Structure factor

In physics, in the area of crystallography, the structure factor of a crystal is a mathematical description of how the crystal scatters incident radiation. The structure factor is a particularly useful tool in the interpretation of interference patterns obtained in X-ray, electron and neutron diffraction experiments.

Scattering from a crystal

A crystal is a periodic arrangement of atoms in a particular pattern. Each of the atoms may scatter incident radiation such as X-rays, electrons and neutrons. Because of the periodic arrangement of the atoms, the interference of waves scattered from different atoms may cause a distinct pattern of constructive and destructive interference to form. This is the diffraction pattern caused by the crystal.

In the kinematical approximation for diffraction, the intensity of a diffracted beam is given by:

$I_{\mathbf {K} }=\left|\psi _{\mathbf {K} }\right|^{2}\propto \left|S_{\mathbf {K} }\right|^{2}$

Here $\psi _{\mathbf {g} }$ is the wavefunction of the diffracted beam and $S_{\mathbf {g} }$ is the so called structure factor which is given by:

$S_{\mathbf {K} }=\sum _{i}f_{i}e^{i\mathbf {K} \cdot \mathbf {r} _{i}}$

where $\mathbf {K}$ is a reciprocal lattice vector (since diffraction only occurs for $\Delta \mathbf {k} =\mathbf {K}$), $\mathbf {r} _{i}$ is the position of an atom $i$ in the unit cell, and $f_{i}$ is the scattering power of the atom, also called the atomic form factor. The sum is over all atoms in the unit cell.

The structure factor describes the way in which an incident beam is scattered by the atoms of a crystal unit cell, taking into account the different scattering power of the elements through the term $f_{i}$. Since the atoms are spatially distributed in the unit cell, there will be a difference in phase when considering the scattered amplitude from two atoms. This phase shift is taken into account by the complex exponential term.

The atomic form factor, or scattering power, of an element depends on the type of radiation considered. Because electrons interact with matter though different processes than for example X-rays, the atomic form factors for the two cases are not the same.

Structure Factors for specific lattice types

To compute structure factors for a specific lattice, compute the sum above over the atoms in the unit cell. Since crystals are often described in terms of their Miller_indices, it is useful to examine a specific structure factor in terms of these.

Body-Centered Cubic As a convention, the body-centered cubic system is described in terms of a simple cubic lattice with primitive vectors $a{\hat {x}},a{\hat {y}},a{\hat {z}}$, with a basis consisting of $r_{0}={\vec {0}}$ and $r_{1}=(a/2)({\hat {x}}+{\hat {y}}+{\hat {z}})$. The corresponding reciprocal lattice is also simple cubic with side $2\pi /a$. The equation above then gives:

${\begin{matrix}S_{\mathbf {K} }&=&e^{i\mathbf {K} \cdot {\vec {0}}}+e^{i\mathbf {K} \cdot (a/2)({\hat {x}}+{\hat {y}}+{\hat {z}})}\\&=&1+e^{i\mathbf {K} \cdot (a/2)({\hat {x}}+{\hat {y}}+{\hat {z}})}\\&=&1+e^{i\pi (n_{1}+n_{2}+n_{3})}\\&=&1+(-1)^{n_{1}+n_{2}+n_{3}}\\\end{matrix}}$

Where we have used that $\mathbf {K} =(2\pi /a)(n_{1}{\hat {x}}+n_{2}{\hat {y}}+n_{3}{\hat {z}})$ where the $n_{i}$ are Miller_indices. Finally, $S_{\mathbf {K} }={\begin{cases}2,&n_{1}+n_{2}+n_{3}\ \ {\mbox{even}}\\0,&n_{1}+n_{2}+n_{3}\ \ {\mbox{odd}}\end{cases}}$