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Local icosahedral packing energetics for d-band metals
Materials Science and Engineering, 99 (1988) 349-352
349
Local Icosahedral Packing Energetics for d-band Metals* ROBERT B. PHILLIPS and A. E. CARLSSON
Department of Physics, Washington University, St. Louis, Missouri 63130 (U.S.A.)
Abstract
Local icosahedral order in solid environments is modelled by polytope {3, 3, 5} (a packing of atoms on S 3, the surface of the unit sphere in four dimensions, with perfect icosahedral symmetry) and by a 147 atom extension of the Mackay icosahedron. The bonding energies of each are calculated in a one-electron d-band, tight binding model, as a function of d-band filling. The energies per site on the polytope, and on the central atom of the extended Mackay icosahedron, are then compared with those in the f.c.c, structure to reveal chemical trends in the relative stability of local icosahedral packing. The calculated structural energy differences show that even in fairly close-packed metals, local icosahedral packing is preferred only over a limited range of d-band fillings, roughly between 2 and 5 delectrons per atom, even when frustration effects are artificially turned off by the use of the polytope. It is also observed that the magnitude of these energy differences is of the same order of magnitude as the structural energy differences between f.c.c, and b.c.c, structures and typically exceed the elastic energy differences which arise as a result of atomic size mismatches. These phase stability results are in marked contrast with those calculated in pair potential models. 1. Introduction
The discovery of quasi-crystals by Shechtman et al. [1], as well as recent work on the structure of metallic glasses [2] has raised significant questions concerning the relative stability of icosahedral packing in metallic alloys. Frank showed that using simple pair potentials, thirteen-atom icosahedral clusters are energetically favored over their cubo-octahedral counterparts [3]. Molecular dynamics simulations based on pair potentials have also revealed an abundance of icosahedral bond orientational order in supercooled liquids further suggesting energetic or entropic favorability for icosahedral packing [4]. Models in which angle dependent forces are introduced via electronic
*Paper presented at the Sixth International Conference on Rapidly Quenched Metals, Montr6al, August 3-7, 1987. 0025-5416/88/$3.50
structure calculations have demonstrated that small icosahedral nickel clusters are energetically more stable than nickel clusters with cubo-octahedral geometry [5]. On the basis of such calculations, it has often been assumed that for metals in the solid environment local icosahedral packing is nearly universally preferred, but is not always observed because it cannot be propagated indefinitely. It is therefore essential to establish the stability of local icosahedral packing relative to other types, without the complications due to, for example, the spurious inclusion of surface effects in small cluster calculations. Nelson and Widom [9, 10] and Sadoc and coworkers [11] have argued that curved-space models provide a useful way of modelling local icosahedral packing. One of the attractive features of such models is that they allow us to circumvent the frustration effects associated with attempting to propagate icosahedral packing in fiat space [2]. In fact, recent Landau-type theories have assumed that in the absence of such frustration effects, icosahedral configurations could be states of minimum free energy [12-14]. In the present work, we examine the electronic structure, moments of the density of states, and bonding energies of a model elemental transition metal on polytope {3, 3, 5} as well as a 147 atom extension of the Mackay icosahedron [ 15]. Although, by far, most structures containing icosahedral sites are obtained only in alloy systems, a great deal about the relative magnitude of angle dependent forces and the universality (or lack of) of the energetic stability of local icosahedral packing can be learned from these calculations. Following earlier work on bulk solids [ 16], we calculate the low-order moments of the densities of states, and use the recently implemented principle of maximum entropy [ 17-19] to reconstruct the site projected density of states (DOS). From the DOS, we then compute the bonding energy per site as a function of d-band filling, and compare this with the bonding energy per site in the f.c.c, structure. We adopt this procedure instead of using the exact eigenvalues associated with the clusters because it utilizes only information concerning the local environment, in keeping with the aim of our calculations. For the same reason, we consider only the DOS on the central atom © Elsevier Sequoia/Printed in The Netherlands
350 of the Mackay icosahedron, which is the only one having exact icosahedral symmetry (all sites of the polytope are equivalent). 2. Method Polytope {3, 3, 5} is a 120 atom cluster on S3 in which every atom has perfect icosahedral coordination. The mathematical structure of the polytope was analyzed in detail by Coxeter [20]. By contrast, the Mackay icosahedron is a geometric structure in R 3 with 55 atoms. We extended the Mackay icosahedron so that atoms in the second, third and fourth coordination shells would have 12-fold coordination. The primary difference in geometry between the polytope and the extended Mackay icosahedron first shows up in the second-neighbor environment, which in the polytope is built up by packing atoms above the faces of the icosahedron, while the extended Mackay icosahedron is built by packing atoms above the edges of the icosahedron (see Fig. 1). In the A15 structure Frank-Kasper phases [21], the second-neighbor shell around the icosahedral sites is built up above the faces of the icosahedron, as in the polytope. The electronic structure of these models is calculated in a one-electron, d-band, tight-binding model where for the flat-space calculations the Slater-Koster table [22-23] is used to compute angular dependence of the couplings. We include only nearest-neighbor couplings at a fixed interatomic spacing, so it is not necessary to obtain the radial dependence of the couplings. The ratio of the couplings were similar to those obtained from canonical band theory [24] and the overall magnitude was chosen to reproduce a typical d-band width of 7-8 eV. The numerical values are Vdd~= -- 1.230 eV, V d d l t = 0.664 eV, and Vdd6 ------0.049 eV. The proper definition of the couplings and their angular dependence in curved space is a subtle ~aestion, and the answer is not unique. Our
(b)
(o) /
/
\~.. \ "-.
