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By means of a Green's function method we discuss the energy-spectrum of an electron in a one-dimensional lattice containing only one type of atom, represented as a Dirac δ-function. Limiting the infinite crystal model at one side, we get not only a potential step at the surface, but we assume also that the first n atoms are displaced. We find that the localization of the surface states depends not only on the potential strengths, the lattice constant and the height of the potential barrier at the surface, but also on the different displacements of the n atoms, i.e. on the penetration of the surface potential into the crystal. In the case that the surface potential acts only up to the second crystal cell, we prove the existence of true surface states or Tamm-levels and of “subsurface” states.

We consider a semi-infinite crystal model composed of two kinds of atoms. The case of both ionic and valence crystals is considered. We assume that the potential of the atoms at the surface is asymmetric. It is proved that the number and the position of the localized surface states is affected by the lattice constant, by the atomic potential strength of the two different kinds of atoms and by the step in potential energy at the surface. The position of the levels is calculated by the scattering matrix method of Saxon and Hutner7) for various numerical values of the parameters mentioned above. Generally only one level is found in the first forbidden energy gap; exceptionally two or none. We restricted our calculations to a crystal with a clean surface.

In this paper, a general theory is established for the low-energy electron diffraction by a semi-infinite crystal model. The Green function method of Kohn and Rostoker, which seems to be a powerful method in order to calculate energy bands in crystals, is modified to be applicable in LEED theory. The inner potential is also taken into account and the results, which are to some extent similar to those of Capart, are discussed.

We calculate the surface energy bands in a semi-infinite simple cubic lattice in which the potential is assumed to be constant within non-overlapping spheres surrounding the atoms, and zero in the interspace. Considering the case that the crystal is limited by a (100) plane, numerical calculations showed that the existence of surface states is strongly dependent on the choice of the crystal surface. An analogous conclusion could be made by investigating the case that the crystal boundary is a (110) plane.

The forbidden bands of an imperfect, one-dimensional crystal are examined. The existence and number of solutions in each forbidden band is found explicitly for impurity states arising from various types of single impurities, and for Shockley-type surface states arising from models having perfectly terminated surfaces. Also crystals whose only imperfection is an additional attractive or repulsive potential at the surface are examined, and the solutions classified as Tamm-type or Shockley-type surface states.