M sc sem iv u ii

4PHY-3(ii) : CONDENSED MATTER PHYSICS-II
Dr. WS Barde, Shri Shivaji Science College, Amravati 1
Unit-II
Dislocation reaction, Partial Dislocations and stacking faults in close packed
structures, Thompson Tetrahedron. Experimental methods of observing dislocation
and stacking fault.
Book Referred :
Dislosations in Crystals
By W. T. Read
&
Introdustion to Dislosations
Fourth Edition
by D. Hull and D. J. Bacon

2.1 Dislocation Reactions
Different dislocations can react and produce a new dislocation. For example two parallel
dislocations of Burger’s vectors b1 and b2 on same slip plane or lies on two intersecting planes can
interact with each other gives rise to new dislocation of Burger vector b3. The diagrammatic
representation of reaction is shown below.
The reaction can be written as,
𝒃 𝟏 + 𝒃 𝟐 = 𝒃 𝟑…………………………….[1]
Reaction among larger number of dislocations is also possible, such as,
𝒃 𝟏 + 𝒃 𝟐 ⇌ 𝒃 𝟑 + 𝒃 𝟒…………………………[2]
In the dislocation reaction certain amount of energy is always liberated. Let Q be the quantity of
energy liberated. As dislocation energy is proportional to square of Burger’s vector, the balanced
energy equation can be written as,
𝑏1
2
+ 𝑏2
2
= 𝑏3
2
+ 𝑄
Using equation [1] we get,
𝑏1
2
+ 𝑏2
2
= (𝒃 𝟏 + 𝒃 𝟐)2
+ 𝑄
(𝒃 𝟏 + 𝒃 𝟐)2
= 𝑏1
2
+ 𝑏2
2
− 𝑄…………………..[3]
As, (𝒃 𝟏 + 𝒃 𝟐)2
= 𝑏1
2
+ 𝑏2
2
+ 2𝒃 𝟏. 𝒃 𝟐 ……………..[4]
Comparing equations [3] and [4] we have,
𝑄 = −2𝒃 𝟏.. 𝒃 𝟐 = 2𝑏1 𝑏2 𝑐𝑜𝑠𝜃 ………..[5], where θ is the angle between Burger’s vector.
Equation [5] can be considered as criterion for stability of dislocation. That is whether the reaction
( either association or dissociation reaction) between dislocations is favourable or not.
Frank’s criterion for stability :
1. Two parallel dislocation of Burger’s vector 𝒃 𝟏& 𝒃 𝟐 may combine (attract) to form a new
dislocation of Burger vector 𝒃 𝟑 only if the following reaction is satisfied.
𝑏1
2
+ 𝑏2
2
> 𝑏3
2
. Thus the dislocations attract each other when Burgers vectors form an
obtuse angle and repel when they form acute angle.

2. Similarly, a dislocation of Burger’s vector 𝒃 𝟏 may dissociate into two parallel dislocations of
Burger’s vectors 𝒃 𝟐 & 𝒃 𝟑 only if the following reaction is satisfied.
𝑏1
2
> 𝑏2
2
+ 𝑏3
2
.
The favourable slip process can be understood from following examples.
Let us consider a slip occurring in fcc crystal on slip planes {111} along slip directions
〈110〉. As shown in figure the B layer of atoms, instead of moving from one B site to next B
site, prefer to move to nearby site C along the valley between two A atoms and then to B site
via a second valley.
2.2 Dislocation in FCC structures
Perfect Dislocation
In fcc the slip vector is
1
2
〈110〉 , so it is a translation vector for the lattice along that direction. Now,
when the glide of a dislocation takes place by the Burger’s vector equal to
1
2
〈110〉 , it leaves behind
the perfect crystal and the dislocation is called as perfect dislocation.
Partial Dislocation
If a stacking fault ends inside the crystal, its boundary is either a Frank partial (also called sessile)
dislocation or a Shockley partial. There are two types of partial dislocations in FCC.
1. Shockley partial
2. Frank partial
Shockley partial :
A partial dislocation whose Burger’s vector lies in the
plane of the fault is called as Schokley partial.
As shown in figure the (111) layers in fcc are stacked
in a sequence ABCABC… and the lattice is a perfect
lattice.
Now, if A layer atoms have slipped in [12̅1] direction
to B atoms positions, B atoms to C atoms positions
and C atoms to A atoms position then stacking fault is
created and hence the partial dislocation has taken
place. Thus the fault vector in [12̅1] direction is
1
6
[12̅1] which lie in (111) plane. For a perfect
dislocation the dislocation vector is
1
2
[110] with
magnitude
a
√2
while that of fault vector is
a
√6
.
That is ,
Schokley (𝑏 =
1
6
[12̅1]) : 𝑏2
=
𝑎2
36
(12
+ 2̅2
+ 12)=
𝑎2
6

