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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. D. ch. 3.1, pp. 355-360

Section 3.1.3.3. Tables of equitranslational phase transitions associated with irreducible representations

V. Janovecb* and V. Kopskýe

3.1.3.3. Tables of equitranslational phase transitions associated with irreducible representations

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The first systematic symmetry analysis of Landau-type phase transitions was performed by Indenbom (1960[link]), who found all equitranslational phase transitions that can be accomplished continuously. A table of all crystallographic point groups G along with all their physically irreducible representations, corresponding ferroic point groups F and related data has been compiled by Janovec et al. (1975[link]). These data are presented in an improved form in Table 3.1.3.1[link] together with corresponding principal tensor parameters and numbers of ferroic, ferroelectric and ferroelastic domain states. This table facilitates solving of the following typical problems:

  • (1) Inverse Landau problem (Ascher & Kobayashi, 1977[link]) of equitranslational phase transitions: For a given equitranslational symmetry descent [{\cal G}\Downarrow^t{\cal F}] (determined for example from diffraction experiments), find the representation [\Gamma_{\eta}] of [\cal G] that specifies the transformation properties of the primary order parameter. Solution: In Table 3.1.3.1[link], one finds a physically irreducible representation [\Gamma_{\eta}] of the point group G of [\cal G] with epikernel F (point group of [\cal F]). For some symmetry descents from cubic point groups [G=432], [\bar4 3m] and [m\bar3m], the inverse Landau problem has two solutions, which are given in Table 3.1.3.2[link].

    Table 3.1.3.1 | top | pdf |
    Point-group symmetry descents associated with irreducible representations

    Property tensors that appear in this table: [\varepsilon] enantiomorphism, chirality; [P_i] dielectric polarization; [u_{\mu}] strain; [g_{\mu}] optical activity; [d_{i\mu}] piezoelectric tensor; [A_{i\mu}] electrogyration tensor; [\pi_{\mu\nu}] piezo-optic tensor ([i=1,2,3]; [\mu,\nu=1,2,\ldots,6]). Applications of this table to symmetry analysis of equitranslational phase transitions and to changes of property tensors at ferroic transitions are explained in Section 3.1.3.3[link].

    (a) Triclinic parent groups

    R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
    [F_1] [n_F] [n_f] [n_a] [n_e]
    Parent symmetry [\bi G]: [\quad {\bf 1}\quad {\bi C}_{\bf 1}]
    No ferroic symmetry descent
    Parent symmetry [\bi G]: [\quad{\bf\overline 1}\quad {\bi C}_{\bi i}]
    [A_{u}] [{\sf x}_1^-] [1] [C_1] [1] All components of odd parity tensors 2 1 2

    (b) Monoclinic parent groups

    R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
    [F_1] [n_F] [n_f] [n_a] [n_e]
    Parent symmetry [\bi G]: [\quad{\bf 2}_{\bi z} \quad {\bi C}_{\bf 2{\bi z}}]
    B [{\sf x}_3] 1 [C_1] 1 [P_{1}], [P_{2}]; [u_{4}, u_{5}] 2 2 2
    Parent symmetry [\bi G]: [\quad{\bi m}_{\bi z}\quad {\bi C}_{\bi sz}]
    [A''] [{\sf x}_3] 1 [C_1] 1 [\varepsilon]; [P_{3}]; [u_{4}, u_{5}] 2 2 2
    Parent symmetry [\bi G]: [\quad {\bf 2}_{\bi z}{\bf /}{\bi m}_{\bi z}\quad {\bi C}_{\bf 2{\bi hz}}]
    [B_g] [{\sf x}_3^+] [{ {\overline 1}}] [C_i] 1 [u_{4}, u_{5}] 2 2 0
    [A_u] [{\sf x}_1^-] [2_z] [C_{2z}] 1 [\varepsilon]; [P_{3}] 2 1 2
    [B_u] [{\sf x}_3^-] [m_z] [C_{sz}] 1 [P_{1}, P_2] 2 1 2

    (c) Orthorhombic parent groups

    R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
    [F_1] [n_F] [n_f] [n_a] [n_e]
    Parent symmetry [\bi G]: [\quad {\bf 2_{\bi x}2_{\bi y}2_{\bi z}\quad {\bi D}_{2}}]
    [B_{1g}] [{\sf x}_{2}] [2_{z}] [C_{2z}] 1 [P_{3}]; [u_{6}] 2 2 2
    [B_{3g}] [{\sf x}_{3}] [2_{x}] [C_{2x}] 1 [P_{1}]; [u_{4}] 2 2 2
    [B_{2g}] [{\sf x}_{4}] [2_{y}] [C_{2y}] 1 [P_{2}]; [u_{5}] 2 2 2
    Parent symmetry [\bi G]: [\quad{\bi m}_{\bi x}{\bi m}_{\bi y}{\bf 2}_{\bi z}\quad {\bi C}_{\bf 2{\bi vz}}]
    [A_{2}] [{\sf x}_{2}] [2_{z}] [C_{2z}] 1 [u_{6}] 2 2 1
    [B_{2}] [{\sf x}_{3}] [m_{x}] [C_{sx}] 1 [P_2]; [u_{4}] 2 2 2
    [B_{1}] [{\sf x}_{4}] [m_{y}] [C_{sy}] 1 [P_{1}]; [u_5] 2 2 2
    Parent symmetry [\bi G]: [\quad{\bi m_{x}m_{y}m_{z}\quad D_{\bf 2{\bi h}}}]
    [B_{1g}] [{\sf x}_{2}^{+}] [2_{z}/m_{z}] [C_{2hz}] 1 [u_{6}] 2 2 0
    [B_{3g}] [{\sf x}_{3}^{+}] [2_{x}/m_{x}] [C_{2hx}] 1 [u_{4}] 2 2 0
    [B_{2g}] [{\sf x}_{4}^{+}] [2_{y}/m_{y}] [C_{2hy}] 1 [u_{5}] 2 2 0
    [A_{1u}] [{\sf x}_{1}^{-}] [2_{x}2_{y}2_{z}] [D_{2}] 1 [\varepsilon]; [g_{1}], [g_{2}], [g_{3}]; [d_{14}], [d_{25}], [d_{36}] 2 1 0
    [B_{1u}] [{\sf x}_{2}^{-}] [m_{x}m_{y}2_{z}] [C_{2vz}] 1 [P_{3}] 2 1 2
    [B_{3u}] [{\sf x}_{3}^{-}] [2_{x}m_{y}m_{z}] [C_{2vx}] 1 [P_{1}] 2 1 2
    [B_{2u}] [{\sf x}_{4}^{-}] [m_{x}2_{y}m_{z}] [C_{2vy}] 1 [P_{2}] 2 1 2

