Rotational relaxation in HD–inert gas mixturesP. M. Agrawal and M. P. Saksena Citation: The Journal of Chemical Physics 65, 550 (1976); doi: 10.1063/1.433080 View online: http://dx.doi.org/10.1063/1.433080 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/65/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Vibrational relaxation of HCl with inert gas collision partners J. Chem. Phys. 58, 1796 (1973); 10.1063/1.1679434 Rotational Relaxation in Dilute Gas Mixtures J. Chem. Phys. 57, 3421 (1972); 10.1063/1.1678775 Absorption and Dispersion of Ultrasonic Waves in Inert, Monatomic Gas Mixtures J. Acoust. Soc. Am. 44, 708 (1968); 10.1121/1.1911165 Sound Absorption in Substituted Methane— InertGas Mixtures J. Chem. Phys. 42, 2982 (1965); 10.1063/1.1703283 Sound Dispersion in Substituted Methane—Inert Gas Mixtures J. Chem. Phys. 39, 2902 (1963); 10.1063/1.1734122

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.189.93.16 On: Wed, 10 Dec 2014 01:31:16

Rotational relaxation in HD-inert gas mixtures P. M. Agrawal

Department of Physics, MLV Govt. College, Bhilwara, Rajasthan, India

M. P. Saksena

Department of Physics, University of Rajasthan, Jaipur, India (Received 7 July 1975)

An expression for the inelastic collision cross section cr( I m-l /' m') has been derived for the rotational transition in the HD molecule caused by a collision with an inert gas atom. The interaction potentials for the HD systems have been derived from the theoretical potentials for the H2 systems by shifting the center of mass of the molecule. The rotational relaxation number values have been calculated for HD-He, HD-Ne, and HD-Ar systems in the temperature range from 10 to loooK. A good agreement has been obtained between theory and experiment. Temperature dependence and the effect of replacement of an isotope in a homonuclear molecule have been discussed in detail.

I. INTRODUCTION

In rotational-translational energy exchange processes the behavior of an HD molecule is very much different from that of H2 or D2. The rotational relaxation number Zrot for hydrogenl

- 4 or deuterium3- 5 is a few hundred, whereas for HD6 it is about 10. The main reason for this difference is that for rotational transitions in homo­nuclear molecules (H2 or D2) the rotational quantum num­ber changes by 2(~ 1 = 2), whereas for rotational transi­tions in heteronuclear molecules (HD) ~ 1 = 1 is also al­lowed. This reason is, however, not sufficient. It has been observed7 that for 14N15N Zrot is nearly the same as that for 14N2 or 15N2, i. e., here the replacement of an isotope in the formation of a heteronuclear molecule does not appreciably affect the Zrot value. The study of rota­tional relaxation in HD is thus of considerable impor­tance as it is expected to elucidate the behavior of het­eronuclear molecules.

Experimentally, Prangama et al. 6 recently carried out sound absorption measurements for pure HD, HD­He, and HD-Ne systems and obtained the rotational re­laxation times at 20.4-42.6 OK.

Theoretically, the rotational relaxation in the HD-HD system has been investigated by Takayangi, 8,9 and the HD-He system has been studied by Itikawa and Takaya­nagi. 10 The interaction potential for the HD-He system has been derived by these authors from the theoretical potential given by Gordon and Secrestll for the H2-He system by shifting the center of mass of the molecule. However, HD-Ne and HD-Ar systems have not yet been studied theoretically. In this paper we have quantum mechanically calculated the Zrot values for the HD-He, HD-Ne, and HD-Ar systems by using the distorted wave approximation. In addition, calculations for the HD-He system have been performed, for the potentials obtained from the H2-He potentials given by Roberts, 12 Krauss and Mies13 and Gordon and Secrest, 11 to study the poten­tials' relative behavior.

