I cannot get the correct units for the magnetic susceptibility and the Fermi energy for a free electron gas!
According to "Introduction to Solid State Physics", 8th edition by Charles Kittel, the total magnetization of a free electron gas is (page 317):
$M=\dfrac{N\mu_B^2}{\epsilon_F}B$
where:
- $M$ is the magnetization (SI units: A/m)
- $N$ is the number of electrons,
- $\mu_B$ is the Bohr magneton constant (SI units: J/T),
- $\epsilon_F$ is the Fermi energy (SI units: J), and
- $B$ is the magnetic field strength (SI units: A/m).
The Fermi energy $\epsilon_F$ of the electron gas is given by:
$\epsilon_F=\dfrac{\hbar^2}{2m_e}\left(\dfrac{3\pi^2N}{V}\right)^{2/3}$
where:
- $\hbar$ is the reduced Planck constant (SI units: Js),
- $m_e$ the rest mass of each electron (SI units: kg/(number of electrons)), and
- $V$ the volume (SI units: m$^3$).
According to the above equation the Fermi energy is expressed in Joules$\times$(number of electrons)$^{5/3}$ instead of only Joules. But most importantly, the magnetic susceptibility, $\chi = \dfrac{N\mu_B^2}{\epsilon_F}$, is expressed in units of $\dfrac{\text{Joules}}{\text{Tesla}^2\text{(number of electrons)}^{2/3}}$ instead of being dimensionless!
Where is my understanding wrong? Thank you.
