2.6.1.1. embeam.stem.probe.ModelParams

class ModelParams(lens_model_params=None, gun_model_params=None, convergence_semiangle=15, defocal_offset_supersampling=9, skip_validation_and_conversion=False)

Bases: PreSerializableAndUpdatable

The model parameters of a probe.

Note that embeam currently does not support the modelling of finite source effects that lead to spatially incoherent probes.

As discussed in greater detail in e.g. Ref. [Kirkland1], the \(k\)-space wavefunction of a coherent probe is well-described by the following model:

(2.6.1.1.1)\[\Phi_{\text{probe}}\left(k_{x},k_{y};\delta_{f}\right)= \frac{1}{\sqrt{\pi k_{xy,\max}^{2}}} \Theta\left(k_{xy,\max}-k_{xy}\right) e^{-i\chi\left(k_{x},k_{y};\delta_{f}\right)},\]

where \(k_{x}\) and \(k_{y}\) are the Fourier coordinates;

(2.6.1.1.2)\[k_{xy}=\sqrt{k_{x}^{2}+k_{y}^{2}},\]

(2.6.1.1.3)\[k_{xy,\max}=\frac{\alpha_{\max}}{\lambda},\]

with \(\alpha_{\max}\) being the convergence semiangle, and \(\lambda\) being the electron beam wavelength [given by Eq. (2.1.1)]; \(\delta_f\) is the defocal offset; \(\Theta\left(u\right)\) is the Heaviside step function; and \(\chi\left(k_{x},k_{y};\delta_f\right)\) is given by Eq. (2.2.1.1). The corresponding real-space wavefunction of the coherent probe can be expressed as

(2.6.1.1.4)\[\psi_{\text{probe}}\left(x,y;\delta_{f}\right)= \int_{-\infty}^{\infty}dk_{x}\,\int_{-\infty}^{\infty}dk_{y}\, \Phi_{\text{probe}}\left(k_{x},k_{y};\delta_{f}\right) e^{2\pi ik_{x}x+2\pi ik_{y}y},\]

where \(x\) and \(y\) are the real-space coordinates. The \(k\)-space fractional intensity of the coherent probe is simply:

(2.6.1.1.5)\[\begin{split}p_{\text{probe}}\left(\left.K_{x}=k_{x}\right|K_{y}=k_{y}; \delta_{f}\right)&=\left|\Phi_{\text{probe}}\left(k_{x},k_{y}; \delta_{f}\right)\right|^{2}\\&=\frac{1}{\pi k_{xy,\max}^{2}} \Theta\left(k_{xy,\max}-k_{xy}\right)\\ &=p_{\text{probe}}\left(\left.K_{x}=k_{x}\right|K_{y}=k_{y}; \delta_{f}=0\right),\end{split}\]

and the corresponding real-space fractional intensity of the coherent probe is:

(2.6.1.1.6)\[p_{\text{probe}}\left(\left.X=x\right|Y=y;\delta_{f}\right)= \left|\psi_{\text{probe}}\left(x,y;\delta_{f}\right)\right|^{2}.\]

In scenarios where the probe forming lens is subject to chromatic aberrations, and there are small fluctuations in the electron beam energy over time, there will be correspondingly small fluctuations in the defocus of the electron beam over time. In such cases, the \(k\)-space fractional intensity of the incoherent probe is well-described by the following model:

(2.6.1.1.7)\[\begin{split}p_{\text{probe}}\left(\left.K_{x}=k_{x}\right|K_{y}=k_{y}\right) &=\int_{-\infty}^{\infty}d\delta_{f}\, p_{\sigma_{f}}\left(\delta_{f}\right) p_{\text{probe}}\left(\left.K_{x}=k_{x}\right|K_{y}=k_{y}; \delta_{f}\right)\\ &=p_{\text{probe}}\left(\left.K_{x}=k_{x}\right|K_{y}=k_{y}; \delta_{f}=0\right)\int_{-\infty}^{\infty}d\delta_{f}\, p_{\sigma_{f}}\left(\delta_{f}\right)\\ &=p_{\text{probe}}\left(\left.K_{x}=k_{x}\right|K_{y}=k_{y}; \delta_{f}=0\right),\end{split}\]

where \(p_{\sigma_{f}}\left(\delta_{f}\right)\) is the distribution of \(\delta_{f}\):

