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

International Tables for Crystallography (2013). Vol. D, ch. 3.1, pp. 380-381

Section Examples

V. Janovecb* and V. Kopskýe Examples

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Example Phase transition in triglycine sulfate (TGS).  Assume that the space groups of both parent (high-symmetry) and ferroic (low-symmetry) phases are known: [{\cal G}=P2_1/c \ (C_{2h}^5)], [{\cal F}_1=] [P2_1 \ (C_2^2)]. The same number of formula units in the primitive unit cell in both phases suggests that the transition is an equitranslational one. This conclusion can be checked in the lattice of equitranslational subgroups of the software GI[\star]KoBo-1. There we find for the low-symmetry space group the symbol [P112_1({\bf b}/4)], where the vector in parentheses expresses the shift of the origin with respect to the conventional origin given in IT A (2005[link]).

In Table[link], one finds that the corresponding point-group-symmetry descent [2_z/m_z\Downarrow 2_z] is associated with irreducible representation [\Gamma_{\eta}=A_u]. The corresponding principal tensor parameters of lowest rank are the pseudoscalar [\varepsilon] (enantiomorphism or chirality) and the vector of spontaneous polarization with one nonzero morphic component [P_3] – the transition is a proper ferroelectric one. The non-ferroelastic ([n_a=1]) full ferroelectric phase has two ferroelectric domain states ([n_f=n_e=2]). Other principal tensor parameters (morphic tensor components that transform according to [\Gamma_{\eta}]) are available in the software GI[\star]KoBo-1: [ g_{1}], [g_{2}], [g_{3}], [g_{6}]; [d_{31}], [d_{32}], [d_{33}], [d_{36}], [d_{14}], [d_{15}], [d_{24}], [d_{25}]. Property tensors with these components are listed in Table[link]. As shown in Section 3.4.2[link] , all these components change sign when one passes from one domain state to the other. Since there is no intermediate group between G and F, there are no secondary tensor parameters.

Example Phase transitions in barium titanate (BaTiO3).  We shall illustrate the solution of the inverse Landau problem and the need to correlate the crystallographic system with the Cartesian crystallophysical coordinate system. The space-group type of the parent phase is [{\cal G}=Pm{\overline 3}m], and those of the three ferroic phases are [ {\cal F}_{1}^{(1)}=P4mm], [{\cal F}_{1}^{(2)}=Cm2m], [ {\cal F}_{1}^{(3)}=R3m], all with one formula unit in the primitive unit cell.

This information is not complete. To perform mode analysis, we must specify these space groups by saying that the lattice symbol P in the first case and the lattice symbol R in the third case are given with reference to the cubic crystallographic basis ([{\bf a},{\bf b},{\bf c}]), while lattice symbol C in the second case is given with reference to crystallographic basis [[({\bf a}-{\bf b}),({\bf a}+{\bf b}),{\bf c}]]. If we now identify vectors of the cubic crystallographic basis with vectors of the Cartesian basis by [{\bf a}=a{\bf e}_{x}], [{\bf b}=a{\bf e}_{y}], [{\bf c}=a{\bf e}_{z}], where [{\bf e}_{x}], [{\bf e}_{y}], [{\bf e}_{z}] are three orthonormal vectors, we can see that the corresponding point groups are [ F_{1}^{(1)}=4_{z}m_{x}m_{xy}], [F_{1}^{(2)}=m_{x{\overline y}}2_{xy}m_{z}], [F_{1}^{(3)}=3_{p}m_{x{\overline y}}].

Notice that without specification of crystallographic bases one could interpret the point group of the space group [Cm2m] as [m_{x}2_{y}m_{z}]. Bases are therefore always specified in lattices of equitranslational subgroups of the space groups that are available in the software GI[ \star]KoBo-1, where we can check that all three symmetry descents are equitranslational.

In Table[link], we find that these three ferroic subgroups are epikernels of the R-irep [\Gamma_{\eta}=T_{1u}] with the following principal tensor components: [P_{3}], [P_{1}=P_{2}], [P_{1}=P_{2}=P_{3}], respectively. Other principal tensor parameters can be found in the main tables of the software GI[ \star]KoBo-1. The knowledge of the representation [\Gamma_{\eta}] allows one to perform soft-mode analysis (see e.g. Rousseau et al., 1981[link]).

For the tetragonal ferroelectric phase with [F_1=4_zm_xm_y], we find in Fig.[link] an intermediate group [L_1=4_z/m_zm_xm_{xy}]. In Table[link], we check that this is an epikernel of the R-irep [E_g] with secondary tensor parameter [\delta u_3]. This phase is a full (proper) ferroelectric and partial ferroelastic one.

More details about symmetry aspects of structural phase transitions can be found in monographs by Izyumov & Syromiatnikov (1990[link]), Kociński (1983[link], 1990[link]), Landau & Lifshitz (1969[link]), Lyubarskii (1960[link]), Tolédano & Dmitriev (1996[link]) and Tolédano & Tolédano (1987[link]). Group–subgroup relations of space groups are treated extensively in IT A1 (2004[link]).


International Tables for Crystallography (2004). Vol. A1. Symmetry relations between space groups, edited by H. Wondratschek & U. Müller. Dordrecht: Kluwer Academic Publishers.
International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn. Heidelberg: Springer.
Izyumov, Yu. A. & Syromiatnikov, V. N. (1990). Phase transitions and crystal symmetry. Dordrecht: Kluwer Academic Publishers.
Kociński, J. (1983). Theory of symmetry changes at continuous phase transitions. Warsaw: PWN – Polish Scientific Publishers; Amsterdam: Elsevier.
Kociński, J. (1990). Commensurate and incommensurate phase transitions. Warsaw: PWN – Polish Scientific Publishers; Amsterdam: Elsevier.
Landau, L. D. & Lifshitz, E. M. (1969). Course in theoretical physics, Vol. 5, Statistical physics, 2nd ed. Oxford: Pergamon Press.
Lyubarskii, G. Ya. (1960). The application of group theory in physics. Oxford: Pergamon Press.
Rousseau, D. L., Bauman, R. P. & Porto, S. P. S. (1981). Normal mode determination in crystals. J. Raman Spectrosc. 10, 253–290.
Tolédano, J.-C. & Tolédano, P. (1987). The Landau theory of phase transitions. Singapore: World Scientific.
Tolédano, P. & Dmitriev, V. (1996). Reconstructive phase transitions. Singapore: World Scientific.

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