Debye law

This result is called the Debye law. Figure A2.2.4 compares the experimental and Debye model values for... 

FIGURE 1.8 Graphic representation of the Debye law of specific heat. 

This result is called the Debye law. Figure A2.2.4 compares the experimental and Debye model values for the heat capacity C. It also gives Debye temperatures for various solids. One can also evduate Cy for the... 

Nonconformity with the third law also arises when a substance can exist in more than one state of such low energy that the distribution among these states is not influenced by the falling temperature down to the lowest attainable T. These states are frequently due to interactions of electronic or nuclear magnetic dipoles. An extrapolation to zero using the Debye law would reduce the system to a state of zero vibrational entropy, but the rotational entropy due to nuclear spin or the entropy associated with random orientation of magnetic dipoles in... 

It is well known since the beginning of the last century that acoustic waves (longitudinal and transverse) are generated by thermal agitation even at very low temperatures. The Debye law of specific heat is but one of the numerous consequences. These waves of frequency v and wave vector q = 2jt//v, have a linear dispersion ... 

However, Lewis and Gibson [1], Eucken [2], and others showed that there are a large number of substances including elements, for which the temperature dependence of the specific heat is not described by the Debye function over a wide temperature range. For example, as can be seen from Fig. 1, deviations from the Debye law are particularly noticeable for elements whose p-electron shells are being occupied the symmetries and lattice coordination numbers z of these elements depend on the extent to which the p shells are occupied. 

The deviations from the Debye law are, undoubtedly, connected with the structure of the electron shells of atoms and their shape. By the shape of the atom or ion, one may understand the shape of the surface of equal electron density of the outer-shell electrons of the atom. From this point of view, the ions whose outer electrons are the s electrons have spherical symmetry, whereas the p electrons and d electrons form dumb-bells, the shapes of which are shown in Fig. 

The effect of the nature of the covalent, ionic, or van der Waals bond is clearly apparent in the deviations of the temperature dependences of the specific heat from the Debye law. The bond type causes a change in the anisotropy mainly in the vibration component of specific heat. 

Fig. 3. Temperature dependences of specific heat (a) and characteristic temperature (b) of germanium (the dashed curve represents the specific heat according to the Debye law).

As we have previously shown [5], there is also a definite relationship between the Debye characteristic temperature and the heat of activation of diffusion U, etc. However, an analysis of the experimental curves of the specific heat as a function of temperature shows that, for all bodies, including those whose specific heat is more or less satisfactorily described by the Debye equation, the characteristic temperature of the whole crystal is essentially a function of temperature (Fig.3). In addition, it is observed that the greater the deviation of the curve of from the horizontal straight line, the more the temperature dependence of the specific heat deviates from the Debye law. Despite the fact that the temperature dependence of the specific heat is comparatively insensitive to the form of the phonon spectrum, an evaluation of the trend of the specific heat curves already indicates that the vibration spectrum of ion vibrations in real solids differs essentially from the Debye law. 

Watts (KWW) relaxation function. At a=l expression (1.20) transforms into conventional exponent leading to Debye law for dynamic dielectric permittivity of ordered ferroelectrics [11] ... 

Similarly to above glassy systems, the disordered ferroelectrics, polymers and composites are also characterized by slow relaxation processes. Their quantitative measure is complex dielectric permittivity, which can be described by generalized Debye law [29-31] in the form of the following empirical formulas ... 

Fig. 1.9 Relaxation time distribution functions calculated on the basis of Debye law (D), Cole-Cole law (CC), Davidson-Cole law (DC) and...

It has also been established, that only Debye law with single relaxation time in Fig. 1.9 describes the systems with long-range order, while the distribution of relaxation times generates the other laws, which are related to coexistence of short- and long-range polar orders. This coexistence corresponds to mixed ferroglass phase. 

In fact, two expressions of this type are very well known, the Stokes-Einstein law for the self-diffusion coefficient, and the Debye law for the rotational diffusion coefficient. Both of these laws for the appropriate diffusion coefficient D are derived by hydrodynamics and have Darj, where 17 is the coefficient of shear viscosity, a transport coefficient. The Stokes-Einstein and Debye laws were reconciled with formal theory with the use of mode-mode... 

A plot of CpT versus gives initial slope y and intercept A. Secondly, at higher temperatures, where Cn < 1% of the total heat capacity, or is known from the first step, and where the Debye law is assumed to hold, which for the lanthanides means up to about 5 or 6 K, we can consider... 

The real and imaginary parts of the susceptibility of HAT6 are shown in Fig. 7 (a and b), and the temperature dependence of e is shown in Fig. 7(c) [38]. The frequency response can be explained by assuming a Debye law with, surprisingly, a single relaxation time (=5 x 10 s). This process is most probably due to the rotation of the side chain around the oxygen bond. Numerical simulations have shown this mode to have a relatively low activation energy (effective overall thermal dipole can appear. 

The Cggif follows the Debye law, being the number of monomers per blob times the self-correlation of blobs... 

EPR study of rotational diffusion in viscous ionic liquids Analysis by a fractional Stokes-Einstein-Debye law. Chem. Lett, Vol. 38, No. 2, 124-125, ISSN 0366-7022... 

This is the Debye law, which gives an accurate temperature dependence of the heat capacity of solids. 

The well-known classical Debye law for bulk crystals aT is disobeyed, and, as the calculations show, rather significantly. Once more, this confirms the fact that nanoparticles really are new materials, with physicochemical characteristics greatly different from the characteristics of materials with the same chemical composition but... 

In the second case (T < 0d) calculation of AQ is simplified by the fact that we can use the limited property of the Debye law where the heat capacity is... 

---