Vibrational densities

If K is adiabatic, a molecule containing total vibrational-rotational energy E and, in a particular J, K level, has a vibrational density of states p[E - EjiJ,K). Similarly, the transition state s sum of states for the same E,J, and Kis [ -Eq-Ef(J,K)]. The RRKM rate constant for the Kadiabatic model is... 

NIS provides an absolute measurement of the so-called normal mode composition factors that characterize the extent of involvement of the resonant nucleus in a given normal mode. On the basis of the analysis of experimental NIS data, one can therefore construct a partial vibrational density of states (PVDOS) that can be... 

Here, L(v) is a lineshape function that integrates to unity, v is the frequency,/ is the Lamb-Mossbauer factor, and the desired side bands have an area fraction / that is proportional to which hence determines the relative peak heights in a NIS spectrum. More details are provided in Appendix 2 (Part III, 3 of CD-ROM). An equivalent and often more suggestive display of the NIS spectrum is the PVDOS approach, which describes the NIS signal in terms of the partial vibrational density of states ... 

By using NFS, information on both rotational and translational dynamics can be extracted. In many cases, it would be favorable to obtain separate information about either rotational or translational mobility of the sensor molecule. In this respect, two other nuclear scattering techniques using synchrotron radiation are of advantage. Synchrotron radiation-based perturbed angular correlations (SRPAC) yields direct and quantitative evidence for rotational dynamics (see Sect. 9.8). NIS monitors the relative influence of intra- and inter-molecular forces via the vibrational density of states (DOS) which can be influenced by the onset of molecular rotation (see Sect. 9.9.5). 

Fig. 9.41 Partial (vibrational) densities of states (PDOS) at 60 K for [n-Bu4]N2[ Fe4S4(SPh)4] with S solid line) i (Taken from [106])...

An alternative approach is to use the fact that an MD calculation samples the vibrational modes of the polymer for a period of time, f, from 0 to fmax and to calculate from the trajectory, the mass weighted velocity autocorrelation function. Transforming this function from the time domain into the frequency domain by a Fourier transform provides the vibrational density of states g(v). In practice this may be carried out in the following way ... 

In three dimensions, transverse and longitudinal optic and acoustic modes result. The dispersion curve for CuCl along [100] of the cubic unit cell [3] is shown in Figure 8.11(a) as an example. The number of discrete modes with frequencies in a defined interval can be displayed as a function of the frequency. This gives what is termed the density of vibrational modes or the vibrational density of states (DoS). The vibrational DoS of CuCl is given in Figure 8.11(b). 

Figure 8.12 Experimental heat capacity of Cu at constant pressure compared with the Debye and Einstein Cv m calculated by using 0p = 244 K and p> = 314 K. The vibrational density of states according to the two models is shown in the insert.

In contrast to crystalline solids characterized by translational symmetry, the vibrational properties of liquid or amorphous materials are not easily described. There is no firm theoretical interpretation of the heat capacity of liquids and glasses since these non-crystalline states lack a periodic lattice. While this lack of long-range order distinguishes liquids from solids, short-range order, on the other hand, distinguishes a liquid from a gas. Overall, the vibrational density of state of a liquid or a glass is more diffuse, but is still expected to show the main characteristics of the vibrational density of states of a crystalline compound. 

Entropies and heat capacities can thus now be calculated using more elaborate models for the vibrational densities of states than the Einstein and Debye models discussed in Chapter 8. We emphasize that the results are only valid in the quasiharmonic approximation and can only be as good as the accuracy of the underlying force-field calculation of such properties can thus be a very sensitive test of interatomic potentials. 

The superionic phase has been explored with more extensive CPMD simulations.69 Calculated power spectra (i.e., the vibrational density of states or VDOS) have been compared with measured experimental Raman spectra68 at pressures up to 55 GPa and temperatures of 1500 K. The agreement between theory and experiment was very good. In particular, weakening and broadening of the OH stretch mode at 55 GPa was found both theoretically and experimentally. 

