1164
J. Phys. Chem. 1982,86,1164-1174
Kinetk Cross Sectlons in the Infinite Order Sudden Approximation+ R. F. Snlder' and D. A. Coombet Department of Chemlstty, Unlversky of Erltlsh Columbia, Vancouver, Canada V6T IY6 (Ffecelved: August 18, 1981; I n Flnal F m : December 2, 198 1)
The translational-internal coupling scheme allows a factorization of energy-dependent and kinetic cross sections into internal and translational parts. WKB-IOS phase shifts for the N2-Ar potential of Kistemaker and deVries are used to calculate the translational factors. A T-matrix version of the theory based on calculating only downward collisions is used as an attempt to lessen the effect of ignoring energy inelastic effects in the phase shift computation. All kinetic cross sections associated with the viscosity Senftleben-Beenakker effect are estimated in this manner.
Introduction The computational efficiency of the infinite order sudden (10s)approximation1*2has made it a popular method for calculating collision cross sections. I t has also been s h o ~ to n be ~ reasonably ~ ~ accurate when compared to close coupling calculations. It is with this success in mind that it is reasonable to use the 10s approximation to answer certain questions about kinetic theory cross sections, especially those required to describe the Senftleben-Beenakker effects. Previously, only the distorted wave Born (DWB) approximation has been used"7 to give some guidance to the kinetic theory modeling of this phenomena. Specifically of interest is the j dependence of relaxation and production cross sections. Other work in this areag has been aimed at the evaluation of certain average cross sections. The aim of the present work is twofold. Firstly is the shorter ranged goal of demonstrating the efficiency of using the translational-internal coupling scheme in the computation of collision cross sections. Since close coupling computations are more efficiently carried out in the total-J coupling scheme, this has been usedlo as an argument against use of the translational-internal coupling scheme in the 10sapproximation. But as long as the object is to calculate cross sections themselves rather than intermediate results, it seems to the authors that one should use the coupling scheme that is most efficient for the approximation method being employed. This, in the authors' view, involves using the translational-internal coupling scheme for 10s computations. The longer ranged aim of the present work is to obtain a better understanding of kinetic cross sections than have been obtained so far by using the DWB approximation. Only preliminary computations on the Ar-N2 system are reported here, using the atom-atom potential of Kistemaker and devries" and with its Legendre function (P2) expansion. For simplicity, thermal averages over a Maxwellian are approximated by assuming the energy-dependent cross sections to be velocity independent. Comparison with experiment is encouraging, and indicates relatively good agreement for relaxation cross sections, but points out the problem of assessing energetically inelastic collisions. For the production cross sections, which are completely dependent on energetically inelastic processes, *Computer Modelling Group, 3512 - 33 Street N.W., Calgary, Alberta, Canada T2L 2A6. 'We dedicate this work to Professor J. 0. Hirschfelder on the occasion of his 70th birthday. R. F. Snider has had the opportunity to enjoy the Hirschfelder hospitality, which is well-known for its academic stimulation and for its personal warmth. 0022-3654/82/2086-? 164$O?.25/O
a variety of methods of treating the energy inelasticity leads to a large spread of results, but all within an order of magnitude of experiment. Finally, all methods of treating the energy inelasticity seem to indicate the same qualitative behavior of the j dependence of the production cross sections, which corresponds to neither12J3of the popular kinetic theory modeling procedures. Clearly, the 10sapproximation has again in part succeeded, but there is a definite need to obtain a better understanding about how to treat energetically inelastic collision processes. A general review of the sudden approximations has been given by K 0 ~ i . lThe ~ 10s approximation is, in fact, the combination of two sudden approximations. The first of these is the energy sudden (ES) approximation which assumes, for the purpose of estimating the collision dynamics, that all rotational states have the same energy. This leads to great computational simplicity but causes problems when trying to use these dynamics in the estimation of energetically inelastic collision cross sections. The second basic approximation is the centrifugal sudden (CS) approximation, derived independently by Pack15 and by McGuire and Kouri.16 In this approximation, the centrifugal potential is assumed to be governed by a constant orbital angular momentum (quantum number A). A combination of both approximations is extremely efficient for computations, reducing the dynamical problem to the evaluation of a set of angle-dependent phase shifts. But if the orbital angular momentum changes in the collision, (1) T. P. Tsien and R. T. Pack, Chem. Phys. Lett., 6,54 (1970);8,579 (1971). (2) R. T. Pack, Chem. Phys. Lett., 14,393 (1972). (3) R. T. Pack, J. Chem. Phys., 62, 3143 (1975). (4) V. Khare, D. E. Fitz, and D. J. Kouri, J. Chem. Phys., 73, 2802 (1980). (5) F. M. Chen, H. Moraal, and R. F. Snider, J . Chem. Phys., 57,542 (1972). (6) R. F. Snider. Phvsica. 78.387 (1974). (7) R. F. Snider,'J. R.Coope, and B. C. Sanctuary, Physica, 103A, 379 (1980). (8)J. J. M. Beenakker, J. A. R. Coope, and R. F. Snider, Phys. Reu. A , 4, 788 (1971). (9) See, e.g., W. K. Liu, F. R. McCourt, D. E. Fitz, and D. J. Kouri, J. Chem. Phys., 71, 415 (1979). (10) W. K. Liu and F. R. McCourt, J. Chem. Phys., 71, 3750 (1971). (11) P. G. Kistemaker and A. E. deVries, Chern. Phys., 7,371 (1975). (12) J. A. R. Coope and R. F. Snider, J. Chern. Phys., 56, 2049, 2056 (1972). (13) F. Baas, J. N. Breunese, H. F. P. Knaap, and J. J. M. Beenakker, Physica, 88A, 1 (1977). (14) D. J. Kouri in 'Atom-Molecule Collision Theory: A Guide for the Experimentalist", R. B. Bernstein, Ed., Plenum, New York, 1979. (15) R. T. Pack, J. Chem. Phys., 60, 633 (1974). (16) P. McGuire and D. J. Kouri, J . Chem. Phys., 60, 2488 (1974).
A.
0 1982 American Chemical Society
10s Approximation for Calculating Kinetic Cross Sections
The Journal of Physical Chemlstry, Vol. 86, No. 7, 1982 1165
TABLE I: Nz
Figwe 1. Parametersdescribtng the relative orientation of the Ar atom with respect to the N2 molecule.