    (d) Tetragonal parent groups

    R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
    [F_1] [n_F] [n_f] [n_a] [n_e]
    Parent symmetry [\bi G]: [\quad{\bf 4}_{\bi z}\quad {\bi C}_{\bf 4{\bi z}}]
    B [{\sf x}_{3}] [2_{z}] [C_{2z}] 1 [\delta u_{1}=-\delta u_{2}], [u_{6}] 2 2 1
    [^{1}E\, \oplus\, ^{2}\kern-1pt E] [(x_{1},y_{1})] [1] [C_{1}] 1 [(P_{1},P_{2})]; [(u_{4},-u_{5})] 4 4 4
    (Li)                
    Parent symmetry [\bi G]: [\quad{\bf\overline 4}_{\bi z}\quad {\bi S}_{\bf 4{\bi z}}]
    B [{\sf x}_{3}] [2_{z}] [C_{2z}] 1 [\varepsilon]; [P_{3}]; [\delta u_{1}=-\delta u_{2}], [u_{6}] 2 2 2
    [^{1}E\, \oplus \, ^{2}\kern-1pt E] [(x_{1},y_{1})] [1] [C_{1}] 1 [(P_{1},-P_{2})]; [(u_{4},-u_{5})] 4 4 4
    Parent symmetry [\bi G]: [\quad {\bf 4_{\bi z}/{\bi m}_{\bi z}\quad {\bi C}_{4{\bi hz}}}]
    [B_{g}] [{\sf x}_{3}^{+}] [2_{z}/m_{z}] [C_{2hz}] 1 [\delta u_{1}=-\delta u_{2}], [u_{6}] 2 2 0
    [A_{u}] [{\sf x}_{1}^{-}] [4_{z}] [C_{4z}] 1 [\varepsilon]; [P_{3}] 2 1 2
    [B_{u}] [{\sf x}_{3}^{-}] [{\overline 4}_{z}] [S_{4z}] 1 [g_{1}=-g_{2}], [g_{6}]; [d_{31}=-d_{32}], [d_{36}], [d_{14}=d_{25}], [d_{15}=-d_{24}] 2 1 0
    [^{1}E_{g}\, \oplus \, ^{2}\kern-1pt E_{g}] [(x_{1}^{+},y_{1}^{+})] [{\overline 1}] [C_{i}] 1 [(u_{4},-u_{5})] 4 4 0
    [^{1}E_{u} \, \oplus \, ^{2}\kern-1pt E_{u}] [(x_{1}^{-},y_{1}^{-})] [m_{z}] [C_{sz}] 1 [(P_{1},P_{2})] 4 2 4
    Parent symmetry [\bi G]: [\quad{\bf 4_{\bi z}2_{\bi x}2_{\bi xy}\quad {\bi D}_{4{\bi z}}}]
    [A_{2}] [{\sf x}_{2}] [4_{z}] [C_{4z}] 1 [P_{3}] 2 1 2
    [B_{1}] [{\sf x}_{3}] [2_{x}2_{y}2_{z}] [D_{2}] 1 [\delta u_{1}=-\delta u_{2}] 2 2 0
    [B_{2}] [{\sf x}_{4}] [2_{x{\overline y}}2_{xy}2_{z}] [{\hat D}_{2z}] 1 [u_{6}] 2 2 0
    E [(x_{1},0)] [2_{x}] [C_{2x}] 2 [P_{1}]; [u_{4}] 4 4 4
      [(x_{1},x_{1})] [2_{xy}] [C_{2xy}] 2 [P_{1}=P_{2}]; [u_{4}=-u_{5}] 4 4 4
    (Li) [(x_{1},y_{1})] [1] [C_{1}] 1 [(P_{1},P_{2})]; [(u_{4},-u_{5})] 8 8 8
    Parent symmetry [\bi G]: [\quad {\bf 4_{\bi z}{\bi m}_{\bi x}{\bi m}_{\bi xy}\quad {\bi C}_{4{\bi vz}}}]
    [A_{2}] [{\sf x}_{2}] [4_{z}] [C_{4z}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 2 1 1
    [B_{1}] [{\sf x}_{3}] [m_{x}m_{y}2_{z}] [C_{2vz}] 1 [\delta u_{1}=-\delta u_{2}] 2 2 1
    [B_{2}] [{\sf x}_{4}] [m_{x{\overline y}}m_{xy}2_{z}] [{\hat C}_{2vz}] 1 [u_{6}] 2 2 1
    E [(x_{1},0)] [m_{x}] [C_{sx}] 2 [P_{2}]; [u_{4}] 4 4 4
      [(x_{1},x_{1})] [m_{xy}] [C_{sxy}] 2 [P_{2}=-P_{1}]; [u_{4}=-u_{5}] 4 4 4
      [(x_{1},y_{1})] [1] [C_{1}] 1 [(P_{2},-P_{1})]; [(u_{4},-u_{5})] 8 8 8
    Parent symmetry [\bi G]: [\quad{\bf\overline 4}_{\bi z}{\bf 2}_{\bi x}{\bi m}_{\bi xy}\quad {\bi D}_{\bf 2{\bi dz}}]
    [A_{2}] [{\sf x}_{2}] [{\overline 4}_{z}] [S_{4z}] 1 [g_{6}]; [d_{31}=-d_{32}], [d_{15}=-d_{24}] 2 1 0
    [B_{1}] [{\sf x}_{3}] [2_{x}2_{y}2_{z}] [D_{2}] 1 [\varepsilon]; [\delta u_{1}=-\delta u_{2}] 2 2 0
    [B_{2}] [{\sf x}_{4}] [m_{x{\overline y}}m_{xy}2_{z}] [{\hat C}_{2vz}] 1 [P_{3}]; [u_{6}] 2 2 2
    E [(x_{1},0)] [2_{x}] [C_{2x}] 2 [P_{1}]; [u_{4}] 4 4 4
      [(x_{1},x_{1})] [m_{xy}] [C_{sxy}] 2 [P_{1}=-P_{2}]; [u_{4}=-u_{5}] 4 4 4
      [(x_{1},y_{1})] [1] [C_{1}] 1 [(P_{1},-P_{2})]; [(u_{4},-u_{5})] 8 8 8
    Parent symmetry [\bi G]: [\quad{\bf\overline 4}_{\bi z}{\bi m}_{\bi x}{\bf 2}_{\bi xy}\quad {\hat {\bi D}}_{\bf 2{\bi dz}}]
    [A_{2}] [{\sf x}_{2}] [{\overline 4}_{z}] [S_{4z}] 1 [g_{1}=-g_{2}]; [d_{36}], [d_{14}=d_{25}] 2 1 0
    [B_{2}] [{\sf x}_{3}] [m_{x}m_{y}2_{z}] [C_{2vz}] 1 [P_{3}]; [\delta u_{1}=-\delta u_{2}] 2 2 2
    [B_{1}] [{\sf x}_{4}] [2_{x{\overline y}}2_{xy}2_{z}] [{\hat D}_{2z}] 1 [\varepsilon]; [u_{6}] 2 2 0
    E [(x_{1},0)] [m_{x}] [C_{sx}] 2 [P_{2}]; [u_{4}] 4 4 4
      [(x_{1},x_{1})] [2_{xy}] [C_{2xy}] 2 [P_{2}=P_{1}]; [u_{4}=-u_{5}] 4 4 4
      [(x_{1},y_{1})] [1] [C_{1}] 1 [(P_{2},P_{1})]; [(u_{4},-u_{5})] 8 8 8
    Parent symmetry [\bi G]: [\quad{\bf 4_{\bi z}/{\bi m}_{\bi z}{\bi m}_{\bi x}{\bi m}_{\bi xy}\quad {\bi D}_{\bf 4{\bi hz}}}]
    [A_{2g}] [{\sf x}_{2}^{+}] [4_{z}/m_{z}] [C_{4hz}] 1 [A_{31}=A_{32}], [A_{33}], [A_{15}=A_{24}] 2 1 0
    [B_{1g}] [{\sf x}_{3}^{+}] [m_{x}m_{y}m_{z}] [D_{2h}] 1 [\delta u_{1}=-\delta u_{2}] 2 2 0
    [B_{2g}] [{\sf x}_{4}^{+}] [m_{x{\overline y}}m_{xy}m_{z}] [{\hat D}_{2hz}] 1 [u_{6}] 2 2 0
    [A_{1u}] [{\sf x}_{1}^{-}] [4_{z}2_{x}2_{xy}] [D_{4z}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 2 1 0
    [A_{2u}] [{\sf x}_{2}^{-}] [4_{z}m_{x}m_{xy}] [C_{4vz}] 1 [P_{3}] 2 1 2
    [B_{1u}] [{\sf x}_{3}^{-}] [{\overline 4}_{z}2_{x}m_{xy}] [D_{2dz}] 1 [g_{1}=-g_{2}]; [d_{14}=d_{25}], [d_{36}] 2 1 0
    [B_{2u}] [{\sf x}_{4}^{-}] [{\overline 4}_{z}m_{x}2_{xy}] [{\hat D}_{2dz}] 1 [g_{6}]; [d_{31}=-d_{32}], [d_{15}=-d_{24}] 2 1 0
    [E_{g}] [(x_{1}^{+},0)] [2_{x}/m_{x}] [C_{2hx}] 2 [u_{4}] 4 4 0
      [(x_{1}^{+},x_{1}^{+})] [2_{xy}/m_{xy}] [C_{2hxy}] 2 [u_{4}=-u_{5}] 4 4 0
      [(x_{1}^{+},y_{1}^{+})] [{\overline 1}] [C_{i}] 1 [(u_{4},-u_{5})] 8 8 0
    [E_{u}] [(x_{1}^{-},0)] [2_{x}m_{y}m_{z}] [C_{2vx}] 2 [P_{1}] 4 2 4
      [(x_{1}^{-},x_{1}^{-})] [m_{x{\overline y}}2_{xy}m_{z}] [C_{2vxy}] 2 [P_{1}=P_{2}] 4 2 4
      [(x_{1}^{-},y_{1}^{-})] [m_{z}] [C_{sz}] 1 [(P_{1},P_{2})] 8 8 8

