beams. B. MEASUREMENTS OF INTEGRAL

CROSS SECTIONS

I . Absolute Measurements

Figure 36 shows the arrangement used for the determination of integral cross sections. The primary beam issuing from the oven (1) is collimated by the slits (2), (3), and (4). After passing through a velocity selector (7) and a scattering chamber ( 5 ) it is detected at (6). The resolving power of such an arrangement has to be of the order of minutes for atomic particles in order to measure the scattering cross section without significant error (see Section 11, B, 2).

THE STUDY OF INTERMOLECULAR POTENTIALS

297

FIG.36. Experimental arrangement for the determination of integral cross sections.

For the absolute determination of the cross section the particle density n in the scattering chamber (more precisely, the quantity J n(1) dl integrated over the path I of the primary beam) must be known. The particle density is generally derived from a pressure measurement on the gas in the scattering chamber, and a correction is made for the gas flowing out of the entrance and exit slits of the scattering chamber. The pressure measurement and the calibration factors are the main sources of error in the absolute determination of integral cross sections. These sources of error are discussed in more detail in Section 11, B, 3.

As discussed in Section IV the van der Waals constant C can be determined from the absolute value of the integral cross section [Eq. (IV-41)]. From Eq. (IV-44) it follows that AO/O I / Ji.

-

For heavy particles, therefore, the part of the integral cross section that oscillates with changing relative velocity will be small. This method has the disadvantage that the error in the integral cross section leads to a larger error in C because of C a5'2.Table XV summarizes recent experimental results and the derived C values together with theoretical C values. From this comparison it appears that the empirical C values are generally larger than the theoretical values. Table XVI shows, on the other hand, that the relative C values from theory and experiment are in reasonable agreement. Very likely, a large part of the discrepancy in the absolute values is due to a systematic error in the pressure measurement. Recent experiments on the absolute calibration of pressure measuring devices seems to support this possibility (Bennewitz and Dohmann, 1965a). The most recent values of Duren and Florin (Table XV) obtained with these newly measured calibration factors are indeed considerably smaller than the older values. The clarification of this important problem will require reliable absolute measurements as well as a re-examination of the theoretical approximations (Fontana, 1963; Fontana and Bernstein, 1964).

-

TAB= XV VANDER WAALS~

C X X O C0"Is N (1W* erg an@)

s:

Li

K

cs

Ne

Ar

Kr

Ga

Theoret. Expl.

Theoret. Expl.

Theoret. Expl.

Theoret. Expl.

Theoret. Expl.

Theoret. Expl.

Theoret. Expl.

He

0.210

0.46"

0.32"

0.53c 0.40d

0.36'

0.W

0.140

0.461 0.200

0.16k

0.07k

Ne

0.46"

0.59"

0.65"

0.69c

0.73"

1.W

0.310

0.W

0.29

0.14k

Ar

1.9"

3.5"

2.60

3.04

6.2c 4.0d

0.20. 0.29

0.51' 0.525

1.W

1.7f

1.21L

1.7k

Kr

2.9"

5.1"

4.0"

4.6"

9.4c

0.318

0.54j

1.7f

1.83k

2.6k

Xe

4.6"

8.7b 6.2n

6.5"

7.4"

14.6c

3.2'

3.0k

4.2k

6.0c 4.1' 2.8' 9.9C 10.6h 10.9' 13.6c 10.7' 1.4'

0.85c 0.40d

Dalgarno and Kingston (1959). Rothe et ul. (1963a). (Measurements with velocity selection.) c Rothe and Eernstein (1959). Measurements with a thermal velocity distribution in the primary beam.) d Helbing and Pauly (1963). (Measurements with a thermal velocity distribution in the Primary beam.) c Dalgarno and Kingston (1961). f Rothe et ul. (1962a). (Measurements with a thermal velocity distribution in the primary beam.) 0 Slater-Kirkwood approximation (see Section I). [For more recent theoretical results see Kingston (1964).] a

0.W 0.040

0.095= 0.21'

0.118 O.2Oc 0.258 0.65. 0.79 0.56m 1.W 1.450

0.17j 0.120 0.51' 0.525 1.51

00

Y

Beck (1962). (From measurements of rainbow scattering.) * Rothe et ul. (1963b). (Measurements with velocity selection.) f Calderon (1959). (Measurements with a thermal velocity distribution in the primary beam.) Ic Berkling et al. (1962a). Helbing and Pauly (1964). (Measurements of small angle scattering.) m Fontana (1961a). n Florin (1964). O Diiren (1965).

3

3

8 ' b

299

THE STUDY OF INTERMOLECULAR POTENTIALS

TABLE XVI THEORETICAL AND EXPERIMENTAL RELATIVE INTERACTION CONSTANTS FOR THE RARE GASES Collision partners

ClCAr

Theoret." K-He Ne Ar Kr Xe Cs-He Ne Ar Kr Xe Ar-He Ne Ar Kr Xe

0.12 0.24 1 .oo 1.5 2.4 0.1 1 0.24 1 .oo 1.48 2.34

Theoret.*

0.13 0.33 1 .oo 1.33 1.93

ExpLC 0.09 0.11 1 .oo 1.6 2.2 0.13 0.16 1 .oo 1.52 2.28

Exp1.d

0.14 0.34 1 .oo 1.13 2.13

Dalgamo and Kingston (1959). Slater-Kirkwood approximation (see Section I). c Rothe and Bernstein (1959). (Measurements with a thermal velocity distribution in the primary beam.) d Rothe et nl. (1962a). (Measurements with a thermal velocity distribution in the primary beam.) a

2. Relative Measurements of Integral Total Cross Sections

Relative measurements of integral cross sections can be made much more precisely since an absolute pressure determination is not required. Integral cross sections at different primary beam velocities have been measured for a series of atom-atom collision partners. The exponent s is determined by plotting the logarithm of the measured effective cross section divided by the appropriate correction function (for the definition of the effective cross section see Section 11, B, I), against the logarithm of the primary beam velocity. The slope of the straight line, averaged over the oscillations, is equal to -2/(s - 1). An example of scattering results for heavy particles (K-Xe) is shown in Fig. 37. In the cases investigated so far (K-Ar, K-Kr, K-Xe) the experiments give the value s = 6 with an accuracy of 1-2 % (Florin, 1964; Rothe et al., 1962b). In the case of atom-molecule collisions the same result has been obtained (Pauly, 1960; Schoonmaker, 1961).

H . Pauly and J. P . Toennies

300

\ .!.

Li-Xo

‘-A ., !\ 6.w,

-a;,

750

KMO

300

a0

*.

6

*

0

‘

1250

1500

2000

2500

3oa 3500

500

6W

800

I000

I200 1400 1600

4000

v)

C

1

Prlmory beam velocily v, [mlsec]

FIG.37. Measured integral cross section for the systems Li-Xe (Rothe et al., 1962b), Na-Xe (Florin, 1964), and K-Xe (Rothe et al., 1962b) as a function of the primary beam velocity.

For light beam particles (small reduced masses) the undulatory behavior of the integral cross section is expected to be more noticeable, except in those cases in which the velocity averaging leads to a smearing over the maxima. Figure 37 shows two examples in a double logarithmic plot for Li-Xe and Na-Xe scattering (Rothe et al., 1962b; Florin, 1964) with well-resolved extrema in a(g). As pointed out in Section IV, C, 4, this oscillatory behavior of a(g) may be interpreted in terms of the broad maximum in the phase shift versus , provides a significant number of nonangular momentum curve, ~ ( l )which random phases. The maximum phase shift q0 increases with decreasing g, and, assuming the potential well is sufficiently deep, can pass successively through -&n,n, etc., yielding positive and negative deviations, respectively, from

THE STUDY OF INTERMOLECULAR POTENTIALS

30 1

(TMM or oLL. From Eq. (IV-41) for Aa it turns out that maxima in a are expected if n 2 q o - -4= 2 n ( N - + ) with N = 1 , 2 , 3 ,..., W-1)

whereas the condition for minima is

n 21, - 4 = 2n(N - +)

with N = +,$,

3, ....

w-2)

For any assumed potential of the form discussed in Section I, D, the velocity dependence of the maximum phase shift [see Eq. (IV-45)] may be evaluated numerically, and thus the locations of the extrema in a(g) determined. For small N , it is possible to use the high energy approximation for these calculations. For instance for the Lennard-Jones (n, 6) potential, the high energy approximation yields, for the maximum phase shift, Er,

'lo=---

371

n

Together with Eqs. (V-1) and (V-2), the following condition for the Nth extremum is obtained :

(V-4) where gN is the velocity corresponding to the Nth extrema. A plot N - 4 versus g; yields a straight line in this approximation. From the initial slope of this line, the product

may be determined. The available results for lithium-rare gas collisions have been evaluated in this way, using a Lennard-Jones (12, 6) potential. Table XVII lists the potential parameters obtained from the measurements. With increasing N , deviations from the straight line occur since the conditions for the high energy approximation are no longer fulfilled. Since for experimental reasons (thermal velocity distribution of the target particles) the extrema corresponding to higher values of N are difficult to analyze, the high energy approximation will be sufficient in most cases. To analyze the experimental data, the product of v2/5a(u) is plotted as a function of l/u, yielding an undulatory curve, which is symmetrical about a horizontal mean line and in which the extrema are nearly uniformly spaced. From this plot the extrema velocities g N can be estimated and plotted versus N - 8.

