## 2 Analytical Methodology
### Mcn  
The MCN method is utilized for simulating the neutron generations approximately using the discrete batches. In each batch, the computationally pragmatic numbers of neutrons are initialized. Each neutron's history from birth (because of the external source or a fission event) to death (due to absorption or leakage out of the calculation domain) is simulated independently. The user can state the number of batches and the number of histories in a batch. Based on the Monte Carlo spirit, sampling via employing the proper probability density functions (cross sections) with generated random numbers can determine the history of each neutron in a batch with a nature of probabilistic occurrences, such as whether the neutron scattering or absorption events occur when the incident neutron collides with the medium, whether the fission event is induced if the neutron absorption event happens, how many numbers of neutrons are generated, and what the new directions will be and how far they would travel before their subsequent collision events happened in a fission event. The positions where the fission events happen at the end of a batch will serve as the new neutron sources' distribution for the next batch. Meanwhile, the number size of the fission neutrons will be normalized as the neutron weight during the calculation to preserve each batch's initiating fission neutrons number. The specific contributions of the histories in a batch can be tallied further to reckon the \(k_{eff}\) eigenvalue of that batch. MVP3 provides two methods to evaluate \(k_{eff}\) to solve the eigenvalue problems of a system [33]. The first method is based on the definition of the neutron multiplication, as follows:  
\[k_{eff}=\frac{F}{N}, \tag{20}\]  
where \(F\) [-] is the contribution from the weight of neutrons created by the fission events, and \(N\) [-] is from the weight of source neutrons. The other method is based on the equation of neutron balance, as below:  
\[k_{eff}=\frac{F}{L+NA}, \tag{21}\]  
where \(L\) [-] is the contribution from the weight of neutrons leaking out of the calculation domain, and \(NA\) [-] means the weight of the neutrons under the net absorption events. In MVP3, there are three alternative estimators (Collision estimator, Analog estimator, and Track length estimator [33]) to evaluate the contributions related to the incident neutrons colliding with the medium of a batch (\(F\) and \(NA\)). The ways to evaluate \(F\) and \(NA\) via the Collision estimator are as below:  
\[F=\sum_{n}W_{n}\frac{(\overline{v}_{n}\Sigma_{f,n})}{\Sigma_{t,n}}, \tag{22}\]  
\[NA=\sum_{m}W_{m}\frac{\Sigma_{NA,m}}{\Sigma_{t,m}}, \tag{23}\]  
where \(n\) and \(m\) express the indexes for the neutrons inducing the fission and the net absorption events in a batch, respectively. \(W_{n}\) [-] and \(W_{m}\) [-] are the weight of the neutron \(n\) and \(m\). \(\Sigma_{f,n}\) [1/m] and \(\overline{v}_{n}\) [-] are the macroscopic fission cross section and the average number of generated neutrons in the neutron \(n\)'s fission event. \(\Sigma_{t,n}\) [1/m] and \(\Sigma_{t,m}\) [1/m] are the macroscopic total cross sections for the neutron \(n\) and \(m\), respectively. \(\Sigma_{NA,m}\) [1/m] stands for the macroscopic net absorption cross section in the net absorption event of neutron \(m\), which can be expressed as below:  
\[\sum_{NA,m}=(\sum_{a}-\sum_{m,2n}-2\sum_{m,3n}-3\sum_{m,4n})_{m}, \tag{24}\]where \(\sum_{a}\) [1/m] is the macroscopic absorption cross section. \(\sum_{n,2n}\) [1/m], \(\sum_{n,3n}\) [1/m], and \(\sum_{n,4n}\) [1/m] are the macroscopic cross sections for (\(n\), 2\(n\)), (\(n\), 3\(n\)), and (\(n\), 4\(n\)) reactions, respectively. The forms used to estimate \(F\) and \(NA\) by the Analog estimator are based on the Collision estimator, just substituting the microscopic types for the macroscopic cross sections [33].