    (e) Trigonal parent groups

    R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
    [F_1] [n_F] [n_f] [n_a] [n_e]
    Parent symmetry [\bi G]: [\quad\bf 3_{\bi z}\quad {\bi C}_{3}]
    E [(x_{1},y_{1})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]) 3 3 3
              ([u_{1}-u_{2}], [-2u_{6}]), ([u_{4}], [-u_{5}])      
    (La, Li)         [\delta u_{1}=-\delta u_{2}]      
    Parent symmetry [\bi G]: [\quad\bf{\overline 3}_{\bi z}\quad {\bi C}_{3{\bi i}}]
    [A_{u}] [{\sf x}_{1}^{-}] [3_{z}] [C_{3}] 1 [\varepsilon]; [P_{3}] 2 1 2
    [E_{g}] [(x_{1}^{+},y_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([u_{1}-u_{2}], [-2u_{6}]), ([u_{4}], [-u_{5}]) 3 3 0
    (La)         [\delta u_{1}=-\delta u_{2}]      
    [E_{u}] [(x_{1}^{-},y_{1}^{-})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]) 6 3 6
    Parent symmetry [\bi G]: [\quad\bf 3_{\bi z}2_{\bi x}\quad {\bi D}_{3{\bi x}}]
    [A_{2}] [{\sf x}_{2}] [3_{z}] [C_{3}] 1 [P_{3}] 2 1 2
    E [(x_{1}, 0)] [2_{x}] [C_{2x}] 3 [P_{1}]; [\delta u_{1}= -\delta u_{2}], [u_{4}] 3 3 3
    (La, Li) [(x_{1}, y_{1})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]); ([u_{1}-u_{2}], [-2u_{6}]), ([u_{4}], [-u_{5}]) 6 6 6
    Parent symmetry [\bi G]: [\quad\bf 3_{\bi z}{\bi m_{x}\quad C}_{3{\bi vx}}]
    [A_{2}] [{\sf x}_{2}] [3_{z}] [C_{3}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{11}=-d_{12}=-d_{26}], [d_{14}=-d_{25}] 2 1 1
    E [(x_{1}, 0)] [m_{x}] [C_{sx}] 3 [P_{2}]; [\delta u_{1}=-\delta u_{2}], [u_{4}] 3 3 3
    (La) [(x_{1}, y_{1})] [1] [C_{1}] 1 ([P_{2}], [-P_{1}]); ([u_{1}-u_{2}], [-2u_{6}]), ([u_{4}], [-u_{5}]) 6 6 6
    Parent symmetry [\bi G]: [\quad\bf{\overline 3}_{\bi z}{\bi m_{x}\quad D}_{3{\bi dx}}]
    [A_{2g}] [{\sf x}_{2}^{+}] [{\overline 3}_{z}] [C_{3i}] 1 [A_{22}=-A_{21}=-A_{16}], [A_{31}=A_{32}], [A_{33}], [A_{15}=A_{24}] 2 1 0
    [A_{1u}] [{\sf x}_{1}^{-}] [3_{z}2_{x}] [D_{3x}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{11}=-d_{12}=-d_{26}], [d_{14}=-d_{25}] 2 1 0
    [A_{2u}] [{\sf x}_{2}^{-}] [3_{z}m_{x}] [C_{3vx}] 1 [P_{3}] 2 1 2
    [E_{g}] [(x_{1}^{+}, 0)] [2_{x}/m_{x}] [C_{2hx}] 3 [\delta u_{1}=-\delta u_{2}], [u_{4}] 3 3 0
    (La) [(x_{1}^{+},y_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([u_{1}-u_{2}], [-2u_{6}]), ([u_{4}], [-u_{5}]) 6 6 0
    [E_{u}] [(0, y_{1}^{-})] [m_{x}] [C_{sx}] 3 [P_{2}] 6 3 6
      [(x_{1}^{-}, 0)] [2_{x}] [C_{2x}] 3 [P_{1}] 6 3 6
      [(x_{1}^{-},y_{1}^{-})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]) 12 6 12
    Parent symmetry [\bi G]: [\quad\bf 3_{\bi z}2_{\bi y}\quad {\bi D}_{3{\bi y}}]
    [A_{2}] [{\sf x}_{2}] [3_{z}] [C_{3}] 1 [P_{3}] 2 1 2
    E [(0, y_{1})] [2_{y}] [C_{2y}] 3 [P_{2}]; [\delta u_{1}=-\delta u_{2}], [u_{5}] 3 3 3
    (La, Li) [(x_{1}, y_{1})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]); ([2u_{6}], [u_{1}-u_{2}]), ([u_{4}], [-u_{5}]) 6 6 6
    Parent symmetry [\bi G]: [\quad\bf 3_{\bi z}{\bi m}_{\bi y}\quad {\bi C}_{3{\bi vy}}]
    [A_{2}] [{\sf x}_{2}] [3_{z}] [C_{3}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{22}=-d_{21}=-d_{16}], [d_{14}=-d_{25}] 2 1 1
    E [(0, y_{1})] [m_{y}] [C_{sy}] 3 [P_{1}]; [\delta u_{1}=-\delta u_{2}], [u_{5}] 3 3 3
    (La) [(x_{1}, y_{1})] [1] [C_{1}] 1 ([P_{2}], [-P_{1}]); ([2u_{6}], [u_{1}-u_{2}]), ([u_{4}], [-u_{5}]) 6 6 6
    Parent symmetry [\bi G]: [\quad\bf{\overline 3}_{\bi z}{\bi m}_{\bi y}\quad {\bi D}_{3{\bi dy}}]
    [A_{2g}] [{\sf x}_{2}^{+}] [{\overline 3}_{z}] [C_{3i}] 1 [A_{11}=-A_{12}=-A_{26}], [A_{31}=A_{32}], [A_{33}], [A_{15}=A_{24}] 2 1 0
    [A_{1u}] [{\sf x}_{1}^{-}] [3_{z}2_{y}] [D_{3y}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{22}=-d_{21}=-d_{16}], [d_{14}=-d_{25}] 2 1 0
    [A_{2u}] [{\sf x}_{2}^{-}] [3_{z}m_{y}] [C_{3vy}] 1 [P_{3}] 2 1 2
    [E_{g}] [(0, y_{1}^{+})] [2_{y}/m_{y}] [C_{2hy}] 3 [\delta u_{1}=-\delta u_{2}], [u_{5}] 3 3 0
    (La) [(x_{1}^{+}, y_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([2u_{6}], [u_{1}-u_{2}]), ([u_{4}], [-u_{5}]) 6 6 0
    [E_{u}] [(0, y_{1}^{-})] [2_{y}] [C_{2y}] 3 [P_{2}] 6 3 6
      [(x_{1}^{-}, 0)] [m_{y}] [C_{sy}] 3 [P_{1}] 6 3 6
      [(x_{1}^{-}, y_{1}^{-})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]) 12 6 12

    (f) Hexagonal parent groups

    Covariants with standardized labels and conversion equations:[\displaylines{{g}_{1}^{-}=g_{1}+g_{2}; \quad g_{2x}^{-}=g_{1}-g_{2}, \quad g_{2y}^{-}=2g_{6}\cr g_{1}=\textstyle{{1}\over{2}}({g}_{1}^{-}+g_{2x}^{-}), \quad g_{2}=\textstyle{{1}\over{2}}({g}_{1}^{-}-g_{2x}^{-}); \quad \delta g_{1}=-\delta g_{2}= \textstyle{{1}\over{2}}g_{2x}^{-}\cr {d}_{1}^{-}=d_{14}-d_{25}; \quad d_{2x,2}^{-}=d_{14}+d_{25}, \quad d_{2y,2}^{-}=d_{24}-d_{15}\cr {d}_{2,1}^{-}=d_{31}+d_{32}; \quad d_{2x,1}^{-}=2d_{36}, \quad d_{2y,1}^{-}=d_{32}-d_{31}\cr d_{14}=\textstyle{{1}\over{2}}({d}_{1}^{-}+d_{2x,2}^{-}), \quad d_{25}=\textstyle{{1}\over{2}}(-{d}_{1}^{-}+d_{2x,2}^{-}); \quad \delta d_{14}=\delta d_{25}=\textstyle{{1}\over{2}}d_{2x}^{-}\cr d_{36}=\textstyle{{1}\over{2}}d_{2x,1}^{-}, \quad d_{31}=\textstyle{{1}\over{2}}({d}_{2,1}^{-}-d_{2y,1}^{-}); \quad d_{32}=\textstyle{{1}\over{2}}({d}_{2,1}^{-}+d_{2y,1}^{-}).}]

