Double-differential inelastic cross sections

In inelastic or quasi-elastic scattering experiments, the quantity of interest is the double differential scattering cross-section which is also often called dynamic structure... 

We start the discussion of the theory of inelastic scattering by first quoting the basic equation that relates the double differential scattering cross section to the structure and motion of nuclei in the scattering system. Unfortunately the derivation of this fundamental equation requires the use of quantum mechanics much beyond the scope of this book. The scattering system is characterized by the positions rft) of all the nuclei (j = 1,. . . , N) in it as a function of t. The equation we take as the starting point of our discussion is then... 

Inelastic scattering of neutrons is caused by an oscillatory motion. An example is the inelastic scattering of neutrons by phonons. Following a textbook [10], the coherent double-differential scattering cross-section of neutrons for one phonon process is given by... 

In inelastic scattering the scattered intensity needs to be determined as a function of both the scattering angle Q and the magnitude of energy exchange fico. This function, called the double (or partial) differential scattering cross section d2o/dQ dco, is a... 

The scattering from a molecule will be more complicated than for a single atom because the other molecular motions of rotation and vibration come into play. If there are no inelastic features in the measured energy transfer range studied, the vibrational term will only affect the measured intensities in the QENS domain through a Debye-Waller factor. On the other hand, the influence of the rotation on the observed profiles has to be treated in more detail. Sears has derived analytical expressions for the total differential cross-section of a molecular system, where the rotational motion is isotropic [12]. From his work, a simplified expression (Eq. 22) for the double-differential cross-section can be obtained it is spht into three terms ... 

The strength of the inelastic magnetic response is directly proportional to the static susceptibility, but only in special cases is it proportional to the localized magnetic moment (Holland-Moritz 1985). The latter can be extracted only by a complete integration over the double differential cross section... 

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