CRYSTAL STRUCTURE, MECHANICAL BEHAVIOUR ......Module 1 1. Crystal Structure Pruthvi Loy, Chiranth B.P. 4 SJEC, Mangalore 1.2.3 Hexagonal Close Packed (HCP) No of atoms per unit cell

Pruthvi Loy, Chiranth B.P. 1 SJEC, Mangalore

CRYSTAL STRUCTURE

Metallic crystal structures; BCC, FCC and HCP

Coordination number and Atomic Packing Factor (APF)

Crystal imperfections: point, line and surface imperfections

Atomic Diffusion: Phenomenon, Fick’s laws, factors affecting diffusion.

1.1 INTRODUCTION

The engineering materials (either metallic or non-metallic) can be identified as crystalline or

amorphous structured. But most of the metals assume crystalline form with a systematic and

regular arrangement of atoms as compared to amorphous structure that lacks regular atomic

arrangement.

A crystalline material is thus comprised of group of atoms with a specific atomic arrangement

which repeats in a 3D pattern; the small group of atoms that repeats over a 3D array is termed as

a unit cell. The geometry and the atomic positions of a unit cell define the crystal structure.

Figure 1.1: Unit cell lattice parameters

Based on unit cell geometry for different possible combinations of a, b, c and , , seven

crystal systems were identified. Further, considering atomic arrangement within a crystal system

A. Bravais showed that the seven crystal systems could be arranged in 14 independent ways to

obtain the 14 Bravais lattices.

MODULE ONE

CRYSTAL STRUCTURE, MECHANICAL

BEHAVIOUR & FAILURE OF MATERIALS

Module 1 1. Crystal Structure


Table 1.1: Crystal systems and Bravais lattices



1.2 METALLIC CRYSTAL STRUCTURES

Inspite of the different possible crystal structures three relatively simple structures are found for

most of the common metals.

Body Centered Cubic (BCC)

Face Centered Cubic (FCC)

Hexagonal Close Packed (HCP)

1.2.1 Body Centered Cubic (BCC)

A cubic unit cell with atoms located at all eight corners and a single atom at the cubic center.

No of atoms per unit cell = 2

i.e., 8x ⁄ (8 corner atoms each shared by 8 neighboring unit

cells) + 1 (1atom at the cubic center) = 2

Example: Chromium, Tungsten, Iron, etc.

1.2.2 Face Centered Cubic (FCC)

It also has a cubic geometry with atoms located at each of the corners and the centers of all the

cube faces.


i.e., 8x ⁄ (8 corner atoms each shared by 8neighboring unit

cells) + 6 x ⁄ (6 atoms at the cube faces shared by 2 unit

cells) = 2

Example: Copper, Aluminium, Silver, Gold, etc.



1.2.3 Hexagonal Close Packed (HCP)


i.e., 12 x ⁄ (12 corner atoms each shared by 6

neighboring unit cells) + 2 x ⁄ (2 atoms at the

hexagonal faces each shared by 2 unit cells) + 3 whole

atoms = 6

Example: Titanium, Zinc, Cobalt, Magnesium, etc.

1.3 CHARACTERISTICS OF CRYSTAL STRUCTURE

The type of structure and its characteristics has a profound influence on the material properties.

Two of the important characteristics of a crystal structure are the coordination number and the

atomic packing factor (APF); apart from these the other characteristic features of interest is the

stacking of planes.

1.3.1 Coordination number

For metals, each atom has the same number of nearest-neighbor or touching atoms, which is the

coordination number.

