Anisotropy Hamiltonian

By diagonalizing the y-matrix, the principal magnetic axes can be determined. Ignoring the uninteresting constant term and performing a little bit of algebra, the more common form of the anisotropy Hamiltonian in the rotated coordinate system is given by ... 

The anisotropies that lead to line broadening in isotropic ESR spectra influence solid-state spectra more directly. Accordingly a more complex spin Hamiltonian is required to interpret such spectra ... 

From the EPR spectroscopist s viewpoint the spin-trap experiment is next to trivial the molecular mass of the radical adduct is small enough to guarantee the molecule to tumble sufficiently rapidly at ambient temperatures in aqueous solution to ensure complete averaging away of any anisotropy in the spin Hamiltonian ... 

Empirically corrected DFT theories almost invariably go back to second-order perturbation theory with expansion of the interaction Hamiltonian in inverse powers of the intermolecular distance, leading to R 6, R x, and R 10 corrections to the energy in an isotropic treatment (odd powers appear if anisotropy is taken into account [86]). 

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... 

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

A quantitative evaluation of the relaxivities as a function of the magnetic field Bo requires extensive numerical calculations because of the presence of two different axes (the anisotropy and the external field axis), resulting in non-zero off-diagonal elements in the Hamiltonian matrix (15). Furthermore, the anisotropy energy has to be included in the thermal equilibrium density matrix. Figures 7 and 8 show the attenuation of the low field dispersion of the calculated NMRD profile when either the crystal size or the anisotropy field increases. 

In an organic solid representative broadenings are 150 ppm for aromatic carbon chemical shift anisotropy and 25 kHz (full width at half-height) for a rather strong carbon-proton dipolar interaction. At a carbon Larmor frequency of 15 MHz, the shift anisotropy corresponds to 2.25 kHz. In high magnetic fields the forms of the respective Hamiltonians are... 

The expressions for the various parts of the Hamiltonian (equation 1) are well documented and for our purpose and the following discussion it suffices to summarize the results for axially symmetric situations in angular frequency units with the equations 2-6, where and Ashielding tensor and the shielding anisotropy, respectively, D is the dipole coupling, eq or V is the electric field gradient at the nucleus, eQ is the nuclear quadrupole moment and the other symbols have their usual meaning ... 

The Hamiltonian of a single isolated nanoparticle consists of the magnetic anisotropy (which creates preferential directions of the magnetic moment orientation) and the Zeeman energy (which is the interaction energy between the magnetic moment and an external field). In the ensembles, the nanoparticles are supposed to be well separated by a nonconductive medium [i.e., a ferrofluid in which the particles are coated with a surfactant (surface-active agent)]. The... 

The Hamiltonian of a noninteracting nanoparticle with uniaxial anisotropy is given by... 

The bistable character of the zero-field Hamiltonian will be destroyed by a sufficiently large field. The critical field for n h is called the anisotropy field, and is expressed by... 

We will consider dipolar interaction in zero field so that the total Hamiltonian is given by the sum of the anisotropy and dipolar energies = E -TEi. By restricting the calculation of thermal equilibrium properties to the case 1. we can use thermodynamical perturbation theory [27,28] to expand the Boltzmann distribution in powers of This leads to an expression of the form [23]... 

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