Dipole moment, Fermi contact term

Figure L Dipole moment, Fermi-contact term, and total spin-spin coupling constant for HF as a function of field strength.

In these expressions the index i runs over electrons and a runs over nuclei. The Fermi contact term describes the magnetic interaction between the electron spin and nuclear spin magnetic moments when there is electron spin density at the nucleus. This condition is imposed by the presence of the Dirac delta function S(rai) in the expression. The dipole-dipole coupling term describes the classical interaction between the magnetic dipole moments associated with the electron and nuclear spins. It depends on the relative orientations of the two moments described in equation (7.145) and falls off as the inverse cube of the separations of the two dipoles. The cartesian form of the dipole-dipole interaction to some extent masks the simplicity of this term. Using the results of spherical tensor algebra from the previous chapter, we can bring this into the open as... 

The first term is the interaction of the nuclear magnetic moment (spin I) with the current loops made by the orbiting electrons with angular momentum I. The second term is the dipole-dipole interaction between the nuclear (I) and electronic (st) spin magnetic moments. The last term is the Fermi contact term which describes the interaction of the nuclear... 

Figure 1 shows graphically the behavior of the dipole moment, the Fermi-contact term, and total J as a function of the strength of the imposed electric field. The value of the coupling constant is 515.0 Hz at zero field, decreases to its minimum value of 328.3 Hz at a field of 0.100 au, and then subsequendy increases, with the largest value of J (692.6 Hz) found at a field of 0.175 au. At the same time, the dipole moment of HF is reduced from -0.744 au at zero field to approximately zero at a field between 0.115 and 0.120 au, and then increases to +0.50 au at a field of 0.175 au. Since we generally describe nonpolar molecules as those which are stabilized by "covalent bonds, we would expect that if J is a measure of covalency, the largest J values should occur when... 

There are three components of a coupling constant arising from nucleus-electron interactions. First, the magnetic moment of one nucleus interacts with the field produced by orbital motion of the electrons, which in turn interacts with a second nuclear moment. Secondly, there is a dipole interaction involving the electron spin magnetic moments. The final contribution arises from the spins of the electrons in the orbitals that have a non-zero probability of being at the nucleus (and are therefore derived from the s atomic orbitals). This last term, known as the Fermi contact term, is by far the most important for proton-proton couplings, but for other nuclei the situation is not so simple. 

In addition to the isomer shift and the quadrupole splitting, it is possible to obtain the hyperfine coupling tensor from a Mossbauer experiment if a magnetic field is applied. This additional parameter describes the interactions between impaired electrons and the nuclear magnetic moment. Three terms contribute to the hyperfine coupling (i) the isotropic Fermi contact, (ii) the spin—dipole... 

The terms in equation (4) are generally referred to as the orbital-dipolar interaction (o) between the orbital magnetic fields of the electrons and the nuclear spin dipole, the spin-dipolar interaction (D) between the spin magnetic moments of the electrons and nucleus and the Fermi contact interaction (c) between the electron and nuclear spins, respectively. Discussion of the mathematical forms of each of these three terms appears elsewhere. (3-9)... 

In this Hamiltonian (5) corresponds to the orbital angular momentum interacting with the external magnetic field, (6) represents the diamagnetic (second-order) response of the electrons to the magnetic field, (7) represents the interaction of the nuclear dipole with the electronic orbital motion, (8) is the electronic-nuclear Zeeman correction, the two terms in (9) represent direct nuclear dipole-dipole and electron coupled nuclear spin-spin interactions. The terms in (10) are responsible for spin-orbit and spin-other-orbit interactions and the terms in (11) are spin-orbit Zeeman gauge corrections. Finally, the terms in (12) correspond to Fermi contact and dipole-dipole interactions between the spin magnetic moments of nucleus N and an electron. Since... 

Both terms originate from the same set of dipoles. Bdip is the direct dipolar field of the moments surroimding the muon. We will return to it shortly. The term Bcon is called the Fermi contact field and is produced by the net spin density of conduction electrons in contact with the muon. The spin polarization of the conduction electrons in turn is induced by the dipole moments on lattice sites. One finds... 

The first term results from the Fermi contact interaction, while the second represents the long-range dipole-dipole interaction. In the equations above, ge is the free-electron g factor, /Xe the Bohr magneton, gi the nuclear gyromagnetic ratio, and /xi the nuclear moment. Moreover, the nucleus is located at position R, and the vector r has the nuclear position as its origin. Finally, p (r) = p (r) — p (r) is the electron spin density. The only nontrivial input into these equations is precisely this last quantity, i.e. Ps(r), which can be computed in the LSDA or another DFT approximation. The resulting Hamiltonian can be used to interpret the hyperfine structure measured in experiments. A recent application to metal clusters is reported in Ref. [118]. 

Abstract - The temperature dependence of the proton nmr spectra of dithiocarbamato iron(III) complexes is markedly solvent dependent. A study is made of the temperature dependence of the nmr shifts for the N-CH2 protons in tris(N,N-dibutyldithiocar-bamato) iron(III) in acetone, benzene, carbon disulfide, chloroform, dimethyIformamide, pyridine and some mixed solvents. This contribution shall outline first how the nmr shifts may be interpreted in terms of the Fermi contact interaction and the dipolar term in the multipole expansion of the interaction of the electron orbital angular momentum and the electron spin dipol-nuclear spin angular momentum. This analysis yields a direct measure of the effect of the solvent system on the environment of the transition metal ion. The results are analysed in terms of the crystal field environment of the transition metal ion with contributions from (a) the dithiocarbamate ligand (b) the solvent molecules and (c) the interaction of the effective dipole moment of the polar solvent molecule with the transition metal ion complex. The model yields not only an explanation for the unusual nmr results but gives an insight into the solvent-solute interactions in such systems. 

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