# 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: Total neutron structure factor for glassy GeS$$_2$$ at 300 K - A Evaluation using the atomic correlations [Eq. [s2q_2]], B Evaluation using the pair correlation functions [Eq. [s2q_7]].

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. Figure 5.5: Example of various distribution functions neutron-weighted in glassy GeS$$_2$$ at 300 K.

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

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

• $$S(q)$$ and $$Q(q)\ =\ q[S(q)-1.0]$$ ,  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.