Quantum mechanics of electrons in crystal lattices

Abstract

Introduction .–Through the work of Bloch our understanding of the behaviour of electrons in crystal lattices has been much advanced. The principal idea of Bloch’s theory is the assumption that the interaction of a given electron with the other particles of the lattice may be replaced in first approximation by a periodic field of potential. With this model an interpretation of the specific heat, the electrical and thermal conductivity, the magnetic susceptibility, the Hall effect, and the optical properties of metals could be obtained. The advantages and limitations inherent in the assumption of Bloch will be much the same as those encountered when replacing the interaction of the electrons in an atom by a suitable central shielding of the unclear field, as in the work of Thomas and Hartree. In the papers quoted a number of general results were given regarding the behaviour of electrons in any periodic field of potential. To obtain a clearer idea of the details of this behaviour with a view to the application in special problems, however, it appeared worth while to investigate the mechanics of electrons in periodic fields of potential somewhat similar to those met with in practice and of such nature that the energy values W and eigenfunctions Ψ of the wave-equation can actually be computed. It is the purpose of this article to discuss a case where the integration is possible. In Section 1 the energy values and in Section 2 the wave-functions in their dependence on the binding introduced by the potential field are discussed for the one dimensional problem. In Section 3 the matrix elements of the linear momentum, which furnish the electric current associated with the various stationary states, are well as the probability of radiative transitions between these states, are evaluated. In Section 4 the results are extended to the three dimensional case and those features considered which one may expect to find in the case of more general periodic fields of potential. Section 5 deals with some applications to physical problems.

High-frequency homogenization for periodic media

“…This dispersion relation was seemingly first obtained by Kronig & Penney (1931) for electrons in crystal lattices; it also naturally appears in many guises in onedimensional photonic and phononic crystals (with layers of infinite height), e.g. Movchan et al (2002) and Adams et al (2008).…”

Entrainment coefficient and effective mass for conduction neutrons in neutron star crust: simple microscopic models

“…The neutron band structure with a square well potential (whose analytic solution was found a long time ago by Kronig and Penney [18,19]) was the only case discussed by Oyamatsu and Yamada [2]. The single particle wavefunction can be factored in the form 10) in which the reduced wave funtion φ k z is the solution of a one dimensional equation 11) satisfying the boundary conditions…”

Таммовские Связанные Состояния Электронов На Поверхности Кристалла И Поверхностные Колебания Атомов Решётки

“…Таким образом, в ограниченном кри-сталле возможны разрешённые дискретные значения энергии элек-трона в той области энергий, которая в неограниченном кристалле является запрещённой зоной. Соответствующие этим дискретным уровням волновые функции электрона в области вакуума монотонно затухают по мере удаления от поверхности кристалла, как видно из формулы (3). В области же кристалла Еолновые функции зату-хают осциллируя, по мере углубления в кристалл.…”

“…Последнее легко видеть из формулы (2), если учесть, что k комплексно и что при члене, растущем с х, коэффициент полагается равным нулю. В качестве простейшего примера, допускающего легкий количе-ственный расчёт, И. Е. Тамм рассмотрел одномерную модель Кро-нига и Пеннея 3 , в которой периодический потенциал задаётся, как ряд равноотстоящих друг от друга прямоугольных одинаковых по-тенциальных барьеров (рис. 1); затем высота каждого барьера U x стремится к бесконечности, а ширина барьера Ъ стремится к нулю УФН, т. LVI, вьв.…”