Tables for
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.6, p. 291

Section Source emission profile

M. Leonia*

aDepartment of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy
Correspondence e-mail: Source emission profile

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For X-rays, the source emission profile at an energy El can be well described by a Lorentzian of energy width Γl (Hölzer et al., 1997[link]; Deutsch et al., 2004[link]),[I_l(E) = {2 \over {\Gamma_l \pi}} \left[1 + 4 \left({{E - E_l} \over {\Gamma _l}} \right)^2 \right]^{-1}. \eqno (3.6.14)]As dE/E = dλ/λ = ds/s, the function can also be represented as a function of s:[I_{hkl,l}^{\rm IP} (s,d_{hkl}^*) = {2 \over \pi } {{E_l} \over {d_{hkl}^* \Gamma_l}} \left[1 + 4 \left({{s_{hkl}} \over {d_{hkl}^* \Gamma_l/E_l}} \right)^2 \right]^{-1}. \eqno (3.6.15)]For a laboratory tube emitting simultaneously a set of Nλ wavelengths, we have[I_{hkl}^{\rm IP}(s,d_{hkl}^*) = \textstyle\sum\limits_{l = 1}^{N_\lambda} w_l I_{hkl,l}^{\rm IP} (s,d_{hkl}^*), \eqno (3.6.16)]where wl is the relative intensity of the lth wavelength component (referred, for example, to w1 = 1). The corresponding Fourier transform entering (3.6.13)[link] can be written as[\eqalignno {&T^{\rm IP} (L) &\cr &\quad= {\textstyle \sum\limits_{l = 1}^{N_\lambda}} \exp \left[2\pi id_{hkl}^*\left(1 - {{\Gamma_l} \over {E_l}}\right)L\right] \exp \left(-2\pi s_{hkl}{{\Gamma_l} \over {E_l}}L \right) \cr & \quad= {\textstyle \sum\limits_{l = 1}^{N_\lambda}} \bigg\{ \cos \left [2\pi d_{hkl}^*\left(1 - {{\Gamma_l} \over {E_l}} \right)L \right] + i\sin \left [2\pi d_{hkl}^*\left(1 - {{\Gamma _l} \over {E_l}}\right) L \right] \bigg\}&\cr&\quad\quad\times \exp \left(-2\pi s_{hkl} {{\Gamma_l} \over {E_l}}L \right). & (3.6.17)}]The complex term in (3.6.17)[link] accounts for the shift of each emission component with respect to the reference one. For more flexibility (for example to consider the non-ideal behaviour of the instrument), we can use a pseudo-Voigt (pV) in place of the Lorentzian in equation (3.6.14)[link].


Deutsch, M., Forster, E., Holzer, G., Hartwig, J., Hämäläinen, K., Kao, C.-C., Huotari, S. & Diamant, R. (2004). X-ray spectrometry of copper: new results on an old subject. J. Res. Natl Inst. Stand. Technol. 109, 75–98.Google Scholar
Hölzer, G., Fritsch, M., Deutsch, M., Härtwig, J. & Förster, E. (1997). 1,2 and Kβ1,3 X-ray emission lines of the 3d transition metals. Phys. Rev. A, 56, 4554–4568.Google Scholar

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