abtem.parametrizations

abtem.parametrizations#

Module for describing analytical potential parametrizations.

Subpackages#

Package Contents#

Classes#

EwaldParametrization

KirklandParametrization

Potential parametrization by Earl J Kirkland.

LobatoParametrization

Potential parametrization by Ivan Lobato and Dirk Van Dyck.

Parametrization

Base class for potential parametrizations.

PengParametrization

Potential parametrization by Lian-Mao Peng.

Functions#

validate_parameters

validate_parametrization

validate_sigmas

API#

class abtem.parametrizations.EwaldParametrization(...)[source]#

Bases: abtem.parametrizations.Parametrization

scaled_parameters(...) → ndarray[source]#
property width#
class abtem.parametrizations.KirklandParametrization(...)[source]#

Bases: abtem.parametrizations.Parametrization

Potential parametrization by Earl J Kirkland.

Parameters:

  • parameters (str or dict) – A given string must be either the full path to a valid .json file or the name of a valid .json file in the abtem/parametrizations/data/ folder. A given dictionary must be a mapping from chemical symbols to anything that can converted to a NumPy array of shape (4, 3).

  • sigmas (dict or float) – The standard deviation of isotropic displacements for each element as a mapping from chemical symbols to a number. If given as a float the standard deviation of the displacements is assumed to be identical for all atoms.

References

E.J. Kirkland. Advanced computing in electron microscopy. Springer, 2. edition, 2010

Initialization

fit(...)[source]#
scaled_parameters(...) → ndarray[source]#
class abtem.parametrizations.LobatoParametrization(...)[source]#

Bases: abtem.parametrizations.Parametrization

Potential parametrization by Ivan Lobato and Dirk Van Dyck.

Parameters:

  • parameters (str or dict) – A given string must be either the full path to a valid .json file or the name of a valid .json file in the abtem/parametrizations/data/ folder. A given dictionary must be a mapping from chemical symbols to anything that can converted to a NumPy array of shape (2, 5).

  • sigmas (dict or float) – The standard deviation of isotropic displacements for each element as a mapping from chemical symbols to a number. If given as a float the standard deviation of the displacements is assumed to be identical for all atoms.

References

Ivan Lobato and Dirk Van Dyck. Acta Crystallographica Section A, 70:636-649, 2014.

Initialization

fit(...)[source]#

Fit the scattering factor functional form to data.

Parameters:

  • Z (int) – Atomic number the fit is stored under.

  • k (ndarray) – Sample points in reciprocal space [1/Å].

  • f (ndarray) – Scattering factor values at k.

  • guess (ndarray, optional) – Initial parameter guess. Defaults to this element’s own tabulated parameters if already present, else the reference lobato.json values.

  • regularization (float, optional) – Weight of an L2 penalty pulling the fit towards guess. The 10-parameter functional form has near-degenerate parameter directions – pairs of large, nearly-cancelling terms that barely affect the fitted curve in k-space but can shift the reconstructed real-space potential substantially near r = 0. An unregularized fit (the default, 0.0) is appropriate when f is essentially exact (e.g. refitting tabulated values through the same functional form); a small positive value (e.g. 0.05) stabilizes those directions when f carries its own numerical noise (e.g. from GPAWParametrization), at the cost of biasing the fit towards guess.

scaled_parameters(...) → ndarray[source]#
class abtem.parametrizations.Parametrization(...)[source]#

Bases: abtem.core.utils.EqualityMixin

Base class for potential parametrizations.

Parameters:

  • parameters (str or dict) – A given string must be either the full path to a valid .json file or the name of a valid .json file in the abtem/parametrizations/data/ folder. A given dictionary must be a mapping from chemical symbols to anything that can converted to a NumPy array with the correct shape for the given parametrization.

  • sigmas (dict or float) – The standard deviation of isotropic displacements for each element as a mapping from chemical symbols to a number. If given as a float the standard deviation of the displacements is assumed to be identical for all atoms.

Initialization

charge(...) → callable[source]#

Radial charge distribution for given chemical symbol and charge.

Parameters:

  • symbol (str) – Chemical symbol of element.

  • charge (float) – Charge of element. Given as elementary charges.

Returns:

charge – Function describing charge parameterized by the radial distance to the core [Å].

Return type:

callable

finite_projected_potential(...) → callable[source]#

X-ray scattering factor for given chemical symbol and charge.

Parameters:

  • symbol (str) – Chemical symbol of element.

  • charge (float) – Charge of element. Given as elementary charges.

