Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.2, p. 11   | 1 | 2 |

Section 1.2.5. Scattering of thermal neutrons

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260–3000, USA
Correspondence e-mail:

1.2.5. Scattering of thermal neutrons

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The scattering of neutrons by atomic nuclei is described by the atomic scattering length b, related to the total cross section [\sigma_{\rm total}] by the expression [\sigma_{\rm total} = 4\pi b^{2}]. At present, there is no theory of nuclear forces which allows calculation of the scattering length, so that experimental values are to be used. Two types of nuclei can be distinguished (Squires, 1978[link]). In the first type, the scattering is a resonance phenomenon and is associated with the formation of a compound nucleus (consisting of the original nucleus plus a neutron) with an energy close to that of an excited state. In the second type, the compound nucleus is not near an excited state and the scattering length is essentially real and independent of the energy of the incoming neutron. In either case, b is independent of the Bragg angle θ, unlike the X-ray form factor, since the nuclear dimensions are very small relative to the wavelength of thermal neutrons.

The scattering length is not the same for different isotopes of an element. A random distribution of isotopes over the sites occupied by that element leads to an incoherent contribution, such that effectively [\sigma_{\rm total} = \sigma_{\rm coherent} + \sigma_{\rm incoherent}]. Similarly for nuclei with nonzero spin, a spin incoherent scattering occurs as the spin states are, in general, randomly distributed over the sites of the nuclei.

For free or loosely bound nuclei, the scattering length is modified by [b_{\rm free} = [M/(m + M)]b], where M is the mass of the nucleus and m is the mass of the neutron. This effect is of consequence only for the lightest elements. It can, in particular, be of significance for hydrogen atoms. With this in mind, the structure-factor expression for elastic scattering can be written as [F({\bf H}) = {\textstyle\sum\limits_{j}} b_{j, \, {\rm coherent}} \exp 2\pi i(hx_{j} + ky_{j} + lz_{j}) \eqno(]by analogy to ([link]. Magnetic scattering

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The interaction between the magnetic moments of the neutron and the unpaired electrons in solids leads to magnetic scattering. The total elastic scattering including both the nuclear and magnetic contributions is given by [|F({\bf H}){|^{2}_{\rm total}} = |F_{N} ({\bf H}) + {\bf Q}({\bf H})\cdot {\hat\boldlambda}|^{2}, \eqno(]where the unit vector [\hat{{\boldlambda}}] describes the polarization vector for the neutron spin, [F_{N}({\bf H})] is given by ([link] and Q is defined by [{\bf Q} = {mc \over eh} {\int} {\bf \widehat{H}} \times [{\bf M}({\bf r}) \times {\bf \widehat{H}}] \exp (2\pi i{\bf H}\cdot {\bf r})\ \hbox{d}{\bf r}. \eqno(][{\bf M}({\bf r})] is the vector field describing the electron-magnetization distribution and [\widehat{{\bf H}}] is a unit vector parallel to H.

Q is thus proportional to the projection of M onto a direction orthogonal to H in the plane containing M and H. The magnitude of this projection depends on [\sin \alpha], where α is the angle between Q and H, which prevents magnetic scattering from being a truly three-dimensional probe. If all moments [{\bf M}({\bf r})] are collinear, as may be achieved in paramagnetic materials by applying an external field, and for the maximum signal (H orthogonal to M), ([link] becomes [{\bf Q} = {\bf M}({\bf H}) = {mc \over eh} {\int} {\bf M}({\bf r}) \exp (2\pi i{\bf H}\cdot {\bf r})\ \hbox{d}{\bf r} \eqno(]and ([link] gives [{|F|^{2}}_{\!\!\!\rm total} = |F_{N}({\bf H}) - M({\bf H})|^{2} \eqno(]and [|F{|^{2}_{\rm total}} = |F_{N} ({\bf H}) + M({\bf H})|^{2}]for neutrons parallel and antiparallel to [{\bf M} ({\bf H})], respectively.


Squires, G. L. (1978). Introduction to the Theory of Thermal Neutron Scattering. Cambridge University Press.

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