Crystal structures & Packing Fraction

Presentation on crystal structures and packing factor

1. Presentation on Crystal structures

2. “A crystal is a solid in which atoms are arranged in some regular repetition pattern in all directions.” “Aggregation of molecules with a definite internal structure and the external form of a solid enclosed by symmetrically arranged plane faces.” “Structure of anything is defined as the framework of its body.” CRYSTAL STRUCTURES

3. Lattice  T he underlying periodicity of the crystal Basis  Entity associated with each lattice points Lattice  how to repeat Motif  what to repeat Crystal = Lattice+Base Motif or basis : Typically an atom or a group of atoms associated with each lattice point. Translationally periodic arrangement of motifs. Translationally periodic arrangement of points. Lattice Crystal

4. Crystal = Lattice (Where to repeat) + Motif ( What to repeat) = + a a Lattice Motif Note: all parts of the motif do not sit on the lattice point Crystal

5. Let us construct the crystal considered before starting with an infinite array of points spaced a/2 apart Put arrow marks pointing up and down alternately on the points: What we get is a crystal of lattice parameter ‘a’ and not ‘a/2’! And the motif is: +

6. A strict 1D crystal = 1D lattice + 1D motif The only kind of 1D motif is a line segment. Lattice Motif Crystal = + An unit cell is a representative unit of the structure (finite part of a infinite structure) .  Which when repeated gives the whole structure. 1-D Crystal

7. 2D crystal = 2D lattice + 2D motif Lattice + Motif  2-D Crystal

8.                                                         Crystal =

9. 3D crystal = 3D lattice + 3D motifs 3-D Crystal CRYSTAL OR SPACE LATTICE It is defined as an array of points in 3 dimensions in which every point has surroundings identical to every other point in array. According to BRAVAIS there are 14 possible types of space lattice in 7 basic crystal system

10. THE 7 CRYSTAL SYSTEM

11. a = b= c  =  =  = 90 º Simple Cubic (P) - SC Body Centred Cubic (I) – BCC Face Centred Cubic (F) - FCC Elements with Cubic structure → SC : F, O BCC : Cr, Fe, Nb, K, W FCC : Al, Ar, Pb, Ni, Ge Cubic Crystal

12. • Cubic unit cell is 3D repeat unit Rare (only Po has this structure ) • Coordination No. = 6 (# nearest neighbors) SIMPLE CUBIC STRUCTURE

13. • APF for a simple cubic structure = 0.52 ATOMIC PACKING FACTOR contains 8 x 1/8 = 1 atom/unit cell Adapted from Fig. 3.19, Callister 6e. Lattice constant close-packed directions a R=0.5a

14. BODY CENTERED CUBIC STRUCTURE • Coordination No. = 8 (# nearest neighbors)

15. • APF for a body-centered cubic structure = p  3/8 = 0.68 Adapted from Fig. 3.2, Callister 6e. ATOMIC PACKING FACTOR: BCC

16. Atoms are arranged at the corners and center of each cube face of the cell. Atoms are assumed to touch along face diagonals FACE CENTERED CUBIC STRUCTURE

17. • Coordination No. = 12 (# nearest neighbors)

18. • APF for a body-centered cubic structure = p /(3  2) = 0.74 ATOMIC PACKING FACTOR: FCC

19. • ABCABC... Stacking Sequence • FCC Unit Cell FCC STACKING SEQUENCE • 2D Projection

20. HEXAGONAL CLOSE-PACKED STRUCTURE (HCP) Ideally, c/a = 1.633 for close packing However, in most metals, c/a ratio deviates from this value

21. • Coordination NO.= 12 • ABAB... Stacking Sequence • APF = 0.74, for ideal c/a ratio of 1.633 • 3D Projection • 2D Projection

22. Close Packed Structures

23. Close packed crystals A plane B plane C plane A plane …ABCABCABC… packing [Face Centered Cubic (FCC)] …ABABAB… packing [Hexagonal Close Packing (HCP)]

24. Examples of elements with Cubic Crystal Structure Po n = 1 n = 2 n = 4 Fe Cu BCC FCC/CCP SC C (diamond) n = 8 DC

25. a = b  c  =  =  = 90 º Simple Tetragonal Body Centred Tetragonal -BCT Elements with Tetragonal structure → In, Sn Tetragonal Crystal

26. Example of an element with Body Centred Tetragonal Crystal Structure B C T

27. a  b  c  =  =  = 90 º Simple Orthorhombic Body Centred Orthorhombic Face Centred Orthorhombic End Centred Orthorhombic Elements with Orthorhombic structure → Br, Cl, Ga Orthorhombic Crystal

28. Element with Orthorhombic Crystal Structure

29. a = b  c  =  = 90 º =120 º Elements with Hexagonal structure → Be, Cd, Co, Ti, Zn Hexagonal Crystal Simple Hexagonal

30. Element with Hexagonal Crystal Structure

31. a = b = c  =  =   90º Elements with Trigonal structure → As, B, Bi, Hg Trigonal/Rhombohedral Crystal Rhombohedral (simple)

32. Element with Simple Trigonal Crystal Structure

33. a  b  c  =  = 90º   Elements with Monoclinic structure → P, Pu, Po Monoclinic Crystal Simple Monoclinic End Centred (base centered) Monoclinic

34. a  b  c      Simple Triclinic Triclinic Crystal

35. Crystal System Shape of UC Bravais Lattices P I F C 1 Cubic Cube    2 Tetragonal Square Prism (general height)   3 Orthorhombic Rectangular Prism (general height)     4 Hexagonal 120  Rhombic Prism  5 Trigonal Parallopiped (Equilateral, Equiangular)  6 Monoclinic Parallogramic Prism   7 Triclinic Parallopiped (general)  14 Bravais Lattices divided into 7 Crystal Systems P Primitive I Body Centred F Face Centred C A/B/C- Centred A Symmetry based concept ‘Translation’ based concept

36. + Face Centred Cubic (FCC) Lattice Two Carbon atom Motif (0,0,0) & (¼, ¼, ¼) = Diamond Cubic Crystal