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Non-Continuum Energy Transfer: Phonons. The Crystal Lattice. simple cubic. body-centered cubic. hexagonal. a. Ga 4 Ni 3. tungsten carbide. NaCl. The crystal lattice is the organization of atoms and/or molecules in a solid

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the crystal lattice
The Crystal Lattice

simple cubic

body-centered cubic




tungsten carbide


  • The crystal lattice is the organization of atoms and/or molecules in a solid
  • The lattice constant ‘a’ is the distance between adjacent atoms in the basic structure (~ 4 Å)
  • The organization of the atoms is due to bonds between the atoms
    • Van der Waals (~0.01 eV), hydrogen (~kBT), covalent (~1-10 eV), ionic (~1-10 eV), metallic (~1-10 eV)
the crystal lattice1
The Crystal Lattice

potential energy

  • Each electron in an atom has a particular potential energy
    • electrons inhabit quantized (discrete) energy states called orbitals
    • the potential energy V is related to the quantum state, charge, and distance from the nucleus
  • As the atoms come together to form a crystal structure, these potential energies overlap  hybridize forming different, quantized energy levels  bonds
  • This bond is not rigid but more like a spring
phonons overview
Phonons Overview
  • Types of phonons
    • mode different wavelengths of propagation (wave vector)
    • polarization direction of vibration (transverse/longitudinal)
    • branches related to wavelength/energy of vibration (acoustic/optical) heat is conducted primarily in the acoustic branch
  • Phonons in different branches/polarizations interact with each other scattering
  • A phonon is a quantized lattice vibration that transports energy across a solid
  • Phonon properties
    • frequency ω
    • energy ħω
      • ħis the reduced Plank’s constant ħ= h/2π (h = 6.6261 ✕ 10-34Js)
    • wave vector (or wave number) k =2π/λ
    • phonon momentum = ħk
    • the dispersion relation relates the energy to the momentum ω = f(k)
phonons energy carriers
Phonons – Energy Carriers

approximate the potential energy in each bond as parabolic

Because phonons are the energy carriers we can use them to determine the energy storage  specific heat

We must first determine the dispersion relation which relates the energy of a phonon to the mode/wavevector

Consider 1-D chain of atoms

phonon dispersion relation
Phonon – Dispersion Relation

- we can sum all the potential energies across the entire chain

- equation of motion for an atom located at xna is

nearest neighbors

  • this is a 2nd order ODE for the position of an atom in the chain versus time: xna(t)
    • solution will be exponential of the form

form of standing wave

  • plugging the standing wave solution into the equation of motion we can show that

dispersion relation for an acoustic phonon

phonon dispersion relation1
Phonon – Dispersion Relation
  • it can be shown using periodic boundary conditions that

smallest wave supported by atomic structure

- this is the first Brillouinzone or primative cell that characterizes behavior for the entire crystal

  • the speed at which the phonon propagates is given by the group velocity

speed of sound in a solid

  • at k = π/a, vg = 0 the atoms are vibrating out of phase with there neighbors
phonon modes
Phonon – Modes






λmin= 2a

λmax= 2L

note: k = Mπ/L is not included because it implies no atomic motion

  • As we have seen, we have a relation between energy (i.e., frequency) and the wave vector (i.e., wavelength)
  • However, only certain wave vectors k are supported by the atomic structure
    • these allowable wave vectors are the phononmodes
phonon density of states
Phonon: Density of States

more available seats (N states) in this energy level

fewer available seats (Nstates) in this energy level

The density of states does not describe if a state is occupiedonly if the state existsoccupation is determined statistically

simple view: the density of states only describes the floorplan & number of seats not the number of tickets sold

  • The density of states (DOS) of a system describes the number of states (N) at each energy level that are available to be occupied
    • simple view: think of an auditorium where each tier represents an energy level

phonon density of states1
Phonon – Density of States

more available modes k

(Nstates) in this dωenergy level

fewer available modes k

(N states) in this dωenergy level



Density of States:

For 1-D chain: modes (k) can be written as 1-D chain in k-space

phonon occupation
Phonon - Occupation

The total energy of a single mode at a given wave vector kin a specific polarization (transverse/longitudinal) and branch (acoustic/optical) is given by the probability of occupation for that energy state

This in general comes from the treatment of all phonons as a collection of single harmonic oscillators (spring/masses). However, the masses are atoms and therefore follow quantum mechanics and the energy levels are discrete (can be derived from a quantum treatment of the single harmonic oscillator).

number of phonons

energy of phonons

Phononsare bosons and the number available is based on Bose-Einsteinstatistics

phonons occupation
Phonons – Occupation

The thermodynamic probability can be determined from basic statistics but is dependant on the type of particle.



