External magnetic field spin Hamiltonian

While all contributions to the spin Hamiltonian so far involve the electron spin and cause first-order energy shifts or splittings in the FPR spectmm, there are also tenns that involve only nuclear spms. Aside from their importance for the calculation of FNDOR spectra, these tenns may influence the FPR spectnim significantly in situations where the high-field approximation breaks down and second-order effects become important. The first of these interactions is the coupling of the nuclear spin to the external magnetic field, called the... 

With this spin Hamiltonian and the appropriate wave function it is relatively easy to determine (Appendix B) that the spin interactions give rise to four energy levels which are a function of the external magnetic field ... 

The spin Hamiltonian for the hydrogen atom will be used to determine the energy levels in the presence of an external magnetic field. As indicated in Section II.A, the treatment may be simplified if it is recognized that the g factor and the hyperfine constant are essentially scalar quantities in this particular example. An additional simplification results if the z direction is defined as the direction of the magnetic field. For this case H = Hz and Hx = Hv = 0 hence,... 

Wangsness and Bloch16>17 were the first to give a quantum mechanical treatment of spin relaxation using the density matrix formalism. The system considered is a spin interacting with an external magnetic field (which we suppose here to be constant) and with a heat bath. The corresponding Hamiltonian is... 

In order to discuss the origin of these terms we need to allow the spins to have anisotropic shielding tensors. Molecular tumbling in solution makes the chemical shielding in the direction of the external magnetic field a stochastic function of time and acts therefore as a relaxation mechanism, called the chemical shielding anisotropy (CSA) mechanism. The Hamiltonian for each of the two spins, analogous to Eq. (5), contains therefore two... 

Consideration of the spin-orbit interaction and the effect of an external magnetic field on the electronic ground state of an ion in a CF allows evaluation of the various terms in the spin-Hamiltonian of Equation (31). In addition, the interaction of the nucleus of the paramagnetic ion and ligand nuclei with the d-electron cloud must be considered. In this way the experimentally determined terms of the spin-Hamiltonian may be related to such parameters as the energy differences between levels of the ion in the CF and amount of charge transfer between d-electrons and ligands. 

When magnetic fields are present, the intrinsic spin angular momenta of the electrons S (j) and of the nuclei I(k) are affected by the field in a manner that produces additional energy contributions to the total Hamiltonian H. The Zeeman interaction of an external magnetic field (e.g., the earth s magnetic field of 4. Gauss or that of a NMR... 

The nuclear Zeeman term describes the interaction of the nuclear spins with the external magnetic field. Just as the hyperfine splitting, this term is not incorporated in the original purely electronic Breit-Pauli Hamiltonian as presented in Eqs. (59) and (60) but becomes relevant for ESR spectroscopy. 

Before considering an unpaired electron in a molecule, we shall look at the interaction of an isolated electron having spin magnetic moment fis with an applied external magnetic field B the Hamiltonian is [(1.268) and (1.246)]... 

In such a system, the external magnetic field defines the molecular z axis. If we rotate the molecule with respect to Bo, the spin and its magnetic moment are not affected (Fig. 1.14B). However, in the molecule of Fig. 1.14A, a molecular z axis can be defined. When rotating the molecule, the orbital contribution to the overall magnetic moment changes, whereas the spin contribution is constant. The total Zeeman Hamiltonian is... 

The Hamiltonian that describes the interaction of the single magnetic center with the external magnetic field involves the spin-Zeeman term, the orbital Zeeman term, and the operator of the spin-orbit coupling, i.e.,... 

S2 possesses the eigenvalues S(S + 1). For a spin-1/2 dimer (S can be 0 or 1), the same values are obtained from the derivation of the Bleaney-Bowers equation. For spin systems in an external magnetic field, the Zeeman operator Hmag = -g/ BB S accounts for Zeeman splitting. The isotropic Heisenberg Hamiltonian for multiple spin centers can be expanded by adding the individual coupling pairs ... 

A. Mn(II) EPR. The five unpaired 3d electrons and the relatively long electron spin relaxation time of the divalent manganese ion result in readily observable EPR spectra for Mn2+ solutions at room temperature. The Mn2+ (S = 5/2) ion exhibits six possible spin-energy levels when placed in an external magnetic field. These six levels correspond to the six values of the electron spin quantum number, Ms, which has the values 5/2, 3/2, 1/2, -1/2, -3/2 and -5/2. The manganese nucleus has a nuclear spin quantum number of 5/2, which splits each electronic fine structure transition into six components. Under these conditions the selection rules for allowed EPR transitions are AMS = + 1, Amj = 0 (where Ms and mj are the electron and nuclear spin quantum numbers) resulting in 30 allowed transitions. The spin Hamiltonian describing such a system is... 

The unpaired electron with its spin S = 1/2 in a sample disposed into the resonator of the EPR spectrometer interacts magnetically a) with the external magnetic field H (Zeeman interaction) b) with the nuclear spin of the host atom or metal ion / (hyperfine interaction) c) with other electron spins S existing in the sample (dipole-dipole interaction). In the last case, electrons can be localized either at the same atom or ion (the so called fine interaction), for example in Ni2+, Co2+, Cr3+, high-spin Fe3+, Mn2+, etc., or others. These interac-tions are characterized energetically by the appropriate spin-Hamiltonian... 

The first attempt to explain semi-classical interactions between magnetic spins in a lattice was made by Ising10) who considered an atom of spin S to have (2S + 1) possible orientations in a magnetic field, the interactions being proportional to S Sf which is the product of the a components of the spins. In this case, the Hamiltonian describing the system in an external magnetic field is... 

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