Multiplication constant effective

The treatment of non-additivity has also been applied to a large variety of multiple substituent effects on various reactions (Argile et al., 1984) and, in particular, to the bromination of X,Y-disubstituted benzenes where two substituents on the same ring interact strongly (Dubois et al, 1972b) the interaction constant q = — 7.98, associated with a very negative p-value, —12.05, is much higher than those found for the bromination of arylolefins. 

Table 15 Non-additivity of multiple substituent effects p-dependence on X for a substituent Y and interaction constants in arylolefin bromination in methanol at 25°C.

Equations (37)—(39), where the non-additivity of multiple substituent effects is described by a cross-term, express correctly the rate data for bromination and other reactions of polysubstituted substrates. The question arises, therefore has the interaction constant, q, any physicochemical meaning in terms of mechanism and transition state charge To reply to this question, selectivity relationships (42) that relate the p-variation to the reactivity change and not to any substituent constant, have been considered (Ruasse et al., 1984). 

Note that all of these objective functions differ from one another only by a multiplicative constant this constant has no effect on the values of the independent variables at the optimum. For simplicity, we therefore use/i to determine the optimal values of D and L. Implicit in the problem statement is that a relation exists between volume and length, namely the constraint... 

The origin of these multiple temperature effects lies in the dependence of the kinetic rate constants on only the translational (or more generally the bath ) temperature. For vibrational states, these results were first derived for diatomic anharmonic oscillators and used to explain laser action in CO via anharmonic pumping. Multiple vibrational temperature concepts were later applied to the polyatomic molecule SF to explain discrepancies between relaxation rates measured by ultrasonic and laser techniques. The close correspondence between the temperature picture for vibrational levels weakly coupled to a thermal bath and that for nuclear or electron spin levels also weakly coupled to a thermal bath will be immediately obvious to those familiar with spin temperature concepts. ... 

The Monte Carlo code MONK was originally written to replace the Monte Carlo code GEM (7-i). GEM was primarily written to help in the assessment of criticality in chemical and metallurgical plant processing fissile materials, and also in the storage and transport of these materials. MONK not only will perform these tasks but can also perform calculations of interest to the reactor physicist. In criticality assessment work, boundary tracking is quite often used, especially for problems involved in the transport of fissile materials, since this option enables interactions between several similar items to be calculated as a subsidiary calculation. Fission to fission tracking has been incorporated as an optional choice in MONK to enable the effective multiplication constant to be calculated directly from the number of neutrons in successive generations. 

Szilard, also working at Columbia, became interested around this time in what is now called the fast effect. The fast effect, is the increase in the multiplication constant obtained by the emission of neutrons by which is induced to fission by the fission neutrons before they are slowed down. Szilard measured both the cross section of such fission neutrons to induce fast fission and also their inelastic cross section, i.e., the probability for their being slowed down below the fast fission threshold by an inelastic collision with uranium. He concluded on the basis of these measurements that one may obtain an increase of as much as 6-8% in the multiplication constant by using large and metallic lumps of uranium. Szilard was also somewhat discouraged by the low multiplication constant which Fermi s experiment gave but was far from giving up hope. 

The fast effect discovered a short time before by Szilard was omitted in these calculations and this proved to be a serious error, although we now know that the fast effect is less than half as great as Szilard calculated. I believed at that time that it is even smaller than that. The lattice which I proposed for the oxide proved later on to have a multiplication constant only about 2% higher than that of Fermi s Columbia lattice. The reason that it was so much smaller (about 3%) than I expected was mainly due to the fact that the fast effect was much smaller in it than in Fermi s lattice. However I did not quite believe in the fast effect and argued furthermore that if it existed we would recover the loss incurred by the transition to the smaller lattice when we replace the oxide by metal. 

