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Lecture 2 Crystal Structure 1 MSEG 608 September 2, 2021 Prof. LaShanda Korley ([email protected]) Prof. Xi Wang ([email protected]) Material categories wikipedia • Crystalline materials • have long range order (atoms, molecules, ions, etc) – Salt, gemstones, snowflakes Polycrystalline • Amorphous materials have materials are pockets of order disordered – Ice, metals, ceramics (usually) – Plastics, wax, frequently polymers Crystals • Google “nature crystals” Polycrystalline Polycrystalline monocrystalline Silicon wikipedia Diamond http://www.superabrasivespowder.com/products/polypolycrystalline-diamond-powder.html Amorphous solids • Amorphous solid: no order on any “significant” length scale • Blurry line between amorphous solids and nanocrystalline materials http://www.sunflower-solar.com/index.php?act=content&scheduler_id=429 Amorphous • Glasses, liquids, colloids, gels, etc • Polymers tend to easily form glassy states • Glasses are an amorphous state – SiO2 glass • Usually contents other components – fused silica/fused quartz Amorphous semiconductors • Silicon, germanium – Primarily for solar cells • Chalcogenides – AsS, GeS, GeSbTe, AgInSbTe – Infrared detectors, modulatle IR optics, IR fibers, rewritable disks • More scattering • Less efficient • Still carry current • Cheaper, easier to process “The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of glass and the glass transition. The solution of the (...) important and puzzling glass problem may also have a substantial intellectual spin-off. Whether or not it will help make better glass is questionable.” P.W. Anderson, Science (1995) Crystal Structures • “Easy” modelling • Historically Uvarovite – External appearance • Today – Internal structure • Flat facets • Fixed angles between facets • Experience: Crystal formation (crystallization) is from liquid to solid. • Modelling: Something internal? Early study • Snowflakes: always showed perfect six-cornered symmetry and never showed five or seven corners • Material structure modeling since 17th century • Particle packings http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html Bravais Lattices • concluded that the crystals were made by the ordered stacking of small bricks, or unit cells, all of them identical. René Just Haüy (1743-1822) • In 1848 the French physicist Auguste Bravais discovered that in the threedimensional space, periodic repetitions by translation can only be made in 14 different modes (the so-called 14 Bravais lattices), as they must to be compatible with the 32 crystal classes. Auguste Bravais (1811-1863) http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html Crystal structures • 50 years later, the 14 Bravais lattices and the 32 crystal classes were the limitations used to independently deduce the 230 space groups, which are the 230 possible ways to restrict distributions of repetitive structural units of the crystals (atoms, ions and molecules). Mercury • Cif files • https://www.ccdc.cam.ac.uk/solutions/csdsystem/components/mercury/ • http://www.crystallography.net/cod/index.ph p Key Concepts • • • • Lattice Basis Primitive cell Unit cell Lattice • A periodical set of mathematical points 𝐫′ = 𝐫 + 𝑙𝐚1 + 𝑚𝐚2 + 𝑛𝐚3 l, m, n are arbitrary integers; 𝐚1 , 𝐚2 , 𝐚3 are primitive vectors • The environment of any given point is equivalent to the environment of any other given point. • Primitive lattice vectors – Any two points can be translated – Smallest 𝐚1 ⋅ (𝐚2 × 𝐚3 ) 3D 𝐚1 × 𝐚2 2D Basis • Infinite repetition of identical groups of atoms • Crystal structure = Basis + Lattice – Number of atoms in a basis: >=1 – Position of an atom j r j x j a1 y j a 2 z j a3 0<= xj, yj, zj <=1 Primitive cell • The volume/area defined by primitive vectors – Many ways – One lattice point per primitive cell • Number of atoms in a primitive cell is always the same for a given crystal structure – Minimum number of atoms among basis • Wigner-Seitz cell Unit cell • Conventional cells – Often a nonprimitive cell – A more obvious relation with the point symmetry operations – 2D Unit cell • Conventional cells – Often a nonprimitive cell – A more obvious relation with the point symmetry operations – 3D sc bcc Fcc Volume a3 a3 a3 Lattice points per cell 1 2 4 Volume, primitive cell a3 a3/2 a3/4 Number of nearest neighbors 6 8 12 The fourteen (3D) Bravais Lattices
