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Reconstruction possibility of fcc (111) metallic surfaces at room temperature
Materials Letters 59 (2005) 1907 – 1909 www.elsevier.com/locate/matlet
Reconstruction possibility of fcc (111) metallic surfaces at room temperature J.C. Li, W. Liu, Q. JiangT Key Laboratory of Automobile Materials (Jilin University), Ministry of Education Department of Materials Science and Engineering, Jilin University, Changchun 130025, People’s Republic of China Received 15 September 2004; received in revised form 1 February 2005; accepted 10 February 2005 Available online 17 March 2005
Abstract According to known analytical models for surface stress f and surface energy c, f and c of some metals have been calculated. In terms of the calculations, the reconstruction possibility associated with the quantity of ( fc) for (111) facet of nine fcc metals is predicted. The results are in agreement with other theoretical and the experimental results. D 2005 Elsevier B.V. All rights reserved. PACS: 68.35.Bs; 68.35.Gy; 68.35.Md; 68.47.De Keywords: Surface reconstruction; Surface stress; Surface energy; Metallic surface
Atomic coordination number (CN) imperfection induced surface reconstruction (in the plane) and relaxation (in the normal) are significant as they relate to performance of the surface [1,2]. It is well known that clean surfaces of some pure materials undergo reconstruction, a process in which the atoms in the top monolayer(s) are redistributed to create a surface region with a different (and often more complex) structure from the ideally terminated 1 1 surface. Reconstructions occur on a variety of semiconductor and metal surfaces both in the clean state and in the presence of an adsorbate. On low-index metal surfaces reconstructions are often characterized by a change in the atomic density of the top monolayer at the surface. In particular, certain clean, metal fcc (111) and (001) surfaces feature this type of reconstruction. These are parts of a larger class of surface structural transitions termed commensurate–incommensurate (C–I) phase transitions. These tend to include cases for which neither phase is strictly incommensurate. In all cases an increase in the density is observed, with a concomitant uniaxial or biaxial elastic contraction of the surface layer.
T Corresponding author. Fax: +86 431 5095876. E-mail address: [email protected] (Q. Jiang). 0167-577X/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2005.02.008
The surfaces that exhibit these reconstructions belong exclusively to the 4d and 5d transition metals and feature filled or nearly filled d-shells. These structure reconstructions are not restricted to the transition metals, however. For example, the second layer of Cu grown on top of a pseudomorphic first layer Cu on a Ru (0001) substrate has shown a similar reconstruction behavior [3]. Frenkel–Kontorova (FK) model has been widely used with some success in describing the incorporative type of reconstructions found on clean, metallic fcc surfaces, especially on (111) facet of Au [4,5]. For application to two-dimensional systems, the FK model assumes that the physical system can be adequately described by a planar array of atoms that interact harmonically with each other and rest on a static, corrugated potential provided by the atoms of the substrate. The structure of the overlayer is then determined from a balance of the interlayer and intralayer interactions. By considering the overall change in energy per unit area due to the transition, a stability criterion was derived [4]. However, application of this criterion to fcc (111) metals failed to predict a stable reconstruction of any of the surfaces [6]. To further describe surface reconstruction, a bulk continuum elastic model was originally proposed by Herring
1908
J.C. Li et al. / Materials Letters 59 (2005) 1907–1909
