Curie's Law Calculator
Enter a Curie constant and temperature to find the magnetic susceptibility and magnetization of a paramagnetic material, or switch modes to back-solve for any variable. Choose a preset ion such as Fe³+ or Gd³+, or enter a custom Curie constant. The live chart shows how susceptibility falls as temperature rises, and the steps panel walks through every calculation.
What is Curie's Law?
Curie's Law states that the magnetic susceptibility (chi) of a paramagnetic material is inversely proportional to its absolute temperature: chi = C / T, where C is the material-specific Curie constant and T is the temperature in kelvin. Pierre Curie discovered the relationship empirically in 1895 while studying the temperature dependence of magnetism in paramagnetic salts. The law captures the competition between two effects: the applied magnetic field tries to align the tiny magnetic dipoles (atomic or molecular) of the material, while thermal energy tries to randomise them. At higher temperatures, thermal agitation wins and the susceptibility falls.
How to use this calculator
Use the 'Solve for' menu to choose which quantity you want to find. In the default mode the calculator finds the magnetic susceptibility given the Curie constant and temperature. Switch to 'Magnetization' to get M = chi * H by supplying the applied field as well. Choose 'Temperature' or 'Curie constant' to back-solve from a measured susceptibility. The 'Effective magnetic moment' mode derives mu_eff (in Bohr magnetons) from a measured chi and T, allowing experimental identification of the spin state of a paramagnetic centre. The material preset menu autofills the Curie constant for eight common paramagnetic species from Cu²+ to Dy³+. The live chart below the results shows how susceptibility varies with temperature from 10 K to 1000 K for your current Curie constant.
The Curie constant and its quantum mechanical origin
The Curie constant is not an empirical fitting parameter. Quantum mechanics gives C = (mu_0 * N * mu_eff^2) / (3 * k_B), where N is the number of paramagnetic centres per cubic metre, mu_eff is the effective magnetic moment in SI units, k_B is Boltzmann's constant, and mu_0 is the vacuum permeability. For a molar quantity (N = Avogadro's number), this yields values in K*m^3/mol. The effective magnetic moment depends on whether the atom is in a spin-only regime (mu_eff = 2*sqrt(S*(S+1)) * mu_B) or a full Russell-Saunders regime that includes orbital angular momentum (mu_eff = g_J * sqrt(J*(J+1)) * mu_B). Transition-metal ions typically fall somewhere between these limits because crystal fields partially quench orbital contributions.
Curie-Weiss Law and the limits of validity
Curie's Law works well when the material is well above any magnetic ordering temperature. Near a ferromagnetic or antiferromagnetic phase transition the law breaks down, and the more general Curie-Weiss Law is needed: chi = C / (T - theta), where theta is the Weiss constant. A positive theta (Curie-Weiss temperature) signals ferromagnetic exchange interactions; a negative theta signals antiferromagnetic interactions. The ratio T/|theta| > 10 is a common rule of thumb for when the simpler Curie form is adequate. For most paramagnetic transition-metal salts above about 100 K, Curie's Law gives agreement with experiment to within a few percent.
Practical applications of paramagnetism
Curie's Law underpins several important technologies. In cryogenic thermometry, susceptibility measurements of paramagnetic salts such as cerium magnesium nitrate (CMN) provide accurate temperature readings below 1 K, where gas thermometers fail. In analytical chemistry and coordination chemistry, chi*T plots distinguish spin states and reveal the number of unpaired electrons in transition-metal complexes - a standard tool in magnetochemistry. Oxygen sensors in medical and industrial settings exploit the fact that O2 is paramagnetic while N2 is not: the strong susceptibility difference is proportional to O2 partial pressure. Magnetic resonance imaging (MRI) uses paramagnetic contrast agents to shorten relaxation times and enhance tissue contrast.