(2.6.1.1.8)\[p_{\sigma_{f}}\left(\delta_{f}\right)=\frac{1}{\sigma_{f}\sqrt{2\pi}} \exp\left(-\frac{1}{2}\frac{\delta_{f}^{2}}{\sigma_{f}^{2}}\right),\]

with \(\sigma_{f}\) being the defocal spread:

(2.6.1.1.9)\[\sigma_{f}=C_{c}\sqrt{\left(\frac{\sigma_{E}/e}{V}\right)^{2} +\left(2\frac{\sigma_{I}}{I}\right)^{2} +\left(\frac{\sigma_{V}}{V}\right)^{2}},\]

\(C_c\) being the chromatic aberration coefficient, \(I\) being the mean current of the probe forming lens, \(\sigma_I\) being the standard deviation of the current of the probe forming lens, \(V\) being the mean accelerating voltage, \(e\) being the elementary charge, \(\sigma_V\) being the standard deviation of the accelerating voltage, and \(\sigma_E\) being the standard deviation of the electrons in the gun when operating a voltage supply that does not fluctuate [i.e. \(\sigma_E\) is the intrinsic energy spread of the gun]. The corresponding real-space fractional intensity of the incoherent probe is:

(2.6.1.1.10)\[p_{\text{probe}}\left(\left.X=x\right|Y=y\right)= \int_{-\infty}^{\infty}d\delta_{f}\, p_{\sigma_{f}}\left(\delta_{f}\right) p_{\text{probe}}\left(\left.X=x\right|Y=y;\delta_{f}\right).\]

In embeam, the integral in Eq. (2.6.1.1.10) is numerically evaluated using Gauss-Hermite quadrature:

(2.6.1.1.11)\[p_{\text{probe}}\left(\left.X=x\right|Y=y\right)\approx \frac{1}{\sqrt{\pi}}\sum_{l=1}^{N_{f}}w_{f;l} p_{\text{probe}}\left(\left.X=x\right|Y=y; \delta_{f}=\sqrt{2}\sigma_{f}x_{f;l}\right),\]

where \(x_{f;l}\) is the \(l^{\text{th}}\) root of the Physicists’ version of the \(N_{f}^{\text{th}}\) Hermite polynomial \(H_{N_{f}}\left(x\right)\); and \(w_{f;l}\) is the \(l^{\text{th}}\) quadrature weight:

(2.6.1.1.12)\[w_{f;l}=\sqrt{\pi} \frac{\tilde{w}_{f;l}}{\sum_{l=1}^{N_{f}}\tilde{w}_{f;l}},\]

with

(2.6.1.1.13)\[\tilde{w}_{f;l}=\frac{2^{N_{f}-1}N_{f}!\sqrt{\pi}}{N_{f}^{2} \left\{ H_{N_{f}-1}\left(x_{l}\right)\right\} ^{2}}.\]

Note that the right-hand-side (RHS) of Eq. (2.6.1.1.11) equals the RHS Eq. (2.6.1.1.10) in the limit that \(N_{f}\to\infty\).

Parameters:

lens_model_paramsembeam.lens.ModelParams | None, optional

The model parameters of the probe forming lens. If lens_model_params is set to None [i.e. the default value], then no chromatic or coherent lens aberrations are assumed to be present.

gun_model_paramsembeam.gun.ModelParams | None, optional

The electron gun model parameters. If gun_model_params is set to None [i.e. the default value], then the parameter will be reassigned to the value embeam.gun.ModelParams().

convergence_semianglefloat, optional

The convergence semiangle in mrads. Must be positive.

defocal_offset_supersamplingint, optional

The number of points \(N_f\) to use in the Gauss-Hermite quadrature scheme of Eq. (2.6.1.1.12). Must be positive.

skip_validation_and_conversionbool, optional

Let validation_and_conversion_funcs and core_attrs denote the attributes validation_and_conversion_funcs and core_attrs respectively, both of which being dict objects.

