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A correction to some invariant imbedding equation of transport theory obtained by ’particle counting’. (English) Zbl 0186.58601

In preceding papers [cf. R. Bellman, R. Kalaba and G. M. Wing, J. Math. Mech. 8, 575–584 (1959; Zbl 0087.23501); J. Math. Phys. 1, 280–308 (1960; Zbl 0105.23202)] an invariant imbedding technique has been applied to the diffuse reflection problem of a conical flux of neutrons by a spherical medium; however, the result given by the elementary particle-counting procedure was found to be in error as compared with that obtained rigorously from the Boltzmann formulation [P. B. Bailey, J. Math. Anal. Appl. 8, 144–169 (1964; Zbl 0186.58503) (see preceding review)]. In this paper it is shown how elegantly the particle-counting technique, applied with more careful consideration, gives rise to the additional term \[ [(1+\mu^2)/x\mu^2] R(x,\mu,\mu_0), \] a term which was unsuspected in past derivations (cf. R. Bellman, R. Kalaba and G. M. Wing, loc. cit.). In the above expression \(x\) is the radius of the sphere, \(\mu\), (or \(\mu_0\)) is the cosine of the angle of reflection (or incidence), and \(R\) is the reflection coefficient, i.e. the emergence flux of particles. The source of discrepancy seems to be due to the effect of geometrical convergence which changes the form of the source strength incident on the spherical medium under consideration, when the particles pass through the spherical shell towards the center without collision. In rod and slab geometries no such correction is to be expected.
Reviewer: S. Ueno

MSC:

82C70 Transport processes in time-dependent statistical mechanics
82D75 Nuclear reactor theory; neutron transport
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References:

[1] Bailey, P. B., A rigorous derivation of some invariant imbedding equations of transport theory, J. Math. Anal. Appl., 8, 149-174 (1964) · Zbl 0186.58503
[2] Bellman, R.; Kalaba, R.; Wing, G. M., Invariant imbedding and neutron transport theory IV. Generalized transport theory, J. Math. Mech., 8, 575-584 (1959) · Zbl 0087.23501
[3] Bellman, R.; Kalaba, R.; Wing, G. M., Invariant imbedding and mathematical physics I, Particle processes, J. Math. Phys., 1, 280-308 (1960) · Zbl 0105.23202
[4] Wing, G. M., An Introduction to Transport Theory (1962), Wiley: Wiley New York
[5] Bellman, R.; Kalaba, R., On the fundamental equations of invariant imbedding I, (Proc. Nat. Acad. Sci. U.S., 47 (1961)), 336-338 · Zbl 0103.30604
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