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Orientation dependence of screw dislocation motion in b.c.c. transition metals
Solid State Communications, VoL 29, pp. 545-.-~48. Pergainon Press Ltd. 1979. Printed in Great Britain. ORIENTATION DEPENDENCE OF SCREW DISLOCATION MOTION IN b.c.c. TRANSITION METALS A. Sato and K. Masuda Department of Materials Science and Engineering, Tokyo Institute of Technology, Ookayama, Meguro, Tokyo 152, Japan (Received 18 November 1978 by Y. Toyozawa)
A tight-binding type electronic theory is used to calculate the Peierls stress of the screw dislocation in b.c.c. transition metals. The repulsive core—core interaction energies are simulated by Born—Mayer type potentials as well as the modified Born—Mayer potentials. It is shown that there are essential differences in the orientation dependence of the screw dislocation motion among the transition metals, in agreement with experiments. IT IS WELL KNOWN that there is a striking difference in the orientation dependence of the Peierls stress r, between the b.c.c. transition metals such as tungsten [1] molybdenum [2] and tantalum [31(Nd ~ 4.5 ,N~,is the number of d.electrons; hereafter referred to as Type I transition metal) and the b.c.e. transition metal like a-Fe [4] (Nd ~ 7; referred to as Type II transition metal). The Peierls stress r~of the former (Type I) b.c.e. transition metals behaves as r~,(x = 30 ) > r~,(x = 0 )> ,
r,, (x
that the empirical pair interaction potentials do not take account of redistribution of valence electrons due to the introduction of the lattice defects. In this communication, we present a simple dcctronic theory for d-electronsaround the dislocation and calculate the Peierls stress r~,of a (a/2)(1 11) screw dislocation in a-iron and tungsten for 300 ~ x ~ 30°.We demonstrate that the above mentioned difference in the onentation dependence of r1, between the Type I and Type II transition metals can be explained by the present electronic theory. In view of the fact that the d-electron contribution to the cohesive energy of transition metals accounts for the general trend and magnitude of the corresponding experimental results [141,we do not consider the influence of s-electrons and s—d mixing in the present calculation. For d-band of the transition metals, a tight-binding approximation and the method of moments are used, methods which have been shown to reproduce correctly many integral properties of the density of states [11—13]. The local density of states for d-electronsp,(E) on atom I is approximated by a Gaussian curve [11—13,15, 19,21] fitted to the second moment P21 obtained for example by a walk counting method —
= 30°),while the r~,of the latter (Type II) transition metal exhibits the behavior, r~,(x = 30°)> r, (x = 30°)> r~,(x = 0°),where x is the angle between the maximum shear stress plane and the (110) plane. In order to understand the mechanism of the plastic anisotropy in these b.c.e. transition metals, several model calculations of r~,have been carried out using mainly the empirical pair interaction potentials. Chang [5], using an anharmonic pairwise potential,has estimated the magnitudes of the Peierls stress and the core energies of a (a/2)(111> screw dislocation in a-iron. Minami, Kuramoto and Takeuchi [6, 7] have used the inter-atom-row potentials and calculated the Peierls stress of the screw dislocation in b.c.e. transition metals for the range, —30°~ x ~ 30°.The similar model calcu2 lations [8—101using the empirical pair interaction p1(E) = (10/v21rp2 ~)exp (— E /2p2,). (1) potentials have also been performed. However, these Here, the free atom energy level is taken as the origin of model calculations based on the empirical pair interthe energies and the factor 10 is for the degeneracy of action potentials have not succeeded in explaining the the d-band. The second moment P21 of atom i can be systematic difference in the orientation dependence of calculated by r,, between the Type I and Type II transition metals. This seems to result from the fact that these pair poten- P21 = E (2) tial methods are incapable of taking explicitly into J account the effect of the filling of the d-bands, which where t3~,is an effective resonanceintegral which is plays the leading role in the calculations of the surface related to the usual two-center integrals dda, ddö and tension [111,vacancy formation energy [12],stacking dd~by fault energy [13], and the formation energies of other 2(ij) + 2ddir2(ij) + 2ddS2(if)] (3) lattice defects. Furthermore, one has to bear in mind = (l/5)[dda 545 —
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Fig. 1. Cohesive energy (per atom) vs interatomic distance r curves for a-Fe and W. ris given in units of the equilibrium interatomic distance r0 of each transition metal.
