polarization in elastic scattering: close-coupling studies on arn2

Chemical Physics 27 (1978) 229-236 0 North-Holland Publishing Company

POLARIZATION IN ELASTIC SCATTERING: CLOSE-COUPLING STUDIES ON AI--N2 ’

Millard H. ALEXANDER &partment of &?mistry, tikwsity of h&&and,

C&&e Park, Maryhzd 20742. USA

Received 11 August 1977

We present the results of close-coupling calculations of m,-dependcnt differential and integral ROSS sections forjl = 2 --+ jz = 2 rotationally elastic Ar-N2 collisions. Two potcnhal surfaces were used with differing long- and short-range aniso- tropies. If the anisotropy is long-ranged the scattering of an isotropic beam results in a significant angle dependent pohrization of the elastically scattered products. To a certain extent this reflects a selective loss of m,%tate population due to rotationally inelastic transitions. For quantization along the initial relative velocity vector or perpendicular to the scattering plane, the depolarization of an initially m-state selected beam vanishes in the forward direction and is significantly less than

the statistical limit at all angles, which m . d- rcates a dynamical conservation of the direction of the molecular rotational angu-

lar momentum. By contrast, in the helicity frame depolarization is much more pronounced. The oscilhtory structure present in the rotationally inelastic differential cross section does not appear IO be quenched by the interference between various m -+ m’ transitions.

1. Introduction

A collision between an atom and a diatomic mole-

cule with non-zero rotational angular momentum, j, can result in a reorientation ofjwithout a change in its magnitude. This type of collision, which can be related to the t2 type of buik reiaxation process *+, is distinct from rotationally inelastic collisions which involve a change in the magnitude ofj. It is the latter which are responsible for the tl type of relaxation [1,2 1. With the development of electric quadrupole lenses and tunable lasers as state selectors and detec- tors, one can now begin to determine directly these rotationally elastic, reorientation cross sections [3-51. New theoretical work is needed, to help in the interpretation of these experiments as well as those which measure the m,-dependence of total (elastic +inelastic) cross sections [6].

* Research supported by the National Science Foundation. Grant CHE-7611359, and by the Computer Science Center, University of Maryland, USA.

**Detailed, recent reviews of the theory of bulk rotational relaxation are given by Pickett [l] and I&I and Marcus-

r21.

The formalism for the quantum treatment of fully- oriented cross sections was presented by Reuss and Stolte [7] and later extended by Alexander et al. [8] to arbitrary orientations of the quantization axis. Similar expressions for inelastic transitions were derived within the sudden approximation by Bernstein and co-workers [9]_ Reuss and Stolte [7] and Miller [IO] used the distorted wave Born approximation to develop closed form expressions for the middependence of total cross sections. Recently, Jacobs [ 1 l] has determined from a quantum close-coupling calculation the full Inidependence of the differential and integrali = 1 +& = I cross sections for Kr-Hz collisions. Similar studies on the il = 0 + j2 = 0, 1 + 1, 2 + 2, and 3 + 3 transitions in He-HCl have been carried out by Monchick [121, without however explicitly investigating the dependence of the differential cross sections on the fmal projection quantum number.

In a previous paper [ 13 ] we have reported a detailed close-coupling study of the orientation dependence of rotationally inelastic Ar-N2 collisions. Two potential surfaces were used which differed only in the range of the anisotropic component. The present paper isan extension to the rotationally elastic

230 M.H. AlexnnderlPolnrizatioion in elastic scattering

jl = 2 + j2 = 2 process. We shall specIficalIy try to assess the dependence of reorientation on the strength of the potential anisotropy at long range, on the Initial 9 state, and on the scattering angle. Also, we shall attempt to determine to what extent inelastic collisions affect the polarization of an elastical- Iy scattered beam. Finally, we shah explore the quenching of structure in the differential cross section intro- duced by the averaging over ml + mz transitions.

The next two sections contain a brief outline of the scattering formulation and of the actual caIcuIa- tions. This is followed by three sections containing an analysis of the important resuIts. A short conclusion follows.

