The Debye Temperature

According to the Debye model, acoustic specttum of a solid consists of 3 N independent waves. A maximal frequency Vm of lattice vibration exists. The characteristic Debye temperature 0 corresponds to this frequency  

Diamond has the highest Debye temperature of 1860 K. The semiconductors silicon and germanium have 625 and 290 K, respectively. For metals, the Debye temperature ranged from 100 K for potassium to 470 K for iron. 

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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 upper limit of the dimensionless variable Vp, is typically written in tenns of the Debye temperature 9, as... 

Figure 3.6 The heat capacity of a solid as a function of the temperature divided by the Debye temperature...

It is noteworthy that it is the lower cross-over temperature T 2 that is usually measured. The above simple analysis shows that this temperature is determined by the intermolecular vibration frequencies rather than by the properties of the gas-phase reaction complex or by the static barrier. It is not surprising then, that in most solid state reactions the observed value of T 2 is of order of the Debye temperature of the crystal. Although the result (2.77a) has been obtained in the approximation < ojo, the leading exponential term turns out to be exact for arbitrary cu [Benderskii et al. 1990, 1991a]. It is instructive to compare (2.77a) with (2.27) and see that friction slows tunneling down, while the q mode promotes it. 

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

Table 3 Room-temperature elastic constants, density, and the Debye temperature p of a number of Pd-Ni-P and Pd-Cu-P bulk amorphous alloys. The elastic moduli are in units of GPa and the density p is in units of g/cm. ...

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.

Figure 1. Scaled thermal conductivity (k) data for several amorphous materials is shown. The horizontal axis is temperature in units of the Debye temperature Tjj. The vertical axis scale K = The value of To is somewhat uncertain, but its choice made by Freeman and...

The Debye temperature is usually high for metallic systems and low for metal-organic complexes. For metals with simple cubic lattices, for which the model was developed, is found in the range from 300 K to well above 10 K. The other extreme may be found for iron in proteins, which may yield d as low as 100-200 K. Figure 2.5a demonstrates how sharply/(T) drops with temperature for such systems. Since the intensity of a Mossbauer spectrum is proportional to the... 

The spectrum recorded at 230 K was discarded in the fit procedure because above 200 K the effective thickness decreases drastically because of a significant softening of protein-specific modes [16]. From the simultaneous fit of the spectra in the temperature range 3.2-200 K, the Debye temperature was determined as do = 215 K. AEq proved to be a temperature-dependent quantity, which is discussed later (see Sect. 9.4.2). 

The three inelastic processes (flow, twinning, and phase changes) all require the shearing of atomic neighbors, so they all tend to occur at the same critical elastic strain (at low temperatures i.e., temperatures below the Debye temperature of the specimen material). As they occur, they interfere with one another, thereby increasing the stress needed for further deformation. 

When the stress (compressive) rises to a value approaching G/10 near the Debye temperature, motion of gliding dislocations tends to be replaced by the formation of phase transformation dislocations. The crystal structure then transforms to a new one of greater density. This occurs when the compressive stress (the hardness number) equals the energy band gap density (gap/molecular volume). 

The discussion so far is for low temperatures that is, temperatures below the Debye temperatures of each crystal type. There is little excitation of individual atoms below the Debye temperature. Above the Debye temperature, the temperature is associated with thermal activation and plays a much more important role, as will be discussed later. 

The behavior of covalent semiconductors is quite different below and above the Debye temperature of a crystal. This was first shown by Trehlov and his colleagues (Gridneva, Mil man and Trehlov, 1972). Figure 5.12 illustrates the... 

Figure 5.11 Hardness vs. temperature for Ge and Si (after Gridneva, Mil man, and Trefilov, 1972). Showing the two regimes above and below the Debye temperatures.

For covalent crystals temperature has little effect on hardness (except for the relatively small effect of decreasing the elastic shear stiffness) until the Debye temperature is reached (Gilman, 1995). Then the hardness begins to decrease exponentially (Figure 5.14). Since the Debye temperature is related to the shear stiffness (Ledbetter, 1982) this softening temperature is proportional to C44 (Feltham and Banerjee, 1992). 

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. 

In the following, we shall describe separately the temperature dependence of the contributions to the thermal conductivity for the two heat carriers . In the case of phonons, the Debye temperature 0D will be taken as a reference in analysing the temperature dependence of the thermal conductivity. 

In the case of a perfect crystalline solid, for temperature well below the Debye temperature 0D, a c/T3 versus T graph would give a constant value c/T3 a 1 /dD. However, most crystals show deviations from the Debye s law, in particular c/T3 versus T presents a maximum. This behavior is present also in amorphous solids where the maximum is more evident and appears at temperatures higher than in crystals [40],... 

Since in our temperature range, the Debye temperature of Ge is 370K [47], the phonon contribution to the heat capacity can be neglected. Hence, the heat capacity of our samples is expected to follow the equation ... 

When Wqi / Wq2 the magnetization recovery may appear close to singleexponential, but the time constant thereby obtained is misleading [50]. The measurement of 7) of quadrupolar nuclei under MAS conditions presents additional complications that have been discussed by comparison to static results in GaN [50]. The quadrupolar (two phonon Raman) relaxation mechanism is strongly temperature dependent, varying as T1 well below and T2 well above the Debye temperature [ 119]. It is also effective even in cases where the static NQCC is zero, as in an ideal ZB lattice, since displacements from equilibrium positions produce finite EFGs. 

Finally, it can be shown from the quantum theory of vibrational energy in the solid state that, at temperatures above the Debye temperature 6D, the density of phonons, np, is inversely related to 0D according to the equation... 

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