Einstein characteristic temperature

The Einstein characteristic temperature e, equal to hcw/kf where < > is the mean frequency, is smaller than the Debye which involves the maximum frequency. 

The same considerations apply to the Einstein theory. An Einstein characteristic temperature Og may be defined by the relation... 

The values given are vibrational frequencies rather than the Einstein characteristic temperatures these frequencies can conveniently be compared with those of IR and Raman bands. The conversion of frequencies to characteristic temperatures is performed using a multiplier of 1.43879 K cm. 

These heat capacity approximations take no account of the quantal nature of atomic vibrations as discussed by Einstein and Debye. The Debye equation proposed a relationship for the heat capacity, the temperature dependence of which is related to a characteristic temperature, Oy, by a universal expression by making a simplified approximation to the vibrational spectimii of die... 

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))... 

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. 

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). 

For solids the matter is not quite so simple, and the more exacting theories of Einstein, Debye, and others show that the atomic heal should be expected to vary with the temperature. According lo Debye, there is a certain characteristic temperature lor each crystalline solid at which its atomic heal should equal 5.67 calories per degree. Einstein s theory expresses this temperature as hv /k. in which h is Planck s constant, k is Bolizmanns constant, and r, is a frequency characteristic of ihe atom in question vibrating in the crystal lattice. 

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

Particular characteristic temperatures are denoted with subscripts, e.g. rotational <9r = hcB/k, vibrational <9v = hcv/k, Debye 0D = hcvD/k, Einstein <9E = hcvE/k. 

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. 

Here m is the mass of the rattling atom, and d a term for static disorder (further details can be found in [71]). Neutron powder diflfaction data collected at different temperatures were used to calculate the Einstein temperatures [72] or both, Einstein as well as Debye temperatures [73, 74], from ADPs. Also single crystals were exposed to neutron diffraction at different temperatures in order to provide 6e [75] and Go [73, 76-79]. Another frequently used technique to extract Gd and Ge is X-ray diffraction. In some cases room temperature data, gained from powder samples measured on conventional diffractometers, were used to calculate the characteristic temperatures [16, 43, 80-85], and also temperature dependent measurements using synchrotron radiation and powder samples were carried out [69, 86-94]. Similarly, X-ray diffraction measurements on single crystals were performed to gain atomic displacement parameters. Whereas in a few cases room temperature data were used to evaluate Einstein and Debye temperatures [72, 95, 96], for the majority of investigations, temperature dependent measurements were carried out [49, 52, 53, 55, 59, 64, 68, 69, 78, 79, 97-109]. 

If we sum the three contributions to calculate the value for the heat capacity at constant volume, as the characteristic temperatures of rotation are often lower than the Einstein temperature (see Table 7.3), the variation in the heat capacity, for example of a diatomic molecule, with temperature takes the form of the curve in Figure 7.12(c). At low temperatures, the only contribution is that of translation, given by 3R/2. Then if the temperature increases, the contribution of rotation, is added according to the curve in Figure 7.12(a) until the limiting value of this contribution is reached, then the vibrational contribution is involved until the molecule dissociates which makes the heat capacity become double that of the translational contribution of monoatomic molecules. The limiting value of the vibrational contribution is sometimes never reached. This explains why the values calculated in Table 7.2 are too low if we do not take into account the vibration and too high in the opposite case. 

Table 7.3. Values of Einstein temperature and characteristic temperature rotation for...

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

The example of experimental data processing for LuCla from (Brunetti et al., 2005) considered above substantiates our views on the optimum selection of the Debye characteristic temperature and Einstein characteristic frequencies used to calculate the temperature dependences of heat capacities. 

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. 

It is tempting to identify the superfluid and normal fluid components of He II with the atoms of the condensate and excited states, respectively, of an ideal Bose-Einstein gas, but this, too, turns out to be a gross oversimplification. Nonetheless, it is clear that Bose-Einstein statistics do play an essential role in determining the properties of the liquid. If one calculates the Bose-Einstein condensation temperature of an ideal gas with the same density and atomic mass as liquid " He, one finds 7 3.1 K, and the difference between this value and 7 may plausibly be ascribed to the influence of the interatomic forces. The presence of a condensate of " He atoms in the zero momentum state has recently been confirmed by inelastic neutron scattering measurements. As shown in Fig. 14, the condensate fraction rises from zero at Tx towards an asymptotic limit of about 14% at low temperatures. This departure from the approximately 100% condensate characteristic of an ideal gas for r 7c is an expected consequence of the non-negligible interatomic forces. Of couse, as already noted in Fig. 13, the superfluid component does form 100% of the liquid at 7 = 0 and it cannot, therefore, be directly identified with the condensate fraction. 

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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