it is necessary to make an assignment of the CS decoupling parameter A. Initial, final, and average parameterizations have been ~ o n s i d e r e d . ~ J ~ Previous work1’J8 by the present authors have stressed the initial parameterization. This was in large measure motivated by consequent simplifications that could be obtained by performing certain sums over angular momentum quantum numbers. Recently, however, it has been disco~ered’~ that the position and momentum directional representations of an angular momentum state must satisfy the condition (in reasonably standard notation) (PI@) = C(PlXS)(XSI@) = CYXB(P)WYXB@)* (1)
Molecular Parameters
Kistemaker and deVries potential: A = 41.97 K a, = 0.20 B = 59.82 K b , = 0.61 D = 139.5 K
p = 1.51 A R , = 3.94A d = 0.541 A characteristic rotational temperature for N, : ha/21k, = 2.89 K thermal wavenumber at 300 K : K = 14.27 A - l
mentum quantum number X governing the centrifugal potential energy. Rather than solve the radial Schriidinger equation, it is sufficient to use the WKB approximation
for the phase shift. Here the impact parameter b is identified as (A 1 / 2 ) / ~ while the reduced mass is p , the potential is V(R,e),and F~ is the largest classical turning This means that there must be a different phase factor point. For each phase shift calculated, rc was found by a relating (PIXS) to the spherical harmonic YXB(?), than that simple Newton-Raphson root search and then an eightwhich relates (PlXS) to Y&). On accounting for this point Gauss-Mehler quadrature of the integral was perphase difference, the 10s differential cross section is in formed, following C o h e n ’ ~method ~ ~ with a = -1/2 almost perfect agreementmwith close coupled calculations, The computations reported here are for the atom-atom at least for the j = 0 to j = 1transition in He40 collisions. Morse potential, Vu, of Kistemaker and deVries,ll namely Such a phase difference had not been realized in our previous work on the sudden a p p r o x i m a t i o n ~ , ~and ~ J ~ * ~ ~vu(r,e) ~ ~ ~ = 4exp[-2P(R1 - R o ) ~ + exp[-28(R2 - Rd1J also influences the interpretation of some of our work on 2Bbp[-P(R1 - Roll + exp[-B(Rz - RO)ll (3) kinetic theory cross sections, specifically that which inwith R1 and R2 being the distances between the Ar and the volves orbital angular momentum expansions. The results two nitrogen atoms presented here take into account these changes. In general, no simplification is obtained by choosing the initial or final R 1 2 ~ = ( R ~ d c o s e ) ’ + d Z ( 1 - c o s Z e ) (4) R,Z CS decoupling parameterization, so that Secrest’s average parameterization has been generally adopted because of associated with a center of mass distance R, nitrogen its symmetry between initial and final states. But for the separation 2 4 and angle 8, see Figure 1. A truncated specific production cross sections being calculated here, version, V , is also considered it was also noticed that an initial parameterization would allow significant simplification. Thus the results of estiVAp(R,e)= D[1 + bzP2(cose)] exp[-2P(R - RO)lmating these cross sections with both the initial and av2D[1 + azP2(cose)] exp[-B(R - R,)] (5) erage parameterizations are presented. The potential parameters used are listed in Table I, being those given by Kistemaker and deVries. WKB-IOS Dynamics Inherently because N2 is a homonuclear diatomic, the The 10s approximation for atom-rigid rotor collisions potential and the 10s phase shift are even functions of cos assume^^^*^^ that all directions of the atomic framework 8. A representation of the angle dependence of ?(cos e) remain unchanged during the collision process. Specifically can thus be obtained by expanding in even functions of then, the orientation of the rigid rotor and the unit vector cos 0. Here the Chebyshev polynomials are used as the associated with the relative position of the center of mass set of basis functions, so that ?(cos e) is expressed as of rotor and atom are both fixed. Any rotationally inv X r ( e) ~ = ~ ~ A + BT,(COS e) + CT,(~OS e) + DT~(COS e) variant quantity must then depend on orientations only (6) through the scalar product of these vectors or, equivalently, on the cosine of the angle 6 between the vectors, see Figure To obtain this fit, the phase shifts were needed at only four 1. In particular, the phase shift ?(cos 0) and the S matrix angles. The expansion parameters A , B , C, and D are S(0) = exp[2iq(cos e)] are angle dependent. For the calplotted in Figure 2 for the atom-atom potential as a culation of the phase shift (and S matrix), it is necessary function of A. It is seen that C and D are almost negligible to specify a wave number K and “orbital” angular moso that the phase shift has really a very simple angle dependence. This would further indicate that the Pz potential, eq 5, should have properties very similar to the full (17)D.A. Coombe and R. F. Snider, J. Chem. Phys., 71,4284(1979). atom-atom potential, which is borne out reasonably well (18)D.A. Coombe and R. F. Snider, J. Chem. Phys., 72,2445(1980). in practise. (19)R. F. Snider, J. Chem. Phys., in press. (20)R. F. Snider. J. Chem. Phvs.. in Dress. Exponentiation of the angle-dependent phase shift gives (21)R. F. Snider, D. A. Coombe; and M. G. Parvatiyar, J. Chem. an angle-dependent S matrix whose matrix elements can Phys., 74,1750 (1981). be obtained through angle integration. Use of rotational (22)R. F. Snider and M. G. Parvatiyar, J. Chem. Phys., 74, 5572 XB
.
XB
+
(1981). -. - -,.
(23)D.Secrest, J. Chem. Phys., 62, 710 (1975). (24)L. W.Hunter, J . Chem. Phys., 62, 2855 (1975).
(25)J. S. Cohen, J. Chem. Phys., 68, 1841 (1978).
The Journal of Physical,Chemistry, Vol. 86, No. 7, 1982
1166
Snider and Coombe
01
0
71 0
I
12
I
24
1
36
I
I
48
60
x
I
72
I
84
I
96
I
108
1
120
Figure 2. Chebyshev polynomial expansion coefficients of ?(cos 6 ) as a function of orbital angular momentum A.
invariance simplifies the details of the computation both of the S matrix elements and of the resulting cross sections. For this purpose it is natural to expand Sxll(0)in terms of Legendre functions
sxl.(e)= exp[2i.rlkK(cos e)] = C(2L + i)1/2SLkKPL(cos e) L
(7)
with rotationally invariant expansion coefficients
SLx”= (2L + 1)1/211PL(x)exp[2i.rlk,(x)]dx 0
? ( E ; l q j j ~l’q’j’j’lk = cT(E;lqjjll’q’j’j‘) k (2q t 1)(2q’ + 1)]17(2l + 1)(21’ + l)]”’ ‘jj‘[ n(kqq’) R (kll’)
X
(8)
In the last formula, the evenness of .rlxr(x) in x is taken into account to restrict the integration to positive x values. Both 34- and 48-point Gauss-Chebyshev quadratures of the [0,1] integration were used to obtain the SLh, yielding negligible differences. Because of the evenness in x , only even L values can be nonzero. The expansion parameter L has the physical significance of being the magnitude of the angular momentum transfer of the translation-internal coupling scheme,” that is, it gives the amount of angular momentum which is transferred between the translational and the internal state degrees of freedom.
