Transport Methods

Table of Contents

• Introduction
• Energy Regimes
• Unbound Targets
• Bound Targets
• Continuous Temperature

Introduction

MiniMC currently supports two transport methods: surface tracking and intracell delta tracking. Surface tracking (implemented in SurfaceTracking) searches for the nearest surface intersection at each step. Intracell delta tracking (imlemented in CellDeltaTracking) also searches for the nearest surface intersection at each step but uses a majorant cross section within a Cell to find a distance to collision. Regular delta tracking does not search for the nearest surface interseaction at all but uses a majorant cross section across all Cell objects in the World to find a distance to collision.

Transport Methods and Features

Transport Method	\( T \) Dependence in Cell	Continuous \( T \) Thermal Scattering?	Supported?
Surface Tracking	Constant	No	Yes
		Yes
	Continuous	No	No, surface tracking assumes constant cross section within a Cell
		Yes
Intracell Delta Tracking	Constant	No	Yes, but somewhat pointless
		Yes
	Continuous	No	Yes, see Continuous Temperature
		Yes

The above table roughly corresponds to logic implemented in the definition of TransportMethod::Create.

Energy Regimes

When the temperature in a Cell is constant but thermal scattering data is provided through continuous-in-temperature \( S(\alpha, \beta, T) \) data, neutron transport must consider the following:

There are three regimes defined by two incident neutron threshold energies:

1. \( E_\text{thermal} \approx 5 \text{eV} \) below which neutrons are considered to be at thermal energies so bound targets must be handled using a thermal scattering law and above which targets are considered to be a free gas, and
2. \( E_{\text{static}} \approx 500 k T / A \) above which neutrons are considered to be moving fast enough relative to their target that the target can be treated as stationary.

Unbound Targets

For unbound targets and incident neutron energies \( E < E_{\text{static}} \), the target velocity is non-negligible so free gas scattering adjustments must be made to the scattering (and consequently, total) cross section

\[ \Sigma_s(E) = \Sigma_{\text{fr}} \left[ \left( 1 + \frac{1}{2 A x^{2}} \right) \text{erf} \left( \sqrt{A} x \right) + \frac{e^{-A x^{2}}}{\sqrt{\pi A} x} \right] \]

where \( x = \sqrt{E / k T} \) and \( A \) is the atomic weight ratio of the target. When performing delta tracking, a majorant \( \Sigma_m \) can be obtained by observing that

\[ \frac{\partial \Sigma_s}{\partial x} = -\Sigma_{\text{fr}} \frac{\text{erf}\left( \sqrt{A} x \right)}{A x^{3}} \leq 0 \]

so the maximum value of \( \Sigma_s \) occurs at the minimum value of \( x \) or at the maximum temperature in the cell.

Bound Targets

For bound targets, if \( E_{\text{thermal}} \leq E \), then the target is treated as a free gas and the same rules for unbound targets apply. If \( E < E_{\text{thermal}} \), then thermal scattering is valid and adjustments must be made to the scattering (and consequently, the total) cross section. The scattering cross section for a thermal neutron with incident energy \( E \) on a target at temperature \( T \) is

\[ \sigma_{s} (E \mid T) = \frac{\sigma_b A k T}{4 E} e^{-\beta / 2} \int_{\beta_\text{min}}^{\beta_{\text{max}}} \int_{\alpha_\text{min}}^{\alpha_{\text{max}}} S(\alpha, \beta, T) \, d\alpha \, d\beta \]

Though in principle this integral could be performed on-the-fly, this is currently evaluated by choosing some energy and temperature grid and precomputing the integral at specific values. Actual values of \( E \) and \( T \) encountered during transport are used to interpolate values from the grid.

Note

Address situations where linear interpolation is insufficient for approximating \( \sigma_{s} (E \mid T) \) for certain values of \( E \) and \( T \).

Continuous Temperature

If there is continuous temperature dependence within in a Cell, then transporting a neutron within that Cell will require the use of delta tracking. A majorant cross section for that Cell will have to account for all possible temperatures \( T \) within the Cell. Moreover, the majorant cross section \( \Sigma_{m}(E) \) for a particular nuclide will have to include adjustments due to free gas scattering and thermal scattering where applicable. Specifically, if

• the target is a bound to a molecule and \( E < E_{\text{thermal}} \),

then we require that

\[ \forall T, \quad \Sigma_{t, \text{tsl}}(E \mid T) \leq \Sigma_{m}(E) \]

where \( \Sigma_{t, \text{tsl}} \) is the total cross section after the scattering cross section has been adjusted by integrating \( S(\alpha, \beta, T) \) (usually during data processing). Otherwise, if

• the target is not bound to a molecule and \( E < E_{\text{static}} \), or
• the target is bound to a molecule and \( E_{\text{thermal}} \leq E < E_\text{static} \),

then we require that

\[ \forall T, \quad \Sigma_{t, \text{free gas}}(E \mid T) \leq \Sigma_{m}(E) \]

where \( \Sigma_{t, \text{free gas}} \) is the total cross section after the scattering cross section has been adjusted due to free gas scattering. If neither case applies, then it must be the case that \( E_\text{thermal} \leq E \) and \( E_\text{static} \leq E \) so no thermal scattering adjustment or free gas adjustment is necessary, respectively.