Simple molecular lattice vibrations

The lattice vibrations for the simple tetrahedral lattice wore studied in Section 9-A. The state of the distortion of the lattice was specified by giving the displacement (5r, of each atom. We then made a transformation to normal coordinates u., each corresponding to a normal mode frequency w(k), and these were plotted as a function of k in Fig. 9-2. There were three curves for each atom in the primitive cell. We see immediately that there will be difficulties in complex structures in quartz there arc 27 sets of modes, and oven in the simple molecular lattice there are 9. This complexity suggests that one should proceed by computer. One such approach was taken by Bell, Bird, and Dean (1968). They took a large cluster... 

From Fig. 1 we propose that the water molecule has temporarily tetrahedral-like structure in a short time, because if the water has been constructed by a simple H2O (C2v) molecule there should be only three molecular vibration modes (vi, V2, V3). In Fig. 1 we can see that between 1600 cm l and 4000 cm"l more than three molecular vibrations. They can be classiHed into essentially four kinds molecular vibrations (vi, V2, V3, V4). Besides three or four vibration components in the viAts modes region there exists an extra broad mode at about 2200 cm i. We had better to interpret this spectral pattern as the molecular vibradons of tetrahedral-like C2v symmetry which is composed by two O-H bonds and two 0---H hydrogen bonds in each oxygen. Although the conventional explanation of 2200 cm mode is the combination mode between the molecular vibration V2 and the lattice vibration v, there is no direct experimental evidence. Rather the tetrahedral-like C2v local structure can produce the four molecular vibration modes (Ai, Ai, Bi, B2) in the viA S frequency region and three molecular vibration modes (Ai, Bi, B2) which are bundled in the V4 frequency region. This latter modes correspond to the broad 2200 cm l mode. The above picture is consistent with the pentamer model of liquid water which is stressed in the interpretation of the low-frequency Raman specnal pattern. 

The application of high polymers as insulating materials at low temperatures, as is well known, is limited by their brittleness. Consequently, it is highly desirable to develop methods of checking quickly and simply if a certain polymer can be used at low temperatures or not. In this paper, we propose the use of spectroscopy as a testing procedure. From simple considerations of the molecular structure of polymers, it appears that spectroscopy in the low frequency range of lattice vibrations, i.e., the far infrared (FIR), or the low frequency Raman spectrum, is an appropriate tool for this purpose. 

In a crystal, displacements of atomic nuclei from equilibrium occur under the joint influence of the intramolecular and intermolecular force fields. X-ray structure analysis encodes this thermal motion information in the so-called anisotropic atomic displacement parameters (ADPs), a refinement of the simple isotropic Debye-Waller treatment (equation 5.33), whereby the isotropic parameter B is substituted by six parameters that describe a libration ellipsoid for each atom. When these ellipsoids are plotted [5], a nice representation of atomic and molecular motion is obtained at a glance (Fig. 11.3), and a collective examination sometimes suggests the characteristics of rigid-body molecular motion in the crystal, like rotation in the molecular plane for flat molecules. Lattice vibrations can be simulated by the static simulation methods of harmonic lattice dynamics described in Section 6.3, and, from them, ADPs can also be estimated [6]. 

In molecular crystals or in crystals composed of complex ions it is necessary to take into account intramolecular vibrations in addition to the vibrations of the molecules with respect to each other. If both modes are approximately independent, the former can be treated using the Einstein model. In the case of covalent molecules specifically, it is necessary to pay attention to internal rotations. The behaviour is especially complicated in the case of the compounds discussed in Section 2.2.6. The pure lattice vibrations are also more complex than has been described so far . In addition to (transverse and longitudinal) acoustical phonons, i.e. vibrations by which the constituents are moved coherently in the same direction without charge separation, there are so-called optical phonons. The name is based on the fact that the latter lattice vibrations are — in polar compounds — now associated with a change in the dipole moment and, hence, with optical effects. The inset to Fig. 3.1 illustrates a real phonon spectrum for a very simple ionic crystal. A detailed treatment of the lattice dynamics lies outside the scope of this book. The formal treatment of phonons (cf. e(k), D(e)) is very similar to that of crystal electrons. (Observe the similarity of the vibration equation to the Schrodinger equation.) However, they obey Bose rather than Fermi statistics (cf. page 119). 

The book covers a variety of questions related to orientational mobility of polar and nonpolar molecules in condensed phases, including orientational states and phase transitions in low-dimensional lattice systems and the theory of molecular vibrations interacting both with each other and with a solid-state heat bath. Special attention is given to simple models which permit analytical solutions and provide a qualitative insight into physical phenomena. 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

The temperatureHlepeodence data give effective lattice temperatures of = 166 K and = 85 K, showing how markedly molecular solids differ from simple lattice theories. The four vibrational frequencies of Snl4 arc known from I.R./Raman data to be 47, 63,149, and 216 cm, the tin participating only in the 216 cm mode. Thus in the temperature range used for the measurements ( 80-200 K) the iodine 47 cm (68 K) vibration is almost fully excited whereas the tin 216 cm" (311 K) mode is not. This accounts for the lower lattice temperature of iodine. More detailed molecular-dynamical calculations for SnU have introduced the intermolecular translational and rotational vibrations [100]. 

We have established an effective and simple method for preparing a variety of organic microcrystals in water. The solid-state polymerization of 4BCMU microcrystals was estimated to proceed from one end to the other end. The possibility of preparing PDAs with controlled molecular weight was qualitatively demonstrated. In the case of microcrystals of 14-8ADA, size-dependent conversion is found and can be explained by the looseness or thermal vibrations of the crystal lattice. 

In solid-state physics, the vibrations of very simple lattices containing a small concentration of simple defects have been treated with sophisticated analytical treatments using Green s functions [92]. Even if some authors have bravely tackled an analytical solution of the dynamical problem of disordered polymers [93], they were forced to introduce into the molecule such drastic structural simplifications that the flavor of chemistry has been lost and the theoretical molecular models have become, again, too unrealistic. 

First conclusion. The combination of quasiharmonic lattice dynamics in the quantum regime, for T < 0jjo3> together with molecular dynamics in the classical regime, T > 0j(o3> provides a simple and reasonably accurate representation of the vibrational thermodynamics of a nonquantum solid. 

We consider a mixture of two simple liquids 1 and 2 are small and spherically symmetric and the ratio of their sizes is close to unity. We suppose that the arrangement of the molecules in each pure liquid is that of a regular array all the molecules are situated on lattice points that are equidistant from one another. Molecular motion is limited to vibration about the equilibrium positions and is not affected by the mixing process. We suppose further that for a fixed temperature, the lattice spacing for the two pure liquids and for the mixture is the same, independent of composition. These assumption having been accepted as required, we consider the total number of ways of arranging the Ui identical molecules of the solvent and U2 identical molecules of the solute on the lattice comprising =72 + n. cells. This just the... 

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