Cross section of interaction in recharge model
==============================================

The effective cross section is a physical quantity characterizing the probability of
transition of a system of two interacting particles to a certain final state, a
quantitative characteristic of the acts of collision of particles of a stream hitting a
target with target particles. The effective cross-section has the dimension of the area.

**Notation:**

#. :math:`a_0` (:code:`a_0`) is :attr:`~symplyphysics.quantities.bohr_radius`.
#. :math:`\mathrm{IE}_\text{H}` (:code:`IE_h`) is :attr:`~symplyphysics.quantities.hydrogen_ionization_energy`.
#. :math:`k_\text{B}` (:code:`k_B`) is :attr:`~symplyphysics.quantities.boltzmann_constant`.
#. :math:`e` (:code:`e`) is :attr:`~symplyphysics.quantities.elementary_charge`.

..
    TODO: find link
    TODO: move to `ionization` folder?

.. py:currentmodule:: symplyphysics.laws.chemistry.cross_section_of_interaction_in_recharge_model

.. py:data:: cross_section

    :attr:`~symplyphysics.symbols.chemistry.cross_section` of interaction of particles.

Symbol:
    :code:`sigma`

Latex:
    :math:`\sigma`

Dimension:
    :code:`area`


.. py:data:: ionization_energy

    Ionization :attr:`~symplyphysics.symbols.basic.energy` of the particles.

Symbol:
    :code:`E_i`

Latex:
    :math:`E_\text{i}`

Dimension:
    :code:`energy`


.. py:data:: molecular_mass

    :attr:`~symplyphysics.symbols.basic.mass` of a single gas particle.

Symbol:
    :code:`m`

Latex:
    :math:`m`

Dimension:
    :code:`mass`


.. py:data:: pressure

    Gas :attr:`~symplyphysics.symbols.classical_mechanics.pressure`.

Symbol:
    :code:`p`

Latex:
    :math:`p`

Dimension:
    :code:`pressure`


.. py:data:: temperature

    Gas :attr:`~symplyphysics.symbols.thermodynamics.temperature`.

Symbol:
    :code:`T`

Latex:
    :math:`T`

Dimension:
    :code:`temperature`


.. py:data:: electric_field_strength

    :attr:`~symplyphysics.symbols.electrodynamics.electric_field_strength`.

Symbol:
    :code:`E`

Latex:
    :math:`E`

Dimension:
    :code:`voltage/length`


.. py:data:: law

    :code:`sigma = pi * a_0^2 * IE_h / E_i * log(sqrt(3 * k_B * T / m) * sqrt(E_i / IE_h) * sigma * p * m / (2 * k_B * T * e * E))^2`


    Latex:
        .. math::
            \sigma = \pi a_0^{2} \frac{\mathrm{IE}_\text{H}}{E_\text{i}} \log \left( \sqrt{\frac{3 k_\text{B} T}{m}} \sqrt{\frac{E_\text{i}}{\mathrm{IE}_\text{H}}} \frac{\sigma p m}{2 k_\text{B} T e E} \right)^{2}