Magnetic Moment Calculator - Atomic mu in Bohr Magnetons

A magnetic moment calculator turns the quantum numbers S, L, and J of an atomic electron configuration into the magnitude of the magnetic moment mu, in both Bohr magnetons and SI units of joules per tesla, so you can compare different atomic states without re-deriving the Lande g-factor each time.

• • Spin-only check: Confirm that a single unpaired electron contributes mu = 2.0023 * sqrt(0.75) mu_B ~= 1.7340 mu_B regardless of the orbital configuration around it.
• • Orbital-only check: Estimate the orbital contribution mu = sqrt(L(L+1)) mu_B for an electron in an s, p, d, or f shell.
• • Full atomic term: Compute mu = gJ * mu_B * sqrt(J(J+1)) for an LS-coupled term symbol such as ^3P_2 or ^2D_5/2.
• • Bohr magneton vs SI: Convert an atomic moment from the natural Bohr magneton units used in spectroscopy to SI J/T for engineering or condensed-matter work.

Atomic magnetic moments drive paramagnetism, define the splitting of spectral lines in a magnetic field, and set the magnitude of the torque an atom feels inside a magnetic trap. Picking the right value of mu is the first step in nearly every atomic-physics homework problem, every electron-spin-resonance calculation, and every estimate of magnetic anisotropy in a solid. A reliable magnetic moment calculator also prevents the most common mistakes, such as forgetting the sqrt(S(S+1)) factor that arises from the uncertainty principle and treating J as if it were the classical total angular momentum.

The calculator evaluates one of three formulas depending on the mode you select, each built on the same product of a g-factor, the Bohr magneton mu_B, and a square root of a quantum number times itself plus one.

• S: Total spin quantum number (dimensionless). 0.5 for one unpaired electron, integer multiples of 0.5 for several.
• L: Total orbital quantum number (dimensionless, integer). 0 for s, 1 for p, 2 for d, 3 for f.
• J: Total angular momentum quantum number (dimensionless). Allowed range is |L - S| to L + S in integer steps.
• g: Effective g-factor: g_S = 2.0023 for spin, g_L = 1 for orbital, g_J = 3/2 + (S(S+1) - L(L+1))/(2 J (J+1)) for LS-coupled.
• mu_B: Bohr magneton, 9.2740100657 times 10 to the negative 24 J/T per NIST 2022 CODATA.

For the LS-coupled mode the calculator checks that J lies within the allowed range from |L - S| to L + S and returns a flag of 0 with zeroed outputs if it does not, so an impossible term symbol never produces a misleading number.

According to NIST 2022 CODATA, the Bohr magneton equals 9.2740100657 times 10 to the negative 24 joules per tesla with a relative standard uncertainty of 3.1 times 10 to the negative 10.

Total angular momentum J enters the LS-coupled moment the same way L and omega enter the rigid-body L = I * omega expression used in the Angular Momentum Calculator.

Four ideas come up every time you derive an atomic magnetic moment, and each one shows up directly in the calculator inputs and outputs.

The Lande g-factor controls the torque and energy of a moment in an external field the same way the cross product in the Lorentz Force Calculator controls the force on a moving charge.

Pick the mode that matches your atomic term symbol, enter the quantum numbers and (for spin or orbital mode) a g-factor, and read the magnetic moment in the units you need.

1. 1 Choose the mode: Use Spin Only for an intrinsic spin moment, Orbital Only for an orbital contribution, or LS-Coupled for an atomic term symbol with both.
2. 2 Enter the total spin S: Use 0.5 for a single electron. Use the sum of all unpaired electron spins, in half-integer steps, otherwise.
3. 3 Enter the total orbital L: Use 0 for s states, 1 for p states, 2 for d states, 3 for f states; whole numbers only.
4. 4 Enter J if LS-coupled: Pick J in the allowed range from |L - S| to L + S, in integer steps. The calculator flags out-of-range values.
5. 5 Adjust the g-factor if needed: The default 2.0023 covers a free electron. For orbital mode use 1. For LS-coupled mode the field is ignored because gJ is computed from S, L, and J.
6. 6 Read mu in Bohr magnetons and SI: Use the Bohr magneton value to compare with spectroscopy tables and the SI J/T value when feeding downstream physics like Zeeman or torque calculations.

For a ^2D_5/2 state with S = 0.5 and L = 2, the allowed J values are 1.5 and 2.5. Pick J = 2.5, switch to LS-Coupled mode, leave S = 0.5 and L = 2, and the calculator returns gJ = 1.2 and mu = 1.2 * sqrt(2.5 * 3.5) mu_B = 1.2 * sqrt(8.75) = 3.5495 mu_B.

If you need the classical current-loop version of mu for a coil instead of an atom, switch to the Magnetic Dipole Moment Calculator which uses mu equals N I A.

Working through atomic magnetic moments by hand is repetitive, especially when you need the Lande g-factor for several term symbols in a row. This calculator removes the friction.

• • Three modes in one tool: Spin-only, orbital-only, and LS-coupled modes cover every introductory atomic-physics problem without switching calculators.
• • Built-in Lande g-factor: For LS-coupled mode the calculator derives gJ from S, L, J directly so you never have to memorize the formula or risk dropping a sign.
• • Two output units: Reading mu in Bohr magnetons and SI J/T lets you cross-check against spectroscopy data and physics-engineering results with one click.
• • Range check on J: The calculator flags J values outside the |L - S| to L + S range, so an impossible term symbol never silently produces a wrong answer.
• • CODATA constant: The Bohr magneton value is pinned to NIST 2022 CODATA, so the SI output stays consistent with the most recent fundamental constants.

Both effects share the same plasma magnetic field: the per-atom moment this calculator reports couples to B through the Zeeman energy U = -mu dot B, while the Alfven Velocity Calculator gives the bulk Alfven wave speed v_A = B / sqrt(mu_0 rho) through the same field.

Four things determine the numeric value of mu for an atomic state, and a couple of important limitations apply.

• • The Lande g-factor formula assumes LS coupling, which breaks down for heavy atoms where spin-orbit coupling is strong. For Z > 30 or so, jj coupling is more accurate.
• • The calculator reports magnitudes only. The sign convention for mu follows the negative g-factor convention, which is consistent with spectroscopy but not always the convention used in materials physics.

According to NIST CODATA electron g-factor, the free-electron g-factor equals 2.00231930436, the value the calculator rounds to 2.0023 in spin mode.

The same Zeeman coupling U = -mu dot B applies when B is the Earth's field instead of a laboratory source; the Magnetic Declination Calculator gives the local heading angle needed to orient that dot product for compass and geophysics work.