Temperature Einstein

Jg is called the Einstein temperature = N. Above the ground-state occupancy is not a macroscopic... 

The quantity (hcCb/k) has the units of temperature. It is often written as the Einstein temperature, in which case... 

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. 

Both the Einstein and Debye theories show a clear relationship between apparently unrelated properties heat capacity and elastic properties. The Einstein temperature for copper is 244 K and corresponds to a vibrational frequency of 32 THz. Assuming that the elastic properties are due to the sum of the forces acting between two atoms this frequency can be calculated from the Young s modulus of copper, E = 13 x 1010 N m-2. The force constant K is obtained by dividing E by the number of atoms in a plane per m2 and by the distance between two neighbouring planes of atoms. K thus obtained is 14.4 N m-1 and the Einstein frequency, obtained using the mass of a copper atom into account, 18 THz, is in reasonable agreement with that deduced from the calorimetric Einstein temperature. 

Table 3.1 Entropy values obtained by application of Einstein (E), Debye (D), and Kieifer (K) models, compared with experimental data at three diiferent temperatures. Data are expressed in J/(mole X K). Values in parentheses are Debye temperatures (d, ) and Einstein temperatures (9 ) adopted in the respective models (from Kieifer, 1985).

The temperature dependence of the gap is next calculated, based on an Einstein model of the lattice vibrations, described by the Einstein temperature 0 which is of the Debye temperature, 0j,... 

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 temperature (0 ) - In the Einstein theory of the heat capacity of a crystalline solid, 0 = hvik, where h is Planck s constant, k is the Boltzmann constant, and v is the vibrational frequency of the crystal. 

G = shear or torsional modulus / = specimen length 5 = parameter in Varshni equation t = transit time, Einstein temperature T = temperature, Kelvin... 

In order to fill the knowledge gap on the mechanical performance of clathrates, the aim of this work is, to provide experimental data on hardness and elastic properties of intermetallic clathrates, covering a wide range of different compositions, and to compile, evaluate and discuss all data hitherto available in the literature on (i) hardness, (ii) elastic properties, (iii) Debye and Einstein temperatures, and (iv) on thermal expansion. 

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