"""
Defines the Thomson scattering analysis module as part of
`plasmapy.diagnostics`.
"""
__all__ = [
"spectral_density",
"spectral_density_model",
]
__lite_funcs__ = ["spectral_density_lite"]
import numbers
import warnings
from collections.abc import Callable
from typing import Any, Optional, Union
import astropy.constants as const
import astropy.units as u
import numpy as np
from lmfit import Model
from plasmapy.formulary import (
permittivity_1D_Maxwellian_lite,
plasma_frequency_lite,
thermal_speed_coefficients,
thermal_speed_lite,
)
from plasmapy.particles import Particle, ParticleLike
from plasmapy.particles.exceptions import ChargeError
from plasmapy.particles.particle_collections import ParticleList
from plasmapy.utils.decorators import (
bind_lite_func,
preserve_signature,
validate_quantities,
)
__all__ += __lite_funcs__
c_si_unitless = const.c.si.value
e_si_unitless = const.e.si.value
m_p_si_unitless = const.m_p.si.value
m_e_si_unitless = const.m_e.si.value
# TODO: interface for inputting a multi-species configuration could be
# simplified using the plasmapy.classes.plasma_base class if that class
# included ion and electron drift velocities and information about the ion
# atomic species.
[docs]
@preserve_signature
def spectral_density_lite(
wavelengths,
probe_wavelength: numbers.Real,
n: numbers.Real,
T_e: np.ndarray,
T_i: np.ndarray,
efract: np.ndarray,
ifract: np.ndarray,
ion_z: np.ndarray,
ion_mass: np.ndarray,
electron_vel: np.ndarray,
ion_vel: np.ndarray,
probe_vec: np.ndarray,
scatter_vec: np.ndarray,
instr_func_arr: Optional[np.ndarray] = None,
notch: Optional[np.ndarray] = None,
) -> tuple[Union[np.floating, np.ndarray], np.ndarray]:
r"""
The :term:`lite-function` version of
`~plasmapy.diagnostics.thomson.spectral_density`. Performs the same
thermal speed calculations as
`~plasmapy.diagnostics.thomson.spectral_density`, but is intended
for computational use and thus has data conditioning safeguards
removed.
Parameters
----------
wavelengths : (Nλ,) `~numpy.ndarray`
The wavelengths in meters over which the spectral density
function will be calculated.
probe_wavelength : real number
Wavelength of the probe laser in meters.
n : `~numpy.ndarray`
Total combined number density of all electron populations in
m\ :sup:`-3`\ .
T_e : (Ne,) `~numpy.ndarray`
Temperature of each electron population in kelvin, where Ne is
the number of electron populations.
T_i : (Ni,) `~numpy.ndarray`
Temperature of each ion population in kelvin, where Ni is the
number of ion populations.
efract : (Ne,) `~numpy.ndarray`
An `~numpy.ndarray` where each element represents the fraction
(or ratio) of the electron population number density to the
total electron number density. Must sum to 1.0. Default is a
single electron population.
ifract : (Ni,) `~numpy.ndarray`
An `~numpy.ndarray` object where each element represents the
fraction (or ratio) of the ion population number density to the
total ion number density. Must sum to 1.0. Default is a single
ion species.
ion_z : (Ni,) `~numpy.ndarray`
An `~numpy.ndarray` of the charge number :math:`Z` of each ion
species.
ion_mass : (Ni,) `~numpy.ndarray`
An `~numpy.ndarray` of the mass of each ion species in kg.
electron_vel : (Ne, 3) `~numpy.ndarray`
Velocity of each electron population in the rest frame (in m/s).
If set, overrides ``electron_vdir`` and ``electron_speed``.
Defaults to a stationary plasma ``[0, 0, 0]`` m/s.
ion_vel : (Ni, 3) `~numpy.ndarray`
Velocity vectors for each electron population in the rest frame
(in m/s). If set, overrides ``ion_vdir`` and ``ion_speed``.
Defaults to zero drift for all specified ion species.
probe_vec : (3,) float `~numpy.ndarray`
Unit vector in the direction of the probe laser. Defaults to
``[1, 0, 0]``.
scatter_vec : (3,) float `~numpy.ndarray`
Unit vector pointing from the scattering volume to the detector.
