Curie–Weiss Law Calculator
Compute the paramagnetic susceptibility above the Curie temperature: χ = C / (T − Tc). Solve for χ, the Curie constant C, temperature T, or Curie temperature Tc — with a live χ-vs-T hyperbola, the classic 1/χ-vs-T Curie–Weiss plot, material comparison, and step-by-step algebra.
Thermal disordering of spins
In the paramagnetic regime, thermal energy kBT overwhelms the exchange coupling between neighboring spins, randomizing their orientation. The applied field produces only a weak net magnetization, giving χ ∝ 1/T (Curie's law). The Curie–Weiss extension adds the mean-field correction T → T − Tc.
Tc is the onset of order
As T approaches Tc from above, χ diverges — the susceptibility becomes infinitely sensitive to applied field. This signals a second-order phase transition into the ferromagnetic state, where the spins spontaneously align. Tc is a fingerprint of the exchange interaction strength J.
The 1/χ vs T plot is the signature
Plotting 1/χ against T gives a straight line of slope 1/C that extrapolates to zero at T = Tc. Fitting this line to experimental data is the standard way to extract both C and Tc from high-temperature susceptibility measurements.
C encodes the moment
The Curie constant is proportional to the square of the effective magnetic moment per atom: C = μ_eff²/8 in cgs units. For Fe²⁺ (S = 2) theory predicts μ_eff = 4.9 μ_B and C ≈ 3.0; for Gd³⁺ (S = 7/2) μ_eff = 7.94 μ_B and C ≈ 7.9. Comparing measured C to theory reveals spin state and orbital contributions.
Curie–Weiss Formulas
For the paramagnetic regime T > Tc.
| Quantity | Formula | Units |
|---|---|---|
| Susceptibility | χ = C / (T − Tc) | — |
| Curie constant | C = χ · (T − Tc) | K·cm³/mol |
| Temperature | T = C/χ + Tc | K |
| Curie temperature | Tc = T − C/χ | K |
| Inverse susceptibility | 1/χ = (T − Tc) / C | — |
| Effective moment (cgs) | μeff = √(8C) μB | μB |
| Curie's law (Tc = 0) | χ = C / T | — |
| Mean-field χ (below Tc) | χ ∝ (Tc − T)−1 for T < Tc | — |
Typical Ferromagnetic Materials
| Material | Tc (K) | C (emu·K/mol) | μeff (μB) | Notes |
|---|---|---|---|---|
| Iron (Fe) | 1043 | 4.4 | 5.9 | Classic itinerant ferromagnet |
| Cobalt (Co) | 1394 | 3.2 | 5.1 | Highest Tc of pure metals |
| Nickel (Ni) | 631 | 0.65 | 2.3 | Weaker exchange coupling |
| Gadolinium (Gd) | 293 | 7.9 | 7.9 | Localized 4f moment, S = 7/2 |
| Magnetite (Fe₃O₄) | 858 | 4.4 | 5.9 | Mixed-valence iron oxide |
| Chromium dioxide (CrO₂) | 386 | 1.4 | 3.3 | Half-metallic, spintronics |
| Europium oxide (EuO) | 69 | 2.1 | 4.1 | Low-Tc ferromagnetic semiconductor |
| Dysprosium (Dy) | 88 | 14.6 | 10.6 | Heavy rare earth, large moment |
Frequently Asked Questions
The Curie–Weiss law states that the magnetic susceptibility above the ordering temperature varies as χ = C/(T − Tc), where C is the Curie constant and Tc is the Curie temperature. It is the mean-field approximation for a ferromagnet above Tc and reduces to Curie's simpler χ = C/T when Tc = 0 (a pure paramagnet).
As T → Tc⁺, the thermal energy becomes small compared to the exchange coupling, so the spins become increasingly correlated over long distances. The susceptibility — the linear response to an applied field — grows without bound because an arbitrarily small field can align a macroscopically correlated spin system. This divergence is the hallmark of a continuous phase transition.
C is proportional to the square of the effective magnetic moment per magnetic ion: C = μ_eff²/8 in cgs units (with C in cm³·K/mol). So measuring C tells you μ_eff, which in turn tells you the spin S and the g-factor. For example, C = 4.4 corresponds to μ_eff = √(8·4.4) = 5.9 μ_B, matching the Fe²⁺/Fe³⁺ moment in iron.
Because χ = C/(T − Tc) rearranges to 1/χ = (T − Tc)/C = T/C − Tc/C, which is linear in T with slope 1/C and x-intercept Tc. This is why experimentalists always plot inverse susceptibility — the linear region above Tc is easy to fit, and the intercept directly gives the Curie temperature. Deviations from linearity (curvature) indicate that the simple Curie–Weiss model is incomplete — you may need to account for multiple magnetic sublattices (ferrimagnets like magnetite) or short-range correlations.
Below Tc the material becomes ferromagnetic — the spins order spontaneously and the susceptibility is no longer given by the Curie–Weiss law. Instead, χ depends on the magnetic history (initial magnetization curve, hysteresis) and on the demagnetizing factor of the sample. The initial susceptibility in the ferromagnetic state typically follows a different power law, and the sample shows permanent magnetization even in zero applied field.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| chi | Magnetic susceptibility (dimensionless) | |
| C | Material-specific Curie constant | K |
| T | Absolute temperature | K |
| Tc | Curie temperature | K |
What it means
The Curie‑Weiss law describes the magnetic susceptibility χ of a ferromagnetic material above its Curie temperature T_c (in the paramagnetic state). It is expressed as χ = C/(T − T_c), where C is the Curie constant (proportional to the effective magnetic moment). As the temperature approaches T_c from above, the susceptibility diverges, indicating the onset of ferromagnetic ordering. This law is derived from the mean‑field theory of magnetism. It is used to determine the Curie temperature from susceptibility measurements and to understand magnetic phase transitions. Understanding this law is important in magnetism research, magnetic materials development, and applications like magnetic recording and sensors.
Worked example
Curie‑Weiss Law – Two Examples
Real‑World| Parameter | Value |
|---|---|
| C | 1.0 K |
| T | 400 K |
| Tc | 300 K |
| Parameter | Value |
|---|---|
| C | 2.0 K |
| T | 1200 K |
| Tc | 1043 K |
Common mistakes
- Curie‑Weiss law: χ = C / (T − T_c) – magnetic susceptibility of a ferromagnetic material above the Curie temperature.
- Curie constant C: Material‑specific – in K (or the same units as T).
- Curie temperature T_c: The critical temperature above which the material becomes paramagnetic – in Kelvin.
- Validity: Valid only for T > T_c – near T_c, the law fails due to critical fluctuations.
- Units: χ is dimensionless (in SI), but often given in emu or other systems – ensure consistency.
Applications
The Curie‑Weiss law (χ = C/(T − Tc)) describes the magnetic susceptibility of ferromagnetic materials above the Curie temperature. It is used to characterise magnetic phase transitions, to determine the Curie temperature, and to understand the magnetic behaviour of materials. Engineers apply it in the design of magnetic materials for transformers, motors, sensors, and data storage. The Curie temperature is critical for applications where magnetic properties change with temperature, such as in thermomagnetic switching and magnetic refrigeration. By fitting experimental data to this law, materials scientists can extract fundamental magnetic parameters for alloy development.
- Characterisation of ferromagnetic and ferrimagnetic materials
- Determination of Curie temperature for magnetic alloys
- Design of magnetic sensors and actuators
- Development of high‑temperature magnetic materials
- Study of magnetic phase transitions and critical phenomena
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