STRESS FIELD OF DISLOCATION

The Burger's Vector: The Burger's vector b denotes actually the dislocation-displacement vector. A dislocation can be very well described by a closed loop surrounding the dislocation line. This loop, called the Burger's circuit is formed by proceeding through the undisturbed region surrounding a dislocation in steps which are integral multiples of a lattice translation. The loop is completed by going an equal number of translation in a positive sense and negative sense in a plane normal to the dislocation line. Such a loop must close upon itself if it does not enclose a dislocation, or fail to do so by an amount called a Burger's vector

s = n_aa + n_bb + n_cc

Where n_a, n_b, n_c are equal to integers or zero and a, b, c are the three primitive lattice translations.

 

 

Fig. 23

Defects in Crystal

The Burger's circuit S1234F is shown by dark line in fig. 23 for a screw dislocation. Starting at some lattice point S at the front of the Fig. 23 crystal, the loop fails to close on itself by one unit translation parallel to the dislocation line. This is the Burger's vector which always points in a direction parallel to the screw dislocation. If the loop is continued, it will describe a spiral path around the Burger's dislocation just like the thread of a screw. In the figure, the height of the step on the top surface is one lattice spacing i.e., b, thus b is a vector giving both the magnitude and the vector of the dislocation. It must be some multiple of the lattice spacing so that an extra plane of atoms could be inserted to produce a dislocation. The dilatation ∆ at a point near an edge dislocation can now be described to be given by

where b is the Burger's vector which measures the strength of the distortion caused by the dislocation, r is the radial distance from the point to the dislocation line and  is the angle between the radius vector and the slip plane .as shown in fig. 20. Similarly, the atoms which are on a sheared lattice in a screw dislocation being on a spiral ramp, are displaced from their original positions in the perfect crystal according to the equation of a spiral ramp i.e.

 

where the z-axis lies along the dislocation and u, is the displacement in that direction. The angle  is measured from one axis perpendicular to the dislocation. Thus when  increases by  the displacement increases by a quantity b, the Burger's vector, which measures the strength of the dislocation. The Burger’s vector of a screw dislocation is parallel to the dislocation line while that of an edge dislocation, it is Perpendicular to the dislocation line and lies in the slip plane. In general cases, the Burger's vector may have other directions with respect to the dislocation and for these cases the dislocation is a mixture of both edge and screw types. Thus the mixed dislocation is defined in terms of the direction of the Burger's vector.

Defects in crystals and Elements of Thin Films

Stress Fields around Dislocations : Stress field of a screw dislocation: We know that the core of dislocations is a region within a few lattice constants of the centre of dislocation and that it is a "bad" region where the atomic arrangement of the crystal is severely changed from the regular state. The regions outside the core are "good" regions and the strains in these regions are elastic strains and so these can be treated by the theory of elasticity as an elastic continuum and the core region can be added later as a proper correction term. The calculation of dislocation energy is simple for a straight screw dislocation but similar results are obtained for edge dislocation.

Let us have a cylindrical shell of a material surrounding an axial screw dislocation. Let the radius of the shell be r and the thickness dr, The circumference of the shell is  and let it be sheared by an amount b, so that the shear strain

 

Fig. 24

                                                                                                        (11)               

and the corresponding shear stress in the good region is,

                                                                               (12)

where G is the shear modulus or modulus of rigidity of the material. A distribution of forces is exerted over the surface of the cut for producing a displacement b and the work done by the forces to do it gives the energy E_s of the screw dislocation.

Hence,

Defects in Crystal

                                                                              (13)

where F is the average force per unit area at a point on the surface during the displacement and the integral extends over the surface area of the cut. The average value is to be taken because the force at a point builds up linearly from zero to a maximum value as the displacement is produced. Thus the average force

is half the final value when the displacement is b i.e.,

                                                                                    (14)

Putting it in (13), we get

But dA = dz sr and so for a dislocation of length l, we have

                                                                                      

Thus, total elastic energy per unit length of a screw dislocation is given by

                                                                       (17)

where R and r₀ the proper upper and lower limits of r. The energy depends upon the values taken for R and r₀ is suitable when it is equal to about the Burger's vector b or equal to one or two lattice constants and the value of R is not more than the size of the crystal. Actually, however in most cases K is very much smaller than the size of the crystal. The value of R/r_0­ is not important as it occurs in  the logarithmic term.

