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Spin Quantum Number

31st Jan 2020 @ 4 min read

Physical Chemistry

Spin quantum number

The spin quantum number is the quantum number that describes the orientation of the intrinsic angular momentum of an elementary particle. It is denoted by s. Spin is an intrinsic property of an elementary particle, which is responsible for the spin angular momentum. Spin can be imagined as a particle rotating around its axis, just like Earth around its axis. The value of s defers from particle-to-particle, for example electrons have spin 12 (or 2 precisely) while photons have spin 1 (or ℏ precisely).

The spin angular momentum of an electron can have two orientations: 12 and −12. These are only two possible spins of the electron and represented by the arrows: ↑ (spin up) and ↓ (spin down). An orbital can have maximum two electrons; when an orbital is completely filled with two electrons, both must have opposite spins, i.e. if one is 12, the other is −12.

The Spins of the electron
The two spins of the electron

The spin angular momentum is a vector-like quantity and is quantized by the spin quantum number s.

The spin angular momentum

Here, ℏ is the reduced Planck constant.

Besides the spin angular momentum S, there is the orbital angular momentum L that contributes to the total angular momentum J. The orbital angular momentum is quantized by the azimuthal quantum number. S is results of the rotation of an electron around its axis while L is the results of the revolution of an electron around its nucleus. The total angular momentum is the vector addition of both: J = L + S. J is conserved in a closed system.

The classical interpretation of the spin and orbital angular momentum
The classical interpretation of the spin and orbital angular momentum

The spin quantum number is one of the four quantum numbers that identify an electron in an atom. The other three are the principal quantum number, azimuthal quantum number, and magnetic quantum number.

Stern-Gerlach's experiment

The idea of spin was originated from the Stern-Gerlach experiment. In the early 1920s, Otto Stern and Walther Gerlach, both German physicists, conducted an experiment. The experiment demonstrated the quantized nature of the intrinsic angular momentum. In the experiment, silver atoms were vaporized in an oven. These particles were passed through a spatially varying (or inhomogeneous) magnetic field. A uniform disposition of Ag atoms was expectation. However, the results were contrary. The beam split into two as shown in the diagram below.

The Stern-Gerlach experiment
Stern-Gerlach's experiment

This showed that the existence of an intrinsic magnetic moment of the electron. The splitting of the beam into two implied the presence of the two spin quantum numbers associated with the electrons.

A silver atom has 47 electrons. In a completely filled orbital of an atom, the electrons are always paired, i.e. both have opposite spins. In silver, 23 electrons get paired with other 23 while one electron remains unpaired. This unpaired electron is responsible for the spin magnetic moment of the Ag atom in the experiment. In simple words, the unpaired electron acts as a small magnet. As the Ag atoms were passed through the magnetic field, the atoms were deflected either up or down based the two spins of the unpaired electron.

When s = +12, atoms were repelled and were attracted to the field when s = −12.

Electrons in an external magnetic field
Electrons in an external magnetic field


Scientists carried of the similar experiments with elements having one unpaired electron in the valence shell in the late 1920s and the similar outcomes were observed. Some of them are hydrogen (Z = 1), sodium (Z = 11), potassium (Z = 19) copper (Z = 29), gold (Z = 79).

In 1925, Ralph Kronig was the first to introduce the idea of electronic spins based on the earlier findings of other physicists. But he received severe criticisms from Heisenberg and Pauli. Consequently, he abandoned it and never published it. A few months after him, George Uhlenbeck and Samuel Goudsmit, both American-Dutch physicists, published the discovery.

George Uhlenbeck (left) and Samuel Goudsmit (right) along with Hendrik Kramers (middle) in Ann Arbor c. 1928


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