Let params_to_be_mapped_to_core_attrs denote the dict representation of the constructor parameters excluding the parameter skip_validation_and_conversion, where each dict key key is a different constructor parameter name, excluding the name "skip_validation_and_conversion", and params_to_be_mapped_to_core_attrs[key] would yield the value of the constructor parameter with the name given by key.

If skip_validation_and_conversion is set to False, then for each key key in params_to_be_mapped_to_core_attrs, core_attrs[key] is set to validation_and_conversion_funcs[key] (params_to_be_mapped_to_core_attrs).

Otherwise, if skip_validation_and_conversion is set to True, then core_attrs is set to params_to_be_mapped_to_core_attrs.copy(). This option is desired primarily when the user wants to avoid potentially expensive deep copies and/or conversions of the dict values of params_to_be_mapped_to_core_attrs, as it is guaranteed that no copies or conversions are made in this case.

Attributes:

core_attrs

dict: The “core attributes”.

de_pre_serialization_funcs

dict: The de-pre-serialization functions.

is_azimuthally_symmetric

bool: A boolean variable indicating whether the probe model is

is_coherent

bool: A boolean variable indicating whether the probe model is

pre_serialization_funcs

dict: The pre-serialization functions.

validation_and_conversion_funcs

dict: The validation and conversion functions.

Methods

de_pre_serialize([serializable_rep, ...])	Construct an instance from a serializable representation.
dump([filename, overwrite])	Serialize instance and save the result in a JSON file.
dumps()	Serialize instance.
get_core_attrs([deep_copy])	Return the core attributes.
get_de_pre_serialization_funcs()	Return the de-pre-serialization functions.
get_pre_serialization_funcs()	Return the pre-serialization functions.
get_validation_and_conversion_funcs()	Return the validation and conversion functions.
load([filename, skip_validation_and_conversion])	Construct an instance from a serialized representation that is stored in a JSON file.
loads([serialized_rep, ...])	Construct an instance from a serialized representation.
pre_serialize()	Pre-serialize instance.
update(new_core_attr_subset_candidate[, ...])	Update a subset of the core attributes.

execute_post_core_attrs_update_actions

Methods

de_pre_serialize	Construct an instance from a serializable representation.
dump	Serialize instance and save the result in a JSON file.
dumps	Serialize instance.
execute_post_core_attrs_update_actions
get_core_attrs	Return the core attributes.
get_de_pre_serialization_funcs	Return the de-pre-serialization functions.
get_pre_serialization_funcs	Return the pre-serialization functions.
get_validation_and_conversion_funcs	Return the validation and conversion functions.
load	Construct an instance from a serialized representation that is stored in a JSON file.
loads	Construct an instance from a serialized representation.
pre_serialize	Pre-serialize instance.
update	Update a subset of the core attributes.

Attributes

core_attrs	dict: The "core attributes".
de_pre_serialization_funcs	dict: The de-pre-serialization functions.
is_azimuthally_symmetric	bool: A boolean variable indicating whether the probe model is azimuthally symmetric.
is_coherent	bool: A boolean variable indicating whether the probe model is coherent.
pre_serialization_funcs	dict: The pre-serialization functions.
validation_and_conversion_funcs	dict: The validation and conversion functions.

property core_attrs

dict: The “core attributes”.

The keys of core_attrs are the same as the attribute validation_and_conversion_funcs, which is also a dict object.

