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Chapter 37: Problem 60

What fraction of conduction electrons in a metal at absolute zero have energies less than half the Fermi energy?

Short Answer

Expert verified
The fraction of conduction electrons in a metal at absolute zero that have energies less than half the Fermi energy is 1/2.

Step by step solution

01

Understanding the Fermi–Dirac distribution

At absolute zero temperature, all energetic states from the vacuum state, up to a maximum energy, are filled with one electron each (2 electrons per state, 1 in each, + or - spin state). The maximum energy is the Fermi energy \[E_F\]. Thus all energetic states lower than \[E_F\] are filled and higher than \[E_F\] are vacant. This state of matters is known as Fermi Sea.
02

Application of the Principles to the Problem

The problem is asking for the fraction of conduction electrons in a metal at absolute zero that have energies less than half Fermi Energy i.e., \[E_F/2\]. Given every energy level up to the Fermi Energy is filled, this translates into saying any energy less than the Fermi Energy will also be filled. Therefore, at absolute Zero, all states with energy less than \[E_F/2\] will be filled and none of the states with energy greater than \[E_F/2\] but less than \[E_F\] will be occupied.
03

Calculation

Since we know that the metal is at absolute zero, we know that electrons occupy every energy state up to and including the Fermi energy. Hence, the total number of electron states filled up to an energy \[E_F\] is also the total number of conduction electrons. Now, all energetic states below \[E_F/2\] are filled and those above are vacant. Therefore, the fraction of all occupied states (or electrons) with energy less than half the Fermi Energy (\[E_F / 2\]) is simply 1/2.

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Most popular questions from this chapter

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