Random spin system

An alternative picture was first introduced by Aharony and Pytte in the context of random magnets. In this picture the order parameter correlation function exhibits algebraic decay with distance instead. This situation, intermediate between SRO and LRO, has come to be known as quasi-long-range order (QLRO). The most well-known example of QLRO, due to Berezinsky and to Kosterlitz and Thouless occurs in the low temperature phase of the two-dimensional XY model. A number of recent theoretical and computational studies have supported this point of view in random spin systems in a higher dimensionality . 

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. ... 

The various special ENDOR techniques summarized in Sect. 4 widen the field of applications considerably. They allow investigations either of complex, oriented spin systems, or of paramagnetic centers in randomly oriented large molecules. The ENDOR techniques are particularly useful to study biochemical systems, which are often characterized by very poorly resolved powder EPR spectra. 

The behavior of the dispersion mode derivative is also consistent with a proton dilute spin system, since if only a small fraction of the possible lattice sites are randomly occupied the majority of the protons will experi-... 

Fig. 30. Energy distribution of the spin system as obtained from randomly generated sequences of length 1024 and 16384 and after 1-spin-flip optimization with and without simulated annealing...

Relaxation of the Spin System. After the film is deposited, the interactions are turned on between the spins as well as between the spin and the substrate. For each MC cycle, the molecules are interrogated randomly as to whether their spins flip. After reaching equilibrium, the ideal final state [shown in Figure 1.30(b)] should be achieved that is, all the spins in the first layer are pinned down to the surface, and the spins of molecules in the upper layer are antiparallel to those of molecules in the adjacent layers. 

Fig. 8.7 presents our experimental results plotted by the data published in [128, 129] obtained for polycrystalline Ti02 (rutile) lattice doped with vanadium (IV) ions at different content. The linear dependence of V4+ amount (in spin/g) on total vanadium content (in at.%) shows that all vanadium ions in these samples are in (+4) state while the non-linear graph of Cioc allows to assume that some part of V4+ centers is distributed in the lattice not randomly. Such systems will be discussed in detail in section 8.5. One can see from Fig. 8.7 that Cloc and (r) values can be easily estimated by such a simple approximation as Eqs. (8.7) and (8.8). 

There have been a few interesting fundamental papers on ergodicity 42 45 The distinction was emphasized between true equilibrium and quasiequilibrium phenomena, which now are frequently observed in NMR. An isolated finite system should not be expected to become ergodic. Two other fundamentally interesting processes that have been demonstrated with solid-state NMR are dephasing caused by randomization of geometric phase,46 47 and the possibility of the chaotic behaviour of spin systems.48,49... 

In previous chapters we have seen that the Hamiltonian describing a nuclear spin system is considerably simplified when molecules tumble rapidly and randomly, as in the liquid state. However, that simplicity masks some fundamental properties of spins that help us to understand their behavior and that can be applied to problems of chemical interest. We turn now to the solid state, where these properties often dominate the appearance of the spectra. Our treatment is limited to substances such as molecular crystals, polymers, and glasses, that is, solids in which there are well-defined individual molecules. We do not treat metals, ionic crystals, semiconductors, superconductors, or other systems in which delocalization of electrons is of critical importance. 

Now we wish to pursue the pictorial presentation of Fig. 2.3 in a mathematical manner that permits us to retain explicitly the quantum features that tend to be obscured in the graphical presentation. In more explicit terms, this corresponds to an incoherent superposition of the magnetizations of individual spins or of individual sets of N interacting spins (spin systems). Incoherent or random motions are commonly treated by statistical methods that deal with an ensemble of molecules, each containing N interacting spins. 

The first experiments to analyze EPR correlations used polarized light beams rather than electronic spin systems. The results obtained by Aspect [44] are especially relevant since the systems for study were prepared to be separated space-like. Aspect analyzed the polarization of pairs of photons emitted by a single source toward separate detectors. Measured independently, the polarization of each set of photons fluctuated in a seemingly random way. However, when two sets of measurements were compared, they displayed an agreement stronger than could be accounted for by any local realistic theory. 

Figu re 1.9 (a) Absorption and (b) first derivative EPR lineshape for a randomly oriented S = 1 /2 spin system with axial symmetry. The angular dependence curve (0 vs field) is shown in (c). 

Replication degenerates to a random production of sequences in the limit q- j and corresponds to the limit r->oo, the case of maximum disorder. Direct and complementary replication (see also part C) are the analogs of ferro- and antiferromagnetic cases of the spin system. In the range q < 1 we have K<0, which corresponds to the condition pj<0 for ferromagnetic interaction. For complementary or plus-minus replication, we have... 

All spin systems were enumerated, as illustrated in Fig. 34 for polyradical 51. Plots of the weighed average S and number average Msat as functions of q and p for polyradicals 51 and 52 are shown in Fig. 35.112 Most importantly, for any random distribution of coupling defects and limited density of chemical defects, the values of average S should scale with the molecular weight (or number of radical sites) of polyradical.112... 

We have numerically diagonalized small spin systems containing up to 5 by 5 spins subjected to a small random field hf j flatly distributed in the interval (—(5/2, (5/2). We see that indeed the gap closes rather fast away from the special Jx = Jy point (Fig. 3) but remains significant near Jz = Jx point where it clearly has a much weaker size dependence. Interestingly, the gap between the lowest 2ra states and the rest of the spectrum expected in the limits Jz 33> Jx or Jz -C Jx appears only at Jx/ Jz > jc with a practically size independent jc 1.2. We also see that the condition l 1 eliminates all low lying states in the Jz lowest excited state in l 1 sector... 

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