Sommerfeld’s atomic model: characteristics, postulates, advantages and disadvantages

The atom model proposed by the Danish physicist Niels Bohr describes the simplest atom of all, hydrogen, but could not explain why electrons in the same energy state could present different energy levels in the presence of electromagnetic fields.

In the theory proposed by Bohr, the electron orbiting the nucleus can only have certain values ​​of its orbital angular momentum L, and therefore cannot be in any orbit.

Bohr also considered these orbits to be circular and a single quantum number called the principal quantum number n = 1, 2, 3… served to identify the allowed orbits.

A circle is described by its radius, but for an ellipse two parameters must be given: semi-major axis and semi-minor axis, in addition to its spatial orientation. With this he introduced two more quantum numbers.

Sommerfeld’s second major modification was to add relativistic effects to the atomic model. Nothing is faster than light, however Sommerfeld had found electrons with appreciably close speeds, therefore it was necessary to incorporate relativistic effects into any description of the atom.

Sommerfeld atomic model postulates

Electrons follow circular and elliptical orbits

The electrons in the atom follow elliptical orbits (circular orbits are a particular case) and their energy state can be characterized by 3 quantum numbers: the principal quantum number n , the secondary quantum number or azimuthal number l and the magnetic quantum number L .

But ellipses with the same semi-major axis can have different semi-minor axes, depending on the degree of eccentricity. An eccentricity equal to 0 corresponds to a circle, so it does not rule out circular paths. In addition, ellipses can have different inclinations in space.

Therefore Sommerfeld he added to its model number quantum secondary l to indicate the minor axis and the magnetic quantum number m L . Thus it indicated what are the allowed spatial orientations of the elliptical orbit.

Figure 2. The orbits corresponding to the energy level n = 5 are shown for different values ​​of angular momentum l that have full de Broglie wavelengths. Source: wikimedia commons.

Note that it does not add new principal quantum numbers, so the total energy of the electron in elliptical orbit is the same as in the Bohr model. Therefore there are no new energy levels, but a doubling of the levels given by the number n.

Zeeman effect and Stark effect

In this way it is possible to fully specify a given orbit, thanks to the 3 quantum numbers mentioned and thus explain the existence of two effects: the Zeeman effect and the Stark effect.

And so he explains the doubling of the energy that appears in the normal Zeeman effect (there is also an anomalous Zeeman effect), in which a spectral line is divided into several components when it is in the presence of a magnetic field.

This doubling of the lines also occurs in the presence of an electric field, which is known as the Stark effect, which led Sommerfeld to think about modifying the Bohr model to explain these effects.

The atomic nucleus and electrons move around their center of mass

After Ernest Rutherford discovered the atomic nucleus and the fact that almost all the mass of the atom is concentrated there, scientists believed that the nucleus was more or less stationary.

However, Sommerfeld postulated that both the nucleus and the orbiting electrons move around the center of mass of the system, which of course is very close to the nucleus. His model uses the reduced mass of the electron – nucleus system, rather than the mass of the electron.

In elliptical orbits, as with the planets around the Sun , there are times when the electron is closer, and other times farther from the nucleus. Therefore its speed is different at each point in its orbit.

Figure 3.- Arnold Sommerfeld. Source: Wikimedia Commons. GFHund [CC BY 3.0 (].

Electrons can reach relativistic speeds

Sommerfeld introduced into his model the fine structure constant, a dimensionless constant related to the electromagnetic force:

α = 1 /137.0359895

It is defined as the quotient between the electron charge e  squared, and the product between Planck’s constant h and the speed of light c in a vacuum, all multiplied by 2π:

α = 2π (e 2 / hc) = 1 /137.0359895

The fine structure constant relates to three of the most important constants in atomic physics. The other is the mass of the electron, which is not listed here.

In this way electrons are linked with photons (which move at speed c in vacuum), and thus explain the deviations of some spectral lines of the hydrogen atom from those predicted by the Bohr model.

Thanks to relativistic corrections, energy levels with equal n but different l are separated, giving rise to the fine structure of the spectrum, hence the name of the constant α.

And all the characteristic lengths of the atom can be expressed in terms of this constant.

Advantages and disadvantages


-Sommerfeld showed that a single quantum number was insufficient to explain the spectral lines of the hydrogen atom.

-It was the first model to propose a spatial quantization, since the projections of the orbits in the direction of the electromagnetic field are, in effect, quantized.

-The Sommerfeld model successfully explained that electrons with the same principal quantum number n differ in their energy state, because they can have different quantum numbers l and m L .

-Introduced the constant α to develop the fine structure of the atomic spectrum and explain the Zeeman effect.

-Included relativistic effects, since electrons can move with speeds quite close to that of light.


-Your model was only applicable to atoms with one electron and in many respects to alkali metal atoms such as Li 2+ , but it is not useful in the helium atom, which has two electrons.

-It did not explain the electronic distribution in the atom.

-The model allowed to calculate the energies of the allowed states and the frequencies of the radiation emitted or absorbed in the transitions between states, without giving information about the times of these transitions.

-Now it is known that electrons do not follow paths with predetermined shapes such as orbits, but occupy  orbitals , regions of space that correspond to solutions of the Schrodinger equation.

-The model arbitrarily combined classical aspects with quantum aspects.

-It failed to explain the anomalous Zeeman effect, for this the Dirac model is needed, which later added another quantum number.

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