Atomic Paramagnetism

Consider a collection of N identical noninteracting atoms, each of which has total angular momentum J. The system is in thermal equilibrium at temperature τ and is in the presence of an applied magnetic field H = Hẑ. The magnetic dipole moment associated with each atom is given by μ = -_gμBJ where g is the gyromagnetic ratio and μ_B is the Bohr magneton. Assume the system is sufficiently dilute so that the local magnetic field at each atom may be taken as the applied magnetic field.

a) For a typical atom in this system, list the possible values of  μ, the magnetic moment along the magnetic field, and the corresponding magnetic energy associated with each state.

b) Determine the thermodynamic mean value of the magnetic moment μ and the magnetization of the system M, and calculate it for J = 1/2 and J = 1.

c) Find the magnetization of the system in the limits H → ∞ and H → 0, and discuss the physical meaning of the results.

SOLUTION

a) The energy associated with the magnetic field is

(1)

where m is an integer varying in the range –J ≤ m ≤ J.

b) From (1) we may find the partition function Z:

(2)

where we define x ≡ gμ_BH/τ. The sum (2) may be easily calculated:

(3)

The mean magnetic moment per dipole ⟨μ⟩ is given by

(4)

Since the atoms do not interact,

(5)

For J = 1/2,

(6)

This result can be obtained directly from (3) and (4):

(7)

For J = 1,

(8)

(9)

c) For large H the magnetization saturates (coth x → 1, x → ∞):

(10)

It is convenient to define the so-called Brillouin function B_J(x)  in such a way that

So,

For small H, H → 0, we can expand coth x:

So,

(11)

The saturation value (10) corresponds to a classical dipole gμ_BJ per atom, where all the dipoles are aligned along the direction of H, whereas the value at small magnetic field H (11) reflects a competition between order (H) and disorder (τ).