PDielec.Calculator

Calculator module.

Module Contents

Functions

_brug_iter_error(epsbr, eps1, eps2, shape, L, f1, size)	Routine to calculate the error in the Bruggeman method.
_brug_minimise_scalar(variables, eps1, eps2, shape, L, ...)	Bruggeman method using scalar quantities (suitable for powell minimisation method).
_brug_minimise_tensor(variables, eps1, eps2, shape, L, ...)	Bruggeman method using tensor quantities.
absorption_from_mode_intensities(f, modes, ...)	Calculate the absorption from the frequencies and intensities using a Lorentzian.
anisotropic_mie_scattering(dielectric_medium, ...)	Calculate the effective constant permittivity using a Mie scattering approach.
average_tensor(t)	Return the averaged tensor.
averaged_permittivity(dielectric_medium, ...)	Calculate the effective constant permittivity using the averaged permittivity method.
balan(dielectric_medium, crystal_permittivity, shape, ...)	Calculate the effective constant permittivity using the method of Balan.
bruggeman_iter(eps1, eps2, shape, L, f2, size, epsbr)	Calculate an iteration of the Bruggeman method..
bruggeman_minimise(eps1, eps2, shape, L, f2, size, epsbr)	Calculate the effective constant permittivity using the method of Bruggeman (minimisation).
calculate_angle(a, b, c)	Calculate the angle between a, b and c in degrees.
calculate_bubble_refractive_index(v_cm1, ri_medium, ...)	Calculate the scattering from bubbles embedded in a possibly complex dielectric at a given frequency.
calculate_centre_of_mass(xyzs, masses)	Calculate centre of mass.
calculate_distance(a, b)	Calculate the distance between a and b.
calculate_energy_distribution(cell, frequencies, ...)	Calculate energy distribution in the phonon modes.
calculate_permittivity(refractive_index[, debug])	Calculate the permittivity from the refractive index.
calculate_refractive_index(dielectric[, debug])	Calculate the refractive index from the dielectric constant.
calculate_refractive_index_scalar(dielectric_scalar[, ...])	Calculate the refractive index from the dielectric constant.
calculate_size_factor(x)	Calculate a size effect using Equations 10.38 and 10.39 in Sihvola.
calculate_torsion(a, b, c, d)	Calculate the torsion angle between a, b, c and d in degrees.
cleanup_symbol(s)	Return a true element from the symbol.
coherent(dielectric_medium, crystal_permittivity, ...)	Calculate the effective constant permittivity using the Coherent method.
construct_projection_operator(atoms, xyzs, masses, nats)	Construct the projection operator for the molecule defined by atoms, xyz, masses.
determineEulerAngles(R)	Determine the euler angles of a rotation matrix.
direction_from_shape(data, reader)	Determine the unique direction of the shape from the shape data.
euler_rotation(vector, theta, phi, psi)	Apply a passive Euler rotation to a vector.
get_pool(ncpus, threading[, initializer, initargs, ...])	Return a pool of processors given the number of cpus and whether threading is requested.
hodrick_prescott_filter(y, damping, lambda_value, niters)	Apply a Hodrick Prescott filter to the spectrum in x, y.
infrared_intensities(oscillator_strengths)	Calculate the IR intensities from the trace of the oscillator strengths.
initialise_complex_diagonal_tensor(reals)	Initialise a complex 3x3 tensor with the given diagonal components.
initialise_diagonal_tensor(reals)	Initialise a real 3x3 tensor with the given diagonal components.
initialise_ellipsoid_depolarisation_matrix(unique, aoverb)	Initialise a 3x3 tensor with the ellipsoid depolarisation matrix.
initialise_needle_depolarisation_matrix(unique)	Initialise a 3x3 tensor with the needle depolarisation matrix.
initialise_plate_depolarisation_matrix(normal)	Initialise a 3x3 tensor with the plate depolarisation matrix, returns a tensor.
initialise_sphere_depolarisation_matrix()	Initialise a 3x3 tensor with the sphere depolarisation matrix, returns a tensor.
initialise_unit_tensor()	Initialise a 3x3 tensor to a unit tensor.
ionic_permittivity(mode_list, oscillator_strengths, ...)	Calculate the low frequency permittivity or zero frequency permittivity.
longitudinal_modes(frequencies, normal_modes, ...)	Apply the nonanalytic correction to the dynamical matrix and calculate the LO frequencies.
maxwell(dielectric_medium, crystal_permittivity, ...)	Calculate the effective constant permittivity using the Maxwell Garnett method.
maxwell_sihvola(dielectric_medium, ...)	Calculate the effective constant permittivity using the Maxwell Garnett method.
mie_scattering(dielectric_medium, ...)	Calculate the effective constant permittivity using a Mie scattering approach.
normal_modes(masses, mass_weighted_normal_modes)	Transform from mass weighted coordinates to xyz.
orthogonalise_projection_operator(ps)	Orthogonalise the list of projection operators ps.
oscillator_strengths(normal_modes, born_charges)	Calculate oscillator strengths from the normal modes and the born charges.
reflectance_atr(ns, n0, theta, atrSPolFraction)	Calculate the atr s and p reflectance.
rodridgues_rotations(efield)	Take the field directions in efield and use each direction to calculate a random rotation about that axis.
set_affinity_on_worker()	When a new worker process is created, the affinity is set to all CPUs.
set_no_of_threads(nthreads)	Set default number of threads.
solve_effective_medium_equations(method, vf, size_mu, ...)	Solve the effective medium equations.
waterman_truell_scattering(lambda_vacuum_nm, N_nm, ...)	Calculate the effective wavenumber based on Waterman-Truell scattering model.

