Landé g-factor

In physics, the Landé g-factor is a particular example of a g-factor, namely for an electron with both spin and orbital angular momenta. It is named after Alfred Landé, who first described it in 1921.

In atomic physics, the Landé g-factor is a multiplicative term appearing in the expression for the energy levels of an atom in a weak magnetic field. The quantum states of electrons in atomic orbitals are normally degenerate in energy, with these degenerate states all sharing the same angular momentum. When the atom is placed in a weak magnetic field, however, the degeneracy is lifted.

Description
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,


 * $$g_J= g_L\frac{J(J+1)-S(S+1)+L(L+1)}{2J(J+1)}+g_S\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}.$$

The orbital $$g_L $$ is equal to 1, and under the approximation $$g_S = 2 $$, the above expression simplifies to


 * $$g_J(g_L=1,g_S=2) = 1+\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}.$$

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

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


 * $$\begin{align}

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{align}$$

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

A derivation
The following working is a common derivation.

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
 * $$\vec \mu_L= -\vec L g_L \mu_{\rm B}/\hbar$$
 * $$\vec \mu_S= -\vec S g_S \mu_{\rm B}/\hbar$$
 * $$\vec \mu_J= \vec \mu_L + \vec \mu_S$$

where
 * $$g_L = 1$$
 * $$g_S \approx 2$$

Note that negative signs in the above expressions are because an electron carries negative charge, and the value of $$g_S$$ can be derived naturally from Dirac's equation. The total magnetic moment $$\vec \mu_J$$, as a vector operator, does not lie on the direction of total angular momentum $$\vec J = \vec L+\vec S$$, 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 $$\vec J$$ 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
 * $$\langle J,J_z|\vec \mu_J|J,J'_z\rangle = -g_J\mu_{\rm B}\langle J,J_z|\vec J|J,J'_z\rangle$$

Therefore,
 * $$\langle J,J_z|\vec \mu_J|J,J'_z\rangle\cdot\langle J,J'_z|\vec J|J,J_z\rangle = -g_J\mu_{\rm B}\langle J,J_z|\vec J|J,J'_z\rangle\cdot\langle J,J'_z|\vec J|J,J_z\rangle$$
 * $$\sum_{J'_z}\langle J,J_z|\vec \mu_J|J,J'_z\rangle\cdot\langle J,J'_z|\vec J|J,J_z\rangle = -\sum_{J'_z}g_J\mu_{\rm B}\langle J,J_z|\vec J|J,J'_z\rangle \cdot\langle J,J'_z|\vec J|J,J_z\rangle$$
 * $$\langle J,J_z|\vec \mu_J\cdot \vec J|J,J_z\rangle = -g_J\mu_{\rm B}\langle J,J_z|\vec J\cdot\vec J|J,J_z\rangle = -g_J\mu_{\rm B} \quad \hbar^2 J(J+1) $$

One gets
 * $$\begin{align}

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)}+g_S  \frac{J(J+1)-L(L+1)+S(S+1)} \end{align}$$