Energy-Dependent Cross Sections For atom-rigid rotor collision processes at a fixed relative energy E (independent of the center of mass momentum), the set of collision integrals1s126C’(E;lqjjll’q’j’j?k is sufficient for the description of most relaxation and e ~ ~ki-~ ~ transport phenomena. Here 1 and q d e ~ i g n a t the netic theory tensorial weightings of the relative velocity and (molecular) angular momentum directions after the collision with the rigid rotor ending up in rotational state j . l‘, q‘, and j ’ denote the corresponding initial state parameterization. The index k describes the amount of tensorial coupling5 between the kinetic theory velocity directional weightings and the angular momentum directional weightings. Thus, necessarily, kqq ’and kll’ must each satisfy the triangular inequality of angular momenD. A. Coombe, R. F. Snider, and B. C. Sanctuary, J. Chem. Phys., 63,3015(1975). Equation 8.12 has two typographical errors, the second s u m should be over Il and Tl (I1 is repeated), and just after this second sum there is a which should not appear. (26)
tum coupling theory. For practical purposes, the further restriction will be imposed that q + q’ must be even. Essentially this restricts the discussion to kinetic phenomena in which collisions preserve time reversal symmetry, which is sufficient for the present purposes. It is the purpose of the present section to give explicit 10s approximate formulas for _these collision integrals. An explicit formula for C’in terms of the reduced 10s S matrix of eq 8 has been given; see eq 23 and 27 of ref 18. This has to be modified to account for the phase difference as discussed in the Introduction. Since the 10s approximation essentially treats all j states as being degenerate,’v2J7the distinction between energetically elastic and inelastic processes is eliminated. In particular, certain conservation rules obeyed by the exact cross section are violated in the 10%approximation. As well, the dependence of S and of C‘on wavenumber K is incorrect, especially for those j transitions exciting states to near threshold. One way to minimize this effect is to consider (with thermally averaged cross sections in mind) only downward 0’ decreasing) collision^^^*^^ and associate the wavenumber K with the highest j state. An adiabaticity factor as suggested by DePristo et al.28 has not been included in the present computations. To carry out such a computation, requires expressing C’ in terms of transition cross sections, equivalently, to give a T matrix version for C’. The exact expression for C’ in terms of translational-internal reduced T matrices is discussed in the Appendix. With the restriction that q + q’be even, the result can be written as
C T ( E ; O O ~ i k k j j ) k( 9 )
This represents the gain into state j from state j ’, and total loss, for the diagonal element j = j’, from state j . The transition cross section C?‘(...) is generally given by eq A3. The notation used follows that of ref 18. Besides the n - j symbols, Q(kl1’) is a normalization factor for 3 - j tensors.29 In the ES approximation, the j dependence of the transition matrix is determined solely by a 3 - j symbol. Consequently, the cross section CT becomes a sum of factored quantities c ~ s ( E ; l q j j l l ’ q ’ j ’ j= ’ ) (~ i ) ~ - ~ t ’I)(%’ ( a + 1) x
[ ( 2 +~ 1)(2Z
i1
) ] 1 ’ 4 ~ , T s ( k ’ , ~ ~ f(10) ~i;~)~
The “dynamical cross section” C&(k’,11 is the T matrix analogue of eq 22 of ref 18. After modification to account for the phase differences associated with whether an orientation refers to a position or a momentum (see the discussion in the Introduction), this is given by eq A5 of the Appendix. In the computations performed here, k’is (27)R. Ramaswamy, A. E. DePristo, and H. Rabitz, Chem. Phys. Lett., 61,495 (1979). (28)A. E.DePristo, S. Augustin, R. Ramaswamy, and H. Rabitz, J. Chem. Phys., 71,850 (1979). (29)J. A. R. Coope, J.Math. Phys., 11, 1591 (1970).
10s Approximation for Calculating Kinetic Cross Sections
The Journal of Physical Chemlstry, Vol. 88, No. 7, 1982 1187
TABLE 11: Basic Cross Sections for the Kistemaker-deVries Atom-Atom Potential for K = 14.27 A-l in Units of A av CS parameterization initial L CO(L) Cl(L) CALI 0 204.6 194.2 181.7 9.23 2 21.75 15.61 7.57 6.09 4 12.03 3.92 6 9.49 5.70 3.81 -0.11 8 8.52 10 8.32 0.28 -1.36 6.16 -1.61 -0.49 12 14 3.16 -1.35 0.10 16 1.17 -0.62 0.15 18 0.33 -0.20 0.07 20 0.07 -0.05 0.02 22 0.02 -0.01 0.00
TABLE 111: Basic Cross Sections for the Kistemaker-deVries P, Potential for K = 14.27 A-I in Units of A
av CS parameterization
POG) PAL) PAL) PAL) -1.50 -0.61 -0.43 -0.17 0.40 0.60 0.39 0.16 0.05 0.01 0.00
-1.49 0.39 -2.46 0.22 -0.02 0.02 0.20 0.17 0.14 0.16 0.96 -0.08 -0.07 1.24 -0.36 -0.32 0.82 -0.39 -0.33 0.35 -0.24 -0.19 0.11 -0.10 -0.07 0.03 -0.03 -0.02 0.01 -0.01 -0.01 0.00 0.00
replaced by K , the wavenumber associated with the thermal energy. Laqtly, the reduced T matrix elements are evaluated according to the 10s scheme, eq A6, with TLAK related to the S matrix elements of eq 8 according to TLAK= 8Lo - SLAK
(11)
The computational procedure that has been followed is to tabulate those qos(K;1l which are subsequently needed to compute those C’cro89 sections of interest. Two choices of centrifugal decoupling parameter A are considered, Secrest’s average A = (A A’)/2 value and an initial choice, A = A’. For certain cross sections that are investigated, the initial choice does lead to significant simplifications, see eq 13, but for the average choice, even the special cases considered require the evaluation of 6 - j symbols, see eq 12 and 14. Because of the symmetry of the Ar-N2 system, only even L appear. Parity requires that Z and l’must be either both even or both odd. With the constraint that only those cross sections will be examined that have q + q’even, it follows that the cross sections are symmetric in L and L. This has the advantage that only the symmetric (in L,L) part of C$O&;Zl ILL)k need be-calculated, equivalently, only the real part of C ; r o s ( ~ ; ~ ~ l ~ ~ ) k . For the predominant contributionsm to the viscosity Senftleben-Beenakker (or viscomagnetic) effect, the index q (and q’) is either 0 or 2. There are essentially two types of collision cross section that need be considered, the relaxation cross sections specified by k = 0, and the production cross sections in which two units of orientational anisotropy (k = 2) are transferred from the velocity distribution function to the internal state rotational density matrix. The latter also has q’ = 0 and 1’ = 2. On the basis of the 9 - j symbol appearing in eq 10, there is thus a constraint that, for the above cross sections, either L = L or = L f 2. In particular, for the production cross sections, it is necessary to evaluate only
+
I
P ( z , L ) E Re CFos(K;021~,L),= 7rK-2(5)-1’2(i)2+L-L x (2h t 1)(2h” + 1) x [(2L t 1)(22+ l)]“‘
(
e
hh”h“’
(2h”’ t l)(i)*“+h”’ Re ‘i$”K
‘“K*)(;”
i
:‘t’)x
with L = L and L = L - 2. Tables I1 and I11 list these cross sections for the atom-atom and Pz potentials of Kistemaker and de Vries using the A-average decoupling pa(30)J. Korving, Physica, 60, 27 (1970).
