Phonon long-wavelength

As already observed for some isotropic polynuclear clusters [30 - 32], slow relaxation of the magnetization in an external magnetic field can occur because of the inefficient transfer of energy to the environment, for example, the helium bath, and consequent reabsorption of the emitted phonon by the spin system. The phenomenon, also known as phonon bottleneck (PB), was first introduced by Van Vleck [33]. It is characteristic of low temperatures, where relaxation is dominated by the direct process between closely spaced levels, and results from the low density of phonons with such a long wavelength to match the small energy separation... 

At very low temperatures, the thermal excitations in liquid 4He (the long-wavelength phonons) are scattered only from the walls of the container with a ballistic movement. 

The dynamical behaviour of the atoms in a crystal is described by the phonon (sound) spectrum which can be measured by inelastic neutron spectroscopy, though in practice this is only possible for relatively simple materials. Infrared and Raman spectra provide images of the phonon spectrum in the long wavelength limit but, because they contain relatively few lines, these spectra can only be used to fit a force model that is too simple to reproduce the full phonon spectrum of the crystal. Nevertheless a useful description of the bond dynamics can be obtained from such force constants using the methods described by Turrell (1972). 

Fig. 3. Dispersion curves of the long-wavelength optical phonons, photons, and polaritons in the centre of BZ 1. In order to demonstrate the connection with the dispersion effects in the region 107 < k < 10 cm-1, the branches of an LO and an LA phonon in this region have been added in a different linear scale. The figures correspond to a cubic lattice with two atoms in the unit cell 3S)...

Table 3.4. Frequencies of long-wavelength optical phonon modes of ZnO bulk samples (b) and ZnO thin films (f)a...

P(co, k) peaks sharply at low k under normal experimental conditions i.e. long wavelength phonons are predominantly excited because the electric field associated with such phonons extends outside of the solid further than short... 

When the lattice becomes infinitely long, N oo, k becomes a continuous parameter. In the long wavelength (small A) limit these phonons should become the familiar sound waves. In this limit, A 0, we can expand Eq. (4.25)... 

The fact that a quantum oscillator of frequency u> does not interact effectively with a bath of temperature smaller than hu>/kg implies that if the low temperature behavior of the solid heat capacity is associated with vibrational motions, it must be related to the low frequency phonon modes. The Debye model combines this observation with two additional physical ideas One is the fact that the low frequency (long wavelength) limit of the dispersion relation must be... 

We now consider a more quantitative model of the vibrational density of states which makes a remarkable linkage between continuum and discrete lattice descriptions. In particular, we undertake the Debye model in which the vibrational density of states is built in terms of an isotropic linear elastic reckoning of the phonon dispersions. Recall from above that in order to effect an accurate calculation of the true phonon dispersion relation, one must consider the dynamical matrix. Our approach here, on the other hand, is to produce a model representation of the phonon dispersions which is valid for long wavelengths and breaks down at... 

Now the equipartion theorem implies that the elastic energy of long wavelength phonons is (L2/2)Bk2 u ) = (1/2)k%T, where B is a constant, and hence eq. (69) becomes... 

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