The Art of Simplification in Materials Science
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MRS BULLETIN/MAY 1997
WC form of tungsten carbide cannot exist with less than the stoichiometric amount of carbon because its simple hexagonal sublattice of tungsten atoms (Figure 1) would, under its own interactions, be mechanically unstable. Such general understanding is a form of extreme simplification in which the fine detail is smothered in order to reveal a general dominating feature. Simplification is an art rather like that of the cartoonist who captures the key features of a familiar face in a few deft strokes to make it instantly recognizable.
Electron Theories The theory of materials began at the beginning of this century with what is perhaps the most brilliant—certainly one of the most audacious—of simplifications: the free-electron theory of metals. Here the assumption is that each conduction electron moves as if free from all electrical interactions, despite the immensely strong electrical fields exerted on it from the nearby atoms and other conduction electrons. The boldness paid off, for the theory was immediately successful in explaining various properties of metals and was still more so when it was later recast in quantized form. However people worried about its drastic simplifications for years, until two developments showed that nature also seems to make the same ones. First the pseudopotential theory revealed that when a free electron passes through a lattice atom, the strong electrostatic interaction from the field of this atom is, in a good metal, almost exactly offset by the quantum-mechanical interaction, through the Pauli principle, of the electron with those of the atom. Second it was shown that the free electrons mutually correlate their positions in such a way that each, with its negative charge, appears wrapped inside a positively charged hole in the general electron distribution, so that the combination of the electron with its hole behaves jointly as one electrically neutral quasiparticle. One of the great simplifications of recent times stems from the recognition that an atom or electron in a material is strongly influenced only by its nearest neighbors. Of course the chemists, led by Pauling, would have said, "What is the surprise about this? We knew it all along!" However the physicists, guided by the free-electron theory and needing to explain properties such as electrical conductivity, were for many years concerned with long-range macroscopic effects. Perhaps the crowning glory of this approach was the Bloch theory of nearly free electrons in a peri-
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The Art of Simplification in Materials Science
Figure 2. Embedding energy of hydrogen in electron gas. A = molecular binding energy, B = interstitial density in a transition metal, and C = interstitial density in a transition metal at the center of a vacancy.
odic lattice field, with the ensuing theory of Brillouin zones, which received full endorsement by the observation of Bragg reflection in electron-diffraction experiments. The shift to the modern view of concentrating as far as possible on local, short-range interaction
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