Scattering density

We have seen that the intensities of diffraction of x-rays or neutrons are proportional to the squared moduli of the Fourier transfomi of the scattering density of the diffracting object. This corresponds to the Fourier transfomi of a convolution, P(s), of the fomi... 

In order to estimate the phonon scattering strength and thus the heat conductivity, we need to know the effective scattering density of states, the transition amplitudes, and the coupling of these transitions to the phonons. 

We now calculate the density of the phonon scattering states. Since we have effectively isolated the transition amplitude issue, the fact of equally strong coupling of all transitions to the lattice means that the scattering density should directly follow from the partition function of a domain via the... 

Figure 20. The predicted low T heat conductivity for three different values of Tg/(Ori = 5,3.5,2 for a simpler model of the scattering density as explained in text. The a-Si02 is the same as in Fig. 19.

In crystals, the scattering densities are periodic and the Bragg amplitudes are the Fourier components of these periodic distributions. In principle, the scattering density p(r) is given by the inverse Fourier series of the experimental structure factors. Such a series implies an infinite sum on the Miller indices h, k, l. Actually, what is performed is a truncated sum, where the indices are limited to those reflections really measured, and where all the structure factors are noisy, as a result of the uncertainty of the measurement. Given these error bars and the limited set of measured reflections, there exist a very large number of maps compatible with the data. Among those, the truncated Fourier inversion procedure selects one of them the map whose Fourier coefficients are equal to zero for the unmeasured reflections and equal to the exact observed values otherwise. This is certainly an arbitrary choice. 

In neutron diffraction studies, the Fourier map does not show significant negative scattering density around the terminal Au-oxo oxygen 035, a result similar to the previous neutron diffraction on the terminal Pt-oxo complex 1, thus ruling out the possibility that a... 

The type of measurements that are needed to accomplish the task should also be considered. Refractive index values are generally needed for measurements based on light scattering. Densities are often needed for techniques based on acoustics and sedimentation. Further, most approaches require the samples to be dissolved or suspended in a liquid. Thus, information related to how the liquid affects particle shape and association is also important. 

When the model used for Fcalc is that obtained by least-squares refinement of the observed structure factors, and the phases of Fca,c are assigned to the observations, the map obtained with Eq. (5.9) is referred to as a residual density map. The residual density is a much-used tool in structure analysis. Its features are a measure for the shortcomings of the least-squares minimization, and the functions which constitute the least-squares model for the scattering density. 

The "phase problem" in crystallography arises because in the usual experiment (Eq. () the magnitudes of the complex structure factors arc obtained, but not the phases. Yet in order to obtain the scattering density. 

There are two approaches to the solution of the phase problem that have remained in favor. The first is based on the tremendously important discovery or Patterson in the 1930s ihal the Fourier summation of Eq. 3. with (he experimentally known quantities F2 (htl> replacing F(hkl) leads nol to a map of scattering density, but to a map of all interatomic vectors. The second approach involves the use of so-called direct methods developed principally by Karie and Hauptman of the U.S. Naval Research Laboratory and which led to the award of the 1985 Nobel Prize in Chemistry. Building upon earlier proposals that (he relative intensities of the spots in a diffraction pattern contain information about a crystal phase. Hauptman and Karie developed a mathematical means of extracting the information. A fundamental proposition of (heir direct method is that if thrice intense spots in the pattern have positions whose coordinates add up to zero, their relative phases will cancel out. Compulations done with many triads of spots yield probable phases for a significant number of diffracted waves and further mathematical analysis leads lo a likely solution for the structure of the molecule as a whole. 

PS P4VP Toluene (selective for PS) Determination of scattering density profile and comparison to scaling theory SANS Forster et al. (1996)... 

For typical compact proteins this plot has a positive slope, as the hydrophilic residues on the outside of the dissolved protein have a higher scattering density than the hydrophobic residues on the inside. For casein sub-micelles, the slope is negative (Stothart and Cebula, 1982) (Figure 2). This seems surprising at first sight, but the sub-micelles are so highly hydrated that all the constituent protein... 

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