Crystal Geometry <ul><li>Crystals </li></ul><ul><li>Lattice </li></ul><ul><li>Lattice points, lattice translations </li></...
Matter Crystalline Amorphous Solid Liquid Gas
Crystal?
A 3D translationally periodic arrangement of atoms in space is called a crystal.
Air, Water and Earth A two-dimensional periodic pattern by a Dutch artist M.C. Escher
Lattice?
A 3D translationally periodic arrangement of points in space is called a lattice.
A 3D translationally periodic arrangement of  atoms Crystal A 3D translationally periodic arrangement of  points Lattice
What is the relation between the two? Crystal = Lattice + Motif Motif or basis: an atom or a group of atoms associated wit...
Crystal=lattice+basis Lattice:  the underlying periodicity of  the crystal, Basis:  atom or group of atoms  associated wit...
+ Love Pattern Love Lattice + Heart =
Formula for Love Potion? Mix one molecule of potassium iodide with two molecules of sulfur KI + 2S = KISS
Space Lattice <ul><li>A  discrete  array of points in  3-d space  such that every point has  identical surroundings </li><...
Lattice Finite or infinite?
Primitive cell Primitive cell Nonprimitive cell
Cells <ul><li>A cell is a finite representation of the  infinite lattice </li></ul><ul><li>A cell is a parallelogram (2D) ...
Lattice Parameters Lengths of the three sides of the parallelopiped :  a, b and c. The three angles between the sides:  ...
<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...
The six lattice parameters  a ,  b ,  c ,   ,   ,   The cell of the lattice lattice crystal + Motif
 
 
 
<ul><li>In order to define translations in 3-d space, we need 3 non-coplanar vectors </li></ul><ul><li>Conventionally, the...
<ul><li>With the help of these three vectors, it is possible to construct a parallelopiped called a   CELL </li></ul>
<ul><li>The smallest cell with lattice points at its eight corners has effectively only one lattice point in the volume of...
Bravais  Space Lattices <ul><li>Conventionally, the finite representation of space lattices is done using  unit cells  whi...
<ul><li>Considering </li></ul><ul><li>Maximum Symmetry, and </li></ul><ul><li>Minimum Size </li></ul><ul><li>Bravais concl...
Arrangement of lattice points in the unit cell <ul><li>8 Corners  (P) </li></ul><ul><li>8 Corners and 1 body centre (I) </...
<ul><li>5.   Hexagonal unit cell has 12 corners of the hexagonal prism 2 centres of hexagonal faces </li></ul>
<ul><li>Cubic Crystals </li></ul><ul><li>Simple Cubic (P) </li></ul><ul><li>Body Centred Cubic (I) – BCC </li></ul><ul><li...
<ul><li>Tetragonal Crystals </li></ul><ul><li>Simple Tetragonal </li></ul><ul><li>Body Centred Tetragonal </li></ul>
<ul><li>Orthorhombic Crystals </li></ul><ul><li>Simple Orthorhombic </li></ul><ul><li>Body Centred Orthorhombic </li></ul>...
<ul><li>Hexagonal Crystals </li></ul><ul><li>Simple Hexagonal or most commonly HEXAGONAL </li></ul><ul><li>Rhombohedral Cr...
<ul><li>Monoclinic Crystals </li></ul><ul><li>Simple Monoclinic </li></ul><ul><li>End Centred Monoclinic (A/B) </li></ul><...
Crystal Structure <ul><li>Space Lattice  +  Basis (or Motif) </li></ul><ul><li>Basis consists of a group of atoms located ...
<ul><li>Atoms are assumed to be  hard spheres </li></ul>
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Presentation1

  1. 2. 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>
  2. 3. Matter Crystalline Amorphous Solid Liquid Gas
  3. 4. Crystal?
  4. 5. A 3D translationally periodic arrangement of atoms in space is called a crystal.
  5. 6. Air, Water and Earth A two-dimensional periodic pattern by a Dutch artist M.C. Escher
  6. 7. Lattice?
  7. 8. A 3D translationally periodic arrangement of points in space is called a lattice.
  8. 9. A 3D translationally periodic arrangement of atoms Crystal A 3D translationally periodic arrangement of points Lattice
  9. 10. 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
  10. 11. 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
  11. 12. + Love Pattern Love Lattice + Heart =
  12. 13. Formula for Love Potion? Mix one molecule of potassium iodide with two molecules of sulfur KI + 2S = KISS
  13. 14. Space Lattice <ul><li>A discrete array of points in 3-d space such that every point has identical surroundings </li></ul>
  14. 15. Lattice Finite or infinite?
  15. 16. Primitive cell Primitive cell Nonprimitive cell
  16. 17. 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>
  17. 18. Lattice Parameters Lengths of the three sides of the parallelopiped : a, b and c. The three angles between the sides:  ,  , 
  18. 19. <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>
  19. 20. The six lattice parameters a , b , c ,  ,  ,  The cell of the lattice lattice crystal + Motif
  20. 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>
  21. 25. <ul><li>With the help of these three vectors, it is possible to construct a parallelopiped called a CELL </li></ul>
  22. 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>
  23. 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>
  24. 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>
  25. 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>
  26. 30. <ul><li>5. Hexagonal unit cell has 12 corners of the hexagonal prism 2 centres of hexagonal faces </li></ul>
  27. 31. <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>
  28. 32. <ul><li>Tetragonal Crystals </li></ul><ul><li>Simple Tetragonal </li></ul><ul><li>Body Centred Tetragonal </li></ul>
  29. 33. <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>
  30. 34. <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>
  31. 35. <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>
  32. 36. 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>
  33. 37. <ul><li>Atoms are assumed to be hard spheres </li></ul>

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