00 Crystallography Basics

8/14/2019 00 Crystallography Basics

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Brief introduction to Crystallography

Crystallography Structure Symmetry d-spacing Defines results one achieves with XRD, TEM, SEM, etc.


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Crystalline vs. Amorphous Materials

Crystalline material atoms are situated in aperiodic array over large distances.

Amorphous (or non-crystalline) material where

long range order is absent. Therefore no realsymmetry.


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Close packed directions are cube edges.

SIMPLE CUBIC STRUCTURE (SCC)

Coordination # = 6


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Coordination # = 8

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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(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)

Adapted from Fig. 3.1(a),

Callister 6e.

Coordination Number = 12 Atomic packing factor

(APF) = 0.74 (ideal)


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ABCABC... Stacking Sequence along cube diagonal 2D Projection:

A sites

B sites

C sites

B B

B

BB

B BC C

C

A

A

AB

C

FCC STACKING SEQUENCE

Coordination #= 12


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Coordination # = 12

ABAB... Stacking Sequence

Only difference from FCC is in the stacking (and the

orientation we generally consider when describing/drawing

the stacking):

- ABAB along z-axis, instead of ABCABC along cube

diagonal

3D Projection 2D Projection

A sites

B sites

A sites Bottom layer

Middle layer

Top layer

Adapted from Fig. 3.3,

Callister 6e.

HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)


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Lattices and Unit Cells

What if we want to consider more complicated crystals?

Take advantage of symmetry and well known structures

A lattice is a repeating array of special points in space that haveidentical surroundings (nearest neighbors, etc).

A unit cell defines a repeating unit within the lattice strictly the unit cell is the smallest repeating cell (though this is

sometimes totally impractical and thus more convenient, but larger,

unit cells are employed).

A lattice does not define the crystal, only its symmetry.

Atoms (or ions, molecules, etc.)will be placed at any number of

basis points at given distances/directions (i.e. vectors) from thelattice points. ONLY THEN do we have a crystal.


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2-d Lattices

A lattice repeats in all directions,

and defines the symmetry.

The unit cell is the simplest

repeating structure.

Square

unit cell


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Basis

BUT, a lattice only identifies repeating points and

their symmetry.

Whats the lattice for the wall or floor in the

classroom?

Must be combined with a basis to know where

atoms are in the crystal

What kind of bricks are hung at each lattice point?

Are there smaller stones patterned regularly in

between the bricks?

Only by knowing a lattice AND a basis can a crystal

structure (or anything else symmetric) truly be built.


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2-d Lattice and Basis

Anions at lattice positions (0,0) Cations at (,0)

Cations at (0,)

squareLattice:

Basis:

tetragonal


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Lattice:

A minimized pattern that will be followed

Basis:

a) A vector that defines the position of hooks onwhich something (atoms, molecules, bricks) will behung with relation to the underlying pattern(lattice)

and b) What exactly will be hung on each hook(atom, molecule)

Rules:

At least one hook (basis) per lattice point (but can be more than 1)

Basis vectors must put atoms within the unit cell(not in an adjacent cell)

Each basis vector applies to every single latticepoint


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Seven Crystal Systems

Cubic a=b=c ===90

Tetragonal a=bc ===90

Orthorhombic abc ===90

Hexagonal a=bc ==90 =120

Monoclinic abc ==90

Triclinic abc

Rhombohedral a=b=c ==90

In 3 dimensions, there are only 7 basic crystal systems.Easily defined by: 3 edge lengths (a, b and c) and 3

interaxial angles ( and).

This

class


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Cubic Bravais Lattices

Why no base centered cubic?


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/

Tetragonal

a=bc

/

Why no base centered tetragonal?

Tetragonal Bravais Lattices


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Orthorhombic Bravais Lattices

/ /

/ // /

b

/ /

/ /


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Other Bravais Lattices

a=bc

90

=120


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Fourteen Bravais Lattices (13-14)


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Defining Basis Positions Using

Fractional Co-ordinates

Note that positions are indicated as ( , , ).To locate atom positions in cubic unit cells we use

rectangular x, y, and zaxes.


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The FCC Structure

Note: All atoms are not always

shown because technically thereare only 4/unit cell (for FCC)

Lattice=Face Centered Cubic

Basis=red atoms at (0,0,0)

-of course it looks like there are more atoms, but these simply result

from the repeating lattice and symmetry; ie they are not distinct

Each position might be provided, but probably the fewest

number of positions will be giventhe rest fill in as long asyou know the Bravais lattice (ie the basic symmetry).