Maxwell-Boltzmann statistics

boltzons: gas distinguishable particles



Bose-Einstein statistics

bosons: phonons

indistinguishable particles



fermions: electrons

indistinguishable particles and limited occupancy (Pauli exclusion)

Fermi-Dirac statistics

phonons specific heat of a crystal
Phonons – Specific Heat of a Crystal

total energy in the crystal

specific heat

  • Thus far we understand:
    • phonons are quantized vibrations
    • they have a certain energy, mode (wave vector), polarization (direction), branch (optical/acoustic)
    • they have a density of states which says the number of phonons at any given energy level is limited
    • the number of phonons (occupation) is governed by Bose-Einstein statistics
  • If we know how many phonons (statistics), how much energy for a phonon, how many at each energy level (density of states) total energy stored in the crystal! SPECIFIC HEAT
phonons specific heat
Phonons – Specific Heat
  • As should be obvious, for a real. 3-D crystal this is a very difficult analytical calculation
    • high temperature (Dulong and Petit):
    • low temperature:
  • Einstein approximation
    • assume all phonon modes have the same energy  good for optical phonons, but not acoustic phonons
    • gives good high temperature behavior
  • Debye approximation
    • assume dispersion curve ω(k) is linear
    • cuts of at “Debye temperature”
    • recovers high/low temperature behavior but not intermediate temperatures
    • not appropriate for optical phonons
phonons thermal transport
Phonons – Thermal Transport

G. Chen

  • Now that we understand, fundamentally, how thermal energy is stored in a crystal structure, we can begin to look at how thermal energy is transportedconduction
  • We will use the kinetic theory approach to arrive at a relationship for thermal conductivity
    • valid for any energy carrier that behaves like a particle
  • Therefore, we will treat phonons as particles
    • think of each phonon as an energy packet moving along the crystal
phonons thermal conductivity
Phonons – Thermal Conductivity

Fourier’s Law

what is the mean time between collisions?

Recall from kinetic theory we can describe the heat flux as

Leading to

phonons scattering processes
Phonons – Scattering Processes

There are two basic scattering types collisions

  • elastic scattering (billiard balls) off boundaries, defects in the crystal structure, impurities, etc …
    • energy & momentum conserved
  • inelastic scattering between 3 or more different phonons
    • normal processes: energy & momentum conserved
      • do not impede phonon momentum directly
    • umklapp processes: energy conserved, but momentum is not – resulting phonon is out of 1stBrillouin zone and transformed into 1stBrillouin zone
      • impede phonon momentum  dominate thermal conductivity
phonons scattering processes1
Phonons – Scattering Processes

Molecular description of thermal conductivity

  • When phonons are the dominant energy carrier:
  • increase conductivity by decreasing collisions (smaller size)
  • decrease conductivity by increasing collisions (more defects)
  • Collision processes are combined using Matthiesen rule  effective relaxation time
  • Effective mean free path defined as
phonons what we ve learned
Phonons – What We’ve Learned
  • Phonons are quantized lattice vibrations
    • store and transport thermal energy
    • primary energy carriers in insulators and semi-conductors (computers!)
  • Phonons are characterized by their
    • energy
    • wavelength (wave vector)
    • polarization (direction)
    • branch (optical/acoustic)  acoustic phonons are the primary thermal energy carriers
  • Phonons have a statistical occupation, quantized (discrete) energy, and only limited numbers at each energy level
    • we can derive the specific heat!
  • We can treat phonons as particles and therefore determine the thermal conductivity based on kinetic theory