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Which one of the following curves best represents the E vs. f(E) behavior of the hot end of a metal rod demonstrating Seebeck Effect? ( f(E) is the probability of electron occupancy at an energy state E ; EF is the Fermi energy)

  • A.E against f(E): smooth curve with f rising as E rises, with EF marked at f = 0.5 and dashed guides at f = 1
  • B.E against f(E): smooth curve falling from high E at f = 0 to low E at f = 1, passing through EF at f = 0.5
  • C.E against f(E): a sharp step at EF, f = 1 below EF and f = 0 above it, with a dashed line at f = 0.5
  • D.E against f(E): a vertical solid line at f = 0.5 with EF marked and a dashed guide at f = 1

Answer: B

The correct curve is (B).

Electron occupancy follows the Fermi-Dirac function \(f(E) = 1/(1 + e^{(E - E_F)/kT})\). It is close to 1 well below \(E_F\), close to 0 well above, and exactly 0.5 at \(E_F\). At the hot end of the rod, T is higher, so the change from 1 to 0 is spread over a wider energy range of a few kT. Plot (B) shows f falling smoothly from 1 through 0.5 at \(E_F\) towards 0 as E rises.

Plot (A) has occupancy increasing with energy, which is backwards. Plot (C) is the sharp step of absolute zero, appropriate to a cold end rather than a hot one, and (D), with f = 0.5 at every energy, is not a Fermi distribution.

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