12.4 Anisotropic Pair Correlation Function and Static Structure Factor 225
Fig. 12.1Schematic
scattering geometry with
wave vectorskinandkscfor
the incoming and the
scattered beams
The scattering is elastic, i.e. one hask^2 sc=k^2 in. The quantityS(k)−1 is essentially
the spatial Fourier transform ofg(r)−1, viz.
S(k)= 1 +n∫
exp[−ik·r](g(r)− 1 )d^3 r. (12.108)Due tog(r)= g(−r), the exponential function in (12.108) can be replaced by
cos(k·r). At first glance, curves of the isotropic part ofSas function ofklook similar
to those ofgas function ofr. Typically, there is a first maximum atk≈ 2 π/rnn,
wherernnis the nearest neighbor distance associated with the first maximum ofg.
The behavior ofSforkapproaching zero is fundamentally different from that of
g( 0 )=0. More specifically, one hasS( 0 )=〈(δN)^2 〉/N, whereN is the number
of scatters in the open scattering volume andδN =N −〈N〉is its fluctuation.
The mean square fluctuation of the number of particles is related to the isothermal
compressibilityκT=n−^1 (∂n/∂p)T. More specifically, one hasS( 0 )=nkBTκT.This
quantity is small, but finite, in dense liquids. Close to the critical point, however,S( 0 )
becomes very large, this underlies the critical opalescence.
12.4.4 Expansion ofgðrÞ..........................
In thermal equilibrium, the pair correlation function of a fluid composed of spherical
particles is isotropic, i.e. it depends onrof the vectorr, but not on its direction
specified by the unit vectorrˆ. In general, however, in particular in non-equilibrium
situations,gis a function of bothrandˆr. The angular dependence, also called
directional dependence, ofgcan be taken into account explicitly by an expansion
with respect to irreducible tensors of rankconstructed from the components of the
unit vectorrˆ. Sincegis an even function ofr, only even values= 0 , 2 , 4 ,...occur
in the expansion. The first terms are
g(r)=gs+gμν̂rμ̂rν+gμνλκ̂rμ̂rν̂rλ̂rκ +..., (12.109)