Stress field of an edge dislocation: The calculation of the stress field is done on the assumption that the medium is isotropic having a shear modulus G and Poisson's ratio ʋ Let us consider the cross-section of a cylindrical material of radius R whose axis is along the z-axis and in which a cut has been in the plane y = 0, which becomes the slip plane. The portion above the cut is now slipped to the left by an amount b, the Burger's vector along the x-axis so that the new position assumes the shape shown dotted in fig. 25. Thus, a positive edge dislocation has been produced along the z-axis. Let σ_rr be the radial tensile stress, ie, compression or tension along the radius r and let σ_θθ be the circumferential tensile stress i.e., compression or tension acting in a plane perpendicular to r. Let τ r_θ denote the shear stress acting in a radial direction. As seen from fig. 20, it is an odd function of x, considering the plane y = 0 and is found to be proportional to (cos /r). In an isotropic elastic continuum σ_rr and σ_θθ compression or tension acting in a plane perpendicular to r. are found to be proportional to (sin /r) because we require a function which varies as 1/r and which changes sign when y changes sign. Also it can be shown dimensionally that the constants of proportionality in the stress vary as G and b.

Defects in crystals and Elements of Thin Films

Without giving the details of calculations here, the stress field of the edge dislocation in terms of r and  are given by the following : .

                                           ……(18)

       and                                                                   …..(19)

where the positive values of σ are for tension and negative values for compression. Above the slip plane σ_rr is negative giving a compression, below the slip plane, it corresponds to a tensile stress. It may be noted that for r = 0, the stresses become infinite and so a small cylindrical region of radius r_o around the dislocation must be excluded. This is necessary because in the bad region, the theory of elasticity

 

 

Fig. 25

does not hold as the stresses near a dislocation are very large. To know the value of r_o, let us put r_o = b, the magnitude of the strain there is then of the order of 1/2π(1-v)≈1/4 which is too much large to be treated by Hooke's law. We shall now calculate the energy of formation of an edge dislocation of unit length. The final shear stress in the plane y = 0 is given by (19) by putting θ = 0. For a cut along z-axis in a unit length, the strain energy for edge dislocation will be given by

Defects in Crystal

             ….(20)

 

This shows that the energy of formation becomes infinite if R becomes infinite. But even in large crystals the stress field are actually displaced some distance by other dislocation so that R = 10^-3 cm.

Assuming r₀ = 5 ´10^-8 cm. for a dislocation in copper

       E_e = 3´10^-4 erg/cm = eV/atom plane

Since    G = 4 ´10¹¹ dynes/cm²

         b = 2.5´10^-8 cm. and v = 0.34

In the case of screw dislocations its value is about (2/3) of this. The core energy of edge dislocation should be added to the elastic strain energy but it is of the order of 1eV per atom plane which is much less than the elastic strain energy and can be neglected. For a screw dislocation in the Z-direction in a cylindrical material, the stress field is given by a shear stress, according to (12).

                                                                                          …….(21)

 

 

 

 

Fig. 26 Low angle grain boundary (a) Two crystals Joined Together

(b) Grain boundary formed with 2 rows of dislocations.

There is no tensile and compress ional stress in this expression and this is perhaps due to the fact that there is no extra half plane in a screw dislocation. Also in this case the stresses are independent of θ expecting thereby that the stress field is cylindrically symmetric.

Defects in crystals and Elements of Thin Films

We can also explain the free energy of a dislocation. The contribution to the free energy by entropy, in a dislocation, is very small as compared to the strain energy and so the free energy in crystals of ordinary size at room temperature can be assumed to be nearly equal to the strain energy. Since the strain energy is positive, the free energy increases by the formation of dislocation. Hence no dislocation can exist as a thermodynamically stable lattice defect

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. 26 (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. 26 (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

                                                        …….(22)

Where b is the Burger’s vector of the dislocations and q is small

Burger’s model of low angle grain boundary has been was confirmed experimentally by Vogel and co-workers for germanium single crystals. A germanium crystal was grown from a seeded melt along <100> direction. When the surface of this crystal was etched with a suitable chemical (acid), the terminus of a dislocation at the surface become a nucleus of the etching. action and a row of each pits was formed. It is shown diagrammatically in fig. 27. On examining these boundaries under very high optical magnification they were found to consist of regularly spaced conical pits. By counting the number of these etch pits, we are able to find out the number of dislocations in the crystal grain boundaries. The distance D between  the pits is obtained by counting. The relative inclination angle q was also measured by means of X –ray diffraction experiments. From this value of q and knowing the value of b = 4.0 A⁰ in germanium, the value of D was calculated theoretically. This was found to be in very good agreement with the experimental etch pit interval.

Defects in Crystal

 

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