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Crystal structures & Packing Fraction
CRYSTAL

“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.”

                   STRUCTURES

 “Structure of anything is defined as the framework of
 its body.”
Crystal = Lattice+Base
Motif or basis:
 Typically an atom or a group of atoms associated with
each lattice point.

  Lattice  The underlying periodicity of the crystal
  Basis  Entity associated with each lattice points




            Lattice                                      Crystal
       Translationally periodic                  Translationally periodic
       arrangement of points.                    arrangement of motifs.
Crystal = Lattice (Where to repeat)
            +
             Motif (What to repeat)

   Crystal
                a
                         =
   Lattice
                 a
                         +
   Motif                        Note: all parts of the motif do not sit on the lattice
                         a      point
                         2
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: +
A strict 1D crystal = 1D lattice + 1D motif
 The only kind of 1D motif is a line segment.

 An unit cell is a representative unit of the structure
  (finite part of a infinite structure) .
   Which when repeated gives the whole structure.

Lattice

                                +
                        Motif
                                =

Crystal
 2D crystal = 2D lattice + 2D motif

        Lattice




                           + 
                                    Motif

b
    
    a
Crystal

                      
                      
                      
=                     
                      
                      
                      
                      
 3D crystal = 3D lattice + 3D motifs

          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
Crystal structures & Packing Fraction
a = b= c
              = = = 90º

     • 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
SIMPLE CUBIC STRUCTURE

• Cubic unit cell is 3D repeat unit
• Rare (only Po has this structure)
                                 • Coordination No. = 6
                                   (# nearest neighbors)
a
                                   R=0.5a

    close-packed directions
    contains 8 x 1/8 =
         1 atom/unit cell
         Adapted from Fig. 3.19,
         Callister 6e.


                                                                0.52
Lattice constant
                         • APF for a simple cubic structure =
BODY CENTERED CUBIC STRUCTURE




                  • Coordination No. = 8
                    (# nearest neighbors)
Unit cell c ontains:
                               1 + 8 x 1/8
                              = 2 atoms/unit cell
                     R
     Adapted from        a
     Fig. 3.2,
     Callister 6e.




• APF for a body-centered cubic structure =         3/8 =   0.68
FACE CENTERED CUBIC STRUCTURE


   Atoms are arranged at the corners and center
    of each cube face of the cell.
    ◦ Atoms are assumed to touch along face diagonals
• Coordination No. = 12
  (# nearest neighbors)
Unit cell c ontains:
                              6 x 1/2 + 8 x 1/8
                             = 4 atoms/unit cell
          a




• APF for a body-centered cubic structure = /(3 2) =   0.74
• FCC Unit Cell




• ABCABC... Stacking Sequence
• 2D Projection
                      A
                          B       B
                              C
                  A
     A sites          B       B       B
                          C       C
     B sites              B       B
     C sites
Ideally, c/a = 1.633 for close packing
However, in most metals, c/a ratio deviates from this valu
• ABAB... Stacking Sequence


  • 3D Projection             • 2D Projection

               A sites

               B sites

               A sites



  • Coordination NO.= 12
  • APF = 0.74, for ideal c/a ratio of 1.633
Crystal structures & Packing Fraction
Close packed crystals




                              A plane
                              B plane



                               C plane

                               A plane
  …ABCABCABC… packing                        …ABABAB… packing
[Face Centered Cubic (FCC)]             [Hexagonal Close Packing (HCP)]
Examples of elements with Cubic Crystal Structure




                                Fe                         Cu
 Po

                             n = 2 BCC                    n = 4 FCC/CCP
n = 1 SC




                            n = 8 DC
                         C (diamond)
a=b c
     = = = 90º

   Simple Tetragonal
   Body Centred Tetragonal -BCT

      Elements with Tetragonal structure → In, Sn
Example of an element with Body Centred Tetragonal Crystal Structure




                                 B
                                 C
                                 T
a    b c
         = = = 90º
    Simple Orthorhombic
    Body Centred Orthorhombic
    Face Centred Orthorhombic
    End Centred Orthorhombic



Elements with Orthorhombic structure → Br, Cl, Ga
Element with Orthorhombic Crystal Structure
a=b c
    = = 90º
   =120º

             Simple Hexagonal
 Elements with Hexagonal structure → Be, Cd, Co, Ti, Zn
Element with Hexagonal Crystal Structure
a=b=c
    = =              90º

             Rhombohedral (simple)
Elements with Trigonal structure → As, B, Bi, Hg
Element with Simple Trigonal Crystal Structure
a    b c
     = = 90º

  Simple Monoclinic
  End Centred (base centered) Monoclinic

Elements with Monoclinic structure → P, Pu, Po
a   b    c




        • Simple Triclinic
14 Bravais Lattices divided into 7 Crystal Systems
 A Symmetry based concept                                      „Translation‟ based concept

