18 Dislocations and planar defects

Prof. P. N. Kotru

CONTENTS

18.1	Dislocations and  Crysta l Growth.
18.2	Pla nar Defects
18.3	Twinning in crystals
18.4	Observatio n of Str uctur a l Defects.
18.5	Summary

18.1   Dislocations and Crys tal Growth.

In addition to explanation o f the nature of slip, yield strength, work hardening and other mechanical properties, dislocations play an important role in the theory of crystal growth.

Crystals can be grown either from solution or from pure melt or from vapour phase. Growth from a liquid solution is determined by the degree of supersaturation,

while growth from a melt from determined by the degree of super cooling. In a similar way, the driving force for the growth of a crystal from vapour is determined by the degree of supersaturation which is defined as (p-p0) /p0, where p is pressure of the vapour and p0 is the vapour pressure of the solid.

In spite of the fact that lattice structure is considered as perfect, the crystal may have imperfect faces. An imperfect face is the one which may consists of terraces of steps and kink sites (re- entrant corners). The step covers only a part of the face. If the crystal is to grow bigger then an extra plane of atoms or ions must get attached to the surface of a crystal. If a single atom or ion (say the one marked 1 in figure 18.1) gets attached to the surface of the crystal, it will not be held on to the crystal very firmly and there is every probability, that it will get detached from surface of the crystal again. However, if the same atom gets attached at the kink site, it will be held more firmly. Growth can occur on these imperfect faces, even when the supersaturation is small because atoms or ions can keep on getting deposited along steps or kinks and get firmly attached to the surface by bonding with two or more of its atoms.

Figure 18.1: Schematic diagram showing kink sites as preferred sites for atoms to get attached.

The growth of this kind cannot continue indefinitely, because as the steps advance they leave behind themselves smooth crystal faces. Further progress in growth can occur only by the deposition of atoms or ions on these faces to form a nucleus for the next layer.

Gibbs, Volmer, Becker and Doring and others have developed the theory of growth of a crystal with a perfect lattice structure. The nucleus is said to be stable only if it were greater than about 100 atoms in diameter and that there is no chance for such a nucleus to form unless the degree of supersaturation is more than 50 %. In other words, there is least possibility of a crystal of perfect lattice structure to continue growing indefinitely in a solution of low supersaturation.

Turning point in the understanding of crystal growth came after VolmerSchultze observed growth of iodine crystals at vapour supersaturations less than 1%. This brought into focus a substantial disagreement between experimental result and the theory.

As explained above, for continued growth to take place there is need of a structure in the crystal that makes it to always have steps on its faces, however big it may grow. It was Frank who offered solution by suggesting that this kind of structure exists if the crystals contain screw dislocations. In the presence of a screw dislocation, it is not necessary to nucleate a new layer. Figure 18.2 illustrates explanation given by Frank in 1949.

Figure 18.2: Crystal with a screw dislocation perpendicular to the top surface, providing a spiral ramp and a perpetual step

In this figure, screw dislocation (defined by slip vector parallel to axis of dislocation) emerges on the surface of the crystal at right angles. It offers a step on the surface connecting the end of the screw dislocation with the edge of the crystal. If any extra atom attaches itself to the end of a screw dislocation (i.e., to the step), it is much more firmly held by the crystal surface than is the case if it is simply attached to a flat surface of the crystal. In terms of the model in figure 18.2, if the atom goes into the position shown by the arrow it is attached to the crystal by three of its eight faces instead of just by only one. Thus, in general, it is very much easier for a crystal to grow by deposition of extra atoms to the step of a screw dislocation than by deposition of atoms to perfectly flat faces. Significantly, the step provided by screw dislocation does not end by the addition of these extra atoms to it; it simply sweeps round the actual point at which the screw dislocation intersects the crystal face. In other words, it is never ending step or one can say it is a perpetual step. Figure 18.3 illustrates the continuous development of a spiral step due to the intersection of a screw dislocation with the surface of crystal.