H . Pauly and J. P. Toennies TABLE XVII FROM THE VELOCITY UNDULATIONS POTENTIAL PARAMETERS OBTAINED OF THE INTEGRAL CROSSSECTION

Collision partners Erm x loz2erg cm Li-He" Ne Ar Kr Xe K-Krb K-Arc Na-Xed a

x 1014 erg rm x 108 cm

E

(0.05) 0.18 0.81 1.32 2.02 1.3 0.92 1.95

(0.32) 0.89 4.28 6.81 10.60 7.65 4.72 8.78

Rothe et al. (1963a).

* Rothe et al. (1963b).

(5.86) 5.04 5.26 5.16 5.26 5.88 5.14 4.50

Strunck (1964). Florin (1964).

The absolute mean value of o(g) is given by CTLL and can be used to determine the interaction constant C, which may be written for a Lennard-Jones (n,6) potential in the form n

6

C=&rm n-6

3

yielding

Equations (V-4) and (V-5) are not sufficient to determine the three parameters of the Lennard-Jones (n, 6) potential. In order to obtain all three parameters, it is necessary to take account of the amplitude. According to Eq. (IV-41) the amplitude of Ao is given by

This amplitude can also be determined in the high velocity limit by the high energy approximation with the result

I A a m a x I - 16\1* nr,'

-

(32(n - l)f(n))g'(n-6)

15ncrm

5nn

From Eqs. (V-5), (V-6),and (V-7) it follows that E and r, may be determined separately, provided that absolute measurements of the total cross section are made. These equations show, however, that the method is hardly applicable to the determination of a third potential parameter, such as n, because

THE STUDY OF INTERMOLECULAR POTENTIALS

303

of the only very weak dependence of the cross sections on this parameter. Numerical calculations of the oscillations have been performed using the JWKB phases Eq. (IV-45) for a Kihara potential (Diiren and Pauly, 1964). For very high velocities corresponding to qo < 5n/8 maxima in the velocity dependence of the integral cross section are no longer anticipated. The velocity dependence of the cross section in this region depends largely on the repulsive potential. For light atoms (high beam velocities at thermal energy and small attractive potentials) this high velocity behavior is observable with thermal beams (Harrison, 1962; Rothe et al., 1963a), whereas for heavier atoms (low beam velocities, large attractive potential well) high velocity beam sources are necessary in order to make this region accessible (Hollstein and Pauly, 1964). In order to obtain accurate results on the repulsive part of the potential from a(g) a high degree of measuring precision is required [see Eq. (II-9)]. Figure 38 shows the apparatus being used by von Busch et al. for accurate measurements of the velocity dependence of integral cross sections (von Busch et al., 1964). A molecular beam oven, with a long slit that is divided into two parts, is used to produce two slightly convergent beams. In the region after the oven in which the beams are still separated each beam passes through its own velocity selector. Both beams pass through the same slit into the scattering chamber and from there on have the same geometry. Each beam can be chopped independently by remote controlled beam stops (flags). The velocity dependence is derived directly by alternately measuring the attenuated intensity at the detector for the two beams with different settings of the velocity selectors. This procedure is then repeated for various scattering gas pressures. The high precision is achieved by use of an automatic program, which permits repeated digital measurement of the two beams and rotor I

liquid air cooled scattering chamber detector

FIG.38. Apparatus for accurate measurements of the velocity dependence of integral cross sections.

304

H. Pauly and J. P . Toennies

the background (both beams off) in rapid succession. This is repeated until the desired accuracy is achieved.33 As mentioned previously the angular resolving power of the apparatus must correspond to a certain minimum angle in order to measure the integral cross section which is limited to finite values by quantum mechanical effects. If the resolving angle is considerably larger than this minimum angle, the measured cross section corresponds to the classical cross section with a velocity dependence at thermal energies corresponding to S --t gp4Is.The velocity dependence of this cross section has also permitted a determination of the exponent s for the collision partners K-N, and K-Kr (Pauly, 1958; Beck, 1962). The result s = 6 is in agreement with the results obtained with a small angle of resolution. These measurements also confirm the semiclassical theory discussed in Section IV, according to which the differential cross section approaches the classical behavior for angles considerably larger than the limiting angle.

3. Measurements of Integral Cross Sections with Atoms in De$ned Quantum States Berkling et al. (1962b) were the first to actually measure the difference in the scattering cross sections for atoms in different Zeeman states, i.e., in states of different values of the projection m, of the total angular momentum quantum number J. In their experiment they measured the relative cross section for the two components m, = +$ and m, = -3 for a thermally excited Ga - 2P312primary beam. This difference is of the order of 1 %, and in order to measure it to an accuracy of several percent a relative precision in the measured cross section of was required. The high precision was achieved by alternately comparing the attenuation of the two Zeeman components. By switching the magnetic field of a Stern-Gerlach magnet both components were made to have the same trajectories. As in the velocitydependence measurements described earlier the component intensities and the background were automatically measured in rapid succession and added up over long periods of time. Table XVlII shows the results of these measurements. According to a quantum mechanical calculation for the anisotropy in the potential, Aola should, to a first approximation, be independent of the rare gas scattering partner. This will be true so long as the amplitude in the oscillations in the integral cross sections remain sufficiently small. With the light scattering gases as noted earlier the oscillations will become large and the ratio of the two cross sections will be strongly velocity dependent. Thus 33 The experimental technique involved in the alternate measuring of two atomic beams is described by Berkling et al. (1962b).

THE STUDY OF INTERMOLECULAR POTENTIALS

305

Toschek (1964) observed a negative value for Aa/a in scattering experiments with neon. An analysis of the results for these light scattering gases is therefore extremely complicated since it involves an assumption concerning the anisotropy of the repulsive potential. Analogous experiments have been carried out for atom molecule collisions and are described in Section VI. TABLE XVIII THERELATIVE DIFFERENCE IN INTEGRALTOTAL CROSS SECTIONS FOR THE Two ZEEMAN COMPONENTS OF A Ga ATOM BEAM Scattering partner

Ga- Xe Ga- Ar Ga- He

C.

IumJ=t

- UrnJ-tI U

Meas.

Theoret.

94.9 9.3 19.4 & 6.6 195 5

140 140 140

*

DIFFERENTIAL SCATTERING CROSSSECTIONS

I . General Considerations For measurements of integral cross sections it is usually satisfactory to use a scattering gas in a scattering chamber. For measurements of differential cross sections, on the other hand, it is necessary to use the method of crossed molecular beams in order to have a well-defined scattering region. For theoretical and experimental reasons it is convenient to distinguish between differential cross sections at small angles and at large angles. The small angle scattering is almost entirely governed by the attractive long range forces, whereas the large angle scattering comes from the repulsive part of the potential. Measurements of the small angle differential cross sections require an extremely high angular resolving power if the scattering behavior at angles smaller than the limiting angle is to be examined. For reasons of intensity, messurements of the large angle scattering are, however, usually carried out with less resolving power. 2. Measurements of Small Angle Scattering

Results of small angle scattering experiments are available for the following scattering partners: Na-Hg, K-Hg, K-Ar, K-Xe (Pauly, 1959; Helbing and Pauly, 1964). These results are in agreement with the theory of Section IV,

H. Pauly and J. P. Toennies

306

according to which the effect of the long range forces predominates at small angles for collisions between heavy particles. Down to angles of the order of the limiting angle, the differential cross section is well described by classical mechanics. In the velocity realm of classical mechanics, in which for small angles I($) $ - ( z s + z ) l s it is possible to determine s from the differential cross section. At angles smaller than the limiting angle, the measured differential cross sections deviate from the classical behavior and remain finite in accord with the discussion of Section IV. Figure 39 shows measurements of the small angle scattering for the partners K-Xe, in which the measured beam intensity is plotted against the detector position, which is proportional to the scattering angle. The solid curve was calculated at small angles using the quantum mechanical result [Eq. (IV-32)] and at large angles using the classical result [Eq. (IV-6)]. The velocity distribution and the beam and detector geometry were accounted for in the theoretical curves. The deviation from the classical behavior is clearly apparent at small angles. At larger angles oscillations about the classical behavior are expected. These oscillations are averaged out in these

K-Xe

05

1

2

3

4

5

10

20

30 40 50

Deflection angle in the laboratory systam €I[minutes of arc]

Fro. 39. Measured small angle differential cross section for the system K-Xe. The straight line was calculated from classical mechanics for a V = C/rspotential: positive angles; 0, negative angles.

