Resonance narrow-------approximation

In the narrow resonance approximation (NR), the assumption is made that the absorption is large only in an interval small compared to aoE, the maximum energy loss in an absorber collision. In this case, the quantities inside the integral in Equation (3.10) can be replaced by their asymptotic values (Equation (3.11) and os = and one obtains... 

The theoretical development shows further that because of overlap effects it is a good approximation to include in ,( ) only the resonances of the particular sequence for which the Doppler effect is being evaluated rather than ail resonances, i.e., ,( ) is replaced by Zp + Zc ( ). This represents an additional simplification beyond the usual narrow resonance approximation. It is, however, demonstrated to be a good approximation rather than being taken as a basic assumption or postulate. 

Aside from the narrow resonance approximation and the errors introduced by statistical fluctuations, (46) is a rigorous result. Each of the brackets is dependent upon only one of the two individual sequences, and it is possible by numerical means to evaluate as many terms in the series as required for a desired accuracy. In practice, one will wish to introduce further approximations to simplify and minimize the numerical effort required. All of the succeeding results of the present work stem from various approximations introduced into (46). 

Thus, a formal result for the effective cross section has been obtained in terms of the ratio between two series, and each term can be evaluated in terms of an individual sequence of levels. Up to this point, no major approximations have been introduced other than the narrow resonance approximation. 

As an example of the resulting spectrum, a simple approximation of the slow-ing-down region neutron balance (through the narrow resonance approximation) and a simple spatial approximation of a lump of resonance absorber surrounded by an infinite "sea" of moderator results in a simplified neutron balance relation of... 

In this development we apply the various approximations for F(u) derived in the preceding analysis to the calculation of for both homogeneous and heterogeneous systems, using Eqs. (10.123) and (10.21). We begin with the homogeneous system. In the narrow resonance approximation, the appropriate form for F is (10.110). Substitution into (10.123) yields the result previously given in (10.20) in the present notation, we have... 

Following the heat-load monochromator, the X-ray bandwidth is narrowed to approximately 1 eV and centered on the nuclear resonance energy (14.4 kev for Fe). The high-resolution monochromator further reduces the X-ray bandwidth to about 1 meV and motorized scanning of this monochromator tunes the energy over a range (typically within 100 meV of the resonance) adequate to explore excitation or annihilation of vibrational quanta. The X-ray flux at the sample is about 10 photons/s ( 10 tW), which is very low compared to typical milliwatt beam powers in laser-based Raman experiments see Vibrational Spectroscopy). Additional X-ray optics may reduce the beam size. The cross section of the beam at the sample point is currently about 0.5 x 0.5 mm at station D of beam line 3ID at APS. 

We note that, although the spacing between the resonances differing only in J is very small, the INR approximation need not be invalid for this reason because its validity only requires narrow resonances well separated from others with the same total angular momentum. 

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