Notes7 - ECE215A/Materials206A Fundamentals of Solids for...

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ECE215A/Materials206A Fundamentals of Solids for Electronics E.R. Brown/Winter 2008 1 NOTES 7: Energy of Lattice Waves and their Quantization (Phonons) Classical Analysis So far we have addressed only the mechanical behavior of lattice waves – the description of the wave phenomenology consistent with Newton’s law and the kinematics of the crystal lattice. This led to two different types of plane waves, one (the longitudinal) polarized along the direction of wave propagation, and the other (transverse) polarized in one of the lateral directions in the perpendicular plane. Now we take the analysis one step further to understand the wave energy. In addition to being essential to the statistical mechanics, the energy also leads to a quantum mechanical description of lattice waves and the introduction of the particle dual of lattice waves – the phonon . We start with a monatomic linear lattice of atomic mass m, spacing a, and nearest neighbor interaction (spring constant) C. We consider the longitudinal wave () cos rA t s k a s ω ∆= . The total energy of the wave is the energy of each atom summed over all atoms. From mechanics we know that the instantaneous energy of a mass-spring system has a kinetic energy term associated with each mass and a potential energy term associated with each spring. The kinetic energy depends only on the velocity of the individual masses in motion. The potential term depends only on the force constant and the displacement of the atoms from their equilibrium positions. The kinetic energy associated with the sth mass is 2 11 2 22 dr s KE M M ss dt υ ⎛⎞ →= ⎜⎟ ⎝⎠ The potential energy associated with the sth spring is PE C r r C r r s s s −∆ = ∆ −∆ ++ By summing over all atoms we get the total energy: 2 2 1 s Ut M C r r dt =+ + ∑∑ (instantaneous) (1) For longitudinal wave cos t s k a s , we have sin s A ts k a dt ωω =− and
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ECE215A/Materials206A Fundamentals of Solids for Electronics E.R. Brown/Winter 2008 2 () ( ) { } 2 11 22 2 2 sin cos cos 22 S U t M A t ska CA t ska t ska ka ωω ω =− + The last term has the form 2 cos cos , with t ska ka αβ α α ω β ⎡⎤ −− = − = ⎣⎦ . So we can use the trigonometric identity ( ) cos cos cos sin sin βα −= + 2 2 sin cos cos 1 sin sin S Ut M A C A ωα =+ + ⎢⎥ ( ) 1 1 2 2 2 2 sin cos cos 1 2cos cos 1 sin sin sin sin 2 2 s MA C A +− + + = To get the average, we integrate over the period of wave, τ 21 , o UU t f dt τ π == 2 sin ; cos ; 2 sin cos sin 2 0 oo o dt dt dt dt ττ ααα = = ∫∫ So, 12 1 2 2 2 cos 1 sin 2 cos 2cos 1 sin 44 2 1c o s 42 UM A C A A C A A C A ωβ + + + for linear monatomic lattice we have 2
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Notes7 - ECE215A/Materials206A Fundamentals of Solids for...

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