2. Formalism

If the Initial and fial axes of quantization are rotated, with respect to the Initial wavevector, through polar angles 8,, ql and “2, &, respectively, then the scattering amplitude for a rotationally elasticjnl +sjpl collision is 18,131

where 0, Q are the center-of-mass scattering angles and the rotation matrices follow the convention of Edmonds [14]. To avoid confusion the m-indices indicate quantization along the initial wavevector k,; the /.&dices, along an arbitrarily oriented z-axis. The scattering amphtude which appears in the RHS of eq. (1) refers to the standard ‘space-frame” expression [8,13,15],

where Jis the total angular momentum, I, and Z2 are the initial and fmal orbital angular momenta, (: : :) is a 3/’ symbol [14], [x] = 2x + 1, Ylrn is a spherical ~XIIIO& [14], and 7’$, jr2 is a rotationally elastic T-matrix element. The ifferential cross section Is then given by

d~g,+j~21do = l$~r~~L2(~,~)12/k~. (3)

By integrating over B and I$, one can obtain an analytic expression for the jpl + jl-12 integral cross section, which has been given previously [8].

In the present article we shall consider explicitly the following three choices of quantization axes:

(1) “Space-fixed” or k, frame. Both the initial and fmal quamization refer to the initial relative velocity vector. The angles in eq. (1) are el = & = $2 =& =O.

(2) “Helicity” frame [16,17]. The initial quantization refers to the initial relative velocity vector; the fmal quantization, to the final relative velocity vector. lneq.(l)Fl =$l =O,& =eJ2 =#.

(3) “Perpendicular” or kl X k2 frame. Both the initial and fmal quantization refer to an axis perpen- dicuIar to the scattering plane. ln eq. (1) e1 = & = a2 =q2 =ri/2_

The vector polarization resulting from the elastic scattering of an upolarized beam is defmed as [IS]

which can range in value from -1 to +I. To deal with colhsional reorientation, one can also define a dif- ferentiaI “depolarization ratio”

which is just the probability of changing p1 during the collision. It is similarly possible to defme an integral depolarization ratio in terms of the corresponding integral cross sections. Since Y1,(O,O) vanishes unless m = 0, the scattering amplitude in eq. (2) will go to zero in the forward direction unless ml = m2_ Con- sequently, as pointed out by Jacobs [l l] and by Kouri and Shimoni [19], in the space-fixed frame reorientation does not occur in purely forward scattering..

MN. Alexander/Polarization in chic scattering 231

3. Scattering calculations

The rigid-rotor Ar-N2 potential adopted here was a Lennard-Jones 612 spherically symmetric term with a modified 612 anisotropy

V&Y) = E{(R&2 [1+ @&(COS r)l

- 2@l+$U +cQP,(cos $11, (6)

where the parameters have their usual meaning [13,20] Two sets of anisotropy parameters were used. The first, a1 = 0.5 and a2 = 0.13, corresponds to the potential of PattengiIl et al. [2 I], which will be de- ‘noted buy the initials SR to indicate that the anisotropy is predominantly short-ranged. The second set, “1 =cyz = 0.3 15, corresponds to a longer-ranged anisotropy [20,22] and will therefore be denoted by the initials IR. For both the LR and SR potentials e = 83.05 cm-’ and R, = 7.424 bohr [22].

The T-matrices in eq. (2) were obtained from a full close-coupling treatment of the collision dynamics [15,23). The total energy was 208.5 cm-’ and the N, molecu!e was treated as a rigid rotor with a rotational constant of 2.010 cm-l [13,20,223. The channel expansion included the j = 0,2 and 4 states, which is suffkient to allow for the major inelastic corrections to the jl i= 2 +jZ = 2 elastic scattering. T-matrix elements were computed for every value of J up to 150; extrapolation of the large J values indicates that the integral degeneracy-averaged 2 + 2 cross section are converged to within 0.5%.

The results of our scattering calculations are discussed in the next three sections.

4. Scattering of an unpolarized beam

As is well known in nuclear physics [24], spin- dependent forces can result in collisional polarization of an unpolarized beam, if quantization is mea- sured perpendicular to the scattering plane. We have already observed a similar phenomenon in rotationally inelastic Ar-NZ collisions governed by the LR potential [13J. Fig. I reveak that for this potential thejl = 2 +jZ = 2 elastically scattered products also display a significant degree of polarization, although the magnitude is less than for the 0 + 2, 0 + 4, and 2 + 4 inelastic pracesses. For the SR aniso-

LR Anisotropy SR Llniaottopy I-.

