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Nuclear Fuel and Control Rods - Oper Experience, Design,

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1 - 60) - (D0/v)]M(E) / (l/v)M(E) dE dE (100) It is seen that Eqs. (99) and (100) are identical with those of the isotropie scattering case, Eqs. (96) and (97), respectively, if instead of S s (l-bi) = Str, one uses the scattering cross section, 2S, only. The foregoing analysis is based on the consideration of an infinite medium and consequently it is adequate for sufficiently large samples. However, in applying infinite-medium results to the decay of a thermalized pulse in a finite assembly, care should be taken in the use of the extrapolated end point, especially in the case of an energydependent transport mean free path.

If now we write Ψφη = -£η2φη(Γ), with Bn2 being the geometric buckling associated with the nth mode, Eq. (Ε')Φη(Ε', t)F(E' -> E) dE'\ = 0, (51) Bn being the Fourier transform variable. Equation (51) may be re­ duced to an eigenvalue equation if an expansion of Φη(Ε, t) of the form Φη(Ε, t) = Σ Φηΐ{Ε) exp ( - W ) (52) i is performed. In this representation, Φηί(Ε) is the ith energy eigenfunction associated with the nth spatial mode, and ληι is the corre- THE PHYSICS OF PULSED NEUTRONS 33 sponding eigenvalue.

Purohit expanded each eigenfunction in a complete sum of the associated Laguerre poly­ nomials of order one, weighted by the Maxwellian distribution, *m(E) = Σ AunUl\E)M(E). (54) The next step is to substitute this expansion into Eq. (53), multiply through Lm{E)i and integrate over all energies as done earlier by Singwi {18). \E)M{E) dE, lj-i/2 = Vo/v (VQ = speed corresponding to most probable energy), 2 and (DBn )mi is defined similarly as (So) mi. The energy eigenvalues associated with the nth spatial mode for the special case of a nonmultiplying medium are obtained from the determinant I __ni—m_ ^0 _j_ (2) a ) m j _|_ (pBn2)ml ~" Fml\ I = 0.

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