.. _plasma:

******
Plasma
******


The role of the plasma module is to determine the ionisation and excitation states of the elements of the 
supernova ejecta, given the basic structure, including the elemental abundances, densities and radiation temperature.
After the calculation of the plasma state, the :math:`\tau_{\textrm{sobolev}}` values can be calculated.

The TARDIS plasma structure inherits from the `BasePlasma` class. The code currently uses the `LegacyPlasmaArray`
for generating a plasma from the information provided by `model`. A variety of different plasmas can be generated
depending on the options selected in the plasma section of the TARDIS config. file. The options currently considered
by the Legacy Plasma when creating the plasma calculation structure include:

plasma:
 * ionization: lte/nebular
 * excitation: lte/dilute-lte
 * line_interaction_type: scatter/downbranch/macroatom
 * helium_treatment: dilute-lte/recomb-nlte
 * nlte: [can provide list of ion species to be treated in NLTE, as well as specifying the use of the coronal_approximation/classical_nebular settings.

`LegacyPlasmaArray` uses these options to construct a map of the necessary plasma parameters that demonstrates how these parameters are dependent on one another (using `NetworkX <https://networkx.github.io/>`_). Each time a particular parameter of the plasma is updated, all of the parameters dependent (directly or indirectly) on that particular one can be easily updated automatically, without requiring that all the plasma calculations are repeated.

Properties, Inputs and Outputs
------------------------------
Each TARDIS plasma possesses an array of plasma properties, which are used to calculate plasma parameter values. Most plasma properties have a single output, e.g.
 * `GElectron`: (`g_electron`,)
 * `HeliumNLTE`: (`helium_population`,)

but some have two or more, e.g.
 * `IonNumberDensity`: (`ion_number_density`, `electron_densities`)
 * `Levels`: (`levels`, `excitation_energy`, `metastability`, `g`)

Every property has a `calculate` function that returns the values of its outputs. The arguments required for that function become the property inputs. TARDIS will raise an error if it does not have all of the required inputs for a particular property. It will also raise an error if there is an output loop, i.e. if two properties are dependent on each other. Some different properties share output names; for example, `PhiSahaLTE` and `PhiSahaNebular` both have an output called `phi`. That is because the `phi` value is calculated differently depending on the ionization method selected, but once calculated, both values interact in the same way with the rest of the plasma. TARDIS will import only one of the `phi` properties when initialising the plasma.

The Plasma Graph
----------------
If the necessary Python modules (`PyGraphviz <https://pygraphviz.github.io/>`_ and `dot2tex <https://dot2tex.readthedocs.io/en/latest/>`_) are available, TARDIS can output a .dot and a .tex file at the beginning of each run that can be compiled to produce a PDF image of the plasma module graph via the :code:`write_to_dot()` and :code:`write_to_tex()` functions, respectively. The nodes on this graph are the names of plasma properties, e.g. `Levels`, `TauSobolev`, `IonNumberDensity`, along with a list of outputs from those properties. These nodes are connected by edges linking them with the sources of their inputs. The .tex file contains the name of the input/output linking the properties (e.g. `levels`, :math:`\tau_{\textrm{sobolev}}`, :math:`n_{e}`), as well as the equations used to calculate them, written in LaTeX. See the tutorial below!

.. toctree::
    :maxdepth: 2

    ../../../io/output/plasma_graph

Updating the Plasma
-------------------
During each iteration of the main code, TARDIS updates the plasma using the `update_radiationfield` function. This requires, at minimum, new values for `t_rad` (the radiation temperature), `w` (the dilution factor) and `j_blues` (the intensity in the blue part of each line).

.. _plasma_calculations:

Plasma Calculations
-------------------

.. note::
    In this documentation we use the indices :math:`i, j, k` to mean atomic number, ion number and level number respectively.

`BasePlasma` serves as the base class for all plasmas and can just calculate the atom number densities for a given input of abundance fraction.

.. math::
    N_{atom} = \rho_\textrm{total} \times \textrm{Abundance fraction} / m_\textrm{atom}

In the next step the line and level tables are purged of entries that are not represented in the abundance fractions are saved in `BasePlasma.levels` and `BasePlasma.lines`. Finally, the function `BasePlasma.update_t_rad` is called at the end of initialization to update the plasma conditions to a new :math:`T_\textrm{radiation field}` (with the give t_rad). This function is the same in the other plasma classes and does the main part of the calculation. In the case of `BasePlasma` this is only setting `BasePlasma.beta_rad` to :math:`\frac{1}{k_\textrm{B}T_\textrm{rad}}`.

The next more complex class is `LTEPlasma` which will calculate the ionization balance and level populations in Local Thermal Equilibrium conditions (LTE). The :class:`NebularPlasma`-class inherits from `LTEPlasma` and uses a more complex description of the BasePlasma.

.. toctree::
    :maxdepth: 2

    lte_plasma
    nebular_plasma

TARDIS also allows for NLTE treatments of specified species, as well as special NLTE treatments for Helium.

.. note::
    The NLTE treatment of specified species is currently incompatible with the NLTE treatment for helium and cannot be used simultaneously.

.. toctree::
    :maxdepth: 2

    nlte
    helium_nlte
    


.. _tau_sobolev:

Sobolev optical depth
---------------------

After the above calculations, TARDIS calculates the Sobolev optical depth :math:`\tau_\textrm{Sobolev}` with the following formula:


.. math::
    C_\textrm{Sobolev} = \frac{\pi e^2}{m_e c}

    \tau_\textrm{Sobolev} = C_\textrm{Sobolev}\,  \lambda\, f_{\textrm{lower}\rightarrow\textrm{upper}}\,
        t_\textrm{explosion}\, N_\textrm{lower}
        (1 - \frac{g_\textrm{lower}}{g_\textrm{upper}}\frac{N_\textrm{upper}}{N_\textrm{lower}})
        

Macro Atom Line Interaction Treatment
-------------------------------------

The following page describes the macro atom treatment of line interactions:

.. toctree::
    :maxdepth: 2
    
    macroatom