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Fig. 2. Changes in the core configuration with increasing the shear stress for a-Fe. x = 0°,(a) r = 0, (b) r = 0.045p, (c) r = 0.05p.
Vol. 29, No.7
SCREW DISLOCATIONMOTION IN b.c.c. TRANSITION METALS
547
In order to reproduce the experimental values of the Table 1. CakulatedPeierls stresses for a-Fe and W, in bulk modulus (or the elastic constants) with the criterion units of the shear modulusp qr0 ~ 2—3 it is necessary to use the modified Born—
reproduce the order of magnitude of(6). these values the Mayer potential (n *0) of equation However, to Born—Mayer potential with pr 0 9 may be used [15]. In addition to the condition, qro ~ 2—3, the parameters appearing in the present calculation are chosen so as to satisfy the equilibrium condition and to reproduce the experimental values of the cohesive energy E~and the bulk modulus K: 112 — Co{4(n + pro) Fq(4E1 + 3aE2X4E1 + 3E2) — +3a ~n+apr 0ji~2j — v ~
2+ C —F(4E1 + 3E2)i~ 0(4R1 + 3&”R2) = E~, I !ot r_ Fq 2i~ ~ \2 ~a,aj~ -“-‘2)~-3/2 1~”-’1 “-‘-‘2) 2E 2 + n)Ri -n — a) 1E2} 2 + Co{4((n + pr0) + 24(1 + 3a ((n + apro) + n)R 2)] = K, .j
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tl 12Vl 11) ff)jshear 11(111) Hardfl 12V1 11> r /p for different systems I. J L Soft I. J a-Iron 0.049 Tungsten 0.018
where F = I 0j3~~ exp (— E~/p2),Ei = exp (— 2qro), E2 = exp (— 2aqr0), R1 = exp (— pr0), R2 = exp (— apr0), a = 2/.../~, and K is expressed assuming the unit of r0 = I. In Fig. 1 we present the cohesive energy vs interatomic distance curves for a-iron and tungsten (b.c.e. structure) using the parameters; qr0 = 2.0, n = 3.0 and pr0 = 3.0 for a-Fe and qr0 = 3.0, n = 6.0 and pr0 = 0.4 for W. Note that these curves do not mean the pairwise interaction potentials between the transition metal atoms. We now apply the above mentioned electronic theory to the calculation of the Peierls stress of a (a/2)( 111> screw dislocation in a-iron and tungsten. First, we take a cylindrical crystal region with radius l0~’~ b/3 (b is the magnitude of the Burgers vector b), which consists of 96 atomic rows. At the center of the cylindrical crystal region, a screw dislocation is introduced according to the strain field calculated by the linear elasticity theory.The cylindrical crystal region is sheared homogeneously in z-direction so that a designed shear stress is applied to the crystal. The crystal is then relaxed along the screw dislocation (z-axis) with a fixed boundary condition to attain equilibrium configuration by iterative calculation. For the sake of computational time,we do not consider the dilational (x andy components) relaxation around the dislocation. The convergence condition for the equilibrium atomic configuration is that the reduction of the total energy during one iterative process becomes less than a certain amount, 10_6 eV in the present calculation. The calculated relaxation in the atomic configuration near the core of a screw dislocation in a-Fe is shown in Fig. 2(a) by the differential displacement representation originally introduced by Vitek et a!. [16].
0.069 0.034
.
The arrows mdicate the magmfied (x 5) view of the relaxation occurred m the relative displacements along the z-axis (normal to the paper) neighbouring atomic rows with respect to the between elastic solution. The changes of core configuration with increasing the shear •
stress are shown m Figs. 2(b) and 2(c). As shown m Fig. 2(b),whenthe shear stress of 0.045p (j.i is the shear modulus) is applied in -the (011) plane so that the force coreis configuration rb actmg m the [211] becomes direction asymmetric on thebut dislocation, the force the is not enough to move the dislocation. When the stress is .
