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Chap 13 Phonons classical theory of vibration 1-dim, 3-dim quantum theory of vibration

Dept of Phys. M.C. Chang. Chap 13 Phonons classical theory of vibration 1-dim, 3-dim quantum theory of vibration phonon specific heat Einstein model, Debye model thermal expansion neutron scattering. One dimensional vibration consider only longitudinal motion

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Chap 13 Phonons classical theory of vibration 1-dim, 3-dim quantum theory of vibration

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  1. Dept of Phys M.C. Chang • Chap 13 Phonons • classical theory of vibration • 1-dim, 3-dim • quantum theory of vibration • phonon specific heat • Einstein model, Debye model • thermal expansion • neutron scattering

  2. One dimensional vibration • consider only longitudinal motion • consider only NN coupling PBC:

  3. k -/a /a Dispersion curve (redundant) The waves with wave numbers k and k+2π/adescribe the same atomic displacement Therefore, we can restrict k to within the first BZ [-π/a, π/a]

  4. Displacement of the n-th atom • Pattern of vibration: • k ～ 0, exp(ikXn) ～1. • Every atom move in unison. Little restoring force. • k ～π/a,exp(ikXn)～(-1)n. • Adjacent atoms move in opposite directions.Maximum restoring force. • Velocity of wave: • k ～ 0, ω= (ωMa/2)k • Linear dispersion, phase velocity = group velocity • k～π/a, group velocity～ 0 PBC Number of “normal modes”: △k=2π/Na Each k describes a normal mode of vibration (i.e. a vibration with a specific frequency) Within the 1st BZ, there are Nk-points.

  5. Vibration of a crystal with2 atoms in a unit cell a

  6. Two branches of dispersion curves (assume M2 > M1) a d c b Patterns of vibration similar See a nice demo at http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html

  7. Three dimensional vibrationAlong a given direction of propagation, there are 1 longitudinal wave and 2 transverse waves, each may have different velocities Sodium (bcc)

  8. cm-1 3D crystal with atom basis FCC lattice with 2-atom basis • Rules of thumb • For a 3-dim crystal, if each unit cell has p atoms, then there are • 3acoustic branches, • 3(p-1) optical branches • If a crystal has N unit cells, then each branch has N normal modes。 • As a result, the total number of normal modes are 3pN (= total DOF of this system)

  9. Quantum theory of vibration Review:1D simple harmonic oscillator (DOF=1) • Classically, it oscillates with a single freq ω=(α/m)1/2 • The energy ε can be continuously changed. Quantization: Note: define then • After quantization, the energy becomes discrete • The number n of energy quanta depends on the amplitude of the oscillator.

  10. Quantization of a 1-dim vibrating lattice (DOF=N) • Classically, for a given k, it vibrates with a single frequency ω(k). • The amplitude ( and hence energy ε) can be continuously changed. Quantization: Fourier transf. Note: then k=2πm/L, L=Na. A collection of N independent oscillators !

  11. Similarly, define eigenstate • The number of energy quanta (called phonons) for the k-mode is nk. • There are no interaction between phonons → “free” phonon gas. • If there are p-atoms in a unit cell (p branches), then thetotal vibrational energy of the lattice is

  12. Specific heat approaches 3R (per mole) at high temperature (Dulong-Petit law) Specific heat drops to zero at low temperature Specific heat: experimental fact After rescaling the temperature by θ(Debye temperature), which differs from material to material, a universal behavior emerges:

  13. Debye temperature In general, a harder material has a higher Debye temperature

  14. Specific heat: theoretical framework • Internal energy Uof a crystal is the summation of vibrational energies (consider an insulator so there’s no electronic energies) • For a crystal in thermal equilibrium, the average phonon number is Bose-Einstein distribution • Therefore, • Specific heat • Density of states (similar to electron energy band) Ex: In 3D

  15. Einstein model (1907) Assume that each atom vibrates independently of each other, and every atom has the same vibration frequency ω0 DOS 3 dim  number of atoms (Activation behavior)

  16. Debye model (1912) Atoms vibrate collectively in a wave-like fashion. • Debye assumed a simple dispersion relation: ω = vsk. Therefore, Also, the 1st BZ is approximated by a sphere with the same volume

  17. Debye temperature = π4/15 as T→ 0 solid Argon (θ=92 K) At low T, Debye’s curve drops slowly because long wavelength vibration can still be excited.

