Incoherent superposition

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. 

Fig. 9.16 Time-dependent NFS of [Fe(tpa)(NCS)2] recorded at 108 K. The two curves represent comparison of a coherent vs incoherent superposition of the scattering from 50 % LS and 50 % HS iron(II) characterized by their corresponding quantum beat pattern. The effective thickness of the sample was =18. (Taken from [42])...

Note that the above equations for the radiation of a dipole refer to a single dipolar source of radiation. For our spectroscopic interests, we more commonly are dealing with an assembly of many identical such systems. Usually, the total radiation of such an assembly of sources may be assumed to be the incoherent superposition of the individual intensities (see, however, the discussions of the intercollisional effect below, Chapters 3 and 5). We may then simply multiply the contributions of one source by the number iV of sources in the (initial) state i. 

This breakdown of the linear relationship between the absorption coefficient a and the product of densities, Q1Q2, indicates that the observed absorption is not a binary process. Specifically, for the case at hand, one can no longer assume that the measured absorption consists of an incoherent superposition of the pair contributions. Rather, the correlations of the dipoles that are induced in subsequent binary collisions lead to a partially destructive interference, an absorption defect that occurs if the product of the time T12 between Ne-Xe collisions, and microwave frequency, /, approaches unity [404], We note that for the spectra shown above, Figs. 3.1 and 3.2, the product fx 2 is substantially greater than unity at all frequencies where experimental data are shown and, consequently, incoherent superpositions of the waves arising from different induced dipoles occur. The intercollisional absorption defect is limited to low frequencies (Lewis 1980). 

We end this section with a comparison of the basic concepts of laser control and traditional temperature control. This discussion includes an elementary explanation and definition of concepts such as incoherent superpositions of stationary states versus coherent superpositions of stationary states and quantum interference. 

Now we wish to pursue the pictorial presentation of Fig. 2.3 in a mathematical manner that permits us to retain explicitly the quantum features that tend to be obscured in the graphical presentation. In more explicit terms, this corresponds to an incoherent superposition of the magnetizations of individual spins or of individual sets of N interacting spins (spin systems). Incoherent or random motions are commonly treated by statistical methods that deal with an ensemble of molecules, each containing N interacting spins. 

The nonlinear relationship due to the final term causes the wave packet to spread as it propagates. Dropping it assumes that W is so small that the detector can be placed close enough to the scattering target to neglect the spread. Note that only for a photon wave packet is E strictly proportional to k E = tick. The physical situation that we will ultimately consider is that W tends to zero. In section 3.2.2 we showed that the absence of time resolution in an experiment results in the experiment being equivalent to an incoherent superposition of independent experiments, each with an incident plane wave, i.e. an incident wave packet of zero width. 

States of unimolecular reactants prepared by collisional energization and chemical activation, reactions (1) and (2), can also be viewed as incoherent superposition states. The most specific excitation will occur when the collision partners are in specific vibrational/rotational states and the relative translational energy is highly resolved. However, even for this situation it is difficult to avoid preparing a superposition state since the collisions have a distribution of orbital angular momentum. 

The MLDAD vanishes in normal emission if s polarized light is used keeping the other parameters fixed. Unpolarized light can be described by an incoherent superposition of s and p polarized light. Therefore, for the absorption of unpolarized light the effect is two times smaller, but it is described by exactly the same parameters... 

If the different atoms have slightly different absorption frequencies a> = ( 2 -Ei)/h (for example, because of the different velocities in a gas), the phase factors for different atoms are different for t 2T, and the macroscopic fluorescence intensity is the incoherent superposition (7.30) from all atomic contributions. However, for r = 2T the phase factor is zero for all atoms, which implies that all atomic dipole moments are in phase, and super-radiance is observed. 

It is impossible by means of any instrument to distinguish between various incoherent superpositions of wave fields, having the same frequency, that may together form a beam with the same Stokes parameters. This is known as the principle of optical equivalence. 

We now pass from this finite set of trials of the particular membrane sample to the infinite set of trials, whose density distribution is assumed to be independent of the azimuthal angle ip and can be written h(a) sin (a). In the literature this function is frequently called mosaic spread, in this paper we prefer the more illustrative name/lower bunch statistics. The set of all stacks of our particular membrane sample having an axis in the solid angle element (a, da, tp, d ) can be numbered by the index s. Owing to the random centre-to-centre distances, the scattering intensity of this set of stacks can be written as incoherent superposition of the intensities I, of its individual elements ... 

With D 1 mm as the primary beam diameter of conventional LEED optics on the sample, we have D t. As a consequence, the diffraction pattern is an incoherent superposition of the contribution of each coherence area, which means that their intensities just add. This is of htde importance if the structure in each domain is the same but has implications if not In particular, adsorption systems can have symmetrically equivalent but rotationaUy different small domains, so that the total diffraction pattern is a superposition of different single domain patterns. 

Consider radiation incident on an element of area da with unit normal vector n as shown in Fig. 1.8.1. The radiation can be regarded as an incoherent superposition of plane waves, each with an associated Poynting vector S(v, k) where v is the wavenumber of the wave and k is a unit vector defining the direction of propagation. Let N S, V, k) d5 dv dco be the number of plane waves with Poynting vector... 

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