Unit Cell Coordination number

1. Simple Cubic 06

2. Body Centered Cubic 08

3. Face Centered Cubic 12

4. Hexagonal Close Packed 12

1.3.2 Atomic Packing Factor

The Atomic Packing Factor (APF) is the fraction of volume in a crystal structure that is occupied

by the atoms.

i.e., APF =

=

=



a) Simple Cubic:

Number of atoms per unit cell, = 1

Volume of each atom, =

Volume of a unit cell, =

Therefore,

APF =

⁄

=

⁄

(because, = 2r)

APF = 0.52

i.e., only 52 % of the space available inside a unit cell of simple cubic structure is occupied

by atoms.

b) BCC:

From ADC,

AC2 = AD

2 + DC

2

AC2 = 2 + 2

AC = √

From ABC,

AB2=AC

2+ BC

2

2 = (√ )2 + 2 = 3 2

= √

⁄






Therefore,

APF = ⁄

=

⁄

( √

⁄ )

APF = 0.68

i.e., 68 % of the space available in a BCC unit cell is occupied by atoms.

c) FCC:

From ABC,

AC2+BC

2= AB

2

2 + 2 = 2

= √

⁄




Therefore,

APF = ⁄

=

⁄

( √

⁄ )

APF = 0.74

i.e., 74 % of the space available in a FCC unit cell is occupied by atoms.



d) HCP:

From ABD,

2 = h2 + ( ⁄ )2

h2 = 2 -

⁄ =

⁄

h = √ ⁄



Volume of a unit cell, = Area of hexagon x c

= 6 x Area of ABC x c

= 6 x ( ⁄ ) x c

= 6 x ( ⁄ √

⁄ ) x c

= 6 √

⁄ c

Also,

= 2r and

c = 1.633

Where, c is lattice constant, its value can be calculated as shown below(considering the

atoms to be spherical in shape),



From AMN,

AM2

= AN2

+ MN2

2 = ( √

)2 + ( )2

2 =

⁄ +

⁄

2 =

⁄

c = √ ⁄ = 1.633

Therefore,

APF = ⁄

√ ⁄

= ⁄

√ ⁄

APF = 0.74

i.e., 74 % of the space available in a HCP unit cell is occupied by atoms.

cos 30 =

𝑎 ⁄

AN = √ ⁄

AN=𝑎

√



Note: It is possible to compute the theoritical density of a metallic solids having the knowledge

of its crystal ctructure.

=

=

Where,

n = number of atoms per unit cell

A = atomic weight

VC = volume of the unit cell

NA = Avogadro’s number (6.023 x 1023

atoms/mol)

Table 1.2: BCC, FCC and HCP Unit cell parameters

Atoms/unit cell,

n

Coordination

number, Z

edge length,

lattice constant,

c

Unit cell

volume, Vc APF

BCC 2 8

√ - 0.68

FCC 4 12

√ - 0.74

HCP 6 12 2r 1.633 6 √

⁄ c 0.74

Example Problem:

Copper has an atomic radius of 0.128 nm (1.28 Å), 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.

SOLUTION:

Given, n = 4, A= 63.5 g/mol, r = 0.128 x 10-9

m = 0.128 x 10-7

cm

Volume of unit cell, = = (

√ )

=

√ = 16 √ = 16 (0.128x10-7)3√ = 4.75x10-23 cm3

Therefore,

Density, =

=

= 8.89 ⁄

From periodic table of elements, the literature value of density of copper is 8.96 ⁄ .



1.3.3 Stacking of Planes

During solidification the atoms within the solid pack together as tightly as possible, i.e., a layer

of atoms stack one above the other to make up the solid material.Although layers of atoms are

stacked one above the other, their sequence of stacking varies for different crystal structures.The

stacking sequence of few crystal strucures are as shown below.

Figure 1.2: Stacking of Planes in a crystal structure



1.4 CRYSTAL IMPERFECTIONS

For the study of crystal structures we have assumed a perfect or ideal crystal. However, such an

idealized crystal does not exist; all contain large number of various defects or imperfections. The

properties of most of the metals are profoundly influenced by the presence of imperfections.

Thus specific characteristics can be obtained in crystals by introducing crystalline defects.

Adding alloying elements to the metal is one way of introducing a crystal defect.