    R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
    [F_1] [n_F] [n_f] [n_a] [n_e]
    Parent symmetry [\bi G]: [\quad \bf 6_{\bi z}\quad {\bi C}_{6}]
    B [{\sf x}_{3}] [3_{z}] [C_{3}] 1 [d_{11}=-d_{12}=-d_{26}], [d_{22}=-d_{21}=-d_{16}] 2 1 1
    [E_{2}] [(x_{2},y_{2})] [2_{z}] [C_{2z}] 1 ([u_{1}-u_{2}], [2u_{6}]) [\delta u_{1}=-\delta u_{2}] 3 3 1
    (La, Li)                
    [E_{1}] [(x_{1},y_{1})] [1] [C_{1}] 1 ([P_{1}],[P_{2}]) 6 6 6
    (Li)         ([u_{4}], [-u_{5}])      
    Parent symmetry [\bi G]: [\quad\bf{\overline 6}_{\bi z}\quad {\bi C}_{3{\bi h}}]
    [A''] [{\sf x}_{3}] [3_{z}] [C_{3}] 1 [\varepsilon]; [P_{3}] 2 1 2
    [E'] [(x_{2},y_{2})] [m_{z}] [C_{sz}] 1 ([P_{2}], [P_{1}]) 3 3 3
    (La)         ([u_{1}-u_{2}], [2u_{6}]) [\delta u_{1}=-\delta u_{2}]      
    [E''] [(x_{1},y_{1})] [1] [C_{1}] 1 ([u_{4}], [-u_{5}]) 6 6 6
    Parent symmetry [\bi G]: [\quad\bf 6_{\bi z}/{\bi m}_{\bi z}\quad {\bi C}_{6{\bi h}}]
    [B_{g}] [{\sf x}_{3}^{+}] [{\overline 3}_{z}] [C_{3i}] 1 [A_{11}=-A_{12}=-A_{26}], [A_{22}=-A_{21}=-A_{16}] 2 1 0
    [A_{u}] [{\sf x}_{1}^{-}] [6_{z}] [C_{6}] 1 [\varepsilon]; [P_{3}] 2 1 2
    [B_{u}] [{\sf x}_{3}^{-}] [{\overline 6}_{z}] [C_{3h}] 1 [d_{11}=-d_{12}=-d_{26}], [d_{22}=-d_{21}=-d_{16}] 2 1 0
    [E_{2g}] [(x_{2}^{+},y_{2}^{+})] [2_{z}/m_{z}] [C_{2hz}] 1 ([u_{1}-u_{2}], [2u_{6}]) [\delta u_{1}=- \delta u_{2}] 3 3 0
    (La)                
    [E_{1g}] [(x_{1}^{+},y_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([u_{4}], [-u_{5}]) 6 6 0
    [E_{2u}] [(x_{2}^{-},y_{2}^{-})] [2_{z}] [C_{2z}] 1 ([g_{1}-g_{2}], [2g_{6}]) [g_{1}=-g_{2}], [g_{6}] 6 3 2
              ([2d_{36}], [d_{32}-d_{31}]) [d_{32}=-d_{31}], [d_{36}]      
              ([d_{14}+d_{25}], [d_{24}-d_{15}]) [d_{14}=d_{25}], [d_{24}=-d_{15}]      
    [E_{1u}] [(x_{1}^{-},y_{1}^{-})] [m_{z}] [C_{sz}] 1 ([P_{1}], [P_{2}]) 6 3 6
    Parent symmetry [\bi G]: [\quad\bf 6_{\bi z}2_{\bi x}2_{\bi y}\quad {\bi D}_{6}]
    [A_{2}] [{\sf x}_{2}] [6_{z}] [C_{6}] 1 [P_{3}] 2 1 2
    [B_{1}] [{\sf x}_{3}] [3_{z}2_{x}] [D_{3x}] 1 [d_{11}=-d_{12}=-d_{26}] 2 1 0
    [B_{2}] [{\sf x}_{4}] [3_{z}2_{y}] [D_{3y}] 1 [d_{22}=-d_{21}=-d_{16}] 2 1 0
    [E_{2}] [(x_{2},0)] [2_{x}2_{y}2_{z}] [D_{2}] 3 [\delta u_{1}=-\delta u_{2}] 3 3 0
    (La, Li) [(x_{2},y_{2})] [2_{z}] [C_{2z}] 1 ([u_{1}-u_{2}], [2u_{6}]) 6 6 2
    [E_{1}] [(x_{1},0)] [2_{x}] [C_{2x}] 3 [P_{1}]; [u_{4}] 6 6 6
      [(0,y_{1})] [2_{y}] [C_{2y}] 3 [P_{2}]; [u_{5}] 6 6 6
    (Li) [(x_{1},y_{1})] [1] [C_{1}] 1 ([P_{1}], [P_{2}]); ([u_{4}], [-u_{5})] 12 12 12
    Parent symmetry [\bi G]: [\quad\bf 6_{\bi z}{\bi m_{x}m_{y}\quad C}_{6{\bi v}}]
    [A_{2}] [{\sf x}_{2}] [6_{z}] [C_{6}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 2 1 1
    [B_{2}] [{\sf x}_{3}] [3_{z}m_{x}] [C_{3vx}] 1 [d_{22}=-d_{21}=-d_{16}] 2 1 1
    [B_{1}] [{\sf x}_{4}] [3_{z}m_{y}] [C_{3vy}] 1 [d_{11}=-d_{12}=-d_{26}] 2 1 1
    [E_{2}] [(x_{2},0)] [m_{x}m_{y}2_{z}] [C_{2vz}] 3 [\delta u_{1}=-\delta u_{2}] 3 3 1
    (La) [(x_{2},y_{2})] [2_{z}] [C_{2z}] 1 ([u_{1}-u_{2}], [2u_{6}]) 6 6 1
    [E_{1}] [(x_{1},0)] [m_{x}] [C_{sx}] 3 [P_{2}]; [u_{4}] 6 6 6
      [(0,y_{1})] [m_{y}] [C_{sy}] 3 [P_{1}]; [u_{5}] 6 6 6
      [(x_{1},y_{1})] [1] [C_{1}] 1 ([P_{2}], [-P_{1}]); ([u_{4}], [-u_{5}]) 12 12 12
    Parent symmetry [\bi G]: [\quad\bf{\overline 6}_{\bi z}2_{\bi x}{\bi m_{y}\quad D}_{3{\bi h}}]
    [A_{2}'] [{\sf x}_{2}] [{\overline 6}_{z}] [C_{3h}] 1 [d_{22}=-d_{21}=-d_{16}] 2 1 0
    [A_{1}''] [{\sf x}_{3}] [3_{z}2_{x}] [D_{3x}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 2 1 0
    [A_{2}''] [{\sf x}_{4}] [3_{z}m_{y}] [C_{3vy}] 1 [P_{3}] 2 1 2
    [E'] [(x_{2},0)] [2_{x}m_{y}m_{z}] [C_{2vx}] 3 [P_{1}]; [\delta u_{1}=-\delta u_{2}] 3 3 3
    (La) [(x_{2},y_{2})] [m_{z}] [C_{sz}] 1 ([P_{1}],[-P_{2}]); ([u_{1}-u_{2}], [2u_{6}]) 6 6 6
    [E''] [(x_{1},0)] [2_{x}] [C_{2x}] 3 [u_{4}] 6 6 3
      [(0,y_{1})] [m_{y}] [C_{sy}] 3 [u_{5}] 6 6 6
      [(x_{1},y_{1})] [1] [C_{1}] 1 ([u_{4}], [-u_{5}]) 12 12 12
    Parent symmetry [\bi G]: [\quad{\bf\overline 6}_{\bi z}{\bi m}_{\bi x}{\bf 2}_{\bi y}\quad {\hat {\bi D}}_{{\bf3}{\bi h}}]
    [A_{2}'] [{\sf x}_{2}] [{\overline 6}_{z}] [C_{3h}] 1 [d_{11}=-d_{12}=-d_{26}] 2 1 0
    [A_{2}''] [{\sf x}_{3}] [3_{z}m_{x}] [C_{3vx}] 1 [P_{3}] 2 1 2
    [A_{1}'] [{\sf x}_{4}] [3_{z}2_{y}] [D_{3y}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 2 1 0
    [E'] [(x_{2},0)] [m_{x}2_{y}m_{z}] [C_{2vy}] 3 [P_{2}]; [\delta u_{1}=-\delta u_{2}] 3 3 3
    (La) [(x_{2},y_{2})] [m_{z}] [C_{sz}] 1 ([P_{2}], [P_{1}]); ([u_{1}-u_{2}], [2u_{6}]) 6 6 6
    [E''] [(x_{1},0)] [m_{x}] [C_{sx}] 3 [u_{4}] 6 6 6
      [(0,y_{1})] [2_{y}] [C_{2y}] 3 [u_{5}] 6 6 3
      [(x_{1},y_{1})] [1] [C_{1}] 1 ([u_{4}], [-u_{5}]) 12 12 12
    Parent symmetry [\bi G]: [\quad\bf 6_{\bi z}/{\bi m_{z}m_{x}m_{y}\quad D}_{6{\bi h}}]
    [A_{2g}] [{\sf x}_{2}^{+}] [6_{z}/m_{z}] [C_{6h}] 1 [A_{31}=A_{32}], [A_{33}], [A_{15}=A_{24}] 2 1 0
    [B_{1g}] [{\sf x}_{3}^{+}] [{\overline 3}_{z}m_{x}] [D_{3dx}] 1 [A_{11}=-A_{12}=-A_{26}] 2 1 0
    [B_{2g}] [{\sf x}_{4}^{+}] [{\overline 3}_{z}m_{y}] [D_{3dy}] 1 [A_{22}=-A_{21}=-A_{16}] 2 1 0
    [A_{1u}] [{\sf x}_{1}^{-}] [6_{z}2_{x}2_{y}] [D_{6}] 1 [\varepsilon]; [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 2 1 0
    [A_{2u}] [{\sf x}_{2}^{-}] [6_{z}m_{x}m_{y}] [C_{6v}] 1 [P_{3}] 2 1 2
    [B_{1u}] [{\sf x}_{3}^{-}] [{\overline 6}_{z}2_{x}m_{y}] [D_{3h}] 1 [d_{11}=-d_{12}=-d_{26}] 2 1 0
    [B_{2u}] [{\sf x}_{4}^{-}] [{\overline 6}_{z}m_{x}2_{y}] [{\hat D}_{3h}] 1 [d_{22}=-d_{21}=-d_{16}] 2 1 0
    [E_{2g}] [(x_{2}^{+},0)] [m_{x}m_{y}m_{z}] [D_{2h}] 3 [\delta u_{1}=-\delta u_{2}] 3 3 0
    (La) [(x_{2}^{+},y_{2}^{+})] [2_{z}/m_{z}] [C_{2hz}] 1 ([u_{1}-u_{2}], [2u_{6}]) 6 6 0
    [E_{1g}] [(x_{1}^{+},0)] [2_{x}/m_{x}] [C_{2hx}] 3 [u_{4}] 6 6 0
      [(0,y_{1}^{+})] [2_{y}/m_{y}] [C_{2hy}] 3 [u_{5}] 6 6 0
      [(x_{1}^{+},y_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([u_{4}], [-u_{5}]) 12 12 0
    [E_{1u}] [(x_{1}^{-},0)] [2_{x}m_{y}m_{z}] [C_{2vx}] 3 [P_{1}] 6 3 6
      [(0,y_{1}^{-})] [m_{x}2_{y}m_{z}] [C_{2vy}] 3 [P_{2}] 6 3 6
      [(x_{1}^{-},y_{1}^{-})] [m_{z}] [C_{sz}] 1 ([P_{1}], [P_{2}]) 12 6 12
    [E_{2u}] [(x_{2}^{-},0)] [2_{x}2_{y}2_{z}] [D_{2}] 3 [\delta g_{1}=-\delta g_{2}]; [d_{36}], [\delta d_{14}=\delta d_{25}] 6 3 0
      [(0,y_{2}^{-})] [m_{x}m_{y}2_{z}] [C_{2vz}] 3 [g_{6}]: [d_{32}=-d_{31}], [d_{24}=-d_{15}] 6 3 2
      [(x_{2}^{-},y_{2}^{-})] [2_{z}] [C_{2z}] 1 ([g_{1}-g_{2}], [2g_{6}]); [(2d_{36}, d_{32}-d_{31})], [(d_{14}+d_{25}, d_{24}-d_{15})] 12 6 2

