bluemira.plasma_physics.collisions
==================================

.. py:module:: bluemira.plasma_physics.collisions

.. autoapi-nested-parse::

   Tokamak plasma collision formulae.



Functions
---------

.. autoapisummary::

   bluemira.plasma_physics.collisions.debye_length
   bluemira.plasma_physics.collisions.reduced_mass
   bluemira.plasma_physics.collisions.thermal_velocity
   bluemira.plasma_physics.collisions.de_broglie_length
   bluemira.plasma_physics.collisions.impact_parameter_perp
   bluemira.plasma_physics.collisions.coulomb_logarithm
   bluemira.plasma_physics.collisions.spitzer_conductivity


Module Contents
---------------

.. py:function:: debye_length(temperature: float, density: float) -> float

   Debye length

   :param temperature: Temperature [K]
   :param density: Density [m^-3]

   :returns: Debye length [m]

   .. rubric:: Notes

   Debye length is given by the formula:

   .. math::
       \lambda_D = \sqrt{\frac{\varepsilon_0 k_B T}{n e^2}}

   where:

   - :math:`\varepsilon_0` is the vacuum permittivity,

   - :math:`k_B` is the Boltzmann constant,

   - :math:`T` is the temperature in Kelvin,

   - :math:`n` is the number density of particles,

   - :math:`e` is the elementary charge.

   The conversion of the temperature from Kelvin to Joules
   implicitly includes the Boltzmann constant (:math:`k_B`).


.. py:function:: reduced_mass(mass_1: float, mass_2: float) -> float

   Calculate the reduced mass of a two-particle system

   :param mass_1: Mass of the first particle
   :param mass_2: Mass of the second particle

   :returns: Reduced mass

   .. rubric:: Notes

   Reduced mass of a two-particle system :

   .. math::

       \mu_{AB} = \frac{m_A m_B}{m_A + m_B}

   where :math:`m_A` and :math:`m_B` are the masses
   of the particles.


.. py:function:: thermal_velocity(temperature: float, mass: float) -> float

   :param temperature: Temperature [K]
   :param mass: Mass of the particle [kg]

   :returns: Thermal velocity [m/s]

   .. rubric:: Notes

   The thremal velocity is calculated as

    .. math::

       \sqrt{\frac{2 k_B T}{m}}

   The conversion of the temperature from Kelvin to Joules
   implicitly includes the Boltzmann constant (  :math:`k_B` ).

   The  :math:`\sqrt 2` term is for a 3-dimensional system and the
   most probable velocity in the particle velocity distribution.


.. py:function:: de_broglie_length(velocity: float, mu_12: float) -> float

   Calculate the de Broglie wavelength

   :param velocity: Velocity [m/s]
   :param mu_12: Reduced mass [kg]

   :returns: De Broglie wavelength [m]

   .. rubric:: Notes

   The de Broglie length is given by

   .. math::

       \lambda_{th} = \frac{h}{2 \cdot \mu_{12} \cdot velocity}

   where  :math:`h` is the Planck Constant.


.. py:function:: impact_parameter_perp(velocity: float, mu_12: float) -> float

   Calculate the perpendicular impact parameter, a.k.a. b90

   :param velocity: Velocity [m/s]
   :param mu_12: Reduced mass [kg]

   :returns: Perpendicular impact parameter [m]

   .. rubric:: Notes

   .. math::

       b_{90} = \frac{e^2}{4 \pi \epsilon_0 \mu_{12} v^2}

   where:

   - :math:`e` is the elementary charge (absolute charge of an electron)

   - :math:`\epsilon_0` is the vacuum permittivity,

   - :math:`\mu_{12}` is the reduced mass,

   - :math:`v` is the relative velocity.


.. py:function:: coulomb_logarithm(temperature: float, density: float) -> float

   Calculate the value of the Coulomb logarithm for an electron hitting a proton.

   :param temperature: Temperature [K]
   :param density: Density [1/m^3]

   :returns: Coulomb logarithm value

   .. rubric:: Notes

   The Coulomb logarithm is calculated using the formula:

   .. math::

       \ln{\Lambda} = \ln{\left(1 + \left(\frac{\lambda_{Debye}}
       {b_{min}}\right)^2\right)^{1/2}}

   where:

   :math:`\lambda_{Debye}` is the Debye length,

   :math:`b_{min}` is the minimum impact parameter, which
   is defined as the maximum value between :math:`b_{90}`
   (the perpendicular impact parameter) and the de Broglie
   wavelegth :math:`\lambda_{th}`


.. py:function:: spitzer_conductivity(Z_eff: float, T_e: float, ln_lambda: float) -> float

   Formula for electrical conductivity in a plasma as per L. Spitzer.

   :param Z_eff: Effective charge [dimensionless]
   :param T_e: Electron temperature on axis [keV]
               The equation takes in temperature as [eV], so an in-line conversion is used here.
   :param ln_lambda: Coulomb logarithm value

   :returns: Plasma resistivity [1/Ohm/m]

   .. rubric:: Notes

   Spitzer and Haerm, 1953

       :math:`\sigma = 1.92e4 (2-Z_{eff}^{-1/3}) \dfrac{T_{e}^{3/2}}{Z_{eff}ln\Lambda}`