initial
L
CO(L) Cl(L) CALI Po(L) P d L ) 0 221.4 209.6 194.9 2 24.08 15.92 11.82 -1.42 -1.74 4 13.31 8.99 4.45 -0.78 0.52 6 12.56 3.25 -0.37 0.15 0.30 8 9.48 -0.91 -0.65 0.67 -0.23 4.85 - 1.39 -0.07 10 0.50 -0.41 1.83 -0.73 12 0.08 0.22 -0.27 0.55 -0.26 14 0.05 0.07 -0.11 0.14 -0.07 16 0.02 0.02 -0.04 0.03 -0.02 18 0.01 0.00 -0.01 0.01 -0.00 20 0.00 0.00 0.00
Po(L) PAL) 0.05 -2.33 0.49 ’ 0.45 1.49 -0.20 1.22 -0.63 0.57 -0.49 0.19 -0.22 0.05 -0.07 0.01 -0.02 0.00 0.00 0.00
0.00
+
rameters A” = (A + A”)/2 and A”’ = (A A’”)/,. For the A-initial choice (A’’ = A”, A’’’ = A’”), the A summation may be explicitly performed to give
z (2h” t 1)(2h”‘ t l)(i)h”-*“’
(”0’ i ;‘“y
h“h“’
R~
&“K
(
L
h”’K*)
(13)
TL
These are also listed in Tables I1 and 111. This has exactly the same form as found earlier (eq 48 of ref 18) except for the phase factor (i)x”-x”’arising from the distinction between position and momentum orientations. For gas mixtures, the ordinary viscosity coefficient requires q = q ‘ = k = 0 with both 1’= 1 = 1 and Z’= 1 = 2 contributing. Internal state relaxation processes involve 1 = 1’ = It = 0. The spin-rotational mechanism of nuclear magnetic relaxation involves31q’ = q = 1,while the dipolar mechanism3I and depolarized Rayleigh s ~ a t t e r i n ginvolves ~~ q’ = q = 2. Ordinary rotational relaxation (internal energy relaxation) requires q’ = q = 0. In summary then, for the relaxation processes mentioned it is sufficient to tabulate
(2h” t 1)(2h”’
0
0
(14)
for 1 = 0, 1, and 2. These are given in Tables I1 and I11 and were obtained by using Secrest’s A-average decoupling parameterization [A’’ = (A A”)/2, A”’ = (A’ A”’)/2], While there is no simplification of these formulas if the A-initial centrifugal sudden decoupling parameterization is used instead, it is worthy to note that previous A-initial calculations, ignoring the position vs. momentum orientational phase differences, gave almost identical results. I t follows that kinetic relaxation processes are little affected by the choice of CS decoupling parameter. For the A-initial decoupling parameterization, unitarity of the S matrix requires that
+
+
The right-hand side of this expression is evaluated as (31)F. M.Chen and R. F. Snider, J. Chem. Phys., 48,3185 (1968). (32)R. G.Gordon, J. Chem. Phys., 44,3083 (1966).
1188
Snider and Coombe
The Journal of Physicel Chemlstry, Vol. 86, No. 7, 1982
TABLE IV: Internal State Weighting Factors relaxation type
qt
particle conservation internal energy angular momentum vector angular momentum alignment direction
00 01 10
angular momentum alignment
20
kinetic theory terminology
Scoooo~
At’Q’O.)
1
$0
j ( j t 1)[4jG + 1)- 31 l ) [ 4 j ( j 1)- 31)
S(0200)
+
Q(j’
+
property of the 10s approximation.
X FIgw 9. Ccmtr<kns to the basic aoss sections from different orbital angular momentum X.
CIos(0) and agrees with the left-hand sum to 3 in lo4. Contributions from different partial waves to CIos(0) and CIos(L) = C&) for L = 2,4,6, and 8 are plotted in Figure 3 for the atom-atom potential. While this plot is for the A-initial parameterization, the plot using the X-average parameterization is almost identical. The sums in eq 15 are not required to be equal for the X-average parameterization, yet the sums were found to be equal within 4 in 103, for both atom-atom and Pz potentials. The vanishing of all basic cross sections at A = 67 corresponds to the vanishing of the Chebyshev expansion coefficient B as shown in Figure 2. This simply reflects the angle independence of ?(cos 0) for X = 67 as B goes from negative (effectively a repulsive anisotropy) to positive (attractive anisotropy) with increasing orbital angular momentum. The vanishing of Co(L)contributions at smaller X values arise from the same mechanism33 as that which causes rotational rainbows in differential cross sections, namely, on those interferences between the scattering occurring at different molecular orientations. The quantity Co(L)for L # 0 is also related to the degeneracy averaged cross section
Relaxation Cross Sections Those cross sections for which there is no coupling of velocity and angular momentum directions (technically, those for which k = 0) are referred696to as relaxation cross sections. In this paper, only very specific relaxation cross sections are considered and these either have no velocity weighting, 1 = 1’ = 0, so deal exclusively with relaxation of the internal state degrees of freedom, or are purely velocity relaxation cross sections having q = q’ = 0. In kinetic theory, there is ultimately a thermal average over velocities and rotational states. While it is sometimes c ~ n v e n i e n t ~to J ~leave * ~ ~ the j dependence explicit, the velocity averaging always needs to be done. Such kinetic cross sections are d e n ~ t e d ~in* the ~ ’ literature by a capital German S, but here, for printing reasons, by S and, for relative coordinate cross sections, are obtained by a symmetric averaging over initial y’and final y reduced relative velocities [y2 = (E - Ej)/kBTwith rotational energy Ej, Boltzmann’s constant kg, and absolute temperature T] according to18g26 Prel (Iq njj 11 ’q ’n’j’j3 =
Here Yinl(y)denote certain weight functions of the magnitude of y and y and y’are related by energy conservation. For no velocity weightings, n = 1 = n’ = 1’ = 0, Roo is the square root of a Boltzmann factor %X(,)
= 2 ( W 4 exp(-llzy2)
(18)
Thus purely internal state relaxation cross sections are given by S’(jjlqjj’j? -2Jm
S’(OqOjj(OqOj’j?, =
exp[-’/(y2 + y’2)]e’(E;Oqjj10qj’j30y’3 dy’ = -exp[(Ej - E~)/2kBT]e’(E;Og.jjlOqj’j?o(19)
Here energy conservation has been used to get the Boltzmann factor involving j states and the energy-dependent cross section C has been assumed to be velocity independent. It is this latter approximation that is used in this paper to make estimates of kinetic cross sections from phase shifts calculated at only the thermal wave number. We expect such an approximation to be not bad, but in need of further exploration. a(L 0) = (2L + 1)-1Co(L) = (2L + l)-’qo~(K,oo~LL)o Averaging over j states can be accomplished in a number of ways. Here only certain simple weightings are consid(16) ered. Because we have chosen5 to keep cross sections for the deexcitation of rotational state L to 0. That all symmetric between initial and final states, it is natural to internal state relaxation processes are determined by this have an overall ( p j ~ f ) ’ / weighting ~ factor, where p j = ( 2 j set of collision cross sections is a well-knownNfactorization
-
(33)R.Schinke, Chern. Phys., 34,6.5,(1978). (34)R.Goldflam, S.Green, and D. J. Kouri, J.Chern. Phys., 67,4149 (1977).
(35) J. A. R. Coope and R. F. Snider, J.Chern. Phys., 57,4266(1972). (36)L.J. F.Hermans, A. Schutte,H. F. P. Knaap, and J. J. M. Beenakker, Physica, 46,491 (1970). (37)F. R. McCourt and H. Moraal, Chern. Phys. Lett., 9,39 (1971).
