Einstein crystal model

Rgure28.7 The Heat Capacity of Diamond Fit to the Einstein Crystal Model Result. The horizontal line corresponds to the law of Dulong and Petit. From J. S. Blakemore, Solid State Physics, 2nd ed.. W. B. Saunders. Philadelphia, 1974, p. 121. 

There is a difficulty with the pressure of the Einstein crystal model. The model does not include any simple way to evaluate the derivative in Eq. (28.2-7e). We might try to evaluate the pressure by finding the difference between G and A, since G = A + PV. For a one-component system, G is given Eq. (26.1-29) as... 

The parameter e has the dimensions of temperature and is called the Einstein temperature or the characteristic temperature. Figure 28.7 shows the heat capacity of diamond as a function of temperature as well as the heat capacity of the Einstein crystal model with an Einstein temperature of 1320K, which gives the best fit to the experimental data. 

This model is a physically motivated improvement over the Einstein crystal model. Debye sought a realistic way to assign different frequencies to the vibrational normal modes. He assumed that the normal modes could be represented by standing waves that vanish at the surfaces of the crystal. The quanta of energy of these waves are called phonons since the waves are essentially sound waves. Consider a cubic crystal with side L. The amplitude of a standing wave that vanishes at the boundaries of the cube is represented by... 

The value of e that fits the Einstein crystal model heat-capacity formula to data for aluminum is 240 K. 

Consider a modified Einstein crystal model with two... 

The integral in the expression (20) can be calculated exactly only in the absence of the frequency dispersion of the phonons, i.e. for Einstein s model of the crystal cos — a>. Then, the expression for the rate constant of multiphonon transition results from the formula (20) ... 

The adsorbed molecules are assumed to vibrate normal to the surface and the interaction with the sohd is presumed to fall off rapidly enough so that only a monolayer exists at low temperatures. The model describes, therefore, only the monolayer adsorption of a 2D fluid on a model solid which is energetically homogeneous and has a plane surface. The partition function is then calculated by considering that there are regions of phase space that give most of the contribution and neglecting the rest of phase space. The molecular partition function for the 2D solidhke structure is obtained for a 2D Einstein crystal, whereas the gas structure is treated as an ideal 2D gas. 

The Einstein and Debye crystal models describe vibrations in crystals. 

The above picture of slowly cooled SCLs allows considering the liquid cell model of Lennard-Jones and Devonshire [34] (Figure 10.1 and its various elaborations [35]. In the figure, we show a cell representation of a dense liquid in (a) and of a crystal in (b). Each cell is occupied by a particle in which the particle vibrates. A defect in the cell representation corresponds to some empty cells. The regular lattice in (b) is in accordance with Einstein s model of a crystal. In the liquid state, this regularity is absent. We consider the conjiguratiorud partition JunctionZ T, V) (Appendix lO.A),... 

Fig,2.11. Dispersion and densities of states for the acoustic and optic modes of the linear NaCl crystal, a) Qualitative behaviour for nearest-neighbour interactions note the Van Hove singularities at the critical points w, o) and a)j . b) Einstein model, c) Debye model, d) Hybride Einstein-Debey model... 

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...

Although Einstein s model correctly captures the heat capacity at the limits of very low and very high temperatures, it agrees poorly with the observed heat capacity of solids for a wide range of temperatures. This is because of the assumption that all vibrations in the crystal have the same frequency. 

P. Debye modified the Einstein s model by introduction of inter-atomic forces in a crystal model. This is equivalent as to take phonons into account (refer to Section 9.3.1). To each elastic wave (phonon) the Bom-Karman atomic chain was attracted spread out in a three-dimensional array (Figure 9.13 and 9.15). As a result of reflection from external crystal borders, standing waves with various values co and k (refer to Section 2.9.2 and 2.9.3) are formed. There is the certain relationship between the wavelength of standing waves X and the size of the crystal L expressed by the eqn (2.9.8). Phase speed of running... 

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

To verify effectiveness of NDCPA we carried out the calculations of absorption spectra for a system of excitons locally and linearly coupled to Einstein phonons at zero temperature in cubic crystal with one molecule per unit cell (probably the simplest model of exciton-phonon system of organic crystals). Absorption spectrum is defined as an imaginary part of one-exciton Green s function taken at zero value of exciton momentum vector... 

This idea that the heat was transfered by a random walk was used early on by Einstein [21] to calculate the thermal conductance of crystals, but, of course, he obtained numbers much lower than those measured in the experiment. As we now know, crystals at low enough T support well-defined quasiparticles—the phonons—which happen to carry heat at these temperatures. Ironically, Einstein never tried his model on the amorphous solids, where it would be applicable in the / fp/X I regime. 

The collective modes of vibration of the crystal introduced in the previous paragraph involve all the atoms, and there is no longer a single vibrational frequency, as was the case in the Einstein model. Different modes of vibration have different frequencies, and in general the number of vibrational modes with frequency between v and v + dv are given by... 

Even though the Einstein and Debye models are not exact, these simple one-parameter models illustrate the properties of crystals and should give reliable estimates of the volume dependence of the vibrational entropy [15]. The entropy is given by the characteristic vibrational frequency and is thus related to some kind of mean interatomic distance or simpler, the volume of a compound. If the unit cell volume is expanded, the average interatomic distance becomes larger and the... 

The important message from Einstein or Debye models is that vibrations of atoms in a crystal contribute to Entropy S and to Heat Capacity C therefore they affect the thermodynamic equilibrium of a crystal by modifying both the Eree energy F, which... 

Since Eq. (5) is equivalent to that of the Einstein model according to which every molecule in the crystal lattice moves freely in the volume Vf of the constant potential [Pg.65]

Table 3.13. Parameters of the temperature dependence of Eq according to the Varshni and Bose-Einstein model for ZnO single crystal bulk samples...

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