Open-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]). 

Marcelis, B., van Kempen, E.G.M., Verhaar, B.J., and Kokkehnans, S.J.J.M.F., Fesh-bach resonances with large background scattering length Interplay with open-channel resonances, Phys. Rev. A, 70, 012701, 2004. 

Some of the earliest applications of MQDT dealt with vibrational and rotational autoionization in H2 [21-25]. One concept that emerged from these studies is that of complex resonances [26], which are characterized by a broad resonant distribution of photoionization intensity with an associated rather sharp fine structure. These complex resonances cannot be characterized by a single decay width they are the typical result of a multichannel situation where several closed and open channels are mutually coupled. The photoionization spectrum of H2 affords a considerable number of such complex resonances. 

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... 

Resonances unassociated with eigenstates of Feshbach s QHQ are often associated with the shape of some effective potential in an open channel, normally a combination of short-range attractive and long-range repulsive potentials, forming a barrier, within which a large part of the wavefunction is kept. These resonances are called "shape resonances" or "potential resonances." They occur at energies above and usually close to the threshold of that open channel. 

Conversely, a coherent superposition of continuum states with a population closely reproducing an isolated peak in the density of states, which corresponds to a resonance, can be built in such a way to give rise to a localized state. From this localized state, there will be an outward probability density flux, i.e., it will have a finite lifetime. In the limit of a resonance position far from any ionization threshold and a narrow energy width, the decay rate will be exponential with the rate constant T/ft. The decay is to all the available open channels, in proportion to their partial widths. 

Magnetic resonance imaging permitted direct observation of the liquid hold-up in monolith channels in a noninvasive manner. As shown in Fig. 8.14, the film thickness - and therefore the wetting of the channel wall and the liquid hold-up -increase nonlinearly with the flow rate. This is in agreement with a hydrodynamic model, based on the Navier-Stokes equations for laminar flow and full-slip assumption at the gas-liquid interface. Even at superficial velocities of 4 cm s-1, the liquid occupies not more than 15 % of the free channel cross-sectional area. This relates to about 10 % of the total reactor volume. Van Baten, Ellenberger and Krishna [21] measured the liquid hold-up of katapak-S . Due to the capillary forces, the liquid almost completely fills the volume between the catalyst particles in the tea bags (about 20 % of the total reactor volume) even at liquid flow rates of 0.2 cm s-1 (Fig. 8.15). The formation of films and rivulets in the open channels of the structure cause the further slight increase of the hold-up. 

Figure 1 (a) Illustration of the unimolecular dissociation of a reactant molecule ABC into products A and BC. p( ), Rts and Nts are the density of reactant states, the intermolecular distance of the transition state (TS) and the number of open channels at the transition state, respectively. The vertical axis on the left-hand side shows a spectrum dominated by sharp resonances, (b) Schematic representation of the energy dependence of the micro-canonical rate constant k E) (solid line). The dots represent the state-specific quantum mechanical rates kn-... 

Since resonances correspond to poles of the S-matrix (see 2.1), TrQ( ) has a familiar Lorentzian shape in the vicinity of each isolated resonance. The positions and widths can be determined from a non-linear fit to the Breit-Wigner form, Eq. (7) [40]. Another option was chosen by Dobbyn et al. in studies of the dissociation of the HO2 radical [60]. They overlapped the scattering state in each open channel a with some arbitrary wave packet 4 0) localized in the interaction region of the potential, and constructed an artificial photo-absorption spectrum (t E), which is a sum of partial contributions (Ta E) [20], i.e.,... 

Several conclusions can be drawn from Eqs. (76) and (77). First, the influence of fluctuations is the largest when the number of open channels u is of the order of unity, because then the distribution Q k) is the broadest. Second, the effect of a broad distribution of widths is to decrease the observed pressure dependent rate constant as compared to the delta function-like distribution, assumed by statistical theories [288]. The reason is that broad distributions favor small decay rates and the overall dissociation slows down. This trend, pronounced in the fall-of region, was clearly seen in a recent study of thermal rate constants in the unimolecular dissociation of HOCl [399]. The extremely broad distribution of resonances in HOCl caused a decrease by a factor of two in the pressure-dependent rate, as compared to the RRKM predictions. The best chances to see the influence of the quantum mechanical fluctuations on unimolecular rate constants certainly have studies performed close to the dissociation threshold, i.e. at low collision temperatures, because there the distribution of rates is the broadest. 

I started to make sounds I had never heard before. Every shift in resonance and tone led to strange significance and new, textured sounds. The sound was not a separate thing but an open channel of the biological, temporal, emotional, the light, color, and form. 

Complex adiabatic energies (a) compared to resonance energies (b) for a model of a closed and an open channel described in reference (30). The zeroth order energy has been subtracted out. Unit cm , Interchannel coupling 200 cm . Crossing diabatic energy equal to the energy of the level with v = 18. 

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)... 

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