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Chapter 37: Problem 10

The Fermi energy in metals is much higher than the thermal energy at typical temperatures. Why does this make the mean speed of conduction electrons nearly independent of temperature?

Short Answer

Expert verified
The Fermi energy in metals is substantially higher than the thermal energy at typical temperatures. As a result, thermal fluctuations usually can't provide enough energy to excite electrons to a significantly higher energy level where they could move faster. Hence, the mean speed of conduction electrons remains nearly independent of temperature.

Step by step solution

01

Define Fermi Energy

Fermi energy, denoted as \(E_F\), is the highest occupied single-particle energy state in a fermion system at absolute zero temperature. For metals, this refers to the highest energy level that an electron can have at absolute zero temperature.
02

Connect Fermi Energy with Thermal Energy

At low temperatures, the thermal energy \(kT\) (where \(k\) is Boltzmann's constant and \(T\) is the temperature) is much less than the Fermi energy \(E_F\). Therefore, most electrons are not excited to higher levels, and so their energy and, by extension, their speed remains nearly unaffected by changes in temperature. This results in the Fermi-Dirac distribution becoming a step function at low temperatures as only electrons with energies very near the Fermi level can absorb the limited available thermal energy to move to an empty state above the Fermi level.
03

Final Understanding

Any change in electron speed due to temperature would be insignificant because of the vast energy gap between the Fermi energy and the energy of most of the electrons. Unless the temperature is raised drastically, it typically does not provide enough thermal energy to excite electrons to a higher energy level, and therefore, the mean speed of conduction electrons remains independent of the temperature.

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