Normal-mode analysis of the structures of perovskites with tilted octahedra. Erratum

There is an error in the mode assignment for hettotype 9, [a⁻b⁺a⁻], discussed in the paper by Darlington [Acta Cryst. (2002). A58, 66–71 ], which has been pointed out by Dr Kevin Knight, Rutherford Appleton Laboratory, Didcot, Oxon, England. In this paper, a mode involving displacements of the anions of hettotype 9 was labelled [(½, 0, ½), M₁] rather than [(½, 0, ½), M₂]. Both modes involve plus-like distortion of the octahedra. In the corrected Tables 1–4 shown below, this mode, which is only found in hettotype 9, has been labelled K₂ rather than H₂. Therefore, there are not seven but eight normal modes of the cubic phase required to describe the displacements found in the nine hettotypes considered. The weights of K₂ in all the materials examined in the original paper with the structure of hettotype 9 [labelled W(H₁) in the original Table 4] are correct, unaltered by the change in the labelling of the mode. It should be noted that [(½, 0, ½), M₂] is a longitudinal mode – the seven other modes are all transverse. The weights of K₂ are not significantly different from zero in the 15 structures examined.

Table 1 The nine hettotypes Number +/- notation [Mi Rj] Multiplicity True unit cell Space group 1 a0a0c− R3 2 × 2 × 2 21/2 × 21/2 × 2 I4/mcm (140) 2 a−b0a− R1 = R3 2 × 2 × 2 21/2 × 2 × 21/2 Imma (74) 3 a−a−a− R1 = R2 = R3 2 × 2 × 2 21/2 × 21/2 × 21/2 (167) 4 a0a0c+ M3 2 × 2 × 1 21/2 × 21/2 × 1 P4/mbm (127) 5 a+a+c0 M1 = M2 2 × 2 × 2 2 × 2 × 2 I4/mmm (139) 6 a+a+a+ M1 = M2 = M3 2 × 2 × 2 2 × 2 × 2 Im (204) 7 a0b−c+ R2, M3 2 × 2 × 2 2 × 2 × 2 Cmcm (63) 8 a+a+c− M1 = M2, R3 2 × 2 × 2 2 × 2 × 2 P42/nmc (137) 9 a−b+a− R1 = R3, M2 2 × 2 × 2 21/2 × 2 × 21/2 Pnma (62)

Table 2 Atomic displacements in the seven normal modes, the symbol used in the construction of the Landau potential, and character of each mode Normal mode Displacement Symbol Character (, , ), Γ25 OI(y) = −OII(z) R1 Octahedral minus tilt (, , ), Γ25 OI(x) = −OIII(z) R2 Octahedral minus tilt (, , ), Γ25 OII(x) = −OIII(y) R3 Octahedral minus tilt (, , ), Γ15 OI(y) = OII(z) GO1 Octahedral minus distortion (, , ), Γ15 OI(x) = OIII(z) GO2 Octahedral minus distortion (, , ), Γ15 OII(x) = OIII(y) GO3 Octahedral minus distortion (, , ), Γ15 A(x) GA1 A cation displacement (, , ), Γ15 A(y) GA2 A cation displacement (, , ), Γ15 A(z) GA3 A cation displacement (0, , ), M3 OI(y) = −OII(z) M1 Octahedral plus tilt (, 0, ), M3 OI(x) = −OIII(z) M2 Octahedral plus tilt (, , 0), M3 OII(x) = −OIII(y) M3 Octahedral plus tilt (0, , ), M1 OI(y) = OII(z) H1 Octahedral plus distortion (, 0, ), M1 OI(x) = OIII(z) H2 Octahedral plus distortion (, , 0), M1 OII(x) = OIII(y) H3 Octahedral plus distortion (0, , ), M2 OII(y) = −OI(z) K1 Octahedral plus distortion (, 0, ), M2 OI(z) = −OIII(x) K2 Octahedral plus distortion (, , 0), M2 OIII(x) = −OII(y) K3 Octahedral plus distortion (, 0, 0), M5′ OIII(y) XO12 Octahedral distortion (, 0, 0), M5′ OIII(z) XO13 Octahedral distortion (0, , 0), M5′ OII(z) XO23 Octahedral distortion (0, , 0), M5′ OII(x) XO21 Octahedral distortion (0, 0, ), M5′ OI(x) XO31 Octahedral distortion (0, 0, ), M5′ OI(y) XO32 Octahedral distortion (, 0, 0), M5′ A(y) XA12 Cation displacement (, 0, 0), M5′ A(z) XA13 Cation displacement (0, , 0), M5′ A(z) XA23 Cation displacement (0, , 0), M5′ A(x) XA21 Cation displacement (0, 0, ), M5′ A(x) XA31 Cation displacement (0, 0, ), M5′ A(y) XA32 Cation displacement

