i. structural aspects sphere packingswells, pp. 141-161 densest packing of spheres two-dimensions:...

I. Structural Aspects Sphere Packings Wells, pp. 141-161

Densest Packing of Spheres

Two-Dimensions: Unit Cell

Hand-Outs: 7

I. Structural Aspects Sphere Packings: Geometry Wells, pp. 141-161


Two-Dimensions: PERIODICa

a

a = side of unit cell (Å, pm) = angle between 2 sides =120

“Coordinate System”

Unit Cell

Hand-Outs: 7



Two-Dimensions: PERIODIC

“A”

aa

Unit Cell

A: (0, 0)


Hand-Outs: 7




“A”

aa

Unit Cell

A: (0, 0)

B: (1/3, 2/3)

“B”


Hand-Outs: 7




“A”

aa

Unit Cell

A: (0, 0)

B: (1/3, 2/3)

C: (2/3, 1/3)

“Fractional Coordinates”

“B”

“C”


Hand-Outs: 7

I. Structural Aspects Sphere Packings: Closest Packings Wells, pp. 141-161


Three-Dimensions: Tetrahedron, ca. 79% Efficiency

70.5

Hand-Outs: 7




70.5

CANNOT fill 3D space with just tetrahedra!

Hand-Outs: 7




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


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


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


La (DHCP):

B

CA

A

A ABACABAC

Sm: hhc

Examples:

Hand-Outs: 8


La (DHCP):

B

CA

A

A ABACABAC

hchchc

“ hc ”

Sm: hhc = BABACACBCBAB

Examples:

Hand-Outs: 8

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


Exercise: Fill in the Blanks, atLeast for 5-8 layers

Hand-Outs: 9

I. Structural Aspects Sphere Packings: Packing Efficiencies Wells, pp. 141-161

Body-Centered Cubic Packing

Unit Cell

Efficiency = 2Vsphere / Vcell

Hand-Outs: 10




Vcell = a3

Rsphere = (3/4)a

Vsphere = (4/3)(Rsphere)3 = (3/16)a3

Unit Cell

a

2a

Hand-Outs: 10


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


Unit Cell

a

2a

CoordinationEnvironment


= (3/8) = 0.6802Vcell = a3

Rsphere = (3/4)a

Vsphere = (4/3)(Rsphere)3 = (3/16)a3

Hand-Outs: 10

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


How to Quickly Draw a Closest Packing:

Hand-Outs: 11


Octahedral “Holes” (Voids):

2 closest packed layers: 1 octahedral void / 2 atoms

closest packed layers : 1 octahedral void / 1 atom

Hand-Outs: 11


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


Tetrahedral “Holes” (Voids):

2 closest packed layers: 2 tetrahedral voids / 2 atoms

closest packed layers : 2 tetrahedral void / 1 atom

Hand-Outs: 11


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

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

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



(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)


Hand-Outs: 12



(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.


Hand-Outs: 12

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


Hand-Outs: 12

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.


Hand-Outs: 13



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


Hand-Outs: 12-13



CN = 8: R8 = 0.97 R12

CN = 6: R6 = 0.96 R12

CN = 4: R4 = 0.88 R12


(1) Constant Vatom

(2) Use alloys that show close packed structures, e.g., Ag3Sb (HCP) – provides R12(Sb)


Hand-Outs: 12-13



CN = 8: R8 = 0.97 R12

CN = 6: R6 = 0.96 R12

CN = 4: R4 = 0.88 R12


(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.


Hand-Outs: 12-13

i. structural aspects sphere packingswells, pp. 141-161 densest packing of spheres two-dimensions:...

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