Free · Physics models
LS-Coupled Landé Factor and Effective Moment
Calculate the ideal LS-coupled Landé factor and effective magnetic moment in Bohr magnetons.
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
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Formula and method
g factor = 1.3333333; magnetic moment = 2.5819889.
gJ = 3/2 + [S(S+1) − L(L+1)]/[2J(J+1)]; μeff/μB = gJ √[J(J+1)].
Worked example
Enter these known values and leave the other values blank.
- Orbital (L)
- 1
- Spin (S)
- 0.5
- Total angular momentum (J)
- 1.5
- g-factor
- 1.3333
- Magnetic moment
- 2.582 μB
Assumptions and limitations
- Calculate the ideal LS-coupled Landé factor and effective magnetic moment in Bohr magnetons.
- This model uses ideal orbital g = 1 and spin g = 2, neglecting anomalous and configuration corrections. Use nonnegative integer or half-integer quantum numbers and positive J.
- Orbital L must be a nonnegative integer, while S and J are nonnegative integer or half-integer quantum numbers with positive J. J must run from |L−S| through L+S in whole steps. These discrete conditions are enforced.
- Inputs and inverse outputs share the displayed unit selectors. Clear a known field to solve for it, and lock quantities that must remain fixed. Singular inverses may have no finite result.
- Only computed quantum inverses within absolute 10⁻⁵⁰ of an allowed integer or half-integer are recovered. This arithmetic remainder tolerance does not round manual quantum numbers or represent measurement uncertainty.
- The Landé-only J inverse uses the nonnegative quadratic root. At g = 1.5 and L = S, the factor alone does not determine J: the J and moment outputs remain blank unless another independent input such as effective moment determines them. At g = 1.5 and unequal L/S, the supplied state is inconsistent.
Common questions
Is this the saturated moment?
No. The effective moment uses √[J(J+1)], whereas a saturated projection uses J.
What happens at J = 0?
The Landé formula has a zero denominator. This calculator’s Landé branch requires J greater than zero.
Why does g = 1.5 leave J undetermined?
For equal L and S, their difference S(S+1)−L(L+1) is zero. The Landé expression then equals 1.5 for every allowed positive J. Supply another independent quantity, such as effective moment, to identify J.
References
Enforce the explicitly stated physical domain: The Landé branch requires positive J. Orbital quantum number must be a nonnegative integer. Spin must be a nonnegative integer or half integer. J must be an integer or half integer. J must advance from the coupling minimum in integer steps. This effective moment branch requires nonnegative g. Effective moment magnitude cannot be negative. A Landé factor of 1.5 requires equal orbital and spin angular momenta in this positive-J model. Outside the g = 1.5 degenerate case, the supplied Landé factor must imply positive J(J+1). Use the nonnegative quadratic root J = (sqrt(1 + 4[S(S+1) - L(L+1)]/(2g - 3)) - 1)/2 for the Landé-only J inverse, repairing the captured negative-root branch. When g = 3/2 this inverse is underdetermined for L = S and is skipped; a known effective moment can still determine J through the separate moment relation. Recover computed integer L and half-integer S/J only within absolute 1e-50 numerical remainder. Manual state inputs remain exact; ordinary fractional inverse outputs remain invalid.