+,

THE STUDY OF INTERMOLECULAR POTENTIALS

307

experiments because of the relatively high beam and detector slits (heightto-width ratio : 1400). Consequently, different portions of the undulations are observed at a given angle, and only the classical curve is measured. The velocity distributions of primary (a velocity selector was not used) and secondary beams lead to an additional equally serious averaging. The measurements confirm the s = 6 long range potential. An estimate of C , independent of pressure calibration, is also possible from the angle at which the differential cross section deviates from the classical form (Helbing and Pauly, 1964). For light atoms the distance between extrema increase, and they have been observed in the scattering of lithium atoms on mercury (Bernstein, 1963~). In the region of small angles the distance between the extrema depends essentially on the parameter A (Bernstein, 1961) [see the discussion of Eq. (V-31)], and r, can be determined. Figure 40 shows an example of such oscillations in the differential scattering cross section for the partners Li-Hg. 1(9)94/3sin 9 has been plotted versus 9 in order to reduce the large rise in the differential cross section at small angles and thereby facilitate location of the extrema. Measurements of differential cross sections at small angles also permit an empirical evaluation of the effect of the resolving power on measured total cross sections (Pauly, 1959; Helbing and Pauly, 1964) (see Section 11, B, 2).

.-C

bl

s3

FIG.40.Measured differential cross sections (weighted with sin$) for the collision partners Li-Hg. The measured differential cross sections have been divided by the classical value for s = 6 to remove the increase at small angles (Bernstein, 1963~).

H . PauIy and J. P . Toennies

308

3. Measurements of Large Angle Scattering

Of primary interest at large angles is the location and the shape of the scattering cross section in the vicinity of the rainbow maximum from which a determination of E and rm is possible (see Section IV). In these experiments, secondary beams of the heaviest possible particles at the lowest possible source temperature are chosen for kinematic reasons. In order to illustrate the connection between the quantities that are observable in rainbow scattering and the parameters of a Lennard-Jones (n, 6) potential, the high energy phases and the semiclassical equivalence relationship may be used to calculate the rainbow scattering. This approximation yields for the rainbow angle

---(32K

R-

30n

whereas the quantity 4 is given by 45n n 32f(n)(n 30n 16 K

q=-~

(

(V-10)

As pointed out in Section IV, the maximum of the differential cross section occurs at angles smaller than the classical rainbow angle QR. QR is located on the large angle slope of the differential cross section at that point where the intensity has dropped to 44 % of its peak value. This point is very near the inflection point of the differential cross section curve. Thus it is possible to determine 9,(E) from measurements of the differential cross section at different energies. Since the angular distance between QR and Q,,, depends on q/rn,2 [Eq. (IV-39)], this quantity can also be determined experimentally. Equations (V-9) and (V-10) are not sufficient to determine the three potential parameters E , r,,,, and n. If a value of n is assumed, E and r,,, may be determined. If absolute measurements of the differential cross section are available, a third equation, involving the intensity in the rainbow maximum [Eq. (IV-38)], can be used, which allows the determination of all three parameters. Beck (1961) was the first to observe the rainbow maximum in scattering experiments of potassium atoms on krypton. Since then other systems have also been investigated (Morse et al., 1962; Morse and Bernstein, 1962; Hundhausen and Pauly, 1964b). Figure 41 shows a recent measurement of wellresolved rainbow scattering for the collision partners Na-Hg (Hundhausen and Pauly, 1964b). At angles smaller than the rainbow angle, additional oscillations can be observed. These are attributed to interference between the partial waves scattered from the two sides of the potential minimum.

THE! STUDY OF INTERMOLECULAR POTENTIALS

I

5

,

10

15

20

25

30

Deflection angle in the center of mass system

309

35

3

FIG.41. Measured differential cross sections for the partners Na-Hg. The maxima are attributed to interference effects, associated with rainbow scattering.

A high degree of homogenity of relative motion is required to make the measurement of these additional oscillations possible. This explains the fact that the same oscillations are less well resolved in the scattering of potassium (Fig. 42) since the mercury velocity spread is more important here. Table XIX summarizes the available results of rainbow scattering. The relative accuracy of E and rm values for different systems may be as high as 1 % and 3 %, respectively [see Hundhausen and Pauly (1964b)l. This high accuracy does not apply to the absolute values of these parameters since the absolute values are also functions of the unknown repulsive term in the potential. The values listed in Table XIX were obtained using a LennardJones (12,6) potential, or, in some cases, using a Buckingham potential with c1= 12. Since the repulsive term is not sufficiently well known from measurements of differential cross sections, it would appear desirable to measure the velocity dependence of the integral cross section at velocities preferably above thermal energies. From these additional results it should then be possible to establish absolute values for potential parameters with an accuracy of 1 %

310

H. Pauly and J. P. Toennies

from relative measurements. At the present time the accuracy of the absolute values listed in Table XIX is estimated to be 10 % for E, whereas, for r,, which is more sensitive to the repulsive term, it is estimated to be 20 %. In conclusion a word of caution concerning the meaning of quantities such as E and r, seems appropriate. These quantities depend on the potential model used and the introduction of additional potential parameters will, in general, lead to revisions in the magnitude of the presently used parameters.

K-Hg

10

15

20

25

L

3!

30

Deflection angle in the center of mass system

J

FIG.42. Measured differential cross sections for the partners K-Hg. The maxima are attributed to interference effects associated with rainbow scattering.

THE STUDY OF INTERMOLECULAR POTENTIALS

311

TABLE XIX

RESULTSOF POTENTIAL PARAMETERS FROM AVAILABLE RAINBOW SCATTERING Collision partners Na-Hg K-Hg K-Kr Rb-Hg Cs-Hg K-HBr K-HCI K-HJ K-CH3Br Na-Xe K-Xe Na-HBr Na-CH3J K-CHsJ Na-(CHaBr)z

E

x 1014 erg 7.89a 7.51a 7.46b 1.2c 7.33L" 7.72b 3.70 3.14d 4.63d 2.22e 1.71f 1.76f 2.74f 2.79f 3.02f 2.82f

rm x 108 cm 7.3' 7.6a 6.8c 5.9 3.2d 4.2d 5.3f 5.6, 5.0f

Hundhausen and Pauly (1964b) Morse and Bernstein (1962). c Beck (1962). d Ackermann et al. (1964). e Ackermann et al. (1963). f Hundhausen (1964a). a b

VI. Scattering Experiments Involving Molecules A. INTRODUCTION Scattering experiments in which at least one of the partners is a molecule are discussed in this section. With molecules, information about the potential can be obtained from several types of experimental cross sections: total, inelastic rotational, inelastic reactive, and differential cross sections at wide angles. Compared with the interpretation of atom-atom scattering the interpretation of these results is made more complicated by the anisotropy of the potential and the presence of chemical forces. At the present time only fragmentary information on molecular potentials is available for some special cases. Nevertheless, it is hoped that a review of these results will be of use in stimulating new work along these lines.

312

H . Pauly and J. P . Toennies

B. MEASUREMENTS OF NONREACTIVE INTEGRAL TOTALCROSS SECTIONS WITHOUT ROTATIONAL STATESELECTION Although molecules were occasionally used as a scattering gas in the early experiments (Pauly, 1961) no effort was made to interpret these results in terms of the angle-dependent potential. To obtain information of this nature, account must be taken of the averaging of the potential over the rotational motion of the molecules. Several attempts at finding out more about molecular interactions, by changing the relative veldcity of the collision partners, but without using rotational state selection, have been reported. In the first of these Schumacher et al. (1960) studied scattering of CsCl (dipole moment = 10.5 D) on a number of other molecules. This experiment was preceded by a series of measurements of the integral cross sections for K on seventy-seven different gases of varying complexity and reactivity (Rothe and Bernstein, 1959). In the CsCl experiment a velocity selector was not used, but, instead, the temperature of the scattering gas was varied between 200" and 735"K, and in this way it was possible to change the distribution over rotational states of the target molecules as well as the relative velocity. Using an approximate method for averaging over the velocity distribution of the scattering gas, the temperature dependence of the cross section of CsCl with the nonpolar scattering partners, Ar, CH,, and CF,, could be explained entirely in terms of changes in the relative velocity and an r - 6 potential. The same method was then used to analyze the results obtained with the polar scattering gases, NO, CHF,, CH,F, cis-C,H,Cl,, and NH,, assuming an orientation averaged dipole-dipole potential defined by

where

11@e-"lkT do,do, '*) = 1s eCeikT do,do,

N

2p12p2, -~ 3kTr6 '

(VI-1)

do,= sin 'P d'P d Z . Since the statistical equilibrium assumed in deriving Eq. (VI-1) does not exist in the short time of a collision, the use of a temperature-dependent potential appears questionable. With the additional potential term, Eq. (VI-1), the effective van der Waals constant becomes (VI-2) where CT= ,contains the temperature-independent dispersion and induction terms. The observed temperature coefficient of the van der Waals constant was of the order of a factor 2 larger than predicted by Eq. (VI-2). NH, showed an anomalous behavior in that the observed cross section was largely

THE STUDY OF INTERMOLECULAR POTENTIALS

313

independent of the temperature, and could therefore not be explained in terms of Eq. (VI-2). Schoonmaker (1961) was the first to measure the velocity dependence of the total cross section in a similar system, K C I + N 2 . He obtained the interesting result: s = 5.30 0.23, which he checked by measuring K + N, for which he found s = 6.18, in agreement with the expected behavior. Recently Kydd (1962) reported total cross sections for H,O -+ H,O and NH, + H,O measured in a cross beam apparatus, but without a velocity selector. Surprisingly, he found that H,O-H,O cross section to be larger than the NH,-H,O cross section, but was unable to offer an explanation for the difference. A discussion of these results follows a description of experiments using state selected beams.