    (g) Cubic parent groups

    Covariants with standardized labels and conversion equations:[\displaylines{ u_{3x}=u_{3x}^{+}=u_{3}-a(u_{1}+u_{2}); \quad u_{3y}=u_{3y}^{+}=b(u_{1}-u_{2})\cr \delta u_{1}=-\textstyle{{1}\over{3}}u_{3x}^{+}+ {\textstyle{1}\over{\sqrt 3}}u_{3y}^{+}; \quad \delta u_{2}=-\textstyle{{1}\over{3}}u_{3x}^{+}-\textstyle{{1}\over{\sqrt 3}}u_{3y}^{+}; \quad \delta u_{3}=\textstyle{{2}\over{3}}u_{3x}^{+}\cr {g}_{1}^{-}=g_{1}+g_{2}+g_{3}; \quad g_{3x}^{-}=g_{3}-a(g_{1}+g_{2}); \quad g_{3y}^{-}=b(g_{1}-g_{2}) \cr g_{1}=\textstyle{{1}\over{3}}{g}_{1}^{-}-\textstyle{{1}\over{3}}g_{3x}^{-}+ \textstyle{{1}\over{\sqrt 3}}g_{3y}^{-}; \quad g_{2}=\textstyle{{1}\over{3}}{g}_{1}^{-}-\textstyle{{1}\over{3}}g_{3x}^{-}- \textstyle{{1}\over{\sqrt 3}}g_{3y}^{-}; \quad g_{3}=\textstyle{{1}\over{3}}{g}_{1}^{-}+\textstyle{{2}\over{3}}g_{3x}^{-}\cr {d}_{1}^{-}=d_{14}+d_{25}+d_{36}; \quad d_{3x}^{-}=b(d_{14}-d_{25}), \quad d_{3y}^{-}=a(d_{14}+d_{25})-d_{36}\cr d_{14}=\textstyle{{1}\over{3}}{d}_{1}^{-}+\textstyle{{1}\over{\sqrt 3}}d_{3x}^{-}+ \textstyle{{1}\over{3}}d_{3y}^{-}; \quad d_{25}=\textstyle{{1}\over{3}}{d}_{1}^{-}-\textstyle{{1}\over{\sqrt 3}}d_{3x}^{-}+ \textstyle{{1}\over{3}}d_{3y}^{-}; \quad d_{36}=\textstyle{{1}\over{3}}{d}_{1}^{-}-\textstyle{{2}\over{3}}d_{3y}^{-}\cr d_{1x}=d_{13}-d_{12}; \quad d_{1y}=d_{21}-d_{23}; \quad d_{1z}=d_{32}-d_{31}\cr d_{2x}=d_{13}+d_{12}; \quad d_{2y}=d_{21}+d_{23}; \quad d_{2z}=d_{32}+d_{31}\cr d_{13}=\textstyle{{1}\over{2}}(d_{1x}+d_{2x}); \quad d_{21}=\textstyle{{1}\over{2}}(d_{1y}+d_{2y}); \quad d_{32}=\textstyle{{1}\over{2}}(d_{1z}+d_{2z})\cr d_{12}=\textstyle{{1}\over{2}}(d_{2x}-d_{1x}); \quad d_{23}=\textstyle{{1}\over{2}}(d_{2y}-d_{1y}); \quad d_{31}=\textstyle{{1}\over{2}}(d_{2z}-d_{1z})}]

    [a=\textstyle{{1}\over{2}}], [b=\textstyle{{\sqrt{3}}\over {2}}], [\pi^a_{\mu\nu} = (\pi_{\mu\nu}-\pi_{\nu\mu})], [\mu=1,2,\ldots,6], [\nu=1,2,\ldots,6].

