Density of Vibrational States and Neutron Scattering

We will introduce the basic principle and use of neutron-scattering spectroscopy by first introducing an important dynamical quantity, namely the density of vibrational states g(v). 

Let g(v) be the fraction of normal modes with frequencies in the interval v and V -1- dv with dv — 0. g(v) can be calculated analytically only for very simple atomic chains which are never encountered when reaUstic polymeric materials must be studied. 

Let us take a very large molecule and let 3N = C be the number of vibrations of the whole system if is the number of frequencies in the interval Av = v -t- dv then 

The total density of states is evaluated as the sum over all the r branches of the dispersion relation (Eq. (3-43)). Nmnerical methods such as the root sampling method have been proposed for simple one- or tridimensional lattices [51]. When the systems become more complex, as in the cases discussed in this chapter, the easiest numerical method is that proposed by Dean based on the application of the Negative Eigenvalues Theorem (NET) [79]. The method is extremely useful and has been extensively used in the laboratory of tlie writer for many polymeric cases. The details of the method are discussed in the next section. 

NET is a way to calculate g(v), i.e., it provides the number (or the fraction) of vibrational energy states or normal modes in a chosen (p (or k) interval. g(v) can be plotted as an histogram whose accuracy depends on the frequency interval Av chosen in the numerical calculation (numerical resolution of the computer experiment). This will be used later in this chapter in the study of disordered systems. 

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