Editing Neutron transport (section)

==Neutron transport equation==

The neutron transport equation is a balance statement that conserves neutrons. Each term represents a gain or a loss of a neutron, and the balance, in essence, claims that neutrons gained equals neutrons lost. It is formulated as follows:<ref name=Adams>{{cite book |last=Adams |first=Marvin L. |title=Introduction to Nuclear Reactor Theory |year=2009 |publisher=Texas A&M University}}</ref>
:<math>\left(\frac{1}{v(E)}\frac{\partial}{\partial t}+\mathbf{\hat{\Omega}}\cdot\nabla+\Sigma_t(\mathbf{r},E,t)\right)
\psi(\mathbf{r},E,\mathbf{\hat{\Omega}},t)=\quad</math>

:<math>\quad\frac{\chi_p \left( E \right)}{4\pi}\int_0^{\infty} \mathrm dE^{\prime}\nu_p \left( E^{\prime} \right) \Sigma_f \left(\mathbf{r}, E^{\prime}, t \right) \phi \left( \mathbf{r}, E^{\prime}, t \right) + \sum_{i=1}^N \frac{\chi_{di}\left( E \right)}{4\pi} \lambda_i C_i \left( \mathbf{r}, t \right)+\quad</math>

:<math>\quad\int_{4\pi}\mathrm d\Omega^\prime\int^{\infty}_{0}\mathrm dE^\prime\,\Sigma_s\!\!\left(\mathbf{r},E^\prime\rightarrow E,\mathbf{\hat{\Omega}}^\prime\rightarrow \mathbf{\hat{\Omega}},t\right)\psi(\mathbf{r},E^\prime,\mathbf{\hat{\Omega}^\prime},t)+s(\mathbf{r},E,\mathbf{\hat{\Omega}},t)</math>

Where:
{| class="wikitable" border="1"
|-
! Symbol
! Meaning
! Comments
|-
| <math>\mathbf{r}</math>
| Position vector (i.e. x, y, z)
|
|-
|<math>E</math>
|Energy
|
|-
|<math>\mathbf{\hat{\Omega}}=\frac{\mathbf{v}(E)}{|\mathbf{v}(E)|}=\frac{\mathbf{v}(E)}{{v(E)}}</math>
|Unit vector ([[solid angle]]) in direction of motion
|
|-
|<math>t</math>
|Time
|
|-
|<math> \mathbf{v}(E) </math>
|Neutron velocity vector
|
|-
|<math>\psi(\mathbf{r},E,\mathbf{\hat{\Omega}},t)\mathrm dr\,\mathrm d\!E\,\mathrm d\Omega</math>
|Angular neutron flux <br /> Amount of neutron track length in a differential volume <math>\mathrm dr</math> about <math>r</math>, associated with particles of a differential energy in <math>\mathrm dE</math> about <math>E</math>, moving in a differential solid angle in <math>\mathrm d\Omega</math> about <math>\mathbf{\hat{\Omega}},</math> at time <math>t.</math>
|Note integrating over all angles yields [[Neutron flux|scalar neutron flux]]<br /> <math>\phi \ = \ \int_{4\pi}\mathrm d\Omega\psi</math>
|-
|<math>\phi(\mathbf{r},E,t)\mathrm dr\,\mathrm dE</math>
|[[Neutron flux|Scalar neutron flux]] <br /> Amount of neutron track length in a differential volume <math>\mathrm dr</math> about <math>r</math>, associated with particles of a differential energy in <math>\mathrm dE</math> about <math>E</math>, at time <math>t.</math>
|
|-
|<math>\nu_p</math>
|Average number of neutrons produced per fission (e.g., 2.43 for U-235).<ref>{{cite web|title=ENDF Libraries|url=https://www-nds.iaea.org/exfor/endf.htm}}</ref>
|
|-
|<math>\chi_p(E)</math>
|Probability density function for neutrons of exit energy <math>E</math> from all neutrons produced by fission
|
|-
|<math>\chi_{di}(E)</math>
|Probability density function for neutrons of exit energy <math>E</math> from all neutrons produced by delayed neutron precursors
|
|-
|<math>\Sigma_t(\mathbf{r},E,t)</math>
|Macroscopic total [[Nuclear cross section|cross section]], which includes all possible interactions
|
|-
|<math>\Sigma_f(\mathbf{r},E^{\prime},t)</math>
|Macroscopic fission [[Nuclear cross section|cross section]], which includes all fission interactions in <math>\mathrm dE^{\prime}</math> about <math>E^{\prime}</math>
|
|-
|<math>\Sigma_s\!\!\left(\mathbf{r},E'\rightarrow E,\mathbf{\hat{\Omega}}'\rightarrow \mathbf{\hat{\Omega}},t\right)\mathrm dE^\prime \mathrm d\Omega^\prime</math>
|Double differential scattering cross section<br />Characterizes scattering of a neutron from an incident energy <math>E^\prime</math> in <math>\mathrm dE^\prime</math> and direction <math>\mathbf{\hat\Omega^\prime}</math> in <math>\mathrm d\Omega^\prime</math> to a final energy <math>E</math>  and direction <math>\mathbf{\hat{\Omega}}.</math>
|
|-
|<math>N</math>
|Number of delayed neutron precursors
|
|-
|<math>\lambda_i</math>
|Decay constant for precursor ''i''
|
|-
|<math>C_i \left( \mathbf{r}, t \right)</math>
|Total number of precursor ''i'' in <math>\mathbf{r}</math> at time <math>t</math>
|
|-
|<math>s(\mathbf{r},E,\mathbf{\hat{\Omega}},t)</math>
|Source term
|
|-
|}

The transport equation can be applied to a given part of phase space (time ''t'', energy ''E'', location <math>\mathbf{r},</math> and direction of travel <math>\mathbf{\hat{\Omega}}.</math>) The first term represents the time rate of change of neutrons in the system. The second terms describes the movement of neutrons into or out of the volume of space of interest. The third term accounts for all neutrons that have a collision in that phase space.  The first term on the right hand side is the production of neutrons in this phase space due to fission, while the second term on the right hand side is the production of neutrons in this phase space due to delayed neutron precursors (i.e., unstable nuclei which undergo neutron decay).  The third term on the right hand side is in-scattering, these are neutrons that enter this area of phase space as a result of scattering interactions in another. The fourth term on the right is a generic source.  The equation is usually solved to find <math>\phi(\mathbf{r},E),</math> since that will allow for the calculation of reaction rates, which are of primary interest in shielding and dosimetry studies.