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 r.l.u.=2π/a and a is the lattice parameter.

Some notes

  • ki 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=2kisin(θ/2) where θ 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 , such that p=ki and KE=2k22m. Denoted in the table as ki=2π/λ (or q=2π/λ).
    • The crystallographer convention which uses the ordinary Planck’s constant h so that p=hk. Denoted in the table as ki=1/λ (or q=1/λ).
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