Lattice vibrations spectrum, Debye

To interpret these observations we must consider the spectrum of lattice vibrations in an ice crystal, since it is in deviations of this spectrum from that of a simple Debye continuum that their explanation will be found. Experimental information about the lattice vibration spectrum can be found from infrared and Raman... 

DEBYE THEORY OF SPECIFIC HEAT. The specific heal of solids is attributed to the excitation of thermal vibrations of the lattice, whose spectrum is taken to be similar to that of an elastic continuum, except that it is cut off at a maximum frequency in such a way that the total number of vibrational modes is equal to the total number of degrees of freedom of the lattice. 

Two of the more direct techniques used in the study of lattice dynamics of crystals have been the scattering of neutrons and of x-rays from crystals. In addition, the phonon vibrational spectrum can be inferred from careful analysis of measurements of specific heat and elastic constants. In studies of Bragg reflection of x-rays (which involves no loss of energy to the lattice), it was found that temperature has a strong influence on the intensity of the reflected lines. The intensity of the scattered x-rays as a function of temperature can be expressed by I (T) = IQ e"2Tr(r) where 2W(T) is called the Debye-Waller factor. Similarly in the Mossbauer effect, gamma rays are emitted or absorbed without loss of energy and without change in the quantum state of the lattice by... 

At or near absolute zero the elastic constants determine the thermal spectrum of lattice vibrations and, consequently, the Debye temperature. Since elastic constants can be determined with high precision, it is useful to calculate the Debye temperature from acoustic wave velocities, i.e. from elastic constants measured near absolute zero. In the Debye theory the characteristic temperature at absolute zero is given (Anderson, 1963) by... 

Variations in lattice vibrations in fine particles with respect to the bulk may arise from (i) the reduced volume leading to lattice softening with resultant decrease of the Debye temperature, (ii) surface effects since the surface atoms are probably more weakly bound than the atoms in the interior, or (iii) changes in the lower and upper cut-off frequencies of the phonon spectrum. The first two phenomena should decrease / while the latter could increase /. In general, one observes a recoil-free fraction in fine particle systems that is much smaller than that of bulk materials. However, most often this is not due to effects of the lattice vibrations but to the motion of the particle as a whole, which indeed drastically lowers the / factor. 

Dulong-Petit limit can be reproduced. In the Debye model, lattice vibrations are approximated as a continuous elastic body considering only acoustic modes, and therefore, phonon spectrum is treated linearly. In... 

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