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mcreased to O.05p, the dislocation moves over the next low energy position as shown in Fig. 2(c) and it stops because of the imposed boundary condition in the calculation with a similar core configuration as in Fig. 2(b). This is in contrast to the results obtained by using the empirical inter-atomic [17] or inter-atomic row [6, 7] potentials where the dislocation is shown to move along a zigzag pass for the {Oll}(lll) shear. When other shear system such as hard or soft {1 12}<1 11> is employed the dislocation motion occurs along a zigzag pass as wifi be shown elsewhere [18]. Thus, the results obtained are very sensitive to the model for the b.c.e. transition metals. The calculated Peierls stresses for the three different shear systems are shown in Table 1. The results of r,,, for a-iron and tungsten are generally 10 times larger than those obtained by experiments. Fortunately, however, it is expected that theycould be reduced substantially when the dilational relaxation around the dislocation are taken into account and the much larger cylindrical crystal region is used. Our calculations in Table 1 clearly show that there are essential differences in the orientation dependence of r, between the Type I (tungsten) and Type II (a-iron) transition metals, in agreement with experiments. In addition, our results of r,, in Table 1 qualitatively account for the relative magnitude of the observed Peierls stresses of W and Fe, in agreement with our previous results [191calculated from the equipotential energy curves [20] for the screw dislocation. It is interesting to note that the observed difference of r~ between the Type I and Type II transition metals can be attributed mainly to the filling of the d-band, i.e., the number of d-electrons Nd of each transition metal. This conclusion is in agreement with the fact that the
548
SCREW DISLOCATIONMOTION IN b.c.c. TRANSITION METALS
magnitudes of the atomic relaxation around a vacancy [12] or near the surface [21]of the transition metals strongly depend on the filling of the d-bands. We hope that these exploratory calculations and discussions will stimulate more theoretical and experimental works on the plastic behavior of b.c.e. transition metals and the electronic structure of dislocations. Acknowledgement One of the authors (K.M.) gratefully acknowledges the fmancial support provided by the Sakkokai Foundation. —
8. 9. 10.
11. 12. 13.
REFERENCES I. 2. 3. 4. 5. 6. 7.
AS. Argon & S.R. Maloof,Acta Met. 14, 1449 (1966). S.S. Lan &J.E.Dom,Phys. Status Solidi 2a,825 (1970). S. Takeuchi, E. Kuranioto & T. Suzuki,Acta Met. 20,909(1972). W.A. Spitzig & A.S. Keh,ActaMet. 18, 611 (1970). R. Chang,Phil. Mag. 16, 1021 (1967). F. Minami, E. Kuramoto & S. Takeuchi,Phys. Status Solidi 12a, 581 (1972); Phys. Status Solidi 22a, 81(1974). E. Kuramoto, F. Minami & S. Takeuchi, Phys. Status Solidi 22a,411 (1974).
14. 15. 16. 17. 18. 19. 20. 21.
Vol. 29, No.7
R. Heinrich & W. Schellenberger, Phys. Statu& Solidl 4Th, 81(1971); G. Diener, R. Heinrich & ~~~ellenberger,Phys. Status Solidi 44b, 403 Z.A. Basinski, M.S. Duesbery & R. Taylor, Can. J. Phys. 49,2160(1970). M. Doyama & R.MJ. Cotterili, Bull. Amer. Phys. Soc. 11,460 (1966); J.Th.M. de Hosson, A.W. Sleeswyk, LM. Caspers, W. van Heugten & A. van Veen,Solid State Commun. 18,479(1976). F. Cyrot-Lackmann, J. Phys. C7zem. Solids 29, 1235 (1968);Surf Sci 15,535 (1969). G. Allan & M. Lannoo,J. Phys. Chem. SolIds 37, 699 (1976). F. Ducastelle & F. Cyrot-Lackmann, /. Phys. Chem. Solids 32, 285 (1971). J. Friedel, The Physics ofMetals (Edited by J.M. Ziman), p.340. Cambridge University Press (1969). F. Ducastelle,J. dePhys. (Paris) 31, 1055 (1970). V. Vitek, R.C. Perrin & D.K. Bowen,Phil. Mag. 21, 1049(1970). M.Yamaguchi&V.Vitek,J. Phys. F5, 11(1975); V. Vitek,P~oc.Roy. Soc. London A352, 109 (1976). A. Sato & K. Masuda (to be published). K. Masuda & A. Sato,Phil. Mag. 37B, 531 (1978). H. Suzuki, Dislocation Dynamics (Edited by A. Rosenfield, G.T. Hahn, A.L. Bement & R.I. Jaffee) p. 679. McGraw-Hill, New York (1968). G.Allan &M. Lannoo,Surf Sci 40,375 (1973).