    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)                       

                                                                        P       Primitive
                                                                        I       Body Centred
                                                                        F       Face Centred
                                                                        C       A/B/C- Centred
Face Centred Cubic (FCC) Lattice   Two Carbon atom Motif
                                     (0,0,0) & (¼, ¼, ¼)


                              +
                               =


               Diamond Cubic Crystal

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Crystal structures & Packing Fraction

  • 2. CRYSTAL “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.” STRUCTURES “Structure of anything is defined as the framework of its body.”
  • 3. Crystal = Lattice+Base Motif or basis: Typically an atom or a group of atoms associated with each lattice point. Lattice  The underlying periodicity of the crystal Basis  Entity associated with each lattice points Lattice Crystal Translationally periodic Translationally periodic arrangement of points. arrangement of motifs.
  • 4. Crystal = Lattice (Where to repeat) + Motif (What to repeat) Crystal a = Lattice a + Motif Note: all parts of the motif do not sit on the lattice a point 2
  • 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.  An unit cell is a representative unit of the structure (finite part of a infinite structure) .  Which when repeated gives the whole structure. Lattice + Motif = Crystal
  • 7.  2D crystal = 2D lattice + 2D motif Lattice +  Motif  b  a
  • 8. Crystal                      =                                   
  • 9.  3D crystal = 3D lattice + 3D motifs 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
  • 11. a = b= c = = = 90º • 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
  • 12. SIMPLE CUBIC STRUCTURE • Cubic unit cell is 3D repeat unit • Rare (only Po has this structure) • Coordination No. = 6 (# nearest neighbors)
  • 13. a R=0.5a close-packed directions contains 8 x 1/8 = 1 atom/unit cell Adapted from Fig. 3.19, Callister 6e. 0.52 Lattice constant • APF for a simple cubic structure =
  • 14. BODY CENTERED CUBIC STRUCTURE • Coordination No. = 8 (# nearest neighbors)
  • 15. Unit cell c ontains: 1 + 8 x 1/8 = 2 atoms/unit cell R Adapted from a Fig. 3.2, Callister 6e. • APF for a body-centered cubic structure = 3/8 = 0.68
  • 16. FACE CENTERED CUBIC STRUCTURE  Atoms are arranged at the corners and center of each cube face of the cell. ◦ Atoms are assumed to touch along face diagonals
  • 17. • Coordination No. = 12 (# nearest neighbors)
  • 18. Unit cell c ontains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell a • APF for a body-centered cubic structure = /(3 2) = 0.74
  • 19. • FCC Unit Cell • ABCABC... Stacking Sequence • 2D Projection A B B C A A sites B B B C C B sites B B C sites
  • 20. Ideally, c/a = 1.633 for close packing However, in most metals, c/a ratio deviates from this valu
  • 21. • ABAB... Stacking Sequence • 3D Projection • 2D Projection A sites B sites A sites • Coordination NO.= 12 • APF = 0.74, for ideal c/a ratio of 1.633
  • 23. Close packed crystals A plane B plane C plane A plane …ABCABCABC… packing …ABABAB… packing [Face Centered Cubic (FCC)] [Hexagonal Close Packing (HCP)]
  • 24. Examples of elements with Cubic Crystal Structure Fe Cu Po n = 2 BCC n = 4 FCC/CCP n = 1 SC n = 8 DC C (diamond)
  • 25. a=b c = = = 90º  Simple Tetragonal  Body Centred Tetragonal -BCT  Elements with Tetragonal structure → In, Sn
  • 26. Example of an element with Body Centred Tetragonal Crystal Structure B C T
  • 27. a b c = = = 90º  Simple Orthorhombic  Body Centred Orthorhombic  Face Centred Orthorhombic  End Centred Orthorhombic Elements with Orthorhombic structure → Br, Cl, Ga
  • 28. Element with Orthorhombic Crystal Structure
  • 29. a=b c = = 90º =120º Simple Hexagonal  Elements with Hexagonal structure → Be, Cd, Co, Ti, Zn
  • 30. Element with Hexagonal Crystal Structure
  • 31. a=b=c = = 90º Rhombohedral (simple) Elements with Trigonal structure → As, B, Bi, Hg
  • 32. Element with Simple Trigonal Crystal Structure
  • 33. a b c = = 90º  Simple Monoclinic  End Centred (base centered) Monoclinic Elements with Monoclinic structure → P, Pu, Po
  • 34. a b c • Simple Triclinic
  • 35. 14 Bravais Lattices divided into 7 Crystal Systems A Symmetry based concept „Translation‟ based concept 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)  P Primitive I Body Centred F Face Centred C A/B/C- Centred
  • 36. Face Centred Cubic (FCC) Lattice Two Carbon atom Motif (0,0,0) & (¼, ¼, ¼) + = Diamond Cubic Crystal