Zero-field resonance

This leads to distinguish two types of resonances the resonances induced by the field that occur beyond a threshold of the field, and resonances that occur for an arbitrary small value of the field. These are called, respectively, (i) the dynamical resonances (or equivalently field induced resonances or nonlinear resonances) and (ii) the zero-field resonances. 

The zero-field resonances can be identified with respect to the system energy levels and the field frequency when the field is off. They are usually one- or two-photon resonances. The one-photon resonance is of first order with respect to the field amplitude in the sense that the degeneracy of the eigenvalues is lifted linearly with the field amplitude. The two-photon resonance is of second order since the degeneracy of the eigenvalues is lifted quadratically with the field amplitude. Multiphoton resonances (more than two-photon) are more complicated since they are generally accompanied by dynamical shifts of second order... 

As we have stated, the Floquet Hamiltonian (113) has no terms that are resonant if we take small enough e, and the iteration of the KAM procedure converges. However, if we take e large enough, we encounter new resonances that are not present at zero or small fields that is, they are not related to degeneracies of the unperturbed eigenvalues of Kq that lead to the zero-field resonances we have discussed in the previous subsection. These new resonances are related to degeneracies of the new effective unperturbed operator K 0(e), which appear at some specific finite values of e. These are the dynamical resonances. 

Adiabatic passage can result in a robust population transfer if one uses adiabatic variations of at least two effective parameters of the total laser fields. They can be the amplitude and the detuning of a single laser (chirping) or the amplitudes of two delayed pulses [stimulated Raman adiabatic passage (STIRAP) see Ref. 69 for a review]. The different eigenenergy surfaces are connected to each other by conical intersections, which are associated with resonances (which can be either zero field resonances or dynamical resonances appearing beyond a threshold of the the field intensities). The positions of these intersections determine the possible sets of paths that link an initial state and the... 

When the processes involve a zero-field resonance, one has to add the ingredient of lifting of degeneracy. This means that we have to consider the dynamics starting (or ending) near the conical intersection in a direction not parallel to the Cl = 0-plane. This can be seen in Fig. 6 where the surfaces of Fig. 2 have been redrawn for positive detunings (case of a one-photon resonance). When the dynamics starts this way, it is characterized by two adiabatic paths, one on each surface. They will lead in general to coherent superpositions of states. 

In Figs. 12 and 13 we compare the results of the present theory for quadrupole moments with the experimental data obtained for the l) line of rubidium [Budker 2002 (b)]. The calculations for our simpler system reproduce many of the qualitative aspects of the experimental data for Rb. As seen, at the center of the in-phase plots of Figs. 12 and 13, zero-field resonances arise, while resonances centered at the magnetic-field values for which = 1/2 and... 

The first experiments with ESR on electronically excited molecular triplet states were published by Hutchison et al. [1]. A few years later, the first optical detection of ESR in a magnetic field followed [2], as well as the first optical detection of the zero-field resonance [3]. Eor these experiments, molecular crystals (host crystals doped with guest molecules) were investigated. The host crystal in these mixed-crystal systems serves mainly as an orienting matrix and the result of these ESR measurements are values for the fine structure and the dynamic properties of the Ti state of the guest molecules. 

Fig. 7.6 Above zero-field splitting (Bq = 0) of a triplet state for non-spherical symmetry. s = energy, D, = fine-structure constants, lhf>. Ey>, and 7z> = spin functions, x, y,z = polarisation of the magnetic high-frequency field Bi cosfflt for the three ESR transitions with quantum energies D-E, D+E and 2 in zero field (zero-field resonances). Below the three zero-field resonance lines from naphthalene molecules in a biphenyl crystal at T= 83 K. After [5].

The three transitions between the three terms are referred to as zero-field transitions or the zero-field resonances. In an applied high-frequency field Bi cos rot, all three transitions are allowed. This follows immediately by computation of the transition matrix elements (g/UB/h)(Tu Bi S T > using Table 7.2 for example, the transition matrix element for the zero-field transition T -(- Tz (Fig. 7.5), the transition matrix element (Txl(BixSx+ BiySy+ Bi2S2) T2> is nonzero only for the y component. For this transition, Bi must thus have a component in the y direction. From Fig. 7.5, this becomes intuitively clear for this transition, aU the spins must process around the y axis. The other two zero-field transitions follow by cyclic permutations. 