Returns:

x_ray_scattering_factor

Return type:

callable

finite_projected_scattering_factor(...) → callable[source]#

Analytical infinite projection of radial scattering factor for given chemical symbol and charge.

Parameters:

  • symbol (str) – Chemical symbol of element.

  • charge (float) – Charge of element. Given as elementary charges.

Returns:

projected_scattering_factor – Function describing projected scattering parameterized by the squared reciprocal radial distance to the core [1/Å^2].

Return type:

callable

from_json(...)[source]#
get_function(...) → callable[source]#

Returns the line profiles for a parameterized function for one or more element.

Parameters:

  • name (str) – Name of the function to return.

  • symbol (str) – Chemical symbol of element.

  • charge (float) – Charge of element. Given as elementary charges.

line_profiles(...) → RealSpaceLineProfiles | ReciprocalSpaceLineProfiles[source]#

Returns the line profiles for a parameterized function for one or more element.

Parameters:

  • symbol (str or list of str) – Chemical symbol(s) of atom(s).

  • cutoff (float) – The outer radial cutoff distance of the line profiles in units of Ångstrom for potentials and units of reciprocal Ångstrom for scattering factors.

  • sampling (float, optional) – The radial sampling of the line profiles in units of Ångstrom for potentials and units of reciprocal Ångstrom for scattering factors. Default is 0.001.

  • name (str) – Name of the line profile to return.

property parameters: dict[str, ndarray]#
potential(...) → callable[source]#

Radial electrostatic potential for given chemical symbol and charge.

Parameters:

  • symbol (str) – Chemical symbol of element.

  • charge (float) – Charge of element. Given as elementary charges.

Returns:

electrostatic_potential – Function describing electrostatic potential parameterized by the radial distance to the core [Å].

Return type:

callable

projected_potential(...) → Callable[source]#

Analytical infinite projection of radial electrostatic potential for given chemical symbol and charge.

Parameters:

  • symbol (str) – Chemical symbol of element.

  • charge (float) – Charge of element. Given as elementary charges.

Returns:

projected_potential – Function describing projected electrostatic potential parameterized by the radial distance to the core [Å].

Return type:

callable

projected_scattering_factor(...) → callable[source]#

Analytical infinite projection of radial scattering factor for given chemical symbol and charge.

Parameters:

  • symbol (str) – Chemical symbol of element.

  • charge (float) – Charge of element. Given as elementary charges.

Returns:

projected_scattering_factor – Function describing projected scattering parameterized by the squared reciprocal radial distance to the core [1/Å^2].

Return type:

callable

abstract scaled_parameters(...) → ndarray[source]#

The parameters of the parametrization scaled to the abTEM units for a given function.

Parameters:

  • symbol (str) – Chemical symbol to get the scaled parameters for.

  • name (str) – Name of the function to get the scaled parameters for.

Returns:

scaled_parameters

Return type:

ndarray

scattering_factor(...) → Callable[source]#

Radial scattering factor for given chemical symbol and charge.

Parameters:

  • symbol (str) – Chemical symbol of element.

  • charge (float) – Charge of element. Given as elementary charges.

Returns:

scattering_factor – Function describing scattering parameterized by the squared reciprocal radial distance to the core [1/Å^2].

Return type:

callable

property sigmas#

The standard deviation of isotropic displacements.

to_json(...)[source]#
x_ray_scattering_factor(...) → callable[source]#

X-ray scattering factor for given chemical symbol and charge.

Parameters:

  • symbol (str) – Chemical symbol of element.

  • charge (float) – Charge of element. Given as elementary charges.

Returns:

x_ray_scattering_factor

Return type:

callable

class abtem.parametrizations.PengParametrization(...)[source]#

Bases: abtem.parametrizations.Parametrization

Potential parametrization by Lian-Mao Peng.

Parameters:

  • parameters (str or dict) – A given string must be either the full path to a valid .json file or the name of a valid .json file in the abtem/parametrizations/data/ folder. A given dictionary must be a mapping from chemical symbols to anything that can converted to a NumPy array of shape (2, 5).

  • sigmas (dict or float) – The standard deviation of isotropic displacements for each element as a mapping from chemical symbols to a number. If given as a float the standard deviation of the displacements is assumed to be identical for all atoms.

References

Peng, L. Micron, 30(6):625–648, 1999.

Initialization

scaled_parameters(...) → ndarray[source]#
abtem.parametrizations.validate_parameters(...) → dict[source]#
abtem.parametrizations.validate_parametrization(...) → Parametrization[source]#
abtem.parametrizations.validate_sigmas(...) → dict[source]#