Source code for plasmapy.diagnostics.thomson

"""
Defines the Thomson scattering analysis module as part of
`plasmapy.diagnostics`.
"""

__all__ = [
    "spectral_density",
    "spectral_density_model",
]
__lite_funcs__ = ["spectral_density_lite"]

import warnings
from collections.abc import Callable
from typing import Any

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: float, n: float, 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: np.ndarray | None = None, notch: np.ndarray | None = None, ) -> tuple[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: Callable | None = None, notch: u.m = None, ) -> tuple[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], strict=False): 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, )