Resonances Feshbach

The theory of isolated resonances is well understood and is discussed below. Mies and Krauss [79, ] and Rice [ ] were pioneers m treating unimolecular rate theory in temis of the decomposition of isolated Feshbach resonances. 

Jolicard G, Leforestier C and Austin E J 1988 Resonance states using the optical potential model. Study of Feshbach resonances and broad shape resonances J. Chem. Phys. 88 1026... 

Fig. 1. A schematic diagram of the relationship between adiabatic potential curves and reactive resonances, (a) shows the conventional Feshbach resonance trapped in a well of an adiabatic curve, (b) illustrates barrier trapping, which occurs near the energy of the barrier maximum of an adiabatic curve.

The SQ method extracts resonance states for the J = 25 dynamics by using the centrifugally-shifted Hamiltonian. In Fig. 20, the SQ wavefunc-tion for a trapped state at Ec = 1.2 eV is shown. The wavefunction has been sliced perpendicular to the minimum energy path and is plotted in the symmetric stretch and bend normal mode coordinates. As anticipated, the wavefunction shows a combination of one quanta of symmetric stretch excitation and two quanta of bend excitation. The extracted state is barrier state (or quantum bottleneck state) and not a Feshbach resonance. 

The other group consists of the so-called Feshbach resonances that are formed when the incident electron first excites the molecule directly and then is captured by the molecule which is now in an excited state, provided that by doing so the molecule benefits in energy. Very often this excited state of the molecule is one of the Rydberg states, since in that case a positive electron affinity is preferable. 

Key words Feshbach resonance Shape resonance Diborides Heterostructure at atomic limit. 

The superconducting gaps in the second, A2, and third, A3, subband in a superlattice of quantum wires as a function of the reduced Lifshitz parameter z are shown in Fig. 7a, where EF is tuned at Ec for z =0, the energy cut off for the pairing interaction is fixed at 500K. The increase of the gap A2 is driven only by the Feshbach resonance in the interband pairing since the partial DOS of the second subband has not peaks. 

Probably the most accurate positron-hydrogen s-wave phase shifts are those obtained by Bhatia et al. (4974), who avoided the possibility of Schwartz singularities by using a bounded variational method based on the optical potential formalism described previously. These authors chose their basis functions spanning the closed-channel Q-space, see equation (3.44), to be of essentially the same Hylleraas form as those used in the Kohn trial function, equation (3.42), and their most accurate results were obtained with 84 such terms. By extrapolating to infinite u in a somewhat similar way to that described in equation (3.54), they obtained phase shifts which are believed to be accurate to within 0.0002 rad. They also established that there are no Feshbach resonances below the positronium formation threshold. 

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.

Feshbach resonances is purely model dependent since trapping well may exist on one type of adiabatic potential, say in hyperspherical coordinates, while only a barrier may exist on another type, say in natural collision coordinates. However, this is not correct since there are fundamental differences between QBS and Feshbach states. First, the pole structure of the S-matrix is intrinsically different in the two cases. A Feshbach resonance corresponds to a single isolated pole of the scattering matrix (S-matrix) below the real axis of the complex energy plane, see the discussion below. On the other hand, the barrier resonance corresponds to an infinite sequence of poles extending into the lower half plane. For a parabolic barrier, it is easy to show that the pole positions are given by... 

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. 

For both the Feshbach resonance in F+HD and the barrier resonance in H+HD, the trapped resonance state was localized close to the saddle point. Thus, the complexes correspond to the intermediates with partially formed (or broken) bonds. The F+HCl-tHF+Cl reaction illustrates a third category of resonance, the prereactive (and postreactive) resonance. In this case, the bonding of the reagent (or product) is only weakly perturbed by the colliding... 

The formal theory of resonances due to Feshbach begins with the decomposition of the Hamiltonian in terms of a projection operator Q [8]. He defines Q as the projection onto the closed-channel space, just like the example of H discussed around Eqs. (4) and (5). Then, QBSs described well by the eigenfunctions Q4> of Eq. (5) with his Q may be called Feshbach resonances." A simplified picture would be that eigenstates Q are supported by some attractive effective potential approaching asymptotically the threshold energy of a closed channel. If this is the case, then the energies EQ of... 

Infinite series of Feshbach resonances due to the Coulomb tail... 

Rydberg states, which are Feshbach resonance states. Their complex energies n, for a fixed angular momentum and labeled by the principal quantum number n, satisfy the Rydberg formula with a complex quantum defect l n + j-Yn/... 

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