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5. Phonons Thermal Properties

5. Phonons Thermal Properties. Phonon Heat Capacity Anharmonic Crystal Interactions Thermal Conductivity. Phonon Heat Capacity. Planck Distribution Normal Mode Enumeration Density of States in One Dimension Density of States in Three Dimensions Debye Model for Density of States

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5. Phonons Thermal Properties

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  1. 5. Phonons Thermal Properties • Phonon Heat Capacity • Anharmonic Crystal Interactions • Thermal Conductivity

  2. Phonon Heat Capacity • Planck Distribution • Normal Mode Enumeration • Density of States in One Dimension • Density of States in Three Dimensions • Debye Model for Density of States • Debye T3 Law • Einstein Model of the Density of States • General Result for D(ω)

  3. where = Thermal expansivity = isothermal compressibility = 1/ B B = Bulk modulus α = linear (1-D) thermal expansivity

  4. Lattice heat capacity: p = polarization Planck distribution:

  5. Planck Distribution System at constant T Canonical ensemble : Boltzmann factor For a set of identical harmonic oscillators Nn = number of oscillators in the nth excited state when system is in thermal equilibrium Probability of an oscillator in the nth excited state: Occupation number:

  6. Normal Mode Enumeration

  7. Density of States in One Dimension Fixed boundary problem of N+1 particles. N = 10 → with Number of allowed K for non-stationary solutions is N–1 = Number of mobile atoms Polarization p : 1 long, 2 trans

  8. Periodic boundary problem of N particles N = 8 → with → Number of allowed K for non-stationary solutions is N = Number of mobile atoms

  9. fixed B.C. → Periodic B.C. →

  10. Density of States in Three Dimensions Periodic B.C. ; N 3 cells in cube of side L → density of states in K-space is Number of modes per polarization lying between ωand ω + d ωis → density of states in ω-space is For isotropic materials, →

  11. Debye Model for Density of States Debye model: v velocity of sound (for a given type of polarization) For a crystal of N primitive cells: →  Debye frequency

  12. Thermal (vibrational) energy  Debye integrals: See Ex on Zeta functions, Arfken where θ Debye temperature for each acoustic branch

  13. Debye model

  14. Debye T3 Law For low T, xD →  : → for each acoustic branch To account for all 3 acoustic branches, we set and so that Good for T < θ /50

  15. Solid Ar, θ = 92K

  16. c.f. Table for Bulk Moduli, Chap 3, p.52

  17. Qualitative Explanation of the T3 Law Of the 3N modes, only a fraction (KT /KD ) 3 = ( T / θ)3 is excited. →

  18. Einstein Model of the Density of States N oscillators of freq ω0. Diamond Classical statistical mechanics: Dulong-Petit valueCV = 3NkB

  19. General Result for D(ω) Si Debye solid vg ~ 0 Van Hove Singularities

  20. Anharmonic Crystal Interactions • Harmonic (Linear) Waves: • Normal modes do not decay. • Normal modes do not interact. • No thermal expansion. • Adiabatic & isothermal elastic constants are equal. • Elastic constants are independent of P and T. • C → constant for T > θ . Deviation from harmonic behavior → Anharmonic effects

  21. Thermal Expansion 1-D anharmonic potential: c = 1 g = .2 f = .05 Boltzmann distribution High T: → Thermal Expansion

  22. Lattice constant of solid argon

  23. Thermal Conductivity For phonons, JQ = JU . Heat current density: κ = Thermal conductivity coefficient • Key features of kinetic theory (see L.E.Reichl, “A Modern Course of Statistical Mechanics”, §13.4 ): • Quantities not conserved in particle collisions are quickly thermalized to (global) equilibrium values. (e.g., velocity directions & magnitudes ) • Conserved quantities can remain out of global equilibrium (e.g., stay in local equilibrium. They get transported spatially in the presence of a “gradient”. • MFP is determined by collisions that do not conserve the total momenta of particles. Net amount of A(z) transported across the x-y plane at z0in the +z direction per unit area per unit time: Δz = distance above/below plane at which particle suffered last collision. n = particle density, l = mean free path, a = some constant. bA = 1/3 is determined from self diffusion

  24. For heat conduction, we set where c = heat capacity per particle = sound velocity C = nc = heat capacity

  25. Thermal Resistivity of Phonon Gas Harmonic phonons: mfp l determined by collisions with boundaries & imperfections. Anharmonic phonons: only U-processes contribute.

  26. Gas: Elastic collisions. No T required. κ =  Gas: No net mass flow. Inelastic collisions with walls sets up T & n gradients. Finite κ. Crystal: N-processes only. κ = . Crystal: U-processes. Finiteκ.

  27. Umklapp Processes 2-D square lattice Normal processes: Umklapp processes: Energy is conserved: Condition: T > θ: all modes excited → no distinction between N- & U- processes → l 1/T. T < θ: probability of U- processes & hence l1 exp(–θ /2T).

  28. Imperfections Low T → umklapp processes negligible. Geometric effect dominates. Size effect: l > D = smallest dimension of specimen. Dielectric crystals can have thermal conductivities comparable to those of metals. Sapphire (Al2O3): κ ~ 200W cm–1 K–1 at 30K. Cu: max κ ~ 100W cm–1 K–1. Metalic Ga: κ ~ 845W cm–1 K–1 at 1.8K. ( Electronic contributions dominate in metals. ) Highly purified c-NaF

  29. Isotope effect on Ge. Enriched: 96% Ge74. Normal: 20% Ge70 , 8% Ge73 , 37% Ge74 , 8% Ge76.

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