Chemical Physics 20 (1977) 83-92 0 North-Holland Publishing Company

THE DECONVOLUTiON OF EXPERIMENTAL V 4 T RATE CONSTANTS: He-Hz, H,H,, AND D2-D2 *

Millard H. ALEXANDER Depnment of Chemistry. University of Maryland. Md. 20742. USA

Received 20 April 1976 Revised manuscript received 25 October 1976

We present a S-parameter hyperbolic representation for V -t T cross sections which is sufficiently ffexible to frt accurately a series of recent theoretical cross sections. This representation is used to deconvolute low-temperature V * T rate constants for Hz-H,, Hz-He, and D,-D,. The scatter in the experimental He-H2 points results in considerable uncertainty in the cross section below E = 0.04 eV. For Hz-H, there is a pronounced minimum in the V -+ T cross section at E= 0.015 eV which is absent for D2-Dz Extrapolation to higher temperatures predicts relaxation times which are significantly less than the values given by shock-tube studies, which suggests that vibrational relaxation occurs faster than the process detected by these experiments. The present results suggest that a log-log, rather than Landau-Teller, plot of experimental V--r T data may allow a more meaningful theoretical interpretation.

1. Introduction

To invert a set of thermal rate constants and obtain a energy-dependent experimental cross section is a longstanding goal of the kineticist. Melton and Gordon [l] , Bernstein and Levine [2,3], as well as Russ, Barnhill, and Woo [4] have recently discussed aspects of this problem. Some of these studies [ 1,4] have in- dicated that one can not extract detailed information about the energy dependence of the cross section from only a finite range of thermal rate constants, especially if these are contaminated by experimental error. How- ever, if the qualitative behavior of the cross section is known, then thermal rate constants can be used to determine the absolute magnitude of the cross section.

In this paper we shall present a novel hyperbolic form for V + T cross sections which is sufficiently flexible to fit accurately the results of a number of recent theoretical studies. This functional form will be used to deconvolute the experimentaI V+ T rates deter- mined by Audibert, Lukasik and Ducuing for Hz-H2

* Research Supported by the Computer Science Center, University of Maryland and by the Office of Naval Research, Contract N00014-67-0239-02.

[S] , H2-He [6,7] and D,-D2 18-J collisions. The energy dependence of the resulting cross sections will be discussed.

2. V + T rate constants

Consider the collision of a diatomic molecule in the initial vibration-rotation state u = 1, j with another diatomic molecule in the state u’ = 0,j’. The thermal rate associated with the vibrationally inelastic process u= 1,j~~=0,j”;j’_tj”‘is [9]

kli,j’~oj”, j”’ = (SP3/lr~) I/2

X s CT~~,~~__,~~~~,~~~~(E) exp (-DE) dE , 0

(1)

where 6 = I/kT, p is the collision reduced mass, and we have suppressed the vibrational index of the second molecule **_ The energy E refers to the initial transla- tional (collision) energy. If an equihbrium distribution

** This will change only for V + V processes, which will not be treated here.

of rotational levels is assumed, then the total V + T rate is

&4j = 5 (z~z~)-~(~P 0 (2; + 1)

i”X’

where Ej andzl.denote, respectively, the rotational energies and rotational partition function of the first molecule (in the vibrational level IJ = 1) and likewise for the second molecule (in IJ’ = 0). .

In a manner closely related to the work of Melton and Gordon [I]-, it is formally possible to define a rotationally summed cross section

ul-+OQ= 3 C2i+ l)(2j’+ l)fj'(E) 9

i”J”’

X ~lj,j’+_~j*s,j”*(E - E_ - of) , (3) where&(E) is a dimensionless factor given by

&(E) = 0 , E<E~++, @a)

= (E - Ej - Ejl)/E 1 E > ej + 4. . (4b)

Th_e reader may show that the total V 4 T rate (eq. 2) can be expressed in terms of or-,0 as

z1z2k,,o(T) = (SP3/rr~)“2 f oI,,&E) E em@ dE .(5) 0

The quantity E represents an effective translational energy in the u = 1 manifold, defined by eqs. (3) and (4) so that the integration in eq. (5), which is formally equivalent to that in eq. (I), will yield the rotationally averaged rate, Ic~-,~(~). It should be pointed out that eqs. (2) and (5) assume equilibrium distributions, characterized by the same temperature ?‘, for both the rotational an~translational degrees of freedom.