The most precise measurements of the fine-structure parameters D and E have in fact been carried out using zero-field resonance. Figure 7.6 shows the three zero-field transitions in the Ti state of naphthalene molecules in a biphenyl crystal at T = 83 K. In these experiments, the absorption of the microwaves was detected as a function of their frequency [5]. The lines are inhomogeneously broadened and nevertheless only about 1 MHz wide. Owing to the small hnewidth of the zero-field resonances, the fine-structure constants can be determined with a high precision. This small inhomogeneous broadening is due to the hyperfine interaction with the nuclear spins of the protons (see e.g. [M2] and [M5]). For triplet states in zero field, the hyperfine structure vanishes to first order in perturbation theory, since the expectation value of the electronic spins vanishes in all three zero-field components (cf Sect. 7.2). The hyperfine structure of the zero-field resonances is therefore a second-order effect [5]. 

As a second example, Fig. 7.7 shows the optical detection of the zero-field resonance (ODMR) from X-traps (cf Sect. 4.1) in an anthracene crystal at T = 1.2 K [6]. Here, both the variation of the intensity of the phosphorescence. Alp, and also the changes in the intensity AIdf of the delayed fluorescence (cf Sect. 6.9.2) were measured as functions of the microwave frequency. With both methods, aU three zero-field resonances were detected. The optical detection of the resonance at 1850MHz requires simultaneous irradiation at one of the other two resonance frequencies. The method of (optical) detection of this resonance is therefore referred to as electron-electron double resonance (FEDOR). From the three zero-field resonances and their structures, the three fine-structure parameters of two different X-traps in the anthracene crystal were found to be... 

A third example of a zero-field resonance is shown in Fig. 7.8 it is the optically-detected D- - E transition of triplet excitons in 1,2,4,5-tetrachlorobenzene crystals at a temperature of T = 4.2 K [7]. In this measurement, the phosphorescence intensity... 

NMR presents a convenient local probe for the temperature dependence of the spontaneous magnetization. Evidently, the measurement of the temperature dependence of the zero-field resonance frequency is much easier than the careful analysis of the variation of the resonance frequency with orientation and strength of an external field and with temperature in a single crystal. It has, however, always to be kept in mind that the zero-field NMR analysis of a multidomain powder sample can only yield information about the phenomena in the Bloch wall if the zero-field signal originates from nuclei in the Bloch wall. This variation should not be compared with the temperature dependence of the magnetization in the domain. Only a careful NMR analysis is thus worthwhile for a discussion of the eventual differences in the variations of h T) = ... 

Optical detection of magnetic resonance (ODMR) was attempted for measurements of the pH effects on the triplet state of purine to investigate the protonation site of purine at low temperatures (78JA7131). The ODMR spectrum did not show the presence of more than one triplet state at liquid helium temperatures. Since the protonated tautomers 1H,9H (3a) and H,1H (3b) have similar bond structures, their triplets should have similar zero-field parameters and are thus not easy to distinguish by ODMR. 

The deuterium line of the deuterated solvent is used for this purpose, and, as stated earlier, the intensity of this lock signal is also employed to monitor the shimming process. The deuterium lock prevents any change in the static field or radiofrequency by maintaining a constant ratio between the two. This is achieved via a lock feedback loop (Fig. 1.10), which keeps a constant frequency of the deuterium signal. The deuterium line has a dispersion-mode shape i.e., its amplitude is zero at resonance (at its center), but it is positive and negative on either side (Fig. 1.11). If the receiver reference phase is adjusted correcdy, then the signal will be exactly on resonance. If, however, the field drifts in either direction, the detector will... 

The spectrum of Mn2+ in zeolites has been used to study the bonding and cation sites in these crystalline materials. This is a 3d5 ion hence, one would expect a zero-field splitting effect. A detailed analysis of this system was carried out by Nicula et al. (170). When the symmetry of the environment is less than cubic, the resonance field for transitions other than those between the + and — electron spin states varies rapidly with orientation, and that portion of the spectrum is spread over several hundred gauss. The energies of the levels are given by the equation... 

In general, no simple, consistent set of analytical expressions for the resonance condition of all intradoublet transitions and all possible rhombicities can be derived with the perturbation theory for these systems. Therefore, the rather different approach is taken to numerically compute all effective g-values using quantum mechanics and matrix diagonalization techniques (Chapters 7-9) and to tabulate the results in the form of graphs of geff,s versus the rhombicity r = E/D. This is a useful approach because it turns out that if the zero-field interaction is sufficiently dominant over... 

This is a pretty unusual expression, and it should warn us that resonance conditions derived via perturbation theory should always be checked for their validity under actual conditions. Suppose that the zero-field intradoublet splitting, /Lni))-ii,(0)... 

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