Defaults to [0, 1, 0] which, along with the default ``probe_vec``,
corresponds to a 90 degree scattering angle geometry.
instr_func_arr : `~numpy.ndarray`, shape (Nwavelengths,) optional
The instrument function evaluated at a linearly spaced range of
wavelengths ranging from :math:`-W` to :math:`W`, where
.. math::
W = 0.5*(\max{λ} - \min{λ})
Here :math:`λ` is the ``wavelengths`` array. This array will be
convolved with the spectral density function before it is
returned.
notch : (2,) or (N, 2) `~numpy.ndarray`, |keyword-only|, optional
A pair of wavelengths in meters which are the endpoints of a notch over
which the output Skw is set to 0. Can also be input as a 2D array
which contains many such pairs if multiple notches are needed.
Defaults to no notch.
Returns
-------
alpha : float
Mean scattering parameter, where ``alpha`` > 1 corresponds to
collective scattering and ``alpha`` < 1 indicates non-collective
scattering. The scattering parameter is calculated based on the
total plasma density :math:`n`.
Skw : `~numpy.ndarray`
Computed spectral density function over the input
``wavelengths`` array with units of s/rad.
"""
scattering_angle = np.arccos(np.dot(probe_vec, scatter_vec))
# Calculate plasma parameters
# Temperatures here in K!
coefs = thermal_speed_coefficients("most_probable", 3)
vT_e = thermal_speed_lite(T_e, m_e_si_unitless, coefs)
vT_i = thermal_speed_lite(T_i, ion_mass, coefs)
# Compute electron and ion densities
ne = efract * n
zbar = np.sum(ifract * ion_z)
ni = ifract * n / zbar # ne/zbar = sum(ni)
# wpe is calculated for the entire plasma (all electron populations combined)
wpe = plasma_frequency_lite(n, m_e_si_unitless, 1)
# Convert wavelengths to angular frequencies (electromagnetic waves, so
# phase speed is c)
ws = 2 * np.pi * c_si_unitless / wavelengths
wl = 2 * np.pi * c_si_unitless / probe_wavelength
# Compute the frequency shift (required by energy conservation)
w = ws - wl
# Compute the wavenumbers in the plasma
# See Sheffield Sec. 1.8.1 and Eqs. 5.4.1 and 5.4.2
ks = np.sqrt(ws**2 - wpe**2) / c_si_unitless
kl = np.sqrt(wl**2 - wpe**2) / c_si_unitless
# Compute the wavenumber shift (required by momentum conservation)
# Eq. 1.7.10 in Sheffield
k = np.sqrt(ks**2 + kl**2 - 2 * ks * kl * np.cos(scattering_angle))
# Normal vector along k
k_vec = scatter_vec - probe_vec
k_vec = k_vec / np.linalg.norm(k_vec)
# Compute Doppler-shifted frequencies for both the ions and electrons
# Matmul is simultaneously conducting dot products over all wavelengths
# and ion populations
w_e = w - np.matmul(electron_vel, np.outer(k, k_vec).T)
w_i = w - np.matmul(ion_vel, np.outer(k, k_vec).T)
# Compute the scattering parameter alpha
# expressed here using the fact that v_th/w_p = root(2) * Debye length
alpha = np.sqrt(2) * wpe / np.outer(k, vT_e)
# Calculate the normalized phase velocities (Sec. 3.4.2 in Sheffield)
xe = np.outer(1 / vT_e, 1 / k) * w_e
xi = np.outer(1 / vT_i, 1 / k) * w_i
# Calculate the susceptibilities
chiE = np.zeros([efract.size, w.size], dtype=np.complex128)
for i, _fract in enumerate(efract): # noqa: B007
wpe = plasma_frequency_lite(ne[i], m_e_si_unitless, 1)
chiE[i, :] = permittivity_1D_Maxwellian_lite(w_e[i, :], k, vT_e[i], wpe)
# Treatment of multiple species is an extension of the discussion in
# Sheffield Sec. 5.1
chiI = np.zeros([ifract.size, w.size], dtype=np.complex128)
