The incoherent approximation

The approximation works very well within the limits originally set for its use [1]. However, in the molecular spectroscopy of hydrogenous molecules a further version of the approximation is made. Here the cross section used in the calculation of the observed intensities is the total bound scattering cross section of each atom and the dynamics are always treated as incoherent, irrespective of the momentum transferred. This is discussed further with specific examples in 11.1. 

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Fig. 4.26 PB NSE spectra in the -relaxation regime a at 205 K for the Q-values indicated b at Q=1.88 A and c at 2.71 A for the temperatures indicated. Solid lines are the fitting curves obtained in the incoherent approximation for the inelastic part (jump distance dp=1.5 A). (Reprinted with permission from [133]. Copyright 1996 The American Physical Society)...

To account for this we invoke the incoherent approximation. This is discussed in more detail in 2.1.3, in essence, it treats the scattering as purely incoherent but uses the total scattering cross section rather than simply the incoherent cross section. 

Hexafluorobenzene, CeFe [3], provides an interesting test of the validity of the incoherent approximation. In principle, CgFg should be an almost purely coherent scatterer. Fig. 11.2 shows a comparison of the experimental INS spectrum and that calculated from a DFT calculation using the incoherent approximation. It can be seen that the agreement with experiment is very good. There are discrepancies between the observed and calculated frequencies, this is because fluorine is computationally pathological. 

Looking back. Fig. 11.1 provides further confirmation of the validity of the incoherent approximation. The ratio, H/D, of the normalised areas of the bands near 400 cm is 9.8. The predicted ratio on the basis of their incoherent cross sections is 39.2 (= 80.27/2.05) in contrast, the ratio of their total scattering cross sections is 10.7 (= 82.03/7.64) in much better agreement with the experimental observation. 

Fig. 11.2 (a) Experimental INS spectrum of (0.053 mole), (b) spectrum calculated from a DFT calculation assuming the incoherent approximation. 

C. Heavy crystalline moderators. For crystalline materials, the dynamics of the atomic motions is well represented in terms of the quantized, simple-harmonic vibrations of the lattice. These excitations are commonly known as phonons, and are of considerable interest to the solid-state physicist. Since the materials of interest as reactor moderators will occur in polycrystalline form, the use of the incoherent approximation to determine the cross... 

Expression (25) describes the smooth background belonging to a crystalline phase due to the incoherent (or Compton) scattering and the TDS or disorder scattering. The last contribution in (25) is very approximate because it is known that the TDS has a very complicated shape with very large peaks centered in the same position as the Bragg ones [56]. 

Note that in this approximation the incoherent scattering measures the time-dependent thermally averaged, mean square displacement <(rd(t) — (O))2). 

Fig. 6.4 SANS response of carbon exposed to water vapour from HDO (i.e. D20 H20 = 1 1) at RH = 0.87 and TD (3) and an equal mixture of TD and TH (THD) (4), compared to that of D O and TD (2) and dry carbon (1). The incoherent background, 0.07 cm , is the same for samples 2 and 3, and only slightly larger than for the dry carbon (0.06 cm ) The incoherent background due solely to the adsorbed (THD), i.e., excluding the background from the carbon (0.11-0.06 cm ), is thus approximately 5 times greater than that due to the adsorbed TD alone (0.07-0.06 cm )...

Depth of field depends on substrate reflectivity, the degree of partial coherence and the minimum feature size (5). In practice, however, the classical depth of field for the incoherent case (X-f2(N.A.)2) gives a reasonable approximation. Two layer resist processes in which the image is formed in a thin, flat, resist layer on top of a much thicker planarizing layer, alleviate the need for a large depth of field and make it easier to form high resolution, high aspect ratio, resist patterns (6,7). Satisfactory results can be obtained at contrast levels as low as 40%. 

If one wanted to predict slow neutron incoherent scattering from CO then, in the Gaussian approximation, all one would need would be the... 

