Crystal Geometry

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???? ????? ?? ????? ???? ???? ???? ???? (??????
???? ) ????? ???? ??? ?? ???? ? ?? ???? ????? ???
????
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???? ?? ??? (??????) ????? ?????? ?? ?? ??????
?????? ???????
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??????
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???? ???? ?? ???? ?? ????
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???? ???? ?? ???? ?? ???? NaCl
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?????? ?????? ???? ????(??? ?????)
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???? ????? ?? ???? ???? ??? ????
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- or if you dont start from an atom
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This is NOT a unit cell even though they are all
the same - empty space is not allowed!
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???? ???? ?? ???????? ???? ?? ???? ???? ????
????
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Why can't the blue triangle be a unit cell?
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Five Bravais Lattices in 2D
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Unit Cell in 3D
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Unit Cell in 3D
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???? ????
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Crystal Systems Some Definitional information
Unit cell smallest repetitive volume which
contains the complete lattice pattern of a
crystal.
7 crystal systems of varying symmetry are
known These systems are built by changing the
lattice parameters a, b, and c are the edge
lengths ?, ?, and ? are interaxial angles
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Wigner-Seitz Method
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Wigner-Seitz Cell - 3D
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Crystal Systems
Crystal structures are divided into groups
according to unit cell geometry (symmetry).
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1-CUBIC
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Simple Cubic Structure (SC)
Rare due to low packing density (only Po
Polonium -- has this structure) Close-packed
directions are cube edges.
Coordination No. 6 ( nearest neighbors)
for each atom as seen
(Courtesy P.M. Anderson)
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1-CUBIC CRYSTAL SYSTEM
a- Simple Cubic (SC)
?? ??? ?? ?? ???? ???? ???? ???? ????? ????
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a- Simple Cubic (SC)
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???? ?? ?? ?????? ?? sc
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b-Body Centered Cubic (BCC)

• ???? ???????
• 8 ?????? ?????
• (Fe,Li,Na..etc) are BCC

c
b
a
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???? ?? ?? ?????? ?? BCC
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Atomic Packing Factor BCC
a
Adapted from Fig. 3.2(a), Callister 7e.
APF for a body-centered cubic structure 0.68
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Face Centered Cubic (FCC)

• 4 ??? ?? ???? ????? ???? ????
• (Cu,Ni,Pb..etc) ?????? fcc. ?????

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Face Centered Cubic Structure (FCC)
Atoms touch each other along face diagonals.
--Note All atoms are identical the
face-centered atoms are shaded differently
only for ease of viewing.
ex Al, Cu, Au, Pb, Ni, Pt, Ag
Coordination 12
Adapted from Fig. 3.1, Callister 7e.
4 atoms/unit cell (6 face x ½) (8 corners x
1/8)
(Courtesy P.M. Anderson)
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Atomic Packing Factor FCC
APF for a face-centered cubic structure 0.74
The maximum achievable APF!
(a 2?2R)
Adapted from Fig. 3.1(a), Callister 7e.
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3 - Face Centered Cubic
Atoms are all same.
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Primitive and conventional cells of FCC
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Primitive and conventional cells of BCC
Primitive Translation Vectors
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Primitive and conventional cells
Body centered cubic (bcc) conventional
¹primitive cell
?????? ???? ????? 000,100, 010, 001, 110,101,
011, 111, ½ ½ ½
Simple cubic (sc) primitive cellconventional
cell
?????? ???? ????? 000, 100, 010, 001, 110,101,
011, 111
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Primitive and conventional cells
Face centered cubic (fcc) conventional ¹
primitive cell
?????? ???? ????? 000,100, 010, 001, 110,101,
011,111, ½ ½ 0, ½ 0 ½, 0 ½ ½ ,½1 ½ , 1 ½ ½ , ½ ½ 1
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2 - HEXAGONAL SYSTEM

• ?? ??? ?? ???? ????? ???? ????.