According to the geometry or dimensionality the defects may be classified as;

Zero dimensional or Point defect

o Vacancy

o Interstitial defect

o Substitutional defect

One dimensional or Line defect

o Edge dislocation

o Screw dislocation

Two dimensional or Surface defect

o External surface

o Grain boundary

o Tilt boundary

o Twin boundary

o Stacking fault

Three dimensional or Volume defect

- Pores, cracks, foreign inclusions and other phases

1.4.1 Point imperfections

Vacancy:

The simplest of the point defect is a vacancy, where in one or more atoms are missing from their

respective location within the crystal lattice. The vacancies may occur as a result of imperfect

packing during crystallization or they may also arise from thermal fluctuation of atoms at high

temperature.

Figure 1.3: Vacancy defect



Impurity:

A pure metal consisting of only one type of atom is highly idealistic; impurity or foreign atoms

will always be present. Most familiar metals are not highly pure rather they are alloys in which

impurity atoms have been intentionally added to impart specific characteristics to the material.

The addition of impurity atoms to a metal will result in the formation of either a solid solution

and/or a new phase. A solid solution forms when as the solute atoms (impurity) are added to the

solvent (host material) and the crystal structure of the parent material is retained with no new

structures being formed.

Impurity point defects in crystals can be,

Interstitial impurity or

Substitutional impurity

Interstitial impurity: In this an interstitial foreign atom occupies a definite position in a non-

lattice site within the crystal.

Example: Addition of carbon atoms (0.071 nm) to iron (0.124 nm) where the carbon atoms

occupy the interstitial space between the iron atoms.

Substitutional impurity: When a foreign atom substitutes the parent atom and occupies its

position in the lattice site, then it is known as a substitutional defect.

Example: Addition of copper to nickel; copper atoms substitute the nickel atoms.

Note: Point defect in ceramics may exist as both vacancies and interstitials. The atomic bonding

is predominantly ionic in ceramics; i.e., their crystal structures may be thought of as being

composed of electrically charged ions instead of atoms. The metallic ions, or cations, are

positively charged, because they have given up their valence electrons to the nonmetallic ions, or

anions, which are negatively charged. An ionic crystal possess electronegativity, i.e., there is

equal number of positive and negative charges from ions; as a consequence, defects in ceramics

do not occur alone rather defect for each ion type may occur; one such defect is Frenkel &

Schottky defect.

Figure 1.4: Impurity defects

Figure 1.5: Frenkel and Schottky defects



Frenkel defect involves a cation vacancy - cation interstitial pair. It might be thought of as

being formed by a cation leaving its normal position and moving into an interstitial site. There is

no change in charge because the cation maintains the same positive charge as an interstitial.

Schottky defect is a cation vacancy - anion vacancy pair. This defect might be thought of as

being created by removing one cation and one anion from the interior of the crystal. Since for

every anion vacancy there exists a cation vacancy, the charge neutrality of the crystal is

maintained.

1.4.2 Line Imperfections

Linear defects in crystalline solids are due to misalignent of atoms during the dislocation of

atomic planes. Dislocations are of two types;

Edge Dislocation and

Screw Dislocation

Figure 1.6: Line imperfections

Edge Dislocation

It is created in a crystal by insertion of an extra plane of atoms i.e., a half plane as shown in

figure. The edge of the half plane terminates within the crystal, this is termed as dislocation line.

The atoms above the dislocation line are squeezed together and are in a state of compression

while the atoms below are pulled apart and are in a state of tension. Edge dislocation is

represented by the symbol ┴ for positive dislocation and ┬ for negetive dislocation.

Screw Dislocation

It is said to be formed in perfect crystal when part of the crystal displaces angularly over the

remaining part under the action of shear stress. The upper front region of the crystal is shifted

one atomic distance to the right relative to the bottom portion. The screw dislocation derives its

name from the spiral or helical path that is traced around the dislocation line by the atomic

planes of atoms. Screw dislocation is represented by the symbol for clockwise or positive

dislocation and for counterclockwise or negtive dislocation.



Table 1.3: Comparison of Edge and Screw dislocation

Edge Dislocation Screw Dislocation

It is created when a half plane of atoms is

inserted in a crystal

It is created when a part of crystal displaces

angularly over the remaining part

It moves in the direction of Burger’s vector It moves in the direction perpendicular to that

of Burger’s vector

Burger’s vector is perpendicular to the

dislocation line

Burger’s vector is parallel to the dislocation

line

Edge dislocation travels faster when loaded Screw dislocation travels comparatively

slower

It requires less force to form and travels faster

under loads.

