Scattering law dynamic

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. 

Fig. 4 Dynamic scattering law 5(2, at) of PS thin films of thickness (a) 400 A and (b) 1,000 A at various temperatures, as indicated...

Fig. 11 (a) Dynamic scattering law S Q, co) of normal Si02 (circles) and densified Si02 (solid line) [49], (b) Dynamic scattering law S(Q, co) of PMMA, not annealed and annealed at 363 K [50]. The solid line corresponds to the observed full spectra and the solid symbols in (b) correspond to magnified sepctra with the scale factor of 100 (x 100). PMMA at 75 K means S(Q, co) of PMMA measured at 75 K... 

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

Double-Differential Scattering Cross-Section, Dynamic Scattering Law, and Intermediate Scattering Function... 

In order to show the differences between the low energy excitations of amorphous and crystalline phases, we focus on the results of polyethylenes (PE) with different degrees of crystallinity [29,34] (see Fig. 3gdi). Assuming that the observed S(Q,(0) is describedby a linear combination of those of the crystalline and amorphous phases, the dynamic scattering laws S(Q,m) of the crystalline and amorphous phases of PE were evaluated and shown in Fig. 4a.In the spectriun of the amorphous phase, a broad peak is observed at about 3 meV. This broad peak is absent in the spectrum of the crystalline phase. It is then directly concluded that the boson peak at around 3 meV is characteristic of the amorphous phase. 

Fig.4.a Dynamic scattering laws S(Q, ) for amorphous and crystalline phases of PE at 10 K. b Density of states G( ) for amorphous and crystalline phases of PE plotted as G(co)lco against co. Solid lines are fitting curves, c Heat capacity C( T) for amorphous and crystalline phases of PE. Solid lines were calculated from the fitting curves of G((o) in (b) and dashed lines are from experimental data [48]. (Reprinted with permission from [34]. Copyright 1994 Elsevier Science B.V., Amsterdam)... 

Fig. 7. Dynamic scattering laws S(Q,(<)) of poly(methyl methacrylate) (PMMA), not annealed (O) and annealed at 90 °C ( ). The measurements were done at 75 K...

First we show how we can experimentally detect the fast process. Figure 11 shows dynamic scattering laws S(Q,co) of cis-l,4-polybutadiene (PB) below and above Tg (=170 K) [32]. They are obtained with an inverted geometry type of a neutron spectrometer, which is the cause for asymmetric spectra about the origin (ft>=0). The elastic scattering intensity integrated within... 

Fig.11. Dynamic scattering law S(Q,co) of cis-1,4-polybutadiene (PB) below and above the glass transition temperature rg=170 K. Dashed lines are the values expected from the Bose factor. (Reprinted with permission from [32]. Copyright 1993 American Institute of Physics, New York)...

The onset temperature of the fast process was investigated for many kinds of polymers, revealing that the onset temperatures are not always the Vogel-Fulcher temperature. In Fig. 15, for example, dynamic scattering laws of atactic polystyrene (PS) [94] are shown in a temperature range well below Tg (=373 K). At a very low temperature such as 21 K, the boson peak is clearly observed while... 

It is well known that the a-process has a wide distribution of the relaxation time and hence there is a possibility that the E-process is a tail of the a-process. Due to the wide distribution of the relaxation time the intermediate scattering function due to the a-process is described by a stretched exponential function [exp(-(t/T) ) 0< < ]. In order to check this possibility, we fitted a dynamic scattering law, Sjjn(Q>(o)> derived from the Havriliak-Negami (HN) function (see Eqs. 32 and 33) to the observed S(Q,co). The HN function (co) and SjjN(Q.ro) are given by... 

This model has been formulated by Singwi and Sjolander [133]. According to them, the dynamic scattering law S(Q,co) of this model under the condition Ti To is given by... 

The dynamic scattering law S(Q,m) is a Lorentzian, which agrees with our observation and the half-width at half-maximum (HWHM) Fof the Lorentzian function is given by... 

The jump diffusion model with damped vibrations was applied to the analysis of the dynamic scattering laws S(Q,co) of the E-process far above the glass transition temperature Tg, where the conformational transitions are mainly dominated by the intramolecular interactions in a polymer chain, so that it is not necessary to take into accounts intermolecular cooperativity effects on the conformational transitions. 

Finally, we formulate relations between static and dynamic scattering functions. According to the definitions, the static scattering law is identical to the dynamic scattering law at zero time... 

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