Note that core_attrs should be considered read-only.

property de_pre_serialization_funcs

dict: The de-pre-serialization functions.

de_pre_serialization_funcs has the same keys as the attribute validation_and_conversion_funcs, which is also a dict object.

Let validation_and_conversion_funcs and pre_serialization_funcs denote the attributes validation_and_conversion_funcs pre_serialization_funcs respectively, the last of which being a dict object as well.

Let core_attrs_candidate_1 be any dict object that has the same keys as validation_and_conversion_funcs, where for each dict key key in core_attrs_candidate_1, validation_and_conversion_funcs[key](core_attrs_candidate_1) does not raise an exception.

Let serializable_rep be a dict object that has the same keys as core_attrs_candidate_1, where for each dict key key in core_attrs_candidate_1, serializable_rep[key] is set to pre_serialization_funcs[key](core_attrs_candidate_1[key]).

The items of de_pre_serialization_funcs are expected to be set to callable objects that would lead to de_pre_serialization_funcs[key](serializable_rep[key]) not raising an exception for each dict key key in serializable_rep.

Let core_attrs_candidate_2 be a dict object that has the same keys as serializable_rep, where for each dict key key in validation_and_conversion_funcs, core_attrs_candidate_2[key] is set to de_pre_serialization_funcs[key](serializable_rep[key]).

The items of de_pre_serialization_funcs are also expected to be set to callable objects that would lead to validation_and_conversion_funcs[key](core_attrs_candidate_2) not raising an exception for each dict key key in core_attrs_candidate_2.

Note that de_pre_serialization_funcs should be considered read-only.

classmethod de_pre_serialize(serializable_rep={}, skip_validation_and_conversion=False)

Construct an instance from a serializable representation.

Parameters:

serializable_repdict, optional

A dict object that has the same keys as the attribute validation_and_conversion_funcs, which is also a dict object.

Let validation_and_conversion_funcs and de_pre_serialization_funcs denote the attributes validation_and_conversion_funcs de_pre_serialization_funcs respectively, the last of which being a dict object as well.

The items of serializable_rep are expected to be objects that would lead to de_pre_serialization_funcs[key](serializable_rep[key]) not raising an exception for each dict key key in serializable_rep.

Let core_attrs_candidate be a dict object that has the same keys as serializable_rep, where for each dict key key in serializable_rep, core_attrs_candidate[key] is set to de_pre_serialization_funcs[key](serializable_rep[key])``.

The items of serializable_rep are also expected to be set to objects that would lead to validation_and_conversion_funcs[key](core_attrs_candidate) not raising an exception for each dict key key in serializable_rep.

skip_validation_and_conversionbool, optional

Let core_attrs denote the attribute core_attrs, which is a dict object.

If skip_validation_and_conversion is set to False, then for each key key in serializable_rep, core_attrs[key] is set to validation_and_conversion_funcs[key] (core_attrs_candidate), with validation_and_conversion_funcs and core_attrs_candidate_1 being introduced in the above description of serializable_rep.

Otherwise, if skip_validation_and_conversion is set to True, then core_attrs is set to core_attrs_candidate.copy(). This option is desired primarily when the user wants to avoid potentially expensive deep copies and/or conversions of the dict values of core_attrs_candidate, as it is guaranteed that no copies or conversions are made in this case.

Returns:

instance_of_current_clsCurrent class

An instance constructed from the serializable representation serializable_rep.

dump(filename='serialized_rep_of_fancytype.json', overwrite=False)

Serialize instance and save the result in a JSON file.

Parameters:

filenamestr, optional

The relative or absolute path to the JSON file in which to store the serialized representation of an instance.

overwritebool, optional

If overwrite is set to False and a file exists at the path filename, then the serialized instance is not written to that file and an exception is raised. Otherwise, the serialized instance will be written to that file barring no other issues occur.

Returns:

dumps()

Serialize instance.

Returns:

serialized_repdict

A serialized representation of an instance.

get_core_attrs(deep_copy=True)

Return the core attributes.

