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. 2.3, p. 70

Section 2.3.2.6. Structure factors

C. J. Howarda* and E. H. Kisia

aSchool of Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
Correspondence e-mail:  chris.howard@newcastle.edu.au

2.3.2.6. 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.

References

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








































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