10s Approximation for Calculating Kinetic Cross Sections
The Journal of Physical Chemistry, Vol. 86, No. 7, 1982 1169
TABLE V: Rotational Relaxation Factors W ( q t ; L )for Degenerate (DG) and Energy-Lowering (EL) Applications of the 1 0 s ADDrOXimatiOn for N, at 300 K
W(40 ;L)
W(10;L) DG
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
0 0.014 0.047 0.096 0.159 0.233 0.313 0.397 0.481 0.562 0.637 0.705 0.765 0.816 0.859
0
0 0
0 0.044 0.115 0.187 0.246 0.285 0.303 0.301 0.285 0.258 0.225 0.190 0.156 0.124 0.096
0.043 0.173 0.270 0.426 0.586 0.733 0.854 0.939 0.987 1.000 0.986 0.953 0.912 0.871
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0.029 0.093 0.188 0.305 0.436 0.572 0.703 0.822 0.923 1.005 1.065 1.105 1.128 1.135
W(2O;L)
EL
DG
EL
DG
EL
0.025 0.073 0.133 0.194 0.249 0.292 0.320 0.333 0.331 0.316 0.291 0.260 0.225 0.190
0 0.150 0.346 0.531 0.683 0.796 0.871 0.917 0.941 0.952 0.954 0.955 0.954 0.955 0.958
0 0.130 0.270 0.366 0.406 0.397 0.356 0.300 0.240 0.184 0.137 0.099 0.070 0.049 0.033
0 0.036 0.115 0.229 0.365 0.511 0.652 0.777 0.879 0.954 1.002 1.025 1.029 1.019 1.003
0 0.041 0.113 0.194 0.269 0.330 0.371 0.392 0.393 0.378 0.350 0.314 0.274 0.232 0.191
0
TABLE VI: Relaxation Cross Sections (A’) atom-atom potential
kinetic theory terminology
relaxation type internal energy angular momentum directions angular momentum alignment directions angular momentum alignment alignment lifetime relative velocity relative velocity alignment viscosity
s (0001)
22.7 17.1 34.9 19.9 28.1 70.1 122.8 54.9
s (0100) 8 (0;OO) S-(0200) (02) ,,1(1000)
,,1(2000)
20001N1>
P, potential
10s-DG 1 0 s - E L 10s-DG 1 0 s - E L 11.8 9.8 19.8 13.3 17.9 44.3 88.6 36.5
15.2 11.0 27.6 13.1 19.8 72.0 125.1 56.1
9.4 7.1 17.9 10.0 14.7 51.3 98.0 41.5
experiment
13.9
2
2.4
35 2 3 (24 2 2) (24 ?r 2) (pure N,) 35
(2oooNi N,-Ar
+
1) exp(-Ej/kBT)Q-’ is the thermal population of the (degenerate) j level (Q is the rotational partition function). Further j dependence of the weighting factors is phenomenon dependent. Such cross sections are calculated according to the formula S(0qOt) = C (Pjpy)’/2At(Q)O‘)At(Q)O’?s’(.jjlqV ’j9 (20) ij
Five particular cases are considered, see Table IV. For = 1. Such cross particle conservation, q = t = 0 and A$Q)(.j) sections vanish exactly but for the 10sapproximation, not necessarily. It depends on how the approximation is applied, see Table V an_d later discussion. For q = 2, there are two weightings, 0200 designating that only the angular momentum directions are relaxing, while 0200 designates that the relaxation of the angular momentum magnitudes are also being considered. Two methods of applying the 10sapproximation to the estimation of O’(.jjlqp’j? are employed here. The first is to literally consider all j states as being degenerate (DG) as far as any selection of states or velocity dependence. Specifically, the Boltzmann factors in eq 19 are ignored and eq 9,10, and 14 literally applied. The result is a sum over the basic cross sections Co(L)
~DGO’&V??
CWDG(~,LL~L~?C&) (21) L
with weight factor W D c i ( q , L , j , j ’ ) =6 j j ‘ A ( q , j , j ) -
+
1)X
A denotes the triangle condition of angular momentum coupling theory. Averages over j as in eq 20, namely
W D G ( ~ ~ ; L )C (pjpj.)1/2At(q)O’)At(Q)O’ ’) WDG(Q,LLJ’J’ 9 (23) ii
with j summed to 100, are presented in Table V. These quantities depend only on the rotational constant of the rigid rotor (N,) and the temperature, besides the obvious dependence on the parameters q, t , and L. What is to be stressed is that these rotational factors W D G ( q t ; L ) are potential independent. It is noted that the particle conservation cross sections, q = t = 0, are all positive and nonzero, thus being inconsistent with exact theory. Essentially this means that the gain and loss parts of the kinetic cross section are being evaluated in different ways. It was because of this inconsistency that the second method of applying the 10s approximation was considered. To complete the presentation of the results of this literal method of interpreting the 10s approximation, Table VI gives the sums ~ D G ( O Q O ~=)
CWDG(qt;L)C&) L
(24)
for both the atom-atom and P2 potential. The other method of applying the 10s approximation to the estimation of collision cross sections is to calculate explicitly only energy lowering cross section^,^'^^^ that is, the final j value must always be less than or equal to the initial J value. Energy raising cross sections may be calculated by detailed balance or, more generally, by the exact, combined parity and time reversal symmetry26
s’,,l(/qnjjll’q’n’j’jj?k= 8’rel(l b’n ’j’jllqnjj)k (25) In using this method it must be recognized that the kinetic loss term (the Sjy term in eq 9 consists of a combination of energy raising and lowering contributions, each one of which must be (after calculating the velocity average) interpreted in terms of a basic cross section evaluated for the thermal wavenumber which is to be associated with the largest j state. This gives rise to extra Boltzmann factors as in eq 19. After sorting out these different con-
1170
The Journal of Physical Chemistty, Voi. 86,No. 7, 1982
tributions, the result can be written as in eq 21 with replaced by (for j’ 1 j )
Snider and Coombe
WDG
r’2)](~’)21+3~r(E;10jj)10j,j?0 dy’ N -exp[(E; -
( - 1 ) 9 ( 2 j t 1)(2j’ t l)(jO
;y{;, ’; ;]
x
eXp[(Ej - E j ’ ) / 2 k ~ T ]( 2 6 )
It is clearly seen that if the Boltzmann factors are ignored, formally that T = a,then W , reduces to W, eq 22, after using a 3 - j sum rule. An average over j states of this energy lowering form must respect the asymmetry between initial and final s t a h . Thus instead of the symmetric sum as in eq 23, the present calculation must be performed according to WEL(qt$) = C C (2 - sjj.)(PjPj.)1/2At’q’O’)At(Q)0’3WEL(qCjj3 (27) j’ jsj‘
These values are also presented in Table V. Roundoff errors of the order of have been equated to zero. It is seen, in particular, that particle conservation is satisfied, i.e., WEL(OO;L) = 0. This indicates that the gain and loss contributions to the kinetic cross section are being treated in a consistent way. Finally, sums of the form of eq 24 are listed in Table VI for both the atom-atom and P2 potential. Another combination of the internal state relaxation cross sections S(jjl2V39 is also calculated. This is obtained by treating 8(jjI2p?’) as a matrix in j states, excluding the j = 0 and/or j ’ = 0 vanishing elements, finding the inverse of the resulting submatrix and its j average, specifically 3”(02)-’ = C @jpj.)”2[s(’121.)-‘]jy
is appropriate for estimating kinetic cross sections in terms of energy-dependent cross sections. Averaging over j states are again calculated for the two methods previously described. The treatment of all j states as being degenerate leads to the estimation formula
while the energy lowering estimation formula is
involving the energy lowering loss factor
exp((Ej - E y ) / 2 k ~ T )-] 1 ( 3 2 )
Prel values are given in Table VI for 1 = 1 and I = 2, as well as the combination18
0.48465”’re1( 1000) ( 3 3 )
(28)
describing the viscous friction of N2 molecules moving through a bath of Ar.
