How to Learn Low-Energy Electron Diffraction (LEED): From Surface Reciprocal Lattices and Diffraction Spots to Reconstructions, I–V Structure Refinement and Micro-LEED

## Wait, What? A Surface Can Have a Different Lattice From the Crystal Underneath It Cut a crystal and surface atoms lose neighbours. They can relax, reconstruct, dimerize or bind ordered adsorbates. LEED is designed to reveal that surface periodicity. Use electrons with tens to hundreds of electronvolts of kinetic energy. Their wavelength is comparable to atomic spacings, and their useful elastic signal comes mainly from the topmost atomic layers. > **LEED is not simply XRD with electrons. Surface symmetry is easy to see, but exact atomic coordinates require a dynamical multiple-scattering inverse problem.** ## The One-Sentence Answer **Learn LEED by tracing low-energy electron → surface elastic scattering → reciprocal-lattice spots → surface unit cell and symmetry, then add electron refraction, multiple scattering, domains, diffuse background and I–V calculations before treating a spot pattern or R-factor minimum as one uniquely solved atomic surface structure.** # Beginner Layer — Electron Waves and Surface Diffraction ## Stage 1: Electrons Behave as Waves **λ = h/p** ## Stage 2: 20–500 eV Electrons Have Atomic-Scale Wavelengths They can diffract strongly from surface lattices. ## Stage 3: Periodic Surface Atoms Scatter Coherently Constructive interference creates diffraction spots. # Why LEED Is Surface Sensitive ## Stage 4: Low-Energy Electrons Have Short Inelastic Mean Free Paths Electrons scattered deep inside the crystal usually lose energy. ## Stage 5: Retarding Optics Select Mainly Elastic Electrons The useful pattern therefore emphasizes the outermost layers. # LEED Optics ## Stage 6: A Monochromatic Electron Gun Illuminates the Surface ## Stage 7: Electrons Backscatter Toward a Hemispherical Detector ## Stage 8: Retarding Grids Suppress Inelastic Background A fluorescent screen or camera displays the pattern. # Two-Dimensional Reciprocal Space ## Stage 9: The Surface Is Periodic in Two Dimensions The crystal truncation breaks long-range periodicity normal to the surface. ## Stage 10: Reciprocal-Lattice Points Become Rods ## Stage 11: The Ewald Sphere Intersects Those Rods This is why surface diffraction conditions remain accessible over a broad energy range. # Reading the Pattern ## Stage 12: Spot Arrangement Gives Surface Symmetry ## Stage 13: Spot Spacing Is Inversely Related to Real-Space Periodicity ## Stage 14: Spots Are Not Real-Space Images of Atoms Many atomic arrangements can share the same surface unit cell. # Reconstructions and Superstructures ## Stage 15: Surface Atoms Can Rearrange to Lower Free Energy ## Stage 16: Fractional-Order Spots Reveal Superperiodicity A (2×1)-type surface doubles periodicity along one direction. ## Stage 17: Adsorbates Can Form New Ordered Lattices Commensurate, incommensurate and rotated phases create different reciprocal patterns. ## Stage 18: Moiré Superstructures Produce Satellite Spots Common in graphene and other 2D heterostructures. # Domain Averaging ## Stage 19: Rotational or Terrace Domains Can Superimpose A large-area pattern can have higher apparent symmetry than any single domain. ## Stage 20: Micro-LEED Can Isolate One Terrace or Domain Recent single-terrace studies show why local diffraction can recover the true underlying surface symmetry. > **Apparent symmetry can be an averaging artifact.** # I–V Curves ## Stage 21: Change Incident Electron Energy Spot intensities vary dramatically. ## Stage 22: Record Intensity Versus Energy for Each Beam Each beam gives an **I(E)** curve. ## Stage 23: I–V Structure Contains Atomic-Position Information Electron paths through several surface layers interfere. # Kinematic Versus Dynamical Diffraction ## Stage 24: Reciprocal Geometry Explains Spot Positions Well ## Stage 25: Single-Scattering Theory Usually Fails for Intensities Low-energy electrons scatter strongly. ## Stage 26: Multiple-Scattering Theory Is Required This is the central leap from qualitative to quantitative LEED. # Structural Refinement ## Stage 27: Propose an Atomic Surface Model Specify coordinates, layer spacings, adsorbate sites and vibration parameters. ## Stage 28: Calculate I–V Curves Use dynamical electron-scattering theory. ## Stage 29: Compare Model and Experiment Adjust the structure iteratively. ## Stage 30: Pendry R-Factor Quantifies Agreement A lower R is better under the chosen metric. ## Stage 31: The Lowest R Is Not Absolute Truth Different models can remain statistically competitive. # Inner Potential and Thermal Motion ## Stage 32: Electron Waves Refract at the Vacuum–Crystal Boundary An effective inner potential enters the model. ## Stage 33: Thermal Motion Reduces Coherent Intensity Debye–Waller-type factors matter. # Diffuse and Spot-Profile LEED ## Stage 34: Disorder Produces Diffuse Intensity Short-range order can remain even without sharp long-range