c. MEASUREMENTS OF NONREACTIVE INTEGRAL TOTAL CROSS SECTIONS WITH ROTATIONAL STATE SELECTION

In order to obtain detailed information on the anisotropy of the potential from measurements of total cross sections it is essential to orient the molecule with respect to the collision trajectory. One way of doing this is to produce a beam of molecules in a defined rotational state ( j , m),where j is the total angular momentum and m is its projection on the applied electric field direction. Using such a beam of molecules it has been possible to observe the ratio of the integral total cross sections for a collision trajectory parallel and perpendicular to the applied electric field direction (Bennewitz et al., 1964). The difference in cross sections provides a direct measure of the anisotropic part of the potential, since the contributions to the cross section coming from the spherically symmetric parts of the potential cancel. The apparatus used in these experiments is shown in Fig. 43. An electrostatic four-pole field was used to focus only the ( j ,m) = (1,O) molecules of a T1F beam into a scattering chamber in which the primary beam was crossed with a secondary beam of rare gas atoms. In the experiment the secondary beam was held fixed while the electric field direction was rotated by 90". An end view of the scattering chamber (Fig. 44) shows the electrode arrangement used to rotate the field direction without changing the scattering chamber geometry or interfering with the secondary beam. Under ideal conditions, in which the primary moleculebeam is stationarywith respect to the secondary beam, the cross sections measured for the field direction EIJg(Fig. 44) is equivalent to that for a molecule in the (1, 0) state; in the other field direction the state (1, is always presented to the secondary beam. 3* The Stark effect energy is a function of m 2only, so that the m state is always doubly degenerate in rn.

314

H. Pauly and J. P. Toennies

FIG.43. Beam trajectories and schematic diagram of the apparatus used for measuring the anisotropy in the molecule-atom potential.

0

Rare gas atom

FIO.44. Schematic diagram of the scattering region showing the relative orientation of the electric field and the relative velocity : (a) E 11g (for ui/ca +. 0), (b) E I8.

THE STUDY OF INTERMOLECULAR POTENTIALS

-arbitrary: v 1 c2

315

ctot (1.1) .

Elg;

Since the ratio vI/c2 was between 0.5 and 2.0, depending on the scattering gas, the results had to be extrapolated to the ideal conditions, indicated above, in order to compare them with calculated cross sections for the specified rotational states. The extrapolated cross sections were found to be independent of the actual value of v1/c2 but did depend on the secondary beam atom used (Table XX). TABLE XX

MEASURED VALUESOF U ~ ~ ; ~ ) / CFOR T ~ TIF ~ ; ~ON ) DIFFERENT SCATTERING GASES

a

b

(1,l)

(1 0 )

lutoi

Secondary beam atom

(weighted average)

Hea Nea Arb Kra Xeb

1.O040 & 0.0015 1.0070 & 0.0022 1.0118 & 0.0007 1.0140 i0.0015 1.0136 0.0030

utot

From Bennewitz et al. (1964). Gengenbach (1964).

The 1 % effect in al~;')/a~~;')corresponded to a ratio of the actually measured intensities of only 0.1 %. The indicated errors are due largely to the large statistical fluctuations of the small primary beam signals in the (1,O) state ( x lo5 molecules/sec). Each value in Table XX represents an average over about 4 runs, where each run took of the order of 4 h of measuring time. These results were interpreted by calculating the ratio c{:/ ')/a!:;') with the help of Eq. (IV-51) and a potential of the type V

c

D26 r6

= - - - -P ~ ( C OY) S

r6

D3 7 -[ + P l ( C O S Y) + +P~(cos Y)], (VI-3) r7

where Y is the angle between the axis of the molecule and the vector connecting the centers of mass and where D are the angle-dependent interaction constants; the indices refer, respectively, to the angle and distance dependence. A trial calculation showed that the terms of odd parity (Piand P3) contribute much less to atotthan the terms with even parity, justifying the omission of

H . Pauly and J. P . Toennies

316

the odd terms, even though in many cases D 3 , , may be larger than D2.6. Consequently, it was found that O { ~ ; ~ ) / O ! : ; ' )depended almost linearly on the ratio D,,,/C, varying from 1.00 to 1.05 for D , , 6 / c = 0 and co, respectively, the latter corresponding to a pure dipole-induced dipole induction potential. A comparison with the experimental values for Ar, Kr, and Xe leads to D 2 , 6 / c = 0.24 i -0.04. The van der Waals constant, C, was obtained from the absolute value of the total cross section. D2,6 (in the case of the heavy rare gases) can be shown to result from two contributions: the dipole-induced dipole (up2) induction and the anisotropic part (all - al)/(all 2aJ of the dispersion potential. Subtracting the former from the measured value of D2,6 leads to a result for all and cl, for TlF (assuming ti 6.3A3 where = +(a,, 2aL)7

+

N

+

all = 1.8A3,

ctL = 6.5A3.

The interpretation of the results for the lighter scattering gases is more difficult since in this case the total scattering cross section for each rotational state is an oscillating function of the primary beam velocity (see Section V, B, 2). This implies that the ratios of the cross sections O ~ : ; ' ) / ~ ~ ; will ~ ) O also be strongly velocity dependent and, moreover, will be a complicated function of the repulsive potential.

D. MEASUREMENTS OF INELASTIC CROSS SECTIONS FOR ROTATIONAL EXCITATION The apparatus used in the previous experiments was modified to make possible a rotational state analysis of the scattered molecules. In the modified setup, a second four-pole field was mounted behind a gas-filled scattering chamber. The second four-pole field was then used to refocus the scattered molecules and analyze them for their rotational state after the collision. By setting the first four-pole field to transmit only molecules in the ( J m ) state and the second four-pole field to transmit molecules in the ( j ' , m') a direct measurement of the inelastic cross section for ( j ,m) + ( j ' , m') was possible (Toennies, 1962, 1965). Figure 45 shows the operating principle of the apparatus. Since the change in velocity accompanying a rotational transition was only Av/v which is small compared to the velocity resolution of the rotor, Av/v the position of the rotor with respect to the scattering chamber is unimportant in these experiments. Sample beam trajectories, exaggerated in the radial direction, are shown in Fig. 46. In the example used, only the ( 3 , O ) state molecules are allowed to enter the scattering chamber. The second state selector is set to refocus only the (2,O) molecules originating from inelastic scattering, and molecules in other states are either stopped by the obstacle in the analyzer or not sufficiently

--

317

THE STUDY OF INTERMOLECULAR POTENTIALS Molecular Rotational : Thermal Beam Oven (-4OOOC) TL F

State

Vibrational:

Velocity

p( J )

-----------

LA

Mechanical Velocity Seledor (after Fizeau )

-----------

unchanged

unchanged

unchanged

unchonged

unchanged

unchanged

----------Gas f i l l e d %otterirg Chamter

------------

1

2

3

J

Ekctrastatic Rotational Stote Analyzer eg. J-2 M,-0

-----------

1

2

3

J

( Langmuir-Taylor

Detector-Mass Filter-Multiplier )

FIG.45. Block diagram of molecular beam apparatus for measuring inelastic cross sections. The distribution functions for various degrees of freedom at different positions in the apparatus are shown at the right.

deflected to be able to arrive at the detector. By using a heavy molecule for the primary beam and light scattering gases the acceptance angle of the second four-pole field in the center-of-mass system was considerably larger than the angle in the laboratory system of about 4'. Furthermore, for a strongly attractive potential most molecules will be scattered only through small angles, and on the average it has been estimated that approximately 50% of all scattered molecules are accepted for analysis by the second four-pole field.

318

H . Pauly and J. P . Toennies

FIG.46. Molecule trajectories and side view of the apparatus used for measuring inelastic cross sections. Only molecules in the (3.0) state can enter the scattering chamber, whereas only (2.0) molecules can arrive at the detector.

An inelastic cross section for the apparatus is obtained from the pressure dependence of the intensity in the excited state (see Section 11, C). Figure 47 illustrates the agreement between measured points and best fit curves according to Eq. (11-12) for TlF ( 3 , O ) NH3 + T1F (2,O). Values for o ~ ~ ~ , + y o ) for the scatteriog gases: H,, air, N,O, H,O, CF,Cl,, and NH3 wereobtained directly from the best fits such as those shown in Fig. 47. A correction for the attenuation of the background intensity, however, had to be made for scattering gases with a small inelastic cross section (He, Ne, Ar, Kr, 0,, CH,, SF,). In order to compare these results with theoretical cross sections it was necessary to consider the following:

+

(1) The average limiting acceptance angle of the apparatus. From measurements of the total cross section (with both state selectors set for the same rotational state) this was estimated to be 4.17 mrad. (2) An averaging over the velocity distribution of the scattering gas molecules. (3) An averaging over the various orientations of the TIF rotational state with respect to the scattering trajectory. (4) An averaging over the rotational motion and rotational states of the scattering molecule. These corrections make an interpretation of these results more difficult than the previously discussed total cross section data, the chief difficulty being that an accurate estimate of 0,or 6, [see Eq. (IV-63)] is necessary in

319

THE STUDY OF INTERMOLECULAR POTENTIALS lOOX h

I

t

I

0

I

10

8

20

I

30

I

40

I

50

I

60

'

70

I

I

80

I

90

Pressure in the scattering chamber

'

1M)

[ W 6 Torr]

FIG.47. Pressure dependence of total and inelastic scattering of TIF on NH3: 0 , TlF(30) + NH3%TlF(30); 0, TlF(30) + N H 3 2 T l F ( 2 0 ) ; I, TIF(30) N Hs%$' TIF(10).

