## 2 Generalities and Particular Features of FK Phases  
FK phases are generally characterized by the association of (at least) two elements of different size and different electronic properties (for a review see [2; 3; 4]). They are defined from the structural point of view by the sole presence of tetrahedral interstices. This consequently results in a limited number of possible coordination polyhedra, classified following their coordination number (CN) into low coordination site CN12 (icosahedron), and high coordination sites CN14, CN15, CN16. The CN13 polyhedron, for example, present in the \(\chi\) phase, is strictly speaking not a FK polyhedron, but this phase has properties so close to those of other FK phases that it may be classified in this group. Single elements may also crystallize in one of the FK structures. For example, this is the case for \(\beta\)-U (structure of the \(\sigma\) phase), \(\alpha\)-Mn (structure of the \(\chi\) phase) and \(\beta\)-Mn. This is also the case for different metastable structures of Ta (\(A15\)[5], \(\sigma\)[6]). While the size ratio between the two elements may be quite flexible in certain phases (Laves phases), it is restricted to a very small range for other phases (\(\sigma\)). It seems to play a key role for the site preference, with the general rule being that the larger element prefers the sites with higher coordination numbers. Very often the more electropositive atom corresponds also to the atom with larger atomic radius, so it may be difficult to deconvoluate the electronic effect from the size effect. In the following, the larger radius element will be designated \(A\) as opposed to the smaller element which will be designated \(B\). Figure 1 presents the homogeneity ranges of different FK phases. It can be seen that, while several phases depart only slightly from their ordered composition (e.g., \(AB_{2}\) for \(C14\) and \(C15\) Laves phases), other phases appear at very distinct \(A/B\) ratios depending on the \(A\)-\(B\) system (\(\sigma\) and \(\chi\) phases, and to a lesser extent the \(\mu\) phase). Additionally, in a given system, the homogeneity range may be quite large (e.g., the \(\sigma\) phase in W-Re system extends from 29 to 57 at.% W). From what is known experimentally, the major structural defect responsible for the stoichiometry deviation from the ideal, ordered crystal structure is substitution. In the case where the deviation is  
Figure 1: Homogeneity ranges of various Frank-Kasper (FK) phases in the combined systems in which they crystallize represented as a function of the composition of the larger (\(A\)) element. The ordered or ideal composition(s) for each phase is indicated. very limited, this can be called an anti-site defect since the major element at each given site is clearly identified (e.g., Laves phases). For other systems for which the non-stoichiometry is much larger, it may not be easy to identify the stoichiometric ordered composition, and some sites may completely switch as a function of composition from the occupation by one atom to the occupation by the other atom (see e.g., the case of the \(\upmu\) phase in Section 3.2). A common idea concerning the site occupancies of \(A\) atoms within the homogeneity domain (_i.e._, as a function of \(A\) composition) originates from [PERSON] [7, 8], though it is not expressed as clearly in their original papers. Considering the tendency of \(A\) atom to prefer the occupancy of high CN sites (with larger atom volume to fit it), it is supposed that \(A\) would occupy sequentially the sites from high to low coordination as a function of \(A\) composition. Different sites of equal CN would be occupied simultaneously. This scheme will be called Kasper rules in the following paragraphs. An example of this ideal behavior is represented in the case of the \(\sigma\) phase in Figure 2.  
**Figure 2.** Ideal site occupancies following Kasper rules for the \(\sigma\) phase. As a function of \(A\) composition, sites of decreasing CN are occupied sequentially.