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Magnetism: Paramagnetism and Diamagnetism

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Quantum Angular MomentumFermi Liquid Theory→→Ferromagnetism and Heisenberg ModelKondo Effect
paramagnetism diamagnetism magnetic-susceptibility langevin pauli

Core Idea

All materials exhibit some magnetic response to an applied field. Diamagnetism (χ < 0) is the universal tendency of orbital electron motion to oppose an applied field, present in all materials but typically weak (~10-5). Paramagnetism (χ > 0) arises from alignment of permanent magnetic moments: in insulators with localized moments, the Curie law χ = C/T describes the competition between field alignment and thermal disorder. In metals, Pauli paramagnetism gives a temperature-independent χ = μ_B2 g(E_F), reflecting that only Fermi-surface electrons contribute to the spin response. The net susceptibility of a material is the sum of all contributions.

Explainer

Magnetism in condensed matter begins with two universal but weak effects. Diamagnetism is present in every material: an applied magnetic field induces tiny orbital currents (Lenz's law at the atomic scale) that produce a moment opposing the field. The resulting susceptibility chi_dia = -e2 N <r2> / (6mc2) is negative, small (~10-5), and temperature-independent. It is the dominant magnetic response only in materials with no unpaired electrons — noble gases, many ionic crystals, and organic molecules.

Paramagnetism occurs when atoms or ions carry permanent magnetic moments (from unpaired electrons). In insulators with localized moments, each moment independently tries to align with the field while thermal agitation randomizes it. The Langevin/Brillouin theory gives the Curie law: chi = C/T, where the Curie constant C depends on the magnitude of the atomic moment. This 1/T dependence is the signature of thermal demagnetization: at high temperature, moments are randomly oriented and the susceptibility is small; at low temperature, alignment is easier and chi grows. The saturation magnetization is reached only when mu B >> k_BT.

In metals, paramagnetism takes a different form. Conduction electrons have spin-1/2 magnetic moments, but the Fermi-Dirac distribution restricts which electrons can respond to a field. Only those within ~k_BT of the Fermi level have access to empty states with opposite spin. This produces Pauli paramagnetism: chi_Pauli = mu_B2 g(E_F), which is temperature-independent and much smaller than Curie paramagnetism would predict for the same number of spins. The ratio chi_Pauli / chi_Curie is of order k_BT/E_F ~ 1/100 at room temperature. Additionally, the orbital motion of conduction electrons contributes Landau diamagnetism, which is exactly -1/3 of the Pauli susceptibility for free electrons (and modified by band structure effects in real metals).

The magnetic properties of a material are the sum of all contributions: core-electron diamagnetism, Pauli spin paramagnetism, Landau orbital diamagnetism, and (if present) Curie paramagnetism from localized moments. Van Vleck paramagnetism — a temperature-independent correction from virtual transitions to excited states — can also contribute. In most simple metals, the net susceptibility is weakly paramagnetic (Pauli > Landau + core diamagnetism). These weak magnetic responses are the "background" on which cooperative phenomena — ferromagnetism, antiferromagnetism, and spin glass behavior — are built.

Practice Questions 4 questions

Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → Counting to 20 → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Number Bonds to 10 → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Dividing Integers → Unit Rates → Proportions → Percent Concept → Converting Between Fractions, Decimals, and Percents → Operations with Rational Numbers → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Sine, Cosine, and Tangent Ratios → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Indefinite Integrals → Basic Integration Rules → Riemann Sums → Definite Integral Definition → Double Integrals: Definition and Setup → Iterated Integrals and Fubini's Theorem → Double Integrals over Rectangular Regions → Double Integrals over General Regions → Applications of Double Integrals: Area, Mass, and Moments → Triple Integrals in Cartesian Coordinates → Triple Integrals in Cylindrical and Spherical Coordinates → Change of Variables and the Jacobian Determinant → Applications of Triple Integrals: Volume and Mass → Vector Fields and Their Representations → Line Integrals of Vector Fields → Work and Circulation → Line Integrals of Scalar and Vector Functions → Fundamental Theorem for Line Integrals → Conservative Vector Fields → Conservative Vector Fields and Potential Functions → Curl and Divergence of Vector Fields → Curl and Divergence → Divergence Theorem → Electric Flux and Divergence Theorem → Gauss's Law: Integral Form and Meaning → Solving Problems with Gauss's Law → Conductors in Electrostatic Equilibrium → Capacitance and Capacitors → Dielectrics → Dielectric Constant and Relative Permittivity → Electric Field Inside Dielectric Materials → Dielectric Materials and Polarization → Dielectric Susceptibility and Permittivity → Energy Density in Electric Fields → Electric Current and Current Density → Electrical Resistance and Resistivity → Ohm's Law and Circuit Elements → Electromotive Force (EMF) and Batteries → Kirchhoff's Circuit Laws: Voltage and Current → DC Circuit Network Analysis Methods → Transient Response in RC Circuits → RC Circuits → LC and RLC Circuits → AC Circuits: Fundamentals → Impedance and Reactance → AC Power and Resonance → Electromagnetic Waves → Postulates of Special Relativity → Time Dilation → Length Contraction → Lorentz Transformation → Relativistic Velocity Addition → Relativistic Momentum and Energy → Mass-Energy Equivalence and E=mc² → Photons as Particles with Energy and Momentum → Planck-Einstein Relation: Energy and Frequency → Photoelectric Effect → The Photon: Light as Quanta → Compton Scattering → Wave-Particle Duality → de Broglie Wavelength → The Schrödinger Equation → State Vectors and Wavefunctions → Quantum Superposition → Quantum Entanglement → Bell Theorem and Bell Inequalities → Postulates of Quantum Mechanics → Observables and Quantum Operators → Commutators and Commutation Relations → Quantum Angular Momentum → Quantum Mechanical Treatment of Hydrogen → Solving the Schrödinger Equation for Hydrogen Atom → Quantum Numbers → Electron Configuration → Periodic Trends → Covalent Bonding → Electronegativity and Bond Polarity → Ionic Bonding → Lewis Structures → VSEPR Theory and Molecular Geometry → Molecular Geometry and Electron Pair Geometry → Molecular Polarity and Dipole Moments → Intermolecular Forces → States of Matter and Phase Changes: Melting, Boiling, and Sublimation → Gas Laws and the Ideal Gas Equation → Gas Stoichiometry and Volume-Volume Calculations → Thermochemistry and Enthalpy → Heat Capacity and Calorimetry → Entropy and Molecular Disorder → Spontaneity and ΔG → Entropy and Gibbs Free Energy → Chemical Equilibrium → Statistical Mechanics: Ensembles and the Boltzmann Distribution → Partition Function: Definition and Properties → Einstein Model of Solids → The Debye Model of Lattice Vibrations → Debye Model of Solids → Debye Temperature → Phonon Statistics and Dispersion Relations → Quantum Statistics: Fermions vs Bosons → Fermi-Dirac Distribution and Fermi Energy → The Ideal Fermi Gas: Ground State and Excitations → Fermi Liquid Theory → Magnetism: Paramagnetism and Diamagnetism

Longest path: 185 steps · 1112 total prerequisite topics

Prerequisites (2)

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