+

order to compare absolute values with theory. Nevertheless, by comparing with calculated values, some conclusions may be drawn, and these are summarized below according to the interaction potential postulated : (a) Induced dipole-quadrupole (.,up) potential [see Eq. (Z-7)]. This is the only known term in the attractive potential that can explain the observed Aj = 1 transition induced by atoms or molecules without dipole and quadrupole moments (e.g., spherical tops such as CH, and SF,). Using the Born approximation and applying the above four corrections, the calculated values,

H . Pauly and J. P . Toennies

320

based on known values for a, p, and Q , were consistently an order of magnitude or more smaller than the measured values. This preliminary result suggests the possible existence of a long range cos Y repulsive force or that forward glory scattering corresponding to small impact parameters (where the transition probability will be large) accounts for the observed cross sections. The Aj = 2 inelastic cross section is expected to be much smaller, and this transition was not observed. (b) Dipole-quadrupole electrostatic potential. For the molecular scattering gases, H,, O,, N,, N20, and H 2 0 , agreement was obtained between experimental and theoretical cross sections based on this potential and the Born approximation [see Eq. (IV-59)]. In the case of these molecules insufficient data were available for selecting the potential term responsible for the angular deflection from the spherically symmetric dispersion and the dipole-quadrupole potentials. Consequently, the total cross section data were analyzed for the limiting impact parameter assuming one or the other of these potentials. Table XXI shows a comparison of measured and calculated inelastic cross TABLE XXI MEASURED AND THEORETICAL INELASTICCROSS SECTIONS FOR THE TRANSITION (20 + 30) ASSUMING A DIPOLE-QUADRUPOLE INTERACTION OF TIF ON DIFFERENT GASES' (r(2D-30)

,"el

Gas

s=4

Hz

7.48 136.0 160.3 88.7

0 2

N2

NzO

HzO

(talc.)

(A?

s=6

15.3 4.12 70.7 82.0 44.8

I.",:;

30'(meas.)

(A?

19.7 7.8 23.6 80.0 70.0

a s refers to the potential responsible for the

angular deflection.

S ctions. Calculated inelastic cross sections are in rough agreem nt with measured cross sections but are definitely too large in the case of N2. This might possibly be due to the fact that the assumed quadrupole moment in this case is too large [Q k 1.5 x esu cm', taken from Buckingham (1959-196O)l. An explanation for the absence of dipole-dipole interactions with the polar molecules N 2 0 and H 2 0 is given below.

-

THE STUDY OF INTERMOLECULAR POTENTIALS

32 1

(c) Dipole-dipole electrostatic potential. This was thought to have been observed in two instances: CF2C12and NH,. The mechanism for the relative alignment was found to be different in the two cases. With CF2C12 it was attributed to the fact that the molecule rotates only slowly during the time of a collision so that, on the basis of the adiabatic criterium, the probability of a transition was large. Since, at the same time, the TlF transition probability was also large, there was a good possibility of a resonance interaction, or of both molecules undergoing a transition. During a collision the ammonia molecule behaves essentially as a symmetric top, since the inversion period (3-6 x l o - ” sec) is considerably longer than the collision time (8 x sec). Taking this into account the expectation value of the dipole moment of the NH, symmetric top molecule averaged over all rotational states was found to be (p) $pslatic.Despite the very short rotational period of NH, (Z,~ < 3 x lo-” sec), therefore, a dipole-dipole interaction was thought to have been observed. Ammonia is also interesting because the scattering potential and the potential causing inelastic transitions are definitely the same, and for this reason the Born approximation could not be applied.35 The inelastic cross section was estimated from Eq. (IV-47) by usinga simplified expression for the phase. The agreement is, nevertheless, quite good for the Aj = 1 cross section: o{f;,+”) = 685 A2 (measured) compared with 800 A’ (calculated), but the predicted Aj = 2 inelastic cross section coming from higher-order terms in the exponential expression was too large by a factor of 3, which is not surprising since this result depends more strongly on the acceptance angle. The calculated total cross sections using the approximations described in Section IV are, however, about twice as large as the measured total cross sections. From this it appears that the theory does not take proper account of the interactions at small distances of approach. The above techniques are restricted to polar molecules. Information on rotational excitation of nonpolar molecules can be obtained by crossing a monoenergetic atom beam with a molecular beam and observing the change in velocity of the atom beam after the collision. In this way, Blythe et al. (1964) recently succeeded in observing the rotational transitionj = 2 -+ 0 for D,-molecules in collisions with K atoms.

-

E. SUMMARY Since relatively few experimental results are available at present it is still too early to draw definite conclusions about the attractive long range potential 35 Dipole-dipole interactions are especially interesting from a theoretical point of view. Since the dipole moment has a zero diagonal matrix element, the so-called distortion term disappears, and the quantum mechanical calculations are simplified (Davison, 1964; Arthurs and Dalgarno, 1960).

322

H . Pauly and J. P . Toennies

between molecules. A few generalizations, however, are possible. Fairly definite evidence for dipole-dipole forces has been found only in two cases : TlF (2,O) + NH, and T1F ( 2 , O ) + CF,Cl,, and the interactions can be attributed to the fact that both dipole moments are nearly stationary during the collision. In most other experimental situations, using nonstate selected molecule beams, it seems probable that either dipole-quadrupole forces (when one molecule rotates slowly in the time of a collision) or quadrupole-quadrupole forces (when both molecules have quadrupole moments and rotate rapidly during the collision) predominate. Thus, there is some reason to believe that by changing the scattering gas temperature the fraction of molecules in rotational states with rotational periods slow compared to the collision time was changed in the experiments of Schumacher et al. Consequently, the relative number of scattering events involving the dipole-quadrupole interactions, the dipole being that of the scattering gas molecules, was changed in these experiments. Similarly, Schoonmaker’s results can be interpreted in terms of a quadrupole-quadrupole intera~tion.,~ If this explanation is correct, then it indicates that generally the dipole moment of the primary beam alkali halide molecule is not as important as the quadrupole moment in such experiments. An estimate of the quadrupole-quadrupole potential indicates that it is indeed quite likely to be larger than the dispersion potential. In the case of a polar molecule interacting with an atom there is at the present time quite convincing evidence for the existence of a P, attractive term and some evidence for either a P, repulsive or an unexpectedly large PI,P, ( a p e ) attractive term.

F. MEASUREMENTS OF CHEMICAL REACTIONS 1. Calculated Reactive Scattering Cross Sections

As early as 1931, Eyring and Polanyi proposed that the rates of chemical reactions could be derived from a knowledge of the multidimensional potential surfaces of the type V(rAB,rBC,rAC) discussed in Section I, C (Glasstone et al., 1941). Calculations of multidimensional potentials based on the semiempirical London-Eyring-Polanyi-Sat0 method appear to give the best agreement with measured activation energies for chemical reactions. Presumably a more crucial test of the semiempirical methods is possible if more detailed information about reactive collisions could be obtained directly from beam scattering experiments. Before discussing experimental progress in this direction it seems appropriate to digress at this point in order to briefly 38 An alternative explanation is that Schoonmaker’s measurements happened to lie in the region of the last minimum at high velocities (see Fig. 32) (Bernstein, 1963b).

THE STUDY OF INTERMOLECULAR POTENTIALS

323

discuss the extent to which reactive collision cross sections depend on the shape and dimensions of V ( r A B , rBC,rAC). In all discussions of reactive collisions the assumption is made that the relative motion of the particles is adiabatic with respect to the electronic motion. Furthermore, it is assumed that the scattering can be treated by classical mechanics. This assumption requires not only that a/A > 1 (A is the wave length of the incident wave and a is of the dimension of the potential), but also that a/6 > 1, where 6 is the uncertainty in the particle position (see Section IV, C, 2). The latter condition is well satisfied for the nearly head-on collisions responsible for chemical reactions. In the case of reactions involving light particles (e.g., H + H,) a/A 1, and quantum effects such as tunneling through the potential barrier can occur. If, as is usually done, such effects are neglected it is only necessary to solve the mechanical equations for the relative motion. Calculations of this type were first carried out for colinear collisions, i.e., rABllrBc llrAc. By appropriately skewing the coordinates it is possible to reduce the collinear problem to a calculation of the motion of a single particle in the potential in the skewed coordinate system. In more recent calculations Hamilton's equations of motion for the internal coordinates of the three particles are solved numerically for different initial conditions. The results may be compared with beam experiments by properly weighting them in accordance with the distributions over initial conditions. For collinear collisions Evans and Polanyi (1939) showed that if the energy is released as A approaches BC, in an exothermic reaction the product AB will have a high degree of vibrational excitation. If, on the other hand, the energy is released while AB and C separate, then there will be little vibrational excitation. In the last few years there has been a resurgence of interest in calculations of this type, which has been stimulated largely by beam experiments with reactive particles and the widespread availability of high speed computer techniques. In most of these calculations the relative motion has been confined to two dimensions and model potentials of varying degrees of complexity have been used (Wall et al., 1958, 1961; Wall and Porter, 1962, 1963; Smith, 1959; Shavitt, 1959; Weston, 1959; Bunker, 1962; Blais and Bunker, 1962, 1963; Polanyi and Rosner, 1963). Most recently these calculations were performed for three-dimensional motion for the reaction H + H, -,H, + H (Karplus et al., 1964). The London-Eyring-PolanyiSat0 method was extended to take account of configurations other than the linear one. From the three-dimensional calculations and from the earlier work the following generalizations concerning the relationship between the multidimensional potential and scattering cross sections may be made:

