In an x-ray diffraction experiment, the reciprocal lattice vectors can be measured directly. From this, the Bravais lattice of the crystal and the volume of the primitive unit cell are known. However, the atoms of the basis are not yet determined. The experiment provides some evidence for the form of the basis because the intensities of the diffraction peaks are proportional to the squares of the Fourier coefficients of the electron density $n_{\vec{G}}^2$. If the electron density could be calculated, then the peaks of the electron density would tell us where the atoms are, and the integral of the electron density around a peak would tell us how many electrons are associated with the peak, and that would indicate the atomic number of the atom. Unfortunately, to plot the electron density, we need to know the real and imaginary parts of $n_{\vec{G}}$, not just its magnitude.

To determine the basis, we take the information that we know, such as the volume of the primitive unit cell and whatever information we have about the chemical composition of the crystal, to guess a crystal structure. We then calculate the electron density of the guessed structure, determine the Fourier coefficients of this calculated electron density, and compare the values of $n_{\vec{G}}^2$ of the calculation and the experiment. If the values of $n_{\vec{G}}^2$ match for all hundreds or thousands of the measurement points, then we can be confident that the guess was correct.

To a good approximation, the electron density of a crystal can be described by a sum of the electron densities of the atoms that make up the crystal. The x-rays scatter off the total electron density, and most of the electrons are tightly bound to the nucleus, so the electron density tends to be a sharply peaked function.

$$n\left(\vec{r}\right)= \sum\limits_{\vec{T}} \sum\limits_j n_j\left(\vec{r}-\vec{r}_j+\vec{T}\right) .$$

Here $\vec{T}$ are the translation vectors of the Bravais lattice, and $n_j\left(\vec{r}\right)$ is the electron density of atom $j$. The sum over $j$ extends over all of the atoms in the basis. The vectors $\vec{r}_j$ specify the positions of the atoms within the unit cell. This approximation neglects the rearrangement of the valence electrons as they form bonds, but it is a good approximation since most electrons are core electrons.

Since the electron density is a periodic function, it can be expressed as a Fourier series.

$$ n\left(\vec{r}\right)= \sum\limits_{\vec{G}} n_{\vec{G}}e^{i\vec{G}\cdot\vec{r}} = \sum\limits_{\vec{T}} \sum\limits_j n_j\left(\vec{r}-\vec{r}_j+\vec{T}\right),$$

where $\vec{G}$ are the reciprocal lattice vectors and $n_{\vec{G}}$ are complex coefficients.

To determine the coefficients, we multiply both sides by $e^{-i\vec{G}'\cdot\vec{r}}$ and integrate over the volume of a unit cell ($\text{u.c.}$). Since it does not matter which unit cell we integrate over, we choose the one at $\vec{T}=0$.

$$ \sum\limits_{\vec{G}} \int\limits_{\text{u.c.}} n_{\vec{G}}e^{i\vec{G}\cdot\vec{r}}e^{-i\vec{G}'\cdot\vec{r}}d\vec{r} = \sum\limits_j \int\limits_{\text{u.c.}} n_j\left(\vec{r}-\vec{r}_j\right)e^{-i\vec{G}'\cdot\vec{r}}d\vec{r}. $$

On the left-hand side, only the term where $\vec{G} = \vec{G}'$ contributes, and the integral evaluates to $n_{\vec{G}}$ times the volume $V$ of the unit cell. On the right-hand side, the integral is over one unit cell, but since the charge density is only non-zero in one unit cell, the integral can extend over all space.

$$ n_{\vec{G}}V = \sum\limits_j \int n_j\left(\vec{r}-\vec{r}_j\right)e^{-i\vec{G}\cdot\vec{r}}d\vec{r}.$$

Make a substitution $\vec{r}' = \vec{r} - \vec{r}_j$.

$$ n_{\vec{G}} = \frac{1}{V}\sum\limits_j e^{-i\vec{G}\cdot\vec{r}_j}\int n_j\left(\vec{r}'\right)e^{-i\vec{G}\cdot\vec{r'}}d\vec{r}'.$$

The function,

$$ f_j\left(G\right) = \int n_j\left(\vec{r}\right)e^{-i\vec{G}\cdot\vec{r}} d\vec{r}, $$

is called the atomic form factor $f_j\left(G\right)$. If the atomic form factors were known for all of the atoms in the basis, the Fourier coefficients of the electron density could be calculated.

$$ n_{\vec{G}} = \frac{1}{V}\sum\limits_j e^{-i\vec{G}\cdot\vec{r}_j}f_j\left(G\right).$$

The atomic form factors for all of the elements are tabulated in the International Tables for Crystallography: http://it.iucr.org/Cb/ch6o1v0001/. In this table, 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,

$$f(|\vec{G}|)=\sum_{i=1}^4 a_i\exp\left( -b_i\left(\frac{G}{4\pi}\right)^2\right)+c,$$

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 $f(G)$  $G$ [Å-1]

$|\vec{G}|=$  Å-1 $f(G)=$

Each diffraction peak corresponds to a particular reciprocal lattice point $\vec{G}$. To calculate the Fourier coefficient 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 coefficient using,

\[ \begin{equation} n_{\vec{G}} = \frac{1}{V}\sum\limits_j f_j\left(G\right)e^{-i\vec{G}\cdot\vec{r}_j} = \frac{1}{V}\sum\limits_j f_j\left(G\right)\left(\cos\left(\vec{G}\cdot\vec{r}_j\right)-i\sin\left(\vec{G}\cdot\vec{r}_j\right)\right). \end{equation} \]

Here $j$ sums over the atoms in the basis, and $\vec{r}_j$ is the position of atom $j$, and $V$ is the volume of the primitive unit cell in real space. The quantity $n_{\vec{G}}^*n_{\vec{G}}$ is proportional to the measured intensity of the diffraction peak.

Element

a1

b1

a2

b2

a3

b3

a4

b4

c