Vibrational Feshbach resonances

The reverse of step 2 corresponds to a vibrational Feshbach resonance, which has never been observed experimentally for heavy-particle systems. The forward process, on the other hand, can be observed quite easily. One generally needs a supersonic molecular beam In order to achieve low enough temperatures to form A B without having macroscopic condensation of the sample, and a detector which... 

Vibrational Feshbach resonances (VFRs) in a vibrational Feshbach resonance, the interaction of a slow electron takes the form of a virtual excitation of a vibrational level of the neutral molecule with capture of the electron (ffotop et al. 2003 Dessent et al. 2000). For the zero point vibration, the maximum probability of interaction of the electron with parent molecule occurs at zero energy, if the dipole moment of the neutral molecule exceeds the critical value of approximately 2 Debye, the impinging electron maybe trapped into the diffuse bound state, which provides a much longer timescale for the electron to stay near the molecule (fiotop et al. 2003 Dessent et al. 2000 lllenberger 1992), and VFRs may appear as shown in O Fig. 34-5. 

Diabatic potential energy surfaces of neutral and dipole-bound anion showing the origin of the vibrational Feshbach resonances (VFRs). An extra electron attaching into the diffuse dipole-bound MO of the parent can excite the nearby vibrational levels of the parent molecule resulting in VFRs... 

To capture the essence of the Feshbach resonance phenomenon, we will need to understand what happens to the ground vibrational state 4>o(R) of the ground electronic state, also depicted in Figure 1.13, because of the interaction with the continuum of states excited electronic state. The physical process described above can be formulated as a two coupled channels problem where the solution irg(R) in the closed channel (the ground state) depends on the solution ire(R) in the open channel (the excited state) and vice-versa. The coupled Schrodinger equations read... 

Figure 3.1 A schematic diagram showing the relationship of reactive resonances to the vibrationally adiabatic potential curve. The upper panel illustrates a Feshbach resonance trapped in a well the lower panel shows a barrier resonance or QBS.

The dynamics of a reaction that proceeds directly over the transition state is expected to be qualitatively different from that of a resonance-mediated reaction. In particular, one expects that the branching ratios into the product rovibrational states will be very different between the direct and the resonant mechanisms. For example, if a given Feshbach resonance corresponds to trapping on the v = 1 vibrationally adiabatic curve, then one might expect that the population of the v = l vibrational state of the product molecule may be greatly enhanced by the resonant mechanism. Similarly, the rotational product distribution resulting from the fragmentation of a resonance molecule may show a quite distinct pattern from that of a direct reaction. Indeed, Liu and coworkers [94], and Nesbitt and coworkers [95] have noted distinct rotational patterns in the F+HD resonant reaction. 

Resonances of the type illustrated in Figure 12.2 are called Feshbach resonances (Child 1974 ch.4 Fano and Rao 1986 ch.8 see also Figure 12.5). The quasi-bound states trapped by the Vn(.R) potential can only decay via coupling to the lower vibrational state because asymptotically the n = 1 channel is closed and therefore cannot be populated. This is different from the dissociation of CH30N0(Si), for example, [see Figure 7.10(a)] where the resonances can either decay via tunneling or alternatively by nonadiabatic coupling to the lower states. 

Fig. 12.5. Zeroth-order potentials Veff(R-,j,Sl,J) defined in (12.7) for fl = 0 and several total angular momentum quantum numbers J. The excited rotational states can decay either by tunneling (shape resonances) or by rotational predissociation ( Feshbach resonances) as indicated by the horizontal arrows. The excitation through the IR photon originates from the vibrational ground state n = 0 which is not shown in the figure.

The chemical Importance of electronic Feshbach resonances derives from essentially the same effects as were given earlier for shape resonances. They allow either free electrons or electron density from a collision partner to give energy to the internal (vibrational, rotational, or electronic) degrees of freedom of the target. 