    R -irep [\Gamma_\eta] Standard variables Ferroic symmetry Principal tensor parameters Domain states
    [F_1] [n_F] [n_f] [n_a] [n_e]
    Parent symmetry [\bi G]: [\quad\bf 23\quad {\bi T}]
    E [(x_{3},y_{3})] [2_{x}2_{y}2_{z}] [D_{2}] 1 [[u_{3}-a(u_{1}+u_{2})], [b(u_{1}-u_{2})]] 3 3 0
    (La)         [\delta u_{1}+\delta u_{2}+\delta u_{3}=0]      
    T [(0,0,z_{1})] [2_{z}] [C_{2z}] 3 [P_{3}]; [u_{6}] 6 6 6
      [(x_{1},x_{1},x_{1})] [3_{p}] [C_{3p}] 4 [P_{1}=P_{2}=P_{3}]; [u_{4}=u_{5}=u_{6}] 4 4 4
    (La, Li) [(x_{1},y_{1},z_{1})] [1] [C_{1}] 1 ([P_{1}], [P_{2}], [P_{3}]); ([u_{4}], [u_{5}], [u_{6}]) 12 12 12
    Parent symmetry [\bi G]: [\quad\bi m{\bf\overline 3}\quad T_{h}]
    [A_{u}] [{\sf x}_{1}^{-}] [23] T 1 [\varepsilon]; [g_{1}=g_{2}=g_{3}]; [d_{14}=d_{25}=d_{36}] 2 1 0
    [E_{g}] [(x_{3}^{+},y_{3}^{+})] [m_{x}m_{y}m_{z}] [D_{2h}] 1 [[u_{3}-a(u_{1}+u_{2})], [b(u_{1}-u_{2})]] 3 3 0
    (La)         [\delta u_{1}+\delta u_{2}+\delta u_{3}=0]      
    [E_{u}] [(x_{3}^{-},y_{3}^{-})] [2_{x}2_{y}2_{z}] [D_{2}] 1 [[g_{3}-a(g_{1}+g_{2})], [b(g_{1}-g_{2})]] 6 3 0
              [\delta g_{1}+\delta g_{2}+\delta g_{3}=0]      
              [[b(d_{14}-d_{25})], [a(d_{14}+d_{25})-d_{36}]]      
              [\delta d_{14}+\delta d_{25}+\delta d_{36}=0]      
    [T_{g}] [(0,0,z_{1}^{+})] [2_{z}/m_{z}] [C_{2hz}] 3 [u_{6}] 6 6 0
      [(x_{1}^{+},x_{1}^{+},x_{1}^{+})] [{\overline 3}_{p}] [C_{3ip}] 4 [u_{4}=u_{5}=u_{6}] 4 4 0
    (La) [(x_{1}^{+},y_{1}^{+},z_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([u_{4}], [u_{5}], [u_{6}]) 12 12 0
    [T_{u}] [(0,0,z_{1}^{-})] [m_{x}m_{y}2_{z}] [C_{2vz}] 3 [P_{3}] 6 3 6
      [(x_{1}^{-},x_{1}^{-},x_{1}^{-})] [3_{p}] [C_{3p}] 4 [P_{1}=P_{2}=P_{3}] 8 4 8
      [(x_{1}^{-},y_{1}^{-},z_{1}^{-})] [1] [C_{1}] 1 ([P_{1}], [P_{2}], [P_{3}]) 24 12 24
    Parent symmetry [\bi G]: [\quad\bf 432\quad {\bi O}]
    [A_{2}] [{\sf x}_{2}] [23] T 1 [d_{14}=d_{25}=d_{36}] 2 1 0
    E [(x_{3},0)] [4_{z}2_{x}2_{xy}] [D_{4z}] 3 [\delta u_{1}=\delta u_{2}= -{{1}\over{2}}\delta u_{3}] 3 3 0
    (La) [(x_{3},y_{3})] [2_{x}2_{y}2_{z}] [D_{2}] 1 [[u_{3}-a(u_{1}+u_{2})], [b(u_{1}-u_{2})]] 6 6 0
              [\delta u_{1}+\delta u_{2}+\delta u_{3}=0]      
    [T_{1}] [(0,0,z_{1})] [4_{z}] [C_{4z}] 3 [P_{3}] 6 3 6
      [(x_{1},x_{1},0)] [2_{xy}] [C_{2xy}] 6 [P_{1}=P_{2}] 12 12 12
      [(x_{1},x_{1},x_{1})] [3_{p}] [C_{3p}] 4 [P_{1}=P_{2}=P_{3}] 8 4 8
    (Li) [(x_{1},y_{1},z_{1})] [1] [C_{1}] 1 ([P_{1}], [P_{2}], [P_{3}]) 24 24 24
    [T_{2}] [(0,0,z_{2})] [2_{x{\overline y}}2_{xy}2_{z}] [{\hat D}_{2z}] 3 [u_{6}] 6 6 0
      [(x_{2}, -x_{2}, z_{2})] [2_{xy}] [C_{2xy}] 6 [u_{4}=-u_{5}], [u_{6}] 12 12 12
      [(x_{2}, x_{2}, x_{2})] [3_{p}2_{x{\overline y}}] [D_{3p}] 4 [u_{4}=u_{5}=u_{6}] 4 4 0
    (La, Li) [(x_{2}, y_{2}, z_{2})] [1] [C_{1}] 1 ([u_{4}], [u_{5}], [u_{6}]) 24 24 24
    Parent symmetry [\bi G]: [\quad\bf{\overline 4}3{\bi m\quad T_{d}}]
    [A_{2}] [{\sf x}_{2}] [23] T 1 [\varepsilon]; [g_{1}=g_{2}=g_{3}] 2 1 0
              [A_{14}=A_{25}=A_{36}]; [\pi_{23}^{a}=\pi_{31}^{a}=\pi_{12}^{a}]      
    E [(x_{3},0)] [{\overline 4}_{z}2_{x}m_{xy}] [D_{2dz}] 3 [\delta u_{1}=\delta u_{2}= -{{1}\over{2}}\delta u_{3}] 3 3 0
    (La) [(x_{3},y_{3})] [2_{x}2_{y}2_{z}] [D_{2}] 1 [[u_{3}-a(u_{1}+u_{2})], [b(u_{1}-u_{2})]] 6 6 0
              [\delta u_{1}+\delta u_{2}+\delta u_{3}=0]      
    [T_{1}] [(0,0,z_{1})] [{\overline 4}_{z}] [S_{4z}] 3 [g_{6}]; [d_{32}=-d_{31}], [d_{24}=-d_{15}] 6 3 0
      [(x_{1},x_{1},0)] [m_{xy}] [C_{sxy}] 6 [g_{4}=g_{5}] 12 12 12
              [d_{13}=-d_{23}], [d_{12}=-d_{21}]      
              [d_{35}=-d_{34}], [d_{26}=-d_{16}]      
      [(x_{1},x_{1},x_{1})] [3_{p}] [C_{3p}] 4 [g_{4}=g_{5}=g_{6}] 8 4 4
              [d_{13}=d_{21}=d_{32}, d_{12}=d_{23}=d_{31}]      
              [d_{35}=d_{16}=d_{24}, d_{26}=d_{34}=d_{15}]      
      [(x_{1},y_{1},z_{1})] [1] [C_{1}] 1 ([g_{4}], [g_{5}], [g_{6}]) 24 24 24
              ([d_{13}-d_{12}], [d_{21}-d_{23}], [d_{32}-d_{31}])      
              ([d_{35}-d_{26}], [d_{16}-d_{34}], [d_{24}-d_{15}])      
    [T_{2}] [(0,0,z_{2})] [m_{x{\overline y}}m_{xy}2_{z}] [{\hat C}_{2vz}] 3 [P_{3}]; [u_{6}] 6 6 6
      [(x_{2}, -x_{2}, z_{2})] [m_{xy}] [C_{sxy}] 6 [P_{1}=-P_{2}], [P_{3}]; [u_{4}=-u_{5}], [u_{6}] 12 12 12
      [(x_{2}, x_{2}, x_{2})] [3_{p}m_{x{\overline y}}] [C_{3vp}] 4 [P_{1}=P_{2}=P_{3}]; [u_{4}=u_{5}=u_{6}] 4 4 4
    (La) [(x_{2}, y_{2}, z_{2})] [1] [C_{1}] 1 ([P_{1}], [P_{2}], [P_{3}]); ([u_{4}], [u_{5}], [u_{6}]) 24 24 24
    Parent symmetry [\bi G]: [\quad\bi m{\bf \overline 3}m\quad O_{h}]
    [A_{2g}] [{\sf x}_{2}^{+}] [m{\overline 3}] [T_{h}] 1 [A_{14}=A_{25}=A_{36}]; [\pi_{23}^{a}=\pi_{31}^{a}=\pi_{12}^{a}] 2 1 0
    [A_{1u}] [{\sf x}_{1}^{-}] [432] O 1 [\varepsilon]; [g_{1}=g_{2}=g_{3}]; 2 1 0
    [A_{2u}] [{\sf x}_{2}^{-}] [{\overline 4}3m] [T_{d}] 1 [d_{14}=d_{25}=d_{36}] 2 1 0
    [E_{g}] [(x_{3}^{+},0)] [4_{z}/m_{z}m_{x}m_{xy}] [D_{4hz}] 3 [\delta u_{3}] 3 3 0
    (La) [(x_{3}^{+},y_{3}^{+})] [m_{x}m_{y}m_{z}] [D_{2h}] 1 [ [\delta u_{3}-a(\delta u_{1}+\delta u_{2})], [b(\delta u_{1}-\delta u_{2})]] 6 6 0
    [E_{u}] [(x_{3}^{-}, 0)] [4_{z}2_{x}2_{xy}] [D_{4z}] 3 [g_{1}=g_{2}], [g_{3}]; [d_{14}=-d_{25}] 12 3 0
      [(0, y_{3}^{-})] [{\overline 4}_{z}2_{x}m_{xy}] [D_{2dz}] 3 [g_{1}=-g_{2}]; [d_{14}=d_{25}=d_{36}] 6 3 0
      [(x_{3}^{-}, y_{3}^{-})] [2_{x}2_{y}2_{z}] [D_{2}] 1 [[g_{3}-a(g_{1}+g_{2})], [b(g_{1}-g_{2})]] 12 6 0
              [[b(d_{14}-d_{25}),a(d_{14}+d_{25})-d_{36}]]      
    [T_{1g}] [(0,0,z_{1}^{+})] [4_{z}/m_{z}] [C_{4hz}] 3 [A_{33}], [A_{32}=A_{31}], [A_{24}=A_{15}, A_{14}=-A_{25}] 6 3 0
      [(x_{1}^{+},x_{1}^{+},0)] [2_{xy}/m_{xy}] [C_{2hxy}] 6 [A_{11}=A_{22}], 12 12 0
              [A_{13}=A_{23}], [A_{12}=A_{21}]      
              [A_{35}=A_{34}], [A_{26}=A_{16}]      
      [(x_{1}^{+},x_{1}^{+},x_{1}^{+})] [{\overline 3}_{p}] [C_{3ip}] 4 [A_{11}=A_{22}=A_{33}] 8 4 0
              [A_{13}=A_{21}=A_{32}], [A_{12}=A_{32}=A_{31}]      
              [A_{35}=A_{16}=A_{24}], [A_{26}=A_{34}=A_{15}]      
      [(x_{1}^{+},y_{1}^{+},z_{1}^{+})] [{\overline 1}] [C_{i}] 1 ([A_{11}], [A_{22}], [A_{33}]) 24 24 0
              ([A_{13}+A_{12}], [A_{21}+A_{23}], [A_{32}+A_{31}])      
              ([A_{35}+A_{26}], [A_{16}+A_{34}], [A_{24}+A_{15}])      
    [T_{2g}] [(0,0,z_{2}^{+})] [m_{x{\overline y}}m_{xy}m_{z}] [{\hat D}_{2hz}] 3 [u_{6}] 6 6 0
      [(x_{2}^{+},-x_{2}^{+},z_{2}^{+})] [2_{xy}/m_{xy}] [C_{2hxy}] 6 [u_{4}=-u_{5}], [u_{6}] 24 12 12
      [(x_{2}^{+},x_{2}^{+},x_{2}^{+})] [{\overline 3}_{p}m_{x{\overline y}}] [D_{3dp}] 4 [u_{4}=u_{5}=u_{6}] 4 4 0
    (La) [(x_{2}^{+},y_{2}^{+},z_{2}^{+})] [{\overline 1}] [C_{i}] 1 ([u_{4}], [u_{5}], [u_{6}]) 24 24 0
    [T_{1u}] [(0,0,z_{1}^{-})] [4_{z}m_{x}m_{xy}] [C_{4vz}] 3 [P_{3}] 6 3 6
      [(x_{1}^{-},y_{1}^{-},0)] [m_{z}] [C_{sz}] 3 [P_{1}],[P_{2}] 24 12 24
      [(x_{1}^{-},x_{1}^{-},0)] [m_{x{\overline y}}2_{xy}m_{z}] [{\hat C}_{2vxy}] 6 [P_{1}=P_{2}] 12 6 12
      [(x_{1}^{-},-x_{1}^{-},z_{1}^{-})] [m_{xy}] [C_{sxy}] 6 [P_{1}=-P_{2}], [P_{3}] 24 12 24
      [(x_{1}^{-},x_{1}^{-},x_{1}^{-})] [3_{p}m_{x{\overline y}}] [C_{3vp}] 4 [P_{1}=P_{2}=P_{3}] 8 4 8
      [(x_{1}^{-},y_{1}^{-},z_{1}^{-})] [1] [C_{1}] 1 ([P_{1}], [P_{2}], [P_{3}]) 48 24 48
    [T_{2u}] [(0,0,z_{2}^{-})] [{\overline 4}_{z}m_{x}2_{xy}] [{\hat D}_{2dz}] 3 [g_{6}]; [d_{32}=-d_{31}], [d_{24}=-d_{15}] 6 3 0
      [(x_{2}^{-},y_{2}^{-},0)] [m_{z}] [C_{sz}] 3 [g_{4}], [g_{5}]; [d_{13}], [d_{12}], [d_{21}], [d_{23}] 24 12 24
              [d_{35}], [d_{26}], [d_{16}], [d_{34}]      
      [(x_{2}^{-},-x_{2}^{-},0)] [m_{x{\overline y}}2_{xy}m_{z}] [{\hat C}_{2vxy}] 6 [g_{4}=-g_{5}]; [d_{13}=d_{23}], [d_{21}=d_{21}] 12 6 12
              [d_{35}=d_{34}], [d_{16}=d_{26}]      
      [(x_{2}^{-},-x_{2}^{-},z_{2}^{-})] [2_{xy}] [C_{2xy}] 6 [g_{4}=-g_{5}], [g_{6}]; [d_{13}=d_{23}], [d_{21}=d_{21}] 24 12 12
              [d_{35}=d_{34}], [d_{16}=d_{26}]      
              [d_{32}=-d_{31}], [d_{24}=-d_{15}]      
      [(x_{2}^{-},x_{2}^{-},x_{2}^{-})] [3_{p}2_{x{\overline y}}] [D_{3p}] 4 [g_{4}=g_{5}=g_{6}]; 8 4 0
              [d_{13}=-d_{12}=d_{21}=-d_{23}=d_{32}-d_{31}]      
              [d_{35}=-d_{26}=d_{16}=-d_{34}=d_{24}=-d_{15}]      
      [(x_{2}^{-},y_{2}^{-},z_{2}^{-})] [1] [C_{1}] 1 ([g_{4}], [g_{5}], [g_{6}]) 48 24 48
              ([d_{13}-d_{12}], [d_{21}-d_{23}], [d_{32}-d_{31}])      
              ([d_{35}-d_{26}], [d_{16}-d_{34}], [d_{24}-d_{15}])      

    Table 3.1.3.2 | top | pdf |
    Symmetry descents [G\Downarrow F_1] associated with two irreducible representations