PDielec.Calculator._brug_iter_error(epsbr, eps1, eps2, shape, L, f1, size)

Routine to calculate the error in the Bruggeman method.

Parameters

epsbrlist of 2 floats

The real and imaginary components of the scalar permittivity

eps1tensor (3x3)

Permittivity of phase 1

eps2tensor (3x3)

Permittivity of phase 2

shapestring

The shape descriptor

L3x3 tensor

Depolarisation tensor

f1float

The volume fraction of phase 1

sizefloat

The size of the particle

Returns

float : The error associated with the current values of the permittivities

PDielec.Calculator._brug_minimise_scalar(variables, eps1, eps2, shape, L, f1, size)

Bruggeman method using scalar quantities (suitable for powell minimisation method).

Parameters

variableslist of 2 floats

The real and imaginary components of the scalar permittivity

eps1tensor (3x3)

Permittivity of phase 1

eps2tensor (3x3)

Permittivity of phase 2

shapestring

The particle shape

L3x3 tensor

Depolarisation tensor

f1float

The volume fraction of phase 1

sizefloat

The size of the particle

Returns

float : The error associated with the current values of the permittivities

PDielec.Calculator._brug_minimise_tensor(variables, eps1, eps2, shape, L, f1, size)

Bruggeman method using tensor quantities.

Parameters

variableslist of 2 floats

The real and imaginary components of the scalar permittivity

eps1tensor (3x3)

Permittivity of phase 1

eps2tensor (3x3)

Permittivity of phase 2

shapestring

The particle shape

L3x3 tensor

Depolarisation tensor

f1float

The volume fraction of phase 1

sizefloat

The size of the particle

Returns

float : The error associated with the current values of the permittivities

PDielec.Calculator.absorption_from_mode_intensities(f, modes, frequencies, sigmas, intensities)

Calculate the absorption from the frequencies and intensities using a Lorentzian.

Parameters

ffloat

The frequency of the absorption in cm-1.

modeslist of ints

A list of the modes

frequencieslist of reals

Mode frequencies (cm-1)

sigmaslist of reals

Mode widths in cm-1

intensitieslist of reals

Mode intensities (D2/A2/amu).