for i, _fract in enumerate(ifract): # noqa: B007
wpi = plasma_frequency_lite(ni[i], ion_mass[i], ion_z[i])
chiI[i, :] = permittivity_1D_Maxwellian_lite(w_i[i, :], k, vT_i[i], wpi)
# Calculate the longitudinal dielectric function
epsilon = 1 + np.sum(chiE, axis=0) + np.sum(chiI, axis=0)
econtr = np.zeros([efract.size, w.size], dtype=np.complex128)
for m in range(efract.size):
econtr[m, :] = efract[m] * (
2
* np.sqrt(np.pi)
/ k
/ vT_e[m]
* np.power(np.abs(1 - np.sum(chiE, axis=0) / epsilon), 2)
* np.exp(-(xe[m, :] ** 2))
)
icontr = np.zeros([ifract.size, w.size], dtype=np.complex128)
for m in range(ifract.size):
icontr[m, :] = ifract[m] * (
2
* np.sqrt(np.pi)
* ion_z[m]
/ k
/ vT_i[m]
* np.power(np.abs(np.sum(chiE, axis=0) / epsilon), 2)
* np.exp(-(xi[m, :] ** 2))
)
# Recast as real: imaginary part is already zero
Skw = np.real(np.sum(econtr, axis=0) + np.sum(icontr, axis=0))
# Apply an instrument function if one is provided
if instr_func_arr is not None:
Skw = np.convolve(Skw, instr_func_arr, mode="same")
# add notch(es) to the spectrum if any are provided
if notch is not None:
# If only one notch is included, create a dummy second dimension
if np.ndim(notch) == 1:
notch = np.array([notch])
for notch_i in notch:
# For each notch, identify the index for the beginning and end
# wavelengths and set Skw to zero between those indices
x0 = np.argwhere(wavelengths > notch_i[0])[0][0]
x1 = np.argwhere(wavelengths > notch_i[1])[0][0]
Skw[x0:x1] = 0
return np.mean(alpha), Skw
[docs]
@validate_quantities(
wavelengths={"can_be_negative": False, "can_be_zero": False},
probe_wavelength={"can_be_negative": False, "can_be_zero": False},
n={"can_be_negative": False, "can_be_zero": False},
T_e={"can_be_negative": False, "equivalencies": u.temperature_energy()},
T_i={"can_be_negative": False, "equivalencies": u.temperature_energy()},
)
@bind_lite_func(spectral_density_lite)
def spectral_density( # noqa: C901, PLR0912, PLR0915
wavelengths: u.Quantity[u.nm],
probe_wavelength: u.Quantity[u.nm],
n: u.Quantity[u.m**-3],
*,
T_e: u.Quantity[u.K],
T_i: u.Quantity[u.K],
efract=None,
ifract=None,
ions: ParticleLike = "p+",
electron_vel: u.Quantity[u.m / u.s] = None,
ion_vel: u.Quantity[u.m / u.s] = None,
probe_vec=None,
scatter_vec=None,
instr_func: Optional[Callable] = None,
notch: u.m = None,
) -> tuple[Union[np.floating, np.ndarray], np.ndarray]:
r"""Calculate the spectral density function for Thomson scattering of
a probe laser beam by a multi-species Maxwellian plasma.
Parameters
----------
wavelengths : `~astropy.units.Quantity`
The wavelengths over which the spectral density function will be
calculated, in units convertible to m.
probe_wavelength : `~astropy.units.Quantity`
Wavelength of the probe laser, in units convertible to m.
n : `~astropy.units.Quantity`
Total combined number density of all electron populations, in
units convertible to m\ :sup:`-3`\ .
T_e : (Ne,) `~astropy.units.Quantity`, |keyword-only|
Temperature of each electron population in units convertible to
K or eV, where Ne is the number of electron populations.
T_i : (Ni,) `~astropy.units.Quantity`, |keyword-only|
Temperature of each ion population in units convertible to K or
eV, where Ni is the number of ion populations.
efract : (Ne,) |array_like|, |keyword-only|, optional
The ratio of the number density of each electron population to
the total electron number density, denoted by :math:`F_e` below.
