Two-group theory

Two-Group Theory. The group approximation was used in Chicago before it was developed anywhere else, though I cannot say exactly who first intro-... 

We shall first treat a case in which the moderator is the same 1 over the pile so that the diffusion constant for both fast and thermal neutrons remains constant throughout the whole system. It has been stated in CP-455 that, for such a system, Fermi s equations can be solved with the two group theory as described in CP-1461 and CP-1554. It will be shown that in this case, the two methods of calculation give the same result. The general case of different moderators in the two parts of the pile will be treated finally by means of the two group theory. 

We now turn to the same problem, except that we shall assmne a sharp boimdary (at x = 0) between the two parts of the pile and use the two group theory described in reports CP-1461 and CP-1554. The notation will be the same as that adopted in CP-1461. 

One obtains the first equation of the two group theory by integrating (7) with respect to r and writing... 

In addition, one has to assume that D is independent of r or, otherwise, substitute a suitable average for it. The last term on the left side causes a decrease in the total number of neutrons. This term is omitted in the two group theory and will be replaced by inserting a factor p in the transcription of the second equation (7 a). Writing r/ for the average r of fission neutrons, one has, approximately... 

All these results can be obtained, of course, also with standard methods. This is not true, however, if one has to deal with composite piles, with piles with reflectors, control rods, etc. In all these cases in which the equations of the two group theory can be solved, the adjoint equation for the adjoint density can be solved equally easily and one obtains the effect for arbitrary perturbations by the simple equation (6). 

This differs from (16 a) only inasmuch as the —DB term must be omitted in the latter. If this is done, (26 a) goes over into (16 a) by virtue of (9 b). As is well known, the two group theory s validity depends on a relatively small k and —DB is small if k is close to 1. Just as in (16a), the term 2(—jB >o + CTa)ISo can ordinarily be neglected against 1. 

In seeking generalizations in two-group theory, analogous to those developed in 2, the equations adjoint to (15) and (16) are needed ... 

It is customary, however, to divide O Eq. (38.24) into two parts (two-group theory), representing the contributions from thermal and epithermal neutrons. 

The criticality calculations for the earlier study were made using the four-fhctor formulation and an approximation to age theory. The results presented in this study for fuel enrichments of 3% U-235 and lower were based on these earlier calculatienriched fuels were made using two-group theory. It was decided that the latter method gave more accurate results at flie higher enrichments, particularly for fuel rods of larger diameters. ... 

Reflector. In order to apply the above results to reflected reactors, it is necessary to determine the reflector savings, which can be obtained from conventional two-group theory. This method could also be used to estimate the critical size of the reactor but, for small cores, two-group theory underestimates the size of graphite-moderated reactors. 

We have found three distinet irredueible representations for the C3V symmetry group two different one-dimensional and one two dimensional representations. Are there any more An important theorem of group theory shows that the number of irredueible representations of a group is equal to the number of elasses. Sinee there are three elasses of operation, we have found all the irredueible representations of the C3V point group. There are no more. 

The Hard-Soft-Add-Base (HSAB) theory was developed by Pearson in 1963. According to this theory, Lewis acids and Lewis bases are divided into two groups on one hand hard acids and bases, which are usually small, weakly polarizable species with highly localised charges, and on the other hand soft acids and bases which are large, polarizable species with delocalised charges. A selection of Lewis acids, ordered according to their hardness in aqueous solution is presented in Table 1.3. 

Simultaneous stimulation of the tongue with the appHcation of different taste stimuli produces an interaction, modification, or blending of the stimuli in some instances but not in others. Warm and cold sensations are reported to act similarly on the tongue in two groups bitter, warm, and sweet and sour, cold, and salty (24). The theory of the specificity of the taste buds may be subject to modification (25). 

The aeeurate ealeulation and proper evaluation of losses within a eentrifugal eompressor is as important as the ealeulation of the blade-loading parameters. If the proper parameters are not eontrolled, effieieney deereases. The evaluation of various losses is a eombination of experimental results and theory. The losses are divided into two groups (1) losses eneountered in the rotor, and (2) losses eneountered in the stator. 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

In some cases, an electrophile preferentially attacks the position between two groups in the meta relationship. For a list of some of these eases and a theory to explain them, see Kruse, L.I. Cha, J.K. J. Chem. Soc., Chem. Commun., 1982, 1333. 

From 1933 85>, several theoretical approaches to the problem of the chemical reactivity of planar conjugated molecules began to appear, mainly by the Huckel molecular orbital theory. These were roughly divided into two groups 36>. The one was called the "static approach 35,37-40)j and the other, the "localization approach 41,42). in 1952, another method which was referred to as the "frontier-electron method was proposed 43> and was conventionally grouped 44> together with other related methods 45 48> as the "delocalization approach". 

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