Returns

float

The molar absorption coefficient at f, in L/mol/cm.

Notes

The number 4225.6 converts the units of D2/A2/amu to L mole-1 cm-1 cm-1.

PDielec.Calculator.anisotropic_mie_scattering(dielectric_medium, crystal_permittivity, shape, L, vf, size, size_mu, size_distribution_sigma)

Calculate the effective constant permittivity using a Mie scattering approach.

Parameters

dielectric_mediumarray_like (3x3)

The dielectric constant tensor of the medium.

crystal_permittivityarray_like (3x3)

The total frequency dielectric constant tensor at the current frequency.

shapestr

The name of the current shape (NOT USED).

Larray_like

The shape’s depolarisation matrix (NOT USED).

sizefloat

The dimensionless size parameter for the frequency under consideration.

size_mufloat

The particle size in microns

size_distribution_sigmafloat

The log normal value of sigma.

vffloat

The volume fraction of filler.

Notes

Mie only works for spherical particles, so the shape, and L parameters are ignored.

Returns

effective_dielectric_constantfloat

The effective dielectric constant.

PDielec.Calculator.average_tensor(t)

Return the averaged tensor.

Parameters

ttensor (3x3)

The tensor

Returns

outputtensor

The averaged tensor.

PDielec.Calculator.averaged_permittivity(dielectric_medium, crystal_permittivity, shape, L, vf, size)

Calculate the effective constant permittivity using the averaged permittivity method.

Parameters

dielectric_mediumarray_like (3x3)

The dielectric constant tensor of the medium.

crystal_permittivityarray_like (3x3)

The total frequency dielectric constant tensor at the current frequency.

shapestr

The name of the current shape.

Larray_like (3x3)

The shape’s depolarisation matrix.

vffloat

Volume fraction

sizefloat

The dimensionless size parameter for the frequency under consideration (not used).

Returns

array_like

The effective dielectric constant.

PDielec.Calculator.balan(dielectric_medium, crystal_permittivity, shape, L, vf, size)

Calculate the effective constant permittivity using the method of Balan.

Parameters

dielectric_mediumarray_like (3x3)

The dielectric constant tensor of the medium.

crystal_permittivityarray_like (3x3)

The total frequency dielectric constant tensor at the current frequency.

shapestr

The name of the current shape.

Larray_like (3x3)

The shape’s depolarisation matrix.

vffloat

Volume fraction

sizefloat

The dimensionless size parameter for the frequency under consideration (not used).

Returns

array_like

The effective dielectric constant.

PDielec.Calculator.bruggeman_iter(eps1, eps2, shape, L, f2, size, epsbr)

Calculate an iteration of the Bruggeman method..

Parameters

eps1array_like (3x3)

The dielectric constant tensor of medium 1.

eps2array_like (3x3)

The dielectric constant tensor of medium 2.

shapestr

The name of the current shape.

Larray_like (3x3)

The shape’s depolarisation matrix.

f2float

The volume fraction of component 2.

sizefloat

The dimensionless size parameter for the frequency under consideration.

epsbrfloat (3x3)

An initial guess at the solution.

Returns

tensor (3x3)

The effective dielectric constant.

Notes

This function applies homogenization formalisms to active dielectric composite materials as discussed in the work of Tom G. Mackay and Akhlesh Lakhtakia.

PDielec.Calculator.bruggeman_minimise(eps1, eps2, shape, L, f2, size, epsbr)

Calculate the effective constant permittivity using the method of Bruggeman (minimisation).

Parameters

eps1array_like (3x3)

The dielectric constant tensor of medium 1.

eps2array_like (3x3)

The dielectric constant tensor of medium 2.

shapestr

The name of the current shape.

Larray_like (3x3)

The shape’s depolarisation matrix.

f2float

The volume fraction of component 2.

sizefloat

The dimensionless size parameter for the frequency under consideration.

epsbrfloat (3x3)

An initial guess at the solution.

Returns

tensor (3x3)

The effective dielectric constant.

