# Atomic Model of Sommerfeld

1) The orbits can be circular or elliptical.

2) When the path is elliptical, then there are two axes – major axis and minor axis. When the length of the majorand minor axes are equalized, then the orbit is circular.

3) The angular momentum of the electron that moves in an elliptical orbit is kh / 2π.

k is an integer except zero.

Value of k = 1,2,3,4 …….

n / k = major axis length / minor axis length

With the increase in the value of k, the ellipticity of the orbit decreases. When n = k, the orbit is circular.

4) Sommerfeld suggested that the orbits are composed of sub-energy levels. These are S, P, D, F. These sub-shells have slightly different energies.

Bohr gave a quantum number ‘n’, which determines the energy of the electron.

Sommerfeld introduced a new quantum number called orbital or azimuthal quantum number (l) that determines the orbital angular momentum of the electron.

Values of l = 0 to (n-1)

For, n = 1; l = 0; 1s his

n = 2; l = 0.1; 2s, 2p subconcha

n = 3; l = 0.1, 2; 3s, 3p, 3d sub shell 3d

n = 4; l = 0, 1, 2, 3; 4s, 4p, 4d, 4f sub-shell

5) When an electron jumps from one orbit to another, the energy difference (ΔE) depends on the sub-energy levels.

6) Explain the division of the individual spectral lines of hydrogen and therefore of the fine spectrum. I could not predict the exact number of lines that are actually present in the fine spectrum.

Table of Contents

## Defects of the Sommerfeld atomic model.

1) This model does not explain the behavior of the system that has more than one electron.

2) This model does not explain the Zeeman & Stark effect.

## Who was Arnold Sommerfeld and his atomic model?

Arnold Sommerfeld was a famous atomic physicist and mathematician who is known primarily for his work in atomic theory in the field of quantum mechanics and for being the mentor of more Nobel Prizes in Physics than any other physicist.

It was at the University of Munich that Arnold Sommerfeld began his research on atomic theory and quantum physics, considered by many to be his most significant contributions to the field of physics.

In 1911, Sommerfeld began investigating the atomic model of Niels Bohr, which was seen as the most accurate description of the atom at that time. The model postulated that electrons were located in circular orbits, which had different energy levels, around the nucleus .

Electrons can change energy levels depending on the energy of the particles that hit the atom.

Sommerfeld proposed that instead of circular orbits, electrons orbited around the nucleus in elliptical orbits, which improved Bohr’s model and became the standard at that time – until physicist Erwin Schrodinger gave the final touch to the modern atomic model when he introduced his quantum atomic model in 1926.

## Quantum number ℓ

German physicist Arnold Sommerfeld reviewed the model of the Danish physicist’s atom Niels Bohr in 1915. Sommerfeld showed that another number is needed to describe the orbits of electrons, in addition to the shell number (n). 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.

#### Electron orbits of the hydrogen atom (for the magnetic quantum number m = 0).

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

The shapes become more complex the higher the number of the crust. The maximum number of electrons that can have the same shape in each layer can be found using the formula:

MAXIMUM NUMBER OF ELECTRONS WITH THE SAME SHAPE = 2 (2ℓ + 1)

This means that 2 electrons can have the form ℓ = 0 (also known as the s-orbital form), 6 can have the form ℓ = 1 (p-orbital), 10 can have the form ℓ = 2 (d-orbital) , 14 can have the form ℓ = 3 (f-orbital), and so on. Figure 11.1 shows the possible forms that electron orbits can take in the first five layers of electrons.

### Quantum number m

In 1920, Sommerfeld realized that another number was needed to describe the orbit of electrons, because the current model still could not explain the Zeeman effect, the division of spectral lines in the presence of a magnetic field.

Sommerfeld realized that the same form of ℓ can have different orientations in space defined by the magnetic quantum number (m). The maximum number of different orientations can be found using the formula:

MAXIMUM NUMBER OF ORIENTATIONS PER FORM = 2ℓ + 1

This means that the form ℓ = 0 has a value m, which is designated as 0. The form ℓ = 1 can have up to three different m values, which are designated as -1, 0 and 1, the form ℓ = 2 can have up to five different m values, which are designated as -2, -1, 0, 1 and 2, and so on. Possible orientations for the first four orbital forms are shown in the following Figure.

### Different orientations of the first four orbital forms.

A magnetic field causes electrons of the same energy, which would otherwise produce a single spectral line, 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 the first layer produces three lines since approximately 1/3 of the gain energy, 1/3 of the loose energy, and 1/3 are not seen affected by the presence of the magnetic field. This explained the normal Zeeman effect.

Sommerfeld could not explain the anomalous Zeeman effect. This is a term used to describe spectra that are divided into more than three lines in the presence of a magnetic field, [6,7] and are more likely to occur on lines made of atoms with an odd number of electrons in their outer shell.

This meant that many still did not accept that electrons had orbits defined by positions and directions quantified in space. British physicist Paul Dirac would later explain the anomalous Zeeman effect using another quantum number, s.

### Sources and references.

Sommerfeld, A., *Annalen der Physik * **1920** , *368* , 221-263.Zeeman, P., *The Astrophysical Journal * **1897** (1896), *5* , 332–347.

Landé, A., *Zeitschrift für Physik * **1921** , *5* , 231–241.

Forman, P., *Historical Studies in the Physical Sciences * **1970** , *2* , 153-261.