    G [\Gamma_{\eta}] [F_1] Proper or improper Domain states Full or partial
    Ferroelectric Ferroelastic [ n_f ] [ n_e ] [ n_a ] Ferroelectric Ferroelastic
    432 [ T_1 ] [2_{xy}] proper improper 12 12 12 full full
    [ T_2 ] improper proper
    [ T_1 ] 1 improper improper 24 24 24 full full
    [ T_2 ] proper proper
    [\bar{4}3m] [ T_1 ] [m_{xy}] improper improper 12 12 12 full full
    [ T_2 ] proper proper
    [ T_1 ] 1 improper improper 24 24 24 full full
    [ T_2 ] proper proper
    [m\bar3m] [ T_{1g}] [2_{xy}/m_{xy}] non improper 12 0 12 non full
    [ T_{2g} ] non proper
    [ T_{1g} ] [\bar1] non improper 24 0 24 non full
    [ T_{2g} ] non proper
    [ T_{1u} ] [m_{x\bar y}2_{xy}m_z] proper improper 12 12 6 full partial
    [ T_{2u} ] improper improper
    [ T_{1u} ] [m_{z}] proper improper 24 24 12 full partial
    [ T_{2u} ] improper improper
    [ T_{1u} ] 1 proper improper 48 48 24 full partial
    [ T_{2u} ] improper improper

    If for a given symmetry descent [{\cal G}\Downarrow^t{\cal F}] no appropriate R-irep exists in Table 3.1.3.1[link], then the primary order parameter [\eta] transforms according to a reducible representation of G. These transitions are always discontinuous and can be accomplished with several reducible representations. Some symmetry descents can be associated with an irreducible representation and with several reducible representations. All these transitions are treated in the software GI[\star]KoBo-1 and in Kopský (2001[link]). All point-group symmetry descents are listed in Table 3.4.2.7[link] and can be traced in lattices of subgroups (see Figs. 3.1.3.1[link] and 3.1.3.2[link]).

    The solution of the inverse Landau problem – i.e. the identification of the representation [\Gamma_{\eta}] relevant to symmetry descent [G \Downarrow F] – enables one to determine the corresponding nomal mode (so-called soft mode) of the transition (see e.g. Rousseau et al., 1981[link]). We note that this step requires additional knowledge of the crystal structure, whereas other conclusions of the analysis hold for any crystal structure with a given symmetry descent [G \Downarrow F]. Normal-mode determination reveals the dynamic microscopic nature of the instability of the crystal lattice which leads to the phase transition (for more details and examples, see Section 3.1.5).

    The representation [\Gamma_{\eta}] further determines the principal tensor parameters associated with the primary order parameter [\eta]. If one of them is a vector (polarization) the soft mode is infrared-active in the parent phase; if it is a symmetric second-rank tensor (spontaneous strain), the soft mode is Raman active in this phase. Furthermore, the R-irep [\Gamma_{\eta}] determines the polynomial in components of [\eta] in the Landau free energy (basic invariant polynomials, called integrity bases, are available in the software GI[\star]KoBo-1 and in Kopský, 2001[link]) and allows one to decide whether the necessary conditions of continuity of the transition (so-called Landau and Lifshitz conditions) are fulfilled.

  • (2) Direct Landau problem of equitranslational phase transitions: For a given space group [\cal G] of the parent phase and the R-irep [{\Gamma}_{\eta}] (specifying the transformation properties of the primary order parameter [\eta]), find the corresponding equitranslational space group [\cal F] of the ferroic phase. To solve this task, one first finds in Table 3.1.3.1[link] the point group F that corresponds to point group G of space group [\cal G] and to the given R-irep [\Gamma_{\eta}]. The point-group symmetry descent [G\Downarrow F] thus obtained specifies uniquely the equitranslational subgroup [\cal F] of [\cal G] that can be found in the lattices of equitranslational subgroups of space groups available in the software GI[\star]KoBo-1 (see Section 3.1.6[link]).

  • (3) Secondary tensor parameters of an equitranslational phase transition [{\cal G}\Downarrow^t{\cal F}]. These parameters are specified by the representation [\Gamma_{\lambda}] of G associated with a symmetry descent [\Gamma \Downarrow L], where L is an intermediate group [see equation (3.1.3.1)[link]]. In other words, the secondary tensor parameters of the transition [G\Downarrow F] are identical with principal tensor parameters of the transition [G\Downarrow L]. To each intermediate group L there corresponds a set of secondary tensor parameters. All intermediate subgroups of a symmetry descent [G\Downarrow F] can be deduced from lattices of subgroups in Figs. 3.1.3.1[link] and 3.1.3.2[link].

    The representation [{\Gamma }_{\lambda}] specifies transformation properties of the secondary tensor parameter [\lambda] and thus determines e.g its infrared and Raman activity in the parent phase and enables one to make a mode analysis. Representation [{\Gamma }_{\lambda}] together with [{\Gamma }_{\eta}] determine the coupling between secondary and primary tensor parameters. The explicit form of these faint interactions (Aizu, 1973[link]; Kopský, 1979d[link]) can be found in the software GI[\star]KoBo-1 and in Kopský (2001[link]).

  • (4) Changes of property tensors at a ferroic phase transition. These changes are described by tensor parameters that depend only on the point-group-symmetry descent [G \Downarrow F]. This means that the same principal tensor parameters and secondary tensor parameters appear in all equitranslational and in all non-equitranslational transitions with the same [G \Downarrow F]. The only difference is that in non-equitranslational ferroic phase transitions a principal tensor parameter corresponds to a secondary ferroic order parameter. It still plays a leading role in tensor distinction of domains, since it exhibits different values in any two ferroic domain states (see Section 3.4.2.3[link] ). Changes of property tensors at ferroic phase transitions are treated in detail in the software GI[\star]KoBo-1 and in Kopský (2001[link]).

We note that Table 3.1.3.1[link] covers only those point-group symmetry descents [G\Downarrow F] that are `driven' by R-ireps of G. All possible point-group symmetry descents [G\Downarrow F] are listed in Table 3.4.2.7[link] . Principal and secondary tensor parameters of symmetry descents associated with reducible representations are combinations of tensor parameters appearing in Table 3.1.3.1[link] (for a detailed explanation, see the manual of the software GI[\star]KoBo-1 and Kopský, 2000[link]). Necessary data for treating these cases are available in the software GI[\star]KoBo-1 and Kopský (2001[link]).

3.1.3.3.1. Explanation of Table 3.1.3.1[link]

| top | pdf |

  • Parent symmetry G: the short international (Hermann–Mauguin) and the Schoenflies symbol of the point group G of the parent phase are given. Subscripts specify the orientation of symmetry elements (generators) in the Cartesian crystallophysical coordinate system of the group G (see Figs. 3.4.2.3[link] and 3.4.2.4[link] , and Tables 3.4.2.5[link] and 3.4.2.6[link] ).

  • R -irep [{\Gamma}_{\eta}]: physically irreducible representation [{\Gamma}_{\eta}] of the group G in the spectroscopic notation. This representation defines transformation properties of the primary order parameter [\eta] and of the principal tensor parameters. Each complex irreducible representation is combined with its complex conjugate and thus a real physically irreducible representation R-irep is formed. Matrices [D^{(\alpha)}] of R-ireps are given explicitly in the the software GI[\star]KoBo-1.

  • (La) below the symbol of the irreducible representation [\Gamma_{\eta}] indicates that the Landau condition is violated, hence the transition cannot be continuous (second order). The Landau condition requires the absence of the third-degree invariant polynomial of the order-parameter components (the symmetrized triple product [[\Gamma_{\eta}]^3] must not contain the identity representation of G). For more details see Lyubarskii (1960[link]), Kociński (1983[link], 1990[link]), Tolédano & Tolédano (1987[link]), Izyumov & Syromiatnikov (1990[link]) and Tolédano & Dmitriev (1996[link]).

  • (Li) below the symbol of the irreducible representation [\Gamma_{\eta}] means that the Lifshitz condition is violated, hence the transition to a homogeneous ferroic phase is not continuous. The Lifshitz condition demands the absence of invariant terms that couple bilinearly the order-parameter components with their spatial derivatives that are not exact differentials (the antisymmetric square [{\{\Gamma}_{\eta}\}^2] has no representation in common with the vector representation of G). For more details see Lyubarskii (1960[link]), Kociński (1983[link], 1990[link]), Tolédano & Tolédano (1987[link]), Izyumov & Syromiatnikov (1990[link]) and Tolédano & Dmitriev (1996[link]).

    If there is no symbol (La) and/or (Li) below the symbol of the R-irep [\Gamma_{\eta}] (i.e. if both Landau and Lifshitz conditions are fulfilled), then the R-irep is called an active representation. In the opposite case, the R-irep is a passive representation (Lyubarskii, 1960[link]; Kociński, 1983[link], 1990[link]).

  • Standard variables : components of the order parameter in the carrier space of the irreducible representation [\Gamma_{\eta}] expressed in so-called standard variables (see the manual of the software GI[\star]KoBo-1). Upper and lower indices and the typeface of standard variables allow one to identify to which irreducible representation [\Gamma_{\eta}] they belong. Standard variables of one-dimensional representations are denoted by [\sf x] (Sans Serif typeface), two- or three-dimensional R-ireps by [x,y] or [x,y,z], respectively. Upper indices + and − correspond to the lower indices g (gerade) and u (ungerade) of spectroscopic notation, respectively. The lower index specifies to which irreducible representation the variable belongs.

    For multidimensional representations, a general vector of the carrier space [V_{\eta}] is given in the last row; this vector is invariant under the kernel of [\Gamma_{\eta}] that appears as a low-symmetry group in column [F_1]. The other rows contain special vectors defined by equal or zero values of some standard variables; these vectors are invariant under epikernels of [\Gamma_{\eta}] given in column [F_1].

