Atomic form factor measurement on beryllium and aluminum using inelastic x-ray scattering

A method has been developed by means of inelastic x-ray scattering for the determination of atomic form factors. This method employs precise measurement of phonon intensities as a function of momentum and energy transfers in the millielectronvolt regime produced by incident x-ray energy of 21.657 keV. The requirements for very high energy resolution and the basic principles of such instrumentation are discussed. In this work, the x-ray atomic form factors of beryllium and aluminum single crystals along the [0 0 ζ] principal symmetry direction are reported to obtain reliable information on the outer electron charge densities in these elements. In contrast to previous studies with elastic x-ray scattering (Bragg reflections), with this method, charge densities can be studied by phonon intensities without any limitation in the momentum space. Hence, complementary to previous experiments, for the first time, atomic form factor measurements are extended to the very low momentum transfer region (down to 1.5 nm −1). Possible problems with this method have been examined and a method to analyze the experimental results has been established. In the case of beryllium, the measured atomic form factor is compared with those predicted by a free-atom Hartree-Fock calculation and by a pseudopotential calculation. The present experimental data are consistent with the form factor for beryllium obtained by the pseudopotential calculation whereas they are substantially higher than the free-atom Hartree-Fock model calculation. On the other hand, in the case of aluminum, the measured experimental data are in overall agreement with the free-atom Hartree-Fock model calculation.