# Pressure of electrons in a metal

Suppose we apply a tensile/compressive stress of, say, 2 atm on a metal rod. If we consider a metal as consisting of electronic and lattice subsystems, could we say the the pressure of the conduction electrons (i.e. the electronic subsystem) is also 2 atm? If not, what is the pressure of the electronic subsystem and what determines that?

I guess that if there is such thing as pressure of conduction electrons it is something different from lattice stress.

## 1 Answer

Stress is a tensor quantity, pressure is a scalar quantity. For non-hydrostatic conditions, pressure is sometimes defined as one third of the trace of the stress tensor (http://www.geol.umd.edu/~candela/press.html). If your metal rod is under tensile/compressive stress, these are not hydrostatic conditions.

The electronic system, say, in metals, is degenerate at room temperature. Its pressure depends on the density (https://en.wikipedia.org/wiki/Electron_degeneracy_pressure).

• Thanks. The relation for "electron degeneracy pressure" in that wikipedia article is derived for a free electron gas, where electrostatic electron-electron and electron-ion interactions are neglected. The article also speaks of the "normal gas pressure" which "at commonly encountered densities" is "so low that it can be neglected". The relation $P=nkT$ is given for this kind of pressure, which is derived for an ideal gas. I'm wondering what happens to these two equations if we take into account the electrostatic interactions. – apadana Nov 18 '17 at 15:56
• @Arham : The free electron gas is a pretty good model of electrons, say, in metals, in spite of the interactions (en.wikipedia.org/wiki/Fermi_liquid_theory), although the explanation of this fact is quite complex. – akhmeteli Nov 18 '17 at 17:51
• A calculation of the compressibility of the free-electron gas (or of the bulk modulus) gives approximately correct values for some of the alkali metals. This may be a bit of a coincidence, but it shows the importance of the Pauli exclusion principle. – Pieter Nov 29 '17 at 13:10