Scattering by a single crystal

Willis, B. T. M.

doi:10.1107/97809553602060000600

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. C. ch. 6.1, pp. 594-595

Section 6.1.3.4. Scattering by a single crystal

B. T. M. Willis^d

6.1.3.4. Scattering by a single crystal

| top | pdf |

The scattering from a single crystal can be either elastic or inelastic. An elastic process is one in which there is no exchange of energy between the neutron and the target nucleus. In an inelastic process, energy exchange occurs, giving rise to the creation or annihilation of elementary excitations such as phonons [see Section 4.1.1 of Volume B (IT B, 2001)]. Here we shall be concerned only with elastic Bragg scattering.

If kinematic scattering conditions are assumed, the differential cross section, $[({{\rm d}\sigma }/{{\rm d}\Omega}) _{\rm coh,el}]$ , giving the probability of coherent elastic scattering by a single crystal into the solid angle $[{\rm d}\Omega]$ , is $[\left ({{\rm d}\sigma \over{\rm d}\Omega}\right) _{\rm coh,el}=N {(2\pi) {}^3\over{V}}\sum _{{\bf h}} | F({\bf h}) | {}^{2}\delta({\bf H}-2\pi {\bf h}). \eqno (6.1.3.3)]$ Here, N is the number of unit cells, each of volume V, 2πh is a reciprocal-lattice vector, and F(h) is the nuclear structure factor for Bragg scattering. H is the scattering vector $[{\bf H}={\bf k}-{\bf k}_{0},]$ where k and $[{\bf k}_{0}]$ (with k = k₀ = 2π/λ) are the wavevectors of the scattered and incident beams, respectively, and the δ function indicates that the coherent elastic scattering is simply Bragg scattering. F(h) is defined by $[F({\bf h}) =\textstyle \sum \limits _{j}b _{\rm coh}^{j}\exp (i{\bf H}\cdot {\bf r}_{j}) \exp (-W_{j}), ]$ in which $[b _{\rm coh}^{j}]$ is the coherent scattering length of the jth atom in the unit cell, $[{\bf r}_{j}]$ is its equilibrium position with respect to the cell origin, and $[\exp \left (W_{j}\right) ]$ its Debye–Waller temperature factor.

The incoherent elastic scattering cross section is given by $[\left ({{\rm d}\sigma }\over{{\rm d}\Omega}\right) _{\rm inc,el}=N\sum _j [\langle b_j^2 \rangle - \langle b_j \rangle ^2] \exp (-2W_j).]$ Apart from the influence of the Debye–Waller temperature factor, this expression shows that the incoherent scattering is distributed uniformly throughout reciprocal space.

References

International Tables for Crystallography (2001). Vol. B, edited by U. Shmueli, 2nd ed. Dordrecht: Kluwer Academic Publishers.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 6.1, pp. 594-595