16.2 Elasticity 311
remarkable, since the computation ofGfluctinvolves not only two-particle, but also
three- and four-particle correlations. LetGpairbe the approximation forGBG+Gfluct
in a fluid state and for low densities, where three- and four-particle correlations can
be disregarded. This quantity can be expressed as an integral over the pair correlation
functiong(r)=χ(r)exp[−βφ(r)]which assumes the form
Gpair=1
30
n^2 kBT∫
r^2 χ(r)′(exp[−βφ(r)])′d^3 r. (16.36)In contradistinction toGBG, the shear modulusGpairvanishes in the small density
limit whereχ(r)′=0 applies.
For pairwise additive interaction, the Born-Green contributions to the bulk mod-
ulusBand to the cubic shear modulusGcare
BBG=
5
3
GBG+ 2 ppot, (16.37)and
GBGc =5
12 V
〈
∑
i>j(
H(^4 )(r)r−^1 (r−^1 φ′)′)ij〉
0, H(^4 )(r)=x^4 +y^4 +z^4 −3
5
r^4 ,(16.38)
whereH(^4 )is a cubic harmonic of order 4, with full cubic symmetry, cf. Sect.9.5.2.
The fluctuation contributions to the elastic moduli are given by
VBfluct=−β(
〈Φiso^2 〉 0 −(〈Φiso〉 0 )^2)
, (16.39)
VGfluct=−1
5
β(
3 〈Φ^2 +〉 0 + 2 〈Φ−^2 〉 0
)
, VGfluctc =−β(
〈Φ+^2 〉 0 −〈Φ^2 −〉 0
)
.
Here the abbreviations
Φiso=
1
3∑i<j(rφ′)ij,Φ+=∑i<j(xyr−^1 φ′)ij,Φ−=
1
2∑i<j(
(x^2 −y^2 )r−^1 φ′)ij
,(16.40)
are used. The elastic moduli are the sum of the Born-Green and fluctuation contri-
butions, e.g.G=GBG+Gfluct.
The total shear modulus tensor also contains the kinetic contributionnkBTδμν
δλκ. This does not affect the shear moduli, but the total Voigt coefficientsctotal 11 and
ctotal 12 are related the coefficientsc 11 andc 12 used here, and to the modulic 110 andc^012 ,
where the pressureP=nkBT+ppotis zero, by
ctotal 11 =c 11 +nkBT=c^011 −P, ctotal 12 =c 12 +nkBT=c^012 +P.