quantum.py

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

"""
__all__ = [
    "chemical_potential",
    "deBroglie_wavelength",
    "Ef_",
    "Fermi_energy",
    "lambdaDB_",
    "lambdaDB_th_",
    "Thomas_Fermi_length",
    "thermal_deBroglie_wavelength",
    "Wigner_Seitz_radius",
]

import astropy.units as u
import numpy as np

from astropy.constants.si import c, e, eps0, h, hbar, k_B, m_e

from plasmapy import particles
from plasmapy.formulary import mathematics
from plasmapy.formulary.relativity import Lorentz_factor
from plasmapy.utils import RelativityError
from plasmapy.utils.decorators import validate_quantities


# TODO: Use @check_relativistic and @particle_input
@validate_quantities(
    V={"can_be_negative": True}, validations_on_return={"can_be_negative": False}
)
def deBroglie_wavelength(V: u.m / u.s, particle) -> 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`, or `~astropy.units.Quantity`
        An instance of `~plasmapy.particles.particle_class.Particle`, or
        an equvalent 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
    --------
    >>> from astropy import 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 not isinstance(particle, u.Quantity):
        try:
            # TODO: Replace with more general routine!
            m = particles.particle_mass(particle)
        except Exception:
            raise ValueError("Unable to find particle mass.")
    else:
        try:
            m = particle.to(u.kg)
        except Exception:
            raise u.UnitConversionError(
                "The second argument for deBroglie_wavelength must be either a "
                "representation of a particle or a"
                " Quantity with units of mass."
            )

    if V.size > 1:

        lambda_dBr = np.ones(V.shape) * np.inf * u.m
        indices = V.value != 0
        lambda_dBr[indices] = h / (m * V[indices] * Lorentz_factor(V[indices]))

    else:

        if V == 0 * u.m / u.s:
            lambda_dBr = np.inf * u.m
        else:
            lambda_dBr = h / (Lorentz_factor(V) * m * V)

    return lambda_dBr


lambdaDB_ = deBroglie_wavelength
""" Alias to :func:`deBroglie_wavelength`. """


@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.K) -> 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}}

    Example
    -------
    >>> from astropy import units as u
    >>> thermal_deBroglie_wavelength(1 * u.eV)
    <Quantity 6.9193675e-10 m>
    """
    lambda_dbTh = h / np.sqrt(2 * np.pi * m_e * k_B * T_e)
    return lambda_dbTh


lambdaDB_th_ = thermal_deBroglie_wavelength
""" Alias to :func:`thermal_deBroglie_wavelength`. """


@validate_quantities(
    n_e={"can_be_negative": False}, validations_on_return={"can_be_negative": False}
)
def Fermi_energy(n_e: u.m ** -3) -> 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

    Example
    -------
    >>> from astropy import units as u
    >>> Fermi_energy(1e23 * u.cm**-3)
    <Quantity 1.2586761e-18 J>
    """
    coeff = (np.pi * hbar) ** 2 / (2 * m_e)
    energy_F = coeff * (3 * n_e / np.pi) ** (2 / 3)
    return energy_F


Ef_ = Fermi_energy
""" Alias to :func:`Fermi_energy`. """


@validate_quantities(
    n_e={"can_be_negative": False}, validations_on_return={"can_be_negative": False}
)
def Thomas_Fermi_length(n_e: u.m ** -3) -> 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
    --------
    Fermi_energy
    plasmapy.formulary.Debye_length

    Example
    -------
    >>> from astropy import units as u
    >>> Thomas_Fermi_length(1e23 * u.cm**-3)
    <Quantity 5.37991409e-11 m>

    """
    energy_F = Fermi_energy(n_e)
    lambda_TF = np.sqrt(2 * eps0 * energy_F / (3 * n_e * e ** 2))
    return lambda_TF


@validate_quantities(
    n={"can_be_negative": False}, validations_on_return={"can_be_negative": False}
)
def Wigner_Seitz_radius(n: u.m ** -3) -> 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
    --------
    Fermi_energy

    Example
    -------
    >>> from astropy import units as u
    >>> Wigner_Seitz_radius(1e29 * u.m**-3)
    <Quantity 1.33650462e-10 m>

    """
    radius = (3 / (4 * np.pi * n)) ** (1 / 3)
    return radius


# TODO: remove NotImplementedError and 'doctest: +SKIP' when the following issues are addressed...
#       https://github.com/PlasmaPy/PlasmaPy/issues/726
#       https://github.com/astropy/astropy/issues/9721
@validate_quantities(
    n_e={"can_be_negative": False},
    T={"can_be_negative": False, "equivalencies": u.temperature_energy()},
)
def chemical_potential(n_e: u.m ** -3, T: u.K) -> 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 given by [1]_:

    .. math::
        χ_a = I_{1/2}(β μ_a^{ideal})

    where :math:`χ` is the degeneracy parameter, :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_a^{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 Fermi_integral are complex, a nonlinear
    Levenberg-Marquardt least squares method is used to iteratively approach
    a value of :math:`μ` which minimizes
    :math:`I_{1/2}(β μ_a^{ideal}) - χ_a`

    This function returns :math:`β μ^{ideal}` the dimensionless
    ideal chemical potential.

    Warning: at present this function is limited to relatively small
    arguments due to limitations in the `~mpmath` package's implementation
    of `~mpmath.polylog`, which PlasmaPy uses in calculating the Fermi
    integral.

    References
    ----------
    .. [1] Bonitz, Michael. Quantum kinetic theory. Stuttgart: Teubner, 1998.

    Example
    -------
    >>> from astropy import units as u
    >>> chemical_potential(n_e=1e21*u.cm**-3,T=11000*u.K)  # doctest: +SKIP
    <Quantity 2.00039985e-12>

    """

    raise NotImplementedError(
        "This function has been temporarily disabled due to a bug.\n"
        "Please refer to https://github.com/PlasmaPy/PlasmaPy/issues/726 \n"
        "and https://github.com/astropy/astropy/issues/9721 "
        "for progress in fixing it."
    )
    # deBroglie wavelength
    lambdaDB = thermal_deBroglie_wavelength(T)
    # degeneracy parameter
    degen = (n_e * lambdaDB ** 3).to(u.dimensionless_unscaled)

    def residual(params, data, eps_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 = (data - model) / eps_data
        return complexResidue.view(np.float)

    # setting parameters for fitting along with bounds
    alphaGuess = 1 * u.dimensionless_unscaled
    try:
        from lmfit import minimize, Parameters
    except (ImportError, ModuleNotFoundError) as e:
        from plasmapy.optional_deps import lmfit_import_error

        raise lmfit_import_error from e

    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
    eps_data = np.array([1e-15])  # numerical error
    minFit = minimize(residual, params, args=(data, eps_data))
    beta_mu = minFit.params["alpha"].value * u.dimensionless_unscaled
    return beta_mu


# TODO: decorate with validate_quantities
# TODO: remove NotImplementedError and 'doctest: +SKIP' when the following issues are addressed...
#       https://github.com/PlasmaPy/PlasmaPy/issues/726
#       https://github.com/astropy/astropy/issues/9721
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.

    Example
    -------
    >>> from astropy import units as u
    >>> _chemical_potential_interp(n_e=1e23*u.cm**-3, T=11000*u.K)  # doctest: +SKIP
    <Quantity 8.17649>

    """
    raise NotImplementedError(
        "This function has been temporarily disabled due to a bug.\n"
        "Please refer to https://github.com/PlasmaPy/PlasmaPy/issues/726 \n"
        "and https://github.com/astropy/astropy/issues/9721 "
        "for progress in fixing it."
    )
    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)