In the methyl-substituted compound (6.20), R = CH3, the rate constants kjj and are approximately the same as for R = FI, though the energy of 3K, level is at least 0.7 kcal/mol greater than that of 3Ej level. Since the transition 3Ej—is endoergic, kXY > kYX [Eisenberger et al., 1991], We emphasized earlier that the tunneling frequencies of coherent transitions are very sensitive to any asymmetry of the potential. However, in this case the rate constant for the incoherent transition is unaffected by the difference in energies of the initial and final states. 

Due to Eq. (A.3) the above theories only permit the excitation of one quantum of vibration at a time (b and b+ connect vibrational states where their populations differ by only one quantum). This is a consequence of the linear approximation because the nuclear coordinates deviate slightly from the equilibrium situation the molecule can only change its vibrational state by the smallest of the allowed quantities one quantum. In order to account for the excitation of several quanta at a step (coherent excitation) one needs to use other kind of theories (see for example [24]). Nevertheless, the presented approaches permit the sequential excitation of quanta in a ladder climbing fashion (incoherent excitation). 

Step (a) is illustrated in Figme 16. In a reflection, the Bragg peak is superimposed upon the lower, but broader, peakofTDS which peaks at the same position, and the incoherent scattering which varies slowly and not periodically through reciprocal space. The latter can be subtracted, in the first approximation, as an averaged fiat background, provided we know where... 

The aqueous LiCl solution with the composition of LiC1.6.0H2O was prepared in the same way as described for the X-ray samples. In the aqueous solution the proton ( H) has a very large incoherent scattering cross section the observed differential scattering cross section can be approximated to the incoherent dynamic structure factor through... 

Incoherent scattering is not essential when the interaction of x-rays with crystal lattices is of concern, and it is generally neglected. When absorption becomes significant, it is usually taken into account as a separate effect. Thus, in the first approximation only coherent scattering results in the diffraction from periodic lattices and will be considered in this chapter. 

In Fig. 4 we show results for a spin-boson system at low temperature and large Kondo parameter where the linearization approximation is expected to do most poorly. Indeed, in this situation we see a significant discrepancy between the results of our linearized calculations and exact values. The linearized path integral approximation overemphasizes the effect of the friction, and underestimates the importance of the coherent dynamics. Thus the exact result oscillates around zero while the linearized approximate result is overdamped and shows slow incoherent decay. 

The single differential scattering cross-section (1) can be split (in the static approximation) into its incoherent and coherent contributions. ... 

To obtain further insight into the meaning of the inelastic neutron spectra, it is necessary to have specific theoretical models with which to compare the experimental results. In the harmonic approximation it is possible to calculate the incoherent inelastic neutron spectrum i.e., the neutron scattering cross section for the absorption or emission of a specific number of phonons can be obtained with the exact formulation of Zemach and Glauber.481 A full multiphonon inelastic spectrum can be evaluated by use of Fourier transform techniques.482 The availability of the normal-mode analysis for the BPTI136 has made possible detailed one-phonon calculations483 for this system the one-phonon spectrum arises from transitions between adjacent vibrational levels and is the dominant contribution to the scattering at low frequencies for typical experimental conditions.483 The calculated one-phonon neutron en-... 

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]. 

Various methods have been developed that interpolate between the coherent and incoherent regimes (for reviews see, e.g. (3)-(5)). Well-known approaches use the stochastic Liouville equation, of which the Haken-Strobl-Reineker (3) model is an example, and the generalized master equation (4). A powerful technique, which in principle deals with all aspects of the problem, uses the reduced density matrix of the exciton subsystem, which is obtained by projecting out all degrees of freedom (the bath) from the total statistical operator (6). This reduced density operator obeys a closed non-Markovian (integrodifferential) equation with a memory kernel that includes the effects of (multiple) interactions between the excitons and the bath. In practice, one is often forced to truncate this kernel at the level of two interactions. In the Markov approximation, the resulting description is known as Redfield theory (7). 

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