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.
Calculation
We can calculate the observable spin magnetic moment, a vector, , for a sub-atomic particle with charge q, mass m, and spin angular momentum,, via: where is the gyromagnetic ratio, g is a dimensionless number, called the g-factor, q is the charge, and m is the mass. The g-factor depends on the particle: it is for the electron, for the proton, and for the neutron. The proton and neutron are composed ofquarks, which have a non-zero charge and a spin of, and this must be taken into account when calculating their g-factors. Even though the neutron has a charge, its quarks give it a magnetic moment. The proton and electron's spin magnetic moments can be calculated by setting and, respectively, where e is the elementary charge unit. The intrinsic electron magnetic dipole moment is approximately equal to the Bohr magnetonμ because and the electron's spin is also : Equation is therefore normally written as: Just like the total spin angular momentum cannot be measured, neither can the total spin magnetic moment be measured. Equations,, 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 ±. However, in order to obtain the magnitude of the total spin angular momentum, be replaced by its eigenvalue, where s is the spin quantum number. In turn, calculation of the magnitude of the total spin magnetic moment requires that be replaced by: Thus, for a single electron, with spin quantum number the component of the magnetic moment along the field direction is, from, while the total spin magnetic moment is, from, or approximately 1.73 μ. The analysis is readily extended to the spin-only magnetic moment of an atom. For example, the total spin magnetic moment of a transition metalion with a single d shell electron outside of closed shells is 1.73 μ since while an atom with two unpaired electrons (e.g. Vanadium V with would have an effective magnetic moment of
