spectral_density

plasmapy.diagnostics.thomson.spectral_density(wavelengths: Quantity, probe_wavelength: Quantity, n: Quantity, *, T_e: Quantity, T_i: Quantity, efract=None, ifract=None, ions: str | Integral | Particle | CustomParticle | Quantity = 'p+', electron_vel: Quantity = None, ion_vel: Quantity = None, probe_vec=None, scatter_vec=None, instr_func: Callable | None = None, notch: Unit('m') = None) → tuple[floating | ndarray, ndarray][source]

Calculate the spectral density function for Thomson scattering of a probe laser beam by a multi-species Maxwellian plasma.

Parameters:

wavelengths (Quantity) – The wavelengths over which the spectral density function will be calculated, in units convertible to m.
probe_wavelength (Quantity) – Wavelength of the probe laser, in units convertible to m.
n (Quantity) – Total combined number density of all electron populations, in units convertible to m^-3.
T_e ((Ne,) 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,) 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 \(F_e\) 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 \(F_i\) 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) 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 \([0, 0, 0]\) m/s.
ion_vel ((Ni, 3) 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 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) 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 (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):

\[S(k,ω) = \sum_e \frac{2π}{k} \bigg |1 - \frac{χ_e}{ε} \bigg |^2 f_{e0,e} \bigg (\frac{ω}{k} \bigg ) + \sum_i \frac{2π Z_i}{k} \bigg |\frac{χ_e}{ε} \bigg |^2 f_{i0,i} \bigg ( \frac{ω}{k} \bigg )\]

where \(χ_e\) is the electron population susceptibility of the plasma and \(ε = 1 + ∑_e χ_e + ∑_i χ_i\) is the total plasma dielectric function (with \(χ_i\) being the ion population of the susceptibility), \(Z_i\) is the charge of each ion, \(k\) is the scattering wavenumber, \(ω\) is the scattering frequency, and \(f_{e0,e}\) and \(f_{i0,i}\) 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 Sheffield et al. [2011].

The number density of the e^th electron populations is defined as

\[n_e = F_e n\]

where \(n\) is the total number density of all electron populations combined and \(F_e\) is the fractional number density of each electron population such that

\[\sum_e n_e = n\]

\[\sum_e F_e = 1\]

The plasma is assumed to be quasineutral, and therefore the number density of the i^th ion population is

\[n_i = \frac{F_i n}{∑_i F_i Z_i}\]

with \(F_i\) defined in the same way as \(F_e\).

For details, see “Plasma Scattering of Electromagnetic Radiation” by Sheffield et al. [2011]. This code is a modified version of the program described therein.

For a summary of the relevant physics, see Chapter 5 of the Schaeffer [2014] thesis.