Coherence phase

A second type of relaxation mechanism, the spin-spm relaxation, will cause a decay of the phase coherence of the spin motion introduced by the coherent excitation of tire spins by the MW radiation. The mechanism involves slight perturbations of the Lannor frequency by stochastically fluctuating magnetic dipoles, for example those arising from nearby magnetic nuclei. Due to the randomization of spin directions and the concomitant loss of phase coherence, the spin system approaches a state of maximum entropy. The spin-spin relaxation disturbing the phase coherence is characterized by T. ... 

M continually decreases under the influence of spin-spin relaxation which destroys the initial phase coherence of the spin motion within they z-plane. In solid-state TREPR, where large inliomogeneous EPR linewidths due to anisotropic magnetic interactions persist, the long-time behaviour of the spectrometer output, S(t), is given by... 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

The simplest scheme that accounts for the destruction of phase coherence is the so-called stochastic interruption model [Nikitin and Korst 1965 Simonius 1978 Silbey and Harris 1989]. Suppose the process of free tunneling is interrupted by a sequence of collisions separated by time periods vo = to do After each collision the system forgets its initial phase, i.e., the off-diagonal matrix elements of the density matrix p go to zero, resulting in the density matrix p ... 

Spin-spin relaxation is the steady decay of transverse magnetisation (phase coherence of nuclear spins) produced by the NMR excitation where there is perfect homogeneity of the magnetic field. It is evident in the shape of the FID (/fee induction decay), as the exponential decay to zero of the transverse magnetisation produced in the pulsed NMR experiment. The Fourier transformation of the FID signal (time domain) gives the FT NMR spectrum (frequency domain, Fig. 1.7). 

Static defects scatter elastically the charge carriers. Electrons do not loose memory of the phase contained in their wave function and thus propagate through the sample in a coherent way. By contrast, electron-phonon or electron-electron collisions are inelastic and generally destroy the phase coherence. The resulting inelastic mean free path, Li , which is the distance that an electron travels between two inelastic collisions, is generally equal to the phase coherence length, the distance that an electron travels before its initial phase is destroyed ... 

At low temperatures, in a sample of very small dimensions, it may happen that the phase-coherence length in Eq.(3) becomes larger than the dimensions of the sample. In a perfect crystal, the electrons will propagate ballistically from one end of the sample and we are in a ballistic regime where the laws of conductivity discussed above no more apply. The propagation of an electron is then directly related to the quantum probability of transmission across the global potential of the sample. 

It turns out that, in the CML, the local temporal period-doubling yields spatial domain structures consisting of phase coherent sites. By domains, we mean physical regions of the lattice in which the sites are correlated both spatially and temporally. This correlation may consist either of an exact translation symmetry in which the values of all sites are equal or possibly some combined period-2 space and time symmetry. These coherent domains are separated by domain walls, or kinks, that are produced at sites whose initial amplitudes are close to unstable fixed points of = a, for some period-rr. Generally speaking, as the period of the local map... 

Suppose the first pulse resulted in the creation of a phase coherence across the Ai transition between the aa and a/3 states (Fig. 1.44). It is possible to transfer this phase information from the a)3 state to the )3/3 state by applying a selective it pulse across the Xi transition. The two successive pulses would therefore transfer the phase of the aa state to the )8)3 state, with the two states now becoming phase coherent with one another. 

In ESR, it is also customary to classify relaxation processes by their effects on electron and nuclear spins. A process that involves an electron spin flip necessarily involves energy transfer to or from the lattice and is therefore a contribution to Tx we call such a process nonsecular. A process that involves no spin flips, but which results in loss of phase coherence, is termed secular. Processes that involve nuclear spin flips but not electron spin flips are, from the point of view of the electron spins, nonsecular, but because the energy transferred is so small (compared with electron spin flips) these processes are termed pseudosecular. 

Lorentzian line shapes are expected in magnetic resonance spectra whenever the Bloch phenomenological model is applicable, i.e., when the loss of magnetization phase coherence in the xy-plane is a first-order process. As we have seen, a chemical reaction meets this criterion, but so do several other line broadening mechanisms such as averaging of the g- and hyperfine matrix anisotropies through molecular tumbling (rotational diffusion) in solution. 

Landauer proposed in 1957 the first mesoscopic theoretical approach to charge transport [176]. Transport is treated as a scattering problem, ignoring initially all inelastic interactions. Phase coherence is assumed to be preserved within the entire conductor. Transport properties, such as the electrical conductance, are intimately related to the transmission probability for an electron to cross the system. Landauer considered the current as a consequence of the injection of electrons at one end of a sample, and the probability of the electrons reaching the other end. The total conductance is determined by the sum of all current-carrying eigenmodes and their transmission probability, which leads to the Landauer formula of a ID system ... 

Bauer R, Neuhauser D (2002) Phase coherent electronics a molecular switch based on quantum interference. J Am Chem Soc 124 4200... 

In a basic pulsed NMR experiment (for I = 1/2), when a sample is placed in the applied magnetic field (B0), the nuclear spins distribute themselves between parallel and antiparallel positions, according to Boltzmann distribution [Eq. (11)] (Figure 21 A). The number of spins in the parallel position is slightly greater than that in the antiparallel position. At equilibrium, the spins are processing randomly (i.e., lack phase coherence). The populations... 

Measurement of a true T2 can be obtained using a spin-echo pulse sequence, such as the Carr-Purcell-Meiboom-Gill (CPMG) sequence, which minimizes the loss of phase coherence caused by inhomogeneities (Kemp, 1986). 

The phase coherence can also be achieved simply by using a linearly phase incremented pulse (PIP) without resorting to additional hardware. Since all the RF pulses utilized for constructing a PIP have the same carrier, no frequency jump is involved. Consequently, phase coherence between the RF pulses applied before and after the PIP is reserved naturally. 

As in the phase coherence in RF pulses, the phase coherence in PIPs -... 

Under such circumstances, the evolution of a spin system has to be calculated in different rotating frames defined by the corresponding PIPs and a special case may arise, where a spin experiences an on-resonance excitation but off-resonance evolution in the conventional rotating frame. Unpredictable results may occur if the phase coherence in PIPs fails. Unfortunately, to date, no... 

NMR instruments take care of this phase coherence in PIPs and therefore human intervention becomes inevitable. 

To understand the concept of phase coherence in PIPs it is quite helpful to define an Eigenframe of a PIP as shown in Fig. 18. For a frequency-shifted... 

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