Spin eigenvalues relaxation

In eq. (18), Mi t) and Mzit) are the magnetizations at time t for sites 1 and 2 respectively, M oc) is the equilibrium value of the magnetization in either site, R is the spin-lattice relaxation rate (= 1/Ti) for the two sites, which are assumed to be equal, and k is the exchange rate. For a larger system, numerical solutions are readily available with standard eigenvalue methods [42-44]. 

The dynamic RIS model developed for investigating local chain dynamics is further improved and applied to POE. A set of eigenvalues characterizes the dynamic behaviour of a given segment of N motional bonds, with v isomeric states available to each bond. The rates of transitions between isomeric states are assumed to be inversely proportional to solvent viscosity. Predictions are in satisfactory agreement with the isotropic correlation times and spin-lattice relaxation times from 13C and 1H NMR experiments for POE. 

Powell et al. give an excellent review of several approaches to interpret the frequency dependence of Tle and T2e in these systems [71]. One convenient approach is that developed by Hudson and Lewis [72], who showed that the eigenvalues of the relaxation matrix R as defined in Bloch-Wangsness-Red-field (BWR) theory [73] are functions of rv and the experimental frequency co, and are related to the relaxation time T2ei of the i-th allowed electron spin transition by the expression ... 

ECP (Effective Core Potential) 171 Effective pair potential 68 EHMO (Extended Huckel Molecular Orbital Model) 130 Eigenvalue 17 Eigenvector 17 Einstein relation 253 Electric dipole moment 100, 265, 282 Electric field gradient 278 Electric moments 184 Electric quadrupole moment 268, 269 Electric second moment 268 Electric susceptibility 256 Electron affinity 147 Electron correlation 186, 273 Electron density 100, 218, 222 Electron relaxation 118 Electron spin 91, 95, 99, 277, 305 Electronic Schrodinger equation 74 Electrostatic field 14 Electrostatic field gradient 271... 

Since the relaxation determines the lifetime, At, of a spin state, the Heisenberg Uncertainty Principle relates it to the uncertainty of the Zeeman eigenvalues, E and Ei, thereby allowing this phenomenon to affect the linewidth of EPR signals, as these depend inversely on T. ... 

The n-site Bloch-McConnell equations describe the evolution of nuclear spin magnetization in the laboratory or rotating frames of reference for molecules subject to chemical or conformational interconversions between n species with distinct NMR chemical shifts. Trott and Palmer used perturbation theory to approximate the largest eigenvalue of the Bloch-McConnell equations and obtain analytical expressions for the rotating-frame relaxation rate constant and for the laboratory frame resonance frequency and transverse relaxation rate constant. The perturbation treatment is valid whenever the population of one site is dominant. The new results are generally applicable to investigations of kinetic processes by NMR spectroscopy. 

F.6.4.2. Lineshape Models. The Mossbauer lineshape can be influenced by all relaxation modes of the Fokker-Planck equation (see Section D.3). Because the relative importance of these modes depends on their population, it should be necessary to know both the eigenvalues of Brown s equation and the amplitudes of the associated modes. In fact, to determine the lineshape, it is necessary to connect the dynamics of the stochastic vector m given by Brown s equation with the quantum dynamics of the nuclear spin. This necessitates the use of superoperator Fokker-Planck equations and, to our knowledge, the problem has not yet been completely solved. 

The relaxation of a spin system, for which a temperature can be defined and whose Hamiltonian H has eigenvalues E, can be described as soon as a "master" equation is introduced. This master equation relates the temporal dependence of the probability Pj on the probability per second that the lattice induces a transition in the spin system from n to m or from m to n... 

---