• Metal Crystal Structures

• Lattice Structure and Crystal

Systems

• Crystallographic Directions and

Planes 1

LEARNING OBJECTIVE By the end of this chapter, students will be able to …..

Describe the difference in atomic/molecular structure between

crystalline and noncrystalline materials.

Draw unit cells for face-centered cubic, body-centered cubic, and

hexagonal close-packed crystal structures.

Recognize and also give the lattice parameter relationships for all

seven crystal systems--i.e., cubic, hexagonal, tetragonal,

rhombohedral, orthorhombic, monoclinic, and triclinic.

Given three index integers, sketch the direction corresponding to

these indices within a unit cell (for all crystal systems).

Given a direction that has been drawn referenced to a unit cell (for all

crystal systems), specify its direction indices.

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Given a unit cell and the Miller indices for a plane, draw the plane

represented by these indices referenced to this unit cell.

Specify the Miller indices for a plane that has been drawn within a

unit cell.

Given the unit cell for some crystal structure, be able to draw the

atomic/ionic packing arrangement for a specific crystallographic

plane.

Distinguish between single crystals and polycrystalline materials.

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FUNDAMENTAL CONCEPTS

Atoms are considered as solid spheres, with well-

defined diameters. This is termed atomic hard

sphere model.

Atoms self-organize in crystals, most of the time

Crystalline materials – a material in which atoms

are situated in a repeating / periodic array over large

atomic distance. Long range order exists

The properties of crystalline solids depend on crystal

structure – the manner in which atoms are spatially

arranged

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Crystalline lattice – a three-dimensional array of

points coinciding with atom positions (or sphere

centers)

When the solid is not crystalline, it is called

amorphous (e.g. glass and most plastics)

Unit cell – small groups of atoms that form a

repetitive pattern (small repeat entities)

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2.1 METAL CRYSTAL STRUCTURES

The most common types of crystal structures found in

metals:

faced-centered cubic (FCC)

body-centered cubic (BCC)

hexagonal close-packed (HCP)

Important characters of a unit cell are:

cell dimensions

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number of atoms per unit cell (n). For an

atom that is shared with m adjacent unit

cells, we only count a fraction of the atom,

1/m

the coordination number, which is the

number of closest neighbours to which an

atom is bonded

the atomic packing factor (APF), which is

the fraction of the volume of the cell

actually occupied by the hard spheres

𝐴𝑃𝐹 =𝑆𝑢𝑚 𝑜𝑓 𝑎𝑡𝑜𝑚𝑖𝑐 𝑣𝑜𝑙𝑢𝑚𝑒

𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑐𝑒𝑙𝑙

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Face-Centered Cubic Crystal Structure

The FCC crystal structure has a unit cell of cubic

geometry

Atoms are located at each of the corners and the

centers of all the cube face

Example of metals having this crystal structure is

copper, aluminum, silver and gold

The cube edge length a and the atomic radius R are

related through

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For the face-centered cubic

crystal structure:

a) a hard sphere unit cell

b) a reduced-sphere unit cell

c) an aggregate of many

atoms

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The number of atoms per unit cell for FCC crystal structure

is 4 atoms (each corner atom is shared among eight unit

cells, a face-centered atom belongs to only two unit cells)

The coordination number is 12 (4 atoms at the corner + 4

face atoms from behind + 4 face atoms at the next unit cell

to the front)

The atomic packing factor (APF) is 0.74 (maximum packing

possible for spheres)

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For the face-centered cubic crystal structure:

a) a hard sphere unit cell

b) a reduced-sphere unit cell

c) an aggregate of many atoms

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The number of atoms per unit cell for BCC crystal

structure is 2 atoms (each corner atom is shared

among eight unit cells, the single center atom is

wholly owned by this unit cells)

The coordination number is 8 (8 atoms at the

corner)

The atomic packing factor (APF) is 0.68

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Hexagonal Close-Packed (HCP)

The HCP crystal structure has a unit cell of hexagonal

The top and bottom faces of the unit cell consist of six

atoms that form regular hexagons and surround a

single atom in the center. Another plane consists of 3

atoms is in between the top and bottom planes.

Example of metals having this crystal structure is

cadmium, magnesium, titanium, zinc

The long (C) and short (a) unit cell dimensions are

related through

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The number of atoms per unit cell for HCP crystal

structure is 6 atoms (one- sixth of the top face

corner atom, one-sixth of the bottom face corner

atom, one-half of each of the 2 center face atoms

and all 3 midplane interior atoms)

The coordination number is 12 The atomic packing

factor (APF) is 0.74

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Polymorphism and Allotropy

Some materials may exist in more than one crystal

structure, this is called polymorphism. If the

material is an elemental solid, it is called allotropy.

The changes of crystal structure depend on both

temperature and pressure. An example of allotropy is

carbon, which can exist as diamond, graphite and

amorphous carbon.

