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PHY.K02UF Molecular and Solid State Physics

## Low Energy Electron Diffraction

In Low Energy Electron Diffraction (LEED), a sample is irradiated with a collimated 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 elastically by the surface atoms. Therefore the diffraction pattern depends only on the 2D crystal structure of the surface. 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.

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

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

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[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 $|n_{\vec{G}}|$ $|n_{\vec{G}}|^2$ $\text{Re}\{n_{\vec{G}}\}$ $\text{Im}\{n_{\vec{G}}\}$