Total neutron structure factor

Fig. 12.5 Total neutron structure factor for liquid GeSe2 solid red lines) obtained by Fourier integration of the calculated PDF within given a integration ranges [O-rd. The black dashed lines correspond to the neutrons S(k) directly calculated in reciprocal space [12]...

The total neutron structure factor Sjik) is defined by... 

Fig. 12.9 Fourier transformed total neutron structure factors of the FPMD models at 300 K for g-GeS4 N = 120 atoms (blue line), N=480 atoms (red line) and g-GeSe4 ...

We examine now structural anomalies that correlate to those detected previously, and follow the total neutron weighted structure factor SN(k) of glassy NS2 with pressure. In Fig. 11.22 is represented the total computed structure factor SN(k), obtained from a linear combination of the partial structure factors Sij (fe), defined by ... 

The data taken is normally presented as the total structure factor, F(Q). This is related to the neutron scattering lengths hi, the concentrations C , and the partial structure factor Sy(Q) for each pair of atoms i and j in the sample, by Equation 4.1-1 ... 

Bowron et al. [11] have performed neutron diffraction experiments on 1,3-dimethylimidazolium chloride ([MMIM]C1) in order to model the imidazolium room-temperature ionic liquids. The total structure factors, E(Q), for five 1,3-dimethylimidazolium chloride melts - fully probated, fully deuterated, a 1 1 fully deuterated/fully probated mixture, ring deuterated only, and side chain deuterated only - were measured. Figure 4.1-4 shows the probability distribution of chloride around a central imidazolium cation as determined by modeling of the neutron data. 

It is noteworthy that the neutron work in the merging region, which demonstrated the statistical independence of a- and j8-relaxations, also opened a new approach for a better understanding of results from dielectric spectroscopy on polymers. For the dielectric response such an approach was in fact proposed by G. Wilhams a long time ago [200] and only recently has been quantitatively tested [133,201-203]. As for the density fluctuations that are seen by the neutrons, it is assumed that the polarization is partially relaxed via local motions, which conform to the jS-relaxation. While the dipoles are participating in these motions, they are surrounded by temporary local environments. The decaying from these local environments is what we call the a-process. This causes the subsequent total relaxation of the polarization. Note that as the atoms in the density fluctuations, all dipoles participate at the same time in both relaxation processes. An important success of this attempt was its application to PB dielectric results [133] allowing the isolation of the a-relaxation contribution from that of the j0-processes in the dielectric response. Only in this way could the universality of the a-process be proven for dielectric results - the deduced temperature dependence of the timescale for the a-relaxation follows that observed for the structural relaxation (dynamic structure factor at Q ax) and also for the timescale associated with the viscosity (see Fig. 4.8). This feature remains masked if one identifies the main peak of the dielectric susceptibility with the a-relaxation. 

Recently, the PDF method was extended to describe the local dynamics of disordered materials (Dmowski W, Vakhrushev SB, Jeong I-K, Hehlen M, Trouw F, Egami T (2006) Abstracts American conference on neutron scattering, St. Charles, IL, 18-22 June 2006, unpublished). The total PDF is obtained by the powder diffraction method so that S(Q) includes both elastic and inelastic intensities. To determine the dynamics we have to use an inelastic neutron scattering spectrometer and measure the dynamic structure factor, S(Q,a>), over a large Q and co space, and Fourier-transform along Q to obtain the dynamic PDF (DPDF). While the interpretation of the DPDF is a little... 

The algorithm described above is specifically for modelling a single set of diffraction data which could be obtained using either X-rays, neutrons or electrons. The fit may be either to the structure factor or to the radial distribution function, though the former is recommended because the distribution of errors in the latter may be highly non-uniform. In practice a fit is normally made first to the radial distribution function, then to a subset of the total structure factor points, and finally to all the structure factor points. This considerably reduces the time required. 

The quantity measured in such an experiment is the differential neutron scattering cross section da/dQ, of which the interference part (or total structure factor) can be expressed as a linear sum of the partial structure factors, S(Q). The structure factor describes the spatial distribution of scattering centres (the atomic nuclei) of the sample in question. Thus, in the total structure factor, all distances between all scatterers are present, weighted according to the concentration of each particular type of atom, c, and their scattering length, b. The differential neutron scattering cross section can be written as ... 

If one wishes to determine aU the three partial radial distribution functions of a two-component material, like water, then three independent total structure factors must be measured for (chemically) the same material. In this way, the coefficients of the partial structure factors Sy(Q), O Eq. (29.33), will be different, because the scattering lengths wiU be different. As shown by Table 29.1, different isotopes of the same element do possess different scattering amplitudes for neutrons. This property made possible the development of the technique called isotopic substitution (North et al. 1968). For a two-component system, three different isotopic samples have to be prepared, so that the coefficients of the partial structure factors should differ sufficiently weU for being able to solve a set of three linear equations, of the form of Eq. (29.33), for the three unknown partial structure factor. 

The total structure factors and total pair correlation functions from the neutron diffraction are shown in Figs. 4.2 and 4.3a. There is a high degree of similarity in the total structure factors for the five samples. However in the total pair correlation functions there appears to be a degree of difference between the five samples occurring at a distance of 2.3 A. 

The total pair correlation functions for the neutron diffraction, MD, and RMC of the all samples are compared in Fig. 4.7. While there is broad similarity between the MD and the neutron data, it is apparent that there are significant interatomic interactions that are not accounted for with the simple Buckingham potentials described in Sect. 4.2. The RMC total structme factors are in very good agreement with the corresponding neutron diffraction structure factors, as shown in Fig. 4.7a. 

Fig. 9. a) Bathia-Thornton type neutron total structure factors of Ni (X 0,... 

In a neutron diffraction experiment on a liquid or glassy MX2 system, the coherent scattered intensity measured with respect to the magnitude of the scattering vector k can be represented by the total structure factor [44]... 

The full set of Sap (k) functions for an MX2 system can be extracted from the measured diffraction patterns by applying the NDIS method, provided that isotopes are available with a sufficiently large neutron scattering length contrast [44, 46, 47], The total structure factor can also be expressed in terms of the Bhatia-Thornton [48] number-number, concentration-concentration and number-concentration partial structure factors denoted by Sccik) and nc) ), respectively. These partial... 

If bu = bx the incident neutrons in a diffraction experiment cannot distinguish between the different scattering nuclei and the measured total structure factor gives Sxxik) directly (see (1.4)). The corresponding Fourier transform nn(c) therefore... 

Comparing structural information from X-ray and neutron diffraction provides a very valuable way to validate MD simulation results of glasses. In some simple systems, the partial pair distribution function or partial structure factors of all atom pairs can be determined experimentally and they provide excellent validations for simulated structures. However, as the composition becomes more complicated and more elements included, larger number of pair contributions will complicate the comparison and the validation becomes more and more difficult in multicomponent glass systems. For example, for binary oxides, e.g. sodium silicate, there are six partial pair distribution functions, but for a four component systems, for example the bioactive glass composition, there are a total of fifteen partials contributions. The overlap between partial contributions makes it very challenging to assign the peaks and to determine the quality of comparison and hence the validation of the simulated structure models. 

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