Single-Mode Model

The temporal evolution of the simulated peak position of the fluorescence spectrum is shown in Fig. 3a The corresponding experimental peak-shift (figure 2 in [5]) consists of an oscillatory and an exponential part. Our model reproduces the weakly damped oscillatory part of the peak shift, but does not describe the large Stokes shift of 1400 cm-1. The reason is that our single-mode model does not take into account other system modes and, what is more important, solvent modes, which contribute to the overall shift of the SE spectrum. The model may be improved by including an additional overdamped solvent mode. 

Since the phonon energy quantum hu> does not explicitly appear in both Eqs. 35 and 36, these formulas for the rate constant in the semiclassical regime should be derived without the single-mode model being relied on. That is, we should not neglect that various phonon modes contributing to the reaction coordinate Q have, in reality, various energy quanta. In this case, the phonon Hamiltonian in the reactant state (where the electron to be transferred is at the donor) consists of various normal modes. 

Note that mosaic artifacts can also occur physically in real spectra when a real powder sample of a model compound exhibits microcrystallinity and thus contains too few different molecular orientations. This phenomenon is rare in X-band EPR and is usually easily solved by grinding the sample in a mortar it is, however, not at all uncommon even for extensively ground samples in high-frequency EPR with single-mode resonators where the sample size is orders of magnitude less than that of an X-band sample. 

In addition to packed catalyst bed, a fluidized bed irradiated by single and multi-mode microwave field, respectively, was also modeled by Roussy et al. [120]. It was proved that the equality of solid and gas temperatures could be accepted in the stationary state and during cooling in a single-mode system. The single-mode cavity eliminates the influence of particle movements on the electric field distribution. When the bed was irradiated in the multimode cavity, the model has failed. Never-... 

As a simple model, we confine our attention just to a single mode Ha(t) of the Hamiltonian (23). Note that neither any instantaneous eigenstate of Ha(t) is an exact quantum state nor e-/3ii W is a density operator. To calculate the thermal expectation value of an operator A, one needs either the Heisenberg operator Ah or the density operator pa(t) = UapaUa Now we use the time-dependent creation and annihilation operators (24), invariant operators, to construct the Fock space. 

As evidenced by the correlation of the BH model with the experimental data, Figure 2.4, the model is only in qualitative accord with the experiment. Clearly, the BH model cannot account for the breadth in the correlation of the rate constants for porton transfer with driving force. The origin of the discrepancy may lie in the single-mode nature of the BH model, which allows only for vibrational excitation in the low-frequency promoting mode. Excitation in the reactant and product modes of the vibration associated with the transferring proton is not taken into account in the BH model. Therefore, the discrepancy between experiment... 

The sum over weighted relaxation times is heavily dominated by the longest time (the reptation time) r gp=L /7T Dp. Because of this the frequency-dependent dissipative modulus, G"(cd) is expected to show a sharp maximum The higher modes do modify the prediction from that of a single-mode Maxwell model, but only to the extent of reducing the form of G"(a>) to the right of the maximum from ccr to In fact, experiments on monodisperse linear polymers... 

Figure 32. Vibronic periodic orbits of a coupled electronic two-state system with a single vibrational mode (Model IVa). All orbits are displayed as a function of the nuclear position x and the electronic population N, where N = Aidia (left) and N = (right), respectively. As a further illustration, the three shortest orbits have been drawn as curves in between the diabatic potentials Vi and V2 (left) as well as in between the corresponding adiabatic potentials Wi and W2 (right). The shaded Gaussians schematically indicate that orbits A and C are responsible for the short-time dynamics following impulsive excitation of V2 at (xo,po) = (3,0), while orbit B and its symmetric partner determine the short-time dynamics after excitation of Vi at (xo,po) = (3, —2.45).

Chin M.K. and Ho, S.T., 1998, Design and modeling of waveguide-coupled single-mode microring resonators, J. Lightwave Technol. 16(8) 1433-1446. 

In papers , unsteady-state regime arising upon propagation of the stationary fundamental mode from linear to nonlinear section of a single-mode step-index waveguide was studied via numerical modeling. It was shown that the stationary solution to the paraxial nonlinear wave equation (2.9) at some distance from the end of a nonlinear waveguide has the form of a transversely stable distribution ( nonlinear mode ) dependent on the field intensity, with a width smaller than that of the initial linear distribution. 

A major development reported in 1964 was the first numerical solution of the laser equations by Buley and Cummings [15]. They predicted the possibility of undamped chaotic oscillations far above a gain threshold in lasers. Precisely, they numerically found almost random spikes in systems of equations adopted to a model of a single-mode laser with a bad cavity. Thus optical chaos became a subject soon after the appearance Lorenz paper [2]. 

The attenuation of die dipole of the repeat unit owing to thermal oscillations was modeled by treating the dipole moment as a simple harmonic oscillator tied to the motion of the repeat unit and characterized by the excitation of a single lattice mode, the mode, which describes the in-phase rotation of the repeat unit as a whole about the chain axis. This mode was shown to capture accurately the oscillatory dynamics of the net dipole moment itself, by comparison with short molecular dynamics simulations. The average amplitude is determined from the frequency of this single mode, which comes directly out of the CLD calculation ... 

A crucial part of these experiments is the preparation of the sodium atoms into the excited state by laser optical pumping. A commercial single-mode Rhodamin 6G continuous wave (cw)-dye laser (Spectra Physics model 580) is used, having 20-40-mW single mode output power when tuned to the sodium resonance line. 

Whereas the dual sorption and transport model described above unifies independent dilatometric, sorption and transport experiments characterizing the glassy state, an alternate model offered recently by Raucher and Sefcik provides an empirical and fundamentally contradictory fit of sorption, diffusion and single component permeation data in terms of parameters with ambiguous physical meanings (28), The detailed exposition of the dual mode model and the demonstration of the physical significance and consistency of the various equilibrium and transport parameters in the model in the present paper provide a back drop for several brief comments presented in the Appendix regarding the model of Raucher and Sefcik,... 

---