Einstein characteristic temperature functions

In consequence, the statistical characteristic temperature of relic radiation is fully determined in terms of relativistic invariant spectrum of the cosmic microwave background radiation and the distribution velocity function of radiating particles, i.e., is described with the following expression (compare with the results of reference (Einstein, 1965))... 

Equation 3.37, known as the Einstein function, is tabulated for various X-values (see, for instance, Kieffer, 1985). In the Einstein function, the characteristic frequency o), (and the corresponding characteristic temperature see, for instance, eq. 3.40) has an arbitrary value that optimizes equation 3.35 on the basis of high-T experimental data. Extrapolation of equation 3.35 at low temperature results in notable discrepancies from experimental values. These discrepancies found a reasonable explanation after the studies of Debye (1912) and Born and Von Kar-man (1913). 

This result was first obtained by Einstein and is often called an Einstein function. Introducing the characteristic temperature from Eq. (4.7), we have... 

In the case of polyatomic molecules that have more than one vibration (for example, H2O, which has three vibrations) there are three distinct frequencies and therefore three distinct characteristic temperatures, so that the heat capacity contains three distinct Einstein functions. 

A rather different sort of calculation was made by Blue (1954), who observed that the main difference between the vibrational spectra of H2O ice and D2O ice arises from the frequency shift of a factor ( )f in the librational modes. In the region between 100 and 200 °K the difference between the two Cp values can, he found, be fitted to within 3 per cent by the difference between two Einstein functions located at 0 = 945 °K (P = 660 cm ) for H2O ice and appropriately shifted for D2O ice. Still better agreement over the full temperature range o to 273 °K was obtained by representing the librational modes by an Einstein function with 0 = 1040 °K (P = 723 cm ) and the translational modes by a Debye function with characteristic temperature 315 °K for HgO ice, using appropriately shifted frequencies for DgO and small corrections for intramolecular modes. 

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. 

In Fig. 9.5, the Einstein model heat capacity is shown as a function of the characteristic temperature. Equation 9.69 correctly predicts the heat capacity at the limits of T 0 and T oo. 

The value 6 is called the characteristic temperature. If one derives a function of the heat capacity upon T/d for a series of simple substances, it will be represented by a single curve (Figure 9.20). It appears that Einstein s characteristic temperature defines the border behind which an essential deviation from the Dulong-Petit law takes place for every element. If we substitute the frequency o) = V Tm (see eq. (2.4.5)) into the expression for the value, we can arrive at... 

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