Consider again the gas of N ultrarelativistic free electrons, within a container of volume V ,

Consider again the gas of N ultrarelativistic free electrons, within a container of volume V ,at temperature T , in the pres- ence  of  a  magnetic  field H .  If  we  neglect  the  effects  of  orbital magnetism, the energy spectrum is given by

ε_p^→,σ = Cp — µBHσ,

where µB is the Bohr magneton and σ =±1.

(a)    Show that the Fermi energy of this system may be written

ε_T = A + BH² + O(H⁴ ).

Obtain expressions for the prefactors A and B.

(b)      Show that the magnetization in the ground state can be written in the form 

M = CH + O( H³ ).

 Obtain an expression for the constant C.

(c) Calculate the susceptibility of the ground state in zero field.