Parabolic well models

Hat-Curved Model as Symbiosis of Rectangular-Well and Parabolic-Well Models... 

In the case of a parabolic well the period is independent on the phase variables, the anharmonicity vanishes, and the bandwidth is nonzero only due to strong collisions. The more a potential profile differs from the parabolic one, the larger the anharmonicity and the wider the absorption band. The intensity of the absorption peak should then decrease since in accord with the Gordon rules (see, e.g., GT, Section III.G or see Section VIIA.4 in the present chapter) in an isotropic medium the integrated absorption does not depend on parameters of the model. 

Thus, evolution of semiphenomenological molecular models mentioned in Section V.A (items 1-6) have led to the hat-curved model as a model with a rounded potential well. This model combines useful properties of the rectangular potential well and those peculiar to the field models based on application of the parabolic, cosine, or cosine-squared potentials. Namely, the hat-curved model retains the main advantage of the rectangular-well model—its possibility to describe both the librational and the Debye-relaxation bands. 

We have used the calculation scheme described in Section VII.B for the HC-CS model. Now we employ the composite HC-EB model. The only difference from the above-mentioned calculations concerns the formula for the spectral function Now we use Eq. (474) instead of Eq. (315). Setting vm to be close to Vr 200 cm-1, we estimate the steepness of the parabolic well as... 

Eg between the valence band and the conduction band. The band structure of a direct II-VI intrinsic semiconductor like CdSe can be represented reasonably well by a parabolic band model like that shown schematically in Fig. 2. Here, k = 7r/ris the wave vector and r is the radial distance from an arbitrary origin in the center of the crystal. The kinetic energy of the electron is proportional to E- and the energy minimum of the conduction band and the maxima of the valence bands occur at k = 0 (corresponding to r = co in a bulk sample). 

In the harmonic approximation in which a parabolic well near the equilibrium bond length is assumed, the frequencies will not be altered, but inclusion of anharmonic (cubic and higher order) terms in the intraionic potential function will further modify the effective force constant of the impurity ion. F was introduced via an expression by Bauer and Magat [26] and modified by Maki and Decius [27] based on Onsager s model of a homogeneous solvent of dielectric constant e with the solute contained in a spherical cavity of radius a. The frequency shift equation is given by... 

The solution flow is nomially maintained under laminar conditions and the velocity profile across the chaimel is therefore parabolic with a maximum velocity occurring at the chaimel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately detemiinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffiision equation for mass transport within the rectangular duct may be described by... 

In the parabolic model the equations for caustics are simply Q+ = Q, and Q- = <2-- The periodic orbits inside the well are not described by (4.46), but they run along the borders of the rectangle formed by caustics. It is these trajectories that correspond to topologically irreducible contours on a two-dimensional torus [Arnold 1978] and lead to the quantization condition (4.47). 

In this case, the bilinear model of eq. (37.10) fits the data as well as the parabolic one. A possible reason for the lack of improvement is that the part of the model which accounts for the higher values of log P is not well covered by the data (Fig. 37.1). The parabolic model yields an optimal value for log P of 5.10 while the optimum of the bilinear model is found at 5.18. 

The theory for classifying linear, second-order, partial-differential equations is well established. Understanding the classification is quite important to understanding solution algorithms and where boundary conditions must be applied. Partial differential equations are generally classified as one of three forms elliptic, parabolic, or hyperbolic. Model equations for each type are usually stated as... 

One-Particle Model with Parabolic Potential-Energy Wells... 

Figure 4.13 Potential energy diagram of the intersection of two parabolic energy wells as a simple model of electron transfer. The reactants R can cross to the products P at the point X of common energy and common coordinate x...

Inside a rectangular well a dipole rotates freely until it suffers instantaneous collision with a wall of the well and then is reflected, while in the field models a continuously acting static force tends to decrease the deflection of a dipole from the symmetry axis of the potential. Therefore, if a dipole has a sufficiently low energy, it would start backward motion at such a point inside the well, where its kinetic energy vanishes. Irrespective of the nature of forces governing the motion of a dipole in a liquid, we may formally regard the parabolic, cosine, or cosine squared potential wells as the simplest potential profiles useful for our studies. The linear dielectric response was found for this model, for example, in VIG (p. 359) and GT (p. 249). 

In our early work33 [50] the constant field model was applied to liquid water, where the harmonic law of particles motion, corresponding to a parabolic potential, was actually employed in the final calculations of the complex permittivity. In this work, qualitative description of only the libration band was obtained, while neither the R-band nor the low-frequency (Debye) relaxation band was described. Moreover, the fitted mean lifetime x of the dipoles, moving in the potential well, is unreasonably short ( ().02 ps)—that is, about an order of magnitude less than in more accurate calculations, which will be made here. 

See description of this collision model in Sections VII.C.2 and VII.C.3 (on the example of a parabolic potential well). 

The main advantage of the hat-curved potential is that it is possible to narrow the width Avor of the librational absorption band by decreasing the form factor /. Indeed, Avor attains its maximum value when/ = 1. Note that / = 1 is just the case of the hat flat or its simplified variant, the hybrid model, both of which were described in Section IV. The latter was often applied before (VIG) and is characterized by a rather wide absorption band, especially in the case of heavy water. In another extreme case, / — 0, the linewidth Avor becomes very low. When / = 0, we have the case of the parabolic potential well, whose dielectric response was described, for example, in GT and VIG. Thus, when the form factor/of the hat-curved well decreases from 1 to 0, the width Avor decreases from its maximum to some minimum value. 

It exhibits maxima at 0 = 0 and 0 = it, with pm being the dimensionless resonance frequency of angular harmonic declinations 0 performed near the bottom of the well. In the EB model, small declinations 0 are assumed. Then Eq. (475) yields the parabolic dependence on 0 ... 

---