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3Hexagonal Close-Packed Str.
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Hexagonal Close-packed Structure
He, Be, Mg, Hf, Re (Group II elements) ABABAB
Type of Stacking 
ab a120, c1.633a,  basis (0,0,0) (2/3a
,1/3a,1/2c)
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Packing
Close pack
Sequence AAAA - simple cubic

• Sequence ABABAB..
• hexagonal close pack

Sequence ABAB - body centered cubic
Sequence ABCABCAB.. -face centered cubic close
pack
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2 - HEXAGONAL SYSTEM
Atoms are all same.
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Hexagonal Close-Packed Structure (HCP)
ex Cd, Mg, Ti, Zn
ABAB... Stacking Sequence
3D Projection
2D Projection
Adapted from Fig. 3.3(a), Callister 7e.
6 atoms/unit cell
Coordination 12
APF 0.74
c/a 1.633 (ideal)
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We find that both FCC HCP are highest density
packing schemes (APF .74) this illustration
shows their differences as the closest packed
planes are built-up
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3 - TRICLINIC 4 - MONOCLINIC CRYSTAL SYSTEM

• ??? ?????? ?????? ????? ????? ?? ??????

Monoclinic (Simple) a g 90o, ß ¹ 90o a ¹ b
¹c
Monoclinic (Base Centered) a g 90o, ß ¹ 90o
a ¹ b ¹ c,
Triclinic (Simple) a ¹ ß ¹ g ¹ 90 oa ¹ b ¹ c
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5 - ORTHORHOMBIC SYSTEM
Orthorhombic (FC) a ß g 90o a ¹ b ¹ c
Orthorhombic (Base-centred)a ß g 90o a ¹
b ¹ c
Orthorhombic (BC) a ß g 90o a ¹ b ¹ c
Orthorhombic (Simple) a ß g 90o a ¹ b ¹ c
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6 TETRAGONAL SYSTEM
Tetragonal (BC) a ß g 90o a b ¹ c
Tetragonal (P) a ß g 90o a b ¹ c
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7 - Rhombohedral (R) or Trigonal
Rhombohedral (R) or Trigonal (S) a b c, a
ß 90,g ¹ 90o
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• Cubic Crystals
• a b c? ? ? 90º
• Simple Cubic (P)
• Body Centred Cubic (I) BCC
• Face Centred Cubic (F) - FCC

Tetrakaidecahedron (Truncated Octahedron)
PyriteCube
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GarnetDodecahedron
Fluorite Octahedron
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1 http//www.yourgemologist.com/crystalsystems.h
tml 2 L.E. Muir, Interfacial Phenomenon in
Metals, Addison-Wesley Publ. co.
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• Tetragonal Crystals
• a b ? c ? ? ? 90º
• Simple Tetragonal
• Body Centred Tetragonal

Zircon
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1
1
1 http//www.yourgemologist.com/crystalsystems.h
tml
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• Orthorhombic Crystals
• a ? b ? c ? ? ? 90º
• Simple Orthorhombic
• Body Centred Orthorhombic
• Face Centred Orthorhombic
• End Centred Orthorhombic

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Topaz
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1 http//www.yourgemologist.com/crystalsystems.h
tml
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• Hexagonal Crystals
• a b ? c ? ? 90º ? 120º
• Simple Hexagonal

Corundum
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1 http//www.yourgemologist.com/crystalsystems.h
tml
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• 5. Trigonal Crystals
• a b c ? ? ? ? 90º
• Trigonal (simple)

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1
Tourmaline
1 http//www.yourgemologist.com/crystalsystems.h
tml
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• Monoclinic Crystals
• a ? b ? c ? ? 90º ? ?
• Simple Monoclinic
• End Centred (base centered) Monoclinic (A/C)

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Kunzite
1 http//www.yourgemologist.com/crystalsystems.h
tml
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• 7. Triclinic Crystals
• a ? b ? c ? ? ? ? ?
• Simple Triclinic

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Amazonite
1 http//www.yourgemologist.com/crystalsystems.h
tml
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Miller Indices

• ????? ??? ???? ???????? ????? ?? ??? ????? ????
  ?? ?? ??????? ???? ?? ????
• ??? ????? ?? ?? ???? ?? ???? ?? ???? ?? ????? ??
  ?? ????? ?? ????? ????? ?? ???? ?????
• ???? ????????? ????? ?? ??? ?? ????? ???? ??????
  ??? ???? ???? ???? ???? ?? ????.

