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Thermal conduction is caused by two different things: Phonon conduction and Electron conduction.

Electrons are thermal conductors because electrons have mass and velocity, ie. they have momentum. Thermal energy can be carried by these electrons. What makes a good thermal conductor through the energy carrying of electrons is good electrical conductor. This is because if the electrons are bouncing all around the inside of the conductor, they are transfering their energy to the molecules/atoms of the conductor and not to the other "side".

More technically, the crystal lattice of the conductor must be free of as many defects as possible, and the lattice atoms must have electrons in the the conduction band.

The other thermal conductor is phonons. Phonons are basically a name we give to the movement of the lattice itself. Think of phonons as waves in a pool. The better the phonon conductivity, the better the thermal conduction. Think of phonons as a rope that is held at noth ends and you whip a wave across it. If there is a lot of tention, the wave will quickly travel to the other side. If it is loose, the wave will travel slowly and maybe not even make it to the other side.

The technical equivalent is high bond strengths in the lattice. High bond strength between the atoms means that the phonons will travel faster because the movement of one atom will greatly affect the movement of the neighboring atoms. This is why diamond is such a good thermal conductor, it has very very strong bonds.

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