Landé g-factor

The factor comes about during the calculation of the first-order perturbation in the energy of an atom when a weak uniform magnetic field (that is, weak in comparison to the system's internal magnetic field) is applied to the system. Formally we can write the factor as,[2]

${\displaystyle g_{J}=g_{L}{\frac {J(J+1)-S(S+1)+L(L+1)}{2J(J+1)}}+g_{S}{\frac {J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}}.}$

The orbital ${\displaystyle g_{L}}$ is equal to 1, and under the approximation ${\displaystyle g_{S}=2}$, the above expression simplifies to

${\displaystyle g_{J}(g_{L}=1,g_{S}=2)=1+{\frac {J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}}.}$

Here, J is the total electronic angular momentum, L is the orbital angular momentum, and S is the spin angular momentum. Because ${\displaystyle S=1/2}$ for electrons, one often sees this formula written with 3/4 in place of ${\displaystyle S(S+1)}$. The quantities g_L and g_S are other g-factors of an electron. For an ${\displaystyle S=0}$ atom, ${\displaystyle g_{J}=1}$ and for an ${\displaystyle L=0}$ atom, ${\displaystyle g_{J}=2}$.

If we wish to know the g-factor for an atom with total atomic angular momentum ${\displaystyle {\vec {F}}={\vec {I}}+{\vec {J}}}$ (nucleus + electrons), such that the total atomic angular momentum quantum number can take values of ${\displaystyle F=J+I,J+I-1,\dots ,|J-I|}$, giving

${\displaystyle {\begin{aligned}g_{F}&=g_{J}{\frac {F(F+1)-I(I+1)+J(J+1)}{2F(F+1)}}+g_{I}{\frac {\mu _{\text{N}}}{\mu _{\text{B}}}}{\frac {F(F+1)+I(I+1)-J(J+1)}{2F(F+1)}}\\&\approx g_{J}{\frac {F(F+1)-I(I+1)+J(J+1)}{2F(F+1)}}\end{aligned}}}$

Here ${\displaystyle \mu _{\text{B}}}$ is the Bohr magneton and ${\displaystyle \mu _{\text{N}}}$ is the nuclear magneton. This last approximation is justified because ${\displaystyle \mu _{N}}$ is smaller than ${\displaystyle \mu _{B}}$ by the ratio of the electron mass to the proton mass.

The following working is a common derivation.[3][4]

Both orbital angular momentum and spin angular momentum of electron contribute to the magnetic moment. In particular, each of them alone contributes to the magnetic moment by the following form

${\displaystyle {\vec {\mu }}_{L}=-{\vec {L}}g_{L}\mu _{\rm {B}}/\hbar }$

${\displaystyle {\vec {\mu }}_{S}=-{\vec {S}}g_{S}\mu _{\rm {B}}/\hbar }$

${\displaystyle {\vec {\mu }}_{J}={\vec {\mu }}_{L}+{\vec {\mu }}_{S}}$

where

${\displaystyle g_{L}=1}$

${\displaystyle g_{S}\approx 2}$

Note that negative signs in the above expressions are because an electron carries negative charge, and the value of ${\displaystyle g_{S}}$ can be derived naturally from Dirac's equation. The total magnetic moment ${\displaystyle {\vec {\mu }}_{J}}$, as a vector operator, does not lie on the direction of total angular momentum ${\displaystyle {\vec {J}}={\vec {L}}+{\vec {S}}}$, because the g-factors for orbital and spin part are different. However, due to Wigner-Eckart theorem, its expectation value does effectively lie on the direction of ${\displaystyle {\vec {J}}}$ which can be employed in the determination of the g-factor according to the rules of angular momentum coupling. In particular, the g-factor is defined as a consequence of the theorem itself

${\displaystyle \langle J,J_{z}|{\vec {\mu }}_{J}|J,J'_{z}\rangle =-g_{J}\mu _{\rm {B}}\langle J,J_{z}|{\vec {J}}|J,J'_{z}\rangle }$

Therefore,

${\displaystyle \langle J,J_{z}|{\vec {\mu }}_{J}|J,J'_{z}\rangle \cdot \langle J,J'_{z}|{\vec {J}}|J,J_{z}\rangle =-g_{J}\mu _{\rm {B}}\langle J,J_{z}|{\vec {J}}|J,J'_{z}\rangle \cdot \langle J,J'_{z}|{\vec {J}}|J,J_{z}\rangle }$

${\displaystyle \sum _{J'_{z}}\langle J,J_{z}|{\vec {\mu }}_{J}|J,J'_{z}\rangle \cdot \langle J,J'_{z}|{\vec {J}}|J,J_{z}\rangle =-\sum _{J'_{z}}g_{J}\mu _{\rm {B}}\langle J,J_{z}|{\vec {J}}|J,J'_{z}\rangle \cdot \langle J,J'_{z}|{\vec {J}}|J,J_{z}\rangle }$

${\displaystyle \langle J,J_{z}|{\vec {\mu }}_{J}\cdot {\vec {J}}|J,J_{z}\rangle =-g_{J}\mu _{\rm {B}}\langle J,J_{z}|{\vec {J}}\cdot {\vec {J}}|J,J_{z}\rangle =-g_{J}\mu _{\rm {B}}\quad \hbar ^{2}J(J+1)}$

One gets

${\displaystyle {\begin{aligned}g_{J}\langle J,J_{z}|{\vec {J}}\cdot {\vec {J}}|J,J_{z}\rangle &=\langle J,J_{z}|g_{L}{{\vec {L}}\cdot {\vec {J}}}+g_{S}{{\vec {S}}\cdot {\vec {J}}}|J,J_{z}\rangle \\&=\langle J,J_{z}|g_{L}{({\vec {L}}^{2}+{\frac {1}{2}}({\vec {J}}^{2}-{\vec {L}}^{2}-{\vec {S}}^{2}))}+g_{S}{({\vec {S}}^{2}+{\frac {1}{2}}({\vec {J}}^{2}-{\vec {L}}^{2}-{\vec {S}}^{2}))}|J,J_{z}\rangle \\&={\frac {g_{L}\hbar ^{2}}{2}}(J(J+1)+L(L+1)-S(S+1))+{\frac {g_{S}\hbar ^{2}}{2}}(J(J+1)-L(L+1)+S(S+1))\\g_{J}&=g_{L}{\frac {J(J+1)+L(L+1)-S(S+1)}{2J(J+1)}}+g_{S}{\frac {J(J+1)-L(L+1)+S(S+1)}{2J(J+1)}}\end{aligned}}}$

• Einstein–de Haas effect
• Zeeman effect