A tight-binding type electronic theory is used to calculate the Peierls stress of the screw dislocation in b.c.c. transition metals. The repulsive core—core interaction energies are simulated by Born—Mayer type potentials as well as the modified Born—Mayer potentials. It is shown that there are essential differences in the orientation dependence of the screw dislocation motion among the transition metals, in agreement with experiments. IT IS WELL KNOWN that there is a striking difference in the orientation dependence of the Peierls stress r, between the b.c.c. transition metals such as tungsten [1] molybdenum [2] and tantalum [31(Nd ~ 4.5 ,N~,is the number of d.electrons; hereafter referred to as Type I transition metal) and the b.c.e. transition metal like a-Fe [4] (Nd ~ 7; referred to as Type II transition metal). The Peierls stress r~of the former (Type I) b.c.e. transition metals behaves as r~,(x = 30 ) > r~,(x = 0 )> ,
r,, (x
that the empirical pair interaction potentials do not take account of redistribution of valence electrons due to the introduction of the lattice defects. In this communication, we present a simple dcctronic theory for d-electronsaround the dislocation and calculate the Peierls stress r~,of a (a/2)(1 11) screw dislocation in a-iron and tungsten for 300 ~ x ~ 30°.We demonstrate that the above mentioned difference in the onentation dependence of r1, between the Type I and Type II transition metals can be explained by the present electronic theory. In view of the fact that the d-electron contribution to the cohesive energy of transition metals accounts for the general trend and magnitude of the corresponding experimental results [141,we do not consider the influence of s-electrons and s—d mixing in the present calculation. For d-band of the transition metals, a tight-binding approximation and the method of moments are used, methods which have been shown to reproduce correctly many integral properties of the density of states [11—13]. The local density of states for d-electronsp,(E) on atom I is approximated by a Gaussian curve [11—13,15, 19,21] fitted to the second moment P21 obtained for example by a walk counting method —
= 30°),while the r~,of the latter (Type II) transition metal exhibits the behavior, r~,(x = 30°)> r, (x = 30°)> r~,(x = 0°),where x is the angle between the maximum shear stress plane and the (110) plane. In order to understand the mechanism of the plastic anisotropy in these b.c.e. transition metals, several model calculations of r~,have been carried out using mainly the empirical pair interaction potentials. Chang [5], using an anharmonic pairwise potential,has estimated the magnitudes of the Peierls stress and the core energies of a (a/2)(111> screw dislocation in a-iron. Minami, Kuramoto and Takeuchi [6, 7] have used the inter-atom-row potentials and calculated the Peierls stress of the screw dislocation in b.c.e. transition metals for the range, —30°~ x ~ 30°.The similar model calcu2 lations [8—101using the empirical pair interaction p1(E) = (10/v21rp2 ~)exp (— E /2p2,). (1) potentials have also been performed. However, these Here, the free atom energy level is taken as the origin of model calculations based on the empirical pair interthe energies and the factor 10 is for the degeneracy of action potentials have not succeeded in explaining the the d-band. The second moment P21 of atom i can be systematic difference in the orientation dependence of calculated by r,, between the Type I and Type II transition metals. This seems to result from the fact that these pair poten- P21 = E (2) tial methods are incapable of taking explicitly into J account the effect of the filling of the d-bands, which where t3~,is an effective resonanceintegral which is plays the leading role in the calculations of the surface related to the usual two-center integrals dda, ddö and tension [111,vacancy formation energy [12],stacking dd~by fault energy [13], and the formation energies of other 2(ij) + 2ddir2(ij) + 2ddS2(if)] (3) lattice defects. Furthermore, one has to bear in mind = (l/5)[dda 545 —
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SCREW DISLOCATIONMOTION IN b.c.e. TRANSITION METALS
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Fig. 1. Cohesive energy (per atom) vs interatomic distance r curves for a-Fe and W. ris given in units of the equilibrium interatomic distance r0 of each transition metal.
