Valence charge density in grey tin: X-ray determination of the (222) "forbidden" reflection and its temperature dependence

The (222) reflection of diamondlike lattices is essentially contributed at room temperature by valence electrons piled up between neighboring atoms. It is therefore a direct experimental proof of the existence of covalent bonding charges in diamond structures. Such a proof has now been established in the case of grey tin. The (222) reflection has been measured in absence of simultaneous reflections on an absolute basis, using Cu $K\alpha$ radiation. Its intensity is four orders of magnitude smaller than the (333), corresponding to a structure factor $F_{222}=1.06\mathrm{}\pm 0.05$ electrons/cell at 200°K. This value is 52% smaller than that calculated using a pseudopotential approach. Similar measurements were repeated at different temperatures between 80 and 223 °K. The nuclear mean-square vibrational amplitude was found to be 0.0319 ± 0.002 ${\mathrm{\AA }}^2$ at 200 °K, very close to the value obtained from lattice-dynamics calculations, corresponding to a Debye temperature of 150 °K. The temperature dependence of the valence bonding charges suggests, however, that their mean-square vibrational amplitudes may be some 20% smaller than that of the core.