Interaction Hamiltonian magnetic dipole

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... 

A nucleus in a state with spin quantum number 7 > 0 will interact with a magnetic field by means of its magnetic dipole moment p. This magnetic dipole interaction or nuclear Zeeman effect may be described by the Hamiltonian... 

Magnetic dipole interaction Hm (4.47) and electric quadmpole interaction //q (4.29) both depend on the magnetic quantum numbers of the nuclear spin. Therefore, their combined Hamiltonian may be difficult to evaluate. There are closed-form solutions of the problem [64], but relatively simple expressions exist only for a few special cases [65]. In Sect. 4.5.1 it will be shown which kind of information can be obtained from a perturbation treatment if one interaction of the two is much weaker than the other and will be shown below. In general, however, if the interactions are of the same order of magnitude, eQV Jl, and... 

The low-temperature Mossbauer spectra of the spinel type oxides, NiCr204 [14,18] (Fig. 7.6b) and NiFe204 [3, 18], have been found to exhibit combined magnetic dipole and electric quadrupole interaction (Fig. 7.7). For the evaluation of these spectra, the authors have assumed a small quadrupolar perturbation and a large magnetic interaction, as depicted in Fig. 7.3 and represented by the Hamiltonian [3]... 

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)... 

Provided that a transition is forbidden by an electric dipole process, it is still possible to observe absorption or emission bands induced by a magnetic dipole transition. In this case, the transition proceeds because of the interaction of the center with the magnetic field of the incident radiation. The interaction Hamiltonian is now written as // = Um B, where is the magnetic dipole moment and B is the magnetic field of the radiation. 

Two distinct sources contribute to the spin Hamiltonian of the form of (6), spin-orbit coupling and spin-spin interaction. The latter, magnetic dipole-dipole interaction between the unpaired electrons, is dominant in organic molecules that do not contain heavy elements. 

Thus suppose we had included the interaction of the radiation s magnetic field B with the atomic or molecular electrons and nuclei. The Hamiltonian for this interaction is [Equation (1.268)] -B , where p is the magnetic dipole-moment operator for the system. This gives additional terms in cm that are proportional to... 

We recall from chapter 4 that there are many individual types of interaction which involve the nuclear magnetic dipole and electric quadrupole moments. Let us take just three of these to exemplify how the effective Hamiltonian is constructed. They are as follows ... 

For molecules that are tumbling rapidly so that magnetic dipole interactions can be neglected, the sum of and W1 is adequate as the complete Hamiltonian to determine energy levels for the nuclear spin system. However, as we noted in Section 5.5, additional terms must be added to account for external perturbations, such as strong rf fields. In this chapter we take the steady-state Hamiltonian as... 

Hyperfine interactions between the electron and any magnetic nuclei (7>0) present (such as a proton, for example) produces hyperfine splitting, as illustrated in a very simple example in Fig. 2.51. This hyperfine interaction may be divided into an isotropic and an anisotropic component. The isotropic part arises from unpaired electron density at the nucleus and can only be nonzero for i -type orbitals. The anisotropic term corresponds to the classical part of the magnetic dipole interaction for which the Hamiltonian is ... 

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 ... 

Static spin echo decay spectroscopy also forms the basis for the measurement of magnetic dipole-dipole interactions between two unlike nuclei I and S. While this interaction is refocused by the Hahn spin echo, it can be recoupled by applying a 7i-pulse to the S-spins during the dipolar evolution period [12]. This manipulation inverts the sign of the heterodipolar Hamiltonian, and thereby interferes with the ability of the Hahn spin echo technique to refocus this interaction. The corresponding pulse sequence, termed SEDOR spin echo double resonance) shown in Fig. 4, compares the I-spin echo intensity as a function of dipolar evolution time (a) in the absence and (b) in the presence of the ti(S) pulses. Experiment (a) produces a decay F(2ti)/Fo, which is dominated by homonuclear dipole-dipole interactions, while experiment (b) results in an accelerated decay, reflecting the contribution from the heteronuclear I-S dipole-dipole interaction, which is now re-introduced into the spin Hamiltonian. For multi-spin systems, a Gaussian decay is expected ... 

The irreducible tensor product between two (spherical) vectors is defined in Eq. (37). An important feature of this Hamiltonian is that it explicitly describes the dependence of the coupling constants J, Am, and T, on the distance vectors rPP between the molecules and on the orientations phenomenological Hamiltonian (139). Another important difference with the latter is that the ad hoc single-particle spin anisotropy term BS2y, which probably stands implicitly for the magnetic dipole-dipole interactions, has been replaced by a two-body operator that correctly represents these interactions. The distance and orientational dependence of the coupling parameters J, A, , and Tm has been obtained as follows. 

Because circular dichroism is a difference in absorption for left and right circularly polarized light, its theoretical description includes subtraction of the transition probabilities induced by left and right circularly polarized radiation. The interaction Hamiltonian that determines transition probability includes electric, , and magnetic, B, fields of electromagnetic circularly polarized radiation, and the electric, /i, and magnetic, m, dipole moments of the molecule. 

One might expect that magnetic interactions arise from dipole-dipole interactions between the magnetic moments, but the fact is that magnetic interactions are largely effects of electrostatic interactions. To see this, we can consider a Hydrogen molecule with the Hamiltonian ... 

The third term in the Hamiltonian is the exchange term and represents the quantum mechanical exchange interaction. The quantity Jij is, in fact, the exchange integral. Finally, the last term is a dipole interaction term and represents the magnetic dipole-dipole interaction of two spins in close proximity. These last two terms, in contrast to the first two, do not depend upon orientation relative to an external coordinate system but rather depend upon the relative orientation of more than one spin... 

Another mechanism of paramagnetic ion coupling is the magnetic dipole-dipole interaction, the Hamiltonian of such interaction is of the form... 

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