Naturally, in a system containing about 1700 tubes, it is wise to guard against the consequences of failure in a few tubes. For this reason, all arrangements discussed provide for the possibility of quickly removing any rods the sheath on which may have become injured. Similarly, if a tube may become injured, the U would be removed from that tube and the water flow in it shut off. This would, of course, reduce the effective multiplication constant of the pile, but as long as only a few tubes are out of commission, the pile would continue to function satisfactorily. 

Comparison with W Pile. One of the favorable features of the present pile is its very great stability. An increase in the rate of production will cause an increase in the temperature of the slurry and a decrease in its density. This automatically decreases the effective multiplication constant. It may be possible to run this pile without regular control rods, although safety rods for stopping the pile must be provided. 

Papers 29 and 30 served as the basis for the earhest calculations of the multiplication constant in an infinite heterogeneous reactor. By early 1942, the significance of fast fission was recognized, and all later calculations included estimates of e, the fast effect. This gave the four-factor formula, k = rjepf rj being the number of neutrons released per neutron absorbed in U. The quantity actually calculated in the following reports was rj = (1/ep/), k = r /rj ... 

Wigner was the flrst to point out, in Paper 36, that the multigroup reactor equations were not self-adjoint. Thus the effect of a poison on the multiplication constant was proportional in general to the product of the neutron flux and its adjoint, not to the square of the flux as had been assumed before Paper 35 appeared. Of course in a large uniform reactor, the adjoint is everywhere proportional to the flux, so the simple recipe was usually valid however in small enriched reactors, the adjoint is no longer proportional to the flux, and the adjoint has to be computed separately. 

It would seem that only the effective multiplication constant has real significance but it turns out that the calculation of k is an almost necessary preliminary for the calculation of kgff. 

It may be desirable to avoid these disadvantages and Mr. Christy will shortly report on other controlling mechanisms which try to avoid these defects. More generally, if one wants to use the limits control in such a way that the intensity varies between no(l — v) and no(l + v) and if one uses upper and lower positions of the control rods which give effective multiplication constants 1 + ke and 1 — fcc the time t during which the control rod will remain in the upper or lower position will be given by the equation... 

K the ke which gives a stifficiently large f should turn out to be so low that one must fear accidental variations of the effective multiplication constant which are of the same order of magnitude, additional controls should be introduced with a more drastic action on the multiplication constant. These should operate if the neutron density goes substantially above or below the permitted limits no(l + v) or no(l — v). 

It was customary so far to express the effect of small perturbations as a change in the multiplication constant. However, this is a possible procedure only in the case of a uniform, bare pile. A composite pile, or even a simple pile with a reflector, has no single multiplication constant and it was thought best to express the effect of perturbations in terms of the change in reciprocal pile period. However, in order to avoid unnecessary complications, the effect of the delayed neutrons was omitted. Their inclusion would not cause any fundamental difficulty but would complicate the formulae and make them more cumbersome. They were omitted for this reason and the results for the change in period apply as they stand, only for very short perturbations (short as compared with the period of the delayed neutrons) or if the infinitely small perturbations cause a very much larger change in some sort of multiplication constant than the about 1% for which the delayed neutrons are responsible. However, as was stated before, it would not be difficult at all to include the delayed neutrons into... 

In these and the above equations, the a are cross sections per imit volume, the a in (8) is scattering cross section, the average loss in r per collision. The are used because the material may contain different types of atoms. The (Ta is the thermal absorption cross section r(r) the resonance absorption cross section per unit volume. The = qef is the multiplication constant divided by the resonance escape probability. The product of thermal utilization / and (Ta is the effective cross section of uranium per unit volume, i.e., its cross section per unit volume multiplied by the thermal neutron density in it and divided by the average thermal neutron density. One can write, therefore, (Tu for f(Ta- If one multiplies this with rj the result is the same as crfU where fission cross section for thermal neutrons per unit volume, p the number of fast neutrons per fission. As a result, the third term in (7) can be written also as e is the multiplication by fast effect)... 

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