# Why do regular lattice elements heat up faster?

For example iron. A metal spoon heats up much quicker than a wooden/plastic one.

Why?

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in physics jargon, metals are better conductors. They're better electrical conductors as well as heat conductors. It's because metals have lots of electrons that are shared by the whole material.

Each atom of the metal contributes some electrons that move in the potential of all atoms. The corresponding solution are Bloch waves that are completely delocalized. While respecting the Pauli exclusion principle, the electrons occupy states $\exp(ipx)$ for various values of the momentum $p$.

For electrical conductivity to exist, it must be true that some states of this kind are occupied, but there are still nearby states with slightly different values of $p$ that remain empty. A small voltage is enough to move electrons a little bit, without making "too big amount of work", from the previously filled to the previously empty states - given by different values of $p$. For a voltage, the electrons will pick an asymmetric distribution in which "positive" values of $p$ are chosen by a higher percentage of electrons that the negative values of $p$ or vice versa (the sign is measured in the direction of the electric field). This asymmetry, which is favored because of the $Q\phi$ interactions with the voltage, will make the electric current flow in the direction correlated with the electric field.

In a related way, the metal is also a good heat conductor because the extra heat (energy) is also quickly transmitted to the chaotic motion of the shared electrons near the "Fermi surface" (the boundary between unoccupied and occupied shared states). Moreover, the very regular lattice is also more efficient in transferring the heat. The lattice of atoms is like a system of coupled harmonic oscillators - springs that excite the nearby springs, and so on. They're pretty strong springs.

On the other hand, the wood and plastic materials are typically (electrical and heat) insulators because there are no shared electrons. The electrons are just sharply associated with the individual molecules and then there is a gap (an energy gap). There are no nearby free electron states which the electrons could occupy. You may still imagine that there is a "band" of shared electrons but this band is always completely filled with the electron, and the closest electron states that remain empty are separated by a nonzero, finite energy from the highest-energy occupied states - so you would need a huge amount of work to transfer an electron from one state to another one.

Consequently, the electricity and heat can't get transferred by these electrons, so the heat ultimately propagates from one molecule of "wood" to the other by the interactions between the relevant nuclei (through their electrons, of course). However, the "springs" between the nearby molecules of "wood" or "plastic" - which are also typically much greater - are much weaker because the molecules are not arranged so nicely and tightly - because of the irregularity, there is a lot of lost space. Because the springs are weaker, it is harder for one molecule to move with the adjacent molecule, and the heat doesn't propagate too efficiently.

Best wishes Lubos

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