Description of a prototype example

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, p. 340

Section 3.1.2.2.1. Description of a prototype example

J.-C. Tolédano^d ^*

3.1.2.2.1. Description of a prototype example

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Fig. 3.1.2.1 represents a unit cell of a speculative crystalline structure with a simple tetragonal Bravais lattice, in which a phase transition is assumed to take place. Negative ions (filled circles) occupy the vertices of the tetragonal cell (lattice constants $[a = b \ne c]$ ). A positive ion $[M{^+}]$ is at the centre of the cell.

Figure 3.1.2.1 | top | pdf |

Model of a structural transition. The filled circles at the vertices of the cell are singly charged negative ions and the empty circle at the centre is a singly charged positive ion. d is an arbitrary displacement of the central ion.

This configuration is assumed to be the equilibrium state of the system above the temperature [T_c] of the transition (see Fig. 3.1.2.2). Below [T_c] , equilibrium is assumed to correspond to a structure that only differs from the high-temperature structure by the fact that [M^+] lies out of the centre of the cell in an unspecified direction. Hence the latter equilibrium is characterized by the magnitude and direction of the displacement $[{\bf d}_0 = (d_x, d_y, d_z)]$ of the central ion. At high temperature, the equilibrium corresponds to $[{\bf d}_0 = 0]$ .

Figure 3.1.2.2 | top | pdf |

(a) Variation of the free energy as function of the amplitude of the displacement of the central ion in Fig. 3.1.2.1. (b) Typical temperature dependence in the vicinity of [T_c] of the coefficient of a second-degree term in the Landau expansion (3.1.2.1) whenever this coefficient is strictly positive at [T_c] : one can see that this positivity is also valid slightly above and below [T_c] .

References

International Tables for Crystallography (2006). Vol. D. ch. 3.1, p. 340