Spin in Atomic Physics

The particle spin and the concept of spin angular momentum came into existence when the spectral lines of elements were studied in detail. When the spectral lines were studied it was noticed that these spectral lines split into a group of lines lying close to one another. The splitting of spectral lines into a group of lines is known as the hyperfine structure. In 1929, Paul Dirac suggested that this hyperfine structure of atoms is related to the properties like spin and angular momentum, which are responsible for the motion of an electron inside the nucleus. 

The spin concept is one of the important topics we study in quantum physics and atomic physics. A nucleus is a collection of two particles, protons and neutrons. We say these particles are of spin half (i.e., spin-12), which means these particles will be possessing an intrinsic spin angular momentum of their own. In atomic physics, the inherent angular momentum of a particular particle is parametrized by a spin quantum number. 

Define Spin:

The spin quantum number is one of the four quantum numbers available and the spin quantum number is designated as the fourth number and is denoted by the letter S. The other three quantum numbers are a principal quantum number, azimuthal quantum number, and magnetic quantum number. The spin quantum number of any particle explains the unique quantum state of an electron.

Now to define spin, we must realise the fact that spin is just an intrinsic property of the particle. Just like the mass and charge of an electron, spin is also an inherent characteristic of the particles and these are responsible for the motion of a particle. When any particle is created or originated, it will be created with a definite amount of internal angular momentum known as the spin angular momentum. The electron spin can be explained as the spinning of the electron around its axis. 

The basic idea of particle spin was we aim derived from the motion of the earth. We know that the earth revolves around the sun in a particular orbit with certain orbital angular momentum, also the earth rotates about an axis through its centre with a definite spin angular momentum. But the only difference between the angular momenta of the earth and the electron is that the earth can have continuous momenta, whereas the electrons and other subatomic particles are quantized, for example, an electron can have spin angular momenta only either + \[\frac{1}{2}\] or - \[\frac{1}{2}\].

Mathematically, the spin of any particle can be calculated using the following equation:

⇒ S = \[\sqrt{s(s + 1)ħ}\]

Where,

S - The Magnitude of the spin angular momentum

s - The spin quantum number

ħ - Correction to Planck’s constant

Therefore, the spin quantum number of a particle is directly related to the quantized spin vector and the correction to Planck’s constant. The spin angular momentum is measured in units of ħ (the reduced Planck’s constant.)

Spin Physics:

According to the Bohr atomic model, he explained that the atomic levels resemble our solar system, just like the sun being at the centre of the solar system and the rest of the planets revolving around it in precise orbits, similarly in atomic level, the nucleus is at the centre and the electrons are revolving around the nucleus in discrete orbits. As the advancement took place in the study of atoms and spin physics, we understood that the electron is not just revolving around the sun, but also is spinning around an axis through its centre, which is associated with definite spin angular momentum. 

In 1925, two physicists Alan Berg and Goldsmith suggested that in addition to the orbital angular momentum of an electron inside the atom, there should be spin angular momentum of the electron. That suggests the fact that the electrons are not only revolving around the orbits but also spinning around an axis through its centre.

In 1928, one of the famous physicists, Paul Dirac on the bass of relativistic quantum mechanics showed that the spin angular momentum or the magnitude of the spin angular momentum associated with one electron can be derived from the given formula.

⇒ S = \[\sqrt{s(s + 1)ħ}\]

Where,

S - The Magnitude of the spin angular momentum

s - The spin quantum number

ħ - Correction to Planck’s constant or reduced Planck’s constant = 1.055 x 10\[^{-34}\] J - s 

After attempting several tedious mathematical calculations, Paul Dirac was able to successfully explain and calculate the value of the spin quantum number of an electron is \[\frac{1}{2}\].

After knowing the value of the spin quantum number of the electron, we can calculate the spin angular momentum of an electron using the above formula, and we found that the spin angular momentum of the electron is given by \[\frac{\sqrt{3}}{2}\]ħ.

The Electron can Spin in Two Directions

The electron present in an atom can in both directions either up or down. The directions of the spin are given by:

1. Spin up

2. Spin down

The spin up (positive direction) and spin down (negative direction) direction are corresponding to the spinning in the +z or –z-direction. The spin up and spin down are of the particles that have spin s equal to 1/2, i.e. for electrons.

In quantum theory, the electron is thought of like the minute bar of the magnet, and its spin points to the north pole of the minute bar. If two proximate electrons have a similar spin direction (i.e., with both electrons are having spin up or spin down), the magnetic field formed by them strengthens each other and therefore a strong magnetic field is obtained. If the proximate electrons have an opposite spin direction (i.e., if one electron is having up spin and the other down or vice versa), the magnetic field formed by them cancels each other and no magnetic field is existent.

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Example:

1. Calculate the spin angular momentum of the proton.

Sol:

Just like neutrons and electrons, the proton is also a spin half particle. The only difference is that the protons belong to baryons whereas the electrons are fermions in nature. Now we aim to calculate the spin angular momentum of the proton.

We know that the magnitude of the spin angular momentum is given by:

⇒ S = \[\sqrt{s(s + 1)ħ}\] ……….(1)

Where,

S - The Magnitude of the spin angular momentum

s - The spin quantum number

ħ - Correction to Planck’s constant or reduced Planck’s constant = 1.055 x 10\[^{-34}\] J - s 

Substituting the value of the spin quantum number in equation (1) we write:

⇒ S = \[\sqrt{-\frac{1}{2}(-\frac{1}{2} + 1)ħ}\] = \[\sqrt{\frac{1}{4}ħ}\]

⇒ S = \[\frac{1}{2}\]ħ

Therefore, the spin angular momentum of the proton is \[\frac{ħ}{2}\].

2. Calculate the Spin Angular Momentum of the Electron.

Sol:

We know that an electron is a spin half particle. The electrons are fermions in nature and hence they always possess half-integral spin (fermions are the particle with half-integral spin). Now we aim to calculate the spin angular momentum of the proton.

We know that the magnitude of the spin angular momentum is given by:

⇒ S = \[\sqrt{s(s + 1)ħ}\] ……….(1)

Where,

S - The Magnitude of the spin angular momentum

s - The spin quantum number

ħ - Correction to Planck’s constant or reduced Planck’s constant = 1.055 x 10\[^{-34}\] J - s 

Substituting the value of the spin quantum number in equation (1) we write:

⇒ S = \[\sqrt{\frac{1}{2}(\frac{1}{2} + 1)ħ}\] = \[\sqrt{\frac{3}{4}ħ}\]

⇒ S = \[\frac{\sqrt{3}}{2}\]ħ

Therefore, the spin angular momentum of the proton is \[\frac{\sqrt{3ħ}}{2}\].