Volume 49A, number 3 PHYSICS LETTERS 9 September 1974

T H E O R Y O F F I E L D - I N D U C E D T R A N S L A T I O N A L A B S O R P T I O N

F. SCHULLER and Ph. MARTEAU Laboratoire des Interactions Moldculaires e t des Hautes Pressions C N.R.S. - (92) Bellevue, France

Received 30 July 1974

We consider the possibility of inducing translational absorption in pure rare gases by means of an external electric field. Our expression of the integrated binary absorption coefficient involves polarizability components of a pair of atoms and their variation with the internuclear distance.

As is well known, pure translational absorption due to collision-induced dipole moments has been observed in the far-infrared region with various rare gas mixtures [1-3] . In pure gases however, where binary collisions are not active and where, according to theory [4, 5 ], multiple collisions produce only a weak effect, no mea- sureable absorption is obtained.

In this paper we want to discuss the possible existence of a translational spectrum in pure monatomic gases ob- tained by applying an external electric field. In this case the field induces dipole moments which are modulated by the translational motion, since the polarizabilities are influenced by atomic interactions. A somewhat si- rnilar phenomenon, but concerning field-induced vibra- tional and/or rotational absorption has been studied extensively with diatomic molecules (H 2 , D 2) in the past few years [6, 7].

We shall restrict ourselves to the calculation of the integrated absorption intensity, assuming that most likely the frequency range of the spectrum is not essen- tially different from that observed in the case of coUi- sion.induced absorption in rare gas mixtures.

It can be shown that the integrated binary absorption coefficient introduced by Poll and Van Kranendonk [8] as a l , is given by the following general expression:

A = K~ 3 Tr (exp (-/3H)ja[K, It] ) , (1)

with r = 2n/3h"2c, h =tf(4rr[makT)ll2, (3 = l[kT, ma= mass of one atom, H and K being the hamiltonian of the translational motion and its kinetic energy part respectively.

The formula (1) is identical with that established in ref. [8] for rare gas mixtures except that now # desig- nates the field induced dipole moment of the pair of atoms.

Introducing irreducible components ajM of the po- larizability tensor associated with a pair of interacting atoms, we write a relation between the spherical com- ponents of/ t and those of the external electric field in the form:

ts m = ~C(Jl l ;M,m-M,m)aJMEm_ M . (2) J M

Let co stand for two polar angles defining the orien- tation in space of the internuclear axis. Then the a j M ' S

can be expressed in terms of spherical harmonics:

"JM = aj(R) YjM(.~), (3) R being the internuclear distance. We calculate the trace in eq. (1) by introducing a complete set of trans- lational eigenfunctions, as given by:

Xnlml(R, co) =/anl(R) rlml(cO ) . (4)

Using the identity:

h 2 u[K,u] = - ~ ( -Om um[a,U_,,,] , (5)

m a m

and expressing the/a m's by means of eq. (2), we get after a lengthy calculation the following result:

E2h 2 ~ 2l +1 exp (_/3En) Tr{...)= ma nl 41r

X f R2dR/a2I(R ) ( ~ (daj/dR)2+6a2/R216). 0

The coefficients a j may now be expressed in terms of the more familiar quantities all and a l , since one can easily show that for diatomic structures the following relations hold:

all + 2a± (8~.~__~1]2 (7) ~t 0 = (4n)1/2 3 ' a2 = X-' I (at-- all).

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Volume 49A, number 3 PHYSICS LETTERS 9 September 1974

Using these and introducing furthermore the radial dis- tribution function defined as:

g(R) ~ h3 ~ 2 1 + 1 exp ( - (3 E n ) #2I (R ) , nl

we obtain for the integrated absorption coefficient the final expression:

87r2E2 ~ ( / [d~l l \2

A-3mac f I ~ - d ~ ~

+ 2 \ - ~ / +-}R--~(ai-al l ) 2 g(R)R2dR. (8)

As in Poll and Van Kranendonk's formula [8], the two terms in the brackets correspond respectively to the radial and the angular part of the translational motion.

We tried to evaluate the quantity A by using for the polarizabilities the expressions:

~± - 2~ = - 2~2/R 3 + k a exp ( A R ) , (9)

o~11- oq. = 6~2 [R 3 + k b exp ( A R ) ,

for which values of the parameters are available from Hart ree-Fock calculations, at least in the case of He. These values are:

1 A - 0 . 7 4 a ' u ' ; k a - ' - - 2 . 0 a . u . ; k b - - - 2 1 a . u . ,

~ = polarizability of the free atom = 1.39 a.u.. Putting g(R) - 1 and taking R ; 4.9 a.u. as the lower bound of

the R-integration (hard-sphere model) we found for helium:

A = 1.7 × 10-50 E 2 cm 5 sec-1 (E in Volts cm -1 ) .

Compared to a typical value o fA = lO-33cmSsec -1 , referring to collision-induced absorption in mixtures, our result appears to be rather small since for experi- mental reasons values of E higher than 3.5 × 105 have not been attained so far [7]. Nevertheless, considering atoms of much higher polarizabilities than He, the value of A can be increased by several orders of mag- nitude. So it seems reasonable to expect that experi- mental observations of field induced translational ab- sorption are possible with heavy atoms like Kr or Xe.

References

[1] Z.J. Kiss and H.L. Welsh, Phys. Rev. Letters 2 (1959) 166. [2] D.R. Bosomworth and H.P. Gush, Can. J. Phys. 43 (1965)

751. [3] Ph. Marteau, H. Vu and B. Vodar, J.Q.S.R.T. 10 (1970)

283. [4] C.G. Gray, preprint (1973). [5] S. Weiss, Chem. Phys. Letters 19 (1973)41. [6] P.J. Brannon, C.H. Church and C.W. Peters, J. Mol. Spec.

27 (1968)44. [71 W.J. Boyd, P.J. Brannon, N.M. Gailar, Appl. Phys. Letters

16 (1970) 135. [8] J.D. Poll and J. Van Kranendonk, Can. J. Phys. 39 (1961)

189. [9] D.F. Heller and W.M. Gelbart, preprint (1973).

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