# Calculation of a structure factor

A structure factor is calculated by summing up scattering factors of each atom with multiplying the phases at each atomic position in a unit cell as described previously in an equation. It is a facile approach to calculate the structure factor; however, the calculation becomes complicated when the unit cell includes the large number of atoms. Furthermore, it is frequently difficult to distinguish the equivalent and non-equivalent sites in this approach.

For example in body-centered cubic (BCC) metal, atom occupy (0, 0, 0), (1/2, 1/2, 1/2) sites, and all of them are equivalent. Thus, these atoms must have exactly the same local/electronic structure. In contrast, in cesium chloride structure, (0, 0, 0) and (1/2, 1/2, 1/2) are occupied by chloride and cesium atoms, respectively, being inequivalent each other; therefore, the local/electronic structure must be different. The “site-distinguished” analysis of the diffraction anomalous fine structure (DAFS) method provides individual XAFS spectra at the inequivalent sites in the material as seen in the latter case. These relations and concepts are comprehensively understood in the framework of Space Group, which is well-described in the International Tables for Crystallography vol. A[1]. Any crystalline materials except for a quasicrystal belong to a certain space group, and the structure factor of each space group is calculated and included in the international table [2].1 Thus, the calculation of the structure factor should be carried out based on the crystallographic site in space group, which is more versatile and convenient.

The table provides the structure factor as values of “$$A(\vec{G}\cdot \vec{r}_{m})$$” and “$$B(\vec{G}\cdot \vec{r}_{m})$$”, whose definition is as follows:

\begin{align}
A(\vec{G}\cdot \vec{r}_{m}) &= \sum_{e} \cos (\vec{G}\cdot \vec{r}_{m}) \\
B(\vec{G}\cdot \vec{r}_{m}) &= \sum_{e} \sin (\vec{G}\cdot \vec{r}_{m}),
\label{Eq:def_A_and_B}
\tag{1}
\end{align}

where $$\sum_{e}$$ denotes the summation of equivalent positions belonging the site in a unit cell. Then, the structure factor is described with these $$A$$ and $$B$$ values as

\begin{align}
F(\vec{G}) = \sum_{j} f_{j}A_{j} + i \sum_{j} f_{j}B_{j},
\label{Eq:A_B_based_structure_factor}
\tag{2}
\end{align}

where $$\sum_{j}$$ is the summation of independent sites, and $$f_{j}$$ is the atomic scattering factor of the atom at site $$j$$. Note that both $$A$$ and $$B$$ values should be multiplied by the ratio of the numbers of atoms at general and special positions. For example, when calculating the structure factor of $$32e$$ site (Wyckoff letter for a special position) in space group $$F d \bar{3} m$$ (No. 227), the factor of 32/192 ($$192i$$ is for general position in this space group) should be multiplied in order to take into account the overlap of the atoms at the special position. Also note that the value of $$B$$ disappears in the space group with centrosymmetry, because $$\sum_{e} \sin (\vec{G}\cdot \vec{r}_{m})$$ becomes 0 when the same element locates at $$\vec{r}_{m}$$ and $$-\vec{r}_{m}$$ due to the nature of $$\sin$$ function.

###### References
[1] International Tables for Crystallography Volume A, 5th ed., T. Hahn, Ed., Springer, 2006.
[Bibtex]
@book{Hahn2006,
edition = {5th},
editor = {Hahn, Th},
file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Unknown/Unknown\_Unknown\_2006.pdf:pdf},
isbn = {0792365909},
publisher = {Springer},
title = {{International Tables for Crystallography Volume A}},
year = {2006}
}
[2] International Tables for X-ray Crystallography Volume I, N. F. M. Henry and K. Lonsdale, Eds., Birmingham: Kynoch Press, 1952.
[Bibtex]
@book{Henry1952,
file = {:C$\backslash$:/Users/Tomoya/Documents/Mendeley Desktop/Unknown/Unknown\_Unknown\_1952.pdf:pdf},
}