0

-+-&n-&T4 -Ii 90 120 0 30 60 90 120

Scoltering Angle Scattering Angle

Fig. 1. Dependence on scattering angle of the vector polarization [(eq. (4)) of scattered products corresponding to the elasticj = 2 --t 2 and inelastic 0 + 2,O + 4 and 2 -+ 4 transitions. The 0 + 4 and 2 -+ 4 results are taken from our earlier calculation 1: 31. The initial beam is unpolarized. Polariza- tion is measui-ed with respect to the ki X k2 axis, perpendicu- iar to the scattering plane. The slowly vary@ curves re- present an atrempt to simulate the resalts of an experimental measurement, where ftite velocity and angular resolution would efface the highly oscillatory structure. The dashed line corresponds to an isotropic distribution of final w,-states. Note the expanded ordinate for the elastic 2 * 2 panel.

tropy the resuhing polarization is virtually insignificant. These results indicate that the experimental measurement of product polarization in the elastic scattekg of unpolarized beams could be a sensitive probe of the anisotropy in the intermolecular potential *.

For the LR potential the overall slow variation in the polariation vector corresponding to elastically scattered products is out of phase with those corresponding to inelastically scattered products. This suggests that the elastic polarization may be the by product of a non-isotropic loss of flux into the various inelastic channels. To explore this point, we carried out an additional series of scattering calculations with the LR potential, retaining only the elastic 0 = 2)

* Such an experiment could, in principle, be performed with laser-induced fluorescence detection modified to allow circularly polarized excitation followed by spectral and circular polarization resolution of thefluorescence. The analysis wpuld followrefs. [4) and [8] _

232 M H. Alexander/Pohrization in elastic scattering

wijh mebxtrc ch~nn~~~ dk=2 .

without inelastic channels -

0 n -0.5 0 30 60 90 120 0 30 60 SO I20

Scattering Angle Scattering Angle

Fig. 2. Dependence on scattering an& of the vector polarize- tion of elastically (i = 2 + 2) scattered products. computed

with the LR potential. The left hand panel refers to a cal-

culation including both elastic (j= 2) and inelastic (j= 0 and j = 4) cbanncls. Only the j = 2 channels were retained in the calculation !mding to the right hand panel. Additional detaiis are contained in the legend to fig. 1.

channels in the close-coupling expansion_ The resulting r&e dependent polarization is compared in fig. 2 with thej = 2 + 2 LR panel from fig. 1. Although the purely elastic calculations do lead to some degree of polarization, this is clearly enhanced by the addi- tion of the inelastic charmets. By way of a spectro- scopic analogy, the 2 -+ 0 and 2 + 4 inelastic transitions have a “hole-burning” effect on the initially isotropic distribution of j = 2 projection states.

This could well have important consequences for the understanding of more highly averaged relaxation experiments, which are often interpreted in terms of a rotation&y inelastic relaxation process with time constant tl and a rotational reorientation with relaxation time r2 [ I,2 J. The discussion of the preceding paragaph, as illustrated in fig_ 2, suggests that the tl and t, type processes may, in fact, be coupled together, rather than representing two independent relaxation modes.

5. Depolarization of state-selected beams

In this section we consider the inverse of the type of effect discussed in the previous paragraph, name- ly the depolarization of an initially mj-state selected beam caused by collisional reorientation. One can anaiyse this effect in terms of the depolarization ratio defied by eq. (5). Fig. 3 displays the dependence on scattering angle of the depolarization ratios for quantization in the space-fixed, helicity, and perpendicular frames, where these are defmed in section 2. In all cases, the initial projection quantum number was -1.

LR Anisotropy SR Anisotropy

0.m I

0 30 60 90 120 Scattering Angle Scattering Angle

Fig. 3. Dependence on scattering angle of the rotationally elastic (j = 2) depolarization ratio [eq. (S)] for the case of an initiai projection quantum number ml = -1; taken with respect to the designated .zL axis. For each potential surface the three vertical paneIs refer, respectively, to the space- tixed, h&city, and perpendicular frames, which arc defined in section 2. The dashed lines indicate the depolarization ratios which would be expected in a purely st&istical limit, where the scattered flux would be distributed equally among the 2j + 1 projection states.

