Problems and Solutions on Thermodynamics and Statistical Mechanics

Statidtical Phyaics 2 79

energy per particle, (c) the pressure, (d) the Pauli spin susceptibility. Show
that in Gaussian units the susceptibility can be written as 3pi N/Zp(O)V,
where p(0) is the chemical potential at zero temperature. Assume each
fermion has interaction with an external magnetic field of the form 2poHS,,
where p~ is the Bohr magneton and S, is the z-component of the spin.
( was co flsifl)

Solution:

As the spin of a fermion is $, its z component has two possible di-

rections with respect to the magnetic field: up (I) and down (I). These

correspond to energies 3=p~H, respectively. Thus the energy of a particle

is

& = - P2 * pgH.

2m

At T = 0 K, the particles considered occupy all the energy levels below

the Fermi energy p(0). Therefore, the kinetic energies of the particles of

negative spins distribute between 0 and p(0) - ~BH, those of positive spins

distribute between 0 and p(0) + ~BH, their numbers being

(a) The total number of particles is

With H = 0, we obtain the chemical potential

p(0) = tL" (32;) 2/3.

2m

1 1

2 2

(b) For particles with z-components of spin, - and --, the Fermi

momenta are respectively