  18. kD kT A simple explanation of the T3 behavior: Suppose that 1. All the phonons with wave vector k<kT are excited with thermal energy kT. 2. All the modes between kT and kD are not excited. Then the fraction of excited modes = (kT/kD)3 = (T/θ)3. energy U～kT3N(T/θ)3, and the heat capacity C～ 12Nk(T/θ)3

  19. Thermal expansion Coeff. Of volume expansion: Bulk modulus: Next page we’ll show that,

  20. Partition function: (Ei are the macroscopic eigen-energies) Grüneisen parameter (γ～1-2 for most materials)

  21. Neutron scattering Why neutron? • Neutron has no charge (can probe bulk properties) • Neutron wavelength comparable to interatomic spacings (1-5 Å) • Neutron energy comparable to phonon’s (5-100 meV) • Neutron has spin (can probe magnetic structure and magnetic excitations) Measure phonon dispersions by neutron scattering More than one phonon mode may be excited out E’,p’ in E, p Conservation of energy Conservation of crystal momentum (for a proof. see App. M of A+M) (momentum of phonon with λ< a must be shifted by G)

  22. k=0 k≠0 One phonon scattering Neutron energy: En=p2/2mn Consider phonon absorption q andωscan be determined from the intersections in the figure.

  23. several solutions at a given direction. → a series of peaks in the data Phonon dispersion curve for Si, comparing experimental data and ab initio calculation (Wei and Chou 1994) • Width of one-phonon peaks due to anharmonic effect • Multi-phonon scatterings give a continuous background

  24. Neutron scattering: formal theory • Fermi golden rule (transition rate) • Neutron states scattered to d3k’ (per unit time) • Differential cross-section • Quantum state of “neutron + crystal” • The energy gained by a neutron due to a phonon in crystal initial: final: Momentum transfer Neutron energy Crystal energy

  25. Neutron-ion interaction potential v(r) is the atomic potential Matrix element Sum over a complete set of phonon states

  26. One can re-write using dynamical structure factor for phonon absorption (always from neutron’s viewpoint) Dynamical structure factor Density operator (for ions) Density correlation function For a crystal at finite temperature

  27. Evaluation of the dynamical structure factor D. Mermin, J Math.Phys. 7,1038 (1966) Translation symmetry of the system Debye-Waller factor Exact so far (for a harmonic crystal) 0-phonon 1-phonon process …

  28. Zero-phonon process • a rough estimate (Kittel. App.A) Debye-Waller factor Elastic scattering Laue’s diffraction condition Differential cross-section Intensity I=I0 e-2W (I0 for a rigid lattice) Scattering length Cf: For more discussion, see A+M, App. N For X-ray scattering (the same S)

  29. A more accurate evaluation (Prob. 7) • In 3D, W weakens the diffraction peaks. • In 2D, W is finite at T=0 but infinite at finite-T. • In 1D, no long-range order even at T=0 Mermin-Wagner theo (Mermin PR1968) There is no long-range crystalline order at finite-T in 2D.

  30. One-phonon process Phonon emission absorption Delta peaks are broadened only if anharmonic effect (phonon-phonon interaction) is included. One-phonon cross-section

  31. Mössbauer effect (1958) 57Fe nucleus (radioactive) 1961 • Natural linewidth 4.65x10-9 (eV) • Recoil energy 1.94x10-3 (eV) Mössbauer found that,by placing emitting and absorbing nuclei in a crystal, you could have (almost) recoilless nuclei with resonant absorption → an extremely sensitive detector for energy shift.

  32. The Pound-Rebka experiment (1959) ν note: precision measurement using atom interferometer ν0 Gravitational freq shift Ref: hyperphysics Müller H et al , Nature 2010

  33. Quinn and Yi, SSP, Sec 2.3 Back to Mössbauer effect Recoilless fraction Put the nucleus at the origin

  34. Using the Debye model Cf: the specific heat calculation Recoilless fraction Recoil energy E(q) ～2x10-3 eV, θ ～ 300 K At low T, Put 57Fe impurities in a diamond (θ=2230 K) Mossbauer Effect in Lattice Dynamics. Chen and Yang 2007

  35. Peierls instability in 1D metal (1955) increase elastic energy, decrease electron energy. Who wins? (Chap 8) For small displacement δ, U～δ ∴ distortion is always favored in 1D

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