Multigroup theory

My own experience with calculating machines is so limited that it would be difficult for me to deal with their use in any detail. This would also be unnecessary, since the use of computers will be discussed by Drs. Ehrlich, Varga, Richtmyer and Carlson along with the mathematical models, such as multigroup theories, which the availability of computers inspired. 

It is possible to develop a proper multigroup theory based upon group cross sections obtained by bilinear flux-adjoint weighting 40). It may even be more accurate in some applications than the normal procedure used here. But, in principle, provided the flux and adjoint spectra are accurately known for the purpose of obtaining group cross sections, either procedure is a valid one, and they will both give the same answer. 

These equations for F, necessarily have the same eigenvalues as the equations for (j>i. Yet the multigroup constants are different, and Fj is the total reaction rate rather than the flux. The are the fractions of the total reaction rate of type x and replace the cross sections of normal multigroup theory. 

D + 6D, and S + SJL. In one group theory, the flux and adjoint flux are identical, but the derivation that follows differentiates between them because it eases the generalization to multigroup theory. However, it is assumed that the perturbations are small enough so that difference between the perturbed and unperturbed flux can be ignored, and so the prime notations will be dropped. 

As before, it is assumed that diffusion theory is valid, and one-group theory is utilized, although the extension to multigroup theory will be plain. 

Although the flux and adjoint flux are identical in one-group theory, the two quantities are retained so that the extrapolation to multigroup theory will be plain.)... 

In the case of some measurements, it is possible to interpret the data in several different ways. For instance, measurements of migration area may be interpreted by age-diffusion, age, one group, or multigroup theory. Numerical values of M as computed by the different methods will vary coMiderably, although each will form a con-slstent set of reactor parameters in its own critical equation, hi addition, there appears to be agreement in values of M as measured by subcritical and by critical. methods. Thus, it is necessary to make any comparison between critical and subcritical experiments with a common theoretical interpretation. 

Comparing calculated and experimentally determined reactivity worth enables verification of the accuracy qf nuclear data and the adequacy of computational methods used. For a meaningful comparison of theory and experiment, it is essential that the perturbation theory expressions used for the calculations apply to exactly the same parameter as that deduced fi-om the experiments, and that these expressions are evaluated accurately. This paper reviews three aspects of accurate determination of reactivity (1) the definition of reactivity, (2) high-order perturbation theory expressions for reactivity, and (3) the accuracy of computational techniques based on the multigroup approximation. 

What about situations—as in heterogeneous reactors or small reactors— where elementary theory is inaccurate In this case one can attempt an analytic solution of the Boltzmann equation—as is proposed by Professor Wigner or one can make the spherical harmonics expansion and try to solve for each component or one can simply apply the multigroup ... 

Derivation of the Fermi age equation. The Boltzmann equation is an integro-dififerential equation involving distance, energy and directional variables. By making suitable approximations we shall now proceed to reduce this equation to a set of coupled differential equations in which the spatial coordinates are the only independent variables. These are the so-called multigroup diffusion equations. But first we discuss the so-caUed age theory. 

R. Ehrlich, One dimensional multigroup calculations Estimation of group constants. Proceedings of Symposium on Nuclear Reactor Theory, vol. 11, 1960, pp. 151-163. 

Continuous models. The extension from the finite models of 2 to multigroup diffusion and transport theory models involving a continuum of possible positions and velocities, involves some very careful technical analysis. 

EXISTENCE THEOREMS AND SPECTRAL THEORY FOR THE MULTIGROUP DIFFUSION MODEL... 

B. J. Garrick, Multigroup-multiregion theory for the consistent approximation to the Boltzmann equation, ORNL-CF-55-8-189, August, 1955 (also ORNL 2081, p. 133) (See Reference 13). 

J. Franklin and E. J. Leshan, A multigroup multiregion one space-dimensi(mal program using neutron diffusion theory, ASAE-4, December, 1956. 

R. N. Stuart, E. H. Canfield, E. E. Dougherty and S. P. Stone, ZOOM—A one-dimensional, multigroup, neutron diffusion theory reactor code for the IBM-104, UCRL-5293, November, 1958. 

R. S. Varga and M. A. Martino, The theory for the numerical solution of time-dependent and time-independent multigroup diffusion equations. Proceedings of the Second International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1958, vol. 16, pp. 570-577. 

For further details regarding transport theory, the multigroup formulation, the physical assumptions involved here, and the problem of constructing appropriate cross sections (a, (jg g), reference is made to [1] and to a recent paper by Wilkins [9]. 

The influence of Doppler effect on reactivity is determined through its effect upon the group cross sections of multigroup diffusion theory. These are defined by... 

Thus, Ij. is the contribution to the effective group cross section for group / due to process x from all resonances of sequence (i). It is shown in Section IV that, if < ( ), the energy-dependent neutron flux, is known exactly, then, as is well known, this definition of the multigroup cross sections leads to exactly the same value of the multiplication constant as the continuous energy diffusion theory. 

A second way to obtain the difference between the two eigenvalues is to use multigroup perturbation theory, which, for a small change in capture and fission cross sections only, states that the change in eigenvalue is given by... 

In the present work, it will be assumed that 5v is to be calculated either from the solution of the two sets of multigroup equations of the form (33) with the effective fission and capture cross sections defined by (27) and (28) or by one such set of multigroup equations and the first-order perturbation theory of (34). We require the same effective cross sections for either procedure. Various appropriate approximations are employed in the application of these formulas. 

Almost all of the practical calculations of the Doppler effect in fast reactors are carried out by the conventional method, which consists of tracing through the expressions for the effective cross sections, as developed in this paper for the different fuel isotopes and energy ranges. Then, one can either run two multigroup diffusion calculations with the effective cross sections at different temperatures and obtain the reactivity changes associated with the Doppler effect as the difference of the two criticality factors, or else one can do only one multigroup calculation and find the Doppler coefficient by the use of perturbation theory. Both ways are, of course, equivalent from a mathematical point of view. 

The numerical work in this paper considers a reactor that can be described in terms of one-group diffusion theory. However, the derivations included in this Appendix indicate the modifications required to utilize multigroup diffusion theory. The diffusion equation is... 

---