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Phonons in nanocrystalline fcc nickel
Surface Science 532–535 (2003) 272–275 www.elsevier.com/locate/susc
Phonons in nanocrystalline fcc nickel Ranber Singh *, S. Prakash
1
Department of Physics, Panjab University, Chandigarh 160014, India
Abstract The enhancement in the phonon DOS of nanocrystalline fcc Ni at low and high energies is due to the anisotropic changes in the interatomic force constants as compared to the bulk phase fcc Ni. The specific heat and vibrational entropy of nanophase is found to be greater than that of the bulk phase. There occurred a peak in the excessive specific heat of nanocrystalline fcc Ni at low temperatures. The excessive entropy of nanocrystalline Ni also has a peak at low temperatures but it becomes constant at high temperatures. The free energy of nanophase is lower than that of the bulk phase. The difference in the free energies of bulk phase and nanophase Ni increases linearly with temperature. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Phonons; Nickel; Clusters
1. Introduction The crystalline structures of a few (1–100 nm) nanometer sizes are known as nanocrystals. These nanocrystalline materials have the same geometrical symmetry as their bulk counterparts but with some changes in the interatomic distances and surface geometry. The quantum size and larger surface to volume ratio of these nanocrystalline materials make their properties unusual as compared to that of their bulk counterparts. The vibrational properties of metallic nanocrystals are strongly affected by their high surface to volume ratio [1]. The phonon density of states (DOS) of nanocrystalline materials exhibits an enhancement at low frequencies and an extension towards higher
* Corresponding author. Tel.: +172-541741/541714; fax: +172-783336. E-mail address: [email protected] (R. Singh). 1 Present address: Jiwaji University, Gwalior 474011, India.
frequencies when compared with their coarsegrained counterparts. Stuhr et al. [2] probed the phonon DOS contribution from the grain-boundary region using the strong incoherent neutron scattering component from the hydrogen atoms induced into the grain boundary of the nanocrystalline palladium samples and found that in the low-frequency limit phonon DOS exhibits gðxÞ x, rather than usual quadratic relation. This suggests that the increase in the low-frequency modes arises from the atoms at the grain boundaries and/ or surfaces. In the simulations of metallic nanoclusters [2] the phonon DOS were investigated with separate contributions from inside the clusters and outer surface layers. The latter region (outer surface layers) did indeed show an enhancement at low frequencies. An investigation of magnetic properties of nanocrystalline Ni films [3] showed an in-plane magnetization anisotropy and perpendicular coercivity anisotropy. The phonon DOS of nanocrystalline Ni (10 nm) is strongly enhanced at
0039-6028/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0039-6028(03)00098-0
R. Singh, S. Prakash / Surface Science 532–535 (2003) 272–275
frequencies below 15 meV [4] as compared to that of its course grained counterpart. Derlet et al. [5] investigated the phonon DOS of model nanocrystalline samples (of fcc Ni and Cu) with separation of contributions from grain, grain boundaries and internal surfaces and showed an enhancement of phonon DOS at low and high phonon energies. They also showed that the grain boundary atoms mainly cause these enhancements. Wassermann and Rieder [6] have attributed the experimental peak of longitudinal optical modes in the high-frequency region in MgO nanocrystals to surface inward relaxation. The specific heat and vibrational entropy of nanocrystalline materials have been reported higher than that of their bulk counterparts [7–9]. Meyer et al. [10] have shown, through their molecular dynamics simulations of metallic nanoclusters, the existence of large capillary pressure at the surface of these nanoclusters. This suggested that the interatomic force constants of nanocrystalline materials are stiffened as compared to those of their bulk counterparts. In this paper we have investigated this suggestion for nanocrystalline fcc Ni (10 nm) by using 5NN Born–von Karman model.
273
done by first changing the force constants of first and second NNs and then the force constants of third, fourth and fifth NNs. The close fit is obtained only when the force constants were increased as compared to that of the bulk phase fcc Ni [12]. The quantum numbers describing the phonon modes in nanocrystalline Ni over which integration is carried out to yield the phonon DOS are the same as in bulk phase but there are anisotropic changes in the effective interatomic force constants in the nanophase as compared to the bulk phase. The same set of force constants are used to calculate the phonon DOS for the intermediate and high energy region. The finally obtained phonon DOS are given in Fig. 1. Since we assume that crystal symmetry of nanophase and bulk phase single crystal to be the same, we compare these force constants with those of bulk phase single crystal [12] in Table 1. The phonon DOS in Fig. 1 are further used to investigate the temperature dependence of specific heat, vibrational entropy and free energy of nanocrystalline fcc Ni. The results are given in Fig. 2.
