By W. B. Pearson and G. V. Raynor (Auth.)
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Extra resources for A Handbook of Lattice Spacings and Structures of Metals and Alloys
N(E) is defined as the number of states per unit volume between £ and (£ + dE). THE BRILLOUTN ZONE CONCEPT 33 For any particular crystal structure there are a large number of specific planes for which a Bragg reflection occurs and so in k space there are a succession of energy discontinuities as the electron wave number increases in any direction. In fact, successive Brillouin zones may be specified as the electron wave number increases. It can be shown that each successive Brillouin zone in k space has the same volume and contains two electrons per atom.
BRILLOUIN (1946) Wave Propagation in Periodic Structures, pp. 152-54, McGraw-Hill, New York. R. L. BERRY, M. B. WALDRON, and G. V. RAYNOR (1950) Research 3, 195. J. F. NICHOLAS (1951) Proc. Phys. Soc. B64, 953. * At the absolute zero; at higher temperatures the energies of the electrons at the top of the band spread out over a range of energy —kT in width, although at all ordinary temperatures this is a very small range of energy compared with the total ("zero-point") energy of the electrons. 34 THE THEORY OF ALLOYS third, and fourth zones.
BERRY, WALDRON, and RAYNOR (1952), however, while admitting the formal correctness of NICHOLAS'S treatment, point out that in order to understand alloy structures it is useful to construct zones from such planes that the volume enclosed may be of interest in the structure under examination, and that any plane leading to an energy discontinuity may be of importance, t They emphasize that this approach has frequently been used successfully. g. Co2Al9). Calculations for the first prominent Brillouin zone of a number of structures can be found in the literature, and systematic methods of calculating the volumes of complex polyhedra corresponding to Brillouin zones have been discussed by SHOEMAKER and HUANG (1954).