# PDF Analysis

## PDF analysis has been attracting a lot of attention

**Why not take structure analysis one step further?**

### Basic information about PDF analysis

The PDF (Pair Distribution Function) can extract information about interatomic distances and coordination numbers from scattering patterns independent of the crystallinity of the material.

The RDF (Radial Distribution Function) may be more familiar to some. It is essentially the same analysis with a different definition of the vertical axis.

Before we get into the explanation of PDF analysis, let's look at an example measurement using the X-ray scattering patterns of crystalline and amorphous carbon shown in Figure 1.

**Figure 1**: X-ray total scattering profiles of crystalline and amorphous carbon (Courtesy of Dr. Kubota, Komaba Laboratory, Tokyo University of Science. Currently, his affiliation is NIMS. Learn more)

X-ray scattering patterns of amorphous carbon show broad peaks compared to the diffraction pattern of crystalline carbon. This happens because C-C bonds of various distances exist in the sample (microscopic view at the atomic bond scale), and scattered X-rays corresponding to each bonding distance interfere with each other. Therefore, the obtained scattering pattern can be Fourier transformed to directly obtain real-space information.

The obtained real-space information can be expressed in terms of coordination number or existence probability on the vertical axis and interatomic distance “r” on the horizontal axis. The profile plotted with RDF (coordination number) on the vertical axis and interatomic distance on the horizontal axis, and the structural model of carbon considered as a plane are shown in Figure 2.

**Figure 2**: Left: RDF of amorphous carbon; Right: Structure of carbon considered as a plane.

The peak position is related to the average interatomic distance and the peak area to the coordination number. Considering the 2D carbon structural model of carbon as a plane (in reality, information in 3-dimensional space is derived), each peak position indicated the 1st, 2nd, 3rd, ⋯ etc. neighbor distances.

This property makes it applicable to short-range structural analysis of amorphous systems and local structural disorder analysis of crystalline systems, and it is expected to be a promising method in a variety of fields.

Next, we will explain the PDF calculation method in more detail. Please refer to the *Rigaku Journal*¹) and books for detailed formulas.

The following is a flow chart of the PDF calculation method:

① Profile measurement (total scattering measurement)

② Calculation of structure factor S(Q)

③ Calculation PDF G(r) by Fourier transform S(Q)

The points to keep in mind when measuring profiles are summarized in the section on total scattering measurement.

The structure factor $S(Q)$ is calculated from the following equation:

\begin{equation} S(Q) = \frac{I_{\mathrm{coh}} - \langle f^{2} \rangle + \langle f \rangle^{2}}{\langle f \rangle^{2}} \nonumber \end{equation} \begin{eqnarray} \langle f \rangle &= \sum_{i}^{n} c_{i}f_{i} \nonumber \\ \langle f^{2} \rangle &= \sum_{i}^{n} c_{i}f_{i}^{2} \nonumber \end{eqnarray}

where $c_{i}$ and $f_{i}$ are the concentration and atomic scattering factor of the $i$-th element and $I_{\mathrm{coh}}$ is the coherent scattering intensity derived from the sample structure extracted from the total X-ray scattering measurement data. To derive the coherent scattering, various corrections are applied to the obtained profile. Specifically, after background subtraction, polarization correction, absorption correction, and the Compton scattering correction, the experimentally profile is normalized by the atomic scattering factor.

PDF $G(r)$ can be obtained from the Fourier transform of the measured structure factor $S(Q)$.

Using SiO₂ glass as an example, we show the definition equations^{2)} and figures for four typical functions. The dashed line in each figure shows the average density in the system.

### Reduced pair distribution function G(r)

\begin{equation} G(r) = \frac{2}{\pi} \int^{Q_{\max}}_{Q_{\min}} {Q\{S(Q)-1\}\sin Qr} \,\mathrm{d}Q\nonumber \end{equation}

### Atomic pair distribution function $g(r)$

\begin{equation} g(r) = \frac{1}{2\pi^2\rho_{0}r} \int^{Q_{\max}}_{Q_{\min}} {Q\{S(Q)-1\}\sin Qr} \,\mathrm{d}Q + 1\nonumber \end{equation}

### Radial distribution function RDF $R(r)$

\begin{equation} R(r) = \frac{2r}{\pi} \int^{Q_{\max}}_{Q_{\min}} {Q\{S(Q)-1\}\sin Qr} \,\mathrm{d}Q + 4\pi r^2\rho_{0}\nonumber \end{equation}

### Total correlation function $T(r)$

\begin{equation} T(r) = \frac{2}{\pi} \int^{Q_{\max}}_{Q_{\min}} {Q\{S(Q)-1\}\sin Qr} \,\mathrm{d}Q + 4\pi r\rho_{0}\nonumber \end{equation}

PDF analysis often uses $g(r)$ for amorphous and $G(r)$ for crystalline. They can also be easily converted to each other.

### References：

[1] Y.Shiramata and M.Yoshimoto, *Rigaku Journal*, **50(1)**, (2019) 1-8.

[2] D. A. Keen. *J. Appl. Cryst.*, **34**, (2001), 172. doi: 10.1107/S0021889800019993.

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