Chem 104A, UC, Berkeley - University of California,...

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Chem 104A, UC, Berkeley


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A Floor-Tiling Problem

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Esher drawing


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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.


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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Esher drawing


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{R = n1 a1 + n2 a2 + n3 a3}

Translationalvector


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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5 Bravais Lattice in 2D

P P NP


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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Conventional cell vs. Primitive CellReflecting the symmetry

Different Basis

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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


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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


4 fold rotation axes TOTAL = 3

3-fold rotation axes(passing through cube

body diagonals) TOTAL = 4


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Copper metal is face-centered cubic

Identical atoms at corners and at face centers

Lattice type F

also Ag, Au, Al, Ni...


-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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Caesium Chloride (CsCl) is primitive cubic

Different atoms at corners and body center. NOT body centered, therefore.

Lattice type P

Also CuZn, CsBr, LiAg


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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Diamond Structure: two sets of FCC Lattices

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Cubic: four 3-fold + three 4-fold


One 4-fold axes

Why not F tetragonal?

Tetragonal: P, I

a1 = a2 a3 = = = 90°

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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


Orthorhombic: P, I, F, C

C F

a1 a2 a3 = = = 90°

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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


Trigonal: P : 3-fold rotation

a1 = a2 = a3 = = < 120° 90°

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Hexagonal

a1 = a2 a3 = = 90° = 120°

Monoclinica1 a2 a3 = = 90°

Triclinic

a1 a2 a3


Unit cell contentsCounting the number of atoms within the unit cell

Many atoms are shared between unit cells

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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)]


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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(0,0,0)(0, ½, ½)(½, ½, 0)(½, 0, ½)

Fractional Coordinates


Cs (0,0,0)Cl (½, ½, ½)

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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


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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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(100) (111)

(200) (110)


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Chem 104A, UC, BerkeleyCrystallographic Directions And Planes

Miller-Bravais indices

[uvtw], (hkil)i=-(h+l)t=-(u+v)


In cubic system,

[hkl] direction perpendicular to (hkl) plane

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2

222

2

1

a

lkh

dhkl

For cubic system

Lattice spacing


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Chem 104A, UC, BerkeleyPowder diffraction

2

222

2

1

a

lkh

dhkl


52.36%


6

)2

(34

%3

3

a

a

a

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-Iron is body-centered cubic

68%

BCC Lattice

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33)43

(34

2%3

3

a

a

a


What is the highest density for sphere packing?

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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).


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Close packing structures: Cubic vs. Hexagonal


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CN=12

(0,0,0)(1/3,2/3,1/2)


CN=12

%05.74)22(

33.14%

3

3

r

rFor BCC; 68.02%

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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)


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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


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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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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


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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


Applicable to ionic solidNot covalent solid

Many exceptions!

ZnS:r+/r-=0.88/1.70=0.5Expect CN=6

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