Notes

This function applies homogenization formalisms to active dielectric composite materials as discussed in the work of Tom G. Mackay and Akhlesh Lakhtakia.

PDielec.Calculator.calculate_angle(a, b, c)

Calculate the angle between a, b and c in degrees.

The bond is a-b-c, b is the central atom

Parameters

alist of reals

Coordinates of a

blist of reals

Coordinates of b

clist of reals

Coordinates of c

Returns

float

The angle between a-b-c in degrees

PDielec.Calculator.calculate_bubble_refractive_index(v_cm1, ri_medium, vf, radius_mu)

Calculate the scattering from bubbles embedded in a possibly complex dielectric at a given frequency.

Parameters

v_cm1float

The frequency in cm-1.

ri_mediumfloat

The refractive index of the medium.

vffloat

The volume fraction of bubbles.

radius_mufloat

The radius of the bubbles in microns.

Returns

effective_dielectric_constantfloat

The effective dielectric constant.

ri_mediumfloat

The refractive index of the medium

Notes

This function calculates the scattering from bubbles embedded in a dielectric medium, which can have a complex refractive index. It considers the frequency of interest, the refractive index of the medium, the volume fraction of bubbles, and the radius of the bubbles to calculate the effective dielectric constant and its associated refractive index.

PDielec.Calculator.calculate_centre_of_mass(xyzs, masses)

Calculate centre of mass.

Parameters

xyzslist of xyz coordinates of the atoms

The xyz coordinates

masseslist of the atomic masses

The list of atomic masses in amu

Returns

mass : float - The total mass cm : vector (3) - the coordinates of the centre of mass

PDielec.Calculator.calculate_distance(a, b)

Calculate the distance between a and b.

Parameters

alist of reals

Coordinates of a

blist of reals

Coordinates of b

Returns

float

The distance between a and b

PDielec.Calculator.calculate_energy_distribution(cell, frequencies, normal_modes, debug=False)

Calculate energy distribution in the phonon modes.

Parameters

cellunit cell object

The unit cell object

frequenciesarray_like

The frequencies in cm-1.

normal_modesarray_like

The mass weighted normal modes.

debugboolean

True for debugging

PDielec.Calculator.calculate_permittivity(refractive_index, debug=False)

Calculate the permittivity from the refractive index.

Parameters

refractive_indexcomplex

The refractive index from which the permittivity is calculated.

debugboolean

True for debugging information

Returns

complex

The calculated permittivity.

PDielec.Calculator.calculate_refractive_index(dielectric, debug=False)

Calculate the refractive index from the dielectric constant.

Calculate the trace of the dielectric and calculate both square roots. Then choose the root with the largest imaginary component. This obeys the Kramers-Konig requirements.

Parameters

dielectriccomplex

The permittivity

debugboolean

True for debugging information

Returns

complex

The refractive index calculated from the dielectric constant.

Notes

The calculation of the refractive index from the dielectric constant involves the trace of the dielectric tensor and the selection of the square root with the largest imaginary component.

PDielec.Calculator.calculate_refractive_index_scalar(dielectric_scalar, debug=False)

Calculate the refractive index from the dielectric constant.

Calculate the trace of the dielectric and calculate both square roots. Then choose the root with the largest imaginary component. This obeys the Konig-Kramer requirements.

Parameters

dielectric_scalarcomplex

The permittivity

debugboolean

True for debugging information

Returns

complex

The refractive index calculated from the dielectric constant.

Notes

The calculation of the refractive index from the dielectric constant involves the trace of the dielectric tensor and the selection of the square root with the largest imaginary component to satisfy the Konig-Kramer conditions.

PDielec.Calculator.calculate_size_factor(x)

Calculate a size effect using Equations 10.38 and 10.39 in Sihvola.

If x is small the result is close to 1

Parameters

xfloat

The size of the particle

Returns

float

Size effect

PDielec.Calculator.calculate_torsion(a, b, c, d)

Calculate the torsion angle between a, b, c and d in degrees.

