# Landé g-factor

List of Landé g-factors for the Lanthanides
Element Landé g-factor
Z Name
57 Lanthanum 0.800 [1]
59 Praseodymium 0.732 [1]
60 Neodymium 0.603 [1] 0.605 [2]
62 Samarium -
63 Europium 1.996 [1] 1.996 [3] 1.9926 [4]
64 Gadolinium 2.653 [1]
65 Terbium 1.326 [1]
66 Dysprosium 1.243 [1]
67 Holmium 1.97 [1]
68 Erbium 1.166 [1] 1.165 [5]
69 Thulium 1.143 [1]
70 Ytterbium -

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,[6]

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

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

$g_J \approx \frac{3}{2}+\frac{S(S+1)-L(L+1)}{2J(J+1)}.$

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.

If we wish to know the g-factor for an atom with total atomic angular momentum F=I+J,

$g_F= g_J\frac{F(F+1)-I(I+1)+J(J+1)}{2F(F+1)}+g_I\frac{F(F+1)+I(I+1)-J(J+1)}{2F(F+1)}$
$\approx g_J\frac{F(F+1)-I(I+1)+J(J+1)}{2F(F+1)}$

This last approximation is justified because $g_I$ is smaller than $g_J$ by the ratio of the electron mass to the proton mass.

## A derivation

The following derivation basically follows the line of thought in [7] and.[8]

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

$\vec \mu_L= \vec L g_L \mu_B$
$\vec \mu_S= \vec S g_S \mu_B$
$\vec \mu_J= \vec \mu_L + \vec \mu_S$

where

$g_L = -1$
$g_S = -2$

Note that negative signs in the above expressions are due to the fact that 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 $\vec J = \vec L+\vec S$. 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

$\langle J,J_z|\vec \mu_J|J,J_{{z'}}\rangle = g_J\mu_B\langle J,J_z|\vec J|J,J_{z'}\rangle$

Therefore,

$\langle J,J_z|\vec \mu_J|J,J_{z'}\rangle\cdot\langle J,J_{z'}|\vec J|J,J_z\rangle = g_J\mu_B\langle J,J_z|\vec J|J,J_{z'}\rangle\cdot\langle J,J_{z'}|\vec J|J,J_z\rangle$
$\sum_{J_{z'}}\langle J,J_z|\vec \mu_J|J,J_{z'}\rangle\cdot\langle J,J_{z'}|\vec J|J,J_z\rangle = \sum_{J_{z'}}g_J\mu_B\langle J,J_z|\vec J|J,J_{z'}\rangle \cdot\langle J,J_{z'}|\vec J|J,J_z\rangle$
$\langle J,J_z|\vec \mu_J\cdot \vec J|J,J_z\rangle = g_J\mu_B\langle J,J_z|\vec J\cdot\vec J|J,J_z\rangle$

One gets

$g_J\langle J,J_z|\vec J\cdot\vec J|J,J_z \rangle = g_L {{\vec L}\cdot {\vec J}}+g_S {{\vec S} \cdot {\vec J}}$
$= g_L {(\vec L^2+\frac{1}{2}(\vec J^2-\vec L^2-\vec S^2))}+g_S {(\vec S^2+\frac{1}{2}(\vec J^2-\vec L^2-\vec S^2))}$
$g_J = g_L \frac{J(J+1)+L(L+1)-S(S+1)}{{2J(J+1)}}+g_S \frac{J(J+1)-L(L+1)+S(S+1)}{{2J(J+1)}}$

## References

1. Quinet, Pascal; Biémont, Emile (2004). "Lande g-factors for experimentally determined energy levels in doubly ionized lanthanides". Atomic Data and Nuclear Data Tables 87 (2): 207–230. doi:10.1016/j.adt.2004.04.001.
2. ^ Bord, D.J. (June 2000). "Ab initio calculations of oscillator strengths and Landé factors for Nd III". Astron. Astrophys. 144: 517. doi:10.1051/aas:2000226.
3. ^ Mashonkina, L. I.; Ryabtsev, A. N.; Ryabchikova, T. A. (2002). "Eu III oscillator strengths and europium abundances in Ap stars". Astron. Lett. 28 (1): 34. doi:10.1134/1.1434452.
4. ^ Baker, J. M.; Williams, F. I. B. (8 May 1962). "Electron Nuclear Double Resonance of the Divalent Europium Ion". Proc. R. Soc. Lond. A. 267 (1329): 283. doi:10.1098/rspa.1962.0098.
5. ^ Wyart, Jean-François; Blaise, Jean; Bidelman, William P; Cowley, Charles R (1997). "Energy levels and transition probabilities in doubly-ionized erbium (Er III)" (PDF). Phys. Scr. 56 (5): 446. doi:10.1088/0031-8949/56/5/008.
6. ^ Nave, C. R. (25 January 1999). "Magnetic Interactions and the Lande' g-Factor". HyperPhysics. Georgia State University. Retrieved 14 October 2014.
7. ^ Ashcroft, Neil W.; Mermin, N. David (1976). Solid state physics. Saunders College. ISBN 9780030493461.
8. ^ Yang, Fujia; Hamilton, Joseph H. (2009). Modern Atomic and Nuclear Physics (Revised ed.). World Scientific. p. 132. ISBN 9789814277167.