13o exp (— qr1j). (4) It has been shown that if r0 is the equilibrium interatomic distance at the middle of the transition series, reasonable values of q is given as qr0 ~ 2—3 [15]. One can now determine the band structure energy ~Ee as a function of the atomic displacements (configurations) from the expression
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Vol. 29, No.7
SCREW DISLOCATIONMOTION IN b.c.c. TRANSITION METALS
547
In order to reproduce the experimental values of the Table 1. CakulatedPeierls stresses for a-Fe and W, in bulk modulus (or the elastic constants) with the criterion units of the shear modulusp qr0 ~ 2—3 it is necessary to use the modified Born—
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tl 12Vl 11) ff)jshear 11(111) Hardfl 12V1 11> r /p for different systems I. J L Soft I. J a-Iron 0.049 Tungsten 0.018
where F = I 0j3~~ exp (— E~/p2),Ei = exp (— 2qro), E2 = exp (— 2aqr0), R1 = exp (— pr0), R2 = exp (— apr0), a = 2/.../~, and K is expressed assuming the unit of r0 = I. In Fig. 1 we present the cohesive energy vs interatomic distance curves for a-iron and tungsten (b.c.e. structure) using the parameters; qr0 = 2.0, n = 3.0 and pr0 = 3.0 for a-Fe and qr0 = 3.0, n = 6.0 and pr0 = 0.4 for W. Note that these curves do not mean the pairwise interaction potentials between the transition metal atoms. We now apply the above mentioned electronic theory to the calculation of the Peierls stress of a (a/2)( 111> screw dislocation in a-iron and tungsten. First, we take a cylindrical crystal region with radius l0~’~ b/3 (b is the magnitude of the Burgers vector b), which consists of 96 atomic rows. At the center of the cylindrical crystal region, a screw dislocation is introduced according to the strain field calculated by the linear elasticity theory.The cylindrical crystal region is sheared homogeneously in z-direction so that a designed shear stress is applied to the crystal. The crystal is then relaxed along the screw dislocation (z-axis) with a fixed boundary condition to attain equilibrium configuration by iterative calculation. For the sake of computational time,we do not consider the dilational (x andy components) relaxation around the dislocation. The convergence condition for the equilibrium atomic configuration is that the reduction of the total energy during one iterative process becomes less than a certain amount, 10_6 eV in the present calculation. The calculated relaxation in the atomic configuration near the core of a screw dislocation in a-Fe is shown in Fig. 2(a) by the differential displacement representation originally introduced by Vitek et a!. [16].
0.069 0.034
.
The arrows mdicate the magmfied (x 5) view of the relaxation occurred m the relative displacements along the z-axis (normal to the paper) neighbouring atomic rows with respect to the between elastic solution. The changes of core configuration with increasing the shear •
stress are shown m Figs. 2(b) and 2(c). As shown m Fig. 2(b),whenthe shear stress of 0.045p (j.i is the shear modulus) is applied in -the (011) plane so that the force coreis configuration rb actmg m the [211] becomes direction asymmetric on thebut dislocation, the force the is not enough to move the dislocation. When the stress is .
(9)
0.046 0.020
•
.