Table 3 The space group, possible condensed modes and non-zero pseudocubic spon­taneous macrostrain in the nine hettotypes Number Space group Allowed modes Macrostrain 1 I4/mcm (140) R3 ∊11 = ∊22; ∊33 2 Imma (74) R1 = R3 ∊11 = ∊33; ∊22; ∊31     GO1 = GO3       GA1 = GA3   3 (167) R1 = R2 = R3 ∊11 = ∊22 = ∊33; ∊23 = ∊31 = ∊12 4 P4/mbm (127) M3 ∊11 = ∊22; ∊33 5 I4/mmm (139) M1 = M2 ∊11 = ∊22; ∊33     H1 = H2 ≠ H3   6 (204) M1 = M2 = M3 ∊11 = ∊22 = ∊33     H1 = H2 = H3   7 Cmcm (63) R2 ∊11; ∊22; ∊33     M3       GO2       GA2       H3       XO32       XA32   8 P42/nmc (137) R3 ∊11 = ∊22; ∊33     M1 = M2       GO3       H1 = H2       XO13 = XO23       XA13 = XA23   9 Pnma (62) R1 = R3 ∊11 = ∊33; ∊22; ∊31     M2       GO1 = GO3       GA1 = GA3       K2       XO21 = XO23       XA21 = XA23

Table 4 Weights of condensed modes in units of u Å2 Hettotype Material W(Ri) W(Mi) W(XAij) W(XOij) W(GAi) W(GOi) W(Hi) W(Ki) Ref.† Entry 1 R3 SrTiO3, 77 K 0.0726               1 1   SrZrO3, 1223 K 1.6971               2 2 2 R1 = R3 BaCeO3, 573 K 6.7294       0.0586 0.1460     3 3 3 R1 = R2 = R3 BaCeO3, 773 K 5.2701               3 4   LaGeO3, 673 K 4.3662               4 5 4 M3 NaNbO3, 888 K   0.9520             5 6   NaTaO3, 878 K   0.8965             5 7 5 M1 = M2 No available data                     6 M1 = M2 = M3 CaCu3Ti4O12   10.2753         0.2031   6 8   Tb0.67Cu3Ti4O12   9.9740         0.1964   6 9   CaCu3Mn4O12   9.2271         0.1323   7 10   Li0.36WO3   5.0358         0.0060   8 11   Na0.73WO3   0.3953         0.0000   8 12   Na0.54WO3   0.3657         0.0000   8 13 7 R2, M3 NaNbO3, 813 K 0.8442 1.2912 0.0082 0.0012 0.0001 0.0002 0.0042   5 14   NaTaO3, 803 K 0.9408 1.1623 0.0808 0.0048 0.0000 0.0067 0.0013   5 15   SrZrO3, 973 K 3.1806 0.8497 0.4539 0.0063 0.0154 0.0118 0.0002   2 16 8 M1 = M2, R3 CaFeTi2O6 4.3399 7.3194 0.3690 0.1987   0.0508 0.0015   9 17 9 R1 = R3, M2 BaCeO3, 473 K 7.2126 0.5108 0.8058 0.0503 0.0775 0.1518   0.0002 3 18   SrZrO3 5.8137 1.2545 1.7364 0.0958 0.0471 0.1209   0.0001 2 19   LaGeO3 5.0130 0.4181 1.1826 0.0162 0.0713 0.1034   0.0002 4 20   PrFeO3 7.4537 1.7425 8.3611 0.2237 0.3453 0.1390   0.0000 10 21   NdFeO3 8.3981 1.9033 10.7150 0.2898 0.4901 0.1649   0.0000 10 22   SmFeO3 9.6041 2.3606 15.1215 0.4215 0.7547 0.1970   0.0005 10 23   EuFeO3 10.0301 2.5231 17.2613 0.5149 0.9157 0.2237   0.0000 10 24   GdFeO3 10.2814 2.6910 19.5499 0.5419 1.0893 0.2683   0.0000 10 25   TbFeO3 11.1054 2.7621 20.4797 0.6507 1.1498 0.2368   0.0000 8 26   DyFeO3 11.5369 2.9724 22.5061 0.7089 1.3311 0.2471   0.0002 8 27   HoFeO3 12.0370 3.0026 23.8465 0.7804 1.4574 0.2704   0.0002 10 28   ErFeO3 12.8142 3.1006 24.9061 0.8218 1.5771 0.3205   0.0000 10 29   TmFeO3 13.1795 3.0947 25.1013 0.9675 1.6745 0.2982   0.0000 10 30   YbFeO3 13.6153 3.2967 26.7549 1.0592 1.7761 0.3007   0.0000 10 31   LuFeO3 14.3837 3.2072 27.5146 1.0463 1.8962 0.2892   0.0000 10 32 †References: (1) Unoki & Sakudo (1967); (2) Kennedy et al. (1999); (3) Knight (1995); (4) Howard & Kennedy (1999); (5) Darlington & Knight (1999); (6) Bochu et al. (1979); (7) Chenevas et al. (1975); (8) Wiseman & Dickens (1976); (9) Leinenweber & Parise (1995); (10) Marezio et al. (1970).