  Recent discoveries of supposedly pure alpha-tetragonal boron require to
revisit its structure. The system is also interesting with respect to a new
type of geometrical frustration in elemental crystals, which was found in
beta-rhombohedral boron. Based on density functional theory calculations, the
present study has resolved the structural and thermodynamic characteristics of
pure alpha-tetragonal boron. Different from beta-rhombohedral boron, the
conditions for stable covalent bonding (a band gap and completely filled
valence bands) are almost fulfilled at a composition B_52 with two 4c
interstitial sites occupied. This indicates that the ground state of pure
alpha-tetragonal boron is stoichiometric. However, the covalent condition is
not perfectly fulfilled because non-bonding in-gap states exist that cannot be
eliminated. The half occupation of the 4c sites yields a macroscopic amount of
residual entropy, which is as large as that of beta-rhombohedral boron.
Therefore, alpha-tetragonal boron can be classified as an elemental crystal
with geometrical frustration. Deviations from stoichiometry can occur only at
finite temperatures. Thermodynamic considerations show that deviations delta
from the stoichiometric composition B_(52+delta) are small and positive. For
reported high-pressure syntheses conditions delta is predicted to be about 0.1
to 0.2. An important difference between pure and C- or N-containing
alpha-tetragonal boron is found in the occupation of interstitial sites: the
pure form prefers to occupy the 4c sites, whereas in C- or N-containing forms a
mixture of 2a, 8h, and 8i sites are occupied. The present article provides
relations of site occupation, delta values, and lattice parameters, which
enable us to identify pure alpha-tetragonal and distinguish the pure form from
other ones.