3. Results and discussion 2. Calculations The phonon DOS of nanocrystalline fcc Ni are calculated under 5NN Born–von Karman model by using the Gilat and RaubenheimerÕs numerical integration method [11]. The Born-von Karman model depends only on the nearest neighbours considered and are independent of the total number of atoms in the crystallite. It has been shown through X-ray diffraction analysis that nanocrystals retain the structural symmetries of their bulk counterparts [7,8]. Thus the symmetry of the nanocrystalline fcc Ni is considered to be the same as that of bulk phase fcc Ni. The force constants for the bulk phase Ni [12] are the initial input data. The force constants for the nanocrystalline fcc Ni are determined by fitting the calculated phonon DOS to the experimental data for phonon DOS of nanocrystalline fcc Ni (10 nm) [4]. The force constants are varied to get a close fit of the experimental data of phonon DOS of nanocrystalline fcc Ni (10 nm) [4] in the low energy region. This is
The Born–von Karman model (up to fifth, sixth and even eighth) has been extensively used to determine the crystal lattice force constants for a long time. In this model the force constants are treated as fitting variables and the fit is done with the experimentally determined phonon frequencies at few k-points (see e.g. [11,12]). These force constants are then used to calculate other vibrational and thermodynamics properties of the crystal. Here we did little bit differently that the force constants are fitted to experimentally determined phonon DOS for the sample of nanocrystalline fcc Ni (10 nm) [4]. It is evident from Table 1 that there are anisotropic changes in the interatomic force constants in the nanophase as compared to the bulk phase fcc Ni. These changes lead to the shrinking and enhancement of phonon DOS at low and high energies in the nanophase Ni. The shrinking has also been observed in Ni nanoclusters by extended X-ray absorption fine structure measurements [13].
274
R. Singh, S. Prakash / Surface Science 532–535 (2003) 272–275 0.4 0.4
(a)
-1
Phonon DOS (meV )
0.3
∆CV(JK-1 mol -1 )
bulk fcc Ni (calc.) nano fcc Ni (calc.) nano fcc Ni (exp. )
0.2
0.0
0.2 -0.2 0.06
0.1 ∆S (k B/atom)
0.04
0.0 100
bulk fcc Ni (calc.) nano fcc Ni (calc.) nano fcc Ni (exp. )
0.00
80
0
0.000
60
∆ F (eV/atom)
-1
Normalized Phonon DOS (eV )
(b)
0.02
40
-0.001
-0.002
20
0
0 0
10
20
30
200
400 600 Temperature (K)
800
1000
40
Phonon Energy (meV)
Fig. 1. Phonon DOS of nanocrystalline and bulk phase fcc Ni.
Table 1 Interatomic force constants for bulk phase and nanophase of fcc Ni (in 104 dyne/cm) Force constants
Bulk phase [12]
Nanophase
1XX 1ZZ 1XY 2XX 2YY 3XX 3ZZ 3XY 3YZ 4XX 4ZZ 4XY 5XX 5YY 5ZZ 5XY
1.7720 )0.1015 1.8735 0.1148 )0.0998 0.0940 0.0182 0.0505 0.1011 0.0459 )0.0153 0.0612 )0.0363 0.0100 0.0421 0.0174
1.7916 )0.1582 2.8768 0.2869 )0.1014 0.1003 0.0439 0.0672 0.1016 0.0964 )0.0189 0.0638 )0.0369 0.0176 0.0428 0.0194
Fig. 2. Excessive values of specific heat (DCV ), entropy (DS) and free energy (DF ) of nanocrystalline fcc Ni relative to those of bulk phase fcc Ni.