Screw dislocation provides a built- in step which never ends from the growth surface on further addition of atoms. When atoms get deposited at a constant rate to the step provided by screw dislocation, all parts of the step achieve the same constant linear growth velocity. This results in an angular growth velocity about the centre which is greater near the dislocation than far from it, as a result of which step tends to assume a spiral form winding around the central dislocation. As already explained, presence of screw dislocation providing a never ending step on the surface of a crystal eliminates necessity to nucleate new layers for growth to continue.

Experimental confirmation of spiral steps on crystal surfaces has been obtained by Griffin (1950), Dawson and Vand (1951), Verma (1951) and several others thereafter. Using optical techniques like multiple-beam interferometry and phase-contrast microscopy spiral growth steps of step height 15 Å or one cell constant on the surfaces of crystals have been observed.

Figure 18.3: Development of a spiral step produced by intersection of a screw dislocation with the surface of a crystal

18.2 Planar Defects

These are two dimensional defects e.g., grain boundaries and twin boundaries.

18.2.1 Simple grain boundary

All planes of a single crystal are in the same orientation of a given crystallographic direction. This is true for a perfect single crystal. However, there are examples of two crystals joined at a boundary containing a crystal axis common to both crystals. The two or more than two grains are misoriented with respect to each other. Bragg (1940) and Burgers (1940) proposed dislocation model of grain boundary. According to Burgers the boundaries of two adjoining grains or two crystallites at a small angle inclination with each other can be thought of in terms of dislocation model which emerges naturally from the crystal geometry.

Figure 18.4: A simple small angle grain boundary (a) two grains have a common axis and angular difference in orientation ө (b) two grains have joined and formed a bicrystal.

Figure 18.4 shows two simple crystals that have a common axis. The two grains are joined to form a bicrystal. These two grains differ in orientation by a very small angle θ about their common axis. According to dislocation model the two low angle grain boundaries between adjoining crystallites or crystal grains consist of arrays of dislocations as shown in figure 18.4 (b).The boundary occupies a (010) plane in a simple cubic lattice and divides two parts of the crystal that have a [001] axis in common. Such a boundary separating the two grains is also called a pure tilt boundary.

The misorientation is defined by a small rotation θ about the common [001] axis of one part of the crystal relative to the other. The tilt boundary is represented as an array of edge dislocations of spacing D = b/ θ, where b is the burgers vector of the dislocations .The vector b which defines the magnitude and direction of the slip is called the burgers vector.

Experiments have substantiated the Burgers model of low angle tilt boundaries. It was Vogel who experimentally measured the angle of tilt θ by using etch pit method and X-ray diffraction. A germanium crystal was etched in a suitable chemical etchant for a specific period. It was observed that a row of equidistant etch pits was formed along the grain boundary. The distance D between the consecutive etch pits along the row was measured with the help of microscope. Using the formula D=b/θ and substituting for D and the Burgers vector b, θ was calculated. The misorientation θ as measured by x- rays and as calculated from etch pit counting method matched very well. This established the Burgers dislocation model for low angle grain boundaries.

If θ is large, the dislocations along the boundary are closer together. When θ is very large, the dislocations are separated by only one or two atomic spacings and almost all the misfit consists of atomic disorder on the boundary. The dislocation model becomes applicable only when θ is so small that the dislocations are many atomic spacing apart as a consequence of which most of the material on the boundary is elastically deformed.

18.3 Twinning in crystals

Twinning is also a type of defect which occurs in crystals. It is quite common that crystals grow in groups, and the individual crystals composing any group may have different relations to each other. In most of such cases these relations are irregular which may depend on variety of reasons including accidental conditions of growth or any other. However, in some specific cases, the individual members of the group bear definite relationship and follow some law. Sometimes, the grouping is such that the two individual crystals of the similar elements have all faces, edges etc. parallel to each other. It is known as parallel growth or grouping. Two individual crystals growing together with only part of their similar faces, edges, etc. in positions are said to form a twin crystal or group. Both the cases are examples of phenomenon of crystal intergrowths. It happens when two or more than two crystals try to occupy the same space during growth as happens in the case of human twins.