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Crystal Planes
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Crystallographic Planes -- families
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Example-1
Axis X Y Z
Intercept points 1 8 8
Reciprocals 1/1 1/ 8 1/ 8
Smallest Ratio 1 0 0
Miller Indices (100) Miller Indices (100) Miller Indices (100) Miller Indices (100)
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Example-2
Axis X Y Z
Intercept points 1 1 8
Reciprocals 1/1 1/ 1 1/ 8
Smallest Ratio 1 1 0
Miller Indices (110) Miller Indices (110) Miller Indices (110) Miller Indices (110)
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Example-3
Axis X Y Z
Intercept points 1 1 1
Reciprocals 1/1 1/ 1 1/ 1
Smallest Ratio 1 1 1
Miller Indices (111) Miller Indices (111) Miller Indices (111) Miller Indices (111)
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Example-4
Axis X Y Z
Intercept points 1/2 1 8
Reciprocals 1/(½) 1/ 1 1/ 8
Smallest Ratio 2 1 0
Miller Indices (210) Miller Indices (210) Miller Indices (210) Miller Indices (210)
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Determine the Miller indices for the plane shown
in the sketch

x
y z

Intercepts
Intercept in terms of lattice parameters
Reciprocals Reductions Enclosure
?a -b c/2
? -1 1/2
0 -1 2
N/A
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Example-5
Axis a b c
Intercept points 1 8 ½
Reciprocals 1/1 1/ 8 1/(½)
Smallest Ratio 1 0 2
Miller Indices (102) Miller Indices (102) Miller Indices (102) Miller Indices (102)
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Example-6
Axis a b c
Intercept points -1 8 ½
Reciprocals 1/-1 1/ 8 1/(½)
Smallest Ratio -1 0 2
Miller Indices (102) Miller Indices (102) Miller Indices (102) Miller Indices (102)
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Crystallographic Planes (HCP)

• In hexagonal unit cells the same idea is used

Adapted from Fig. 3.8(a), Callister 7e.
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Miller Indices
Indices of the plane (Miller) (2,3,3)
Indices of the direction 2,3,3
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Crystallographic Directions
Algorithm
z
1. Vector is repositioned (if necessary) to pass
through the Unit Cell origin.2. Read off line
projections (to principal axes of U.C.) in terms
of unit cell dimensions a, b, and c3. Adjust to
smallest integer values4. Enclose in square
brackets, no commas uvw
y
x
ex 1, 0, ½
gt 2, 0, 1
gt 201
-1, 1, 1
families of directions ltuvwgt
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What is this Direction ?????
0c
a/2
b
0
1
1/2
2
0
1
120
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????? ??? ???? ? ?????? ????? ???? ?? ????
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????? ??? ???? ? ?????? ????? ???? ?? ????
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??? ??? ????? ? ????? ???? ???? ?? ???? ????? ???
???? ?????? ?????.
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HCP Crystallographic Directions
Algorithm
1. Vector repositioned (if necessary) to pass
through origin.2. Read off projections in
terms of unit cell dimensions a1, a2, a3, or
c3. Adjust to smallest integer values4. Enclose
in square brackets, no commas uvtw
Adapted from Fig. 3.8(a), Callister 7e.
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????? ??? ?? ??? ????? ?????? ?? ??? ????? ???
???? u1,v1,w1 u2,v2,w2
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????? ????? ????? ?? ????? ??? ???? ?????
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hn a/OA knb/OB lnc/OC ?? ???????? ?? ?????
??????? dn/h2/ a2 k2/ b2 l2/ c2 0.5
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Example-7
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Indices of a Family or Form
??? hkl ???? ???? ????? ??? ???? ????? ??
????? (hkl) ?? ???? ?? ??? ?? ?????? ???? ??
?????? ?? ??? ?? ????
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3D 14 BRAVAIS LATTICES AND THE SEVEN CRYSTAL
SYSTEM

• ???? 14 ???? ????? ???? ???? ?? ???? ?? ???? ??
  ?? ??????.
• ??? 14 ???? ??? ?? ??? ????? ????? ????? ???
  ??????? ?? ????.
• Cubic Crystal System (SC, BCC,FCC)
• Hexagonal Crystal System (S)
• Triclinic Crystal System (S)
• Monoclinic Crystal System (S, Base-C)
• Orthorhombic Crystal System (S, Base-C, BC, FC)
• Tetragonal Crystal System (S, BC)
• Trigonal (Rhombohedral) Crystal System (S)

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THE MOST IMPORTANT CRYSTAL STRUCTURES

• Sodium Chloride Structure NaCl-
• Cesium Chloride Structure CsCl-
• Hexagonal Closed-Packed Structure
• Diamond Structure
• Zinc Blende

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1 Sodium Chloride Structure

• ??? ???? NaCl ????? ???? ????? ?? ???? ??? ????
  ???? ?? ???
• ????? ????? ?? ?? ??? ??? ?? ???? ?? ????
  ?????? ???? ????? ???.
• ?? ??? ?????? ?? ??? ??? ???? ?????? ??? ???.