The Structure of Crystalline Solids

1
The Structure of Crystalline Solids
2
Atomic Arrangement

• Crystal structures have short-range order and
  long-range order (on a lattice - see CDROM)
• unit cell smallest grouping of atoms that show
  the geometry of the structure and be repeated to
  form the structure
• Seven unit cell geometries possible

3
Atomic Arrangement

• When abc and abg90 the crystal system is
  cubic
• lattice points atom positions within a unit
  cell (e.g., at the corners, center of the cube,
  center of the faces, etc.)
• 14 types of unit cells (Bravis lattices) in 7
  crystal systems are possible

4
Atomic Arrangement
lattice parameter (for cubic crystals)
a0 measured at room T and given in nm 1x10-9 m
you should remember or these unit
conversions Å 1x10-10 m
5
Why do we care?

• To study properties of crystalline materials
  (such as strength or electrical conductivity) we
  work with unit cells.
• Therefore we need to know how many atoms/unit
  cell there are for a particular material

6
Atoms/unit cell

• Depending on where the atoms are located within
  the unit cell they belong to it completely or are
  shared with one or more other cells.
• Remember! These cells are replicated in 3D to
  generate the macroscopic material

Atom in center of cube is not shared with any
other unit cell 1 atom/unit cell
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Atoms/unit cell
Atom in the face of the cube is shared with 1
neighboring unit cell 1/2 atom/unit cell
Atom in the corner of the cube is shared among 8
neighboring unit cells 1/8 atom/unit cell
8
This is called a Simple Cubic (SC) crystal
structure 1 atom/unit cell Volume V
a03 Coordination number 6
a0
Coordination number number of nn of each atom
in a perfect crystal
9
This is called a Body Centered Cubic (BCC or bcc)
crystal structure 2 atom/unit cell V
a03 Coordination number 8 examples Fe, W
a0
10
This is called a Face Centered Cubic (FCC or fcc)
crystal structure 4 atom/unit cell V
a03 Coordination number 12 examples Ag, Pt, Au
a0
11
Close-Packed Directions
In SC
Directions in which atoms are in continuous
contact with one another
Front of the cube
a0 2r lattice constant 2 x atomic radius
12
In BCC ?3 a0 4r a0 4r
?3
a0
13
In FCC ?2 a0 4r a0 4r 2 ?2 r
?2 Derivation given in example problem
3.1 in the book, p. 36
a0
14
This is called a Hexagonal Close Packed (HCP or
hcp) crystal structure 2 atom/unit cell (3 unit
cells shown in figure) Va02 c0 cos 30? per unit
cell Coordination number 12 examples Cd,
Zn Ideally c0/a0 1.633
c0
a0
a0
angles in hexagon 120?
15
Atomic Packing Factor
If we assume each atom is a hard sphere the
atomic packing factor (APF) is the fraction of
the unit cell occupied by atoms APF
Vatoms/Vcell (atoms)(Vatom)/(Vcell) Vatom
4pr3 where r radius of atom 3 Vcell
given for each type of unit cell r and a are
related to each other by close-packed directions
(examples)
16
Density
Mass/Volume Theoretical density of a crystalline
material assumes the unit cell propagate in 3D
space perfectly r (mass of atoms in unit
cell)/(Vcell) (atoms/cell)(atomic mass)
(Vcell)(Avagodros number) (1
cell)(atoms/cell)(g/mol) g/cm3
(cm3)(atoms/mol) (examples)
17
Polymorphic Structures

• Same material, more than one crystal structure
  possible.
• When an elemental material is polymorphic, this
  is called allotropy.

18
Crystalline and Noncrystalline Materials

• Single crystal materials crystalline solid
  where the periodic arrangement of the crystalline
  unit cell extends in all directions without
  interruption
• Polycrystalline materials crystalline solids
  made up of many single crystals separated by
  grain boundaries

19
Crystalline and Noncrystalline Materials
Amorphous materials non-crystalline materials
that have short-range order (to satisfy local
bonding requirements) but no long-range order
20
Crystalline and Noncrystalline Materials
Isotropic materials gt measured properties are
independent of direction of measurement (e.g.,
cubic single crystals, polycrystalline
materials) Anisotropic materials gt measured
properties depend on direction of measurement
(e.g., HCP single crystals)
21
Points, Directions, and Planes
We have already discussed close-packed directions
in some unit cells. Materials cleave, deform,
etc. preferentially along certain planes or in
certain directions. Therefore, we need clear,
unambiguous language to discuss directions and
planes in crystalline unit cells.
22
Points - Cubic Unit Cells
z
y
0, 0, 1
x
1, 0, 1
1, 1, 1
Distance is measured in terms of the number of
lattice parameters to move in x, y, and z to get
to a point from the origin
0, 0, 0
0, 1, 0
1/3, 0, 0
1, 0, 0
23
Miller Indices
Short-hand notation to unambiguously describe
directions and planes in crystalline
materials There is a specific procedure for
finding miller indices for Directions
24
Directions - Cubic Unit Cells

• Determine coordinates of 2 points that form a
  line in the direction of interest
• Subtract the coordinates of the tail point from
  the coordinates of the head point lattice
  parameters traveled along each axis
• Reduce to smallest integers
• Put final numbers in uvw with any negative
  numbers shown as bar numbers (e.g., -2 is
  written as 2)

(examples)
25
Notes

• Because directions are vectors, a direction and
  its negative are not identical. Therefore, 100
  is not the same direction as 100.
• A direction and its multiple are identical. 100
  is the same direction as 200 it has just not
  been reduced to the smallest integers.

(examples)
26
More Notes

• Certain groups of directions are equivalent they
  are only different because of the way we
  constructed the coordinates. In cubic lattices,
  100 is equivalent to 010 if we redefine the
  coordinate system (or rotate the unit cell).
  Equivalent groups of equivalent directions are
  written in ltuvwgt

(examples)
27
Planes - Cubic Unit Cells

• Find the x, y, z intercepts of the plane within
  the unit cell in terms of the number of lattice
  parameters. Note if the plane passes through the
  point you have designated as the origin, the
  origin must be moved.
• Take the reciprocals of the intercepts
• Clear all fractions but do not reduce to lowest
  integers
• Write inside (uvw) negative numbers written as
  bar numbers

(examples)
28
c
Points - Hexagonal Unit Cells
a3
a2
a1
a3 is redundant a1a2 -a3 can use (uvw)
(a1, a2, c) or can use (uvtw) (a1,a2,a3,c)
0, 0, 1
0, 1, 1
0, 0, 0
0, 1, 0
1, 1, 0
1, 0, 0
29
Directions - Hexagonal Unit Cells

• Determine the number of lattice parameters to
  move in each direction to get from tail to head
  and keep u v -t
• Can be expressed in terms of uvw or uvtw
• To convert between 3-coordinate and 4-coordinate
  system, follow these rules
• u 1/3 (2u-v)
• v 1/3 (2v-u)
• t -1/3(uv) -(uv)
• w w

Then clear fractions and reduce to the smallest
integers
(examples)
30
Planes - Hexagonal Unit Cells

• Express in 3-coordinate system or 4-coordinate
  system
• Use u v -t to convert back and forth between
  the two
• Otherwise, no different from planes in cubic
  crystals

(examples)
31
Linear Density
Number of atoms along a particular directions. 1D
version of the packing factor Fraction of the
line length intersected by atoms
(examples)
32
Planar Density
Number of atoms per unit area in the plane. 2D
version of the packing factor Fraction of area of
the plane covered by atoms
(examples)