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

Primitive vectors Of the reciprocal

lattice G1=A=(2π/a)(100) G2=B=(2π/a)(010) G3=C=(2π/a)(001)

Multiple ways to define shape of

unit cell. [Wigner-Seitz Construction

is conventional for the reciprocal

lattice] X=(π/a)(100) or similar

primitive translation vectors a1=a(100) a2=a(010) a3=a(001)

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

“first principles” calculation Density-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, 2006 Lecture 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 motion 2. translational symmetry and Bloch states 3. generalized eigenvalue equation Eq. 2.13 4. monatomic linear chain in d=1 5. diatomic – acoustic versus optical 6. 3 acoustic branches in d=3 7. Lattice sums; Ewald procedure 8. Thermodynamics; lattice specific heat 9. Density of states 10. Debye model 11. Van Hove singularities 12. x-ray diffraction 13. diffraction with lattice vibrations – inelastic scattering 14. one-phonon emission and absorption in diffraction 15. Debye-Waller factor 16. anharmonicity and thermal expansion 17. phonon-phonon (anharmonic) interaction; 18. N and U processes 19. 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 oscillator has a natural scale of time (ω-1) but no natural scale of distance. A is arbitrary. Excitation energy E - V0 = kA2/2 is also.

Harmonic spring const k=72ε/r02

Excitation energy

0.65

Level spacing En+1 – En = ħω

Quantum Harmonic OscillatorQuantum Harmonic Oscillator

2: Thermodynamics of Harmonic Oscillation2: Thermodynamics of Harmonic Oscillation

Lennard-Jones Potential V(x)/ε = 1/x12 - 2/x6 Taylor series (around x=1) First 5 terms V(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 potential will diminish as n increases.

Compute this by perturbation theory. Small parameter:

ħω/ε or kBT/ε

0 1

2

0.65

Predicted oscillator frequency shift

⎟ ⎠ ⎞

⎜ ⎝ ⎛−=Δ

ε ω

ω h

h 64 70E