The depolarization ratios for other initial projection quantum numbers are qualitatively similar, and for brevity are not displayed.

In the limit of purely forward scattering the helicity and space fixed frames are identical. In both cases the depolarization ratios drop to zero at small scattering angles, reflecting the rigorous conservation-of mj discussed in section 2. The perpendicular frame depolarization ratios also vanish in the forward direction.

Although we shall not dwell on this point in detail, this appears to imply that the space-fied scattering amplitude becomes independent of the projection quantum number as the scattering vgle goes to zero *.

At larger angles in the case of both the space-fixed and perpendicular frames the degree of depolarization varies with scattering angle, but is always sign6 icantly less than the purely statistical limit. Physically,

’ If ~~ =& and & = $2, then eq. (1) defines a unitary trans.- formation of the space-fued scattering amplitude. A suffi- cient condition for the preservation of the small-angle dii- gonal (ml = m2) character is that the space-fixed amplitude be independent ofml.

MI% AlexanderiPolarizatiotz ilz elastic scattering 233

Table 1 Integral rotationally-elastic (i= 2) Ar-N2 cross sections (A2); quantizstion along kl.

m2 SR potential LR potential

mt =‘-2 -1 0 ml=-2 -1 0

-2 399 4.19 4.93 404 6.65 9.45 -1 2.41 398 2.78 6.96 383 6.43

0 4.15 1.58 385 10.8 6.91 377 1 1.87 6.91 2.78 7.68 13.9 6.43 2 3.92 2.88 4.93 4.40 7.32 9.45

a)

D u;, 12.4 15.6 15.4 29.8 34.8 31.8

b) 0.030 0.038 0.039 0.069 0.083 0.078

aJ Total reorientation cross section; or= &_+,,rr emr+,,r2. bJ Depolarization ratio, cq. (5).

this implies that the orientation of the rotationai angular momentum, with respect to either the initial relative velocity vector or with respect to an axis perpendicular to the scattering frame, tends to be conserved during rotationally elastic collisions. McCaffety and co-workers [4] have arrived at a similar conclusion from the analysis of their experiments.

The small angle behavior of the perpendicular frame depolarization ratios may be of some relevance to the experiments of Boerkenhagen et al. [S]. These authors are currently studying the collisions of CsF with various noble gases using electric quadrupole state selection and detection in an experimental con- figuration corresponding to small angle scattering with perpendicular quantization. By extrapolating from the present calculations, one would predict that the experimental reorientation cross sections will be much smaller than the purely elastic values.

The h&city frame depolarization ratios deserve special attention. At scattering angles greater than =20”, where the space-fixed and helicity frames begin to differ significantly, the helicity depolarization ratios grow rapidly and reach the statistical limit. Thus the helicity frame, while containing a conceptually satisfying symmetry between the initial and final scattering states 1161, is clearly poorly suited to describing the conservation of rotational orientation which appears to be a distinct dynamical feature of rotationally elastic collisions.

This has important consequences for the interpretation of the quantum coupled-states (CS) approximation

[ 19,25,26]. Many of the original derivations explicitly assumed the dynamical conservation of rrti in either

a helicity [27] or “body-fixed” [25,27,28] frame, which, to within an arbitrary phase factor, were shown to lead to identical scattering amplitudes [19,30]. Recently, in a significant paper, Kouri and Shimoni [ 191 have reanalysed the CS approximation in temrs” of mi-conservation in the space-fixed frame. The present close-coupling results,.as illustrated by fig. 3, suggest that this latter interpretation is clearly more reasonable, at least for elastic scattering *.

The integral cross sections and the corresponding depolarization ratios are listed in table 1. Since most of the rotationally elastic scattering occurs at small angles, where the depolarization ratios are small, the overall reorientation cross sections are much smaller than the purely elastic cross sections. This behavior is entirely comparable to what has been found by Monchick for He-HCI collisions [12] and by Jacobs for Kr-H, [I I]. The LR potential, with a larger degree of anisotropy at long range, results in larger reorientation cross sections, which is what one would expect on intuitive grounds. This is also con- sistent with the work of Jacobs [ 111, who found the magnitude of the &IZ cross sections to be directly related to the degree of anisotropy in the potential.

