# Source code for plasmapy.formulary.quantum

"""
Functions for quantum parameters, including electron degenerate
gases and warm dense matter.

"""

__all__ = [
"chemical_potential",
"deBroglie_wavelength",
"Fermi_energy",
"quantum_theta",
"Thomas_Fermi_length",
"thermal_deBroglie_wavelength",
]

import astropy.units as u
import numpy as np
from astropy.constants.si import c, e, eps0, h, hbar, k_B, m_e
from lmfit import Parameters, minimize

from plasmapy.formulary import mathematics
from plasmapy.formulary.relativity import Lorentz_factor
from plasmapy.particles.decorators import particle_input
from plasmapy.particles.particle_class import ParticleLike
from plasmapy.utils.decorators import validate_quantities
from plasmapy.utils.exceptions import RelativityError

__all__ += __aliases__

# TODO: Use @check_relativistic
[docs] @validate_quantities( V={"can_be_negative": True}, validations_on_return={"can_be_negative": False} ) @particle_input def deBroglie_wavelength( V: u.Quantity[u.m / u.s], particle: ParticleLike, ) -> u.Quantity[u.m]: r""" Return the de Broglie wavelength. The de Broglie wavelength (:math:λ_{dB}) of a particle is defined by .. math:: λ_{dB} = \frac{h}{p} = \frac{h}{γ m V} where :math:h is the Planck constant, :math:p is the relativistic momentum of the particle, :math:γ is the Lorentz factor, :math:m is the mass of the particle, and :math:V is the velocity of the particle. **Aliases:** lambdaDB_ Parameters ---------- V : ~astropy.units.Quantity Particle velocity in units convertible to meters per second. particle : str, ~plasmapy.particles.particle_class.Particle, or |Quantity| An instance of ~plasmapy.particles.particle_class.Particle, or an equivalent representation (e.g., 'e-', 'p+', 'D+', or 'He-4 1+'), for the particle of interest, or the particle mass in units convertible to kg. If a ~plasmapy.particles.particle_class.Particle instance is given, then the particle mass is retrieved from the object. Returns ------- lambda_dB : ~astropy.units.Quantity The de Broglie wavelength in units of meters. Raises ------ TypeError If the velocity is not a ~astropy.units.Quantity and cannot be converted into a ~astropy.units.Quantity. ~astropy.units.UnitConversionError If the velocity is not in appropriate units. ~plasmapy.utils.exceptions.RelativityError If the magnitude of V is larger than the speed of light. Warns ----- : ~astropy.units.UnitsWarning If units are not provided, SI units are assumed. Examples -------- >>> import astropy.units as u >>> velocity = 1.4e7 * u.m / u.s >>> deBroglie_wavelength(velocity, "e-") <Quantity 5.18997095e-11 m> >>> deBroglie_wavelength(V=0 * u.m / u.s, particle="D+") <Quantity inf m> """ V = np.abs(V) if np.any(V >= c): raise RelativityError( "Velocity input in deBroglie_wavelength cannot " "be greater than or equal to the speed of " "light." ) if V.size > 1: lambda_dBr = np.ones(V.shape) * np.inf * u.m indices = V.value != 0 lambda_dBr[indices] = h / ( particle.mass * V[indices] * Lorentz_factor(V[indices]) ) elif V == 0 * u.m / u.s: lambda_dBr = np.inf * u.m else: lambda_dBr = h / (Lorentz_factor(V) * particle.mass * V) return lambda_dBr
lambdaDB_ = deBroglie_wavelength """Alias to ~plasmapy.formulary.quantum.deBroglie_wavelength."""
