X-Ray Powder Diffraction

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XRD Bragg's law diffraction phase identification Rietveld refinement

Core Idea

X-ray powder diffraction (XRPD) is the primary technique for identifying crystalline phases and determining crystal structures from polycrystalline samples. When monochromatic X-rays strike a powdered crystalline sample, they diffract from lattice planes according to Bragg's law: n-lambda = 2d sin(theta). Because a powder contains crystallites in all orientations, every set of lattice planes simultaneously satisfies the Bragg condition at its characteristic angle, producing a unique pattern of peak positions and intensities. Peak positions reveal the unit cell dimensions; peak intensities encode the atomic arrangement; peak shapes carry information about crystallite size and strain.

Explainer

X-ray diffraction is the most important technique in materials chemistry for answering the question: what crystalline phases are present, and what are their structures? The physical basis is straightforward — X-rays have wavelengths comparable to interatomic distances (about 1.5 Angstroms for Cu K-alpha radiation), so they diffract from the regularly spaced planes of atoms in a crystal. Bragg's law gives the condition for constructive interference: the path difference between X-rays reflecting from adjacent planes must equal a whole number of wavelengths.

In a powder diffraction experiment, the sample is a finely ground polycrystalline material. The random orientation of crystallites ensures that for every set of lattice planes, some fraction of crystallites will satisfy the Bragg condition. The detector sweeps through angles, recording intensity as a function of 2-theta. The resulting pattern — a series of peaks at specific angles with specific intensities — is a fingerprint of the crystal structure. Phase identification works by matching the observed pattern against a database (the ICDD Powder Diffraction File contains over 400,000 reference patterns). If your pattern matches entry number 04-0787, your sample contains aluminum.

Beyond identification, XRPD provides quantitative structural information. The peak positions are determined by the unit cell dimensions through Bragg's law and the Miller index relation for d-spacings. By fitting peak positions, you extract the lattice parameters a, b, c, alpha, beta, gamma with high precision. The peak intensities depend on which atoms are at which positions within the unit cell — heavy atoms scatter X-rays more strongly, and the relative intensity of different reflections encodes the atomic arrangement. Rietveld refinement fits a complete structural model (atom types, positions, thermal parameters) to the entire diffraction pattern simultaneously, refining all parameters to minimize the difference between observed and calculated patterns. This method has become the standard approach for structure determination and refinement from powder data.

Peak shapes carry additional information. Broadening beyond the instrumental resolution arises from two main sources: small crystallite size (Scherrer broadening) and microstrain (non-uniform lattice distortions). These can be separated by their different angular dependences. For nanomaterials, where crystallite sizes are below 100 nm, peak broadening analysis is often the quickest way to estimate particle size. For engineering materials, strain broadening reveals residual stresses from processing. The combination of phase identification, structure refinement, and microstructural analysis makes XRPD an indispensable tool across all of materials chemistry.

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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 UncertaintyHeisenberg Uncertainty Principle and Measurement LimitsTime-Independent Schrödinger Equation and EigenvaluesHydrogen Atom in Quantum MechanicsSpectral Lines and Energy TransitionsSelection Rules for Atomic TransitionsLS and jj Coupling Schemes in Multi-Electron AtomsPauli Exclusion Principle and Antisymmetric WavefunctionsElectron Configuration and the Aufbau PrincipleThe Periodic Table and Atomic Electronic StructureThe Periodic TableElectron ConfigurationPeriodic TrendsIonization EnergyIonic BondingLewis StructuresPolar Covalent Bonds and Dipole MomentsClassification of Bonds: Ionic, Covalent, and MetallicMain Group Chemistry OverviewSolid State Chemistry FundamentalsCrystal Structures and Unit CellsCrystal Symmetry and Space GroupsX-Ray Powder Diffraction

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