Ultrarelativistic Electron Gas
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 33
10-9-2016
1632
Ultrarelativistic Electron Gas
Derive the relation between pressure and volume of a free ultrarelativistic electron gas at zero temperature.
SOLUTION
The fact that the gas is ultrarelativistic implies that the energy of the electron is large compared to its rest energy mc2. In this case, the dispersion law is linear: ε = cp. The number of quantum states is the same as for the nonrelativistic case considered:
However, the Fermi energy now is different since ε = cp, and
Hence,
(1)
The total energy is
After substituting εF from (1), we obtain
So, the pressure is
Hence, for an ultrarelativistic gas we have PV = 3/E, the same as for massless particles (e.g., photons), which is not surprising since the dispersion law is the same.
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