- Email: [email protected]
Atomic structure and energetics of constricted screw dislocations in copper
Materials Science and Engineering A234-236 (1997) 544-547
Atomic structure and energetics of constricted screw dislocations in copper T. Rasmussen a,b,*, K.W. Jacobsen a, T. Leffers b, O.B. Pedersen b a Center for Atomic-Scale Materials Physics, Department of Physics, b Materials Research Department, Risg National
Technical Laboratory,
University DK-4000
of Denmark, Roskiide,
DK-2800 Denmark
Lyngby,
Denmark
Received 3 February 1997
Abstract
Atomistic simulations of cross slip of a dissociated screw dislocation have been performed. Shapes and energetics of different dislocation configurations relevant to cross slip in an f.c.c. metal (Cu) are determined. The minimum stress-free activation energy and activation length in the Friedel-Escaig cross-slip mechanism are determined. The simulations reveal a new energetically favourable configuration of a dissociated screw dislocation not previously considered. The importance of this configuration to surface nucleated cross slip is discussed. 0 1997 Elsevier Science S.A. Keywords:
Cross slip; Simulations; Screw dislocation
1. Introduction
Cross slip of a screw dislocation is an intricate mechanism involving both long range elastic interactions and atomistic effects when the Shockley partials recombine. In this paper we describe the results of an atomistic approach to the problem of cross slip in Cu. Cross slip has earlier only been addressed theoretically by methods based on elasticity theory [l-4] mostly within the line
tension
approximation
and
isotropic
elasticity
the-
ory. In the present work we have performed atomic scale simulations of various dislocation configurations relevant to cross slip via the Friedel-Escaig cross-slip mechanism [1,5] and we determine the stress-free activation energy and activation length for this mechanism. We show that the two constrictions needed in this mechanism are not equivalent and that a configuration with just one of them is energetically favourable compared to two parallel partials in a single glide plane. Simulations show [6] that a free surface can act as a centre of nucleation of such a configuration. Hence, surface nucleated cross slip via a configuration with just one constriction is a possibility.
* Corresponding author. e-mail: [email protected]
Tel.:
+ 45 45253177;
fax:
+ 45 45932399;
0921-5093/97/$17.00 0 1997 Elsevier Science S.A. All rights reserved. PIISO921-5093(97)00311-O
The interatomic interactions are described by a many-body potential derived from the effective-medium theory [7]. The potential is slightly modified so as to produce a reasonable value for the intrinsic stackingfault energy and it also produces very reasonable values for the elastic constants as well as the equilibrium splitting width of the partials [6].
2. Results
and
discussion
In the Friedel-Escaig (FE) cross-slip mechanism [1,5] for the b = $ [l lo] screw dislocation sketched in Fig. 1, the two partials must recombine (constrict) in the primary glide plane before the subsequent redissociation in the cross-slip plane. The redissociation creates two nonequivalent twisted constrictions (A and B in Fig. 1) on the dislocation. We shall denote constriction A ‘edgelike’ and constriction B ‘screw-like’ because of the characters of the partials in the vicinity of the constrictions. The activation energy for the FE cross-slip mechanism is the sum of the energies of a pair of infinitely separated edge-like and screw-like constrictions. Noting this, we decided to calculate the activation energy by treating the two kinds of constrictions separately. We define the energy of a constriction as the difference in energy between a system containing a dislocation with a
T. Rasmussen
et al. /Materials
Science
constriction and a reference system of the same length containing two parallel Shockley partials in a single glide plane. The dislocation with B = 4 [110] is centered im a box tit&t 3111) facets as tree sudaces par&e1 to the disIocation. In the reference systems we appIy peric&c boudary conditions in the { 1 lOj dire&i0 u, i.e. along the dislocation line. The two constriction energies are found by first relaxing two reference systems each containing a pair of partials in either of the two possitie glide pknes. We theen extract the tag of one aE t&e reference systems and the bottom of the other reference system and make a sandwich with a system containing an already constricted and twisted dislocation. Keeping the atoms of the top and the bottom from the reference systems fixed in their relaxed positions we are able to mimic a semi infinite dislocation on either side of the constricted pad and the system can then t>e Aaxed using a standard energy minimization algorithm [S]. Using the other set of top and bottom from the reference sysCems and another initially constricted and twisted dislocation the second constriction system can be constructed. We use four (110) planes (equal to 26) with atoms in fixed positions in either end of the sjams containing a constrichm. ln Fig. 2 we show a relaxed configuration of a screw-like constriction. The system has been ctit in tv+wohalves, one part rotated and then cut again parallel to the partials, to display the uMoca%on jn >Is 5ti5 ‘lengIn. The atoms &rose 10 the uMoca%on cafes are co5omeh Ma&, The e@e-%& CconsIr~cIion hoes no1 &Ber %snaY& %rom ‘rhe screw-%e constriction. The length of the dislocation is denoted 1 am2 r);L- w&h 02 t+?esyskm peLqm&cu>ar to the d&w
