Diffusion equation from neutron flux
====================================

The diffusion equation, based on Fick's law, provides an analytical solution of spatial
neutron flux distribution in the multiplying system.

**Links:**

#. `NuclearPower, possible similar formula <https://www.nuclear-power.com/nuclear-power/reactor-physics/neutron-diffusion-theory/diffusion-equation/>`__.

.. py:currentmodule:: symplyphysics.laws.nuclear.diffusion_equation_from_neutron_flux

.. py:data:: diffusion_coefficient

    :attr:`~symplyphysics.symbols.nuclear.neutron_diffusion_coefficient`.

Symbol:
    :code:`D`

Latex:
    :math:`D`

Dimension:
    :code:`length`


.. py:data:: macroscopic_fission_cross_section

    :attr:`~symplyphysics.symbols.nuclear.macroscopic_cross_section` of fission.

Symbol:
    :code:`Sigma_f`

Latex:
    :math:`\Sigma_text{f}`

Dimension:
    :code:`1/length`


.. py:data:: macroscopic_absorption_cross_section

    :attr:`~symplyphysics.symbols.nuclear.macroscopic_cross_section` of absorption.

Symbol:
    :code:`Sigma_a`

Latex:
    :math:`\Sigma_\text{a}`

Dimension:
    :code:`1/length`


.. py:data:: effective_multiplication_factor

    :attr:`~symplyphysics.symbols.nuclear.effective_multiplication_factor`.

Symbol:
    :code:`k_eff`

Latex:
    :math:`k_\text{eff}`

Dimension:
    :code:`dimensionless`


.. py:data:: position

    :attr:`~symplyphysics.symbols.classical_mechanics.position`.

Symbol:
    :code:`x`

Latex:
    :math:`x`

Dimension:
    :code:`length`


.. py:data:: neutron_flux

    :attr:`~symplyphysics.symbols.nuclear.neutron_flux` as a function of :attr:`~position`.

Symbol:
    :code:`Phi(x)`

Latex:
    :math:`\Phi{\left(x \right)}`

Dimension:
    :code:`1/(area*time)`


.. py:data:: neutron_flux_laplacian

    Laplacian of the :attr:`~neutron_flux` as a function of :attr:`~position`.

Symbol:
    :code:`Laplace(Phi)(x)`

Latex:
    :math:`\nabla^{2} \Phi{\left(x \right)}`

Dimension:
    :code:`1/(length**4*time)`


.. py:data:: law

    :code:`-D * Laplace(Phi)(x) + Sigma_a * Phi(x) = nu / k_eff * Sigma_f * Phi(x)`


    Latex:
        .. math::
            - D \nabla^{2} \Phi{\left(x \right)} + \Sigma_\text{a} \Phi{\left(x \right)} = \frac{\nu}{k_\text{eff}} \Sigma_text{f} \Phi{\left(x \right)}