Landau potential

The excess thermodynamic properties correlated with phase transitions are conveniently described in terms of a macroscopic order parameter Q. Formal relations between Q and the excess thermodynamic properties associated with a transition are conveniently derived by expanding the Gibbs free energy of transition in terms of a Landau potential ... 

Obviously, if we know experimentally the behavior of the macroscopic ordering parameter with T, we may determine the corresponding coefficients of the Landau expansion (eq. 2.52). However, things are not so easy when different transitions are superimposed (such as, for instance, the displacive and order-disorder transitions in feldspars). In these cases the Landau potential is a summation of terms corresponding to the different reactions plus a couphng factor associated with the common elastic strain. 

IMmcm transition as a function of composition at 296 K in the CaTiOs-SrTiOs solid solntion. The curve is a fit to the data using a standard 246 Landau potential. From Carpenter et al. (2000b). 

An ideal situation for the observation of twin structures is an array of parallel, non-periodically spaced walls. The diffraction signal has then a dog-bone structure as shown in Figure 2. The equi-intensity surface shows diffraction signals at an intensity of lO " compared with the intensity of the two Bragg peaks which are located inside the two thicker ends of the dog bone (Locherer et al. 1998). The intensity in the handle of the dog bone is then directly related to the thickness and internal structure of the twin wall. In order to calibrate the wall thickness the following theoretical framework is chosen. The specific energy of a wall follows in lowest order theory from a Landau potential... 

Using these energies as reference points we may now estimate the wall energy in LaAlOs at low temperatures. The Landau potential follows from our observation that cc / T-Tc (0s T) with... 

Figure 5. The dependence of the critical exponent p on the Landau coefficients. The critical exponents were calculated by least squares fitting of an equilibrium order parameter evolution from a Landau potential. The transition temperature was kept constant, while the b and c coefficients were varied thus changing the value of the a coefficient. The dark grey regions show critical exponents of less than 0.25, approaching a first order type of behaviour. The light grey regions indicate a zone were the critical exponent is between 0.25 and 0.5, and the transition is continuous.

Palmer DC (1994) Stuffed derivatives of the silica polymorphs. Rev Mineral 29 83-122 Palmer DC, Bismayer U, Salje EKH (1990) Phase transitions in leucite order parameter behavior and the Landau potential deduced from Raman spectroscopy and birefringence studies. Phys Chem Minerals 17 259-265... 

Fig.12. Ginzburg-Landau potential function f(a,t)==at +bt, b>0, the sections of which are shown in Fig.11. The point t=0,a=0 is a parabolic umbilic (cf.Sect.2.6.2)...

In the chapter two approaches to the crystals ordered states description are used. The first is basing on the group theoretical considerations incorporated in theory with the help of Landau potential. The second way of ordering description consist of accounting different models of interatomic interactions. Our goal is to elucidate correlations between... 

Thus, our problem is reduced to the calculation of the dependence of the coefficient in the Landau potential for the squared order parameter on the effective pair interactions. It is this coefficient that determines the temperature of the loss of stability of the high-temperature phase with respect to the stabilization of the order in the 1 arrangement of eations in the... 

It is convenient to represent the Landau potential describing such ordering as a function of the two components of the effective order parameter... 

Repeating the calculations of the previous section, we obtain the part of the Landau potential quadratic in the components in the form ... 

The quadratic part of the effective Landau potential, dependent on... 

As all these functions approximate the energy of the same interaction, in main aspects the differences between them are nonessential. The explicit relationships between some generalizations of these functions are demonstrated in the papers [[43],[44]. To consider general problems which are observed in this chapter it is enough to confine ourselves to investigation of one representative of this class of pair interactions potentials. As a representative it is convenient to choose the (1) potential without the Dyj multiplier, that does not play any role in the comparison of the Landau potential coefficients. 

The second important problem which must be solved to proeeed to the computations is limitation of the accoimting coordinate spheres number. There are two functions expressing the energy of state - Hamiltonian and Landau potential. In the second order respect to generalized coordinates these fimctions can be written as... 

---