Curvature oscillation

L.M. Blinov, G. Durand and S.V. Yablonsky, Curvature oscillations and linear electro-optical effect in a surface layer of a nematic liquid crystal, J. Phys. II France 2(5), 1287-1300, (1992). doi 10.1051/jp2 1992200... 

In the microemulsion the role of A is played by the period of damped oscillations of the correlation functions (Eq. (7)). The surface-averaged Gaussian curvature Ky, = 2t x/ S is the topological invariant per unit surface area. Therefore the comparison between Ryyi = Kyy / in the disordered microemulsion and in the ordered periodic phases is justified. We calculate here R= Since K differs for diffused films from cor-... 

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. 

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))...

Figure 2.9. Measured force F (normalized by the mean radius of curvature R of the surfaces) as a function of the surface separation between crossed mica cylinders coated with an adsorbed bUayer of CTAB and immersed in a micellar solution of CTAB (volume fraction of 0.073). In addition to the depletion attractive minimum, two oscillations due to structural forces turn up. (Reproduced from [21], with permission.)...

The first derivatives of a potential energy function define the gradient of the potential and the second derivatives describe the curvature of the energy surface (Fig. 3.4). In most molecular mechanics programs the potential functions used are relatively simple and the derivatives are usually determined analytically. The second derivatives of harmonic oscillators correspond to the force constants. Thus, methods using the whole set of second derivatives result in some direct information on vibrational frequencies. 

FIGURE 4 1.7 K reflectance spectra for GaN epilayers grown on various C-plane AI2O3 and 6H-SiC substrates. The strains are those measured from the radius of curvature at 293 K and corrected to 1.7 K. The data are taken from Skromme [1], who can also fit the evolution of the oscillator strengths with strain using his zero strain value for the A]s exciton line and a crystal field splitting parameter of 3.7 1.4 meV. 

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