’ For brelastic At-N2 collisions, our previous study [ 131 indicates that tn,. is not conserved in either the space-f&d or h&city frames.

234 M.H. Alexander/Pohrization in elastic scatten~tlg

Although much smaller than the purely elastic values, the integral reorientation cross sections are nevertheless comparable in magnitude to the rotationally inelastic cross sections which were reported previously [13]. Physically, a collision is just as likely to change the magnitude of the rotational angular momentum as it is to change its direction. This would imply, if one is justified in extrapolating to bulk relaxation experiments involving other molecules, that f1 and t2 relaxation processes could be expected to occur on the same time scale, a conclusion which is in good agreement with recent experimental results [31-331. Since the depolarization ratios rise steeply with increasing scattering angle (fig. 3), a beam experiment which is sensitive only to small angle collisions could yield a slower reorientation rate than the corresponding ceil experiment. A more detailed discussion of this point has been given by Wang et al. [34].

From table 1, the total reorientation cross sections for both potential surfaces dispIay little variation with initial mi-state. Again, if one can extrapolate from the present caIculations, then one might expect t2 relaxation times to show only a small (I&-20%) m/-dependence. This conclusion is in satisfactory agreement with recent experimental fiidings 135,361.

Examination oftable 1 reveals a slight degree of alternation in the reorientation cross sections with the even Am values being consistently larger than those associated with odd changes in the projection quantum number, a phenomenon which was also observed by Jacobs [I 11. .4t first glance this might seem to be a consequence of the &(cos 7) angular dependence of the anisotropy; however, the following argument demonstrates that this is not the case here *. . In the j, mi. 1, ml representation, the mi dependence of the coupling between the rotor states ljml) and @n?) is, in the case of a P,(cos 7) anisotropy, contained in the 3j symbol [9]

jj 2 ( “1 -t?z2 12 ) .

For thei = 2 case under consideration in the present article, the variation of this 3i symbol would

’ In very recent work on Ar-CsF collisions, Fitz [37] found that the alternation in thei = 1 + 1 reorientation cross sections was well predicted from symmetry considerations.

correctly predict the ml = 0 + m2 = 52 cross see- tion to be larger th_an the 0 + Cl value and the -1-f 1 cross section to be larger than the -I+ 12 or -1 + 0 values (although the magnitude of these differences would be overestimated). However, by the same argument one would expect the -2 + -1 cross section to be larger than the -2 + 0 value, which is in obvious disagreement with table 1. Also, due to the restric- tions on the range of n in the above 3J‘ symbol (-2drtd2),the-2-t1,-2+2,and-l+2transi- tions would be forbidden in first order, so that the corresponding crdss sections would be expected to be smaller than the others, which is clearly not the case. The variation in magnitude of the reorientation cross sections in table 1 probably reflects a subtle interplay between various dynamical factors and may therefore defy a simplistic explanation.

6. M-state quenching of elastic differentiaI cross section

The differential cross section for the elastic (i+i) scattering of un unpolarized beam can be obtained from eq. (3) by summing over the final m2 quantum numbers and averaging over the initial m 1 values. The resuIt is

For a spherically symmetric potential only Am = 0 transitions would occur and, furthermore, all the jm +-jm cross sections would be equal. This equality as well as the lack of Am #O transitions would be lifted by the presence of an anisotropy in the potential It has been argued [38,39] that the resulting interference among in the (2i + l)* terms on the r.hs. of eq. (7) would result in a marked quenching of the osciliatory pattern characteristic of elastic scattering from an isotropic potential.

On this point recent close-coupling calculations are somewhat ambiguous. No quenching was found for He-HCl [40] ,‘Ar-Hz [41], or &-Hz [I 11. In the case of He-CO Monchick and Green-[421 observed a shift in the oscillatory pattern without significant quenching. However, McGuire [43] found that the. fast osciJ.lations in the Li+--Hz elastic differential cross section were mtich reduced in the i = I+- 1 case when combarid to i = 0 + 0.

