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I. Structural Aspects Sphere Packings Wells, pp. 141-161
Densest Packing of Spheres
Two-Dimensions: Unit Cell
Hand-Outs: 7
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I. Structural Aspects Sphere Packings: Geometry Wells, pp. 141-161
Densest Packing of Spheres
Two-Dimensions: PERIODICa
a
a = side of unit cell (Å, pm) = angle between 2 sides =120
“Coordinate System”
Unit Cell
Hand-Outs: 7
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I. Structural Aspects Sphere Packings: Geometry Wells, pp. 141-161
Densest Packing of Spheres
Two-Dimensions: PERIODIC
“A”
aa
Unit Cell
A: (0, 0)
a = side of unit cell (Å, pm) = angle between 2 sides =120
Hand-Outs: 7
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I. Structural Aspects Sphere Packings: Geometry Wells, pp. 141-161
Densest Packing of Spheres
Two-Dimensions: PERIODIC
“A”
aa
Unit Cell
A: (0, 0)
B: (1/3, 2/3)
“B”
a = side of unit cell (Å, pm) = angle between 2 sides =120
Hand-Outs: 7
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I. Structural Aspects Sphere Packings: Geometry Wells, pp. 141-161
Densest Packing of Spheres
Two-Dimensions: PERIODIC
“A”
aa
Unit Cell
A: (0, 0)
B: (1/3, 2/3)
C: (2/3, 1/3)
“Fractional Coordinates”
“B”
“C”
a = side of unit cell (Å, pm) = angle between 2 sides =120
Hand-Outs: 7
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I. Structural Aspects Sphere Packings: Closest Packings Wells, pp. 141-161
Densest Packing of Spheres
Three-Dimensions: Tetrahedron, ca. 79% Efficiency
70.5
Hand-Outs: 7
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I. Structural Aspects Sphere Packings: Closest Packings Wells, pp. 141-161
Densest Packing of Spheres
Three-Dimensions: Tetrahedron, ca. 79% Efficiency
70.5
CANNOT fill 3D space with just tetrahedra!
Hand-Outs: 7
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I. Structural Aspects Sphere Packings: Closest Packings Wells, pp. 141-161
Densest Packing of Spheres
Three-Dimensions: Tetrahedron, ca. 79% Efficiency
70.5
CANNOT fill 3D space with just tetrahedra!
Three-Dimensions: PERIODIC (Closest Packing: ca. 74% Efficiency)
1st Layer: over “A” sites (0, 0)2nd Layer: over “B” sites (1/3, 2/3)
Hand-Outs: 7
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I. Structural Aspects Sphere Packings: Closest Packings Wells, pp. 141-161
Hexagonally Closest PackedHCP
Cubic Closest PackedCCP = FCC
A
B
A
B
A
B
C
A
ABAB
h “h” = “BAB” or “CBC” or …
ABCABC
c “c” = “ABC” or “BCA” or …
JagodzinskiSymbol
c
CoordinationEnvironments
Hand-Outs: 8
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I. Structural Aspects Sphere Packings: Closest Packings Wells, pp. 141-161
Hexagonally Closest PackedHCP
Cubic Closest PackedCCP = FCC
A
B
A
B
A
B
C
A
ABAB
h “h” = “BAB” or “CBC” or …
Unit Cellc-axis: 2 closest packed layers
c/a = (8/3) = 1.633
A: (0, 0, 0); B: (1/3, 2/3, 1/2)
ABCABC
c “c” = “ABC” or “BCA” or …
Unit Cellc-axis: 3 closest packed layers
c/a = ((8/3)(3/2) = 6 = 3 / (1/2)
A: (0, 0, 0); B: (1/3, 2/3, 1/3); C: (2/3, 1/3, 2/3)
JagodzinskiSymbol
c
Hand-Outs: 8
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I. Structural Aspects Sphere Packings: Closest Packings Wells, pp. 141-161
La (DHCP):
B
CA
A
A ABACABAC
Sm: hhc
Examples:
Hand-Outs: 8
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I. Structural Aspects Sphere Packings: Closest Packings Wells, pp. 141-161
La (DHCP):
B
CA
A
A ABACABAC
hchchc
“ hc ”
Sm: hhc = BABACACBCBAB
Examples:
Hand-Outs: 8
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Number of CP Layers in Unit Cell
Number of Different Sequences
Stacking Sequence
Jagodzinski Symbol
2 1 AB h
3 1 ABC c
4 1 ABAC hc
5 1
6 2
