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PHY.K02UF Molecular and Solid State Physics

## Electron diffraction

In electron diffraction, the intensity of a diffraction peak at reciprocal lattice vector $\vec{G}$ is the square of the structure factor, $n_{\vec{G}}$.

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

Here $V$ is the volume of the unit cell, $j$ sums over the atoms in the basis, $\vec{r}_j$ are the positions of the atoms in the basis, and $f_j\left(\vec{G}\right)$ are the electron atomic form factors evaluated at $\vec{G}$.

The form below calculates the electron structure factors based on this formula. The crystal structure is specified by providing the primitive 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. 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 will calculate the diffraction pattern either for the primitive lattice vectors or the conventional lattice vectors. You just have to modify the basis appropriately.

$H$:  $K$:  $L$:

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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]

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

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

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]

[1] The atomic form factors were taken from the International Tables for Crystallography: // Data from http://it.iucr.org/Cb/ch4o3v0001/sec4o3o2/.

Structure factors in the diffraction pattern

 $hkl$ $|\vec{G}|$ Å-1 $|n_{\vec{G}}|$ $|n_{\vec{G}}|^2$ $\text{Re}\{n_{\vec{G}}\}$ $\text{Im}\{n_{\vec{G}}\}$