Debye solid

Once equation (10.158) has been obtained for relating Cy. m to T for a Debye solid, equations relating (Um - U0 m), (Hm - Uo.m), and Sm to T can be derived. Tables of values, expressed in terms of 9d/T. can be found in Table A4.7, Appendix 4, with more extensive tables found in the literature1 to calculate these thermodynamic properties. 

The zero point energy follows from quantum theory, according to which atoms do not cease to vibrate at the absolute zero point. For a Debye solid (that is, a homogeneous body of N equal particles) the zero point energy is... 

However, for a real crystal at zero temperature it is impossible to group all vibrational motions on the lowest single vibrational mode and, if the crystal behaves as a Debye solid (see later), zero-point energy is expressed as ... 

Nitzan, A., Shugard, M., Tully, J.G. Stochastic classical trajectory approach to relaxation phenomena. 11. Vibrational relaxation of impurity molecules in Debye solids, J. Chem. Phys. 1978,69, 2525. 

The above-mentioned relationships may be due to the form of the initial part of the phonon spectrum, i.e., its acoustical branch for three-dimensional Debye solids the spectrum is g8(v) oc for two-dimensional structures one-dimensional structures... 

The atoms in a Debye solid are treated as a system of weakly coupled harmonic oscillators. Normal modes with wavelengths that are large compared to the atomic spacing do not depend on the discrete nature of the crystal lattice, and consequently these normal modes can be obtained by treating the crystal as an isotropic elastic continuum. In the Debye treatment of a solid all of the normal modes are treated as elastic waves. The partition function for a Debye solid cannot be obtained In closed form, but the thermodynamic functions for a Debsre solid have been tabulated as a function of 9p/T- For the pair of Isotopic metals Li(s)... 

Since we are treating high-temperature properties of 5-Pu, 8 -Pu and s-Pu from calculations at OK, a reliable thermodynamic model is needed. As Pu-3.6 at. % Ga alloy behaves like a Debye solid at ambiant pressure and low temperature, we chose to use the Debye model for plutonium instead of a classical mean-field approach. We first briefly summarize this model. 

For the frequency spectrum of a molecule adsorbed on a Debye solid the frequency shift is determined from Eq. (65) to be... 

The Debye theory is so firmly established that it has become common practice to express specific heat data in terms of an "equivalent Debye temperature". For an experimental measurement or for a calculated value of c (T) via (3.102), we can define an equivalent Debye temperature this is the value of that must be used in (3.107) such that Cp(T/0p) = e (T). Unless the crystal is exactly described by the Debye model (which will never be the case), the equivalent Debye temperature depends on T, since no single value of 0p will satisfy Cp(T) = Cy(T) at all temperatures. Thus, the experimental or theoretical specific heat data are usually presented as a plot of 0p against T and in this way, show the deviations of the crystal from a Debye solid. 

In the general case of a Debye solid the temperature dependence of G is determined via the parameter 0d (see Chapter 3). Hence, the free enthalpy of a perfect sohd is determined for all temperatures by the parameters effective pair bonding energy, co-ordination number and Debye-temperature. While (e) is a meaningful parameter in the case of co radent crystals (for elemental crystals x-x> cf. Section 2.2.4), in the case of ionic crystals charge numbers, lattice constant and Madelung number are the appropriate parameters. Hence... 

---