[7] and later extended by Cammarata [8]. According to that model, a surface should reconstruct under the condition of f c Na ð1Þ b¼ Gh where a i 1 / [4k(1m)] i 0.1 with m being PoissonVs ratio, and b is the stability parameter. The driving force for the reconstruction has been associated with the quantity ( fc) while the opposing force is due to the disregistry with the underlying lattice, which yields a stability criterion depending on f, c, the equilibrium nearest-neighbor atomic distance h and the shear modulus G. In spite of its simplicity, the above model has successfully predicted surface reconstruction condition [6]. Recently, the molecular dynamics (MD) simulations of the surface reconstruction of Au (111) have been performed, which are consistent with the bulk continuum elastic theory for surface reconstructions on clean metal fcc (111) surfaces [6]. Since f and c are experimentally difficult to determine, the utilization of Eq. (1) depends on whether experimental data are available. In this contribution, f and c are calculated in terms of our thermodynamic models. Based on the results, the stabilities of (111) facets of nine fcc metals are predicted, which correspond to experimental and other theoretical results. Based on the bond-broken rule of surface, c has been modeled analytically without any free parameter as follows [9], h i c ¼ 2 ZS =ZB ðZS =ZB Þ1=2 E=2: ð2Þ where Z S and Z B are the coordination number (CN) of surface atoms and that of interior atoms, and E is the cohesion energy per bond. Z S can be determined according to the crystalline structure through determining Z (h k l) by a geometric consideration [9]. For a fcc structure, Z B = 12. For other structure, Ref. [9] has discussed in detail. Z (h k l) can be determined by some known geometrical rules. For any surface of a fcc structure with h z k z l [9],
Zðh
k lÞ
¼ 2h þ k
Zðh
k lÞ
¼ 4h þ 2k
ð3aÞ
for h; k; l being odd;
ð3bÞ
for the rest:
f can be expressed [10] as, f ¼ ½ð3csl D0 Þ=ð8jÞ1=2
ð4Þ
where c sl is the solid–liquid interface energy, D 0 = 3h for particles and wires while D 0 = 2h for thin films. j is the coefficient of compressibility with j = 1 / B where B is the bulk modulus. c sl in Eq. (4) has been deduced according to Gibbs– Thomson equation [10], csl ¼ 2hSvib Hm ðT Þ=ð 3Vm RÞ
ð5Þ
where R is the ideal gas constant, H m is the temperaturedependent melting enthalpy of bulk crystals, S vib is the vibrational part of the overall melting entropy S m, and V m is the molar volume. Although the melting entropy of crystals consists, at least, of three contributions: positional, vibrational and electronic component [11], the melting for metallic and organic crystals is mainly vibrational in nature and S vib c S m = H m / T m is used [11]. According to Helmholtz function, H m(T) = G m(T) TdG m(T) / dT where G m(T) is the Gibbs free energy. For elements, Gm ðT Þ ¼ Hm ðTm T Þð7T Þ [12] and, thus, H m(T) = 49 / (T m / T + 6)2H m Tm ðTm þ6T Þ where H m is the melting enthalpy of bulk crystals at the melting temperature T m. In terms of this relationship, there is, cs1 ¼
98hT 2 Hm Svib 3RðTm þ 6T Þ2 Vm
ð6Þ
:
f and g and b of some (111) facets of fcc metals are calculated in light of Eqs. (4), (2) and (1), respectively, which are shown in Table 1. As shown in the table, although b values obtained from Eq. (1) where f and c values are calculated by Eqs. (4) and (2) differ from bV values with
Table 1 b and bV (literature data) values in terms of Eq. (1) and related parameters for (111) facets of nine fcc metals where h values are obtained by substituting f and c values determined by Eq. (4) with D 0 = 2h for plane surfaces and Eq. (2). c sl in Eq. (4) is calculated in terms of Eq. (6) at T = 300 K Ir
Ni
Cu
Ag
Pd
Pt
Pb
Au
Al
h (2) [18] 2.174 2.492 2.556 2.899 2.751 2.774 3.500 2.885 2.863 V m (cm3/mol) [19] 8.54 6.6 7.1 10.3 8.90 9.10 18.7 10.2 10 T m (K) [19] 2716 1726 1358 1234 1825 2045 600.6 1337 933.3 26.10 17.47 13.05 11.30 17.60 19.60 4.799 12.55 10.79 H m (K J/mol) [19] j (1012/Pa) 2.695 5.640 7.257 9.653 5.348 3.623 21.83 5.848 13.30 G (GPa) [20] 209 76 48.3 30.3 43.6 60.9 5.59 26 26.2 f (J/m2) 4.01 3.04 2.23 1.65 2.91 3.71 0.75 2.27 1.65 3.19 2.44 1.83 1.20 1.85 2.54 0.55 1.52 1.45 c (J/m2) b 0.034 0.032 0.032 0.051 0.088 0.069 0.102 0.100 0.026 bV 0.034 [6] 0.022 [6] 0.007 [6] 0.035 [6] 0.057 [6] 0.087 [6] 0.103 [6] 0.119 [6] 0.041 [13] 0.034 [13] 0.19 [13]T 0.19 [13] Note that the temperature dependence of H m(T) in Eq. (6) is induced by the difference of specific heat between the crystal and the liquid DC p [11]. At T b T k, where T k determined by dG m(T) / dT = 0 is the ideal glass transition temperature or the isentropic temperature, liquid transforms to glass and DC p between the crystal and glass could be neglected [10] and thus H m(T b T k) c H m(T k). j = 1 / B where B is bulk modulus [20]. T Pt (111) reconstructs under certain circumstances (data taken from [15–17]).