-

(1) The reaction takes place during a single pass through the potential in sec. So far there appears to be little evidence for a time of the order of

H . Pauly and J. P. Toennies

324

the existence of an intermediate short-lived molecular state as is assumed in the “theory of absolute reaction rates.” Even for systems with a potential basin at the saddle point, most of the reactive collisions appear to be of the simple pass type, and only rarely do the reactants oscillate with respect to each other before the products fly off (Blais and Bunker, 1962). This result applies only to simple systems, and, in the case of complicated molecules, the reactants may stay together long enough to allow the various degrees of freedom to approach equilibrium before the products are ejected (collision compound model for nuclear reactions). On the basis of the single-pass mechanism the products will be ejected in a preferred direction with respect to the incident atom beam in the center-of-mass system. Beam experiments on chemical reactions have so far provided evidence for both a peaking of the products in the backward direction (rebound mechanism, illustrated in Fig. 48) and a peaking of the products in the forward direction (stripping mechanism). The former have been observed by Hershbach (1962) in reactions of alkali atoms with alkyl iododes, whereas the latter have been recently found in reactions of alkali atoms with halogen molecules (Datz and Minturn, 1964; Wilson et al., 1964) and also in the reaction K HBr (Grosser et al., 1964).

+

I time

FIG. 48. Schematic diagram of the rebound mechanism for a reactive collision in the center-of-masssystem. Note that in the case of a stripping mechanism the products will be exchanged.

(2) In general it appears that the angular distribution of the products gives less information on the shape of the potential than the extent of internal excitation of the products. In addition to confirming the result of Evans and Polanyi the more recent studies indicate that for a small moment of inertia of the product molecule the internal excitation will also be small. Similarly if the reactant atom is light the internal excitation of the product will also be less (Blais and Bunker, 1963). Calculations for collinear collisions suggest furthermore that the vibrational energy of the final molecule is strongly dependent upon the exact location of the saddle point (Wall and Porter, 1963).

THE STUDY OF INTERMOLECULAR POTENTIALS

325

(3) The reactive cross section as a function of the relative velocity g calculated on the basis of a simple classical model appears to be a good approximation to the results of Karplus et al. In the classical model the relative velocity of approach along the line connecting the centers of the two particles has to exceed a certain value in order for a reaction to occur. This simple assumption leads to the following expression for the reactive cross section (Fowler and Guggenheim, 1952; Eliason and Hirschfelder, 1959), (VI-4) where b* is the largest impact parameter and EZc is the minimum relative energy, along the line of centers, for which a reaction occurs. E,*,, is the activation energy for the reaction. Equation (VI-4) is frequently used in interpreting molecular beam experiments on chemical reactions. The result for crreaC1(9) of the recent calculations of Karplus et al. disagrees with earlier trajectory calculations (Wall et al., 1958). At the present time the numerical methods have reached the level of sophistication required to solve the classical equations of motion for multidimensional potential energy surfaces. These calculations neglect the coupling of the over-all rotation with the internal degrees of freedom (Eliason and Hirschfelder, 1959) and, of course, do not take proper account of the quantum nature of the rotational and vibrational transition, which take place during approach and separation. The error due to these approximations is probably small but has not yet been carefully assessed. 2. Chemical Reactions in Crossed Molecular Beams

The application of molecular beam techniques to the study of chemical reactions has been suggested for a long time (Fraser, 1931). It is only as recently as 1954, however, that Bull and Moon (1954) and shortly thereafter Taylor and Datz (1955) succeeded in directly detecting the products of chemical reactions in molecular beam experiments. In their experiments, Taylor and Datz studied the following reaction, I(

+ HBr+

KBr

+ H,

between two crossed beams of K and HBr by measuring the angular distribution of the product KBr. This reaction is illustrative of the type of reaction that can best be studied by molecular beam techniques because of the following features:

H . Pauly and J. P. Toennies

326

(1) Very low activation energies (<0.5 kcal/mole). (2) The product can be detected with a Langmuir-Taylor detector and discriminated from the reactant potassium atom beam. Two hot wires are used: tungsten which ionizes K and KBr and platinum which ionizes only K (see Section 111, E). (3) The detectable product is much heavier than the other product, so that it is expected to move along the direction of the center of mass. Exceptions are discussed later. Despite these features the total expected product intensity is quite small since typical values of n(b*)2 are around 10 A', and, depending on the kinematics, the detector currents are of the order of only A. For this reason it has not been possible to specify the relative velocities and the internal degrees of freedom more precisely by using two velocity selectors and state selectors as shown in the idealized apparatus of Fig. 15. The distributions in relative velocities and internal energies must, therefore, be accounted for in an analysis of the results. Because of these difficulties, another beam technique has been introduced for studying chemical reactions. In experiments on rainbow scattering an anomalous decrease in the differential scattering cross section of potassium was observed at large angles with HBr, whereas the behavior with Kr showed no such decrease (Beck et al., 1962). The drop has been attributed to removal of the potassium by the chemical conversion. This method for measuring reactive cross sections will be referred to as elastic scattering analysis (ESA) as opposed to product analysis (PA). In either case g is usually varied by velocity selecting the atom beam and crossing this beam at right angles with a cold beam of the heavier gas. The resulting change in product or reactant intensity is then used to determine CT,,,,~ as a function of g and from this the activation energy. In the following sections a few of the reactions on which most of the work has been done are discussed in some detail. All reactions for which molecular beam results are available are summarized in Table XXII. 3. Product Analysis Experiments on Chemical Reactions

+

a. K + HBr + KBr H . The early results of Taylor and Datz have been reinterpreted by Datz et a f . (1961a). In the experiment a nonmonoenergetic K atom beam was crossed at right angles with an HBr beam. CT,,,,,(~') was measured by varying the oven temperatures between 51 1" and 837°K and 373" and 460"K, respectively. The two-wire detector described previously (see Section 111, E) was used to measure the angular distribution of KBr in the

TABLE Xxn SUMMARY OF EXPERIMENTAL RESULTSON MMICAL REACTIONS FROM CROSSED BEAMEXPERIMENTS ~

Results

+ +

Do" = (Di" Dz") - (D3" D4')

Reaction

+ + +

(kcal/mole)

+ + + + + + + +

H Dz - t H H D +1.09 D + Hz+D HD -0.83 KCI +0.7 i 1.2 K HCI + H K + H B r +H KBr -4.3 1.2 K HBr H KBr K + HBr + H KBr K + HBr+ H KBr -5.9 f 1.2 K + HI + H + KI K HI + H KI MI M RI+R -17 to -37 e.g. K CH3I+CH3 KI -22 K + CH3Br + CH3 + KBr - 24 c s + cc14 + cc13 + CSCl -25 K + SnI4 + S n h + KI -25 C1 K Clz+KCI Cs BrZ + CsBr Br -40 M Xz + MX X (M = K, Rb, Cs; X = Br, I) 40 M I X & MI x (X= CI, Br) -26 to -50 --f

+

+

+

+ + + +

Q

*

+

+

+ + + +

--

N

-

Method ( v 1 implies velocity selected primary beam) PA PA ESA PA

01

vi

ui

-

PA vi ESA vi PA, ESA vi ESA PA ESA VI PA vi PA, ESA ESA PA vi

Activation energy

Reference9

Ureact

(kcal/mole)

(A3

0.55 f0.1

-

3.O 2.5-3.0 <0.4 0.15 0.2 5 0.1

-

3.6 for E = 2 kcal/mole

34 31

for E = 2.6 kcal/mole for E = 2 kcal/mole

-

-

-

<0.3

0.24

17

10 21

1, 2 2, 3 16 4 8 5, 20 6 16

9, 10, 11, 12

for E = 1.93 kcal/mole

15 13 14

-

-

-

-

> 100

18

PA

-

> 100

19

PA

-

>100

19

17

% 2

26

E

$w

21 2

3

F

0)

References

1. Fite and Brackmann ( I 963)

2. Fite and Datz (1963). 3. Datz and Taylor (1963). 4. Taylor and Datz (1955); Datz and Taylor (1959).

5. 6. 7. 8. 9.

Greene et al. (1960). Beck et af. (1962). Herschbach (1960b). Datz et al. (1961a). Herschbach et al. (1961).

10. Kwei er af. (1961). 11. Kinsey et of. (1962). 12. Herschbach (1962). 13. Bull and Moon (1954). 14. Gersing el al. (1963).

