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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
A Floor-Tiling Problem
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Chem 104A, UC, Berkeley
Esher drawing
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
12
12cos1
m
n
m-1 n
-2 -1 2
-1 -1/2 3
0 0 4
1 1/2 6
2 1 1
n
2cos
t
t t
n
2
n
2
mt
A B C D
nC nC
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Chem 104A, UC, BerkeleyCrystal StructureCrystal Symmetry
The following elements from molecular symmetry are consistentWith 3-dimensional crystal symmetry:
E, C2, C3, C4, C6, S3, S4, S6, i,
All crystals have three additional symmetry elements, each Corresponding to a translational vector:
a, b, c
The collection of the symmetry elements present in a specificCrystal is called : space group
There are 230 different space group.
Chem 104A, UC, Berkeley
The translational symmetry elements in a crystal define a periodic
Array of points called Bravais Lattice.
{n1a+n2b+n3c}
LATTICE = An infinite array of points in space, in which each point has identical surroundings to all others.
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Chem 104A, UC, Berkeley
Esher drawing
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
{R = n1 a1 + n2 a2 + n3 a3}
Translationalvector
Chem 104A, UC, Berkeley
Primitive Cell: simplest cell, contain one lattice pointNot necessary have the crystal symmetry
UNIT CELL = The smallest component of the crystal, which when stacked together with pure translational repetition reproduces the whole crystal
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Chem 104A, UC, Berkeley
5 Bravais Lattice in 2D
P P NP
Chem 104A, UC, Berkeley
Square a=b =90
Rectangular a b =90
Centered Rectangular
a b =90
Hexagonal a=b =120
Oblique a b 90
5 Bravais Lattice in 2D
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
Conventional cell vs. Primitive CellReflecting the symmetry
Different Basis
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
Lattice parameters: a, b, c;
7 Crystal Systems
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Chem 104A, UC, BerkeleyDefinition:Bravais Lattice: an infinite array of discrete points with an arrangement and orientation that appears exactly the same from whichever of the points the array is viewed.
Name Number of Bravais lattices ConditionsTriclinic 1 (P) a1 a2 a3
Monoclinic 2 (P, C) a1 a2 a3
= = 90° Orthorhombic 4 (P, F, I, A) a1 a2 a3
= = = 90°
Tetragonal 2 (P, I) a1 = a2 a3 = = = 90°
Cubic 3 (P, F, I) a1 = a2 = a3 = = = 90°
Trigonal 1 (P) a1 = a2 = a3 = = < 120° 90°
Hexagonal 1 (P) a1 = a2 a3 = = 90° = 120°
3D: 14 Bravais Lattice, 7 Crystal System
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
4 fold rotation axes
(passing through pairs of opposite face centers, parallel
to cell axes)
TOTAL = 3
a1 = a2 = a3 = = = 90°
Unit cell symmetries - cubic
Chem 104A, UC, Berkeley
4 fold rotation axes TOTAL = 3
3-fold rotation axes(passing through cube
body diagonals) TOTAL = 4
Unit cell symmetries - cubic
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Chem 104A, UC, Berkeley
Copper metal is face-centered cubic
Identical atoms at corners and at face centers
Lattice type F
also Ag, Au, Al, Ni...
Chem 104A, UC, Berkeley
-Iron is body-centered cubic
Identical atoms at corners and body center (nothing at face centers)
Lattice type I
Also Nb, Ta, Ba, Mo...
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Chem 104A, UC, Berkeley
Caesium Chloride (CsCl) is primitive cubic
Different atoms at corners and body center. NOT body centered, therefore.
Lattice type P
Also CuZn, CsBr, LiAg
Chem 104A, UC, Berkeley
Sodium Chloride (NaCl) -Na is much smaller than Cs
Face Centered Cubic
Rocksalt structure
Lattice type F
Also NaF, KBr, MgO….
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
Diamond Structure: two sets of FCC Lattices
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Chem 104A, UC, Berkeley
Cubic: four 3-fold + three 4-fold
Chem 104A, UC, Berkeley
One 4-fold axes
Why not F tetragonal?