The extension to atom-molecule collisions is ob- vioUs;z2 is set equal to unity, Ej' is set equal to zero, and-the i and j”’ indices are suppressed. In experi- ments with n-H2 and n-D2 due account must be made of the two rotational isomers. The cross sections Ol j,j’-Clj”,l -,,c vanish unless j(~‘) and j”(j”‘) are of identical parity. Additionally, the partition functions which appear in eqs. (2) and (5) must be replaced by

weighted sums of the rotational partition functions of the ortho and para species:

zn-H, = epH 2 + 3z,-H J/4 9 (64

=n-D2 = (3zp_H, t 6zz,4@’ . (6b)

3_ Hyperbolic cross section

The goal of the present paper is the deconvolution of eq. (S), whereby experimentalSvah_res of k,,, over a range of temperature are used to construct aI_&!?). The functional form of the cross section is suggested by a recent series of fulIy quantum studies of the vibrational de-excitation of H2 and D2 in collisions with He [9-l I] as well as by the results of collinear [12] , distorted wave [ 131 and semi-classical [ 141 calculations. All this theoretical work indicates that a log-log plot of the cross section against the initial energy is linear at moderate energies (0.1 - 1 ev), corresponding to a power-law relation

er+,, (E) = A.?? , (7) with an increasing degree of positive curvature at lower energies [9] - The 1 -+ 0 subscript wilI be dropped for the remainder of the article; all cross sections will refer to the rotationally summed quantity of eq. (3) unless indicated to the contrary.

An obvious functional form for this general behavior (in a log-log representation) is a hyperbola, as sketched in fig. 1. The hyperbola is described in a coordinate slstern (x’,~‘) which is displaced, rotated, and scaled with respect to the original log u, -log E system. The explicit relation between the two coordinate systems is

x’ = h[(log E. -log E) cos B + (log u - log uO) sin 81 ,

and (8)

y’ = A[--(log Eg - log./?) sin B + (log u - log Co) cos 01 , (9)

where h is a scale factor, E,-, and 00 specify the origin of the x’, y’ system and f3 is a rotation angle.

In the x’,y’ system the hyperbola is given by y’ = [(@&‘)2 - 1] U2 . (10) After a little algebra it is possrble to show that the re-

ML?. AlexanderlDeconvolution of experimentcl V+ T rate constants 85

Fig. 1. Pictorial representation of hyperbolic cross section. The hyperbola is described in the x’, y’ system by eq. (10) of the text and in the x, y system by eq. (11). The x’, y’ axes are rotated through an angle 0 and scaled by a factor h with respect to the x, y axes. The parameters E,, and oo defme the origin of the x’,y’ system. The angle between the asymptotes of the hyperbola and the X’ axis is given by the arctangent of the parameter 0.

suiting expression for the cross section is

logo=Iogu~~(asinB+bcos8)/A,

where the plus sign is taken if E>& iO-(cosel/Aa! ,

and

(11)

(12)

a = - {XY + [C&l + X2)- Y2]1’2}/(Y2- a2) ) (13)

b = (CY2P - I)“2 . 04) Here

Y=cot8, X = X log(E/E&in 0 , (15)

and o! is the tangent of the angle between the asymp- totes of the hyperbola and the x’ axis. In terms of cr, the equation for the positive asymptote of the hyper- bola isy’ = ox’, implying that the power B in eq. (7) is given by

B = (atos 0 + sin @)/(a sin 0 - cos 0). (16) An obvious test of this S-parameter hyperbolic

representation is the degree to which it can fit the results of theoretical V + T cross section calculations. To determine this we shall use a “minimax” criterion [ 15]_ Specifically, supposeg(x) is a parameterized function chosen to represent another function f(x)

over the range a <x & b. Then the best “minimax percent” choice of the parameter values is the one which minimizes the maximum percentage deviation

max llOO[(g-_)/fll, aGx=Gb - (17) An attempt was made to fit the fully quantum

He-H2 V + T cross sections of Alexander and McGuire [9] and the distorted-wave (DW) D?-D2 V + T cross sections of Lukasik [ 13]_ Alexander and McGuire determined cross sections only for relaxa- tion of the u = 1, j = 0 level; consequently the summa- tion in eq. (3) is restricted to the indexj”. In this case the quantity E is equivalent to the initial translational energy in the u = 1, j = 0 level. Also, although the DW calculation of Lukasik did not incl.ude the rotational degrees of freedom, for the purposes of the present comparison we shall assume that his DW cross section is equivalent to uI_+~(E)_