Perfect (𝑏 =
1
2
[110]) : 𝑏2
=
𝑎2
4
(12
+ 12
+ 0)=
𝑎2
2
Since, 𝑏2
=
𝑎2
2
for perfect dislocation and 𝑏2
=
𝑎2
6
for Shockley partial, energetically Shockley partial
is favourable. In other words a perfect slip via two Shockley partials is fevourable and can be
explained with the help of following diagram.
In the figure it is shown that B layer atom instead of moving from one B site to other over the top of
A atoms (vector b1), will move first to the nearby C site along the valley between A toms (vector
b2) and then to the new B site via a second valley(vector b3).
This implies that dislocation passes as two Schokley partials one after the other. The perfect
dislocation thus split in to two Schokley partials accroding to the given reaction.
b 𝟏 → b 𝟐 + b 𝟑
Or
1
2
〈110〉 →
1
2
〈211〉 +
1
2
〈12𝟏̅〉
2.3 Thompson Tetrahedron
Partial dislocations can be describe with the help of Thompson notation and the Thompson
tetrahedron. Figure shows all the lattice points in an FCC unit cell, as well as some that are outside
the unit cell. The four lattice points that make up the Thompson tetrahedron are illustrated with a
dark spheres such that one lattice point located at the origin and three at the centers of the three
adjoining faces. These four lattice points form the vertices of the Thompson tetrahedron. The
faces of the Thompson
tetrahedron coincide with the
close packed slip planes in the
FCC lattice while the edges of
the tetrahedron correspond to
the close packed slip directions.
The vectors extending from
one vertex to another are
perfect lattice translations.
From figure it is observed that
ABC corresponds to the plane
(111) and the vector AC
corresponds to the a /2 [0 1 1]
translation. All of the other
planes and directions implied
by the Thompson notation can be determined by reference to the unit cell. The centers of the faces
of the tetrahedron are denoted with Greek symbols corresponding the label of the opposite vertex, (
α opposite to A, β opposite to B, γ opposite to C and δ opposite to D) . These points are not lattice
points; they represent faulted atomic positions. Slip Occurs on the Inside Surface of the Thompson
Tetrahedron, not on the Outside Surface.

Consider a unit amount of slip on the (111) plane (the shaded planes in Fig.) and in the closed-
packed, BC direction. Referring to the unit cell and to the corresponding tetrahedron, one might
be tempted to say that an atom initially located at B would first slide to the faulted position δ and
then to the final lattice position C relative to the atoms in the layer that contains the lattice point D.
But if shear of this kind occurred, the atom initially at B would have to ride over the top of the
atom located at D, a highly energetic process. Instead the atom at B slides to δ and then to C.
Thinking of the tetrahedron as a solid object, we say that atoms cannot slide from B to δ to C on
the outside surface of the tetrahedron because that would result in the atoms riding up over the
tops of the atoms below (in layer D ). Instead, all of the atoms in the tetrahedron slide together
relative to the atoms outside of the tetrahedron. We can think of atoms sliding on the inside
surface of the tetrahedron relative to the atoms outside the tetrahedron. The dotted arrows
indicate the positions taken as atoms slide from B to δ and then to C on the plane outside of the
tetrahedron.
2.4 Frank Partial Dislocation
The Frank partial dislocation is formed by the
inserting or by removing one close-packed (111)
plane. As shown in figure, removal of the close-
packed plane results in the intrinsic stacking fault
with sequence ABCBCABC…. (fig a). It can occure by
condensation of vacancies on octahydral plane.
Where as insertion of a plane produces an extrinsic
fault with sequence ABCBABC…. (fig. b). It can be
produces by interstitial agglomeration.
The Frank partial has a Burger’s vector normal to
the (111) plane of the fault and magnitude is
1
3
[111].
This type of dilocation is
sessile, that is it cannot glide
but it undergoes climb.
In case of Frank partials
insertion or removal of part
of the slip plane could be
occurred by either of the following ways.
(a) by splitting of a total dislocation,
(b) by condensation of vacancies or interstitial atoms,
(c) by combination of a Shockley partial with a total dislocation.
a. Splitting.: splitting of a dislocation into several dislocations is known as a dislocation
reaction. A reaction is represented by an equation relating the Burgers vectors; for example