(a>
I
and Engineering
A234&236
(1997)
5444547
545
+
W
4
Fig. 2. Screw-like constriction. The perfect Burgers vector i IllOJ points upwards in the figure. The system has been cut in two halves and one half has been rotated to display the dislocation in both glide planes. The length of the disIo&iion is I = 50 5 ~$2.7 nmj and Ihe width of the system is pi = 11.3 nm. ML By xqimg ) h Ihe rangfi 3%5t’h j3, b-32-7 n2L-qam3 w iv .a2Pm22gE 9.3 -14. B 22222 IO .w25 pDs>blP ID determine the constriction energies as functions of system dimensions. i”ne results of me simulations are summarized in Fig. 3. 123hDJ2
-
0.0 -
8 Fig. 1. The Friedel-Escaig cross-slip mechanism for a b = f [l lo] screw dislocation (a) Creation of a constriction in the primary plane, (111). (b) Dissociation in the shaded cross-slip plane, (11 l), creating two twisted constrictions A and B. (c) Two non-interacting constrictions The Burgers vectors in the two glide planes are indicated with arrows. Due to the different characters of the Shockley partials, the two constrictions are not equivalent. We denote the constrictions A and B ‘edge-like’ and ‘screw-like’, respectively.
9 10 11 12 13 14 Width of system / nm
15
Fig. 3. Energies in units of eV of the two different constriction types as a function of system dimensions, For each system the energy of the edge-like constriction (top), the energy of the screw-like constriction (bottom) and the sum of the two (middle) are shown. Notice the negative energy of the screw-like constriction.
546
T. Rasmussen
et al. /Materials
1
Q-EI edge-like m-m screw-like
0.4 % . k
Om3
d
0.1
Science
0.2
-0.1’
0
’ 10
I n 20 30 (110) pair/b
.
’ 40
50
Fig. 4. Energy contribution from individual AB (110) plane pairs to the total energy of the two kinds of constrictions. Edge-like constriction above and screw-like constriction below. The contributions from the static layers in either end of the systems are not shown. The length of the dislocations is 1= 50 b (12.7 nm) and the width of the systems is M‘ = 9.5 nm.
A striking feature is the negative energy of the screwlike constriction. This means that it is energetically favourable for the dislocation to adopt a twisted configuration with partials in both possible glide planes connected by a screw-like constriction, compared to two parallel partials in a single glide plane. From Fig. 3 it is possible to estimate the energies of the edge-like and the screw-like constrictions to 4.0 and - 1.0 eV, respectively. Hence, the activation energy for two independent constrictions in an infinite crystal, can be estimated to 3.0 eV. The small scatter of the energies is due to the uncertainty in the determination of the reference energy level, which can only be found within f 0.1 eV. Since the constriction energies have converged as a function of dislocation length I, the minimum separation between two non-interacting constrictions can be estimated to w 50 6. The value obtained for the activation length is in very good agreement with estimates based on elasticity theory [3,4,9]. The activation energy is, however, somewhat higher than the results from these works. As discussedelsewhere [2,6], the estimates of the activation energy from elasticity theory are in fact subject to rather large uncertainty due to the approximations of this theory and none of the mentioned works based on elasticity theory investigate the difference between the two types of constrictions. It is interesting to examine the energetics of the constrictions in more detail. To do so we utilize the fact that the systemscan be seenas an . ..ABAB... stacking of AB pairs of (110) planes along the dislocation line. In Fig. 4 we plot the energies of the AB pairs along the
and Engineering
A234-236
(1997)
544-547