Must sum to one. The default corresponds to a single electron
population.
ifract : (Ni,) |array_like|, |keyword-only|, optional
The fractional number densities of each ion population, denoted
by :math:`F_i` below. Must sum to one. The default corresponds
to a single ion population.
ions : (Ni,) |particle-like|, |keyword-only|, default: "p+"
One or more positively charged ions representing each ion
population.
electron_vel : (Ne, 3) `~astropy.units.Quantity`, |keyword-only|, optional
Velocity vectors for each electron population in the rest frame,
in units convertible to m/s. If set, overrides ``electron_vdir``
and ``electron_speed``. Defaults to a stationary plasma at
:math:`[0, 0, 0]` m/s.
ion_vel : (Ni, 3) `~astropy.units.Quantity`, |keyword-only|, optional
Velocity vectors for each ion population in the rest frame, in
units convertible to m/s. If set, overrides ``ion_vdir`` and
``ion_speed``. Defaults to zero drift for all specified ion
species.
probe_vec : (3,) |array_like|, |keyword-only|, default: [1, 0, 0]
Unit vector in the direction of the probe laser.
scatter_vec : (3,) |array_like|, |keyword-only|, default: [0, 1, 0]
Unit vector pointing from the scattering volume to the
detector. The default, along with the default for ``probe_vec``,
corresponds to a 90° scattering angle geometry.
instr_func : function
A function representing the instrument function that takes a
`~astropy.units.Quantity` of wavelengths (centered on zero)
and returns the instrument point spread function. The
resulting array will be convolved with the spectral density
function before it is returned.
notch : (2,) or (N, 2) `~astropy.units.Quantity`, |keyword-only|, optional
A pair of wavelengths in units convertible to meters which are the
endpoints of a notch over which the output Skw is set to 0. Can also
be input as a 2D array which contains many such pairs if multiple
notches are needed. If the ``notch`` and ``instr_func`` keywords are both
set, the notch is applied after the instrument function such the
instrument function does convolve the values of the theoretical
spectrum originally in the notch region. Defaults to no notch.
Returns
-------
alpha : `float`
Mean scattering parameter, where ``alpha`` > 1 corresponds to
collective scattering and ``alpha`` < 1 indicates
non-collective scattering. The scattering parameter is
calculated based on the total plasma density ``n``.
Skw : `~astropy.units.Quantity`
Computed spectral density function over the input
``wavelengths`` array with units of s/rad.
Notes
-----
This function calculates the spectral density function for Thomson
scattering of a probe laser beam by a plasma consisting of one or
more ion species and one or more thermal electron populations (the
entire plasma is assumed to be quasi-neutral):
.. math::
S(k,ω) = \sum_e \frac{2π}{k}
\bigg |1 - \frac{χ_e}{ε} \bigg |^2
f_{e0,e} \bigg (\frac{ω}{k} \bigg ) +
\sum_i \frac{2π Z_i}{k}
\bigg |\frac{χ_e}{ε} \bigg |^2 f_{i0,i}
\bigg ( \frac{ω}{k} \bigg )
where :math:`χ_e` is the electron population susceptibility of the
plasma and :math:`ε = 1 + ∑_e χ_e + ∑_i χ_i` is the total plasma
dielectric function (with :math:`χ_i` being the ion population of
the susceptibility), :math:`Z_i` is the charge of each ion,
:math:`k` is the scattering wavenumber, :math:`ω` is the scattering
frequency, and :math:`f_{e0,e}` and :math:`f_{i0,i}` are the
electron and ion velocity distribution functions, respectively. In
this function, the electron and ion velocity distribution functions
are assumed to be Maxwellian, making this function equivalent to Eq.
3.4.6 in :cite:t:`sheffield:2011`\ .
The number density of the e\ :sup:`th` electron populations is
defined as
.. math::
n_e = F_e n
where :math:`n` is the total number density of all electron
populations combined and :math:`F_e` is the fractional number
density of each electron population such that
.. math::
\sum_e n_e = n
.. math::
\sum_e F_e = 1
The plasma is assumed to be quasineutral, and therefore the number
density of the i\ :sup:`th` ion population is
.. math::
n_i = \frac{F_i n}{∑_i F_i Z_i}
with :math:`F_i` defined in the same way as :math:`F_e`.
For details, see "Plasma Scattering of Electromagnetic Radiation"
by :cite:t:`sheffield:2011`. This code is a modified version of
the program described therein.
For a summary of the relevant physics, see Chapter 5 of the
:cite:t:`schaeffer:2014` thesis.
