Spin magnetic moment

The idea of a spin angular momentum was first proposed in a 1925 publication by George Uhlenbeck and Samuel Goudsmit to explain hyperfine splitting in atomic spectra.[lower-alpha 3] In 1928, Paul Dirac provided a rigorous theoretical foundation for the concept in the Dirac equation for the wavefunction of the electron.[1]

Spin magnetic moments create a basis for one of the most important principles in chemistry, the Pauli exclusion principle. This principle, first suggested by Wolfgang Pauli, governs most of modern-day chemistry. The theory plays further roles than just the explanations of doublets within electromagnetic spectrum. This additional quantum number, spin, became the basis for the modern standard model used today, which includes the use of Hund's rules, and an explanation of beta decay.

We can calculate the observable spin magnetic moment, a vector, μ→_S, for a sub-atomic particle with mass m, and spin angular momentum (also a vector), S→, via:[2]

${\displaystyle {\vec {\mu }}_{\text{S}}\ =\ g\ {\frac {e}{2m}}\ {\vec {S}}=\gamma {\vec {S}}}$

(1)

where ${\displaystyle \gamma \equiv g{\frac {e}{2m}}}$ is the gyromagnetic ratio, e is the elementary charge unit, g is a dimensionless number, called the g-factor, and m is the mass. The g-factor 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 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:

${\displaystyle \mu _{\text{S}}\approx -2{\frac {e}{2m_{\text{e}}}}{\frac {\hbar }{2}}=-\mu _{\text{B}}}$

(2)

Equation (1) is therefore normally written as:[3]

${\displaystyle {\vec {\mu }}_{\text{S}}={\tfrac {1}{2}}g\mu _{\text{B}}{\vec {\sigma }}}$

(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:

${\displaystyle |{\vec {\mu }}_{\text{S}}|=g\mu _{\text{B}}{\sqrt {s(s+1)}}}$

(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 μ_B.

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.

• Nuclear magneton
• Pauli exclusion principle
• Nuclear magnetic resonance
• Multipole expansion
• Relativistic quantum mechanics
• Magnetic spin vortex disc

• An Introduction to the Electronic Structure of Atoms and Molecules by Dr. Richard F.W. Bader (McMaster University)