Debye crystal theory

Debye was responsible for theoretical treatments of a variety of subjects, including molecular dipole moments (for which the de-bye is a non-SI unit). X-ray diffraction and scattering, and light scattering. His theories relevant to thermodynamics include the temperature dependence of the heat capacity of crystals at a low temperature (Debye crystal theory), adiabatic demagnetization, and the Debye-Huckel theory of electrolyte solutions. In an interview in 1962, Debye said that he... 

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

It is shown that solute atoms differing in size from those of the solvent (carbon, in fact) can relieve hydrostatic stresses in a crystal and will thus migrate to the regions where they can relieve the most stress. As a result they will cluster round dislocations forming atmospheres similar to the ionic atmospheres of the Debye- Huckel theory ofeleeti oly tes. The conditions of formation and properties of these atmospheres are examined and the theory is applied to problems of precipitation, creep and the yield point."... 

Let us now consider a pair of ions in aqueous solution from such a crystal. In the Debye-Hilckel theory it is assumed that in pure solvent, the mutual potential energy is — e2/ r, where e is the macroscopic dielectric constant of the solvent,2 until the ions come into contact with each... 

The condition of electrical neutrality will not apply at the surface of a crystal, and since g+ is not equal to g there will be an excess of one of the defects. This effect, which is referred to in the early literature as the Frenkel-Lehovec space charge layer-results in an electric potential at the surface of the crystal [23-25]. In this instance, the surface will not simply be the external surface but will also include internal surfaces such as grain boundaries and dislocations. The effect decays away in moving from the surface to the bulk, and can be treated by classical Debye-Hiickel theory [26-29]. This leads to a Debye screening length, Lp, given by... 

When referring to a Debye dispersion, it should be kept in mind that Debye s theory applies exclusively to rotational polarization of molecules with permanent dipoles, without net charge transfer (33, 90). The occurrence of an appreciable ionic transfer complicates the picture, as in ice doped with ionic impurities. A further complicating factor is aging. Steinemann observed that thin crystals (0.1 to 0.2 cm.) showed a decrease of both the low-frequency dielectric constant and the low-frequency conductivity, suggesting difiusion of impurity (HF) out of the sample into the electrodes. Thicker samples (of the order of 1 cm.) were not affected. 

It follows from the introduction that the amplitude of oscillations of crystal atoms situated close to the surface is larger by than in the bulk. As a consequence, the Debye temperature of surface layers is lower by than that of the bulk material. Since the heat capacity is related to the values specified in accordance with the Debye quantum theory of heat capacity, the heat capacity of surface layers should be higher than the heat cap>acity of bulk materials. The amplitudes of atomic oscillations are higher in the liquid than in the solid phase, and the temperature dependence of the heat capacity of the liquid phase is in the majority of cases steeper compared with the solid phase. Extending this conclusion to the surface layer, we are led to suggest that the difference between the temp>erature dep>endences of the heat capacities of the surface layer in the liquid and solid state (that is, ACp(T)) increases compared with the bulk material. 

Peter Debye, a Dutch physical chemist after whom the Debye-Hiickel theory is partly named (see Figure 18.5), expanded on Einstein s work. Rather than assume that all atoms in a crystal had the same vibrational frequency (as Einstein had presumed), Debye suggested that the possible vibrational motions of the atoms in a crystal could have any frequency from zero to a certain maximum. That is, he suggested that atoms could have a range, or distribution, of frequencies. 

Debye s theory of heat capacities of monatomic crystals is built on transverse and longitudinal frequencies of oscillation in the whole crystal... 

As the density of a gas increases, free rotation of the molecules is gradually transformed into rotational diffusion of the molecular orientation. After unfreezing , rotational motion in molecular crystals also transforms into rotational diffusion. Although a phenomenological description of rotational diffusion with the Debye theory [1] is universal, the gas-like and solid-like mechanisms are different in essence. In a dense gas the change of molecular orientation results from a sequence of short free rotations interrupted by collisions [2], In contrast, reorientation in solids results from jumps between various directions defined by a crystal structure, and in these orientational sites libration occurs during intervals between jumps. We consider these mechanisms to be competing models of molecular rotation in liquids. The only way to discriminate between them is to compare the theory with experiment, which is mainly spectroscopic. 

Compared with the momentum of impinging atoms or ions, we may safely neglect the momentum transferred by the absorbed photons and thus we can neglect direct knock-on effects in photochemistry. The strong interaction between photons and the electronic system of the crystal leads to an excitation of the electrons by photon absorption as the primary effect. This excitation causes either the formation of a localized exciton or an (e +h ) defect pair. Non-localized electron defects can be described by planar waves which may be scattered, trapped, etc. Their behavior has been explained with the electron theory of solids [A.H. Wilson (1953)]. Electrons which are trapped by their interaction with impurities or which are self-trapped by interaction with phonons may be localized for a long time (in terms of the reciprocal Debye frequency) before they leave their potential minimum in a hopping type of process activated by thermal fluctuations. 

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. 

Tarassov (1955) and also Desorbo (1953) have considered these ideas in relation to a onedimensional crystal in which case the one-dimensional frequency distribution function predicts a T dependence of the specific heat at low temperatures. In the case of crystalline selenium, however, it has been found necessary to combine the one-dimensional theory with the three-dimensional Debye continuum model in order to obtain quantitative agreement with the data below about 40° K. Tem-perley (1956) has also concluded that the one-dimensional specific heat theory for high polymers would have to be combined with a three-dimensional Debye spectrum proportional to T3 at low temperatures. For a further discussion of one-dimensional models see Sochava and TRAPEZNrKOVA (1957). 

We must also consider the conditions that are implied in the extrapolation from the lowest experimental temperature to 0 K. The Debye theory of the heat capacity of solids is concerned only with the linear vibrations of molecules about the crystal lattice sites. The integration from the lowest experimental temperature to 0 K then determines the decrease in the value of the entropy function resulting from the decrease in the distribution of the molecules among the quantum states associated solely with these vibrations. Therefore, if all of the molecules are not in the same quantum state at the lowest experimental temperature, excluding the lattice vibrations, the state of the system, figuratively obtained on extrapolating to 0 K, will not be one for which the value of the entropy function is zero. 

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