Langevin diamagnetism

Langevin diamagnetism
Name	Langevin Diamagnetism
Phenomena	Magnetism
Related	Diamagnetism, Paramagnetism, Ferromagnetism
Theorist	Paul Langevin
Field	Statistical mechanics, Electromagnetism

Langevin diamagnetism. It is a fundamental theory in classical physics describing the weak, temperature-independent diamagnetism present in all materials, arising from the Larmor precession of atomic electrons in an applied magnetic field. Formulated by the French physicist Paul Langevin in 1905, it provided an early classical explanation for the magnetic susceptibility of materials lacking permanent magnetic moments. While superseded by a full quantum mechanical treatment, the theory remains a historically important conceptual model within the broader study of condensed matter physics.

Theoretical foundation

The theory is built upon the principles of classical electrodynamics and statistical mechanics. Langevin's key insight was applying Lenz's law at the atomic scale, where an external magnetic field induces circulating electron currents within atoms. This induction is a consequence of Faraday's law of induction, leading to a change in the angular momentum of the orbital motion. The resulting microscopic current loop creates a magnetic moment that opposes the applied field, as dictated by the law of electromagnetic induction. This opposition is the hallmark of diamagnetic behavior, fundamentally distinct from the alignment mechanisms in paramagnetism.

Derivation of the Langevin formula

Starting with an electron in a circular orbit of radius *r*, the application of a magnetic field **B** alters its angular velocity due to the Lorentz force. This change induces an additional current, and the resulting magnetic moment is calculated. For an atom with *Z* electrons, the total induced diamagnetic moment is summed over all orbits. Using the equipartition theorem and assuming a spherically symmetric charge distribution, the mean square radius ⟨*r²*⟩ is introduced. The final expression for the magnetic susceptibility *χ* is derived as *χ* = –(*N* *Z* *e²* ⟨*r²*⟩) / (6 *m* *c²*), where *N* is the number of atoms, *e* is the electron charge, *m* is the electron mass, and *c* is the speed of light.

Physical interpretation and significance

The negative sign in the susceptibility formula confirms the induced moment opposes the field, a direct manifestation of Lenz's law. A crucial result is the temperature independence of the susceptibility, contrasting sharply with the Curie law governing paramagnetism. This universality implies that all atoms, regardless of their electronic structure, exhibit this weak diamagnetic response. The theory was significant for unifying the understanding of diamagnetism within the framework of classical physics, providing a clear mechanism linked to fundamental constants like the electron charge and electron mass. It established a baseline magnetic behavior against which other forms like paramagnetism and ferromagnetism could be compared.

Comparison with other forms of magnetism

Unlike paramagnetism, which arises from the alignment of permanent magnetic moments (as described by the Langevin function for paramagnetism) and follows the Curie law, this diamagnetism is always present and much weaker. In materials with paramagnetism or ferromagnetism, the stronger positive susceptibility often masks the diamagnetic contribution. The theory of ferromagnetism, later explained by Pierre-Ernest Weiss with the concept of the molecular field, involves strong, cooperative interactions absent in diamagnetic systems. The Van Vleck paramagnetism is another temperature-independent mechanism but yields a positive susceptibility, arising from second-order quantum mechanical perturbations.

Experimental observation and materials

Pure diamagnetic materials, where this effect is the dominant magnetic response, include noble gases like helium and argon, many organic compounds such as benzene, and elements like carbon (in graphite), bismuth, and copper. The effect is famously demonstrated in the Meissner effect in superconductors, though that is a perfect diamagnetism of different origin. Early experimental work by scientists like Pieter Zeeman and Hendrik Lorentz on spectral lines provided indirect evidence. Precise measurements of susceptibility in materials like water or sodium chloride confirm the predicted weak, negative, and temperature-independent value.

Limitations and quantum mechanical correction

The primary limitation is its classical nature, which fails to account for the quantized angular momentum of electrons and the stability of atomic orbits as described by Niels Bohr and Erwin Schrödinger. It incorrectly predicts the magnitude of susceptibility for many atoms because it uses a classical average for ⟨*r²*⟩. A full quantum treatment, provided by John Hasbrouck Van Vleck, uses perturbation theory and the Hamiltonian operator for an electron in a magnetic field. This corrects the susceptibility formula and properly accounts for the contributions from all electron orbitals, reconciling theory with experimental data for atoms and ions.

Category:Magnetism Category:Condensed matter physics Category:Physical phenomena