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I am reading Introduction to Solid State Physics (by Kittel).

When studying the heat capacity of a metal, conformed by $N$ atoms (each providing one valence electron, which is mobile and capable of electrical conduction), it says that classical statistical mechanics predict that the heat capacity should be $ \frac{3}{2}N k_B$. However, it says that experimental results give around 1% of this value.

Then he says that this can by explained as follows:

"When we heat the specimen from absolute zero, not every electron gains an energy $~k_BT$ as expected classically, but only those electrons in orbitals within an energy range $k_BT$ of the Fermi level are excited thermally"

My question is: why only these electrons are excited?

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  • $\begingroup$ You should google Pauli blocking $\endgroup$
    – Lewis Miller
    Commented May 16, 2020 at 22:21

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Pauli exclusion principle. Before heating (at 0K), all electron energy levels are full below the fermi energy. When heated, only electrons within kT can be excited since they're the only ones that have a free state that is within kT of their current state

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  • $\begingroup$ After heated, can't all electrons 'jump' to their next energy level since it will be unoccupied because the electron above would also have 'jumped' to the next state? I know this doesn't make too much sense, but is there a formal explanation for why this doesn't happen? $\endgroup$
    – Ivan
    Commented May 17, 2020 at 2:30
  • $\begingroup$ No, it makes sense. Yes there will be some free energy states that now those electrons at a lower energy than kT from the Fermi energy can jump up to. What Kittel says is more of an approximation because of what you've said. Look at the graph he provides, after a change in temperature the occupied levels don't go down in a straight diagonal line starting from $E_f-kT$, it curves down (the curve starting from before $E_f-kT$) $\endgroup$
    – baker_man
    Commented May 17, 2020 at 12:36
  • $\begingroup$ Just to add, not all electrons will have a free energy level above them. Those at the top will be the first to be excited, and then only those close to them etc etc, but we've only added a finite amount of energy, not enough to free up an energy level for all electrons. The more energy you add though, the flatter the graph in kittel becomes precisely because of this $\endgroup$
    – baker_man
    Commented May 17, 2020 at 13:15

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