Einstein solid

Results of the next section indicate that the three-dimensional lattice acts much like an Einstein solid with a single phonon frequency vE. Viscoelastic experiments (5) on rosin in the glassy state are consistent with the sharp distribution of relaxation times thus predicted. In the dielectric case where each oscillator contains a dipole,... 

At the other extreme we might expect a fluid to have some characteristic of a simple Einstein solid, i.e., a collection of independent oscillators each oscillating at the same frequency %. The linear momentum correlation function and its memory would then simply be... 

We shall consider what is arguably the archetypal example of the NIRM strategy the Einstein Solid Method (ESM) [28]. The ESM provides a simple way of computing the free energies of crystalline phases and therefore addresses questions of the relative stability of competing crystalline structures. We describe its implementation for the simplest case where the interparticle interaction is of hard-sphere form it is readily extended to deal with particles interacting through soft potentials [29]. 

Recalling the definition of the specific heat as the temperature derivative of the average energy (see eqn (3.99)), it is found that for the Einstein solid... 

The ideal system is such that it connects to the real system without going through a phase transition, and its Helmholtz free energy, A(X = 0), must be known at as a function of temperature and pressure. The ideal systems usually chosen for the real liquid and solid are, respectively, the ideal gas and ideal Einstein solid. In order to evaluate the integral in Eq. 

The simplest model of thermal diffuse scattering is one where the crystal is considered to be an Einstein solid . In this approximation all the atoms are vibrating independently as isotropic harmonic oscillators, each atom having the same mean square displacement U. For this model... 

The details of lattice vibrations will not be discussed here. But for the sake of discussion, the main results of one of the simpler models, namely, the Einstein solid, are given below without proof. By assuming the solid to consist of Avogadro s number jVav of independent harmonic oscillators, all oscillating with the same frequency Einstein showed that the thermal entropy per mole is given by... 

If we adopt the simple picture of an Einstein solid such that there is one frequency only given by oje then... 

An Einstein solid is one in which all of the atoms are treated as three-dimensional harmonic oscillators with frequency v. Thus for loonatomic Einstein solids, equation ( ) becomes... 

It is indeed interesting to see that there is a perfect compatibility of the macroscopic definitions of pressure, temperature and even entropy, with the microscopic ones, as may be seen at an extremely elementary level, using a model as the Einstein solid for accounting for the distribution of energy (Costa Pereira, 1990e). 

Slack [25] and Cahill et al. [26] explored the theoretical limits on k for solids within a phonon model of heat transport. Their work utilized the concept of the minimum thermal conductivity, Kj n- At this minimum value the mean free path for all heat carrying phonons in a material approaches the phonon wavelengths [25]. In this limit, the material behaves as an Einstein solid in which energy transport occurs via a random walk of energy transfer between localized vibrations in the solid. Experimentally, K an is often comparable to the value in the amorphous state of the same composition. In principle jc in can be achieved by the introduction of one or more phonon scattering mechanisms that reduce the phonon mean free path to its minimum value over a broad range of frequencies, and therefore reduces Kl over a broad range of temperatures. In practice, there are relatively few crystalline compounds for which this limit is approached. 

Dynamics. There are calculations in which the metal is modeled as an Einstein solid with harmonic vibrations[33j. When surface molecules and ions are strongly adsorbed molecular dynamics becomes an inefEcient way to study surface processes due to the slow exchange between surface and solution. In this case it is possible to use umbrella sampling to compute distribution profiles[34, 35]. Recently the idea underlying Car-Parrinello was used for macroion dynamics[36, 37] in which the solvent surrounding charged macroions is treated as a continuum in a self consistent scheme for the potential controlling ion dynamics. Dynamical corrections from the solvent can be added. There is a need to develop statistical methods to treat the dynamics of complex objects that evolve on several different time scales. 

The Debye model is more appropriate for the acoustic branches of tire elastic modes of a hanuonic solid. For molecular solids one has in addition optical branches in the elastic wave dispersion, and the Einstein model is more appropriate to describe the contribution to U and Cj from the optical branch. The above discussion for phonons is suitable for non-metallic solids. In metals, one has, in addition, the contribution from the electronic motion to Uand Cy. This is discussed later, in section (A2.2.5.6T... 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

The final technique addressed in this chapter is the measurement of the surface work function, the energy required to remove an electron from a solid. This is one of the oldest surface characterization methods, and certainly the oldest carried out in vacuo since it was first measured by Millikan using the photoelectric effect [4]. The observation of this effect led to the proposal of the Einstein equation ... 

