Question

A system contains an ideal gas of atoms with spin 1/2 in a magnetic field B(r). The concentration of the spin up (down) particles is n ↑ (r)(n ↓ (r)). The temperature is T (i) Evaluate the total chemical potential for the spin up and down particles. (ii)These two chemical potentials have to be the same and independent of r. Explain why. (iii)Calculate the magnetic moment of this gas as a function of position.

Answer 1

The energy of an atom with magnetic moment in a magnetic field is
,

where is the angle between and .
In Quantum Mechanics,

,

where is the spin magnetic quantum number.

Hence

.

For spin half particles , for spin up particles and for spin down particles.

Hence total energy of the system is

,

where V is the volume of the system and and are the energies of a spin up atom and a spin down atom respectively.

By substituting , , and , we obtain

.

Note that (total number of spin up particles) and   (total number of spin down particles) must be independent of r .

(i) The chemical potential for the spin up particles is

.

Similarly, the chemical potential for the spin down particles is

.

(ii) The change in energy of the system due to addition or removal of a spin up or a spin down particle must be same and independent of the position r. As a result the absolute values of the two chemical potential should be same and independent of r.

(iii) The magnetic moment of the system can be obtained as

.