Hume-Rothery phases and VEC

An important class of intermetallic phases (generally showing rather wide homogeneity ranges) are the Hume-Rothery phases, which are included within the so-called electron compounds . These are phases whose stable crystal structures may be supposed to be mainly controlled by the number of valence electrons per atom, that is, by the previously defined VEC. 

The Hume-Rothery phases constitute an interesting and ubiquitous group of binary and complex intermetallic substances it was indeed Hume-Rothery who, already in the twenties, observed that one of the relevant parameters in rationalizing compositions and structures of a number of phases is the average number of valence electrons per atom (nJnM). An illustration of this fact may be found in Table 4.6, where a number of the Hume-Rothery structure types have been collected, together with a few more major structure types relevant to transition metal alloys. For each phase the corresponding VEC has been reported as njnai ratio, both calculated on the basis of the s and p electrons and of s, p and d electrons. 

In a first hypothesis (s and p electrons, according to Hume-Rothery), the following valence-electron numbers have been assumed  

In the calculation based on s,p and d electrons the following values are considered  

The Hume-Rothery types listed in Table 4.6 include structures of different complexity ranging from the simple close-packed (fee and hep) and bcc structures to the more complex Mn-type and 7-brass type structures. As for the hexagonal hep structures, notice that for this type several variations are known three branches of this structure are generally considered. The (-hep corresponds to the true closest packing with the value of the axial ratio da close to the ideal one ((8/3) = 1.6329. .. (see 3.7.4 and Fig. 3.16). The e-hex corresponds to da ranging around 1.55 to 1.58 and the r/ phase to c/a values of about 1.77 to 1.88. As observed by Lee and Hoistad (1995), the various hexagonal types can be considered genuinely different. 

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