Characteristic vibrational energy density

Figure 10.3 Carbide hardnesses vs. characteristic vibrational energy densities derived from average force constants (entropic specific heat). After Grimvall and Theissen (1986). The crystal structures are of the NaCi type. The hardness data are fromTeter (1998).

To rationalize the units, g is divided by the lattice parameter, a of each carbide. The final parameter (g /a) = characteristic vibrational energy density has the units of energy per volume (GPa) which is the same as the hardness units. The correlation of this with hardness is shown in Figure 10.3. The correlation is good especially when it is considered that the hardness numbers for carbides scatter as much as 30 percent. 

Figure 21. Vibrational density of states (pv b) versus vibrational energy (Evib) in anthracene calculated using a direct count method and the frequencies of Refs. 64. Given in the Figure are characteristic times for IVR in the absent, restricted, and dissipative regimes.

The reorganization free energy /.R represents the electronic-vibrational coupling, ( and y are fractions of the overpotential r] and of the bias voltage bias at the site of the redox center, e is the elementary charge, kB the Boltzmann constant, and coeff a characteristic nuclear vibration frequency, k and p represent, respectively, the microscopic transmission coefficient and the density of electronic levels in the metal leads, which are assumed to be identical for both the reduction and the oxidation of the intermediate redox group. Tmax and r max are the current and the overvoltage at the maximum. 

Fourier-transform infrared (FTIR) spectroscopy Spectroscopy based on excitation of vibrational modes of chemical bonds in a molecule. The energy of the infrared radiation absorbed is expressed in inverse centimeters (cm ), which represents a frequency unit. For transition-metal complexes, the ligands -C N and -C=0 have characteristic absorption bands at unusually high frequencies, so that they are easily distinguished from other bonds. The position of these bonds depends on the distribution of electron density between the metal and the ligand an increase of charge density at the metal results in a shift of the bands to lower frequencies. 

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