15. Ackerrnan et af. (1963). 16. Ackerman et al. (1964). 17. Gienapp (1961). 18. Datz and Minturn (1964). 19. Wilson et al. (1964). 20. Grosser et ul. (1964).

Et: 4

328

H . Pauly and J. P. Toennies

laboratory plane. The maximum KBr intensity was about 5 x A. The measured angular distribution as well as several theoretical distributions are shown in Fig. 49.

FIG.49. Comparison of observed angular distribution of KBr (circles) with calculated distributions of the center-of-mass for various values of the activation energy E*. @at2 er al., 1961a.)

As the relatively narrow peaking of the product distribution suggests the reaction is an example of case A in Table VIII. For the theoretical analysis of the data, Eq. (VI-4) was used for creac,, and, as the curves indicate, the angular distributions depend significantly on the assumed activation energy. The result for the activation energy of 2.5-3.0 kcal/mole compares favorably with the original result of Datz and Taylor, which was based on an oversimplified theory. Greene, Ross and co-workers (Greene et al., 1960; Beck et al., 1962) reinvestigated this reaction with a monoenergetic K atom beam. The analysis of their data showed that the angular distribution of the products depended

329

THE STUDY OF INTERMOLECULAR POTENTIALS

strongly on the assumed angular spread of the HBr beam. The angular spread was ascertained by fitting the measured distribution with various calculated distributions. With this model it was then possible to interpret the data in terms of a plot of urea=, (E)as a function of E where E = +p[DIz fizz]. These results showed a definite activation energy at less than 0.4 kcal/mole and indicate a rapid rise to a plateau which was constant up to energies of 5 kcal/ mole. This result is confirmed by ESA measurements to be reported on later. The discrepancy with the previous interpretation of Datz, Herschbach, and Taylor would appear to be due to the omission of the angular spread of the HBr beam in the early experiment. Grosser et al. (1964) have recently succeeded in measuring the velocity of the KBr formed in this reaction. From these results it appears that the products are scattered in the forward direction (as in a stripping mechanism) and that most of the released energy appears as internal excitation of the KBr. An analysis of the angular spread of KI formed in the reaction K + H I -+ KI + H (Ackerman et al., 1964) suggests that the distortion in the secondary beam velocity distribution also has a significant effect on calculated product distributions. This observation points out some of the difficulties in attempts to improve the accuracy of reactive scattering experiments. b. K + C H J + KI + CH,. In order to obtain information about I ( $ , cp), Herschbach et al. (1961), (Norris, 1963) chose the above and related reactions, since they have the kinematical features of a Case C reaction (see Table VIII). In addition to methyl iodide the following iodides were studied: ethyl, n-propyl, i-propyl, n-butyl, and i-butyl. The experimental arrangement was similar to that used in the Datz and Taylor experiment (i.e., with nonmonoenergetic beams) except that the angle of crossing between the beams was variable and out-of-plane scattering could also be measured. The main result of these experiments was the observation of a maximum in the KI distribution at 83" measured from the direction of the K atom beam [Fig. 5O(a)]. The circles in Fig. 50(b) correspond to the possible KI product velocities for the indicated values of the final relative kinetic energy. Since the final kinetic energy of the products is not known, two explanations for the observed maximum are possible:

+

(1) The K1 is scattered backward in the direction of the incoming CHJ beam with a final kinetic energy of about 2 kcal/mole [shaded area in Fig. 50(b)]. (2) The KI is scattered over a range of angles with final relative kinetic energies greater than 2 kcal/mole, for example, vector a in Fig. 50(b).

The second possibility was excluded by observing that there was only little scattering out of the plane of the reactant beams. On the basis of the first

330

H . Paub and J. P. Toennies

explanation it appears therefore that the KI molecule leaves the reactive complex in a direction 180" from the direction with which the K atom approached the CH,I molecule, indicating that the reaction complex is extremely short lived (< 5 x sec). The narrow angular spread of the products also supports this conclusion, as can be seen from the dashed curve in Fig. 50(a) which was calculated on the assumption that all the KI recoiled in the K atom direction and that the final relative kinetic energy was 1.6 kcal/mole.

Scattering angle 0 [dog.]

FIG.50. (a) Observed and calculated distribution of KI. (b) Velocity diagram showing two possible center-of-mass velocities which can explain observed angular distributions (Herschbach, 1962).

Furthermore, it was concluded that most (90 %) of the liberated energy was taken up by the internal degrees of freedom rather than in translational energy. These conclusions were confirmed by additional experiments with other alkali metals and alkyl iodides. From these experiments it appears that the average relative kinetic energy after collision decreases with the heavier alkyls, whereas there is no steric influence with increasing size of the alkyl group. In these reactions total cross sections were about 10 A' and the activation energies <0.3 kcal/mole. These conclusions are in agreement with calculations for planar collisions by Blais and Bunker (1962). Recently Karplus and Raff (1964) have carried out extensive two- and three-dimensional calculations for this system. c. D + H2 + HD + H. Preliminary results on this (Datz and Taylor, 1963) and the associated reaction (Fite and Brackmann, 1963) H + D, + HD + D have been reported recently. The atomic beam was modulated, so that the product intensity could be measured with a phase-sensitive amplifier. Liquid helium cooling is essential to keep the background pressure in the scattering chamber sufficiently low. Reactive differential cross sections could only be

THE STUDY OF INTERMOLECULAR POTENTIALS

33 1

measured outside a region of 12" to the left and right of the atomic beam. The region at smaller angles being obscured by a 1 % H D impurity in the incident beam of deuterium atoms. Thus, for the interpretation of the data, Datz and Taylor had to rely on the phase lag of the products and the observed product distribution at large angles. These results appear to be consistent with a linear complex of short life, yielding mostly unexcited products. Additional molecular beam information on this, the most fundamental reaction in chemistry, is lacking at the present time.

4. Information on Chemical Reactions from an Analysis of Elastic Scattering

+

+

a. K HBr + H KBr. Beck et al. (1962) proposed that the probability of reaction could be obtained from the shape of the elastic differential cross section at large angles for the scattering of K on HBr. The procedure used is suggested by the observation that the differential cross section for the large angle scattering of K on Kr is very nearly constant for angles larger than the rainbow angle, whereas the observed cross section for HBr at these angles is considerably smaller in magnitude and decreases with increasing scattering angle. Assuming two-body central forces, similar intermolecular potentials, and a small probability of rotational excitation (vibrational excitation is not possible at the low relative kinetic energies) the authors attribute the difference to the removal of K from the scattered beam by the chemical reaction. The reaction probability is then defined by (VI-5) In actual practice the elastic differential cross section I J E , 9) is estimated from the potential parameters deduced from measurements at small angles where the reaction is not thought to occur. Figure 51 illustrates the method used in evaluating Eq. (VI-5). Unusually small differential cross sections have also been observed for scattering of K on HgC1, and HgI, and on SnI, (Gersing et al., 1963). By way of the classical deflection function it is possible to assign an impact parameter to the observed scattering angle. Then from the assumed potential energy function the repulsive energy at closest approach can be ascertained. The probability of reaction as a function of the potential energy at closest approach is shown in Fig. 52. The threshold energy for reaction is found to be 0.15 kcal/mole, in agreement with the product analysis experiments discussed earlier. Furthermore, it is observed that the probability of reaction rises to a maximum of 0.90 at a potential of 1.2 kcal/mole. For this probability the reaction cross section had a radius of 2.7 A.

Y 20

postulated elastic cross section rneasur curve

60

40

80

loo

Scattering angle 8 [dq.]

FIG.51. General behavior of the differential elastic cross section observed when chemical reactions occur.

0.8

-

0.6

-

0.4

-

0.2

-

Potential energy at dislance of closest approach

[kcal/rnole]

FIG.52. Measured probability of reaction as a function of the repulsive potential energy at the distance of closest approach and the relative initial kinetic energy E.

332

THE STUDY OF INTERMOLECULAR POTENTIALS

333

The interesting result that the probability of reaction decreases with increasing initial energy of relative motion is also in agreement with the assumption made above. This can be understood from the following argument. The distance of closest approach for a repulsive potential will be the same for slow atoms with small impact parameters as for faster atoms but with larger impact parameters. Such collisions differ therefore only in that the total angular momentum (pgb) in the fast collisions is considerably larger. Due to the formation of H the reduced mass of the products is much less than that of the reactants, and the KBr molecule has to take up the excess angular momentum. Consequently, a relatively larger fraction of the total energy is taken up by the rotational motion in the fast collisions, leaving less for the vibrational degrees of freedom. This leads to an inhibition of the reaction in the faster collisions by the restricted phase space available for the products. In a more recent investigation (Ackermann et al., 1964), with HC1 in place of HBr, a reaction was found to occur only when the kinetic energy was sufficient to form a KC1 molecule in the ground vibrational state. The reaction probability was then observed to increase further as the first excited vibrational state of the product molecule became energetically accessible. These conclusions were deduced from ESA experiments since KC1 was not directly observed. In the case of HI however PA and ESA techniques gave reactive cross sections that agreed within a factor 2. b. K + CH,Br + KBr + CH,. In the case of CH,Br (Ackermann et al., 1963), in which no large change in the reduced mass is to be anticipated, it was found that the probability of reaction increased with increasing kinetic energy. Such a behavior is to be expected if barrier penetration has to precede the chemical reaction. Although the internal consistency of these results is exceedingly good, there is still some question concerning the validity of the procedure adopted for calculating the probability of reaction. In particular, it is questionable if the elastic cross section measured under conditions in which a reaction is probable can be interpreted as is done in Eq. (VI-5). In the case of rotational inelastic scattering, which also occurs in these collisions, it was shown (see Section IV, D) that the partial elastic cross section is indeed affected by the fact that transitions can occur. Another explanation for the observed decrease in the differential cross section at large angles has been put forth by Herschbach and Kwei (1963). Using a simplified two-center model like the one shown in Fig. 5 they calculated differential cross sections at large angles. Their results for certain special cases indicate a decreasing differential cross section with humps corresponding to the rainbow maxima and interference effects produced by the additional minimum in the potential. As yet however these additional minima have not been observed in the experiments.

