Total scattering - Debye approach

Neutron or X-ray scattering static structure factor \(S(q)\) is defined as: \[\label{s2q_1} S(q)=\frac{1}{N} \sum_{j,k} b_j\,b_k \left< e^{\displaystyle{iq[{\bf{r}}_j-{\bf{r}}_k]}} \right>\] where \(b_j\) and \({\bf{r}}_j\) represent respectively the neutron or X-ray scattering length, and the position of the atom \(j\). \(N\) is the total number of atoms in the system studied.
To take into account the inherent/volume averaging of scattering experiments it is necessary to sum all possible orientations of the wave vector \(q\) compared to the vector \({\bf{r}}_j-{\bf{r}}_k\). This average on the orientations of the \(q\) vector leads to the famous Debye's equation: \[\label{s2q_2} S(q)\ =\ \frac{1}{N} \sum_{j,k} b_j\,b_k \frac{\sin (q|{\bf{r}}_j-{\bf{r}}_k|)}{q|{\bf{r}}_j-{\bf{r}}_k|}\] Nevertheless the instantaneous individual atomic contributions introduced by this equation [s2q_2] are not easy to interpret. It is more interesting to express these contributions using the formalism of radial distribution functions [Sec. 5.2].
In order to achieve this goal it is first necessary to split the self-atomic contribution (\(j=k\)), from the contribution between distinct atoms: \[\label{s2q_3} S(q)\ =\ \sum_{j}\ c_j b_j^2\ +\ \underbrace{\frac{1}{N} \sum_{j\ne k} b_j\,b_k \frac{\sin (q|{\bf{r}}_j-{\bf{r}}_k|)}{q|{\bf{r}}_j-{\bf{r}}_k|}}_{\displaystyle{I(q)}} \qquad \text{with}\quad c_j=\frac{N_j}{N}\] where \(4\pi\ \sum_{j}\ c_j b_j^2\) represents the total scattering cross section of the material.
The function \(I(q)\) which describes the interaction between distinct atoms is related to the radial distribution functions through a Fourier transformation: \[\label{s2q_4} I(q)\ =\ 4 \pi \rho\ \int_{0}^{\infty}\ dr\ r^{2}\ \frac{\sin qr}{qr}\ G(r)\] where the function \(G(r)\) is defined using the partial radial distribution functions [Eq. [g2r_4]]: \[\label{s2q_5} G(r)\ =\ \sum_{\alpha,\beta}\ c_{\alpha} b_{\alpha}\ c_{\beta} b_{\beta}\ (g_{\alpha\beta}(r) -1)\] where \(c_{\alpha}=\displaystyle{\frac{N_\alpha}{N}}\) and \(b_{\alpha}\) represents the neutron or X-ray scattering length of species \(\alpha\).
\(G(r)\) approaches - \(\displaystyle{-\ \sum_{\alpha,\beta}}\ c_{\alpha} b_{\alpha}\ c_{\beta} b_{\beta}\) for \(r = 0\), and 0 for \(r\to\infty\).
Usually the self-contributions are subtracted from equation [s2q_3] and the structure factor is normalized using the relation: \[\label{s2q_6} S(q)\ -\ 1\ =\ \frac{I(q)}{\displaystyle{\langle b^{2} \rangle}} \quad \text{with} \quad \langle b^{2} \rangle = \left(\sum_{\alpha} c_{\alpha} b_{\alpha} \right)^{2}\] It is therefore possible to write the structure factor [Eq. [s2q_2]] in a more standard way: \[\label{s2q_7} S(q)\ =\ 1\ +\ 4 \pi \rho \int_{0}^{\infty}\ dr\ r^{2}\ \frac{\sin qr}{qr} ({\bf{g}}(r) -1)\] where \({\bf{g}}(r)\) (the radial distribution function) is defined as: \[\label{s2q_8} {\bf{g}}(r)\ =\ \frac{\displaystyle{\sum_{\alpha,\beta}}\ c_{\alpha} b_{\alpha}\ c_{\beta} b_{\beta}\ g_{\alpha\beta}(r) }{\displaystyle{\langle b^{2} \rangle}}\] In the case of a single atomic species system the normalization allows to obtain values of \(S(q)\) and \({\bf{g}}(r)\) which are independent of the scattering factor/length and therefore independent of the measurement technique. In most cases, however, the total \(S(q)\) and \({\bf{g}}(r)\) are combinations of the partial functions weighted using the scattering factor and therefore depend on the measurement technique (Neutron, X-rays ...) used or simulated.

Figure 5.4 presents a comparison between the calculations of the total neutron structure factor done using the Debye relation [Eq. [s2q_2]] and the pair correlation functions [Eq. [s2q_7]]. The material studied is a sample of glassy GeS\(_2\) at 300 K obtained using ab-initio molecular dynamics. In several cases the structure factor \(S(q)\) and the radial distribution function \({\bf{g}}(r)\) [Eq. [s2q_8]] can be compared to experimental data. To simplify the comparison Atomes computes several radial distribution functions used in practice such as \(G(r)\) defined [Eq. [s2q_5]], the differential correlation function \(D(r)\), \({\bf{G}}(r)\), and the total correlation function \(T(r)\) defined by: \[\begin{aligned} \label{s2q_9} D(r)\ =\ 4\pi r \rho\ G(r) \\ \nonumber {\bf{G}}(r)\ =\ \frac{D(r)}{\langle b^{2} \rangle} \\ T(r)\ =\ D(r)\ +\ 4\pi r \rho\ \langle b^{2} \rangle \nonumber\end{aligned}\] \({\bf{g}}(r)\) equals zero for \(r=0\) and approaches 1 for \(r\to\infty\).
\(D(r)\) equals zero for \(r=0\) and approaches 0 for \(r\to\infty\).
\({\bf{G}}(r)\) equals zero for \(r=0\) and approaches 0 for \(r\to\infty\).
\(T(r)\) equals zero for \(r=0\) and approaches \(\infty\) for \(r\to\infty\).
This set of functions for a model of GeS\(_2\) glass (at 300 K) obtained using ab-initio molecular dynamics is presented in figure 5.5.

 can compute, for the case of x-rays and/or neutrons, the following functions:

• \(S(q)\) and \(Q(q)\ =\ q[S(q)-1.0]\) [1], [2] computed using the Debye equation

• \(S(q)\) and \(Q(q)\ =\ q[S(q)-1.0]\) [1], [2] computed using the Fourier transform of \({\bf{g}}(r)\)

• \({\bf{g}}(r)\), \({\bf{G}}(r)\), \(D(r)\) and \(T(r)\) computed using the standard real space calculation

• \({\bf{g}}(r)\), \({\bf{G}}(r)\), \(D(r)\) and \(T(r)\) computed using the Fourier transform of Debye \(S(q)\)

1. M. T. Dove, M. G. Tucker, and D. A. Keen, Eur. Jour. Mat., vol. 14, pp. 331–348, 2002.
2. B. Thijsse, J. App. Cryst., vol. 17, pp. 61–76, 1984.