Parameters

alist of reals

Coordinates of a

blist of reals

Coordinates of b

clist of reals

Coordinates of c

dlist of reals

Coordinates of d

Returns

float

The torsion angle between a-b-c-d in degrees

PDielec.Calculator.cleanup_symbol(s)

Return a true element from the symbol.

Parameters

sstr

The element symbol to be cleaned up

Returns

str

The cleaned symbol

PDielec.Calculator.coherent(dielectric_medium, crystal_permittivity, shape, L, vf, size, dielectric_apparent)

Calculate the effective constant permittivity using the Coherent method.

Parameters

dielectric_mediumtensor (3x3)

The dielectric constant tensor of the medium.

crystal_permittivitytensor (3x3)

The total frequency dielectric constant tensor at the current frequency.

shapestr

The name of the current shape.

Lmatrix (3x3)

The shape’s depolarisation matrix.

sizefloat

The dimensionless size parameter for the frequency under consideration.

vffloat

The volume fraction of filler.

dielectric_apparent3x3 array of floats

The current estimate of the dielectric

Returns

tensor

The effective dielectric constant.

PDielec.Calculator.construct_projection_operator(atoms, xyzs, masses, nats)

Construct the projection operator for the molecule defined by atoms, xyz, masses.

Parameters

atomslist strings

The atom types

xyzslist of atom coordinates vector (3)

The list of coordinates

masseslist of atom masses

The list of atomic masses in amu

natsint

The number of atoms

Returns

type

Description of the returned object.

PDielec.Calculator.determineEulerAngles(R)

Determine the euler angles of a rotation matrix.

Parameters

Rtensor (3,3)

The rotation matrix

Returns

theta, phi, psifloat

The Euler angles

PDielec.Calculator.direction_from_shape(data, reader)

Determine the unique direction of the shape from the shape data.

Parameters

datalist of strings

Data may contain a miller indices which defines a surface, e.g., (1,1,-1), or a direction as a miller direction vector, e.g., [1,0,-1].

readera reader objecy

The reader is used to get the unit-cell

Returns

outvector (3)

Description of the unique direction determined from the data.

PDielec.Calculator.euler_rotation(vector, theta, phi, psi)

Apply a passive Euler rotation to a vector.

Parameters

vectorvector (3)

The vector to be rotated

thetafloat

The angle theta

phifloat

The angle phi

psifloat

The angle psi

Returns

vector (3)

Notes

A passive Euler rotation refers to the rotation of the coordinate system while the vector remains fixed. This operation is often used in physics and engineering to describe the orientation of an object with respect to a reference coordinate system.

PDielec.Calculator.get_pool(ncpus, threading, initializer=None, initargs=None, debugger=None)

Return a pool of processors given the number of cpus and whether threading is requested.

Parameters

ncpusint

the number of processors

threadingbool

true if threading is to be used

initializerfunction

Function to be called before getting the pool

initargsfunction arguments

Any other parameters

debuggera debugger object

A debugger object

Returns

pool : the pool of processors

PDielec.Calculator.hodrick_prescott_filter(y, damping, lambda_value, niters)

Apply a Hodrick Prescott filter to the spectrum in x, y.

Parameters

yarray_like

The experimental absorption data.

dampingfloat

The damping factor used to damp the iterations.

lambda_valuefloat

The chosen smoothing factor.

nitersint

The number of iterations

Returns

list of floats : The new spectrum

Notes

Based on ideas in the thesis of Mayank Kaushik (University Adelaide).

PDielec.Calculator.infrared_intensities(oscillator_strengths)

Calculate the IR intensities from the trace of the oscillator strengths.

Returns

np.array

An array of the calculated IR intensities in units of (D/A)^2/amu.

PDielec.Calculator.initialise_complex_diagonal_tensor(reals)

Initialise a complex 3x3 tensor with the given diagonal components.

