The Physics of Spallation Processes

4.2. NUCLEAR PHYSICS MODELS 37The boundary crossing estimator is the current J.∫ π ∫⃗J i (⃗x, t) = 2π sin θdθ dE cos θ ˙Φ i (⃗x, E, Ω, ⃗ t) (4.4)0ETo estimate the average fluence on a boundary, the factor 1/ cos θ for each particlehas to be added, where θ is the angle between the particle’s direction and the normalto the surface at the crossing point. Therefore the current is equal to the fluenceonly if all particles pass perpendicular to the surface.The density of particles or number of particles per volume element dxdydz is∫n(⃗x, t) = d ⃗ ∫Ω dE ˙Φ i (⃗x, E, Ω, ⃗ t)/v (4.5)4π EThe energy spectrum of particles can be expressed by∫˙Φ i (⃗x, E, t) = dΩ ⃗ ˙Φ i (⃗x, E, Ω, ⃗ t) (4.6)Essentially two classes of numerical procedures and special techniques have emergedfor solving the transport equation and finding expedient solutions to particular problems.On the one hand there are deterministic methods; the transport equation is discretizedusing a variety of methods and than solved directly or iteratively. As thereare the “straight ahead” approximation [Pas62, Als65], the “spherical harmonics” BLapproximation[Ben67, Joa63] and the methods of “discrete ordinates” (S N method)[Car64, Car68]. Secondly Monte-Carlo methods [Car75, Kah54] are found. They constructa stochastic model in which the expected value of a certain random variable isequivalent to the value of the physical quantity to be determined. The expected value isestimated by the average of many independent samples representing the random variable.Particle tracks or histories 1 are generated by simulating the real physical situation. Thereis not even the need to invoke the transport equation for more elementary operations.Only the complete mathematical description of probability relationships is needed thatgovern the track length of individual particles between interaction points, the choice ofinteraction type, the new energies and directions and the possible production of secondaryparticles. Especially for 3-dimensional problems S N methods and Monte-Carlo techniquesas used in the current work turned out to be most advantageous.4.2 Nuclear physics modelsThe predictive power of the models discussed in the following can be judged and rankedonly in comparison to high quality experiments. These experiments likewise serve thecomprehension of the physics implemented in the codes.The main objective is the development of powerful and accurate models for the descriptionof nucleon-nucleus spallation reactions, based on microscopic many-body theory.1 The experience a particle undergoes from the time it leaves its source until it is absorbed or until itleaves the system is called the particle’s history4π