Obtain The Expression For Bohr Magneton

The Bohr magneton is a fundamental physical constant that represents the magnetic moment of an electron due to its orbital motion around the nucleus. It plays a crucial role in quantum mechanics and solid-state physics, particularly in explaining the magnetic properties of atoms and materials.

In this topic, we will derive the expression for the Bohr magneton, explain its significance, and explore its applications in physics.

Table of Contents

What is the Bohr Magneton?

The Bohr magneton ( mu_B ) is defined as the quantum of the electron’s magnetic moment due to its orbital motion in an atom. It provides a standard measure of magnetic dipole moments associated with electrons and is essential in explaining atomic and molecular magnetism.

Mathematically, the Bohr magneton is given by the expression:

mu_B = frac{e hbar}{2m_e}

where:

e = elementary charge of an electron ( $1.602 times 10^{-19} C$ )
hbar = reduced Planck’s constant ( frac{h}{2pi} = 1.055 times 10^{-34} Js )
m_e = mass of an electron ( $9.109 times 10^{-31} kg$ )

Now, let’s derive this expression step by step.

Derivation of the Bohr Magneton Expression

To obtain the Bohr magneton expression, we consider the classical model of an electron revolving around the nucleus in a circular orbit, following Bohr’s atomic model.

1. Magnetic Moment of a Moving Electron

A charged ptopic in motion, such as an electron in a circular orbit, generates a magnetic moment. The magnetic moment ( mu ) is given by:

mu = I A

where:

I = current due to the electron’s motion
A = area enclosed by the orbit

2. Expression for Current

The current I is defined as charge per unit time:

I = frac{text{Charge}}{text{Time Period}}

Since the charge of an electron is e , and the time period of revolution T is:

T = frac{2pi r}{v}

where r is the radius of the orbit and v is the velocity of the electron, we get:

I = frac{e}{T} = frac{e}{frac{2pi r}{v}} = frac{e v}{2pi r}

3. Area of the Orbit

The area of a circular orbit is:

A = pi r^2

Thus, the magnetic moment ( mu ) becomes:

mu = I A = left( frac{e v}{2pi r} right) times (pi r^2)

mu = frac{e v r}{2}

4. Using Bohr’s Quantization Condition

According to Bohr’s model, the angular momentum of an electron is quantized:

m_e v r = n hbar

For the lowest energy level ( n = 1 ):

m_e v r = hbar

Solving for v r :

v r = frac{hbar}{m_e}

5. Expression for the Bohr Magneton

Substituting v r = frac{hbar}{m_e} into the magnetic moment equation:

mu = frac{e}{2} times frac{hbar}{m_e}

mu = frac{e hbar}{2 m_e}

This is the expression for the Bohr magneton ( mu_B ):

mu_B = frac{e hbar}{2 m_e}

Significance of the Bohr Magneton

The Bohr magneton represents the smallest unit of magnetic moment an electron can possess due to its orbital motion. It serves as a reference for:

Electron Magnetic Moments – Electrons in atoms exhibit intrinsic magnetic moments due to both orbital and spin contributions.
Paramagnetism and Diamagnetism – The Bohr magneton helps explain magnetic behaviors of materials based on electron alignment.
Quantum Mechanics and Spectroscopy – It plays a role in Zeeman splitting, where energy levels split in the presence of a magnetic field.
Spin Magnetic Moment – The electron also has a spin magnetic moment, approximately equal to one Bohr magneton.

Numerical Value of the Bohr Magneton

By substituting the known physical constants:

mu_B = frac{(1.602 times 10^{-19} C) (1.055 times 10^{-34} Js)}{2 (9.109 times 10^{-31} kg)}

mu_B = 9.274 times 10^{-24} J/T

This value is widely used in atomic physics to describe the magnetic properties of electrons.

Comparison with Other Magnetic Moments

While the Bohr magneton applies primarily to electrons, other ptopics also have magnetic moments:

Ptopic	Magnetic Moment Expression	Approximate Value
Electron	mu_B = frac{ehbar}{2m_e}	$9.274 times 10^{-24} J/T$
Proton	mu_p = frac{ehbar}{2m_p}	$1.41 times 10^{-26} J/T$
Neutron	Small intrinsic moment	-9.66 times 10^{-27} J/T

These values show that electrons have a much larger magnetic moment than protons and neutrons, making them dominant in atomic magnetism.

Applications of the Bohr Magneton

The concept of the Bohr magneton is essential in various fields of physics, including:

1. Atomic and Molecular Magnetism

The Bohr magneton is used to determine orbital and spin contributions to atomic magnetic moments.
It helps explain electron configurations and magnetic properties of elements.

2. Zeeman Effect

In an external magnetic field, atomic energy levels split due to the interaction of the magnetic moment with the field.
The Bohr magneton defines the strength of this splitting.

3. Electron Spin Resonance (ESR) and Nuclear Magnetic Resonance (NMR)

ESR measures electron magnetic moments, while NMR focuses on nuclear magnetic properties.
The Bohr magneton plays a crucial role in both techniques.

4. Solid-State Physics and Magnetism

Magnetic materials are classified based on their electron magnetic moments.
The Bohr magneton helps describe ferromagnetic and paramagnetic behaviors.

The Bohr magneton is a fundamental concept in quantum physics, providing a measure of the magnetic moment of an electron due to its motion in an atom. By using Bohr’s atomic model and classical electromagnetism, we derived its expression:

mu_B = frac{e hbar}{2 m_e}

This value is critical in understanding atomic magnetism, quantum mechanics, and material science. Its applications extend from basic atomic theory to advanced technologies like ESR and NMR, making it a cornerstone of modern physics.