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.