Modular crystals as modulated structures: the case of the lillianite homologous series

The use of the superspace formalism is extended to the description and refinement of the homologous series of modular structures with two symmetry-related modules with different orientations. The lillianite homologous series has been taken as a study case. Starting from a commensurate modulated composite description with two basic subsystems corresponding to the two different modules, it is shown how a more efficient description can be achieved using so-called zigzag modulation functions. These linear zigzag modulations, newly implemented in the program JANA2006, have very large fixed amplitudes and introduce in the starting model the two orientations of the underlying module sublattices. We show that a composite approach with this type of function, which treats the cations and anions as two separate subsystems forming a misfit compound, is the most appropriate and robust method for the refinements.

Computing details

Program(s) used to refine structure: Jana2006 (Petricek, Dusek & Palatinus, 2006); software used to prepare material for publication: Jana2006 (Petricek, Dusek & Palatinus, 2006).

(I)

Crystal data

PbS	Z = 16
Mr = 239.2	F(000) = 214
Orthorhombic, Bmmm(0β0)s00†	Dx = 63.789 Mg m^−3
q = 0.05556b*	Mo Kα radiation, λ = 0.71069 Å
a = 13.737 Å	µ = 77.24 mm^−1
b = 1.7496 Å	T = 293 K
c = 4.1448 Å	Platy, black
V = 99.62 Å^3	0.10 × 0.05 × 0.02 mm

† Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₁, −x₂, x₃, −x₄; (3) −x₁, x₂, −x₃, x₄+1/2; (4) x₁, −x₂, −x₃, −x₄+1/2; (5) −x₁, −x₂, −x₃, −x₄; (6) x₁, x₂, −x₃, x₄; (7) x₁, −x₂, x₃, −x₄+1/2; (8) −x₁, x₂, x₃, x₄+1/2; (9) x₁+1/2, x₂, x₃+1/2, x₄; (10) −x₁+1/2, −x₂, x₃+1/2, −x₄; (11) −x₁+1/2, x₂, −x₃+1/2, x₄+1/2; (12) x₁+1/2, −x₂, −x₃+1/2, −x₄+1/2; (13) −x₁+1/2, −x₂, −x₃+1/2, −x₄; (14) x₁+1/2, x₂, −x₃+1/2, x₄; (15) x₁+1/2, −x₂, x₃+1/2, −x₄+1/2; (16) −x₁+1/2, x₂, x₃+1/2, x₄+1/2.

Data collection

Radiation source: X-ray tube	756 independent reflections
Absorption correction: empirical (using intensity measurements)	454 reflections with I > 3σ(I)
Tmin = 0.037, Tmax = 0.187

Refinement

Refinement on F	24 parameters
R[F^2 > 2σ(F^2)] = 0.043	Weighting scheme based on measured s.u.'s w = 1/(σ^2(F) + 0.0001F^2)
wR(F^2) = 0.051	(Δ/σ)max = 0.049
S = 1.42	Extinction correction: B-C type 1 Gaussian isotropic (Becker & Coppens, 1974)
756 reflections	Extinction coefficient: 243 (28)

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

	x	y	z	Uiso*/Ueq	Occ. (<1)
Pb0	0.92118 (19)	0.5	0	0.0443 (11)	0.0555
S1	0.5	0.5	0	0.0199 (14)	0.5
Pb1	0	0	0	0.0174 (3)	0.5

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Pb0	0.0341 (16)	0.068 (2)	0.0310 (19)	0	0	0
S1	0.024 (2)	0.018 (3)	0.019 (2)	0	0	0
Pb1	0.0184 (5)	0.0163 (5)	0.0175 (5)	0	0	0