J.C. Li et al. / Materials Letters 59 (2005) 1907–1909
literature f and c values, b and bV values have the same prediction for the surface reconstructions of the concerned nine elements. The results are in agreement with the other theory predicted and experimentally observed results. Namely the surface reconstructions are absent for most elements except Au (111) and Pb (111) since their b values are smaller than 0.1. Note also that although Au and Pb exhibit surface reconstruction at room temperature, T / T m of the elements is different. bV values of Ir (111) and Ni (111) are negative in Ref. [6] as shown in the table, i.e. f b g. It is known that f = g + Bg / BA where A is surface area, g = f for liquids, while g b f for solids is very often [10] although there are exceptions depending on the sign of Bg / BA [14]. Thus, bV values of Ir (111) and Ni (111) should also be positive in our opinion. In addition, according to the CN imperfection theory [1,2], the size of the bond strength of surface atoms and thus that of surface stress in one side increase while the size of the surface energy in another side decreases after the surface relaxation for any metallic elements. These results must lead to g b f or bV N0. In light of Eq. (1), even if this driving force in comparison with the surface stress is large enough, the atoms of the topmost layer may overcome the cohesion to the underlying lattice in order to reduce their distance from each other; the surface reconstruction of metals will occur. Since Eqs. (4) and (2) for surface stress and surface energy can be approximately applied for any surface of any structure, the surface reconstruction can be in general predicted. In summary, the stability of metal surfaces to certain commensurate–incommensurate reconstructive phase transformations is investigated. f and c of (111) facets for fcc metals are calculated, which leads to facility of predicting the stability of the surfaces.
1909
Acknowledgements The financial support from National Key Basic Research and Development Program (Grant No. 20004CB619301) is acknowledged.
References [1] C.Q. Sun, L.K. Pan, H.L. Bai, Z.Q. Li, P. Wu, E.Y. Jiang, Acta Mater. 51 (2003) 4631. [2] C.Q. Sun, H.L. Bai, S. Li, B.K. Tay, C. Li, T.P. Chen, E.Y. Jiang, J. Phys. Chem., B 108 (2004) 2162. [3] K. Meinel, H. Wolter, Ch. Ammer, H. Neddermeyer, Surf. Sci. 401 (1998) 434. [4] R.J. Needs, M.J. Godfrey, M. Mansfield, Surf. Sci. 242 (1991) 215. [5] N. Takeuchi, C.T. Chan, K.M. Ho, Phys. Rev., B 43 (1991) 13899. [6] T.M. Trimble, R.C. Cammarata, K. Sieradzki, Surf. Sci. 531 (2003) 8. [7] C. Herring, in: W.E. Kingston (Ed.), The Physics of Powder Metallurgy, McGraw-Hill, New York, 1951, p. 143. [8] R.C. Cammarata, Surf. Sci. 279 (1992) 341. [9] Q. Jiang, H.M. Lu, M. Zhao, J. Phys. Condens. Mater. 16 (2004) 521. [10] Q. Jiang, D.S. Zhao, M. Zhao, Acta Mater. 49 (2001) 3143. [11] Q. Jiang, H.X. Shi, M. Zhao, J. Chem. Phys. 111 (1999) 2176. [12] J.C. Li, W. Liu, Q. Jiang, Acta Mater. 53 (2005) 1067. [13] W. Haiss, Rep. Prog. Phys. 64 (2001) 591. [14] S. Olivier, A. Sau´l, G. Tre´glia, Appl. Surf. Sci. 212–213 (2003) 866. [15] A.R. Sandy, S.G.J. Mochrie, Phys. Rev. Lett. 68 (1992) 2192. [16] H. Ibach, Surf. Sci. Rep. 29 (1997) 195. [17] V. Fiorentini, M. Methfessel, M. Scheffler, Phys. Rev. Lett. 71 (1993) 1051. [18] H.W. King, in: R.W. Cahn, P. Haasen (Eds.), Physical Metallurgy, North-Holland Physics Publishing, Amsterdam, 1983, pp. 63 – 65. [19] Sargent-Welch Scientific Company, Periodic Table of the Elements, Skokie, Illinois, 1980, p. 1. [20] E.A. Brandes, Smithells Metals Reference Book, 6th edition, Butterworth and Co (Publishers), Ltd, London, 1983, p. 15.