It requires comparatively high force to form

and travels slower under loads

Atomic bonds around the dislocation line

experiences tension and compression

Atomic bonds around the dislocation line

experiences shear force.

Symbolic representation:

┴ for positive dislocation

┬ for negetive dislocation

Symbolic representation:

for positive dislocation

for negtive dislocation

1.4.3 Surface Imperfections

External Surface: One of the most obvious surface defect is the external surface, along which

the crystal structure terminates. Surface atoms are not bonded to the maximum number of nearest

neighbors, and are therefore in a higher energy state than the atoms at interior positions.

Grain Boundary: A grain boundary is formed when two adjoining growing crystals meet at

their surface.The atoms are bonded less regularly along the grain boundary and are at a higher

energy state as a result the impurity atoms preferentially segregate along these boundaries.Also,

grain boundary acts as a barrier for dislocation motion; the smaller the grains, larger is the grain

boundary area and dislocations if any moves only a short distance and stops at the grain

boundary.

A polycrystalline solid contains numerous grains or crystals. Each crystal has nearly the same

crystal structure but different orientations. The grain boundary is few atomic radius thick and

contains crystallographic misalignment between adjacent grains; various degrees of

crystallographic misalignments are possible. When this orientation mismatch is slight (of the

order of few degrees), then it is termed as small angle grain boundary.

A small angle of misorientation (less than 10) with the edge dislocations aligned in the manner

as shown in figure 1.7(b), then it is called a tilt boundary.



Figure 1.7: (a) Grain boundary (b) Tilt boundary

Twin Boundary: A twin plane or boundary is a special type of grain boundary across which

there is a specific morror lattice symmetry; i.e., the atoms on one side of the boundary are

located in mirror image positions of the atoms on the other side. The region of material between

these boundaries is termed as twinned region.

Figure 1.8: Twinning and twin boundary

Stacking Faults: A crystal structure has a specific stacking sequence; any deviations from the

actual stacking sequence of the plane of atoms is termed as a stacking fault.

For example: The stacking sequence of a FCC structure is A, B, C, A, B, C, A, B, C, ….

Sometimes it may appear as A, B, C, A, B, A, B, C, … with a missing C plane which is termed

as a stacking fault.



1.4.4 Volume Imperfections

These are three dimensional imperfections that are formed inside the solid material. These

includes voids, cracks, foreign inclusions and other phases which are normally introduced during

processing and fabrication.

1.5 ATOMIC DIFFUSION

From an atomic perspective diffusion may be defined as the mass flow process by which atoms

or molecules migrate from lattice site to lattice site within a material resulting in the uniformity

of composition as a result of thermal agitation.

The importance and various applications of diffusion phenomenon are;

Diffusion occurs more rapidly with increasing temperature and is the basis for most

metallurgical processes.

Diffusion is fundamental to phase changes and is important aspect in heat treatment of

metals.

It is important in the formation of metallic bonds (soldering, welding, brazing, etc.)

1.5.1 Diffusion Phenomenon

Diffusion in solids can take place by the following methods;

Vacancy diffusion

Interstitial diffusion

Vacancy diffusion involves the movement of an atom from original lattice position to an

adjacent vacant lattice site. The extent to which vacancy diffusion can occur depends on the

number of vacant sites present in the crystal; significant concentrations of vacancies may exist in

metals at elevated temperature.

Figure 1.9: Vacancy diffusion

Interstitial diffusion involves the movement of interstitial atoms from an interstitial site to its

neighbouring site without permanently displacing any of the parent atoms in a crystal lattice.

With interstitial diffusion an activation energy is associated because to move into an adjacent

interstitial site it must squeeze past the parent atoms in the crystal attice with the energy supplied

by the vibrational energy of moving atoms.



