Lattice’Vibrations’’.. simple1Dexamplerossgroup.tamu.edu/416/slides11_416.pdf ·...
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Lattice Vibrations simple 1D example • Small displacements, • Harmonic-limit potential energy: • Classical equation of motion determines normal modes. [ ] 1 1 2 + = i i i i u u u K u M ! ! ( ) 2 sin 2 ka M K = Zone 1 2 2 3 3 4 4 ( ) + = j i j i ij o u u K 2 2 1 i i i i u d R r ! ! ! ! + +
Transcript of Lattice’Vibrations’’.. simple1Dexamplerossgroup.tamu.edu/416/slides11_416.pdf ·...
Lattice Vibrations -‐-‐ simple 1D example
• Small displacements,
• Harmonic-limit potential energy:
• Classical equation of motion determines normal modes.
[ ]1121 2 +- --= iiii uuuKuM !! ( )2sin2 ka
MK=w
Zone 12 2 33 44
( )å¹-+=
ji jiijo uuK 221ee
iiii udRr !!!!++º
Lattice Vibrations -‐-‐ simple 1D example
Classical normal modes.
( )2sin2 kaM
K=w
Zone 12 2 33 44
• Solutions can limit to 1st Brillouin zone.Sampling theorem, k & k + G equivalent.
• Mode counting: # modes = N (atoms in crystal)Need periodic boundary conditions.
• Connection to sound velocity, waves on a string(small wavevectors, w = kc )
Mode Counting
• Mode counting: # modes = N (atoms in crystal)Need periodic boundary conditions.
Crystal size L, & assume u(x + L) = u(x)(details of boundary conditions normally not important for large crystal)
result:
3D result: cell arrangement
yields:
cell i
!´= - )( tkxi ieu w1e 2e
ikae2eikae1e
kaie 21eikae-2e
02
212
212 =+-÷øö
çèæ +
- ikaeMK
MK
MKKw
Note correction; sketches reversed.