We, therefore, define twin crystals as intergrowth of two or more individua l crystals in such a way as to have parallelism of some parts of different individual crystals and at the same time other parts of the different individual crystals are in reverse positions with respect to each other. Externally they give the appearance of two or more crystals symmetrically joined and may in some cases appear to be in the form of cross or star or any other shape of an aggregate. The figures 18.5(a-d) below are some examples which project the external appearance of twinned crystals.

It is rather difficult to give a rigorous and unique definition of twinning viz., Cahn. Quoting some workers viz., Cahn (1954), Friedel (1926) and Bendersky (1988), twinning has been described as follows:

1. “A twin is a polycrystalline edifice , built up of two or more homogeneous portions of the same crystal species or juxta position , and oriented with respect to each other according to well-defined laws”
2. “A twin is a homogeneous crystalline aggregate  in which one observes a large number of identical mutual orientations between crystals that one can rule out a random cause.”
3. “If the diffraction pattern of a polycrystalline edifice of N orientational variants with a fixed, not random, orientational relationship between pairs can be indexed with less than 3N reciprocal- lattice vectors, twinning (or “hypertwinning”) can be said to exist.”

Figure 18.5: Externa l appearance of some twinned crystals

Boundary which separates the two twinned crystals is called as twin boundary. Twin boundary is a planar defect.

In most of the cases a twin crystal grows as such from the beginning. The two (or more ) individual crystals forming a twin crystal have different orientations of their atomic structures but the different positions of the crystal network have certain planes or directions in common such as to make it possible to derive one orientation from another by some simple movement. Therefore, formation of twin crystals is generally as a result of a simultaneous and regular growth according to two interlocking orientations of the same atomic network. The composing parts of a twin crystal are geometrically related to each other. For example, one part could be thought of being derivable from the other by reflection over a plane common to both. It is also possible that one part could be derived from the other by a revolution of 180 0 about some crystal line common to both. The plane and axis that are involved in the said operations are known as twinning plane and twinning axis.

Twinning is a defect which affects structure sensitive properties of a crystal. There are different types of twinning in crystals. We may briefly mention them here without going deep into their details. The different types are:

1. Transformation twins Twinning caused by, or attributable to, phase transitions is called transformation twinning
2. Growth twins Brazil twins of quartz crystals are typical examples of growth twins. It is almost impossible to detwin a quartz crystal having Brazil twins. Several varieties of growth twins include annealing twins, repeated twins, mimetic twins, lamellar twins and polysynthetic twins.
3. Mechanical twins In some crystals it is possible to cause atom movements on application of mechanical stress in such a way that results into a new crystal of different orientation but identical structure.

Thus a part of the parent crystal is changed into its twin, and the two together constitute a mechanical twin. The interface between the parent crystal and its twin produced by shear deformation is called habit plane or twin plane.

Twinning in crystals is revealed by etching and /or by observing crystals under polarizing microscope.

18.4  Obse rvation of Structural Defects.

Having known about various types of imperfections in real crystals, it becomes important to have some knowledge of means (techniques or methods) used to detect them. In other words we have to know about the techniques enabling us to make direct observation of defects in real crystals. In this section we will summarily describe these techniques.

Two main categories of these techniques are (A) Surface Methods and (B) Bulk Methods. Let us describe them one by one.

18.4.1 Surface Methods

Surface methods include microscopic (optical and electron microscopy) examination of habit faces of crystals. Study of microstructures on the surfaces of a crystal can lead to an understanding of deviations from normal growth of a crystal to imperfect surfaces. It is through these techniques that spiral steps were observed on crystal surfaces which supported Frank’s model of crystal growth.

