Collective lattice vibrations

This model, the Einstein model for heat capacity, predicts that the heat capacity is reduced on cooling and that the heat capacity becomes zero at 0 K. At high temperatures the constant-volume heat capacity approaches the classical value 3R. The Einstein model represented a substantial improvement compared with the classical models. The experimental heat capacity of copper at constant pressure is compared in Figure 8.3 to Cy m calculated using the Einstein model with 0g = 244 K. The insert to the figure shows the Einstein frequency of Cu. All 3L vibrational modes have the same frequency, v = 32 THz. However, whereas Cy m is observed experimentally to vary proportionally with T3 at low temperatures, the Einstein heat capacity decreases more rapidly it is proportional to exp(0E IT) at low temperatures. In order to reproduce the observed low temperature behaviour qualitatively, one more essential factor must be taken into account the lattice vibrations of each individual atom are not independent of each other - collective lattice vibrations must be considered. 

A first impression of collective lattice vibrations in a crystal is obtained by considering one-dimensional chains of atoms. Let us first consider a chain with only one type of atom. The interaction between the atoms is represented by a harmonic force with force constant K. A schematic representation is displayed in Figure 8.4. The average interatomic distance at equilibrium is a, and the equilibrium rest position of atom n is thus un =na. The motion of the chain of atoms is described by the time-dependent displacement of the atoms, un(t), relative to their rest positions. We assume that each atom only feels the force from its two neighbours. The resultant restoring force (F) acting on the nth atom of the one dimensional chain is now in the harmonic approximation... 

Phonon Collective lattice vibration Neutral Raman, IR, photoluminescence... 

When water condenses into the liquid or crystalline state, one mole occupies only about 0.018 liters, and this is because a cohesive force (Chapter 4) holds the molecules together much more tightly than in the gas phase. The internal energy now has a substantial contribution from the intermolecular potential energy. In addition, inter-molecular vibrational degrees of freedom provide a large number of extra pockets into which potential energy can be stored in the crystalline state, molecules oscillate in collective lattice vibrations (Section 6.3). 

The Bragg scattering of X-rays by a periodic lattice in contrast to a Mossbauer transition is a collective event which is short in time as compared to the typical lattice vibration frequencies. Therefore, the mean-square displacement (x ) in the Debye-Waller factor is obtained from the average over the ensemble, whereas (r4) in the Lamb-Mossbauer factor describes a time average. The results are equivalent. 

The procedure of Lifson and Warshel leads to so-called consistent force fields (OFF) and operates as follows First a set of reliable experimental data, as many as possible (or feasible), is collected from a large set of molecules which belong to a family of molecules of interest. These data comprise, for instance, vibrational properties (Section 3.3.), structural quantities, thermochemical measurements, and crystal properties (heats of sublimation, lattice constants, lattice vibrations). We restrict our discussion to the first three kinds of experimental observation. All data used for the optimisation process are calculated and the differences between observed and calculated quantities evaluated. Subsequently the sum of the squares of these differences is minimised in an iterative process under variation of the potential constants. The ultimately resulting values for the potential constants are the best possible within the data set and analytical form of the chosen force field. Starting values of the potential constants for the least-squares process can be derived from the same sources as mentioned in connection with trial-and-error procedures. 

As a rule, the density of states for molecular lattice vibrations is negligible as compared to that for crystal phonons. Therefore, the K-mode of a molecular lattice is coupled with the crystal phonons specified by the same wave vector K. Besides, the low-frequency collective mode m of adsorbed molecules can be considered as a... 

In the previous sections, we have considered that the optical center is embedded in a static lattice. In our reference model center ABe (see Figure 5.1), this means that the A and B ions are fixed at equilibrium positions. However, in a real crystal, our center is part of a vibrating lattice and so the environment of A is not static but dynamic. Moreover, the A ion can participate in the possible collective modes of lattice vibrations. 

Ions in the lattice of a solid can also partake in a collective oscillation which, when quantized, is called a phonon. Again, as with plasmons, the presence of a boundary can modify the characteristics of such lattice vibrations. Thus, the infrared surface modes that we discussed previously are sometimes called surface phonons. Such surface phonons in ionic crystals have been clearly discussed in a landmark paper by Ruppin and Englman (1970), who distinguish between polariton and pure phonon modes. In the classical language of Chapter 4 a polariton mode is merely a normal mode where no restriction is made on the size of the sphere pure phonon modes come about when the sphere is sufficiently small that retardation effects can be neglected. In the language of elementary excitations a polariton is a kind of hybrid excitation that exhibits mixed photon and phonon behavior. 

For intramolecular vibrations, each site was considered independently. However, the reorganizations in the surrounding solvent are necessarily properties of both sites since some of the solvent molecules involved are shared between reactants. The critical motions in the solvent are reorientations of the solvent dipoles. These motions are closely related to rotations of molecules in the gas phase but are necessarily collective in nature because of molecule—molecule interactions in the condensed phase of the solution. They have been treated theoretically as vibrations by analogy with lattice vibrations of phonons which occur in the solid state.32,33... 

