354 17 Tensor Dynamics
Fig. 17.1Depolarized
Rayleigh scattering, VH- and
HH-geometries. Thedouble
arrowsindicate the
directions of the electric field
vectors of the incident and of
the scattered light
light with a weaker intensity, whose electric field, however, has also a component
perpendicular to that of the incident light. For this reason, the termdepolarized
scatteringisused.Thename“Rayleigh”indepolarized Rayleigh scatteringindicates,
that the frequency of this contribution to the scattered light is centered about the
frequencyof theincident light, just as theordinaryRayleighscattering. Therotational
Raman scattering, where the frequency is shifted, also has a depolarized component.
Lete′andebe unit vectors parallel to the electric field vectors of the incident and of
the scattered light. The intensity of the scattered light is proportional to
Iscat=e′μeνSμν,λκeλ′eκ. (17.8)The spectral functionS..depends onω=ω 1 −ω 2 andk=k 1 −k 2 , whereω 1 ,k 1
andω 2 ,k 2 are the frequencies and the wave vectors of the incident and of the scat-
tered light. Depolarized scattering means:eis perpendicular toe′. Two scattering
geometries, referred to byVHandHHare sketched in Fig.17.1. The lettersVand
Hstem from ‘vertical’ and ‘horizontal’, with respect to the scattering plane spanned
byk 1 andk 2. The HH-case is for 90◦scattering only.
Orientational fluctuations of molecules cause fluctuations of εμν. Thus the
time-correlation function and consequently the spectral function of the depolarized
Rayleigh scattering can be inferred from relaxation equation of the second rank align-
ment tensoraμνof liquids, cf. (12.19) or of the tensor polarizationaμνT in gases of
linear molecules, cf. (13.64).
In the absence of external fields and when the coupling with the friction pressure
tensor is ignored, the (16.59) and (16.74) describe a simple exponential relaxation
for the alignment tensor:
∂aμν
∂t+τ−^1 aμν= 0 ,with a relaxation timeτ. This equation implies
aμν(t)=Cμν,λκ(t)aλκ( 0 ), Cμν,λκ(t)=Δμν,λκC(t), C(t)=exp[−t/τ],
(17.9)
with an isotropic time-correlation tensor. The resulting spectral function has the
Lorentz line shape