[docs] @validate_quantities( T_e={"can_be_negative": False, "equivalencies": u.temperature_energy()}, validations_on_return={"can_be_negative": False}, ) def thermal_deBroglie_wavelength(T_e: u.Quantity[u.K]) -> u.Quantity[u.m]: r""" Calculate the thermal de Broglie wavelength for electrons. **Aliases:** lambdaDB_th_ Parameters ---------- T_e : ~astropy.units.Quantity Electron temperature. Returns ------- lambda_dbTh : ~astropy.units.Quantity The thermal de Broglie wavelength for electrons in meters. Raises ------ TypeError If argument is not a ~astropy.units.Quantity. ~astropy.units.UnitConversionError If argument is in incorrect units. ValueError If argument contains invalid values. Warns ----- : ~astropy.units.UnitsWarning If units are not provided, SI units are assumed. Notes ----- The thermal de Broglie wavelength is approximately the average de Broglie wavelength for electrons in an ideal gas and is given by .. math:: λ_{dbTh} = \frac{h}{\sqrt{2 π m_e k_B T_e}} Examples -------- >>> import astropy.units as u >>> thermal_deBroglie_wavelength(1 * u.eV) <Quantity 6.9193675e-10 m> """ return h / np.sqrt(2 * np.pi * m_e * k_B * T_e)
lambdaDB_th_ = thermal_deBroglie_wavelength """Alias to ~plasmapy.formulary.quantum.thermal_deBroglie_wavelength."""
[docs] @validate_quantities( n_e={"can_be_negative": False}, validations_on_return={"can_be_negative": False} ) def Fermi_energy(n_e: u.Quantity[u.m**-3]) -> u.Quantity[u.J]: r""" Calculate the kinetic energy in a degenerate electron gas. **Aliases:** Ef_ Parameters ---------- n_e : ~astropy.units.Quantity Electron number density. Returns ------- energy_F : ~astropy.units.Quantity The Fermi energy in joules. Raises ------ TypeError If argument is not a ~astropy.units.Quantity. ~astropy.units.UnitConversionError If argument is in incorrect units. ValueError If argument contains invalid values. Warns ----- : ~astropy.units.UnitsWarning If units are not provided, SI units are assumed. Notes ----- The Fermi energy is the kinetic energy in a degenerate electron gas and is given by .. math:: E_F = \frac{π^2 ℏ^2}{2 m_e} \left( \frac{3 n_e}{π} \right)^{2/3} This quantity is often used in place of thermal energy for analysis of cold, dense plasmas (e.g. warm dense matter, condensed matter). See Also -------- Thomas_Fermi_length Examples -------- >>> import astropy.units as u >>> Fermi_energy(1e23 * u.cm**-3) <Quantity 1.2586761e-18 J> """ coeff = (np.pi * hbar) ** 2 / (2 * m_e) return coeff * (3 * n_e / np.pi) ** (2 / 3)
Ef_ = Fermi_energy """Alias to ~plasmapy.formulary.quantum.Fermi_energy."""
[docs] @validate_quantities( n_e={"can_be_negative": False}, validations_on_return={"can_be_negative": False} ) def Thomas_Fermi_length(n_e: u.Quantity[u.m**-3]) -> u.Quantity[u.m]: r""" Calculate the exponential scale length for charge screening for cold and dense plasmas. Parameters ---------- n_e : ~astropy.units.Quantity Electron number density. Returns ------- lambda_TF : ~astropy.units.Quantity The Thomas-Fermi screening length in meters. Raises ------ TypeError If argument is not a ~astropy.units.Quantity. ~astropy.units.UnitConversionError If argument is in incorrect units. ValueError If argument contains invalid values. Warns ----- : ~astropy.units.UnitsWarning If units are not provided, SI units are assumed. Notes ----- The Thomas-Fermi screening length is the exponential scale length for charge screening and is given by .. math:: λ_{TF} = \sqrt{\frac{2 ε_0 E_F}{3 n_e e^2}} for an electron degenerate gas. This quantity is often used in place of the Debye length for analysis of cold, dense plasmas (e.g. warm dense matter, condensed matter). The electrical potential will drop by a factor of :math:1/e every Thomas-Fermi screening length. Plasmas will generally be quasineutral on length scales significantly larger than the Thomas-Fermi screening length. See Also -------- ~plasmapy.formulary.quantum.Fermi_energy ~plasmapy.formulary.lengths.Debye_length Examples -------- >>> import astropy.units as u >>> Thomas_Fermi_length(1e23 * u.cm**-3) <Quantity 5.37991409e-11 m> """ energy_F = Fermi_energy(n_e) return np.sqrt(2 * eps0 * energy_F / (3 * n_e * e**2))
