Einstein approximation

Comment on the choice of representative values of Vj for the 12 vibrational modes of the crystal. How much would reasonable changes (say, 10 to 20 percent) in these values affect the results of the calculations If possible, conunent on the effect of using the Debye approximation for the acoustic lattice modes instead of the Einstein approximation. 

The calculation of vibration spectra in terms of force constants is similar to the calculation of energy bands in terms of interatomic matrix elements. Force constants based upon elasticity lead to optical modes, as well as acoustical modes, in reasonable accord with experiment, the principal error being in transverse acoustical modes. The depression of these frequencies can be understood in terms of long-range electronic forces, which were omitted in calculations tising the valence force field. The calculation of specific heat in terms of the vibration spectrum can be greatly simplified by making a natural Einstein approximation. 

Distribution of vibralioiial frequencies in GaA.s. The frequencies of the two main peaks arc identified with characteristic frequencies of the spectrum. The figure suggests the Einstein approximation of replacing the distribution by two sharp peaks. [After Dolling and Waugh, 1965, p. 19.]... 

The g((o) can be used to provide a simple approximation to the lattice vibrations, the Einstein approximation. Let us begin by agreeing that single characteristic frequencies, could be chosen to individually represent each of the six types (three translational and three rotational) of external mode. The ( 6)y value is the density of states weighted mean value of all of the frequeneies over which that external mode, j, was dispersed. Normalising as discussed above ... 

We invoke the Einstein approximation, see Eq. (2.57), and assume that (a/Oext is the displacement that stems from Thus there is only a single external vibrational mode and so is isotropic. In a manner analogous to Eq. (2.44) we write ... 

Kamholz A, Yager P (2002) Molecular diffusive scaling laws in pressure-driven microfluidic channels deviation from one-dimensional Einstein approximations. Sens Actuators B Chem 82(1) 117-121... 

The discussion in this section has only been concerned with the enthalpy term. In order to determine the free energy, which is necessary for a calculation of the equilibrium defect concentration, the standard entropy change for the formation of a mole of defects may be estimated as follows. In the simplest case of the Einstein approximation for the limiting case of Dulong-Petit behaviour, the crystal with Nq lattice atoms is considered to be a system of... 

Fig. 43. Upper curve, Debye approximation lower curve, Einstein approximation. 0, cy for silver, dp—215K X, Cp for KCl per gram atom, dp=230 K. N.B. The experimental points have been worked out with the best value of d to fit the Debye curve corresponding points to show the best agreement which can be obtained with the Einstein curve have not been included.

Having adopted this convention we may ask. Does there exist a physical state of a substance for which the conventional entropy is actually zero Now perfect crystals are known to have a very orderly structure, and at very low temperatures the lattice vibrations will all be in their lowest states which correspond to the zero-point energy. Therefore it may be expected that a crystal will have a very low entropy at temperatures approaching the absolute zero, and in one of the original forms (Planck s version) of the third law it was asserted that the entropy of a pure substance is actually zero under such conditions. On the other hand, from (13 51), based on the Einstein approximation, it is seen that... 

Fig.3,9. Phonon dispersion and density of state for a crystal with two atoms in the primitive unit cell, a) Qualitative general behaviour, b) Einstein approximation, c) Debye approximation, and d) Hybride Einstein-Debye model. The corresponding situation for the diatomic linear chain is shown in Fig. 2.11...

At high temperatures (T 0 ) we obtain the classical value (3.64a), namely, C (T) = 3nNkg, while at low temperatures (T 0 ) the Einstein approximation gives... 

These equations allow us to obtain an equation (in Einstein approximation, according to Landau-Lifschitz [248]) describing the time function of viscosity increase Ar (t), which occurs because of the increase of the aggregates size due to the addition of molecules activated by an external influence. 

Expression (2.112) allows us to estimate the aggregate contribution to the viscosity of the system, which is proportional the to overall mass of aggregates according to Einstein approximation ... 

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