a.
1
2
[110] =
1
6
[211] +
1
6
[121̅] represents the splitting of a total dislocation into two
Shockley partials.
b. Condensation of Vacancies or Interstitials : The simplest way to form a Frank partial is to
remove or insert part of a close-packed plane. Physically, this means the condensation of
vacancies or interstitial atoms on a (111) plane.
c. Combination of a Shockley Partial with a Total Dislocation on an Intersecting Slip Plane. :
Frank partials can be formed indirectly by combination of a total dislocation with a
Shockley partial. This is the converse of one of the splitting of dislocation. An example of the
combination in terms of Burgers vectors is
1
2
[110] +
1
6
[1̅1̅2̅] =
1
3
[111̅]
2.5 Dislocation in HCP (Hexagonal Close-packed) structures
In hcp metals (0001) basal plane is close-
packed plane and 〈112̅0〉 is the close-
packed directions. The shortest lattice
vector (Burger’s vector) are
1
3
〈112̅0〉. The
slip ocures on basal plane (0001).
Like wise Thomson tetrahedron in FCC bi-
pyramid is used to explain Burger’s vectors
in HCP.
a) Six Burger’s vector in basal plane
are along the sides AB, BC, CA, BA,
CB and AC of triangular base ABC of
the pyramid
b) Two more perfetc dislocations are
represented by vectors ST and TS.
c) Partial dislocationa of Shockley type are represented by Burger’s vector Aσ, Bσ, Cσ, σA, σB
and σC (
1
3
〈101̅0〉) .
Thus Burger’s vector reaction can be written as-
AB → Aσ+Bσ
Ex. :
1
3
[112̅0] =
1
3
[101̅0] +
1
3
[011̅0]
d) Similarly other possible partial dislocationa of Shockley type are represented by Burger’s
vector Sσ, Tσ, σT and σS.
Again the hexagonal and f.c.c. cases are very similar. Here, in hexagonal, partials are formed by
following ways.
a. Removal of a Basal Plane : This is not analogous to removal of plane in the f.c.c. case because
the planes adjoining the missing plane cannot come together because stacking order would

become an AA sequence. So missing of B plane is always followed by shifting of one A plane
into either the B or the C hollows on the other A plane.
For example, If we remove a B plane and, above the gap, shift 𝐴 → 𝐵 → 𝐶.
Now we have . . . 𝐴𝐵𝐴𝐵𝐴 ↓ 𝐵𝐶𝐵𝐶𝐵𝐶𝐵. ..
b. Inserting a Basal Plane : Now insert a C plane between A and B planes without offset. The
stacking order becomes . . . 𝐴𝐵𝐴𝐵𝐴𝐵𝑪𝐴𝐵𝐴𝐵 . .. ; which has five planes ABCAB in the ABC
order.
2.6 Dislocation in BCC structures
In BCC metals slip occures in clos-packed direction 〈111〉. The Burger vector for perfect dislocation
is of type
1
2
〈111〉. The crystallographic slip planes are {110}, {112} and {123}. Each of these planes
contains 〈111〉 slip direction. It is sinificant that three {110}, three {112} and six {123} planes
intersect each other along the same direction 〈111〉. Thus during cross slip it is easy for screw
dislocation to move in a hapahazard way on different {110} planes or on combination of planes.
That is why the slip lines observed in bcc metals is often wavy.
2.7 GRAIN BOUNDARIES
Burger suggested that the boundaries of two crystallites or crystal grains at a low angle inclination
with each other can be considered to be a regular array of dislocations. Two such crystallites
placed close together at a small angle θ have been shown in fig. (a).
There are simple cubic crystals with
their axes perpendicular to the plane of
the paper and parallel. The crystals have
been rotated by θ /2 left and right of
these axes. The results of joining the two
crystals together is shown in fig. (b).
A grain boundary of the simple example
of Burger's model is formed. The
boundary plane contains a crystal axis
common to the two crystals. Such a
boundary is called a pure tilt boundary.
Crystal orientations on both sides of the
boundary plane are symmetric with
each other such a boundary has a
vertical arrangement of more than two
edge dislocations of same sign. This
arrangement is also stable as that for
two dislocations. From the figure it is seen that the interval D between the dislocations so formed is
given by


