in which the order parametersSmare defined as
Sm¼ D^2 m 0 ðÞ (^) LA, (^) LA,0
,
Sm¼ 1 mSm:
½ 2 : 319
The order parameters in [2.319] are identical to the corresponding
coefficients in the expansion of the probability density [2.316].
The set of five order parameters in [2.318] transform under rotation
like a second-rank tensor and constitute the irreducible representation of
the 33 Saupe order matrix defined by ( 22 )
Sij¼^32 cos (^) icos (^) j
ij=2, ½ 2 : 320
in which (^) k, fork¼{x,y,z}, is the angle between thekth axis of the
alignment frame and thez-axis of the laboratory frame. The spherical
and Saupe order parameters are related by
S 0 ¼Szz,
S 1 ¼
ffiffiffiffiffiffiffiffi
2 = 3
p
Sxz iSyz
,
S 2 ¼
ffiffiffiffiffiffiffiffi
1 = 6
p
SxxSyy
i
ffiffiffiffiffiffiffiffi
2 = 3
p
Sxy:
½ 2 : 321
The Saupe order matrix is a traceless, real, and symmetric Cartesian
tensor of rank 2. Consequently, the alignment frame always can be
defined such that the order matrix is diagonal with principal valuesSxx,
Syy, andSzz. In this frame,S 1 ¼0 and
H¼^13 TrfgCuvþA^02 Szz
X^2
k¼ 2
D^2 k 0 ðÞ (^) AP, (^) APFk 2 ðPASÞ
(
þ
ffiffi
1
6
q
SxxSyy
X^2
k¼ 2
D^2 k 2 ðÞþ (^) AP, (^) AP, (^) AP D^2 k 2 ðÞ (^) AP, (^) AP, (^) AP
Fk 2 ðPASÞ
)
:
½ 2 : 322
This expression is frequently expressed as ( 23 )
H¼^13 TrfgCuvþA^02 Aa
X^2
k¼ 2
D^2 k 0 ðÞ (^) AP, (^) APF 2 kðPASÞ
(
þ
ffiffi
3
8
q
Ar
X^2
k¼ 2
D^2 k 2 ðÞþ (^) AP, (^) AP, (^) AP D^2 k 2 ðÞ (^) AP, (^) AP, (^) AP
F 2 kðPASÞ
)
,
½ 2 : 323
2.8 AVERAGING OF THESPINHAMILTONIANS 109