In addition to the individual and uncorrelated particle motions, we also have collective ones. In a strict sense, the hopping of an individual vacancy is already coupled to the correlated phonon motions. Harmonic lattice vibrations are the obvious example for a collective particle motion. Fixed phase relations exist between the vibrating particles. The harmonic case can be transformed to become a one-particle problem [A. Weiss, H. Witte (1983)]. The anharmonic collective motion is much more difficult to treat theoretically. Correlated many-particle displacements, such as those which occur during phase transformations, are further non-trivial examples of collective motions. 

In atomic-molecular media the damping of plasmon states is due to the interaction of plasmon waves with electrons, lattice vibrations, and impurities. The electron-plasmon interaction is a long-range one. With absorption of a plasmon, the momentum q is transferred to the electron, resulting in a decay of the collective state into a single-particle one. The latter process is identical with absorption of a photon with the same energy. Wolff102 (see also Ref. 103) has shown that in this case the lifetime can be expressed in terms of two optical constants the absorption coefficient k and the refractive index nT, namely,... 

Inelastic neutron scattering (INS) measurements have been successfully used to study dynamical phenomena such as molecular or lattice vibrations in pristine C60 [43] and a variety of fullerides [44-48]. When INS spectra are collected on instruments with a large energy window, it is possible to observe all phonon modes including the molecular vibrations and the generalised phonon density-of-states (GDOS) can be directly calculated. 

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... 

Mossbauer s discovery [49] consisted in the fact that when the nuclei of the emitter and the absorber are included in a solid matrix, they vibrate in a crystal lattice [49,54,56], Therefore, owing to the essential quantum character of solid vibrations (see Section 1.4), the atoms located in a solid matrix are limited to a certain collection of quantized lattice vibration energies [54], Consequently, if the recoil energy is smaller than the lowest quantized lattice vibration energy, Ew, then / v = 0D, in which, k is the Boltzmann constant and 0D is the Debye temperature of the solid. In this case, this... 

Lattice vibrations (Phonons) The vibrations of a crystal are classically described in terms of collective motions in the form of waves called lattice vibrations. 

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. 

Molecular dynamics examines the temporal evolution of a collection of atoms on the basis of an explicit integration of the equations of motion. From the point of view of diffusion, this poses grave problems. The time step demanded in the consideration of atomic motions in solids is dictated by the periods associated with lattice vibrations. Recall our analysis from chap. 5 in which we found that a typical period for such vibrations is smaller than a picosecond. Hence, without recourse to clever acceleration schemes, explicit integration of the equations of motion demands time steps yet smaller than these vibrational periods. 

As far as we know, application of the MD method to ice is based on the lattice-vibration concept. The theoretical spectra represent chaotic collections of peaks [6-8], while the experimental FIR spectra of permittivity g(v) and absorption a(v) are similar to such spectra in liquid water and therefore are rather smooth. 

In 1912 Bom and von Karman [1, 2] proposed a model for the lattice dynamics of crystals which has become the standard description of vibrations in crystals. In it the atoms are depicted as bound together by harmonic springs, and their motion is treated collectively through traveling displacement waves, or lattice vibrations, rather than by individual displacements from their equilibrium lattice sites [3]. Each wave is characterized by its frequency, wavelength (or wavevector), amplitude and polarization. 

The Einstein model gives a good qualitative agreement with the real behavior of solids, but the quantitative agreement is poor (Cranshaw et al. 1985). A more realistic representation of a solid is given by the Debye model. The model describes the lattice vibration of solids as a superposition of independent vibrational modes (i.e., collective wave motion of the lattice, associated with phonons ) with different frequencies. The (normalized) density function p(co) of the vibrational frequencies is monotonically increasing up to a characteristic maximum of cod, where it abruptly drops to zero (Kittel 1968) ... 

Elementary excitations ( dynamics ) in condensed matter systems maybe studied by inelastic neutron scattering. A large proportion of solid-state physics concerns dynamical phenomena ( collective excitations ) in crystalline materials, such as lattice vibrations (phonons), or spin waves (magnons) (see, e.g., Aschroft and Mermin 1976). These phenomena have been... 

In molecular solids the molecules cannot move around freely, but they are trapped in relatively deep potential wells, caused by the intermolecular potential. In these wells they can vibrate and since the vibrations of individual molecules are coupled, again by the intermolecular potential, one obtains collective vibrations of all the molecules in the solid, called lattice vibrations or phonons. Phonons associated with the center of mass motions of the molecules are called translational phonons, phonons associated with their hindered rotations or librations are called librons. The degree of hindrance of the rotations may vary. If the molecules have well-defined equilibrium orientations and perform small amplitude librations about these, one speaks about ordered phases. If the molecular rotations are nearly free or if the molecules can oscillate in several orientational pockets and easily jump between these pockets, then the solid is called orientationally disordered or plastic. Several molecular solids may occur in each of these phases, depending on the temperature and pressure they undergo order/disorder phase transitions. Also the intramolecular vibrations are coupled by the intermolecular potential, via its dependence on the internal coordinates. The excitations of the solid associated with such vibrations are called vibrational excitons or vibrons. 

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