Debye temperature calculation

Although Si02 is a typical inorganic glass, it is also atypical in many ways. Several properties of vitreous silica are known to vary anomalously at low temperatures. Anomaly in the low temperature specific heat is the most notable and well investigated. This is reflected in serious disagreement between Debye temperatures calculated from thermal and acoustic measurements. (thermal) and(elastic) are respectively given by (Anderson and Dienes, 1960),... 

Two nanoparticles with linear particle dimensions Lj = 10a and Lj = 20a (a is the lattice period) are at the same temperature T - 0d/1OO, where 0 is the Debye temperature. Calculate the ratio of the internal energy UJ for the case of the same amonnt of atoms (the zero energy can be neglected). 

The relative intensity of a certain LEED diffraction spot is 0.25 at 300 K and 0.050 at 570 K using 390-eV electrons. Calculate the Debye temperature of the crystalline surface (in this case of Ru metal). 

The Debye temperature of the bulk amorphous alloys was calculated from the relation ... 

The calculated Debye temperatures are also listed in Table 3. From this table, it is clear that the elastic properties of the bulk amorphous Pd-Ni-P and Pd-Cu-P alloys change little with changing composition. The elastic moduli of the Pd-Cu-P alloys are slightly lower than those for the Pd-Ni-P alloys. 

The Debye temperature 9d can be calculated from the slope of the line. The value obtained for Kr is 72 K. This small 9o results from the weak van der Waals forces that hold the Kr atoms together in the solid. 

The Debye temperature, can be calculated from the elastic properties of the solid. Required are the molecular weight, molar volume, compressibility, and Poisson s ratio.11 More commonly, do is obtained from a fit of experimental heat capacity results to the Debye equation as shown above. Representative values for 9o are as follows ... 

Table A4.7 summarizes the thermodynamics properties of monatomic solids as calculated by the Debye model. The values are expressed in terms of d/T, where d is the Debye temperature. See Section 10.8 for details of the calculations. Tables A4.5 to A4.7 are adapted from K. S. Pitzer, Thermodynamics, McGraw-Hill, New York, 1995.

Table 7.8 Summary of results obtained for the four Os Mossbauer transitions studied. The absorber thickness d refers to the amount of the resonant isotope per unit area. The estimates of the effective absorber thickness t are based on Debye-Waller factors / for an assumed Debye temperature of 0 = 400 K. For comparison with the full experimental line widths at half maximum, Texp, we give the minimum observable width = 2 S/t as calculated from lifetime data.

The sigma phases are hard and brittle at below their Debye temperatures, but have some plasticity at higher temperatures. Thus there is some covalent bonding in them, and their glide planes are puckered, making it difficult for dislocations to move in them until they become partially disordered. Their structures are too complex to allow realistic hardness values to be calculated for them. Their shear moduli indicate their relative hardnesses. 

The Debye temperatures of stages two and one were determined by inelastic neutron scattering measurements [33], The total entropy variation using equation 8 is in the order of about 2 J/(mol.K). Although smaller in value, such variation accounts for 10-15% of the total entropy and should not be neglected. We are currently carrying on calculations of the vibrational entropy from the phonon density of states in LixC6 phases. 

Although the Debye model reproduces the essential features of the low- and high-temperature behaviour of crystals, the model has its limitations. A temperature-dependent Debye temperature, d(F), can be calculated by reproducing the heat capacity at each single temperature using the equation... 

An alternative to deriving the Debye temperature from experimental heat capacities is to derive an entropy-based Debye temperature by calculation of the s that reproduces the observed entropy for each single temperature using... 

The second term in (5-4) is the second-order Doppler shift. This is the higher-order term of the Taylor expansion that we ignored in (5-3). Like , it can be calculated in the Debye model. Figure 5.6 shows plots of the second-order Doppler shift for the case of iron and for different values of the Debye temperature. Soft lattice vibrations are expected to decrease the isomer shift, although the effect becomes only significant at temperatures well above 80 K. 

The Debye temperature characterizes the rigidity of the lattice it is high for a rigid lattice but low for a lattice with soft vibrational modes. The mean squared displacement of the atom, , can be calculated in the Debye model and depends on the mass of the vibrating atom, the temperature and the Debye temperature. 

