Abstract
In almost all previous works, the hyperbolic dispersion surfaces of the central proper quadrics have been crudely derived from the degree of reduction from the bi-quadratic equation by use of some roughly indefinable approximate relations. Moreover, neglecting the high symmetry of the hyperbola, both the branches have been approximated on the asymmetric surfaces composed of a pair of a branch of the hyperbola and a vertex of the ellipse without the presentation of reasonable evidence. Based upon the same dispersion surfaces equation, a new original gapless dispersion surfaces could be rigorously introduced without crude omission of even a term in the bi-quadratic equation based upon usual analogy with the extended band theory of solid as the close approximation to the truth.
Highlights
First of all, it could be necessarily considered that the firm establishment of E(energy) vs. k(wave number) curves as the dispersion relation of the electron in solids, which have been used as the usually popular gappy dispersion surfaces by solid line in Figure 1 [1] in almost all works of the dynamical theory of X-ray diffraction (DTXD) [2,3,4,5,6], were carefully introduced from the solutions of the secular equation based upon the experimental and theoretical examinations in the low-energy electron-diffraction by R
Based upon the same dispersion surfaces equation, a new original gapless dispersion surfaces could be rigorously introduced without crude omission of even a term in the bi-quadratic equation based upon usual analogy with the extended band theory of solid as the close approximation to the truth
In the band theory of the solid state physics, the energy gap at the Brillouin zone boundary between the hyperbola and ellipse in Figure 1 could be introduced as a perturbative effect of the Fourier component of the periodic potential in the crystal [7], which is the off-diagonal term in the secular equation shown in the Section 5
Summary
It could be necessarily considered that the firm establishment of E(energy) vs. k(wave number) curves as the dispersion relation of the electron in solids, which have been used as the usually popular gappy dispersion surfaces by solid line in Figure 1 [1] in almost all works of the dynamical theory of X-ray diffraction (DTXD) [2,3,4,5,6], were carefully introduced from the solutions of the secular equation based upon the experimental and theoretical examinations in the low-energy electron-diffraction by R. In the band theory of the solid state physics, the energy gap at the Brillouin zone boundary between the hyperbola and ellipse in Figure 1 could be introduced as a perturbative effect of the Fourier component of the periodic potential in the crystal [7], which is the off-diagonal term in the secular equation shown in the Section 5.
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