# Arnold Sommerfeld's model of the Atom

The history of physics from ancient times to the modern day, focusing on light and matter. In 1915, German physicist Arnold Sommerfeld showed that electrons can be described by a second quantum number, besides shell number. This describes the shape of the orbit, rather than the distance from the nucleus. In 1920, he described a third number that shows which way a shape is facing.

Last updated on 5th June 2017 by Dr Helen Klus

## 1. Azimuthal quantum number l ↑

German physicist Arnold Sommerfeld extended Danish physicist Niels Bohr's model of the atom[1]. In 1915, Sommerfeld showed that another number is needed to describe electron orbits, besides the shell number, n[2][3]. This is known as the azimuthal quantum number, l.

The azimuthal quantum number describes the orbital angular momentum of an electron, which defines the 'shape' of the orbit.

Bohr had assumed that orbits would be circular, but Sommerfeld showed that they could take as many shapes as the shell number i.e. the two electrons in the first shell of helium both have the same shape because the shell number is 1, but the two electrons in the second shell of beryllium can each have different shapes because the shell number is 2.

The shapes become more complex the higher the shell number. The maximum number of electrons that are allowed to have the same shape can be found using the formula:
Maximum number = 2(2l + 1).

This means that 2 electrons can have the l = 0 shape, 6 can have the l = 1 shape, 10 can have the l = 2 shape, and so on. The image below shows the possible shapes that electron orbits can take in the first five electron shells.

Hydrogen atom electron orbitals (for magnetic quantum number ml = 0). Image credit: modified by Helen Klus, original image by Florian Marquardt/CC-SA.

## 2. Magnetic quantum number ml ↑

In 1920, Sommerfeld realised that another number was needed to describe the orbit of electrons[4][5]. This is because the current model could still not explain the Zeeman effect[6], the splitting of spectral lines in the presence of a magnetic field.

Sommerfeld realised that the same l shape can have different orientations in space defined by the magnetic quantum number, ml. The maximum number of different orientations can be found using the formula Maximum number = 2l + 1.

This means that the l = 0 shape has 1 ml value, which is designated 0. The l = 1 shape can have up to 3 different ml values, which are designated -1, 0, and 1, the l = 2 shape can have up to 5 different ml values, which are designated -2, -1, 0, 1, and 2, and so on. Possible orientations for the first four orbital shapes are shown below.

Different orientations of the first four orbital shapes. Image credit: modified by Helen Klus, original image by haade/CC-SA.

A magnetic field causes electrons of the same energy, which would otherwise produce a single spectral line, to have different energies depending on their orientation with respect to the magnetic field.

In the case of a cloud of hydrogen atoms, the transition from the second to first shell produces three lines as about 1/3 gain energy, 1/3 loose energy, and 1/3 are unaffected by the presence of the magnetic field. This explains the normal Zeeman effect.

Sommerfeld was still unable to explain the anomalous Zeeman effect, a term used to describe spectra that split into more than three lines in the presence of a magnetic field[7][8]. This is more likely to occur in lines made from atoms with an odd number of electrons in their outer shell.

Because of this, many did not yet accept that electrons had orbits that were defined by quantised positions and directions in space.

British physicist Paul Dirac would later explain the anomalous Zeeman effect using another quantum number, that associated with spin.

In 1920, Bohr formulated the correspondence principle[9]. This states that the predictions of quantum physics appear to be the same as the predictions of classical physics when quantum numbers, n, l, and ml, are very large.

## 3. References ↑

1. Bohr, N., 1913, 'On the constitution of atoms and molecules', The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 26, pp.1-25.

2. Sommerfeld, A., 1915, 'On the Theory of the Balmer series', Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften Matematisch-physikalische Klasse, 1, pp.425-458.

3. Sommerfeld, A., 1915, 'The fine structure of the hydrogen and the hydrogen-like lines', Verlagd. KB Akad. d. Wiss.

4. Sommerfeld, A., 1920, 'Ein Zahlenmysterium in der Theorie des Zeemaneffektes' ('A number mystery in the theory of the Zeeman effect'), Naturwissenschaften, 8, pp.61-64.

5. Sommerfeld, A., 1920, 'Allgemeine spektroskopische Gesetze, insbesondere ein magnetooptischer Zerlegungssatz' ('General spectroscopic laws, particularly a magneto-optical decomposition theorem'), Annalen der Physik, 368, pp.221-263.

6. Zeeman, P., 1897 (1896), 'On the Influence of Magnetism on the Nature of the Light Emitted by a Substance', The Astrophysical Journal, 5, pp.332-347.

7. Landé, A., 1921, 'Über den anomalen Zeemaneffekt (Teil I)' ('About the anomalous Zeeman effect - Part I'), Zeitschrift für Physik, 5, pp.231-241.

8. Forman, P., 1970, 'Alfred Landé and the anomalous Zeeman effect, 1919-1921', Historical Studies in the Physical Sciences, 2, pp.153-261.

9. Bohr, N., 1920, 'Über die serienspektra der elemente' ('On the series spectra of the elements'), Zeitschrift für Physik A Hadrons and Nuclei, 2, pp.423-469.

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