Curie constants and effective magnetic moments of common paramagnetic ions
| Ion / Material | Electronic config | Spin state | mu_eff (mu_B) | C (K·m³/mol) |
|---|---|---|---|---|
| Cu²+ | d9 | S = 1/2 | 1.73 | 0.375 |
| Ni²+ | d8 | S = 1 | 2.83 | 1.000 |
| Co²+ | d7 | S = 3/2 | 3.87 | 1.875 |
| Fe³+ / Mn²+ | d5 | S = 5/2 | 5.92 | 4.375 |
| Gd³+ | 4f7 | J = 7/2 | 7.94 | 7.875 |
| Dy³+ | 4f9 | J = 15/2 | 10.65 | 14.175 |
| O₂ gas | biradical | S = 1 | 2.83 | 1.003 |
Spin-only effective moments and molar Curie constants for representative transition-metal and rare-earth ions. Real values may differ due to orbital contributions and crystal-field effects.
Frequently asked questions
What is Curie's Law and when does it apply?
Curie's Law (chi = C / T) describes the magnetic susceptibility of paramagnetic materials - those with unpaired electrons that weakly attract magnetic fields but have no long-range magnetic order. It applies when the temperature is well above the magnetic ordering temperature (Weiss constant), typically T >> theta. For most dilute paramagnetic salts and transition-metal compounds above about 50 K, the law gives good results. Below that range, or near a phase transition, the Curie-Weiss Law (chi = C / (T - theta)) is needed.
What are typical values of the Curie constant?
Molar Curie constants (in SI units of K*m^3/mol) range from about 0.375 for Cu^2+ (one unpaired electron, S = 1/2) to roughly 14 for Dy^3+ (many unpaired f electrons). The constant scales with the square of the effective magnetic moment: C is proportional to mu_eff^2. An effective moment of 1 Bohr magneton gives C of approximately 0.125 K*m^3/mol in SI.
How does temperature affect magnetic susceptibility?
Susceptibility is inversely proportional to temperature for a paramagnet. Halving the temperature doubles the susceptibility; raising the temperature 10-fold reduces it by 90%. This happens because thermal fluctuations randomise the orientation of the microscopic magnetic dipoles. At very low temperatures the susceptibility can become very large, eventually saturating when most dipoles are fully aligned - a regime Curie's Law does not describe (the Brillouin function is needed for saturation).
What is the effective magnetic moment and what does it tell me?
The effective magnetic moment mu_eff (measured in Bohr magnetons, mu_B) is the per-atom or per-formula-unit moment that you would infer from the slope of a chi vs 1/T plot. For spin-only systems, mu_eff = 2*sqrt(S*(S+1)) mu_B. Measuring mu_eff from a susceptibility experiment and comparing it to spin-only values is a standard way to count unpaired electrons in transition-metal complexes. A measured value much larger than the spin-only prediction signals significant orbital angular momentum or ferromagnetic coupling.
How is magnetization related to magnetic susceptibility?
Magnetization M is the magnetic moment per unit volume. For a linear paramagnet, M = chi * H, where H is the applied magnetic field in A/m. So susceptibility is the proportionality constant linking the response (M) to the drive (H). In SI units, M and H both have units of A/m, making chi dimensionless. In CGS units the relationship is the same in form (M = chi * H) but the units differ: chi(SI) is about 4*pi times chi(CGS).
What is the difference between Curie's Law and the Curie-Weiss Law?
Curie's Law assumes non-interacting magnetic dipoles: chi = C / T. The Curie-Weiss Law adds an exchange interaction through a Weiss constant theta: chi = C / (T - theta). A positive theta means the dipoles tend to align (ferromagnetic interactions), giving a higher susceptibility than expected from pure Curie behavior. A negative theta means antiparallel alignment is favoured (antiferromagnetic), lowering the susceptibility. The Weiss constant can be extracted by plotting 1/chi vs T: the x-intercept gives theta.
Sources
- Curie, P. (1895). Proprietes magnetiques des corps a diverses temperatures. Annales de Chimie et de Physique, 5, 289-405.
- Kittel, C. Introduction to Solid State Physics (8th ed.). Wiley, 2005. Chapter 14: Diamagnetism and Paramagnetism.
- NIST Reference on Constants, Units, and Uncertainty - Fundamental Physical Constants.