Lecture 20 - Crystals can be represented by a Harmonic...

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Unformatted text preview: Crystals can be represented by a Harmonic Oscillators each atom/molecule is kept at the bottom of the well by the lattice springs V(x) displacement of each particle from its equil.position curvature stifness of spring packing ( ) 2 j i U curvature since atoms remain at the bottom of the well, we can expand in a Taylor series around 0 displacement ( 29 ( 29 = + = + + 1 2 0,0 U , ,.., 0,0,0,... 1 ... 2! j N j i j i j j j i j U U U ( 29 ( 29 distance between atoms f f 2 =k, force constant U at the minimum of the well, =0 U ( 29 ( 29 = + , 1 U , ,.., ; 1 2 2 ij i j i j U k N terms with describe coupling between 2 atoms ij i j k i j linear combination of and potential as a sum of uncoupled modes i j (see lecture in vibrations in polyatomic molecules) NORMAL MODES Normal Modes: collective motion of atoms described by a single parameter = for a large # of particles, # normal modes 3 N ( 29 = + 2 1 U ; 2 m m m U k ( 29 ( 29 = = 2 at the minimum 2 , m m m k k k = m m U ( 29 -- = = ; 3 6 1 U N kT m m Q e q atoms do not exchange places in the lattice for they are distinguishable no need 1 N! o position of q ofvibrational normal mode m -- =- 2 1 m m kT m kT h h e q e ( 29 ---- = = - ; 3 6 2 1 1 m m U N kT kT m kT h h e Q e e = 2 ln to evaluate the energy we use v Q E kT T ( 29 ( 29 ( 29 ---- =-- = = - = - - + - ; 3 6 2 1 3 6 1 2 ; 1 ln ln ln 1...
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Lecture 20 - Crystals can be represented by a Harmonic...

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