Scalar interactions mechanism

Figure 2 The four-level diagram for a system of two interacting spins, in this case an electron (S) and nucleus with a positive gyromagnetic ratio (/). The intrinsic electron and nuclear spin relaxation are given by p and w°, respectively, and the dipolar and/or scalar interactions between the electron and nuclear spin are represented by p, w0, w, and w2. The transition w0 is known as the zero-quantum transition, while w, is the singlequantum transition and w2 is the double-quantum transition. Nuclear and electronic relaxation through mechanisms other than scalar or dipolar coupling are denoted with w° — 1/Tio and p — 1/Tie, where Ti0 and T1e are the longitudinal relaxation times of the nucleus and electron in the absence of any coupling between them. Since much stronger relaxation mechanisms are available to the electron spin, the assumption p>p can be safely made. Adapted with permission from Ref. [24].

In a dilute solution of a paramagnetic solute, the nuclear relaxation is often entirely dominated by the pairwise interaction between an unpaired electron, S, and nucleus, I. This is because the rapid diffusion of the solute in the solvent ensures that all nuclei are equally affected. The strong local fields produced by the electron can be coupled to the nuclei by a dipole—dipole interaction and sometimes by a scalar interaction as well. The scalar interaction is transmitted to the nucleus by similar mechanism to that producing spin-spin multiplets in n.m.r. spectra and the hyperfine structure in e.s.r. spectra. 

When there is unpaired electron density produced at the nucleus being studied, a so-called scalar interaction between the electron and the nucleus can occur. This unpaired electron density is transmitted from the free radical to the nucleus by a similar mechanism to that giving rise to the nuclear hyperfine structure in normal e.s.r. spectra. Since the unpaired electron and the nucleus are usually in different molecules, the rapid molecular motion means that the scalar interaction, which of necessity can only occur when the unpaired electron and the nucleus are close to one another, will be varying rapidly. This rapid switching on and off of the scalar interaction means that although any intermolecular nuclear hyperfine structure in the e.s.r. signal is averaged to zero, the scalar interactions may now provide an efficient... 

There are a couple of special methods of separating the contribution of dipolar relaxation in solution. One is by the NOE factor which is the fractional difference in the signal intensity of one spin with and without irradiation applied to another spin system. For a sample containing protons and carbon-13 in the motionally narrowed limit, this factor should be 2 if the relaxation takes place through the dipolar and the scalar interactions. Thus, the departure from 2 of the NOE factor is an indication of other relaxation mechanisms. Clearly, any other pairs of spin systems with NOE s can be treated this way, with appropriate limiting NOE factors. See, for example, Noggle and Shirmer listed in Appendix A for more details. 

Some of the terms included in the Breit-Pauli Hamiltonian also describe small interactions that can be probed experimentally by inducing suitable excitations in the electron or nuclear spin space, giving rise to important contributions to observable NMR and ESR parameters. In particular, for molecular properties for which there are interaction mechanisms involving the electron spin, also the spin-orbit interaction (O Eqs. 11.13 and O 11.14) becomes important The Breit-Pauli Hamiltonian in O Eqs. 11.5-11.22, however, only includes molecule-external field interactions through the presence of a scalar electrostatic potential 0 (and the associated electric field F) and the appearance of the magnetic vector potential in the mechanical momentum operator (O Eq. 11.23). In order to extract in more detail the interaction between the electronic structure of a molecule and an external electromagnetic field, we need to consider in more detail the form of the scalar and vector potentials. 

Relaxation data of the quadrupolar halogens are directly obtained from pulsed NMR studies, although most data on T2 are from measurements of linewidth or from peak-to-peak distances of the first derivative of absorption curves. Using pulse methods the lower limit of Tj and T2 depends on the dead time of the spectrometer. Presently the practical limits appear to be ca. 10 ys. A number of the available relaxation data on Cl, Br, and I compounds have been determined through the effects of the halogen relaxation on the relaxation of directly bonded spin 1/2 nuclei through modulation of scalar interaction this mechanism is commonly termed "scalar relaxation of the second kind. Very short... 

Lee next generalizes discrete quantum mechanics to the case of a massless scalar field interacting with an arbitrary external current J ([tdlee85a], [tdlee85b]). 

Multi-scalar triadic interactions in differential diffusion with and without mean scalar gradients. Journal of Fluid Mechanics 321, 235-278. 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

Dipolar (or direct) coupling between nuclear dipoles is the second mechanism of spin-spin interaction (see Section 4.1.1.2. for scalar coupling). Acting through space209-212 it is dependent on the distance r between the dipoles as well as the angle between their vectors and the direction of the external field B0. 

The mathematics we shall need is confined to the properties of vector spaces in which the scalar values are real numbers. From a mathematical viewpoint the whole discussion will take place in the context of two vector spaces, an S-dimensional space of chemical mechanisms and a Q-dimen-sional space of chemical reactions, which are related to each other by the fact that each mechanism m is associated with a unique reaction R(m) which it produces. The function R is a transformation of mechanisms to reactions which is linear by virtue of the fact that reactions are additive in a chemical system and that the reaction associated with combined mechanisms mt + m2 is R(m,) + R(m2). All mechanisms are combinations of a simplest kind of mechanism, called a step, which ideally consists of a one-step molecular interaction. Each step produces one of the elementary reactions which form a basis for the space of all reactions. 

The accurate quantum mechanical first-principles description of all interactions within a transition-metal cluster represented as a collection of electrons and atomic nuclei is a prerequisite for understanding and predicting such properties. The standard semi-classical theory of the quantum mechanics of electrons and atomic nuclei interacting via electromagnetic waves, i.e., described by Maxwell electrodynamics, turns out to be the theory sufficient to describe all such interactions (21). In semi-classical theory, the motion of the elementary particles of chemistry, i.e., of electrons and nuclei, is described quantum mechanically, while their electromagnetic interactions are described by classical electric and magnetic fields, E and B, often represented in terms of the non-redundant four components of the 4-potential, namely the scalar potential and the vector potential A. 

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