In atomic physics, the Landé interval rule [1] states that, due to weak angular momentum coupling (either spin-orbit or spin-spin coupling), the energy splitting between successive sub-levels are proportional to the total angular momentum quantum number (J or F) of the sub-level with the larger of their total angular momentum value (J or F). [2] [3]
The rule assumes the Russell–Saunders coupling and that interactions between spin magnetic moments can be ignored. The latter is an incorrect assumption for light atoms. As a result of this, the rule is optimally followed by atoms with medium atomic numbers.[4]
The rule was first stated in 1923 by German-American physicist Alfred Landé.
As an example, consider an atom with two valence electrons and their fine structures in the LS-coupling scheme. We will derive heuristically the interval rule for the LS-coupling scheme and will remark on the similarity that leads to the interval rule for the hyperfine structure.
The interactions between electrons couple their orbital and spin angular momentums. Let's denote the spin and orbital angular momentum as
s
l
L=l1+l2
S=s1+s2
Hs-o=\beta1s1 ⋅ l1+\beta2s2 ⋅ l2
where
\beta1
\beta2
Hs-o
\vertLmLSmS\rangle
L
S
J=L+S
Jz
\vertLmLJmJ\rangle
\vertLmLSmS\rangle
To study this perturbation, we consider the vector model where we treat each
l
l1
l2
L
L
L
l1 → \left[\left(\overline{l1 ⋅ L
|L|2
L(L+1)
\left(\overline{l1 ⋅ L
\left\langlel1 ⋅ L\right\rangle
Applying this change to all the terms in the Hamiltonian, we can rewrite it as
\begin{aligned} Hs-o&=\beta1
\left\langles1 ⋅ S\right\rangle | |
S(S+1) |
S ⋅
\left\langlel1 ⋅ L\right\rangle | |
L(L+1) |
L+\beta2
\left\langles2 ⋅ S\right\rangle | |
S(S+1) |
S ⋅
\left\langlel2 ⋅ L\right\rangle | |
L(L+1) |
L\\ &=\betaLS ⋅ L \end{aligned}
Es-o=\betaL\langleS ⋅ L\rangle.
L ⋅ S=(J ⋅ J-L ⋅ L-S ⋅ S)/2
Es-o=
\betaL | |
2 |
\{J(J+1)-L(L+1)-S(S+1)\}.
J
\DeltaEFS=EJ-EJ-1=\betaLJ
As an example, consider a
{}3P
{}3P0,{}3P1,{}3P2
J=2
J=1
2\beta
J=1
J=0
\beta
As for the spin-spin interaction responsible for the hyperfine structure, because the Hamiltonian of the hyperfine interaction can be written as
HHFS=AHFSI ⋅ J
I
J
EF-EF-1=AHFSF
F
F=I+J
I
J
F
The interval rule holds when the coupling is weak. In the LS-coupling scheme, a weak coupling means the energy of spin-orbit coupling
Es-o
Es-o\llEre
A\gg\DeltaEQuadrupole
For example, in helium, the spin-spin interactions and spin-other-orbit interaction have an energy comparable to that of the spin-orbit interaction.