Ali A. Husain

Ali A. Husain

ألا بذكر الله تطمئن القلوب

Useful Calculator for the Diffraction Kinematics of Relativistic Electrons

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When dealing with electron diffraction within an electron microscope, it is often useful to have a quick converter to convert between the beam energy, angles, and reciprocal lattice units (r.l.u.). Below is a simple calculator that converts between electron beam energies, angles, and momentum transfer. For added benefit, there is also a lattice parameter option to present the momentum transfer in reciprocal lattice units where $1\textrm{ r.l.u.} = 2\pi/a$ and $a$ is the lattice parameter.

Some notes

  • $k_i$ is the incident electron wavevector
  • $q$ is the momentum transfer. The small angle approximation is not assumed, so the momentum transfer is given by $q = 2k_i \sin(\theta/2)$ where $\theta$ is the scattering angle.
  • Momentum (or more accurately the wavevector) is presented according to two different conventions.
    • The physicist convention which uses the reduced Planck’s constant $\hbar$, such that $p=\hbar k_i$ and $\textrm{KE}=\frac{\hbar^2 k^2}{2m}$. Denoted in the table as $k_i = 2\pi/\lambda$ (or $q = 2\pi/\lambda$).
    • The crystallographer convention which uses the ordinary Planck’s constant $h$ so that $p=h k$. Denoted in the table as $k_i = 1/\lambda$ (or $q = 1/\lambda$).
Incident e-BeamMomentum TransferLattice Parameter
keV mrad Å
Å-1 (ki = 2π/λ) Å-1 (q = 2π/λ)
Å-1 (ki = 1/λ) Å-1 (q = 1/λ)
Å (λ) Å (λ)
pm (λ) nm (λ)
β = v/c r.l.u.
γ degrees

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