"""
# Validate efract
if efract is None:
efract = np.ones(1)
else:
efract = np.asarray(efract, dtype=np.float64)
if np.sum(efract) != 1:
raise ValueError(f"The provided efract does not sum to 1: {efract}")
# Validate ifract
if ifract is None:
ifract = np.ones(1)
else:
ifract = np.asarray(ifract, dtype=np.float64)
if np.sum(ifract) != 1:
raise ValueError(f"The provided ifract does not sum to 1: {ifract}")
if probe_vec is None:
probe_vec = np.array([1, 0, 0])
if scatter_vec is None:
scatter_vec = np.array([0, 1, 0])
# If electron velocity is not specified, create an array corresponding
# to zero drift
if electron_vel is None:
electron_vel = np.zeros([efract.size, 3]) * u.m / u.s
# Condition the electron velocity keywords
if ion_vel is None:
ion_vel = np.zeros([ifract.size, 3]) * u.m / u.s
# Condition ions
# If a single value is provided, turn into a particle list
if isinstance(ions, ParticleList):
pass
elif isinstance(ions, str):
ions = ParticleList([Particle(ions)])
# If a list is provided, ensure all values are Particles, then convert
# to a ParticleList
elif isinstance(ions, list):
for ii, ion in enumerate(ions):
if isinstance(ion, Particle):
continue
ions[ii] = Particle(ion)
ions = ParticleList(ions)
else:
raise TypeError(
"The type of object provided to the ``ions`` keyword "
f"is not supported: {type(ions)}"
)
# Validate ions
if len(ions) == 0:
raise ValueError("At least one ion species needs to be defined.")
try:
if sum(ion.charge_number <= 0 for ion in ions):
raise ValueError("All ions must be positively charged.")
# Catch error if charge information is missing
except ChargeError as ex:
raise ValueError("All ions must be positively charged.") from ex
# Condition T_i
if T_i.size == 1:
# If a single quantity is given, put it in an array so it's iterable
# If T_i.size != len(ions), assume same temp. for all species
T_i = np.array([T_i.value]) * T_i.unit
# Make sure the sizes of ions, ifract, ion_vel, and T_i all match
if (
(len(ions) != ifract.size)
or (ion_vel.shape[0] != ifract.size)
or (T_i.size != ifract.size)
):
raise ValueError(
f"Inconsistent number of ion species in ifract ({ifract}), "
f"ions ({len(ions)}), T_i ({T_i.size}), "
f"and/or ion_vel ({ion_vel.shape[0]})."
)
# Condition T_e
if T_e.size == 1:
# If a single quantity is given, put it in an array so it's iterable
# If T_e.size != len(efract), assume same temp. for all species
T_e = np.array([T_e.value]) * T_e.unit
# Make sure the sizes of efract, electron_vel, and T_e all match
if (electron_vel.shape[0] != efract.size) or (T_e.size != efract.size):
raise ValueError(
f"Inconsistent number of electron populations in efract ({efract.size}), "
f"T_e ({T_e.size}), or electron velocity ({electron_vel.shape[0]})."
)
# Create arrays of ion Z and mass from particles given
ion_z = ions.charge_number
ion_mass = ions.mass
probe_vec = probe_vec / np.linalg.norm(probe_vec)
scatter_vec = scatter_vec / np.linalg.norm(scatter_vec)
# Apply the instrument function
if instr_func is not None and callable(instr_func):
# Create an array of wavelengths of the same size as wavelengths
# but centered on zero
wspan = (np.max(wavelengths) - np.min(wavelengths)) / 2
eval_w = np.linspace(-wspan, wspan, num=wavelengths.size)
instr_func_arr = instr_func(eval_w)
if type(instr_func_arr) != np.ndarray:
raise ValueError(
"instr_func must be a function that returns a "
"np.ndarray, but the provided function returns "
f" a {type(instr_func_arr)}"
)
if wavelengths.shape != instr_func_arr.shape:
raise ValueError(
"The shape of the array returned from the "
f"instr_func ({instr_func_arr.shape}) "
"does not match the shape of the wavelengths "
f"array ({wavelengths.shape})."
)
instr_func_arr /= np.sum(instr_func_arr)
else:
instr_func_arr = None
# Valildate notch input
if notch is not None:
notch_unitless = notch.to(u.m).value
if np.ndim(notch_unitless) == 1:
notch_unitless = np.array([notch_unitless])
for notch_i in notch_unitless:
if np.shape(notch_i) != (2,):
raise ValueError("Notches must be pairs of values.")
if notch_i[0] > notch_i[1]:
raise ValueError(
"The first element of the notch cannot be greater than "
"the second element."
