How to Use Magnetic Moment Calculator
The Magnetic Moment Calculator lets you pick one of three formulas — orbital, spin, or total angular momentum — then plug in the relevant quantum number.
- Choose the mode - Orbital (L only), spin (S only), or total J with Landé g-factor. Select the mode that matches the quantum state you are analyzing.
- Enter quantum numbers - Input L, S, and J as integers or half-integers (e.g., S = 1/2 for a single electron, J = 3/2 for a ¹P₃/₂ state).
- Read the result - The Magnetic Moment Calculator shows the moment in Bohr magnetons μ_B and in SI units (J/T), along with the Landé g_J factor.
- Compare with experimental values - Enter different L, S, J combinations to match the measured moment from ESR or susceptibility measurements and identify the ground-state term symbol.
Formula & Theory - Magnetic Moment Calculator
The Magnetic Moment Calculator uses these standard expressions:
μ_L = √(L(L+1)) · μ_B
μ_S = g_S · √(S(S+1)) · μ_B, g_S ≈ 2.0023
μ_J = g_J · √(J(J+1)) · μ_B
g_J = 1 + [ J(J+1) + S(S+1) − L(L+1) ] / [ 2 J(J+1) ]| Symbol | Meaning |
|---|---|
| L, S, J | Orbital, spin, total angular momentum quantum numbers |
| μ_B | Bohr magneton 9.274×10⁻²⁴ J/T |
| g_S, g_J | Spin and Landé g-factors |
Landé g-Factor and LS Coupling
The Landé formula connects the g-factor to the spectroscopic term. For ground-state ions:
| Ion | Term | J | g_J | μ_eff (μ_B) |
|---|---|---|---|---|
| Cu²⁺ | ²D₅/₂ | 5/2 | 1.200 | 1.73 |
| Fe³⁺ | ⁶S₅/₂ | 5/2 | 2.000 | 5.92 |
| Gd³⁺ | ⁸S₇/₂ | 7/2 | 2.000 | 7.94 |
Deviations from spin-only values indicate significant orbital contributions.
Assumptions and Limits
The Landé form assumes LS coupling, valid for light atoms (Z < ~40). For heavy atoms or strong external fields, jj coupling or intermediate coupling schemes are required.
Use Cases for Magnetic Moment Calculator
The Magnetic Moment Calculator is useful when you need a quick, transparent calculation for atomic and condensed-matter magnetism:
- Zeeman effect - Predict line splittings for atoms in modest magnetic fields, comparing orbital and spin contributions to the total splitting.
- Paramagnetic ions - Compare the measured effective moment √(g_J² J(J+1)) μ_B from susceptibility data with the tabulated theoretical value to confirm the ground state.
- ESR/NMR teaching - Demonstrate why electronic moments (~μ_B) are roughly 10³ times larger than nuclear moments (~μ_N), explaining the factor-of-1000 difference in ESR vs NMR frequencies.
- Atomic clock design - Reference state-selection transitions in alkali atoms, where g-factor differences drive the clock frequency in a bias field.
- Magnetic susceptibility fitting - Use g_J and J from the Magnetic Moment Calculator as starting parameters for a Brillouin function fit to magnetization data.
- Spin-orbit coupling analysis - Compare the pure orbital, pure spin, and mixed J moments to quantify the degree of spin-orbit mixing in a given electronic state.
For precision QED-grade values, use the latest CODATA g-factor for the electron and apply radiative corrections explicitly. The Magnetic Moment Calculator provides the first-order LS-coupling result that is accurate enough for most spectroscopic and materials applications.