The Einstein model

The decrease in the heat capacity at low temperatures was not explained until 1907, when Einstein demonstrated that the temperature dependence of the heat capacity arose from quantum mechanical effects [1], Einstein also assumed that all atoms in a solid vibrate independently of each other and that they behave like harmonic oscillators. The motion of a single atom is again seen as the sum of three linear oscillators along three perpendicular axes, and one mole of atoms is treated by looking at 3L identical linear harmonic oscillators. Whereas the harmonic oscillator can take any energy in the classical limit, quantum theory allows the energy of the harmonic oscillator (en) to have only certain discrete values ( )  

The probability that an oscillator at a given temperature occupies a given energy state, en, is given by Bose-Einstein statistics (see e.g. C. Kittel and H. Kroemer, Further reading) and the mean value of n at a given temperature is given by 

In the Einstein model, all the independent oscillators have the same angular frequency, coe, and the average total internal energy is 

The heat capacity is derived by differentiation with respect to temperature  

Let us describe the displacement of the nth atom from its equilibrium rest position by a cosine-wave with amplitude w0, angular frequency x and wave vector q = 2n/X 1 

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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... 

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... 

If optical phonons are responsible for the Raman processes, the Einstein model for the phonon spectrum is more appropriate. In this case, one finds... 

I2H2O as a function of the reciprocal temperature. The points are data obtained from fits of the Mdssbauer spectra (Fig. 6.6). The broken curve is a fit to the Einstein model for a Raman process. The dotted curve corresponds to a contribution from a direct process due to interactions between the electronic spins and low-energy phonons associated with critical fluctuations near the phase transition temperature. (Reprinted with permission from [32] copyright 1979 by the Institute of Physics)... 

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. 

Figure 8.3 Experimental heat capacity of Cu at constant pressure compared with Cv m calculated by the Einstein model using 0E = 244 K. The vibrational frequency used in the Einstein model is shown in the insert.

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... 

The experimental constant-pressure heat capacity of copper is given together with the Einstein and Debye constant volume heat capacities in Figure 8.12 (recall that the difference between the heat capacity at constant pressure and constant volume is small at low temperatures). The Einstein and Debye temperatures that give the best representation of the experimental heat capacity are e = 244 K and D = 315 K and schematic representations of the resulting density of vibrational modes in the Einstein and Debye approximations are given in the insert to Figure 8.12. The Debye model clearly represents the low-temperature behaviour better than the Einstein model. 

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

Here 0O is the characteristic temperature at volume V0. An average value for the volume dependence of the standard entropy at 298 K for around 60 oxides based on the Einstein model is 1.1 0.1 J K-1 cm-3 [15]. A corresponding analysis using the Debye model gives approximately the same numeric value. 

The simplest model to describe lattice vibrations is the Einstein model, in which all atoms vibrate as harmonic oscillators with one frequency. A more realistic model is the Debye model. Also in this case the atoms vibrate as harmonic oscillators, but now with a distribution of frequencies which is proportional to o and extends to a maximum called the Debye frequency, (Oq. It is customary to express this frequency as a temperature, the Debye temperature, defined by... 

Chapter 5, vapor pressure isotope effects are discussed. There, a very simple model for the condensed phase frequencies is used, the Einstein model, in which all the frequencies of a condensed phase are assumed to be the same. From this model, one can derive the same result for the relationship between vapor pressure isotope effect and zero-point energy of the oscillator as that derived by Lindemann. 

Jones developed an equation of the Griin-eisen type, based on the Einstein model of a solid, of the form p=Ae av— B+fRT, where a, A, B and f are constants. Lutzky, however, preferred to use an equation based on the discussion by Zel dovich Kompaneets (Ref 21) of the equation derived by Landau Stanyukovich (Ref 2). In their view, the comparatively stable molecules of the detonation products are in a highly compressed state, being at a density over twice that of the liquid gases. The predominant part of the pressure is due to elastic repulsion. 