To see how Feshbach resonances appear in classical S-matrix theory, consider the collinear H + Cl2 collision as studied by Rankin and Miller.47 Fig. 8 shows the quantum number function n2(q,) for one region of ql the function is smooth, these trajectories being direct . The remaining interval of ql leads to complex trajectories, those which spend a number of additional vibrational periods in the interaction region for this region of qY values the final vibrational quantum number changes dramatically with small changes in q,. The S-matrix for the particular transition indicated in Fig. 8 thus has the form... 

This dependence of the H+ KE on the XUV-IR delay in this case of the longer, 35 fs FWHM, IR pulse can be understood in terms of the adiabatic-ity of the Floquet dynamics underlying the dissociation processes, and the way that the IR intensity affects both the preparation and the propagation of the Floquet components of the wavepackets. More precisely, the IR probe pulse projects the various vibrational components of the wavepacket onto Floquet resonances, whose widths vary with the intensity of the IR pulse. We recall that these resonances are of two types Shape resonances supported by the lower adiabatic potential defined at the one-photon crossing between the dressed (g, n), (u, n ) channels and leading to efficient dissociation through the BS mechanism, or Feshbach resonances, vibrationally trapped in the upper adiabatic potential well. 

Compared to nonoptical routes toward cold molecules discussed in this book, the main advantage of the PA route is the ability to produce a large number of stable ultracold molecules, at the same translational temperature (a few ttK, or even less) than the precursor atoms. In contrast with the halo molecules produced by sweeping magnetic Feshbach resonances [8], the stable molecules are formed in vibrational levels, which can be relatively deeply bound. The drawback is that stabilization through the spontaneous emission process spreads the population into a variety of excited vibrational levels, so that the product molecules are not in a pure vibrational state. Schemes to cool down the vibrational and rotational degrees of freedom have to be implemented. 

After the pulse, there is also population of the two upper vibrational levels of the lower state, v" = 52 and v" = 53 (see Figure 7.4b) stable molecules are thus formed in a one-color scheme, because the time-dependent frequency of the pulse is sweeping an optical Feshbach resonance. In the dressed potential picture, the initial continuum level of the ground potential Vg R, t) is at resonance with a bound level of the excited potential Ve(R, t). Note the efficiency of the process the number of molecules formed in these two levels of the lower state is equivalent to the number of photoassociated molecules in 15 levels in the excited state. The efficiency of various PA pulses for this population transfer has been discussed in Ref [19]. The levels v" = 53 and v = 52 are respectively bound by 5 x 10 and 0.042 cm to be compared with the resonance window of 1.74 cm in the excited state. Due to the very small value of the binding energy, these molecules are halo molecules, as defined by Koehler and colleagues [8] their creation as a byproduct of the PA process should be further investigated. Recently, Kallush and Kosloff [28] have discussed the nonperturbative character of the PA process, where the conservation of the total population requires a... 

FIGURE 7.4 Molecules formed by PA within the resonance window and by an optical Feshbach resonance, (a) Probability of population transfer P (f), during and after the pump pulse PI , into several vibrational levels i> of the excited state. The levels i> = 95-107 (solid line) lie in the resonance window and remain populated after the pulse. In contrast, for the levels outside the resonance window (dash-dotted line), no population remains after the pulse, (b) Population transferred P ° //(f) in the two last vibrational levels v" = 52 (solid line) and... 

During the PA step, there is also an important population transfer to the highest bound vibrational levels of the initial electronic state. The formation of halo molecules by an optically induced Feshbach resonance should be further exploited. The redistribution of population in the continuum is creating pairs of hot atoms. 

The Other reactive resonance is called the trapped-state resonance or Feshbach resonance, shown in Fig. 4.1c. In this case, the ABC complex is dynamically trapped along the reaction coordinate, even the minimum energy path on the BO PES is totally repulsive. The trapping of the short-lived ABC complex is caused by the vibrationally adiabatic potential, which is based on the concept of vibrational adiabaticity [23, 75, 76, 120]. As the vibrational motions along the directions perpendicular to R are fast compared with the motion along R, the vibrational modes should approximately conserve the quantum number n, which is in the spirit of BO separation of motions with different time scale. A typical vibrationally adiabatic potential along the reaction coordinate R is shown in Fig. 4.2b (left), and it can be constructed as... 

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