Electron Spin: Theory, Direction, Spin magnetic Moment of Electron, Applications

In 1925, two great scientists, Samuel Goutsmit and G.E. Uhlenbeck proposed that electrons have an inherent angular momentum which is present in the form of magnetic moment of electron and is termed as spin. (But the first experiment that detected the electron spin was the Stern and Gerlach experiment). In simpler terms, an electron spinning around its axis is known as electron spin.

This article will provide the concepts of electron spin alongwith the electron spin theory and direction and further will discuss about the applications of electron spin in quantum mechanics.

Electron Spin

The electron spin is a fundamental property, i.e., an unvarying property of electrons like its charge and mass. The three quantum numbers are primary, azimuthal and magnetic, and the fourth one that defines the unique quantum state of an electron is the electron spin quantum number. The spin quantum number is denoted by the letter ‘s’.

In quantum physics, where we deal with microparticles that are not visible by the naked eyes, spin plays a vital role in commuting the properties of fundamental particles.

Mathematically it is given as,

Where,

: Spin vector

s: spin angular momentum

h: Planck’s constant

The spin quantum number can be depicted as,

Where, n can be any non-negative integer. Therefore, spin quantum number can take up any value as follow-

and so on.

The intrinsic angular momentum of the electron is .

Electron Spin Theory

In classical theory, an electron is described as a sphere. In contrast, electron spin theory depicts an electron as a quantum particle. This theory focuses on the direction of the electron spin and provides information about it. Electron spin theory also says that the direction of electron spin impacts several atom properties, like the magnetic moment.

The electron spin magnetic moment is essential in the atom’s interaction with the externally applied magnetic field. Electron spin magnetic moment is significant for spin-orbit interaction that splits the atomic energy levels and leads to a fine structure in the spectra of atoms.

Let us talk about the direction of electron spin.

Electron Spin Directions

The electron can spin in two directions, clockwise and counter-clockwise (or anticlockwise). These are also described as spin-up and spin-down, respectively. Here, spin-up corresponds to spinning in +z direction while spin-down corresponds to -z direction.

These two types have the magnitude of and .

The two spins are depicted in the figure given below for better understanding.

As per the quantum theory, an electron is considered a magnetic bar whose spin points can be taken as the two poles of the bar magnet, north and south. Hence, if it can be taken as a magnet, then there will be a magnetic field around it, as shown in the following picture:

We can see that the magnetic field lines are emerging from the north pole and terminating at the south pole, just like in the general bar magnet. Talking about the strength of the magnetic field, if two proximate electrons have similar spin directions, then the magnetic field formed by them will strengthen one another. But if their directions are opposite, their respective magnetic fields will cancel each other. As a result, there will be zero magnetic fields.

Spin Quantum Number

A spin quantum number represents an electron's angular momentum. As an electron rotates around its axis, it has angular momentum and orbital angular momentum. Since angular momentum has both magnitude and direction, it is a vector quantity. An electron orbital can support two electron spins: one "spin up" and one "spin down". Thus, during molecule bonding, electrons prefer filling orbitals before pairing up. The symbol for the spin quantum number is ms.

Spin Magnetic Moment of Electron

The spin magnetic moment of electron links to the electron's spin angular momentum. By custom, we can relate spin magnetic moment and spin angular momentum. The Dirac equation helps derive a connection between spin and magnetic moment. Using the Dirac equation gives:

The relation reveals the spin magnetic moment is double what classical methods predict. So, we must determine the magnetic moment by analyzing the effective magnetic field on a moving electron. This gives:

Here, the g is known as the g-factor and its value is 2.002319

We see a discrepancy between experimental and theoretical values. Quantum theory of electrodynamics explains this. It states that charged particles interact with electromagnetic fields. 

Hence, the spin magnetic moment of the electron

The value of spin magnetic moment, S, is
Using this value, we get

From the above equation, it can be seen that spin magnetic moment of electron is √3 times the Bohr magneton. 

Application of Electron Spin

One of the essential applications of electron spin is in ESR, where ESR stands for electron spin resonance. ESR is a branch of absorption spectroscopy where the radiations of the microwave region are absorbed by the paramagnetic substance to get the transitions between magnetic energy levels of the electron having unpaired spins.

ESR is used in various fields these days. Some of them are given below-

• It is used to determine the rate of catalysis.
• These days ESR is also used in a lot of biological systems.
• A study of free radicals is also done with this technique.
• To learn about inorganic compounds, ESR is proved to be helpful.
• Study of naturally occurring substances like minerals with transition elements, minerals with defects (e.g., quartz), Hemoglobin, Petroleum, Coal, Rubber, etc., is done using ESR.
• Conducting Electrons is being studied with the technique of ESR.

We hope this article has provided the readers with an insight into the topic of Electron spin. For a better understanding of concepts and a detailed explanation of Physics topics, download the Testbook app today.