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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
At absolute zero, all the electrons have energies less than the Fermi energy. Therefore, the fraction of electrons with energies less than half the Fermi energy would also be 1, or all the conduction electrons.

Step by step solution

01

Understanding Fermi-Dirac Distribution

At absolute zero, each energy state up to a certain level which is occupied by one electron and all the higher energy states are empty. This level is the Fermi level. So, at absolute zero, all energy states below the Fermi Energy level are filled.
02

Locating Half the Fermi Energy

Since it is a question about half the Fermi energy, identify this level on the energy scale. At absolute zero, all energy states less than half the Fermi Energy will also be filled.
03

Determining the Fraction of Electrons

The Fermi-Dirac distribution function gives the probability that an energy state at energy level E will be occupied by an electron at temperature T. However, at 0 K, every energy state up to the Fermi energy is occupied by one electron and all states above are empty. Hence the fraction of all electrons having energy less than half the Fermi energy is 1, because all energy states upto the Fermi energy are occupied at absolute zero.

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