The accuracy of the resulting fits is demonstrated graphically in fig. 2, where the precent deviation is plotted as a function of the energy. Clearly, the hyper- bolic representation is capable of quantitative accuracy over the entire range. The fits all display the equiampli- tude oscillations about the exact values which are characteristic of minimax “best approximation” polynomials [ 1.5]_ The fitting parameters for the MO SG He-H, results [9] are listed in table 1.

A second test of the appropriateness of the hyper- bolic cross sections is its ability to deconvolute a set of rate constants generated from a known cross sec-

-I0 ..‘:I 0.001 0.01 1.0

E (eV)

Fig. 2. Plot of the percentage deviation as a function of energy for the best minimax hyperbolic tits to theoretical V + T cross sections_ The solid and dashed lines refer to the MO SG and HO He-Ha cross sections of Alexander and McGuire (91, while the dotted line refers to the distorted wave Da-Da results of Lukasik [ 131.

86. MN. AlexaIlderlDeconvoIurion of experimenlal V -, T rate constants-

Table I Parameters for hyperbolic fit to theoretical He-H, V - T CJOSS sections, MO SG resultsa)

Pammetersh) Fit Ic) Fit IId)

oo (AZ) 1.035 - 8 5.98 - 8 ErJ (ev) 2.74 - 1 3.99 - 2 0 (deg) 56.57 37.86 oz 1.117 2.343

$) 1.128 1.1 3.8 3.8

a) See ref. [9] for details concerning the theoretical cross section calculations.

b) See eqr (ll)-(6) and fg. 1. c) Minimax tit to MO SG cross sections, 215 X 10e3 <E < 0.41 eV 191. As explained in section 3, E represents the initial translational energy in the v = 1,j = 0 state.

d) Minimax fit to MO SG rate constants, 60 < T < 477 K [9J _ d Exponent for high energy limit of cross section; CT - EB_

tion. For this test we chose He-H2 rates geneiated from the MO .SG cross sections of Alexander and McGuire [P] and spanning the temperature range 60 < T <477 K, which is comparable to the range of the recent stimulated Raman experiments of Audibert, Lukasilc, and Ducuing [5--8]_ The five parameters oo, Eo, 6, CY and h were varied until a best minimax tit was obtained. me integral (5) was evaluated using 7-pt Gauss-Laguerre quadrature, which proved to be sufficient for the present study. The best fit corre- sponded to a maximum deviation of <I%; the param- eters appear in table 1.

Fig. 3 displays the percentage deviations between the exact MO SG cross sections and rate cons.tants [9] and the hyperbolic values obtained by (I) directly fitting the MO SG cross sections and (II) by fitting the MO SG rates as described in the preceding para- graph. The comparison between the original and deconvoluted cross sections in fig. 3 illustrates that the temperature range of rate constants used provides re- IativeIy little information on the cross section for E 5 0.01 eV. Consequently, fit II samples the cross over a less extended energy range than fit I and, thus, can achieve a better degree of agreement not only with the higher energy cross sections but also with the rate constants. The fact that a given set of rate constants provides information on the cross section only over a limited energy range has been clearly

0.5

IO

E (eV) 0.i 0.05 091 o.oo_Q45 %)

;-- ---J ,

:_,j yyj

0.5 1.5 2.5 log E (eV)

T VW 500 300 200 100

31

log T (OK)

Fig. 3. Comparison of two tits to the He-H2 MO SG results of ref. [9] _ Plotted are the percentage deviations of the V + T cross sections and rate constants as a function of energy and temperature, respectively. The solid line, fit I, represents a direct miniiax fit to the MO SG cross sections (2.5 x 10e3 <E < 0.415 eV); while the dashed line, fit II, arises from a minimax deconvolution of the MO SC rates (60 < T < 477 K). ?he specific parameter values for fats I and II are contained in table 1.

discussed by Melton and Gordon [l] . As shown by table 1, there are significant differences between the parameters corresponding to the two fits, although, as lllustrated in fig. 2, both reproduce the cross section quite accurately forEZO.O1 eV. It is also worthwhile observing that both fits predict the same power-law exponent (eq. (7)). -