)
else:
notch_unitless = None
alpha, Skw = spectral_density_lite(
wavelengths.to(u.m).value,
probe_wavelength.to(u.m).value,
n.to(u.m**-3).value,
T_e.to(u.K).value,
T_i.to(u.K).value,
efract=efract,
ifract=ifract,
ion_z=ion_z,
ion_mass=ion_mass.to(u.kg).value,
ion_vel=ion_vel.to(u.m / u.s).value,
electron_vel=electron_vel.to(u.m / u.s).value,
probe_vec=probe_vec,
scatter_vec=scatter_vec,
instr_func_arr=instr_func_arr,
notch=notch_unitless,
)
return alpha, Skw * u.s / u.rad
# ***************************************************************************
# These functions are necessary to interface scalar Parameter objects with
# the array inputs of spectral_density
# ***************************************************************************
def _count_populations_in_params(params: dict[str, Any], prefix: str) -> int:
"""
Counts the number of electron or ion populations in a ``params``
`dict`.
The number of populations is determined by counting the number of
items in the ``params`` `dict` with a key that starts with the
string defined by ``prefix``.
"""
return len([key for key in params if key.startswith(prefix)])
def _params_to_array(
params: dict[str, Any], prefix: str, vector: bool = False
) -> np.ndarray:
"""
Constructs an array from the values contained in the dictionary
``params`` associated with keys starting with the prefix defined by
``prefix``.
If ``vector == False``, then values for keys matching the expression
``prefix_[0-9]+`` are gathered into a 1D array.
If ``vector == True``, then values for keys matching the expression
``prefix_[xyz]_[0-9]+`` are gathered into a 2D array of shape
``(N, 3)``.
Notes
-----
This function allows `lmfit.parameter.Parameter` inputs to be
converted into the array-type inputs required by the spectral
density function.
"""
if vector:
npop = _count_populations_in_params(params, f"{prefix}_x")
output = np.zeros([npop, 3])
for i in range(npop):
for j, ax in enumerate(["x", "y", "z"]):
output[i, j] = params[f"{prefix}_{ax}_{i}"].value
else:
npop = _count_populations_in_params(params, prefix)
output = np.zeros([npop])
for i in range(npop):
output[i] = params[f"{prefix}_{i}"]
return output
# ***************************************************************************
# Fitting functions
# ***************************************************************************
def _spectral_density_model(wavelengths, settings=None, **params):
"""
lmfit Model function for fitting Thomson spectra.
For descriptions of arguments, see the `thomson_model` function.
"""
# LOAD FROM SETTINGS
ion_z = settings["ion_z"]
ion_mass = settings["ion_mass"]
probe_vec = settings["probe_vec"]
scatter_vec = settings["scatter_vec"]
electron_vdir = settings["electron_vdir"]
ion_vdir = settings["ion_vdir"]
probe_wavelength = settings["probe_wavelength"]
instr_func_arr = settings["instr_func_arr"]
notch = settings["notch"]
# LOAD FROM PARAMS
n = params["n"]
T_e = _params_to_array(params, "T_e")
T_i = _params_to_array(params, "T_i")
efract = _params_to_array(params, "efract")
ifract = _params_to_array(params, "ifract")
electron_speed = _params_to_array(params, "electron_speed")
ion_speed = _params_to_array(params, "ion_speed")
electron_vel = electron_speed[:, np.newaxis] * electron_vdir
ion_vel = ion_speed[:, np.newaxis] * ion_vdir
# Convert temperatures from eV to kelvin (required by fast_spectral_density)
T_e *= 11604.51812155
T_i *= 11604.51812155
alpha, model_Skw = spectral_density_lite(
wavelengths,
probe_wavelength,
n,
T_e,
T_i,
efract=efract,
ifract=ifract,
ion_z=ion_z,
ion_mass=ion_mass,
electron_vel=electron_vel,
ion_vel=ion_vel,
probe_vec=probe_vec,
scatter_vec=scatter_vec,
instr_func_arr=instr_func_arr,
notch=notch,
)
model_Skw *= 1 / np.max(model_Skw)
return model_Skw
[docs]
def spectral_density_model( # noqa: C901, PLR0912, PLR0915
wavelengths, settings, params
):
r"""
Returns a `lmfit.model.Model` function for Thomson spectral density
function.