II. INTERMOLECULAR POTENTIAL AND SCATTERING CROSS SECTION

The theoretical potential functions given by Roberts, 12 Krauss and Mies, 13 and Gordon and Secrestll for H2-He;

550 The Journal of Chemical Physics, Vol. 65, No.2, 15 July 1976

by Gelb et al. 14 for H2-Ne; and by Pedersen et al. 15 for H2-Ar may be expressed (neglecting vibration coordi­nates) as

V(r', 9') = C exp( - ar')[1 + {3P2 (cos9')] , (1)

where r' is the distance between the center of mass of the two colliding partners, 9' is the angle between r' and the molecular axis, and P2 (cose') is the Legendre poly­nomial.

The HD-He, HD-Ne, and HD-Ar potentials are ob­tained from the corresponding H2-inert gas potentials by shifting the center of mass of the molecule by an amount 0= 0.233 a. u. (Ref. 10). In thermal energy collisions, the second and higher order terms in olr may be ne­glected since the colliding particles do not penetrate much into each other. Here, r is the distance between the center of mass of the HD molecule and the inert gas atom. If the angle between r and the molecular axis is denoted by e then

r' ~ r[1 - (olr) cose], (2)

cose' = cose - (olr)(1 - cos2e) . (3)

Substituting these values of r' and cose' in Eq. (1) we ob­tain the interaction potential for the HD-inert gas sys­tems:

V(r, e) = C exp(- ar)[1 + (olr)( - ~ {3 + ar +~ a{3r)Pl (cose)

+ {3P2 (cose) + (olr)(~ (3 +~ a{3r)P3 (cose)] . (4)

In calculating the cross sections for the transitions ~ I = 1 the use of the distorted wave approximation allows us'to ignore terms with P2 (cose) and P3 (cose). Thus, the potential given by Eq, (4) reduces to the following form:

V(r, e) =Cexp(- ar) + Vi (r)Pl (cose) . (5)

By following the approach used by Agrawal and Sak­sena16- 18 and Brout, 19 the cross section a(lm -I'm') for the transitions in the HD molecule from the rotational state 11m) to the l1'm') state is found to be

Copyright © 1976 American Institute of Physics

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.189.93.16 On: Wed, 10 Dec 2014 01:31:16

P. M. Agrawal and M. P. Saksena: Rotational relaxation in HD-inert gas mixtures 551

Here,

(7)

with

Css' =(_1)S(i)s+s'21Texp(ili!, +ili!)bs's, (8)

81T{l + 1m' I )! limm, [(1' I ' I )15 a1= (21+1)(21'+1)(1-lm')! + m 1'-1,1

_(1' -lm'l +l)li l '+l,I](SliS(s'+l) +s'lis'(s+l» , (9)

a2 = 15m, (m'+1)(a4 +as)(lis(s'+l) -lis'(s+l» , (10)

as = 15m" (m+l)(a4 +as)(lis(s'+l) -lis'(s+l» , (11)

_ 41T(1'+lm'I)! li lm l,lm'I+1 [(l+lm'I)(l+lm'I+1)li • a4 - (21 + 1)(21' + l)(Z' _ 1m' I) I 1-1,1

-(l-lm'I)(l-lm'l +1)15/+1,/'], (12)

_ 41T(1 + 1m I)! lilm'I,lml+l [(1' + 1m 1)(1' + 1m I + 1)15 • as - (21 + 1)(21' + 1)(1- I m I) I I -1,1

- (1' - I m 1)(1' - I m I + 1)151'+1,1] , (13)

N(lm)=[(21+1)(1-lml)I][41T(1+lml)I]-1, (14)

kl = (2IJ.Einc)1/2/1f ,

k l, = {k7 +[1(1 + 1) -l'(l'+ 1)] v./I }1/2 ,

and

bs' s = 100

F!, (r')F!' (r')V1 (r')r'2 dr' , Yo .

where

F!(r) = cosli![j s(k1r)] - sinli![ns(k lr)] .