Since the ryiprocal of a cross section is proportional to a lifetime, 8 is here referred to as the cross section for the alignment lifetime, alignment referring to the second spherical harmonic (q,= 2) of angular momentum directions. Table VI gives 5” for both potentials, calculated by the two (DG and EL) methods of applying the 10s approximation. This cross section governs3s the overall (low-frequency)depolarized Rayleigh scattering phenomenon and it has been argued39that it also governs the size of the Senftleben-Beenakker viscosity effect. Lastly, those k N 2collision cross sections contributing to the viscosity coefficient have also been estimated. Two such relative velocity cross sections are involved, namely, having 1 = 1 and 1 = 2 with k = q = 0. The results are sensitive to the manner in which the velocity averaging is performed. Here the basic cross sections are again assumed to be velocity independent when performing velocity averages as in eq 17. Only the simplest interpretation of these velocity weightings will be considered, specifically the initial y’and final y velocities are taken equal, formally equivalent to treating the j states as being degenerate. From the explicit form for the velocity weight factors Rol(y),it follows that (for j ’ 2 j )
Production Cross Sections Shear viscosity in a gas (mixture) is associated with a velocity distribution that differs from Maxwellian by a second rank tensorial anisotropy. For the SenftlebenBeenakker effect, this anisotropy in the velocity distribution is coupled to (produces) an angular momentum anisotropy via collision processes. The latter precesses in a magnetic field. It is this precessional motion of the angular momentum which is r e ~ p o n s i b l efor ~ ~the phase randomization of the angular momentum anisotropy and, through anisotropic collision processes, for the change in the viscosity coefficient in the presence of a magnetic field. The cross sections describing the coupling of velocity and angular momentum anisotropies are referred to6 as production cross sections. Specific expressions for these cross sections are to be obtained in this section, based on the 10s approximation. Since the relevant cross sections turn out to be dependent on energetically inelastic processes, a variety of ways of interpreting energy inelasticity within the 10s approximation leads to several different numerical values. From the symmetry properties of the experimental results, it is that the polarization that is predom-
(38)R. A. J. Keijser, K. D. van den Hout, M. de Groot, and H.F. P. Knaap, Physica, 76, 515 (1974). (39)R.F.Snider, H. F. P. Knaap, and J. J. M. Beenakker, Chem. Phys. Lett., 28,308 (1974).
(40) H. Hulsman, E. J. van Waasdijk, A. L. J. Burgmans, H. F. P. Knaap, and J. J. M. Beenakker, Physica, 60, 53 (1970). (41)H. HuLsman, F.G. van Kuik, K. W. Walstra, H.F. P. Knaap, and J. J. M.Beenakker, Physica, 57, 501 (1972).
j,j’= 1
10s Approxlmatlon for Calculating Klnetlc Cross Sections
The Journal of Physical Chemistry, Vol. 86, No. 7, 1982 1171
TABLE VII: Production Cross Sections ( A z ) via Average Parameterization. Experiment Is 0.98 A z atom-atom potential P, potential
st:)
4%)
method
Cd,%
DG BN
0.330 0.361 0.287 0.978 0.0957
HT MX EL
71.6 174 10.8 436 61.8
c p , 9%
8( 20 0;)
csd, 9%
278 508 101 1592 153
0.249 0.315 0.176 0.737 0.134
64.9 121 10.8 207 47.4
0.388 0.620 0.202 2.19 0.107
st:)
c p , 9%
0.287 0.471 0.124 1.32 0.135
234 349 101 702 138
TABLE VIII: Production Cross Sections (A,) via Initial Parameterization. Experiment Is 0.98 A z atom-atom potential P , potential ~
method DG BN
HT MX EL
4%)
Cd,%
0.411 0.812 -0.0819 1.91 0.375
149 172 11.0 296 89.0
4:)
c p ,%
" (OS 20)
Csdr 5%
St:)
310 391 104 866 177
0.374 0.639 0.0823 1.40 0.360
99.2 126 10.8 173 68.2
0.525 1.01 0.0578 2.48 0.420
0.716 1.49 -0.0574 4.21 0.492
inantly responsible for the viscous Senftleben-Beenakker effects in N2 and its mixtures42is purely of angular momentum type (having p = 0 and q = 2 in standard terminology). But experiment cannot tell how this polarization is distributed between the different j levels, nor whether there may be some velocity magnitude weighting associated with the angular momentum anisotropy. The latter question is not addressed here, but the j dependence of the angular momentum anisotropy is. For this purpose, the j dependent production cross sections
are to be examined. Rather than to report the values for all the j dependent cross sections, only certain moments of these are given. A deduction mades on the basis of the distorted wave Born approximation and a high temperature expansion was that
cj = s(E;02jj((2000)2(pj)-'/2
(35)
should be approximately independent of j . First and second moments of this quantity are reported in Tables VI1 and VI11 as
8
(ii):
,zjpjCj = ,zjj~(pjpj~)1'*S(E;02jj120j'j'), (36)
The latter is the standard deviation of Cj relative to ita average value. In the older literature it was the angular momentum vector J and its second rank, [J](2),analogue that was assumed to be the orientational anisotropy that was produced. According to that analysis, the relevant production integral is
with Ah2)0given by the last line in Table IV. The relative standard deviation C$y P [C(A62)O')Cj)2pj - (CA62'O')Cj~j)2]1'2(CA62)O')pjCj)-' I
J
I
(39) (42) A. L.J. Burgmans, P.G.van Ditzhuyzen, and H.F. P.Knaap, 2. Naturforsch. A, 28, 849 (1973).
