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Spin quantum number (S) Half-integer or integer ≥ 0

Orbital quantum number (L) Integer ≥ 0

Total angular momentum (J) |L − S| ≤ J ≤ L + S

Landé g-factor (g_J) Typically between 0 and 4

Total angular momentum (J) Half-integer or integer ≥ 0

Ion number density N (m⁻³) Scientific notation accepted (e.g. 6.022e23 for molar)

Temperature T (K) Absolute temperature > 0 K

Presets

Results

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About

Paramagnetic susceptibility follows Curie's law: χ = CT, where C is the Curie constant and T is absolute temperature. The constant encodes the density of magnetic ions, total angular momentum quantum number J, and the Landé g-factor g_J. Errors in J or g_J propagate quadratically into C, so a wrong ground-state term assignment can produce susceptibility values off by a factor of two or more. This matters in Curie-Weiss fitting of experimental data: an incorrect C shifts the extracted Weiss temperature θ and masks the true exchange interaction strength.

This calculator computes C from first principles using N (magnetic ion density), J, and g_J. It optionally derives g_J from Russell-Saunders quantum numbers S, L, J. The tool assumes free-ion behavior with no crystal-field quenching. For rare-earth ions this is a reasonable approximation. For 3d transition metals, orbital angular momentum is often quenched and the spin-only formula (L = 0) should be used instead.

Formulas

The Curie constant for N magnetic ions per unit volume, each with total angular momentum quantum number J and Landé g-factor g_J:

C = N μ₀ g_J² J(J + 1) μ_B²3 k_B

Curie's law gives the magnetic susceptibility at temperature T:

χ = CT

The effective magnetic moment in Bohr magnetons:

p_eff = g_J √J(J + 1)

The Landé g-factor from Russell-Saunders coupling quantum numbers S, L, J:

g_J = 1 + J(J + 1) + S(S + 1) − L(L + 1)2J(J + 1)

where μ₀ = 4π × 10⁻⁷ T⋅m/A is the vacuum permeability, μ_B = 9.2741 × 10⁻²⁴ J/T is the Bohr magneton, k_B = 1.3806 × 10⁻²³ J/K is the Boltzmann constant, N is the number density of magnetic ions in m⁻³, and T is the temperature in K.

Reference Data

Ion	Config	Ground Term	S	L	J	g_J	p_eff (calc)	p_eff (exp)
Ce³⁺	4f¹	²F_5/2	0.5	3	2.5	0.857	2.54	2.4
Pr³⁺	4f²	³H₄	1	5	4	0.800	3.58	3.5
Nd³⁺	4f³	⁴I_9/2	1.5	6	4.5	0.727	3.62	3.5
Sm³⁺	4f⁵	⁶H_5/2	2.5	5	2.5	0.286	0.84	1.5
Eu³⁺	4f⁶	⁷F₀	3	3	0	-	0	3.4
Gd³⁺	4f⁷	⁸S_7/2	3.5	0	3.5	2.000	7.94	8.0
Tb³⁺	4f⁸	⁷F₆	3	3	6	1.500	9.72	9.5
Dy³⁺	4f⁹	⁶H_15/2	2.5	5	7.5	1.333	10.63	10.6
Ho³⁺	4f¹⁰	⁵I₈	2	6	8	1.250	10.61	10.4
Er³⁺	4f¹¹	⁴I_15/2	1.5	6	7.5	1.200	9.59	9.5
Tm³⁺	4f¹²	³H₆	1	5	6	1.167	7.57	7.3
Yb³⁺	4f¹³	²F_7/2	0.5	3	3.5	1.143	4.54	4.5
Fe³⁺	3d⁵	⁶S_5/2	2.5	0	2.5	2.000	5.92	5.9
Co²⁺	3d⁷	⁴F_9/2	1.5	3	4.5	1.333	6.63	4.8
Ni²⁺	3d⁸	³F₄	1	3	4	1.250	5.59	3.2
Cu²⁺	3d⁹	²D_5/2	0.5	2	2.5	1.200	3.55	1.9
Mn²⁺	3d⁵	⁶S_5/2	2.5	0	2.5	2.000	5.92	5.9
Cr³⁺	3d³	⁴F_3/2	1.5	3	1.5	0.400	0.77	3.8
V³⁺	3d²	³F₂	1	3	2	0.667	1.63	2.8
Ti³⁺	3d¹	²D_3/2	0.5	2	1.5	0.800	1.55	1.8

Frequently Asked Questions

Crystal-field interactions in solids partially or fully quench the orbital angular momentum of 3d ions. The 3d orbitals extend spatially and couple strongly to the surrounding ligand field. This reduces the effective L contribution toward zero. For most 3d ions, a spin-only approximation (L = 0, g = 2) gives p_eff = 2√S(S + 1), which matches experiment much better. Set L = 0 in this calculator for those ions.

For a single crystal, N equals the number of magnetic ions per unit cell divided by the unit cell volume. For example, a simple cubic lattice with one ion per cell and lattice parameter a = 4 Å gives N = 1 ÷ (4 × 10⁻¹⁰)³ ≈ 1.56 × 10²⁸ m⁻³. For dilute paramagnetic salts, multiply the molar concentration by Avogadro's number and convert to SI. The default preset uses 10²⁸ m⁻³ as a typical order of magnitude.

Curie's law assumes non-interacting magnetic moments at temperatures well above any ordering temperature. It fails when (1) exchange interactions become significant, requiring the Curie-Weiss form χ = C ÷ (T − θ); (2) the temperature approaches zero, where quantum saturation effects dominate; (3) the applied field is strong enough that μ_BB ÷ k_BT is not small, requiring the full Brillouin function. This calculator assumes the linear (high-temperature, low-field) regime.

When J = 0, the Landé formula has a division by zero and the ground-state Curie constant vanishes. The observed paramagnetism of Eu³⁺ arises from Van Vleck temperature-independent paramagnetism due to mixing with excited J = 1 states. This calculator will flag J = 0 as a special case and return C = 0 with a note about Van Vleck contributions.

Yes. Set N equal to Avogadro's number (6.022 × 10²³ mol⁻¹). Then C has units of K⋅m³/mol (SI molar Curie constant). In CGS-emu, the molar Curie constant is often quoted in emu⋅K/mol, which differs by a factor of 4π ÷ 10. Use the unit system selector in the tool to switch.

The effective moment p_eff = g_J√J(J + 1) measured in Bohr magnetons gives the magnitude of the total magnetic moment of a free ion. It is directly extracted from the slope of a 1÷χ versus T plot. Comparing p_eff from experiment with the calculated value reveals information about crystal-field effects, spin-orbit coupling strength, and whether intermediate coupling corrections are needed.