Scattering laws

As in the case of the Rouse dynamics (see Sect. 3.1.1), the intermediate incoherent scattering law for dominant hydrodynamic interaction (Zimm model) can be... 

From Fig. 35, where the normalized coherent scattering laws S(Q, t)/S(Q,0) are plotted as a function of 2 (Q)t for Zimm as well as for Rouse relaxation, one sees that hydrodynamic interaction results in a much faster decay of the dynamic structure factor. 

The NSE spectra from full stars are well described by the scattering law of the Zimm model in the whole experimental time window (20 ns)... 

Polycrystal-type (rings) electron diffraction patterns (Fig.6) are especially valuable for precision studies - checking on the scattering law, identification of the nature of chemical bonding, and refinement of the chemical composition of the specimen - because these patterns allow the precision measurements of reflection intensities. The reciprocal lattice of a polycrystal is obtained by spherical rotation of the reciprocal lattice of a single crystal around a fixed 000 point it forms a system of spheres placed one inside the other and has the symmetry co oo.m. It is also important for structure... 

Here, the scattering law for the translational motion is expressed by the Lorentzian function as... 

For hydrocarbons in zeolites, only incoherent scattering has to be considered because of the large incoherent cross section of hydrogen. The neutron intensity scattered follows the incoherent scattering law 5i c(Q, (o), which is related to the self-motion of protons, where AQ and Aw denote the neutron momentum transfer and the neutron energy transfer, respectively. 

Assuming for a molecular system the vibrational, rotational, and translational motions as uncoupled, the incoherent scattering law can be expressed as a convolution product of the individual scattering laws for each motion, which can be examined separately ... 

For the quasi-elastic region ( 2 meV) that is of concern here, the vibrational motions affect only the elastic intensity through a Debye-Waller factor, exp(-Q (M )), where (u ) is the mean square hydrogen amplitude in the vibrational modes. The scattering law for rotations can be written as a sum of an elastic peak intensity and a quasi-elastic component ... 

The simplest model of translational motions in three dimensions is the continuous random walk diffusion (Pick s law) giving a Lorentzian-shaped scattering law,... 

The broadening values obtained for methane diffusion can be fitted to a molecular jump model with a Gaussian distribution of jump lengths (25,62). Thus, the scattering law becomes a Lorentzian,... 

S-wave scattering is the only practical outcome since P-wave final neutron states are not accessible to thermal neutrons, because these wave functions have negligible amplitude at the small radial values that are typical of atomic nuclei. It is convenient to rewrite the equation as a dynamical structure factor (or Scattering Law), which emphasises the dynamics of the sample. 

In general, the exact functional form of the scattering law is complex. For most experiments on hydrogeneous polymers at low temperatures, the scattering probability has been taken to be proportional to the one-phonon scattering cross-section 74, 34) for crystals of cubic symmetry, namely ... 

KsQo, identification of optic modes and the nature near-gap scattering law below T, ... 

The light scattering methods provide statistically averaged quantities when applied to polydisperse samples (e.g., micellar or polymer solutions). The case of independent scatterers can be rigorously treated 2 by using the mass distribution function of the particles, f M). By definition, dm =f(M)dM is the mass of particles in the range between M and (M + dM), scaled by the total particle mass. As shown by Zimm, the scattering law in such a system can be presented similarly to the case of monodisperse particles (see Equation 5.405) ... 

Fig. 1.1 INS spectroscopy in relation to optical (photon) spectroscopies. INS spectra are within the wavenumber range 16-4000 cm", the same as the mid-infrared range. The vertical axis is the neutron scattering intensity expressed as the scattering law (or function), see text.

We rewrite the double differential of Eq. (2.28) in terms of a function, S, which emphasises the dynamics of each individual atom in the sample. This is conventionally called the scattering law and is directly related to the observed intensities when summed over all the atoms in the sample, Eq. (A2.49). 

Most modem neutron spectroscopic data are reduced to the scattering law since its relationship to the common model of molecular vibrations is straightforward. The scattering law is the natural meeting point of experimental data and the dynamical models that have been developed for its understanding. However, it must be admitted that the literature is inconsistent on this point and many different functions can be found. Fortunately many of them are simply scaled versions of the scattering law. 

Whilst loosely referring to the scattering law all of the spectroscopic intensities reported in this book are S Q,o. This function is closely related to the amplitude and cross section weighted density of states often referred to in condensed matter physics studies of molecular systems. For a single molecule containing, Adatom, atoms... 

Similarly the number of molecules in the neutron beam, A/moi, is also subsmned into the scaling factor. What remains has all of the properties of the scattering law since the cross section has effectively disappeared from the formulation. Whenever the expression scattering law is used in the text it will refer to SXQ,(ii), unless otherwise stated. In Appendix 2 the calculated intensity in a particular mode, labelled v, involving the atomic displacement is given (for a single crystal sample) by Eq. (A2.80) ... 

The reader will appreciate that the scattering law of a particular vibrational transition, given by Eq. (2.32), is written in terms of the scalar (or inner, or dot) product of the neutron momentum transfer vector, Q, and the atomic displacement vector, . This product is a function of the angle between the vectors and allows some interesting single crystal effects to be examined. 

Occasionally molecular crystals of suitable size are found that have all of their molecules aligned mutually parallel [8], It is worth investigating how the scattering law of one particular atom, vibrating in the normal mode, v, would change as a function of the relative... 

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