.
mcreased to O.05p, the dislocation moves over the next low energy position as shown in Fig. 2(c) and it stops because of the imposed boundary condition in the calculation with a similar core configuration as in Fig. 2(b). This is in contrast to the results obtained by using the empirical inter-atomic [17] or inter-atomic row [6, 7] potentials where the dislocation is shown to move along a zigzag pass for the {Oll}(lll) shear. When other shear system such as hard or soft {1 12}<1 11> is employed the dislocation motion occurs along a zigzag pass as wifi be shown elsewhere [18]. Thus, the results obtained are very sensitive to the model for the b.c.e. transition metals. The calculated Peierls stresses for the three different shear systems are shown in Table 1. The results of r,,, for a-iron and tungsten are generally 10 times larger than those obtained by experiments. Fortunately, however, it is expected that theycould be reduced substantially when the dilational relaxation around the dislocation are taken into account and the much larger cylindrical crystal region is used. Our calculations in Table 1 clearly show that there are essential differences in the orientation dependence of r, between the Type I (tungsten) and Type II (a-iron) transition metals, in agreement with experiments. In addition, our results of r,, in Table 1 qualitatively account for the relative magnitude of the observed Peierls stresses of W and Fe, in agreement with our previous results [191calculated from the equipotential energy curves [20] for the screw dislocation. It is interesting to note that the observed difference of r~ between the Type I and Type II transition metals can be attributed mainly to the filling of the d-band, i.e., the number of d-electrons Nd of each transition metal. This conclusion is in agreement with the fact that the
548
SCREW DISLOCATIONMOTION IN b.c.c. TRANSITION METALS
magnitudes of the atomic relaxation around a vacancy [12] or near the surface [21]of the transition metals strongly depend on the filling of the d-bands. We hope that these exploratory calculations and discussions will stimulate more theoretical and experimental works on the plastic behavior of b.c.e. transition metals and the electronic structure of dislocations. Acknowledgement One of the authors (K.M.) gratefully acknowledges the fmancial support provided by the Sakkokai Foundation. —
8. 9. 10.
11. 12. 13.
REFERENCES I. 2. 3. 4. 5. 6. 7.
AS. Argon & S.R. Maloof,Acta Met. 14, 1449 (1966). S.S. Lan &J.E.Dom,Phys. Status Solidi 2a,825 (1970). S. Takeuchi, E. Kuranioto & T. Suzuki,Acta Met. 20,909(1972). W.A. Spitzig & A.S. Keh,ActaMet. 18, 611 (1970). R. Chang,Phil. Mag. 16, 1021 (1967). F. Minami, E. Kuramoto & S. Takeuchi,Phys. Status Solidi 12a, 581 (1972); Phys. Status Solidi 22a, 81(1974). E. Kuramoto, F. Minami & S. Takeuchi, Phys. Status Solidi 22a,411 (1974).
14. 15. 16. 17. 18. 19. 20. 21.
Vol. 29, No.7
R. Heinrich & W. Schellenberger, Phys. Statu& Solidl 4Th, 81(1971); G. Diener, R. Heinrich & ~~~ellenberger,Phys. Status Solidi 44b, 403 Z.A. Basinski, M.S. Duesbery & R. Taylor, Can. J. Phys. 49,2160(1970). M. Doyama & R.MJ. Cotterili, Bull. Amer. Phys. Soc. 11,460 (1966); J.Th.M. de Hosson, A.W. Sleeswyk, LM. Caspers, W. van Heugten & A. van Veen,Solid State Commun. 18,479(1976). F. Cyrot-Lackmann, J. Phys. C7zem. Solids 29, 1235 (1968);Surf Sci 15,535 (1969). G. Allan & M. Lannoo,J. Phys. Chem. SolIds 37, 699 (1976). F. Ducastelle & F. Cyrot-Lackmann, /. Phys. Chem. Solids 32, 285 (1971). J. Friedel, The Physics ofMetals (Edited by J.M. Ziman), p.340. Cambridge University Press (1969). F. Ducastelle,J. dePhys. (Paris) 31, 1055 (1970). V. Vitek, R.C. Perrin & D.K. Bowen,Phil. Mag. 21, 1049(1970). M.Yamaguchi&V.Vitek,J. Phys. F5, 11(1975); V. Vitek,P~oc.Roy. Soc. London A352, 109 (1976). A. Sato & K. Masuda (to be published). K. Masuda & A. Sato,Phil. Mag. 37B, 531 (1978). H. Suzuki, Dislocation Dynamics (Edited by A. Rosenfield, G.T. Hahn, A.L. Bement & R.I. Jaffee) p. 679. McGraw-Hill, New York (1968). G.Allan &M. Lannoo,Surf Sci 40,375 (1973).