Low Energy Electron Diffraction

In Low Energy Electron Diffraction (LEED), a sample is irradiated with a collimated low-energy electron beam. Because of the low energy of the electrons (20-200 eV), the electrons only penetrate a few atomic layers into the sample and are scattered by the surface atoms. The diffracted electrons exhibit peaks in intensity at the reciprocal lattice vectors of the two-dimensional Bravais lattice of the surface. As with electron diffraction with high-energy electrons, the intensity of the diffraction peaks is proportional to the square of the structure factor $n_{\vec{G}}$.

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

Here $V$ is the volume of the primitive unit cell, $j$ sums over the atoms in the basis, $\vec{r}_j$ are the positions of the atoms in the basis, $\vec{G}$ are the reciprocal lattice vectors of the 2D crystal, and $f_j\left(\vec{G}\right)$ are the electron atomic form factors evaluated at $\vec{G}$.

In the form below, the electron beam energy is adjustable. The 2D crystal structure of the surface is specified by the two primitive lattice vectors $\vec{a}_1$ and $\vec{a}_2$ and by the positions of up to five atoms in the basis. The structure of the crystal is shown on the right.

The script calculates the primitive lattice vectors $\vec{b}_1$ and $\vec{b}_1$ in reciprocal space, the structure factors for the $\vec{G}$'s $\left( \vec{G}_{hk} = h\,\vec{b}_1 + k\,\vec{b}_2 \right)$ and the LEED pattern.

Energy of the electron beam: [eV]

Primitive lattice vectors:

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

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

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

Primitive reciprocal lattice vectors

$\vec{b}_1=2\pi\frac{R\,\vec{a}_2}{\vec{a}_1\cdot R\,\vec{a}_2}=$ $\hat{k}_x+$ $\hat{k}_y$ [m^-1]

$\vec{b}_2=2\pi\frac{R\,\vec{a}_1}{\vec{a}_1\cdot R\,\vec{a}_2}=$ $\hat{k}_x+$ $\hat{k}_y$ [m^-1]

$\text{with} \qquad R = \left( \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix}\right)$

[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

$hk$

$|\vec{G}|$ Å^-1

$|n_{\vec{G}}|$

$|n_{\vec{G}}|^2$

$\text{Re}\{n_{\vec{G}}\}$

$\text{Im}\{n_{\vec{G}}\}$