Collision probability method

In this paper, we review three developments of perturbation theory which have considerably broadened the scope and role it has to play in reactor physics. These three topics are (1) computing characteristics in subcritical systems, (2) perturbation theory for collision probability methods, and (3) computation of ratios in critical systems. 

Conventional perturbation theory will be of no help in the first-order approximation, since not only would the adjoint equation be space independent in the unperturbed homogeneous problem, but no distinction is made between the collisions with different nuclides in respect to the subsequent contribution to the reactivity. Thus, conventional first-order perturbation theory shows no heterogeneity effect. But we shall see that the collision probability method leads to a importance that varies with nuclides in the homogeneous problem and admits the computation of a finite heterogeneity effect even to first order. 

Development work is needed, although already underway, in applying the collision probability method to the problem of Doppler coefficients. Similarly, work in different models on the calculation of ratios such as power peaking factors and perturbations in these ratios would considerably assist design procedures. 

Again, the traditional calculational tool for this step has been integral transport (collision probability) methods, but discrete ordinates and Monte Carlo are seeing increased use. 

Grabejnoi, V.A., Usanov, V.I., A Collision Probability Method Applied to Neutron Shielding Calculations, Preprint-2270, IPPE, Obninsk, 1992. 

The ASMBURN assembly bumup calculation code is based on the neutron flux calculation by the collision probability method and the bumup calculation by interpolations of macro-cross sections [14]. As the bumup proceeds, the compositions of the fuel rods in the assembly start to differ from each other depending on the spatial distribution of the neutron flux. Therefore, a precise modeling would require production and decay calculations for each fuel rod constituting the fuel assembly. However, when the fuel rods are aligned in a regular lattice, the differences in the... 

Ingamm 3, a Code for the Calculation of Self-shielding Factors of Resolved Resonances using Collision Probability Methods. Not published. 

Experimental access to the probabilities P(E ,E) for energy transfer in large molecules usually involves teclmiques providing just the first moment of this distribution, i.e. the average energy (AE) transferred in a collision. Such methods include UV absorption, infrared fluorescence and related spectroscopic teclmiques [11. 28. 71. 72, 73 and 74]. More advanced teclmiques, such as kinetically controlled selective ionization (KCSI [74]) have also provided infonnation on higher moments of P(E ,E), such as ((AE) ). 

In the recollision method the probability of multiple collisions with the reaction site are considered. The reaction probability (]3 ) is expressed in terms of the recollision probability ), which is the probability that a molecule starting at r = f collides with the reaction surface rather than escaping to r=b, and the first collision probability (1 ), which is the probability that a molecule starting at the initiation surface collides with the reaction surface rather than escaping to r = oo. Both these probabilities are obtained from Brownian dynamics. In addition, the surface reaction probability ([Pg.812]

If one hopes to develop detailed, predictive models of plasmas, microscopic information such as electron-molecule collision probabilities clearly is needed. But why obtain that information from theory The short answer is that experimental data are often absent and—given the difficulty of the measurements and the paucity of research groups conducting them—in many cases are likely to remain so indefinitely. A longer answer would add that, as both theoretical methods and computer hardware improve, theory is, at least in some areas, becoming competitive with experiment in terms of accuracy and time to solution. 

The Monte Carlo method is a suitable way of obtaining values of P j which, when associated with estimates of M j, can be used to assess the criticality of an array. A version of MONK known as BLACK MONK has been written which calculates the collision probabilities, P,y, of neutrons arising from each source region in turn, having a first collision with each of the other units. The collision probability matrix is then multiplied by the surface multiplications, and Eq. (44) is solved for X. 

The collision probability is one of several possible formulations of integral transport theory. Three other formulations are the integral equations for the neutron flux, neutron birth-rate density, and fission neutron density. Oosterkamp (26) derived perturbation expressions for reactivity in the birth rate density formulation. The fission density formulation provides the basis for Monte Carlo methods for perturbation calculations (52, 55). 

The other approach, which is used at different laboratories, is the Monte Carlo method, which consists of following a large number of neutron histories, each collision being sampled from the known collision probabilities. 

The point at issue, however, is that any method based on the collision probability model will involve the estimation of the property perturbations the usefulness of having the perturbation expression is the decreased emphasis on having to solve for the perturbed density in order to obtain sufficiently accurate estimates of the perturbation in the eigenvalue. 

In the particular model of collision probability, a more elementary higher-order method is available from which to calculate the new density to considerable accuracy. This method is possible because the model contains only scalar elements in the matrix terms and may therefore easily be inverted. Thus the perturbed form of Eq. (46) can be written as... 

The SMC Method —A modified Monte Carlo treatment is used for the solution of the Boltzmann equation in the resonance region. The energy variation of a neutron history is treated exactly and the space-angle transport is approximated by a flat-source collision probability formulation. 

This does not happen with the clock synchronized method, which keeps transmitting in moments of high collision probability until the clock drifts separate the transmitting instants of the interfering processes, which may take a significant amount of time, potentially creating a long critical period. 

The Andersen thermostat is very simple. After each time step Si, each monomer experiences a random collision with a fictitious heat-bath particle with a collision probability / coll = vSt, where v is the collision frequency. If the collisions are assumed to be uncorrelated events, the collision probability at any time t is Poissonian,pcoll(v, f) = v exp(—vi). In the event of a collision, each component of the velocity of the hit particle is changed according to the Maxwell-Boltzmann distribution p(v,)= exp(—wv /2k T)/ /Inmk T (i = 1,2,3). The width of this Gaussian distribution is determined by the canonical temperature. Each monomer behaves like a Brownian particle under the influence of the forces exerted on it by other particles and external fields. In the limit i —> oo, the phase-space trajectory will have covered the complete accessible phase-space, which is sampled in accordance with Boltzmann statistics. Andersen dynamics resembles Markovian dynamics described in the context of Monte Carlo methods and, in fact, from a statistical mechanics point of view, it reminds us of the Metropolis Monte Carlo method. 

Marcus R A 1970 Extension of the WKB method to wave functions and transition probability amplitudes (S-matrix) for inelastic or reactive collisions Chem. Phys. Lett. 7 525-32... 

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