On A Method of Generating Space Groups and Calculating Structure Factors By Coset Decomposition

Visible Information Center, Inc.* Institute of Applied Beam Science, Graduate School of Science and Engineering, Ibaraki University** Institute of Materials Structure Science, High Energy Accelerator Research Organization (KEK)*** Quantum Beam Science Directorate, Japan Atomic Energy Research Institute (JAERI )****
â—‹RYOKO OISHI* Masao Yonemura** Fumihito Shikanai*** Shuki Torii*** Toru Ishigaki**** Takashi Kamiyama***

As a method of generating space group elements and calculating structure factors, a new algorithm using coset decomposition will be presented. This algorithm is a part of the Rietveld code developed for the versatile neutron diffractometer (IBARAKI Materials Design Diffractometer) in J-PARC(Japan Proton Accelerator Research Complex)that will be built by Ibaraki prefecture government to promote industrial applications for neutron beam.
When H is a subgroup of a space group G, the structure factor formula corresponding to G is decomposed into the term of the structure factor formula corresponding to H and the term expressed by coset representatives of G over H. In this reason, site symmetries, Wyckoff multiplicity, reflection multiplicity and extinction rules can be determined in the same process of generating space groups by coset decomposition. Since every space group and its subgroups have a normal subgroup of index 2 or 3, this algorithm can be executed by a recursive function that replaces its argument with one of the maximal normal subgroups of the previous group repeatedly.
For calculation of structure factors, our program contains tables on space groups as follows; Bravais lattice, all the maximal normal subgroups of 32 point groups, coset representatives for these subgroups, the translation vectors of the elements corresponding to the above coset representatives for the non-symmorphic space groups.