Magnetism: Paramagnetism and Diamagnetism

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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_B^2 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 = -e^2 N <r^2> / (6mc^2) 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_B^2 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

Counting to 10Counting to 20Understanding ZeroThe Number ZeroCounting to FiveOne-to-One CorrespondenceCombining Small Groups Within 5Addition Within 10Addition Within 20Two-Digit Addition Without RegroupingTwo-Digit Addition with RegroupingAddition Within 100Repeated Addition as MultiplicationMultiplication Facts Within 100Division as Equal SharingDivision as Grouping (Measurement Division)Division: Grouping (Repeated Subtraction) ModelDivision: Fair Sharing ModelDivision as Equal SharingDivision as GroupingBasic Division FactsDivision Facts Within 100Two-Digit by One-Digit DivisionDivision with RemaindersRemainders and Quotients in DivisionDivision Word ProblemsIntroduction to Long DivisionFactors and MultiplesPrime and Composite NumbersEquivalent FractionsRelating Fractions and DecimalsDecimal Place ValueReading and Writing DecimalsComparing and Ordering DecimalsAdding and Subtracting DecimalsMultiplying DecimalsDividing DecimalsDividing FractionsMixed Number ArithmeticOrder of OperationsInteger Order of OperationsVariable ExpressionsCombining Like TermsOne-Step EquationsTwo-Step EquationsSolving Multi-Step EquationsEquations with Variables on Both SidesAngle Pairs: Complementary, Supplementary, and VerticalParallel Lines and TransversalsCorresponding AnglesAlternate Interior AnglesTriangle Angle Sum TheoremExterior Angle TheoremTriangle Inequality TheoremSimilar Triangles: AA SimilaritySimilar Triangles: SSS and SAS SimilarityProportions in Similar TrianglesRight Triangle Trigonometry IntroductionTrigonometric Ratios ReviewRadian MeasureConverting Between Degrees and RadiansThe Unit CircleGraphing Sine and CosineGraphing Tangent and Reciprocal Trigonometric FunctionsDerivatives of Trigonometric FunctionsAntiderivativesIterated Integrals and Fubini's TheoremDouble Integrals in Cartesian CoordinatesDouble Integrals over Rectangular RegionsDouble Integrals in Polar CoordinatesDouble Integrals: Definition and SetupIterated Integrals and Fubini's TheoremDouble Integrals over Rectangular RegionsDouble Integrals over General RegionsApplications of Double Integrals: Area, Mass, and MomentsTriple Integrals in Cartesian CoordinatesTriple Integrals in Cylindrical and Spherical CoordinatesChange of Variables and the Jacobian DeterminantApplications of Triple Integrals: Volume and MassVector Fields and Their RepresentationsLine Integrals of Vector FieldsGreen's TheoremSurface Integrals and Flux of Vector FieldsSurface Integrals and Flux of Vector FieldsDivergence Theorem: Flux and OutflowDivergence TheoremElectric FluxGauss's LawConductors in Electrostatic EquilibriumCapacitance and CapacitorsDielectricsDielectric Constant and Relative PermittivityElectric Field Inside Dielectric MaterialsDielectric Materials and PolarizationDielectric Susceptibility and PermittivityEnergy Density in Electric FieldsElectric Current and Current DensityElectrical Resistance and ResistivityOhm's Law and Circuit ElementsElectromotive Force (EMF) and BatteriesKirchhoff's Circuit Laws: Voltage and CurrentDC Circuit Network Analysis MethodsTransient Response in RC CircuitsRC CircuitsLC and RLC CircuitsAC Circuits: FundamentalsImpedance and ReactanceAC Power and ResonanceElectromagnetic WavesThe Electromagnetic SpectrumBlackbody Radiation and Planck's LawPhotoelectric EffectThe Photon: Light as QuantaCompton ScatteringWave-Particle Dualityde Broglie WavelengthHeisenberg Uncertainty PrincipleWavefunction and the Born RuleThe Schrödinger EquationState Vectors and WavefunctionsQuantum SuperpositionQuantum EntanglementBell Theorem and Bell InequalitiesPostulates of Quantum MechanicsScattering TheoryIntroduction to Scattering TheoryPartial Wave Analysis in ScatteringSpin Angular MomentumElectron Spin and Intrinsic Magnetic MomentStern-Gerlach Experiment: Spin Quantization and MeasurementElectron Diffraction and Matter Wave PropertiesDavisson-Germer Experiment: Crystal Diffraction of ElectronsElectron Diffraction and Matter Wave InterferenceWavefunctions and Probability Density InterpretationQuantum Superposition and Linear Combinations of StatesQuantum Operators and ObservablesCanonical Commutation Relations and UncertaintyThe Quantum Harmonic OscillatorThe Debye Model of Lattice VibrationsDebye Model of SolidsDebye TemperaturePhonon Statistics and Dispersion RelationsQuantum Statistics: Fermions vs BosonsFermi-Dirac Distribution and Fermi EnergyThe Ideal Fermi Gas: Ground State and ExcitationsFermi Liquid TheoryMagnetism: Paramagnetism and Diamagnetism

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