(15)

(16)

(17)

(18)

Here j sand ns are spherical Bessel and spherical Neu­mann functions, 20 respectively, and li! is the phase shift. li! is given by

tanli! = (jS(klrO)]![ns(k,ro)] •

ro is to be obtained from the relation

Cexp(- aWol =lf2k l k l ,/21l,

(19)

(20)

where 11 and E inc denote the reduced mass and the rela­tive incident translational kinetic energy of the colliding system, respectively.

Using Eqs. (6)-(20) and assuming16-20 bs's = boo for s or s'::'fk1ro and bs's=O for s or s'>k1ro, we obtain the following easily computable expression for the cross section for the transition from the rotational state 11) to 11'):

a(l- 1') = (21 + 1)-1 L L u(lm _1' m') m m'

(21)

where

C(O, 1) =¥, C(1, 0) =11, C(1, 2) =¥, C(2, 1) =E ' (22)

C(2, 3) =¥, C(3, 2) =¥, C(3, 4) =§,f-, C(4, 3) =¥ ,

and I;(klro) is a function of klro. Some of its values are

1;(6) = 1.361, 1;(8) = 1.269, 1;(10) = 1. 219, and 1;(12) = 1.174.

III. CALCULATION OF Zrot AND DISCUSSION OF THE RESULTS

At low temperatures where the rotational levels 12: 2 are not appreciably populated (at 50 OK more than 99.7% of the HD molecules are in the states 1 =0,1, and even at 100 OK this percentage is more than 94) the transi­tions 0= 1 are only significant for rotational relaxation, and Zrot [more specifically (Zrot)lo] is given bylO, 18

1Ta~ exp( - (/kT) (23)

Here, (To represents the hard sphere diameter of the col­liding system. In the present case of dissimilar inter­acting molecules (To has been taken as the arithmetic mean of the hard sphere diameters of the individual molecules. E1 - Eo = 0.01107 eV. The factor exp( - (/ kT) has been introduced17,18,21,22 to account for long range attractive forces. Here, (/k is the depth of the attrac­tive well of the intermolecular potential and (a(0 -1» is the cross section averaged over Boltzmann distribution of energies at temperature T, i. e.,

«(T(O -1» = (kT)-2 100

a(0 -l)Eine exp( - Einc/kT) dEine • o (24)

By using relations (1)-(4), the potentials for HD-inert gas systems are obtained from the corresponding poten­tials for the H2-inert gas systems. Zrot for these poten­tials have been calculated as a function of temperature. The (To and (/k values used here are the same as those for the H2 systems. Thus, (To values2,6,10 for HD-He, HD-NeJ and HD-Ar systems are 2.745, 2.830, and 3.175 A, respectively. The values of (/k have been taken from Hirschfelder et al., 23 and the geometric mean rule has been used.

The calculated values of Zrot for all the systems have been reported in Table I along with the available experi­mental data of Prangsma et ale 6 and the close coupling calculation results of Itikawa and Takayanagi10 for the HD-He system. It is obvious from the table that the present results are in good agreement with the experi­ment as well as with the theoretical values obtained by close coupling calculations. For HD-Ne and HD-Ar sys­tems, experimental as well as other theoretical data are not available for comparison. However, the present cal­culations predict that out of the three collision partners He, Ne, and Ar, He is the least efficient while Ar is the most efficient in causing rotational transitions in HD. This fact is verified by the similar behavior of these three partners in causing rotational transitions in D2 and H2 as observed by Jonkman et al. 2,S

The present as well as Itikawa and Takayanagi's the­ory predicts the existence of a maximum in Zrot versus temperature curve. As discussed by Itikawa and Takaya­nagi and as shown by expression (23), the dominance of long range attractive forces at low temperatures is re­sponsible for such a maximum. It may be mentioned that the existence of such a maximum for H2 and D2 sys­tems has also been demonstrated. 17 Further, at high

J. Chern. Phys., Vol. 65, No.2, 15 July 1976

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.189.93.16 On: Wed, 10 Dec 2014 01:31:16

