Spin-Half Paramagnet and Curie Law

Worked Example

Consider a spin ensemble where the dimensionless field scale is b = 2 and the thermal scale is tau = 0.25. We calculate the magnetization and probabilities as follows:

1. The ratio of field to thermal energy is x = b / tau = 2 / 0.25 = 8.
2. The Boltzmann factor for the energy difference is q = exp(-2|x|) = exp(-16) ≈ 1.12535 × 10^-7.
3. Since b > 0, the majority spin state is the "plus" state. The minority "minus" state probability is P(-) = q / (1 + q) ≈ 1.12535 × 10^-7.
4. The majority "plus" state probability is P(+) = 1 / (1 + q) ≈ 0.999999887.
5. The reduced magnetization is m = tanh(8) ≈ 0.999999775.
6. The reduced differential susceptibility is χ = 4q / (tau(1+q)²) ≈ 1.80056 × 10^-6.

Experiment

You can interact with the Spin-Half Paramagnet and Curie Law module to observe how temperature and field affect the system.

1. Set Thermal scale to 1 and Field scale to 1. Note the magnetization is about 0.762.
2. Decrease the Thermal scale to 0.25 while keeping the field at 1. The magnetization increases to near 0.999, demonstrating saturation as thermal fluctuations diminish.
3. Set Thermal scale to 0 and observe the zero-temperature limit, where the moments are perfectly aligned with the field, giving a magnetization of 1.

Limitations

This model assumes non-interacting magnetic moments, meaning it does not capture ferromagnetic phase transitions where spontaneous magnetization occurs even at zero external field. Furthermore, the real volume susceptibility of physical materials requires incorporating the density of spins and the fundamental magnetic moment, whereas this module uses dimensionless reduced units.

References

For a comprehensive textbook treatment of two-state paramagnetism, see:

• Thermodynamics and Statistical Mechanics by Richard Fitzpatrick. This resource covers the partition function, spin probabilities, and the derivation of the Curie law in the weak-field limit.