356 17 Tensor Dynamics
The unit vectorhis parallel to the magnetic field. Consider the HH-geometry and
puthperpendicular to bothe′ande. Then one hasW 0 =W± 1 =0 and resulting
spectral line is split by the frequency 4|ωB|, provided that the line widthνis not
larger than about|ωB|. Similarly, for the VH-geometry,W 0 =W± 2 =0 is obtained,
whenhis parallel to eithere′or toe. Then the line splitting is 2|ωB|.
17.1.3 Collisional and Diffusional Line Broadening
The examples of time-correlation and spectral functions considered so far do not
depend on the wave vectork. For depolarized Rayleigh scattering in gases, this
applies when the densitynof the gas is large enough, such thatk1, where
∼n−^1 is the mean free path, i.e. the average distance traveled by a molecule, in free
flight, between two collisions. Under these conditions, the line width is determined
by the collision frequencyνwhich is proportional to the number density. This type
of broadening is calledcollisional broadeningor alsopressure broadening, since
the density increases with increasing pressure. In the opposite limiting case, realized
at low densities wherek 1 applies, the line broadening is determined by the
Doppler broadeningwhere the line shape, reflecting the velocity distribution of the
particles, is Gaussian. For intermediate cases, where one hask≈1, diffusional
processes contribute to the line width. Thisdiffusional broadeningis described by
spatial derivatives in the relevant equations.
For a spatially inhomogeneous system, the alignment tensor obeys the equation
∂aμν
∂t+∇λbλμν+νaμν= 0 , (17.15)wherebλμν∼〈cλJμJν〉is the flux of the tensor polarization,cis the velocity
of a molecule. Equations for the three irreducible parts of the tensorbλμν, which
are tensors of ranks 1, 2 ,3, can be derived by kinetic theory. When the collision
frequencies for these three parts are practically equal to a single collision frequency
νband large compared withν, the approximation
bλμν=−Da∇λaμν, Da=kBT
mνb−^1 (17.16)can be made. Due to collisional changes of the rotational angular momenta, the
diffusion coefficientDais smaller than a self diffusion coefficient. Insertion of the
relation for the flux into the equation for the tensor polarizationaμνleads to
∂aμν
∂t−DaΔaμν+νaμν= 0. (17.17)