Incoherent intensity

Valuable information on the geometry of the proton motion is offered by the Q-dependence of the elastic incoherent intensity (EISF) (see Fig. 4.34). For two site jumps this intensity is described by ... 

We know that the total intensity scattered by the bound electron is equal to the coherent intensity scattered by the free electron, therefore the incoherent intensity is given by the following relation ... 

Figure 1.5. Simplified representation of the coherent and incoherent intensities as a function of the scattering vector...

In (a), two photon waves combine to give a new waveform, which has the same appearance and frequency as the initial separate waves. The photons are said to be coherent, and the amplitude of the waves (light intensity) is simply doubled. In (b), the two photon waves are shown out of step in time (incoherent). Addition of the two waveforms does not lead to a doubling of amplitude, and the new waveform is more complex, composed of a doubled overlapping frequency. In (c), the two waveforms are completely out of step (out of phase) and completely cancel each other, producing darkness rather than light (an interference phenomenon). 

Inelastic Incoherent Scattering Intensity. For a system executing harmonic dynamics, the transform in Eq. (4) can be performed analytically and the result expanded in a power series over the nonnal modes in the sample. The following expression is obtained [26] ... 

The amplitude of the elastic scattering, Ao(Q), is called the elastic incoherent structure factor (EISF) and is determined experimentally as the ratio of the elastic intensity to the total integrated intensity. The EISF provides information on the geometry of the motions, and the linewidths are related to the time scales (broader lines correspond to shorter times). The Q and ft) dependences of these spectral parameters are commonly fitted to dynamic models for which analytical expressions for Sf (Q, ft)) have been derived, affording diffusion constants, jump lengths, residence times, and so on that characterize the motion described by the models [62]. 

The performance of an optical imaging system is quantified by a point-spread function (PSF) or a transfer function. In astronomy we image spatially incoherent objects, so it is the intensity point-spread function that is used. The image is given by a convolution of the object 0, r]) with the PSF... 

Usually we are only interested in mutual intensity suitably normalised to account for the magnitude of the helds, which is called the complex degree of coherence 712 (r). This quantity is complex valued with a magnitude between 0 and 1, and describes the degree of likeness of two e. m. waves at positions ri and C2 in space separated by a time difference r. A value of 0 represents complete decorrelation ( incoherence ) and a value of 1 represents complete eorrelation ( perfect coherence ) while the complex argument represents a difference in optical phase of the helds. Special cases are the complex degree of self coherence 7n(r) where a held is compared with itself at the same position but different times, and the complex coherence factor pi2 = 712(0) which refers to the case where a held is correlated at two posihons at the same time. 

We still need to consider the coherence properties of astronomical sources. The vast majority of sources in the optical spectral regime are thermal radiators. Here, the emission processes are uncorrelated at the atomic level, and the source can be assumed incoherent, i. e., J12 = A /tt T(ri) (r2 — ri), where ()(r) denotes the Dirac distribution. In short, the general source can be decomposed into a set of incoherent point sources, each of which produces a fringe pattern in the Young s interferometer, weighted by its intensity, and shifted to a position according to its position in the sky. Since the sources are incoherent. 

It is important to note that expression (23) can be applied to the crystalline phase intensities only if we include, in the first integral, its own smooth diffuse background and not just the intensity belonging to the crystalline peaks. In fact, a pure crystalline sample also has a smooth background due to the incoherent inelastic scattering (i.e. Compton scattering), the TDS, disorder scattering and, very often, unresolved tails of overlapped peaks. 

For the NFS spectrum of [Fe(tpa)(NCS)2] recorded at 108 K, which exhibits a HS to LS ratio of about 1 1, a coherent and an incoherent superposition of the forward scattered radiation from 50% LS and 50% HS isomers was compared, each characterized by its corresponding QB pattern (Fig. 9.16) [42]. The experimental spectrum correlates much better with a purely coherent superposition of LS and HS contributions. However, this observation does not yield the unequivocal conclusion that the superposition is purely coherent, because in the 0.5 mm thick sample the longitudinal coherence predominates since many HS and LS domains lie along the forward scattering pathway. In order to arrive at a more conclusive result, the NFS measurement ought to be performed with a smaller ratio aJD on a much thinner sample. Such an experiment would require a sample with 100% eiuiched Fe and a much higher beam intensity. 

Two additional feature can be incorporated into Eqs. (7.32)—(7.35) the dipole orientation distribution and the concentration distribution in systems consisting of many dipoles. The orientation of the dipole with respect to the surface, described by angles Q = (8, ), affects E and all the other measurables derived from it.(33) Consider a concentration distribution of dipoles in both orientation and distance from the surface specified by C(0, , z). Since the dipoles all oscillate incoherently with respect to one another, the integrated intensity J due to this distribution is simply ... 

The computation of far-field radiation from a collection of incoherently radiating dipoles is in general quite a complicated problem. To calculate the angular dependence of the far-field intensity, the volume distribution of excited states must first be obtained, which, as we have seen, depends on the volume distribution of the absorbers and the electromagnetic field which stimulates them. The fields in turn depend on the frequency and linewidth of the exciting light source. Then the emission problem for the excited-state distribution (both spatial and frequency) must be solved including reorientation and depolarization effects. 

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