Specific heat 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. 

Horning, R. D., and Staudenmann, J.-L. The Debye-Waller factor for polyatomic solids. Relationship between X-ray and specific-heat Debye temperatures. The Debye-Einstein model. Ada Cryst. A44, 136-142 (1988). 

The important fact that the quantum theory explains the extraordinarily small specific heat of electrons at normal temperatures, as is well known, was pointed out by Sommerfeld. Our model likewise explains this fact, though in a somewhat different way. If, as is permissible here, we work with the spins in the same way as we previously did with the free electrons, it follows by what we have said above that we have to deal with a degenerate Bose-Einstein gas, the specific heat of which at low temperatures is, as we know, given by... 

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.

Einstein showed that the behaviour of the specific heat of solid bodies furnished vafuable support for Planck s bold hypothesis of energy quanta. The crudest model of a solid consisting of N atoms is a system of 3N linear oscillators, each of which more or less represents the vibration of an atom in one of the three directions of space. If the energy content of such a system be calculated on the assumption of a continuous energy distribution, we get from (2)... 

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... 

As mentioned, many experimental results have shown that the specific heat for composites increases sHghtly with temperature before decomposition. In some previous models, the specific heat was described as a Hnear function. Theoretically, however, the specific heat capacity for materials wiU change as a function of temperature, as on the micro level, heat is the vibration of the atoms in the lattice. Einstein (1906) and Debye (1912) individually developed models for estimating the contribution of atom vibration to the specific heat capacity of a sohd. The dimensionless heat capacity is defined according to Eq. (4.32) and Eq. (4.33) and illustrated in Figure 4.12 [25] ... 

Most of the char material was composed of glass fiber and Cp was therefore considered as the specific heat capacity of the glass fibers. The results from Eq. 4.42 are compared with the results from the Einstein model (Eq. (4.33) in Figure 4.13 [12], as well as with the model used in previous studies [4, 5]. A linear function dependent on temperature for the specific heat capacity of fibers was used by Samanta et al. [4] and Looyeh et al. in 1997 [5], however, without direct experimental validation. As shown in Figure 4.13, the theoretical curve based on the Einstein model (Eq. (4.33) gives a reasonable estimation for the specific heat capacity of glass fibers. 

The true specific heat capacity of a composite material was obtained by the mle of mixture and the mass fraction of each phase was determined by the decomposition and mass transfer model. The true specific heat capacity of resin or fiber was derived based on the Einstein or Debye model. The effective specific heat capacity was obtained by assembhng the trae specific heat capacity with the decomposition heat that was also described by the decomposition model. The modeling approach for effective specific heat capacity is useful in capturing the endothermic decomposition of resin and was further verified by a comparison to DSC curves. 

Similar to the elastic properties also Debye and Einstein temperatures decrease in quaternary clathrate-I compounds with varying Si/Ge-ratio displayed in Fig. 10.12 for BaCu5Si4i xGex [78, 98], where 0d and 0e are extracted from single crystal X-ray diffraction at different temperatures, and for SrgGaigSiso-xGex [119], where specific heat data, Cp(T), were fitted to the Debye model using two... 

The specific heat capacities are represented by functions derived from statistical models - e.g. for solids [SOU 15a], the Einstein function ... 

The spin-boson model, its various ramifications, and analysis of its specific heat, when decoupled from its environment, have been the subject of Section 11.3. It is shown that the specific heat for the free Hamiltonian, d la Schottky, exhibits dominant exponential drop to zero temperatures, an aspect that is shared with the property of the Einstein oscillator. This aspect has been considered unsatisfactory vis-d-vis the third law that conjectures a power-law behavior with temperature, as far as the low-temperature specific heat is concerned [18]. 

Einstein himself did not believe seriously in his model in fact, in a second article The Present State of the Problem of Specific Heats presented to the first Solvay Congress in 1911 and later published in the Proceedings, commenting on a proposal of Nemst and Lindemann [7], he wrote, One can therefore suppose that the body behaves as a set of oscillators with different frequencies. [...] A theoretical meaning could only be attributed to a formula in which all the infinite frequency values compare in a sum [8]. However, not only has the Einstein model had a historical and tutorial relevance, but also formula (15) continues to be used for the vibrational specific heat of isolated defects. Nemst and Lindemann had observed that most of the experimentally known specific heats could be adequately represented by taking 8 oo — (o ) + 8 (jo — oo ) for g(a>) [7] in a way, this formula can be considered the first attempt to solve the ab fine problem. 

