Curie Constant Calculator
This calculator finds the Curie constant C for a paramagnetic material from three inputs: the number density of magnetic atoms, the Lande g-factor and the total angular-momentum quantum number J. You also get the effective magnetic moment in Bohr magnetons and, if you supply a temperature, the dimensionless magnetic susceptibility. Choose a common paramagnetic ion from the preset list to fill J and g automatically, or enter your own values. Every result updates instantly as you type.
Formula
Worked example
For Mn2+ with g = 2, J = S = 5/2, and n = 8.49 x 10^28 m^-3 (typical dense lattice): mu_eff = 2 * sqrt(2.5 * 3.5) = 5.92 muB. C = (1.257e-6 * 8.49e28 * (9.274e-24)^2 * 4 * 8.75) / (3 * 1.381e-23) ≈ 4.38 K. At 300 K, chi = 4.38 / 300 ≈ 0.0146.
What is the Curie constant?
The Curie constant C characterises how strongly a paramagnetic material responds to an applied magnetic field relative to the disruptive effect of thermal energy. It appears in Curie's Law, M = C * B / T, which states that the magnetisation M of an ideal paramagnetic solid is proportional to the applied magnetic flux density B and inversely proportional to the absolute temperature T. A larger C means stronger magnetisation for the same field and temperature. The constant depends on the density of magnetic atoms in the material, how many unpaired electrons each atom carries, and the geometry of their angular momenta through the g-factor and quantum number J.
How the formula is derived
Starting from quantum statistical mechanics, each atom is treated as a quantised magnetic dipole whose energy levels in a field B are split by Zeeman splitting. Averaging the thermal population of those levels with a Brillouin function and taking the high-temperature (linear) limit yields M = (n * mu0 * g^2 * muB^2 * J*(J+1)) / (3 * kB) * B / T. Comparing with M = chi * B / mu0 and chi = C / T identifies the Curie constant as C = (mu0 * n * muB^2 * g^2 * J*(J+1)) / (3 * kB). Here mu0 is the permeability of free space, muB is the Bohr magneton (9.274e-24 J/T), kB is the Boltzmann constant, n is the number density of magnetic atoms, g is the Lande g-factor, and J is the total angular-momentum quantum number.
Choosing the right g and J values
For first-row transition metal ions (Fe, Mn, Ni, Cu, Co, Cr), orbital angular momentum is largely quenched by the crystal field, so J = S (spin quantum number alone) and g is close to 2.0023 (usually rounded to 2). For rare-earth ions (Gd, Dy, Er, Nd), spin-orbit coupling is strong, so J is not simply S and the full Lande expression must be used: g = 1 + [J*(J+1) + S*(S+1) - L*(L+1)] / (2*J*(J+1)). The reference table above lists the effective magnetic moment for the most common first-row transition metal ions assuming spin-only behaviour. Gadolinium (Gd3+) is included because its half-filled 4f shell gives S = J = 7/2 with negligible orbital contribution, making it one of the best experimental tests of Curie's Law.
Curie's Law vs the Curie-Weiss Law
Curie's Law (chi = C / T) holds for ideal paramagnets well above any magnetic ordering temperature. Real materials deviate because atoms interact with their neighbours. The Curie-Weiss Law adds a correction term: chi = C / (T - theta), where theta is the Weiss temperature (positive for ferromagnets, negative for antiferromagnets). When theta is small compared to T the two laws are nearly identical, but near the ordering transition the chi vs T curve curves away sharply from the pure Curie prediction. This calculator implements the ideal Curie case; if your material has a known Weiss temperature, subtract it from the temperature before computing susceptibility.
Effective magnetic moment for common paramagnetic ions (spin-only, g = 2)
| Ion | Configuration | Unpaired electrons | S | J = S | mu_eff (muB) |
|---|---|---|---|---|---|
| Cu2+ | 3d9 | 1 | 1/2 | 1/2 | 1.73 |
| Ni2+ | 3d8 | 2 | 1 | 1 | 2.83 |
| Co2+ | 3d7 | 3 | 3/2 | 3/2 | 3.87 |
| Fe2+ | 3d6 | 4 | 2 | 2 | 4.90 |
| Mn2+ | 3d5 | 5 | 5/2 | 5/2 | 5.92 |
| Fe3+ | 3d5 | 5 | 5/2 | 5/2 | 5.92 |
| Cr3+ | 3d3 | 3 | 3/2 | 3/2 | 3.87 |
| Gd3+ | 4f7 | 7 | 7/2 | 7/2 | 7.94 |
Values assume pure spin behaviour (L = 0) with g = 2. First-row transition metal ions often match closely; rare-earth ions may differ due to orbital contributions.
Frequently asked questions
What are the SI units of the Curie constant?
In SI the Curie constant has units of K (Kelvin). This follows from Curie's Law chi = C / T: susceptibility chi is dimensionless in SI (it is M / H, both in A/m), and T is in Kelvin, so C carries the unit of Kelvin. Some older texts use Gaussian units where the formula differs by a factor of 4 pi and C has different dimensions, so always check which convention a source is using.
What is the Lande g-factor and how do I find it?
The Lande g-factor relates the total magnetic moment of an atom to its total angular momentum. For a free electron, g = 2.0023. For atoms with only spin angular momentum (L = 0), g is also about 2. When orbital angular momentum is present, g is given by the Lande formula: g = 1 + [J*(J+1) + S*(S+1) - L*(L+1)] / (2*J*(J+1)). For most first-row transition metal ions the crystal field quenches L, so g is close to 2 and J = S. Values can also be measured directly by electron paramagnetic resonance (EPR) spectroscopy.
How does temperature affect the Curie constant?
It does not: the Curie constant C is temperature-independent. Temperature only appears in Curie's Law through the denominator, so the susceptibility chi = C / T changes with temperature while C itself stays fixed. If the susceptibility you measure does change in a way that cannot be described by 1/T, the material is either not an ideal paramagnet, has a Weiss temperature correction, or is undergoing a phase transition.
What is the effective magnetic moment and how is it related to C?
The effective magnetic moment mu_eff = g * sqrt(J*(J+1)) is measured in units of the Bohr magneton (muB). It is the root-mean-square magnetic moment per atom that enters the thermal average. Once you know mu_eff, the Curie constant follows from C = (mu0 * n * muB^2 * mu_eff^2) / (3 * kB). Experimentally, mu_eff is extracted from a plot of 1/chi vs T: the slope gives 1/C, from which mu_eff = sqrt(3 * kB * C / (mu0 * n)) / muB.
Why is Mn2+ listed with the same J as Fe3+?
Both Mn2+ and Fe3+ have a 3d5 electron configuration with five unpaired electrons, giving S = 5/2, L = 0 (half-filled d shell, Hund's first and second rules), and therefore J = S = 5/2. Because both have the same configuration they have the same spin-only effective moment of 5.92 Bohr magnetons and the same Curie constant for a given number density. They differ in other properties such as charge, ionic radius and bonding behaviour.
When does Curie's Law break down?
Curie's Law fails when: (1) the temperature approaches or falls below the magnetic ordering temperature (Curie point for ferromagnets, Neel temperature for antiferromagnets); (2) the magnetic field is strong enough to saturate the magnetisation (non-linear regime); (3) quantum effects between neighbouring spins are significant (exchange interactions), requiring the Curie-Weiss correction; or (4) the atoms are in a singlet ground state with no thermally accessible magnetic levels. In practice, simple salts of transition metal ions obey Curie's Law quite well down to a few Kelvin above their ordering temperature.