Oscillating depth

Figure 14.24 Schematic of a Unimix section a) the section is used in place of a metering section, and it has three channels with oscillating depths, and b) cross-sectional view perpendicular to the flight tips showing channels at a local position (courtesy of Jeffrey A. Kuhman of Glycon Corporation)...

Fig. 3a shows a measurement of the beat structure in the Di-line of cesium. The two hyperfine splitting frequencies of the ground and excited state clearly show up. The fast oscillation corresponds to the splitting of the ground state, while the envelope with the smaller frequency results frran the splitting in the excited state. The difference between the measured and calculated oscillating depths can be explained by the finite pulse duration. 

The potential energy is often described in terms of an oscillating function like the one shown in Figure 10.9(a) where the minima correspond to the relative orientations in which the interactions are most favorable, and the maxima correspond to unfavorable orientations. In ethane, the minima would occur at the staggered conformation and the maxima at the eclipsed conformation. In symmetrical molecules like ethane, the potential function reflects the symmetry and has a number of equivalent maxima and minima. In less symmetric molecules, the function may be more complex and show a number of minima of various depths and maxima of various heights. For our purposes, we will consider only molecules with symmetric potential functions and designate the number of minima in a complete rotation as r. For molecules like ethane and H3C-CCI3, r = 3. 

The simple harmonic oscillator picture of a vibrating molecule has important implications. First, knowing the frequency, one can immediately calculate the force constant of the bond. Note from Eq. (11) that k, as coefficient of r, corresponds to the curvature of the interatomic potential and not primarily to its depth, the bond energy. However, as the depth and the curvature of a potential usually change hand in hand, the infrared frequency is often taken as an indicator of the strength of the bond. Second, isotopic substitution can be useful in the assignment of frequencies to bonds in adsorbed species, because frequency shifts due to isotopic substitution (of for example D for H in adsorbed ethylene, or OD for OH in methanol) can be predicted directly. 

The structures of the solid-melt interface and the melt confined within a narrow gap are of great significance in diverse areas of research such as lubrication, adhesion, or in future nanometer science. It is well recognized that the melt of n-alkanes, and other simple molecules show anomalous oscillations in density, viscosity, etc. vs. depth from the surface showing the presence of marked layer structures in the melt [40]. Even in polymer melts similar layering phenomena were suggested near the solid surface [41], but no pronounced ordering or the onset of crystallization were reported. 

We first examined the density distribution near the solid-melt interface vs. depth from the surface. Figure 19 shows typical density profiles in a relatively small system of 8000 atoms (80 chains of Cioo). It is readily noticed that even at 500 K marked density oscillation is present near the solid surface, though... 

Apart from inversions, there is another way to determine whether or not there is mixing in the Sun. Any spherically symmetric, localized sharp feature or discontinuity in the Sun s internal structure leaves a definite signature on the solar p-mode frequencies. Gough (1990) showed that changes of this type contribute a characteristic oscillatory component to the frequencies z/ / of those modes which penetrate below the localized perturbation. The amplitude of the oscillations increases with increasing severity of the discontinuity, and the wavelength of the oscillation is essentially the acoustic depth of the sharp-feature. Solar modes... 

Other parameters of the simulation are specified in subroutine SPECS. The quantity solcon is the solar constant, available here for tuning within observational limits of uncertainty. The quantity diffc is the heat transport coefficient, a freely tunable parameter. The quantity odhc is the depth in the ocean to which the seasonal temperature variation penetrates. In this annual average simulation, it simply controls how rapidly the temperature relaxes into a steady-state value. In the seasonal calculations carried out later in this chapter it controls the amplitude of the seasonal oscillation of temperature. The quantity hcrat is the amount by which ocean heat capacity is divided to get the much smaller effective heat capacity of the land. The quantity hcconst converts the heat exchange depth of the ocean into the appropriate units for calculations in terms of watts per square meter. The quantity secpy is the number of seconds in a year. 

Na+ concentrations confirms their common marine origin. However, the concentration variations with depth are much less clear than the sinusoidal oscillations at J-9. At Station Q-13, only 70 km... 

In combination with DFT calculations, the time- and depth-dependent phonon frequency allows to estimate the effective diffusion rate of 2.3 cm2 s 1 and the electron-hole thermalization time of 260 fs for highly excited carriers. A recent experiment by the same group looked at the (101) and (112) diffractions in search of the coherent Eg phonons. They observed a periodic modulation at 1.3 THz, which was much slower than that expected for the Eg mode, and attributed the oscillation to the squeezed phonon states [9]. 

One problem with the above control strategy is that when the tank fails there is a rapid change in the flow rate entering the main treatment plant. Modify the program so that the flow rate leaving the tank is controlled such that the flow is proportional to the depth of liquid in the tank. The result of this should be that although oscillations in the flow rate are damped and not eliminated entirely, the risk of complete and sudden failure is reduced. 

As in the previous case of a single oscillator, the analysis of the ground-state energies does not provide information on the values of (3, and rei (i =1,2), but only relates the depths of the potentials to the algebraic parameters. These depths (or dissociation energies) are given by... 

Fig. 5.1 A schematic projection of the 3n dimensional (per molecule) potential energy surface for intermolecular interaction. Lennard-Jones potential energy is plotted against molecule-molecule separation in one plane, the shifts in the position of the minimum and the curvature of an internal molecular vibration in the other. The heavy upper curve, a, represents the gas-gas pair interaction, the lower heavy curve, p, measures condensation. The lighter parabolic curves show the internal vibration in the dilute gas, the gas dimer, and the condensed phase. For the CH symmetric stretch of methane (3143.7 cm-1) at 300 K, RT corresponds to 8% of the oscillator zpe, and 210% of the LJ well depth for the gas-gas dimer (Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M. /. Phys. Chem. A 105, 9284 (2001))...

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