Parameters
----------
wavelengths : numpy.ndarray
Wavelength array, in meters.
settings : dict
A dictionary of non-variable inputs to the spectral density
function which must include the following keys:
- ``"probe_wavelength"``: Probe wavelength in meters
- ``"probe_vec"`` : (3,) unit vector in the probe direction
- ``"scatter_vec"``: (3,) unit vector in the scattering
direction
- ``"ions"`` : list of particle strings,
`~plasmapy.particles.particle_class.Particle` objects, or a
`~plasmapy.particles.particle_collections.ParticleList`
describing each ion species. All ions must be positive.
and may contain the following optional variables:
- ``"electron_vdir"`` : (e#, 3) array of electron velocity unit
vectors
- ``"ion_vdir"`` : (e#, 3) array of ion velocity unit vectors
- ``"instr_func"`` : A function that takes a wavelength
|Quantity| array and returns a spectrometer instrument
function as an `~numpy.ndarray`.
- ``"notch"`` : A wavelength range or array of multiple wavelength
ranges over which the spectral density is set to 0.
These quantities cannot be varied during the fit.
params : `~lmfit.parameter.Parameters` object
A `~lmfit.parameter.Parameters` object that must contain the
following variables:
- n: Total combined density of the electron populations in
m\ :sup:`-3`
- :samp:`T_e_{e#}` : Temperature in eV
- :samp:`T_i_{i#}` : Temperature in eV
where where :samp:`{i#}` and where :samp:`{e#}` are replaced by
the number of electron and ion populations, zero-indexed,
respectively (e.g., 0, 1, 2, ...). The
`~lmfit.parameter.Parameters` object may also contain the
following optional variables:
- :samp:`"efract_{e#}"` : Fraction of each electron population
(must sum to 1)
- :samp:`"ifract_{i#}"` : Fraction of each ion population (must
sum to 1)
- :samp:`"electron_speed_{e#}"` : Electron speed in m/s
- :samp:`"ion_speed_{ei}"` : Ion speed in m/s
These quantities can be either fixed or varying.
Returns
-------
model : `lmfit.model.Model`
An `lmfit.model.Model` of the spectral density function for the
provided settings and parameters that can be used to fit Thomson
scattering data.
Notes
-----
If an instrument function is included, the data should not include
any `numpy.nan` values — instead regions with no data should be
removed from both the data and wavelength arrays using
`numpy.delete`.
"""
required_settings = {
"probe_wavelength",
"probe_vec",
"scatter_vec",
"ions",
}
if missing_settings := required_settings - set(settings):
raise ValueError(
f"The following required settings were not provided in the "
f"'settings' argument: {missing_settings}"
)
required_params = {"n"}
if missing_params := required_params - set(params):
raise ValueError(
f"The following required parameters were not provided in the "
f"'params': {missing_params}"
)
# **********************
# Count number of populations
# **********************
if "efract_0" not in params:
params.add("efract_0", value=1.0, vary=False)
if "ifract_0" not in params:
params.add("ifract_0", value=1.0, vary=False)
num_e = _count_populations_in_params(params, "efract")
num_i = _count_populations_in_params(params, "ifract")
# **********************
# Required settings and parameters per population
# **********************
for p, nums in zip(["T_e", "T_i"], [num_e, num_i]):
for num in range(nums):
key = f"{p}_{num!s}"
if key not in params:
raise ValueError(
f"{p} was not provided in kwarg 'parameters', but is required."
)
# **************
# ions
# **************
ions = settings["ions"]
# Condition ions
# If a single value is provided, turn into a particle list
if isinstance(ions, ParticleList):
pass
elif isinstance(ions, str):
ions = ParticleList([Particle(ions)])
# If a list is provided, ensure all values are Particles, then convert
# to a ParticleList
elif isinstance(ions, list):
for ii, ion in enumerate(ions):
if isinstance(ion, Particle):
continue
ions[ii] = Particle(ion)
ions = ParticleList(ions)
else:
raise TypeError(
"The type of object provided to the ``ions`` keyword "
f"is not supported: {type(ions)}"
)
# Validate ions
if len(ions) == 0:
raise ValueError("At least one ion species needs to be defined.")
try:
if sum(ion.charge_number <= 0 for ion in ions):
raise ValueError("All ions must be positively charged.")
