Physics 481: Condensed Matter Physics  Homework 4
due date: Friday, Feb 11, 2011
Problem 1: Nematic order parameter
(15 points)
Consider a threedimensional nematic liquid crystal consisting of
N
rodlike molecules. The nematic
order parameter (i.e., a quantity characterizing the degree of nematic order) can be defined as
S
=
N
X
j
=1
¿
cos
2
Θ
j

1
3
where Θ
j
is the angle between the axis of molecule
j
and the director ˆ
n
.
h
. . .
i
indicates the
thermodynamic average.
a) Show that
S
is indeed a nematic order parameter. To this end, show that it vanishes if the rods
point in random directions. Also calculate
S
for a state with perfect nematic order.
b) In a magnet, an order parameter can be defined as
M
=
∑
j
h
cos Θ
j
i
where Θ
j
is the angle
between the spin
j
and the magnetization direction ˆ
n
.
Is this an order parameter for the
nematic, too? Explain your answer.
c) Consider hypothetical molecules shaped like a + sign (symmetric under 90
◦
rotations).
Sug
gest an order parameter characterizing the degree of order in their relative orientations. Hint:
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 Spring '11
 ThomasVojta
 Physics, Work, Liquid crystal, nematic order parameter

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