External mode

The classification of external modes into librations and translations was ensured by group theoretical considerations and by comparison of expected with observed intensities in the vibrational spectra [107]. In addition, it was... 

Up to now, the intermolecular potential models are only fair in reproducing the wavenumbers of the external modes. Although various refinements have been made, none of the models seems to be superior to the others. More recently developed intermolecular potentials have been applied to structural and thermodynamical studies but not to the analysis of the vibrational spectra [122-125]. 

The presence of isotopic impurities causes clear effects in the vibrational spectra. Almost all modes studied so far show frequency shifts on S/ S substitution [81, 107]. The average shift of the internal modes is ca. 0.6 cm , and of the external modes it is 0.1-0.3 cm (Tables 3, 4 and 5). Furthermore, the isotopomers which are statistically distributed in crystals of natural composition can act as additional scattering centers for the phonon propagation. Therefore, in such crystals the lifetime of the phonons is shortened in comparison with isotopically pure crystals and, as a conse-... 

The pressure dependence of wavenumbers has been investigated theoretically by LD methods on the basis of a Buckingham 6-exp potential. In the studies of Pawley and Mika [140] and Dows [111] the molecules were treated as rigid bodies in order to obtain the external modes as a function of pressure. Kurittu also studied the external and internal modes [141] using his deformable molecule model [116]. The force constants of the intramolecular potential (modified UBFF) were obtained by fitting to the experimental wavenumbers. The results of these studies are in qualitative agreement with the experimental findings. 

FIGURE 7.9 Outcoupling in an OLED. (a) Three radiative modes in an OLED (i) external modes, (ii) substrate modes, and (iii) ITO-organic modes, (b) Attaching a lens to the backside of an OLED converts some of the light from substrate to external modes. 

H20(as) 48> and was assigned to a combination mode of >i with a translational external mode. The intensity of the Raman component at 3360 cm-1 seems too large to be due to such a combination mode. 

Using harmonic oscillator partition functions to describe both internal and external modes, the logarithmic Q ratios introduced above, ln(Qg/Qc/QgQc/) = ln(Qc/QcO + ln(Qg7Qg), become... 

In the condensed phase the sum is over all 3n frequencies, but in the ideal vapor phase the six external (zero) frequencies do not contribute to the IE s, the sum is over the remaining 3n — 6 internals. For condensed rare gases the harmonic assumption is highly approximate, and this is also true for the lattice modes of polyatomics. However as molecular size increases the relative contribution of the external modes becomes less and less important relative to internals. 

For a molecular crystal, the description can be simplified considerably by differentiating between internal and external modes. If there are M molecules in the cell, each with nM atoms, the number of external translational phonon branches will be 3M, as will the number of external rotational branches. When the molecules are linear, only 2M external rotational modes exist. For each molecule, there are 3nM — 6 (3nM — 5 for a linear molecule) internal modes, the wavelength of which is independent of q. Summing all modes gives a total number of N M(3nM — 6) + 6M = 3nN, as required, because each of the modes that have been constructed is a combination of the displacements of the individual atoms. 

As the oscillators of the OPP model vibrate independently of each other, the frequencies are dispersionless, that is, independent of a wavevector q. For the internal modes of a molecular crystal, this tends to be a very good approximation. For the external modes, the dispersion can be pronounced, as shown in Figs. 2.1 and 2.2. In order to obtain the mean-square vibrational amplitudes for the latter, a summation over all phonon branches in the Brillouin zone must be performed. 

In molecular crystals, the separation between internal and external modes is of importance. Except for torsional oscillations in some types of molecules, the internal modes have much higher frequencies than the external modes. According to expressions such as Eqs. (2.51) and (2.58), the latter are then the dominant... 

Fig. 3 Experimental heat capacities of benzene [11], Cv is obtained from observed Cp after subtracting the expansion work, computed using the experimentally determined bulk modulus. The Cv estimated from molecular translational and librational lattice modes (obtained from neutron diffraction ADP s) is also plotted. Note that these external modes well reproduce the observed Cv up to ca. 100 K. Above this temperature the internal modes are active and Cv exceeds the classical limit of 3 k T...

Insert the sample probe in the MALDI-TOF mass spectrometer. Calibration is performed in external mode with peptides covering the mass range of 500 Da to 5 kDa (see Note 15). 

Similar results were found by Komeda et al. [51] and Pascual et al. [43] for the diffusion CO and NH3 respectively. In both cases, molecular motion was activated by excitation of internal stretch modes. By modeling theoretically the coupling between internal and external modes, these works gave a magnitude for the relevance of such internal pathways compatible with experimentally measured reaction yields. 

Fig. 4 Phonon density of states at 180 K for the low-frequency modes of C60 at atmospheric pressure and at 0.5 GPa. Reprinted with permission from H Schober and B Renker, Pressure dependence of the external mode spectrum of solid C60 , Phys. Rev. B vol. 59 (1999) 3287-90 [39]. Copyright 1999 The American Physical Society...

The optical spectral region consists of internal vibrations (discussed in Section 1.13) and lattice vibrations (external). The fundamental modes of vibration that show infrared and/or Raman activities are located in the center Brillouin zone where k = 0, and for a diatomic linear lattice, are the longwave limit. The lattice (external) modes are weak in energy and are found at lower frequencies (far infrared region). These modes are further classified as translations and rotations (or librations), and occur in ionic or molecular crystals. Acoustical and optical modes are often termed phonon modes because they involve wave motions in a crystal lattice chain (as demonstrated in Fig. l-38b) that are quantized in energy. 

The unit cell is 2(7 ). The two La atoms sit on a C3h site, and the six chlorine atoms are on a Cs site (see Appendix 4). Since the Hermann-Mauguin nomenclature cites that the unit cell is primitive (Pb6 /m) we need not reduce it. For the two La atoms there are six degrees of freedom (3n,Z ) = 3 x 1 x 2 = 6. The six Cl atoms possess 18 degrees of freedom (3 , Z ) = 3x3x2=18. Since all vibrational modes can be considered external modes, we need only correlate the site group to factor group. For the... 

The external modes are determined as for the LaCl3 case by correlating the site group — factor group. 

---