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1
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84864706703
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edited by, S. Hunklinger, W. Ludwig, G. Weiss, World Scientific, Singapore
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(1990)
PHONONS 89
, pp. 1383
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Sadoulet, B.1
Cabrera, B.2
Maris, H.J.3
Wolfe, J.P.4
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4
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84926929403
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The only exceptions are some of the slow transverse (ST) phonons. The range of wave vectors for these stable phonons is calculated in Sec. IIIC.
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5
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84926910075
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For example, in silicon the time for the average phonon enegy to decrease to 0.1 K via anharmonic processes is of the order of 1 year;
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11
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84926891277
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See Ref. 8, p. 122.
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16
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84926929402
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This discussion assumes that the majority of the recoil energy goes into phonon production, and not into ionization.
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17
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84926910074
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This result can be derived using the expressions connecting the normal mode coordinates and the momentum and displacement of atoms in a crystal lattice which are given in A. A. Maradudin, E. W. Montroll, and G. H. Weiss, Lattice Dynamics in the Harmonic Approximation (Academic, New York, 1963), Chap. 2.
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25
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0001084771
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These excitations are the same as the ``focusons'' originally discussed by
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(1957)
J. Appl. Phys.
, vol.28
, pp. 1246
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Silsbee, R.H.1
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26
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84926948427
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These are dimensionless recoil velocities in the approximate numerical range relevant to dark-matter experiments.
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27
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84926929401
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For a discussion of the Lennard-Jones potential, see C. Kittel Introduction to Solid State Physics (Wiley, New York, 1971), Chap. 3.
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35
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84926910073
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gamma for Ne, Ar, Kr, and Xe is discussed by P. Korpiun and E. Löscher, in Rare Gas Solids, edited by M. L. Klein and J. A. Venables (Academic, New York, 1976), p. 816.
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36
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34248644538
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The results were obtained with a modification of the Brillouin zone-integration method originally devised by G. Gilat and L. J. Raubenheimer [, ] for the calculation of the one-phonon density of states. In the application of this method to the calculation of the decay rate for k vec j -> k vec1vec j1vec + k vec2vec j2 the first Brillouin zone of k vec1vec is divided into small cubic cells with sides of length π /na, where a is the lattice constant and n is typically 30–40. One then finds those cubic cells within which there is a section of the surface on which the quantity ω ( k vec j)- ω ( k vec1vec j1)- ω ( k vec - k vec1vec j2) vanishes. To obtain the contribution to the decay rate from a particular cell one then makes a linear approximation to the variations of ω ( k vec1vec j1) and ω ( k vec - k vec1vec j2) with k vec1vec so that the integral over the cell can be evaluated. The matrix element of the anharmonic potential is taken to be constant throughout each cell.
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(1966)
Phys. Rev.
, vol.144
, pp. 390
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37
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84926929400
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The strong enhancement of the decay rate for longitudinal phonons near to the X point occurs because these phonons have a very large number of possible decay products. This is a special feature of the dispersion relation for phonons in the fcc lattice with nearest-neighbor central forces, and is unlikely to occur in real crystals.
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41
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84926948426
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In this paper the range of wave vectors of the ST phonons that cannot decay was estimated using a very coarse mesh of points in k vec space. Consequently, some fine details of the range were missed in this calculation. For example, the fact that phonons with small wave vectors near to the langle 111 rangle directions can decay was missed.
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47
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If the optical frequencies are higher than this, may still be possible for the optical phonons to decay by a higher-order process.
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By the ``usual sign,'' we mean that is of the same type as occurs in a linear chain of particles interacting via nearest-neighbor forces.
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84926891275
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This follows from the results shown in Fig. 5, together with the parameters listed in Table I. The decay rate has to be less than about 106 s-1.
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53
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See, for example, p. 121 of Ref. 8.
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Note that this is a much stronger condition than the requirement that the elastic scattering rate be less than the anharmonic decay rate for phonons that are allowed to decay.
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84926910071
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There is usually a peak in the density of states in the range near to the highest frequency of the ST phonon spectrum. Consequently, the order of magnitude estimate we have given of the density of states is probably somewhat too low.
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