Einstein oscillator

Figure 2. A schematic of the free energy density of an aperiodic lattice as a function of the effective Einstein oscillator force constant a (a is also an inverse square of the locahzation length used as input in the density functional of the liquid). Specifically, the curves shown characterize the system near the dynamical transition at Ta, when a secondary, metastable minimum in F a) begins to appear as the temperature is lowered. Taken from Ref. [47] with permission.

The thermal energy of an Einstein oscillator is k0e where k = Boltzman s constant, and 0e is the Einstein temperature. The mechanical energy of the oscillator is h0e/2jr where h = Planck s constant. 

Figure 3J Schematic representation of a vibrational spectrum of a crystalline phase. Dotted curves acoustic branches and optical continuum. Solid line total spectrum, and 0) 2 Einstein oscillators. Reprinted with permission from Kieffer (1979c), Review of Geophysics and Space Physics, 17, 35-39, copyright 1979 by the American Geophysical Union.

In our present study To = 298.15 K thus AHo and ASo are respectively, the standard heat of formation and entropy. In our computational model we consider the heat capacity for the liquid phase to be temperamre independent and set it to 99.036 J/mole-K from literature data. For the solid phase, we employ a single Einstein oscillator to compute heat capacity ... 

Incoherent inelastic scattering from a proton in a harmonic potential well the Einstein oscillator model... 

The temperature dependence of the resistivity of amorphous non-magnetic metals has been studied by many authors. The most popular theory is still Ziman s theory for the resistivity of liquid metals and several authors have extended this theory to include a dynamic structure factor. Cote and Meisel (1979) used an isotropic Debye spectrum for the phonons to calculate the dynamic structure factor. They obtained a quadratic dependence of the resistivity on temperature at low temperatures. Frobose and Jackie (1977) used both a Debye model and a model of uncorrelated Einstein oscillators in conjunction with a dynamic structure factor to analyse the resistivity. They found that the second model leads to a satisfactory fit for the temperature dependence of the resistivity of CuSn alloys for T 10 K. Ohkawa and Yosida (1977) predict a T ... 

The densities of vibrational states for the two potentials (Fig. 15) are actually very similar, exeept for a somewhat wider gap for RIMl between the two peaks at higher frequencies, corresponding to inner modes of CO3. As for the Kieffer s model, the cut off frequencies of acoustic modes were derived from elastic constants by the Voigt-Reuss-Hill approximation[38], amounting to 51, 66 and 90 cm"i. An optic continuum ranging from 113 to 287 cm was used, and four Einstein oscillators at 708,867, 1042 and 1470 cm with appropriate weights represented the internal optical modes. The... 

Fig.3.9. aE for a driven Einstein oscillator compared to that for a driven Debye model (3.29,38). The Einstein-os-cillator frequency was taken as v TEojq. Other system parameters were wp = variable, mg = 39.95, m3 = 108, D = 418 K, and a = 1.69 A . Although obscured by the scale, both curves have a shallow minimum near (Dp/co = 0 as a result of the characteristic shape of F(cd)... 

The distribution function G(f2) is generally derived using a specific model. In the Einstein oscillator model there is only one frequency Qg and thus G(Q)=5K(i -i E) where is the Kronekker delta function. Hence, defining an Einstein temperature 0, given by this gives... 

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]. 

If instead of the isotropic exchange interaction (A, S) = (0,1) we consider the aspherical Coulomb interaction (A, 2 ) = (2,0) we obtain fig. 17.15 instead. The negative sign for y indicates pair-enhancement instead of pair-breaking. Since this is a dynamical effect it vanishes for 5 = 0. The maximum effect occurs for 8 = lOTc. It resembles the one of additional Einstein oscillators put into the matrix. The quantity t, in the definition of y has to be replaced by to which is defined in a similar way as Ts (see eq. (17.66)) but now with Hac instead of He. ... 

TABLE 11 Debye characteristic temperature, Einstein oscillator frequencies, and a for YCI3, YbCl3, and LUCI3... 

Attempts to rationalize the heat capacity of polyisoprene have been published by Ichimura (1948) who propos a Debye term of the 0-temperature 110° K and 6 Einstein oscillators with a characteristic... 

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].)...

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