POLYTOPE
O t \ I \
MACKAY ICOSAHEDRON
Fig. 1. Packing schemes for second neighbor environment in polytope and extended Mackay icosahedron.
approach is to consider the polytope as embedded in R 4 and to employ the nine d-hyperspherical harmonics [25], which form a basis for an irreducible representation of SO(4), as the angular part of the orbital wave functions. In this case, these nine harmonic functions ~buL~t(r~)are defined at each site with respect to a local set of axes. L labels the magnitude of the angular momentum with respect to the local axis of quantization. Since we are considering d-electrons, we consider only the five L = 2 orbitals. The matrix elements to be evaluated then have the form
n,(i, L = 2, MIHIj, L' = 2, M')nj
(1)
where i a n d j are labels describing the particular atoms being coupled, and hi and fij refer to the direction of the axis of quantization at sites i and j respectively. The angular dependence of the couplings is contained in the difference vector r -- r~ - r~, and also in the orientation of the quantization axes (~i, hi). For any given pair of nearest neighbor atoms, the angle between the quantization axes h~ and hj defines a one parameter family of quantization axes which preserve the icosahedral symmetry. Our quantization axes were chosen to obtain a value of the third moment of the DOS, reasonably close to that of the icosahedron in flat space. Here the moments on site i are defined [ 16] by
(~,),-
Y. (i,~lH"li,~)=
p,(E)E"dE
(2)
~t=l --oo
where p,(E) is the density of states projected on site i, and ct is an orbital index. Our requirement on the quantization axes is made on the grounds that the third moment is the first that reflects the angles in the first neighbor environment, and at this level we want close agreement between the polytope and the flatspace icosahedron. This is achieved by choosing hi and hj such that h,(fij) is given by the direction of the position vector pointing from the origin to atomj(i). If one chooses h~ and hj to be parallel to one another and perpendicular to the difference vector r = rj - ri, one obtains results equivalent to earlier work by Widom [ 10], but in this case the third moment on the polytope differs both in magnitude and sign from the third moment on the flat-space icosahedron. The interatomic coupling strengths were obtained by first choosing the difference vector r = r j - r i to point along the ( l, 0, 0, 0) axis. In this case the interatomic coupling matrix is diagonal and has three distinct elements, denoted by iTdd,, lYdd. and frdda. Using representation matrices [25] for SO(4), and performing rotations which bring the difference vector into the (1, 0, 0, 0) axis the matrix elements were obtained as
351 linear combinations of l?dd,,, Pdd,~and ITda6.The values of these parameters we used are lYdd~= -- 1.566 eV, ITad~ = 0.730 eV, and Pda6 = --0.049 eV. These were selected so that the dimer in curved space would have the same eigenvalues as the flat-space dimer. To calculate the moments of the density of states, we employed the recursion method [26], as well as path counting [27] and explicit diagonalization of the Hamiltonian matrix. From the moments with n ~< 6, the DOS was obtained using the maximum entropy method [16-19]. This was employed because it is the optimally "unbiased" scheme for reproducing a nonnegative function from its moments (see ref. 28 for a discussion). 3. Results The calculated structural energy differences per atom between the polytope and f.c.c., and the central atom on the extended Mackay icosahedron and f.c.c., are shown in Fig. 2. For both the polytope and the extended Mackay icosahedron, we see that they are significantly more stable than f.c.c, only in the region of d-band filling between about 2 and 5 d-electrons per atom (Nd). This range of Nd does not correlate well with the observed range of average d-band occupation in real alloys exhibiting local icosahedral packing [29], though it does correlate with earlier tight-binding calculations on the phase stability of A15 compounds [30]. The disagreement with observed trends is undoubtedly partly due to the fact that in most alloys having icosahedral sites, many of the other sites have 14-, 15-, and 16-fold coordination [21]. Furthermore, we have not treated the distortions of the icosahedra found in real alloy systems. However, our calculations show that, when angle-dependent forces are included, local icosahedral packing is not necessarily the minimum energy configuration, even when this packing is done in the artificially frustration-free environment of curved space. The magnitude of the structural energy differences is of order 0.5 eV. For convenience we define a scaled energy difference
0.6 0,4-
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,
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Nd Fig. 2. Calculated structural energy differences between polytope and f.c.c. ( ), and extended Mackay icosabedron and f.c.c. ( ) vs. number of d-electrons per atom. AE is energy per atom for polytope and energy projected on central atom for Mackay icosahedron. icosahedral packing and f.c.c, packing, yields (using a typical transition metal bulk modulus B of 2 Mbar) Ae3 = B ( A V ) 2 / ( 2 V x bandwidth) ~ 0.03 The energy scale of the elastic energy is that which would be expected from pair potential calculations without surface effects. It is seen to be smaller than that associated with the angle-dependent forces in our bonding energy calculations. Furthermore the energy scale associated with icosahedral vs. conventional packing schemes is of the same order of magnitude as the f.c.c.-b.c.c, energy difference.
