1. Classical Statistical Mechanics:
Paramagnetism in the Canonical Ensemble

2. Classical Statistical Mechanics:
Paramagnetism in the Canonical Ensemble
Paramagnetism occurs in substances where the
individual atoms, ions or molecules possess a
permanent magnetic dipole moment.
The permanent magnetic moment is due to
the contributions from:
1. The Spin or intrinsic moments of the electrons.
2. Orbital motion of the electrons.
3. Spin magnetic moment of the nucleus.

3. Some Paramagnetic Materials
Metals.
Atoms, & molecules with an odd number
of electrons, such as free Na atoms, gaseous
Nitric oxide (NO) etc.
Atoms or ions with a partly filled inner shell:
Transition elements, rare earth & actinide
elements. Mn2+, Gd3+, U4+ etc.
A few compounds with an even number of
electrons including molecular oxygen.

4. Classical Theory of Paramagnetism
Consider a material with N
magnetic dipoles per unit
volume each with moment .
In the presence of a
magnetic field B, the
potential energy of a
magnetic dipole is:
B = 0, M = 0
B ≠ 0, M ≠ 0
This shows that dipoles tend to line up with B.
The effect of temperature, is to randomize the
directions of dipoles. The effect of these 2 competing
processes is that some magnetization is produced.

5. The Canonical Ensemble Probability of
Suppose that B is applied along the z-axis, so that
 is the angle made by the dipole with the z-axis.
The Canonical Ensemble Probability of
finding the dipole along the  direction is
proportional to the Boltzmann factor f():
The average value of z then has the form:
The integration is carried out over the solid angle,
whose element is d. The integration thus takes into
account all possible orientations of the dipoles.

6. z =  cos & d = 2 sin d So: To do the integral, substitute
which gives:
Let cos = x, then sin d = - dx & limits -1 to +1
So:

7. The mean value of z is thus:
“Langevin
Function” L(a)
So: z

8. Variation of L(a) with a.
The mean value of z is thus:
Langevin Function, L(a) z
In most practical situations, a < < 1, so
The magnetization is
N = Number of
dipoles per unit
volume
Variation of L(a) with a.

9. This is known as the “Curie Law”
This is known as the “Curie Law”. The susceptibility is referred as the Langevin paramagnetic susceptibility.
It can be written in a simplified form as:
Curie constant 9