Phys560Notes-11 - Lattice Waves Thus far, static lattice...

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Lattice Waves Thus far, static lattice model. In reality, atoms vibrate even at 0 T because of zero-point vibration. Monatomic Crystals Basis = 1 atom. Lattice: 1, 2, 3 ii i ni  Ra    ; t  rR R uR Actual atomic position = lattice position + vibration ave 0 u r = ionic velocity ( 10 5 cm/sec, typically) electronic velocities (~10 8 cm/sec) Adiabatic approximation : system always in the electronic ground state; i.e., the electons follow the slow ionic motion. Good at moderate temperatures. Potential energy   U ; no need to specify the electronic quantum numbers.      2 1 2 UU U U U      RR R uR R R  1 st term = constant; set to 0. 2 nd term = 0 because U is a minimum at (equilibrium position). Harmonic approximation (keeping 2 nd order terms only): 2 1 2  R R R Terms  34 , Ou Ou anharmonic effects or phonon-phonon interactions (later).      2 11 , 22 U Uu u u u D       RR RR R R , DD  R R translational symmetry ( U and D are periodic) D inversion symmetry D  because  & inversion symmetry D = 3x3 symmetric tensor (for one atom per unit cell) D u  RR RR R R u R D R R u R Equations of motion:   U Mu F D u u R R R R R 
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     M   R uR FR DR R uR  If = same for all R (uniform translation),   0 . 0  R DR R . Shifting origin 0 R DR N atoms. 3 N coupled equations of motion. Normal-mode coordinate transformation to decouple the motions. First step – apply Bloch theorem (lattice Fourier transform). Bloch theorem for electrons: i ew kr k r For lattice vibrations, rR ,  ww kk RR = constant. So, i ue kR . Solution: it e e ; k in 1 st BZ; e = unit polarization vector . 2 i t Me e  R eD R R e  2 i R    kRR R R e Definition: dynamic matrix :     i e R Dk 3x3 symmetric tensor 2 M Dk e e   2 De M e  k Eigenvalue problem: ek = eigenvector,   2 M k = eigenvalue; 3 modes (one atom cell).    1 2 2 ii i ee e    because     D R & 0 R . 2 1 2s i n 2  R k R real D is real and symmetric; it has three real eigenvalues, 2 s M k , for each k , where s = 1, 2, and 3 ( branch index ). The eigenvalues 2 s M k are positive, because U is a minimum. So,   s k = real. Keep positive root only. With s included, 2 ss s M Dk e k ke k .
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Phys560Notes-11 - Lattice Waves Thus far, static lattice...

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