Crystal Structure - UPRMacademic.uprm.edu/pcaceres/Courses/MatEng3045/EME2-1.pdf · Radius, nm Metal Lattice Constant ... Avogadro’s number (6.023x10 23 atoms/mole) Example Calculate
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Crystal StructureTypes of Solids
� Amorphous: Solids with considerable disorder in their structures
(e.g., glass). Amorphous: lacks a systematic atomic arrangement.
� Crystalline: Solids with rigid a highly regular arrangement of
their atoms. (That is, its atoms or ions, self-organized in a 3D
periodic array). These can be monocrystals and polycrystals.
Amorphous crystalline polycrystalline
To discuss crystalline structures it is useful to consider atoms as being
hard spheres with well-defined radii. In this hard-sphere model, the
shortest distance between two like atoms is one diameter.
Lattice: A 3-dimensional system of points that designate the positions
of the components (atoms, ions, or molecules) that make up the
substance.
Unit Cell: The smallest repeating unit of the lattice.
The lattice is generated by repeating the unit cell in all three
dimensions
Crystal SystemsCrystallographers have shown that only
seven different types of unit cells are
necessary to create all point lattice
Cubic a= b = c ; α = β = γ = 90
Tetragonal a= b ≠ c ; α = β = γ = 90
Rhombohedral a= b = c ; α = β = γ ≠ 90
Hexagonal a= b ≠ c ; α = β = 90, γ =120
Orthorhombica≠ b ≠ c ; α = β = γ = 90
Monoclinic a≠ b ≠ c ; α = γ = 90 ≠ β
Triclinic a≠ b ≠ c ; α ≠ γ ≠ β ≠ 90
Bravais Lattices
Many of the seven crystal systems have variations of the basic unit
cell. August Bravais (1811-1863) showed that 14 standards unit
cells could describe all possible lattice networks.
Principal Metallic Structures
Most elemental metals (about
90%) crystallize upon
solidification into three densely
packed crystal structures: body-
centered cubic (BCC), face-
centered cubic (FCC) and
hexagonal close-packed (HCP)
Structures of Metallic Elements
Ru
H
Li
Na
K
Rb
Cs
Fr
Be
Mg
Ca
Sr
Ba
Ra
Sc
Y
La
Ac
Ti
Zr
H f
V
Nb
Ta
Cr
Mo
W
Mn
Tc
Re
Fe
Os
Co
Ir
Rh
Ni
Pd
Pt
Cu
Ag
Au
Zn
Cd
Hg
B
Al
Ga
In
Tl
C
Si
Ge
Sn
Pb
N
P
As
Sb
Bi
O
S
Se
Te
Po
F
Cl
Br
I
At
Ne
Ar
Kr
Xe
Rn
He
Primitive Cubic
Body Centered Cubic
Cubic close packing(Face centered cubic)
Hexagonal close packing
0.1470.35840.2950Titanium
0.1250.40690.2507Cobalt
0.1600.52090.3209Magnesium
0.1330.56180.2665Zinc
HCP
0.1440.409Silver
0.1250.352Nickel
0.1440.408Gold
0.1280.361Copper
FCC
BCC
Structure
0.1370.316Tungsten
0.1860.429Sodium
0.2310.533Potassium
0.1360.315Molybdenum
0.1240.287Iron
0.1250.289Chromium
0.1430.405Aluminum
c, nma, nm
Atomic
Radius, nm
Lattice ConstantMetal
• Rare due to poor packing (only Po has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6 (# nearest neighbors)
SIMPLE CUBIC STRUCTURE (SC)
• Number of atoms per unit cell= 1 atom
6
APF = Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
• APF for a simple cubic structure = 0.52
APF =
a3
4
3π (0.5a)31
atoms
unit cellatom
volume
unit cell
volumeclose-packed directions
a
R=0.5a
contains 8 x 1/8 = 1 atom/unit cell
Atomic Packing Factor (APF)
• Coordination # = 8
• Close packed directions are cube diagonals.--Note: All atoms are identical; the center atom is shaded differently only for ease of
viewing.
Body Centered Cubic (BCC)
aR
Close-packed directions: length = 4R
= 3 a
Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell
APF =
a3
4
3π ( 3a/4)32
atoms
unit cell atom
volume
unit cell
volume• APF for a BCC = 0.68
• Coordination # = 12
• Close packed directions are face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded differently only for
ease of viewing.
