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. 2.5, p. 133

Section 2.5.4.1.1. Relative intensity

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: bob.he@bruker.com

2.5.4.1.1. Relative intensity

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The integrated intensity diffracted from polycrystalline materials with a random orientation distribution is given by[{{{I}}_{hkl}} = {k_I}{{{p_{hkl}}} \over {{v^2}}}({\rm LPA}){\lambda ^3}F_{hkl}^2{g_{hkl}}(\alpha, \beta)\exp \left({ - 2{M_t} - 2{M_s}} \right),\eqno(2.5.45)]where kI is an instrument constant that is a scaling factor between the experimental observed intensities and the calculated intensity, phkl is the multiplicity factor of the crystal plane (hkl), v is the volume of the unit cell, (LPA) is the Lorentz–polarization and absorption factors, λ is the X-ray wavelength, [F_{hkl}] is the structure factor for the crystal plane (hkl), [{g_{hkl}}(\alpha, \beta)] is the normalized pole-density distribution function and exp(−2Mt − 2Ms) is the attenuation factor due to lattice thermal vibrations and weak static displacements (Warren, 1990[link]; He et al., 1994[link]). Except for the texture effect, all the factors in the above equation are either discussed in the previous sections or have the same definitions and values as in conventional diffraction.

Phase-identification studies by XRD are preferably carried out on powders or polycrystalline samples with a random orientation distribution of crystallites. Preferred orientation causes relative intensities to deviate from theoretical calculations or those reported in reference databases. In practice, a sample with a perfectly random orientation distribution of crystallites is very hard to fabricate and most polycrystalline samples have a preferred orientation to a certain extent. Discrepancies in the relative peak intensities between conventional diffraction and 2D-XRD are largely due to texture effects. For B-B geometry, the diffraction vector is always perpendicular to the sample surface. With a strong texture, it is possible that the pole density of certain reflections in the sample normal direction is very low or even approaches zero. In this case, the peak does not appear in the diffraction pattern collected in B-B geometry. In 2D-XRD, several diffraction rings may be measured with a single incident beam; the corresponding diffraction vectors are not necessarily in the sample normal direction. The diffraction profiles from 2D frames are produced by γ integration, therefore the texture factor [{g_{hkl}}(\alpha, \beta)] should be replaced by the average normalized pole-density function within the γ integration range [\left\langle {{g_{hkl}}(\Delta \gamma)} \right\rangle ]. The relation between (α, β) and (2θ, γ) is given in Chapter 5.4[link] . The chance of having zero pole density over the entire γ-integration range is extremely small. Therefore, phase identification with 2D-XRD is much more reliable than with conventional diffraction.

References

He, B., Rao, S. & Houska, C. R. (1994). A simplified procedure for obtaining relative x-ray intensities when a texture and atomic displacements are present. J. Appl. Phys. 75, 4456–4464.Google Scholar
Warren, B. E. (1990). X-ray Diffraction. New York: Dover Publications.Google Scholar








































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