The internal modes

In Table 5.2 we draw together the optical results and the generally accepted assignment scheme for benzene alongside the results of an ab initio calculation. The likely spectrum will then be that produced by programs like ACLIMAX [16]. Immediately we scale the calculated 

Since we have no information on the relative strengths of the internal and external vibrations the simplest isotropic representation is used and its initial strength is estimated ( 2.6.3). The observed INS spectrum is shown in Fig. 5.5(upper), where it is compared with the calculated spectrum, Fig. 5.5(lower), using program default values. 

The first thing to note is that the overall pattern is good, we have obviously measured a benzene spectrum very similar to that calculated. We can immediately conclude that there are no strong intermolecular interactions in the crystal. 

However, the strength of the external modes has been seriously underestimated, the internal mode band origins are too strong and the phonon wings too weak. The external mode contribution parameter is adjusted by eye so that the observed and calculated distribution of intensities more closely resembles one another in the internal mode region. Most of the sharp peaks observed in the INS correspond to 

Recalling Eq. (5.9), on low-bandpass spectrometers the intensity is a slowly varying function of the energy transferred (°cg ). Therefore, small but seemingly arbitrary changes to the ab initio frequencies will have little impact on the calculated ab initio intensities. 

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For many applications, it may be reasonable to assume that the system behaves classically, that is, the trajectories are real particle trajectories. It is then not necessary to use a quantum distribution, and the appropriate ensemble of classical thermodynamics can be taken. A typical approach is to use a rnicrocanonical ensemble to distribute energy into the internal modes of the system. The normal-mode sampling algorithm [142-144], for example, assigns a desired energy to each normal mode, as a harmonic amplitude... 

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-... 

Internal Bend (Figures 4c and 5b). The internal modes are used to identify the adsorbed species, and a comparison with the isolated molecule values indicates to what extent the molecule is affected by the surface. 

Under these conditions, Eq. (32) indicates the maximum extent to which a particular mode p can reduce S(Q,t) as a function of the momentum transfer Q. Figure 10 presents the Q-dependence of the mode contributions for PE of molecular weights Mw = 2000 and Mw = 4800 used in the experiments to be described later. Vertical lines mark the experimentally examined momentum transfers. Let us begin with the short chain. For the smaller Q the internal modes do not influence the dynamic structure factor. There, only the translational diffusion is observed. With increasing Q, the first mode begins to play a role. If Q is further increased, higher relaxation modes also begin to influence the... 

Obviously, in the case of PS these discrepancies are more and more reduced if the probed dimensions, characterized by 2ti/Q, are enlarged from microscopic to macroscopic scales. Using extremely high molecular masses the internal modes can also be studied by photon correlation spectroscopy [111,112], Corresponding measurements show that - at two orders of magnitude smaller Q-values than those tested with NSE - the line shape of the spectra is also well described by the dynamic structure factor of the Zimm model (see Table 1). The characteristic frequencies QZ(Q) also vary with Q3. Flowever, their absolute values are only 10-15% below the prediction. 

NSE measurements at zero average contrast conditions on a symmetric diblock copolymer of H-PS and D-PS dissolved in an appropriate mixture of proto-nated and deuterated benzene are reported [171,172]. The measurements were performed at different concentrations c > c. For comparison, the interdiffusion of a corresponding blend of H-PS and D-PS homopolymers dissolved in deuterated benzene was studied, too [171]. Owing to the relatively low molecular masses, only the regime Q1/2 < 1 was accessible, and the internal modes could not be probed. 

The interfacial zone is by definition the region between the crystallite basal surface and the beginning of isotropy. Due to the conformationally diffuse nature of this region, quantitative contents of the interphase are most often determined by indirect measures. For example, they have been computed as a balance from one of the sum of the fractional contents of pure crystalline and amorphous regions. The analysis of the internal modes region of the Raman spectrum of polyethylene, as detailed in the previous section of this chapter, was used to quantify the content of the interphase region (ab). 

Diphenyl-l,3,4-oxadiazole crystallization revealed two polymorphic forms (centrosymmetric and non-centrosymmetric) of the substance. Raman spectra of both phases recorded between 15 and 1700 cm-1 showed well-resolved internal modes and the external lattice vibrations below 200 cm-1, offering a fast tool for discrimination between different polymorphs. The internal modes were dominated by two groups, one around 1000 cm-1 and the second one between ca. 1500 and 1600 cm-1 <2003JST219>. 

It is often useful to have an approximate relation for VPIE s, especially when complete vibrational analysis is impossible. The AB approximation serves that purpose, and sometimes gives more physical insight than do detailed, but very complicated calculations using Equation 5.24. It is based on the observation that ordinarily condensed phase vibrations fall in two groups the first containing the high frequencies, m > 1 (most often the internal modes, uj = hcvj/kT), where the zero point (low temperature) approximation is appropriate, and... 

The Bartell mechanical model has also been used to estimate the isotope effect on molar volume due to the over all motion (i.e. hindered translation) of molecules interacting in a Lennard-Jones potential. For C6H6/C6D6 one finds AV/V 4 x 10-5, about two orders of magnitude smaller than the contribution of the internal modes (and experiment). We conclude that for all but very light molecules this contribution can be neglected. 

While the NSE results show that, within the experimental accuracy, in the range Q<0.15 A the Rouse model gives a good account for the internal modes as well as for the diffusion of the chain centre of mass, it is also clear that for higher Q-values the experimental structure factors decay significantly more slowly than the Rouse model would require. These deviations are quantified in fitting the Rouse model to the different spectra separately. This procedure results in a strong dispersion of the elementary Rouse rate. The values determined for at Q>0.15 A" follow a Q-dependence, which can be described by the power law ... 

For a molecular crystal, the internal modes tend to be q independent and thus appear as horizontal lines in Fig. 2.1 n is then equal to the number of molecules M in the cell, leading to a considerable simplification. The resulting dynamical matrix has 6M x 6M elements, considering both translational and rotational motions, and atom-atom potential functions may be used for its evaluation. Dispersion curves obtained in this manner for anthracene and naphthalene, are illustrated in Fig. 2.2. 

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. 

A considerable simplification is achieved when molecules can be treated as rigid bodies, as was done for naphthalene and anthracene (Fig. 2.2), the frequency spectra of which were derived using atom-atom potential functions. The mean-square displacements due to the internal modes can be calculated from the experimental infrared and Raman force constants, and added to the values obtained with Eq. (2.58). The rigid-body model for thermal vibrations is further discussed in section 2.3.3. 

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...

Biirgi generalized the model, assuming a temperature independent high frequency term accounting for displacements due to the internal modes. By means of multi-... 

The internal modes are periodic functions of bead position along the chain the characteristic wavelength of internal mode i is N/i. Thus the lower modes govern the large scale motions and have the longest relaxation times. 

The processes observed in the depolarized Rayleigh spectrum correspond to internal modes of motion. Thus, they may have relaxation times which substantially exceed those obtained from the longitudinal or bulk relaxation alone. Nevertheless they are a part of the a relaxation process as it is normally observed in the creep compliance. All processes with the same shift factors make up the full a relaxation. In liquids with substantial depolarized Rayleigh scattering the slowly relaxing part of the W scattering is also dominated by the orientation fluctuations associated with the internal modes of motion. Each internal mode contributes some intensity, but it is believed that fairly short wavelength modes dominate the scattered intensity. 

The fraction, ff, of the available translational energy that appears in the fragments from the channel that produces molecular products (HX) is low, suggesting that a lot of the excess energy remains in the internal modes of the fragments. 

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