<commit_before>.. _methods_depletion:

=========
Depletion
=========

When materials in a system are subject to irradiation over a long period of
time, nuclides within the material will transmute due to nuclear reactions,
producing other nuclides which may be radioactive. The time-dependent process by
which nuclides transmute and decay is known as *depletion* or *burnup*. To
accurately analyze nuclear systems, it is often necessary to predict how the
composition of materials will change since this change results in a
corresponding change in the solution of the transport equation. The equation
that governs the transmutation and decay of nuclides inside of an irradiated
environment can be written as

.. math::

    \begin{aligned} \frac{dN_i(t)}{dt} = &\sum\limits_j
    \underbrace{\left [ \underbrace{f_{j \rightarrow i} \int_0^\infty dE \;
    \sigma_j (E, t) \phi(E,t)}_\text{transmutation} +
    \underbrace{\lambda_{j\rightarrow i}}_\text{decay} \right ]
    N_j(t)}_{\text{Production of nuclide }i\text{ from nuclide }j} \\
    &- \underbrace{\left [\underbrace{\int_0^\infty dE \; \sigma_i
    (E,t) \phi(E,t)}_\text{transmutation} +
    \underbrace{\sum\limits_j \lambda_{i\rightarrow j}}_\text{decay} \right ]
    N_i(t)}_{\text{Loss of nuclide }i} \end{aligned}

where :math:`N_i` is the density of nuclide :math:`i` at time :math:`t`,
:math:`\sigma_i` is the transmutation cross section for nuclide :math:`i` at
energy :math:`E`, :math:`f_{j \rightarrow i}` is the fraction of transmutation
reactions in nuclide :math:`j` that produce nuclide :math:`i`, and
:math:`\lambda_{j \rightarrow i}` is the decay constant for decay modes in
nuclide :math:`j` that produce nuclide :math:`i`. Note that we have not included
the spatial dependence of the flux or cross sections. As one can see, the
equation simply states that the rate of change of :math:`N_i` is equal to the
production rate minus the loss rate. Because the equation for nuclide :math:`i`
depends on the nuclide density for possibly many other nuclides, we have a
system of first-order differential equations. To form a proper initial value<commit_msg>Revise first paragraph in depletion methods docs (thanks @drewejohnson)<commit_after>.. _methods_depletion:

=========
Depletion
=========

When materials in a system are subject to irradiation over a long period of
time, nuclides within the material will transmute due to nuclear reactions as
well as spontaneous radioactive decay. The time-dependent process by which
nuclides transmute under irradiation is known as *depletion* or *burnup*. To
accurately analyze nuclear systems, it is often necessary to predict how the
composition of materials will change since this change results in a
corresponding change in the solution of the transport equation. The equation
that governs the transmutation and decay of nuclides inside of an irradiated
environment can be written as

.. math::

    \begin{aligned} \frac{dN_i(t)}{dt} = &\sum\limits_j
    \underbrace{\left [ \underbrace{f_{j \rightarrow i} \int_0^\infty dE \;
    \sigma_j (E, t) \phi(E,t)}_\text{transmutation} +
    \underbrace{\lambda_{j\rightarrow i}}_\text{decay} \right ]
    N_j(t)}_{\text{Production of nuclide }i\text{ from nuclide }j} \\
    &- \underbrace{\left [\underbrace{\int_0^\infty dE \; \sigma_i
    (E,t) \phi(E,t)}_\text{transmutation} +
    \underbrace{\sum\limits_j \lambda_{i\rightarrow j}}_\text{decay} \right ]
    N_i(t)}_{\text{Loss of nuclide }i} \end{aligned}

where :math:`N_i` is the density of nuclide :math:`i` at time :math:`t`,
:math:`\sigma_i` is the transmutation cross section for nuclide :math:`i` at
energy :math:`E`, :math:`f_{j \rightarrow i}` is the fraction of transmutation
reactions in nuclide :math:`j` that produce nuclide :math:`i`, and
:math:`\lambda_{j \rightarrow i}` is the decay constant for decay modes in
nuclide :math:`j` that produce nuclide :math:`i`. Note that we have not included
the spatial dependence of the flux or cross sections. As one can see, the
equation simply states that the rate of change of :math:`N_i` is equal to the
production rate minus the loss rate. Because the equation for nuclide :math:`i`
depends on the nuclide density for possibly many other nuclides, we have a
system of first-order differential equations. To form a proper initial value
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