Parameters

realslist

A list of 3 real numbers for the diagonals.

Returns

array

The returned tensor is a complex 3x3 array.

PDielec.Calculator.initialise_diagonal_tensor(reals)

Initialise a real 3x3 tensor with the given diagonal components.

Parameters

realslist

A list of 3 real numbers for the diagonals.

Returns

array

The returned tensor is a real 3x3 array.

PDielec.Calculator.initialise_ellipsoid_depolarisation_matrix(unique, aoverb)

Initialise a 3x3 tensor with the ellipsoid depolarisation matrix.

Parameters

uniquelist of 3 floats

Unique direction for ellipsoid

aoverbfloat

The ratio of a / b (the ratio of the principle axis lengths of the ellipsoid)

Returns

np.array

The Ellipsoid dpolarisation tensor

PDielec.Calculator.initialise_needle_depolarisation_matrix(unique)

Initialise a 3x3 tensor with the needle depolarisation matrix.

Parameters

unique: np.array

The unique direction of the needle

Returns

ndarray

A 3x3 tensor initialized with the plate depolarisation matrix.

PDielec.Calculator.initialise_plate_depolarisation_matrix(normal)

Initialise a 3x3 tensor with the plate depolarisation matrix, returns a tensor.

Parameters

normalnp.array

A 3 vector giving the normal direction of the plate

Returns

ndarray

A 3x3 tensor initialized with the plate depolarisation matrix.

PDielec.Calculator.initialise_sphere_depolarisation_matrix()

Initialise a 3x3 tensor with the sphere depolarisation matrix, returns a tensor.

Parmeters

None

Returns

ndarray

A 3x3 tensor representing the sphere depolarisation matrix.

PDielec.Calculator.initialise_unit_tensor()

Initialise a 3x3 tensor to a unit tensor.

Parameters

None

Returns

array

The returned tensor is a 3x3 array.

PDielec.Calculator.ionic_permittivity(mode_list, oscillator_strengths, frequencies, volume)

Calculate the low frequency permittivity or zero frequency permittivity.

Parameters

mode_listlist

List of integers giving the active modes

oscillator_strengthsarray_like

Oscillator strengths, in atomic units.

frequenciesarray_like

Frequencies, in atomic units.

volumefloat

Volume, in atomic units.

Returns

3x3 np.array

The calculated low frequency permittivity or zero frequency permittivity.

Notes

The calculation of low frequency permittivity or zero frequency permittivity requires oscillator strengths, frequencies, and volume all to be specified in atomic units.

PDielec.Calculator.longitudinal_modes(frequencies, normal_modes, born_charges, masses, epsilon_inf, volume, qlist, reader)

Apply the nonanalytic correction to the dynamical matrix and calculate the LO frequencies.

Parameters

frequenciesarray_like

The frequencies (f) in atomic units.

normal_modesarray_like

The mass weighted normal modes (U).

born_chargesarray_like

The born charges (Z) stored as [[Z1x, Z1y, Z1z], [Z2x, Z2y, Z2z], [Z3x, Z3y, Z3z]], where 1, 2, 3 are the directions of the field and x, y, z are the coordinates of the atom.

massesarray_like

The atomic masses in atomic units

epsilon_inf3x3 array

Epsilon infinity

volumefloat

volume in atomic units

qlistlist

A list of direction vectors.

readera reader object

a reader object

Returns

list

A list of (real) frequencies in atomic units. Any imaginary frequencies are set to 0.

Notes

If projection was requested in the reader, the correction is modified to ensure translational invariance.

PDielec.Calculator.maxwell(dielectric_medium, crystal_permittivity, shape, L, vf, size)

Calculate the effective constant permittivity using the Maxwell Garnett method.

Parameters

dielectric_mediumtensor (3x3)

The dielectric constant tensor of the medium.

crystal_permittivitytensor (3x3)

The total frequency dielectric constant tensor at the current frequency.

shapestr

The name of the current shape.

