Download Advances in Nuclear Science and Technology by V. Protopopescu (auth.), Jeffery Lewins, Martin Becker PDF

By V. Protopopescu (auth.), Jeffery Lewins, Martin Becker (eds.)

Dur earlier quantity 14 used to be dedicated to an exposition of the themes of sensitivity research and uncertainty concept with its improvement and alertness in nuclear reactor physics on the middle of the dialogue. during this quantity, we go back to our wide-spread layout as a variety of themes of present curiosity, authored by means of these operating within the box. those themes variety from the theoretical underpinnings of the (linear) Boltzmann shipping equation to a resume of our ex­ pectations in what nonetheless will be considered twenty-first century know-how, the world's fusion reactor software. within the first article of this quantity, we've got Protopop­ escu's research of the constitution of the Boltzmann equation and its recommendations for power and space-dependent difficulties of an eigenvalue nature. There lengthy has been a curious "folk background" influence during this quarter~ Wigner and Weinberg might de­ scribe it as "what used to be normally identified used to be more often than not untrue". This account of the Boltzmann equation without doubt will convey rigorous foundation for our expectancies of sure options may be well-founded on research. Ely Gelbard's assessment of the equipment of picking out diffusion-type parameters in advanced geometries the place uncomplicated diffusion conception will be welcome has required simply as a lot rigor to figure out how such modeling should be made exact, even if to a extra quick and sensible goal. the 2 articles may be noticeable as attention-grabbing contrasts, elements of an identical underlying challenge displaying it sounds as if diversified points of an analogous valuable core.

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PROTOPOPESCU 26 It is c1ear that HA and H" A will have on the cut the same va1ues and that H" A can be seen as the continuation on another Riemann sheet of the ana1ytic branch HA. One obtains the "eigenva1ues" gi ven by H" A for Re < - 1 by sol ving the equation 1 - 1 k k arctan A+1 - 7T k = o. 4) f, that is, beyond k = ~. y the edge A 1 (see Figure 8). Of course it is evident that the hope of approximating we11 the continuum by a pseudo mode is 1egitimate on1y for sma11 departures of B 2 from the critica1 va1ue B*2; that is, for k > fand that for 1arger va1ues, this procedure comp1ete1y loses i ts sense (observe that for k -+ 7T, A -+ - co) • Natura11y, a simi1ar situation is valid for the anisotropie case also (see Section VI) and we can compute "eigenvalues" starting from the ana1ytic continuation 15 of HA in {AIRe A< - 1J.

Kp7T - 1 -1. a Figure 7. The Spectrum of the Reduced Operator ~ (Periodie Boundary Conditions) C. 20) leading to the same dispersion equation 1 - 1 B B arctan A+1 = o. 21) However, some differences must be pointed out: As already mentioned (122), the buckling parameter B, which is fixed for a given geometry, has no relation with the Fourier parameter k which takes all real values (for infinite geometry) or all integer values (for slab geometry with periodic boundary conditions). Brfo(~) (pulsed source) EIGENVALUE PROBLEMS FOR THE BOLTZMANN OPERATOR 23 and to neglecting the surface effects.

Spectrum of the Bo1tzmann Operator with an Isotropie Therma1ization KerneI," in lAEA Symposium Pulsed Neutron Research, Vienna, 1965, Vo1. ~, Page 259. 87. , Nuc1ear Science and Engineering 26, Page 279, 1966. 88. , Advances in Nuclear Science and Techno1ogy, Volume l, Page 1, Academic Press, 1966. 89. , "Sorne Recent Resu1ts in the Theory of the Transport of Thermal Neutrons," in American Mathematica1 Society Symposium on Transport Theory, New York,1967. 90. Wi11iams, M. M. , Nukleonik 10, Page 252, 1967.

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