Asymptotically, the hyperbolic cross section dis- plays the power law form of eq. (7). This wiu result in a considerable degree of error at high energies, where the true V += T cross section is expected to reach a maximal value on the order of l-10 A2 and then decrease with increasing energy [3,16]. There is sort& evidence that the maxima in V + T cross sections and the onset of deviation from power law behavior coincide with the classical dynamic threshold, which: for light molecules, appears to lie at initial translational

M.H. AlexanderjDeconvolution of experimental V+ T rate constants a7

energies greater than -1 eV (12000 K) 116,173 *. Thus, the hyperbolic representation will most likely be well suited for the deconvolution of rate constant data even at temperatures of several thousand degrees.

The hyperbolic representation contains jive param- eters. It is entirely possible that one could construct a representation containing fewer parameters which would have the same qualitative energy dependence, as, for example, the four-parameter “double power- law” forms

a(E)= AEB+CED,

and

(18)

a(E)=AEB, E>EO,

=CED, E=GE,,. Wb) Unfortunately, both these forms proved to be con- siderably less accurate than the hyperbolic represen- tation in fitting the He-Hz cross sections of Alexander z&d McGuire [9]. Nevertheless, additional effort should be devoted to exploring alternative represen- tations containing fewer than five parameters.

* For example, the parameters in table 1 predict a V-B T cmss section of -0.015 AZ at E = 1 eV, which is still well below ihe expected maximum

Table 2 V+ T rates used in deconvolution

4. Deconvolution of experimental V + T rates .,-

We have used the hyperbolic representation to de- convolute the results of the low-temperature vibrational relaxation studies of Audibert, Lukasik and Duciung on the n-Ha + n-HZ [5], n-H2 f He [6,7] and n-D2 + n-D2 [S] systems. The experimental relaxation times (pi) were converted to rate constants using the relation k = (iVj~r)-~, where N is the particle density. Under ideal gas conditions, this becomes

k (cm3/s) = T/[7.339 X 102’pT(atm s)] . (33

For each system 11 rates were selected which spanned the temperature range of the experiments. The values, multiplied by the appropriate partition functions (eq. (5)), are listed in table 2.

The deconvolution of these rates to yield a V + T cros section using eq. (5) is meaningful only if an equilibrium distribution of rotational levels is established before vibrational relaxation occurs. In an important series of studies Pritchard [18,19] as well as Rabitz and Zarur [20] have used model sets of V + R,T and R + T rates to demonstrate that this criterion is clearly satisfied under the conditions which characterize the experiments of Audibert, Lukasik and Ducuing [S-S]. Also, model kinetic studies 1211 indicate that the ex- perimental relaxation rate may represent the sum or

4He + n-H, a) n-H, + n-Hzb) n-D, + n-D, C)

T(K) kd) T kd) T k“)

50.8 8.09 - 19 82.2 1.47 - 18

111.1 2.46 - 18 lSO.5 5.90 - 18 203.5 1.46 - 17 296.3 7.81 - 17 345.6 3.21 - 16 316.4 6.41 - 16 411.0 8.71 - 16 455.2 1.23 - 15 476.8 3.73 - 15

39 2.16 - 17 49 2.07 - 17 63 2.32 - 17 91 2.84 - 17

138 5.50 - 17 172 9.22 - 17 211 1.94 - 16 270 6.32 - 16 323 1.85 - 15 390 5.03 - 15 471 1.48 - 14

54.8 1.55 - 19 58.0 2.02 - 19 61.8 2.03 - 19 85.8 5.01 - 19

112.5 1.27 - 18 152.5 4.63 - 18 202.0 1.92 - i7 242.0 6.28 - 17 296.8 2.72 - 16 350.0 1.01 - 15 400.0 2.82 - 15

a) Refc [6] and [ 7]- b) Ref. [5] _ C) Ref. [8] _ d) Experimental rates in cm3/s (eq. (20)) multiplied by appropriate partition function <eq. (5)).

