Separation of interactions

The reduced intensity observed in an X-ray diffraction experiment corresponds to the sum over the different partial structure functions, each weighted by the product of the scattering factors for the two atomic species involved. In an aqueous solution of a salt MX, which contains four atomic species M, X, O, and H, the number of different pair interactions is ten and the reduced intensity function can be written  

If two metal ions, M and M, form isostructural solutions, interactions, which do not involve the metal ions, are identical and are eliminated in the difference function A i(s)  

This reduces the ten partial functions to four and of these the last two terms are small and can usually be neglected. 

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The first step towards the development of appropriate expressions is the decomposition of the nonassociative pair potential into repulsive and attractive terms. In this work we apply the Weeks-Chandler-Andersen separation of interactions [117], according to which the attractive part of the Lennard-Jones potential is defined by... 

The separation of interactions by 2D spectroscopy can be compared with 2D chromatography. In a onedimensional thin layer or paper chromatogram, the separation of the constituents by elution with a given solvent is often incomplete. Elution with a second solvent in a perpendicular direction may then achieve full separation. In NMR spectroscopy, the choice of two solvents is replaced by the choice of two suitable (effective) Hamiltonians for the evolution and detection periods which allow unique characterisation of each line. 

Fig. 10. (a) Upper curve shows the RDF for a 1 M tetrachloroplatinate(II) solution before separation of interactions. Lower curve shows the RDF after elimination of nonplatinum interactions. It is compared with a theoretical curve calculated for four Pt—Cl bonds, (b) The part of the RDF involving only nonplatinum interactions. 

More precise structural information can be obtained when a separation of interactions can be made. The RDFs for 2.9 M solutions of erbium(III) and yttrium(III) nitrate, which are isostructural (36), are... 

Lebowitz, J. L., Stell, G., and Baer, S., Separation of interaction potential into two parts in treating many-body systems. I. General theory and applications to simple fluids with short-range and long-range forces. /. Math. Phys. 6, 1282 (1965). 

A discussion of the acceptable range of the Hammett equation, the treatment of deviations, and the meaning of the separation of interaction mechanism is of prime importance in assessing the contribution of electronic effects to biological activity. 

For most substances, the retardation effect in the dispersion interaction energy becomes significant at distances greater than 100 A. The important point is that the Hamaker constant A defined in Eq. (167) is no longer constant but depends on the separation of interacting bodies. 

The effect of dilation of the heart is to augment surface potentials. The potential is augmented by 82.3% when the radius of the heart is increased from 5 to 8 cm (the area occupied by the activation wavefront is proportionately increased as well). This theoretical prediction is opposite to the results of a clinical study (Ishikawa, 1971) of patients with congestive heart failure, which showed a decrease in potentials with increasing cardiothoracic ratio. The model simulations which permit the separation of interactive processes provide a possible explanation to this apparent discrepency. The congestive heart failure is accompanied by pulmonary edema, bringing about an increase in lung conductivity. As discussed above. 

Separation of interactions allows for precise measurements of the small interactions of the observed electron spin with remote spins in the presence of line broadening due to larger contributions. Such techniques are therefore most useful for solid materials or soft matter, where ESR spectra are usually poorly resolved. The most selective techniques for isolating one type of interaction from all the others are pulsed double resonance experiments, such as ENDOR and electron-electron double resonance (ELDOR), which are discussed in more detail in Chapter 2. If the hyper-fine couplings are of the same order of magnitude as the nuclear Zeeman frequency, ESEEM techniques may provide higher sensitivity than ENDOR techniques. In particular, the two-dimensional hyperfme sublevel correlation (HYSCORE) experiment provides additional information that aids in the assignment of ESEEM spectra. These experiments are also discussed in Chapter 2. 

ENDOR techniques work rather poorly if the hyperfine interaction and the nuclear Zeeman interaction are of the same order of magnitude. In this situation, electron and nuclear spin states are mixed and formally forbidden transitions, in which both the electron and nuclear spin flip, become partially allowed. Oscillations with the frequency of nuclear transitions then show up in simple electron spin echo experiments. Although such electron spin echo envelope modulation (ESEEM) experiments are not strictly double-resonance techniques, they are treated in this chapter (Section 5) because of their close relation and complementarity to ENDOR. The ESEEM experiments allow for extensive manipulations of the nuclear spins and thus for a more detailed separation of interactions. From the multitude of such experiments, we select here combination-peak ESEEM and hyperfine sublevel correlation spectroscopy (HYSCORE), which can separate the anisotropic dipole-dipole part of the hyperfine coupling from the isotropic Fermi contact interaction. 

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