X-Ray Determination of the Electron Momentum Density in Diamond, Graphite, and Carbon Black

The Compton profile has been measured with the x-ray scattering vector along five selected crystallographic directions in diamond, in polycrystalline diamond, in carbon black, and with the x-ray scattering vector along three selected directions in pyrolytic graphite. The electron momentum densities in both diamond and graphite are more extended in momentum space than a superposition of Hartree-Fock $1s^{2}2s2p^3$ free-atom momentum densities, with the diamond momentum density more extended than the graphite. The momentum densities of carbon black and graphite are similar, as expected from an earlier x-ray diffraction investigation of their crystal structures. The small anisotropies found in the diamond momentum density can be qualitatively explained by assuming that some momentum density at high momentum values in momentum space has been transferred from the $〈211〉$ directions to the $〈100〉$ directions. The small anisotropy measured in graphite is in marked contrast to the large anisotropy found in positron-annihilation measurements. In addition, the momentum density of graphite deduced from positron annihilation is less extended in momentum space than that deduced from the x-ray measurements. As suggested by Berko, these results give insight into the positron wave function and its perturbation of the annihilated electron. In diamond, the momentum densities deduced from positron annihilation and the Compton profile are similar.