Enter the spin quantum number (s) of a particle to calculate its spin-only magnetic moment using the formula μ = √(4s(s+1)) in units of Bohr magnetons (μB). Choose from common particles like electrons, protons, or enter a custom spin value. You get the magnetic moment in Bohr magnetons along with the value in SI units (J/T).
Spin Magnetic Moment (μB)
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Magnetic Moment (SI)
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Spin-Only Magnetic Moment (ideal, gS=2)
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Equivalent Unpaired Electrons (n)
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The spin-only magnetic moment is the magnetic moment of a particle arising solely from its intrinsic spin, ignoring any orbital angular momentum contribution. It is calculated using the formula μ = gS × μB × √(s(s+1)), where s is the spin quantum number, gS is the spin g-factor, and μB is the Bohr magneton. For a simplified calculation with gS = 2, this becomes μ = √(4s(s+1)) μB.
The spin quantum number (s) describes the intrinsic angular momentum of a particle. Despite its name, spin does not correspond to actual physical spinning — it is an intrinsic quantum property like mass or charge. Fermions (e.g. electrons, protons, neutrons) have half-integer spins (1/2, 3/2, ...), while bosons have integer spins (0, 1, 2, ...).
The spin-only magnetic moment formula is μ = √(4s(s+1)) in units of Bohr magnetons, derived using gS ≈ 2. The more precise form is μ = gS × μB × √(s(s+1)), where gS ≈ 2.0023 for a free electron. For an electron (s = 1/2), this gives μ ≈ 1.732 μB using the simplified formula.
The Bohr magneton is the natural unit for expressing atomic magnetic moments, defined as μB = eℏ/(2me) ≈ 9.274 × 10⁻²⁴ J/T. It represents the magnetic moment of an electron due to its orbital angular momentum of one unit of ℏ. All atomic magnetic moments are conveniently expressed as multiples of μB.
For an electron, the spin quantum number s = 1/2. Substituting into the formula μ = √(4 × 0.5 × (0.5+1)) = √(4 × 0.5 × 1.5) = √3 ≈ 1.732 μB. Using the precise g-factor gS ≈ 2.0023 gives a slightly higher value of approximately 1.7327 μB.
For a system with n unpaired electrons, each contributing spin s = 1/2, the total spin S = n/2. The spin-only magnetic moment formula then simplifies to μ = √(n(n+2)) μB, which is a common expression used in chemistry to relate the number of unpaired electrons to the experimentally measured magnetic moment of transition metal complexes.
The g-factor (gS) is a dimensionless quantity that accounts for the anomalous magnetic moment of the electron. Dirac's relativistic quantum mechanics predicts gS = 2 exactly, but quantum electrodynamics (QED) corrections raise this to approximately 2.0023193. For most practical calculations, gS = 2 is used, and the small correction is significant only in precision measurements.
An atom's magnetic moment has three contributions: (1) the spin magnetic moment from intrinsic electron spin, (2) the orbital magnetic moment from electron orbital motion (μ = gL × μB × √(L(L+1)), gL=1), and (3) the nuclear magnetic moment (typically much weaker and often neglected). The total magnetic moment combines spin and orbital contributions via the total angular momentum quantum number J and the Landé g-factor gJ.