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. 361

Section 3.1.4. Example of a table for non-equitranslational phase transitions

J.-C. Tolédanod*

3.1.4. Example of a table for non-equitranslational phase transitions

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In the preceding Section 3.1.3[link], a systematic tabulation of possible symmetry changes was provided for the class of equitranslational phase transitions. This tabulation derives from the principles described in Section 3.1.2[link], and relates the enumeration of the symmetry changes at structural transitions to the characteristics of the irreducible representations of the space group [\cal G] of the `parent' (highest-symmetry) phase adjacent to the transition. Systematic extension of this type of tabulation to the general case of transitions involving both a decrease of translational and of point-group symmetry has been achieved by several groups (Tolédano & Tolédano, 1976[link], 1977[link], 1980[link], 1982[link]; Stokes & Hatch, 1988[link]). The reader can refer, in particular, to the latter reference for an exhaustive enumeration of the characteristics of possible transitions. An illustration of the results obtained for a restricted class of parent phases (those associated with the point symmetry [4/m] and to a simple Bravais lattice P) is presented here.

In order to clarify the content Table 3.1.4.1[link], let us recall (cf. Section 3.1.2[link]) that Landau's theory of continuous phase transitions shows that the order parameter of a transition transforms according to a physically irreducible representation of the space group [\cal G] of the high-symmetry phase of the crystal. A physically irreducible representation is either a real irreducible representation of [\cal G] or the direct sum of two complex-conjugate irreducible representations of [\cal G]. To classify the order-parameter symmetries of all possible transitions taking place between a given parent (high-symmetry) phase and another (low-symmetry) phase, it is therefore necessary, for each parent space group, to list the various relevant irreducible representations.

Table 3.1.4.1| top | pdf |
Possible symmetry changes across transitions from a parent phase with space group [P4/m], [P4_2/m], [P4/n], [P4_2/n], [I4/m] or [I4_1/a]

Equitranslational symmetry changes are not included (cf. Section 3.1.3[link]). The coordinates of the points in the second column are referred to the primitive unit cell of the reciprocal lattice. The terms used in the fifth column are introduced in Section 3.1.1.[link] The last column is characteristic of non-equitranslational transitions.

