Effective resonance integral

The introduction of the effective resonance integrals makes it possible to define effective absorption cross sections due to the resonances in multigroup calculations by... 

Eric Hellstrand, Measurements of the effective resonance integral in uranium metal and oxide in different geometries, J. Appl. Phys. vol. 28 (1957) p. 1493. 

The values of the effective resonance integral derived from toe p measurement (neglecting resonance flux depression) are 9.61 i 0.35 barns for toe dry case and 11.15... 

E. HELLSTRAND, Measurements of the Effective Resonance Integral in Uranium Metal and Oxide in Different Geometries, J. Appl. Pkys., 28, 1493 (1957). 

For interaction between the MOs and Fy, the numerator is the square of the effective resonance integral between them. This is determined by the degree to which the MOs overlap, i.e., by their product. Using equations... 

If there were a simple n bond between atoms r and s, the overlap density in it would be given by the product [cf. equation (1.11) and the discussion following it], and this would correspond to the resonance integral The overlap of MOs and Py, being times then corresponds to an effective resonance integral The difference in energy between the... 

The effective resonance integral P between the MOs is then given by... 

E here are two distinctive water density effects on the resonance escape prohahllity. Firstly the reduction of the water density makes the lattice even more under-moderated and thus increase the inportance of resonance capture. Secondly the reduction of water density in Ihe coolant internal to the fuel element decreases the effective resonance integral (lower surface-to-mass ratio) and thus decreases the inportance of resonance capture. Ihus the two effects are opposite in sign the former effect is dominant so the net coolant temperature coefflent of p is negative. 

Another interesting observation involves the physical interpretation of the effective resonance integral. According to (10.21)... 

The effective-resonance integrals for the two fertile fuels and Th have been studied in greatest detail, and empirical formulas have been obtained for computing these integrals as functions of fuel concentration. The best available experimental data are plotted in Fig. 10.2. Dresner has computed from resonance parameters, by methods to be discussed... 

Table 10.1 Effective Resonance Integrals for Infinitely Dilute Mi.xtures of Various Nuclear Fuels (in barns)...

In the event that all resonances of the fuel are narrow relative to the average energy loss in a collision with a fuel atom, the asymptotic form (10.110) may be used throughout in evaluating the effective resonance integral. 

The effective resonance integral may be computed by means of the relation (10.21). 

This result is the so-called infinite-dilution effective resonance integral. The sum of over all resonances has been measured for many materials (see, for example, Table 10.1). At the other extreme, when /S 1 — b, as often happens in the low-energy resonances of uranium and thorium or other absorbers in fairly concentrated mixtures, it is seen that is proportional to This rule is roughly verified by experiment. ... 

We found previously that in the NR approximation the effective resonance integral for heterogeneous systems could be written in the form (10.140) which is mathematically identical to (10.125) or (10.23) thus the result (10.155) may be adapted to heterogeneous systems. It follows that in this approximation is given by... 

The heterogeneous system formulas (10.159) and (10.166) for the contribution to based on the NR and NRIA approximations may be used to deduce the dependence of the effective resonance integral upon the surface-to-mass ratio (Ay/My) of the fuel lump. In the cases wherein the NRIA formula applies, it frequently occurs that (< or / 1. Then the form (10.166) is well approximated by... 

Thus for many situations of practical interest the effective resonance integral varies as the square root of the surface-to-mass ratio of the fuel lump in both the NR and NRIA approximations. It is also of interest to note that in mixtures when > 10 both approximations approach... 

The functions (10.188), (10.191), and (10.192) may be used to obtain the generalization of the effective resonance integral in both the NR and NRIA approximations to the case of Doppler broadening. The calculation which follows is developed from the resonance-integral formulas for the homogeneous system and does not include the interference between resonance and potential scattering thus, we take b == 0. This result will be a good approximation for two situations of practical... 

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