Einstein function

This procedure is based on the fact that an Einstein function reaches half its maximum value at the temperature 0.33/3 o. 

Particles that obey Bose-Einstein statistics are called Bose particles or bosons. The probability density of bosons in their energy levels is represented by the Bose-Einstein function as shown in Eqn. 1-2 ... 

Fig. 1-1. Probability density functions of partiele energy distribution (a) Fermi function, (b) Bose-Einstein function, e = particle energy f(i) - probability density function cp = Fermi level sb - Bose-Einstein condensation level.

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). 

Because the dispersed acoustic function 3.69, the optic continuum function 3.71, and the Einstein function 3.73 may be tabulated for the limiting values of undi-mensionalized frequencies (see tables 1, 2, 3 in Kieffer, 1979c), the evaluation of Cy reduces to the appropriate choice of lower and upper cutoff frequencies for the optic continuum (i.e., X/ and limits of integration in eq. 3.71), of the three... 

In the case of polyethylene it is interesting to attempt to calculate the heat capacity per mole of chain atoms using the Einstein function... 

Fig. 3 illustrates a comparison of C calculated by the means of the frequencies of Table 1 and the Einstein function (solid lines) with Cv values of a crystalline high molecular weight monomeric hydrocarbon, C38H66, and with the Cv values of polyethylene used in Fig. 2 plus some additional values taken from the recent papers of Sochava (1960). The agreement is excellent except at temperatures... 

Fig. 3, Comparison of heat capacities of crystalline Q (C ), open circles, and of a low density polyethylene Cv), solid circles, with values calculated from molecular frequencies and the Einstein function...

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

Taylor and Smith (10] reported S (298.15 K) = 32.29 cal K mol" based on S (60 K) 6.68 cal K" mol" obtained from Debye-Einstein functions which represented their C data to only 1.8 percent from 60 to 100 K. A comparison of their extrapolated C data with those which have been measured for SrCl, BaCl, and Cal, (4] indicates that the values for SrBr decrease much more rapidly with temperature below 50 K than would be expected. We have made our own extrapolation to 0 K for... 

Using the combination of Debye and Einstein functions suggested... 

King ( ) measured the heat capacity of the high temperature o-phase NbgOg from 53.24 to 296.64 K and fitted the data (29 data points) with a combination of Debye and Einstein functions. These functions fit the data over the entire measured temperature range with a maximum deviation of 0.6% (14) and are used to calculate S (50 K) 2.42 cal K" mol". ... 

Anderson (1 ) measured the heat capacity of V20g(cr) in the range 57-287 K. The data indicated an anomaly in the region 165-182 K. These heat capacity data are Joined smoothly at 298 K with the high temperature heat capacity values as derived from the enthalpy measurements of 0>ok (JJ.). The adopted C values are based on these two studies (1, 13 ). Using the combination of Debye and Einstein functions as suggested by Anderson (12), we calculate S (50 K) 0.783 cal K mol and H (50 0) 0.0284 kcal mol. There is considerable scatter in the data of Cook (13) the deviations from the adopted values range from -0.8 to 0.6% except for the data point at 369.1 K which is -1.4% low ( 25 cal K mol ),... 

Low temperature heat capacities of ZnS0 (cr, o) have been measured by Weller ( ) from 51.7 - 296.5 K. A small heat capacity maximum was observed at 124.37 K. Our adopted value of S°(298.15 K) = 26.42+0.3 cal K mol obtained from C is based on S (51 K) = 2.27 cal K mol obtained by Weller (1 ) by extrapolation of the measured heat capacity with a combination of Debye and Einstein functions. We have smoothed the data of Weller (H)) by fitting the data with orthogonal polynomials over selected overlapping temperature intervals. 

P is a universal function called the Planck-Einstein function, of which an abbreviated table is given in table 10.1. 

Planck-Einstein function which occurs in c (T), it is possible to define a temperature T such that for most practical purposes... 

Either by insertion of this value in equation (16.32), or by the use of the tables of Einstein functions, it is found that... 

Table V-52. Heat capacity and third law entropy of a-CdSe at 298.15 K. Original values including an estimated entropy term for the temperature range 0 to 50 K are denoted by (a), values corrected using the experimental mean entropy at 50 K from [76PET/KOF] and [92SIR/GAV] are denoted by (b), and values derived from estimated Debye-Einstein functions are denoted by (c).

However that may be, the above considerations suggest the expression of the molecular heats of compounds by means of a suitable combination of a Debye function with one or more Einstein functions as was to be expected, many elements (e.g. sulphur) behave here like compounds. For our present purpose it does not matter whether the above considerations from molecular theory are right or not it is sufficient to establish the fact that the method of calculation mentioned has proved itself convenient in practice examples are given in the lecture cited on page 62. 

The somewhat abstract character of the above calculations is more readily grasped if a definite function is used in making them for example, the increase in the energy of rotation with the temperature may be expressed by means of an Einstein function, which should at least come very near to the truth in this case a simple expression may be obtained for the relation between i and i. We shall frequently have to carry out such calculations. 

Q /Q) was calculated by substituting ro and co for each vibrational mode into Equation (11). For the sake of simplicity, all the vibrational modes, including the acoustic modes, were represented by Einstein functions. More rigorous calculations would treat the acoustic modes using Debye functions. Given a value for (Q /Q),/and 3 values are given by ... 

For the entropy below 50 K, Todd [50TOD] employed the empirical Debye and Einstein functions below 50 K to extrapolate a value for the entropy (1.73 J-K -mor ). 

The entropy at 298 K was determined from the graphical integration of the heat capacity data. The entropy below 50 K was obtained by extrapolation using Debye and Einstein functions, which were found to adequately represent the measured heat capacity data. The calculated entropy was (50.33 0.33) J-K -moP it is assumed that the error is Ict. From this value and an earlier value for the enthalpy of formation, [44KEL] determined a Gibbs energy of formation of - 1021.7 kJ-moP. This latter value is considerably lower than the value recommended in this review. 

There are no measurements of the heat capacity of ThD2(cr) or ThT2(cr), but Flotow et al. [1984FLO/HAS] have estimated the following values by modelling the heat capacity contributions from the lattice vibrations, represented by an Einstein function, conduction electrons and acoustic modes. This method had been shown to be valid for the deuterides of uranium, yttrium and zirconium. 

The function on the right-hand side of Eq. (4.88) is called an Einstein function. The Einstein function is shown as a function of T/O in Fig. 4.13. Thus for a diatomic molecule we have for the heat capacity... 

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