Electronic and Nuclear Magnetic Dipoles

At small distances, the two unpaired electrons will experience a strong dipole-dipole interaction analogous to the interaction between electronic and nuclear magnetic dipoles, and this gives rise to anisotropic hyperfine interactions. The electron-electron interaction is described by the spin-spin Hamiltonian given by ... 

The through-space interaction is a dipolar coupling between the electron and nuclear magnetic moments. When the Zeeman interaction for both the electron and nuclear spins is the dominant term in the spin Hamiltonian of the system, the energy of the dipole-dipole interaction is inversely proportional to the third power of the dipole-dipole distance fis according to... 

One of the most important interactions of the electron spin of the paramagnetic molecule, which influences its line position, is that with the nearby nuclear spins. These are the nuclei of the molecule itself and nuclei in the local surrounding upto a distance of ca. 0.6 nm. There are two different contributions to this interaction an isotropic part (fcrmi-contact interaction), which arises from electron spin density of the unpaired electron at the nucleus, and an anisotropic part, which arises from through-space magnetic dipole-dipole interaction between the electron and nuclear magnetic moments. Whereas for intramolecular nuclei, the isotropic part can be very large, interactions with intermolecular nuclei further apart are mainly anisotropic. This offers the possibility to measure directly the distance to the nucleus by the... 

Electron Current Density Induced by Magnetic Fields and Nuclear Magnetic Dipoles... 

The interaction of the electron spin s magnetic dipole moment with the magnetic dipole moments of nearby nuclear spins provides another contribution to the state energies and the number of energy levels, between which transitions may occur. This gives rise to the hyperfme structure in the EPR spectrum. The so-called hyperfme interaction (HFI) is described by the Hamiltonian... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

The physical interpretation of the anisotropic principal values is based on the classical magnetic dipole interaction between the electron and nuclear spin angular momenta, and depends on the electron-nuclear distance, rn. Assuming that both spins can be described as point dipoles, the interaction energy is given by Equation (8), where 6 is the angle between the external magnetic field and the direction of rn. 

Though the ESR Hamiltonian is typically expressed in terms of effective electronic and nuclear spins, it can, of course, also be derived from the more fundamental Breit-Pauli Hamiltonian, when the magnetic fields produced by the moving nuclei are explicitly taken into account. In order to see this, we shall recall that in classical electrodynamics the magnetic dipole equation can be derived in a multipole expansion of the current density. For the lowest order term the expansion yields (59)... 

In quantum mechanics this then becomes the vector product of the nuclear magnetic dipole moment and the distance vector between an electron i and the field-creating nucleus I (60)... 

Modem structural chemistry differs from classical structural chemistry with respect to the detailed picture of molecules and crystals that it presents. By various physical methods, including the study of the structure of crystals by the diffraction of x-rays and of gas molecules by the diffraction of electron waves, the measurement of electric and magnetic dipole moments, the interpretation of band spectra, Raman spectra, microwave spectra, and nuclear magnetic resonance spectra, and the determination of entropy values, a great amount of information has been obtained about the atomic configurations of molecules and crystals and even their electronic structures a discussion of valence and the chemical bond now must take into account this information as well as the facts of chemistry. 

The ground-state electronic configurations (levels) of neutral and singly ionized berkelium were identified as 5f 7s2 (6H15/2) and Sf s1 (7H8), respectively (82). A nuclear magnetic dipole moment of 1.5 nuclear magnetons (61) and a quadrupole moment of 4.7 barns (83) were determined for 249Bk, based on analysis of the hyperfine structure in the berkelium emission spectrum. 

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 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)... 

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