Debye temperatures

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...

Element Debye temperature Element Debye temperature... 

Finally, it can be shown from dre quantum dreoiy of vibrational energy in dre solid state drat, at temperatures above dre Debye temperature 0d, dre density of phonons, p, is inversely related to 6 according to dre equation... 

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 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.

Debye Temperature 436K Specific Heat( (Cp) 37.4 J/mole K... 

Melting Point 2870°C (decomposes by melting incongruently) WC has a large stability domain but reacts with W or W Debye Temperature 493K... 

Lattice Parameter a = 0.452 nm Space Group Fm3m Pearson Symbol cF8 Composition HfNo 7510 HFNj 12 Molecular Weight 192.497 Color greenish yellow X-ray Density 13.8 g/cm Melting Point 3387°C Debye Temperature 421 K... 

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...

This procedure is applicable if the relaxation between the spin states is fast (t<1 X 10 s) and thus the quadrupole doublets of the two spin states collapse into one. It is found that, in the intermediate temperature range, the widths of the two lines are significantly enlarged. This shows that the assumption of fast relaxation strictly does not apply. In spite of this, the areas of the lines ean be well reproduced within the Debye model employing the same Debye temperature for both spin states, p 123 K. 

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... 

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