Differential cross section time-dependent scattering

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

We will merely content ourselves by examining the significance of the various elements in (8.9) in comparison with those in the analogous equation valid in the static approximation. The differential scattering cross section da/dQ, in the static approximation in which the time dependence of the system is ignored, is obtained by taking the absolute square of the amplitude as given by (1.71) ... 

After a short description of the experimental setup in the previous section, we now focus on the basic mathematical description of the intensity measured in a SAXS experiment. X-rays are electromagnetic waves. Assume an incident plane wave with flux 7o, which is scattered by the electrons in the sample. The scattered spherical waves interfere with each other, resulting in an angle-dependent flux J of scattered radiation. The flux of the incident plane wave corresponds to energy transmitted per unit area per unit time and the flux J of the scattered radiation to energy transmitted per unit solid angle per unit time. Now the differential scattering cross section or scattered intensity is defined as the ratio... 

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