dislocation line in two constriction systems relative to the energy of an AB pair from the reference systems. In this kind of plot the qualitative difference between the constrictions is very apparent. For the screw-like constriction the energy drops towards the centered constriction with a small increase in energy close to the constriction. This behaviour of the energy of the screwlike constriction can be explained in the following way. The self-energy of the partials will, according to dislocation theory [lo], decreaseas the partials acquire more screw character, and the area of the stacking fault between the partials is also reduced by the constriction. Furthermore, it is known that the interaction energy between two screw dislocations inclined at an angle of 45” is zero. The geometry involved here is not very different from this. The increase in energy around the constriction must be attributed to atomistic effects due to core overlap of the partials which of course cannot be included in an elastic description, but together the three energy lowering mechanisms are sufficient to make the constriction energy negative. For the edgelike constriction only the reduction of stacking fault area tends to lower the energy compared to two parallel partials, whereas the acquirement of more edge character increases the self-energy as well as the interaction energy. This explains the shape of the energy projection for this kind of constriction. It is worth noting once again that a configuration with a single screw-like constriction is energetically favourable compared to two parallel partials. For cross slip in the bulk it is difficult to imagine a cross-slip mechanism which would not require a pair of constrictions as in the FE mechanism. However, for a dislocation ending at a free surface it is possible to make the partials in one glide plane recombine at the surface and then redissociate into the other glide plane thereby creating a screw-like constriction on the dislocation. Pushing the screw-like constriction along the dislocation line would then produce cross slip without the edge-like constriction necessary for cross slip initiated in the bulk. The redissociation does not happen spontaneously [6], thus we can deduce that there is an energy barrier for this kind of surface nucleated cross slip. However, further work is required to determine the energy barrier for surface nucleated cross slip and it is therefore too early to draw any final conclusions about this kind of cross slip. As a direct application of such surface nucleated cross slip, it has been possible to perform a simulation of the annihilation of two screw dislocations of opposite signs [6]. The starting configuration was two screw dislocations split in two parallel glide planes separated by just 2.2 nm. The annihilation was initiated by one of the dislocations performing surface nucleated cross slip via the mechanism discussed above and it proceeded through a successionof dislocation reactions including
T. Rasmussen
et al. /Materials
Science
the formation of a sessile stair-rod dislocation as an intermediate step and a glissile dislocation loop comprising Shockley partials moving through the crystal. It was possible to monitor these reactions and thereby to exactly specify a set of possible dislocation reactions leading to annihilation.
and Engineering
A234-236
(1997)
54L.547
547
Acknowledgements Center for Atomic-Scale Materials Physics (CAMP) is sponsored by the Danish National Research Foundation. The present work was done as a collaboration between CAMP and the Engineering Science Centre for Structural Characterization and Modelling of Materials, Rise.
3. Conclusion References By using an atomistic approach to the problem of cross slip, we have determined the activation energy and activation length in the Friedel-Escaig cross-slip mechanism. Our result for the activation length is in very good agreement with earlier results obtained on the basis of elasticity theory. However, our result for the activation energy is higher than estimates based on elasticity theory. The non-equivalence of the two constrictions has not been investigated before, and we show that the screw-like constriction is energetically favoured compared to two parallel partials. Cross slip is also possible via a dislocation configuration with just a screw-like constriction nucleated at a free surface. The energy barrier for this kind of cross slip is still not known, but a simulation showing annihilation of two screw dislocations of opposite signs initiated by such cross slip has been performed.
[I] B. Escaig, in: A.R. Rosenfeld, G.T. Hahn, A.L. Bement Jr, R.I. Jaffee (Eds.), Dislocation Dynamics, Series in Materials Science and Engineering, McGraw-Hill, New York, 1968. B. Escaig, J. de Phys. (France), 29 (1968) 225. [2] G. Saada, Mat. Sci. Eng. Al37 (1991) 177. [3] M.S. Duesbery, N.P. Louat, K. Sadananda, Acta. Metall. Mat. 40 (1) (1992) 149. [4] W. Ptischl, G. Schoeck, Mat. Sci. Eng. Al64 (1993) 286. [5] J. Friedel, in: J.C. Fisher (Ed.), Dislocations and Mechanical Properties of Crystals, Wiley, New York, 1957. [6] T. Rasmussen, K.W. Jacobsen, T. Leffers, O.B. Pedersen, Phys. Rev. B 56 (6) (1997) 2977. [7] K.W. Jacobsen, P. Stoltze, J.K. Norskov, Surf. Sci. 366 (1996) 394. [8] P. Stoltze, Simulation Methods in Atomic Scale Materials Physics, Polyteknisk Forlag, 1997. [9] J. Bonneville, B. Escaig, J.L. Martin, Acta. Metall. 36(8) (1988) 1989. [lo] J.P. Hirth, J. Lothe, Theory of Dislocations, 2nd ed., Krieger, Malabar, FL, 1992.