7 3
8 6
9 7
10 16
11 21
12 43
I. Structural Aspects Sphere Packings: Closest Packings Wells, pp. 141-161
Exercise: Fill in the Blanks, atLeast for 5-8 layers
Hand-Outs: 9
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I. Structural Aspects Sphere Packings: Packing Efficiencies Wells, pp. 141-161
Body-Centered Cubic Packing
Unit Cell
Efficiency = 2Vsphere / Vcell
Hand-Outs: 10
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I. Structural Aspects Sphere Packings: Packing Efficiencies Wells, pp. 141-161
Body-Centered Cubic Packing
Efficiency = 2Vsphere / Vcell
Vcell = a3
Rsphere = (3/4)a
Vsphere = (4/3)(Rsphere)3 = (3/16)a3
Unit Cell
a
2a
Hand-Outs: 10
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I. Structural Aspects Sphere Packings: Packing Efficiencies Wells, pp. 141-161
CN Name Sphere Density
6 Simple Cubic 0.5236
8 Simple Hexagonal 0.6046
8 + 6 Body-Centered Cubic 0.6802
10 Body-Centered Tetragonal 0.6981
11 Tetragonal Close-Packing 0.7187
12 Closest Packing 0.7405
Body-Centered Cubic Packing
Unit Cell
a
2a
CoordinationEnvironment
Efficiency = 2Vsphere / Vcell
= (3/8) = 0.6802Vcell = a3
Rsphere = (3/4)a
Vsphere = (4/3)(Rsphere)3 = (3/16)a3
Hand-Outs: 10
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I. Structural Aspects Sphere Packings: Interstitial Sites Wells, pp. 141-161
How to Quickly Draw a Closest Packing:
Projection of 2 closest packed planes
Hand-Outs: 11
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I. Structural Aspects Sphere Packings: Interstitial Sites Wells, pp. 141-161
How to Quickly Draw a Closest Packing:
Hand-Outs: 11
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I. Structural Aspects Sphere Packings: Interstitial Sites Wells, pp. 141-161
How to Quickly Draw a Closest Packing:
Hand-Outs: 11
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I. Structural Aspects Sphere Packings: Interstitial Sites Wells, pp. 141-161
Octahedral “Holes” (Voids):
2 closest packed layers: 1 octahedral void / 2 atoms
closest packed layers : 1 octahedral void / 1 atom
Hand-Outs: 11
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I. Structural Aspects Sphere Packings: Interstitial Sites Wells, pp. 141-161
Octahedral “Holes” (Voids):
2 closest packed layers: 1 octahedral void / 2 atoms
closest packed layers : 1 octahedral void / 1 atom HCP: share faces, edges AcBcAcBc CCP: share edges, corners AcBaCbAcBaCb
A
Bc
Hand-Outs: 11
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I. Structural Aspects Sphere Packings: Interstitial Sites Wells, pp. 141-161
Tetrahedral “Holes” (Voids):
2 closest packed layers: 2 tetrahedral voids / 2 atoms
closest packed layers : 2 tetrahedral void / 1 atom
Hand-Outs: 11
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I. Structural Aspects Sphere Packings: Interstitial Sites Wells, pp. 141-161
Tetrahedral “Holes” (Voids):
2 closest packed layers: 2 tetrahedral voids / 2 atoms
closest packed layers : 2 tetrahedral void / 1 atom HCP: share faces, edges AbaBabAbaBab CCP: share edges, corners AbaBcbCacAbaBcbCac
A
Ba
b
Hand-Outs: 11
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I. Structural Aspects Sphere Packings: Radius Ratios Wells, pp. 141-161
Coordination Number
Optimum Radius Ratio
Coordination Polyhedron
4 0.225 Tetrahedron
6 0.414 Octahedron
0.528 Trigonal Prism
8 0.732 Cube
9 0.732 Tricapped Trigonal Prism
12 0.902 Icosahedron
1.000 Cuboctahedron (ccp)
1.000 Triangular Orthobicupola (hcp)
Octahedral Hole
Hand-Outs: 11
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I. Structural Aspects Atomic and Ionic Sizes Wells, pp. 287-291, 312-321, 1286-1295
• Atoms and ions are not “hard spheres;”
• What factors inflence “atomic radii”?