552 P. M. Agrawal and M. P. Saksena: Rotational relaxation in HD-inert gas mixtures

TABLE I. Rotational collision numbers for lID-inert gas sys­tems,

HD-He

Other Temp Present calculations theorylO Experi-('K) RP' GSPb KMp· GSPb mental HD-Ne HD-Ar

10 8.3 11 13 19 20 17 20 25 20 10 1.8 27 17.3 30 18 22 28 19 16 5.5 32 16.3 40 17 21 27 17 18.1 17 7.9 50 14 18 24 14 16 9.3 70 11 15 20 14 9.7

100 8.7 10 14 11 7.9

aRoberts potential. lloordon-Secrest potential, cKrauss-Mies potential.

temperatures the higher level transitions will also be of importance and thus (Zrot)Zh (Zrot>SZ'" will also con­tribute to Zrot. Because of larger spacings between higher levels the corresponding rotational relaxation numbers will be larger. 4, 18, Z4 The contribution of (Zrot)z1> (Zrot>Sz • •• to Zrot at high temperatures may offset the decreasing behavior of (Zrot)10 with temperature. Thus, after a certain temperature Zrot will start increasing with increase in temperature. Thus, the Zrot versus temperature curve is expected to have a maximum in low temperature region and a minimum in high tempera­ture region. Sufficient experimental data over a wide temperature range are needed to confirm this behavior.

To account for long range attractive forces we have followed the work of Sather and Dahle~z and 0' Neal and Brokawl and have multiplied the expression for Zrot by exp(- E/kT). Such multiplication has also been done for the Hz and Dz systems. 17,18 For all these systems such a multiplication appears to be satisfactory except at very low temperatures «20 OK). However, at very low tem­peratures this factor becomes very small, resulting in very low values of Zrot> e. g., at 10 OK exp(- E/kT) is as low as 1.14x 10-3 for HD-Ar and Zrot comes out to be 0.09. Hence, at very low temperatures it becomes nec­essary to take into consideration long range forces more rigorously.

Regarding the relative behavior of the different poten­tials used here for HD-He we find that the Zrot values are largest for the Krauss-Mies potential and smallest for the Roberts potential. Qualitatively similar behavior has been observed17 for the Hz-He system. Quantita­tively, for Hz-He the Gordon-Secrest and Roberts poten­tials are found17 to be, respectively, 1. 5 and 3 times more effective than the Krauss-Mies potential in causing rotational transition; for HD-He the results for different potentials do not differ appreciably (Table I). This is mainly owing to the fact that the anisotropy parameters, (3 [Eq. (1)], for the three potentials for the Hz systems vary by 50%, whereas for the corresponding potentials for t!.l = 1 transitions in HD the effective anisotropy pa­rameters, o(a+% a{3-~{3/r) [Eq. (4)], show only 15% variation.

It will be interesting to discuss the fact that Zrot for

HD and Hz differ by a factor of 10 whereas Zrot for 14N2 and 14N15N are nearly the same. As mentioned earlier, in the case of a heteronuclear molecule transitions with t!.l = 1 are allowed, whereas for a homonuclear molecule, transitions t!.l = 1 are forbidden. Thus, in the homo­nuclear molecule, transitions occur between the states with a larger energy gap as compared to that for the cor­responding heteronuclear molecule. One may except that the transition probability should be larger for smaller energy gap transitions (t!.l = 1) as compared to that for t!.l = 2 transitions. But in addition to the energy gap there is another important factor which also governs the transitions probability: the anisotropy parameter in the potential function. If for a homonuclear molecule (say Hz or l4 Nz) the potential function is that given by Eq. (1), then the corresponding potential for the heteronuclear molecule (HD or l4N15N) will be that given by Eq. (4). Thus, the anisotropy parameter for a heteronuclear molecule for the transitions t!.l = 2 is {3 [coefficient of Pz (cose)] and for the transitions t!.Z = 1 it has a value o(a +t a{3 -~ (3/r). For HD, 0 =0. 233 a. u. whereas for l4NI5N it is as small as 0.036 a. u., i. e., the anisotropy parameter for the t!.Z = 1 transition in the 14NI5 N mole­cule is very small, leading to a very small transition probability for t!.l = 1 transitions. Thus for l4N15N the dominating transitions will be again t!.Z = 2 transitions; hence, Zrot for Nz and 14NI5N are nearly the same. It may be mentioned that for the HD-He system corre­sponding to the transition 0= 2, the rotational relaxa­tion number (Zrot>ZO, obtained using relation (36) of Ref. 17, the potential equation (4), and the Gordon-Secrest potential,11 comes out to be as large as 250 at 50 oK, which is nearly the same17 as that for Hz or D2. ThUS, translational-rotational energy exchange in HD is mainly governed by t!.Z = 1 transitions; in 14 N15 N, however, t!.Z =2 transitions will dominate the rotational relaxation be­cause of the small value of O.

