Electron diffraction

Electrons are quantum particles that move like waves. They diffract from crystals in a manner similar to x-rays. The $\vec{k}$ vector of electrons is proportional to the momentum of the electron, $\vec{p} = \hbar\vec{k}$. This is known as the de Broglie relation. Typically, electron diffraction is performed in a transmission electron microscope. A beam of electrons is generated by accelerating them through a voltage, $V$. The kinetic energy of electrons is,

$$eV = \frac{mv^2}{2} = \frac{p^2}{2m} = \frac{\hbar^2k^2}{2m}.$$

Here $e$ is the elementary charge and $m$ is the mass of an electron. The energy is typically tens to hundreds of keV in an electron microscope. The electrons pass through a crystal as waves, and intensity peaks are recorded on a detector behind the sample. It is usually not possible to rotate the sample as is done in x-ray diffraction, so the diffraction peaks that are measured correspond to reciprocal lattice vectors $\vec{G}_{hkl}$ in the plane perpendicular to the incoming electron beam. The diffraction condition for electrons is the same as for x-rays, $\Delta\vec{k} = \vec{G}.$ 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 $\vec{G}$. This is the same formula as for x-rays, but the atomic form factors are different for electrons and x-rays because the scattering mechanism is different. Electrons scatter via a Coulomb interaction.

The form below calculates the electron diffraction pattern for a beam of electrons travelling in the direction of $\vec{G}_{hkl}$. The pattern that is displayed corresponds to the diffraction peaks in the plane perpendicular to $\vec{G}_{hkl}$. The table includes all diffraction peaks in three dimensions as if the detectors could be moved arbitrarily in space, even though this is not typically the case in the experiments. This form can only generate diffraction patterns for a basis of up to five atoms. The script first calculates the primitive reciprocal lattice vectors from the primitive lattice vectors in real space and then 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 indexed either with the primitive lattice vectors or the conventional lattice vectors. You just have to modify the basis appropriately.

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

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

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

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

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