Resonance widths calculations

The ability to measure small index perturbations, (I/Ven. depends on the depth (i.e., the extinction ratio at resonance) and width of these resonances. The depth and width in turn depend on the relative values of the coupling coefficient t and the resonator loss a. Figure 9.16 shows typical ring resonator spectra calculated for two different loss values. The maximum extinction ratio is obtained at critical coupling when t = a, and the output intensity at resonance is exactly zero. The width of the resonances depends on the total round trip resonator loss, at, which is the product of waveguide losses within the ring (a) and coupling loss (t) at the coupler. 

As for the width of NMR resonance lines, it is inversely proportional to mobility of resonating nuclei. The width calculated from the experimental NMR lines of the three samples is of the highest value for the sample CT ODA 5 and there is only a small difference between the linewidths of CT and CT ODA 2. 

Figure 9. Comparison of measured [42] and calculated [33] resonance widths (in cm-1) for selected HCO resonances. The terms vi, u2, and 113 are the H-CO stretch, C-O stretch, and H-C-O bending quantum numbers, respectively. (Redrawn from Ref. 42.)...

P. Froelich, E. Brandas, Variational Principle for Quasibound States, Phys. Rev. A12 (1975) 1 P. Froelich, E. Brandas, Calculation of Resonance Widths via Expansion Techniques, Int. J. Quant. Chem. Symp. 10 (1976) 353. 

S. Skokov, J.M. Bowman, Complex L2 calculation of the variation of the resonance widths of HOC1 with total angular momentum, J. Chem. Phys. Ill (1999) 4933. 

P.W. Langhoff, Stieltjes-Tchebycheff moment-theory approach to molecular photoionization studies, in T. Rescigno, V. McKoy, B. Schneider (Eds.), Electron-Molecule and Photon-Molecule Collisions, Plenum, New York, 1979 A.U. Hazi, Stieltjes-moment-theory technique for calculating resonance widths, in T. Rescigno, V. McKoy, B. Schneider (Eds.), Electron-Molecule and Photon-Molecule Collisions, Plenum, New York, 1979. 

The value for resonance energy calculated using the bi-orthogonal dilated electron propagator is quite reasonable but the width of the 2II CO- using different decouplings is close only to that from the boomerang model. The... 

Except for the negative sign in front of the kinetic energy, which can be interpreted as a negative mass , the Hamiltonian (6.4.6) is the (autonomous) Hamiltonian of a pendulum. The resonance width Wm can immediately be calculated. Using the results derived in Section 5.2 and the approximation (6.1.56) for the derivatives of the Bessel functions we obtain ... 

Figure 16 Calculated (dashed lines and open squares) and measured (solid lines and filled circles) HCO resonance widths F as function of the CO stretching quantum number V2 for three progressions. Reprinted, with permission of the American Institute of Physics, from Ref. 51.

Measurements have also been made for vibrational states (7,0,0) and (8,0,0) [265]. The main observation is that on average the rates increase by two orders of magnitude from = 6 to 7 and again by two orders of magnitude from = 7 to 8. Calculations performed in the same way as for (6,0,0) reproduced this strong increase [69]. The calculated resonance widths are depicted in Fig. 23. Because details like the pronounced maxima caused by mixings with other vibrational states are not satisfactorily described by the calculations, only the ranges of the experimental data are indicated. For example, both the theoretical and the experimental widths... 

The main obstacle for calculating temperature and pressure dependent rate constants according to Eq. (76) or variants of it is the need for state-resolved dissociation rates for high rotational states. To perform exact quantum mechanical calculations for J = 40, for example, is not possible at present time even for a triatomic molecule, especially when it consists of three heavy atoms like O3. Until now, except for very few studies — HCO and HOCl, for example, discussed in Sect. 5 — most studies of resonance widths have been performed for J = 0. However, even at temperatures well below room temperature the molecules with J — 0 form only a small fraction of the ensemble. The common way of evaluating the resonance... 

Several approximate methods for calculating resonance energies and widths for atom-diatom reactive collisions are discussed. In particular, we present resonance energy calculations by semlclassical and quantal vi-bratlonally adiabatic models based on minimum-energy and small-curvature paths, by the semlclassical SCF method, by quantal SCF and configuration-mixing... 

Co11inear quantal calculations on M5 show sharp resonances in the reaction probability as a function of collision energy (5-6). The resonance energies have been shown to correspond to the energies of quasi-bound FH2 states (6,8). The resonance widths are... 

In Fig. 7, we show the cross section for photodetachment of Li via the S Skp channel over photon energies of approximately 5.04-5.16 and 5.29-5.46 eV. These ranges cover the regions below, and including, the Li(42p) and Li(52p) thresholds, respectively. Several Feshbach window resonances are observed to lie below these thresholds. In the figure the present measurements are compared with the result of a recent eigenchannel R-matrix calculation by Pan et al. [28]. The experimental resolution, which is estimated to be about 25 p,eV, is sufficiently high compared to the typical resonance widths that a direct comparison with theory can... 

In early studies of the QCM and the EQCM, only the resonance frequency was determined, and conclusions were drawn, based on the shift of frequency. Unfortunately, in many cases this shift was attributed to mass loading alone, and it was used to calculate the weight added or removed from the surface, disregarding other factors that affect the frequency. In the past decade more and more laboratories expanded such studies to include measurements of the impedance spectrum of the crystal [17-28]. This provides an additional experimental variable that can obviously yield further information and deeper understanding of the structure of the interface. For instance, a variation of the resonance width provides an unambiguous proof that mechanisms other than mass loading are also involved. 

The electronic structure is reformulated in terms of free electrons and a d resonance in order to relate the band width Wj to the resonance width T, and is then reformulated again in terms of transition-metal pseudopotential theory, in which the hybridization between the free-electron states and the d state is treated in perturbation theory. The pseudopotential theory provides both a definition of the d-state radius and a derivation of all interatomic matrix elements and the free-electron effective mass in terms of it. Thus it provides all of the parameters for the LCAO theory, as well as a means of direct calculation of many properties, as was possible in the simple metals. ... 

A.U. Hazi, A purely H method for calculating resonance widths, J. Phys. B 11 (1978) L259. 

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