Kara and Rahman [1] showed that for Ni nanocrystals the average NN distance is shortened by as much as 1.6–2.0% as compared to the bulk values. Note that phonon DOS in the present calculations is for a harmonic system. Anharmonic effects may still lead to lifetime broadening as suggested by Frase et al. [7,14] and Fultz et al. [8]. The results for the phonon DOS as given in Fig. 1(b) are quite comparable with those due to Derlet et al. [5] for model nanocrystalline samples of fcc Ni except a high energy sharp peak in our calculations which is in fact because of the harmonicity of the model used. The calculated phonon DOS is also used to investigate the temperature dependence of specific heat, vibrational entropy and free energy of nanocrystalline fcc Ni. The appearance of a peak in specific heat at low temperatures is in agreement
R. Singh, S. Prakash / Surface Science 532–535 (2003) 272–275
with the experiment results [9,15] and molecular dynamic simulation of model nanocrystals [16]. The excessive vibrational entropy of nanophase fcc Ni also has a peak at low temperatures but becomes constant at higher temperatures. The larger vibrational entropy of nanophase indicates that it is vibrationally more disordered as compared to bulk phase as entropy is the measure of disorderness. The large vibrational entropy of nanocrystalline materials has also been proposed as to stabilize them at moderate temperatures [17–19]. The free energy of nanophase is lower than that of the bulk phase. The difference in the free energies of bulk phase and nanophase Ni increases linearly with temperature.
4. Conclusions The anisotropic changes in the nearest neighbour interatomic force constants of nanophase fcc Ni as compared to the bulk phase fcc Ni leads to the enhancements in the phonon DOS of nanophase at low and high energies. The specific heat and vibrational entropy of nanophase is greater than that of the bulk phase. There occurred a peak in the excessive specific heat of nanophase fcc Ni at low temperatures which is in agreement with the experimental results [9,15] and computer simulations of model nanocrystals [16]. The excessive entropy of nanocrystalline Ni also has a peak at low temperatures but it becomes constant at high temperatures. The free energy of nanophase is lower than that of the bulk phase. The difference in the free energies of bulk phase and nanophase Ni increases linearly with temperature.
275
Acknowledgement The financial support from the Council of Scientific and Industrial Research, New Delhi, Govt. of India, is gratefully acknowledged.
References [1] A. Kara, T.S. Rahman, Phys. Rev. Lett. 81 (1998) 1453. [2] U. Stuhr, H. Wipf, K.H. Andersen, H. Hahn, Phys. Rev. Lett. 81 (1998) 1449. [3] J. Zhang, G.M. Chow, S.H. Lawrence, C.R. Feng, Mater. Phys. Mech. 1 (2000) 11. [4] J. Trampenau, K. Bauszuz, W. Petry, V. Herr, Nanostruct. Mater. 6 (1995) 551. [5] P.M. Derlet, R. Meyer, L.J. Lewis, U. Stuhr, H. Van Swygenhoven, Phys. Rev. Lett. 87 (2001) 205501. [6] B. Wassermann, K.H. Rieder, Phys. Rev. Lett. 88 (2002) 045501. [7] H. Frase, B. Fultz, J.L. Robertson, Phys. Rev. B 57 (1998) 898. [8] B. Fultz, C.C. Ahn, E.E. Alp, W. Sturhahn, T.S. Toellner, Phys. Rev. Lett. 79 (1997) 937. [9] J. Rupp, R. Birriger, Phys. Rev. B 36 (1987) 7888. [10] R. Meyer, S. Prakash, P. Entel, Phase Trans. 75 (2002) 51. [11] G. Gilat, L.J. Raubenheimer, Phys. Rev. 144 (1965) 390. [12] R.J. Birgeneau, J. Cordes, G. Dolling, A.D.B. Woods, Phys. Rev. 136 (1964) A1359. [13] G. Apai, J.F. Hamilton, J. Stohr, A. Thomson, Phys. Rev. Lett. 43 (1979) 165. [14] H.N. Frase, L.J. Nayel, J.L. Robertson, B. Fultz, Philos. Mag. B 75 (1997) 335. [15] A. Tsch€ ope, R. Birringer, Acta Metall. Mater. 41 (1993) 2791; Philos. Mag. B 68 (1993) 2223. [16] D. Wolf, J. Wang, S.R. Phillpot, H. Gleiter, Phys. Rev. Lett. 74 (1995) 4686. [17] H. Gleiter, Prog. Mater. Sci. 33 (1989) 223. [18] H.J. Fecht, Phys. Rev. Lett. 65 (1990) 610. [19] M. Wagner, Acta Metall. Mater. 40 (1992) 957.
