Appendix: Exercises... 413
15.2 Compute the Cubic Order Parameter〈H 4 〉for Systems with Simple Cubic,
bcc and fcc Symmetry(p.294)
Hint: The coordinates of one the nearest neighbors, in 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!
By definition, one hasH 4 =u^41 +u^42 +u^42 −^35 , where theuiare the Cartesian compo-
nents of one of the nearest neighbors. Due to the symmetry of the first coordination
shell in these cubic systems, all nearest neighbors give the same contribution in the
evaluation of the average〈H 4 〉, thus it suffices to consider one of them. The result
then is
〈H 4 〉=
2
5
, −
4
15
, −
1
10
,
for simple cubic, bcc, and fcc symmetry, respectively.
15.3 Renormalization of Landau-de Gennes Coefficients(p.296)
Consider the special case whereΦb=^12 Abbμνbμν,for simplicity put Ab=1.
Determine bμνfrom∂∂Φbμν = 0 with the help of the second equation of (15.52).
Insert this expression into the first equation of(15.52)to obtain a derivative of a
Landau-de Gennes potential with coefficients A,B,C which differ from the original
coefficients Aa,Ba,Cadue to the coupling between the tensors.
Hint: use relation(5.51)fora,viz.aμκaκλaλν =^12 aμνaλκaλκ.
With the assumptions made here,∂∂Φbμν=0, used for the second equation of (15.52),
implies
bμν=−(
c 1 aμν+c 2√
6 aμκaκν)
.
Insertion into the first equation of (15.52) yields
∂Φ
∂aμν=Φμνa −(
c^21 aμν+ 3√
6 c 1 c 2 aμλaλν+ 12 c^22 aμκaκλaλν)
.
Due to
aκλaλν =aκλaλν−1
3
δκνaαβaαβ,the triple product of the tensors in the last term is
aμκaκλaλν =aμκaκλaλν−1
3
aμνaαβaαβ=
(
1
2
−
1
3
)
aμνaαβaαβ=1
6
aμνaλκaλκ.