Hand-Outs: 12
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• Atoms and ions are not “hard spheres;”
• What factors inflence “atomic radii”?
(1) Repulsive Forces: approach of uncharged atoms with filled valence subshells(van der Waals radii)
(2) Attractive Forces: effective nuclear charge; orbital overlap; electrostatic(metallic, covalent or ionic radii)
I. Structural Aspects Atomic and Ionic Sizes Wells, pp. 287-291, 312-321, 1286-1295
Hand-Outs: 12
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• Atoms and ions are not “hard spheres;”
• What factors inflence “atomic radii”?
(1) Repulsive Forces: approach of uncharged atoms with filled valence subshells(van der Waals radii)
(2) Attractive Forces: effective nuclear charge; orbital overlap; electrostatic(metallic, covalent or ionic radii)
Scales of Atomic and Ionic Radii:
Slater, Goldschmidt, Pauling – empirical, based on extensive surveys of interatomic distances.
Some corrected for coordination numbers, ionicity, valence bond types, etc.
I. Structural Aspects Atomic and Ionic Sizes Wells, pp. 287-291, 312-321, 1286-1295
Hand-Outs: 12
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Metallic Radii: CN = 12
For ideal cp structures (CCP, HCP with c/a = 1.63): R12 = d / 2 For distorted cp structures: R12 = d / 2 For lower CN: Relative Metallic Radii (Goldschmidt)
CN = 8: R8 = 0.97 R12
CN = 6: R6 = 0.96 R12
CN = 4: R4 = 0.88 R12
I. Structural Aspects Atomic and Ionic Sizes Wells, pp. 287-291, 312-321, 1286-1295
Hand-Outs: 12
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Met
alli
c R
adii
(A
)
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
4th Period
5th Period
6th Period
Group Number
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Coh
esiv
e E
nerg
y (k
J/m
ol)
0
200
400
600
800
M(s) M(g)Filling M-MBonding States
Filling M-MAntibonding States
MinimumRadii
Maximum Cohesive E.
I. Structural Aspects Atomic and Ionic Sizes Wells, pp. 287-291, 312-321, 1286-1295
Hand-Outs: 13
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Metallic Radii: CN = 12
For ideal cp structures (CCP, HCP with c/a = 1.63): R12 = d / 2 For distorted cp structures: R12 = d / 2 For lower CN: Relative Metallic Radii (Goldschmidt)
CN = 8: R8 = 0.97 R12
CN = 6: R6 = 0.96 R12
CN = 4: R4 = 0.88 R12
Estimation Strategies:
(1) Constant Vatom (How to estimate R12 from BCC elements (R8))
FCC vs. BCC: Vatom = (aFCC)3 / 4 = (aBCC)3 / 2
dFCC = 2R12 = aFCC / 2 dBCC = 2R8 = 3 aBCC / 2
Therefore, R8 = 0.972 R12
I. Structural Aspects Atomic and Ionic Sizes Wells, pp. 287-291, 312-321, 1286-1295
Hand-Outs: 12-13
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Metallic Radii: CN = 12
For ideal cp structures (CCP, HCP with c/a = 1.63): R12 = d / 2 For distorted cp structures: R12 = d / 2 For lower CN: Relative Metallic Radii (Goldschmidt)
CN = 8: R8 = 0.97 R12
CN = 6: R6 = 0.96 R12
CN = 4: R4 = 0.88 R12
Estimation Strategies:
(1) Constant Vatom
(2) Use alloys that show close packed structures, e.g., Ag3Sb (HCP) – provides R12(Sb)
I. Structural Aspects Atomic and Ionic Sizes Wells, pp. 287-291, 312-321, 1286-1295
Hand-Outs: 12-13
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Metallic Radii: CN = 12
For ideal cp structures (CCP, HCP with c/a = 1.63): R12 = d / 2 For distorted cp structures: R12 = d / 2 For lower CN: Relative Metallic Radii (Goldschmidt)
CN = 8: R8 = 0.97 R12
CN = 6: R6 = 0.96 R12
CN = 4: R4 = 0.88 R12
Estimation Strategies:
(1) Constant Vatom
(2) Use alloys that show close packed structures, e.g., Ag3Sb (HCP) – provides R12(Sb)
(3) Linear extrapolation of solid solutions of the element in a close packed metal.
I. Structural Aspects Atomic and Ionic Sizes Wells, pp. 287-291, 312-321, 1286-1295
Hand-Outs: 12-13