[docs] @validate_quantities( n={"can_be_negative": False}, validations_on_return={"can_be_negative": False} ) def Wigner_Seitz_radius(n: u.Quantity[u.m**-3]) -> u.Quantity[u.m]: r""" Calculate the Wigner-Seitz radius, which approximates the inter-particle spacing. This function returns the radius of a sphere whose volume is equal to the mean volume per atom in a solid. This parameter is often used to calculate the coupling parameter. When ion density is used, this is the ion sphere radius, i.e., the space occupied by a single ion with no other ions in that space. Higher density means less space for each ion, so the radius is smaller. Parameters ---------- n : ~astropy.units.Quantity Particle number density. Returns ------- radius : ~astropy.units.Quantity The Wigner-Seitz radius in meters. Raises ------ TypeError If argument is not a ~astropy.units.Quantity. ~astropy.units.UnitConversionError If argument is in incorrect units. ValueError If argument contains invalid values. Warns ----- : ~astropy.units.UnitsWarning If units are not provided, SI units are assumed. Notes ----- The Wigner-Seitz radius approximates the interparticle spacing. It is the radius of a sphere whose volume is equal to the mean volume per atom in a solid: .. math:: r = \left(\frac{3}{4 π n}\right)^{1/3} See Also -------- ~plasmapy.formulary.quantum.Fermi_energy Examples -------- >>> import astropy.units as u >>> Wigner_Seitz_radius(1e29 * u.m**-3) <Quantity 1.33650462e-10 m> """ return (3 / (4 * np.pi * n)) ** (1 / 3)
[docs] @validate_quantities( n_e={"can_be_negative": False}, T={"can_be_negative": False, "equivalencies": u.temperature_energy()}, ) def chemical_potential( n_e: u.Quantity[u.m**-3], T: u.Quantity[u.K] ) -> u.Quantity[u.dimensionless_unscaled]: r""" Calculate the ideal chemical potential. Parameters ---------- n_e : ~astropy.units.Quantity Electron number density. T : ~astropy.units.Quantity The temperature. Returns ------- beta_mu : ~astropy.units.Quantity The dimensionless ideal chemical potential. That is the ratio of the ideal chemical potential to the thermal energy. Raises ------ TypeError If argument is not a ~astropy.units.Quantity. ~astropy.units.UnitConversionError If argument is in incorrect units. ValueError If argument contains invalid values. Warns ----- : ~astropy.units.UnitsWarning If units are not provided, SI units are assumed. Notes ----- The ideal chemical potential is implicitly given by Eq. 1.2 in :cite:p:bonitz:1998\ : .. math:: χ = nΛ^{3} = I_{1/2}(β μ^{ideal}) where :math:χ is the degeneracy parameter, :math:n is the species number density, :math:Λ is the thermal de Broglie wavelength, :math:I_{1/2} is the Fermi integral with order 1/2, :math:β is the inverse thermal energy :math:β = 1/(k_B T), and :math:μ^{ideal} is the ideal chemical potential. The definition for the ideal chemical potential is implicit, so it must be obtained numerically by solving for the Fermi integral for values of chemical potential approaching the degeneracy parameter. Since values returned from the ~plasmapy.formulary.mathematics.Fermi_integral are complex, the Broyden–Fletcher–Goldfarb–Shanno algorithm is used to iteratively approach a value of :math:μ^{ideal} which minimizes :math:\lvert I_{1/2}(β μ^{ideal}) - χ \rvert. This function returns the dimensionless ideal chemical potential :math:β μ^{ideal}. Examples -------- >>> import astropy.units as u >>> chemical_potential(n_e=1e25 * u.cm**-3, T=11000 * u.K) <Quantity 283.43506297> """ # deBroglie wavelength lambdaDB = thermal_deBroglie_wavelength(T) # degeneracy parameter degen = (n_e * lambdaDB**3).to(u.dimensionless_unscaled) def residual(params, data): """Residual function for fitting parameters to Fermi_integral.""" alpha = params["alpha"].value # note that alpha = mu / (k_B * T) model = mathematics.Fermi_integral(alpha, 0.5) complexResidue = abs(data - model) return complexResidue # noqa: RET504 # setting parameters for fitting along with bounds alphaGuess = 1 * u.dimensionless_unscaled params = Parameters() params.add("alpha", value=alphaGuess, min=0.0) # calling minimize function from lmfit to fit by minimizing the residual data = np.array(degen) # result of Fermi_integral - degen should be zero minFit = minimize(residual, params, args=(data,), method="bfgsb") beta_mu = minFit.params["alpha"].value * u.dimensionless_unscaled return beta_mu # noqa: RET504
