Einfluß der thermischen Gitterschwingungen auf die Streuung schneller Elektronen in Kristallen

Abstract

In the present paper the influence of thermal lattice vibrations on the electron diffraction intensities of polycrystalline foils is investigated. It is shown, that the distinction between Bragg-reflections and thermal diffuse scattering is equivalent to splitting the crystal potential under influence of lattice vibrations into two terms: V = 〈V〉 + U, where 〈V〉 is the statistical average. 〈V〉 has the same periodicity as the potential V°(r) of the lattice at rest, but the Fouriercoefficients are reduced with increasing temperature by the Debye-Waller factor: 〈V〉h = V. exp (−Mh). The temperature dependence of the Fouriercoefficients 〈V〉h determines the variation of the reflection intensities with temperature. The time dependent potential U gives rise to the thermal diffuse scattering. Simple theoretical results are obtained from the kinematical approximation and from the two beam approximation of the dynamical theory. Furthermore, the influence of absorption of electrons on the temperature dependence of diffraction intensities is discussed on the basis of these approximations. Experimental investigations of the temperature dependence of Bragg-reflections and of the diffuse scattering have been carried out mainly for aluminium, because in this case the dynamical extinction of the strong reflections can be described by the two beam approximations. The scattered electrons have been detected by a Faraday cage. In order to filter off electrons scattered inelastically with energy losses above ∼ w ev, a retarding field has been used. The experimental results concerning the influence of temperature on the intensities of various reflections can be described by introducing the temperature dependent Fouriercoefficients 〈V〉h into the relevant intensity formulae for elastic scattering. The temperature dependence of the intensities of all reflections, which behave kinematically, can be well described by the Debye-Waller factor exp (−2Mh) within a large range of 2Mh. The mean square displacement of the lattice atoms, which was derived from the measured values of 2Mh at different temperatures, is in good agreement with results from X-ray measurements. For reflections, which are influenced by dynamical extinction, an effective Debye-Waller factor exp (αhMh) has been introduced, in order to describe the temperature dependence of the intensities. The αh-values, measured at 50 keV and for Al-foils with crystal sizes of about 300 A lie between 0 and 2,5 and can be interpreted by the two beam approximation of the dynamical theory. Further the thermal diffuse scattering of Al has been investigated. The measured intensity shows good agreement with the theoretical angular distribution for one- and two-phonon processes. From the integrated intensities of the Debye-Schemer rings (IR) and of the thermal diffuse scattering (ITDS) the mean free paths AR and ATDS for these scattering processes have been determined. The measured value of AR shows good agreement with the calculation based on the two beam approximation and ATDS agrees fairly well with the kinematical value for the Einstein-model of the vibrating crystal. Moreover the transmission of polycrystalline foils for fast electrons has been investigated. Besides of aluminium, also silver and gold were used, in order to study the effect of atomic number on the absorption coefficients μel and μunel due to elastic and inelastic scattering processes respectively. The experimental results show, that μunel decreases only slightly with temperature. On the other hand, the absorption coefficient μel, which is composed of μR = Δ for Bragg-scattering and μTDS = Δ fur thermal diffuse scattering increases with temperature. As μR and μTDS depend on temperature in opposite sense, the resulting increase of μel = μR + μTDS with temperature depends on the extent, to which the temperature dependence of μR is reduced by dynamical extinction. For Al the measured temperature dependence of μel is in good agreement with the theoretical calculation of μTDS for the Einstein-model and of μR according to the two beam approximation. For Ag and Au the temperature dependence of μel is much more pronounced than for Al. This is due to the facts, that for heavy elements firstly the elastic scattering is stronger and secondly dynamical extinction effects are generally more pronounced. A quantitative interpretation however, cannot be based on the two beam approximation. In all cases studied here, the influence of anomalous absorption on the reflection intensities as well as on the transmission of the polycrystalline foils can be neglected because of the small crystal sizes.