# Ohm's Law: Drude Model

For a very long time now, I've been thinking about the Drude Model derivation of Ohm's Law. I know that a rigorous derivation requires a Quantum Mechanical approach. However, the mere fact that the Drude Model churns out the right equation seems to suggest to me that it is at least partially sound from a qualitative viewpoint. A particular assumption strikes me. Electrons are assumed to collide with protons. However, a direct collision doesn't seem possible to me (the kind you think of when you typically think of collisions). And in these electron-proton collisions, the momentum of electrons is assumed to be reset, and protons are assumed to be fixed. The only way I can think of this is that certain electrostatic interactions take place that cause protons to gain all the momentum, and electrons lose it all (for conservation of momentum to hold). And in such a scenario, protons will move very slowly, thus having a negligible effect. Over time, as the momentum on proton builds up, the rate of collisions, and increases. This seems to explain the temperature-resistance relationship at least qualitatively. However, something else that strikes me is how energy is released during the "collisions". I've heard that accelerating charges produce EM waves, and therefore find reason to believe that this is how energy is released.

This also makes Kirchhoff's Loop Rule appear reasonable. I could not accept it at first because it seemed to suggest that a non-conservative "collision" force opposed the work done by the electric force, and I could not think of what would cause such a force at the microscopic level. But, if EM waves are released, this alternate picture of energy loss seems much more reasonable; the work done by the electric force is released as heat. I guess this is "friction" on the macroscopic scale.

I just want to know, does this mechanism seem reasonable, and can I accept it till such time I learn Quantum Mechanics, and learn the true story? Or is the Drude Model simply too flawed to accept?

Also, I want to know if momentum is actually transferred to the proton.

## 4 Answers

Wait a minute. Surrounding the protons are a matrix of electrons. These are the electrons not "cool enough" to exist in the conduction band. I imagine these are what collide with the electrons in the Drude model approach. The closer you get to those electrons, the more they're going to push back by the Coulomb potential. Furthermore, I imagine some Pauli exclusion principle issues might be happening here as well, if the electrons try to get too close. In general, you need to add the inner electrons to your case study.

Interestingly enough, you are right-on with that temperature dependence of the collision rate. You have uncovered why most superconductors are at sub-zero temperatures!

• And yes, as George Harold describes, the momentum is transferred to the heavy protons. This is why shorting a circuit creates more heat. More current=more collisions=more released energy. A qualitative approach to P=VI Commented Jul 22, 2014 at 15:54

The Drude model is fine for thinking about some things. (It is still taught.) The electrons collide with the atoms. (or a bit more precisely, the outer electrons of the atoms.) Since this is a classical picture perhaps it's OK to have a classical picture of the atoms. Imagine they are little sphere's all joined to the other atoms by little springs. When an electron crashes into one it causes the atom to vibrate. (There's not nearly enough energy to free the atom from the lattice (all the springs) nor enough to move inside the outer electrons and hit the nucleus.) Now the vibration of the atom is what we would call heat. The hotter the substance the more the atoms are vibrating. Does this help? Quantum mechanically we have a somewhat different picture of what the scattering looks like. (But maybe that is for a different question.)

• Drude assume that electron collides with the atoms. But it was proved experimentally that it was not the case (determining the mean free path experimentaly). Now, it's prefer to say that electron collide with the vibration of the network, ie phonons, or impurities, etc... ) So the electron do not collide with (every) the atoms network (the mean free path is like 100 times more than distance between two atoms) Commented Jul 22, 2014 at 18:28
• Yeah, I agree with you @sailx, (I did say it was a classical picture.) The big failing (AIUI) of the drude model is that it got the electron density wrong... Instead of all the electrons being involved in transport (both electrical and thermal) only those that where within kT of the Fermi energy were involved. So things like the thermo power/ Seebeck coefficient were way off when calculated with the Drude model. See Chap. 1 of Ashcroft and Mermin. Commented Jul 23, 2014 at 14:00

I can answer that yes, momentum is definitely transferred to the surrounding lattice, though not through a direct scatting of electrons with protons. This is important in increasingly small electronics. When the cross-section of the wire or metal is extremely small, the collisions can be enough to displace the metal, and wreck a circuit. Tungsten is sometimes used as a material, simply because of its large mass, and thus ability to resist displacement.

