Chapter 3: Atomic & Ionic Arrangements

Solids

Large numbers of atoms or molecules held together in an amorphous or crystalline arrangement by metallic, covalent, ionic bonds or Van der Waal’s bonds

What order do monoatomic gases have? Give an example

• No order

• ex. Argon gas

What order do amorphous materials have? Give an example

• Short-range order

• ex. amorphous Si, Glasses, Plastics

What order do liquid crystals have? Give an example.

• Short and long-range order in small values

• ex. LCD polymers

What order do crystalline materials have? Give an example for single crystal and polycrystalline.

• Short and long-range order

• single crystal: Si, GaAs

• polycrystalline: metals, alloys, and most ceramics

Single crystal

crystalline material consisting of only one large crystal

Polycrystalline

composed of many small crystals with varying orientations

Crystallography

study of the arrangements of atoms in solids

Lattice

• collection of point that divide space into smaller equally sized segments

Basis

• group of atoms associated with a lattice point

• actual atom or group of atoms attached to every lattice point

Unit cell

• subdivision of the lattice that still retains the overall characteristics of the entire lattice

• repeat the unit cell in all directions to build entire crystal

Lattice points

• points that make up the lattice

• surroundings of each point in lattice are identical

Define 1D, 2D and 3D lattices

1D: only 1 type of lattice (points separated by equal distance)

2D: more than one type of lattice

3D: 14 unique lattices (Bravais lattice)

What are the 4 out 14 types of Bravais lattices you need to know?

1. Simple cubic (SC)

2. Face-centered cubic (FCC)

3. Body-centered cubic (BCC)

4. Hexagonal

What are the axes and angles between axes in cubic lattices?

a=b=c

all angles are 90 degrees

What are the axes and angles between axes in hexagonal lattices?

a=b≠c

two angles are 90 and the other is 120

Provide an example of FCC lattice

Copper and nickel

Provide an example of hexagonal lattice

Graphite

How many atoms are in a BCC unit cell?

2

How many atoms are in an FCC unit cell?

4

How many atoms are in a simple cubic unit cell?

1

Coordination number

number of nearest neighbours that an atom has

What is the coordination number of an SC unit cell?

6 atoms touch each other

What is the coordination number of an FCC until cell?

12 atoms

What is the coordination number of a BCC unit cell?

8 atoms touch each atom

Identify the lattice and basis and define the unit cell of CsCl, an ionic and crystalline compound.

• BCC lattice

• Basis of 2 ions

• Definition of unit cell:

  • a=b=c

  • all angles are 90 degrees

  • Cl^- ion at (0,0,0) & Cs⁺ ion at (1/2,1/2,1/2)

Atomic Packing Factor (APF)

= fraction of space occupied by atoms

= (number of atoms/cell)(volume of each atom)/volume of until cell

*used to calculate relationships between lattice parameter and radius of atom

Close-packed directions

directions in unit cell along which atoms are in continuous contact

What is the relationship between the atomic radius and the lattice parameter in an SC unit cell?

a =2r OR r=a/2

What is the relationship between the atomic radius and the lattice parameter in an FCC unit cell?

a=2r√2 OR a=4r/√2

What is the relationship between the atomic radius and the lattice parameter in a BCC unit cell?

a=4r/√3

Calculate the volume of an RCC until cell in terms of the atomic radius R

v_cell = 16√2R³

What is the APF for SC?

π/6 = 52%

What is the APF for FCC?

π/3√2 = 74%

What is the APF for BCC?

√3π/8 = 68%

Equation for calculating the theoretical density of a material from the properties of the crystal structure

ρ = nA/V_cN_A

n = # of atoms associated with each unit cell

A = atomic weight (g/mol)

V_c = volume of the unit cell

N_A = Avogadro’s number (6.022 × 10²³ atoms/mol)

Copper has an atomic radius of 0.128 nm, an FCC crystal structure, and an atomic weight of 63.5
g/mol. Compute its theoretical density and compare the answer with its measured density

8.89 g/cm³

What can accompany a transformation of material? What does that mean for ceramics?

• Volume can change

• for ceramics, can cause the material to fracture

What happens when the crystal structure of a metal changes?

Changes in crystal structure cause changes in the metal’s properties (like hardness, strength, and ductility).

Why does heat treatment work for steels and other alloys?

Because heating and cooling cause controlled changes in crystal structure, which change the material’s properties

Allotropy

The characteristic of an element being able to exist in more than one crystal structure depending on temperature and pressure

Polymorphism

compounds (more than one element) exhibiting more than one type of crystal structure

Which elements have a Simple Cubic (SC) crystal structure?

