# How long does it take for a metal to reach equilibrium?

I wonder if there is a measure of how long a piece of metal takes to reach electrostatic equilibrium.

Does it depend on piece's size? Does it depend on the amount of imbalance?

Lots of websites and textbooks report "after a very short time". But how much short?

Thanks a lot

• What do you mean by electrostatic equilibrium? Can you give an example? – John Rennie Nov 15 '13 at 12:42
• What's the input? Depending on how you dump charge onto/into a chunk of metal, the distribution as a function of time will differ. – Carl Witthoft Nov 15 '13 at 12:46
• @JohnRennie when the inner eletric field is zero. – Surfer on the fall Nov 15 '13 at 13:02
• @CarlWitthoft what about conduction? – Surfer on the fall Nov 15 '13 at 13:03

As a first approximation, calculate the drift velocity due to the largest electric field and divide that into the length scale of the object.

## Edit

Actually the drift velocity already assumes a sort of equilibrium, so what you need is the relaxation time for a charge distribution inside a conductor. That link gives $\sim 1.5 10^{-19}s$ for copper.

• Could you just give me some examples? I haven't enough physical and mathematical skills, I just want a rough approximation :) – Surfer on the fall Nov 15 '13 at 13:01
• Mhm, this very short relaxation time surprises me a bit... anyway, I think the question is ill-posed, as John Rennie pointed out, I interpreted it in the sense of electron thermalization, which of course depends on the source too. – Mattia Nov 15 '13 at 15:32
• That relaxation time looks way too short to me. Electron motion over slightly longer timescales ($10^{-18}-10^{-16}$ s) is indeed possible on a single atom/molecule scale. Anything involving solid state will trail that by at least a few orders of magnitude, and macroscopic objects by a few more. – Emilio Pisanty Nov 15 '13 at 15:50
• Well the derivation looks solid and it's the one I remember from school. Remember that it is the time taken for the charge imbalance to decay by a factor of $1/e \approx 0.36$, – lionelbrits Nov 15 '13 at 17:10
• Yeah, but unfortunately nature usually doesn't care for our models ;-) I think your time is right in the context of response to elecrostatic field. The scale to which I and probably Emilio are referring to is for the response to high frequency perturbation. – Mattia Nov 15 '13 at 22:14

Usually the electron thermalization time scale in metals is in the range of femtoseconds to picoseconds (in gold you have a value of ~ 400 fs, for not so strong excitations). To study it you need excitation that are even faster, for example ultrashort laser pulses, which nowadays can be made as short as 3-4 fs at 800 nm wavelength.

The main (or faster) mechanism responsible for this thermalization, or relaxation, is electron-electron scattering. Electron-phonon scattering can also contribute, but on a tipically longer time-scale (few picoseconds).

The size of the object plays an important role when it affects the mean-free-path of the electrons inside the metal. This can happen in very small nanoparticles and in cluster, where the surface potential barriers changes the electron wavefunction. Otherwise what matters are the bulk properties of the metal, in primis its electron density. And the excitation strength, depending on which you can be in a "perturbative regime" (weak), or a non-perturbative one (strong).