Questions: Magnetism: Paramagnetism and Diamagnetism

Question 1

Pauli paramagnetism in metals is temperature-independent, while Curie paramagnetism in insulators follows χ ∝ 1/T. What causes this fundamental difference?

• A. Metals have stronger magnetic moments than insulators
• B. In metals, the Pauli exclusion principle restricts spin flips to the ~k_BT energy shell near E_F; the number of available spins grows as T but each contributes less by 1/T, and these effects cancel. In insulators, all localized moments are free to reorient, so thermal disorder (∝ T) directly competes with field alignment (∝ 1/T)
• C. The crystal structure of metals suppresses the temperature dependence
• D. Insulators have more unpaired electrons per atom

Answer: (B)

In Curie paramagnetism, N independent moments each contribute μ²B/3k_BT to the susceptibility, giving χ = Nμ²/(3k_BT) — pure competition between magnetic energy and thermal energy. In Pauli paramagnetism, the Fermi sea blocks most spin flips. Only electrons within ~k_BT of E_F can respond, but the density of these electrons (proportional to g(E_F)) is temperature-independent at leading order. The result is χ_Pauli = μ_B² g(E_F), independent of T. The ratio χ_Pauli/χ_Curie ~ k_BT/E_F ~ 1/100 at room temperature — Pauli paramagnetism is much weaker.

Question 2

Lenz's law applied at the atomic level explains diamagnetism: an applied field induces orbital currents that oppose the field. Why is diamagnetic susceptibility typically so small?

• A. Diamagnetism is small because the induced currents are on the atomic scale — the induced moment per atom is proportional to <r²>, the mean square orbital radius, which is tiny (~Ų), and the proportionality constant involves e²/mc², which is very small
• B. Diamagnetism is only present in superconductors
• C. The diamagnetic response cancels with the paramagnetic response in most materials
• D. Only core electrons contribute to diamagnetism

Answer: (A)

The Langevin diamagnetic susceptibility per atom is χ_dia = -e²N<r²>/(6mc²), where <r²> is the mean square distance of electrons from the nucleus. With <r²> ~ 1 Ų and N ~ 10-30 electrons, the susceptibility is of order -10^{-5} to -10^{-6} in CGS units. This is universally present but easily overwhelmed by paramagnetism when unpaired spins exist. Materials with no unpaired electrons (noble gases, many ionic crystals, bismuth) show measurable diamagnetism.

Question 3

Superconductors are 'perfect diamagnets' with χ = -1. This is qualitatively different from ordinary diamagnetism.

• T. True
• F. False

Answer: True

Ordinary (Langevin/Larmor) diamagnetism gives tiny susceptibilities (χ ~ -10^{-5}) from atomic-scale induced currents. Superconducting diamagnetism (the Meissner effect) gives χ = -1 (perfect screening) from macroscopic persistent currents that flow on the surface and completely expel the magnetic field from the interior. The physical mechanisms are completely different: Larmor diamagnetism is a perturbative response of individual atoms, while the Meissner effect is a collective quantum phenomenon involving the macroscopic coherence of the superconducting condensate.

Question 4

Why do rare earth ions often have much larger paramagnetic moments than transition metal ions, despite both having unpaired f or d electrons?

Model answer: In rare earth ions, the 4f electrons are deep inside the atom, well-shielded from the crystal electric field by the outer 5s and 5p shells. Spin-orbit coupling is strong and acts on the full J = L + S multiplet, and the crystal field is too weak to quench the orbital angular momentum. The moment is μ = g_J√(J(J+1)) μ_B with the full J value. In transition metal ions, the 3d electrons are the outermost shell and experience strong crystal fields that typically quench the orbital angular momentum (L is frozen). The moment is approximately μ ≈ 2√(S(S+1)) μ_B with g ≈ 2 and only the spin contribution. Since J (with both L and S) can be much larger than S alone, rare earth moments are often larger.

This is the origin of Hund's rules applied to solids: rare earths follow the free-ion J values closely, while transition metals are often 'spin-only' due to crystal field quenching of L.