Spin Magnetic Moment - Calculation

Calculation

We can calculate the observable spin magnetic moment, a vector, μ_S, for a sub-atomic particle with charge q, mass m, and spin angular momentum (also a vector), S, via:

(1)

where g is a dimensionless number, called the g-factor. This number depends on the particle: it is g = −2.0023 for the electron, g = 5.586 for the proton, and g = −3.826 for the neutron. The proton and neutron are composed of quarks, which have a non-zero charge and a spin of ħ/2, and this must be taken into account when calculating their g-factors. Even though the neutron has a charge q = 0, its quarks give it a magnetic moment. The proton and electron's spin magnetic moments can be calculated by setting q = +e and q = −e, respectively, where e is the elementary charge.

The intrinsic electron magnetic dipole moment is approximately equal to the Bohr magneton μ_B because g ≈ −2 and the electron's spin is also ħ/2:

(2)

Equation (2) is normally written as

(3)

Just like the total spin angular momentum cannot be measured, neither can the total spin magnetic moment be measured. Equations (1), (2), (3) give the physical observable, that component of the magnetic moment measured along an axis, relative to or along the applied field direction. Assuming a Cartesian coordinate system, conventionally, the z-axis is chosen but the observable values of the component of spin angular momentum along all three axes are each ±ħ/2. However, in order to obtain the magnitude of the total spin angular momentum, S be replaced by its eigenvalue, √s(s + 1), where s is the spin quantum number. In turn, calculation of the magnitude of the total spin magnetic moment requires that (3) be replaced by:

(4)

Thus, for a single electron, with spin quantum number s = 1/2, the component of the magnetic moment along the field direction is, from (3), |μ_{S, z}| = μ_B, while the (magnitude of the) total spin magnetic moment is, from (4), |μ_S| = √3 μ_B, or approximately 1.73 Bohr magnetons.

The analysis is readily extended to the spin-only magnetic moment of an atom. For example, the total spin magnetic moment (sometimes referred to as the effective magnetic moment when the orbital moment contribution to the total magnetic moment is neglected) of a transition metal ion with a single d shell electron outside of closed shells (e.g. Titanium Ti3+) is 1.73 μ_B since s = 1/2, while an atom with two unpaired electrons (e.g. Vanadium V3+) with S = 1 would have an effective magnetic moment of 2.83 μ_B.