9 Crystal Defects: Line Defects

Amarjeet Singh

    Learning Outcomes

 

After studying this module, you shall be able to

  • The classification of one dimensional line defects
  • How to draw a Burger’s circuit
  • How to differentiate between Edge dislocation and Screw dislocation
  • How to relate Burger’s vector with edge dislocation and screw dislocation
  • How to understand the motion of defects

   LINE DEFECTS: DISLOCATIONS

 

Dislocations are linear defects; they are lines through the crystal along which crystallographic registry is lost. Their principle role in the microstructure is to control the yield strength and subsequent plastic deformation of crystalline solids at ordinary temperatures. Dislocations also participate in the growth of crystals and in the structures of interfaces between crystals. They act as electrical defects in optical materials and semiconductors, in which they are almost always undesirable. The concept of a dislocation in a solid was introduced by Volterra in the nineteenth century. However, it was not until much later that their relevance to the deformation of crystals was recognized; the notion of a dislocation as the carrier of plastic deformation did not appear until 1934. Since the 1950’s it has been possible to observe and study dislocations directly using such techniques as transmission electron microscopy and x-ray topography. While dislocations influence many aspects of physical behavior, they are studied almost exclusively in Materials Science. Most of you are encountering the concept of a dislocation for the first time.

 

The edge dislocation

 

The simplest way to grasp the idea of a dislocation is to imagine how you might go about creating one. We begin by making an edge dislocation, which is the easiest type of dislocation to visualize in a crystal, and follow the recipe laid down by Volterra. Consider the solid body that is drawn in Fig. 8.1 For the present purpose it does not matter whether the body is crystalline; it may be easier to imagine that it is rubber.

 

Figure 8.1: A block of material with a planar cut indicated by the shading.

 

To create an edge dislocation in this body we first make a planar cut part way through it, as illustrated by the shaded region in the figure. We then fix the part of the body below the cut, and apply a force to the body above the cut that tends to displace it in the direction of the cut, as illustrated in Fig. 8.2. The upper part slides, or slips over the lower by the vector distance b, which is the relative displacement of the two lips of the cut. The plane of the cut, where slip occurs, is called the slip plane. The cut is finite and constrained at its end, so material accumulates there. The end of the cut, or equivalently, the boundary of the planar region of slip, is a linear discontinuity in the material. The situation after the slip is shown in Fig. 8.2.

 

 

Figure 8.2: The upper part of the body has been slipped by the vector b over the shaded area. The terminal line is a discontinuity, marked by the heavy line.

 

Now suppose that we have a mechanism for welding the material back together which is so efficient that it is impossible to tell after the fact that the weld was ever made. If we match the material above the shaded plane to that below so that there is no physical discontinuity across the plane and weld the lips together, the shading disappears since it is impossible to tell that the material was ever separated. However, matching the material across the plane of slip requires that excess material be gathered at the line at which slip terminates. This line is, therefore, a linear defect in the material. It is called an edge dislocation. It is an isolated defect, as shown in Fig. 8.3 since after the material has been re-welded there is no unique way to determine how the dislocation was created. For example, the dislocation would be exactly the same if the material to the right of it on the slip plane were slipped by the vector – b on the same plane. Only the exterior step indicates the origin of the dislocation, and this may be removed, or may have pre-existed the formation of the dislocation.

 

However it was created, the edge dislocation in Fig. 8.3 has the property that it defines an element of slip, b, where the vector b is called the Burgers vector of the dislocation. We can always identify the slip plane of a dislocation like that shown in Fig. 8.3. It is the plane that contains both the Burgers vector, b, and the line of the dislocation. However the dislocation actually came to its present position, its net effect is that the material above the slip plane to the left of the dislocation (in the direction of – b)   has been displaced by b relative to that below the slip plane. The dislocation is a linear defect whose location is defined by its line and whose nature is characterized by its Burgers vector, b. In the case shown in Fig. 8.3 the Burgers vector is perpendicular to the dislocation line. This perpendicularity is characteristic of an edge dislocation.

 

Figure 8.3: isolated edge dislocation after the cut surface has been rejoined.

 

If a dislocation moves the area that has been slipped grows or shrinks accordingly. Imagine that the dislocation is initially created at the left edge of the slip plane in Fig. 8.3, and is then gradually moved to the right edge. Applying the construction in Fig. 8.3 to the initial and final positions of the dislocation, it follows that the motion of the dislocation through the body causes the whole volume of material above the slip plane to be displaced by the vector b with respect to that below it, as shown in Fig. 8.4.