Phonons in nanocrystalline fcc nickel Ranber Singh *, S. Prakash
1
Department of Physics, Panjab University, Chandigarh 160014, India
Abstract The enhancement in the phonon DOS of nanocrystalline fcc Ni at low and high energies is due to the anisotropic changes in the interatomic force constants as compared to the bulk phase fcc Ni. The specific heat and vibrational entropy of nanophase is found to be greater than that of the bulk phase. There occurred a peak in the excessive specific heat of nanocrystalline fcc Ni at low temperatures. The excessive entropy of nanocrystalline Ni also has a peak at low temperatures but it becomes constant at high temperatures. The free energy of nanophase is lower than that of the bulk phase. The difference in the free energies of bulk phase and nanophase Ni increases linearly with temperature. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Phonons; Nickel; Clusters
1. Introduction The crystalline structures of a few (1–100 nm) nanometer sizes are known as nanocrystals. These nanocrystalline materials have the same geometrical symmetry as their bulk counterparts but with some changes in the interatomic distances and surface geometry. The quantum size and larger surface to volume ratio of these nanocrystalline materials make their properties unusual as compared to that of their bulk counterparts. The vibrational properties of metallic nanocrystals are strongly affected by their high surface to volume ratio [1]. The phonon density of states (DOS) of nanocrystalline materials exhibits an enhancement at low frequencies and an extension towards higher
* Corresponding author. Tel.: +172-541741/541714; fax: +172-783336. E-mail address: [email protected] (R. Singh). 1 Present address: Jiwaji University, Gwalior 474011, India.
frequencies when compared with their coarsegrained counterparts. Stuhr et al. [2] probed the phonon DOS contribution from the grain-boundary region using the strong incoherent neutron scattering component from the hydrogen atoms induced into the grain boundary of the nanocrystalline palladium samples and found that in the low-frequency limit phonon DOS exhibits gðxÞ x, rather than usual quadratic relation. This suggests that the increase in the low-frequency modes arises from the atoms at the grain boundaries and/ or surfaces. In the simulations of metallic nanoclusters [2] the phonon DOS were investigated with separate contributions from inside the clusters and outer surface layers. The latter region (outer surface layers) did indeed show an enhancement at low frequencies. An investigation of magnetic properties of nanocrystalline Ni films [3] showed an in-plane magnetization anisotropy and perpendicular coercivity anisotropy. The phonon DOS of nanocrystalline Ni (10 nm) is strongly enhanced at
0039-6028/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0039-6028(03)00098-0
R. Singh, S. Prakash / Surface Science 532–535 (2003) 272–275
frequencies below 15 meV [4] as compared to that of its course grained counterpart. Derlet et al. [5] investigated the phonon DOS of model nanocrystalline samples (of fcc Ni and Cu) with separation of contributions from grain, grain boundaries and internal surfaces and showed an enhancement of phonon DOS at low and high phonon energies. They also showed that the grain boundary atoms mainly cause these enhancements. Wassermann and Rieder [6] have attributed the experimental peak of longitudinal optical modes in the high-frequency region in MgO nanocrystals to surface inward relaxation. The specific heat and vibrational entropy of nanocrystalline materials have been reported higher than that of their bulk counterparts [7–9]. Meyer et al. [10] have shown, through their molecular dynamics simulations of metallic nanoclusters, the existence of large capillary pressure at the surface of these nanoclusters. This suggested that the interatomic force constants of nanocrystalline materials are stiffened as compared to those of their bulk counterparts. In this paper we have investigated this suggestion for nanocrystalline fcc Ni (10 nm) by using 5NN Born–von Karman model.
273
done by first changing the force constants of first and second NNs and then the force constants of third, fourth and fifth NNs. The close fit is obtained only when the force constants were increased as compared to that of the bulk phase fcc Ni [12]. The quantum numbers describing the phonon modes in nanocrystalline Ni over which integration is carried out to yield the phonon DOS are the same as in bulk phase but there are anisotropic changes in the effective interatomic force constants in the nanophase as compared to the bulk phase. The same set of force constants are used to calculate the phonon DOS for the intermediate and high energy region. The finally obtained phonon DOS are given in Fig. 1. Since we assume that crystal symmetry of nanophase and bulk phase single crystal to be the same, we compare these force constants with those of bulk phase single crystal [12] in Table 1. The phonon DOS in Fig. 1 are further used to investigate the temperature dependence of specific heat, vibrational entropy and free energy of nanocrystalline fcc Ni. The results are given in Fig. 2.