def _chemical_potential_interp(n_e, T): r""" Fitting formula for interpolating chemical potential between classical and quantum regimes. See [1]_, [2]_ for more information. Parameters ---------- n_e : ~astropy.units.Quantity Electron number density. T : ~astropy.units.Quantity Temperature in units of temperature or energy. Returns ------- beta_mu : ~astropy.units.Quantity The dimensionless chemical potential, which is a ratio of chemical potential energy to thermal kinetic energy. Raises ------ TypeError If argument is not a ~astropy.units.Quantity. ~astropy.units.UnitConversionError If argument is in incorrect units. ValueError If argument contains invalid values. Warns ----- : ~astropy.units.UnitsWarning If units are not provided, SI units are assumed. Notes ----- The ideal chemical potential is given by [1]_: .. math:: \frac{μ}{k_B T_e} = - \frac{3}{2} \ln Θ + \ln \frac{4}{3 \sqrt{π}} + \frac{A Θ^{-b - 1} + B Θ^{-(b + 1) / 2}}{1 + A Θ^{-b}} where .. math:: Θ = \frac{k_B T_e}{E_F} is the degeneracy parameter, comparing the thermal energy to the Fermi energy, and the coefficients for the fitting formula are :math:A = 0.25945\ , :math:B = 0.0072\ , and :math:b = 0.858\ . References ---------- .. [1] Ichimaru, Statistical Plasma Physics Addison-Wesley, Reading, MA, 1991. .. [2] Gregori, G., et al. "Theoretical model of x-ray scattering as a dense matter probe." Physical Review E 67.2 (2003): 026412. Examples -------- >>> import astropy.units as u >>> _chemical_potential_interp(n_e=1e23 * u.cm**-3, T=11000 * u.K) <Quantity 8.17649> """ A = 0.25945 B = 0.072 b = 0.858 theta = k_B * T / Fermi_energy(n_e) term1 = -3 / 2 * np.log(theta) term2 = np.log(4 / (3 * np.sqrt(np.pi))) term3num = A * theta ** (-b - 1) + B * theta ** (-(b + 1) / 2) term3den = 1 + A * theta ** (-b) term3 = term3num / term3den beta_mu = term1 + term2 + term3 return beta_mu.to(u.dimensionless_unscaled)
[docs] @validate_quantities( T={"can_be_negative": False, "equivalencies": u.temperature_energy()}, n_e={"can_be_negative": False}, ) def quantum_theta( T: u.Quantity[u.K], n_e: u.Quantity[u.m**-3] ) -> u.Quantity[u.dimensionless_unscaled]: r""" Compare Fermi energy to thermal kinetic energy to check if quantum effects are important. The quantum theta (:math:θ) of a plasma is defined by .. math:: θ = \frac{E_T}{E_F} where :math:E_T is the thermal energy of the plasma and :math:E_F is the Fermi energy of the plasma. Parameters ---------- T : ~astropy.units.Quantity The temperature of the plasma. n_e : ~astropy.units.Quantity The electron number density of the plasma. Examples -------- >>> import astropy.units as u >>> quantum_theta(1 * u.eV, 1e20 * u.m**-3) <Quantity 127290.619...> >>> quantum_theta(1 * u.eV, 1e16 * u.m**-3) <Quantity 59083071...> >>> quantum_theta(1 * u.eV, 1e26 * u.m**-3) <Quantity 12.72906...> >>> quantum_theta(1 * u.K, 1e26 * u.m**-3) <Quantity 0.00109...> Returns ------- theta : ~astropy.units.Quantity Notes ----- The thermal energy of the plasma (:math:E_T) is defined by .. math:: E_T = k_B T where :math:k_B is the Boltzmann constant and :math:T is the temperature of the plasma. See Also -------- ~plasmapy.formulary.quantum.Fermi_energy """ fermi_energy = Fermi_energy(n_e) thermal_energy = k_B * T return thermal_energy / fermi_energy