Chemistry Reference
In-Depth Information
phase transition, the total magnetization is an order parameter, for example. In
general, the order parameter can be a scalar, a vector, or even a tensor. Landau
theory can be understood as a unification of earlier mean-field theories such
as the van der Waals theory of the liquid-gas transition
2
or Weiss's molecular
field theory of ferromagnetism.
13
It is based on the crucial assumption that the
free energy is an analytic function of the order parameter
m
and can thus be
expanded in a power series:
rm
2
wm
3
um
4
m
5
F
¼
F
L
ð
m
Þ¼
F
0
þ
þ
þ
þ
O
ð
Þ
½
1
Close to the phase transition, the coefficients
r
,
w
,
u
vary slowly with respect to
the external parameters such as temperature, pressure, and electric or mag-
netic field. For a given system, the coefficients can be determined either by a
first-principle calculation starting from a microscopic model, or, phenomeno-
logically, by comparison with experimental data. The correct equilibrium
value of the order parameter
m
for each set of external parameter values is
found by minimizing
F
L
with respect to
m.
Let us now discuss the basic properties of phase transitions that result from
the Landau free energy, Eq. [1]. If the coefficient
r
is sufficiently large, the mini-
mum of
F
L
is located at
m
0, i.e., the system is in the disordered phase. In con-
trast, for sufficiently small (negative)
r
, the minimum is at some nonzero
m
,
putting the system into the ordered phase. Depending on the value of
w
, the
Landau free energy describes a first-order or a continuous transition. If
w
¼
6¼
0, the order parameter jumps discontinuously from
m
¼
0to
m
6¼
0, i.e.,
the transition is of first order. If
w
0 (as is often the case due to symmetry),
the theory describes a continuous transition or critical point at
r
¼
¼
0. In this
case,
r
can be understood as the distance from the critical point,
r
T
c
.
Within Landau theory, the behavior close to a critical point is superuniversal,
meaning that all continuous phase transitions display the same behavior.
For instance, the order parameter vanishes as
m
/
T
1
=
2
when the critical
¼ð
r
=
2
u
Þ
point
r
0 is approached from the ordered phase, implying that the criti-
cal exponent
¼
, which describes the singularity of the order parameter at the
critical point via
m
b
T
c
j
b
, always has the mean-field value
2
.
In experiments, the critical exponent values are usually different fromwhat
Landau theory predicts; and while they show some degree of universality, it is
weaker than the predicted superuniversality. For instance, all three-dimensional
Ising ferromagnets (i.e., ferromagnets with uniaxial symmetry and a scalar order
parameter) fall into the same universality class with
j
b
/j
/j
r
T
b
0
:
32 while all two-
1
dimensional
8
. All three-dimensional Heisenberg
magnets [for which the order parameter is a three-component vector with
O
(3) symmetry] also have a common value of
Ising magnets have
b
35, but this value for
Heisenberg magnets is different from that in Ising magnets.
The failure of Landau theory to describe the critical behavior correctly
was the central puzzle in phase transition theory over many decades. It was
b
0
:
