Lattice modes of vibration

The vibrations of molecules in crystals can be divided into two groups. The external or lattice modes of vibration are oscillations in which the entire (rigid) molecule moves, and, in so doing, involves neighboring molecules. The internal or molecular modes of vibration are the... 

External (or lattice) modes of vibration Modes of vibration involving the entire molecule moving with respect to others in the crystal structure. 

Calculated phonon spectrum for phase I NH3 showing (a) the internal modes of vibration and (b) the external lattice modes of vibration. Note the calculated band intensities are not indicative of the experimental values. Adapted with permission from [10]. Copyright 2008 American Chemical Society. 

The vibrational motions of a crystalline polymer may be considered as having two origins, internal and lattice. Lattice modes of vibration are those due to polymer chains moving relative to each other and occur at low wavenumbers, generally below 150cm (above 66.67 pm). Internal vibrational modes are those due to the motions of the atoms of a chain relative to each other and, in general, these occur in the region 4000-150 cm (5.00-66.67 pm). 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

The observation of most of the lines reported in Table IV is not correlated with the doping of the material. At least two possibilities exist to explain them they are either due to the neutralization by hydrogen of accidental impurities or to the local mode of vibration of hydrogen at a lattice defect site. 

Because of the lattice damage, the absorptions due to the local modes of vibration are usually broader in implanted materials than, for instance, in plasma diffused samples. For proton energies around 1 MeV, the line-widths are in the range 5-100 cm-1 (as compared with 0.1-5 cm-1 for plasma hydrogenation). 

Table V summarizes all the sharp absorptions due to local modes of vibration in proton and deuteron implanted GaP, GaAs and InP. It has to be noted that the results depend upon the reports. For instance, for GaP implanted with protons, Newman and Woodhead (1980) observed only one line at 1849 cm-1 whereas Sobotta et al. (1981) observed only one line at 2204 cm-1. These differences probably come from the differences in implantation conditions. However, unfortunately, these conditions are not always well described in the literature the ion energy and dose are usually given, but the ion current is specified only by Tatarkiewicz et al. (1987, 1988). This parameter is of importance as it contributes to control local temperature and therefore the defect creation and the binding of hydrogen to the lattice. 

Figure 5.4. Resonance Raman spectra of [Fe2S2] centers in A. vinelandii. A, NifU as isolated. B, D37A NifU-1 as isolated. C, NifU-1 repnrified after NifS-mediated cluster assembly. D, IscU containing two [Fc2S2] clnsters per dimer pnrified fraction after IscS-mediated clnster assembly. All spectra were recorded nsing 457-nm excitation at 17 K and with 6cm resolntion. Vibrational modes resnlting from lattice modes of ice have been snbtracted.

Lattice vibrations are described as follows [9, 10]. If the deviation of an atom from its equilibrium position is u, then <(u2> is a measure for the average deviation of the atom (the symbol < ) represents the time average note that <(u> = 0). This so-called mean-squared displacement depends on the solid and the temperature, and is characteristic for the rigidity of a lattice. Lattice vibrations are a collective phenomenon they can be visualized as the modes of vibration... 

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 vibrational frequencies of the so-called lattice modes of aluminosilicate zeolites (stretching and bending modes of the T-0 linkages, plus specific vibrations of discrete structural units) were first studied in detail by Flanigen more than 20 years ago [21], The lattice modes are sensitive to both the composition of the lattice and the structure. For example, Jacobs et al. showed that the T-0 stretching... 

In the weakly anharmonic molecular crystal the natural modes of vibration are collective, with each internal vibrational state of the molecules forming a band of elementary excitations called vibrons, in order to distinguish them from low-frequency lattice vibrations known as phonons. Unlike isolated impurities in matrices, vibrons may be studied by Raman spectroscopy, which has lead to the establishment of a large body of data. We will briefly attempt to summarize some of the salient experimental and theoretical results as an introduction to some new developments in this field, which have mainly been incited by picosecond coherent techniques. 

The unit cell group description of the normal modes of vibration within a unit cell, many of which are degenerate, given above is adequate for the interpretation of IR or Raman spectra. The complete interpretation of vibronic spectra or neutron inelastic scattering data requires a more generalized type of analysis that can handle 30N (N=number of unit cells) normal modes of the crystal. The vibrations, resulting from interactions between different unit cells, correspond to running lattice waves, in which the motions of the elementary unit cells may not be in phase, if ky O. Vibrational wavefunctions of the crystal at vector position (r+t ) are described by Bloch wavefunctions of the form [102]... 

Translational (lattice) modes of hydroxides display TO/LO splittings in the same order of magnitude as other salt-like compounds. Those of pure OH -type vibrations are small [60]. 

---