88 M.H. Alexander/Deconvolution of ex&imentaI V* T mte constants

Table 3 Rotationally summed cross section parameters for best minimax deconvoluti&n of %-Hz, DZ-DZ, and Hz-He V +T rates

Parameters a) 4iie+ n-% h) n-H, + II-% ‘=I wD, + n-D, d)

Qo (A*) 1.719 - 8 3.398 - 7 2.075 - 7 E, (ev) 1.272 - 1 5.500 - 1 6.998 - 1 0 (deg) 46.812 64.109 21.50

f: 11.164 0.714 4.135

0.1 1.7 1.0 maxC%) e)

.%I 36.0 6.7 10.5

9.3 5.9 7.2

a) Se& eqs (1 l&(16) and tig- 1. b)Raiesfromrefs.[6] and[7].50CTC476K. C) Rates from ref. [5], 30 c T t471 Ii. d) Rates from ref. [S] ,154 < T < 400 R. e) Resulting maximal percentage deviation, see eq. (17). n Esponent for high-energy limit behavior of cross section; D = AEB. As discussed in section 2. E represents an effective transla-

tional energy in the u = 1 manifold.

T VW 50 100 200 500 1000 3000

-9. * * ’ 4 I

-19 I I

I.5 2.0 2.5 310 3.5

!og T (“KI

Fig. 4. Best minimax tits of experimental He + n-H, [6,7], n-H, + n-H, [S] , and n-D2 + n-D2 [8,13] V-+ T rates (fdled circles). For clarity of presentation the He-% rates have been multiplied by 100. The bifurcation in tbe‘tbeoretical He -II, rates at low temperature is a measure of the variation result- ing from equaliy acceptable minimax fits. The high-temperature rates (- H,-,H,, --- D,-D,, -*- He-H,) refer to shock- tube studies [26,27] _

the difference, depending on whether one adopts a “two state” or “harmonic oscillator” model [21], of the relaxation (1 + 0) and excitation (0 + 1) sates. For the systems under consideration here simple microreversibility considerations reveal that the Hz and D2 vibrational spacings are so large that the 0 + 1 rates will be far smaller than the 1 + 0 rates at the temperatures sampled in the studies of Audibert, Lukasik and Ducuing [S-S].

-In the deconvolution procedure, the hyperbolic parameters were varied until a minimax fit of the rates in table 2 was attained. For fxed values of X and B, a relatively unsophisticated algorithm was used to ob- tain a best fit in the variables oo, E. and 0. The final two-dimensional minimization in A and B was carried out graphically in a real-time operation via terminal. Typically, for a given set of rate constants, a complete minimization required - 5 min of Univac 1108 cpu time and -1 h of connect time. Attempts at exploiting more sophisticated minimum-seeking algorithms, although not pursued at great length, were unsuccessful, due to the occurrence of multiple relative minima.

Table 3 lists the parameter values which result in. the best minimax tits. The agreement between the computed and experimental rates is demonstrated graphically in fig. 4. Examining first the results for the self-relaxation of n-H2 and n-D2, we see that the hyper- bolic cross section is fully capable of reproducing the experimental rates with quantitative accuracy. The maximum deviation associated with the best fits (6.7

MN. AIexander/Deconvolution of experimental V - T rate constants 89

and 10.5%, respectively) lies well within the reported experimental error [5,8,13].

For He-HZ, there is considerably more scatter in the experimental results, which reflects the greater difficulty of the experiment [7]. As a consequence, it is not possible to obtain as close a fit. Nevertheless, -the maximum deviation of the best fit (36%) again lies within the estimated experimental uncertainty [7,22]_ Additionally, several different choices of the parameters were found to give maximal deviations less than 1% greater than the best value (36%). As is clear from fig. 4, these tits, which must be considered to be equally acceptable, predict a significant degree of variation in the low-temperature rates.

5. Deconvoluted cross sections

Fig. 5 displays the energy dependence of the He-H,, Hz-H2 and DZ-D2 rotationally summed cross sections corresponding to the parameters of table 3. The error bars are not to be taken as rigorous bounds but rather as indications of the degree of variation in the cross

‘T 1-10-3

140-4

110-S

&- “3 1-10-E

g

110-7

1.10-e

140‘9 -I I 01 0.01 ODOI

E (eVl

Fig. 5. Deconvoluted, rotationally-averaged V + T cross sec- tions (eq. (3)) for *I+ + rrH,, 4He + rtI$ and n-D, + n-D,. The error bars represent the variation resulting from equally acceptable minimax fits

sections resulting from the use of other minimax fits whose maximum deviations were within 10% (relative) - of the best values listed in table 3. Due to the larger degree of scatter in the experimental He-H, rates, the cross section is determined with less precision than for the Hz-H, and D,-D, self-relaxation processes, es- pecially at energies below -0.04 eV. As discussed in section 2, the energy which appears as the abscissa in fig, 5 represents and effective translational energy in the u = 1 level.