Figure 1.10: Interstitial diffusion

Interstitial diffusion occurs more rapidly than vacancy diffusion since interstitial atoms are

smaller and as more empty interstitial positions are present than the vacancies.

1.5.2 Fick’s Law of Diffusion

Diffusion is a time dependent process; i.e., the quantity of an element that is transported within

another is a function of time. Often it is necessary to know how fast diffusion occurs, or the rate

of mass transfer. This rate is expressed as a diffusion flux (J), which is defined as the mass (m)

diffusing through and perpendicular to a unit cross-sectional area of solid (A) per unit time (t).

i.e., J =

or J =

kg/m

2-s or atoms/m

2-s

The diffusion flux may or may not vary with time and accordingly we have two laws of

diffusion:

i. Fick’s first law for steady state diffusion

ii. Fick’s second law for unsteady state diffusion

Fick’s first law of diffusion:

It states that the flux of atoms (J), moving across a unit area in unit time is proportional to

concentration gradient under steady state.

i.e., J

or J = - D

Where,

J – diffusion flux, atoms/m2-s

⁄ – concentration gradient

D – diffusivity or diffusion coefficient, m2/s

The negetive sign indicates that the direction of diffusion is down the concentration gradient, i.e.,

from a region of higher concenration to a region of lower concentration.



Concentration gradient is obtained as,

=

Fick’s second law of diffusion:

Most practical diffusion situations are usually of unsteady state. i.e., the diffusion flux and the

concentration gradient at some particular point in solid vary with time resulting in net

accumulation or depletion of diffusing species.

Therefore,

=

*

+

Where,

= rate of composition change

= concentration gradient

D = diffusivity, m2/s

Figure 1.12: Concentration profile for unsteady state diffusion

Figure 1.11: Steady state diffusion



If diffusion coefficient is independent of concentration;

= D

i.e., the rate of composition change is equal to the diffusivity times the rate of concentration

gradient.

The solution to the above equation can be obtained by applying appropriate boundary conditions.

For t = 0, C = C0 (0 x ∞)

For t > 0, C = Cs (at x = 0) and C = C0 (at x = ∞)

Applying the above boundary conditions the solution can be obtained as,

= 1 – erf (

√ )

From the above equation may be determined at any time and position if the parameters

and D are known.

1.5.3 Factors Affecting Diffusion

The various factors affecting diffusion are:

1. Crystal Structure

2. Grain size

3. Atomic radius

4. Temperature

5. Concentration

Crystal Structure: The ease with which the atoms diffuse increases with decreasing density of

packing. Example: Atoms have higher diffusion coefficients in BCC iron than FCC iron because

the former has low atomic packing factor.

Grain size: As we know grain boundary diffusion is faster than diffusion within the grains, it is

to be expected that overall diffusion rate would be higher in fine grained material due to

increased grain boundary.

Atomic radius: Diffusion occurs more rapidly when the size of the diffusing atom is small.

Example: diffusion of carbon atoms in iron.

Concentration: a higher concentration gradient results in faster diffusion rates.

Temperature: It has most profound influence on the coefficient and diffusion rate. The diffusion

coefficient (D) is related to temperature by Arrhenius type of equation as shown below,



D = D0

⁄

Where,

D0 – temperature independent pre-exponential, m2/s

Q – activation energy, J/mol

R – gas constant ( R = 8.314 J/mol-K)

T – absolute temperature, K

When the temperature increases, the diffusion coefficient increases and therefore the flow atoms

also increase.

Problems (Diffusion):

1. Calculate the diffusion coefficient for magnesium in aluminium at 570 C given that,

Do = 1.2 x 10-4

m2/s and Q = 131 kJ/mole.