Figure 8.4: Final state of the body after an edge dislocation with Burgers vector b has crossed the whole of the slip plane shown.

 

Fig. 8.4 illustrates the connection between the motion of a dislocation (in this case, an edge dislocation) and the plastic, or permanent deformation of a material. As we shall discuss in more detail when we consider the mechanical properties of materials, plastic deformation changes the shape of a body without changing its volume, since the total number of atoms and the crystal structure remain the same. A change in shape that occurs at constant volume can always be represented geometrically as the sum of elementary deformations of a type known as simple shear. A simple shear is the kind of deformation that deforms a cube into a parallelogram; it changes the angles between initially perpendicular directions in the cube. The shear due to the passage of an edge dislocation is illustrated in Fig. 8.5. While the dislocation translates the top of the crystal rigidly over the bottom to create a discrete step, the Burgers’ vector has atomic dimensions, so the step is invisible. Macroscopic deformation is the sum of the slip caused by many dislocations.

 

Figure 8.5: Figure (b) is obtained from (a) by a simple shear of the top over the bottom. Figure (c) shows how the same shear can be caused by an edge dislocation.

 

An edge dislocation in a simple cubic crystal

 

The procedure that was used to create the edge dislocation that appears in Fig. 8.4 made no reference to the structure of the solid, and can be used to form an edge dislocation in any material. However, when the material is crystalline the ordered pattern of atoms restricts the values that the Burgers vector, b, can have. The restriction is introduced by the welding step that is used to change the configuration shown in Fig. 8.1 to that in Fig. 8.2. The welding must be so perfect that it is impossible to tell that the two surfaces were ever separated. If the solid is crystalline this can only be true if the crystal structure is continuous across the slip plane after the weld is made. It follows that the relative displacement across the slip plane must equal a lattice vector so that atoms can re-bond without changing their local atomic configurations. Since the relative displacement is equal to the Burgers vector, b, of the dislocation, b must be a lattice vector. If the dislocation is an edge dislocation, b must also be perpendicular to the dislocation line.

 

The geometry of an edge dislocation is relatively easy to visualize when the crystal has a simple cubic crystal structure. The atomic configuration around the dislocation line is more complicated in real crystal structures, but it is not necessary to deal with that complexity to understand the behavior of dislocations at the level we shall need in this course. Whenever we need to consider the crystallography of the dislocation we shall assume that the crystal structure is simple cubic.

 

An edge dislocation in a simple cubic structure is drawn in Fig. 8.6, which shows both a two-dimensional view and a three-dimensional section along the dislocation line. The dislocation can be created by making a cut in the crystal on the dashed plane that terminates at the dislocation line, displacing the material above the cut plane to the left of the dislocation by one lattice spacing, and allowing the atoms to re-bond across the slip plane. This recipe recreates the simple cubic unit cell everywhere except on the dislocation line itself (ignoring the small elastic distortion of the cells that border the dislocation line). Hence the Burgers vector, b, of the dislocation that is drawn in the figure is b = a [100], where a is a vector along the edge of the cubic unit cell.

 

 

Figure 8.6: An edge dislocation in a simple cubic structure. The dotted plane is the slip plane.

 

The process that creates the edge dislocation shown in Fig. 8.6 leaves one extra vertical half-plane of atoms above the slip plane. This extra half-plane terminates at the dislocation line, and is compressed there, as shown in the figure. The distortion at the dislocation line is local. The simple cubic arrangement of atoms is essentially restored a few atom spacings away from the dislocation line. The influence of the dislocation on the atomic configuration rapidly decays into a small displacement that decreases in magnitude with the inverse cube of the distance from the dislocation line. The local distortion near the dislocation line (or dislocation core) is indicated in the figure. In principle, the Burgers vector of a crystal dislocation can be any lattice vector; for example, it is geometrically possible for an edge dislocation to be the termination of any integral number of lattice planes. In reality, however, the Burgers vector is almost invariably equal to the shortest lattice vector in the crystal.

 

The Burgers circuit

 

While it is always possible to find the Burgers vector, b, of a dislocation by determining the slip that would be required to make it, this is often inconvenient. A simpler method uses a geometric construction known as the Burgers circuit.

 

To construct the Burgers circuit, choose a direction for the dislocation line and draw a clockwise closed circuit in the perfect crystal by taking unit steps along the lattice vectors. An example is drawn for a {100} plane in a simple cubic crystal in Fig. 8.7. If the same circuit is drawn so that it encloses a dislocation, it fails to close. The vector (from the starting position) that is required to complete the circuit is the Burgers vector, b, of the dislocation, and measures the net displacement experienced by an imaginary observer who completes a circuit around the dislocation that would be closed in a perfect crystal.

