Zero-order coherence

Organic Solids A few organic compounds decompose before melting, mostly nitrogen compounds azides, diazo compounds, and nitramines. The processes are exothermic, classed as explosions, and may follow an autocatalytic law. Temperature ranges of decomposition are mostly 100 to 200°C (212 to 392°F). Only spotty results have been obtained, with no coherent pattern. The decomposition of malonic acid has been measured for both the solid and the supercooled liquid. The first-order specific rates at 126.3°C (259.3°F) were 0.00025/min for solid and 0.00207 for liquid, a ratio of 8 at II0.8°C (23I.4°F), the values were 0.000021 and 0.00047, a ratio of 39. The decomposition of oxalic acid (m.p. I89°C) obeyed a zero-order law at 130 to I70°C (266 to 338°F). 

According to the coherent averaging theory,3,4,53 the zero-order average Hamiltonian can be obtained straightforwardly... 

The method relies on the measurement of cross-correlated relaxation rates in a constant time period such that the cross-correlated relaxation rate evolves during a fixed time r. In order to resolve the cross-correlated relaxation rate, however, the couplings need to evolve during an evolution time, e.g. tt. The first pulse sequence published for the measurement of the cross-correlated relaxation rate between the HNn and the Ca j,Ha i vector relied on an HN(CO)CA experiment, in which the Ca chemical shift evolution period was replaced by evolution of 15N,13C double and zero quantum coherences (Fig. 7.20). 

Coherence Order. The coherence order, p, is zero for z magnetization and zero-quantum coherence, 1 or -1 for single-quantum coherence, and 2 or -2 for doublequantum coherence. The coherence order is useful for diagraming the coherence pathway in a pulse sequence and for predicting the effect of gradient pulses on the sample magnetization. 

Evolution of multiple quantum coherence (of order p) is treated as successive evolution of each of the p chemical shifts. As expected, double quantum coherence precesses at wt + cos, and zero quantum coherence precesses at (i)j — 0)s. As we noted in Section 11.5, such coherences are not affected by spin coupling. 

A rough, zero-order, estimate of the extent to which a Fermi liquid description would be viable in the normal phase is provided by the scale of given by band calculations. Consider the temperature range Tthermal fluctuations are suffieiently weak to lower the uncertainty on the transverse band wave vector to a range of values l/dj, that is small compared to the size of Brillouin zone. The band wave vector is therefore a good quantum number so the transverse band motion and the curvature of the Fermi surface are coherent. Otherwise, when one has which is large enough for... 

This modification reflects the fact that a p-quantum coherence dephases at a rate proportional to p. Thus, double-quantum coherences dephase twice as fast as single-quantum coherences yet zero-quantum coherences are insensitive to field gradients. When the eoherence involves different nuclear species, allowance must be made for the magnetogyric ratio and coherence order for each, such that ... 

For double- and zero-quantum coherence in which spins / and j are active it is convenient to define the following set of operators which represent pure multiple quantum states of given order. The operators can be expressed in terms of the Cartesian or raising and lowering operators. 

Coherences, of which transverse magnetization is one example, can be classified according to a coherence order, p, which is an integer taking values 0, 1, 2. .. Single quantum coherence has p = 1, double has p = 2 and so on z-magnetization, "zz" terms and zero-quantum coherence have p = 0. This classification comes about by considering the phase which different coherences acquire is response to a rotation about the z-axis. 

In general, a radiofrequency pulse causes coherences to be transferred from one order to one or more different orders it is this spreading out of the coherence which makes it necessary to select one transfer among many possibilities. An example of this spreading between coherence orders is the effect of a non-selective pulse on antiphase magnetization, such as 27j/2z, which corresponds to coherence orders 1. Some of the coherence may be transferred into double-and zero-quantum coherence, some may be transferred into two-spin order and some will remain unaffected. The precise outcome depends on the phase and flip angle of the pulse, but in general we can see that there are many possibilities. 

In this equation we have chosen to consider a near prolate symmetric top (like anthracene). The symbols J and Ka have their usual meanings as the rotational quantum numbers of such a molecule and refer in the equation to rotational levels in the manifold of the a> zero-order vibrational level. As for the other factors in Eq. (3.36), T is the rotational temperature of the sample, W(J, Ka, T) is a weighting factor for each ro-vibrational level a,J,Ka, and Iy(J,Ka,t) is the y-type fluorescence decay which arises from the coupling of a,J, Ka) with the same rotational levels of the other zero-order vibrational states. Iy(J, Ka, t) is just given by Iy(t) of Eq. (3.12). [Note that in deriving Eq. (3.36) we have neglected the possibility of any coherence effects arising from the coherent preparation of rotational levels within the same vibrational state. This possibility is the subject of Section III D 2.]... 

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