Lmatrix (3x3)

The shape’s depolarisation matrix.

sizefloat

The dimensionless size parameter for the frequency under consideration.

vffloat

The volume fraction of filler.

Returns

tensor (3x3)

The effective dielectric constant.

PDielec.Calculator.maxwell_sihvola(dielectric_medium, crystal_permittivity, shape, L, vf, size)

Calculate the effective constant permittivity using the Maxwell Garnett method.

Parameters

dielectric_mediumtensor (3x3)

The dielectric constant tensor of the medium.

crystal_permittivitytensor (3x3)

The total frequency dielectric constant tensor at the current frequency.

shapestr

The name of the current shape.

Lmatrix (3x3)

The shape’s depolarisation matrix.

sizefloat

The dimensionless size parameter for the frequency under consideration.

vffloat

The volume fraction of filler.

Returns

tensor

The effective dielectric constant.

PDielec.Calculator.mie_scattering(dielectric_medium, crystal_permittivity, shape, L, vf, size, size_mu, size_distribution_sigma)

Calculate the effective constant permittivity using a Mie scattering approach.

Parameters

dielectric_mediumarray_like (3x3)

Dielectric constant tensor of the medium.

crystal_permittivityarray_like (3x3)

Total frequency dielectric constant tensor at the current frequency.

shapestr, optional

The name of the current shape (NOT USED).

Larray_like, optional

The shape’s depolarisation matrix (NOT USED).

sizefloat

The dimensionless size parameter for the frequency under consideration.

size_mufloat

The particle size in microns

size_distribution_sigmafloat

The log normal value of sigma.

vffloat

The volume fraction of filler.

Returns

float or array_like

The effective dielectric constant.

Notes

In this method, the MG method is used to calculate the averaged effective permittivity. Then, the permittivity of the isotropic sphere that would give the same average permittivity is calculated. Finally, the Mie scattering of that sphere is calculated. The routine returns the effective dielectric constant.

PDielec.Calculator.normal_modes(masses, mass_weighted_normal_modes)

Transform from mass weighted coordinates to xyz.

Note this returns an array object. The returned normal modes have NOT been renormalized. The input masses are in atomic units. The output normal modes are in atomic units.

Parameters

massesarray of floats

The masses in atomic units

mass_weighted_normal_modesarray of floats

The mass weighted normal modes

Returns

np.array

An array of xyz coordinates derived from mass-weighted coordinates.

Notes

The transformation to xyz coordinates does not involve renormalization of the normal modes. Both the input masses and the output normal modes are in atomic units.

PDielec.Calculator.orthogonalise_projection_operator(ps)

Orthogonalise the list of projection operators ps.

Use Gramm Schmidt orthogonalisation to perform the operation

Parameters

pslist of operators

The projection operators to orthogonalise.

Returns

outlist of operators

The orthogonalised projection operators.

Notes

This function orthogonalises a set of projection operators.

PDielec.Calculator.oscillator_strengths(normal_modes, born_charges)

Calculate oscillator strengths from the normal modes and the born charges.

Parameters

normal_modesarray_like

Normal modes are in the mass weighted coordinate system and normalized.

born_chargesarray_like

Born charges are in electrons, so in atomic units.

Returns

array_like

Oscillator strengths calculated from the given normal modes and born charges.

PDielec.Calculator.reflectance_atr(ns, n0, theta, atrSPolFraction)

Calculate the atr s and p reflectance.

Parameters

nscomplex

The complex permittivity of the effective medium.

n0complex

The permittivity of atr material.

thetafloat

The angle of incidence in degrees.

atrSPolFractionfloat

The fraction of S wave to be considered. The amount of P wave is 1 - atrSPolFraction.

Returns

rsfloat

The s-wave Fresnel amplitude.

rpfloat

The p-wave Fresnel amplitude.

PDielec.Calculator.rodridgues_rotations(efield)

Take the field directions in efield and use each direction to calculate a random rotation about that axis.

