Scattering by a single atom

Willis, B. T. M.

doi:10.1107/97809553602060000600

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 6.1, p. 594

Section 6.1.3.3. Scattering by a single atom

B. T. M. Willis^d

6.1.3.3. Scattering by a single atom

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For a single element containing several isotopes, each isotope has its own characteristic scattering length(s). The mean value of the scattering length of the atom is obtained by averaging (where necessary) over the two spin states of the isotope: $[ \langle b \rangle _{\rm isotope}=w_{+}b_{+}+w_{-}b_{-},]$ where the angle brackets indicate a mean and $[w_{+}]$ and $[w_{-}]$ are given by (6.1.3.2); $[\left \langle b\right \rangle _{\rm isotope}]$ is then averaged over all isotopes, taking into account their relative abundance. The resultant quantity, $[\left \langle b\right \rangle _{\rm isotopes}^{\rm all}]$ , is known as the coherent scattering length of the atom, denoted b_coh. b_coh plays the same role in neutron scattering as the atomic scattering factor f in X-ray scattering. Table 4.4.4.1 lists the coherent scattering lengths for the atoms in the Periodic Table.

The coherent scattering cross section of an atom is $[\sigma _{\rm coh}=4\pi b_{\rm coh}^2.]$

It represents that part of the total scattering cross section, σ_tot, that gives interference effects with other atoms. The total cross section is $[\sigma _{\rm tot}=4\pi \langle b^2 \rangle _{\rm isotopes}^{\rm all},]$ and the incoherent scattering cross section, σ_inc, is the difference between σ_tot and σ_coh: $[\sigma _{\rm inc}=4\pi [\langle b^2 \rangle - \langle b\rangle ^2] . ]$ In incoherent scattering, there is no phase relationship between the waves scattered by different atoms. σ_inc for hydrogen is 40 times larger than σ_coh, but the proportion of coherent scattering is substantially increased by deuteration. The scattering from vanadium is almost entirely incoherent, and so it is useful as a container of polycrystalline samples.

References

International Tables for Crystallography (2006). Vol. C. ch. 6.1, p. 594