Tensors for Physics

12.4 Anisotropic Pair Correlation Function and Static Structure Factor 233

For=2 this relation implies, see also (12.1),

Sμν=

15

2

1

4 π

∫

̂kμ̂kνS(k)d^2 ̂k, (12.132)

the case=4 is analogous to (12.112).
The static structure factor and the pair correlation function are related to each other
by a spatial Fourier transformation, see (12.108), so there exist also interrelations
between the tensorsSμ 1 ,μ 2 ···μandgμ 1 ,μ 2 ···μ. The key for this connection is the
Rayleigh expansion

exp[−ik·r]=

∑∞

= 0

(−i)( 2 + 1 )j(kr)P(̂k·̂r), j(kr)=

(π

2 kr

)^12

J+ 1

2

(kr),

(12.133)

where theJ..areBessel functionsand thejare referred to asspherical Bessel
functions, [66], sometimes also called Sommerfeld’s Bessel functions. The Legendre
polynomialPis the scalar product of theth rank irreducible tensors constructed
from the components of the unit vectorŝkand̂r, cf. Sect.9.3. The integral relation
analogous to (12.1) leads to

1

4 π

∫

̂kμ 1 ̂kμ 2 ···̂kμP′(̂k·̂r)d^2 ̂k=δ′( 2 + 1 )−^1 ̂rμ 1 ̂rμ 2 ···̂rμ. (12.134)

As a consequence, insertion of (12.108)into(12.131), use of the relations just stated
and of

∫

...d^3 r=

∫

...d^2 drd^2 ̂rleads to

Sμ 1 μ 2 ···μ(k)=(−i)

∫

j(kr)gμ 1 μ 2 ···μ(r)r^2 dr, (12.135)

for≥1. The interrelation between the spherical parts, corresponding to the case
=0, is

Ss(k)− 1 =

∫

j 0 (kr)(gs(r)− 1 )r^2 dr.

The first few of the spherical Bessel functions are

j 0 (x)=x−^1 sinx, j 1 (x)=x−^1 (x−^1 sinx−cosx)

j 2 (x)= 3 x−^2 (x−^1 sinx−x/ 3 −cosx).

The spherical Bessel functions obey the recursion relation

j+ 1 (x)=



x

j−

d

dx

j=−x

d

dx

(x−j).