Soild 2011 Autumn
-
Upload
ayanchatterjee -
Category
Documents
-
view
219 -
download
0
Transcript of Soild 2011 Autumn
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 12/76
Crystal Structure
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 13/76
Can pack with irregular shapes
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 14/76
-
Most efficient way of packing equal sized spheres.
In 2D, have close packed layers
Coordination number
(CN) = 6. This is the2D packing.
Can stack close packed (c.p.) to give 3D structures.
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 15/76
If we start with one c la er two ossible wa s of addin a
second layer (can have one or other, but not a mixture) :
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 16/76
If we start with one c la er two ossible wa s of addin a
second layer (can have one or other, but not a mixture) :
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 17/76
Two possibilities:
.regular sequence …ABABABA…..Hexagonal close packing (hcp)
(2) Can have layer in C position, followed by the samerepeat, to g ve … …Cubic close packing (ccp)
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 18/76
Hexagonal close packed Cubic close packed
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 19/76
o matter w at type o pac ng, t e coor nat onnumber of each equal size sphere is always 12
possible for non-equal size spheres
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 22/76
FCC structure has a-b-c-a-b-c
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 24/76
Build up ccp layers Add construction lines(ABC… packing) - can see fcc unit cell
c.p layers are oriented perpendicular to the bodydiagonal of the cube
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 30/76
Lattices
By definition, crystals are periodic in three dimensions. A lattice is aregular infinite arrangement of points in which every point has the sameenvironment as any other point. A lattice in 2 dimensions is called a net
and a regular stacking of nets gives us a 3-dimensional lattice.
2-D net Stacks of 2-D nets
produce 3-D lattices.
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 31/76
Symmetry The unit cell in three dimensions.The unit cell is defined by three
Crystals are regular periodic arrays,i.e. they have long range translationalsymmetry. Crystals are often
edge vectors a, b, and c, with , , , corresponding to the anglesbetween b-c, a –c, and a-b,
cons ere to ave essent a y n n te
dimensions.a
respectively.
bOne Unit Cell
a
c
Unit cell = The smallest volume from whichthe entire crystal can be constructed bytranslation onl . All cr stals have
Unit cells are defined in terms of thelengths of the three vectors and the
translational symmetry, with the translationalvectors equal to edges of the unit cell.
ree ang es.For example, a=94.2Å, b=72.6Å,
c=30.1Å, =90°, =102.1°, =90°.
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 32/76
Translations (Lattices)A property at the atomic level, not of crystal
shapes
Symmetric translations involve repeatdistancesThe origin is arbitrary
1-D translations = a rowa
a is the repeat vectora is the repeat vector
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 33/76
Translations (Lattices)- =
a
b
, ,, ,
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 34/76
Translations (Lattices)- =
b
a
Every point that is exactly n repeats from that point is an equipoint to the originalEvery point that is exactly n repeats from that point is an equipoint to the original
T l ti l S tT l ti l S t
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 35/76
Translational SymmetryTranslational Symmetry
Every translation has a distance and a direction.
Translations are not rotated or reflected…the shape remains thesame size.
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 36/76
Examples of TranslationalExamples of Translational
SymmetrySymmetry
The objects simply move from one position to another retaining size and shape.
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 37/76
Crystal: Periodic Arrays of Atoms
ranslation Vectorsa3
Atoma1
a2a1, a2 ,a3
Primitive Cell:
• Smallest building block forthe crystal structure.
• Repetition of the primitive cell
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 38/76
-(sodium chloride, NaCl)
e e ne a ce po n s ; ese are po n swith identical environments
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 39/76
-atoms - but unit cell size should always be the same.
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 40/76
-it doesn’t matter if you start from Na or Cl
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 41/76
- or if you don’t start from an atom
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 42/76
same - empty space is not allowed!
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 43/76
. . sc er wor s c or on r - aarn- e e er an s.All rights reserved.
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 44/76
Seven unit cell shapes
• Cubic a=b=c == =90°• Tetragonal a=bc == =90°• Orthorhombic abc == =90°• Monoclinic abc = =90°, 90°• r c n c a c
• Hexagonal a=bc ==90°, =120°• = = = = °
Think about the sha es that these define - look at themodels provided.
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 45/76
Centering in Unit Cells
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 46/76
Centering in Unit Cells
Note that to best indicate the symmetry of the crystal lattice, it is
necessary to choose a unit cell that contains more than one lattice point.Unit cells that contain only one lattice point are called “Primitive” and areindicated with a “P ”. There are only a limited number of unique ways toc oose centere ce s an t e num er an compos t on o poss e
centered cells depend on the crystal system.
Remember that only 1/8of an atom at the cornerof the cell is actually in
.is primitive.
Centering in Unit Cells
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 47/76
Centering in Unit Cells
For monoclinic cells, no other form of centering provides a different
solution, however for other crystal systems different types of centering arepossible. These include centering of specific faces (“A”, “B” or “C”),centering of all faces “F” or body centering “I”, in which there is a latticepo nt at t e geometr c center o t e un t ce .
All of the differentpossible lattices,including unique
centering schemeswere eterm ne y .Bravais and thus the14 unique lattices are
“
Lattices”
Primitive, P Body-centered, I Face-centered, F
Bravais Lattices
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 48/76
Bravais Lattices
These 14 lattices arethe unique scaffolds inw c atoms or
molecules may bearranged to form.
However there is a
symmetry in crystalswhich must beconsidered to describe
the actual arrangementof atoms in a crystal.These are the variouscrystallogaphic Point
Groups .
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 49/76
Name axes angles
Triclinic a b c 90o
= o
o
-
Orthorhombic a b c = 90o
Tetragonal a1 = a2 c = 90o
Hexagonal
++cc
Hexagonal (4 axes) a1 = a2 = a3 c = 90o 120o
Rhombohedral a1 = a2 = a3 90o
Isometric a1 = a2 = a3 = 90o
++aa
++bb
Axial convention:Axial convention:
“right“right--hand rule”hand rule”
c cc
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 50/76
b b
a P a I = Ca
b
Ponoc n c
a
b
c
a b c
c
b
POrthorhombic
a b c
C F I
c c
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 51/76
c c
a2
a1
P Tetragonal I
a2
a1
P or C RHexagonal Rhombohedral
a1 = a2 c
a1a2c
a1 = a2 = a3
a3
a
a2
Isometric a1 = a2 = a3
F I
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 52/76
Three Cubic Lattices
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 53/76
Three Cubic Lattices
1. Simple Cubic (SC)
a3a1 a2 a3
=
a1
2
Add one atom at the Add one atom
at the center of each face
2. Body-Centered Cubic (BCC) 3. Face-Centered Cubic (FCC)
Conventional Cell Primitive Cell
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 54/76
Primitive Cell of FCC
• ng e e ween a1, a2, a3:
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 55/76
a
aa
Body-Centered
Unit Cell Primitive Cell
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 56/76
a
a
a
Face-Centered
Cubic (F)
Primitive Cell
a
Unit Cell
Rotated 90º
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 66/76
When silver crystallizes, it forms face-centered cubic
8/3/2019 Soild 2011 Autumn
http://slidepdf.com/reader/full/soild-2011-autumn 67/76
. .the density of silver.
m -=V
= = = .
4 atoms/unit cell in a face-centered cubic cell
m = 4 Ag atoms107.9 g
mole A
x1 mole Ag
6.022 x 1023 atoms
x = 7.17 x 10-22 g
m 7.17 x 10-22 g 3 V
6.83 x 10-23
cm3
.