Slow passage conditions

In a continuous wave (CW) magnetic resonance experiment, the radiation field B is continuous and BQ is changed only slowly compared with the relaxation rates (so-called slow passage conditions). Thus a steady-state solution to eqns... 

To satisfy slow passage conditions, cnm must be sufficiently small such that there are several cycles of the modulation frequency during the passage between the halfamplitude points of the resonance line, i.e. 

Write G as the sum of real and imaginary functions, Gj = Uj + an then separate Equation 7. Since most experiments are done under slow passage conditions, set (d/dt)Gj = +0. Finally, scale Uj and Vj by the factor MofyHx to arrive at the working equations ... 

The McConnell equations are also suitable for describing transient N.M.R. experiments when the rate of change of magnetization cannot be set equal to zero (McConnell and Thompson, 1959) and differential equations must be solved. The case of transfer between two sites of very different T1 and T2 values can be accommodated easily and McConnell has shown that under slow passage conditions if... 

The optical lineshape under slow passage conditions may be directly derived from (40) through the relation... 

Under these slow passage conditions, the solutions of the Bloch equations are (steady-state response to the rf-field) ... 

The saturation curves must be recorded under slow passage conditions, that is, conditions such that the time between successive field modulation cycles is sufficiently long for each spin packet to relax between cycles. The spin system is then continually in thermal equilibrium and the true line shape is observed. A convenient formulation of the slow passage condition is given by the expression (9.16) [79] ... 

The Bloch equations can be solved analytically under the condition of slow passage, for which the time derivatives of Eq. 2.48 are assumed to be zero to create a steady state. The nuclear induction can be shown to consist of two components, absorption, which is 90° out of phase with B, and has a Lorentzian line shape, and dispersion, which is in phase with B,. The shapes of these signals are shown in Fig. 2.10. By appropriate electronic means (see Section 3.3), we can select either of these two signals, usually the absorption mode. 

The solution of equations (13) is simplified under the ideal experimental conditions of slow passage . The frequency of the radiation <0, equal in this treatment to the angular velocity of the rotating frame ofreference, is changed very slowly so that the u, v and z magnetizations come to equilibrium at each frequency as the resonant condition co = w0 is passed. In view of the direct proportionality between frequency and field, the... 

When discussing the general aspects of FTNMR, we have to remember that all principal statements about Fourier methods have been introduced for a strictly linear system (mechanical oscillator) in Chapter 1. In Chapter 2, on the other hand, we have seen that the nuclear spin system is not strictly linear (with Kramer-Kronig-relations between absorption mode and dispersion mode signal >). Moreover, the spin system has to be treated quantummechanically, e.g. by a density matrix formalism. Thus, the question arises what are the conditions under which the Fourier transform of the FID is actually equivalent to the result of a low-field slow-passage experiment Generally, these conditions are obeyed for systems which are at thermal equilibrium just before the initial pulse but are mostly violated for systems in a non-equilibrium state (Oberhauser effect, chemically induced dynamic nuclear polarization, double resonance experiments etc.). 

The expression for (S/N)s in the steady-state experiment is available in the literature (Ernst and Anderson, 1966). A singlescan, true slow passage experiment is assumed. It is also assumed that the balanced bridge is perfectly balanced and that Tj = T2. To reduce the noise bandwidth, a low-pass filter is used. The maximum S/N ratio is obtained with a linear matched filter and under conditions of partial saturation. 

Figure C2.5.9. Examples of folding trajectories iT=T derived from the condition = 0.21. (a) Fast folding trajectory as monitored by y/t). It can be seen that sequence reaches the native state very rapidly in a two-state manner without being trapped in intennediates. The first passage time for this trajectory is 277 912 MCS. (b) Slow folding trajectory for the same sequence. The sequence becomes trapped in several intennediate states with large y en route to the native state. The first passage time is 11 442 793 MCS. Notice that the time scales in both panels are dramatically different.

R to P is slow, even when the isoenergetic conditions in the solvent allow the ET via the Franck-Condon principle. The TST rate for this case contains in its prefactor an electronic transmission coefficient Kd, which is proportional to the square of the small electronic coupling [28], But as first described by Zusman [32], if the solvation dynamics are sufficiently slow, the passage up to (and down from [33]) the nonadiabatic curve intersection can influence the rate. This has to do with solvent dynamics in the solvent wells (this is opposed to the barrier top description given above). We say no more about this here [8,11,32-36]. 

Adiabatic passage schemes are particularly suited to control population transfer between states, since the adiabatic following condition assesses the stability of the dynamics to fluctuations in the pulse duration and intensity [3]. The time evolution of the wave function does not depend on the dynamical phase, and is therefore slow in comparison with the vibrational time scale. This fact guarantees that the time variation of the observables during the controlled dynamics will be slow. Adiabatic methods can therefore be of great utility to control dynamic observables that do not commute with the Hamiltonian. We are interested in the control of the bond length of a diatomic molecule [4]. 

Under special conditions in which a reverse hydrodynamic flow was imposed to slow the passage of analytes through the capillary, up to 17 million plates were observed in the separation of small molecules 27... 

Bunsen 5 showed that the passage of a rapid stream of hydrogen sulphide through a hot solution of an alkali arsenate acidified with hydrochloric acid produced a precipitate of arsenic pentasulphide, and that this was a satisfactory method of determining arsenic quantitatively. These results were confirmed by McCay,6 and led to more systematic investigation of the subject,7 the result of which showed that the conditions favourable for the formation of arsenic pentasulphide, when hydrogen sulphide acts on aqueous arsenic acid or acid solutions of arsenates, are (a) a considerable excess of hydrochloric acid present, (b) a rapid passage of the gas, and (c) a comparatively low temperature—the liquid should be warm, as precipitation is extremely slow in the cold. Under these conditions arsenic pentasulphide alone is formed ... 

---