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Simple unit cells

Based on Simple Cubic

Lattice with a basis of:

1) anions at the SC points

2) cations at the center

[offset by (, , )]

Based on FCC Lattice with a basis of:

1) anions at the FCC points

2) cations at the center

[offset by (, 0, 0)]


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Consider CaF2 : based on the CsCl structure.

BUT: there are half as many Ca2+ as F- ionshow can

we accommodate this?

CsCl structure w/only half of cation sites occupied.

The initial unit cell is no longer valid since every lattice

site doesnt have a basis attached to it (some are void).

Instead, we use a supercell. 2 possibilities here:

AmXp STRUCTURES


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AmXp STRUCTURES Consider CaF

2

: Based on the CsCl structure,

BUT: there are half as many Ca2+ as F- ions

This is accommodate with empty cubes.

i.e. CsCl based structure w/only half of cation sites

occupied. Thus, the proposed CsCl unit cell is no longer valid

since every Simple Cubic lattice site doesnt have a

basis attached to it (half are void!). Instead, use

a supercell.

Other

possibilities:

m is not equal to p

Adapted from Fig. 12.5, Callister 6e.


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Crystallographic Directions and Planes

Terminology

Positions (,,,)

Directions [] note no commas

Planes () note no commas

Families of directions note no

commas

Families of planes {} note no commas

100)=bar one zero zero and NOT one bar zero zero


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Directions in Cubic Unit Cells

Position co-ordinates of vector OM = 1, 0.5, 0 (but these arent

integers)

Position co-ordinates must be multiplied by 2 to obtain integers.

Direction indices of OM = 2 (1, 0.5, 0) = [2, 1, 0]

A negative index direction (e.g. see vector ON) is written with a

bar over the index

For cubic crystals, the crystallographic direction indicesare the vector components of the direction resolved along

each of the co-ordinate axes (x,y,z)

and reduced to the smallest integers.


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[Directions]

[100] [110]

[111]

[021] [011]

[200]

[210]

x

y

z

1. Draw cell + origin

2. Draw vector

3. reduce to

smallest integer

values

4. [xyz]

Group exercise

[Di ti ]


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more [Directions]

[100]

[011]

[011]

xy

z

For negative directions:

a. Add more unit cells.

OR

b. Shift the origin.


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Directions in Cubic Unit Cellsz

y

x

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

[001 ] [01 1 ][11 1]

[112 ][2 1 2]

[22 1 ]

The origin may be shifted. Thus:

a) If all positive, origin is at(0,0,0) b) If z is negative, origin will be at (x,y,1)

c) If y is negative, origin will be at (x,1,z) d) If x is negative, origin will be at (1,y,z)

Group exercise

f il f di ti


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=family of directions

[100]

[010]

[001]

[100]

[010]

[001]

xy

z

A includes all possible

directions with the same basic coordinates.

1 D h i i ll


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(Planes)

(xyz)

(100)

(110)

(111) (100)

(020)

(040) (04/30)

()

x

y

z1. Draw the origin, cell,

and normal vector.

2. Draw the plane at adistance from the origin

of 1/sqrt(a2+b2+c2).


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{Planes} =family of planes

{xyz}

{100}

{110}

{111}

{100}

x

y

z

How many

equivalentplanes are there

in each family

(multiplicity)?

multiplici

ty

3

6

?

?


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For a simple cubic lattice and (0,0,0) basis:

Along (xyz) plane

(100)

(110)

xy

z

Should be able to draw atoms in, and above/below,

a given plane in a crystal.

For CsCl (simple cubic lattice

Group exercise


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For CsCl (simple cubic lattice,

basis of Cl-(0,0,0) and Cs-

(,)) Along (xyz)plane

(100)

(110)

xy

z

Group exercise


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Crystallographic Planes Miller Indices

Intercept; 1, ,

Reciprocal; 1/1, 1/, 1/

Simplify; 1, 0, 0

Miller indices; (1 0 0)

1,1,

1/1,1/1,1/

1,1,0

(1 1 0)

1,1,1

1/1, 1/1, 1/1

1,1,1

(1 1 1 )


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Crystallographic Planes Miller

Indices

When planes are more complicated, follow these

rules:

1. Choose a plane that does not

pass through the origin at

(0,0,0)

2. Determine the intercepts of the

plane in terms of the x, y, and z

axes of the cube. 1/3,2/3,1

3. Form the reciprocals of these

intercepts 3, 3/2, 1

4. Bring to smallest integers by

multiplying or dividing through

by a common factor. 6, 3, 2

5. Enclose integers in parentheses

(6 3 2)

Group exercise


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Planes in Cubic Unit Cellsz

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

x

y

(010)(11 0)

(111 )

(012 ) (112) (342 )

Group exercise

00 Crystallography Basics

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