Crystal Dynamics of Lead. I. Dispersion Curves at 100°K

Abstract

Frequency/wave vector $\ensuremath{\nu}(\mathrm{q})$ dispersion curves for lead at 100\ifmmode^\circ\else\textdegree\fi{}K have been measured by neutron spectrometry along the lines [$\ensuremath{\zeta}00$], [$\ensuremath{\zeta}\ensuremath{\zeta}\ensuremath{\zeta}$], and [$\ensuremath{\zeta}10$] in the reduced zone. The experiments were performed with the triple-axis crystal spectrometer, making extensive use of the constant Q method. The results show many interesting features. The dispersion relations have local minima at the point (1,0,0), leading to extra critical points in the frequency distribution. The dispersion curves are analyzed into Fourier components (within the estimated errors), according to the equation $M{\ensuremath{\omega}}^{2}={{\ensuremath{\Sigma}}_{n=1}}^{N}{\ensuremath{\Phi}}_{n}[1\ensuremath{-}cos(\frac{\ensuremath{\pi}\mathrm{nq}}{{q}_{M}})]$ with $N\ensuremath{\le}12$. The existence of high Fourier components is definitely established. These high Fourier components imply the existence of very long range forces between the atoms in lead. In some cases the forces are of alternating sign. The dispersion curves show small anomalies which are believed to arise from the effect predicted by Kohn. The positions in reciprocal space at which these anomalies occur agree well with the quasi-free model for the electrons in lead, proposed by Gold. In the [111] direction (extended zone scheme) the Fermi radius is less than 1% greater than the free electron value of $1.24(\frac{2\ensuremath{\pi}}{a})$; in the [110] direction the Fermi radius is $(1.19\ifmmode\pm\else\textpm\fi{}0.01)(\frac{2\ensuremath{\pi}}{a})$. The Kohn effect is discussed in terms of theoretical work of Bardeen and Toya and the reasons for its observability in Pb are elucidated.