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Bohr MagnetonEdit

The Bohr magneton is the natural scale for the magnetic moment of an electron, serving as a fundamental yardstick in quantum mechanics, atomic physics, and solid-state physics. Named for Niels Bohr, it encapsulates how a charged, spinning, or orbiting electron couples to magnetic fields. The constant is defined by μ_B = e ħ / (2 m_e), tying together the elementary charge, the reduced Planck constant, and the electron mass. In practice, magnetic moments of atoms, ions, and materials are often expressed in units of μ_B, making it easier to compare different systems at the microscopic level. The Bohr magneton underpins a broad range of phenomena, from Zeeman splittings in spectroscopy to the magnetic properties of metals and insulators.

Definition

  • The Bohr magneton is the magnitude of the magnetic moment associated with a single electron moving in a quantum state. It is given by μ_B = e ħ / (2 m_e), where:
  • Numerically, μ_B ≈ 9.274 009 994 × 10^-24 joule per tesla (J/T). The electron’s actual magnetic moment is negative in sign relative to its spin angular momentum, so μ ≈ −g μ_B S/ħ for spin, with g ≈ 2 for the electron’s spin in the Dirac limit.
  • In practice, magnetic moments of atoms or ions are often written in units of μ_B per atom, commonly denoted as μ = n μ_B for some integer or fractional n depending on the electronic configuration and coupling scheme. See also the concept of the Landé g-factor for how total angular momentum couples to magnetic fields.

Origin and historical context

  • The Bohr magneton emerges from early quantum theory that linked angular momentum to magnetic moments. In the Bohr model, quantization of orbital motion implied a specific magnetic moment, and when the theory was extended to include spin and spin–orbit coupling, μ_B became the natural unit for those moments.
  • The broader quantum-mechanical framework, including the Stern-Gerlach experiment and the Dirac equation, showed that electrons possess intrinsic spin and a gyromagnetic response characterized by g-factors near 2. The combination of charge, mass, and angular momentum that enters μ_B makes it a fundamental scale for magnetism at the atomic scale.
  • The Bohr magneton sits at the intersection of atomic physics, spectroscopy, and magnetism, appearing in formulas that describe how magnetic fields split energy levels (the Zeeman effect) and how spins align in materials.

Physical significance

  • Magnetic energy in a field: When a magnetic moment μ interacts with a magnetic field B, the energy shifts by ΔE = −μ · B. For an electron in a given quantum state, this leads to characteristic splittings of spectral lines and level shifts that are described by μ_B, the g-factor, and the magnetic quantum numbers.
  • Atomic and molecular spectra: Zeeman splitting, hyperfine structure, and related phenomena depend on magnetic moments expressed in μ_B. The Bohr magneton thus provides a universal scale to compare how different atoms and ions respond to magnetic fields.
  • Spin and orbital contributions: The total magnetic moment of an electron in an atom is the sum of orbital and spin contributions. Orbital moments scale with the orbital angular momentum L, while spin moments scale with the intrinsic spin S. The total moment is often written as μ = −g_J μ_B J/ħ, where J is the total angular momentum and g_J is the Landé factor. See angular momentum and g-factor for more on these couplings.
  • Nuclear magnetism in contrast: The nuclear magneton μ_N = e ħ / (2 m_p) is the corresponding unit for nuclear magnetic moments, with m_p the proton mass. Since m_p ≈ 1836 m_e, μ_N is about 1/1836 of μ_B, which explains why nuclear magnetic effects are typically much weaker than electronic ones in many contexts. See Nuclear magneton for details.

Applications and examples

  • Spectroscopy and magnetism in atoms: Researchers use μ_B as a standard to describe how energy levels shift in a magnetic field (the Zeeman effect) and to interpret fine and hyperfine structure in spectra. See Zeeman effect.
  • Electron spin resonance and related techniques: Electron paramagnetic resonance (EPR) or ESR measurements probe the magnetic environment of unpaired electrons, often expressed in terms of μ_B and related g-factors. See Electron paramagnetic resonance and g-factor.
  • Solid-state magnetism: In ferromagnets, antiferromagnets, and paramagnets, the magnetic moments of electrons contribute to macroscopic magnetization. It is common to express magnetic moments per atom in units of μ_B, which provides a direct link between microscopic states and bulk properties. See magnetism and ferromagnetism.
  • Nuclear magnetic resonance context: While NMR primarily involves nuclear magnetic moments, the related nuclear magneton μ_N provides the natural unit for those moments. The contrast between μ_B and μ_N underlines the hierarchy of magnetic scales in matter. See Nuclear magnetic moment.
  • Metrology and fundamental constants: The 2019 redefinition of SI units anchored several constants, including the Planck constant and the elementary charge, influencing how μ_B is realized in high-precision measurements. The combination e, ħ, and m_e remains central to accurately describing electronic magnetism across laboratories. See Planck constant and elementary charge.

See also