The reader should note that the precision of the deconvoluted cross sections depends implicitly on our initial assumption that the qualitative behavior of the cross sections can be adequately described by the hyper- bolic representation of section 3. If no assumptions are made concerning the energy dependence of the cross sections then, as discussed by Melton and Gordon il] , the bounds on the deconvoluted V + T cross sections would be significantly larger.

Since the experimental V + T rate constants ana- lysed here are available only at temperatures greater than -40 K, the deconvoluted cross sections can not be extrapolated to infinitely low energy. From fig_ 3 it is easy to see that the deconvoluted cross sections are not expected to be accurate for energies lower than 0.005-0.01 eV.

Of particular interest is the minimum in the HZ-HZ cross section at E = 0.015 eV, which is not present for the D,D, system. From the discussion in the preceding paragraph we expect this minimum to be a real feature, not an artifact of the deconvolu- tion procedure. This is also supported by the distorted- wave Born approximation calculations of Lukasik [ 131, which predict the Hz-Hz V += T cross section to dis- play a broad minimum at E i= 0.005 eV and then rise to a maximum at E = 4 X 10~~ eV.

Within the Born-Oppenheimer approximation the Hz--HZ. and DZ-D2 interaction potentials are identi- cal; therefore an argument based only on consideration of the static force field would be unable to explain the striking difference in the low-energy cross sections. Clearly, the origin of this difference is a dynamic ef- fect, reflecting the role that the change in isotopic mass plays in the collision kinematics. The Dz-D, collision mass is 1.5 times greater than that for Hz--Hz, while, at the same time, both the rotational and vibrational spacings for D, are considerably less. Although it is reasonable to expect that the D,---D,

90 MH. Alexa~~e~/~ecoonvoZuiion of experimental V-+ T rate constants

cross section will also display a minimum 1131, it is apparent from fig. 5 that the available relaxation data (8,131 do not extend to a temperature low enough to give evidence of this feature. As discussed by Audibert et al. [23], a major test of any theoretical treatment of the self-relaxation of Hz should be its ability to predict the large difference in the low- enew behavior of the Hz-Hz and Dz-Dz cross

sections.

6. Extrapolation to high temperature

Ct wouId be most interesting to use the deconvoluted cross sections to predict relaxation rates at shock-tube temperatures (1000-3000 I()- Before doing so, one must first ascertain the accuracy of such an extrapola- tion. Errors arise from the fact that power-law depen- dence (eq. (7)) is inappropriate at high energy. If E mpr denotes the energy beyond which this relation is no longer valid, then, since ul._,o must be positive definite, one can use eq. (S) to establish the following bound

The relative error in a rate constant computed by using the hyperbolic cross section at all energies will then be bounded by

i .AE!Ee-@dE

where z,,(T) denotes the extrapolated rate constant. In this result we have explicitly introduced the high energy power-law behavior of the hyperbolic cross section.

At temperatures above -1000 K one can evaluate the integral in the denominator of eq. (22) accurately from the power-law limit of the hyperbolic cross sec-

Table 4 Percentage error in high-temperature rate constants computed using hyperbolic cross section

Bb) ACT) X 100 a)

T= 1500 K 2000 K 3000 K E max= 1 eve)

6 0.01 0.8 16 7 0.02 1.0 21 8 0.03 1.3 27 9 0.04 1.6 35

E max= 1.5 eVc)

6 0.0 0.003 0.8 7 0.0 0.004 1.0 8 0.0 0.004 1.3 9 0.0 0.005 1.6

a) I& (23) of text. b) dower law exponent; eq. (7) OF text. C) Energy beyond which power law behavior is not expected

to be valid.

tion, since the low energy deviation from power-law behavior will have an insignificant effect. The result is

A(T) G r@ + 2, N&,,,)Ir@ + 2, P&,.& > (23)

where ~(a, x) and r(Q, x) are incomplete gamma func- tions [24,25] . Table 4 illustrates the bounds to the relative error, A(T), for various values of B which are characteristic of those in table 2 and for two values of Enlax (1 .O and 1.5 eV) which are reasonable in light of recent V + T cross section calculations. The values of T span the temperature range sampled in past shock- tube studies of the vibrational relaxation of H2 and D2 [26,27] - The numbers in table 4 indicate that the extrapolation of the deconvoluted 1 + 0 cross sections from table 2 will no! introduce an unreasonable degree of error into the calculated rate constants even at tem- peratures of 3000 K.