 b
Dor
D
b
==
22
tan …….(1)
Where b is the Burger’s vector of the dislocations and  is small angle.
The relative inclination angle  can be measured by means of X –ray diffraction experiments. From
this value of  and knowing the value of b, the value of D can be calculated theoretically.
2.8 Interfaces within crystals
The surface defects that commonly appear in the interior of a crystal are of two types:
1. stacking faults and 2. Antiphase boundaries.
We have already encountered the stacking fault, which is a defect in the sequence of planes of atom
positions in the crystal. An antiphase boundary is a defect in the sequence of atom distributions
over the planes. Anti-phase boundaries can only occur in ordered compounds.
Stacking faults
The simplest examples of stacking faults are found in close-packed structures. A stacking of close-
packed planes in the order ...ABCABC... generates the FCC structure. But when stacking fault occurs
this pattern ...ABCABC... in FCC structure is broken.
For example, in a crystal with the local stacking sequence ...ABCBABC... the extra B plane in the
middle constitutes a stacking fault. It is sometimes useful to distinguish three kinds of stacking
fault.
An intrinsic stacking fault is one that can be
created by removing a plane of atoms. In the
sequence ...ABCA||CAB.. the defect marked ||
corresponds to a missing A-plane, and is hence
an intrinsic stacking fault. An intrinsic stacking
fault is created by the condensation of vacancies
onto a close-packed plane. It is also created by a
partial slip of the crystal to the right of the fault
that carries A-sites to B-sites, B-sites to C-sites,
and C-sites to A-sites.
An extrinsic stacking fault is one that can be
created by inserting an extra plane of atoms into
the structure. In the ...ABCA|C|BCA... pattern
given above the plane marked |C| is an extrinsic
stacking fault. Extrinsic stacking faults appear
after an irradiation treatment creates many interstitials that subsequently gather together on close-
packed planes.
In the sequence ...ABCA|C|BCA... the plane marked C is a twin boundary; on either side the crystal
has a perfect FCC structure.

Note that if a stacking fault is repeated with perfect periodicity it does not constitute a fault, but
creates a new crystal structure. For example, the sequences ABCAB, ABCABCAB, and ABABABC are
regular mixtures of FCC and HCP packing that generate new structures if they are repeated
indefinitely. Such mixed structures occur in nature. They are usually found in materials that exhibit
both FCC and HCP-type structures . Mixed structures are particularly common in SiC, ZnS and CdS,
called polytypes.
2.9 VOLUME DEFECTS
Volume defects in crystals are three-dimensional aggregates of atoms or vacancies. It is grouped
into four classes. The four categories are:
• precipitates, which are a fraction of a micron in size and decorate the crystal;
• second phase particles or dispersants, which vary in size from a fraction of a micron to the
normal grain size (10-100μm), but are intentionally introduced into the microstructure;
• inclusions, which vary in size from a few microns to macroscopic dimensions, and
• voids, which are holes in the solid formed by trapped gases or by the accumulation of
vacancies.
Precipitates are small particles that are introduced into the matrix by solid state reactions. While
precipitates are used for several purposes, their most common purpose is to increase the strength
of structural alloys by acting as obstacles to the motion of dislocations. Their efficiency in doing this
depends on their size, their internal properties, and their distribution through the lattice.
Dispersants are larger particles that behave as a second phase as well as influencing the behavior of
the primary phase. They may be large precipitates, grains, or polygranular particles distributed
through the microstructure.
Inclusions are foreign particles or large precipitate particles. For example, inclusions have a
deleterious effect on the useful strength of structural alloys. They are also often harmful in
microelectronic devices since they disturb the geometry of the device by interfering in
manufacturing, or alter its electrical properties by introducing undesirable properties of their own.
Voids (or pores) are caused by gases that are trapped during solidification or by vacancy
condensation in the solid state. They are almost always undesirable defects. Their principal effect is
to decrease mechanical strength.
2.10 Observationof dislocations
Introduction
A wide range of techniques has been used to study the distribution, arrangement and density of
dislocationsandtodeterminetheirproperties. The techniques can be divided into five main groups.

