Magnetic Moment Calculator — Spin, Orbital & Total Magnetic Moment of an Atom

Magnetic moment calculator

The Magnetic moment calculator helps you compute the magnetic moment of an atom. The text below explains the origin of the electron magnetic moment and provides the magnetic moment of an atom formula.

Origin of the magnetic moment of an atom

The magnetic moment of an atom has three origins:

• The intrinsic magnetic moment of electrons (spin magnetic moment),
• The magnetic moment related to the orbital motion of electrons (orbital magnetic moment),
• The magnetic moment of the nucleus.

The contribution from the magnetic moment of the nucleus is usually much weaker than from the first. Because of this, we only focus on the electronic part of the magnetic moment of an atom.

Electron magnetic moment - spin

An electron possesses an intrinsic magnetic moment due to the spin. Contrary to what the name suggests, the spin is not related to an actual spinning of an electron. Spin is just another characteristic of an elementary particle, such as mass or charge. The magnetic moment of a single electron is

μ = −√(3/4) × g_S × μ_B,

where

• g_S ≈ 2.0023 is the g-factor for an electron's spin,
• μ_B = 9.274 × 10⁻²⁴ J/T is the Bohr magneton.

The g-factor generally relates the magnetic moment to the angular momentum. Such a relation holds even if the angular moment is an intrinsic spin, not an actual angular motion.

Orbital electron moment

The second contribution to the magnetic moment of an atom comes from the actual orbital movement. In quantum mechanics, not every circular motion is allowed. We say that the orbital motion is quantized.

Angular momentum quantum number L parametrizes allowed orbital motions. L can take only positive integer or 0 values. The corresponding magnetic moment is

μ = −g_L × μ_B × √(L·(L+1)),

where g_L = 1 is the g-factor for the orbital motion. The total magnetic moment of an atom comes from adding the spin and orbital contributions.

Magnetic moment of an atom formula

The total magnetic moment of an atom is the sum of the spin and orbital magnetic moments. However, under the rules of quantum mechanics, it is not a simple sum. Instead, the resulting equation is

μ = −g_J × μ_B × √(J·(J+1)),

where

• g_J is the g-factor for the spin and orbital factors taken together,
• J is the quantum number describing together the spin and orbital contributions.

Quantum number J can take only limited values: |L−S| ≤ J ≤ L+S. The formula for the g_J factor is

g_J = 3/2 + (S(S+1) − L(L+1)) / (2·J·(J+1)).

In these equations, S is the sum (ordinary) of spins of all electrons (a single one takes values ± 1/2). Similarly, L is the sum of orbital quantum numbers of all electrons. Note that the spin and orbital contributions to the g-factor g_J differ in sign. This difference is related to the effects of paramagnetism (caused by the spin part of the g_J factor) and diamagnetism (caused by the orbital part).

Units of the magnetic moment

The natural atomic unit of magnetic moment is the Bohr magneton (μ_B). This calculator also reports the result in SI units (J/T, equivalent to A·m²) and in CGS-Gaussian units (erg/G), where 1 J/T = 1000 erg/G. The quantum numbers L, S and J are dimensionless, so they are identical in every unit system.