The recoilless fraction, /, has been calculated (13) for monotomic lattices using the Debye approximation. When the specific heat Debye temperatures of the alkali iodides are inserted in the Debye-Waller factor, a large variation of f follows (from 0.79 in Lil to 0.15/xCsI). It is not... 

The dynamic calculations include all beams with interplanar distances dhki larger than 0.75 A at 120 kV acceleration voltage and thickness between 100 A and 300 A for the different zones. The structure factors have been calculated on the basis of the relativistic Hartree - Fock electron scattering factors [14]. The thermal difiuse scattering is calculated with the Debye temperature of a-PbO 481 K [15] at 293 K with mean-square vibrational amplitude

= 0.0013 A following the techniques of Radi [16]. The inelastic scattering due to single-electron excitation (SEE) is introduced on the base of real space SEE atomic absorption potentials [17]. All calculations are carried out in zero order Laue zone approximation (ZOLZ). 

The excess term should allow the total Gibbs energy to be fitted to match that of Eq. (6.3) while at the same time incorporating a return to the inclusion of f 6) and f i) in the lattice stabilities. With the increased potential for calculating metastable Debye temperatures and electronic specific heats from first principles (Haglund et al. 1993), a further step forward would be to also replace Eq. (6.5) by some function of Eq. (6.8). 

The increasing availability of electron energy calculations for lattice stabilities has produced alternative values for enthalpy differences between allotropes at 0 K which do not rely on the various TC assumptions and extrapolations. Such calculations can also provide values for other properties such as the Debye temperature for metastable structures, and this in turn may allow the development of more physically appropriate non-linear models to describe low-temperature Gibbs energy curves. 

The four sites of Aujj exhibit different line intensities, and from the relative site occupations, the Mossbauer f-factors for the different sites could be calculated [24], using standard techniques [91]. These, in turn, could be related to effective Einstein (or Debye) temperatures 0 (or 0 ) associated with the vibrations of the individual sites. An unexpected consequence was that the three surface sites could not be described by a single meaning that the use of... 



Figure 9.15 Dielectric function of water at room temperature calculated from the Debye relaxation model with r = 0.8 X 10 11 sec, eQcl = 77.5, and e0l, = 5.27. Data were obtained from three sources Grant et al. (1957), Cook (1952), and Lane and Saxton (1952).

Consider a mixture of acoustic-mode (rL) and ionized-impurity (r,) scattering. For tL t, we would expect r 0 = 1.18 and for r, tl, rn0 = 1.93. But for intermediate mixtures, r 0 goes through a minimum value, dropping to about 1.05 at 15% ionized-impurity scattering (Nam, 1980). For this special case (sL = i, s, = — f), the integrals can be evaluated in terms of tabulated functions (Bube, 1974). For optical-mode scattering the relaxation-time approach is not valid, at least below the Debye temperature, but rn may still be obtained by such theoretical methods as a variational calculation (Ehrenreich, 1960 Nag, 1980) or an iterative solution of the Boltzmann equation (Rode, 1970), and typically varies between 1.0 and 1.4 as a function of temperature (Stillman et al., 1970 Debney and Jay, 1980). 



Fig. 55. Debye temperature, d, and density of states at the Fermi level, N(Ep), for Y(Ni xCox)2B2C and Y(Ni xCux )2B2C as a function of the Co/Cu substitution level x. Symbols results derived from a relativistic band calculations in the atomic sphere approximation. Curves (in lower panel) rigid band model. After Ravindran...

In order to discuss thermodynamic properties in dilute aqueous solutions at temperatures other than 298.15 K, it is necessary to have the standard enthalpies of the species involved. Over narrow ranges of temperature, calculations can be based on the assumption that Af// values are independent of temperature, but more accurate calculations can be made when Cpm(i) values are known. It is also necessary to take into account the temperature dependencies of the numerical coefficients in equations 3.6-4 to 3.6-6. Clarke and Glew (1980) calculated the Debye-Hiickel slopes for water between 0 and 150°C. They were primarily concerned with electrostatic deviations from ideality of the solvent osmotic... 



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