334

H. Pauly and J. P . Toennies

A somewhat different behavior than that reported by Beck, Greene, and Ross has been observed in the large angle scattering of K on SnI, (Gersing et al., 1963). In this case the reaction product KI was found to be fairly evenly distributed over the entire available angular region (1 to 35"). From these results it would appear that further experimental and theoretical work is necessary before the wide angle scattering of reactive systems is fully understood.

VII. Concluding Remarks The advances of the last five years in the execution and interpretation of molecular beam experiments have shown that the method is capable of yielding detailed information on intermolecular potentials. The inherent difficulties of these experiments arising from the extremely small intensities available are more than compensated for by the relative directness with which the information may be extracted from the measurements. Up to the present time most of the measurements have been aimed at studying the spherically symmetric atom-atom potential. For this potential it has been possible to measure not only the size of the attractive well (van der Waals potential) but also its shape. This information has been obtained from measured differential cross sections and from the velocity dependence of the integral cross section. Two regions of the intermolecular potential still remain to be explored. The predicted weakening of the attractive potential at relatively large distances (> 50 A) arising from the finite time for photon exchange has not yet been observed in molecular beam scattering experiments. Secondly, there exists little information on the potential in the intermediate region where the potential passes over from an attractive to a repulsive potential. Study of this region is important for an understanding of transport properties. In addition to the work on the atom-atom potential it has been possible to obtain some preliminary data on the anisotropic force fields surrounding simple diatomic molecules. Finally, a number of pioneering experiments on crossed beam reactions have demonstrated the power of the technique for acquiring an understanding of chemical reactions. The future trend of molecular beam experiments will undoubtedly be to explore more carefully the intricacies of the forces responsible for chemical reactions and the way in which these forces determine the angular distributions of the products as well as the distributions of energy among the translational and internal degrees of freedom. It seems certain that, once chemical reactions are understood, this knowledge will influence other branches of science such as biology and will have a considerable impact on the many aspects of technology in which chemical reactions are either harnessed as a source of energy or are used for providing molecular conversion.

THE STUDY OF INTERMOLECULAR POTENTIALS

335

The study of chemical forces will most likely involve the following four basic types of experiments: (1) Measurements of inelastic cross sections for rotational excitation. They will provide information on anisotropic forces. (2) Measurements of inelastic cross sections for vibrational excitation. Quite difficult to perform, they will, nevertheless, eventually yield results on the missing link in the potential at intermediate distances. (3) Direct studies of crossed beam reactions using state selectors to specify internal quantum states before and after reactive collisions. These experiments will show how the potential influences the detailed kinetics. (4) Measurements of elastic and inelastic scattering of beams on surfaces (not discussed in this review). They will lead to an understanding of gassurface interactions, which are always present in practical applications of chemical reactions (catalysis) and are of special importance to modern aerodynamics.

Together with the more classical experiments these scattering methods will eventually shed light on a problem that has long resisted detailed study. ACKNOWLEDGMENTS Theauthorswould liketo take thisopportunity to thank Professor W. Paul forhisconstant encouragement, which made their research possible. Funds were generously provided by theDeutsche Forschungsgemeinschaft.Thanks arealsodue to Professor C. Schlier(Freiburg), Professor E. F. Greene (Providence), and Dr. H. Harrison (Seattle) for reading the manuscript and calling a number of errors to our attention.

LIST OF a Dimension of the potential; selector slit width a0 Sound speed b Impact parameter bo Impact parameter corresponding to 00 c Most probable velocity in a gas; velocity of light d Diameter of thin-walled orifice; diameter of a single channel e Electron charge f Fraction of ions that enters the mass spectrometer of a universal ionization detector; angulardependent part of electrostatic potential

SYMBOLS

f ( v ) Velocity distribution function f(8) Scattering amplitude g Relative velocity; reduction factor for background resulting from a mass spectrometer h Planck’s constant j Rotational angular momentum quantum number k Boltzmann constant; wave number (k = 27rpg/h); fraction of beam molecules ionized in a universal ionization detector

H. Pauly and J. P. Toennies

336

I Angular momentum quantum number; channel length; path of primary beam rn Mass; projection along field direction of rotational state quantum number n Particle density (particles/cnP); exponelit in power law dependence of the repulsive part of the potential p Multipole, e.g., dipole; pressure q Anisotropy factor q Direction of the molecular axis in the laboratory system 4' Second derivative of the classical deflection function at the rainbow angle r Distance between centers of mass; radius of velocity selector disks rn, Potential parameter (position of the potential minimum) s Exponent in the power dependence of the attractive part of the potential ( V = - C/r*) t Time; exponent in power law dependence of inelastic potential term rcol Time for a collision (rcol = 2abIg) v Velocity vc Velocity of center of mass

fi Average velocity x Coordinate; v&vz = v i / ( 2 k T a / m ~ ) ~ l ~ y cilcz = dTimz/Tzrni

z Coordinate; collision path di-

rection (z = g r ) A krm (dimensionless wave number)

B(T) Second virial coefficient C(T) Third virial coefficient C, Interaction constant ( V = Cs/rs) C van der Waals constant for a s = 6 potential CqkRacah's notation for normalized spherical harmonics, Cqk= [ 4 ~ / ( 2 k 1)11" Y&", E) D A ,Interaction ~ constant for angledependent potential

+

D Coefficient of diffusion E Kinetic energy of relative motion ( E = &pg2);detector efficiency Fao(s, x ) Correction function for correcting for motion of scattering gas F Total source area Fo Area of thin-walled orifice Gao(s,y) Correction function for correcting for motion of scattering gas and for velocity distribution in the primary beam 4 Beam intensity I Ionizadon potential I(@ Differential cross section I ( @ = lf(W Jv(x) Bessel function of order v J Total angular momentum quantum number K Reduced kinetic energy (K = El&) L Length of the scattering chamber; length of the velocity selector M Inelasticity parameter ( M = c z C ) ; molecular weight N Number of particles; noise of beam intensity NO Avogadro's number P Reaction probability Q Quadrupole moment [Q= 4 J (3z2-rz)p(r) dr] R Velocity resolving power of a selector R(r) Radial wave function S Total cross section for low angular resolution (5' = 2 a J:(a) b db); scattering matrix T Temperature; transmission U Reduced potential [ V(r)/&] V Intermolecular potential Yt,&',E) Normalized spherical harmonic a' Potential parameter (Kihara potential) a Potential parameter (Buckingham potential); polarizability; angular resolution in the centerof-mass system ao Limiting angle in the center-ofmass system

THE STUDY OF INTERMOLECULAR POTENTIALS

r Gamma function A Energy term A Mean free path in molecular beam source chamber 0 Angle of deflection in the laboratory system 00Angular acceptance of inelastic scattering apparatus E Azimuthal angle (defined in Fig. 2) Work function; azimuthal angle in the laboratory system; electrostatic potential Y Angle between molecular axis and the vector connecting the centers of mass of the colliding particles R Solid angIe in laboratory system

/3 Reduced impact parameter

(6 = b/rm)

-

y Ratio of specific heats ( y c p / c v ) ;reduced relative velocity (g/c = g d M / 2 k T = d ~ m ) angle ; between crossed beams; factor in interaction constant for molecule-molecule collisions. 6 Uncertainty in the particle position e Potential well depth; internal energy; relative ionization efficiency 5 Imaginary component of phase shift 7 Coefficient of viscosity; angular resolving eficiency of the detector qt Phase shift TI* Reduced phase shift

(7r*= ? L / 4 8 Angle of deflection in the center-

of-mass system A Wavelength; coefficient of heat conductivity p Reduced mass; dipole moment 6 Real component of phase shift p Reduced distance (p = r/rtn) u Integral scattering cross section T Characteristic time; transparency of a multichannel oven 9, Azimuthal angle in the centerof-mass system Diamagnetic susceptibility 11, Wave function o Solid angle in the center-of-mass system; radial velocity of a rotor A Angular resolution in the laboratory system; limiting angle in the laboratory system

x

337

INDICES Subscripts 1 Primary beam 2 Secondary beam 3,4 New molecules after reactive collision rot Rotation el Elastic inel Inelastic tot Total c Center-of-mass system disp Dispersion ind Induction (electrostatic)

Superscripts i Initial state

f Final state

p Quantum state

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