Space lattices


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Space lattices

  1. 1. Crystallography
  2. 2. <ul><li>In 1669 Nicolaus Steno found angles between adjacent prism faces of quartz crystal (interfacial angle), to be 120 °. </li></ul><ul><li>In 1780 Carangeot invented the goniometer, a protactor like device used to measure interfacial angles on crystals. </li></ul><ul><li>Law of “constancy of interfacial angels”: angles between equivalent faces of crystals of the same mineral are always the same.The law acknowledges that the size and shape of the crystal may vary. </li></ul>
  3. 3. <ul><li>In 1784 Rene’Hauy hypothesized the existence of basic building blocks of crystals called integral molecules and argued that large crystals formed when many integral molecules bonded together. </li></ul>
  4. 4. <ul><li>Old view </li></ul><ul><li>Crystals are made of small building blocks </li></ul><ul><li>The blocks stack together in a regular way, creating the whole crystal. </li></ul><ul><li>Each block contains a small number of atoms </li></ul><ul><li>All building blocks have the same atomic composition </li></ul><ul><li>The building block has shape and symmetry of the entire crystal. </li></ul><ul><li>We now accept that: </li></ul><ul><li>Crystals have basic building blocks called unit cells </li></ul><ul><li>The unit cells are arranged in a pattern described by points in a lattice. </li></ul><ul><li>The relative proportions of elements in a unit cell are given by the chemical formula of a mineral. </li></ul><ul><li>Crystals belong to one of the seven crystal systems. Unit cells of distinct shape and symmetry characterize each crystal system. </li></ul><ul><li>Total crystal symmetry depends on Unit cell symmetry and lattice symmetry. </li></ul>
  5. 5. Crystal Geometry <ul><li>Crystals </li></ul><ul><li>Lattice </li></ul><ul><li>Lattice points, lattice translations </li></ul><ul><li>Cell--Primitive & non primitive </li></ul><ul><li>Lattice parameters </li></ul><ul><li>Crystal=lattice+motif </li></ul>
  6. 6. Matter Crystalline Amorphous Solid Liquid Gas
  7. 7. Crystal?
  8. 8. A 3D translationally periodic arrangement of atoms in space is called a crystal.
  9. 9. A two-dimensional periodic pattern by a Dutch artist M.C. Escher
  10. 10. Lattice?
  11. 11. A 3D translationally periodic arrangement of points in space is called a lattice.
  12. 12. A 3D translationally periodic arrangement of atoms Crystal A 3D translationally periodic arrangement of points Lattice
  13. 13. What is the relation between the two? Crystal = Lattice + Motif Motif or basis: an atom or a group of atoms associated with each lattice point
  14. 14. Crystal=lattice+basis Lattice: the underlying periodicity of the crystal, Basis: atom or group of atoms associated with each lattice points Lattice: how to repeat Motif: what to repeat
  15. 15. + Love Pattern Love Lattice + Heart =
  16. 16. Space Lattice <ul><li>A discrete array of points in 3-d space such that every point has identical surroundings </li></ul>
  17. 17. Lattice Finite or infinite?
  18. 18. Primitive cell Primitive cell Nonprimitive cell
  19. 19. Cells <ul><li>A cell is a finite representation of the infinite lattice </li></ul><ul><li>A cell is a parallelogram (2D) or a parallelopiped (3D) with lattice points at their corners. </li></ul><ul><li>If the lattice points are only at the corners, the cell is primitive. </li></ul><ul><li>If there are lattice points in the cell other than the corners, the cell is nonprimitive. </li></ul>
  20. 20. Lattice Parameters Lengths of the three sides of the parallelopiped : a, b and c. The three angles between the sides:  ,  , 
  21. 21. <ul><li>Convention </li></ul><ul><li>a parallel to x -axis </li></ul><ul><li>b parallel to y -axis </li></ul><ul><li>c parallel to z -axis </li></ul><ul><li>Angle between y and z </li></ul><ul><li>Angle between z and x </li></ul><ul><li> Angle between x and y </li></ul>
  22. 22. The six lattice parameters a , b , c ,  ,  ,  The cell of the lattice lattice crystal + Motif
  23. 24. <ul><li>In order to define translations in 3-d space, we need 3 non-coplanar vectors </li></ul><ul><li>Conventionally, the fundamental translation vector is taken from one lattice point to the next in the chosen direction </li></ul>
  24. 25. <ul><li>With the help of these three vectors, it is possible to construct a parallelopiped called a CELL </li></ul>
  25. 26. <ul><li>The smallest cell with lattice points at its eight corners has effectively only one lattice point in the volume of the cell. </li></ul><ul><li>Such a cell is called PRIMITIVE CELL </li></ul>
  26. 27. Bravais Space Lattices <ul><li>Conventionally, the finite representation of space lattices is done using unit cells which show maximum possible symmetries with the smallest size. </li></ul><ul><li>Symmetries: 1.Translation </li></ul><ul><li>2. Rotation </li></ul><ul><li>3. Reflection </li></ul>
  27. 28. <ul><li>Considering </li></ul><ul><li>Maximum Symmetry, and </li></ul><ul><li>Minimum Size </li></ul><ul><li>Bravais concluded that there are only 14 possible Space Lattices (or Unit Cells to represent them). These belong to 7 Crystal Classes </li></ul>
  28. 29. Arrangement of lattice points in the unit cell <ul><li>8 Corners (P) </li></ul><ul><li>8 Corners and 1 body centre (I) </li></ul><ul><li>8 Corners and 6 face centres (F) </li></ul><ul><li>8 corners and 2 centres of opposite faces (A/B/C) </li></ul><ul><li>Effective number of l.p. </li></ul>
  29. 30. <ul><li>Cubic Crystals </li></ul><ul><li>Simple Cubic (P) </li></ul><ul><li>Body Centred Cubic (I) – BCC </li></ul><ul><li>Face Centred Cubic (F) - FCC </li></ul>
  30. 31. <ul><li>Tetragonal Crystals </li></ul><ul><li>Simple Tetragonal </li></ul><ul><li>Body Centred Tetragonal </li></ul>
  31. 32. <ul><li>Orthorhombic Crystals </li></ul><ul><li>Simple Orthorhombic </li></ul><ul><li>Body Centred Orthorhombic </li></ul><ul><li>Face Centred Orthorhombic </li></ul><ul><li>End Centred Orthorhombic </li></ul>
  32. 33. <ul><li>Hexagonal Crystals </li></ul><ul><li>Simple Hexagonal or most commonly HEXAGONAL </li></ul><ul><li>Rhombohedral Crystals </li></ul><ul><li>Rhombohedral (simple) </li></ul>
  33. 34. <ul><li>Monoclinic Crystals </li></ul><ul><li>Simple Monoclinic </li></ul><ul><li>End Centred Monoclinic (A/B) </li></ul><ul><li>Triclinic Crystals </li></ul><ul><li>Triclinic (simple) </li></ul>
  34. 35. Crystal Structure <ul><li>Space Lattice + Basis (or Motif) </li></ul><ul><li>Basis consists of a group of atoms located at every lattice point in an identical fashion </li></ul><ul><li>To define it, we need to specify </li></ul><ul><li>Number of atoms and their kind </li></ul><ul><li>Internuclear spacings </li></ul><ul><li>Orientation in space </li></ul>
  35. 36. <ul><li>Atoms are assumed to be hard spheres </li></ul>