Parameters:

deep_copybool, optional

Let core_attrs denote the attribute core_attrs, which is a dict object.

If deep_copy is set to True, then a deep copy of core_attrs is returned. Otherwise, a shallow copy of core_attrs is returned.

Returns:

core_attrsdict

The attribute core_attrs.

classmethod get_de_pre_serialization_funcs()

Return the de-pre-serialization functions.

Returns:

de_pre_serialization_funcsdict

The attribute de_pre_serialization_funcs.

classmethod get_pre_serialization_funcs()

Return the pre-serialization functions.

Returns:

pre_serialization_funcsdict

The attribute pre_serialization_funcs.

classmethod get_validation_and_conversion_funcs()

Return the validation and conversion functions.

Returns:

validation_and_conversion_funcsdict

The attribute validation_and_conversion_funcs.

property is_azimuthally_symmetric

bool: A boolean variable indicating whether the probe model is azimuthally symmetric.

See the summary documentation of the classes embeam.coherent.PhaseDeviation and embeam.coherent.Aberration for additional context.

A probe model is azimuthally symmetric if the coherent aberrations of the probe-forming lens are azimuthally symmetric, i.e. if if \(\sum_{m=0}^{\infty}\sum_{n=0}^{\infty} n \left(C_{m,n}^{\text{mag}} C_{m,n}^{\text{ang}}\right)^2=0\).

Note that is_azimuthally_symmetric should be considered read-only.

property is_coherent

bool: A boolean variable indicating whether the probe model is coherent.

See the summary documentation of the class embeam.stem.probe.ModelParams for additional context.

If is_coherent is set to True, then the probe model is coherent. Otherwise, the probe model is not coherent.

Note that is_coherent should be considered read-only.

classmethod load(filename='serialized_rep_of_fancytype.json', skip_validation_and_conversion=False)

Construct an instance from a serialized representation that is stored in a JSON file.

Users can save serialized representations to JSON files using the method fancytypes.PreSerializable.dump().

Parameters:

filenamestr, optional

The relative or absolute path to the JSON file that is storing the serialized representation of an instance.

filename is expected to be such that json.load(open(filename, "r")) does not raise an exception.

Let serializable_rep=json.load(open(filename, "r")).

Let validation_and_conversion_funcs and de_pre_serialization_funcs denote the attributes validation_and_conversion_funcs de_pre_serialization_funcs respectively, both of which being dict objects as well.

filename is also expected to be such that de_pre_serialization_funcs[key](serializable_rep[key]) does not raise an exception for each dict key key in de_pre_serialization_funcs.

Let core_attrs_candidate be a dict object that has the same keys as de_pre_serialization_funcs, where for each dict key key in serializable_rep, core_attrs_candidate[key] is set to de_pre_serialization_funcs[key](serializable_rep[key])``.

filename is also expected to be such that validation_and_conversion_funcs[key](core_attrs_candidate) does not raise an exception for each dict key key in serializable_rep.

skip_validation_and_conversionbool, optional

Let core_attrs denote the attribute core_attrs, which is a dict object.

Let core_attrs_candidate be as defined in the above description of filename.

If skip_validation_and_conversion is set to False, then for each key key in core_attrs_candidate, core_attrs[key] is set to validation_and_conversion_funcs[key] (core_attrs_candidate), , with validation_and_conversion_funcs and core_attrs_candidate being introduced in the above description of filename.

Otherwise, if skip_validation_and_conversion is set to True, then core_attrs is set to core_attrs_candidate.copy(). This option is desired primarily when the user wants to avoid potentially expensive deep copies and/or conversions of the dict values of core_attrs_candidate, as it is guaranteed that no copies or conversions are made in this case.

Returns:

instance_of_current_clsCurrent class

An instance constructed from the serialized representation stored in the JSON file.

classmethod loads(serialized_rep='{}', skip_validation_and_conversion=False)

Construct an instance from a serialized representation.

