Magnetic Moment Calculator

Use the magnetic moment calculator to compute the magnetic moment of an atom.

Effective magnetic moment7.93725 µ_Bg√(J(J+1)) Bohr magnetons — note the √(J(J+1)), NOT J: the moment vector cannot point entirely along the measurement axis, so its magnitude always exceeds its largest component

Fermi energy7.04383 eVcopper's is 7.00 eV — this is the energy of the highest occupied state at absolute zero, and it is nothing to do with temperature

Fermi velocity1.5741e+6 m/s0.525% of the speed of light — electrons at the Fermi surface move this fast even at absolute zero, purely because the Pauli principle forbids them from all sitting still

Fermi temperature81740 K272.5× room temperature, which is why a metal's electrons contribute so little to its heat capacity: only the tiny fraction within k_BT of the Fermi level can absorb energy at all

Fermi wavevector1.3597e+10 m⁻¹

Thermal energy k_BT at this temperature0.025852 eV0.367% of the Fermi energy

Fraction of electrons thermally active0.367%roughly T/T_F — the rest are locked in by the exclusion principle with no empty state to move to

Maximum measurable component7 µ_BgJ — smaller than the magnitude above, which is a purely quantum effect with no classical analogue

Curie constant1.6439 Knµ₀µ²/3k_B — the slope of 1/χ against T, which is how the effective moment is measured experimentally

Magnetic susceptibility0.0054797C/T — paramagnetic susceptibility falls as 1/T because thermal agitation randomises the moments faster than the field can align them

Susceptibility at half the temperature0.010959doubled — the Curie law is exactly inverse in T, as long as the moments do not interact with each other

Bohr magneton9.27401e-24 J/Teħ/2m_e — the natural unit of atomic magnetism

The formula

E_F = (ħ²/2m)(3π²n)^⅔; C = nµ₀µ²/3k_B; µ = g√(J(J+1))µ_B

The Fermi energy has nothing to do with heat

At absolute zero the electrons in a metal are not at rest. The Pauli principle forbids two of them from occupying the same state, so they stack up into every available level, and the topmost occupied one sits several electron-volts above the bottom. Electrons at that surface are moving at around 1% of the speed of light with the metal at absolute zero — entirely because there is nowhere slower for them to go.

This resolves a classical embarrassment. Classically, every free electron should contribute to a metal's heat capacity, and the prediction comes out roughly a hundred times too large. In fact only the small fraction within k_BT of the Fermi level has an empty state to move into, and since the Fermi temperature is tens of thousands of kelvin, that fraction is around 1% at room temperature.

Why √(J(J+1)) and not J

The magnitude of a magnetic moment is g√(J(J+1)) Bohr magnetons, but the largest component measurable along any axis is only gJ. The magnitude always exceeds the biggest component, meaning the moment can never point entirely along the direction you measure. That has no classical analogue — a classical vector can always be aligned with an axis — and it is why the Curie constant, which depends on the magnitude squared, uses J(J+1).