The Bohr magneton (symbol μ_B) is a physical constant that represents the natural unit of magnetic moment for an electron due to its orbital or spin angular momentum. It is defined as
$$ \mu_B = \frac{e\hbar}{2m_e}, $$
where e is the elementary charge, ħ (h-bar) is the reduced Planck constant, and mₑ is the electron rest mass. In the International System of Units (SI), its value is
$$ \mu_B = 9.274,010,0783(28) \times 10^{-24}\ \text{J·T}^{-1}, $$
equivalently about $5.788 \times 10^{-5}$ eV T⁻¹.
Historical background
The constant is named after the Danish physicist Niels Bohr, who introduced the concept in the context of his early atomic model (1913). Bohr derived the expression for the magnetic moment associated with an electron moving in a circular orbit, which later was generalized to include electron spin.
Physical significance
- Atomic spectroscopy: The Bohr magneton serves as the scale for the magnetic moments of electrons in atoms and ions, influencing fine and hyperfine splitting of spectral lines via the Zeeman effect.
- Solid‑state physics: In magnetic materials, magnetic moments of electrons are often expressed in multiples of μ_B, aiding comparison across different systems.
- Fundamental constants: The Bohr magneton is related to other constants such as the fine‑structure constant (α) through the relation $\mu_B = \alpha \frac{e\hbar}{2m_ec}$, where c is the speed of light.
Usage in equations
The magnetic moment μ of an electron due to its spin is commonly written as
$$ \boldsymbol{\mu}_s = -g_s ,\mu_B ,\frac{\boldsymbol{S}}{\hbar}, $$
where g_s ≈ 2.002 319 304 is the electron spin g‑factor and S is the spin angular momentum vector. For orbital motion, the magnetic moment is
$$ \boldsymbol{\mu}_L = -\mu_B ,\frac{\boldsymbol{L}}{\hbar}, $$
with L the orbital angular momentum vector.
Related constants
- Nuclear magneton (μ_N): Analogous constant for nuclear magnetic moments, defined with the proton mass instead of the electron mass.
- Magneton (general term): The term “magneton” can refer generically to any elementary magnetic moment unit, of which the Bohr magneton is the most widely used for electrons.
References
- CODATA recommended values of the fundamental physical constants.
- N. Bohr, “On the Constitution of Atoms and Molecules,” Philosophical Magazine 26 (1913): 1–25.
- P. A. M. Dirac, The Principles of Quantum Mechanics (4th ed., 1958).