  • [F_1]: short international (Hermann–Mauguin) and Schoenflies symbol of the point group [F_1] which describes the symmetry of the first single domain state of the ferroic (low-symmetry) phase. The subscripts define the orientation of symmetry elements (generators) of [F_1] in the Cartesian crystallophysical coordinate system of the group G (see Figs. 3.4.2.3[link] and 3.4.2.4[link] , and Tables 3.4.2.5[link] and 3.4.2.6[link] ). This specifies the orientation of the group [F_1], which is a prerequisite for domain structure analysis (see Chapter 3.4[link] ).

  • [n_{F}]: number of subgroups conjugate to [F_1] under G. If [n_{F}=1], the group [F_1] is a normal subgroup of G (see Section 3.2.3[link] ).

  • Principal tensor parameters : covariant tensor components, i.e. linear combinations of Cartesian tensor components that transform according to the same matrix R-irep [D^{(\eta)}] as the primary order parameter [\eta]. Principal tensor parameters are given in this form in the software GI[\star]KoBo-1 and in Kopský (2001[link]).

    This presentation is in certain situations not practical, since property tensors are usually described by numerical values of their Cartesian components. Then it is important to know morphic Cartesian tensor components and symmetry-breaking increments of nonzero Cartesian components that appear spontaneously in the ferroic phase. The bridge between these two presentations is provided by the conversion equations that express Cartesian tensor components as linear combinations of principal and secondary covariant components (for more details on tensorial covariants and conversion equations see Appendix E of the manual for GI[\star]KoBo-1 and Kopský, 2001[link]).

    We illustrate the situation on a transition with symmetry descent [4_z2_x2_{xy} \Downarrow 2_x2_y2_z]. In Table 3.1.3.1[link], we find that the principal tensor parameter transforms according to irreducible representation [B_1] with standard variable [{\sf x}_3]. The corresponding covariant [{\sf u}_{3}=u_{1}-u_{2}] can be found in Appendix E of the manual of GI[\star]KoBo-1 (or in Kopský, 2001[link]), where one also finds an invariant containing [u_1] and [u_2]: [{\sf u}_{1,1}= u_{1}+u_{2}]. The corresponding conversion equations are: [u_{1}={{1}\over{2}}({\sf u}_{1,1}+{\sf u}_{3})], [u_{2}={{1}\over{2}}({\sf u}_{1,1}-{\sf u}_{3})]. In the parent phase [{\sf u}_{3}=u_{1}^{(p)}-u_{2}^{(p)}=0], hence [u_{1}^{(p)}=u_{2}^{(p)}={{1}\over{2}}{\sf u}_{1,1}], whereas in the ferroic phase [u_{1}^{(f)}=] [{{1}\over{2}}({\sf u}_{1,1}+{\sf u}_{3})=] [u_{1}^{(p)}+{{1}\over{2}}{\sf u}_{1,1}=] [ u_{1}^{(p)}+{\delta}u_1], [u_{2}^{(f)}=] [u_{2}^{(p)}-{{1}\over{2}}{\sf u}_{1,1}=] [u_{2}^{(p)}+{\delta}u_2=] [u_{1}^{(p)}-{\delta}u_1]. The symmetry-breaking increments [{\delta}u_1=-{\delta}u_2] describe thus the changes of the Cartesian components that correspond to the nonzero principal tensor component [u_{1}-u_{2}].

    An analogous situation occurs frequently in trigonal and hexagonal parent groups, where [u_{1}-u_{2}] (or [g_{1}-g_{2}]) transforms like the first or second component of the principal tensor parameter. In these cases, the corresponding symmetry-breaking increments of Cartesian components are again related: [\delta u_{1}=-\delta u_{2}] (or [\delta g_{1}=-\delta g_{2}]).

    We note that relations like [A_{11}=-A_{12}=-A_{26}] do not imply that these components transform as the standard variable. Though these components are proportional to the principal tensor parameter in the first domain state, they cannot be transformed to corresponding components in other domain states as easily as covariant tensor components of the principal tensor parameter.

    In general, it is useful to consider a tensor parameter as a vector in the carrier space of the respective representation. Then the Cartesian components are projections of this vector on the Cartesian basis of the tensor space.

    The presentation of the principal tensor parameters in the column Principal tensor parameters of this table is a compromise: whenever conversion equations lead to simple relations between morphic Cartesian components and/or symmetry-breaking increments, we present these relations, in some cases together with corresponding covariants. In the more complicated cases, only the covariants are given. The corresponding conversion equations and labelling of covariants are given at the beginning of that part of the table which covers hexagonal and cubic parent groups G. In the main tables of the software GI[\star]KoBo-1, the principal tensor parameters and the secondary tensor parameters up to rank 4 are given consistently in covariant form. Labelling of covariant components and conversion equations are given in Appendix E of the manual.

    The principal tensor parameters presented in Table 3.1.3.1[link] represent a particular choice of property tensors for standard variables given in the second column. To save space, property tensors are selected in the following way: polarization P and strain u are always listed; if none of their components transform according to [D^{(\eta)}], then components of one axial and one polar tensor (if available) appearing in Table 3.1.3.3[link] are given. Principal parameters of two different property tensors are separated by a semicolon. If two different components of the same property tensor transform in the same way, they are separated by a comma.

    Table 3.1.3.3 | top | pdf |
    Important property tensors

    [i=1,2,3]; [\mu, \nu=1,2,\ldots,6].

    Tensor components Property Tensor components Property
    [\varepsilon] enantiomorphism   chirality
    [P_i] polarization [p_i] pyroelectricity
    [u_{\mu}] strain [\varepsilon_{ij}] dielectric permittivity
    [ g_{\mu}] optical activity    
    [d_{i{\mu}}] piezoelectricity [r_{i{\mu}}] electro-optics
    [A_{i{\mu}}] electrogyration    
    [\pi_{{\mu}{\nu}}] piezo-optics [Q_{{\mu}{\nu}}] electrostriction

    As tensor indices we use integers [1,2,3] instead of vector components [x,y,z] and contracted indices [1,2,3,4,5,6] in matrix notation for pairs [xx, yy, zz, yz \approx zy, zx \approx xz, xy \approx yx], respectively

    Important note : To make Table 3.1.3.1[link] compatible with the software GI[\star]KoBo-1 and with Kopský (2001[link]), coefficients of property tensors in matrix notation with contracted indices 4, 5, 6 do not contain the numerical factors 2 and 4 which are usually introduced to preserve a compact form (without these factors) of linear constitutive relations [see Chapter 1.1[link] , Nye (1985[link]) and especially Appendices E and F of Sirotin & Shaskolskaya (1982[link])]. This explains the differences in matrix coefficients appearing in Table 3.1.3.1[link] and those presented in Chapter 1.1[link] or in Nye (1985[link]) and in Sirotin & Shaskolskaya (1982[link]). Thus e.g. for the symmetry descent [6_z2_x2_y\Downarrow 3_z2_x], we find in Table 3.1.3.1[link] the principal tensor parameters [d_{11}=-d_{12}=-d_{26}], whereas according to Chapter 1.1[link] or e.g. to Nye (1985[link]) or Sirotin & Shaskolskaya (1982[link]) these coefficients for [F_1=3_z2_x] are related by equations [d_{11}=-d_{12}=-2d_{26}].

    Property tensors and symbols of their components that can be found in Table 3.1.3.1[link] are given in the left-hand half of Table 3.1.3.3[link]. The right-hand half presents other tensors that transform in the same way as those on the left and form, therefore, covariant tensor components of the same form as those given in the column Principal tensor parameters. Principal and secondary tensor parameters for all property tensors that appear in Table 3.1.3.3[link] are available in the software GI[\star]KoBo-1.

  • [n_ f]: number of ferroic single domain states that differ in the primary order parameter [\eta] and in the principal tensor parameters.

  • [n_{a}]: number of ferroelastic single domain states. If [n_a=n_f], [n_a \,\lt\, n_f] or [n_a=1], the ferroic phase is, respectively, a full, partial or non-ferroelastic one.

  • [n_{e}]: number of ferroelectric single domain states. If [n_e=n_f], [n_e \,\lt\, n_f] or [n_e=0,1], the ferroic phase is, repectively, a full, partial or non-ferroelectric one ([n=0] or [n=1] correspond to a non-polar or to a polar parent phase, respectively) (see Section 3.4.2[link] ).

References

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 First citation Ascher, E. & Kobayashi, J. (1977). Symmetry and phase transitions: the inverse Landau problem. J. Phys. C: Solid State Phys. 10, 1349–1363.Google Scholar
 First citation Indenbom, V. L. (1960). Phase transitions without change in the number of atoms in the unit cell of the crystal. Sov. Phys. Crystallogr. 5, 105–115.Google Scholar
 First citation Izyumov, Yu. A. & Syromiatnikov, V. N. (1990). Phase transitions and crystal symmetry. Dordrecht: Kluwer Academic Publishers.Google Scholar
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 First citation Kopský, V. (2000). The change of tensor properties at ferroic phase transitions. Ferroelectrics, 237, 127–134.Google Scholar
 First citation Kopský, V. (2001). Tensor parameters of ferroic phase transitions. I. Theory and tables. Phase Transit. 73, No. 1–2, 1–422.Google Scholar
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 First citation Nye, J. F. (1985). Physical properties of crystals. Oxford: Clarendon Press.Google Scholar
 First citation Rousseau, D. L., Bauman, R. P. & Porto, S. P. S. (1981). Normal mode determination in crystals. J. Raman Spectrosc. 10, 253–290.Google Scholar
 First citation Sirotin, Yu. I. & Shaskolskaya, M. P. (1982). Fundamentals of crystal physics. Moscow: Mir Publishers.Google Scholar
 First citation Tolédano, J.-C. & Tolédano, P. (1987). The Landau theory of phase transitions. Singapore: World Scientific.Google Scholar
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