232 12 Integral Formulae and Distribution Functions
Fig. 12.3 The cubic pair correlation functiong 4 , variablesr^2 andtare as in Fig.12.2. The sign
changes from one coordination shell to next
Fig. 12.4The cubic pair correlation functiong 6 , variablesr^2 andtare as in Fig.12.2
12.4.8 Anisotropic Structure Factor.
By analogy to (12.109), the static structure factorS=S(k)can be expanded with
respect to the unit vector̂kspecifying the direction of the scattering wave vectork:
S(k)=Ss+Sμν̂kμ̂kν+Sμνλκ̂kμ̂kν̂kλ̂kκ+..., (12.130)where it is understood, that the isotropic or spherical partSsofS,aswellasthe
expansion tensorsS···are function ofk, viz. of the magnitude of the vectork.In
general, they depend also on the timet.
On account of the orthogonality relation (12.5), the quantitiesS···are tensorial
moments ofS(k), thus
1
4 π∫
̂kμ 1 ̂kμ 2 ···̂kμS(k)d^2 ̂k= !
( 2 + 1 )!!Sμ 1 μ 2 ···μ. (12.131)