Strictly speaking, this extrapolation is justilied only by the assumption that the V + T cross sections obey a power-law dependence at moderate energies up to E = Emax. As discussed in section 3, this asump- tion is supported by all the available accurate V + T cross section calculations_ In the absence of this assump tion then the only justifiable extrapolation procedure would be the bounded extrapolation method of Melton and Gordon [I), which is the best one can do without any knowledge of the energy dependence of the cross section.

M.H. Alexander/Deconvolution of experimental V+ T rate constants 91

From fig. 4 it is clear that if the best fit hyperbolic cross sections are used to extrapolate to higher tem- perature (1000-3000 K), then the predicted V + T rates are significantly faster than the “vibrational” relaxation rate obtained from shock-tube studies [26,27], especially in the He-Hz case. This implies that at shock-tube temperatures the vibrational relaxa- tion of the translationally and rotationally equilibrated u = 1 manifold occurs faster than the process which is actually being observed. The informative simulation studies of Pritchard [19] indicate that at high tempera- tures the slowest HZ relaxation process may be asso- ciated with multiquanta pure rotational transitions. Although Pritchard’s conclusions are possibly biased by this chosen set of model transition rates [28], his studies, along with the discrepancy between our extrapolated rates and the experimental values, strongly suggest tbat one can not extract a rate for vibrational relaxation under translational and rotational equilibrium from shock-tube experiments.

7. Conclusion

The preceding sections demonstrate that the hyper- bolic representation is fully capable both of fitting the results of theoretical calculations of V + T cross sections as well as accurately reproducing experi- mental rate constants. Although this representation appears to be extremely well adapted to vibrational relaxation, additional effort should be devoted to exploring alternative representations, particuiarly those containing fewer than five variable parameters.

Using the hyperbolic cross section to deconvolute the low-temperature He-Hz, HZ-H2 and D,-D, V + T rates of Audibert, Lukasik and Ducuing [5,8 J led to two important results. First, the HZ--H2 cross section goes through a distinct minimum, which is absent in the D, self-relaxation cross sections_ This distinction obviously reflects the dynamical impor- tance of the various reduced masses. Secondly, the high temperature V + T rate constants generated from those cross sections which best reproduce the low-temperature data are significantly larger than the values obtained from shock-tube experiments. This suggests that these experiments may be measuring a relaxation process which is not primarily vibrationaL in character, a hypothesis which is further confirmed

by the recent modelling studies of Pritchard [ 19]_ Another important observation is that a V + T cross

section characterized by a power-law behavior at moderate energies yields a high-temperature rate constant proportional to a power of the temperature. This can be easily seen by substitution of eq. (7) into eq. (5) to give

z1z2 k(T) = .4(8/7r#‘2 (kTp+“2 _ (24)

If one uses the high-temperature limit of the rotational partition function, then it is easy to show that at high temperatures the relaxation time (eq. (20)) wig behave as pr _ +/2-B _ (25)

This equation is not the usual Landau-Teller (LT) relation 1291

logpr- T-“3, (26)

which is somewhat surprising, since the LT relation has been adopted by several generations of experi- mentalists. This should not, however, be interpreted as evidence that the hyperbolic representation is in error. In the first place the power-law behavior at moderate energies is supported by theoretical calcula- tions [9-l l] far more sophisticated and realistic than those of Schwartz, Slawsky and Herzfeld 1301, which were used to give credence to the LT relation [2,29, 30]_ Secondly, as in manifest in fig. 4, experimental data in the shock-tube regime (1000 < T < 3000 K) can, to within experimental error, be equally well fitted by a straight line in a log-log representation, which would be predicted by eq. (25), as in an LT Qogk versus T-ti3) plot. Clearly, future experimental results should be presented on a log-log, rather than LT, plot in order to allow a more direct and meaningful theoretical interpretation.