lA. Van Itterbeek and H. Zink, Physica (Utrecht) 29, 370 (1963).

2R • M, Jonkman, G. J. Prangsma, I. Ertas, H. F. P. Knaap, and J. J. M. Beenakker, Physic a, (Utrecht) 38, 441 (1968).

3C• G. Sluijter, H. F. P. Knaap, and J. J. M. Beenakker, Physica, (Utrecht) 31, 915 (1965).

4T • Winter and G. L. Hill, J. Acoust. Soc. Am. 42, 848 (1967). OR. M. Jonkman, G. J. Prangsma, R. A. J. Keijser, R. A.

Aziz, and J. J. M. Beenakker, Physica (Utrecht) 38, 451 (1968).

6G• J. Prangsma, J. P. J. Heemskerk, H. F. P. Knaap, and J. J. M. Beenakker, Physica (Utrecht) 50, 433 (1970).

7p. G. Kistemaker, A. Tom, and A. E. DeVries, Physica (Utrecht) 48, 414 (1970).

BK• Takayanagi, Sci. Rept. Saitama Univ. Ser. A 3 Univ. Tokyo 3, 87 (1959).

9K• Takayanagi, lnst. Space Aeronaut. Sci. Univ. Tokyo Rep. 36, 229 (1971).

lOy. Itikawa and K. Takayanagi, J. Phys. Soc. Japan 32, 1605 (1972).

11M• D. Gordon and D. Secrest, J. Chern. Phys. 52, 120 (1970). 12C. S. Roberts, Phys. Rev. 131, 203 (1963). 13M• Krauss and F. H. Mies, J. Chern. Phys. 42, 2703 (1965). 14Alan Gelb, C. W. WilSOn, R. Kapral, and G. Burns, J.

Chern. Phys. 57, 3421 (1972). 15L. Pedersen, K. Gammon, and D. Hoskins, Chern. Phys.

J. Chern. Phys., Vol. 65, No.2, 15 July 1976

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.189.93.16 On: Wed, 10 Dec 2014 01:31:16

P. M. Agrawal and M.P. Saksena: Rotational relaxation in HD-inert gas mixtures 553

Lett. 11, 407 (1971). l6p • M. Agrawal and M. P. Saksena, J. Chern. Phys. 61, 848

(1974). 17p . M. Agrawal and M. P. Saksena, J. Phys. B 8, 1575

(1975). l8 p • M. Agrawal and M. P. Saksena, Chern. Phys. Lett. 32,

123 (1975). 19R• Brout, J. Chern. Phys. 22, 934 (1954).

20L. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968), p. 85.

21C. O'Neal and R. S. Brokaw, Phys. Fluids 5, 567 (1962). 22N. F. Sather and J. S. Dahler, J. Chern. Phys. 37, 1947

(1962). 23J • O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecu­

lar Theory of Gases and Liquids (Wiley, New York, 1964). 24L • M. Raff, J. Chern. Phys. 46, 520 (1967).

J. Chern. Phys., Vol. 65, No.2, 15 July 1976 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.189.93.16 On: Wed, 10 Dec 2014 01:31:16