3. Results and discussion 2. Calculations The phonon DOS of nanocrystalline fcc Ni are calculated under 5NN Born–von Karman model by using the Gilat and RaubenheimerÕs numerical integration method [11]. The Born-von Karman model depends only on the nearest neighbours considered and are independent of the total number of atoms in the crystallite. It has been shown through X-ray diffraction analysis that nanocrystals retain the structural symmetries of their bulk counterparts [7,8]. Thus the symmetry of the nanocrystalline fcc Ni is considered to be the same as that of bulk phase fcc Ni. The force constants for the bulk phase Ni [12] are the initial input data. The force constants for the nanocrystalline fcc Ni are determined by fitting the calculated phonon DOS to the experimental data for phonon DOS of nanocrystalline fcc Ni (10 nm) [4]. The force constants are varied to get a close fit of the experimental data of phonon DOS of nanocrystalline fcc Ni (10 nm) [4] in the low energy region. This is
The Born–von Karman model (up to fifth, sixth and even eighth) has been extensively used to determine the crystal lattice force constants for a long time. In this model the force constants are treated as fitting variables and the fit is done with the experimentally determined phonon frequencies at few k-points (see e.g. [11,12]). These force constants are then used to calculate other vibrational and thermodynamics properties of the crystal. Here we did little bit differently that the force constants are fitted to experimentally determined phonon DOS for the sample of nanocrystalline fcc Ni (10 nm) [4]. It is evident from Table 1 that there are anisotropic changes in the interatomic force constants in the nanophase as compared to the bulk phase fcc Ni. These changes lead to the shrinking and enhancement of phonon DOS at low and high energies in the nanophase Ni. The shrinking has also been observed in Ni nanoclusters by extended X-ray absorption fine structure measurements [13].
274
R. Singh, S. Prakash / Surface Science 532–535 (2003) 272–275 0.4 0.4
(a)
-1
Phonon DOS (meV )
0.3
∆CV(JK-1 mol -1 )
bulk fcc Ni (calc.) nano fcc Ni (calc.) nano fcc Ni (exp. )
0.2
0.0
0.2 -0.2 0.06
0.1 ∆S (k B/atom)
0.04
0.0 100
bulk fcc Ni (calc.) nano fcc Ni (calc.) nano fcc Ni (exp. )
0.00
80
0
0.000
60
∆ F (eV/atom)
-1
Normalized Phonon DOS (eV )
(b)
0.02
40
-0.001
-0.002
20
0
0 0
10
20
30
200
400 600 Temperature (K)
800
1000
40
Phonon Energy (meV)
Fig. 1. Phonon DOS of nanocrystalline and bulk phase fcc Ni.
Table 1 Interatomic force constants for bulk phase and nanophase of fcc Ni (in 104 dyne/cm) Force constants
Bulk phase [12]
Nanophase
1XX 1ZZ 1XY 2XX 2YY 3XX 3ZZ 3XY 3YZ 4XX 4ZZ 4XY 5XX 5YY 5ZZ 5XY
1.7720 )0.1015 1.8735 0.1148 )0.0998 0.0940 0.0182 0.0505 0.1011 0.0459 )0.0153 0.0612 )0.0363 0.0100 0.0421 0.0174
1.7916 )0.1582 2.8768 0.2869 )0.1014 0.1003 0.0439 0.0672 0.1016 0.0964 )0.0189 0.0638 )0.0369 0.0176 0.0428 0.0194
Fig. 2. Excessive values of specific heat (DCV ), entropy (DS) and free energy (DF ) of nanocrystalline fcc Ni relative to those of bulk phase fcc Ni.