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2.2 LATTICE STRUCTURE AND CRYSTAL

SYSTEMS

Crystal Systems

Crystal structures are divided

into groups according to unit

cell configurations and/or

atomic arrangements

Based on unit cell geometry 6

lattice parameters are

assign to unit cells (the three

edge lengths a, b, and c & the

three interaxial angles α, β,

and γ )

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Based on the lattice parameters crystals have seven

different possible combination which represents a

distinct crystal system (cubic, tetragonal, hexagonal,

orthorhombic, rhombohedral, monoclinic and triclinic)

Cubic system (a = b = c and α = β = γ = 90°) – the

greatest degree of symmetry

Triclinic system (a ≠ b ≠ c and α ≠ β ≠ γ) – the least

symmetry

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Question: What is the difference between crystal

structure and crystal system?

Answer: A crystal structure is described by both the

geometry of, and atomic arrangements within, the

unit cell, whereas a crystal system is described only

in terms of the unit cell geometry. For example, face-

centered cubic and body-centered cubic are crystal

structures that belong to the cubic crystal system.

Question : What is Fourteen Bravais Lattice?

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2.3 CRYSTALLOGRAPHIC DIRECTIONS AND

PLANES

Crystallographic Directions

Crystallographic direction – line between two points

or a vector

The steps to determine crystallographic direction are:

A vector of convenient is positioned such that it passes

through the origin of the coordination system. Any vector may

be translated throughout the crystal lattice without

alteration, if parallelism is maintained

The length of the vector projection on each of the three axes is

determines; these are measured in terms of the unit cell

dimensions a, b, and c

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These three numbers are multiplied or divided by a common

factor to reduce them to the smallest integer values

The three indices, not separated by comma, are enclosed in

square brackets, thus: [uvw]. The u,v,w integers correspond

to the reduced projections along the x, y and z axes,

respectively

Negative indices are possible, represented by a bar over

appropriate index e.g. [1 1 1] has –y direction

Changing the signs of all indices results in antiparallel

direction e.g. [1 1 1 ] opposite to [1 1 1]

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Crystallographic Planes

Crystallographic planes are specified by three Miller

indices as (hkl)

Any two planes parallel to each other are equivalent

and have identical indices

The steps to determine Miller indices are:

If the plane passes the selected origin, either another parallel

plane must be constructed within the unit cell by an

appropriate translation, or new origin must be established at

the corner of another unit cell

At this point the crystallographic plane either intersects or

parallels each of the three axes; the length of the planar

intercept for each axis is determined in terms of the lattice

parameters a, b and c

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The reciprocals of these numbers are taken. A plane that

parallels an axis may be considered to have an infinite

intercept, and, therefore, a zero index

If necessary, these three numbers are changed to the set of

smallest integers by multiplication or division by a common

factor

Finally, the integer indices, not separated by commas, are

enclosed within parentheses, thus (hkl)

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Representation of a series each of (a) (001), (b) (110), and (c) (111) crystallographic planes

Atomic Arrangements

The atomic arrangement for a crystallographic plane

depends on the crystal structure

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(a) Reduced sphere FCC unit cell with (110) plane.

(b) Atomic packing of an FCC (110) plane

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(a) Reduced sphere BCC unit cell with (110) plane. (b) Atomic packing of an BCC (110) plane

Crystalline and Non-Crystalline Materials

Single Crystals

Single crystal – crystalline solid with perfect

arrangement of atoms or atoms extends throughout

the entirety of the specimen without interruption. It

has a regular geometric structure with flat faces

Polycrystalline Materials

A solid can be composed of many crystalline grains, not

aligned with each other. It is called polycrystalline.

The grains can be more or less aligned with respect to

each other. Where they meet is called a grain

boundary

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Various stages in the solidification of a

polycrystalline material

1) small crystals or nuclei form at various positions with

random crystallographic orientations

2) small grains grow by addition from the surrounding

liquid of atoms to the structure

3) the grains impinge on one another at the end of

solidification

4) The grains meet at grain boundary

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Schematic diagrams of the various stages in the solidification of a polycrystalline material; the square grids represent unit cells. (a) Small crystallite nuclei. (b) Growth of the crystallites; the obstruction of some grains that are adjacent to one another is

also shown. (c) Upon completion of solidification, grains having irregular shapes have formed. (d) The grain structure as it would appear under the microscope; dark

lines are the grain boundaries

Noncrystalline Solids / Amorphous

In amorphous solids, there is no long-range order but

amorphous does not mean random

The formation of either crystalline or amorphous solid

depends on the ease of atoms to arrange themselves

during solidification

e.g. rapid cooling through the freezing temperature

favours the formation of a noncrystalline solid since

little time is allowed for the ordering process

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X-ray Diffraction and Bragg’s Law

X-ray diffraction is a powerful tool used for the

determination of Crystal Structure

Bragg’s Law is a formula that relates x-ray

wavelength, interplanar spacing, and angle of

diffraction for constructive interference.

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X-ray diffractometer

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(Explanation: Chapter 3, page 66, Reference 1)