outlook - eth zn.ethz.ch/~nielssi/download/4. semester/ac ii/unterlagen/basic... · 4/16/2013 ac...
TRANSCRIPT
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
1 4/16/2013 AC II| L. Viciu| Basic Crystallography
Solids classification summary
2
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
4/16/2013 AC II| L. Viciu| Basic Crystallography
3 4/16/2013 AC II| L. Viciu| Basic Crystallography
Crystallography = the science of the arrangement of atoms in ordered solids
Greek term “krystallos” = clear ice
4
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
4/16/2013 AC II| L. Viciu| Basic Crystallography
Early crystallography
5
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
4/16/2013 AC II| L. Viciu| Basic Crystallography
Tiling with regular shapes in 2D
6
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
4/16/2013 AC II| L. Viciu| Basic Crystallography
7
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
4/16/2013 AC II| L. Viciu| Basic Crystallography
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
4/16/2013 AC II| L. Viciu| Basic Crystallography 8
*Red line in the pictures represents the 3-fold axis
a
b
c
R-primitive rhombohedral
P-primitive trigonal
4/16/2013 AC II| L. Viciu| Basic Crystallography 9
= = = 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
10
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.:
4/16/2013 AC II| L. Viciu| Basic Crystallography
The reasons why a particular metal prefers a particular structure are still not well understood
11 4/16/2013 AC II| L. Viciu| Basic Crystallography
Metal Elements from the Periodic System
12
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)
4/16/2013 AC II| L. Viciu| Basic Crystallography
13
Cubic system: Counting Atoms in 3D Cells
a
V = a3
4/16/2013 AC II| L. Viciu| Basic Crystallography
14
Cubic system: Counting Atoms in 3D Cells
a
V = a3
18
18 21
8
18 4
2
16
8
18
4/16/2013 AC II| L. Viciu| Basic Crystallography
15
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+
4/16/2013 AC II| L. Viciu| Basic Crystallography
16
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+
4/16/2013 AC II| L. Viciu| Basic Crystallography
Density of a crystal (D)
17
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
4/16/2013 AC II| L. Viciu| Basic Crystallography
Packing density
18
- 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
4/16/2013 AC II| L. Viciu| Basic Crystallography
19
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)
4/16/2013 AC II| L. Viciu| Basic Crystallography
Characteristic of Cubic systems
20
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
4/16/2013 AC II| L. Viciu| Basic Crystallography
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
21 4/16/2013
AC II| L. Viciu| Basic Crystallography
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)
22 4/16/2013 AC II| L. Viciu| Basic Crystallography
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
23 4/16/2013 AC II| L. Viciu| Basic Crystallography
Characteristic of Cubic systems
24
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
4/16/2013 AC II| L. Viciu| Basic Crystallography
Atom at (½, ½, 0)
Atom at (½, 0, ½)
Atom at (0, ½, ½)
Atoms description in crystal structures
25
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,)
4/16/2013 AC II| L. Viciu| Basic Crystallography
26
Each atom’s height within the cell is indicated as a fraction of the cell height
Atom Projections
4/16/2013 AC II| L. Viciu| Basic Crystallography
(0,1)
27
TiO2 SrTiO3 ZnS
Atom Projections
4/16/2013 AC II| L. Viciu| Basic Crystallography
George Braque: Houses at L’Estaque 1908
28
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
4/16/2013 AC II| L. Viciu| Basic Crystallography