Librational frequencies

The interpretation of the lattice vibrations for scheeUte type molybdates or tungstates with relatively light cations, Ca or Sr, has indicated that the lowest translational vibrations are produced by Mo—Mo or W—W motions respectively, while those at higher frequency are from cation-cation motions 98). This has not been found, however, in the case of the barium or lead compounds. The librational frequencies have been found to decrease hnearly with the ionic radius of the cation for AMO4 type compounds, where A = Ca, Sr, Ba, or Pb and M =Mo or W 98). 

The period of librations Tub = 27ir Alibl corresponds to the estimated librational frequency. 

TABLE 5. B3LYP (BP) harmonic librational frequencies (w in cm"l) for the minima ofthe... 

In He4 C84, the He4 vibrations with frequencies of 175-215 cm-1 are virtually not coupled to the cage vibrations and are roughly half as large as those in He4 C6oH36 (Table 7). The translational and librational frequencies of He4 inside the Cs4 cage, 90-95 cm-1, are reduced by a factor of... 

As compared to CO2 and BeF2, the translational frequencies for the bulkier guest molecules increase (50 and 175 cm-1 for Si02,173 and 229 cm-1 for CS2) and are coupled to deformations of the XMX bond angle. The librational frequency e [ for CS2 is considerably lower than its translational frequencies, whereas, for SiC>2, e [ is intermediate between e and a ... 

Falk and Giguere frequencies the librational frequencies of H30+ were assumed to be the same as those in water. When using the Rudolph and Zimmermann frequencies the cited values were used for the libra-tions. In both cases the restricted translations were assumed to be isotopically insensitive. The even more simplified approximations of Bunton and Shiner (1961a) give a value of 0-73 for l. 

The librational frequencies are much higher than the NQR frequencies ... 

For further confirmation of the mode-softening and a possible identification of the molecular nature of the over-damped mode, we used the rigid-body motion analysis of the thermal- parameters of the room temperature x-ray diffraction study. A thermal-motion analysis (TMA) program was used to calculate the components of the librational (L) and the translational (T) tensors with a least-square fit of the published thermal parameters ( ) of all nonhydrogen atoms of the molecule. The librational frequencies were calculated by the method of Cruickshank (7), using the appropriate eigenvalues of the L-tensor and the corresponding moments of inertia. 

This potential was subsequently used in self-consistent phonon lattice dynamics calculations [115] for a and y nitrogen crystals. And although the potential—and its fit— were crude by present day standards, lattice constants, cohesion energy and frequencies of translational phonon modes agreed well with experimental values. The frequencies of the librational modes were less well reproduced, but this turned out to be a shortcoming of the self-consistent phonon method. When, later [ 116,117], a method was developed to deal properly with the large amplitude librational motions, also the librational frequencies agreed well with experiment. 

Such a decrease of the librational frequency is demonstrated in Section IV in a wide temperature interval 20-60°C (except the break-of-continuity point near 27°C). 

We now examine actual instances. In Table I we have listed the parameters pertinent to the preceding discussion for a number of solids the lowest two librational frequencies are considered. We see that COj, NjO, Clg, C2N3, OCS, and benzene all have k larger than 10(X), and many of them several times that value. We conclude therefore that the harmonic oscillator result is probably quite accurate for the librations in these solids. On the other hand, a-Ng, CO, CgH, and the hydrogen halides (HCl, HBr) have low values of k and therefore the harmonic oscillator results may be inaccurate. It is difficult, on the other hand, to assess the errors... 

Librational Frequencies and Intensity Ratios for a-N [Frequencies are in units of cnv the... 

It appears at this time that there is disagreement over the librational frequency assignment in a-Na- From a purely experimental point of view, the discrepancy between the observations of Brith, Ron, and Schnepp (1969) and Cahill and Leroi (1969) on the one hand and those of Anderson, Sun, and Donkersloot (1970) on the other cannot be resolved. This is due to the fact that the method of sample preparation was different, which could have led to small differences in the observations. Anderson investigated a solid layer prepared by vapor deposition on a cold metal surface, whereas the other workers used solids prepared by cooling liquid in closed cells. Strains in solids prepared by nonequilibrium means could cause the appearance of weak Raman lines and the line in dispute is... 

In addition, serious basic theoretical problems are encountered with the application of a harmonic expansion of the potential in terms of angular displacements to the calculation of the librational frequencies of a-Na (Section IJC.2). It is expected that quantum mechanical calculations now in progress (Jacobi and Schnepp, 1971) will clarify this problem. Further experimental work also is required to settle the question of the dependence of the intensity of the 60 cm Raman line reported by Anderson and co-workers on sample preparation. Only such additional work can contribute further to the assignments of the q = 0 librational frequencies. 

The rotational potential functions for [BHJ" ions embedded in alkali metal halides have been derived from atom-atom potentials. The results allow the computation of librational frequencies of the anion in alkali metal matrices and they compare well with experimental data [12]. 

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