fcc
I >q9 1.5
~.r.o
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/
/ I
A~2 = [AE(b.c.c.-f.c.c.)[/(bandwidth) ~ 0.07.
/
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Ae~ = [AE(polytope-f.c.c.)l/(bandwidth ) ~ 0.06 Earlier calculations [31] of the f.c.c.-b.c.c, structural energy difference, and the bandwidth of a standard transition metal such as ruthenium (bandwidth, approximately 7.5 eV) [32], yield the result
/
-6
/
Polytope /
/
/
I
,I
-4
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4
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E (eV) However, an estimate of the elastic energy difference resulting from the atomic size mismatch between
Fig. 3. Densities of states for polytope, extended Mackay icosahedron, and f.c.c.
352
Acknowledgments 4,
0.5
The authors are grateful to Nick Papanicolaou and Lawrence Mead for the use of their maximum entropy code M A X E N T . This work was supported by the Department of Energy under G r a n t Number DE-FG0284ER45130.
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Fig. 4. Difference in local densities of states between icosahedral environments and f.c.c. ( , polytope-f.c.c.; , Mackay-f.c.c.). Figure 3 shows the site projected density of states for the polytope, the extended Mackay icosahedron and the f.c.c, structure based on the sixth-moment calculation. The differences between the DOS are shown in Fig. 4. Both the polytope and the Mackay icosahedron difference DOS display a peak near the lower band edge, which appears to be due to a degeneracy associated with the high symmetry of the icosahedral environment. This peak is followed by a dip and a smaller peak. In a moment analysis, this type of difference between the icosahedral DOS and the f.c.c. DOS first appears in #4 (the second moments are identical for all the structures). This is significantly smaller in the icosahedral structures even though these have a larger number of paths contributing to #4. The region of relative stability for the icosahedral structures (compare Fig. 2) corresponds closely to Fermi level positions residing in the dip in the difference DOS. In summary, this work shows that even in a curved-space background, where frustration effects are artificially turned off, local icosahedral packing is not energetically favored for all d-band fillings. Though the calculated range (2-5 d-electrons per atom) of energetic favorability is not in good agreement with the observed range of average d-band occupation in icosahedral alloys (4.5-7.5 d-electrons per atom), we believe that our calculations demonstrate the important role played by angle-dependent forces in such packing schemes, in comparison with atomic size mismatch effects. Future work will take into account higher and/or lower coordination polyhedra and distortions of the ideal polyhedral environments, and will treat alloying effects explicitly.