Face Centered Cubic (FCC)
a
APF =
a3
4
3π ( 2a/4)34
atoms
unit cell atom
volume
unit cell
volume
Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell
Close-packed directions: length = 4R
= 2 a
• APF for a FCC = 0.74
Hexagonal Close-Packed (HCP)The APF and coordination number of the HCP structure is the same
as the FCC structure, that is, 0.74 and 12 respectively.
An isolated HCP unit cell has a total of 6 atoms per unit cell.
12 atoms shared by six cells = 2 atoms per cell2 atoms shared by two cells = 1
atom per cell
3 atoms
Close-Packed StructuresBoth the HCP and FCC crystal structures are close-packed structure.
Consider the atoms as spheres:
�Place one layer of atoms (2 Dimensional solid). Layer A
�Place the next layer on top of the first. Layer B. Note that there are
two possible positions for a proper stacking of layer B.
A B A : hexagonal close packed A B C : cubic close packed
The third layer (Layer C) can be placed in also teo different
positions to obtain a proper stack.
(1)exactly above of atoms of Layer A (HCP) or
(2)displaced
A B C : cubic close pack
A
B C
A
120°
90°A
A
B
A B A : hexagonal close pack
Interstitial sitesLocations between the ‘‘normal’’ atoms or ions in a crystal into
which another - usually different - atom or ion is placed.
o Cubic site - An interstitial position that has a coordination number
of eight. An atom or ion in the cubic site touches eight other atoms
or ions.
o Octahedral site - An interstitial position that has a coordination number of six. An atom or ion in the octahedral site touches six
other atoms or ions.
o Tetrahedral site - An interstitial position that has a coordination number of four. An atom or ion in the tetrahedral site touches four
other atoms or ions.
Crystals having filled Interstitial Sites
FCC Lattice has:
3 [=12(¼)] Oh sites at edge centers
+ 1 Oh site at body center
Octahedral, Oh, Sites
FCC Lattice has:
8 Th sites at ¼, ¼, ¼ positions
Tetrahedral, Th, Sites
Interstitial sites are important because we can derive more structures from these basic
FCC, BCC, HCP structures by partially or completely different sets of these sites
Density CalculationsSince the entire crystal can be generated by the repetition of the
unit cell, the density of a crystalline material, ρ = the density of the
unit cell = (atoms in the unit cell, n ) × (mass of an atom, M) / (the
volume of the cell, Vc)
Atoms in the unit cell, n = 2 (BCC); 4 (FCC); 6 (HCP)
Mass of an atom, M = Atomic weight, A, in amu (or g/mol) is
given in the periodic table. To translate mass from amu to grams
we have to divide the atomic weight in amu by the Avogadro
number NA = 6.023 × 1023 atoms/molThe volume of the cell, Vc = a3 (FCC and BCC)
a = 2R√2 (FCC); a = 4R/√3 (BCC)where R is the atomic radius.
Density Calculation
ACNV
nA=ρ
n: number of atoms/unit cell
A: atomic weight
VC: volume of the unit cell
NA: Avogadro’s number
(6.023x1023 atoms/mole)
Example 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=
×××=ρ
8.94 g/cm3 in the literature
Example
Rhodium has an atomic radius of 0.1345nm (1.345A) and a density of 12.41g.cm-3.
Determine whether it has a BCC or FCC crystal structure. Rh (A = 102.91g/mol)
Solution
ACNV
nA=ρ
n: number of atoms/unit cell A: atomic weight
VC: volume of the unit cell NA: Avogadro’s number
(6.023x1023 atoms/mole)
structure FCC a has Rhodium
01376.04
)1345.0(627.22
627.22)2
4( and 4 n then FCC is rhodium If
0149.02
)1345.0(316.12
316.12)3
4( and 2 n then BCC is rhodium If
01376.0103768.1.10023.6.41.12
.91.102
333
333
333
333
3323
1233
13
nmnmx
n
a
rra
nmnmx
n
a
rra
nmcmxmoleatomsxcmg
molg
N
A
n
a
n
V
A
c
==
===
==
===
===== −
−−
−
ρ
Linear And Planar Atomic Densities
Linear atomic density:
= 2R/Ll
Planar atomic density:
Ll
Crystallographic direction
= 2π R2/(Area A’D’E’B’)
3
4RaLl == = 0.866
A’ E’
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