Outlook

• Crystal systems, Baravais lattices, unit cell

• Characteristics of cubic systems: lattice points, nearest and next nearest neighbors, packing density

• Counting atoms, atom coordinates, atom projections

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Solids classification summary

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A. Based on bond type

1. Ionic solids 2. Covalent solids

3. Metals 4. Molecular solids

B. Based on atomic arrangements I. Ordered solids = Crystalline II. Disordered solids = Amorphous

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Crystallography = the science of the arrangement of atoms in ordered solids

Greek term “krystallos” = clear ice

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Early Crystallography

the packing of spherical particles (balls) should give crystals the regular shape

1660: Robert Hooke: cannon balls

all crystals have the same angles between corresponding faces

1669: Niels Steensen: quartz crystals

Figure from “Micrographia”, 1665

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Early crystallography

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there are only 7 distinct space-filling volume elements 7 CRYSTALOGRAPHIC SYSTEMS

1781: René-Just Haüy: cleavage of calcite (CaCO3- rhombohedral)

System Parameters Angles

1.

2.

3.

4.

5.

6.

7.

Cubic

Tetragonal

Orthorhombic

Hexagonal

Trigonal

Monoclinic

Triclinic

a = b = c

a = b c

a b c

a = b c

a = b = c

a b c

a b c

= = = 90ᵒ

= = = 90ᵒ

= = = 90ᵒ

= = 90ᵒ, = 120ᵒ

= = 90ᵒ

= = 90ᵒ, 90ᵒ

90ᵒ

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Tiling with regular shapes in 2D

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In 2D, one can do tilling with squares, rectangles, rhombs, hexagons, but not with pentagons

Ex. Tilling with a pentagon

The space is not filled by pentagons only the white gap remaining

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1848: August Bravais: 14 Bravais lattices

there are 14 distinct ways to arrange points in space

Early crystallography (continued)

colored balls are motifs/basis

I = BCC F = FCC

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Trigonal system 1. Hexagonal with only a 3-fold axis on ab plane

(no 6-fold axis!):

a = b c

= = 90ᵒ, = 120ᵒ

2. Rhombohedral with 3-fold axis on body diagonal

a = b = c

= = 90ᵒ

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*Red line in the pictures represents the 3-fold axis

a

b

c

R-primitive rhombohedral

P-primitive trigonal

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= = = 90ᵒ = = 90ᵒ

A rhombohedral lattice is realized by “stretching” a cubic cell with equal cell dimensions, a=b=c and equal cell angles, ==90.

*Red line in the pictures represents the 3-fold axis

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Bravais lattice

Basis or Motif

+ Crystal

structure

•A crystal system describes only the geometry of the unit cell i.e. cubic, tetragonal, etc •A crystal structure describes both the geometry of, and the atomic arrangements within, the unit cell i.e. face centered cubic, body centered cubic, etc.

1. single atom: Au, Al, Cu, Pt 2. molecule: solidCH4 3. ion pairs: Na+/Cl- 4. atom pairs: Carbon, Si, Ge

1. FCC 2. FCC 3. Rock salt 4. diamond

+

i.e.:

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The reasons why a particular metal prefers a particular structure are still not well understood

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Metal Elements from the Periodic System

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SIMPLE CUBIC (P)

-Rare, due to low packing density -Close pack direction is the cube edges -Ex. Po

BODY CENTERED CUBIC (BCC)

-Atoms touch with each other along face diagonal Ex: Li, Na, K, Rb, Cs, V, Cr, Fe

FACE CENETRED CUBIC (FCC)

-Atoms touch with each other along face diagonal Ex: Ca, Sr, Cu, Ag, Au, Ni, Pd, Pt, Rh, (most noble metals)

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Cubic system: Counting Atoms in 3D Cells

a

V = a3

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Cubic system: Counting Atoms in 3D Cells

a

V = a3

18

18 21

8

18 4

2

16

8

18

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Atoms in different positions in a cell are shared by different numbers of unit cells

•Corner atom shared by 8 cells 1/8 atom per cell

•Edge atom shared by 4 cells 1/4 atom per cell

•Face atom shared by 2 cells 1/2 atom per cell

•Body unique to 1 cell 1 atom per cell

Counting Atoms in 3D Cells

Cl- ions at the corners and face center positions

Na+ ions at the edge centers and body center

i.e.: NaCl

The smallest number of different lattice points in a unit cell is called formula unit, Z (the unit cell contains at least one formula unit!)

