Elastic line

Specification of. S SkCG, CO) requires models for the diffusive motions. Neutron scattering experiments on lipid bilayers and other disordered, condensed phase systems are often interpreted in terms of diffusive motions that give rise to an elastic line with a Q-dependent amplitude and a series of Lorentzian quasielastic lines with Q-dependent amplitudes and widths, i.e.. 

As in QMS, the degree of coherence is changed if the scattering atom itself jumps (with a frequency 1/r which is on the order of i/t , where t is determined by the time resolution of the spectrometer). The jumping atom transfers energy either to or from the scattered neutrons. As mentioned before, the result is a broadening (8v) of the elastic line. We expect Sv to vary exponentially with temperature. 

The elastic line is dressed with the full spectrum of phonons in all their orders, as indeed is each individual internal transition but it remains to be determined what form, or shape, this dressing takes. 

The elastic line, which appears at an energy transfer of zero for zero momentum transfer, is now displaced by 422 cm to higher energies. A similar effect is seen on all the internal transitions. Thus, for example, the doubly degenerate deformation mode of methane, V4, appearing at 1534 cm in the infrared, would be centred at 1956 cm (=1534 + 422) in the INS (if measured at the momentum transfer value of 9 A ). The standard deviation of the Gaussian, F (cm ), is given by [12] ... 

An example of FDS output, for potassium hydrogen maleate at 15 K is shown in Fig. 3.17. The improved resolution of the (Be—BeO) spectrum, over the simple Be filter method, is marked. This results from two factors the narrower bandpass used in the difference spectrum and the removal of the long-time tails of the incident pulse. The long-time tails are seen illustrated in the insets in Fig. 3.17, which show the shapes of the elastic lines. The energy transfer range is 50—5000 cm , the... 

Fig. 3.17 Treatment of FDS (LANSCE) data INS time-of-flight spectrum of potassium hydrogen maleate recorded with (a) Be filter, (b) BeO filter and (c) the difference (Be-BeO). The inset in each spectrum shows the lineshape of the elastic line associated with each measurement. Reproduced from [20] with permission from Elsevier.

Fig. 3.22 (a) The raw time-of-flight data of 2,5-diiodothiophene (notice the strong elastic line at 13600 ps), (b) the resulting INS spectrum. 

The increased incident flight-path has a second consequence. The intense feature at 13600 ps, in Fig. 3.22a, is the elastic line for TFXA, d = 12.1 m. As the flight path is increased, the elastically scattered neutrons reach the sample later and the elastic line occurs at later times. On TOSCA where di = 17.5 m, it occurs at 22600 ps. However, ISIS operates at 50 Hz and each time frame is only 20000 ps long. Thus the elastic line from one pulse of neutrons would occur in the next time frame frame overlap. An alternative viewpoint is that fast neutrons from the current frame have overtaken the slow neutrons from the previous frame. Frame overlap is catastrophic because the uniqueness of the time-stamp, so crucial to the analysis of the spectrum, is lost. 

The results of QENS measurements for TaV2H [76] are consistent with this microscopic picture of H motion. First, on the frequency scale of tf the measured QENS spectra S(Q, co) are well described by the sum of a narrow elastic line and a broader quasielastic line having Q-dependent intensity, but Q-independent width. These features are typical of the case of spatially-confined (localized) motion [14]. 

The quasi-elastic line width obtained by means of inelastic neutron scattering for UCu5 is virtually unaffected by the antiferromagntic transition. The HWHM of the Lorentzian line of = 10 meV considerably exceeds /cBTN. A high-resolution low-energy study, however, displays an additional Gaussian line which can be accounted for as the response of the antiferromagnetic system. Its intensity is maximal around... 

The results of two inelastic neutron scattering experiments on U2Zn17 are rather controversial (Walter et al. 1987, Broholm et al. 1987a). Common features shared by both works are the absence of spin-wave excitations and the presence of a broad Lorentzian quasi-elastic line with T of the order of 10 meV, persisting across Tn. 

Figure 6.4. Temperature dependence of the quasi-elastic line (underneath the central inelastic peak) for IrClH2(H2)(P Pr3)2 at 100,175,210, and 250 (in upward direction). Broadening is evident above 175 in the base of the line. Reprinted with permission from Li et al.34 Copyright 2000 American Chemical...

If this specimen were behaving in a linear viscoelastic pattern, the loop would be an ellipse whose long axis lies on the linear elastic line indicated by the dotted line. The experimental hysteresis loop in Figure 3, however, is bent downward considerably. The extent of this downward bending is such that even the upper curve of the loop which represents the stretching phase is sometimes located below the linear elastic line. 

Fig. 4 Brillouin spectra recorded at atmospheric pressure at Tc 5 K in right-angle geometry along the direction (101), nearly parallel to the anthracene long axis (left), and the (101) direction almost perpendicular to this long axis (right). The R. T, OT, and QL indicate, respectively, the Rayleigh elastic line, the transverse mode polarized along the b monoclinic axis, and the quasitransverse and quasi longitudinal modes polarized in the (a,c) plane. The tree spectral range is 39.15 GHz. (From Ref. [18].)...

Recent analyses of contact angle hysteresis has treated surface heterogeneity in terms of random fluctuations see the excellent review by deGennes The approach is to treat both surface roughness and variations in surface composition as weak fluctuations, i.e. deviations from the ideally smooth surface, du/dy (Fig. 11) and from the difference in solid/liquid and solid/vapor surface energies, — yj - Both fluctuations are considered to be equivalent in perturbing the contact line and are analyzed in terms of their effect on the elastic line energy. 

Fig. 15. Scattered intensity of time-of-flight spectra plotted against energy transfer, E, for polycrystalline CeAlj. Thick continuous line shows result of fit constructed from various components indicated by fainter lines flat background, nuclear elastic line, inelastic magnetic lines at = 5.2meV and 7.5 meV, respectively (Murani et al. 1977).

In the elastic chain model of polymers, the polymer is described by a continuous elastic line or space curve r (n) where n denotes the distance to the origin along the backbone (fig. ib). 

Figure 8 (a) The variation of the measured P parameter and of the fraction of immobile water q versus T for the fully hydrated C-phycocyanin (A = 0.5 g water/g protein) (Ref. 58). (b) Half width at half maximum of the Lorentzian quasi-elastic line (r) versus Q, for T = 293 K, for the fully hydrated C-phycocyanin and for the bulk water (Ref. 58). 

Fig- 18. Dynamic magnetic susceptibility Im x(o3), normalized to its maximum value, as a function of frequency a>. Solid lines result from the NCA, the dashed line corresponds to the quasi-elastic line shape eq, (25), and the dots are experimental data for CePda (solid circles at 5 K, open circles at 280 K from Galera et al. (1985a,b)) (Bickers et al. 1987). 

Fig. 23. Inelastic neutron spectra of YPdj and CePdj taken with q = 3.5 meV at a constant scattering angle of 20 = 20° and two different temperatures. The solid line is a fit with one broad quasi-elastic line (Holland-Moritz et al. 1982).

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