Equation (45) shows that as long as balances, volumetric flasks, and viscometers are available, [17] can be determined. All that is required is to measure viscosity at a series of concentrations, work up the data as (l/c)[(ij/i70) — 1], and extrapolate to c = 0. If the experimental value of [17] turns out to be 2.5 (V2/M2), then the particles are shown to be unsolvated spheres. If [17] differs from this value, the dispersed units deviate from the requirements of the Einstein model. In the next section we examine how such deviations can be interpreted for lyophobic colloids. 

The Einstein theory is based on a model of dilute, unsolvated spheres. In this section we examine the consequences on intrinsic viscosity of deviations from the Einstein model in each of the following areas ... 

The viscosity of a polymer solution is one of its most distinctive properties. The spatial extension of the molecules is great enough so that the solute particles cut across velocity gradients and increase the viscosity in the manner suggested by Figure 4.8. In this regard they are no different from the rigid spheres of the Einstein model. What is different for these molecules is the internal structure of the dispersed units, which are flexible and swollen with solvent. The viscosity of a polymer solution depends, therefore, on the polymer-solvent interactions, as well as on the properties of the polymer itself. 

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]

The Einstein model for the molar heat capacity of a solid at constant volume, Cy, yields the formula ... 

At tet eratures where the Einstein model of a solid is valid, the matrices A and appearing in Eqn. (71) are considerably simplified, and for the... 

Fig. 5.12. Plots of the dimensionless average energy and specific heat resulting from the Einstein model. The average energy is scaled by 3NEe, where Ee = ha>E, while the specific heat is reported in units of 3Nk. The temperature is also plotted in units of the Einstein temperature given by Te = hcoE/k.

In this problem, elaborate on the Einstein model for structural change given in the chapter by replacing the delta function densities of states by rectangular densities of states. Assume that the phase with the lower internal energy has the broader vibrational density of states. Compute the transformation temperature as a function of the difference in the widths of the two rectangular bands. 

The characteristic frequency, previously determined ( 2.6.4) as 34 cm is here found to be 38 cm, whilst the calculated effective mass of the bifluoride is 44 amu, close to the molecular mass of 39 amu. The two calculations are clearly self-consistent, as they must be since the phonon wings are simply the start of the molecular recoil in the lattice. However, the extreme naivety of the Einstein model is not usually successful at modelling the lattice dynamics of even simple systems. 

In Fig. 13, we compared our experimental results on Ti02-water with the results of Masuda et al. [18], He et al. [46] and Murshed et al. [61] and also to Einstein model. All results are well above the prediction of the Einstein model. 

FIGURE 8,4 Thermal conductivity of quartz and glassy silica as a function of temperature [7]. The quartz thermal conductivity exhibits a T behavior at low temperature, a peak at about 10 K, then reduction at higher temperatures. This is typical of a crystalline solid. For amorphous glass the thermal conductivity increases as T2 plateaus between 1 to 10 K and then increases monotonically with temperature. Also plotted are the predictions of the Cahill-Pohl and Einstein models. The Cahill-Pohl model provides accurate predictions for temperatures higher than 50 K but cannot predict the low temperature behavior. The Einstein model predictions are much lower than the measured values. 

Cahill and Pohl [8,9] recently developed a hybrid model which has the essence of both the localized oscillators of the Einstein model and coherence of the Debye model. In the Cahill-Pohl model, it was assumed that a solid can be divided into localized regions of size A./2. These localized regions were assumed to vibrate at frequencies equal to to = 2kvsIX where v, is the speed of sound. Such an assumption is characteristic of the Debye model. The mean free time of each oscillator was assumed to be one-half the period of vibration or x = it/to. This implies that the mean free path is equal to the size of the region or XI2. Using these assumptions, they derived the thermal conductivity to be... 

Although in principle detailed knowledge of the actual phonon phase space is necessary in order to predict accurately the temperature dependence of the line width, a simple approximation can suffice. We simply assume a variation of the Einstein model for the specific heat dependence of solids, namely, that there exists a single phonon mode of energy It is then straightforward to show that the fwhm cr(r) of the band must vary with temperature as... 

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