15.4 Cubatics and Tetradics 291
andcubatics. Here, as in [92, 93], the term ‘tetradic’ is meant as an abbreviation for
‘tetrahedratic’, indicating a tetrahedral symmetry. Due to the closer resemblance of
the theoretical treatment to that of the second rank case, cubatics are discussed first.
15.4.1 Cubic Order Parameter
Consider a reference particle in a dense liquid or solid and letube a unit vector
pointing to a nearest neighbor. The fourth rank irreducible tensor
aμνλκ=ζ〈uμuνuλuκ〉 (15.40)is the lowest rank order parameter tensor which distinguishes a state with local cubic
symmetry from an isotropic state. The numerical factorζcan be chosen conveniently.
The bracket〈...〉indicates an average evaluated with an orientational distribution
functionf(u), just as in Sect.12.2.1. To indicate that hereudoes not specify the
direction of a particle but rather the relative positions of particle neighbors, the term
bond orientational orderis used.
When the order parameter tensor has the full cubic symmetry, as in cubic crystals,
and the coordinate axes are chosen parallel to the symmetry axes, the order tensor is
proportional to the fourth rank cubic tensor defined in Sect.9.5.1:
aμνλκ=√
5
6
aHμνλκ(^4 ) ,Hμνλκ(^4 ) =∑^3
i= 1e(μi)e(νi)e(λi)eκ(i)−1
5
(δμνδλκ+δμλδνκ+δμκδνλ). (15.41)Thee(i), withi= 1 , 2 ,3 are unit vectors parallel to the cubic symmetry axes. Notice
thatHμνλκ(^4 ) Hμνλκ(^4 ) = 6 /5 and consequentlyaμνλκaμνλκ=a^2. The order parameter
ais essentially the average of a cubic harmonic, cf. (9.3.2),
a=√
5
6
ζ〈H 4 〉, H 4 =Hμνλκ(^4 ) uμuνuλuκ =u^41 +u^42 +u^43 −3
5
. (15.42)
The choiceζ=
√
6 /5 impliesa=〈H 4 〉. On the other hand, withζ=( 9 !!)/( 4 !), one
hasa=〈K 4 〉, whereK 4 ≡^54
√
21 H 4 has the normalization( 4 π)−^1∫
K 42 d^2 u=1,
cf. (9.33). In this case, the expansion of the distribution function reads f(u)=
( 4 π)−^1 ( 1 +aK 4 +...). The dots...stand for components of the fourth rank tensor
which do not have the full cubic symmetry and for terms involving tensors of higher
ranks= 6 , 8 ,...