Atomic structure and energetics of constricted screw dislocations in copper T. Rasmussen a,b,*, K.W. Jacobsen a, T. Leffers b, O.B. Pedersen b a Center for Atomic-Scale Materials Physics, Department of Physics, b Materials Research Department, Risg National
Technical Laboratory,
University DK-4000
of Denmark, Roskiide,
DK-2800 Denmark
Lyngby,
Denmark
Received 3 February 1997
Abstract
Atomistic simulations of cross slip of a dissociated screw dislocation have been performed. Shapes and energetics of different dislocation configurations relevant to cross slip in an f.c.c. metal (Cu) are determined. The minimum stress-free activation energy and activation length in the Friedel-Escaig cross-slip mechanism are determined. The simulations reveal a new energetically favourable configuration of a dissociated screw dislocation not previously considered. The importance of this configuration to surface nucleated cross slip is discussed. 0 1997 Elsevier Science S.A. Keywords:
Cross slip; Simulations; Screw dislocation
1. Introduction
Cross slip of a screw dislocation is an intricate mechanism involving both long range elastic interactions and atomistic effects when the Shockley partials recombine. In this paper we describe the results of an atomistic approach to the problem of cross slip in Cu. Cross slip has earlier only been addressed theoretically by methods based on elasticity theory [l-4] mostly within the line
tension
approximation
and
isotropic
elasticity
the-
ory. In the present work we have performed atomic scale simulations of various dislocation configurations relevant to cross slip via the Friedel-Escaig cross-slip mechanism [1,5] and we determine the stress-free activation energy and activation length for this mechanism. We show that the two constrictions needed in this mechanism are not equivalent and that a configuration with just one of them is energetically favourable compared to two parallel partials in a single glide plane. Simulations show [6] that a free surface can act as a centre of nucleation of such a configuration. Hence, surface nucleated cross slip via a configuration with just one constriction is a possibility.
* Corresponding author. e-mail: [email protected]
Tel.:
+ 45 45253177;
fax:
+ 45 45932399;
0921-5093/97/$17.00 0 1997 Elsevier Science S.A. All rights reserved. PIISO921-5093(97)00311-O
The interatomic interactions are described by a many-body potential derived from the effective-medium theory [7]. The potential is slightly modified so as to produce a reasonable value for the intrinsic stackingfault energy and it also produces very reasonable values for the elastic constants as well as the equilibrium splitting width of the partials [6].
2. Results
and
discussion
In the Friedel-Escaig (FE) cross-slip mechanism [1,5] for the b = $ [l lo] screw dislocation sketched in Fig. 1, the two partials must recombine (constrict) in the primary glide plane before the subsequent redissociation in the cross-slip plane. The redissociation creates two nonequivalent twisted constrictions (A and B in Fig. 1) on the dislocation. We shall denote constriction A ‘edgelike’ and constriction B ‘screw-like’ because of the characters of the partials in the vicinity of the constrictions. The activation energy for the FE cross-slip mechanism is the sum of the energies of a pair of infinitely separated edge-like and screw-like constrictions. Noting this, we decided to calculate the activation energy by treating the two kinds of constrictions separately. We define the energy of a constriction as the difference in energy between a system containing a dislocation with a
T. Rasmussen
et al. /Materials
Science
constriction and a reference system of the same length containing two parallel Shockley partials in a single glide plane. The dislocation with B = 4 [110] is centered im a box tit&t 3111) facets as tree sudaces par&e1 to the disIocation. In the reference systems we appIy peric&c boudary conditions in the { 1 lOj dire&i0 u, i.e. along the dislocation line. The two constriction energies are found by first relaxing two reference systems each containing a pair of partials in either of the two possitie glide pknes. We theen extract the tag of one aE t&e reference systems and the bottom of the other reference system and make a sandwich with a system containing an already constricted and twisted dislocation. Keeping the atoms of the top and the bottom from the reference systems fixed in their relaxed positions we are able to mimic a semi infinite dislocation on either side of the constricted pad and the system can then t>e Aaxed using a standard energy minimization algorithm [S]. Using the other set of top and bottom from the reference sysCems and another initially constricted and twisted dislocation the second constriction system can be constructed. We use four (110) planes (equal to 26) with atoms in fixed positions in either end of the sjams containing a constrichm. ln Fig. 2 we show a relaxed configuration of a screw-like constriction. The system has been ctit in tv+wohalves, one part rotated and then cut again parallel to the partials, to display the uMoca%on jn >Is 5ti5 ‘lengIn. The atoms &rose 10 the uMoca%on cafes are co5omeh Ma&, The e@e-%& CconsIr~cIion hoes no1 &Ber %snaY& %rom ‘rhe screw-%e constriction. The length of the dislocation is denoted 1 am2 r);L- w&h 02 t+?esyskm peLqm&cu>ar to the d&w
