For a reactor with no external sources the diffusion eqn. (4.39) takes the form

(4.59)

Let us now consider the case of a homogeneous medium with energy independent boundary conditions of the type

(4.60)

where дфідп is the derivative in the direction perpendicular to the boundary, and d the so-called extrapolation distance. This condition is equivalent to ф = 0 at the extrapo­lated boundary (except in the case d ->» in which the condition is rather equivalent to дфідп = 0 at the boundary).

In this case the wave equation

V2f(r) + Bffi (r) = 0

determines a complete system of eigenfunctions on the considered domain so that one can in general expand the flux

(4.62)

One can substitute this expansion in the diffusion equation (4.59). If the equation so obtained is multiplied by f (r) and integrated over the whole volume, remembering that

these eigenfunctions are orthogonal, i. e. and

V2f,=-B2ft

we have

DB2<Pi(E) + X,(Pi(E) = J 2s0(E’^ £)<?,(£’) d£’ +

This equation can only be satisfied by one value of В from eqn. (4.62)

<Hr, E) = <p(E)f(r).

As ф(г, Е) must be positive throughout the reactor, f(r) must be the fundamental solution of the wave equation (4.61) corresponding to the smallest eigenvalue B2.

This is only valid if the coefficients D, X,, Xf, etc., are space independent and can be extracted from the integral over the reactor volume.

Under the above-mentioned restrictions (homogeneous medium with energy inde­pendent boundary conditions) is valid what is usually called

First Fundamental Theorem of Reactor Theory (see ref. 1, p. 382):

The stationary neutron distribution in a critical bare reactor is separable in space and energy

ф (r, E) = <p (E)f(r) (4.63)

where f(r) is the fundamental solution of the wave equation

V2/(r) + B2/(r) = 0 (4.64)

that is that solution which is positive throughout the reactor and vanishes at the extrapolated boundary.

The equation for cp(E) is then

DB2<p(E) + X,<p{E) = jXs0(E’^E)<p(E’)dE’+ f v(E’)X,(E’)<p(E’) dE’ (4.65)

and the leakage term is expressed as function of В2 which is called geometrical buckling and is the smallest eigenvalue of (4.64).

DV2</> (r, E) = D<p(E)V2f(r) = DB2<p(E)f(r) = — DB2<p(rE) (4.66)

it is also possible to obtain this theorem starting from the Boltzmann equation, for the flux integrated over the angle.<23)

This first fundamental theorem is of limited practical use because it applies only to a homogeneous bare reactor. In a reflected reactor, or in any region of a multi-region reactor, the boundary conditions (4.60) are energy dependent, for example there is a leakage of fast neutrons towards the reflector and a return of thermal neutrons to the core. In practice the problem is solved in multi-group formalism introducing group dependent bucklings B2 for each group i.<24)

Within each group і eqns. (4.63) and (4.64) are valid if the boundary conditions are supposed to be energy independent in each of these energy ranges. In this way it is possible in eqn. (4.40) to substitute

-ДУ2фі(г) = ПВі2фі(г). (4.67)

The leakage term takes then the form of an absorption term and the total losses are

(QB,2 + 2„)«fr.

This substitution can be used to reduce a space-dependent to a space-independent problem, e. g. in the case of spectrum calculations (see § 8.2) provided the В2 are known. In order to obtain the B2 it is theoretically necessary to solve the multi-group space-dependent problem, but a rough approximation of the Bf is sufficient for many applications. Energy-dependent bucklings can also be used to reduce the number of space dimensions. This implies the assumption of separability of the space dependence of the flux along the different coordinates. It is then possible to solve a three­dimensional problem treating explicitly the space dependence in one or two dimen­sions, representing the leakage in the other directions by means of energy-dependent bucklings.

For example,

-DV20(x, У, z) = — DV2[0(x) • ф(у) • ф(2)] = — D~ ф + DBІуф.

This separation of the x, у and z dependence constitutes, of course, an approximation which has to be justified from case to case.

References

1. A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors, University of Chicago Press, 1958.

2. G. E. Bell and S. GlaSSTOne, Nuclear Reactor Theory, Van Nostrand Reinhold Co., 1970.

3. B. G. Carlson and K. D. Lathrop, Transport theory—the method of discrete ordinates. In Computing Methods in Reactor Physics ; edited by H. Greenspan, C. Kelber and D. Okrent, Gordon & Breach, 1968.

4. M. M. Merrill, Nuclear design methods and experimental data in use at Gulf General Atomic, Gulf-Ga-A 12652 (GA-LTR-2), July 1973.

5. K. D. Lathrop, DTF-IV, a Fortran-IV program for solving the multigroup transport equation with anisotropic scattering, LA-3373, 12 Nov. 1965.

6. K. D. Lathrop and F. W. Brinkley, Theory and use of the general geometry TWOTRAN program, LA-4432, May 1970.

7. S. FranceSCON, The Winfrith DSN programme, AEEW-R273.

8. L. Massimo, A programme for the solution of the monoenergetical transport equation in the P 5 approximation for a multiregion cylindrical geometry, EUR-2584e, 1965.

9. G. C. PomraninG, Reduction of transport theory to multigroup diffusion theory. Journal of Nuclear Energy, Parts A/B, 18, 497-512 (1964).

10. G. D. JoanOU and J. S. Dudek, GAM-II, a B3 code for the calculation of fast neutron spectra and associated multigroup constants, GA-4265, Sept. 1963.

11. M. M. R. Williams, The Slowing Down and Thermalization of Neutrons, North-Holland Publishing Co., 1966.

12. H. C. HonECK, THERMOS—a thermalization transport theory code for reactor lattice calculations, BNL-5826 (1961).

13. J. E. BeardwOOD, The solution of the transport equation by collision probability methods, ANL-7050, p. 93, 1965.

14. J. R. Askew, Review of the status of collision probability methods, IAEA-SM-154/69-A.

15. В. K. Chesterton, “MINOTAUR” TNPG/PDI 378, unpublished.

16. R. Bonalumi, Neutron first collision probabilities in reactor physics. Energia Nucleare, 8, 326 (1961).

17. J. R. Askew, Some boundary condition problems arising in the application of collision probability methods, IAEA-SM-154/69-B.

18. J. R. Askew, A coarse mesh correction for collision probabilities, AEEW-M889, June 1969.

19. J. R, Askew, Mesh requirements for neutron transport calculations, AEEW-M760.

20. R. H. W. Stace, A review of the use of collision probabilities in lattice calculations. Bournemouth Symposium on the Physics of Graphite Moderated Reactors, 4 to 7 April, 1962.

21. A. Jansson, THESEUS—a one group collision probability routine for annular systems, AEEW-R253, 1963.

22. M. H. Kalos, F. R. Nakache and J. Celnik, Monte Carlo Methods in Reactor Computations. In Computing Methods in Reactor Physics, edited by H. Greenspan, C. Kelber and D. Okrent, Gordon & Breach, 1968.

23. S. Corno, Report to be published.

24. T. D. Beynon, On the concept of energy dependent buckling in multi-group diffusion theory. Journal of Nuclear Energy, 25, 503-511 (1971).