simple cubic latticeThe Brillouin zone (BZ) is the unit cell of “reciprocal space”

Primitive vectorsOf the reciprocal

latticeG1=A=(2π/a)(100)G2=B=(2π/a)(010)G3=C=(2π/a)(001)

Multiple ways todefine shape of

unit cell.[Wigner-SeitzConstruction

is conventionalfor the reciprocal

lattice]X=(π/a)(100)or similar

primitive translation vectorsa1=a(100) a2=a(010) a3=a(001)

Phy 555 lecture 4 Wednesday Sept. 12, 2007 – HW # 1 due today

“first principles” calculationDensity-functional theory

C. Bungaro, thesis, SISSA, 1995 (with Baroni and Gironcoli)

Phonon dispersion in tungsten metal

Huge elastic peak:

VLab tutorial on Computational Materials/Mineral Physics May 23, 2006Lecture 1: vibrations in molecules and solids, and Bloch’s Theorem

P. B. Allen, SUNY Stony Brook

Ziman Chapter 2 “Lattice Waves” Vlab coverage

1. Harmonic equation of motion2. translational symmetry and Bloch states3. generalized eigenvalue equation Eq. 2.134. monatomic linear chain in d=15. diatomic – acoustic versus optical6. 3 acoustic branches in d=37. Lattice sums; Ewald procedure8. Thermodynamics; lattice specific heat9. Density of states10. Debye model11. Van Hove singularities12. x-ray diffraction13. diffraction with lattice vibrations – inelastic scattering14. one-phonon emission and absorption in diffraction15. Debye-Waller factor16. anharmonicity and thermal expansion17. phonon-phonon (anharmonic) interaction;18. N and U processes19. local modes of vibration at lattice imperfections

Lennard-Jones interatomic potential VLJ(r)

If a particle in this potential has energy E>0, it is unbound. If the energy obeys –ε

Classical harmonic oscillatorhas a natural scale of time (ω-1)but no natural scale of distance.A is arbitrary. Excitation energy E - V0 = kA2/2 is also.

Harmonicspring constk=72ε/r02

Excitationenergy

0.65

Level spacing En+1 – En = ħω

Quantum Harmonic OscillatorQuantum Harmonic Oscillator

2: Thermodynamics of Harmonic Oscillation2: Thermodynamics of Harmonic Oscillation

Lennard-Jones PotentialV(x)/ε = 1/x12 - 2/x6Taylor series (around x=1)First 5 termsV(x)/ε ≈ -1 + 0 + 72(x-1)2/2!

-1512(x-1)3/3!+20664(x-1)4/4!

Level spacing in the L-J potentialwill diminish as n increases.

Compute this by perturbationtheory. Small parameter:

ħω/ε or kBT/ε

01

2

0.65

Predicted oscillatorfrequency shift

⎟⎠⎞

⎜⎝⎛−=Δ

εω

ωh

h 6470E