~~
c p ,% 252 306 101 484 164
of this quantity is also reported in Tables VI1 and VIII. Computation of velocity weighted cross sections is affected by the motion of the center of mass. A factorla of 5ma(m, md)-l for the lab frame to relative frame motion, together with the introduction of the necessary relative velocity factors and the transformation to the T matrix form, eq 9, of the energy-dependent cross sections, leads to the exact equation
+
S(020j((2000)2= 5m,(ma + md)-1S',,1(020jl12000)2=
y'2)/2]CT(E;02jj(20j'j?2y'5dy' exp[(Ef - E j ) / 2 k ~ m Jexp(-r2)CP(E;OOj'j122jj)2y5 dy) (40)
As in writing eq 9, the choice of how the loss term is expressed reflects the choice that this is the loss from state j . When applying the 10s approximation to make an estimate of 5"(020j112000)2it is necessary to choose a scheme for specifying the orbital angular momentum and the wavenumber used for calculating the phase shifts. Here both the average and the initial orbital angular momentum parameterizations are considered, with the initial state (for the loss term) specified as written in eq 40. In either case, only energetically inelastic collisions contribute and a variety of wavenumber associations are considered. As already emphasized, the production cross sections are by nature dependent on energetically inelastic collisions. Consequently, any method that treats all j states as degenerate will give a null result. What would seem to be closest in spirit to the energy sudden approximation would be to calculate all velocity averages of the energy-dependent cross sections CT as if the j states are degenerate (so y' = y) and to obtain the energy inelasticity from the differences in the j-state Boltzmann factors in the gain and loss terms. The result of this procedure can be written in the general form
[(2L + 1)(2Z + l ) ] " ' P ( Z , L ) ~ X j ' j ( Z i+ 1)(2j' I
with X y j given in this approximation as
+ 1) x
1172
The Journal of Physlcal Chetnistty, Vol. 86, No. 7, 1982
XgG = 2[[pi’/(2j’+ 1)]1/2- kj/(2j+ I ) ] ~ / = ~] 2Q-’/’[eXp(-gy/.!2k~n - exp(-Ej/2k~?“)](42) This method is labeled DG as being the closest to treating the j states as being degenerate (DG). If the difference between the Boltzmann factors of eq 42 is expanded to give
X y y = (pj./(2j’+ I))l/’(Ej- ET)/kBT
(43)
terms then the j dependence is governed by the same @lo‘) as introduced in ref 6, namely S(02Oj112000), =
(44) On expanding the opposite way, so that the Boltzmann factor (pj/2j l)l/z comes out of the j’sum, that is
+
x,”j’= (pj/(2j + l))”’(Ej - E f ) / k B T
(45)
gives the high-temperature (HT) result. It was this form that was arrived at6,bya distorted wave Born (BN) calculation and used for previous deductions about the qualitative j dependence of the production cross sections. Rather than treating all j states as being degenerate when calculating velocity avera es, an alternative is to associate the wavenumber K in with the initial j state, initial being assigned as in eq 40. The Maxwellian used to average CT should be for the velocity associated with K. This leads to the Maxwell (MX) form of eq 41, wherein
8
x,y= 2Q-1/2(exp[-(Ei’- l/zEi)/kBT] - eXp(-Ej/zk~T)) (46) Other alternatives for associating K with y and/or y’ have also been considered.18,22In ref 22, K is associated with 7 [1/2(y2 + y’2)]1/2.Equation 7.17 of that reference leads, after assuming the CIos(~,021~L), dynamical cross section to be velocity independent, to a production cross section that is 3/2 times the above discussed high-temperature formula. A factor of 1/2 must be included in eq 7.17 since that formula is for molecule-molecule collisions, where both direct and transfer collisions contribute equally to the production cross section, see ref 22. The method of treating the energy inelasticity as given by eq 65 of ref 18 leads to production cross sections that are 2 times the high-temperature formula. In the methods presented so far, no account has been taken of the velocity magnitude integration limits. Actually, both y and y’ must be positive, so it is the velocity associated with the highest j value that goes from zero to infinity, while the other velocity has a nonzero lower limit determined by the energy inelasticity. Following the association of K with the velocity for the highest j state, as stressed by DePristo et al.= and whose better comparison with close coupling calculations has been noticed by Pack? it is this velocity which is naturally integrated from zero to infinity. After expressing the velocity associated with the lowest j state in terms of the velocity associated with the highest j state and the energy inelasticity, the energy lowering (EL) form for the production cross section can be expressed as in eq 41 with Ej - ET I x#J= (eXp(-Ej/%~n eg’ - j ? eXp[(E,kBTQ1l2 2 E j . ) / 2 k ~ q eG’- j ) i (47) Here 6 ( x ) = 0 if x < 0, and e ( x ) = 1 if x > 0, is the
Snider and Coombe
Heaviside step function. The results of the computations are summarized in Table VI1 by using Secrest’s average CS decoupling parameter, and in Table VI11 by using the initial orbital angular momentum as the CS decoupling parameter.
Discussion The angular momentum transfer magnitude L of the translational-internal coupling scheme arises naturally in the Legendre function expansion of the 10s angle-dependent S matrix. It has been shown how to use these expansion coefficients to compute kinetic cross sections, without reference to either the total-J coupling scheme, or to m-dependent S matrix elements. Essentially the whole computation is done by using rotationally invariant quantities. Moreover, the translational motion factors from the rotational degrees of freedom for each pair of L,L. values. Such properties are convenient for the computation-and storage of intermediate results, which can then be used in a variety of ways. Relaxation cross sections for the viscomagnetic effect have been estimated in two ways, see Table VI. The first treats all j states as if they were degenerate, while the second computes only j decreasing cross sections and uses symmetry to evaluate j increasing cross sections. For the former, particle conservation is not maintained. Comparison with experiment is of mixed agreement. For the atom-atom potential, a choice of degenerate or energy lowering procedures for calculating kinetic cross sections can get reasonable agreement, but the preferred choice depends on the relaxation type, see Table VI. In greater detail, the listed experimental internal energy relaxation cross section is that deduced via S = ( n f i ~ ) from -l Kistemaker and devries’ll range of rotational relaxation times, 4.1 X 10-lo-5.8 X s, at 308.4 K. The cross section S(O200)is that reported by Keijser et al.43in their study of the pressure broadening of the depolarized Rayleigh line. No attempt has been made to more accurately compare the calculated ordinary viscosity coefficient with experiment, than to compare it with the pure Nz value as given by Thijsse et a1.44in their excellent tabulation of Senftleben-Beenakker cross sections. The experimentally listed value, in Table VI, of 24 f 2 A2is the value reported by Burgmans et al.42from their analysis of the magnetic field dependence of the viscosity. Their interpretation of this quantity, which is consistent with the early literature, is that this cross section is S(0200). In contrast, deductions6 fro? the distorted wave Born approximation imply that it is S to which this value should be more closely a ~ s o c i a t e d .Neither ~~ association is excellent, if one is to believe the quantities listed in Table VI. A related question is the j dependence of the production cross sections. To get an association of S with the viscomagnetic relaxation rate requires that Cj, eq 35, be constant. What is found computationally, see Tables VI1 and VIII, is that C. is far from being a constant, having relative standard deviation greater than 60% for most methods of treating the energy inelasticity. If, on the other hand, the older association is correct and it is the [J](2) polarization rather than the WZ)[ J] polarization that is produced, then Cj should be roughly proportional to jz. The calculations reported in Tables VI1 and VI11 for the relative standard deviation C$) of Cj having this j dependence is even (much) larger than Csd. We conclude that (43) R. A. 3. Keijser, K. D. van den Hout, and H. F. P. Knaap, Physica, 76, 577 (1974). (44) B. J. Thijsse, W. A. P. Denissen, L. J. F. Hermans, H. F. P. Knaap, and J. J. Beenakker, Physica, 97A,467 (1979).
10s Approximation for Calculating Kinetic Cross Sections
treating Cj as a constant is not good but is better than polarization. The deductiona leading to assuming a [J]@) a constant Cj h equivalent to the HT calculation done here and is verified by the small percent standard deviation for this method of treating the energy inelasticity. But as an approximation to any of the more realistic treatments of the energy inelasticity, the high-temperature approximation is qualitatively wrong. The DG, BN, and MX all seem to give, for the ,A CS parameterization, fairly constant values up to j = 12 and then increase_withvarying rapidity. The numerical size of the MX S(02/20) value seems associated more with the rapid increase in larger j contributions than with its almost constant value throughout the region of the most populated j states. Further analysis of the j dependence and its implications in analyzing experimental data needs to be done. A better understanding of why the computational results are as they are would also be of great use. Our qualitative interpretation of the relative sizes of these collision processes is at a very rudimentary stage. Except for the MX calculation method, the production cross section is almost uniformly predicted to be smaller than the experimental value of 0.98 f 0.09 A2 reported by Burgmans et al.42 The large variation of results associated with different ways of interpreting the energy inelasticity and CS decoupling parameter is consistent with the cancellation between different total-J contributions found by Curtiss& and by Kouri and co-workers.4 For the atomatom potential and initial CS parameterization, the H T method even gives us the wrong sign, see Table VIII. Comparison with experiment not only depends on the validity of the 10s approximation with its attendant questions of how to treat the energy inelasticity and CS decoupling parameter, but also on what may be the correct j dependence of the production and relaxation cross sections. Acknowledgement. This work has been supported by the Natural Sciences and Engineering Research Council of Canada. The authors thank Dr. B. Shizgal for the use of his efficient WKB phase shift program. D.A.C. thanks Dr. D. W. Sabo for many useful suggestions on the computing aspects of this work.