Polonium (Po), α‑Manganese (α‑Mn)

Which elements have a Body‑Centred Cubic (BCC) crystal structure?

Ferrite (Fe), Titanium (Ti), Tungsten (W), Molybdenum (Mo), Niobium (Nb), Tantalum (Ta), Potassium (K), Sodium (Na), Vanadium (V), Zirconium (Zr), Chromium (Cr)

Which elements have a Face‑Centred Cubic (FCC) crystal structure?

Austenite (Fe), Copper (Cu), Gold (Au), Platinum (Pt), Silver (Ag), Lead (Pb), Nickel (Ni)

Which elements have a Hexagonal Close‑Packed (HCP) crystal structure?

Titanium (Ti), Magnesium (Mg), Zinc (Zn), Beryllium (Be), Cobalt (Co), Zirconium (Zr), Cadmium (Cd)

In what direction is deformation easier?

In the direction where atoms are packed most closely

True or False: Magnetic properties are different in different crystallographic directions

True

How do you find Miller indices for directions?

1. Determine coordinates of 2 points that lie in the direction

2. Subtract the coordinated of the tail point from the head to obtain the number of lattice parameters traveled in the direction of each axis of the coordinate system

3. Clear fraction and/or reduce the results to the lowest integers

4. Enclose the numbers in square brackets []. *No commas

5. Negatives are represented by bar above number

Where does deformation usually occur?

on the planes where the atoms are most tightly packed

True or False: semiconductors are grown and cut along specific crystallographic planes

True

How do you find the Miller indices of planes?

1. Identify the points at which the plane intercepts the x, y and z coordinates in terms of the number of lattice parameters (If the plane passes through the origin, the origin of the coordinate system must be moved)

2. Take reciprocals of the intercepts

3. Clear the fractions but DO NOT reduce to lowest integers

4. Enclose results in () *No commas

5. Negatives should be written with a bar over the number

True or False: Planes and their negatives are not identical (ex. (020) ≠ (0-20))

False, planes are their negatives ARE identical

True or False: In cubic systems, a direction that has the same indices as a plane is perpendicular to that plane

True, for example, (0-10) and [0-10] are perpendicular.

A result of symmetry in the cubic system is:

a family of directions

True or False: A family of directions is represented by square brackets

True

True or False: A family of planes is represented by square brackets.

False, a family of planes is represented by curly brackets

Linear atomic density (LD)

describes how much of a line is covered by atoms (considering only atoms whose centers are on the line)

expressed as [atoms/m]

What is the equation for LD?

LD = equivalent length of atoms centered on the direction vector/length of direction vector

Planar atomic density (PD)

the fraction of the area of the plane is covered by atoms whose centers lie in the plane

expressed as [atoms/m²]

What is the equation for PD?

PD = area of atoms centered on a plane/area of plane

Why is it important to know LD and PD?

Slip (mechanism by which metals plastically deform) occurs:

• Along directions with greatest linear density " ("closed-packed directions”)

• On planes with highest planar density (“closed-packed planes”)

Close-packed direction

the direction where atoms are lined up most tightly and touch along a row

(ex. in FCC, close-packed directions are <110>)

Close-packed plane

the plane with the highest atomic packing density

(ex. in FCC, close-packed planes are {111})

True or False: Close-packed means every atom touches every surrounding atom

False, it means atoms touch their nearest neighbours

What is the coordination number for HCP?

12

True or False: A direction and its (-) are identical

False

True or False: A direction and its multiples are identical

True

True or False: certain groups of directions are equivalent

True

When do close-packed planes occur?

When ideal packing density is achieved (does not happen in the SC or BCC systems)

When is SC in close-packed direction and plane?

Direction: <100>

Plane: None

When is BCC in close-packed direction and plane?

Direction: <111>

Plane: None

When is FCC in close-packed direction and plane?

Direction: <110>

Plane: {111}

When is HCP in close-packed direction and plane?

Directions: <100>, <110>, <1120>

Planes: (0001), (0002) → Basal Plane

Interplanar spacing (d_hkl)

• distance between two adjacent parallel planes of atoms with the same Miller indices

• Important for x-rays to identify materials and crystals

How to calculate interplanar spacing (d_hkl)

d_khl = a₀ / √(h² + k² + l²)

Interstitial sites

the spaces between the atoms/ions of a crystal

(ex. SC until cell has 1 interstitial site)

How do you determine the coordination number of an interstitial atom/ion?

It is equal to the number of atoms/ions it touches

How many tetrahedron and octahedron interstitial sites does a BCC unit cell have?

24 tetrahedron & 18 octahedron

How many tetrahedron and octahedron interstitial sites does a FCC unit cell have?

8 tetrahedron & 13 octahedron