Brillouin Function Calculator

Compute the Brillouin function B_J(x) from total angular momentum J and argument x, with saturation and paramagnetism context.

Input

Enter the total angular momentum J (integer or half-integer) and the parameter x to compute the Brillouin function B_J(x).

Angular momentum J

Integer or half-integer (steps of 0.5)

Parameter x

Dimensionless. Example: x = g μ_B J B / (k_B T)

B_J(x) at J = 0.5, x = 2

0.96402758

Saturation (x → ∞)

Low-field slope (J+1)/(3J)

Remaining to saturation

0.03597242

Graph of B_J(x)

The Brillouin function is defined as B_J(x) = ((2J+1)/(2J)) coth((2J+1)x/(2J)) - (1/(2J)) coth(x/(2J)). Here J is the total angular momentum quantum number, taking integer or half-integer values in steps of 0.5.
The argument x is a dimensionless quantity, typically x = g μ_B J B / (k_B T), relating the external magnetic field B to the temperature T. Larger x drives the magnetization toward saturation.
As x → 0 the function approaches a straight line with slope (J+1)/(3J). This linear regime is the origin of the Curie law, where paramagnetic susceptibility is inversely proportional to temperature.
As x → ∞ the function approaches 1, corresponding to saturation magnetization where all spins align with the field.
For J = 1/2 the Brillouin function reduces to tanh(x), and in the limit J → ∞ it reduces to the classical Langevin function coth(x) - 1/x.
For small x, directly subtracting the two coth terms loses precision, so this tool switches to a series expansion to stay accurate. It is intended for learning and quick checks; verify independently for rigorous research use.

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