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. 3.2, p. 254

Section 3.2.2.2.2. Neutrons and nuclear scattering

P. W. Stephensa*

aDepartment of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794–3800, USA
Correspondence e-mail: peter.stephens@stonybrook.edu

3.2.2.2.2. Neutrons and nuclear scattering

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Neutrons interact with the sample in two ways: the strong interaction with the atomic nuclei, and the magnetic interaction between the neutron's dipole moment and magnetization density in the sample. We consider here only unpolarized neutrons; the use of polarized neutrons permits separation between nuclear and magnetic scattering as well as direct observation of the interference between the two; details are beyond the scope of this chapter. Nuclear scattering is very similar to the X-ray case discussed above, except that the atomic scattering amplitude refn is replaced by the nuclear coherent scattering length [\bar b_n] (given in International Tables for Crystallography, Volume C, Table 4.4.4.1), which is generally independent of neutron energy.

Unlike X-rays, the strength of the neutron–nucleus interaction is not a smooth function of atomic number. This creates opportunities to use neutrons to distinguish atoms with nearly identical X-ray scattering amplitudes, but it also makes certain elements very difficult to study with neutrons. The interaction between neutrons and the nuclei in the sample depends on the isotope and possibly the spin angular momentum of the neutron–nucleus system. This means that incoherent scattering can be significantly larger than the (coherent) diffracted signal for certain atoms, notably hydrogen (1H); see Chapter 2.3[link] of this volume for further details. For wavelengths of interest in crystallography, the nucleus is essentially a point, and so there is no atomic form factor. This generally leads to greater intensity relative to X-rays at increasing scattering vector (decreasing d-spacing).

Neutron diffractometers operate in one of two ways: angle dispersive or energy dispersive. The configuration for angle-dispersive diffraction measurements is conceptually similar to that used for X-rays; a monochromatic beam of neutrons impinges on the sample and a detector measures the distribution of neutrons versus scattering angle. For Bragg neutron diffraction from nuclei,[{{{\rm d} \sigma}\over{{\rm d} \Omega}}= {{N \lambda^3}\over{16 \pi V} }{{m_{hkl}}\over{\sin \theta \sin 2\theta}} |A_{hkl}^{(n)}|^2 \delta \left(2 \theta - 2 \sin^{-1} {{\lambda}\over{2 d}} \right),\eqno(3.2.8)]where the neutron nuclear structure factor is defined as[A_{hkl}^{(n)} = \textstyle\sum\limits_n \bar{b}_n \exp(2 \pi i {\bf G} \cdot {\bf r}_n) \exp(-W).\eqno(3.2.9)]

Time-of-flight neutron diffractometers, generally based at pulsed spallation sources, operate by measuring the time from the creation of the pulse of neutrons at the target until they appear in a given detector. If the total path length from source to detector is L and the detector is situated at an angle 2θ, a neutron with time of flight t had speed L/t and wavelength λ = ht/mnL. Here h is Planck's constant and mn is the mass of the neutron. This provides a measurement of the d-spacing within the sample, d = ht/(2mnL sin θ). Another change of variables from equation (3.2.2)[link] yields[{{{\rm d} \sigma}\over{{\rm d} \Omega}} = {{N}\over{64 \pi V}} \left({{h}\over{m_n L}} \right)^3 {{t^4}\over{\sin^2 \theta}} m_{hkl} |A_{hkl}|^2 \delta (t - 2 d \sin \theta L m_n / h).\eqno(3.2.10)]In practice, a large number of detectors surround the sample and counts from the same d-spacing (appropriately normalized for incident-beam intensity and detector solid angle) are binned together. In convenient units, mn/h = 253 µs m−1 Å−1.








































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