Parent space groupIrreducible representationPossible low-symmetry space groupsMacroscopic characteristics of the transitionChange in the number of atoms per primitive unit cell
Brillouin zone pointDimension of the order parameter
[P4/m] [{1\over 2}, {1\over 2}, 0] 2 [P2/m]; [P2/b] Ferroelastic 2
1 [P4/m]; [P4/n] Non-ferroic 2
[0, 0, {1\over 2}] 2 [P2_1/m] Ferroelastic 2
1 [P4/m]; [P4_2/m] Non-ferroic 2
[{1\over 2}, {1\over 2}, {1\over 2}] 2 [B2/m] Ferroelastic 2
1 [I4/m] Non-ferroic 2
[0, {1\over 2}, {1\over 2}] 2 [B2/m] Ferroelastic 2
1 [I4/m] Non-ferroic 4
[0, {1\over 2}, 0] 2 [P2/m]; [P2/b] Ferroelastic 2
1 [P4/m]; [P4/n] Non-ferroic 4
[P4_2/m] [{1\over 2}, {1\over 2}, 0] 2 [P2/m]; [P2/b] Ferroelastic 2
1 [P4_2/n]; [P4_2/m] Non-ferroic 2
[0, 0, {1\over 2}] 2 [P2_1/m] Ferroelastic 2
2 [P4_1]; [P4_3] Ferroelectric 2
[{1\over 2}, {1\over 2}, {1\over 2}] 2 [B2/m] Ferroelastic 2
1 [I4/m] Non-ferroic 2
[0, {1\over 2}, {1\over 2}] 2 [B2/m] Ferroelastic 2
2 [P4_1/a] Non-ferroic 2
[0, {1\over 2}, 0] 2 [P2/m]; [P2/b] Ferroelastic 2
2 [P4_2/m]; [P4_2/n] Non-ferroic 2
[P4/n] [{1\over 2}, {1\over 2}, 0] 2 [P2/b] Ferroelastic 2
2 [P4] Ferroelectric 2
[0, 0, {1\over 2}] 2 [P2_1/b] Ferroelastic 2
1 [P4/n]; [P4_2/n] Non-ferroic 2
[{1\over 2}, {1\over 2}, {1\over 2}] 2 [B2/b] Ferroelastic 2
2 [I4] Ferroelectric 2
[P4_2/n] [{1\over 2}, {1\over 2}, 0] 2 [P2/b] Ferroelastic 2
[0, 0, {1\over 2}] 2 [P2_1/b] Ferroelastic 2
2 [P4_1]; [P4_3] Ferroelectric 2
[I4/m] [{1\over 2}, -{1\over 2}, -{1\over 2}] 2 [P2_1/m]; [P2_1/b] Ferroelastic 2
1 [P4/m]; [P4_2/m]; [P4/n]; [P4_2/n] Non-ferroic 2
[{1\over 2}, {1\over 2}, 0] 2 [B2/m]; [B2/b] Ferroelastic 2
2 [P4/m]; [P4_2/m]; [P4/n]; [P4_2/n] Non-ferroic 4
[{1\over 2}, 0, 0] 4 [B2/m] Ferroelastic 2
4 [P{\bar 1}] Ferroelastic 8
4 [I4/m]; [I4_1/a] Non-ferroic 8
[{3\over 4}, {1\over 4}, -{1\over 4}] 2 [I4/m]; [I4_1/a] Non-ferroic 4
[I4_1/a] [{1\over 2}, -{1\over 2}, -{1\over 2}] 2 [P2_1/b] Ferroelastic 2
[{1\over 2}, 0, 0] 4 [I{\bar 4}] Higher-order ferroic 8
4 [P{\bar 1}] Ferroelastic 2
4 [P{\bar 1}]; [B2/b] Ferroelastic 4

Each irreducible representation of a given space group can be denoted [\Gamma_n(k^\ast)] and identified by two quantifies. The star [k^\ast], represented by a vector linking the origin of reciprocal space to a point of the first Brillouin zone, specifies the translational symmetry properties of the basis functions of [\Gamma_n(k^\ast)]. The dimension of [\Gamma_n(k^\ast)] is equal to the number of components of the order parameter of the phase transition considered. A given space group has an infinite number of irreducible representations. However, physical considerations restrict a systematic enumeration to only a few irreducible representations. The restrictions arise from the fact that one focuses on continuous (or almost continuous) transitions between strictly periodic crystal structures (i.e. in particular, incommensurate phases are not considered), and have been thoroughly described previously (Tolédano & Tolédano, 1987[link], and references therein).

References

Stokes, H. T. & Hatch, D. M. (1988). Isotropy groups of the 230 crystallographic space groups. Singapore: World Scientific.
Tolédano, J.-C. & Tolédano, P. (1980). Order parameter symmetries and free-energy expansions for purely ferroelastic transitions. Phys. Rev. B, 21, 1139–1172.
Tolédano, J.-C. & Tolédano, P. (1987). The Landau theory of phase transitions. Singapore: World Scientific.
Tolédano, P. & Tolédano, J.-C. (1976). Order parameter symmetries for ferroelectric nonferroelastic transitions. Phys. Rev. B, 14, 3097–3109.
Tolédano, P. & Tolédano, J.-C. (1977). Order parameter symmetries for the phase transitions of nonmagnetic secondary and higher order ferroics. Phys. Rev. B, 16, 386–407.
Tolédano, P. & Tolédano, J.-C. (1982). Non-ferroic phase transitions. Phys. Rev. B, 25, 1946–1964.








































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