Appendix. A T-Matrix Version of 6' For the purpose of treating energy-lowering (or j decreasing) collisions different from energy-raisingcollisions, it is necessary to separate the gain and loss contributions to the generalized collision cross sections into contributions attributable to specific j transitions. This is equivalent to writing a T-matrix version for the generalized cross section C'whose S-matrix version was first given by eq 8.12 of ref 26, and specialized to atom-diatom collisions as eq 1 of ref 18. An exact T-matrix version is obtained, which is specialized to the ES and 10s approximations. In doing so, the phase factors mentioned in the Introduction as arising from the distinction of position and momentum directional representations of an angular momentum state have been ^appropriately introduced. Having a T-matrix version of C 'has the advantage of making the j dependence of the gain and loss terms explicit, which is useful when assessing the uniformity of treating approximation schemes. Alternately, the S-matrix version of the 10s approximate translational cross sections could easily be written in terms of T matrices. But this would not bring (45)C. F. Curtiss, private communication. (46)See, for example: D. E. Fitz, D. J. Kouri, W. K. Liu, F. R. McCourt, D. Evans, and D. K. Hoffman, J.Phys. Chem., article in this
issue.
The Journal of Physical Chemistry, Vol. 86,No. 7, 1982 1173
out the role of j transitions in the loss term in an explicit manner, and so this alternative procedure is not followed here. S and on-the-energy-shell T matrices are related26by S(E),,= ,a, - T(E),@ (AI) or after reduction according to the translational internal coupling scheme4' (written here only for atom-diatom collisions) T@;L;jX) = 8;jSxxSLo - S($i;LiX) (A2)
cr
The quadratic in S expression for then becomes, on using eq A2, a sum of a quadratic and two linear terms in T. The quadratic in T contribution is designated by CT and can be written directly by comparison with eq 1of ref 18, namely CT(E;iq$ll'qfj;j')k = nhz(pt)-2(i)q+1+l'-q'($-j'+j-?(2q + 1 ) 1 / 2 ( 2 q+' l ) l ~ z ~ ( k q q ' ) - ' ~ z ~ ( k l lx' ) ~ l ~ Z
+ 1 ) [ ( 2 L+ 1)(21;+ 1)(21+ 1)(21' + 1 ) ( 2 h + ""') 1)(2h' + 1 ) ( $ + 1)(2j' + 1)1"2(0I'
1)(2h"
TO'' ' " ;L;ih')* ( A 3 ) Before proceeding to a discussion of the linear in T terms, it is remarked that the 10s form of eq A3 is obtained in a manner exactly parallel to the reduction of the S matrix version described in ref 18. Two steps can be recognized, first is the ES approximation
(i)L+
h'Tk'( h' & ; A ) (A4)
of the reduced T matrix which, for J = j and 3 = j', leads directly to eq 10 of the text and the dynamical cross section
(2h"
r,
+ 1)(2h"' + 1 ) ( 2 h +
1)'/2(2h' + 1)'iZ x
hh'h"h"'
(2Z
+ I)]"'
( - l ) h ' + h " T h ' ( h " ~ ; h ) T k ' ( h " ' & ; h ' )(A5) *
The second step is to apply the centrifugal sudden approximation
(A6)
of the ES T matrix to give the 10s approximation, where A is chosen to depend on X and X" in some manner. In general, there is no simplification of eq A5, no matter what choice of A is made. The linear in T terms are simplified because of the Kronecker deltas, to reduce 9 - j.s to 6 - j ' s and also to evaluate L , and one X sum. These obvious manipulations plus the interchange of labels X A' and X" A"' in the T* term allows the linear in T contributions to 6' to be written in the form
e,
-
-
(47)L.W.Hunter and R. F. Snider, J.Chem. Phys., 61,1151(1974).
1174
The Journal of Physical Chemlstry, Vol. 86, No. 7, 1982
Snider and Coombe
(XlW) = (wltlex)
W0)
With the natural phase choice used here,17J8the symmetry property50
{p p' ,, ( 2 h ' t 1)(2h" t 1 ) ( 2 h t hh h
l)l/ZfO
;"
A")
0
@mhs) = (-~)j+~+A+s[j - mA - s ) X
( i ) h ~ * " ( - l ) " + q T ( j h " ; k ; i h ) * ]( A 7 )
Now the A' sum can be explicitly performed by using properties of n - j symbols, specifically a special case of eq 16 of ref 18 and orthogonality properties of 3 - j symbols, so that
Unitarity of the S matrix implies the conditions (1.7) and (1.8) of ref 17. The first of these, written in terms of the reduced T matrix of eq A2, expresses the optical theorem in the form (- 1 Y'+"(jh ;L";i'h')
[ ( 2 L " + 1)(2j' + 1 ) ( 2 h ' t l)]"' (- 1 y"'h'T(j'* ;ih )* -[ ( 2 L " + 1 ) ( 2 j t 1 ) ( 2 h + 1)11'2 $1
is applicable. As a consequence, the matrix elements of T satisfy the time reversal symmetry condition (j'mlh'slmmhs) = (-1)j'+A'-j-A( j - mX - slTb'- mlh' - s') (A12) which on reduction according to the translational-internal coupling scheme, eq 1.5 of ref 17, gives the time reversal symmetry
Parity requires j + h - j ' - A' to be even in eq A12 so the phase factor appearing there actually vanishes. As consequences, both in eq A7 and-in the restriction of eq A3 to diagonal in j observables (j = j, 3' = j?,1 l'must be even is an exact parity condition. On restricting the class of collision integrals to those for which q + q'is even, the factors (-1)q = (-1)g can be taken outside the square bracket in eq A7. Because parity requires 1 + l'to be even, the factor (-1)l = (-1)l' may also be taken outside the bracket in eq A7. Thus, with j' = j , eq A9 and A13 can be combined to give the T-matrix combination appearing in eq A7. Note also that k is even by a 3 - j symbol symmetry, because 1 $ l'is even. After putting the pieces together, the part of C' which was linear in T can be written in the form
+
h
While the combination of linear terms in eq A9 is not exactly the required grouping of eq A7, for a restricted class of collision integrals, the time reversal symmetry properties of Tare sufficient to bring these combinations into coincidence. For Hamiltonians that are time reversal invariant, OHBl = H, where O is the anti-unitary time reversal operator,48 the abstract transition operator t sati~fies,4~ on the energy shell (48) E.P. Wigner, "Group Theory and Ita Application to the Quantum Mechanics of Atomic Spectra", Academic Press, New York, 1959. (49) R. F. Snider and B. C. Sanctuary, J. Chem. Phys., 55,1555 (1971). (50) A. R. Edmonds, "Angular Momentum in Quantum Mechanics", Princeton University, Princeton, 1960.
(All)
C$in = - 6 j j '
(2q t 1)(2q' t
"1"
[,21 t 1)(21' + n (kll')
"3"z
T(jh";L$)*
(A14)
which, on comparing with eq A3, can be identified as a s u m over j of particular cases of CT. The result is given in eq 9 of the text.
