Calculations in stochastic neutron transport
||Calculations in stochastic neutron transport|
||SNIC Small Compute|
||Imre Pazsit <firstname.lastname@example.org>|
||Chalmers tekniska högskola|
||2020-08-01 – 2021-08-01|
Detecting and characterising fissile material in nuclear safeguards, and determining the reactivity (criticality or subcriticality) of a reactor is based on the statistical properties of the detection of neutrons generated in or emitted from the system. The underlying reason is that fission chains constitute a branching process, which exhibits non-trivial statistical properties, such that each statistical moment carries independent information. Hence, measuring several moments (mean, variance, skewness etc). makes it possible to unfold important parameters of the system, in addition in an absolute way, i.e. without calibration.
In order to extract the desired information (parameters), which is a so-called inverse problem, it is necessary to have a solution of the direct problem, i.e. to calculate how the various moments depend on the parameters of the system. This necessitates the solution of the stochastic neutron transport problem. In a deterministic setting, neutron transport is described with the neutron transport equation, which is an integro-differential (or purely integral) equation. Due to the difficulties associated with the properties of integral equations, and the boundary conditions, this transport equation cannot be solved analytically even for the simplest cases. When a transport equation is needed not for the expectation, but for the whole probability distribution (or at least for the higher order moments), the situation will be even more complicated. The higher order moments are also integral equations, but have a non-linear combination of the lower order moments as an inhomogeneous part of the equation.
In our recent work we have developed a theoretical extension of the calculations of the first few order moments of the number of neutrons from the traditional space- and energy-independent framework (which leads to simple nonlinear algebraic equations) to a space- and angularly dependent description, leading to transport equations. The first three moments of the process were calculated in a homogeneous sphere with an isotropic distribution of fission neutrons (both spontaneous and induced. For the quantitative calculations the numerical options of the symbolic computation code Mathematica were used. The numerical solution of the integral equations was based on a Neumann-series expansion, which took several hours on a laptop (Macbook Pro) for the number of different system sizes that we want to study.
A new theoretical development is the observation that the moment equations can be solved by the Green's function technique, where all moments can be calculated with the same Green's function. However, it introduces new variables, hence the Green's function must be calculated by a Neumann series expansion for more cases than the number of moments. On the other hand, these calculations can be made parallel to each other, but these require parallel computing capabilities, which are provided by the SUPR resources.