Use the field (which is a random unit vector in xyz space) to generate an orthogonal rotation matrix using the Rodrigues rotation formula A = I3.cos(theta) + (1-cos(theta)) e . eT + ex sin(theta), where I3 is a unit matrix, e is the direction, and ex is the cross product matrix:

\[\begin{split}\begin{align} &\text{ex} = \begin{pmatrix} 0 & -e3 & e2 \\ e3 & 0 & -e1 \\ -e2 & e1 & 0 \end{pmatrix} \end{align}\end{split}\]

Parameters

efieldarray_like

The field directions, where each direction is used to calculate a random rotation about that axis. Assumes the field is real.

Returns

list of ndarray

A list of rotation matrices for each direction in efield.

Notes

• The input field is assumed to be real.

• Output is a list of 3x3 rotation matrices.

PDielec.Calculator.set_affinity_on_worker()

When a new worker process is created, the affinity is set to all CPUs.

PDielec.Calculator.set_no_of_threads(nthreads)

Set default number of threads.

Parameters

nthreadsint the number of threads to be used

The number of threads to be applied

The environment is modified to set the most common environment variables for the number of threads a BLAS implementation will use. BLAS implementations include: MKL, OPENBLAS, OMP, NUMEXPR, BLIS and VECLIB

PDielec.Calculator.solve_effective_medium_equations(method, vf, size_mu, size_distribution_sigma, matrixPermittivityFunction, shape, L, concentration, atrPermittivity, atrTheta, atrSPol, bubble_vf, bubble_radius, previous_solution_shared, atuple)

Solve the effective medium equations.

Parameters

methodstr

The method to be used, options include bruggeman, balan, maxwell, maxwell-garnet, averagedpermittivity, maxwell-sihvola, coherent, bruggeman-minimise, mie, anisotropic-mie.

vffloat

The volume fraction of dielectric.

size_mufloat

The particle size in micron

size_distribution_sigmafloat

The width of the size distribution.

matrixPermittivityFunctionfunction

Function returning the matrix permittivity at a frequency.

shapestr

The shape of the particles.

Larray

The depolarisation matrix.

concentrationfloat

The concentration of particles.

atrPermittivityfloat

The permittivity of the ATR substrate.

atrThetafloat

The ATR angle of incidence.

atrSPolstr

The ATR polarisation.

bubble_vffloat

Volume fraction of bubbles.

bubble_radiusfloat

The radius of bubbles.

previous_solution_sharedbool

Use the previous solution to speed up iterations in the case of Bruggeman and coherent methods.

atupletuple

A tuple containing frequency in cm-1 (v_cm1) and a rank 3 tensor of the permittivity of the crystal at a given frequency (crystalPermittivity).

Returns

Tuple of results

v_cm1 method size_mu size_distribution_sigma shape data trace absorption_coefficient molar_absorption_coefficient spatr

PDielec.Calculator.waterman_truell_scattering(lambda_vacuum_nm, N_nm, radius_nm, ri_medium)

Calculate the effective wavenumber based on Waterman-Truell scattering model.

Parameters

lambda_vacuum_nmfloat

Wavelength of the incident light in vacuum, in nanometers.

N_nmfloat

Number density of scatterers, in inverse cubic nanometers.

radius_nmfloat

Radius of a scatterer, in nanometers.

ri_mediumfloat

Refractive index of the medium.

Returns

new_kcomplex

The complex effective wavenumber calculated based on the Waterman-Truell scattering model, which accounts for the multiple scattering effects among the particles in the medium.

Notes

The Waterman-Truell scattering model is a method to calculate the effective wavenumber of a medium containing scatterers. This calculation takes into account the size of the scatterers, their number density, and the refractive index of the medium.

The model utilizes the Mie scattering solutions to evaluate the forward scattering amplitudes, which are then used to approximate the effective wavenumber for wave propagation in the medium. This model is particularly useful in the study of wave scattering in composite materials and biological tissues.