Electronic Energy Bands in Metallic Tungsten

Calculations of the electronic energy bands in metallic tungsten are carried out by the Wigner-Seitz-Slater method. All numerical integrations were carried out on the M. I. T. differential analyzer. It is found that the $d$ band is broken up into five sub-bands. Some of these $d$ bands are found to be about fifteen electron volts in width. One is about two electron volts in width. The occupied energy range extends about five electron volts. The $s$ band starts at a higher energy than the $d$ bands and is occupied by much less than one electron per atom at the equilibrium interatomic distance. From the results of a previous paper, curves of $E \mathrm{vs}.\mathrm{} k$ are plotted for the principal directions of propagation. From these, curves of the number of energy levels per unit energy range were determined by numerical and graphical methods which are described in an appendix. It is assumed that the $n\left(E\right)\mathrm{}$ curve for tantalum is not greatly different from that for tungsten except that there is one less electron per atom. From the $n\left(E\right)\mathrm{}$ curves the electronic contribution to the specific heat is calculated for the two metals and the results found to be in good agreement with the excess specific heat at high temperatures for both metals. The computed value does not agree with low temperature data on tantalum. There are no low temperature data for tungsten. Qualitative discussions of the differences in electrical resistance, temperature coefficient of resistance, and thermoelectric power of the two metals are given. The contribution of exchange effects to the paramagnetic susceptibility is treated by a rough model and it is shown that the assumption of the same value of the exchange integral for both metals gives satisfactory agreement with observed data.