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Sommerfeld's Extension to Bohr's Model

Bridging the gap: How elliptical orbits explained the fine structure of spectral lines.

By chemca Team • Updated Jan 2026

While Bohr's atomic model successfully explained the main spectral lines of the hydrogen atom, it failed to account for the fine structure—the splitting of spectral lines into multiple closely spaced lines observed under high-resolution spectroscopes. In 1916, Arnold Sommerfeld modified Bohr's model to address this limitation.

1. The Need for Extension

Bohr assumed electrons move in circular orbits. However, high-resolution spectroscopy revealed that single spectral lines (like $H_\alpha$) actually consisted of several very close components. This "fine structure" suggested that the energy levels were split into sublevels, which circular orbits alone could not explain.

2. Key Postulates of Sommerfeld's Model

A. Elliptical Orbits

Sommerfeld proposed that electrons revolve around the nucleus in elliptical orbits, with the nucleus situated at one of the foci. Circular orbits are just a special case of elliptical orbits.

B. Introduction of Two Quantum Numbers

To describe an electron in an elliptical orbit, two quantum numbers are required (unlike just one in Bohr's model):

1. Principal Quantum Number ($n$): Determines the size of the orbit (major axis).
2. Azimuthal Quantum Number ($k$): Determines the shape of the orbit (minor axis).
   (Note: In modern notation, $k$ relates to $l$ as $k = l + 1$).

Shape Relationship:

The shape of the ellipse is defined by the ratio of the length of the minor axis ($b$) to the major axis ($a$):

$$ \frac{b}{a} = \frac{k}{n} = \frac{l+1}{n} $$

• If $n = k$, then $b = a$, and the orbit is Circular.
• If $k < n$, then $b < a$, and the orbit is Elliptical.

3. Visualization of Orbits

For a given value of $n$, $k$ can take integer values from $1$ to $n$.

Example: For n = 3

Possible values for $k$ are 1, 2, and 3.

• $k=3$ ($n=k$): Circular orbit (Modern 3d).
• $k=2$: Elliptical orbit.
• $k=1$: Narrower elliptical orbit (Modern 3s).

This implies that the 3rd shell is composed of 3 subshells with slightly different energies.

4. Relativistic Variation of Mass

Why do different shapes result in different energies?

In an elliptical orbit, the distance of the electron from the nucleus varies. Consequently, its velocity varies (moving faster near the nucleus). According to Einstein's theory of relativity, mass increases with velocity.

$$ m = \frac{m_0}{\sqrt{1 - (v/c)^2}} $$

This relativistic change in mass causes the orbit to precess (a rosette path). This slight difference in energy between circular and elliptical paths of the same shell creates the fine spectral lines.

5. Limitations

• It could not explain the spectra of atoms with more than one electron (e.g., Helium).
• It failed to explain the Zeeman Effect (splitting of lines in a magnetic field) and the Stark Effect (splitting in an electric field).
• It did not provide the correct number of spectral lines for anomalous Zeeman effect.
• It was eventually superseded by the quantum mechanical model (Schrödinger).

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