 

Figure 8.7: A Burgers circuit closes in a {100} plane of a cubic crystal, but fails to close by the Burgers vector, b, when the same circuit encloses an edge dislocations.

 

Motion of an edge dislocation: glide and climb

 

The reason that dislocations control the plastic deformation of crystalline solids is that it is relatively easy to move dislocations to produce shear deformation of the sort that is pictured in Fig. 8.8. It would be enormously difficult to shear a crystal by forcing the glide of rigid planes of atoms over one another; one would have to force the simultaneous reconfiguration of every crystal bond that crossed the slip plane. The same result is more easily achieved by moving dislocations stepwise through the crystal. Stepwise dislocation motion requires a much smaller force since each elementary step can be accomplished by reconfiguring only the bonds that neighbor the dislocation line. The stepwise motion of an edge dislocation in a simple cubic crystal is illustrated in Fig. 8.8. In order for the dislocation to move one lattice spacing to the right it is only necessary to break the bond indicated by the long dash in 8.9a and establish the bond indicated by the short dash.

 

The new configuration is shown in Fig. 8.8b. Of course one bond must be broken for each plane through which the dislocation threads, so a significant force is still required. But the force is small compared to that required to slip the upper part of the crystal as a rigid body. If the dislocation moves through the crystal in a sequence of individual steps like that shown in Fig. 8.9 it causes a net slip of the material above its plane of motion by the Burgers vector, b, and hence causes a rigid displacement of the whole upper part of the crystal.

 

Figure 8.8: Glide of an edge dislocation. Only a single bond must be broken per plane for each increment on glide.

 

The type of motion that is illustrated in Fig. 8.8 is called dislocation glide, and is relatively easy to accomplish. However, an edge dislocation cannot glide in an arbitrary direction. It can only glide in a particular plane, the slip plane or glide plane, which contains both the Burgers vector and the dislocation line.

 

 

Figure 8.9: Climb of an edge dislocation. Movement up out of the plane requires the elimination of atoms by vacancies. Movement down requires the addition of atoms.

 

When an edge dislocation moves out of its glide plane its motion is called climb. The climb of a dislocation is difficult at ordinary temperatures since it requires that atoms must be absorbed on or liberated from the extra half-plane of atoms that defines the dislocation line. The climb of an edge dislocation is illustrated in Fig. 8.9. The mechanism is slightly different depending on whether the dislocation moves up, which contracts the extra half-plane, or down, which extends it.

 

If the dislocation climbs up atoms must be liberated from the edge of the extra half-plane. Since the number of atoms is conserved, this requires the absorption of vacancies from the lattice. One vacancy is needed per plane the dislocation threads. If the dislocation climbs down it must add atoms to the extra half-plane, and can only do this by liberating one vacancy per plane into the matrix, as shown in Fig. 8.9b. Both processes are difficult except at high temperature when, as we shall see, the equilibrium concentration of vacancies is high and the exchange of vacancies and atoms is relatively easy. Because of the difficulty of climb at ordinary temperatures the plastic deformation of real crystals tends to occur through the motion of dislocations on well-defined planes that are the glide planes of the active dislocations. Under a microscope one can often see discrete slip steps on the surface of a crystal that has been deformed. These result from the glide of many dislocations on closely spaced, parallel planes. At high temperature climb becomes possible and the slip planes are less well-defined. For this reason most solids are relatively soft at high temperature.

 

Screw dislocations

 

Our discussion to this point has dealt only with edge dislocations, that is, dislocations in which the Burgers vector is perpendicular to the dislocation line. Dislocations in real crystals rarely have a pure edge character. Their Burgers vectors lie at various angles to their line directions. In the extreme case the Burgers vector is parallel to the dislocation line, which is the characteristic of a screw dislocation. A screw dislocation is difficult to visualize in a crystal, but can be created by a method suggested by Volterra that closely resembles the way the edge dislocation was formed. A screw dislocation of the general Volterra type is shown in Fig. 8.10.

 

Figure 8.10: A method for forming a screw dislocation in a solid.

 

To introduce a screw dislocation we slice the solid part-way through in the direction of its width, as shown in Fig. 8.10. But instead of displacing the material above the cut toward the dislocation line we displace it by a vector “b” that lies parallel to the dislocation line, as shown in the figure. The direction of the force required to do this is also indicated. The material is then re-welded so that it is continuous across the plane of slip. The residual distortion is concentrated at the dislocation line, which then constitutes an isolated linear defect.

