2.5. Multigroup Neutron Data

This discusses the formation of the neutron data used in ANTS, as well as different coarsening processes for the total, scattering, and fission data.

2.5.1. Neutron Velocity

The neutron velocity \(v\) was calculated from the energy grid at the group centers \(E\) using the relativistic speed. The constants (neutron rest mass \(m\) and speed of light \(c\)) are found in the ants.constants.py file.

The classical energy to velocity equation is \(E = 0.5 m v^2\). However, since there are energies close to the speed of light, the relativistic equation and derivation is accomplished below. The original equation, from [Ber64], is

(1)\[ E = m c^2 \left( \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} - 1 \right)\]

which can be reformed as

(2)\[ \frac{E}{m c^2} + 1 = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}.\]

Setting the left hand side to \(\gamma\), we can solve for \(v\) as

(3)\[\begin{split}\begin{aligned} \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} &= \gamma \\ 1 - \frac{v^2}{c^2} &= \frac{1}{\gamma^2} \\ v^2 &= -\frac{c^2}{\gamma^2} + c^2 \\ v &= \sqrt{c^2 -\frac{c^2}{\gamma^2}} \end{aligned}\end{split}\]

This is simplified so

(4)\[ v = \frac{c}{\gamma} \sqrt{\gamma^2 - 1} \qquad \text{where} \qquad \gamma = \frac{E}{m c^2} + 1\]