Single transition operators

At the end of the mixing period at a given time f p, the magnetization vector in the single-transition operator system is transformed into the lateral magnetization vector of the S-system. The corresponding complex value for the rotating... 

For the evolution of the difference magnetization it is, in the absence of r.f. irradiation, sufficient to consider the evolution of the subspace spanned by the (2,3) single-transition operators [15]. We obtain, therefore, the following reduced Hamiltonian ... 

When selectively exciting single transitions, it is often convenient to introduce single transition operators defined as... 

Here we have assumed exact MAS in the second step single-transition operators and the DQ nutation frequency... 

If we choose I = 0 and use single-transition operators for ease of comparison, we can rewrite this as... 

The theoretical basis for such features can be obtained by analysis of the time evolution in the rotating frame under action of the Hamiltonian (2.9.8). This requires the diagonaliza-tion of such Hamiltonian and the solution of the Liouville-von Neumann equation (Equation (2.5.3)). Usually, the results are properly described using the fictitious spin-1/2 formalism or the single-transition operator approach [18,20,21]. As example, we give below the matrix representation of the rotation operator corresponding to the selective excitation of the central transition in the case / = 3/2 for a r/2 pulse [22] ... 

A. Wokaun, R.R. nst. Selective excitation and detection in multilevel spin systems Application of single transition operators, J. Chem. Phys. 67 (1977) 1752. 

Suppose a selective excitation experiment is considered using a 90° — r — 180° spin echo sequence with the 90° pulse selectively irradiating the 1 2 ( 1) —> 0)) transition at o/q — u)q (see Fig. 2.1). The single transition (or fictitious spin- ) operators may be used in the eigenbase of Hq [5.3]. The complete set of single transition operators for a deuteron [5.11] is represented by the nine generalized Pauli spin matrices ... 

The usual way of solving eqn (7) requires its transformation into the interaction representation (Dirac picture) that is often called rotating frame for a particular case, when static part of the spin Hamiltonian is restricted to the electron Zeeman interaction. In the Dirac picture only the stochastic dipolar interaction is left in the spin Hamiltonian, its matrix elements get additional oscillatory factors due to the static Hamiltonian transitions. The integral on each matrix element of the double commutator in eqn (7) thus evolves into the Fourier transform /(co ) of the correlation function for the corresponding stochastic process. This Fourier transform is often called spectral density of the stochastic process and it is to be taken at a frequency co of a particular transition of the static Hamiltonian operator, driven by a single transition operator ki ... 

Fig. 70.3 En ergy level diagram of a two spin J4 system / and S showing the identification of com ponents of the 2D multiplet expressed via single transition basis operators, Z13, I24, S12, S34. Contour plots of a N- H backbone moiety with...

Figure 2.8 A schematic diagram of the gain spectral profile, G(v), of a laser transition (solid line), together with the axial resonator modes (dotted line) of a cavity in which the frequency separation between adjacent modes is A v. (a) Multimode and (b) single-mode operation. The frequencies of those modes for which the gain exceeds the losses have been marked.

Let us consider a laser oscillating at a single frequency (single-mode operation) and gas molecules inside the laser resonator which have absorption transitions at this frequency. Some of the molecules will be pumped by the laser-light into an excited state. If the relaxation processes (spontaneous emission and collisional relaxation) are slower than the excitation rate, the ground state will be partly depleted and the absorption therefore decreases with increasing laser intensity. 

However, because of the correlated motion of the electrons, many-electron processes will also occur. (Looking at the many-particle effects in this way, the photon operator is a single-particle operator and electron-electron interactions have to be incorporated explicitly into the wavefunction. It is, however, also possible to describe the combined action of the electrons as an induced field which adds to the external field of the photoprocess, i.e., the transition operator becomes modified. Generally, the influence of the electron-electron interaction can be represented by modifying the wavefunction or the operator or by modifying both the wavefunction and the operator [DLe55, CWe87].) Of all the possible processes, only the important two-electron processes restricted to electron emission will be considered here. In many cases they can be divided into two different classes (see Fig. 1.3) t... 

All three forms of the dipole matrix element are equivalent because they can be transformed into each other. However, this equivalence is valid only for exact initial- and final-state wavefunctions. Since the Coulomb interaction between the electrons is responsible for many-body effects (except in the hydrogen atom), and the many-body problem can only be solved approximately, the three different forms of the matrix element will, in general, yield different results. The reason for this can be seen by comparing for the individual matrix elements how the transition operator weights the radial parts R r) and Rf(r) of the single-particle wavefunction differently ... 

The light source is a home made CM ring LD 700 dyo laser, pumped by a Kr+ laser. In the range 730-780 nm (wavelength of the two-photon 2S-nD transitions for n 8 it provides a power of about 1W on single mode operation. The frequency stabilization is made by locking the laser to an external auxiliary Fabry-Perot cavity indicated FPA in Fig.2 the resulting... 

Transition metal-catalyzed [2 + 2 + 2] cocyclization of two molecules of an alkyne with an alkene is a powerful method for forming 1,3-cyclohexa-dienes [29-31]. These compounds are of course valuable partners for Diels-Alder reactions [32]. Through the right choice of substrates, both [2 + 2 + 2] and [4 + 2] cycloadditions can be performed in a single chemical operation [33]. Indeed, the reaction of electroneutral diyne 14 with electron-deficient maleimide 15 in the presence of lmol% of a Ru(I) catalyst exclusively afforded the highly symmetrical 1 2 adduct 17 in 74% yield (Scheme 9). 

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