 

If the body shown in Fig. 8.10 is crystalline then the cut surface shown in the figure is a plane of atoms. In order for the crystal to be continuous across the slip plane after it is rejoined, the displacement, b, must be such that this plane of atoms joins continuously onto a crystallographically identical plane. It follows that a closed circuit (Burgers circuit) that encloses a screw dislocation not only fails to close, but produces a net translation by b along the dislocation line, where b is a lattice vector. A circuit that starts on one plane of atoms finishes on another a distance b below. Continuing the circuit causes a displacement by b at each revolution, without the circuit ever leaving the atom plane. The effect of a screw dislocation is to join a set of parallel atom planes so that they become a single plane like one that would be created by extending a plane outward from the thread of a screw. Hence the name: screw dislocation.

 

As in the case of an edge dislocation the line energy of a screw dislocation is proportional to the square of its Burgers vector. Hence the Burgers vector of the screw dislocation is ordinarily the smallest lattice vector that is compatible with the direction of its line. A screw dislocation differs from an edge not only in its geometry but in the way it accomplishes plastic deformation. The most important qualitative differences concern its direction of motion under an applied force and its relative freedom of movement. Figure 8.10 suggests the connection between slip and dislocation motion for a screw dislocation. As the screw dislocation is displaced through the width of the body the material above its plane is slipped in the direction of the Burgers vector, hence along the length of the body. It follows that the longitudinal force shown in the figure acts to drive the screw dislocation sideways. If a screw dislocation is passed through the full width of the body it causes the shear shown in Fig. 8.4, which is same as that caused by the passage of an equivalent edge dislocation through the length.

 

In contrast to an edge dislocation, a screw dislocation can glide in any plane. Since the Burgers vector lies parallel to the dislocation line both are in any plane that contains the dislocation line, and the screw dislocation can move in any direction perpendicular to its line.

 

Value Addition:

 

Do You Know?

 

The important discovery of transmission electron microscope based on the fundamental principle of quantum mechanics, that is, wave-particle duality. It implies that the beam of particles can be treated as waves. The wavelength can be tuned depending on the kinetic energy of the particle and therefore it became possible to visualize ‘atoms’ in a solid. The Nobel Prize in Physics 1986 was divided, one half awarded to Ernst Ruska “for his fundamental work in electron optics, and for the design of the first electron microscope”, the other half jointly to Gerd Binnig and Heinrich Rohrer “for their design of the scanning tunneling microscope”.

 

 

TEM today is one of the most important analytical tool for material scientists. Direct observation of atomic planes, defects, vacancies at such a small length scale is boon to the solid state physics. Its scope has gone beyond solid state physics and biologists and chemist also often use it for analysis of their specimen.

 

Crystal Defects: Line Defects

  1. Suggested Reading

    For More Details (on this topic and other topics discussed in Text Module) See

  1. Neil W. Ashcroft and N. David Mermin, Solid State Physics, Thomson Brooks/Cole, Eastern Press Bangalore (India) 2005
  2. Charles Kittel, Introduction to Solid State Physics, John Wiley & Sons, Singapore 1999
  3. Some of the materials is taken from internet where the proper reference was not available, so we could not provide reference to such material. However it was first verified that the information was genuine. Some pictures were taken from internet which were freely available and are used in good faith.
  4. Wikipedia

    Glossary:

 

Adsorption: It is a phenomenon where atoms and molecules are physically bonded to the interface.

 

Interfacial tension: On an interface which is the boundary between two phases or two materials, the atoms have higher free energy that gives to interfacial tension.

 

Anisotropy of surface tension: surface tension is direction dependent on the crystal surface.

 

Burger’s vector: When we move around a dislocation in the steps of unit vectors with equal number of left and right movement as well as equal number of up and down movement to enclose dislocation, a vector is needed to enclose the loop. This extra length or vector is called Burger’s vector.

 

Burger’s circuit: A close circuit enclosing the dislocation is called Burger’s circuit.

 

Dislocation line: The line along which the imperfection is visible. The dislocation line is perpendicular to the plane of Burger’s circuit.

 

Slip plane: A slip plane (or glide plane) for a dislocation must contain both the Burgers vector and Line vector of the dislocation. Dislocation motion in this plane is said to be “conservative” as no atoms are added or subtracted from the dislocation line.