ssp_12y - 4 Continued Fig. 4.11. The density of states Z()...

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4 Continued
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4-15 Fig. 4.11. The density of states Z ( ω ) is given by the number of states between the surfaces ( q ) = const. and ( q + dq ) = const. Density of states: Z ( ) d = number of states in ( , ω + d ) () Z d Zq dq Zq V q qd q ωω π == + 3 3 2 , cf. eq. (4.33) (4.34) dq dS d q dg r a d d q dq dS grad d q q 3 3 =⋅ = = Z vd S grad q const = = 2 3 . (4.35)
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4-16 Fig. 4.12. Phonon density of states of Si. The dashed line is the density of states in the Debye approximation with θ = 640K (see below). The high peak near 15 x 10 12 Hz is caused by optical phonons (cf. Fig. 4.9). () Z Vd S grad q const ω π = = 2 3 . Z ( ) maximum for grad q = 0 ( van Hove singularity ) particularly at Brillouin zone boundaries (cf. Fig. 4.6, 4.9) Elastic isotropic medium Longitudinal waves : = c L q Transversal waves : = c T q 2 degenerate branches for isotropic medium q surfaces ( q ) = const. spheres: grad c i L T d S q qi const ωπ == = = ,, ; . 4 2 ⇒= = Z Vq c V c i i i ππ 22 1 2 2 2 3 2 Z v cc LL =+ 2 12 2 33 2 (4.36) Z ( ) 2 , final number of modes, e.g. 3 N for monatomic basis cutoff frequency !
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4-17 Fig. 4.13. Phonon density of states () Z h ω for an amorphous metallic alloy (metallic glass) Mg 70 Zn 30 (solid circles) and after crystallization (crosses). The sharp van Hove singularities in the crystalline state are broadened by the resolution of the inelastic neutron scattering technique (see below) used to obtain the data. Distribution of interatomic distances (see Fig. 2.20) Distribution of overlap integrals Distribution of coupling constants f f M sharp frequencies of crystal smeared out in glass. No periodicity No Brillouin zones No van Hove singularities.
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4-18 Thermal energy of an harmonic oscillator Energy eigenvalues (spectrum): En n =+ h ω 1 2 Contact with heat bath at temperature T Probability of exitation of a vibration with energy E n : Pe n E kT n (Boltzmann distribution) (4.37) Average number of phonons of frequency ⇒= n e T kT 1 1 h (Bose distribution * ) (4.38) (derivation see practical course) Average thermal energy of a lattice vibration of frequency () εω , Tn e T kT hh h 1 2 1 2 1 1 (4.39) * For those who are familiar with the subject we mention that eq. (4.38) is the Bose distribution for noninteracting particles which can occupy a given energy state in an unlimited number. The wave quanta (phonons) behave as Bose particles (bosons).
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ssp_12y - 4 Continued Fig. 4.11. The density of states Z()...

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