(1) Surface methods, in which the point of emergence of a dislocation at the surface of a crystal is
observed and studied.
(2) Decoration methods, in which dislocations in bulk specimens which are transparent to light are
decorated with precipitate particlestoshowuptheirposition.
(3) Transmission electron microscopy in which the dislocations are studied at very high
magnification inspecimens 0.1 to 4.0 µm thick. This is the most widely applied t e c h n i q u e .
(4) X-ray diffraction, in which local differences in the scattering of X-rays are used to show up the
dislocations.
(5) Field ion microscopy, which reveals the position of individual atoms.
Computer-based methods have also been employed extensively in recent years. Computer
simulation has been used, for example, to assist in the analysis of images obtained by transmission
electronmicroscopy and to predict the atomic configuration around dislocations in model crystals.
1. SurfaceMethods
If a crystal containing dislocations is subjected to an environment which removes atoms from the
surface, the rate of removal of atoms around the point at which a dislocation emerges at the surface
maybedifferent from that for the surrounding part of the crystal.
The difference in the rate of removal is because of different properties of the dislocation such (1)
lattice distortion and strain field of the
dislocation (2) geometry of planes
associated with a screw dislocation (3)
concentration of impurity atoms at the
dislocation . If the rate ofremoval is more
rapid around the dislocation, pits are
formed at these sites and if less rapid
small hillock are formed.
The most common and useful methods to
form pits are chemical and electrolytic
etching. Other methods include thermal
etching, in which the atoms are removed
by evaporation when the crystal is
heated in a low pressure.
In figure 1 formation of etch pits at the site where a dislocation meets the surface is shown.
Figures1(a) and 1(b) show formation pits at edge dislocation. Figures 1 (c) and (d) show
formation of spiral pit at a screw dislocation.

2. Decoration Method
The dislocations in transparent crystals are not normally visible. However, it is possible to
decorate the dislocations by precipitating foreign atoms along the line of the dislocation. Sites along
the dislocation are favoured for precipitation by the lattice distortion there. The position of the
dislocation is revealed by the scattering of the light at the precipitates and can be observed in an
optical microscope. The procedure with the widest range of application, which has been used to
study dislocations in alkali halide and semiconductor crystals, is to dope with impurity atoms. By
suitable heat treatment, the precipitation of the foreign atoms can be induced along the
dislocations. By a similar method, dislocations in silicon can be decorated and then observed with
infrared radiation by diffusing in metallic elements such as copper or aluminium. These decoration
techniques can be used in combination with etch-pit studies to demonstrate the one-to-one
correspondence between dislocations and etch pits.
2.11 Electron Microscopy
General Principles
Electron microscopy is the most
widely used technique for the
observation of dislocations and other
crystal defects, such as stacking
faults,twinandgrainboundaries,and
voids. Transmission electron
microscopy is applicable to a wide
range of materials, the only
conditions arethat the specimens can
be prepared in very thin section and
that they remain stable when
exposed to a beam of high-energy
electrons within a high vacuum.
The electrons in a conventional
transmission electron microscope
have an energy of typically 100
keV. The electrons behave as de
Broglie waves, with a wavelength of
3.7 pm at 100 keV. In practice, lens
aberrations and electrical and
mechanical stabilities place the limit at typically 0.2 to 0.4nm.
Schematicraydiagramsforaconventionaltransmissionmicroscope are shown in Figure.
Electrons emerge from the condenser lens onto the specimen, which is usually a disc a few mm in
diameter. The diffracted electron beams are brought to focus in the back focal plane of the
objective lens (Fig. a), which is the plane of the diffraction pattern. When the microscope is

operated in the diffraction mode, the diffraction lens is focused on the back focal plane and the
subsequent lenses project a magnified diffraction pattern on the fluorescent screen. The objective
lens also produces an inverted image of the specimen in the first image plane, however, and if the
diffraction lens is focused on this plane, the microscope is in the imaging mode and produces a
magnified image (Fig. b).
An aperture is placed in the objective back focal plane to permit only one beam to form the image.
The image reveals the variation in intensity of the selected electron beam as it leaves the
specimen. If the specimen is perfectly flat, uniformly thick and free of defects, the image is
homogeneous with no variations in intensity. Image contrast only arises if variations in beam
intensity occur from one part of the specimen to another. For example, if the specimen is bent
becauseofdislocations and other crystal defects then strong variations in intensity can result. Thus
from intensity variations in diffracted beam of electrons one can calculate density of defects
present in the specimen.

M sc sem iv u ii

Recommended

Recommended

More Related Content

Similar to M sc sem iv u ii

Similar to M sc sem iv u ii (20)

Recently uploaded

Recently uploaded (20)

M sc sem iv u ii