APPLICATION OF A SIX-DIMENSIONAL TWIN REFINEMENT TECHNIQUE TO MULTIPLE-TWINNED CRYSTALS OF Cu2SnS3, Cu8GeS6 and Ag7TaS6

M. Onoda, H. Wada, X.-a. Chen, A. Sato and M. Ishii

Advanced Materials Laboratory, National Institute for Material Science,

Namiki 1-1, Tsukuba, Ibaraki, 305-0044, Japan (ONODA.Mitsuko@nims.go.jp)

The twinned structure could be described by a six-dimensional formalism and refined using the multi-dimensional refinement program FMLSM [1,2]. When the reflections can be indexed based on two sets of basic vectors, a, b, c and a’, b’, c’, each reflection is expressed by q=ha*+kb*+lc*+h’a’*+ k’b’*+l’c’*. Symmetry operations are also expressed in the (3+3)-dimensional formalism [2]. Those for the major domains (a, b, c) are given by the scheme g=(( R 0 | 0 R) | (v|v)), where R and v are the matrix and the vector of the corresponding three-dimensional symmetry operations. The symmetry operations for the minor domains (a’, b’, c’) are obtained by multiplying the six-dimensional twinning operation t={ T | 0 } into the major domain symmetry operations g where T is the matrix (000100 | 000010 | 000001 | 100000 | 010000 | 001000) [2].

The X-ray diffraction data of low-symmetry Cu₂SnS₃ were obtained from a twinned monoclinic crystal (a=6.653, b=11.537, c=6.665 Å, b=109.39°) [3]. The reflections were indexed on the basis of a, b, c and a’, b’, c’, where a’*=-a*, b’*=b*, c’*=-(2/3)a*+c*. The reflections are of three groups, the first is assigned by both of hkl000 and 000h’k’l’ and considered to be common to two twin domains, while the second (hkl000) comes from the major domain and the third (000h’k’l’) comes from the minor domain. The structure was described based on a, b, and c with space group Cc. Refinement was performed using all reflections through FMLSM. The agreement was satisfactory with 52 structural parameters and 2 scale factors; R_F=0.036 and wR_F=0.038.

Observed reflections of Cu₈GeS₆ [4] seemed to be from a twinned rhombohedral crystal (a~9.9 Å and a~90°), the major ones based on a, b, c and minor ones based on a’=-a, b’=-b, c’=-c. The model proposed using powder data was that with A=7.0445, B=6.9661, C=9.8699Å and Pmn2₁. After selecting new bases a=A+B, b=B-A, c=C, the lattice constants were a=b= 9.9073, c=9.8703Å, a=b=90°, g=90.642°. Symmetry operations were also converted into the formula based on C-lattice: generator sets are 1/2+x,1/2+y, z; y,x,z; 3/4-y,1/4-x,1/2+z. The major group reflections and six major domains with the twin operations x,y,z, y,z,x, z,x,y, x,-y,-z, -y,-z,x, and -z,x,-y were considered. Then the minor group reflections and six minor domains were added. Besides structural parameters, 12 scale factors were considered as parameters in refinement. The agreement was satisfactory for 1804 reflections; R_F=0.083 and wR_F=0.091.

In a similar manner, the structure of the monoclinic low-temperature phase of Ag₇TaS₆ was examined on the basis of X-ray diffraction data from a 24-fold twinned crystal with 12 major domains and 12 minor domains.

References

1 Kato, K. (1994) Acta Crystallogr. A50, 351-357.

2 Kato, K. (1997) Z. Kristallogr. 212, 423-427.

3 Onoda, M., Chen X.-a., Sato, A. and Wada, H., (2000) Mat. Res. Bull. 35, 1563-1570.

4 Onoda, M., Chen X.-a., Kato, K., Sato, A. and Wada, H., (1999) Acta Crystallogr. B55, 721-725.