Scattering of X-rays by thermal vibrations

Willis, B. T. M.

doi:10.1107/97809553602060000563

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 4.1, pp. 403-404 | 1 | 2 |

Section 4.1.3. Scattering of X-rays by thermal vibrations

B. T. M. Willis^a ^*

^aChemical Crystallography Laboratory, University of Oxford, 9 Parks Road, Oxford OX1 3PD, England
Correspondence e-mail: bertram.willis@chemcryst.ox.ac.uk

4.1.3. Scattering of X-rays by thermal vibrations

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The change of frequency, or energy, of X-rays on being scattered by thermal waves is extremely small. The differential scattering cross section, $[{\rm d}\sigma/{\rm d}\Omega]$ , giving the probability that X-rays are scattered into the solid angle $[{\rm d}\Omega]$ is then $[{{\rm d}\sigma \over {\rm d}\Omega}=\langle|Y(Q)|^2\rangle,]$ where $[Y(Q)=\textstyle \sum \limits_{\kappa{}l}f_{\kappa}({\bf Q})\exp\{i{\bf Q}\cdot[{\bf r}(\kappa{}l)+{\bf u}(\kappa{}l)]\}.\eqno(4.1.3.1)]$ The angled brackets indicate an average value over a period of time much longer than the period of oscillation of an atom. Q is the `scattering vector' defined by $[{\bf Q}={\bf k}-{\bf k}_0,]$ where k and k ₀ are the wavevectors (each of magnitude $[2\pi/\lambda]$ ) of the scattered and incident beams, respectively. The magnitude of Q is $[4\pi(\sin\theta)/\lambda]$ , where $[2\theta]$ is the angle between k and k ₀. $[f_\kappa({\bf Q})]$ is the scattering factor of the $[\kappa]$ th atom in the unit cell.

The cross section $[{\rm d}\sigma/{\rm d}\Omega]$ can be expanded as a power series: $[{{\rm d}\sigma \over {\rm d}\Omega}= \left({{\rm d}\sigma \over {\rm d}\Omega}\right)^{(0)}+\left({{\rm d}\sigma \over {\rm d}\Omega}\right)^{(1)}+\left({{\rm d}\sigma \over {\rm d}\Omega}\right)^{(2)}+\ldots.\eqno(4.1.3.2)]$ The individual terms on the right-hand side refer to the cross sections for zero-order, first-order, second-order scattering …, i.e. for processes involving no exchange of energy between the incident radiation and the crystal (Bragg scattering), the exchange of one quantum of lattice vibrational energy (one-phonon scattering), the exchange of two quanta (two-phonon scattering) ….

The instantaneous thermal displacement $[{\bf u}(\kappa{}l)]$ of the atom $[(\kappa{}l)]$ can be expressed, using equations (4.1.2.3), (4.1.2.6), (4.1.2.7) and (4.1.2.8), as a superposition of the displacements of the 3nN (∼10²³) independent normal modes of vibration: $[\eqalignno{{\bf u}(\kappa{}l)&=[Nm(\kappa)]^{-1/2}\sum_{j{\bf q}}{{E_j^{-1/2}({\bf q})}\over{\omega_j^2({\bf q})}}{\bf e}(\kappa|j{\bf q})&\cr&\quad\times\exp\{i[{\bf q}\cdot{\bf r}(\kappa{}l)-\omega_j({\bf q})t]\}. & (4.1.3.3)\cr}]$ Explicit expressions can now be given for the partial cross sections in equation (4.1.3.2). The cross section for Bragg scattering is $[\left({\rm d}\sigma\over{\rm d}\Omega\right)^{(0)}=N|F({\bf Q})|^2{(2\pi)^3\over{\bf v}}\sum_{\bf h}\delta({\bf Q}-{\bf Q}_{\bf h}),\eqno(4.1.3.4)]$ where v is the cell volume. The delta function requires that $[{\bf Q}={\bf Q}_{\bf h},]$ where $[{\bf Q}_{\bf h}]$ is a reciprocal-lattice vector, so that scattering is restricted to the points h of the reciprocal lattice. The structure factor F(Q) is $[F({\bf Q})=\textstyle \sum \limits_{\kappa}f_{\kappa}({\bf Q})\exp[i{\bf Q}\cdot{\bf r}(\kappa)]\exp(-W_{\kappa}),]$ where the exponent $[W_{\kappa}]$ of the temperature factor of the atom $[\kappa]$ is calculated by summing over the normal modes: $[W_{\kappa}={1 \over 2Nm(\kappa)}\sum_{j{\bf q}}|{\bf Q}_{h}\cdot{\bf e}(\kappa|j{\bf q})|^2{E_j({\bf q}) \over \omega_j^2({\bf q})}.\eqno(4.1.3.5)]$ The last equation shows that the acoustic modes, with frequencies approaching zero as $[{\bf q}\rightarrow0]$ , make the largest contribution to the temperature factor.

