The Bohr magneton \(( \mu_B )\) is a fundamental physical constant that represents the quantum of magnetic moment associated with the intrinsic angular momentum of an electron due to its spin or orbital motion. It serves as the basic unit for measuring the magnetic moment of electrons, atoms, and molecules, particularly in quantum mechanics and magnetism.
The Bohr magneton is defined as the magnetic moment produced by an electron when it moves in a circular orbit with a quantum of angular momentum equal to \( \hbar \) (the reduced Planck constant).
Formula for Bohr Magneton
The Bohr magneton \( \mu_B \) is given by the expression:
\[
\mu_B = \frac{e \hbar}{2 m_e}
\]
where:
- \( e \) = charge of the electron = \( 1.602 \times 10^{-19} \, \text{C} \) (Coulombs),
- \( \hbar \) = reduced Planck constant = \( 1.055 \times 10^{-34} \, \text{J·s} \),
- \( m_e \) = mass of the electron = \( 9.109 \times 10^{-31} \, \text{kg} \).
Calculation of Bohr Magneton
Let’s calculate the value of \( \mu_B \) using the above values.
\[
\mu_B = \frac{(1.602 \times 10^{-19} \, \text{C}) (1.055 \times 10^{-34} \, \text{J·s})}{2 (9.109 \times 10^{-31} \, \text{kg})}
\]
First, calculate the numerator:
\[
1.602 \times 10^{-19} \times 1.055 \times 10^{-34} = 1.69071 \times 10^{-53}
\]
Now calculate the denominator:
\[
2 \times 9.109 \times 10^{-31} = 1.8218 \times 10^{-30}
\]
Finally, divide the numerator by the denominator:
\[
\mu_B = \frac{1.69071 \times 10^{-53}}{1.8218 \times 10^{-30}} \approx 9.274 \times 10^{-24} \, \text{J/T}
\]
Thus, the value of the Bohr magneton is:
\[
\mu_B \approx 9.274 \times 10^{-24} \, \text{J/T}
\]
Here, \( \text{J/T} \) stands for Joules per Tesla, which is the unit of magnetic moment.
Physical Significance of Bohr Magneton
- Magnetic Moment of Electrons: The Bohr magneton represents the magnetic moment of an electron due to its spin or orbital motion in a magnetic field. It serves as the basic unit for expressing the magnetic moments of particles like electrons and protons.
- Quantum Mechanics: In quantum mechanics, the Bohr magneton is important because it provides a scale for the magnetic interactions of atomic systems. The magnetic moments of atoms and molecules are often measured in multiples of \( \mu_B \).
- Magnetic Properties of Materials: The Bohr magneton plays a critical role in understanding the magnetic properties of materials. In paramagnetism and ferromagnetism, the magnetic moments of individual atoms or ions are often expressed in terms of Bohr magnetons.
- Electron Spin and Orbital Contributions: The Bohr magneton helps quantify the contributions of electron spin and orbital angular momentum to the total magnetic moment of an atom, aiding in the study of electron configurations and magnetic behavior.
The Bohr magneton is a fundamental constant representing the magnetic moment of an electron due to its orbital and spin angular momentum. Its value is approximately \( 9.274 \times 10^{-24} \, \text{J/T} \). The Bohr magneton plays a key role in understanding the magnetic properties of particles, atoms, and materials in both classical and quantum physics.