• Thermal energy changes with temperature gives an understanding of the heat energy which necessary to raise the temperature of the material

  
  

• The increase in energy of a system is equal to the amount of heat absorbed by the system minus the amount of work done by the system

  
  

  dU = dQ - dW

• Heat capacity- defined as the amount of heat absorbed by a system per unit change in temperature

Heat Capacity

• for infinitesimal process at constant volume

  
  

  

  
  

• at constant pressure

  
  

  

Relation of heat capacity at constant pressure and volume;

can be written in terms of experimentally measured quantities

where the volume thermal expansion coefficient

and the volume compressibility of a solid

• Dulong and Petit law: The value of the heat capacity is
  3Nk = 3R per mole = 25Jmol^-1deg^-1
  (At room temperature)

• Other half of Cv comes from the potential energy which holds the atoms at their equilibrium positions in the solid (the energy depends on the 3N positions)

  
  

• A potential the atoms in a crystal will oscillate with amplitudes small compared to the internuclear distance

  
  

• Energy of the oscillator

  
  

  

• 
  independent of temperature, and classical physics offers no possibility of a temperature dependence.(Agree with the Dulong-Petit law at high temperatures)
• But failed to explain Cv at low temperatures

• Einstein assumed that the atoms are vibrating as harmonic oscillators.
• Assumed Planck's(1990) quantization rule for each oscillator (vibrating oscillators (atoms) in a solid have quantized energies)
  

• Occupational of energy level n: (probability of oscillator being in level n)
  

• Average total energy of a solid :

  
  

  

  
  

• The prediction obtained :

  
  

  

Limiting Behavior of Cv(T)

• High T limit :

  
  

  

  
  

• Low T limit :

  
  
  

  

• Atoms are considered as harmonic oscillator that produce elastic waves with varying frequencies from ω = 0 to ωmax

• Treat solid as continuous elastic medium (ignore details of atomic structure)

• 3N normal modes (patterns) of oscillations

The relations between the energy of a phonon E, the angular frequency ω and the wave vector q are:

• Debye temperature :
  

Specific Heat

• At high temperature limit
  

• At low temperature limit
  
  

• Thermal conductivity is thermal heat/energy being transferred in a material
• When there is a temperature gradient across a solid, thermal energy is transferred from the hotter to the cooler end.

• When heated, electrons, holes and phonon obtain energy larger than the average energy.
• In metals, electrons, holes and phonons can transfer or conduct thermal energy from the hotter areas to the cooler parts.
• In insulations (dielectric materials), only phonon plays a role in delivering energy.

According to Debye
If the vibration of the fixed lattice in the normal mode:

• Phonon distribution does not change with time
• Thermal waves also remain constant and travels in solids at the speed of speed
• Thermal conductivity of lattices becomes infinite

In reality:

• Edge scattering: because of specimen size.
• scattering due to phonon and defects: Lattice defects such as point defects, isotope inhomogeneity and other types of defects.
• Phonon-phonon scattering: crystals becomes anharmonic at temperatures higher than absolute zero.
• Phonon scattering mechanisms causes thermal resistance and cause thermal conductivity to be infinite.

• There is no thermal expansion.
• Adiabatic and isothermal elastic constants are equal.
• The elastic constants are independent of pressure and temperature
• The heat capacity becomes constant at high temperatures, T>θ
• Two lattice waves do not interact; a single wave does not decay or change form with time

• Harmonic approximation, phonons do not interact with each other, in the absence of boundaries, lattice defects and impurities, the thermal conductivity is infinite.
• Anharmonic effects, phonons collide with each other and these collisions limit thermal conductivity which is due to the flow of phonons.

• In general, heat(energy) can be transmitted through a crystal by phonons, photons, free e- or holes, and e-holes pair
• in metals the free e- are the best conductors of heat.
• In insulators phonons are the principal transporters of heat

• Consider the potential energy of 1D monoatomic linear chain of identical atoms
• Forces between only nearest neighbors are assumed.
• Eqn:
  
  

• Consider the three phonon process where phonons with wave vectors k1 and k2 collide, annihilate one another, and create a phonon with wave vector k3
• before the interaction, energy was flowing to the right, and after the interaction, the same amount of energy is still flowing to the right.
• N-process does not alter the direction of energy flow, so it cannot contribute to the thermal resistance of a crystal.

• It is important to appreciate that normal processes do help to maintain thermal equilibrium.

• k not equal to 0
• This process can provide a thermal resistance to phonon flow (provides for a finite thermal conductivity)
• The energy flow for k1 and k2 was to the right, the energy flow for k3 is to the left
• Energy flow has been reversed