# Magnetic Moment Calculator

Created by Miłosz Panfil, PhD
Reviewed by Dominik Czernia, PhD candidate and Adena Benn
Last updated: Oct 03, 2022

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:

1. The intrinsic magnetic moment of electrons (spin magnetic moment),
2. The magnetic moment related to the orbital motion of electrons (orbital magnetic moment),
3. 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 × gS × μB,

where

• gS ≈ 2.0023 is the g-factor for an electron's spin,
• μB = 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. You can check other consequences of quantization of the orbital motion in the Hydrogen energy levels calculator.

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

μ = - gL * μB * √(L*(L+1)),

where

• gL = 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 = - gJ × μB × √(J×(J+1)),

where

• gJ is the g-factor for a 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 gJ factor is

gJ = 3/2 + (S(S+1) - L(L+1))/(2J×(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 gJ differ in sign. This difference is related to the effects of paramagnetism (caused by the spin part of the gJ factor) and diamagnetism (caused by the orbital part). You can learn more about the relation between the paramagnetism and the magnetic moment by checking the Curie law calculator

## Magnetic moment calculator

The computation of the magnetic moment of an atom is simple with our calculator. It is enough to specify the total quantum numbers S, L, and J. The calculator then computes the g-factor and the magnetic moment. We express the magnetic moment in Bohr magneton units, a convenient unit to describe magnetism in the micro-world.

Miłosz Panfil, PhD
Spin
Orbital
Total angular momentum
g-factor
Magnetic moment
μB
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