Global Neutron Calculations

Nuclear energy production. Self-sustaining chain reaction. The outline of the energy production. Reactor models in large. Applications of reactor models. Validation and verification. Regulation, surveillance.

Binding energy of nucleons. Isotopes. Nuclear reactions, XSs (in nuclear physics and in reactor physics). Scattering and fission XS with energy and Ω and v. Potential scattering. Energy production by fission. Possible fissionable isotopes. Fission spectrum. Nucleus-neutron reactions. Fission products, yields, fissionable products, highly absorbing products, resonances (single level and many level processes). Nuclear physics. Evaluated nuclear data file. Enrichment. Self sustaining chain reaction. Reaction rate. Cross-section, absorption, fission, capture, mean free path. Basic features of the neutron gas: flux, current, partial current. The linearized Boltzmann equation. Neutron flux, current, partial current (partial, net), XS, reaction rates. Neutron balance, leakage, removal, scattering, fission. The Boltzmann equation. Boundary conditions, initial conditions. Criticality, keff . Integral form of the transport equation. The adjoint flux. Phase space.

Green’s function. Neumann series. Case method. Perturbation theory. Variational methods. Response matrices. Heat conductance. Coolant flow. Conservation laws.

Need for approximations. Asymptotic theory. Finite lattice. Heterogeneous lattices. Homogenization methods. Heterogeneous core calculation. Energy groups. Pn method. Sn method. Collision probability method. Boundary conditions. Space variable: finite difference and nodal method. Randomness. Monte Carlo method. Statistical errors.

Approximations in the cell calculation, assembly calculation, and full core calculation. Reflector models, and burnup chains.

Pn method and diffusion theory (Fick-l w); diffusion equation, boundary conditions; solutions of diffusion equation (diffusion kernel and external sources, Helmholtz-equation), buckling, criticality conditions. Discretization. Solution methods (one group and multigroup, scattering matrices).

Fast neutrons. Slowing down by elastic scattering. Placzek transients. Resonances. The Doppler effect. Thermalization. Wigner- Wilkins, Nelkin model, Cadilhac model, influence of leakage, THERMOS. Validation and verification.

Weak form of an equation. Reduction of operators to finite matrices. Acceleration. Numerical methods: finite difference, finite elements, nodal methods. Thermal hydraulics. Lattice Boltzmann method. Computational fluid dynamics.

Delayed neutrons, their effect on the structure of the neutron balance. Delayed neutron groups. Reactor kinetics. Approximate solution. Deterministic and stochastic reactivity measurement.

In core instrumentation. Core surveillance. Measured fields. Measurement techniques. Core monitoring. Measurement and safety. Core monitoring in Generation III reactors. Core surveillance at VVER- 440. Core monitoring principles.

How do we know that a given core is operating safely? Power distribution limits. Role of computer models. Calculational model and reactor operation. Core optimization. Equilibrium fuel cycle. Reactivity coefficients and feedback. Regulation of the reactor. Safety issues. Limitations and their respective natures. Maneuvering with the reactor.

Nuclear reactors for electricity production. Nuclear fuel. Power plant types and generations. PWR technology. Safety design features.

A brief summary of applied mathematics. Vector, matrix, tensor notation. The nabla operator. Helmholtz theorem. Bickley functions. Fredholm alternative theorem. Bessel functions. Cauchy principal value integral. Walsh functions.

Random neutron number. Generating function. Moments from the generating function. Autocorrelation, cross-correlation. Autocovariance. Power spectrum. Cross-power spectrum. Kolmogorov forward and backward equation, Pál-Bell equation.

A simple but effective mathematical model has been elaborated by Casti [158]. The resulting equation (D.8) for the number of neutron multiplication is a variant of the diffusion equation.

Basic mathematical terms for the numerical methods. The equation under consideration is regarded as an operator acting on the solution.

By exploiting the symmetries of the problem, effective numerical methods can be worked out. We investigate the symmetries of the transport operator.