This disadvantage of the former hybrid model versions [14, 21, 50] was one of the main motivations for performing here (in Appendix A) a comprehensive analytical study aiming at the construction of more practicable, integral-free formulas for the integrals (10) of Debye and non-Debye type.
Consider first the integral (10) for the low-energy component of Debye type, [[kappa].sub.Ck=1] ([x.sub.1] (T)) (see Figure 1).
The asymptote of the k =1 component is of Debye type , whereas the asymptote of the k = 2 component is responsible for the onset of the typical non-Debye behaviour.
Furthermore we would still like to observe that, in many cases, one knows at least approximate values for the T [right arrow] 0 limiting values of Debye temperatures, [[THETA].sub.D] (0) (either as results of former analyses of cryogenic heat capacity data [13, 34] or from independent estimations on the basis of elastic constants [35, 73-77]).
We have given in Table 2, for a series of selected T-values, a list of smoothed isobaric lattice heat capacity values, [C.sup.(L).sub.p] (T) = [C.sub.p] (T) - [c.sub.1]T, including the corresponding (effective) lattice Debye temperatures, [[THETA].sup.(L).sub.D] (T), which have been calculated on the basis of precision formulas to be presented below (in Section 5).
Transformation of Heat Capacities into Debye Temperatures
Expressive visualizations of typical non-Debye features of the temperature dependencies of heat capacities of solids are based above all on the results of point-by-point transformations of isobaric heat capacity data, [C.sub.p] (T), into the respective (effective) Debye temperature values, [[THETA].sub.D] (T).
The [x.sub.D] [right arrow] [infinity] asymptote of Debye's low-temperature expansion [1, 3], [kappa]([x.sub.D]) [right arrow] (4[[pi].sup.4]/5[x.sup.3.sub.D]) (cf.
Accordingly, the temperature dependence of the Debye temperature, [[THETA].sub.D] (T) [equivalent to] T x [x.sub.D] (T), for moderate-to-high heat capacities, is clearly confirmed to be given by the approximate algebraic expression 
Couple of High-Precision Formulas for Effective Debye Temperatures.
alternatively, for high-accuracy calculations of the "true" Debye temperatures, [[THETA].sub.Dh](T), which are visualizing the characteristic non-Debye features of the harmonic parts of lattice heat capacity curves, [C.sub.Vh] (T) (see (4); Figures 4 and 5).
By means of this couple of the high-accuracy formulas (27) and (28) we have performed, first of all, the point-by-point transformations, [C.sub.p] (T) [??] [[THETA].sub.D](T), for all those isobaric heat capacity data points, [C.sub.p] (T), with respect to which the concept of effective Debye temperatures is actually applicable, that is, for [C.sub.p] (T) < [C.sub.Vh] ([infinity]) = 6R (cf.