Abstract
For a long time we have shared the belief that the physics of the Hume-Rothery electron concentration rule can be deepened only through thorough investigation of the interference phenomenon of itinerant electrons with a particular set of lattice planes, regardless of whether d-states are involved near the Fermi level or not. For this purpose, we have developed the FLAPW-Fourier theory (Full potential Linearized Augmented Plane Wave), which is capable of determining the square of the Fermi diameter, ( 2 k F ) 2 , and the number of itinerant electrons per atom, e/a, as well as the set of lattice planes participating in the interference phenomenon. By determining these key parameters, we could test the interference condition and clarify how it contributes to the formation of a pseudogap at the Fermi level. Further significant progress has been made to allow us to equally handle transition metal (TM) elements and their compounds. A method of taking the center of gravity energy for energy distribution of electrons with a given electronic state has enabled us to eliminate the d-band anomaly and to determine effective ( 2 k F ) 2 , and e/a, even for systems involving the d-band or an energy gap across the Fermi level. The e/a values for 54 elements covering from Group 1 up to Group 16 in the Periodic Table, including 3d-, 4d- and 5d-elements, were determined in a self-consistent manner. The FLAPW-Fourier theory faces its limit only for elements in Group 17 like insulating solids Cl and their compounds, although the value of e/a can be determined without difficulty when Br becomes metallic under high pressures. The origin of a pseudogap at the Fermi level for a large number of compounds has been successfully interpreted in terms of the interference condition, regardless of the bond-types involved in the van Arkel-Ketelaar triangle map.
Highlights
The establishment of the empirical Hume-Rothery electron concentration rule dates back to 1926 when Hume-Rothery [1] pointed out, for the first time, that CuZn, Cu3Al and Cu5Sn crystallize into bcc structure with a common valence equal to 3/2, as given by a composition average of valence electrons per atom
While the metallurgist Hume-Rothery and crystallographers Westgren and Phragmén took a composition average of valences of constituent elements in a compound, Mott and Jones apparently treated it as an average number of free electrons per atom defined from the diameter of the Fermi sphere in the reciprocal space
After the great success by Mott and Jones [3], a consensus has been gradually built in such a way that alloys or compounds obeying the Hume-Rothery electron concentration rule are limited to those in which the electronic structure can be described in terms of the nearly free electron (NFE) model
Summary
The establishment of the empirical Hume-Rothery electron concentration rule dates back to 1926 when Hume-Rothery [1] pointed out, for the first time, that CuZn, Cu3Al and Cu5Sn crystallize into bcc structure with a common valence equal to 3/2, as given by a composition average of valence electrons per atom. In 1936, Mott and Jones [3] could successfully interpret the Hume-Rothery electron concentration rule in terms of the contact of the Fermi sphere with the set of Brillouin zone planes specific to a given phase. While the metallurgist Hume-Rothery and crystallographers Westgren and Phragmén took a composition average of valences of constituent elements in a compound, Mott and Jones apparently treated it as an average number of free electrons per atom defined from the diameter of the Fermi sphere in the reciprocal space.
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