Functional methods in three-dimensional neutron transport theory

Summary In this work, we are concerned with the stationary neutron transport Boltzmann equation (in its integral form) in a parallelepiped. Functional methods allow us to prove that the integral transport operator, which is defined in L² space, has eigenvalues depending continuously and monotonically on geometrical and physical parameters. We show that the eigenfunctions are continuous with respect to set of the spatial variables and the optical parameters. Finally, we remark that the same results are valid if the study is carried out in the Banach space C.

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Sommario In questo lavoro consideriamo l'equazione stazionaria di Boltzmann (nella forma integrale) per neutroni monoenergetici nel caso di un sistema tridimensionale a forma di parallelepipedo. L'uso di alcuni metodi dell'analisi funzionale ci permette di provare che l'operatore integrale del trasporto, definito nello spazio L², ha autovalori che dipendono continuamente dai parametri geometrici e fisici. Si prova che le autofunzioni sono continue rispetto all'insieme delle variabili spaziali e dei parametri ottici. Infine, si osserva che gli stessi risultati sono validi se l'operatore del trasporto agisce nello spazio di Banach C.

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Work performed under contract C.N.R. (Gruppo Nazionale per la Fisica Matematica).     

Some applications of functional analysis to problems of neutron transport

Ritz method is used to obtain an approximate solution of the stationary neutron transport Boltzmann equation in its integral form in plane and spherical symmetry. Such a method is based on the maximum property of the quadratic form corresponding to a symmetric transformation in a finite dimensional subspace spanned by the firstn functions of a complete orthonormal set. In order to justify some numerical results, we also show that the transport operators are compact as acting both on the spaceC and on the spaceL ₂. Moreover, we investigate some properties of the solution and we prove that the neutron distribution is not increasing as a function of the spatial coordinate. Finally, a series of calculations have been performed for various values of the system-dimensions measured in mean-free-paths and the number of secondaries per conllision to maintain the system critical has been found.     

Some remarks on neutron transport in homogeneous slabs and spheres

Summary In this work we are concerned with the stationary neutron transport Boltzmann equation (in its integral form) in plane and spherical symmetry; both the study of the properties of the exact and approximate solution are based on functional analysis. We have taken advantage of the fact that the kernel of the integral equation is even, to semplify the general discussion; among the results, we quote that relating to the continuity of the solution with respect to the optical thickness. Moreover we emphasized that the Ritz method seems the most convenient one for the approximate evaluation of the solution; in this connexion, some observations have been made about the convergence of the approximate solutions to the exact one.

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Sommario In questo lavoro consideriamo l'equazione stazionaria di Boltzmann (nella forma integrale) in simmetria piana e sferica; l'uso di alcuni metodi dell'analisi funzionale ci consente di studiare sia le proprietà della soluzione esatta che di quella approssimata. La semplice osservazione che il nucleo dell'equazione integrale è pari, ci ha permesso di semplificare la discussione generale; tra i risultati segnaliamo quello riguardante la continuità della soluzione rispetto allo spessore ottico. Inoltre abbiamo sottolineato che il metodo di Ritz ci sembra il più opportuno punto di partenza per il calcolo approssimato della soluzione; in relazione a ciò, vengono fatte alcune osservazioni a proposito della convergenza delle soluzioni approssimate verso quella esatta.

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Work performed under contract C.N.R.     

Outgassing and contamination: Strict solution of a Boltzman-like model

We consider a Boltzmann-like model of a problem of outgassing and contamination in a region ?. For simplicity, we assume that ? = ?₁ ∪ ?₂ ∪ ?₃, where ?₁ is the slab {x: ?a < x < 0}, ?₂ = {x:0 < x < b}, ?₃ = {x: b < x < b₁}. ?₁ is the region where the contaminant particles are produced, ?₂ is the ‘cavity’ where such particles migrate and interact with some inert gas, usually at low pressure, and ?₃ is the region which is contaminated by the particles coming from ?₂. In each of the three regions, the behaviour of the contaminant particles is represented by means of a Boltzmann-like equation. We prove that such a mathematical problem has a unique positive strict solution, belonging to a suitable Banach space Y. A system of ordinary differential equations is also derived, which gives the global balances of the contaminant particles in each of the three regions.