Users can generate serialized representations using the method dumps().

Parameters:

serialized_repstr | bytes | bytearray, optional

The serialized representation.

serialized_rep is expected to be such that json.loads(serialized_rep) does not raise an exception.

Let serializable_rep=json.loads(serialized_rep).

Let validation_and_conversion_funcs and de_pre_serialization_funcs denote the attributes validation_and_conversion_funcs de_pre_serialization_funcs respectively, both of which being dict objects as well.

serialized_rep is also expected to be such that de_pre_serialization_funcs[key](serializable_rep[key]) does not raise an exception for each dict key key in de_pre_serialization_funcs.

Let core_attrs_candidate be a dict object that has the same keys as serializable_rep, where for each dict key key in de_pre_serialization_funcs, core_attrs_candidate[key] is set to de_pre_serialization_funcs[key](serializable_rep[key])``.

serialized_rep is also expected to be such that validation_and_conversion_funcs[key](core_attrs_candidate) does not raise an exception for each dict key key in serializable_rep.

skip_validation_and_conversionbool, optional

Let core_attrs denote the attribute core_attrs, which is a dict object.

If skip_validation_and_conversion is set to False, then for each key key in core_attrs_candidate, core_attrs[key] is set to validation_and_conversion_funcs[key] (core_attrs_candidate), with validation_and_conversion_funcs and core_attrs_candidate_1 being introduced in the above description of serialized_rep.

Otherwise, if skip_validation_and_conversion is set to True, then core_attrs is set to core_attrs_candidate.copy(). This option is desired primarily when the user wants to avoid potentially expensive deep copies and/or conversions of the dict values of core_attrs_candidate, as it is guaranteed that no copies or conversions are made in this case.

Returns:

instance_of_current_clsCurrent class

An instance constructed from the serialized representation.

property pre_serialization_funcs

dict: The pre-serialization functions.

pre_serialization_funcs has the same keys as the attribute validation_and_conversion_funcs, which is also a dict object.

Let validation_and_conversion_funcs and core_attrs denote the attributes validation_and_conversion_funcs and core_attrs respectively, the last of which being a dict object as well.

For each dict key key in core_attrs, pre_serialization_funcs[key](core_attrs[key]) is expected to yield a serializable object, i.e. it should yield an object that can be passed into the function json.dumps without raising an exception.

Note that pre_serialization_funcs should be considered read-only.

pre_serialize()

Pre-serialize instance.

Returns:

serializable_repdict

A serializable representation of an instance.

update(new_core_attr_subset_candidate, skip_validation_and_conversion=False)

Update a subset of the core attributes.

Parameters:

new_core_attr_subset_candidatedict, optional

A dict object.

skip_validation_and_conversionbool, optional

Let validation_and_conversion_funcs and core_attrs denote the attributes validation_and_conversion_funcs and core_attrs respectively, both of which being dict objects.

If skip_validation_and_conversion is set to False, then for each key key in core_attrs that is also in new_core_attr_subset_candidate, core_attrs[key] is set to validation_and_conversion_funcs[key] (new_core_attr_subset_candidate).

Otherwise, if skip_validation_and_conversion is set to True, then for each key key in core_attrs that is also in new_core_attr_subset_candidate, core_attrs[key] is set to new_core_attr_subset_candidate[key]. This option is desired primarily when the user wants to avoid potentially expensive deep copies and/or conversions of the dict values of new_core_attr_subset_candidate, as it is guaranteed that no copies or conversions are made in this case.

property validation_and_conversion_funcs

dict: The validation and conversion functions.

The keys of validation_and_conversion_funcs are the names of the constructor parameters, excluding skip_validation_and_conversion if it exists as a construction parameter.

Let core_attrs denote the attribute core_attrs, which is also a dict object.

For each dict key key in core_attrs, validation_and_conversion_funcs[key](core_attrs) is expected to not raise an exception.

Note that validation_and_conversion_funcs should be considered read-only.