# Q & A: measuring free electron density

Most recent answer: 12/03/2014
Q:
i am doing my science project about delocalized electorns. is that the more free electrons a metal has, the more heat it transfer? but i realized that the big factor is how far the electrons go before it hit something, so my idea is wrong. Can you show me a way to find the free electrons? and how to set up a lab to find the number of it. Thankyou so much.
- Tam (age 16)
Herndon, VA, USA
A:
Great question, but the answer can get complicated. For those metals which are well described by free-electron theory, however, you might track down information using two key search phrases: "Drude model" and "Hall effect".

The key idea is this. In a magnetic field at right angles to the electric field,  electron paths curve due to magnetic forces before they hit anything and scatter. The result is some sideways current or voltage, depending on whether the circuit lets that sideways current flow. That's called the Hall effect.  If you have a few electrons with long free paths between scattering, the curvature is important and you get a big Hall effect. If you have the same conductivity but from many electrons with shorter free paths, the Hall effect is reduced.

If the current is carried by 'holes' (missing electrons) it acts like the carriers are positively charged, and the Hall effect has the opposite sign. If there are both electrons and holes, it gets much messier.

So the basic technique is something like this:

Take a thin foil (or evaporated film) of the metal in question. Cut it into a sort of cross shape. Run current one way through the cross. Measure the voltage at right angles in the cross. Now apply a measured magnetic field pointing into the plane of the metal. An extra voltage will appear on the right-angle part of the cross. That's the Hall voltage.

You can calculate the effective concentration of electrons or holes from that, using the more detailed sources I'm sure you can find with the search terms I mentioned.

Good luck on a nice project.

Mike W.

(published on 12/25/2009)

## Follow-Up #1: free electrons and conductivity

Q:
I have a question for my science fair project. I need to know how the number of free electrons per atom affect how well it conducts electricity. Thank you for answering.
- Noah (age 13)
Mannassas, VA, U.S.
A:

The conductivity just goes up proportional to the density of free electrons, if other things remain equal. Usually, however, other things don't remain equal.

For example, the extra free electrons in a semiconductor may come from dopant atoms added to the semiconductor. Those dopant atoms also scatter the free electrons, so the conductivity doesn't go up as much as you might expect when there are lots of them

The electrons also interact with each other. That causes all sorts of interesting effects. For example, in the new high-temperature superconductors there's typically an optimum concentration of free electrons to get superconductivity at high temperatures. Either adding or subtracting free electrons (via doping levels( can destroy the superconductivity.

Mike W.

(published on 12/03/2014)