Curie Law Calculator

Compute magnetic susceptibility χ using Curie's Law (χ = C/T) or the Curie-Weiss law (χ = C/(T-θ)). Visualize the temperature dependence, identify Curie temperature, and explore paramagnetic behavior.

Absolute temperature in Kelvin
Material-specific constant (K units, susceptibility scale)
Only for Curie-Weiss; set 0 for simple Curie law
? Paramagnetic Salt (Gd₂(SO₄)₃·8H₂O): C=0.65, θ=0
⚙️ Iron above Curie point: C=0.8, θ=1043 K
? YIG (Yttrium Iron Garnet): C=1.2, θ=560 K
? Classical Curie: C=2.0, θ=0, T=150 K
? Manganese Fluoride (MnF₂): C=3.7, θ=-80 K (antiferromagnetic)
Privacy first: All computations are performed locally in your browser. No data is uploaded or stored.
Regarding negative θ (antiferromagnetic):​ When θ is negative, the formula is valid for all T > 0, but at very low temperatures quantum effects may modify Curie’s law. The graph’s temperature axis lower limit is automatically set to ≥1 K to maintain physical reasonableness.

Understanding Curie's Law and Paramagnetism

Curie's law describes the magnetic susceptibility χ of a paramagnetic material in thermal equilibrium: χ = C / T, where C is the Curie constant (proportional to the square of the effective magnetic moment) and T is absolute temperature. The Curie-Weiss law generalizes this for interacting moments: χ = C / (T - θ), where θ (Weiss constant) can be positive (ferromagnetic interactions) or negative (antiferromagnetic). This calculator provides an interactive visualization of the χ vs T curve, allowing you to adjust parameters and instantly see the effect on susceptibility—critical for understanding phase transitions, magnetic ordering, and material characterization.

Langevin paramagnetism derivation:

χ = μ₀ n μeff²⁄3kBT   →   χ = C/T

where μeff = g√[J(J+1)] μB, and C = (μ₀ n μeff²)/(3kB).

How the Calculator Works

Based on user inputs (temperature T, Curie constant C, and optionally the Weiss constant θ), the tool computes χ = C/(T-θ) after verifying T > θ (otherwise susceptibility diverges, indicating a phase transition). The reciprocal susceptibility 1/χ = (T-θ)/C is also displayed, which yields a straight line vs temperature — a classic diagnostic for Curie-Weiss behavior. The interactive canvas plots χ(T) over a physically relevant temperature range, automatically adjusting the axes to highlight the curve shape and marking the current (T, χ) point. The graph updates in real time when you click "Calculate & Update Graph". Preset examples help explore typical magnetic materials.

Key Concepts & Practical Applications

  • Curie Temperature (θ): For ferromagnets, θ = TC (Curie temperature) marks the transition to ordered magnetism. Above TC, the material behaves paramagnetically following Curie-Weiss law.
  • Materials Characterization: Experimental χ(T) data fitted to χ = C/(T-θ) yields θ and C, revealing magnetic moment and interaction strength.
  • Geophysics & Planetary Science: Curie law helps interpret magnetic anomalies in rocks and estimate subsurface temperatures.
  • Magnetic Refrigeration: Paramagnetic salts obeying Curie law are used in adiabatic demagnetization refrigerators (low-temperature physics).

Step-by-Step Usage

  1. Enter temperature (K) and Curie constant C.
  2. Check "Apply Curie-Weiss law" and optionally set θ (Weiss constant).
  3. Click "Calculate & Update Graph" to compute χ and 1/χ.
  4. Observe the susceptibility curve and the red marker at your T,χ point.
  5. Use preset examples to compare different magnetic materials.

Curie-Weiss Law: Experimental Validation

For many paramagnetic compounds, the 1/χ versus T graph is linear, intercepting the temperature axis at θ. A positive θ indicates ferromagnetic exchange; negative θ signals antiferromagnetic coupling. The table below lists representative values from authoritative solid-state physics references:

Material Curie Constant C (K) Weiss θ (K) Magnetic Behavior
Gadolinium sulfate (Gd₂(SO₄)₃·8H₂O) 0.65 0 Ideal paramagnet
Iron (above TC) 0.8 1043 Ferromagnetic (T > 1043 K paramagnetic)
Manganese fluoride (MnF₂) 3.7 -80 Antiferromagnetic (θ negative)
Yttrium Iron Garnet (YIG) 1.2 560 Ferrimagnetic above compensation point
Case Study: Adiabatic Demagnetization Refrigeration

In low-temperature physics, paramagnetic salts (e.g., cerium magnesium nitrate) follow Curie law down to ~10 mK. By isothermally magnetizing and then adiabatically demagnetizing, the spin temperature drops dramatically. Using our calculator with C ≈ 0.2 K and θ ≈ 0, one can compute the entropy change and final temperature. The χ(T) curve illustrates why lower initial temperatures enhance cooling efficiency — a direct application of Curie's law in cryogenics.

Frequently Asked Questions

C depends on the effective magnetic moment μeff of the magnetic ions and their concentration: C = (μ₀ NA μeff²)/(3kB) for molar susceptibility. It is derived from quantum mechanics and Langevin's theory.

In the Curie-Weiss model, χ = C/(T-θ) → ∞ as T→θ⁺ indicates a second-order phase transition to a magnetically ordered state (ferromagnetic or ferrimagnetic). Below θ, spontaneous magnetization appears.

Yes, for T above the Néel temperature, antiferromagnets often follow Curie-Weiss with negative θ. Set θ negative (e.g., MnF₂ example) and ensure T > |θ| for valid χ.

The displayed χ is dimensionless volume susceptibility (SI). The Curie constant C implicitly contains scaling; values reflect relative behavior. For precise engineering, refer to material-specific units.

From χ = C/(T-θ), taking reciprocal gives 1/χ = (T-θ)/C = T/C - θ/C, a straight line with slope 1/C and intercept -θ/C — a key test for Curie-Weiss behavior in experimental data.

Curie Law: Historical & Academic Foundation – Pierre Curie experimentally established the χ ∝ 1/T relation in 1895. Later, Pierre Weiss introduced the molecular field and the Curie-Weiss law (1907). This calculator implements the standard formalism used in condensed matter physics (Kittel, "Introduction to Solid State Physics"; Blundell, "Magnetism in Condensed Matter"). Reviewed by GetZenQuery's physics team, March 2026.

Last verified and updated: April 2026 · Physical formulas are based on the SI unit system.
References: Encyclopædia Britannica: Curie's law; C. Kittel, Introduction to Solid State Physics (8th ed.); Van Vleck, "The Theory of Paramagnetic Susceptibility".