Debye integral

Heberle J (1971) The Debye integrals, the thermal shift and the Mossbauer fraction. In Mossbauer Effect Methodology Vol 7. Gruverman IJ (ed), Plenum, p 299-308 Herzberg G (1945) Molecular Spectra and Molecular Structure. II. Infrared and Raman Spectra of Polyatomic Molecules. Von Nostrand Reinhold, New York Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev 136 B864-871... 

Heberle J (1971) The Debye integrals, the thermal shift and the Mossbauer fraction. In IJ Gruverman (ed) Mossbauer Effect Methodology, Vol 7. Plenum Press, New York, p 299-308... 

The formulas for the Debye approximation are given by Menzel and Tarassov °. The Debye integral is done numerically using Simpson s Rule and 1000 intervals. 

Here T is the actual and 6 the Debye temperature. The last term is the Debye integral, which can be found in mathematical tabulations. In the limit of T-> 0 one may approximate eq. (7) by... 

Low-temperature behaviour. In the Debye model, when T upper limit, can be approximately replaced by co, die integral over v then has a value 7t /15 and the total phonon energy reduces to... 

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

If the coefficients dy vanish, dy = 28y, we recover the exact Debye-Huckel limiting law and its dependence on the power 3/2 of the ionic densities. This non-analytic behavior is the result of the functional integration which introduces a sophisticated coupling between the ideal entropy and the coulomb interaction. In this case the conditions (33) and (34) are verified and the... 

To integrate equation (10.149) we must know how dn is related to v. Debye assumed that a crystal is a continuous medium that supports standing (stationary) waves with frequencies varying continuously from v = 0 to v = t/m. The situation is similar to that for a black-body radiator, for which it can be shown that... 

Intermediate values for C m can be obtained from a numerical integration of equation (10.158). When all are put together the complete heat capacity curve with the correct limiting values is obtained. As an example, Figure 10.13 compares the experimental Cy, m for diamond with the Debye prediction. Also shown is the prediction from the Einstein equation (shown in Figure 10.12), demonstrating the improved fit of the Debye equation, especially at low temperatures. 

In the second approximation, Debye and Hiickel introduced the idea that the centers of the ions cannot come closer than a certain minimum distance a, which depends on ion size the ions were now treated as entities with a finite radius. The mathematical result of this assumption are charge densities Qy, which are zero for r[Pg.120]

Thus we have found that the screening should be more efficient than in the Debye-Hiickel theory. The Debye length l//c is shorter by the factor 1 — jl due to the hard sphere holes cut in the Coulomb integrals which reduce the repulsion associated with counterion accumulation. A comparison with Monte Carlo simulation results [20] bears out this view of the ion size effect [19]. 

So far in our revision of the Debye-Hiickel theory we have focused our attention on the truncation of Coulomb integrals due to hard sphere holes formed around the ions. The corresponding corrections have redefined the inverse Debye length k but not altered the exponential form of the charge density. Now we shall take note of the fact that the exponential form of the charge density cannot be maintained at high /c-values, since this would imply a negative coion density for small separations. Recall that in the linear theory for symmetrical primitive electrolyte models we have... 

The Debye-Hiickel limiting law is the least accurate approximation to the actual situation, analogous to the ideal gas law. It is based on the assumption that the ions are material points and that the potential of the ionic atmosphere is distributed from r = 0 to r->oo. Within these limits the last equation is integrated by parts yielding, for constant k, the value ezk/Aite. Potential pk is given by the expression... 

With the proper definitions of ex and k0, this equation is applicable to the metal as well as to the electrolyte in the electrochemical interface.24 Kornyshev et al109 used this approach to calculate the capacitance of the metal-electrolyte interface. In applying Eq. (45) to the electrolyte phase, ex is the dielectric function of the solvent, x extends from 0 to oo, and x extends from L, the distance of closest approach of an ion to the metal (whose surface is at x = 0), to oo, so that kq is replaced by kIo(x — L). Here k0 is the inverse Debye length for an electrolyte with dielectric constant of unity, since the dielectric constant is being taken into account on the left side of Eq. (45). For the metal phase (x < 0) one takes ex as the dielectric function of the metal and limits the integration over x ... 

Coefficients cl-c4 are used to approximate the integral function "J" aphi is the Debye-Huckel constant at 25 C /... 

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