Y R Alexander/Pohization in elastic scatter@ 235

LR Potential SR Potentiot The above discussion applies only to quenching

arising from different ml + m, rotationally elastic transitions. In experiments where one detects all products irrespective of their rotational state, quenching can also occur because of interference between elastic and rotationally inelastic processes. McGuire [44] has recently discussed the latter effect in Ar-N, collisions.

7. Conclusion

Scattering Angle Scattering Angle

Fig. 4. Differentiali = 2 -f 2 Ar-N2 cross sections. In each panel the upper curve corresponds to the unpolarized result [eq. (711; the middle curve, to the m = 0 + 0 transition; and the lower curve. to the m = 2 -+ 2 transition. For clarity, the top curve has been multiplied by a scale factor of 10; the bottom curve, by a scale factor of 0.1. The cross sections were determined at 0.5” intervals for 0.5” d B 4 20”, and at lo intervals for.9 z 21°.

The results of the present study of orientation effects in rotationally elastic Ar-N2 collisions may be summarized as follows:

In the present section we explicitly consider the effect of mrstate quenching by comparing the differential cross sections corresponding to the fully state- selected j = 2, ml =O+m, =Oandml =2+m2=2 transitions with the unpolarized cross section defmed in eq. (7). This comparison appears in fig. 4. Since the oscillatory structure of the unpolarized cross section is so similar to that of the fully state-selected cases,

it is clear that both the position of the primary and secondary rainbows as well as the amplitude, period, and phase of the fast oscillations are virtually independent of the initial and fmal nz,-state. There is, however, a slight quenching in the amplitude of the fast oscillations at f3 = 50” for the LR potential as well as the small differences in the structure at the joint between the primary and secondary rainbows.

Also, in the case of the LR potential there is a slight variation in the curvature of the primary rainbow.

(i) For an initially isotropic beam the presence of a significant long-range anisotropy can result in a substantial polarizatioh of the elastically scattered products. Part of this effect arises from inelastic collisions, which selectively deplete the initial nz,-state distribution. This could be indicative of a coupling between rotational relaxation (rl processes) and otien- tation relaxztion (t2 processes) in other systems.

(ii) Cross sections for collisional reorientation are much smaller than those for purely elastic (j, = j2, ml = m?) scattering, but comparable in magnitude to those for rotationally inelastic scattering. As a function of scattering angle, the probability for collisional reorientation vanishes in the forward direction and

is small at all angles, if quantization refers to either the initial relative velocity vector or to an axis perpendicular to the scattering plane. Substantial mj depolarization does occur for helicity frame quantization.

(iii) It appears, at least for the Ar-N2 system with the two potential surfaces used here, that interference between different ml + m2 transitions does not quench the oscillaloly structure of the purely elastic differential cross section.

For both potentials the rotationally inelastic Obviously, much additional work needs to be scattering appears to be dominated by the spherically done before we will possess a sufficiently general theo- symmetric part of the potential. Any reorientation retitil understanding of reorientation in rotationally effects are small and do not significantly affect the elastic scattering. Specifically, it is necessary to features of the differential cross section. Nerlertheless, determine to what extent the qualitative effects in systems with a more pronounced degree of aniso- uncovered here change as a function of the collision tropy, quenching of the oscillatory pattern could energy. Likewise, it would be informative to invest-’ occur, which would explain the results obtained by McGuire 1431 for Li’-HZ.

igate other atom-molecule or molecule-molecule systems, especially those where the intermolecular

236 M.H. Alexander/Pohizdtion in elastic smttpring

forces are stro&ly anisotrbpic at long range. Since close-cotiplhig calctdations are expensive, particularly for moiecule-molecuie collisions, it is important to develop approximation techniques which can deal accurately ‘&th the m,-dependence of differential and integral cross sections.

We-expect that orientation effects in elastic and inelastic collisions will receive increasing attention in the future as both experiment and theory become better able to resolve these fie details of molecular encounters.

Acknowledgement

The author is grateful to L. Monchick, D. Kouri and D. Fitz for preprints of their work prior to publication and to J. Reuss for a copy of ref. [ 1 I] -

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polarization in elastic scattering: close-coupling studies on arn2

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