Fractography is another means of looking at defects in crystals. Several crystals exhibit easy cleavage along certain crystallographic planes. Defects in crystals, if any, do affect normal propagation of cleavage crack leading to disruption of cleavage. Examination of cleaved surfaces can lead to some clue about the presence of certain type of structural defects like impurities etc.

Very popular technique of surface methods is etching (chemical or thermal). If a crystal is put in a chemical solvent under suitable conditions and parameters of growth like concentration, temperature and time, it produces etch pits at the sites of defects emerging on  the crystal surface. Through this technique one can get defects like dislocations, superficial defects, grain boundaries, twin boundaries and cracks in crystals exposed. Under suitable conditions of etching which include chemical solvent, temperature, concentration and time, the shape and structure of etch pits reveal symmetry of the crystal face under consideration. It is through this method supplemented by x-ray diffraction that Vogel et al were able to confirm the dislocation model for low angle tilt boundaries in crystals. It is an inexpensive, simple and rapid method for determining dislocation density in crystals.

The details of this method will be discussed separately in the relevant sections.

18.4.2   Bulk Methods

This   category   of   techniques   includes   decoration;   x-ray   diffraction   topography and  transmission electron microscopy. Decoration is a technique in which dislocations in transparent crystals are decorated by causing a precipitate to form on them so that dislocations are delineated by the precipitate particles. As for example, decoration of dislocations in AgBr crystals is done by exposing these crystals to light when photolytic silver is deposited on the dislocations.

The more conventional decoration technique is to precipitate impurities on to the dislocations. Dislocations in ionic crystals have been studied by this method. In this technique the crystal is heated to relatively high temperature so that impurities can diffuse to dislocations and decorate them.

The other technique with the help of which one can observe defects in the bulk is electron microscopy of thin foils. This technique consists of thinning a specimen to several thousand angstroms thickness, so that it can be examined under transmission electron microscopy.

A versatile technique amongst bulk methods is X-ray diffraction topographic techniques. These techniques can easily be used to reveal low angle boundaries, inclusions of foreign material, dislocations and other defects like growth facets, growth bands etc. There are different experimental arrangements for recording X-ray topographs like Berg-Barrett (Newkirk) techniques and Lang technique. Topographs may be obtained either in reflection or in transmission mode. Advances in recent years have been made to improve the efficiency and sensitivity of X-ray topographic techniques. It includes high resolution X-ray diffraction abbreviated as HRXRD. X-ray topographic techniques, using synchrotron radiation, has also supplemented the advance of X-ray diffraction techniques for making direct observation of dislocations and other defects.

Detailed discussion on direct observation of defects in crystals particularly through application of x-rays, electrons and etching will be done separately under the relevant sections that follow. All the techniques outlined above have their advantages and limitations .It would, however, be of interest to make a comparison.

The same is summarized in the table given below:

Table 18.1 

18.5 SUMMARY:

References.

1.Read,W.T. “ Dislocations In Crystals”, McGraw-Hill Book Co.,Inc.,N.Y.,1953.

2. Kittel,C. : “ Introduction to Solid State Physics”, Wiley Eastern Ltd.,N.Delhi,1985.

3. Dekker,A.J. “ Solid State Physics”, Macmillan,London,1979.

4. Johnston,W.G.: “ Progress in Ceramic Science”, Pergamon Press Ltd.,Oxford,Vol.2,1962.

5. Amelinckx,S.” The Direct Observation Of Dislocations,Supplement 6 in Solid State Physics,

Ed.Seitz,F.,Turnbull,D. Academic Press 1964.

For extensive discussion on dislocations:

1. Friedel,J. : “ Dislocations”, Addison- Wesley Publication Co.,1964.

For historical account.

Mitchell, J.W. :in”Direct observation of Imperfections in Crystals” ed.Newkirk and Wernick, J.H., Interscience

Publishers, 1962.