Magnetic Moment and Zeeman Effect Calculator

A magnetic moment is a measure of how strongly a particle interacts with a magnetic field, arising from its orbital motion or its intrinsic spin. For a quantum particle, the magnetic moment along a chosen axis is quantized, μ = g μ_B m, where g is the g-factor, μ_B is the Bohr magneton, and m is the relevant magnetic quantum number.

The Zeeman effect is the splitting of an atomic energy level (and its spectral line) into several sub-levels when the atom is placed in an external magnetic field. Each sub-level corresponds to a different value of the magnetic quantum number m, and its energy shift is ΔE = g μ_B B m, proportional to the field strength B.

The orbital g-factor is exactly 1, describing the magnetic moment from a charged particle's orbital motion around the nucleus. The electron's spin g-factor is approximately 2.0023, not exactly 2 as simple theory predicts, the small extra 0.0023 is a celebrated correction from quantum electrodynamics, one of the most precisely tested numbers in physics.

The Bohr magneton, μ_B ≈ 9.274 x 10^-24 joules per tesla, is the natural unit of magnetic moment for an electron, defined as μ_B = eħ/(2m_e). It plays the same role for magnetic moments that the electronvolt plays for atomic energies, giving conveniently sized numbers.

Without a field, all states with the same total angular momentum but different m values have the same energy (they are degenerate). Turning on a magnetic field breaks this degeneracy, each m state shifts by a different amount (ΔE = g μ_B B m), so transitions between split levels now occur at several closely-spaced, slightly different photon energies instead of just one.

It is the physical basis of magnetic resonance imaging (MRI) and nuclear magnetic resonance (NMR) spectroscopy, where the splitting of nuclear spin energy levels in a strong magnetic field is probed with radio waves. It is also used in astrophysics to measure magnetic fields on the Sun and other stars from the splitting of their spectral lines.

For an orbital angular momentum quantum number l, m ranges from −l to +l in integer steps (2l+1 values). For spin 1/2, m (usually written ms) is either +1/2 or −1/2. This calculator accepts any half-integer or integer value of m to cover both orbital and spin cases.

No. The size of the splitting depends on the g-factor and total angular momentum of the states involved, so different spectral lines from the same atom can split by different amounts in the same magnetic field. This more complex, multi-line pattern (the anomalous Zeeman effect) was historically one of the first hints that electron spin needed to be included in atomic theory.

The normal Zeeman effect, seen in transitions between spin-zero states, splits a line into a simple, evenly-spaced triplet. The anomalous Zeeman effect, seen whenever electron spin is involved (which is most real atoms), produces more complex, unevenly-spaced splitting patterns, historically 'anomalous' only because it was observed before spin was understood, it is now the more commonly encountered case.

No, the energy splitting ΔE = g μ_B B m depends only on the g-factor, the field strength, and the magnetic quantum number, not on temperature. Temperature does affect how the split levels are populated (via Boltzmann statistics), which changes the relative intensity of the resulting spectral lines, but not their energy spacing.