First order coherences

Case 2. The particles rotate in small packets ( coherently or in phase ). Obviously, the first-order rate law no longer holds. In chapter B2.1 we shall see that this simple consideration has found a deeper meaning in some of the most recent kinetic investigations [21]. 

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). 

Figure 1.45 Coherence transfer pathways in 2D NMR experiments. (A) Pathways in homonuclear 2D correlation spectroscopy. The first 90° pulse excites singlequantum coherence of order p= . The second mixing pulse of angle /3 converts the coherence into detectable magnetization (p= —1). (Bra) Coherence transfer pathways in NOESY/2D exchange spectroscopy (B b) relayed COSY (B c) doublequantum spectroscopy (B d) 2D COSY with double-quantum filter (t = 0). The pathways shown in (B a,b, and d) involve a fixed mixing interval (t ). (Reprinted from G. Bodenhausen et al, J. Magn. Resonance, 58, 370, copyright 1984, Rights and Permission Department, Academic Press Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887.)...

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. 

This term is independent of the coherence order p. Furthermore, it vanishes for all symmetric transitions (q = 0), and thus can be disregarded when describing the SQ CT coherence or the symmetric MQ coherence. Note, however, that this is not the case for the STs (q 0), which are therefore strongly affected by the first-order quadrupolar interaction, except when the sample is spun at the magic angle. 

If j Rf is exactly the magic angle and infinite spinning speed is assumed, the first-order anisotropic terms are zero for both single and DQ coherence (33). This does not hold true for finite spinning speed, but a complete averaging of the first-order effect occurs at the exact rotor cycles. Therefore, the x evolution time has to match exactly a multiple of the rotor period. The second-order anisotropy refocusing occurs for... 

Experimental verification of the ISRS generation can be primarily given by the pump polarization dependence. The coherent phonons driven by ISRS (second order process) should follow the symmetry of the Raman tensor, while those mediated by photoexcited carriers should obey the polarization dependence of the optical absorption (first order process). It is possible, however, that both ISRS and carrier-mediated generations contribute to the generation of a single phonon mode. The polarization dependence is then described by the sum of the first- and second-order processes [20-22], as shown in Fig. 2.3. 

The first order term in A/p comes from the difference of the potential energy and the higher order terms should be included when AIP/UP is not small enough. The phases, which the freed electrons accumulate during their different quantum paths, are transferred to the harmonics through the coherent process of HHG and lead to the interferences (Fig. 4.1). 

In the past two decades, a variety of semiclassical initial-value representations have been developed [105-111], which are equivalent within the semiclassical approximation (i.e., they solve the Schrodinger equation to first order in H), but differ in their accuracy and numerical performance. Most of the applications of initial-value representation methods in recent years have employed the Herman-Kluk (coherent-state) representation of the semiclassical propagator [105, 108, 187, 245, 252-255], which for a general n-dimensional system can be written as... 

In linear superposition, the method is literally that of adding components. When treating the optics of coherent light, for example, the instantaneous values of the field vectors are superimposed. Incoherent light, on the other hand, requires us to deal with the time-averaged square of the field. In nonlinear optics, superposition breaks down as it does in other nonlinear systems. Even when it does not hold exactly, however, superposition is often useful as a first-order approximation. 

We note here that gel is a coherent solid because its structure is characterized by a polymer network, and hence, the above theoretical considerations on crystalline alloys should be applicable to gels without essential alteration. It is expected that the curious features of the first-order transition of NIPA gels will be explained within the concept of the coherent phase equilibrium if the proper calculation of the coherent energy and the elastic energy of the gel network is made. This may be one of the most interesting unsolved problems related to the phase transitions of gels. 

Moreover, the coherent scatter interaction is also elastic, and hence photons have identical energy before and after the interaction. Hence, a first-order attenuation correction simply normalizes the coherent scatter spectrum against the spectrum of transmitted rays. Further refinement of the first-order correction is possible, but its discussion is beyond the scope of this chapter. 

Coherent light sources are characterized by a spectral intensity distribution E(oj) and a frequency-dependent phase 0(w). According to first-order perturbation theory, linear absorption probabilities are given by the overlap between the spectrum of the light source E(w) and the optical transition, and are independent of the phase function In non-linear processes (2nd... 

To maintain thermal stability, hence a condition EB/kBT= In (for) needs to be fulfilled. For z = 10 years storage, 109-10u Hz [28] and ignoring dispersions, i.e. assuming monodisperse particles, this becomes Es/kBT= 40-45. Reversal for isolated, well-decoupled grains to first order can be described by coherent rotation over EB. This simple model, as first discussed by Stoner and Wohlfarth in 1948 [29], considers only intrinsic anisotropy and external field (Zeeman) energy terms. For perpendicular geometry one obtains the following expression ... 

Fig. 14. The pulse sequence for recording the double-quantum 2H experiment.37 The entire experiment is conducted under magic-angle spinning. This two-dimensional experiment separates 2H spinning sideband patterns (or alternatively, static-like 2H quadrupole powder patterns) according to the 2H double-quantum chemical shift, so improving the resolution over a single-quantum experiment. In addition, the doublequantum transition frequency has no contribution from quadrupole coupling (to first order) so, the double-quantum spectrum is not complicated by spinning sidebands. Details of molecular motion are then extracted from the separated 2H spinning sideband patterns by simulation.37 All pulses in the sequence are 90° pulses with the phases shown (the first two pulses are phase cycled to select double-quantum coherence in q). The r delay is of the order 10 gs. The q period is usually rotor-synchronized.

For CqS in the MHz range dynamical effects on the single-quantum (SQ) and double-quantum (DQ) 14N coherences were compared. Hereby it was demonstrated that the DQ lineshape was not broadened as much as the SQ lineshape as the DQ transition is not affected by the first-order quadrupolar Hamiltonian. 

Both experiments are applicable for acquisition of 2H and 6Li spectra and for these nuclei only first-order EFG- and CSA-terms in the Hamiltonian are required. For CqS below 750 kHz, both 14N QCPMG and MAS experiments are applicable at 14.1 T but above this Cq limit the hardware demands make it very difficult to employ the QCPMG experiment and single-pulse MAS must be the method of choice for such applications. For larger CqS, indirect detection of either 14N SQ or DQ coherences using rotor-synchronized acquisition is suggested. In this context, the DQ lineshape is not as severely broadened as the SQ lineshape as it is not affected by the first-order quadrupolar Hamiltonian. 

---