Closed-channel resonances

In the above orthogonal separation into localized (Zoc) and asymptotic (as) correlations, cTas(pi) is zero for closed channel resonances (the so-called Feshbach resonances). This term is non-zero and physically significant in the cases of open channel resonances (the so-called shape resonances), see Section 8.3. 

Since the 1960s, it has become customary among many researchers studying and/or discussing resonances to divide them into two categories 1) "Shape or "open channel" resonances and, 2) "Feshbach," or "closed channel" resonances (e.g., see the reviews by Burke [15] and by Buckman and Clark [19]). 

Resonant enhancements of scattering cross-sections in multichannel collision physics are often described in terms of the Feshbach theory of closed-channel resonance states [57], Feshbach s general formalism involves projecting the stationary Schrddinger equation onto complementary subspaces associated with the open and closed scattering channels. This theory has been applied in the context of the nearthreshold collision physics of ultracold gases consisting of alkali-metal atoms in a variety of different approaches (e.g.. Refs. [9,30,58]). 

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. 

Feshbach-type resonances [51], also known as Fano resonances [52] and Floquet resonances [22] depending on the system studied, are formed in a different manner. We encounter this type of metastable states whenever a bound system is coupled to an external continuum. In the same spirit as before, one can define a reference Hamiltonian in which the closed channel containing the bound states is uncoupled from the open channel through which the asymptote can be reached. When the coupling is introduced, the previously bound state decays into the continuum of the open channel. The distinction from shape-type resonances, described above, is that the resonance state decays into a different channel of the reference Hamiltonian. 

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... 

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... 

The formal scattering theory for describing cosipound-state resonances such as the vibrationally predissociaCing states of interest here, is well established (see, e.g., (32-33) and references therein). For an isolated narrow resonance associated with closed channel m, the S-matrix element between (open) channels j and j is given by (33)... 

The phase-space model has been extended by Miller (1970) in order to incorporate the effect of closed channels. He made use of a parametrized form of the S matrix previously developed for compound-state resonances in atom-molecule collisions (Micha, 1967). Indicating with Sd the S-matrix for direct scattering, i.e. in the absence of coupling to closed channels, one can write (omitting the index J). 

In this formulation the effects of closed channels can be straightforwardly incorporated into the 5-matrix. Resonances in the scattering clearly correspond to the complex solutions of... 

Note that the interaction between the entrance and closed channels shifts the point of singularity of a(B) from Be to Bq. Such magnetically tunable Feshbach resonances are characterized by four parameters, namely, the background scattering length abg, the magnetic moment difference 8p, the resonance width A, and position Bq. 

FIGURE 6.7 The lower panel shows an expanded view of Ei, B) near Sq for th resonance with Bq = 54.693 mT (546.93 G) in Figure 6.6. The solid line comes from a coupled-channels calculation that includes all 12 channels with the same —7/2 projection quantum number. The dashed and dotted hues respectively show the universal energy of Equation 6.3 and the van der Waals corrected energy of Equation 6.11. The upper panel shows the closed channel norm Z(B). The width A = O.SlOmT (3.10G), a g = —191no. h i/h =... 

The wavefunction of such a weakly bound molecular state has only a small admixture of the closed channel, and the size of the molecule is The characteristic momenta of the atoms in the molecule are of the order of and in this respect the inequality 10.4 represents the criterion of a wide resonance for the molecular system. 

CLASSIFICATION OF RESONANCES 11.5.1 Closed-Channel Dominated Resonances... 

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