Hume-Rothery theories

Understanding alloys in terms of electron theory. The band theory of solids had no impact on the thinking of metallurgists until the early 1930s, and the link which was eventually made was entirely due to two remarkable men - William Hume-Rothery in Oxford and Harry Jones in Bristol, the first a chemist by education and the second a mathematical physicist. 

Hume-Rothery s ideas and their theoretical development by Mott and Jones stimulated much consequential research around the world. The most impressive early eonvert was a Freneh physicist, Jacques Friedel, who should have been mentioned in connection with dislocations, in the theory of whieh he played an early part (see the Corrigenda). After a very disturbed war, which ranged from study at the... 

Meanwhile, electron theory was revived effectively in Hume-Rothery s own base of Oxford, and is now led by a distinguished mathematical physicist, David Pettifor. 

Hume-Rothery (1946) Atomic Theory for Students of Metallurgy (The Institute of Metals, London). 

W. Hume-Rothery, The Engel-Brewer Theories of Metals and Alloys , Prog. Mat. Sci. 13 (5), 229-265 (1967). 

Hume-Rothery was to prove a fair, if demanding, editor, and the result was an important review on the stability of metallic phases as seen from the CALPHAD viewpoint (Kaufman 1969). The relevant correspondence provides a fascinating insight into his reservations concerning the emerging framework fiiat Kaufman had in mind. Hume-Rothery had spent most of his life on the accurate determination of experimental phase diagrams and was, in his words (Hume-Rothery 1968), ... not unsympathetic to any theory which promises reasonably accurate calculations of phase boundaries, and saves the immense amount of work which their experimental determination involves . 

Hume-Rothery, W. (1962). Atomic theory for students of metallurgy (Fourth revised reprint). Institute of Metals, London. 

Discuss the origin of the Hume-Rothery electron phases within the framework of Jones original rigid-band analysis. How does second-order perturbation theory help quantify Mott and Jones earlier supposition on the importance of the free electron sphere touching a Brillouin zone boundary ... 

Hume-Rothery (12,13) has pointed out that in some alloys the structure of the intermetallic phases are determined by the electron concentration (E.C.). The work of Hume-Rothery and others has shown that the series of changes (i.e. a phase —> (3 phase —> 7 phase — phase), which occurs as the composition of an alloy is varied continuously, takes place at electron-atom ratios of 3/2, 21/13, and 7/4, respectively. The interpretation of these changes in terms of the Brillouin zone theory has been made by H. Jones (14) and can be understood from the A (E)-curves for typical face centered cubic (a) and body centered cubic (6) structures as... 

Fig. 5. Interpretation of Hume-Rothery alloys in terms of Brillouin zone theory.

P.Caldirola, jChemPhys 16, 846-7(1948) (Detonation wave in nuclear explosions) 12)W.Hume-Rothery, Atomic Theory for Students of Metallurgy, Institute of Metals, London (1948) 13)G.Gamow C.L.Critch-field, "Theory of Atomic Nucleus and Nuclear Energy Sources, Clarendon Press, Oxford... 

The classical theory of Hume-Rothery states that a difference in atomic diameters of solute and solvent atoms of more than 15% produces restricted solid solubility. The closest distance of approach of the atoms in the crystals of the element is taken as a measure of the atomic size. Substitution of a larger atom into a lattice requires a high amount of energy due to the concomitant disorganization of the parent lattice. However, the size factor becomes less important [221] when the difference in size is 8% or less. It is desirable (though not essential) that the size factor and the crystal structure of the elements producing a solid solution in all proportions be favourable. It is, however, apparent that if elements forming alloys did not possess the same crystal structure, a continuous series of solid solutions would be impossible. 

The correlation between the valence electron counts and the stabilities of intermetallic phases and stmctures were also espoused by others, like the physical chemists Neds N. Engel (b. 1904) and Leo Brewer (1919-2005), although Hume-Rothery found their result somewhat controversial. The Engel-Brewer theory asserts that the crystal stmctures of transition metals and their intermetallic compounds are determined solely by the number of valence s and p electrons. For example, Engel suggested in 1949 that the BCC stmcture correlated with (where n is the total number of valence... 

Seitz, F. The Modern Theory of Solids New York, 1940 Hume-Rothery, W, The Metallic State Oxford, 1931... 

Coulson, C. A. Valence. Oxford Clarendon Press. 2nd ed. 1961. Hume-Rothery, W. Atomic Theory for Students of Metallurgy. London Institute of Metals (Monograph 3). 4th revised reprint. 1962. 

DFT studies of binary hard-sphere mixtures predate the simulation studies by several years. The earliest work was that of Haymet and his coworkers [221,222] using the DFT based on the second-order functional Taylor expansion of the Agx[p]- Although this work has to some extent been superceded, it was a significant stimulus to much of the work that followed both with theory and computer simulations. For example, it was Smithline and Haymet [221] who first analyzed the Hume-Rothery rule in the context of hard sphere mixture behavior and who first investigated the stability of substitutionally ordered solid solutions. The most accurate DFT results for hard-sphere mixtures have come from the WDA-based theories. In particular the results of Denton and Ashcroft [223] and those of Zeng and Oxtoby [224] give qualitatively correct behavior for hard spheres forming substitutionally disordered solid solutions. 

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