peaks. ## Stage 35: Spot Width Carries Correlation-Length Information Finite domains and steps broaden spots. ## Stage 36: SPA-LEED Measures Spot Profiles Precisely Terrace size, roughness and domain boundaries can be constrained. # UHV, Charging and Beam Damage ## Stage 37: LEED Requires a Clean, Ordered Surface Monolayer contamination can alter the pattern. ## Stage 38: Sputtering and Annealing Can Create the Surface Being Measured “Clean” is not always “original.” ## Stage 39: Poor Conductors Can Charge ## Stage 40: Molecular Adsorbates Can Be Beam Sensitive Dose dependence is a necessary control. # Neighboring Diffraction Methods ## Stage 41: LEED Versus XRD XRD is generally bulk sensitive; LEED is surface sensitive. ## Stage 42: LEED Versus RHEED RHEED uses high-energy grazing electrons and is common during thin-film growth. ## Stage 43: LEED Versus EBSD EBSD maps grain orientation and strain in an SEM rather than clean-surface reconstructions. ## Stage 44: LEED + STM Is Powerful LEED gives long-range reciprocal order; STM gives local real-space structure. # 2026 Frontier ## Stage 45: LEED I–V Remains Active for 2D and Epitaxial Structure Recent work combines quantitative LEED and DFT to solve complex surface structures such as intercalated bismuthene systems. ## Stage 46: Micro-LEED Corrects Ensemble Symmetry Errors Single-terrace measurements can reveal symmetry hidden by large-area averaging. # Machine-Learning Layer ## Stage 47: Computer Vision Can Detect Spots and Domains ## Stage 48: Automated I–V Fitting Can Search Structural Parameters ## Stage 49: ML Cannot Replace Multiple-Scattering Physics A learned structure should reproduce spot geometry, I–V curves, symmetry and independent STM/DFT evidence. # Professional Layer ## Stage 50: Separate Five Objects 1. true surface atomic structure; 2. elastic scattering amplitudes; 3. multiple-scattering paths; 4. measured spot/I–V pattern; 5. refined structural model. ## Stage 51: Professional LEED Is a Surface–Multiple-Scattering Inverse Problem > **Which reconstruction or adsorbate geometry remains identifiable after domain averaging, electron refraction, multiple scattering, thermal motion, diffuse disorder, beam damage and alternative structural models are all allowed to explain the same LEED data?** # Evidence: What Makes a LEED Claim Strong? Strong evidence combines several electron energies, indexed reciprocal lattices, equivalent beams, reproducible preparation, dose tests, quantitative I–V refinement, competing structural models, STM/DFT/XPS/AES evidence and micro-LEED where domains matter. # Misconceptions Worth Hunting – LEED is simply XRD with electrons. – Diffraction spots are a real-space image of atoms. – Spot brightness directly equals atom count. – A (2×1) pattern uniquely gives atomic positions. – Quantitative LEED can use simple single scattering. – A sixfold pattern always proves sixfold surface symmetry. – LEED samples the bulk because electrons penetrate the crystal. – UHV preparation always preserves the original surface. – The lowest R-factor makes all other models impossible. # Transfer Check A large-area surface looks sixfold but micro-LEED on one terrace looks threefold. Is that contradictory? **No. Domain averaging can superimpose orientations.** A half-order spot appears after adsorption. Does it prove one unique adsorption site? **No. It proves new periodicity.** A molecular pattern fades during repeated exposure. Did thermal disorder necessarily rise? **No. Beam damage can destroy order.** # Model Limits LEED works best on ordered, crystalline, vacuum-compatible, reasonably conductive surfaces. Amorphous, rough, strongly charging and buried structures are difficult. Professional LEED keeps **electron energy + dose + surface preparation + reciprocal indexing + domain structure + I–V curves + multiple-scattering model + inner potential + R-factor + orthogonal real-space evidence** visible together. # Teaching Guide Teach in this order: **electron wave → low energy → surface sensitivity → LEED optics → 2D reciprocal lattice → spots → reconstruction → domains → I–V → multiple scattering → structural refinement → R-factor → diffuse/spot-profile LEED → UHV/damage → micro-LEED → validation.** # Connect This to the eduKate Learning Estate – X-Ray Diffraction — bulk crystallography. – EBSD — SEM orientation and strain mapping. – Vacuum Science — surface preparation and growth. – STM/STS — local real-space surface structure. – AES — nanoscale surface elemental/chemical analysis. # The Quiet Ending The beginner asks, “Where are the diffraction spots?” The developing crystallographer asks, “What two-dimensional reciprocal lattice created them?” The advanced learner asks, “Which reconstruction and domain population reproduce the full energy-dependent pattern?” And the professional asks: > **Which atomic surface structure survives after multiple scattering, domain averaging, beam perturbation and every competing I–V model are made explicit?**