Line shape function homogeneous

This Lorentzian line-shape function has been sketched in Figure 1.4(b). The natural broadening is a type of homogeneous broadening, in which all the absorbing atoms are assumed to be identical and then to contribute with identical line-shape functions to the spectrum. There are other homogeneous broadening mechanisms, such as that due to the dynamic distortions of the crystalline environment associated with lattice vibrations, which are partially discussed in Chapter 5. 

As expected, we find that the total response function Xa=i Ri = Xa=4 Ri = J2i=1 Ri = 0 (i.e., for each possible time ordering) vanishes exactly in the harmonic case, defined by A = 0 and /x2i 2 = 2 /r10 2. Furthermore, it can be easily seen that in the case of a strict separation of time scales of homogeneous and inhomogeneous broadening, the line shape function becomes g(t) = t/T2 + a212/2 and the total response function reduces exactly to the result obtained within a Bloch picture (see, for example Refs. 52 and 75), e.g.,... 

Homogeneous interaction (Lorentzian line shape function)... 

The homogeneous line shape function the perfect crystal case... 

The free induction decay following 90° pulse has a line shape which generally follows the Weibull functions (Eq. (22)). In the homogeneous sample the FID is described by a single Weibull function, usually exponential (Lorentzian) (p = 1) or Gaussian (p = 2). The FID of heterogeneous systems, such as highly viscous and crosslinked polydimethylsiloxanes (PDMS) 84), hardened unsaturated polyesters 8S), and compatible crosslinked epoxy-rubber systems 52) are actually a sum of three... 

The effective S values were determined from the homogeneous absorption line-shapes calculated using a thermally averaged Green s function approximation method... 

Kamack offered the following solution to equation (8.12). If Q is plotted as a function of v, = rJSy with f = o9-t as parameter, a family of curves is obtained whose shape depends on the particle size distribution function. The boundary conditions are that Q - 1 when f = 0 for all r, (i.e. the suspension is initially homogeneous) and = 0 for r, = S when f>0 (i.e. the surface region is particle free as soon as the centrifuge bowl spins). Hence all the curves, except for f>0, pass through the point Q = 0, S, and they will all be asymptotic to the line / = 0, which has the equation Q = - Furthermore, from equation (8.12), the areas under the curves are each equal to F(r/., ). 

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