(a>
I
and Engineering
A234&236
(1997)
5444547
545
+
W
4
Fig. 2. Screw-like constriction. The perfect Burgers vector i IllOJ points upwards in the figure. The system has been cut in two halves and one half has been rotated to display the dislocation in both glide planes. The length of the disIo&iion is I = 50 5 ~$2.7 nmj and Ihe width of the system is pi = 11.3 nm. ML By xqimg ) h Ihe rangfi 3%5t’h j3, b-32-7 n2L-qam3 w iv .a2Pm22gE 9.3 -14. B 22222 IO .w25 pDs>blP ID determine the constriction energies as functions of system dimensions. i”ne results of me simulations are summarized in Fig. 3. 123hDJ2
-
0.0 -
8 Fig. 1. The Friedel-Escaig cross-slip mechanism for a b = f [l lo] screw dislocation (a) Creation of a constriction in the primary plane, (111). (b) Dissociation in the shaded cross-slip plane, (11 l), creating two twisted constrictions A and B. (c) Two non-interacting constrictions The Burgers vectors in the two glide planes are indicated with arrows. Due to the different characters of the Shockley partials, the two constrictions are not equivalent. We denote the constrictions A and B ‘edge-like’ and ‘screw-like’, respectively.
9 10 11 12 13 14 Width of system / nm
15
Fig. 3. Energies in units of eV of the two different constriction types as a function of system dimensions, For each system the energy of the edge-like constriction (top), the energy of the screw-like constriction (bottom) and the sum of the two (middle) are shown. Notice the negative energy of the screw-like constriction.
546
T. Rasmussen
et al. /Materials
1
Q-EI edge-like m-m screw-like
0.4 % . k
Om3
d
0.1
Science
0.2
-0.1’
0
’ 10
I n 20 30 (110) pair/b
.
’ 40
50
Fig. 4. Energy contribution from individual AB (110) plane pairs to the total energy of the two kinds of constrictions. Edge-like constriction above and screw-like constriction below. The contributions from the static layers in either end of the systems are not shown. The length of the dislocations is 1= 50 b (12.7 nm) and the width of the systems is M‘ = 9.5 nm.
A striking feature is the negative energy of the screwlike constriction. This means that it is energetically favourable for the dislocation to adopt a twisted configuration with partials in both possible glide planes connected by a screw-like constriction, compared to two parallel partials in a single glide plane. From Fig. 3 it is possible to estimate the energies of the edge-like and the screw-like constrictions to 4.0 and - 1.0 eV, respectively. Hence, the activation energy for two independent constrictions in an infinite crystal, can be estimated to 3.0 eV. The small scatter of the energies is due to the uncertainty in the determination of the reference energy level, which can only be found within f 0.1 eV. Since the constriction energies have converged as a function of dislocation length I, the minimum separation between two non-interacting constrictions can be estimated to w 50 6. The value obtained for the activation length is in very good agreement with estimates based on elasticity theory [3,4,9]. The activation energy is, however, somewhat higher than the results from these works. As discussedelsewhere [2,6], the estimates of the activation energy from elasticity theory are in fact subject to rather large uncertainty due to the approximations of this theory and none of the mentioned works based on elasticity theory investigate the difference between the two types of constrictions. It is interesting to examine the energetics of the constrictions in more detail. To do so we utilize the fact that the systemscan be seenas an . ..ABAB... stacking of AB pairs of (110) planes along the dislocation line. In Fig. 4 we plot the energies of the AB pairs along the
and Engineering
A234-236
(1997)
544-547