Polyakov 1997). Because the second-order Doppler shift is not the only factor controlling Mossbauer absorption frequencies, it is generally necessary to process data taken at a variety of temperatures, and to make a number of assumptions about the invariance of other factors with temperature and the form and properties of the vibrational density of states of the Mossbauer atom. Principles involved in analyzing temperature dependencies in Mossbauer spectra are extensively discussed in the primary literature (Hazony 1966 Housley and Hess 1966 Housley and Hess 1967) and reviews (e.g., Heberle 1971). 

Every example of a vibration we have introduced so far has dealt with a localized set of atoms, either as a gas-phase molecule or a molecule adsorbed on a surface. Hopefully, you have come to appreciate from the earlier chapters that one of the strengths of plane-wave DFT calculations is that they apply in a natural way to spatially extended materials such as bulk solids. The vibrational states that characterize bulk materials are called phonons. Like the normal modes of localized systems, phonons can be thought of as special solutions to the classical description of a vibrating set of atoms that can be used in linear combinations with other phonons to describe the vibrations resulting from any possible initial state of the atoms. Unlike normal modes in molecules, phonons are spatially delocalized and involve simultaneous vibrations in an infinite collection of atoms with well-defined spatial periodicity. While a molecule s normal modes are defined by a discrete set of vibrations, the phonons of a material are defined by a continuous spectrum of phonons with a continuous range of frequencies. A central quantity of interest when describing phonons is the number of phonons with a specified vibrational frequency, that is, the vibrational density of states. Just as molecular vibrations play a central role in describing molecular structure and properties, the phonon density of states is central to many physical properties of solids. This topic is covered in essentially all textbooks on solid-state physics—some of which are listed at the end of the chapter. 

If greater precision is required in describing entropic contributions, the vibrational density of states for each material can be calculated as briefly described in Section 5.5. The entropy due to lattice vibrations can then be calculated. For more details on this concept, see the references associated with Section 5.5. 

Figure 1. Intramolecular vibrational density redistribution IVR of Na3

Figure 4. Tunneling characteristics of an Al-AlOx-4-pyridine-COOH-Ag junction run at 1.4 K with a 1 mV modulation voltage, (a) Modulation voltage Vu across the junction for a constant modulation current Iu. This signal is proportional to the dynamic resistance of the sample, (b) Second harmonic signal, proportional to d2V/dI2. (c) Numerically obtained normalized second derivative signal G , dG/ dfeVJ, which is more closely related to the molecular vibrational density of states, (d) Normalized G0 dG/d(eVJ with the smooth elastic background subtracted out...

Figure 6. Full width at half-maximum of the IETS spectral lines, for an infinitely narrow vibrational density of states, vs. modulation voltage, for two temperatures, and for the counter-electrode (--------) normal or (---) superconducting...

Fig. 8. Difference in the inelastic neutron scattering data between LaFe4Sb 2 and CeFe4Sb 2 vs. energy loss (Keppens et al., 1998). CeFe4Sbi2 was used as a reference compound since the neutron scattering cross section of Ce is much smaller than that of La. The difference spectra therefore reflect the vibrational density of states (DOS) associated with the La atoms. The peak at 7 meV (78 K) corresponds to the quasi-localized La mode. The second broader peak at about 15 meV corresponds to the hybridization of La and Sb vibrational modes. Both peaks can be accounted for using lattice dynamic models based on first-principles calculations (Feldman et al., 2000).

The electronic densities of states (EDOS) calculated for LDA and HDA Si (Fig. 17) confirm that HDA is metallic, as suggested by the experimental results [264]. There is no gap at the Fermi energy in EDOS for HDA, as Fig. 17 shows clearly. The calculated vibrational densities of states (VDOS) are also consistent with the previous experimental results [263, 264]. The LA and LO bands ( 300 and 420 cm-1, respectively) in LDA are broadened, and the TO band ( 500 cm-1) shifts to a lower frequency after the transition to HDA. This results in broad intensity in the range 20CM-50 cm-1 in VDOS for HDA (Fig. 17). The overall profile is consistent with previous experimental findings [263, 264],... 

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