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

Measurement of the conductivity / resistivity units

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

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).

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

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

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.

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

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

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

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.