PX262 - H5 - metals in a magnetic field

in a magnetic field, the energy of electrons depends on the state:

\begin{matrix} E (\vec{k}, ↑) = \frac{ℏ^{2} k^{2}}{2 m} + γ \\ E (\vec{k}, ↓) = \frac{ℏ^{2} k^{2}}{2 m} - γ \end{matrix}

where, $γ = μ_{B} B$ , and $μ_{B} = \frac{e ℏ}{2 m_{e}}$ is the bohr magneton

need to separate $n_{↑} (E)$ and $n_{↓} (E)$

system reduces energy by polarising, ie. having more $↓$ than $↑$

δ N_{↓} = - δ N_{↑} ≃ n_{↓} (E_{F}) γ

N_{↓} - N_{↑} = δ N_{↓} - δ N_{↑} = 2 δ N_{↓} ≃ 2 n_{↓} (E_{F}) γ = n (E_{F}) γ

the magnetisation, ie. the total magnetic moment:

M = (N_{↓} - N_{↑)} μ_{B} = n (E_{F}) μ_{B}^{2} B

the susceptibility:

χ = \frac{d M}{d B} = μ_{B}^{2} n (E_{F})

summary

the thermodynamic properties of metals are determined by states close to $E_{F} :$
- heat capacity: $C_{V} \sim n (E_{F}) k_{B}^{2} T$
- magnetic susceptibility: $χ = μ_{B}^{2} n (E_{F})$
- also, conductivity
the reason for this is because $k_{B} T ≪ E$ , only excitations close to $E_{F}$ are possible