Brillouin Spin Paramagnetism

The magnetic properties of a paramagnet composed of non-interacting spins are fundamentally tied to quantum mechanics. Each particle with total angular momentum quantum number J (or "spin") has 2J + 1 possible states, determined by its magnetic quantum number m, which ranges from -J to J.

Worked Example

Let us calculate the partition sum and magnetization for a spin J=1/2 particle in a field x=1.

Identify the possible states for J=1/2. The magnetic quantum number m can be -1/2 or +1/2.
Calculate the statistical weight for each state. The weight is given by exp(x · m / J).
- For m = -1/2: weight = exp(1 · (-0.5) / 0.5) = exp(-1) ≈ 0.367879
- For m = +1/2: weight = exp(1 · (0.5) / 0.5) = exp(1) ≈ 2.718281
Calculate the dimensionless partition sum Z by adding the weights:
```
Z = exp(-1) + exp(1) ≈ 3.0861612696304874
```

Calculate the normalized magnetization ⟨ m ⟩ / J:

M = [ (-1) * exp(-1) + (1) * exp(1) ] / Z M = (2.718281 - 0.367879) / 3.086161 ≈ 0.7615941559557649

Experiment

Explore the interactive module to see how the Brillouin function changes with different spin values and external fields. Follow these steps:

Use the Spin select control to set the spin to 1.
Use the Field over temperature select control to choose a field of 4.
Observe the Normalized magnetization approach 1, showing saturation.

Brillouin Spin Paramagnetism

Limitations

This model only considers non-interacting spins. It does not account for exchange interactions or ferromagnetic phase transitions. Real materials often have complex interactions between neighboring magnetic moments that lead to spontaneous magnetization below a critical temperature.

References

For more information on thermal physics, see the reference below.

Paramagnetism - Farside Physics

Brillouin Spin Paramagnetism