# Source code for plasmapy.formulary.ionization

"""Functions related to ionization states and the properties thereof."""

__all__ = ["ionization_balance", "Saha", "Z_bal_"]
__aliases__ = ["Z_bal_"]

import astropy.units as u
import numpy as np
from astropy.constants import a0, k_B

from plasmapy.utils.decorators import validate_quantities

[docs]
@validate_quantities(
T_e={"can_be_negative": False, "equivalencies": u.temperature_energy()}
)
def ionization_balance(
n: u.Quantity[u.m**-3], T_e: u.Quantity[u.K]
) -> u.Quantity[u.dimensionless_unscaled]:
r"""
Return the average ionization state of ions in a plasma assuming that
the numbers of ions in each state are equal.

Z_bal is the estimate average ionization level of a plasma in thermal equilibrium
that balances the number density of ions in two different ionization states.
Z_bal is derived from the Saha equation with the assumptions that
the atoms are of a single species,
are either hydrogenic or completely ionized,
and that there is a balance between ionization and recombination,
meaning that the number of atoms in either state are equal.
The Saha equation and therefore Z_bal are more accurate when the plasma
is at a high density and temperature.

.. math::

Z\_bal = \sqrt{\frac{k_B T_e}{E_H}} \sqrt{\ln{\frac{1}{4 n a_{0}^3}
(\frac{k_B T_e}{π E_H})^{3/2}}} - \frac{1}{2}

Where :math:k_B is the Boltzmann constant,
:math:a_0 is the Bohr radius, and
:math:E_H is the ionization energy of Hydrogen

**Aliases:** Z_bal_

Parameters
----------
T_e : ~astropy.units.Quantity
The electron temperature.

n : ~astropy.units.Quantity
The electron number density of the plasma.

Warns
-----
: ~astropy.units.UnitsWarning
If units are not provided, SI units are assumed.

Raises
------
TypeError
If either of T_e or n is not a ~astropy.units.Quantity
and cannot be converted into a ~astropy.units.Quantity.

~astropy.units.UnitConversionError
If either of T_e or n is not in appropriate units.

Examples
--------
>>> import astropy.units as u
>>> T_e = 5000 * u.K
>>> n = 1e19 * u.m**-3
>>> ionization_balance(n, T_e)
<Quantity 0.274...>
>>> T_e = 50 * u.eV
>>> n = 1e10 * u.m**-3
>>> ionization_balance(n, T_e)
<Quantity 12.615...>

Returns
-------
Z : ~astropy.units.Quantity
The average ionization state of the ions in the plasma
assuming that the numbers of ions in each state are equal.

"""
E_H = 1 * u.Ry

A = np.sqrt(k_B * T_e / E_H)
B = np.log(1 / (4 * n * a0**3) * (k_B * T_e / (np.pi * E_H)) ** (3 / 2))

return A * np.sqrt(B) - 1 / 2

Z_bal_ = ionization_balance
"""Alias for ~plasmapy.formulary.ionization.ionization_balance."""

[docs]
@validate_quantities(
T_e={"can_be_negative": False, "equivalencies": u.temperature_energy()}
)
def Saha(
g_j, g_k, n_e: u.Quantity[u.m**-3], E_jk: u.Quantity[u.J], T_e: u.Quantity[u.K]
) -> u.Quantity[u.dimensionless_unscaled]:
r"""
Return the ratio of populations of two ionization states.

The Saha equation, derived in statistical mechanics, gives an
approximation of the ratio of population of ions in two different
ionization states in a plasma. This approximation applies to plasmas
in thermodynamic equilibrium where ionization and recombination of
ions with electrons are balanced.

.. math::
\frac{N_j}{N_k} = \frac{1}{n_e} \frac{g_j}{4 g_k a_0^{3}} \left(
\frac{k_B T_e}{π E_H}
\right)^{\frac{3}{2}}
\exp\left( \frac{-E_{jk}}{k_B T_e} \right)

Where :math:k_B is the Boltzmann constant, :math:a_0 is the Bohr
radius, :math:E_H is the ionization energy of hydrogen, :math:N_j
and :math:N_k are the population of ions in the :math:j and
:math:k states respectively. This function is equivalent to Eq.
3.47 in Drake_.

Parameters
----------
T_e : ~astropy.units.Quantity
The electron temperature in units of temperature or thermal
energy per particle.

g_j : int
The degeneracy of ionization state :math:j\ .

g_k : int
The degeneracy of ionization state :math:k\ .

E_jk : ~astropy.units.Quantity
The energy difference between ionization states :math:j and
:math:k\ .

n_e : ~astropy.units.Quantity
The electron number density of the plasma.

Warns
-----
: ~astropy.units.UnitsWarning
If units are not provided, SI units are assumed.

Raises
------
TypeError
If any of T_e, E_jk, or n_e is not a ~astropy.units.Quantity
and cannot be converted into a ~astropy.units.Quantity.

~astropy.units.UnitConversionError
If any of T_e, E_jk, or n is not in appropriate units.

Returns
-------
ratio : ~astropy.units.Quantity
The ratio of population of ions in ionization state :math:j to
state :math:k\ .

Examples
--------
>>> import astropy.units as u
>>> T_e = 5000 * u.K
>>> n = 1e19 * u.m**-3
>>> g_j = 2
>>> g_k = 2
>>> E_jk = 1 * u.Ry
>>> Saha(g_j, g_k, n, E_jk, T_e)
<Quantity 3.299...e-06>
>>> T_e = 1 * u.Ry
>>> n = 1e23 * u.m**-3
>>> Saha(g_j, g_k, n, E_jk, T_e)
<Quantity 1114595.586...>

Notes
-----
the Saha equation, see chapter 3 of R. Paul Drake's book,
"High-Energy-Density Physics: Foundation of Inertial Fusion and
Experimental Astrophysics" (DOI: 10.1007/978-3-319-67711-8_3_).

.. _Drake: https://doi.org/10.1007/978-3-319-67711-8
.. _DOI: 10.1007/978-3-319-67711-8_3: https://doi.org/10.1007/978-3-319-67711-8_3
"""
E_h = 1 * u.Ry

degeneracy_factor = (1 / n_e) * g_j / (4 * g_k * a0**3)
physical_constants = (k_B * T_e / (np.pi * E_h)) ** (3 / 2)
boltzmann_factor = np.exp(-E_jk / (k_B * T_e))

return degeneracy_factor * physical_constants * boltzmann_factor