The Rydberg formula is one of the most important results in atomic spectroscopy. It predicts the wavelengths of the
spectral lines emitted or absorbed by hydrogen and hydrogen-like atoms. In a hydrogen-like atom, a single electron
moves in the Coulomb field of a nucleus of charge \(+Ze\), so the energy levels follow a simple quantized pattern.
When the electron changes from a higher shell to a lower shell, the atom emits a photon whose wavelength is fixed by
the difference between the two energy levels.
The Rydberg formula
For a transition between an upper level \(n_2\) and a lower level \(n_1\), with \(n_2 > n_1\), the wavelength is
determined by
Hydrogen-like transition formula.
\[
\begin{aligned}
\frac{1}{\lambda}
&= R_\infty Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right).
\end{aligned}
\]
Here \(R_\infty\) is the Rydberg constant and \(Z\) is the atomic number of the hydrogen-like ion. For ordinary
hydrogen, \(Z = 1\). For helium ion He⁺, \(Z = 2\), so the inverse wavelength is multiplied by \(4\). That means the
wavelengths become much shorter as the nuclear charge increases.
The formula is usually written with \(n_2 > n_1\), because that makes the quantity in parentheses positive. The same
wavelength also describes the corresponding absorption transition in the reverse direction.
Energy-level interpretation
The Rydberg formula can be understood from the hydrogen-like shell energies
Hydrogen-like energy levels.
\[
\begin{aligned}
E_n &= -\frac{13.6 Z^2}{n^2}\ \mathrm{eV}.
\end{aligned}
\]
These levels are negative because the electron is bound to the nucleus. The higher the value of \(n\), the closer the
energy is to zero. A transition from \(n_2\) to \(n_1\) produces a photon with energy
\[
\begin{aligned}
E_\gamma &= \left|E_{n_2} - E_{n_1}\right|.
\end{aligned}
\]
Since photon energy also satisfies \(E_\gamma = hf\) and \(f = c/\lambda\), the Rydberg formula is consistent with the
quantized energy-level picture of the atom.
Named spectral series
Spectral lines are grouped into series according to the lower level \(n_1\):
| Series |
Lower level |
Typical region |
Example line |
| Lyman |
\(n_1 = 1\) |
Ultraviolet |
Lyα: \(2 \to 1\) |
| Balmer |
\(n_1 = 2\) |
Visible / near UV |
Hβ: \(4 \to 2\) |
| Paschen |
\(n_1 = 3\) |
Infrared |
Paα: \(4 \to 3\) |
| Brackett |
\(n_1 = 4\) |
Infrared |
\(5 \to 4\) |
This is why the lower level is so important: it determines the series name and often the spectral region in which the
line appears.
Sample calculation: Balmer Hβ
A classic example is the Balmer-series transition with \(n_1 = 2\), \(n_2 = 4\), and \(Z=1\). This is the Hβ line of
hydrogen.
Step 1. Write the Rydberg formula.
\[
\begin{aligned}
\frac{1}{\lambda}
&= R_\infty \left(\frac{1}{2^2} - \frac{1}{4^2}\right).
\end{aligned}
\]
Step 2. Simplify the bracket.
\[
\begin{aligned}
\frac{1}{2^2} - \frac{1}{4^2}
&= \frac{1}{4} - \frac{1}{16} \\
&= \frac{3}{16}.
\end{aligned}
\]
Step 3. Compute the inverse wavelength.
\[
\begin{aligned}
\frac{1}{\lambda}
&= R_\infty \cdot \frac{3}{16}.
\end{aligned}
\]
Step 4. Invert to find the wavelength.
\[
\begin{aligned}
\lambda &\approx 486.1\ \mathrm{nm}.
\end{aligned}
\]
This value lies in the visible part of the spectrum, which is why Balmer lines are so important in astronomy and in
introductory discussions of atomic spectra.
Frequency and photon energy
Once the wavelength is known, the corresponding photon frequency and energy follow from
\[
\begin{aligned}
f &= \frac{c}{\lambda}, \\
E_\gamma &= hf.
\end{aligned}
\]
These conversions are useful because spectroscopy is reported in different units depending on context. Optical
spectroscopy often uses wavelength in nanometers, while atomic and quantum calculations often use energy in electron
volts.
Series limit and convergence
For a fixed lower level \(n_1\), the series converges as \(n_2 \to \infty\). In that limit, the term \(1/n_2^2\)
becomes negligible, so the series limit is
\[
\begin{aligned}
\frac{1}{\lambda_{\mathrm{limit}}}
&= R_\infty Z^2 \frac{1}{n_1^2}.
\end{aligned}
\]
This explains why the spectral lines crowd closer together at the short-wavelength end of each series.
Advanced note
The standard Rydberg formula uses the infinite-nuclear-mass Rydberg constant \(R_\infty\). At university level, one
improves the prediction slightly by including the reduced mass of the electron-nucleus system. That correction changes
the wavelengths by a small amount, but the simple formula remains the standard first approximation and is usually the
most useful form for educational calculations.
| Concept |
Main relation |
Meaning |
| Rydberg formula |
\(1/\lambda = R_\infty Z^2(1/n_1^2 - 1/n_2^2)\) |
Transition wavelength for a hydrogen-like atom |
| Energy levels |
\(E_n = -13.6 Z^2/n^2\ \mathrm{eV}\) |
Hydrogen-like shell energies |
| Photon frequency |
\(f = c/\lambda\) |
Frequency associated with the transition wavelength |
| Photon energy |
\(E_\gamma = hf\) |
Energy carried by the emitted or absorbed photon |
| Series naming |
Named by \(n_1\) |
Lyman, Balmer, Paschen, Brackett, and higher series |
| Series limit |
\(1/\lambda_{\mathrm{limit}} = R_\infty Z^2/n_1^2\) |
Short-wavelength convergence point of the series |