Cl-

Na+

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Atoms in different positions in a cell are shared by different numbers of unit cells

•Corner atom shared by 8 cells 1/8 atom per cell

•Edge atom shared by 4 cells 1/4 atom per cell

•Face atom shared by 2 cells 1/2 atom per cell

•Body unique to 1 cell 1 atom per cell

Counting Atoms in 3D Cells

Cl- ions at the corners and face center positions

Na+ ions at the edge centers and body center

Cl-: 8 x 1/8 + 6 x ½ = 1 + 3 = 4 ions per unit cell Na+: 12 x ¼ + 1 x 1 = 3 + 1 = 4 ions per unit cell

Z = 4

i.e.: NaCl

The smallest number of different lattice points in a unit cell is called formula unit, Z (the unit cell contains at least one formula unit!)

Cl-

Na+

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Density of a crystal (D)

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Nunitformulaofvolume

FW

volumemolar

weightformula

volume

massD

N = Avogadro’s number (6.023x1023 mol-1) V = volume of one formula unit (Å3 =10-24cm3) x Z

)/(66.1 3cmg

V

ZFWD

V = 5.663 = 181.321 Å3; FW = 22.99 +

35.453 = 58.44 g; Z = 4

3/14.2321.181

66.1444.58cmgD

ArraClNa

66.581.1202.1222

i.e.: NaCl

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Packing density

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- 4 atoms per unit cell (volume)

- Volume of an atom: (4/3) r3 4 x (4/3) r3

-Face diagonal = 4 r = a2a = 2r 2

the next nearest neighbor is at (a2)/2

-Total Volume = a3= 16 2r3

volumetotal

atomsofvolumedensitypaking

Ex: for FCC structure

74.023216

3

44

3

3

r

rx

densitypacking FCC

a

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SIMPLE CUBIC (P)

-Rare, due to low packing density -Close pack direction is the cube edges -Ex. Po

BODY CENTERED CUBIC (BCC)

-Atoms touch with each other along face diagonal Ex: Li, Na, K, Rb, Cs, V, Cr, Fe

FACE CENETRED CUBIC (FCC)

-Atoms touch with each other along face diagonal Ex: Ca, Sr, Cu, Ag, Au, Ni, Pd, Pt, Rh, (most noble metals)

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Characteristic of Cubic systems

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Simple BCC FCC

Unit cell volume

Lattice points per cell (Z)

Nearest neighbor distance

Number of nearest neighbors (C.N.)

Second nearest neighbor distance

Number of second neighbors

Packing density

a=f(r)

a3

1

a

6

a2

12

0.52

2r

a3

2

8

a

6

0.68

4r/3

a3

4

12

a

6

0.74

2r2

2

3a

2

2a

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Atom neighbors in Simple Cubic (SC)crystal: counting the neighbors of the red atom

6 blue nearest neighbors at distance a (cube edge)

12 green neighbors at a2 distance (face diagonal)

The transparent atoms in the bottom picture appear twice when the two pictures are glued together

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Atom neighbors in BCC crystal: counting the neighbors of the red atom

8 black nearest neighbors at (a3)/2 distance (half of

the body diagonal)

6 blue neighbors at distance a (two blue neighbors not

shown: one in front cube and one in the back cube of

the red atom)

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Atom neighbors in FCC crystal: counting the neighbors of the red atom

12 blue nearest neighbors at a distance (a2)/2 (half of the

face diagonal)

6 green second neighbors at distance a (cube edge)

The transparent atoms in the bottom picture appear twice when the two pictures are glued together

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Characteristic of Cubic systems

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Simple BCC FCC

Unit cell volume

Lattice points per cell (Z)

Nearest neighbor distance

Number of nearest neighbors (C.N.)

Second nearest neighbor distance

Number of second neighbors

Packing density

a=f(r)

a3

1

a

6

a2

12

0.52

2r

a3

2

8

a

6

0.68

4r/3

a3

4

12

a

6

0.74

2r2

2

3a

2

2a

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Atom at (½, ½, 0)

Atom at (½, 0, ½)

Atom at (0, ½, ½)

Atoms description in crystal structures

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The atoms are given in terms of fractional coordinates x, y, z which describe fractions of the lattice constants a, b and c.

Atom at (0, 0, 0)

Atom at (½, ½, ½)

Atom at (½, 0, 0)

Atom at (0, 0, ½,)

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Each atom’s height within the cell is indicated as a fraction of the cell height

Atom Projections

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(0,1)

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TiO2 SrTiO3 ZnS

Atom Projections

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George Braque: Houses at L’Estaque 1908

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Pablo Picasso: Landscape with Bridge 1909

Pablo Picasso: Houses with Trees 1907

Pablo Picasso: Reservoir Horta 1909

“Cubism is based much less on the expression of emotion than it is an intellectual experiment with structure.”

The visual language of cubic shapes

http://www.robinurton.com

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