Why don't the delocalised electrons in a metal emit light when they hit an atom and change their velocity very quickly (i.e. accelerate)

Question

We know that in metals there is a sea of delocalised electrons which can freely travel around the lattice of metal ions and that these delocalised electrons move around at large speeds, sometimes hitting the ions and bouncing off, thereby changing their velocity very quickly and accelerating.

We also know that if we have a charged particle accelerating then it emits electromagnetic radition. Hence we would expect the bouncing electrons to continually emit this radiation and to be able to detect it. However this doesn't appear to be the case empirically.

I know for lone atoms you can make an argument that since emitting electromagnetic radiation impies losing energy and the electrons in an atom have discrete energy levels you can't get an electron to emit light as it orbits around the nucleus. However for metals you have whole bands of permitted energy levels instead of exact discrete numbers and it's certainly possible for an electron at the top of a band to lose energy and drop to the bottom of its band. So what's the reason behind why we don't see metals emit EM radiation?

Answer 1

Why don't the delocalised electrons in a metal emit light when they hit an atom...

And why don't the electrons emit light when they hit other electrons? The reason is because we neglect the electron-electron interaction in most theories of solids. If that sounds crazy to you, welcome to condensed matter physics.

Similarly, the electron-atom interactions are usually elided into a periodic potential of fixed atomic-like potentials. This is basically the Born-Oppenheimer approximation. Some atomic vibrations can be accounted for via phonons, but higher energy excitations of the atoms are usually neglected (which is usually reasonable unless you shoot in some high energy electron from a transmission electron microscope or something like that). See below for more discussion.

... these delocalised electrons move around at large speeds, sometimes hitting the ions and bouncing off,

The ions are typically accounted for as a periodic potential. Each ion is treated as a fixed potential with no dynamics. This is effectively the Born-Oppenheimer approximation, which (along with neglecting the electron-electron interactions) reduces the many-body physics problem down to a single-particle problem (sometime considered in an "effective" or "mean" or "density functional" potential field).

The single-particle electron wavefunctions, e.g., Bloch waves, are the stationary state solutions for the periodic potential--they don't "hit" or "bounce off" anything. The time-dynamics of a single Bloch wavefunction is a trivial phase factor.

thereby changing their velocity very quickly and accelerating

Nope, this does not happen in the single particle picture of electrons. Or, to the extent it does, any changes in momentum are already accounted for in the periodic potential.