How this calculator works
The LS-coupling approximation adds total orbital L and spin S to form J. Using orbital gL = 1 and spin gS = 2 gives the usual Lande factor. The reported effective magnitude is |gJ| sqrt[J(J+1)] Bohr magnetons, the quantity commonly used in Curie-law estimates. It differs from the maximum field projection |gJ|J. The moment-only inverse chooses a nonnegative g because magnitude cannot determine its sign. Nuclear moments, anomalous electron spin corrections and strong spin-orbit or crystal-field effects are omitted. J = 0 has zero first-order moment but no defined g factor.
Formula
gJ = 1 + [J(J+1)+S(S+1)-L(L+1)]/[2J(J+1)]; mu_eff/muB = |gJ| sqrt[J(J+1)]. J ranges in integer steps from |L-S| to L+S.
Worked example
For L = 0, S = J = 1/2, g = 2 and the effective moment is sqrt(3), about 1.73205 Bohr magnetons. For S = 0 and L = J = 1 it is sqrt(2) Bohr magnetons.
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Frequently asked questions
Is the effective moment the maximum projection?
No. The magnitude uses sqrt[J(J+1)], while the saturated projection uses J.
Why do some inverses fail?
Not every g and angular-momentum combination describes an allowed multiplet, and some inverse equations are underdetermined.
What happens at J = 0?
The first-order effective moment is zero. Dividing to define g at J = 0 is not meaningful.