PHY.K02UF Molecular and Solid State Physics

Structure factor

In a crystal, the electron density is a periodic function. To a good approximation, it can be described by a sum of the electron densities of the atoms that make up the crystal.

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

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 the Fourier transform of the electron density of atom $j$. This is called the atomic form factor $f_j\left(G\right)$. The atomic form factors for all of the elements are tabulated in the International Tables for Crystallography: Plots of the atomic form factors, can be found here. One can look up the atomic form factors and calculate the structure factors,

\( S_{\vec{G}} =Vn_{\vec{G}} = \sum\limits_j f_j\left(G\right)e^{-i\vec{G}\cdot\vec{r}_j} = \sum\limits_j f_j\left(G\right)\left(\cos\left(\vec{G}\cdot\vec{r}_j\right)-i\sin\left(\vec{G}\cdot\vec{r}_j\right)\right) \),

The form below calculates the x-ray structure factors based on this last formula. The crystal structure is specified by providing the lattice vectors and the positions of the atoms in the basis. A basis of up to five atoms can be calculated. The script first calculates the reciprocal lattice vectors and from them calculates the reciprocal lattice vectors $\vec{G}_{hkl}=h\vec{b}_1+k\vec{b}_2+l\vec{b}_3$.

Often in experiments, the directions are given in terms of the conventional lattice vectors and the reciprocal lattice vectors are labeled as the reciprocal lattice vectors of the conventional unit cell. When this is done, a bcc lattice is described as simple cubic with a basis of atoms at (0,0,0) and (0.5,0.5,0.5) while fcc is described as simple cubic with a basis of atoms at (0,0,0), (0,0.5,0.5), (0.5,0,0.5), and (0.5,0.5,0). The program below will calculate the diffraction pattern either for the primitive lattice vectors or the conventional lattice vectors. You just have to modify the basis appropriately.

Primitive lattice vectors:

 $\vec{a}_1=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [m] 
 $\vec{a}_2=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [m]
 $\vec{a}_3=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [m]

 The positions of the atoms are given in fractional coodinates between -1 and 1.

$\vec{a}_1+$  $\vec{a}_2+$  $\vec{a}_3$ 
$\vec{a}_1+$  $\vec{a}_2+$  $\vec{a}_3$ 
$\vec{a}_1+$  $\vec{a}_2+$  $\vec{a}_3$ 
$\vec{a}_1+$  $\vec{a}_2+$  $\vec{a}_3$ 
$\vec{a}_1+$  $\vec{a}_2+$  $\vec{a}_3$ 
$\vec{a}_1+$  $\vec{a}_2+$  $\vec{a}_3$ 
$\vec{a}_1+$  $\vec{a}_2+$  $\vec{a}_3$ 
$\vec{a}_1+$  $\vec{a}_2+$  $\vec{a}_3$ 

Primitive unit cells:

Conventional unit cells:

Primitive reciprocal lattice vectors

$\vec{b}_1=2\pi\frac{\vec{a}_2\times\vec{a}_3}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}=$ $\hat{k}_x+$ $\hat{k}_y+$ $\hat{k}_z$ [m-1]
$\vec{b}_2=2\pi\frac{\vec{a}_3\times\vec{a}_1}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}=$ $\hat{k}_x+$ $\hat{k}_y+$ $\hat{k}_z$ [m-1]
$\vec{b}_3=2\pi\frac{\vec{a}_1\times\vec{a}_2}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}=$ $\hat{k}_x+$ $\hat{k}_y+$ $\hat{k}_z$ [m-1]

Structure factors

The value of $|S_{\vec{G}}|$ for the 000 diffraction peak is the total number of electrons in the primitive unit cell. The intensities of the peaks in an x-ray diffraction experiment are proportional to $|S_{\vec{G}}|^2$. Note that elements with more electrons produce stronger diffraction intensities.


 $|\vec{G}|$ Å-1