Solution:

Do = 1.2 x 10-4

m2/s

Q = 131 kJ/mole = 131000 J/mole

T = 570 C = 570 + 273 = 843 K

R = 8.314 J/mol-K

D = D0

⁄ = 1.2 x 10-4

*

+

D = 9.075 x 10-13

m2/s

2. It is proposed to enhance the surface wear resistance of a steel gear by carburizing

treatment. The initial carbon content of steel is 0.15 wt%. After the treatment the surface

concentration is to be maintained at 0.95 wt%. For the treatment to be effective a carbon

content of 0.55 wt% must be established at 0.75mm below the surface. Specify appropriate

heat treatment in terms of temperature and time for temperature 900 C to 1050 C. Take

Do = 2.3x10-5

m2/s, Q = 148000 J/mole.

Solution:

Do = 2.3x10-5

m2/s

Q = 148000 J/mole

C0 = 0.15 wt %

Cs = 0.95 wt %

Cx = 0.55 wt %

X = 0.75 mm = 0.00075m



WKT,

= 1 – erf (

√ )

= 1 – erf , where, z =

√

erf =

Table 1.4: Error-function values

From error-function value table, the value of z can be obtained

by interpolation as follows,

0.45 0.4755

z 0.5

0.5 0.5205

i.e.

=

Therefore, z = 0.4722

But z =

√ =

√ = 0.4722

Dt = *

+

= 6.17 x 10-7

m2

Also,

D = D0 *

+

D = D0 *

+ x

For 900 C (1173 K)

6.17 x 10-7 = 2.3 x 10 -5 *

+ x

t = 29.6 hrs

Similarly calculate for 950, 1000 and 1050 C

z erf (z)

0.35 0.3794

0.4 0.4284

0.45 0.4755

0.5 0.5205

0.55 0.5633



The following heat treatment parameters were calculated

Temperature, C Time, hrs

900 29.6

950 15.9

1000 9.0

1050 5.3

3. Steel gear, having carbon content of 0.2% is to be gas carburized to achieve carbon

content of 0.9% at the surface and 0.4% at 0.5mm depth from the surface. If the process is

to be carried out at 927 °C, find the time required for carburization. Take diffusion

coefficient of carbon in given steel = 10.28x10-11

m2/s. Given data:

Z erf(Z)

0.75 0.7112

Z 0.7143

8 0.7421

Solution:

D = 10.28x10-11

m2/s

C0 = 0.2 %

Cs = 0.95 %

Cx = 0.4 %

x = 0.5 mm = 0.0005m

WKT,

= 1 – erf (

√ )

= 1 – erf (

√

)

0.2857 = 1 – erf (

√ )

0.7143 = erf (

√ )

Let z =

√ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Equation (1)

Therefore, erf(z) = 0.7143



From the given error-function value table, the value of z can be obtained by interpolation as

follows,

i.e.

=

Therefore, z = 0.755

Substituting the value on z in equation (1),

z =

√

t =

t = 8566.35 sec = 142.8 min

References:

1. Fundamentals of Materials Science & Engineering – William D. Callister

2. Material Science and Metallurgy – K. R. Phaneesh

3. Material Science and Metallurgy – Kesthoor Praveen

4. Mechanical Metallurgy – G. E. Dieter

CRYSTAL STRUCTURE1.1 INTRODUCTION1.2 METALLIC CRYSTAL STRUCTURES1.2.1 Body Centered Cubic (BCC)1.2.2 Face Centered Cubic (FCC)1.2.3 Hexagonal Close Packed (HCP)

1.3 CHARACTERISTICS OF CRYSTAL STRUCTURE1.3.1 Coordination number1.3.2 Atomic Packing Factor1.3.3 Stacking of Planes

1.4 CRYSTAL IMPERFECTIONS1.4.1 Point imperfections1.4.2 Line Imperfections1.4.3 Surface Imperfections1.4.4 Volume Imperfections

1.5 ATOMIC DIFFUSION1.5.1 Diffusion Phenomenon1.5.2 Fick’s Law of Diffusion1.5.3 Factors Affecting Diffusion

Problems (Diffusion):

Documents

CRYSTAL STRUCTURE, MECHANICAL BEHAVIOUR ......Module 1 1. Crystal Structure Pruthvi Loy, Chiranth B.P. 4 SJEC, Mangalore 1.2.3 Hexagonal Close Packed (HCP) No of atoms per unit cell