Tetragonal: P, I
a1 = a2 a3 = = = 90°
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Chem 104A, UC, Berkeley
CaC2 - has a rocksalt-like structure but with non-spherical carbides
2-C
C
Carbide ions are aligned parallel to c
c > a,b tetragonal symmetry
Chem 104A, UC, Berkeley
Orthorhombic: P, I, F, C
C F
a1 a2 a3 = = = 90°
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Chem 104A, UC, Berkeley
Another type of centering
Side centered unit cell
Notation:
A-centered if atom in bc plane
B-centered if atom in ac plane
C-centered if atom in ab plane
Chem 104A, UC, Berkeley
Trigonal: P : 3-fold rotation
a1 = a2 = a3 = = < 120° 90°
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Chem 104A, UC, Berkeley
Hexagonal
a1 = a2 a3 = = 90° = 120°
Monoclinica1 a2 a3 = = 90°
Triclinic
a1 a2 a3
Chem 104A, UC, Berkeley
Unit cell contentsCounting the number of atoms within the unit cell
Many atoms are shared between unit cells
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Chem 104A, UC, Berkeley
Atoms Shared Between: Each atom counts:corner 8 cells 1/8face center 2 cells 1/2body center 1 cell 1edge center 4 cells 1/4
lattice type cell contentsP 1 [=8 x 1/8]I 2 [=(8 x 1/8) + (1 x 1)]F 4 [=(8 x 1/8) + (6 x 1/2)]C 2 [=(8 x 1/8) + (2 x 1/2)]
Chem 104A, UC, Berkeley
e.g. NaCl Na at corners: (8 1/8) = 1 Na at face centres (6 1/2) = 3Cl at edge centres (12 1/4) = 3 Cl at body centre = 1
Unit cell contents are 4(Na+Cl-)
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Chem 104A, UC, Berkeley
(0,0,0)(0, ½, ½)(½, ½, 0)(½, 0, ½)
Fractional Coordinates
Chem 104A, UC, Berkeley
Cs (0,0,0)Cl (½, ½, ½)
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Chem 104A, UC, Berkeley
Density Calculation
AC NV
nA
n: number of atoms/unit cell
A: atomic mass
VC: volume of the unit cell
NA: Avogadro’s number (6.023x1023 atoms/mole)
Calculate the density of copper.
RCu =0.128nm, Crystal structure: FCC, ACu= 63.5 g/mole
n = 4 atoms/cell, 333 216)22( RRaVC
3
2338/89.8
]10023.6)1028.1(216[
)5.63)(4(cmg
8.94 g/cm3 in the literature
2R
Chem 104A, UC, Berkeley
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Chem 104A, UC, BerkeleyPowder diffraction
Chem 104A, UC, BerkeleyCrystallographic Directions And Planes
Lattice DirectionsIndividual directions: [uvw]Symmetry-related directions: <uvw>
Miller Indices:1. Find the intercepts on the axes in terms of the lattice
constant a, b, c2. Take the reciprocals of these numbers, reduce to the
three integers having the same ratio(hkl)
Set of symmetry-related planes: {hkl}
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Chem 104A, UC, Berkeley
(100) (111)
(200) (110)
Chem 104A, UC, Berkeley
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Chem 104A, UC, BerkeleyCrystallographic Directions And Planes
Miller-Bravais indices
[uvtw], (hkil)i=-(h+l)t=-(u+v)
Chem 104A, UC, Berkeley
In cubic system,
[hkl] direction perpendicular to (hkl) plane
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Chem 104A, UC, Berkeley
2
222
2
1
a
lkh
dhkl
For cubic system
Lattice spacing
Chem 104A, UC, Berkeley
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Chem 104A, UC, BerkeleyPowder diffraction
2
222
2
1
a
lkh
dhkl
Chem 104A, UC, Berkeley
52.36%
Unit cell symmetries - cubic
6
)2
(34
%3
3
a
a
a
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Chem 104A, UC, Berkeley
-Iron is body-centered cubic
68%
BCC Lattice
24
33)43
(34
2%3
3
a
a
a
Chem 104A, UC, Berkeley
What is the highest density for sphere packing?
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Chem 104A, UC, Berkeley
In 1611 the German astronomer Johannes Kepler stated that no packing could be denser than that of the face-centred cubic (f.c.c.) lattice arrangement favored by grocers for stacking oranges, which fills about 0.7405 of the available space.
It took mathematicians some 400 years to prove him right.
Kepler's Conjecture
Hales, T. C. Discrete Computational Geom. 17, 1-51 (1997); 18, 135-149 (1997).
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
Close packing structures: Cubic vs. Hexagonal
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
CN=12
(0,0,0)(1/3,2/3,1/2)
Chem 104A, UC, Berkeley
CN=12
%05.74)22(
33.14%
3
3
r
rFor BCC; 68.02%
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
Rare Gas: Ne, He, Ar, Kr, Xe (ccp; fcc)
Metal: Cu, Ag, Au, Ni, Pd, Pt (ccp)
Mg, Zn, Cd, Ti (hcp)
Fe, Cr, Mo (bcc)
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
Caesium Chloride (CsCl) is primitive cubic
Different atoms at corners and body center. NOT body centered, therefore.
Lattice type P
Also CuZn, CsBr, LiAg
Simple Cubic Lattice
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
Ionic structures:
Can be considered as close packing of large anions with Cation filling in the interstitial sites.
For every anion, there are
1 Octahedral site2 tetrahedral sites.
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
What's the Numerical Value of a specific Ionic Radius?
High resolution X-Ray Diffraction
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Chem 104A, UC, BerkeleyRadiusRatioRule
Red line:Contacting
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
Chem 104A, UC, Berkeley
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Chem 104A, UC, Berkeley
Limiting Radius Ratios - anions in the coordination polyhedron of cation are in contact with the cation and with each other
Radius Ratio Coordination no. Binary (AB) Structure-type
r+/r- = 1 12 none known
1 > r+/r- > 0.732 8 CsCl
0.732 > r+/r- > 0.414 6 NaCl
0.414 > r+/r- > 0.225 4 ZnS
Chem 104A, UC, Berkeley
Applicable to ionic solidNot covalent solid
Many exceptions!
ZnS:r+/r-=0.88/1.70=0.5Expect CN=6