# Scattering from a crystal

$$\newcommand{\diff}{\mathrm{d}} \renewcommand{\deg}{^{\circ}} \newcommand{\e}{\mathrm{e}} \newcommand{\A}{\unicode{x212B}}$$
In a crystal, atoms or molecules form a periodic structure with the translational symmetry, which is frequently called long range order. Thus, scatterings from each atom are interfered and eventually create a diffraction pattern, which reflects the crystalline structure. Since each diffraction has a different contribution from each atom, the diffraction anomalous fine structure (DAFS) method is capable to distinguish the crystalline site-specific spectroscopic information by measuring an energy dependence of the diffraction intensity. This article will provide a brief review about the description of the conventional x-ray diffraction.

As seen in the discussion of the scattering from an atom, the scattering amplitude from plural atoms, i.e., a crystal, are similarly calculated from the summation of phases multiplied by atomic scattering factors of each atom. First, the position of an atom in the crystal is described by taking the translational symmetry into account as

\begin{align}
\label{Eq:position_vector}
\tag{1}
\vec{r}_{l}= \vec{R}_{n} + \vec{r}_{m},
\end{align}

with

\begin{align}
\label{Eq:atomic_position_vector_definition}
\tag{2}
\vec{R}_{n} &\equiv n_{1}\vec{a}_{1}+n_{2}\vec{a}_{2}+n_{3}\vec{a}_{3} \\
\vec{r}_{m} &\equiv x_{m}\vec{a}_{1}+y_{m}\vec{a}_{2}+z_{m}\vec{a}_{3},
\end{align}

where $$\vec{R}_{n}$$ is a vector specifying the number $$n$$-th unit cell, $$\vec{r}_{m}$$ is a vector indicating $$m$$-th atomic position in a unit cell, $$\vec{a}_{1}$$, $$\vec{a}_{2}$$ and $$\vec{a}_{3}$$ are lattice vectors along with $$a$$, $$b$$ and $$c$$ axes, and $$x_{m}$$, $$y_{m}$$ and $$z_{m}$$ are the fractional coordinate of $$m$$-th atom. Thus, the scattering amplitude from the crystal can be described as

\begin{align}
\label{Eq:scat_amp_all_atom}
\tag{3}
F^{\mathrm{crystal}}(\vec{Q}) = \sum^{\mathrm{All\ atoms}}_{l}f_{l}(\vec{Q})\e^{i\vec{Q}\cdot \vec{r}_{l}},
\end{align}

where $$f_{l}(\vec{Q})$$ is the atomic form factor1 of the atom placed at position $$\vec{r}_{l}$$. The scattering amplitude is readily decomposed into two contributions from the lattice and that from the inside of the unit cell with Eq. \eqref{Eq:position_vector} as

\begin{align}
\label{Eq:decomposition_of_scattering_amplitude}
\tag{4}
F^{\mathrm{crystal}}(\vec{Q}) = \sum^{\mathrm{All\ atoms}}_{\vec{R}_{n}+\vec{r}_{j}} f_{j}(\vec{Q})\e^{i\vec{Q}\cdot(\vec{R}_{n}+\vec{r}_{j})} = \overbrace{\sum_{n}\e^{i\vec{Q}\cdot\vec{R}_{n}}}^{\mathrm{Lattice}} \overbrace{\sum_{m}f_{j}(\vec{Q})\e^{i\vec{Q}\cdot\vec{r}_{m}}}^{\mathrm{Unit\ cell}}
\end{align}

The first term of Eq. \eqref{Eq:decomposition_of_scattering_amplitude} is the sum of the scattering from the lattice, while the second term is the sum of atoms in the unit cell, which is known as structure factor.

The diffraction from a crystal is observed under the diffraction condition, which is conventionally described as $$2d\sin \theta = \lambda$$. The equivalent diffraction condition in the vector form is

\begin{align}
\label{Eq:laue_condition}
\tag{5}
\vec{Q} = \vec{G},
\end{align}

where $$\vec{G}$$ is a vector pointing reciprocal lattice defined as

\begin{align}
\label{Eq:def_G}
\tag{6}
\vec{G} = h\vec{a}^{*}_{1}+k\vec{a}^{*}_{2}+l\vec{a}^{*}_{3}.
\end{align}

$$h$$, $$k$$ and $$l$$ are all integers and called “Miller indices”. $$\vec{a}^{*}_{1}$$, $$\vec{a}^{*}_{2}$$ and $$\vec{a}^{*}_{3}$$ are basis vectors of reciprocal space defined as2

