Stable state below Tc and physical anomalies induced by the transition

Tolédano, J.-C.

doi:10.1107/97809553602060000642

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

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International Tables for Crystallography (2006). Vol. D. ch. 3.1, pp. 342-343

Section 3.1.2.2.5. Stable state below T_c and physical anomalies induced by the transition

J.-C. Tolédano^d ^*

3.1.2.2.5. Stable state below T_c and physical anomalies induced by the transition

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We have seen that either [d_z] or the couple ( [d_x,d_y] ) of components of the displacement constitute the order parameter of the transition and that the free energy needs only to be expanded as a function of the components of the order parameter. Below the transition, the corresponding coefficient $[\alpha_i]$ is negative and, accordingly, the free energy, limited to its second-degree terms, has a maximum for $[{\bf d} = 0]$ and no minimum. Such a truncated expansion is not sufficient to determine the equilibrium state of the system. The stable state of the system must be determined by positive terms of higher degrees. Let us examine first the simplest case, for which the order parameter coincides with the [d_z] component.

The same symmetry argument used to establish the form (3.1.2.1) of the Landau free energy allows one straightforwardly to assert the absence of a third-degree term in the expansion of F as a function of the order parameter [d_z] , and to check the effective occurrence of a fourth-degree term. If we assume that this simplest form of expansion is sufficient to determine the equilibrium state of the system, the coefficient of the fourth-degree term must be positive in the neighbourhood of [T_c] . Up to the latter degree, the form of the relevant contributions to the free energy is therefore $[F = F_o(T, p) + {\alpha(T-T_c)\over 2}d_z^2 + {\beta\over 4}d_z^4. \eqno (3.1.2.2)]$

In this expression, $[\alpha_1]$ , which is an odd function of [(T - T_c)] since it vanishes and changes sign at [T_c] , has been expanded linearly. Likewise, the lowest-degree expansion of the function $[\beta (T - T_c)]$ is a positive constant in the vicinity of [T_c] . The function [F_0] , which is the zeroth-degree term in the expansion, represents the normal `background' part of the free energy. It behaves smoothly since it does not depend on the order parameter. A plot of [[F(d_z) - F_0]] for three characteristic temperatures is shown in Fig. 3.1.2.4.

Figure 3.1.2.4 | top | pdf |

Plots of the Landau free energy as a function of the order parameter, for values of the temperature above or below [T_c] or coincident with [T_c] . The shape of the plot changes qualitatively from a one-minimum plot to a two-minimum plot.

The minima of F, determined by the set of conditions $[{\partial F \over \partial d_z} = 0\semi \quad {\partial^2 F \over \partial^2 d_z}> 0, \eqno (3.1.2.3)]$ occur above [T_c] for [d_z =0] , as expected. For $[T \,\lt\, T_c]$ they occur for $[d_z^0 = \pm \sqrt{\alpha{(T_c - T)\over \beta}}.\eqno (3.1.2.4)]$

This behaviour has a general validity: the order parameter of a transition is expected, in the framework of Landau's theory, to possess a square-root dependence as a function of the deviation of the temperature from [T_c] .

Note that one finds two minima corresponding to the same value of the free energy and opposite values of [d_z^0] . The corresponding upward and downward displacements of the [M^+] ion (Fig. 3.1.2.1) are distinct states of the system possessing the same stability.

Other physical consequences of the form (3.1.2.2) of the free energy can be drawn: absence of latent heat associated with the crossing of the transition, anomalous behaviour of the specific heat, anomalous behaviour of the dielectric susceptibility related to the order parameter.

The latent heat is $[L = T \Delta S]$ , where $[\Delta S]$ is the difference in entropy between the two phases at [T_c] . We can derive S in each phase from the equilibrium free energy [F(T, p, d_z^0(T,p))] using the expression $[S = - \left.{{\rm d}F\over {\rm d}T}\right\vert d_z^0 = -\left [{\partial F\over \partial T}\left\vert d_z^0 + {\partial F\over \partial d_z}{{\rm d}(d_z)\over{\rm d}T} \right\vert d_z^0\right].\eqno (3.1.2.5)]$ However, since F is a minimum for [d_z = d_z^ 0] , the second contribution vanishes. Hence $[S = -{\alpha\over 2}\left(d_z^0\right)^2 - {\partial F_0\over \partial T}. \eqno (3.1.2.6)]$

Since both [d_z^0] and ( $[\partial F_0/\partial T]$ ) are continuous at [T_c] , there is no entropy jump $[\Delta S = 0]$ , and no latent heat at the transition.

