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Principles of Atomic Packing p. 5.1

Many crystal structures can be describedas stacking arrangements of two-dimensional lattices,

or nets , like these:

16 lattice pointsin gray square

Square netClose-packed net,

a.k.a. hexagonal net

18.475 lattice pointsin gray square

×B

×C

×B

×C×A

×B

×C

a

a

a

a

∴ For equal lattice parameters, the close-packed net con-

tains more lattice points per unit area than the square net

⇒more efficient packing

EMSE 201 - Introduction to Materials Science & Engineering © 2003 Mark R. De Guire rev. 01/29/03

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STRUCTURES BASED ON THE SQUARE NET

• Simple cubic (SC) — Stack square net layersso that lattice points lie directly above each other

• One atom per unit cell (⇔ one atom per lattice point

in the SC Bravais lattice)

• Atoms touch along cell edge ⇒ a = 2r (r: atomic radius

• Atomic packing factor (APF)

• ≡ volume of atoms in unit cellvolume of unit cell

• For SC, APF = 52%

=4πr3/3(2r)3

• Each atom has six nearest neighbors

• Interstice, halfway between layers, is surrounded byeight atoms in cubic coordination

• Important as basis for other structures

Callister, Fig. 3.19 — simple cubic (hard sphere)


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STRUCTURES BASED ON THE SQUARE NET (cont.)

• Body-centered cubic (BCC) — Stack alternate layersover “holes” in bottom square net (× in left diagram, p. 5.1)

• Two (total) atoms per unit cell(⇔

one atom per lattice point in BCC Bravais lattice)

• Atoms touch along <111> (i.e., the body diagonals

— not the cell edges!) ⇒ a =4r

3

• APF = 68%

=

2 × 4πr3/3

(4r/ 3)3

• Each atom has eight nearest neighbors

• Examples:

-Fe (low-T form), Mo, W (also V, Cr, Nb, Ta, K, Na)


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STRUCTURES BASED ON THE SQUARE NET (end)

Callister, Fig. 3.2 — BCC (3 versions)


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STRUCTURES BASED ON THE CLOSE-PACKED NET:

Hexagonal Close-Packed (HCP) & Cubic Close-Packed (CCP)

1) Bottom layer: “A”

2) Place successive layers overthe “B” or “C” positions

(labeled on p. 5.1, right diagram)

• Characteristics shared by HCP and CCP:

• Every atom has twelve nearest neighbors

• 6 in same layer, forming a hexagon

• 3 in layer above

• 3 in layer below

• APF = 74% ← maximum packing efficiencyfor equal-sized spheres

• Interstices per atom :

• Two tetrahedral(4 neighbors)

• One octahedral(6 neighbors)


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STRUCTURES BASED ON THE CLOSE-PACKED NET(cont.)

A

A

A

A A A

A A A A

B B B

A A A A A

B B B B

CC C C

C C C C C

(Callister, Figure 3.13)

A

A

A

A A A

A A A A

A A A A A

B B B

B B B B

CC C C

C C C C C

B

B

B

B

B B B BB BB

Layers stacked ABABAB… ⇒ hexagonal close-packed (HCP)

Layers stacked ABCABC… ⇒ cubic close-packed (CCP)


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Interstices between the atoms in close-packed structures:

• Tetrahedral (surrounded by four atoms)

• Octahedral (surrounded by six atoms)

Tetrahedralinterstice

Octahedralinterstice

Callister, Figure 12.7 — oct’l & tet’l interstices

• Exist in both HCP and in CCP structures• Ratio of interstices to lattice points is always the same:

• 2 tetrahedral interstices

• 1 octahedral interstice per lattice point


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• Hexagonal close-packed (HCP)

• Stacking sequence: ABABAB…

• Stacking direction: c-axis of the hexagonal unit cell

• Close-packed planes ≡ basal planes of the unit cell

• 2 atoms per unit cell:0,0,0 (corner positions)

23,

13,

12 (interior position)

(Bravais lattice: hexagonal; 2 atoms per lattice point)

• Interstices (per cell):

• 4 tetrahedral:

0,0, 38 0,0,58

23, 1

3, 18

23, 1

3, 78

• 2 octahedral:

13,

23,

14

13,

23,

34

• In ideal HCP, with atomic radius r,

• a = 2r • c = 2 23a

• Examples: Mg, Ti, Zn, Be, Co, Zr, Cd


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STRUCTURES BASED ON THE CLOSE-PACKED NET (cont.)

• Cubic close-packed (CCP)

• Stacking sequence: ABCABC…

• Stacking direction: [111] in FCC lattice

• Close-packed planes: {111}

• 4 atoms per unit cell:

0,0,012,

12,0

12,0,

12 0,

12,

12

Bravais lattice: face-centered cubic, one atom perlattice point

• Interstices (per CCP cell):

• 8 tetrahedral:

14,

14,

14

34,

14,

14

14,

34,

14

34,

34,

14

14,

14,

34

34,

14,

34

14,

34,

34

34,

34,

34

• 4 octahedral

• One at cell center: 12

,12

,12

• Three at middle of cell edges:

12

,0,0 0,12

,0 0,0,12

• Atoms touch along face diagonals ⇒ a = 2 2r

• Examples:-Fe (high-T form), Al, Ni, Cu, Ag, Pt, Au, Pb


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Callister, Fig. 3.1 — FCC (3 views)

aa

a

octahedral interstices tetrahedral interstices


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STRUCTURES BASED ON THE CLOSE-PACKED NET (end)

• Diamond cubic • Bravais lattice: FCC with alternate tetrahedral

interstices filled, e.g.:14,

14,

34

34,

14,

14

14,

34,

14

34,

34,

34

• Each atom has four nearest neighbors

• 8 atoms per unit cell

• Atoms touch along body diagonal ⇒ a = 83

r• APF = 34%

=8×4πr3/3

(8r/ 3)3

• Examples: C (diamond), Si, Ge, -Sn (gray tin)

Callister, Fig. 12.15 — diamond cubic

aa

a


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