Strictly speaking, the above results are limited to the vibrational relaxation of HZ and D2_ In addition, it is worthwhile repeating that our entire deconvolu- tion procedure is based on the assumption that the hyperbolic representation can mimic, both qualitatively as well as quantiatively, the exact V + T cross sections, at least for the systems studied. Although a rigorous justification is impossible, all the available theoretical evidence (as described above) supports this assump- tion. If one is unwilling to assume a qualitatively form for the cross section, then one must accept the ctin-

92 hl.H. AIexander/Deconvolution of experimental V- Trate constants

elusion of Melton and Gordon [ 11, that without addi- tional information, rate constant data alone do not supply enough information to determine a cross sec- tion to within any reasonable degree of precision.

[8] J. Lukasik and J. Ducuing, Chem. Phys. Letters 27 (1974) 203.

[9] M.H. AIexander and P. McGuire, J. Chem Phys. 64 (1976) 452.

[lo] MH Alexander, Chem Phys. 8 (1975) 86. [ll] M-H_ Alexander, Chem Phyr Letters 38 (1976) 417;

Acknowledgement

Part of this work was accomplished during a visit to the Laboratoire d’optique Quantique at the Ecole Polytechnique in Palaiseau, France. This author wishes to thank Professor J. Ducuing and Drs. M. Audibert and J. Lukasik for their hospitality and for helpful discussions on various aspects of this study. He is also indebted to Dr. A. DePristo for critical comments on the choice of the hyperbolic cross sec- tion and to Professor H. Pritchard for preprints of his recent work.

References

(l] L.A. Melton and R.G. Gordon, J. Chem. Phys. 51 (1969) 5449.

[21 RD. Levine and R.B. Bernstein J. Clrem. Phys 56 (1972) 2281.

[ 31 R.B. Bernstein and R.D. Levine, Adv. Atom. Mol. Phys. ll(1975) 215.

[4] C. Russ, M.V. Barnhill, HI and S.B. Woo, J. Chem. Phys. 62 (1975) 4420.

[5] M.M. Audibert, C Joffrin and J. Ducuing, Cbcm. Phys Letters 25 (1974) 158.

[6] M.A. Audibert, C. Joffrin and J. Ducuing, J. Chem Phys 61 (1974) 4357.

[ 71 M.M. Audibert, R. Vihseca, J. Luknsik and J. Ducuing,

Chem. Phys. Letters 37 (1976) 408.

40 (1976) 101. [ 121 M-H. Alexander, J. Chem. Phys 60 (1974) 4274. [13] J. Lukasik, Thesis, Univ. of Paris&rd (1975). [ 141 KK. Shin, Chem Phys Letters 37 (1976) 143. [ 151 E. lsaacson and H.B. Keller. AnaIvsis of Numerical

1161

rr71

1181

1191 [201 WI

[221

[231

[241

1251

L261

[271

WI [291 1301

Methods (Wiley, New York, 1966) pp. 221-226. SC Cohen and M.H. Alexander, J. Chem. Phys. 61 (1974) 3967. W.H. Miller and A-W_ Raczkowski, Disc. Faraday Sot. 55 (1973) 45. H-0. Pritchard and N-1. Labrb, Canad. J. Chem. 54 (1976) 329. H.O. Pritchard, Canad. J. Chem. 54 (1976) 2372. H. Rabitz and G. Zarur, J. Chem. Phys. 62 (1975) 1425. K.F. Herzfeld and T.A. Litovitz, Absorption and Dis- persion of Ultrasonic Waves (Academic, New York, 1959) pp. 83-90. M.M. Audrbert and J. Lukasik, private communication (1975). M.M. Audibert. R. Vilaseca, J. Lukasik and J. Ducuing, Chem. Phys Letters 31 (1975) 232. M. Abramowitz and I. Stegun, Natl. Bur. Std. (U.S.) Appl. Math. Ser. 55 (1965) 260. K. Pearson, Tables of the Incomplete r-Function (Cambridge Univ. Press Cambridge, England, 1965). J.H. Kiefer and R.W. Lutz, J. Chem. Phys. 44 (1966) 668. J.E. Dove and H. Teitelbaum, Chem. Phys. 6 (1974) 431. D. TruhIar and N. Blah. to be published. D. Rapp and T. Kassal, Chem. Rev. 69 (1969) 61. R.N. Schwartz, 2.1. Slawsky and K.F. Herzfeld, J. Chem. Phys. 20 (1952) 1591.