dislocation line in two constriction systems relative to the energy of an AB pair from the reference systems. In this kind of plot the qualitative difference between the constrictions is very apparent. For the screw-like constriction the energy drops towards the centered constriction with a small increase in energy close to the constriction. This behaviour of the energy of the screwlike constriction can be explained in the following way. The self-energy of the partials will, according to dislocation theory [lo], decreaseas the partials acquire more screw character, and the area of the stacking fault between the partials is also reduced by the constriction. Furthermore, it is known that the interaction energy between two screw dislocations inclined at an angle of 45” is zero. The geometry involved here is not very different from this. The increase in energy around the constriction must be attributed to atomistic effects due to core overlap of the partials which of course cannot be included in an elastic description, but together the three energy lowering mechanisms are sufficient to make the constriction energy negative. For the edgelike constriction only the reduction of stacking fault area tends to lower the energy compared to two parallel partials, whereas the acquirement of more edge character increases the self-energy as well as the interaction energy. This explains the shape of the energy projection for this kind of constriction. It is worth noting once again that a configuration with a single screw-like constriction is energetically favourable compared to two parallel partials. For cross slip in the bulk it is difficult to imagine a cross-slip mechanism which would not require a pair of constrictions as in the FE mechanism. However, for a dislocation ending at a free surface it is possible to make the partials in one glide plane recombine at the surface and then redissociate into the other glide plane thereby creating a screw-like constriction on the dislocation. Pushing the screw-like constriction along the dislocation line would then produce cross slip without the edge-like constriction necessary for cross slip initiated in the bulk. The redissociation does not happen spontaneously [6], thus we can deduce that there is an energy barrier for this kind of surface nucleated cross slip. However, further work is required to determine the energy barrier for surface nucleated cross slip and it is therefore too early to draw any final conclusions about this kind of cross slip. As a direct application of such surface nucleated cross slip, it has been possible to perform a simulation of the annihilation of two screw dislocations of opposite signs [6]. The starting configuration was two screw dislocations split in two parallel glide planes separated by just 2.2 nm. The annihilation was initiated by one of the dislocations performing surface nucleated cross slip via the mechanism discussed above and it proceeded through a successionof dislocation reactions including
T. Rasmussen
et al. /Materials
Science
the formation of a sessile stair-rod dislocation as an intermediate step and a glissile dislocation loop comprising Shockley partials moving through the crystal. It was possible to monitor these reactions and thereby to exactly specify a set of possible dislocation reactions leading to annihilation.
and Engineering
A234-236
(1997)
54L.547
547
Acknowledgements Center for Atomic-Scale Materials Physics (CAMP) is sponsored by the Danish National Research Foundation. The present work was done as a collaboration between CAMP and the Engineering Science Centre for Structural Characterization and Modelling of Materials, Rise.
3. Conclusion References By using an atomistic approach to the problem of cross slip, we have determined the activation energy and activation length in the Friedel-Escaig cross-slip mechanism. Our result for the activation length is in very good agreement with earlier results obtained on the basis of elasticity theory. However, our result for the activation energy is higher than estimates based on elasticity theory. The non-equivalence of the two constrictions has not been investigated before, and we show that the screw-like constriction is energetically favoured compared to two parallel partials. Cross slip is also possible via a dislocation configuration with just a screw-like constriction nucleated at a free surface. The energy barrier for this kind of cross slip is still not known, but a simulation showing annihilation of two screw dislocations of opposite signs initiated by such cross slip has been performed.
[I] B. Escaig, in: A.R. Rosenfeld, G.T. Hahn, A.L. Bement Jr, R.I. Jaffee (Eds.), Dislocation Dynamics, Series in Materials Science and Engineering, McGraw-Hill, New York, 1968. B. Escaig, J. de Phys. (France), 29 (1968) 225. [2] G. Saada, Mat. Sci. Eng. Al37 (1991) 177. [3] M.S. Duesbery, N.P. Louat, K. Sadananda, Acta. Metall. Mat. 40 (1) (1992) 149. [4] W. Ptischl, G. Schoeck, Mat. Sci. Eng. Al64 (1993) 286. [5] J. Friedel, in: J.C. Fisher (Ed.), Dislocations and Mechanical Properties of Crystals, Wiley, New York, 1957. [6] T. Rasmussen, K.W. Jacobsen, T. Leffers, O.B. Pedersen, Phys. Rev. B 56 (6) (1997) 2977. [7] K.W. Jacobsen, P. Stoltze, J.K. Norskov, Surf. Sci. 366 (1996) 394. [8] P. Stoltze, Simulation Methods in Atomic Scale Materials Physics, Polyteknisk Forlag, 1997. [9] J. Bonneville, B. Escaig, J.L. Martin, Acta. Metall. 36(8) (1988) 1989. [lo] J.P. Hirth, J. Lothe, Theory of Dislocations, 2nd ed., Krieger, Malabar, FL, 1992.