PHY.F20 Molecular and Solid State Physics

Atomic form factors

The atomic form factor is the Fourier transform of the electron density of an atom. It is assumed that the electron density is spherically symmetric so that the value of the Fourier transform only depends on the distance from the origin in reciprocal space. The diffraction condition is $\Delta \vec{k}=\vec{q}=\vec{G}$ where $\vec{q}$ is called the scattering vector. In the range of scattering vectors between 0 < $q$ < 25 Å-1, the atomic form factor is well approximated by a sum of Gaussians of the form, [1]

\[ \begin{equation} f(q)=\sum_{i=1}^4 a_i\exp\left( -b_i\left(\frac{q}{4\pi}\right)^2\right)+c, \end{equation} \]

where the values of $a_i$, $b_i$, and $c$ are tabulated below. The different atomic form factors for the elements can be plotted using the form below.

Atomic form factor for H

$f(q)$ 

$q$ [Å-1]



$G=$  Å-1 $f(G)=$ 

Each diffraction peak corresponds to a particular reciprocal lattice point $\vec{G}$. To calculate the structure factor for a diffraction peak, first calculate $G = |\vec{G}|$ and use the form above to calculate $f(G)$ for all the atoms in the basis. Then calculate the structure factor using,

\[ \begin{equation} S_{\vec{G}} = \sum\limits_j f_j\left(G\right)e^{-i\vec{G}\cdot\vec{r}_j} = \sum\limits_j f_j\left(G\right)\left(\cos\left(\vec{G}\cdot\vec{r}_j\right)-i\sin\left(\vec{G}\cdot\vec{r}_j\right)\right). \end{equation} \]

Here $j$ sums over the atoms in the basis and $\vec{r}_j$ is the position of atom $j$. The quantity $S_{\vec{G}}^*S_{\vec{G}}$ is proportional to the intensity of the diffraction peak.

 Element 

a1

b1

a2

b2

a3

b3

a4

b4

c


[1] The atomic form factors were taken from the International Tables for Crystallography: http://it.iucr.org/Cb/ch6o1v0001/.