Unified (3 + 1)-dimensional superspace description of the 2212-type stair-like [Bi₂Sr₃Fe₂O₉]_m[Bi₄Sr₆Fe₂O₁₆] family of compounds

The (3 + 1)-dimensional superspace approach is applied to describe and refine a series of sheared compounds related to layered high T_c superconducting oxides. Two commensurate members (m = 4, 5) of the 2212 stair-like [Bi₂Sr₃Fe₂O₉]_m[Bi₄Sr₆Fe₂O₁₆] family of compounds, previously studied using single-crystal diffraction data, are analyzed. A common average unit cell has been identified and a composition-dependent modulation wavevector is proposed. The model is built using only three independent atomic domains, one for the metal atoms and two for the O atoms. The three Sr, Bi and Fe species are described using close-connected crenel-like functions forming a continuous atomic domain along the internal space. The two oxygen domains are represented by crenel functions and the displacive modulation functions are built up by Legendre polynomials recently implemented in the program JANA2006. Surprisingly, the results of the refinements show a striking similarity of the displacive modulations for the two compounds analyzed, indicating that a unique model can be used to describe the correlations between the atomic displacements of the 2212 stair-like series. This final model is then applied to predict the structure of new members of the family.

Computing details

(I)

Crystal data

Bi12Fe10O52Sr18	β = 81.2438°
Mr = 5475.3	V = 67.87 Å^3
Monoclinic, X2/m(α0γ)0s†	Z = 8
q = 0.275000a* + -0.300000c*	F(000) = 118
a = 3.6174 Å	Dx = 1071.354 Mg m^−3
b = 5.48 Å	µ = 9851.70 mm^−1
c = 3.4641 Å	T = 293 K

† Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₁, x₂, −x₃, −x₄+1/2; (3) −x₁, −x₂, −x₃, −x₄; (4) x₁, −x₂, x₃, x₄+1/2; (5) x₁+1/2, x₂+1/2, x₃+1/2, x₄+1/2; (6) −x₁+1/2, x₂+1/2, −x₃+1/2, −x₄; (7) −x₁+1/2, −x₂+1/2, −x₃+1/2, −x₄+1/2; (8) x₁+1/2, −x₂+1/2, x₃+1/2, x₄.

Data collection

Radiation source: X-ray tube	2671 reflections with I > 3σ(I)
2674 independent reflections

Refinement

Refinement on F	131 parameters
R[F^2 > 2σ(F^2)] = 0.096	0 restraints
wR(F^2) = 0.121	789 constraints
S = 3.00	Weighting scheme based on measured s.u.'s w = 1/(σ^2(F) + 0.0001F^2)
2674 reflections	(Δ/σ)max < 0.001

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq	Occ. (<1)
Sr	0	0	0	0.0097 (3)	0.225
Bi	0	0	0	0.0097 (3)	0.075
Fe	0	0	0	0.0097 (3)	0.125
O1	0	0.451 (3)	0	0.024 (3)*	0.375
O2	0.238 (4)	0.243 (3)	0.266 (4)	0.018 (3)*	0.1375

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Sr	0.0113 (5)	0.0084 (7)	0.0098 (5)	0	−0.0032 (3)	0
Bi	0.0113 (5)	0.0084 (7)	0.0098 (5)	0	−0.0032 (3)	0
Fe	0.0113 (5)	0.0084 (7)	0.0098 (5)	0	−0.0032 (3)	0

(II)

Crystal data

Bi14Fe12O61Sr21	β = 81.8989°
Mr = 6411.8	V = 68.22 Å^3
Monoclinic, X2/m(α0γ)0s†	Z = 8
q = 0.276596a* + -0.297872c*	F(000) = 59
a = 3.5905 Å	Dx = 1248.213 Mg m^−3
b = 5.4903 Å	T = 293 K
c = 3.4954 Å

† Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₁, x₂, −x₃, −x₄+1/2; (3) −x₁, −x₂, −x₃, −x₄; (4) x₁, −x₂, x₃, x₄+1/2; (5) x₁+1/2, x₂+1/2, x₃+1/2, x₄+1/2; (6) −x₁+1/2, x₂+1/2, −x₃+1/2, −x₄; (7) −x₁+1/2, −x₂+1/2, −x₃+1/2, −x₄+1/2; (8) x₁+1/2, −x₂+1/2, x₃+1/2, x₄.

Data collection

Radiation source: X-ray tube	6846 reflections with I > 3σ(I)
11954 independent reflections

Refinement

Refinement on F	0 restraints
R[F^2 > 2σ(F^2)] = 0.086	981 constraints
wR(F^2) = 0.104	Weighting scheme based on measured s.u.'s w = 1/(σ^2(F) + 0.0001F^2)
S = 2.60	(Δ/σ)max < 0.001
11954 reflections	Δρmax = 35.75 e Å^−3
158 parameters	Δρmin = −12.86 e Å^−3

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq	Occ. (<1)
Sr	0	0	0	0.01011 (9)	0.2234
Bi	0	0	0	0.01011 (9)	0.0745
Fe	0	0	0	0.01011 (9)	0.1277
O1	0	0.4494 (8)	0	0.0215 (8)*	0.3723
O2	0.2395 (15)	0.2462 (7)	0.2645 (15)	0.0127 (7)*	0.1383

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Sr	0.01350 (15)	0.00468 (14)	0.01352 (16)	0	−0.00666 (11)	0
Bi	0.01350 (15)	0.00468 (14)	0.01352 (16)	0	−0.00666 (11)	0
Fe	0.01350 (15)	0.00468 (14)	0.01352 (16)	0	−0.00666 (11)	0