The spectral lines of atoms are not perfectly single lines when examined at high resolution. Two important sources of
splitting are the fine structure and the Zeeman effect. Fine structure comes from
relativistic corrections and spin-orbit coupling inside the atom, while Zeeman splitting appears when an external
magnetic field interacts with the magnetic moments associated with angular momentum.
Hydrogen-like base energy
For a hydrogen-like atom, the Bohr-model energy of the \(n\)-th level is
Nonrelativistic level energy.
\[
\begin{aligned}
E_n &= -\frac{13.6 Z^2}{n^2}\ \mathrm{eV}.
\end{aligned}
\]
This depends only on the principal quantum number \(n\) in the simplest model. However, once relativistic effects are
included, the total energy also depends on the total angular momentum quantum number \(j\).
Fine-structure correction
A common approximate formula for the hydrogen-like fine-structure energy is
Approximate fine-structure energy.
\[
\begin{aligned}
E_{n,j} &\approx E_n \left[1 + \frac{(Z\alpha)^2}{n^2}\left(\frac{n}{j+\tfrac{1}{2}} - \frac{3}{4}\right)\right].
\end{aligned}
\]
Here \(\alpha\) is the fine-structure constant. The fine-structure shift is then
\[
\begin{aligned}
\Delta E_{\text{fs}} &= E_{n,j} - E_n.
\end{aligned}
\]
This correction is small compared with the main Bohr energy, but it is large enough to split levels that would
otherwise be degenerate in the simple hydrogen model. For a fixed \(n\) and \(\ell > 0\), the two allowed values
\(j=\ell \pm \tfrac{1}{2}\) usually produce slightly different energies.
Landé \(g\)-factor and the Zeeman effect
When a magnetic field \(B\) is applied, a level with total angular momentum \(j\) splits into sublevels labeled by
\(m_j = -j, -j+1, \dots, j\). The size of the splitting depends on the Landé \(g_J\) factor:
Landé factor.
\[
\begin{aligned}
g_J &= 1 + \frac{j(j+1)+s(s+1)-\ell(\ell+1)}{2j(j+1)},
\qquad s=\frac{1}{2}.
\end{aligned}
\]
The corresponding Zeeman energy shift is
Zeeman shift.
\[
\begin{aligned}
\Delta E_Z &= \mu_B B m_j g_J,
\end{aligned}
\]
where \(\mu_B\) is the Bohr magneton. This formula shows that the shift is linear in the magnetic field strength,
in the magnetic quantum number \(m_j\), and in the Landé factor. That is why equally spaced Zeeman components often
appear for a fixed \(j\)-manifold.
Sample estimate
Take a simple case with \(B = 1\ \mathrm{T}\). Since
\[
\begin{aligned}
\mu_B &\approx 5.788 \times 10^{-5}\ \mathrm{eV\,T^{-1}},
\end{aligned}
\]
the Zeeman scale at one tesla is already of order
\[
\begin{aligned}
\mu_B B &\approx 5.8 \times 10^{-5}\ \mathrm{eV}.
\end{aligned}
\]
This is exactly the scale mentioned in many introductory examples. The actual shift for a chosen sublevel is then
obtained by multiplying by \(m_j g_J\). For example, if \(g_J \approx 1\) and \(m_j = 1\), the shift is about
\(5.8 \times 10^{-5}\ \mathrm{eV}\).
How the preview should be interpreted
The calculator first computes the hydrogen-like base energy \(E_n\), then applies the approximate fine-structure
correction to get \(E_{n,j}\). After that, it computes the Landé factor and shows how the magnetic field splits that
\(j\)-level into different \(m_j\) components. The resulting plot is therefore best understood as a level-structure
preview rather than as a full precision spectroscopy package.
In a more advanced university treatment, one would distinguish the normal and anomalous Zeeman effects more carefully,
include full spectroscopic term symbols, and handle multi-electron atoms with LS or jj coupling. For hydrogen itself,
additional shifts such as the Lamb shift can also be important at high precision. Still, the standard fine-structure
and Zeeman formulas are the correct starting point for understanding why spectral lines split.
| Concept |
Main relation |
Meaning |
| Base hydrogen-like level |
\(E_n = -13.6 Z^2 / n^2\ \mathrm{eV}\) |
Nonrelativistic Bohr-model energy |
| Fine-structure total energy |
\(E_{n,j} \approx E_n[1 + (Z\alpha)^2/n^2 (n/(j+1/2)-3/4)]\) |
Approximate relativistic and spin-orbit corrected energy |
| Fine-structure shift |
\(\Delta E_{\text{fs}} = E_{n,j} - E_n\) |
Small correction relative to the Bohr level |
| Landé factor |
\(g_J = 1 + [j(j+1)+s(s+1)-\ell(\ell+1)]/[2j(j+1)]\) |
Coupling factor for magnetic splitting |
| Zeeman shift |
\(\Delta E_Z = \mu_B B m_j g_J\) |
Magnetic-field energy shift of the sublevel |
| Magnetic sublevels |
\(m_j = -j, \dots, j\) |
Split components of a given \(j\)-level |