Electron nuclear spin interaction energy

Electron orbital moment-nuclear spin interaction energy ... 

The origin of postulate (iii) lies in the electron-nuclear hyperfine interaction. If the energy separation between the T and S states of the radical pair is of the same order of magnitude as then the hyperfine interaction can represent a driving force for T-S mixing and this depends on the nuclear spin state. Only a relatively small preference for one spin-state compared with the other is necessary in the T-S mixing process in order to overcome the Boltzmann polarization (1 in 10 ). The effect is to make n.m.r. spectroscopy a much more sensitive technique in systems displaying CIDNP than in systems where only Boltzmann distributions of nuclear spin states obtain. More detailed consideration of postulate (iii) is deferred until Section II,D. 

The indirect spin-spin coupling is independent of molecular rotation. The coupling mechanism is known to involve the electron spins of the bonding electrons and is the result of a weak electron polarization. The interaction energy AX is proportional to the scalar product of the nuclear spins / of A and X, according to the following expression ... 

The next step is to add terms representing the potential energy, the electron spin interactions and the nuclear spin interactions. The total Hamiltonian Xj can then be subdivided into electronic and nuclear Hamiltonians,... 

In our subsequent development we shall take the origin of coordinates to be at the centre of mass of the two nuclei, although we could equally well have chosen the molecular centre of mass as origin. Setting aside the translational motion of the molecule, we use equation (2.28) to represent the kinetic energy of the electrons and nuclei. To this we add terms representing the potential energy, electron spin interactions, and nuclear spin interactions. We subdivide the total Hamiltonian Xx into electronic and nuclear Hamiltonians,... 

Electron spin echo spectroscopy (ESE) monitors the spontaneous generation of microwave energy as a function of the timing of a specific excitation scheme, i.e. two or more short resonant microwave pulses. This is illustrated in Fig. 7. In a typical two-pulse excitation, the initial n/2 pulse places the spin system in a coherent state. Subsequently, the spin packets, each characterized by their own Larmor precession frequency m, start to dephase. A second rx-pulse at time r effectively reverses the time evolution of the spin packet magnetizations, i.e. the spin packets start to rephase, and an emission of microwave energy (the primary echo) occurs at time 2r. The echo ampHtude, as a fvmction of r, constitutes the ESE spectrum and relaxation processes lead to an irreversible loss of phase correlation. The characteristic time for the ampHtude decay is called the phase memory time T. This decay is often accompanied by a modulation of the echo amplitude, which is due to weak electron-nuclear hyperfine interactions. The analysis of the modulation frequencies and ampHtudes forms the basis of the electron spin echo envelope modulation spectroscopy (ESEEM). 

In a simplified model the energy of an electron and a nuclear spin interacting with each other is obtained by the point dipole approximation as shown in Fig. 2.31 below. The result is anticipated from the classical expression for the energy of the spin magnetic moments [is and /ii when the electron and nuclear spins are both oriented along the applied field that makes an angle 6 with the line joining the spins S and I. 

The fine structure of atomic line spectra and the hyperfine splittings of electronic Zeeman spectra are non-symmetric for those atomic nuclei whose spin equals or exceeds unity, / > 1. The terms of the spin Hamiltonian so far mentioned, that is, the nuclear Zeeman, contact interaction, and the electron-nuclear dipolar interaction, each symmetrically displace the energy, and the observed deviation from symmetry therefore suggests that another form of interaction between the atomic nucleus and electrons is extant. Like the electronic orbitals, nuclei assume states that are defined by the total angular momentum of the nucleons, and the nuclear orbitals may deviate from spherical symmetry. Such non-symmetric nuclei possess a quadrupole moment that is influenced by the motion of the surrounding electronic charge distribution and is manifest in the hyperfine spectrum (Kopfer-mann, 1958). 

High-field energy levels for a system with electron spin S = 1/2 and with a magnetic nucleus with I = 1/2. The levels on the left are the zero-order energies, and the levels on the right include the first-order corrections due to electron spin-nuclear spin interaction. The two vertical arrows show the allowed transitions that would be observed in an ESR experiment. Not shown are the allowed transitions for which AMj = 0 since these are the NMR transitions and occur at very different energies. 

The occurrence of an electron-nuclear hyperfine interaction (HFI) involving coupling between the electron, of spin S, and nucleus, of spin /, increases the total number of energy levels by a factor of 2 / + 1. When... 

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... 

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... 

The electron-spm echo envelope modulation (ESEEM) phenomenon [37, 38] is of primary interest in pulsed EPR of solids, where anisotropic hyperfme and nuclear quadnipole interactions persist. The effect can be observed as modulations of the echo intensity in two-pulse and three-pulse experiments in which x or J is varied. In liquids the modulations are averaged to zero by rapid molecular tumbling. The physical origin of ESEEM can be understood in tenns of the four-level spin energy diagram for the S = I = model system... 

The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic, vibrational, rotational, and nuclear spin). Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions. 

Often the electronic spin states are not stationary with respect to the Mossbauer time scale but fluctuate and show transitions due to coupling to the vibrational states of the chemical environment (the lattice vibrations or phonons). The rate l/Tj of this spin-lattice relaxation depends among other variables on temperature and energy splitting (see also Appendix H). Alternatively, spin transitions can be caused by spin-spin interactions with rates 1/T2 that depend on the distance between the paramagnetic centers. In densely packed solids of inorganic compounds or concentrated solutions, the spin-spin relaxation may dominate the total spin relaxation 1/r = l/Ti + 1/+2 [104]. Whenever the relaxation time is comparable to the nuclear Larmor frequency S)A/h) or the rate of the nuclear decay ( 10 s ), the stationary solutions above do not apply and a dynamic model has to be invoked... 

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