Electron Spin

Introduction

We know that the structure of atoms is the same as the solar system. In the solar system, the sun is the center and other planets revolve around it in a definite path same as The atom nucleus at the center and electrons revolve around it. However, we found that planets also rotate at their fixed position. This discovery leads to the rotation and spinning of electrons at a fixed position. That’s why scientists started to research and found that electrons also rotate. In 1925 two scientists Goutsmit and Ulhenbeck derived the idea that an electron also has angular momentum known as spin.

What is Electron Spin?

According to an experiment in 1920, an electron has two types of movement; one is around the nucleus and another one is around a fixed axis. Around the nucleus, electrons revolve but about the axis, they rotate and spin. Hence, the fourth quantum number is the electron spin quantum number which represents the rotation of the electron around a fixed axis. This electron spin is a quantum property of the electron and the notation of this spin is ms. This property is a fundamental quantity and has a fixed magnitude. The spin is classified into two types; spin up which is denoted by $\mathrm{+\frac{1}{2}}$ and spin down which is denoted by $\mathrm{-\frac{1}{2}}$. The formula of electron spin is $\mathrm{S=\sqrt{s(s+1)h}}$. Here, s is the quantized spin vector and h is the Plank constant.

Fig:1 Spin of electron

Ashurov sindor, Elektronlar-spinlarining-yo'nalishlari, CC BY-SA 4.0

As we all know that charge in motion is a source of the magnetic field. Thus when an electron rotates or spins it produces a magnetic field around itself. When it rotates in a clockwise direction it is called spin up and has south polarity on the upside and north on the downside while in the anti-clockwise direction it is spin down and polarity is reciprocal of spin up.

Electron Spin Theory

Electron spin theory shows that an electron is not the same as a planet and spherical but it is a quantum particle. Also, this theory represents the information regarding the direction of magnetic field and spin and also explains the polarity of electron spin. With the help of this theory, we can calculate and assume the magnetic property of an atom.

Spin Quantum Number

A spin quantum number can be stated as the angular momentum of an electron. When an electron moves to spin on its axis then it associates two types of momentum; angular momentum and orbital angular momentum. As we know very well that angular momentum has both direction and magnitude so it is a vector quantity. An electron orbital has two positions two hold two spins; one is spin up and one is pin down. Thus during a bonding between two molecules, the electron prefers the filling of the orbital before starting the pairing. The symbol of the spin quantum number is ms.

Spin Magnetic Moment of Electron

The spin magnetic moment is the magnetic moment arising because of the spin of an electron. By convention, we may say that the spin magnetic moment is somewhat related to the spin angular moment. So, we can derive a relation between spin and magnetic moment with the help of the Dirac equation. By using the Dirac equation we get that

$$\mathrm{\mu _{s}=2\gamma S}$$

From the above relation, we find that the magnetic moment because of spin is the twotime value expected by the classical method. So, we need magnetic moment which can be determined by analyzing the effective magnetic field on an electron that is in motion. After this we get,

$$\mathrm{\mu _{s}=g\gamma S}$$

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

We can notice that there is a difference in experimental or observed value and expected value. The difference can be defined by the quantum theory of electrodynamics, which state that charged particle can work with an electromagnetic field.

Hence, the spin magnetic moment of the electron

$$\mathrm{\mu _{s}=-g\frac{e}{2m}S}$$

Here, we know that S is the spin magnetic moment and its value is $\mathrm{\sqrt{s(s+1)h}}$. By using the value of S, we get

$$\mathrm{\left|\mu _{s} \right|=-g\mu _{B}\sqrt{s(s+1)}}$$

$$\mathrm{\left|\mu _{s} \right|\simeq \sqrt{3\mu _{B}}}$$

Thus above equation state that the spin magnetic moment of the electron is $\mathrm{\sqrt{3}}$ times the Bohr Magneton.

Consequences of Electron Spin Theory

Electron spin theory stated that an electron is not a complete sphere in shape but it is a quantum particle. Some of the important results of electron spin theory.

• We can know the magnetic properties of atoms

• It also helps us to find the direction of spin and magnetic field

• We can also analyze the bonding mechanism and magnetic effect on the bonding of two orbitals.

Solved Question

Q1. what is the spin magnetic moment of $\mathrm{Fe^{2+}}$?

Sol. We know that spin magnetic moment of $\mathrm{Fe^{2+}}$. The configuration is $\mathrm{1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}3d^{6}}$. There are 4 unpaired electrons. Thus, the magnetic moment

$$\mathrm{\mu _{B}=\sqrt{n(n+2)}}$$

We know that n=4

$$\mathrm{\mu _{B}=\sqrt{4(4+2)}}$$

$$\mathrm{\mu _{B}=\sqrt{4\times 6}}$$

$$\mathrm{\mu _{B}\approx 5\:BM}$$

Now, spin magnetic moment $\mathrm{\left|\mu _{s} \right|\simeq \sqrt{3}\times 5}$

FAQs

Q1. What do you mean by Quantization of Angular Momentum?

Ans. The quantization of angular momentum state that if we take the value of angular momentum in the form of orbital quantum momentum then its value will be in the form of $\mathrm{L=\sqrt{l(l+1)h}}$

Q2. What do you mean by Quantum Number?

Ans. The quantum number is a set or group of four numbers to determine the various information like location, shape, energy, orientation, etc of an electron.

Q3. What do you mean by the Azimuthal quantum number?

Ans. The azimuthal quantum number is the second name of the angular momentum quantum number and it shows the value of orbital angular momentum. It is denoted by l

Q4. What do you mean by Bohr’s Magneton?

Ans. when an electron orbits with the orbital angular momentum of h then the value of its magnetic dipole moment is known as Bohr’s Magneton.

Q5. What is the value of Bohr’s Magneton?

Ans. The value of Bohr’s Magneton is $\mathrm{9.27\times 10^{-24}Joule/Tesla}$.