13.6 Rotational Angular Momentum of Linear Molecules, Tensor Operators 255
13.6.5 Anisotropic Dielectric Tensor of a Gas
of Rotating Molecules
The anisotropic partεμν of the dielectric tensor, cf. Sect.5.3.4,ofagasofrotating
linear molecules is related to the average of the anisotropic part of the molecular
polarizability tensor, cf. Sect.5.3.3,by
εμν =n(α‖−α⊥)〈(uμuν)diag〉=−1
2
n(α‖−α⊥)〈(
J^2 −
3
4
)− 1
JμJν〉
.
(13.64)
Hereα‖andα⊥are the polarizability for an electric field parallel and perpendicular
tou, respectively, andnis the number density of the gas. The relation (13.56)was
used to obtain the second equality in (13.64).
The density operator needed for the evaluation of the averages is given by
ρ=ρeq( 1 +aTμνφTμν+···),where the dots stand for terms involving higher rank tensors, foraTμνandφTμνsee
(13.61) and (13.63). The resulting average needed for the dielectric tensor is
εμν =εTaaμνT,εTa=−1
2
n(α‖−α⊥)√
2
15
ξT,ξT=〈c(J^2 )^2 〉−eq^1〈
c(J^2 )√
J^2
J^2 −^34
〉
eq. (13.65)
Forc=1 and rotational statesj=4 and higher, the factorξTapproaches 1.
The interrelation (13.65) between the anisotropic part of the dielectric tensor and
the tensor polarizationaTplays a key role in the kinetic theory for the depolarized
Rayleigh scattering and the flow birefringence of molecular gases [17, 22, 62–64].
13.6.6 Non-diagonal Tensor Operators
Spherical tensor operators are defined by
T
jj′
m=∑
m′∑
m′′(− 1 )j−m′′
(j′m′,j−m′′|m)|jm′′〉〈j′m′|, (13.66)where the symbol(.., ..|..)indicates a Clebsch-Gordan coefficient These coefficients
govern the coupling of two angular momentum states|j 1 m 1 〉and|j 2 m 2 〉to a state
|jm〉according to, cf. [65],