Closest Packing. Closest Packing

Closest Packing Atoms can be treated approximately as hard spheres for many types of bonding. – + d Covalent bond d Ionic bond d Metallic bond Di...
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Closest Packing Atoms can be treated approximately as hard spheres for many types of bonding. –

+

d Covalent bond

d Ionic bond

d Metallic bond

Different types of bonding

The absence of preferred directional bonding leads to closest packing. packing.

Closest Packing CloseClose-packing – Most efficient way of packing equalequal-sized spheres

Primitive packing # near neighbors = 4

Close packing # near neighbors = 6

Which packing is most efficient?

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Closest Packing Packing spheres in 3D Space B

A t+ t o + t– t– t+ t+ o o t– t+ t+

o t+

t+ t–

o t– t+

t+

What is the ratio of tetrahedral to octahedral sites? 2(t) to 1(o) per packing sphere (2t is 1 t+ + 1 t–) Tetrahedral cavity

Octahedral cavity

Closest Packing Packing spheres in 3D Space – 3rd layer B c a

a c

c a

a c

A

c a

The next layer can pack on top of either the a or c sites If a, the ABAB pattern is called Hexagonal Close Packed (hcp) hcp) If c, the ABCABC pattern is called Cubic Close Packed (ccp) ccp)

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Closest Packing The Unit Cell of Cubic Close Packing (ABCABC)

[111] [110] a [001] A [100] c C B Face centered cubic

[010] b

The close packed layers stack along the body diagonal of the cubic unit cell - [111] direction

Closest Packing The Unit Cell of Hexagonal Close Packing (ABAB) A

B c [010] [001] b A Hexagonal

[100] a

The close packed layers stack along the [001] direction

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In-Class Problem Set Calculate the shortest UU-U distance for the 2D projection of the structure of U3Si2 (a = b = 7.33Å):

y x

Positions: (x, y) U = 0, 0 0.5, 0.5 0.182, 0.682 0.818, 0.318 0.318, 0.182 0.682, 0.818

The last four positional coordinates are related via a 44-fold rotational axis. Give the operation upon the general position (x,y) Æ (±x,± x,±y) which leads to each coordinate. coordinate.

Structures With CCP Packing The Unit Cell of Cubic Close Packing (ABCABC)

Octahedral sites in the faceface-centered cubic (ccp (ccp)) unit cell are located in the center and on every edge What is the # of octahedral sites per unit cell? 1 + ¼(12) = 4 What is the coordination number of each atom? Na = 6, Cl = 6

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Structures With CCP Packing The Unit Cell of Cubic Close Packing (ABCABC) (1,1,1)

(0,0,1)

(½,½, ½) (½,½,0)

c b (0,0,0)

a

(½,0,0)

(1,0,0) Atomic positions are described by fractional coordinates. If all octahedral sites are filled, the structure is NaClNaCl-type. type. If every other layer of octahedral sites are filled, the structure structure is CdCl2-type. type.

Structures With CCP Packing Some compounds with the NaCl structure type, a (Angstroms) MgO CaO SrO BaO

4.213 4.8105 5.160 5.539

MgS CaS SrS BaS

5.200 LiF 5.6948 LiCl 6.020 LiBr 6.386 LiI

4.0270 5.1396 5.5013 6.00

KF KCl KBr KI

5.347 6.2931 6.5966 7.0655

Unit cell increases with: 1) Increasing cation size 2) Increasing anion size size

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Structures With CCP Packing The Unit Cell of Cubic Close Packing (ABCABC)

Polyhedral drawing Tetrahedral sites in the faceface-centered cubic (ccp (ccp)) unit cell are located in each of the eight octants. What is the # of tetrahedral sites per unit cell? 4 T+, 4 T– What is the coordination number of each atom? Yellow, Blue = 4; Red = 8

Structures With CCP Packing The Unit Cell of Cubic Close Packing (ABCABC)

Polyhedral drawing What ½ of the tetrahedral sites (either all T+ or all T–) are filled, the structure is ZnS (sphalerite) sphalerite)-type. type. When all of the tetrahedral sites are filled, the structure is K2Otype (or CaF2; antianti-type)

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Structures With CCP Packing Some compounds with the ZnS (sphalerite) sphalerite) structure type, a (Angstroms) CuF 4.255 CuCl 5.416

BeS BeSe

4.862 5.070

CdS 5.818 CdSe 6.077

BN 3.616 GaAs 5.653

Some compounds with the K2O structure type, a (Angstroms) SrF2 5.800 SrCl2 6.977

PbO2 5.349 Li2O 4.611 K2O 6.449 CeO2 5.411 Li2S 5.710 K2S 7.406

C (diamond) is fcc

Structures With HCP Packing Octahedral sites of Hexagonal Close Packing (ABAB) A B

t+

B t+

c a

b

o

t–

A

t+

t+

A View from above

Unit cell Octahedral sites (2) are located in each unit cell -- They form faceface-sharing chains What is the coordination number of the packing sphere?

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Structures With HCP Packing Octahedral sites of Hexagonal Close Packing (ABAB) A

B c b

a

A Polyhedral drawing

Unit cell If all octahedral sites are filled, the structure is NiAs structurestructure-type If ½ octahedral sites are filled, the structure is CdI2 structurestructure-type What is the coordination geometry of the anions? Cations? Cations?

Structures With HCP Packing Tetrahedral sites of Hexagonal Close Packing (ABAB) A B

t+

B t+ a

b

o

t–

A

t+

t+

A View from above

Unit cell Tetrahedral sites are located within the unit cell (1T+, 1T–) and on each edge (1T+, 1T– per edge). -- A face/vertex sharing chain is formed when all tetrahedral sites are filled

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Structures With HCP Packing Tetrahedral sites of Hexagonal Close Packing (ABAB) A

B

b

a

A

Polyhedral drawing

Unit cell Filling ½ the tetrahedral sites (either T+ or 1T–) gives the ZnS (wurtzite) wurtzite) structure type. Fill all tetrahedral sites results in unfavorable cation/ cation/cation repulsion Æ No analogue to ccp K2O type.

Structures With HCP Packing Some compounds with the NiAsNiAs-type, a c (Angstroms) NiS NiSn FeSe FeTe

3.439 5.348 4.048 5.123 3.637 5.958 3.800 5.651

PtB PtSn CoSb MnSb

3.358 4.103 3.866 4.120

4.058 5.428 5.188 5.784

Solids with this structure type tend to be metallic Ideally c/a = 1.633; A smaller ratio will have shorter metalmetalmetal bonds Some compounds with the ZnS (wurtzite) wurtzite) structure type, a ZnO ZnTe BeO MnS

3.250 4.270 2.698 3.976

5.207 6.990 4.380 6.432

AgI 4.580 AlN 3.111 GaN 3.180 SiC 3.076

7.494 4.978 5.166 5.048

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Closest Packing (Summary) Common Structures and Types Anion Packing ccp

hcp

Interstitial Sites T+ T– O

Coordination Cation Anion

Examples

0

0

1

6

6

NaCl

1

0

0

4

4

ZnS (sphalerite)

1

1

0

4

8

K2O, CaF2

0

0

1/2

6

3

CdCl2

0

0

1

6

6

NiAs

1

0

0

4

4

ZnS (Wurtzite)

0

0

1/2

6

3

CdI2

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