International
Tables for
Crystallography
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 3.6.2.6.2. Source emission profile

M. Leonia*

aDepartment of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy
Correspondence e-mail: Matteo.Leoni@unitn.it

3.6.2.6.2. 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].

References

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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