Structure refinement of the layered composite crystal Sc₂B_1.1C_3.2 in a five-dimensional formalism

The crystal structure of a layered compound Sc₂B_1.1C_3.2, scandium boride carbide (M_r = 140.43), has been re-refined as a commensurate composite crystal using 1795 single-crystal X-ray diffraction intensities with I > 2(I) collected by Shi, Leithe-Jasper, Bourgeois, Bando & Tanaka [(1999), J. Solid State Chem. 148, 442-449]. The crystal is composed of two layered subsystem structures, i.e. Sc-C-Sc sandwiches and graphite-like layers of the composition B_1/3C_2/3. The structure refinement was performed in a five-dimensional formalism based on the trigonal superspace group . The unit cell and other crystal data are a = b = 3.387 (1), c = 6.703 (2) Å, V = 66.59 (1) Å³, = (9/7 0 0), = (0 9/7 0), Z = 1, D_x = 3.501 Mg m^-1. Two different three-dimensional sections through the superspace were analyzed, corresponding to two different superstructure models, one with and the other with . A random distribution of B and C was assumed in the graphite-like layer and 41 structural parameters were introduced. R_F/wR_F} were 0.0533/0.0482 and 0.0524/0.0476, respectively, for the first and second models. Although the difference between these R_F or wR_F values was too fine to exclude one of the models definitely, the advantages of using a superspace group were obvious. It not only brought about better convergence of refinement cycles by virtue of fewer parameters, but also gave an insight into the problem of symmetry of the superstructure.

Computing details

Program(s) used to refine structure: FMLSM (Kato, 1994).

Figures

(ScBC)

Crystal data

Sc2B54/49C157/49	The structural model can be described based on a trigonal cell with lattice constants A=B=23.710(9), C=6.703(2). In the present refinement, the structure is treated as an interpenetration of two layered subsystem structures, and refined in a five-dimensional formalism based on the superspace-group approach for composite crystals. New basis selected are a=a1=A/7, b=b1=B/7, c=C, as the strong reflections could be indexed using two sets of hexagonal axis with different a and common c: a1=b1=A/7=3.387 and c=6.703; a2=b2=A/9=2.634 and c=6.703. Each reflection is expressed by ha1*+kb1*+lc*+ma2*+nb2*. The new cell constants are a=b=a1=b1=3.387(1), c=6.703(2), sigma1=(9/7 0 0), sigma2=(0 9/7 0). The possible superspace group can be derived from the possible space groups of two subsystems. If the space groups of the two parts are assumed to be both P-3m1, the symmetry operations of the probable superspace group are listed as equivalent positions in xyzx4x5 as follows.
Mr = 140.43	Dx = 3.501 Mg m^−3
Trigonal, P3m1(p00)(0p0)0m0†	Mo Kα radiation, λ = 0.70930 Å
q1 = 1.286a*; q2 = 1.286b*‡	µ = 4.79 mm^−1
a = 3.387 (1) Å	T = 293 K
c = 6.703 (2) Å	Platelike, silver metallic
V = 66.59 (1) Å^3	0.20 × 0.10 × 0.06 mm
Z = 1

† Symmetry operations: (1) x, y, z, x₄, x₅; (2) −y, x−y, z, −x₅, x₄−x₅; (3) −x+y, −x, z, −x₄+x₅, −x₄; (4) y, x, −z, x₅, x₄; (5) x−y, −y, −z, x₄−x₅, −x₅; (6) −x, −x+y, −z, −x₄, −x₄+x₅; (7) −x, −y, −z, −x₄, −x₅; (8) y, −x+y, −z, x₅, −x₄+x₅; (9) x−y, x, −z, x₄−x₅, x₄; (10) −y, −x, z, −x₅, −x₄; (11) y−x, y, z, x₅−x₄, x₅; (12) x, x−y, z, x₄, x₄−x₅.

‡ ;

Data collection

Enraf-Nonius CAD4 diffractometer	1795 reflections with I > 2σ(I)
Radiation source: xray tube	Rint = 0.050
Graphite monochromator	θmax = 37.5°, θmin = 1.7°
ω–2θ scans	h = ?→?
Absorption correction: gaussian ?	k = ?→?
Tmin = ?, Tmax = ?	l = ?→?
8284 measured reflections	3 standard reflections
8284 independent reflections	intensity decay: none

Refinement

Refinement on F	43 parameters
Least-squares matrix: full	0 restraints
R[F^2 > 2σ(F^2)] = 0.053	Calculated w = 1
wR(F^2) = .048
1795 reflections

Special details

Refinement. The crystal is composed of two layered subsystem structures, i.e. Sc—C—Sc sandwiches (a1=b1=3.387 and c=6.703) and graphite-like layers of the composition B1/3 C2/3 (a2=b2=2.634 and c=6.703). The structure refinement was performed in a five-dimensional formalism based on the trigonal superspace group P-3m1(p 0 0)(0 p 0)0m0 through FMLSM (Kato, K., Acta Cryst.(1994) A50, 351–357) using the data indexed by five integers, h, k, l, m and n. On account of the commensurate feature of the lattice geometry, the summations instead of integrals in the structure-factor formula were used with summation points 49 (subsystem1) and 81 (subsystem2). In the model, all atoms are located at special positions, that is Sc in (1/3 2/3 z) and C in (0 0 1/2) of the first subsystem and B1/3 C2/3 in (1/3 2/3 z) of the second system. Besides one scaling factor and one parameter for extinction correction, atomic coordinates and thermal parameters, the Fourier amplitudes of the modulation functions were considered as structural parameters. For Sc in subsystem 1, displacive modulations up to the third order and modulations in anisotropic thermal parameters up to the first order were adopted. For C in subsystem 1, displacive modulations up to the first order were adopted. For B and C in subsystem2, random distribution of B and C were assumed and displacive modulations of B1/3 C2/3 up to the first order were adopted. Isotropic and unmodulated thermal parameters were adopted for C in subsystem 1 and B1/3 C2/3 in subsystem 2.

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

	x	y	z	Uiso*/Ueq	Occ. (<1)
Sc	0.333333	0.666667	0.32221 (6)	0.00775 (7)
C	0.00	0.00	0.500000	0.0089 (4)	0.500000
B1/3C2/3	0.333333	0.666667	−0.0004 (3)	0.0093 (3)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Sc	0.0074 (1)	0.0074	0.0084 (2)	0.0037	0.00	0.00