Frank-Kasper alloy structures

Then, we can derive a very simple rule on selecting crystal structures of the clathrate hydrates by employing Yarmolyuk-Kripyakevich s empirical rule on the coordination numbers of a Frank-Kasper alloy structure [79]. The rule denotes the relation between the number of 12-, 14-, 15-, and 16-coordinated atoms in an alloy structure. That is, a set of such numbers in any FK crystal structure is given by a linear combination of the three representative Frank-Kasper crystal structures, A15, C15, and Z. For example, the number of 12-, 14-, 15-, and 16-coordinated atoms in a structure, say, 4, , and Ajg, is 20,32,8, and 0,... 

TS-I) 20/43. In this viewpoint, any FKCH structure can be regarded as the mixture of three basic types. We, therefore, term these constants compositions. The compositions for various FKCH structures including the nonexistent ones, which are composed from the Frank-Kasper alloy structures, are illustrated as a ternary diagram in Fig. 6. 

Table 1.4 Structural relations between Frank-Kasper alloys and IV, V and H clathrate types...

F. C. Frank, J. S. Kasper, Complex alloy structures regarded as sphere packings. I Definitions and basic principles, Acta Crystallogr. 11 (1958) 184. II Analysis and classification of representative structures, Acta Crystallogr. 12 (1959) 483. 

F. C. Frank and J. S. Kasper, Complex Alloy Structures Regarded as Sphere Packings. II. Analysis and Classification of Representative Structures, Acta Cryst., 12, 483 (1959). 

Frank, F.C. Kasper, J.S. (1958). Complex Alloy Structures Regarded as Sphere Packings. I. Definitions and Basic Principles. Acta Crystallogr. 11,184-190. 

This idea of defective vertices is not new but was used by Frank and Kasper more than 40 years ago to study polyhedra found in metal alloy structures. In such systems the defective vertices are vertices of degree 6. Frank and Kasper showed that there are only four polyhedra (Figure 11) with only degree 5 and 6 vertices and with isolated degree 6 vertices, i.e., no pair of degree 6 vertices is connected by an edge. The four Frank—Kasper polyhedra are... 

---