Chemistry Reference
In-Depth Information
f
ð
is a dimensionless scaling function. This can also be used to find how the
critical point shifts as a function of
L
in geometries that allow a sharp transi-
tion at finite
L
(e.g., layers of finite thickness). The finite-
L
phase transition
corresponds to a singularity in the scaling function at some nonzero argument
x
c
. The transition thus occurs at
r
c
L
1=
n
¼
x
Þ
x
c
, and the transition temperature
of the finite-size system is shifted from the bulk value
T
c
by
T
c
ð
L
Þ
T
c
/
x
c
L
1=
n
T
c
ð
L
Þ
r
c
¼
½
9
Note that the simple form of finite-size scaling summarized above is only valid
below the upper critical dimension
d
c
of the phase transition. Finite-size scal-
ing can be generalized to dimensions above
d
c
, but this requires taking dan-
gerously irrelevant variables into account. One important consequence is that
the shift of the critical temperature,
T
c
ð
Þ
T
c
/
L
j
is controlled by an
L
exponent
.
Finite-size scaling has become one of the most powerful tools for analyz-
ing computer simulation data of phase transitions. Instead of treating finite-
size effects as errors to be avoided, one can simulate systems of varying size
and test whether or not homogeneity relations such as Eq. [8] are fulfilled.
Fits of the simulation data to the finite-size scaling forms of the observables
then yield values for the critical exponents. We will discuss examples of this
method later in the chapter.
j
that in general is different from 1
=
n
Quenched Disorder
Realistic systems always contain some amount of quenched (frozen-in)
disorder in the form of vacancies, impurity atoms, dislocations, or other types
of imperfections. Understanding their influence on the behavior of phase transi-
tions and critical points is therefore important for analyzing experiments. In this
section, we focus on the simplest type of disorder (sometimes called weak dis-
order, random-
T
c
disorder, or, from the analogy to quantum field theory,
random-mass disorder) by assuming that the impurities and defects do not
change qualitatively the bulk phases that are separated by the transition. They
only lead to spatial variations of the coupling strength and thus of the local critical
temperature. In ferromagnetic materials, random-
T
c
disorder can be achieved,
for example, by diluting the lattice, which means by replacing magnetic atoms
with nonmagnetic ones. Within a LGW theory such as Eq. [3], random
T
c
dis-
order can be modeled by making the parameter
r
(which measures the distance
from the critical point) a random function of spatial position,
r
.
The presence of quenched disorder naturally leads to the following ques-
!
r
þ
d
r
ð
x
Þ
tions:
.
Will the phase transition remain sharp or will it be rounded?
.
Will the order of the transition (first order or continuous) remain the
same as in the clean case?
