Empirical Methods of Solving Crystal Structures

In addition to reciprocal and direct space techniques considered in the earlier sections, a large variety of approaches may be employed to create a model of the crystal structure in direct space. One of these, that is, the geometrical method has been implicitly employed in Chap. 16, where the location of a single La atom in the unit cell was established from a simple analysis of the unit cell dimensions and from the availability of low multiplicity sites in the space-group symmetry P6/ mmm. Here we consider a more complex example, that is, the solution of several crystal structures occurring in the series of Gd₅(Si_xGe_1−x)4 alloys.¹ These examples illustrate the power of the powder diffraction method in detecting subtle details of the atomic distribution in the unit cell in addition to highlighting how structural information is an enabling step in establishing critical structure — properties correlations.₂ It is worth noting that empirical techniques nearly always require extensive literature searches to find out as much as possible about the crystal structures of closely related materials.³

According to Smith, Tharp and Johnson,⁴ both the silicide and germanide of Gd at 5:4 stoichiometries belong to the same type of crystal structure; the distributions of atoms in their unit cells are essentially identical to the orthorhombic Sm₅Ge₄-type