Matrix spin relaxation

This conclusion is supported by the experimental result " given by the pulsed-NMR measurement that the spin-spin relaxation time T2 is considerably shorter for the gel than that for the matrix mbber vulcanizate, which of course, indicates that the modulus is considerably higher for the gel than for the matrix mbber. More quantitatively, Maebayashi et al. measured the acoustic velocity of carbon gel by acoustic analysis and concluded that the compression modulus of the gel is about twice that of matrix mbber. Thus, at present, we can conclude that the SH layer, of course without cross-linking, is about two times harder than matrix cross-linked mbber in the filled system. 

It is an unfortunate fact that several preexisting theories have tried to explain complicated mechanical phenomena of CB-reinforced rubbery materials but they have not been so successful." " However, a recent report might have a capability of explaining them collectively," when the author accepted the existence of the component whose molecular mobility is different from that of matrix mbber component in addition to the existence of well-known bound rubber component. The report described that this new component might be the most important factor to determine the reinforcement. These mbber components have been verified by spin-spin relaxation time 2 by pulsed nuclear magnetic resonance (NMR) technique, ° while the information obtained by NMR is qualitative and averaged over the sample and, therefore, lacking in the spatial... 

Nuclear spin relaxation is considered here using a semi-classical approach, i.e., the relaxing spin system is treated quantum mechanically, while the thermal bath or lattice is treated classically. Relaxation is a process by which a spin system is restored to its equilibrium state, and the return to equilibrium can be monitored by its relaxation rates, which determine how the NMR signals detected from the spin system evolve as a function of time. The Redfield relaxation theory36 based on a density matrix formalism can provide... 

Wangsness and Bloch16>17 were the first to give a quantum mechanical treatment of spin relaxation using the density matrix formalism. The system considered is a spin interacting with an external magnetic field (which we suppose here to be constant) and with a heat bath. The corresponding Hamiltonian is... 

The kinetic matrix K reflects the fact that only corresponding spins in the free and bound state are connected by exchange. The complete matrix governing relaxation is simply the sum of R and K ... 

The Redfield matrix elements are defined in full analogy with the case where the conventional electron spin relaxation processes in a non-equilibrium ensemble are considered, but the rates are in general different from... 

The spin relaxation of radical pairs also occurs through the dipole-dipole interaction (Ho) of the component radicals (a and b). Its matrix element between T,) and To ) is similar to that of two protons given by Eq. (66) in Chapter 11 of Ref. [22]... 

The conventional approach to the theory of electron spin relaxation is to use a density matrix approach developed by Redfield. (32) However, this method is only valid when x Xg2- Thus, cases of very fast electronic relaxation leading to sharp NMR lines, which are in general of particular interest, are strictly excluded from this theoretical approach. Doddrell et al. (33) have developed a more general theory for... 

Spin-spin relaxation is handled by second order perturbation theory [ Abr 1 ] of the density matrix equation of motion (2.2.62) in the rotating frame (RCF) [Abrl],... 

Density matrix p" of the quasi-stationary state (p" = p"gf when = 0), formed at times T2 (T2 being the time of the spin-spin relaxation) equals... 

An unusually good energy match is required before these intersystem relaxation paths compete with more rapid processes such as spin-relaxation within a multiplet state, vibrational relaxation, or spontaneous fluorescence (NO B—>X). It is interesting that the B2IIi/2 —> a4n5/2 ACl = 2 process seems to require a better energy match than ACl = 1 or 0 processes even though the quantum numbers Cl, A, and E are destroyed by the noncylindrical matrix site. 

By the same experimental technique, the temperature dependence of the nuclear spin relaxation rates was investigated for the radical cations of dimethoxy- and trimethoxybenzenes [89], The rates of these processes do not appear to be accessible by other methods. As was shown, l/Tfd of an aromatic proton in these radicals is proportional to the square of its hyperfine coupling constant. This result could be explained qualitatively by a simple MO model. Relaxation predominantly occurs by the dipolar interaction between the proton and the unpaired spin density in the pz orbital of the carbon atom the proton is attached to. Calculations on the basis of this model were performed with the density matrix formalism of MO theory and gave an agreement of experimental and predicted relaxation rates within a factor of 2. 

However, in a magnetically dilute system it is possible in principle to slow the spin-spin relaxation to the point where hyperfine structure is seen, and in this case the matrix elements involving S+ and S (i.e. a mixing of different electronic states) cause an unequal spacing of the hyperfine levels. 

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