Einstein Solid Method

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]. 

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. 

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 ... 

In the case of determining the freezing temperature, a more robust calculation for the solid phase must be completed, as the solid lattice free energy calculation must consider factors, such as the Pauling entropy. For this reason, the latticecoupling-expansion method, which incorporates such factors, is employed for these types of simulations. For the Einstein lattice used in the simulations, a 6 x 4 x 4 unit cell was used, which consists of 768 water molecules. This simulations size was... 

Some general comments might be useful, however, before considering the individual methods. First, the techniques may be divided into (i) macroscopic methods, which are used to measure the effect of long-range motion of atoms and (ii) microscopic methods, which are used to measure the effect of jump frequencies of atoms [210, 212]. In principle, for a simple jump process via point defects in a solid, the two are interconnected by the classical Einstein-Smoluchowski equation [204] ... 

For ionic solids, measurement of the ionic conductivity, <7 , has long provided a method for studying their atomic diffusion [25, 209, 225, 226] (see also Chapter 3). The measurements are usually made with an alternating current (AC) bridge operating at a fixed frequency, f (typically >1 kHz), to avoid polarization effects. The early studies were restricted to measurements on single crystals, and in this case (7i and the tracer diffusion coefficient were seen to be related by the Nernst-Einstein equation [25] ... 

Photoelectron spectroscopy (PES) is a technique based on the photoelectric effect, which was first documented in 1887 by Hertz and explained in 1905 by Einstein. The use of soft x-ray sources led to the development of x-ray photoelectron spectroscopy (XPS), originally known as electron spectroscopy for chemical analysis (ESCA) [1], indicating the applicability of the method to studies of chemical properties. In parallel with the development of XPS, ultraviolet photoelectron spectroscopy (UPS) [2], i.e., PES based on ultraviolet photon sources, emerged as a tool for studying the valence electronic structure of gaseous and solid samples. However, the increasing use of the continuous spectral distribution of synchrotron radiation [3,4] as a photon source has made the historical terminology less... 

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