Einstein specific heat function

When integrated numerically, the function (3.4) proves to be not unlike an Einstein specific heat curve, except at low temperatures. To facilitate calculations with the Debye function, we give in Table XIV-1... 

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

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... 

In Fig.1.1, is shown as a function of T/e where the Einstein temperature 0g is an abbreviation for hw /kg is zero at T = 0 and rises asymptotically to the Dulong-Petit value 3R when T >>0. For high temperatures, therefore, quantization is unimportant and the specific heat has the same value as if each degree of freedom of the system had energy kgT/2 at lower temperatures, however, there is a pronounced deviation from the law of equipar-tition. According to PAIS [1.4], Fig.1.1 is the first graph dealing with the quantum theory of solids. 

Fig.3.10. Einstein and Debye specific heat as a function of reduced temperature. 0 is either the Einstein or the Debye temperature, depending on which curve is being examined. The Debye specific heat is compared with the observed specific heats of Ag( ), 0n = 215 K A1(A), 0p = 394 K C(diamond)( ),...

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