Chapter 1 discusses the modeling of pure solids. Oscillator models (Einstein s and Debye s) are used to calculate canonical partition functions for four types of solid atomic, ionic, molecular and metallic. These canonical partition functions can be employed, first to calculate the specific heat capacities at constant voliune, and second to determine the expansion coefficients with the Griineisen parameters. 

An excellent way of deducing the specific heat of a crystal is to model it as an empty box containing a gas of phonons. The phonons are bosons so Bose-Einstein statistics must be used. It should be noted, however, that the total number of phonons is not conserved (so that a Bose-Einstein condensation is not to be expected), quite unlike a gas of, for example, helium atoms. This approach yields the famous Debye specific heat curve shown in Fig. 5. It is plotted, not as a function of T directly, but as a function of T/ D where = is the Debye characteristic... 

The theory of lattice specific heat was basically solved by Einstein, who introduced the idea of quantized oscillation of the atoms. He pointed out that, because of the quantization of energy, the law of equipartition must break down at low temperatures. Improvements have since been made on this model, but all still include the quantization of energy. Einstein treated the solid as a system of simple harmonic oscillators of the same frequency. He assumed each oscillator to be independent. This is not really the case, but the results, even with this assumption, were remarkably good. All the atoms are assumed to vibrate, owing to their thermal motions, with a frequency v, and according to the quantum theory each of the three degrees of freedom has an associated energy of which replaces the kT as postulated by... 

Figure 1.31 Fit to the pure ZnO data (a) usingthe Schottky model (with just Schottky and Debye contributions yielding a Debye temperature of 395.5 K) and (b) using the Einstein model. is the excess specific heat (ideally the Einstein contribution) obtained by removing the Debye and Schottky contributions from the total specific heat. The Einstein oscillators are identified as the Zn interstitials. (After Ref [154].)...

Another defect of the cell model that we have already mentioned is the neglect of correlations. This appears very clearly in the limit of high densities and low temperatures. In this case we may use a harmonic oscillator cell potential. All molecules perform independent harmonic oscillations in their cells and the result is a specific heat curve of the Einstein type with an exponential decrease for T- 0 instead of a specific heat curve of the Debye type with a characteristic 7 law. Furthermore quantum statistical effects (exchange effects) cannot be introduced in a one-partide model. For this reason the cell model cannot be applied to problems in which such exchange effects play a dominant role (e.g. the X point of liquid He ). However, multiple cell occupations do take account of these correlations and exchange effects. IDgher multiple cell occupations prodde a continuous transition from a one-particle model to the correct 2V-partide model. 

On the contrary if we are interested in the excited states the one particle model fails completely since the excitation energy increases with V (cf. 18.1.3). At sufficiently low temperatures only the collective modes are excited and contribute to the entropy as well as to the specific heat. For this reason the Einstein model underestimates by far tfie specific heat and one has to use the Debye model to obtain a correct orda of magnitude. 

In 1907, EINSTEIN [1.3] proposed a simple model to account for the decrease of the specific heat at low temperatures. He took the atoms of a cys-tal to be independent oscillators, each having the same frequency and able to vibrate isotropically. He then quantized the energy of the oscillators in accordance with the results obtained by PLANCK for radiation oscillators. 

In 1910 Frederick Lindemann proposed a simple idea solids melt when the amplitude of atomic thermal vibrations exceeds a fraction of the interatomic spacing. His quantitative model made use of Einstein s explanation of the low temperature specific heats of solids, which proposed that the atoms vibrate as quantized harmonic oscillators. Einstein s theory had been published only three years earlier, and its adoption by Lindemann appears to be the first application of quantum theory to condensed matter after Einstein s own paper. 

Determine the specific heat of a two-dimensional crystal according to the Einstein model. 

---