Neutron scattering function

Alben and Boutron suggest that the peak in the X-ray and neutron scattering functions at 1.7 A-1 is indicative of an anisotropic layer structure extending over at least 15 A in Polk type continuous random network models. To show this better Fig. 52 displays the radial distribution function of the Alben-Boutron modified... 

A difference in volume between cyclic and linear polymers leads to different solution properties of the cyclic polymers compared to the linear polymers, such as higher GPC elution volume [233], lower intrinsic viscosity [234], lower translational friction coefficient [235], neutron scattering functions more rapid decrease of second virial coefficient with molecular weight [236], and higher critical solution temperature [237]. 

It is worth noting that it is the radial distribution of core electrons in an atom, which is responsible for the reduction of the intensity when the diffraction angle increases. Thus, it is a specific feature observed in x-ray diffraction from ordered arrangements of atoms. If, for example, the diffraction of neutrons is of concern, they are scattered by nuclei, which may be considered as points. Hence, neutron scattering functions (factors) are independent of the diffraction angle and they remain constant for a given type of nuclei (also see Table 2.2). 

Figure 4. a - Landscape and b - isocontour map representations of the incoherent neutron scattering function for the proton harmonic oscillator in one dimension. The intensity is a maximum along the recoil line labelled H. Recoil lines for oscillators with masses corresponding to D, C and O atoms are shown. For a fixed incident energy, only momentum transfer values inside the parabolic area (dashed lines) can be measured. 

The incoherent elastic neutron scattering function for a pair of coupled bosons obtained from Eqs. (1) and (12)... 

The incoherent fraction of the inelastic neutron scattering function is given by... 

The easiestexample for establishing the relationship between the structure of a sample and its scattering behaviour is the case of monoatomic liquid [93]. In this section, the static neutron scattering function (static structure factor) will be derived. In a simple homogeneous monoatomic liquid the probability to find a certain atom in volume element d3r at the position r is given by P(r)dir. Since the liquid is homogeneous P(r = and... 

Figure 4.8 Calculated coherent (ST) and incoherent (Ss) neutron scattering functions for a-Agl and plot of logarithm of incoherent...

Here (Q,(a) is the neutron scattering function which can be directly extracted from the measured cross sections, (j), (r) and (jy (r) are the initial and final wave functions of the proton, e, and are the initial and final energies and the summation is over all possible final states where the delta function ensured energy conservation. We thus have a framework for testing the ab initio calculations against experiment in a very direct way, and, having established its accuracy, we have a theory that can be extended to predict a number of other important properties of the system. In the following sections, we describe the current state of our work in this direction. 

S q,m) (neutron) scattering function electron magnetic moment... 

Conductivity can be deduced from vibrational spectra in IR spectroscopy, the absorption coefficient a(co) is related to tr(co) a(o)) = 4no(o))/nc, n being the refractive index and c the velocity of light. In Raman spectroscopy, the scattered intensity /(m) is related to conductivity by a(o ) oc o)I (o)/n(a)) + 1, n(co) being the Bose-Einstein population factor . Finally, the inelastic incoherent neutron scattering function P(o)) is proportional to the Fourier transform of the current correlation function of the mobile ions. P co) is homogeneous with a) /(cu) formalism. However, since P(co) reflects mainly single particle motions, its comparison with ff(co) could provide a method for the evaluation of correlation effects. (For further discussion, see also Chapter 9 and p. 333.)... 

An attempt to explain experimental data for the neutron scattering function S(o), T) of the y- and a-phases of metallic Ce on the basis of CF and spin-orbit effects (without taking into account the interaction between 4f and conduction electrons), is due to Orlov (1988). The effect of spin-orbit splitting on the dynamic susceptibility of VF systems within the framework of the NCA has been considered by Cox et al. (1986). 

Flory s contention that bulk configurations of polymers are unperturbed by long-range correlations is confirmed for Cie and Cjj n-alkane chains, by the agreement with experiment of neutron scattering functions calculated with a... 

The assumptions (7.106) are the only limitations to be imposed in the derivation of the Boltzmann equation, and it is important to note that no speciAc relation has been selected to describe the neutron-scattering function. Thus the results of this analysis will be generally applicable to many systems of practical importance. In order to demonstrate the application of these results a particular scattering law is later selected, namely, isotropic scattering in the center-of-mass system of coordinates. However, this specific calculation will not be introduced until the general relations have been developed. 

Fig. 2. Neutron scattering function. The function was calculated according to formula (109) at the values of the parameters 5 = 100 x = 10 E = 10, while values of are shown at the curves. Adapted from the paper of Kokorin and Pokrovskii [77]...

Fig. 2.29 (a) Wide-angle X-ray scattering curve for a sample of perdeuterated poly (vinylchloride) (b) Neutron scattering function for the sample subjected to X-ray scattering in the left hand image (Reproduced from Mitchell 2011)... 

At lower temperatures [66, 67, 190-196], the techniques measure the other relaxation processes which are faster than the primary and secondary relaxations. Measurements have found such a fast relaxation process or processes. From the incoherent-neutron-scattering function, S(q, a>, T), the elastic part of the scattering, Se (q, Aco, T), is operationally defined by the integral of S q, co, T) over CO within —Aco < co < Aco, where q is the momentum transfer and Aco is the resolution frequency width of the spectrometer. After normalizing 5ei(, Aco, T) measured at temperature T by its value at r = 0, Se q, Aco, T = 0), one defines a Debye-Waller factor W q, Aco, T) and a mean-square displacement u T)) of the fast relaxation by... 

Figure 9.1 shows the simulated total neutron scattering functions obtained from both the compressed and quenched configurations and compared to those obtained from high pressure neutron scattering experiments [67]. The (Ashcroft-Langreth) structure factors are calculated directly from the atom positions via =... 

Price, D.L., Saboungi, M.L., Susman, S. etal. (1991) Neutron scattering function of vitreous and molten zinc chloride. J. Phys. Condens. Matter, 3, 9835-9842. 

Following Figure 11, we now analyze in detailed the NSE instmment and the connection of measured intensities with the neutron scattering function. The precession angle before and after, the sample and the -flipper, respectively, is proportional to the time the neutron stays in the precession magnetic field. Each infinitesimal section of the path Af hosts the neutron during At = Af/v with the neutron velocity v = (hjm X and thereby contributes AT = y B(l) At with the Larmor constant y = 2 r X 2913.06598 X lO rads f and B(() the local magnetic field (induction). The full path (1=1, 2) then contrihutes... 

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