Landau theory
The Landau theory is a theory in physics for the description of phase transitions. It is called after the Russian physicist Lev Landau. This theory is based on a polynomial expansion of the free energy as a function of a parameter, the so-called ordering parameter, in the vicinity of the phase transition.
This theory is applied to phase transitions, which are characterized by the loss of certain symmetry elements. The shape of the Landau potential is determined by the symmetry of the phases and can therefore be determined by group-theoretical methods. In fact, the Landau theory is the first application of group theory in thermodynamics.
The basic principles of this theory were presented by Landau in 1937. As a result, this general theory by various working groups has been applied to special cases which therefore designated with slightly different names: Landau - Ginzburg theory of superconductors, Landau Devenshire theory of ferroelectrics, etc.
- 3.1 Superconductivity: Landau - Ginzburg theory
General characteristics of the theory
The Landau theory is a "local" theory. It was intended as an approximation in the environment around the phase transition point, ie for small values of the order parameter. Nevertheless, it happens that the scope of this theory includes a much wider range.
The Landau theory is a " phenomenological " theory: using thermodynamic methods it is capable of all the phenomena that occur in the context of a phase transition to describe in a unified model, but it makes no representations about the microscopic origin of this phase transition. In practice, the expansion coefficients of the Landau theory can be determined by experiment.
Moreover, this theory is a " mean-field " theory: the underlying microscopic interactions are not considered separately, but it is averaged over them. Therefore, this theory can not account for the fluctuations of the order parameter around its equilibrium value. However, this can play a significant role, especially in the vicinity of the phase transition.
Concepts of the Landau theory
Symmetry breaking
The properties of a body are closely related to its symmetry, which can be described in many cases by an appropriate space group. In a second-order phase transition, the symmetry of the system, and thus its properties changes. So it may come among other things, to a spontaneous formation of additional variables, such as magnetization, dielectric polarization or deformation.
As opposed to a phase transition of first order of the state of the system changes continuously in a second-order phase transition. At the point of the phase transition the states of the high - or low-temperature phase match. It follows that a space group must be a subset of the other. In most cases, the phase of higher symmetry of the high temperature phase and the lower the symmetry of the low-temperature phase corresponds to. But this is not a thermodynamic law and therefore allows exceptions, such as the lower Curie point of the Seignettesalzes.
Order parameter Q
In the high-symmetry phase, it is possible according to the postulates of thermodynamics to characterize the whole system by specifying a small number of state variables (such as pressure and temperature). At the phase transition disappear some symmetry properties. The indication of pressure and temperature is no longer sufficient to characterize the state. One therefore needs an additional variable: the order parameter Q. The order parameter is a priori an abstract concept. It describes the process that is causally responsible for the phase transition. In many cases, you can identify him, therefore, with a concrete microscopic process. The order parameter is generally a tensor quantities. The order parameter is defined so that it has a value in the higher symmetry phase is zero and in the lower symmetric phase equal to zero. In addition, for the theory to be symmetry behavior is important.
Thus, the Landau potential leads to a second order phase transition, the three Landau Landau - Ginzburg conditions and the criterion must be met:
Overall, the conditions result in powers of odd order do not appear in the Landau potential. The fourth criterion restricts the possible locations in the Brillouin zone, where the phase transition can take place, strong one.
The Landau potential
To describe the phase transition of the order parameter Q is taken into account as an additional variable in the Gibbs free energy G (P, T). It must be noted that Q is not equivalent to P and T in a certain sense can be arbitrarily during pressure and temperature must be the equilibrium value of Q from the condition that the free energy is to adopt a minimum, be determined.
In the vicinity of the phase transition order of the parameter takes small values. Therefore, the free energy can be expanded in a series in powers of Q. Terms of first and third order in Q are not taken into account, otherwise the high temperature phase or the phase transition point would not be thermodynamically stable states. From this requirement, the Landau conditions ( see above ) result. The free enthalpy thus has the following form:
Wherein the expansion coefficients can depend on the pressure and temperature in principle. The value of the order parameter is determined from
And
The possible solutions to these equations, together with the conditions on the coefficients and the significance of the corresponding phase are summarized below:
Where Tc is the phase transition temperature. Landau under the theory, the simplest assumptions are made for the development of coefficients A and B which satisfy this requirement:
Where a is constant and greater than zero. Substituting these into the free enthalpy, we obtain the Landau potential:
For the order parameter is valid in the high temperature and the low temperature phase.
The phase transition
To study the behavior of the system at the phase transition, the Landau potential is treated as a normal thermodynamic potential. For the entropy S is true:
The entropy of the system without phase transition is. Substituting for Q is the equilibrium values yields:
The entropy remains constant in the phase transition. It is in a low-temperature phase is less than the extrapolated low-temperature phase into the entropy of the high-temperature phase. The specific heat capacity Cp is given by:
Here too is the specific heat capacity of the system without phase transition. The specific heat capacity has a jump at Tc. Since a and b are positive quantities, the heat capacity in the low-temperature phase is higher than in the high temperature phase.
The fact that the first derivative of the Landau potential steadily, the 2nd derivative but discontinuous at Tc means that the Landau potential in this form in fact a 2nd order phase transition describes. In turn, this also shows that a phase transition, which is connected to a change in the symmetry of the system, it must be of at least second order.
The Ordnungsparametersuszeptibilität
An important also for the experimental study size is the Ordnungsparametersuszebtibilität. Its inverse is the second derivative of the Landau potential for the order parameter:
The Ordnungsparametersuszebtibilität therefore in both phases of the form:
Within the framework of Landau theory so for the Ordnungsparametersuszebtibilität follows a Curie -Weiss law with the Curie constant C = 1 / a (high-temperature ) and C = -1/2a ( low temperature phase ).