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

Darlington, C.N.W.

doi:10.1107/S0108767302004695

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	[M_i R_j]	Multiplicity	True unit cell	Space group
1	a⁰a⁰c⁻	R₃	2 × 2 × 2	2^1/2 × 2^1/2 × 2	I4/mcm (140)
2	a⁻b⁰a⁻	R₁ = R₃	2 × 2 × 2	2^1/2 × 2 × 2^1/2	Imma (74)
3	a⁻a⁻a⁻	R₁ = R₂ = R₃	2 × 2 × 2	2^1/2 × 2^1/2 × 2^1/2	$[R\bar3c]$ (167)
4	a⁰a⁰c⁺	M₃	2 × 2 × 1	2^1/2 × 2^1/2 × 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 $[\bar 3]$ (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	2^1/2 × 2 × 2^1/2	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
( $[\,{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}\,]$ ), Γ₂₅	OI(y) = −OII(z)	R₁	Octahedral minus tilt
( $[\,{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}\,]$ ), Γ₂₅	OI(x) = −OIII(z)	R₂	Octahedral minus tilt
( $[\,{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}\,]$ ), Γ₂₅	OII(x) = −OIII(y)	R₃	Octahedral minus tilt
( $[\,{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}\,]$ ), Γ₁₅	OI(y) = OII(z)	G_O1	Octahedral minus distortion
( $[\,{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}\,]$ ), Γ₁₅	OI(x) = OIII(z)	G_O2	Octahedral minus distortion
( $[\,{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}\,]$ ), Γ₁₅	OII(x) = OIII(y)	G_O3	Octahedral minus distortion
( $[\,{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}\,]$ ), Γ₁₅	A(x)	G_A1	A cation displacement
( $[\,{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}\,]$ ), Γ₁₅	A(y)	G_A2	A cation displacement
( $[\,{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}\,]$ ), Γ₁₅	A(z)	G_A3	A cation displacement
(0, $[{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}\,]$ ), M₃	OI(y) = −OII(z)	M₁	Octahedral plus tilt
( $[\,{\textstyle{{1\over2}}}]$ , 0, $[{\textstyle{{1\over2}}}\,]$ ), M₃	OI(x) = −OIII(z)	M₂	Octahedral plus tilt
( $[\,{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}]$ , 0), M₃	OII(x) = −OIII(y)	M₃	Octahedral plus tilt
(0, $[{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}\,]$ ), M₁	OI(y) = OII(z)	H₁	Octahedral plus distortion
( $[\,{\textstyle{{1\over2}}}]$ , 0, $[{\textstyle{{1\over2}}}\,]$ ), M₁	OI(x) = OIII(z)	H₂	Octahedral plus distortion
( $[\,{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}]$ , 0), M₁	OII(x) = OIII(y)	H₃	Octahedral plus distortion
(0, $[{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}\,]$ ), M₂	OII(y) = −OI(z)	K₁	Octahedral plus distortion
( $[\,{\textstyle{{1\over2}}}]$ , 0, $[{\textstyle{{1\over2}}}\,]$ ), M₂	OI(z) = −OIII(x)	K₂	Octahedral plus distortion
( $[\,{\textstyle{{1\over2}}}]$ , $[{\textstyle{{1\over2}}}]$ , 0), M₂	OIII(x) = −OII(y)	K₃	Octahedral plus distortion
( $[\,{\textstyle{{1\over2}}}]$ , 0, 0), M₅′	OIII(y)	X_O12	Octahedral distortion
( $[\,{\textstyle{{1\over2}}}]$ , 0, 0), M₅′	OIII(z)	X_O13	Octahedral distortion
(0, $[{\textstyle{{1\over2}}}]$ , 0), M₅′	OII(z)	X_O23	Octahedral distortion
(0, $[{\textstyle{{1\over2}}}]$ , 0), M₅′	OII(x)	X_O21	Octahedral distortion
(0, 0, $[{\textstyle{{1\over2}}}\,]$ ), M₅′	OI(x)	X_O31	Octahedral distortion
(0, 0, $[{\textstyle{{1\over2}}}\,]$ ), M₅′	OI(y)	X_O32	Octahedral distortion
( $[\,{\textstyle{{1\over2}}}]$ , 0, 0), M₅′	A(y)	X_A12	Cation displacement
( $[\,{\textstyle{{1\over2}}}]$ , 0, 0), M₅′	A(z)	X_A13	Cation displacement
(0, $[{\textstyle{{1\over2}}}]$ , 0), M₅′	A(z)	X_A23	Cation displacement
(0, $[{\textstyle{{1\over2}}}]$ , 0), M₅′	A(x)	X_A21	Cation displacement
(0, 0, $[{\textstyle{{1\over2}}}\,]$ ), M₅′	A(x)	X_A31	Cation displacement
(0, 0, $[{\textstyle{{1\over2}}}\,]$ ), M₅′	A(y)	X_A32	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
		G_O1 = G_O3
		G_A1 = G_A3
3	$[R\bar3c]$ (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	$[Im\bar3]$ (204)	M₁ = M₂ = M₃	∊₁₁ = ∊₂₂ = ∊₃₃
		H₁ = H₂ = H₃
7	Cmcm (63)	R₂	∊₁₁; ∊₂₂; ∊₃₃
		M₃
		G_O2
		G_A2
		H₃
		X_O32
		X_A32
8	P4₂/nmc (137)	R₃	∊₁₁ = ∊₂₂; ∊₃₃
		M₁ = M₂
		G_O3
		H₁ = H₂
		X_O13 = X_O23
		X_A13 = X_A23
9	Pnma (62)	R₁ = R₃	∊₁₁ = ∊₃₃; ∊₂₂; ∊₃₁
		M₂
		G_O1 = G_O3
		G_A1 = G_A3
		K₂
		X_O21 = X_O23
		X_A21 = X_A23

Table 4
Weights of condensed modes in units of u Å²

Hettotype	Material	W(R_i)	W(M_i)	W(X_Aij)	W(X_Oij)	W(G_Ai)	W(G_Oi)	W(H_i)	W(K_i)	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
	Tb_0.67Cu₃Ti₄O₁₂		9.9740					0.1964		6	9
	CaCu₃Mn₄O₁₂		9.2271					0.1323		7	10
	Li_0.36WO₃		5.0358					0.0060		8	11
	Na_0.73WO₃		0.3953					0.0000		8	12
	Na_0.54WO₃		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