Chemistry Reference
In-Depth Information
where
is called the correlation length critical exponent. This divergence was
observed in 1873 in a famous experiment by Andrews:
1
A fluid becomes milky
when approaching its critical point because the length scale of its density fluc-
tuations reaches the wavelength of light. This phenomenon is called critical
opalescence.
Close to the critical point,
n
is the only relevant length scale in the sys-
tem. Therefore, the physical properties must be unchanged, if all lengths in the
system are rescaled by a common factor
b
, and at the same time the external
parameters are adjusted in such a way that the correlation length retains its old
value. This gives rise to a homogeneity relation for the free energy density
f
x
¼ð
k
B
T
=
V
Þ
log
Z
,
b
d
f
rb
1=
n
;
hb
y
h
ð
;
Þ¼
ð
Þ
½
5
f
r
h
The scale factor
b
is an arbitrary number, and
y
h
is another critical exponent.
Analogous homogeneity relations for other thermodynamic quantities can be
obtained by taking derivatives of
f.
These homogeneity laws were first
obtained phenomenologically
14
and are sometimes summarily called the scal-
ing hypothesis. Within the framework of the modern renormalization group
theory of phase transitions,
3,4
the scaling laws can be derived from first
principles. The diverging correlation length is also responsible for the above-
mentioned universality of the critical behavior. Close to the critical point, the
system effectively averages over large volumes rendering microscopic system
details irrelevant. As a result, the universality classes are determined only by
symmetries and the spatial dimensionality.
In addition to the critical exponents
and
y
h
defined above, other expo-
nents describe the dependence of the order parameter and its correlations on
the distance from the critical point and on the field conjugate to the order
parameter. The definitions of the most commonly used critical exponents
are summarized in Table 1. These exponents are not all independent from
each other. The four thermodynamic exponents
n
a
,
b
,
g
,
d
all derive from the
Table 1
Definitions of Critical Exponents
Exponent
Definition
a
Conditions
j
a
Specific heat
a
/j
!
0
;
¼
0
c
r
r
h
Þ
b
Order parameter
b
m
/ð
r
r
!
0
;
h
¼
0
j
g
Susceptibility
g
w
/j
r
r
!
0
;
h
¼
0
j
d
sgn
Critical isotherm
d
h
/j
m
ð
m
Þ
r
¼
0
;
h
!
0
x
/j
r
j
n
Correlation length
n
r
! 0;
h
¼ 0
G
ð
x
Þ/j
x
j
d
þ2
Z
Correlation function
Z
r
¼ 0;
h
¼ 0
z
Dynamical
z
x
t
/
x
ln
x
t
/
x
c
a
m
is the order parameter, and
h
is the conjugate field. The variable
r
denotes the distance from
the critical point, and
d
is the space dimensionality. The exponent
y
h
defined in Eq. [5] is related to
d
Activated dynamical
c
via
y
h
¼
d
d
=ð
1
þ
d
Þ
.
