Study of temperature factors in cubic crystals by high-voltage electron diffraction

The critical voltages for systematic reflections and splits of Kikuchi lines were measured using a high-voltage electron microscope to investigate the atomic temperature factors in cubic crystals. The split of the Kikuchi line at the intersection with the forbidden 222 Kikuchi line as well as the critical voltage of the 333 reflection for Si and Ge decreased steeply with temperature. The temperature dependence showed that the anharmonic contribution to the atomic-temperature factor for Si and Ge is extremely weak in the temperature range 300 approximately 1078 K. On the contrary, the B factors obtained from the measured critical voltages for Al, Cu, and Fe varied nonlinearly with temperature, suggesting the importance of the anharmonic effect in the vibration of atoms. The observed temperature dependence of the critical voltages for the metals were compared with calculations based on harmonic, quasi-harmonic, and anharmonic approximations. The quasi-harmonic approximation that takes into account the thermal expansion modification reproduces well the observed values for Fe but not those for Al and Cu. The effect of intrinsic anharmonic vibration should be considered for reproducing the results for Al and Cu. Fitting the measured critical voltages with the calculated ones, we estimated the values for coefficients of the isolated atom potentials. The results are in good agreement with those obtained by neutron and X-ray diffraction.