Amaneh Tasooji

MSE420/514: Session 1Crystallography & Crystal Structure

(Review)

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Crystal Classes & Lattice Types

7 C

ryst

al C

lass

es

4 Lattice Types

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• Rare due to poor packing (only Po has this structure)• Close-packed directions are cube edges.

• Coordination # = 6(# nearest neighbors)

(Courtesy P.M. Anderson)

SIMPLE CUBIC STRUCTURE (SC)

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• APF for a simple cubic structure = 0.52

Adapted from Fig. 3.19,Callister 6e.

ATOMIC PACKING FACTOR

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• Coordination # = 8

7

Adapted from Fig. 3.2,Callister 6e.

(Courtesy P.M. Anderson)

• 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 STRUCTURE (BCC)

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aR

8

• APF for a body-centered cubic structure = 0.68

Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell

Adapted fromFig. 3.2,Callister 6e.

ATOMIC PACKING FACTOR: BCC

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• Coordination # = 12

Adapted from Fig. 3.1(a),Callister 6e.

(Courtesy P.M. Anderson)

• 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 STRUCTURE (FCC)

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Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell

a

10

• APF for a Face-centered cubic structure = 0.74

Adapted fromFig. 3.1(a),Callister 6e.

ATOMIC PACKING FACTOR: FCC

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• Coordination Number (CN)– Number of nearest neighboring atoms, e.g.,

• 8 for inside atom of a BCC• 6 for corner atoms• 12 for each FCC & HCP atoms

• Number of Atoms Per Unit Cell– Determine Total Number of Atom Fraction Shared by

Unit Cell, e.g., • SC: 8 (corner atoms)/8 (shared by 8 unit cells) = 1• BCC: [8/8] + [1 (atom inside unit cell) /1 (shared by 1 unit cell)] = 2• FCC: [8/8] + [6 (atom on unit cell faces) /2 (shared by 1 unit cell)] = 4• HCP: [12/12] + [2 /2] + [3/1] = 6

Summary: Coordination Number & Atoms/Unit Cell

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Crystal Structure & Unit Cell• Densely Packed Atoms are in Lower & More

Stable Energy arrangement– SC: Densely packed along cube axis– BCC: Densely packed along cube body diagonal– FCC: Densely packed along face diagonal

Inter-atomicSpacing, r

Ene

rgy

+ +

Equilibrium r

SC BCC

FCC

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Summary: Atomic Packing Factor (APF)

• Unit Cell Space Occupied (by atoms)Volume of atoms in each cell

Total volume of unit cellExample:

• SC: [1. (4πr3/3)]/[a3]= p/6 = 0.53 ; (a=2r)• BCC: [2 . (4πr3/3)]/[(4/√3 r)3] = 0.68 ; (a= 4/√3 r)• FCC: [4 . (4πr3/3)]/[(2√2 r)3]= 0.74 ; (a=2√2 r)• HCP: 0.74

APF =

SC BCC FCCa 2r 4/√3 r 2√2 rFD √2 a √2 a √2 aBD √3 a √3 a √3 aat/UC 1 2 4CN 6 8 12APF 0.53 0.68 0.74

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Interstitial Sites

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Interstitial Sites

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Crystal Notations

https://www.doitpoms.ac.uk/tlplib/miller_indices/printall.php

https://www.doitpoms.ac.uk/tlplib/crystallography3/index.php

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Crystallographic Notations: Coordinates

• Atom Coordinates– Locating Atom Position in Unit Cell– Point in space, coordinates in ref. to

origin(1,1,1)

(1,0,0)

O

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Crystallographic Notations: Direction Indices

• [uvw] & <uvw>– Id. coordinate w.r.t. origin

– Transform to integers

* Lattice vector in a,b,c direction

– All parallel direction vectors have the same direction indices

– “Crystallographically equivalent” directions (same atom spacing along each direction) are designated with <uvw> direction family

0,1,½[021]

1,0,0[100]

-1,-1,0[110]- -

0,0,0

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Crystal directions

Crystal directions are defined in the following way, relative to the unit cell.

1) Choose a beginning point (X1, Y1, Z1) and an ending point (X2, Y2, Z2), with the position defined in terms of the unit cell dimensions.Beginning point: (X1, Y1, Z1): (1, 1, 0)Ending point: (X2, Y2, Z2): (1/2, 0, 1)

2) Calculate the differences in each direction, ΔX, ΔY, ΔZ. ΔX, ΔY, ΔZ : (-1/2, -1, 1)

3) Multiply the differences by a common constant to convert them to the smallest possible integers u, v, w (u, v, w) : (-1, -2, 2)

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Crystallographic Notations: Plane Indices

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Crystallographic Notations: Plane Indices

• Miller Indices, (hkl) & {hkl} family

1. Select an appropriate origin2. Id. Intercept with axis (INTERCEPT)3. Determine Reciprocal of intercept (INVERT)4. Clear fractions to smallest set of whole numbers (INTEGER)

• Equivalent lattice planes related by symmetry of the crystal system are designated by {hkl}family of planes

• In cubic system:- [abc] direction ⊥ (abc) plane

•

∞,-1,½0,-1,2(012)-

∞,-1, ∞ (or ∞,1, ∞)0 -1 0 (or 0 1 0)(010) (or (010) )

Reciprocal of Intercepts

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Miller Plane IndicesPlane: (hkl), or plane family {hkl}Methodology for determining Miller

Indices• Identify an Origin• Id. Plane Intercept with 3-Axis• Invert the Intercepts• Clear Fractions to Lowest IntegersOverbars Indicate < 0, e.g. (214)

1 1 1x y zercept ercept erceptint int int

x

y

z

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Hexagonal Closed Packed (HCP)

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HCP Indices

previous chart

previous chart

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Hexagonal Closed Packed (HCP)

_ _[2120]

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Plane Indices in Hexagonal System

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Summary• Many materials form crystalline structure• Material properties are influenced by Atomic Packing &

Crystalline structure• Crystal notations (direction indices & plane/Miller

indices), a communication tool for material scientists & engineers– Directions: [uvw] ; except [uvtw] for HCP– Planes: (hkl) ; except [hkil] for HCP– Families of direction (<>) and planes ({}) are associated with

groups of direction & planes which are crystallographically equivalent (similar atomic arrangement