  The repulsive-potential Kronig-Penney (KP) model for a one-dimensional band
structure is well known. However, real metals contain positively-charged ions
resulting in attractive potential wells seen by the metallic electrons. Here we
consider the latter case in detail. The square-well version of the KP model is
considered first, for which the band structure and wave functions for different
potential-well depths are derived. Then an extended treatment of the attractive
Dirac-comb version of the KP model is presented. For the nearly-free-electron
case, the band structure exhibits a negative-energy band in addition to
positive-energy bands. The wave functions, electron densities of states,
effective masses, and group velocities are derived for the positive-energy band
states. The wave functions of the negative-energy band states are also
calculated and found to be quite different from the sinusoidal wave functions
for the positive-energy band and band-gap states. High-degeneracy bound states
are found at negative energies and their wave functions are derived.