Reconstruction possibility of fcc (111) metallic surfaces at room temperature J.C. Li, W. Liu, Q. JiangT Key Laboratory of Automobile Materials (Jilin University), Ministry of Education Department of Materials Science and Engineering, Jilin University, Changchun 130025, People’s Republic of China Received 15 September 2004; received in revised form 1 February 2005; accepted 10 February 2005 Available online 17 March 2005
Abstract According to known analytical models for surface stress f and surface energy c, f and c of some metals have been calculated. In terms of the calculations, the reconstruction possibility associated with the quantity of ( fc) for (111) facet of nine fcc metals is predicted. The results are in agreement with other theoretical and the experimental results. D 2005 Elsevier B.V. All rights reserved. PACS: 68.35.Bs; 68.35.Gy; 68.35.Md; 68.47.De Keywords: Surface reconstruction; Surface stress; Surface energy; Metallic surface
Atomic coordination number (CN) imperfection induced surface reconstruction (in the plane) and relaxation (in the normal) are significant as they relate to performance of the surface [1,2]. It is well known that clean surfaces of some pure materials undergo reconstruction, a process in which the atoms in the top monolayer(s) are redistributed to create a surface region with a different (and often more complex) structure from the ideally terminated 1 1 surface. Reconstructions occur on a variety of semiconductor and metal surfaces both in the clean state and in the presence of an adsorbate. On low-index metal surfaces reconstructions are often characterized by a change in the atomic density of the top monolayer at the surface. In particular, certain clean, metal fcc (111) and (001) surfaces feature this type of reconstruction. These are parts of a larger class of surface structural transitions termed commensurate–incommensurate (C–I) phase transitions. These tend to include cases for which neither phase is strictly incommensurate. In all cases an increase in the density is observed, with a concomitant uniaxial or biaxial elastic contraction of the surface layer.
T Corresponding author. Fax: +86 431 5095876. E-mail address: [email protected] (Q. Jiang). 0167-577X/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2005.02.008
The surfaces that exhibit these reconstructions belong exclusively to the 4d and 5d transition metals and feature filled or nearly filled d-shells. These structure reconstructions are not restricted to the transition metals, however. For example, the second layer of Cu grown on top of a pseudomorphic first layer Cu on a Ru (0001) substrate has shown a similar reconstruction behavior [3]. Frenkel–Kontorova (FK) model has been widely used with some success in describing the incorporative type of reconstructions found on clean, metallic fcc surfaces, especially on (111) facet of Au [4,5]. For application to two-dimensional systems, the FK model assumes that the physical system can be adequately described by a planar array of atoms that interact harmonically with each other and rest on a static, corrugated potential provided by the atoms of the substrate. The structure of the overlayer is then determined from a balance of the interlayer and intralayer interactions. By considering the overall change in energy per unit area due to the transition, a stability criterion was derived [4]. However, application of this criterion to fcc (111) metals failed to predict a stable reconstruction of any of the surfaces [6]. To further describe surface reconstruction, a bulk continuum elastic model was originally proposed by Herring
1908
J.C. Li et al. / Materials Letters 59 (2005) 1907–1909
