At low temperatures ( T is far less than Debye temperature ), the relaxation time is determined by scattering of fixed impurities, defects, sample boundaries, etc . and is roughly constant.
12.
Umklapp scattering requires production of a phonon beyond the Brillouin zone boundary; because of the high Debye temperature of diamond and graphite, the peak in the thermal conductivity of these materials is near 100 K, significantly higher than for most other materials.
13.
However, as shown below, kT _ D is roughly equal to the phonon energy of the minimum wavelength mode, and so we can interpret the Debye temperature as the temperature at which the highest-frequency mode ( and hence every mode ) is excited.
14.
In the low temperature limit, the limitations of the Debye model mentioned above do not apply, and it gives a correct relationship between ( phononic ) heat capacity, temperature, the elastic coefficients, and the volume per atom ( the latter quantities being contained in the Debye temperature ).
15.
Both are usually found by fitting the models to the experimental data . ( The Debye temperature can theoretically be calculated from the speed of sound and crystal dimensions . ) Because the two methods approach the problem from different directions and different geometries, Einstein and Debye scales are "'not "'the same, that is to say
16.
For scattering from acoustic phonons, for temperatures well above Debye temperature, the estimated cross section ? ph is determined from the square of the average vibrational amplitude of a phonon to be proportional to T . The scattering from charged defects ( ionized donors or acceptors ) leads to the cross section { \ Sigma } _ { def } \ propto { \ left \ langle v \ right \ rangle } ^ {-4 }.
17.
Where \ rho ( 0 ) is the residual resistivity due to defect scattering, A is a constant that depends on the velocity of electrons at the Fermi surface, the Debye radius and the number density of electrons in the metal . \ Theta _ R is the Debye temperature as obtained from resistivity measurements and matches very closely with the values of Debye temperature obtained from specific heat measurements . n is an integer that depends upon the nature of interaction:
18.
Where \ rho ( 0 ) is the residual resistivity due to defect scattering, A is a constant that depends on the velocity of electrons at the Fermi surface, the Debye radius and the number density of electrons in the metal . \ Theta _ R is the Debye temperature as obtained from resistivity measurements and matches very closely with the values of Debye temperature obtained from specific heat measurements . n is an integer that depends upon the nature of interaction:
19.
Where ?M?is the mean atomic weight of the atoms in the primitive cell, " V a " = 1 / " n " is the average volume per atom, " T D, " " is the high-temperature Debye temperature, " T " is the temperature, " N " o is the number of atoms in the primitive cell, and ?? 2 G ?is the mode-averaged square of the Gr�neisen constant or parameter at high temperatures.
