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 × gS × μB,
gS ≈ 2.0023is the g-factor for an electron's spin,
μB = 9.274 × 10^(−24) J/Tis 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)),
gL = 1is 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)),
gJis the g-factor for a spin and orbital factors taken together,
Jis the quantum number describing together the spin and orbital contributions.
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
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.