$\alpha$-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.'
author:
- Naoki Uemura
- Koun Shirai
- Hagen Eckert
- Jens Kunstmann
title: 'Structure, non-stoichiometry, and geometrical frustration of $\alpha$-tetragonal boron'
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Introduction {#sec:intro}
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The existence of $\alpha$-tetragonal ($\alpha$-T) boron has long been a controversial issue. Historically, $\alpha$-T boron was first isolated in 1943 [@Laubengayer43], and tentatively identified as B$_{50}$, which is composed of four tetrahedrally coordinated B$_{12}$ icosahedra and two interstitial B atoms (see Fig. \[fig:struct\]).[@Hoard51] But soon this structural model was doubted by theorists. Longuet-Higgins and Roberts showed that the B$_{50}$ structure is unstable due to electron deficiency.[@LH55] Later, experimental groups showed that the actual crystals contained C or N atoms [@Amberger71; @Ploog72; @Will76] and the chemical compositions were approximately B$_{50}$N$_{2}$ or B$_{50}$C$_{2}$. This point of view was further supported by DFT calculations.[@Morrison92; @Lee92a] Since then, it is almost generally accepted that pure 