Incoherent scattering law

As in the case of the Rouse dynamics (see Sect. 3.1.1), the intermediate incoherent scattering law for dominant hydrodynamic interaction (Zimm model) can be... 

For hydrocarbons in zeolites, only incoherent scattering has to be considered because of the large incoherent cross section of hydrogen. The neutron intensity scattered follows the incoherent scattering law 5i c(Q, (o), which is related to the self-motion of protons, where AQ and Aw denote the neutron momentum transfer and the neutron energy transfer, respectively. 

Assuming for a molecular system the vibrational, rotational, and translational motions as uncoupled, the incoherent scattering law can be expressed as a convolution product of the individual scattering laws for each motion, which can be examined separately ... 

It is convenient to rewrite this equation in terms of a van Hove response funetion, S, which emphasises the strueture and d5mamics of the sample, also called the incoherent Scattering Law , which has units of... 

Rotation of the water molecules. Since this rotational motion is probably nearly isotropic, the incoherent scattering law is given to a good approximation by the Sears formula [19]. 

The INS intensity, 5 (g,fo), as calculated from the Scattering Law, Eq. (2.41), is related to the mean square atomic displacements, weighted by the incoherent scattering cross sections. What is required to calculate this quantity is the mean square atomic displacement tensor, Bi, and this can be obtained from the crystalline equivalent of L/ " ( A2.3), the normalised atomic displacements in a single molecule Eq. (4.20). This is and was introduced above, in Eq. (4.55). We have seen how... 

Direct geometry instruments also have advantages in this area. For many elements, the coherent cross section is larger than the incoherent cross section and with ( -resolved data it is possible to see additional structure [29,81] beyond that predicted by the scattering law, Eq. 2.32. These arise from coherent inelastic scattering and offer further possibilities for investigating the dynamics in such systems. 

The basic formulation of this problem was given by Van Hove [25] in the form of his space-time correlation functions, G ir, t) and G(r, t). He showed that the scattering functions, as defined above, for a diffusing system are given by the Fourier transformation of these correlation functions in time and space. Incoherent scattering is linked to the self-correlation function, Gs(r, t) which provides a full definition of tracer diffusion while coherent scattering is the double Fourier transform of the full correlation function which is similarly related to chemical or Fick s law diffusion. Formally the equations can be written ... 

Let us examine first the incoherent scattering. For isotropic diffusion, the motion of a given atom (or single molecule) is represented by the diffusion equation (Pick s second law), for long enough times (and distances)... 

Furthermore, incoherent and coherent dynamic scattering laws 5i c(6, to) and 5coh(6. to) are described by the time—space Fourier transformations of the time-space self- and pair-correlation functions Gs(r, t) and G(r, t) ... 

The time-dependent mean square displacement can be obtained from the incoherent scattering measurements. It is also pointed out that the frequency-dependent mean square displacement can be obtained from the incoherent dynamic scattering law within the Gaussian approximation. 

In this experiment, we observed dynamics in the frequency region hence, the incoherent elastic scattering intensity of dynamic scattering law S Q,

The EMT can be considered as a basic tool for the analysis of IR spectta of inhomogeneous thin films, and it has been proven valid in many cases, provided that the working formula has been chosen properly (Section 3.9). However, the EMT is not sensitive to the positions or sizes of individual inclusions in the film. Moreover, in contrast to the Rayleigh law, it is inadequate in predicting the color and brightness of the sky, because it neglects incoherent scattering. 

The scattered intensity is analysed as a function of the momentum transfer and the energy transfer ho) = E - Eq (E is the energy of the scattered neutrons). It is proportional to the scattering law S(g,o)) which contains all information about the structure and dynamics of the sample . In the case of hydrogenated samples, the scattering law is almost purely incoherent and reflects mainly the motion of the hydrogen atoms. It is denoted Ss(Q,o)). 

Figure 10. Incoherent quasi-elastic scattering law for a particle undergoing simple translational diffusion (D(= 1.5 X10" ° m s" ). This is normally measured as a series of constant Q spectra.

Figure 12. Incoherent quasti-elastic scattering law for a particle undergoing uniaxial rotational diffusion (Dr=1.5xl0 rad s ) on a circle of radius ZA (0.2 nm) with Q perpendicular to the axis of rotation. The elastic part of the scattering is represented by a triangular peak with the correct area rather than a delta function.

For translational diffusion, the simplest model is to consider that the diffusing particle starts at the origin at time zero. Since this is the dynamics of a single particle, the scattering is incoherent. In this case the scattering law takes the form ... 

Figure 37 Incoherent scattering data from PE in a representation of -6fn [Sseif(0, tj](f which is the mean square displacement (fl (O) as long as the Gaussian approximation holds. Solid lines describe the asymptotic power laws

Figure 11 MCT P-scaling for the amplitudes of the von Schweidler laws fitting the plateau decay in the incoherent intermediate scattering function for a R value smaller than the position of the amorphous halo, q = 3.0, at the amorphous halo, q = 6.9, and at the first minimum, q = 9.5. Also shown with filled squares is the P time scale. All quantities are taken to the inverse power of their predicted temperature dependence such that linear laws intersecting the abscissa at Tc should result. 

Figure 20 Temperature dependence of the a-relaxation time scale for PB. The time is defined as the time it takes for the incoherent (circles) or coherent (squares) intermediate scattering function at a momentum transfer given by the position of the amorphous halo (q — 1.4A-1) to decay to a value of 0.3. The full line is a fit using a VF law with the Vogel-Fulcher temperature T0 fixed to a value obtained from the temperature dependence of the dielectric a relaxation in PB. The dashed line is a superposition of two Arrhenius laws (see text).

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