The one-phonon cross section is $[\eqalignno{\left({\rm d}\sigma\over{\rm d}\Omega\right)^{(1)}&={(2\pi)^3\over{\bf v}}\sum\limits_{j{\bf q}}{|G_j({\bf Q},{\bf q})|^2E_j({\bf q})\over\omega^2_j({\bf q})}&\cr&\quad\times\sum\limits_{\bf h}\delta({\bf Q}\pm{\bf q}-{\bf Q}_{\bf h}), &(4.1.3.6)\cr}]$ where $[G_j({\bf Q},{\bf q})]$ is the `structure factor for one-phonon scattering' by the mode (j q) and is given by $[\eqalign{G_j({\bf Q},{\bf q})&=\sum\limits_{\kappa}{{\bf Q}\cdot{\bf e}(\kappa|j{\bf q})\over[m(\kappa)]^{1/2}}\cdot{}f_{\kappa}({\bf Q})\cr&\quad\times\exp[i{\bf Q}\cdot{\bf r}(\kappa)]\exp(-W_{\kappa}).}]$ The delta function in equation (4.1.3.6) implies that $[{\bf Q}={\bf Q}_{\bf h}\pm{\bf q},\eqno(4.1.3.7)]$ so that the scattering from the 3n modes with the same wavevector q is restricted to pairs of points in reciprocal space which are displaced by $[\pm{\bf q}]$ from the reciprocal-lattice points. (These satellite reflections are analogous to the pairs of `ghosts' near the principal diffraction maxima in a grating ruled with a periodic error.) There is a huge number, N, of q vectors which are uniformly distributed throughout the Brillouin zone, and each of these vectors gives a cross section in accordance with equation (4.1.3.6). Thus the one-phonon TDS is spread throughout the whole of reciprocal space, rising to a maximum at the reciprocal-lattice points where $[\omega\rightarrow{}0]$ for the acoustic modes.

For two-phonon scattering, involving modes with wavevectors q₁ and q₂, the scattering condition becomes $[{\bf Q}={\bf Q}_{\bf h}\pm{\bf q}_1\pm{\bf q}_2.\eqno(4.1.3.8)]$ The intensity at any point in reciprocal space is now contributed by a very large number of pairs of elastic waves with wavevectors satisfying equation (4.1.3.8). These vectors span the entire Brillouin zone, and so the variation of the two-phonon intensity with location in reciprocal space is less pronounced than for one-phonon scattering.

Expressions for $[({\rm d}\sigma/{\rm d}\Omega)^{(2)}]$ and for higher terms in equation (4.1.3.2) will not be given, but a rough estimate of their relative magnitudes can be derived by using the Einstein model of the crystal. All frequencies are the same, $[\omega_j({\bf q})=\omega_E]$ , and for one atom per cell ( [n=1] ) the exponent of the temperature factor, equation (4.1.3.5), becomes $[W={Q^2k_BT\over2m\omega^2_E},\eqno(4.1.3.9)]$ assuming classical equipartition of energy between modes: $[E_j({\bf q})=k_BT]$ . The cross sections for zero-order, first-order, second-order … scattering are then $[\eqalign{\left({{\rm d}\sigma\over{\rm d}\Omega}\right)^{(0)}&=Nf^2\exp(-2W)\cr\left({{\rm d}\sigma\over{\rm d}\Omega}\right)^{(1)}&=Nf^2\exp(-2W)2W\cr\left({{\rm d}\sigma\over{\rm d}\Omega}\right)^{(2)}&=Nf^2\exp(-2W){1 \over 2}(2W)^2\cr\vdots&\cr}]$ and the total cross section is $[\eqalign{{{\rm d}\sigma\over{\rm d}\Omega}&=Nf^2\exp(-2W)\biggl\{1+2W\cr&\quad+{1 \over 2}(2W)^2+{1 \over 6}(2W)^3+\ldots\biggr\}.}]$ The expression in curly brackets is the expansion of $[\exp(2W)]$ . The nth term in the expansion, associated with the nth-order (n-phonon) process, is proportional to [W^n] or to $[Q^{2n}T^n]$ . The higher-order processes are more important, therefore, at higher values of $[(\sin\theta)/\lambda]$ and at higher temperatures.

Our treatment so far applies to the TDS from single crystals. It can be extended to cover the TDS from polycrystalline samples, but the calculations are more complicated as the first-order scattering at a fixed value of $[(\sin\theta)/\lambda]$ is contributed by phonon wavevectors extending over the whole of the Brillouin zone. For a fuller discussion of the TDS from powders see Section 7.4.2 in IT C (2004).

References

International Tables for Crystallography (2004). Vol. C. Mathematical, physical and chemical tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 4.1, pp. 403-404