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Fix spelling errors from second round of review comments
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nglaser3 committed May 31, 2024
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## Multigroup Neutron Diffusion

The neutron diffusion equation is an approximation to the Boltzmann transport equation, and is derived by taking the zeroth and first moment with respect to $\hat\Omega$, the direction of neutron travel.
A full derivation of the diffusion equation will not be presented here for conciceness. Importantly, when taking the first moment of the transport equation the angular flux is assumed to be linearly anisotropic.
A full derivation of the diffusion equation will not be presented here for conciseness. Importantly, when taking the first moment of the transport equation the angular flux is assumed to be linearly anisotropic.
The approximations reduce the phase space through the elimination of angular dependence, but the resulting equation also has reduced fidelity when compared to the transport equation, particularly in regions where the neutron flux has strong angular dependence.
A non-exhaustive list of regions in which the neutron flux strongly depends on angle is near material interfaces between materials with highly dissimilar neutronic properties, within strong absorbers, and within near-void regions.
As a dertiministic method, the neutron diffusion method also requires discretization of the continuous energy dependence into energy groups consisting of non-overlapping, finite energy ranges across the entire energy spectrum.
As a deterministic method, the neutron diffusion method also requires discretization of the continuous energy dependence into energy groups consisting of non-overlapping, finite energy ranges across the entire energy spectrum.
This energy discretization creates a system of equations referred to as the multigroup neutron diffusion equations:

!equation
\frac{1}{v_g} \frac{\partial\phi_g}{\partial t}-\nabla \cdot D_g \nabla \phi_g +\Sigma^R_{g} \phi_g =\sum^G_{g \neq g'} {\Sigma^s_{g'\rightarrow g} \phi_{g'}}+ \chi^p_{g} \sum^G_{g'=1} {\left(1- \beta \right) \nu_{g'} \Sigma^f_{g'}\phi_{g'} }+\chi^d_{g} \sum^I_i {\lambda_i C_i}

where the precursor distributions are governed by:
The delayed neutron precursor distributions are governed by:

!equation
\frac{\partial C_i}{\partial t}=\sum^G_{g'=1}{\beta_i \nu \Sigma^f_{g'}\phi_{g'}}-\lambda_i C_i-\vec{u} \cdot \nabla C_i


Notably, there are two production terms of neutrons, the prompt fission source and delayed neutron precursor decay source.
The first term describes the neutrons immediately born from fission, and the second term describes the neutrons born from the radioactive decay of neutron-emitting radionuclides, commonly called delayed neutron precursors.
The multigroup neutron diffusion equations are generally impossible to solve analytically for realistic problems and are therefore typically solved with numerical methods.
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