Debye-Einstein Model Research Articles

Extended X-ray absorption fine structure and vibrational dynamics in AgI

Abstract The results of extended X-ray absorption fine-structure measurements performed at the IL, edge in AgI in the temperature range 50–292 K are reported. The temperature dependence of the Debye-Waller factors of the first and second coordination shells of iodine is satisfactorily described by a hybrid Debye-Einstein model, which reproduces the main features of the density of states of vibrational modes.

The Debye–Waller factor for polyatomic solids. Relationships between X-ray and specific-heat Debye temperatures. The Debye–Einstein model

The Debye-Waller factor for a polyatomic crystal is derived in the Debye approximation. If the crystal has a basis of p atoms per lattice point, it is shown that the specific-heat Debye temperature, ΘD, and the X-ray Debye temperature, ΘM, are related by ΘD ≃ ΘMP1/2 in the classical limit. The Debye and Einstein theories are then combined to yield an expression for the Debye-Waller factor of a polyatomic solid. The acoustic phonon modes are described with a Debye approximation, and the optic modes with an Einstein model. For temperatures above the Debye temperature, the Debye and Einstein parts of the Debye-Waller factor have the same dependence on temperature and diffraction vector. Thus, the two parts cannot be distinguished.

Rotational-diffusion anomalies in dye solutions from transient-dichroism experiments

Subnanosecond transient-dichroism experiments have been performed to investigate the rotational diffusion of dyes in solution. Dyes and solvents were chosen in a way to obtain information on the influence of size, shape and hydrogen-bonding abilities either of the solute or the solvent molecules. One finds slow orientational relaxation of di-anionic xanthene dyes in alcohols, while oblate cationic dyes rotate faster in spite of their comparable size. The rotational diffusion times for alcohol solutions exceed the theoretical values predicted by the Debye-Einstein model except for prolate molecules. For a solute molecule with internal mobility the rotational diffusion exhibits a partial slip behaviour. It is shown that the deviations from the Debye-Einstein model are restricted to alcohols since for other solvents either with or without strong hydrogen-bonding abilities the experimental values agree with the hydrodynamic model including the stick-boundary condition. Experiments on erythrosine B reveal the influence of size and shape of the attached solvent molecules.

Lattice dynamical aspect of oxygen isotope partition function ratio for alpha quartz.

A full lattice dynamical method has been applied to the calculation of the oxygen isotopic partition function ratio for α-quartz in order to examine the result of the Debye-Einstein model calculation. The modified Urey-Bradley force field was used as a force field model of α-quartz. The phonon dispersion curves and elastic constants of α-quartz have also been calculated. These calculated values are in fairly good agreement with the available experimental data. The agreement gives additional grounds for the use of the modified Urey-Bradley force field. The wave vector-dependence of Infqtz(q) has been examined along the x, y, and z directions in the Brillouin zone. It has been found that the Infqtz(q) little changes with the wave vector. The changes of the Infqtz(q) along the three directions are only less than 0.5% at 0°C, although the value of Infqtz(q) becomes a minimum at the Brillouin zone center. By virtue of the small wave-vector dependence of Inqtz(q), the reduced partition function ratio for α-quartz given by the direct summation technique converges very rapidly. The reduced partition function ratio for α-quartz has been evaluated in the temperature range between 0 and 550°C by means of the direct summation technique combined with the perturbation method. The obtained values of the oxygen isotope fractionation factor between α-quartz and liquid water below 100°C can be expressed as 103Inαqtz-H2O(1) = -17.287+8.6913(103/T) + 1.8459(103/T)2. This temperature scale gives only 1-1.5 ‰ higher fractionations than that of the earlier Debye-Einstein model calculation. When the small differences of the fractionations between the present and earlier calculations and the error associated with the theoretical calculation are taken into consideration, the Debye-Einstein approximation in evaluating the isotopic partition function ratio for α-quartz can be concluded not to bias seriously the result.