1. Thermal conduction in equilibrium heating heat bath T A
______
heat on one side, heat bath on another side
in the middle deltaT and deltax, kappa is the thermal conductivity
Similarity to electrical conductivity, Ohm’s law

2. Measurement of the conductivity / resistivity
units

3. Thermal conduction in equilibrium heating heat bath T current density
1/field
the total thermal conductivity is the sum of phonon (vibrational) and electron contribution
electrical

4. Thermal conduction (at room temperature)
Not only the electrons are important.
A few insulators are also found to be good thermal conductors.
The range of electrical conductivity of materials is much bigger than the range of thermal conductivity (24 vs 5 orders of magnitude)
κ (Wm-1K-1)
diamond
2000
copper
400
gold
310
aluminium
230
silicon
160
sodium
140
glass
1.0
polystyrene
0.02
for electrical conductivity the electrons dominate.
Kind of obvious because the phonons have no charge!
tea spoons should not be made out of diamond
diamond on tongue for testing if it is genuine (feels cold, don’t swallow).

5. Thermal conduction by phonons
scattering from defects or impurities
scattering from the sample boundaries
at any temperature the mean free path is limited by
direct backscattering is even worse!
even if there are no impurities, scattering will be present

6. Thermal conduction by phonons
at high temperature the mean free path is limited by
scattering from other phonons (this an anharmonic effect).
harmonic waves don't scatter from each other, see water waves, light waves (not laser), principle of superposition

7. Thermal conduction by phonons
the mean free path increases at low T
the heat capacity decreases at low T
there is a maximum in the conductivity at about 10% of θD
the curve is for Si. At 60 K even higher kappa than diamond at RT. Very similar to diamond, very pure. Curve also very similar to diamond but Theta_D much lower (645 K), so maximum at lower T.

8. Thermal expansion the dimension for both a K-1.
volume expansion
α (10-5 K-1)
Lead
2.9
Aluminium
2.4
Brass
1.9
Copper
1.7
Steel
1.1
Glass
0.9
Invar
0.09
linear expansion
the dimension for both a K-1.
1/3 only valid for isotropic solids
because V=L^3=L_0^3 (1+alpha T)(1+alpha T)(1+alpha T)
vanishes at 0 which is an exercise
alpha very small but sometimes important, in particular if long enough or detaT big enough

9. Thermal expansion on the atomic scale (classical)
sufficient to look at the classic harmonic oscillator.
For zero energy (at 0K) the oscillator is at the bottom, no movement. This is where we define the energy zero.
For finite temperature, the energy is increased by kT.
Quantum: almost the same, non-equidistant levels non-symmetric prob. distr.
again: note that for the harmonic solid which we usually like to look at, there is no expansion

10. Thermal expansion on the atomic scale (classical)
sufficient to look at the classic harmonic oscillator.
For zero energy (at 0K) the oscillator is at the bottom, no movement. This is where we define the energy zero.
For finite temperature, the energy is increased by kT.
Quantum: almost the same, non-equidistant levels non-symmetric prob. distr.
again: note that for the harmonic solid which we usually like to look at, there is no expansion

11. Thermal expansion on an atomic scale
The bottom line: for the harmonic solid, there is no thermal expansion. Thermal expansion is caused by anharmonicity.
For a quantum treatment, we get the same qualitative result.