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. 2.3, p. 70

Section Structure factors

C. J. Howarda* and E. H. Kisia

aSchool of Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
Correspondence e-mail: Structure factors

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The locations of the Bragg peaks for neutrons are calculated as they are for X-rays7 (Section 1.1.2[link] ), and the intensities of these peaks are determined by a structure factor, which in the nuclear case is [cf. Chapter 1.1, equation (1.1.56)[link] ][{ F}_{hkl}^{\rm nuc}=\textstyle \sum \limits_{i=1}^{m}{b}_{i}{T}_{i}\exp(2\pi i{\bf h}\cdot{{\bf u}}_{i}), \eqno(2.3.7)]where bi here denotes the coherent scattering length, Ti has been introduced to represent the effect of atomic displacements (thermal or otherwise, see Section 2.4.1 in Kisi & Howard, 2008[link]), h is the scattering vector for the hkl reflection, and the vectors ui represent the positions of the m atoms in the unit cell.

For coherent magnetic scattering, the structure factor reads[{ F}_{hkl}^{\rm mag}=\textstyle \sum \limits_{i=1}^{m}{p}_{i}{{\bf q}}_{i}{T}_{i}\exp(2\pi i{\bf h}\cdot{{\bf u}}_{i}), \eqno(2.3.8)]where pi is the magnetic scattering length. The vector qi is the `magnetic interaction vector' and is defined by a triple vector product (Section 2.3.4 in Kisi & Howard, 2008[link]), and has modulus sin α as already mentioned. In this case the sum needs to be taken over the magnetic atoms only.

As expected by analogy with the X-ray case, the intensity of purely nuclear scattering is proportional to the square of the modulus of the structure factor [|{F}_{hkl}^{\rm nuc}|^{2}]. In the simplest case of a collinear magnetic structure and an unpolarized incident neutron beam, the intensity contributed by the magnetic scattering is proportional to [{|{F}_{hkl}^{\rm mag}|}^{2}], and the nuclear and magnetic contributions are additive.


Kisi, E. H. & Howard, C. J. (2008). Applications of Neutron Powder Diffraction. Oxford University Press.Google Scholar

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