Double differential scattering cross

The observable is the rate of change of the cross section with respect to the final energy, Eu and solid angle, d 2. This is the double differential scattering cross section, (d o/d f d/2 ), with units, bam eV steradian, where ... 

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... 

Here, ko and k are the neutron velocities before and after the scattering process. Looking at Eq. (2.75) immediately reveals the relation to G(r, t). The double differential scattering cross-section is directly related to the Fourier transform of the van Hove pair correlation function, which is given by... 

For coherent neutron scattering, the double-differential scattering cross section is... 

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... 

The double differential scattering cross section, according to (8.15), is proportional to the space-time Fourier transform of G(r,f). This is analogous to the fact, discussed in Section 1.5.2, that in the static approximation the intensity I(q) (or the differential scattering cross section ds/dQ) is given by the spatial Fourier transform of the density-density autocorrelation function defined in (1.79). In the special case of t = 0, G(r,0) denotes the probability of finding a particle at r when there is already a particle at position 0. G(r,0) is therefore related to the pair distribution function g(r) discussed in Section 4.1.1, as in... 

In view of (8.28) and from the definition of Gs(r,f) in Section 8.1.2, we find the incoherent component of the double differential scattering cross section to be... 

In presenting the experimentally observed spectrum, i.e., either the double differential scattering cross section d2a/dQdco or the dynamic structure factor S(q,a)) as a function of the energy exchange co, different workers often use different energy units for co. To facilitate conversion between these energy units, Table 8.1 gives the relationships between them. 

Double-Differential Scattering Cross Section, Dynamic... 

Double-Differential Scattering Cross-Section, Dynamic Scattering Law, and Intermediate Scattering Function... 

For a system consisting of one kind of atom the double-differential scattering cross-section d a/dOdE is simply related to the scattering law... 

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... 

The incoherent double-differential scattering cross-section for the one phonon process is given by... 

Figure 2 Setup for deriving a double-differential scattering cross-section...

The procedure for obtaining the double-differential scattering cross-section is to now sum over possible final states f) of the scatterer, subject to the condition that the energy of the combined system of radiation -i- target is conserved. The latter is accomplished by attaching the Dirac delta function, d( — ), to each term in the sum. Finally, we can introduce... 

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