Several values of the specific heat can be considered for a system, depending on the quantity that is maintained constant. In the above example, the displacement d of a positive ion determines the occurrence of an electric dipole (or of a macroscopic polarization P). The quantity $[\varepsilon]$ , which is thermodynamically conjugated to [d_z] , is therefore proportional to an electric field (the conjugation between quantities $[\eta]$ and $[\zeta]$ is expressed by the fact that infinitesimal work on the system has the form $[\zeta \,{\rm d}\eta]$ – cf. Sections 1.1.1.4 and 1.1.5 ). Let us show that the specific heat at constant electric field has a specific type of anomaly.

This specific heat is expressed by $[c_\varepsilon = \left.T{\partial S\over \partial T}\right\vert \varepsilon. \eqno (3.1.2.7)]$ Using (3.1.2.6), we find $[ T> T_c: \ \ c_\varepsilon^0 = -{\partial ^2 F_0(T,p)\over \partial T^2}, \eqno (3.1.2.8)]$ $[\eqalignno{ T \,\lt\, T_c: \ \ c_\varepsilon &= -{\partial ^2 F_0(T,p)\over \partial T^2} -{\alpha\over 2}{{\rm d}(d_z^0)^2\over {\rm d}T} T = c_\varepsilon ^0 + {\alpha^2\over 2\beta}.&\cr &&(3.1.2.8a)}]$ Hence above and below [T_c] the specific heat is a different, smoothly varying function of temperature, determined by the background free energy [F_0(T, p)] and by the smooth variation of the $[\beta]$ coefficient. Fig. 3.1.2.5(a) reproduces the anomaly of the specific heat, which, on cooling through [T_c] , has the form of an upward step.

Figure 3.1.2.5 | top | pdf |

(a) Qualitative temperature dependence of the specific heat at a continuous transition. (b) Temperature dependence of the susceptibility at a continuous transition.

Finally, let us consider the anomaly of the susceptibility $[\chi]$ , which, in the case considered, is proportional to the dielectric susceptibility of the material. It is defined as $[\chi =\left. {\rm lim} _{\varepsilon\rightarrow0} {\partial d_z\over\partial \varepsilon}\right\vert d_z^0. \eqno (3.1.2.9)]$

In order to calculate $[\chi]$ , it is necessary to examine the behaviour of the system in the presence of a small field, $[\varepsilon]$ , conjugated to the order parameter. In this case, the appropriate thermodynamical potential whose minimum determines the equilibrium of the system is not F but $[G = F - d_z \varepsilon]$ . Minimizing G with respect to [d_z] leads to $[d_z \left\{\alpha(T - T_c) + \beta d_z^2\right\}= \varepsilon. \eqno (3.1.2.10)]$

For small values of $[\varepsilon]$ , the solution of this equation must tend towards the equilibrium values [d_z = 0] . Deriving these solutions with respect to $[\varepsilon]$ , we obtain $[\chi(T>T_c) = {1\over \alpha(T - T_c)}; \quad \chi(T \,\lt\, T_c) = {1\over 2\alpha(T - T_c)}. \eqno (3.1.2.11)]$ The susceptibility goes to infinity when $[T \rightarrow T_c]$ from either side of the transition (Fig. 3.1.2.5b). The set of anomalies in $[c_\varepsilon]$ and $[\chi]$ described in this paragraph represents the basic effects of temperature on quantities that are affected by a phase transition. They constitute the `canonical signature' of a phase transition of the continuous type.

Certain complications arise in the cases where the transition is not strictly continuous, where the order parameter is coupled to other degrees of freedom, and where the order parameter is not one-dimensional. We consider one of these complications in Section 3.1.2.3.

References

International Tables for Crystallography (2006). Vol. D. ch. 3.1, pp. 342-343

Section 3.1.2.2.5. Stable state below Tc and physical anomalies induced by the transition

3.1.2.2.5. Stable state below Tc and physical anomalies induced by the transition

References

Section 3.1.2.2.5. Stable state below T_c and physical anomalies induced by the transition

3.1.2.2.5. Stable state below T_c and physical anomalies induced by the transition