Normal Vibrations ofβBrass

Abstract

The normal modes of vibration of the ordered alloy of copper and zinc, $\ensuremath{\beta}$ brass, have been extensively studied at 296\ifmmode^\circ\else\textdegree\fi{}K by means of the coherent one-phonon scattering of slow neutrons from single-crystal specimens. The frequencies of normal modes propagating along the high-symmetry directions [$00\ensuremath{\zeta}$], [$\ensuremath{\zeta}\ensuremath{\zeta}0$], [$\ensuremath{\zeta}\ensuremath{\zeta}\ensuremath{\zeta}$], and [$\frac{1}{2}\frac{1}{2}\ensuremath{\zeta}$] have been measured by means of a triple-axis crystal spectrometer for two specimens mounted in different orientations. These results may be satisfactorily described in terms of a restricted Born-von K\'arm\'an model ($4E$) with interatomic forces extending out to fourth-nearest-neighbor atoms, although Fourier analysis of sums of squares of certain normal-mode frequencies (taken in pairs) indicates the probable existence of nonzero forces extending at least to seventh neighbors. A feature of interatomic-force models for $\ensuremath{\beta}$ brass is the large difference between the second-nearest-neighbor Cu-Cu and Zn-Zn forces. The experimental evidence does not, however, permit an unambiguous determination of which force is to be identified as Cu-Cu and which as Zn-Zn. A fairly successful attempt has been made to interpret the results in terms of more physically realistic models, in which the effective long-range forces are represented by an oscillatory potential arising from the mild singularity in the dielectric function at the Fermi level. The precise form of the coefficient of the oscillatory potential has been chosen on an empirical basis. Model 4E has been used as an interpolation formula for computing the frequency distribution function and its moments, and also the heat capacity of $\ensuremath{\beta}$ brass. A selection of normal modes has been studied at several temperatures above 296\ifmmode^\circ\else\textdegree\fi{}K, particularly in the vicinity of the order-disorder phase transition at about 727\ifmmode^\circ\else\textdegree\fi{}K. The over-all structure of the dispersion curves appears to be substantially unchanged in the disordered phase, although certain "splittings" observed at 296\ifmmode^\circ\else\textdegree\fi{}K become blurred into apparently continuous bands of frequencies at elevated temperatures. In all cases, the frequencies decrease and energy widths increase as the temperature increases. Two particular longitudinal-optic modes display a sharp increase in energy width at the transition temperature, in contrast to the generally smooth behavior of the other modes. No satisfactory explanation of these effects has yet been found.