Use of Tables 4.3.1.1 and 4.3.1.2

Cowley, J. M.

doi:10.1107/97809553602060000593

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 261-262

Section 4.3.1.7. Use of Tables 4.3.1.1 and 4.3.1.2

J. M. Cowley^b ^‡

4.3.1.7. Use of Tables 4.3.1.1 and 4.3.1.2

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In order to calculate the Fourier coefficients V(h) of the potential distribution $[\varphi ({\bf r})]$ , for insertion in the formulae used to calculate intensities [such as (4.3.1.6), (4.3.1.20), (4.3.1.21)], or in the numerical methods for dynamical diffraction calculations , use $[{V}({\bf h})({\rm in\ volts})=47.87801\Phi ({\bf h})/\Omega, \eqno (4.3.1.31)]$ where $[\Phi ({\bf h})=\textstyle\sum\limits_i f_i\exp \{2\pi i\ {\bf h}\cdot {\bf r}_i\}. \eqno (4.3.1.32)]$ The [f_i] values are obtained from Tables 4.3.1.1 and 4.3.1.2, and $[\Omega]$ is the unit-cell volume in Å³. The V(h) and the [f_i] tabulated are properties of the crystal structure and the isolated atoms, respectively, and are independent of the particular scattering theory assumed.

Expressions for the calculation of intensities in the kinematical approximation are given for powder patterns and oblique texture patterns in Section 2.5.4 , and for thin crystal plates in Section 2.5.2 of Volume B (IT B, 2001). Since the formulas for kinematical scattering, such as (4.3.1.19) and (4.3.1.20), include the parameter K = σ /λ, which varies with the energy of the electron beam through relativistic effects, it may be considered that the electron scattering factors for kinematical calculations should be multiplied by relativistic factors.

For high-energy electrons, the relativistic variations of the electron mass, the electron wavelength and the interaction constant, σ, become significant. The relations are $[\eqalignno{ m &=m_0(1-\beta ^2)^{-1/2}, \cr \lambda &=h\left [2em_0E\left (1+{eE \over 2m_0c^2}\right) \right] ^{-1/2} \cr &=\lambda _c {(1-\beta ^2)^{1/2}\over\beta}, & (4.3.1.33)}]$ where [m_0] is the rest mass, $[\lambda _c]$ is the Compton wavelength, [h/m_0c ] , and $[\beta =v/c]$ . Consequently, $[\sigma ]$ varies with the incident electron energy as $[\eqalignno{ \sigma &=2\pi /\{\lambda E[1+(1-\beta ^2){}^{1/2}]\} \cr &=2\pi e/hc\beta . & (4.3.1.34)}]$

For the calculation of intensities in the kinematical approximation, the values of [f^B(s)] listed in Tables 4.3.1.1 and 4.3.1.2, which were calculated using [m_0] , must be multiplied by $[m/m_0=(1-\beta ^2){}^{-1/2}]$ for electrons of velocity v. Values of λ, 1/λ, [m/m_0] , β = v/c, and σ are listed for various values of the accelerating voltage, E, in Table 4.3.2.1.

References

International Tables for Crystallography (2001). Vol. B, 2nd ed. Dordrecht: Kluwer Academic Publishers.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 261-262