Tensors for Physics

224 12 Integral Formulae and Distribution Functions

The factor^12 stems from the fact that the interaction between two particlesiandjhas
to be counted only once whereas the summation overiandjcontains the equivalent
pairs{i,j}and{j,i}. The average potential energy per particle isupot=〈Φ〉/N.The
corresponding integral over the pair correlation function is

upot=

1

2

n

∫

φ(r)g(r)d^3 r, (12.105)

wheren=N/Vis the average number density.
The force associated with the binary interactionφis

Fμ=Fμ(r)=−

∂

∂rμ

φ(r).

The potential contribution to the pressure tensorppotνμthen is given by the integral

ppotνμ=

1

2

n^2

∫

rνFμ(r)g(r)d^3 r. (12.106)

For a substance composed of spherical particles, the forceFis parallel tor, thus the
pressure tensor (12.106) is symmetric. It can be decomposed into its isotropic part
pδμνand its symmetric irreducible (traceless) partpμν, cf. Chap. 6

ppotνμ=ppotδμν+ppotμν ,

with

ppot=

1

6

n^2

∫

rλFλ(r)g(r)d^3 r, ppotμν =

1

2

n^2

∫

rνFμ(r)g(r)d^3 r. (12.107)

The pressure tensor is the sum of its kinetic and potential contributions.

12.4.3 Static Structure Factor

The static structure factorS=S(k)can be measured in experiments where electro-
magnetic radiation or particles, like neutrons, are scattered. The scattered intensity
is determined by theform factor, which reflects the shape of the scatters, and by the
static structure factor, which contains the information on the correlations between the
positions of the particles. Thescattering wave vectork=ksc−kinis the difference
between the wave vectorskscandkinof the detected scattered and the incident beam,
see Fig.12.1.