Curie’s Law Calculator (paramagnetic susceptibility χ = C/T and magnetization)

Understanding the Curie's law calculator — concept and principle

Curie's law describes how a paramagnetic material magnetises in a field: the weaker the temperature's scrambling effect, the stronger the magnetisation. This calculator finds the Curie constant, susceptibility and magnetisation from the density of magnetic ions, their g-factor and angular momentum, and checks the result against the exact Brillouin function.

At room temperature thermal jiggling wins, so 1 T aligns only 0.67% of the moments: M = 2,180 A/m.

How it is calculated

Curie constant C = μ₀ n gJ² J(J + 1) μB² ÷ (3kB), with n the number of magnetic ions per m³.
Susceptibility χ = C ÷ T (SI, volume); magnetisation M = χB ÷ μ₀ in weak fields.
Effective moment = gJ √(J(J + 1)) μB.
Saturation Ms = n gJ J μB; exact M = Ms BJ(x) with x = gJ J μB B ÷ (kB T) and the Brillouin function BJ.
The Curie law is valid when x ≪ 1.

The principle

Each magnetic ion is a tiny magnet with quantised orientations. A field favours alignment; thermal energy kBT randomises it. When the magnetic energy is much smaller than kBT, the net alignment is proportional to B ÷ T, which is Curie's law. At low temperature or strong fields, alignment approaches its maximum, the moments saturate and the Brillouin function is needed. Interactions between moments, which cause ferromagnetism or Curie–Weiss behaviour, are ignored here.

Worked example

Default inputs: n = 0.5 × 10²⁸ per m³, gJ = 2, J = 3.5 (as in Gd³⁺), T = 300 K, B = 1 T.

Curie constant C ≈ 0.822 K; susceptibility χ = 0.822 ÷ 300 ≈ 0.00274.
M = 0.00274 × 1 ÷ μ₀ ≈ 2,180 A/m.
Effective moment = 2 × √(3.5 × 4.5) ≈ 7.937 μB.
Ms = 5 × 10²⁷ × 2 × 3.5 × 9.274 × 10⁻²⁴ ≈ 324,600 A/m; M is only 0.6717% of saturation.
x ≈ 0.01567, so the Brillouin result is also 2,180 A/m: the Curie law is valid.

Each point is the calculator re-run with only “Landé g-factor g_J” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Landé g-factor g_J	Magnetization M, Curie law (A/m)
0.5	136.3
1	545.1
1.5	1,226
2	2,180
2.5	3,407
3	4,906
4	8,721

How to read the result

At room temperature even a 1 T field aligns well under 1% of the moments, which is why paramagnets are only weakly magnetic. Cooling to a few kelvin or applying many tesla pushes x towards 1 and the magnetisation towards saturation, which is exploited in magnetic refrigeration and low-temperature physics.

Common mistakes

Confusing SI and CGS susceptibilities (they differ by 4π).
Using the ion density per cm³ instead of per m³.
Applying Curie's law at low temperature or high field without checking x.
Using g = 2 for ions with strong orbital contributions.

Keep in mind: Assumes non-interacting magnetic moments. Real materials may show crystal-field effects and interactions that change the result.

Frequently asked questions

What is Curie's law?

χ = C ÷ T: the susceptibility of an ideal paramagnet is inversely proportional to temperature.

When does Curie's law fail?

At low temperature or strong field, when moments saturate, and in materials with strong interactions.

What is the Brillouin function?

The exact quantum expression for the average alignment of moments with angular momentum J in a field.

Sources: Kittel, C., Introduction to Solid State Physics, 8th ed. Wiley (Curie law, Brillouin function) · Blundell, S., Magnetism in Condensed Matter. Oxford University Press

Checked 9 Nov 2026. P. Hadley, Advanced Solid State Physics, “Paramagnetism” (TU Graz): Curie law χ_m = C/T, χ_m ≈ nμ₀g_J²J(J+1)μ_B²/(3k_BT), Brillouin function B_J(x), saturation M_s = ng_Jμ_BJ. Constants: standard values (CODATA 2022); k_B exact.

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