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Working with physical units¤

Here, convenience methods for working with physical units are described.

cryojax.constants.wavelength_from_kilovolts(voltage_in_kilovolts: Float[Array, ''] | Float[ndarray, ''] | float) -> Float[Array, ''] ¤

Get the relativistic electron wavelength at a given accelerating voltage. For reference, see Equation 2.5 in Section 2.1 from Spence, John CH. High-resolution electron microscopy. OUP Oxford, 2013..

Arguments:

  • voltage_in_kilovolts: The accelerating voltage given in kilovolts.

Returns:

The relativistically corrected electron wavelength in Angstroms corresponding to the energy energy_in_keV.

cryojax.constants.lorentz_factor_from_kilovolts(voltage_in_kilovolts: Float[Array, ''] | Float[ndarray, ''] | float) -> Float[Array, ''] ¤

Get the Lorentz factor given an accelerating voltage.

Arguments:

  • voltage_in_kilovolts: The accelerating voltage given in kilovolts.

Returns:

The Lorentz factor.

cryojax.constants.interaction_constant_from_kilovolts(voltage_in_kilovolts: Float[Array, ''] | Float[ndarray, ''] | float) -> Float[Array, ''] ¤

Get the electron interaction constant given an accelerating voltage.

The interaction constant is necessary to compute the object phase shift distribution from an electrostatic potential integrated on the plane.

Info

In the projection approximation in cryo-EM, the phase shifts in the exit plane are given by

\[\eta(x, y) = \sigma_e \int dz \ V(x, y, z),\]

where \(\sigma_e\) is typically referred to as the interaction constant. However, in cryojax, the potential is rescaled to units of inverse length squared as

\[U(x, y, z) = \frac{2 m_0 e}{\hbar^2} V(x, y, z).\]

With this rescaling of the potential, the defined as with the equation

\[\eta(x, y) = \sigma_e \int dz \ U(x, y, z)\]

with

\[\sigma_e = \lambda \gamma,\]

where \(\lambda\) the relativistic electron wavelength \(\gamma\) is the lorentz factor.

References:

  • For the definition of the rescaled potential, see Chapter 69, Page 2003, Equation 69.6 from Hawkes, Peter W., and Erwin Kasper. Principles of Electron Optics, Volume 4: Advanced Wave Optics. Academic Press, 2022.
  • For the definition of the phase shifts in terms of the rescaled potential, see Chapter 69, Page 2012, Equation 69.34b from Hawkes, Peter W., and Erwin Kasper. Principles of Electron Optics, Volume 4: Advanced Wave Optics. Academic Press, 2022.

See the documentation on atom-based scattering potentials for more information.

Arguments:

  • voltage_in_kilovolts: The accelerating voltage given in kilovolts.

Returns:

The electron interaction constant.