Supercritical Mixtures Dehenedetti-Reid showed that conven-tionaf correlations based on the Stokes-Einstein relation (for hquid phase) tend to overpredict diffusivities in the supercritical state. Nevertheless, they observed that the Stokes-Einstein group D g l/T was constant. Thus, although no general correlation ap es, only one data point is necessaiy to examine variations of fluid viscosity and/or temperature effects. They explored certain combinations of aromatic solids in SFg and COg. 

The oxide solid elecU olytes have elecuical conductivities ranging from lO Q cm to 10 cm at 1000°C and these can be converted into diffusion coefficient data, D, for die oxygen ions by the use of the Nernst-Einstein relation... 

If we postulate diat die chemical potentials of all species are equal in two phases in contact at any interface, dieii Einstein s mobility equation may be simply applied, in Pick s modified form, to describe die rate of a reaction occun ing dirough a solid product which separates die two... 

When Max Planck wrote his remarkable paper of 1901, and introduced what Stehle (1994) calls his time bomb of an equation, e = / v , it took a number of years before anyone seriously paid attention to the revolutionary concept of the quantisation of energy the response was as sluggish as that, a few years later, whieh greeted X-ray diffraction from crystals. It was not until Einstein, in 1905, used Planck s concepts to interpret the photoelectric effect (the work for which Einstein was actually awarded his Nobel Prize) that physicists began to sit up and take notice. Niels Bohr s thesis of 1911 which introduced the concept of the quantisation of electronic energy levels in the free atom, though in a purely empirical manner, did not consider the behaviour of atoms assembled in solids. 

Following the general trend of looldng for a molecular description of the properties of matter, self-diffusion in liquids has become a key quantity for interpretation and modeling of transport in liquids [5]. Self-diffusion coefficients can be combined with other data, such as viscosities, electrical conductivities, densities, etc., in order to evaluate and improve solvodynamic models such as the Stokes-Einstein type [6-9]. From temperature-dependent measurements, activation energies can be calculated by the Arrhenius or the Vogel-Tamman-Fulcher equation (VTF), in order to evaluate models that treat the diffusion process similarly to diffusion in the solid state with jump or hole models [1, 2, 7]. 

An estimation of the multiphase viscosity is a preliminary necessity for convenient particle processing. For particle-doped liquids the classical Einstein equation [20] relates the relative viscosity to the concentration of the solid phase ... 

The value of the first coefficient b, for the dispersion of spherical particles is well known and generally accepted. This is Einstein coefficient b, = 2.5, taking into account the viscosity variation of the dispersion medium upon introducing noninteracting solid particles of spherical form into it. Thus, for tp [Pg.83]

Einstein (f,) remarked that this point of view can be carried over to the theory of the energy content of a solid body if we suppose that the positive ions of Drude s theory ( 198) may be looked upon as the vibrating resonators, and the seat of the heat content of the body (Korperwarme). He calculated the expression ... 

According to Joule s law ( 9), the molecular heat of a compound is the sum of the atomic heats of its components, and since this holds good even when the atomic heats are irregular, i.e., not equal to 6 4, it seems that the heat content of a solid resides in its atoms, and not in the molecular complexes as such. This agrees with Einstein s theory. Hence the molecular heat of a compound should be calculable by means of the formula ... 

A solid body, the molecules of which are monatomic, and all vibrating with a constant frequency, v, is called an Einstein s solid, since it formed the subject of Einstein s application of the theory of ergonic distribution considered in 222—24. The equation for the vibrational energy... 

But at low temperatures, equation (10.148) does not quantitatively predict the shape of the CV. m against T curve. For example, Figure 10.12 compares the experimental value of CV m for diamond with that predicted from equation (10.148). It is apparent that the Einstein equation predicts that CV.m for diamond will decrease too rapidly at low temperatures. Similar results would be obtained for Ag and other atomic solids. 

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