Tensors for Physics

15.4 Cubatics and Tetradics 293

By analogy with the description of the alignment tensor elasticity, cf. Sect.15.3.3,

the potential function

Φ=ΦL+

1

2

ξ 02 (∇τaμνλκ)(∇τaμνλκ), (15.47)

is used for a spatially inhomogeneous situation. The pertaining equilibrium state

obeys the relationΦμνλκ≡∂a∂Φμνλκ=0, with

Φμνλκ=Aaμνλκ−

√

30 Baμνσ τaστλκ

+Caμνλκ(aμ′ν′λ′κ′aμ′ν′λ′κ′)−ξ 02 Δaμνλκ. (15.48)

When the order parameter tensor has the full cubic symmetry as described by (15.41),

the potential function reduces to

Φ=

1

2

Aa^2 −

1

3

Ba^3 +

1

4

Ca^4 +

1

2

ξ 02 (∇σa)(∇σa).

The pertaining equilibrium condition is

∂Φ

∂a

=Aa−Ba^2 +Ca^3 −ξ 02 Δa. (15.49)

For the spatially homogeneous situation, the equilibrium conditiona(A−Ba+

Ca^2 )=0 is equal to that one discussed for nematics, cf. Sect.15.2.2. In particular,

the equilibrium transition between the isotropic and cubic phases occurs at the tem-

peratureTs, whereA(Ts)= 2 B^2 /( 9 C). There the order parameter isas= 2 B/( 3 C).

At the temperatureT∗, whereA(T∗)=0, one hasa(T∗)=B/C. The sign of the

equilibrium value of the order parameteraeq=^12 BC−^1 ( 1 +

√

1 − 4 AC/B^2 ), with

T<Ts, is determined by the sign ofB. The cubic order parameter for particles in

the first coordination shell ofsimple cubiccrystal is positive, that one ofbccandfcc

crystals is negative.

15.2 Exercise: Compute the Cubic Order Parameter〈H 4 〉for Systems with
Simple Cubic, bcc and fcc Symmetry
Hint: The coordinates of one the nearest neighbors, in the first coordination shells,
are( 1 , 0 , 0 )for simple cubic,( 1 , 1 , 1 )/

√

3 for bcc and( 1 , 1 , 0 )/

√

2 for fcc. Use

symmetry arguments!

15.4.3 Order Parameter Tensor for Regular Tetrahedra

Consider a fluid composed of regular tetrahedra or of practically spherical particles

which have first coordination shell with tetrahedral symmetry. Letui,i= 1 , 2 , 3 , 4