[7] and later extended by Cammarata [8]. According to that model, a surface should reconstruct under the condition of f c Na ð1Þ b¼ Gh where a i 1 / [4k(1m)] i 0.1 with m being PoissonVs ratio, and b is the stability parameter. The driving force for the reconstruction has been associated with the quantity ( fc) while the opposing force is due to the disregistry with the underlying lattice, which yields a stability criterion depending on f, c, the equilibrium nearest-neighbor atomic distance h and the shear modulus G. In spite of its simplicity, the above model has successfully predicted surface reconstruction condition [6]. Recently, the molecular dynamics (MD) simulations of the surface reconstruction of Au (111) have been performed, which are consistent with the bulk continuum elastic theory for surface reconstructions on clean metal fcc (111) surfaces [6]. Since f and c are experimentally difficult to determine, the utilization of Eq. (1) depends on whether experimental data are available. In this contribution, f and c are calculated in terms of our thermodynamic models. Based on the results, the stabilities of (111) facets of nine fcc metals are predicted, which correspond to experimental and other theoretical results. Based on the bond-broken rule of surface, c has been modeled analytically without any free parameter as follows [9], h i c ¼ 2 ZS =ZB ðZS =ZB Þ1=2 E=2: ð2Þ where Z S and Z B are the coordination number (CN) of surface atoms and that of interior atoms, and E is the cohesion energy per bond. Z S can be determined according to the crystalline structure through determining Z (h k l) by a geometric consideration [9]. For a fcc structure, Z B = 12. For other structure, Ref. [9] has discussed in detail. Z (h k l) can be determined by some known geometrical rules. For any surface of a fcc structure with h z k z l [9],
Zðh
k lÞ
¼ 2h þ k
Zðh
k lÞ
¼ 4h þ 2k
ð3aÞ
for h; k; l being odd;
ð3bÞ
for the rest:
f can be expressed [10] as, f ¼ ½ð3csl D0 Þ=ð8jÞ1=2
ð4Þ
where c sl is the solid–liquid interface energy, D 0 = 3h for particles and wires while D 0 = 2h for thin films. j is the coefficient of compressibility with j = 1 / B where B is the bulk modulus. c sl in Eq. (4) has been deduced according to Gibbs– Thomson equation [10], csl ¼ 2hSvib Hm ðT Þ=ð 3Vm RÞ
ð5Þ
where R is the ideal gas constant, H m is the temperaturedependent melting enthalpy of bulk crystals, S vib is the vibrational part of the overall melting entropy S m, and V m is the molar volume. Although the melting entropy of crystals consists, at least, of three contributions: positional, vibrational and electronic component [11], the melting for metallic and organic crystals is mainly vibrational in nature and S vib c S m = H m / T m is used [11]. According to Helmholtz function, H m(T) = G m(T) TdG m(T) / dT where G m(T) is the Gibbs free energy. For elements, Gm ðT Þ ¼ Hm ðTm T Þð7T Þ [12] and, thus, H m(T) = 49 / (T m / T + 6)2H m Tm ðTm þ6T Þ where H m is the melting enthalpy of bulk crystals at the melting temperature T m. In terms of this relationship, there is, cs1 ¼
98hT 2 Hm Svib 3RðTm þ 6T Þ2 Vm
ð6Þ
:
f and g and b of some (111) facets of fcc metals are calculated in light of Eqs. (4), (2) and (1), respectively, which are shown in Table 1. As shown in the table, although b values obtained from Eq. (1) where f and c values are calculated by Eqs. (4) and (2) differ from bV values with
Table 1 b and bV (literature data) values in terms of Eq. (1) and related parameters for (111) facets of nine fcc metals where h values are obtained by substituting f and c values determined by Eq. (4) with D 0 = 2h for plane surfaces and Eq. (2). c sl in Eq. (4) is calculated in terms of Eq. (6) at T = 300 K Ir
Ni
Cu
Ag
Pd
Pt
Pb
Au
Al
h (2) [18] 2.174 2.492 2.556 2.899 2.751 2.774 3.500 2.885 2.863 V m (cm3/mol) [19] 8.54 6.6 7.1 10.3 8.90 9.10 18.7 10.2 10 T m (K) [19] 2716 1726 1358 1234 1825 2045 600.6 1337 933.3 26.10 17.47 13.05 11.30 17.60 19.60 4.799 12.55 10.79 H m (K J/mol) [19] j (1012/Pa) 2.695 5.640 7.257 9.653 5.348 3.623 21.83 5.848 13.30 G (GPa) [20] 209 76 48.3 30.3 43.6 60.9 5.59 26 26.2 f (J/m2) 4.01 3.04 2.23 1.65 2.91 3.71 0.75 2.27 1.65 3.19 2.44 1.83 1.20 1.85 2.54 0.55 1.52 1.45 c (J/m2) b 0.034 0.032 0.032 0.051 0.088 0.069 0.102 0.100 0.026 bV 0.034 [6] 0.022 [6] 0.007 [6] 0.035 [6] 0.057 [6] 0.087 [6] 0.103 [6] 0.119 [6] 0.041 [13] 0.034 [13] 0.19 [13]T 0.19 [13] Note that the temperature dependence of H m(T) in Eq. (6) is induced by the difference of specific heat between the crystal and the liquid DC p [11]. At T b T k, where T k determined by dG m(T) / dT = 0 is the ideal glass transition temperature or the isentropic temperature, liquid transforms to glass and DC p between the crystal and glass could be neglected [10] and thus H m(T b T k) c H m(T k). j = 1 / B where B is bulk modulus [20]. T Pt (111) reconstructs under certain circumstances (data taken from [15–17]).
