Static spin Hamiltonian

This section gives an explanation of the different terms of the static spin Hamiltonian. The concept of orientation selection by selective m.w. excitation, which is central to many pulse EPR experiments on disordered systems, is explained. 

The static spin Hamiltonian is used to describe the energies of states of a paramagnetic species in the ground state with an effective electron spin S and m nuclei with spins I. 

In this review all interactions are given in angular frequency units unless stated 

JEFFREY HARMER, GEORGE MITRIKAS, and ARTHUR SCHWEIGER 

The information obtained from the spin Hamiltonian, the 3x3 matrices g, D, A, and P, is very sensitive to the geometric and electronic structure of the paramagnetic center. The electron Zeeman interaction reveals information about the electronic states the zero-field splitting describes the coupling between electrons for systems where S Vi the hyperfine interactions contain information about the spin density distribution [8] and can be used to evaluate the distance and orientation between the unpaired electron and the nucleus the nuclear Zeeman interaction identifies the nucleus the nuclear quadrupole interaction is sensitive to the electric field gradient at the site of the nucleus and thus provides information on the local electron density. 

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In order to analyze these data, the frequency shift of geffectivecan be calculated by averaging over all orientations the anisotropic shift derived from a static spin Hamiltonian [67]. This treatment is based on the assumptions that molecular motion neither changes the spin precession rate nor perturbs the states and, thus, that the center of gravity of the spectrum is invariant even in presence of some motional averaging. For the allowed 11/2) <-> <-l/2 transition under perturbation theory, with expressions valid up to the third order, this shift is given by [47] ... 

Electron Spin Echo Envelope Modulation (ESEEM) and pulse Electron Nuclear Double Resonance (ENDOR) experiments are considered to be two cornerstones of pulse EPR spectroscopy. These techniques are typically used to obtain the static spin Hamiltonian parameters of powders, frozen solutions, and single crystals. The development of new methods based on these two effects is mainly driven by the need for higher resolution, and therefore, a more accurate estimation of the magnetic parameters. In this chapter, we describe the inner workings of ESEEM and pulse ENDOR experiments as well as the latest developments aimed at resolution and sensitivity enhancement. The advantages and limitations of these techniques are demonstrated through examples found in the literature, with an emphasis on systems of biological relevance. 

A new treatment for S = 7/2 systems has been undertaken by Rast and coworkers [78, 79]. They assume that in complexes with ligands like DTPA, the crystal field symmetry for Gd3+ produces a static ZFS, and construct a spin Hamiltonian that explicitly considers the random rotational motion of the molecular complex. They identify a magnitude for this static ZFS, called a2, and a correlation time for the rotational motion, called rr. They also construct a dynamic or transient ZFS with a simple correlation function of the form (BT)2 e t/TV. Analyzing the two Hamiltonians (Rast s and HL), it can be shown that at the level of second order, Rast s parameter a2 is exactly equivalent to the parameter A. The method has been applied to the analysis of the frequency dependence of the line width (ABpp) of GdDTPA. These results are compared to a HL treatment by Clarkson et al. in Table 2. 

The theoretical framework for a discussion of the hyperfine interactions in radicals is given by the so called spin Hamiltonian, which describes the interaction between the unpaired electrons and the magnetic nuclei (I>l/2) in the sample. When the radical is placed in a static magnetic field, the electrons and the magnetic nuclei will interact with the field. These interactions give rise to the electronic and nuclear Zeeman terms,... 

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 ... 

To compute the energy of the three triplet components in an appUed static magnetic field Bo, the spin Hamiltonian of the dipole-dipole interaction, TYf Eq. (7.4), must be added to the spin Hamiltonian of the Zeeman interaction, TYf = (/XB/h)BogS. 

In the absence of significant molecular motion the spectra can be calculated simply as a powder-like super-imposition of the individual molecular static lines of Lorentzian shape from all over the sample. These lines are then positioned into the spectrum according to Eq. (5) as in Ref. [6j. To include also dynamic effects, such as fluctuations of molecular long axes (defining the scalar order parameter S and the director n) and translational molecular diffusion, it is convenient to use a semi-classical approach with the time-dependent deuteron spin Hamiltonian [25] where the H NMR line shape I cj) is calculated as the Fourier transform of... 

There are many experimental techniques for the determination of the Spin-Hamiltonian parameters g, Ux, J. D, E. Often applied are Electron Paramagnetic or Spin Resonance (EPR, ESR), Electron Nuclear Double Resonance (ENDOR) or Triple Resonance, Electron-Electron Double Resonance (ELDOR), Nuclear Magnetic Resonance (NMR), occasionally utilizing effects of Chemically Induced Dynamic Nuclear Polarization (CIDNP), Optical Detections of Magnetic Resonance (ODMR) or Microwave Optical Double Resonance (MODR), Laser Magnetic Resonance (LMR), Atomic Beam Spectroscopy, and Muon Spin Rotation (/iSR). The extraction of data from the spectra varies with the methods, the system studied and the physical state of the sample (gas, liquid, unordered or ordered solid). For these procedures the reader is referred to the monographs (D). Further, effective magnetic moments of free radicals are often obtained from static... 

The evolution of the density operator is described by Eq. (2.10). Thus far the spin Hamiltonian in this equation has been considered to be static or time-independent. A spin system with the Hamiltonian given by... 

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