International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2006). Vol. D. ch. 3.1, p. 341

Section 3.1.2.2.2. Basic assumptions and strategy

J.-C. Tolédanod*

3.1.2.2.2. Basic assumptions and strategy

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Our aim is to determine the above displacement as a function of temperature. Landau's strategy is to determine [{\bf d}_0] by a variational method. One considers an arbitrary displacement d of the [M^+] ion. For given temperature T and pressure p (or volume V), and specified values of the components of d, there is, in principle, a definite value [F(T, p, d_x, d_y, d_z)] for the free energy F of the system. This function is a variational free energy since it is calculated for an arbitrary displacement. The equilibrium displacement [{\bf d}_0(T, p)] is defined as the displacement that minimizes the variational free energy F. The equilibrium free energy of the system is [F_{\rm eq}(T,p) = F (T, p, {\bf d}_0)]. Note that, strictly speaking, in the case of a given pressure, one would have to consider a variational Gibbs function ([F + pV]) in order to determine the equilibrium of the system. We will respect the current use in the framework of Landau's theory of denoting this function F and call it a free energy, though this function might actually be a Gibbs potential.

The former strategy is not very useful as long as one does not know the form of the variational free energy as a function of the components of the displacement. The second step of Landau's theory is to show that, given general assumptions, one is able to determine simply the form of [F(T, p, {\bf d})] in the required range of values of the functions' arguments.

The basic assumption is that of continuity of the phase transition. It is in fact a dual assumption. On the one hand, one assumes that the equilibrium displacement [{\bf d}_0(T, p)] has components varying continuously across the transition at [T_c]. On the other hand, one assumes that F is a continuous and derivable function of [(T, p, {\bf d})], which can be expanded in the form of a Taylor expansion as function of these arguments.

Invoking the continuity leads to the observation that, on either side of [T_c], [|{\bf d}_0|] is small, and that, accordingly, one can restrict the determination of the functional form of [F(T, p, {\bf d})] to small values of [(d_x, d_y, d_z)] and of [|T - T_c|]. F will then be equal to the sum of the first relevant terms of a Taylor series in the preceding variables.








































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