Kara and Rahman [1] showed that for Ni nanocrystals the average NN distance is shortened by as much as 1.6–2.0% as compared to the bulk values. Note that phonon DOS in the present calculations is for a harmonic system. Anharmonic effects may still lead to lifetime broadening as suggested by Frase et al. [7,14] and Fultz et al. [8]. The results for the phonon DOS as given in Fig. 1(b) are quite comparable with those due to Derlet et al. [5] for model nanocrystalline samples of fcc Ni except a high energy sharp peak in our calculations which is in fact because of the harmonicity of the model used. The calculated phonon DOS is also used to investigate the temperature dependence of specific heat, vibrational entropy and free energy of nanocrystalline fcc Ni. The appearance of a peak in specific heat at low temperatures is in agreement
R. Singh, S. Prakash / Surface Science 532–535 (2003) 272–275
with the experiment results [9,15] and molecular dynamic simulation of model nanocrystals [16]. The excessive vibrational entropy of nanophase fcc Ni also has a peak at low temperatures but becomes constant at higher temperatures. The larger vibrational entropy of nanophase indicates that it is vibrationally more disordered as compared to bulk phase as entropy is the measure of disorderness. The large vibrational entropy of nanocrystalline materials has also been proposed as to stabilize them at moderate temperatures [17–19]. The free energy of nanophase is lower than that of the bulk phase. The difference in the free energies of bulk phase and nanophase Ni increases linearly with temperature.
4. Conclusions The anisotropic changes in the nearest neighbour interatomic force constants of nanophase fcc Ni as compared to the bulk phase fcc Ni leads to the enhancements in the phonon DOS of nanophase at low and high energies. The specific heat and vibrational entropy of nanophase is greater than that of the bulk phase. There occurred a peak in the excessive specific heat of nanophase fcc Ni at low temperatures which is in agreement with the experimental results [9,15] and computer simulations of model nanocrystals [16]. The excessive entropy of nanocrystalline Ni also has a peak at low temperatures but it becomes constant at high temperatures. The free energy of nanophase is lower than that of the bulk phase. The difference in the free energies of bulk phase and nanophase Ni increases linearly with temperature.
275
Acknowledgement The financial support from the Council of Scientific and Industrial Research, New Delhi, Govt. of India, is gratefully acknowledged.
References [1] A. Kara, T.S. Rahman, Phys. Rev. Lett. 81 (1998) 1453. [2] U. Stuhr, H. Wipf, K.H. Andersen, H. Hahn, Phys. Rev. Lett. 81 (1998) 1449. [3] J. Zhang, G.M. Chow, S.H. Lawrence, C.R. Feng, Mater. Phys. Mech. 1 (2000) 11. [4] J. Trampenau, K. Bauszuz, W. Petry, V. Herr, Nanostruct. Mater. 6 (1995) 551. [5] P.M. Derlet, R. Meyer, L.J. Lewis, U. Stuhr, H. Van Swygenhoven, Phys. Rev. Lett. 87 (2001) 205501. [6] B. Wassermann, K.H. Rieder, Phys. Rev. Lett. 88 (2002) 045501. [7] H. Frase, B. Fultz, J.L. Robertson, Phys. Rev. B 57 (1998) 898. [8] B. Fultz, C.C. Ahn, E.E. Alp, W. Sturhahn, T.S. Toellner, Phys. Rev. Lett. 79 (1997) 937. [9] J. Rupp, R. Birriger, Phys. Rev. B 36 (1987) 7888. [10] R. Meyer, S. Prakash, P. Entel, Phase Trans. 75 (2002) 51. [11] G. Gilat, L.J. Raubenheimer, Phys. Rev. 144 (1965) 390. [12] R.J. Birgeneau, J. Cordes, G. Dolling, A.D.B. Woods, Phys. Rev. 136 (1964) A1359. [13] G. Apai, J.F. Hamilton, J. Stohr, A. Thomson, Phys. Rev. Lett. 43 (1979) 165. [14] H.N. Frase, L.J. Nayel, J.L. Robertson, B. Fultz, Philos. Mag. B 75 (1997) 335. [15] A. Tsch€ ope, R. Birringer, Acta Metall. Mater. 41 (1993) 2791; Philos. Mag. B 68 (1993) 2223. [16] D. Wolf, J. Wang, S.R. Phillpot, H. Gleiter, Phys. Rev. Lett. 74 (1995) 4686. [17] H. Gleiter, Prog. Mater. Sci. 33 (1989) 223. [18] H.J. Fecht, Phys. Rev. Lett. 65 (1990) 610. [19] M. Wagner, Acta Metall. Mater. 40 (1992) 957.