\begin{align}
\tag{7}
\vec{a}^{*}_{1} &= 2\pi \frac{\vec{a}_{2}\times \vec{a}_{3}}{v^{*}_{c}}\\
\vec{a}^{*}_{2} &= 2\pi \frac{\vec{a}_{3}\times \vec{a}_{1}}{v^{*}_{c}}\\
\vec{a}^{*}_{3} &= 2\pi \frac{\vec{a}_{1}\times \vec{a}_{2}}{v^{*}_{c}},
\end{align}

where the volume of the unit cell in the reciprocal space, $$v^{*}_{c}$$, is calculated as $$v^{*}_{c} = \vec{a}_{1}\cdot (\vec{a}_{2}\times \vec{a}_{3}) = \vec{a}_{2}\cdot (\vec{a}_{3}\times \vec{a}_{1}) = \vec{a}_{3}\cdot (\vec{a}_{1}\times \vec{a}_{2})$$.
These vectors fulfill a condition that

\begin{align}
\label{Eq:relation_between_real_and_reciprocal_vector}
\tag{8}
\vec{a}_{i} \cdot \vec{a}_{j} = 2\pi \delta_{ij},
\end{align}

where $$\delta_{ij}$$ is the Kronecker’s delta function. This diffraction condition is derived from the feature of the first term of right hand side in Eq. \eqref{Eq:decomposition_of_scattering_amplitude}, i.e., the scattering from the lattice. That is

\begin{align}
\label{Eq:laue_function}
\tag{9}
\left| \sum_{n}\e^{i\vec{Q}\cdot\vec{R}_{n}} \right|^{2} \to Nv^{*}_c \sum_{\vec{G}}\delta (\vec{Q}-\vec{G}) \qquad \text{as} \quad N \to \infty
\end{align}

where $$\delta$$ is the Dirac’s delta function, $$N$$ is the total number of unit cell, i.e., $$N_{1}\times N_{2}\times N_{3}$$. $$N_{1}$$, $$N_{2}$$ and $$N_{3}$$ are the number of unit cell along with $$\vec{a}_{1}$$, $$\vec{a}_{2}$$ and $$\vec{a}_{3}$$, respectively. Therefore, the diffraction intensity, $$\left| F^{\mathrm{crystal}}(\vec{Q}) \right|^{2}$$, is observed only when $$\vec{Q} = \vec{G}$$ as described in Eq. \eqref{Eq:laue_condition}, since the Dirac’s delta function, $$\delta(x)$$, shows the non-zero value only at $$x=0$$.

Under the diffraction condition, the structure factor is reduced as follows:

\begin{align}
\sum_{m}f_{m}(\vec{Q})\exp \left( i\vec{Q}\cdot\vec{r}_{m} \right)
&= \sum_{m}f_{m}(\vec{G})\exp \left( i\vec{G}\cdot\vec{r}_{m} \right) \notag \\
&= \sum_{m}f_{m}(\vec{G})\exp \left\{ 2\pi i \left( x_{m}h + y_{m}k + z_{m}l \right) \right\},
\end{align}

where $$\sum_{m}$$ is a summation of all atoms in a unit cell3 Therefore, the scattering amplitude from a crystal under diffraction condition is finally derived to be

\begin{align}
\label{Eq:derived_scattering_amplitude}
\tag{10}
I(\vec{Q} = \vec{G}) \propto \left| \sum_{m}f_{m}(\vec{G})\exp \left\{ 2\pi i \left( x_{m}h + y_{m}k + z_{m}l \right) \right\} \right|^{2}.
\end{align}

The essence of the site-distinguished analysis by the DAFS method is this structure factor, where the atomic scattering factor is multiplied by the phase factor, $$\exp \left\{ 2\pi i \left( x_{m}h + y_{m}k + z_{m}l \right) \right\}$$. This factor causes the difference in the contributions from a certain element at each diffraction line, consequently enabling us to site-selectively analyze the energy dependence of the atomic scattering factor4 of the same element at the different sites.

###### For further study…(This article was written based on the following book)
• J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics, 2nd ed., John Wiley & Sons, 2011.
[Bibtex]
@book{Als-Nielsen2011,
author = {Als-Nielsen, J and McMorrow, D},
edition = {2nd},
publisher = {John Wiley \& Sons},
title = {{Elements of Modern X-ray Physics}},
year = {2011}
}