Reduction of the number of relevant degrees of freedom: order parameter

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. 341-342

Section 3.1.2.2.4. Reduction of the number of relevant degrees of freedom: order parameter

J.-C. Tolédano^d ^*

3.1.2.2.4. Reduction of the number of relevant degrees of freedom: order parameter

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Let us now derive the key result of the theory, namely, that either the component [d_z] or the pair of components ( [d_x,d_y] ) will take nonzero values below [T_c] (but not both). The meaning of this result will be clarified by symmetry considerations.

The derivation of this result relies on the fact that one, and one only, of the two coefficients $[\alpha_i]$ in equation (3.1.2.1) must vanish and change sign at [T_c] , and that the other coefficient must remain positive in the neighbourhood of [T_c] .

(a) Before establishing the latter property in (b) hereunder, let us show that its validity implies the stated key result of the theory. Indeed, if one $[\alpha_i]$ coefficient is strictly positive (e.g. $[\alpha_1>0]$ ), then the minimum of F with respect to the components of d (e.g. ) multiplying this coefficient in (3.1.2.1) occurs for zero equilibrium values of these components (e.g. ) in the vicinity of , above and below this temperature. Hence, depending on the coefficient $[\alpha_i]$ which remains positive, either or the pair () can be omitted, in the first place, from the free-energy expansion. The remaining set of components is called the order parameter of the transition. At this stage, this fundamental quantity is defined as the set of degrees of freedom, the coefficient of which in the second-degree contribution to F vanishes and changes sign at . The number of independent components of the order parameter (one in the case of , two in the case of the pair ) is called the dimension of the order parameter.

Note that the preceding result means that the displacement of the ion below cannot occur in an arbitrary direction of space. It is either directed along the z axis, or in the () plane.

(b) Let us now establish the property of the $[\alpha_i]$ postulated above.

At [T_c] , the equilibrium values of the components of d are zero. Therefore, at this temperature, the variational free energy (3.1.2.1) is minimum for [d_x, d_y, d_z = 0] . Considering the form (3.1.2.1) of F, this property implies that we have (Fig. 3.1.2.2) $[\alpha_i(T_c) \ge 0]$ ( [i = 1, 2] ).

Note that these inequalities cannot be strict for both coefficients $[\alpha_i]$ , because their positiveness would hold on either side of [T_c] in the vicinity of this temperature. Consequently, the minimum of F would correspond to $[{\bf d} =0]$ on either side of the transition while the situation assumed is only compatible with this result above [T_c] . Using the converse argument that the equilibrium values of the components of d are not all equal to zero below [T_c] leads easily to the conclusion that one, at least, of the two coefficients $[\alpha_i]$ must vanish at [T_c] and become negative below this temperature.

Let us now show that the two coefficients $[\alpha_i]$ cannot vanish simultaneously at [T_c] . This result relies on the `reasonable' assumption that the two coefficients $[\alpha_i]$ are different functions of temperature and pressure (or volume), no constraint in this respect being imposed by the symmetry of the system.

Fig. 3.1.2.3 shows, in the [(T, p)] plane, the two lines corresponding to the vanishing of the two functions $[\alpha_i]$ . The simultaneous vanishing of the two coefficients occurs at an isolated point [(T_0, p_0)] . Let us consider, for instance, the situation depicted in Fig. 3.1.2.3. For [p> p_0] , on lowering the temperature, $[\alpha_1]$ vanishes at [T'] and $[\alpha_2]$ remains positive in the neighbourhood of [T'] . Hence, the equilibrium value of the set ( [d_x,d_y] ) remains equal to zero on either side of [T'] . A transition at this temperature will only concern a possible change in [d_z^0] .

Figure 3.1.2.3 | top | pdf |

Plots representative of the equations $[\alpha_1(p, T)=0]$ and $[\alpha_2(p, T)=0]$ . The simultaneous vanishing of these coefficients occurs for a single couple of temperature and pressure ( [p_0] , [T_0] ).

Likewise for p below [p_0] , a transition at $[T^{\prime\prime}]$ will only concern a possible change of the set of components ( [d_x^0,d_y^0] ), the third component [d_z] remaining equal to zero on either sides of $[T^{\prime\prime}]$ . Hence an infinitesimal change of the pressure (for instance a small fluctuation of the atmospheric pressure) from above [p_0] to below [p_0] will modify qualitatively the nature of the phase transformation with the direction of the displacement changing abruptly from z to the ( [x, y] ) plane. As will be seen below, the crystalline symmetries of the phases stable below $[T^{\prime}]$ and $[T^{\prime\prime}]$ are different. This is a singular situation, of instability, of the type of phase transition, not encountered in real systems. Rather, the standard situation corresponds to pressures away from [p_0] , for which a slight change of the pressure does not modify significantly the direction of the displacement. In this case, one coefficient $[\alpha_i]$ only vanishes and changes sign at the transition temperature, as stated above.

References

International Tables for Crystallography (2006). Vol. D. ch. 3.1, pp. 341-342