One thing that the drude model doesn't predict is that electrons collide with crystal defects, rather than the bulk of the crystal. Increases in temperature increase the number of crystal defects, causing the increase in resistance for metals with temperature. I believe the drude model might be accurate or correct "on average" or in the classical limit of quantum mechanics, but don't have the math to back up this claim.

To be honest, the understanding of exactly how electrical signals propagate is a mess. Certain models predict certain phenomena, while failing to predict others. In power line transmissions and high frequency circuits, it's understood that power is carried by propagating electromagnetic waves. This explains how electrons don't have to travel the full length of a wire for signals to transfer. This also explains reflection of electrical signals at impedance mismatches. In explaining electrical devices small enough for quantum mechanics to play a part, it becomes more useful to talk about allowed energy states of electrons, and their energy distribution.

Since other models are also very messy, I think it's worth accepting the drude model at face value, as good enough, even if not necessarily accurate.

Source/Disclaimer: The statement about momentum transfer in microelectronics comes from a professor with extensive industry experience. My knowledge of solid state physics is not nearly as extensive as some of the other people posting here. It's possible that quantum mechanical models of scattering and reflection become more satisfying and predictive with a better background in the material.

• Hi @David, I think it's a bit misleading to say momentum is transferred to the nucleus. (To me that implies that the nucleus is directly involved in the scattering... the electron is hitting it.) Better is to think of a collision with an atom, and that is still not really correct, it's even better to call it a phonon, or collective motion of the entire atom lattice. The Drude model was somewhat successful, but also a complete mistake. The mistakes helped point the way to our current QM understanding. Commented Jul 23, 2014 at 14:10
• I'll try and fix that, so it's not misleading. Commented Jul 23, 2014 at 15:49
• Also, the temperature dependence of the resistance is usually due to phonons, not due to crystal defects. Scattering from crystal defects are actually quite temperature-independent. Commented Aug 1, 2014 at 1:55

In drude model, Electron don't collide with protons. The best way to figure this is that the electron collides with phonons and with impurities. But the correct Drude model do not involve collision with the nucleus.

The collision gives excite phonons, and can be considered as heat from a macroscopic view point.

The Drude model give the right formula only for the conductiviy, but gives other resultats that are wrong (heat capacity). And it's just a coincidence that the formula is the same as with quatum mechanics

• What impurities can possibly be there other than protons and electrons? Yes, even I'm saying they do not collide in the traditional sense. I'm suggesting that they interact electrostatically to give heat. Am I right? Commented Jul 22, 2014 at 14:10
• you've a network of copper for instance (the crystal-model of the solid) In this crystal you can have impurities like other atoms, lack of atoms. you can have defect in the network... and so on. The electron "collide" on thoses things and yes, they are probably electrostatics. The momentum is those given into the phonons which are the oscillations of the network (heat). Commented Jul 22, 2014 at 14:17
• Phonons are vibrations which are caused by collisions. Colliding "with" a phonon is like punching a sound wave perhaps? What caused the sound wave is the source of the sound, what causes a phonon is an actual collision of an electron with other ions. Commented Jul 22, 2014 at 17:28
• Also @user46268, if you want a basic quantum treatment of the electrons with all the Fermi-level nonsense here: en.wikipedia.org/wiki/Degeneracy_pressure#Degenerate_gases. It was eye-opening personally just as I read it. All electrons can only have so much kinetic energy taken before the Pauli exclusion principle is an issue. Commented Jul 22, 2014 at 17:33
• @Andres : Phonons are not created by collisions. Phonons are oscillations of atoms around their equilibrium position. If I have no free electron in my media (like insulator) I still have phonons. However, there is absolutly no collision (except collision of phonons, phonons/impurities ...). Commented Jul 22, 2014 at 18:16