J.C. Li et al. / Materials Letters 59 (2005) 1907–1909
literature f and c values, b and bV values have the same prediction for the surface reconstructions of the concerned nine elements. The results are in agreement with the other theory predicted and experimentally observed results. Namely the surface reconstructions are absent for most elements except Au (111) and Pb (111) since their b values are smaller than 0.1. Note also that although Au and Pb exhibit surface reconstruction at room temperature, T / T m of the elements is different. bV values of Ir (111) and Ni (111) are negative in Ref. [6] as shown in the table, i.e. f b g. It is known that f = g + Bg / BA where A is surface area, g = f for liquids, while g b f for solids is very often [10] although there are exceptions depending on the sign of Bg / BA [14]. Thus, bV values of Ir (111) and Ni (111) should also be positive in our opinion. In addition, according to the CN imperfection theory [1,2], the size of the bond strength of surface atoms and thus that of surface stress in one side increase while the size of the surface energy in another side decreases after the surface relaxation for any metallic elements. These results must lead to g b f or bV N0. In light of Eq. (1), even if this driving force in comparison with the surface stress is large enough, the atoms of the topmost layer may overcome the cohesion to the underlying lattice in order to reduce their distance from each other; the surface reconstruction of metals will occur. Since Eqs. (4) and (2) for surface stress and surface energy can be approximately applied for any surface of any structure, the surface reconstruction can be in general predicted. In summary, the stability of metal surfaces to certain commensurate–incommensurate reconstructive phase transformations is investigated. f and c of (111) facets for fcc metals are calculated, which leads to facility of predicting the stability of the surfaces.
1909
Acknowledgements The financial support from National Key Basic Research and Development Program (Grant No. 20004CB619301) is acknowledged.
References [1] C.Q. Sun, L.K. Pan, H.L. Bai, Z.Q. Li, P. Wu, E.Y. Jiang, Acta Mater. 51 (2003) 4631. [2] C.Q. Sun, H.L. Bai, S. Li, B.K. Tay, C. Li, T.P. Chen, E.Y. Jiang, J. Phys. Chem., B 108 (2004) 2162. [3] K. Meinel, H. Wolter, Ch. Ammer, H. Neddermeyer, Surf. Sci. 401 (1998) 434. [4] R.J. Needs, M.J. Godfrey, M. Mansfield, Surf. Sci. 242 (1991) 215. [5] N. Takeuchi, C.T. Chan, K.M. Ho, Phys. Rev., B 43 (1991) 13899. [6] T.M. Trimble, R.C. Cammarata, K. Sieradzki, Surf. Sci. 531 (2003) 8. [7] C. Herring, in: W.E. Kingston (Ed.), The Physics of Powder Metallurgy, McGraw-Hill, New York, 1951, p. 143. [8] R.C. Cammarata, Surf. Sci. 279 (1992) 341. [9] Q. Jiang, H.M. Lu, M. Zhao, J. Phys. Condens. Mater. 16 (2004) 521. [10] Q. Jiang, D.S. Zhao, M. Zhao, Acta Mater. 49 (2001) 3143. [11] Q. Jiang, H.X. Shi, M. Zhao, J. Chem. Phys. 111 (1999) 2176. [12] J.C. Li, W. Liu, Q. Jiang, Acta Mater. 53 (2005) 1067. [13] W. Haiss, Rep. Prog. Phys. 64 (2001) 591. [14] S. Olivier, A. Sau´l, G. Tre´glia, Appl. Surf. Sci. 212–213 (2003) 866. [15] A.R. Sandy, S.G.J. Mochrie, Phys. Rev. Lett. 68 (1992) 2192. [16] H. Ibach, Surf. Sci. Rep. 29 (1997) 195. [17] V. Fiorentini, M. Methfessel, M. Scheffler, Phys. Rev. Lett. 71 (1993) 1051. [18] H.W. King, in: R.W. Cahn, P. Haasen (Eds.), Physical Metallurgy, North-Holland Physics Publishing, Amsterdam, 1983, pp. 63 – 65. [19] Sargent-Welch Scientific Company, Periodic Table of the Elements, Skokie, Illinois, 1980, p. 1. [20] E.A. Brandes, Smithells Metals Reference Book, 6th edition, Butterworth and Co (Publishers), Ltd, London, 1983, p. 15.