On a particle level what happens when thermal conduction occurs within a solid

There are 2 modes of heat transfer in solid materials - atomic movements and electrons.

Like you said, we can have atoms vibrating and transferring their energy to each other. In crystalline materials this happens in a wavelike manner and is called a phonon.

In noncrystalline materials (i.e., polymers, amorphous materials, random alloys, etc.) the vibrations are less wavelike and more random. Wavelike vibrations carry energy better than non wavelike vibrations, which is why crystalline materials have higher thermal conductivity.

In metallic materials, the atoms can still transfer vibrational energy to each other, but the electrons actually make up the majority of heat conduction. Electrons gain kinetic energy and transfer energy to each other by colliding, similar to a gas.

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