# Catch error if charge information is missing
except ChargeError as ex:
raise ValueError("All ions must be positively charged.") from ex
# Create arrays of ion Z and mass from particles given
settings["ion_z"] = ions.charge_number
settings["ion_mass"] = ions.mass
# **************
# efract and ifract
# **************
# Automatically add an expression to the last efract parameter to
# indicate that it depends on the others (so they sum to 1.0)
# The resulting expression for the last of three will look like
# efract_2.expr = "1.0 - efract_0 - efract_1"
if num_e > 1:
nums = ["1.0"] + [f"efract_{i}" for i in range(num_e - 1)]
params[f"efract_{num_e - 1}"].expr = " - ".join(nums)
if num_i > 1:
nums = ["1.0"] + [f"ifract_{i}" for i in range(num_i - 1)]
params[f"ifract_{num_i - 1}"].expr = " - ".join(nums)
# **************
# Electron velocity
# **************
electron_speed = np.zeros(num_e)
for num in range(num_e):
k = f"electron_speed_{num}"
if k in params:
electron_speed[num] = params[k].value
else:
# electron_speed[e] = 0 already
params.add(k, value=0, vary=False)
if "electron_vdir" not in settings:
if np.all(electron_speed == 0):
# vdir is arbitrary in this case because vel is zero
settings["electron_vdir"] = np.ones([num_e, 3])
else:
raise ValueError(
"Key 'electron_vdir' must be defined in kwarg 'settings' if "
"any electron population has a non-zero speed (i.e. any "
"params['electron_speed_<#>'] is non-zero)."
)
norm = np.linalg.norm(settings["electron_vdir"], axis=-1)
settings["electron_vdir"] = settings["electron_vdir"] / norm[:, np.newaxis]
# **************
# Ion velocity
# **************
ion_speed = np.zeros(num_i)
for num in range(num_i):
k = f"ion_speed_{num}"
if k in params:
ion_speed[num] = params[k].value
else:
# ion_speed[i] = 0 already
params.add(k, value=0, vary=False)
if "ion_vdir" not in list(settings.keys()):
if np.all(ion_speed == 0):
# vdir is arbitrary in this case because vel is zero
settings["ion_vdir"] = np.ones([num_i, 3])
else:
raise ValueError(
"Key 'ion_vdir' must be defined in kwarg 'settings' if "
"any ion population has a non-zero speed (i.e. any "
"params['ion_speed_<#>'] is non-zero)."
)
norm = np.linalg.norm(settings["ion_vdir"], axis=-1)
settings["ion_vdir"] = settings["ion_vdir"] / norm[:, np.newaxis]
if "instr_func" not in settings or settings["instr_func"] is None:
settings["instr_func_arr"] = None
else:
# Create instr_func array from instr_func
instr_func = settings["instr_func"]
wspan = (np.max(wavelengths) - np.min(wavelengths)) / 2
eval_w = np.linspace(-wspan, wspan, num=wavelengths.size)
instr_func_arr = instr_func(eval_w * u.m)
if type(instr_func_arr) != np.ndarray:
raise ValueError(
"instr_func must be a function that returns a "
"np.ndarray, but the provided function returns "
f" a {type(instr_func_arr)}"
)
if wavelengths.shape != instr_func_arr.shape:
raise ValueError(
"The shape of the array returned from the "
f"instr_func ({instr_func_arr.shape}) "
"does not match the shape of the wavelengths "
f"array ({wavelengths.shape})."
)
instr_func_arr *= 1 / np.sum(instr_func_arr)
settings["instr_func_arr"] = instr_func_arr
warnings.warn(
"If an instrument function is included, the data "
"should not include any `numpy.nan` values. "
"Instead regions with no data should be removed from "
"both the data and wavelength arrays using "
"`numpy.delete`."
)
if "notch" not in settings:
settings["notch"] = None
# TODO: raise an exception if the number of any of the ion or electron
# quantities isn't consistent with the number of that species defined
# by ifract or efract.
# Create and return the lmfit.Model
return Model(
_spectral_density_model,
independent_vars=["wavelengths"],
nan_policy="omit",
settings=settings,
)