# A Are there phonons in a free electron gas?

#### bjnartowt

Summary
Is there emission and absorption of phonons within an electron gas lacking an underlying lattice?
It is well-known that the electron gas of volume V has an equation of state p=p(V) and thus has a bulk modulus $$B=-V(dp/dV)$$. Suppose the electron gas had no underlying lattice but was confined. Do phonons emit and absorb in such an electron gas at finite temperature?

The reason I ask is because I have found that if I try to make a Carnot engine with an electron gas as the working fluid, I get above-Carnot efficiencies if I consider the electrons alone to carry thermal energy. I can only conclude that a separate system of phonons exists within the electrons that carry thermal energy.

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#### lsume

My guess is that the light one might see is from traveling photons. I am unaware of photon energy being held in captivity. I may be very wrong but that’s the extent of my knowledge.

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#### ZapperZ

Staff Emeritus
2018 Award
My guess is that the light one might see is from traveling photons. I am unaware of photon energy being held in captivity. I may be very wrong but that’s the extent of my knowledge.
The OP is asking about PHONONS, not PHOTONS. It is why this is the Condensed Matter forum.

Zz.

#### Henryk

Gold Member
It would appear that an electron gas would support sound wave, just like an ordinary gas, however, the electron gas is charged. So in addition to gas pressure, you have to include electric field effects. Therefore, the oscillation of electron gas do exist but they are known as plasma oscillation.
Summary: Is there emission and absorption of phonons within an electron gas lacking an underlying lattice?

I get above-Carnot efficiencies
That, according to the sate of thermodynamics, is impossible. You can derive Carnot cycle efficiency without any reference to any physical system. All you need is the concept of energy and entropy.

#### f95toli

Gold Member
Indeed, a gas of that type would support plasmons (not phonons).

#### SFXMan

Indeed, a gas of that type would support plasmons (not phonons).
Incidentally, in low-dimensional cases, these plasmons afford us a nice description when the Fermi liquid renormalisation scheme stops working, as a highly correlated 1D electron liquid can be described as a gas of non-interacting plasmons.

#### bjnartowt

Thank you all for your replies.

My next question is, naturally, if plasmons can carry thermal energy.

#### Henryk

Gold Member
My next question is, naturally, if plasmons can carry thermal energy
In principle yes, however, typical plasmon energies are in a few eV range that is much larger than kT at ambient temperature (~ 0.025 eV), so probability of plasmon excitation at ambient temperature is very low.
To get some contribution you have to either increase the temperature or lower plasmon frequency (this by lowering plasma density).

"Are there phonons in a free electron gas?"

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