1 D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, Phys. Rev. Lett., 53(1984) 1951. 2 D. R. Nelson, Phys. Rev. Lett., 50(1983) 982; Phys. Rev. B, 28(1983) 5515. 3 F. C. Frank, Proc. Roy. Soc. (London)Ser. A, 215(1952) 43. 4 P. J. Steinhardt, D. R. Nelson and M. Ronchetti, Phys. Rev. B, 28 (1983) 784. 5 F. Cyrot-Lackmann and M. B. Gordon, J. Phys. Colloq., 41-C3 (1980) 273. 6 M. B. Gordon, F. Cyrot-Lackmann and M. C. Desjonqu6res, Surf. Sci., 80(1979) 159. 7 J. P. Gaspard, C. H. Hodges and M. B. Gordon, J. Phys. Colloq., 38-C2 (1977) 63. 8 M. E. McHenry, M. E. Eberhart, R. C. O'Handley and K. H. Johnson, Phys. Rev. Lett., 56(1986) 81. 9 D. R. Nelson and M. Widom, Nucl. Phys. B, 240(1984) 113. 10 M. Widom, Phys. Rev. B, 31 (1985) 6456. 11 M. Kl6man and J. F. Sadoc, J. Phys. Lett., 40(1979) L569; J. Sadoc, J. Phys. Colloq., 41-C8 (1980) 326; J. F. Sadoc and R. Mosseri, Philos. Mag. B, 45(1982) 467; J. F. Sadoc, J. Non-Cryst. Solids, 75(1985) 103. 12 J. P. Sethna, Phys. Rev. B, 31 (1985) 6278. 13 S. Sachdev and D. R. Nelson, Phys. Rev. B, 32(1985) 1480. 14 M. Widom, K. J. Strandburg and R. H. Swendsen,Phys. Rev. Lett., 58 (1987) 706. 15 A. L. Mackay, Acta Crystallogr., 15(1962) 916. 16 R. H. Brown and A. E. Carlsson, Phys. Rev. B, 32 (1985) 6125. 17 E. T. Jaynes, Phys. Rev., 106(1957) 620; 108 (1957) 171. 18 L. R. Mead and N. Papanicolaou, J. Math. Phys., 25(1984) 2404. 19 A. E. Carlsson and P. A. Fedders, Phys. Rev. B, 34(1986) 3567. 20 H. S. M. Coxeter, Regular Polytopes, Dover Publications, New York, 1973. 21 F. C. Frank and J. S. Kasper, Acta Crystallogr., 11 (1958) 184; 12 (1959) 483. 22 J. C. Slater and G. F. Koster, Phys. Rev., 94(1954) 1498. 23 W. A. Harrison, Electronic Structure and the Properties of Solids, Freeman, New York, 1980. 24 O. K. Anderson and O. Jepsen, Phys. Rev. Lett., 53(1984) 2571. 25 J. D. Talman, SpecialFunctions, Benjamin,New York, 1968. 26 R. Haydock, V. Heine and M. J. Kelly, J. Phys. C, 5(1972) 2845; 8 (1975) 2591. 27 F. Cyrot-Lackmann, J. Phys. Chem. Solids, 29 (1968) 1235. 28 E.T. Jaynes, in R. D. Rosenkrantz (ed.), Papers on Probability, Statistics, and Statistical Physics, Reidel, Dordrecht, 1983. 29 R. E. Watson and L. H. Bennett, Acta Metall., 32(1984) 477. 30 P. Turchi, G. Treglia and F. Ducastelle,J. Phys. F, 13(1983) 2543. 31 H. L. Skriver, Phys. Rev. B, 31 (1985) 1909. 32 V. L. Moruzzi, J. F. Janak and A. R. Williams, Calculated Electrons Properties of Metals, Pergamon, Oxford, 1978.
349
Local Icosahedral Packing Energetics for d-band Metals* ROBERT B. PHILLIPS and A. E. CARLSSON
Department of Physics, Washington University, St. Louis, Missouri 63130 (U.S.A.)
Abstract
Local icosahedral order in solid environments is modelled by polytope {3, 3, 5} (a packing of atoms on S 3, the surface of the unit sphere in four dimensions, with perfect icosahedral symmetry) and by a 147 atom extension of the Mackay icosahedron. The bonding energies of each are calculated in a one-electron d-band, tight binding model, as a function of d-band filling. The energies per site on the polytope, and on the central atom of the extended Mackay icosahedron, are then compared with those in the f.c.c, structure to reveal chemical trends in the relative stability of local icosahedral packing. The calculated structural energy differences show that even in fairly close-packed metals, local icosahedral packing is preferred only over a limited range of d-band fillings, roughly between 2 and 5 delectrons per atom, even when frustration effects are artificially turned off by the use of the polytope. It is also observed that the magnitude of these energy differences is of the same order of magnitude as the structural energy differences between f.c.c, and b.c.c, structures and typically exceed the elastic energy differences which arise as a result of atomic size mismatches. These phase stability results are in marked contrast with those calculated in pair potential models. 1. Introduction
The discovery of quasi-crystals by Shechtman et al. [1], as well as recent work on the structure of metallic glasses [2] has raised significant questions concerning the relative stability of icosahedral packing in metallic alloys. Frank showed that using simple pair potentials, thirteen-atom icosahedral clusters are energetically favored over their cubo-octahedral counterparts [3]. Molecular dynamics simulations based on pair potentials have also revealed an abundance of icosahedral bond orientational order in supercooled liquids further suggesting energetic or entropic favorability for icosahedral packing [4]. Models in which angle dependent forces are introduced via electronic
*Paper presented at the Sixth International Conference on Rapidly Quenched Metals, Montr6al, August 3-7, 1987. 0025-5416/88/$3.50
structure calculations have demonstrated that small icosahedral nickel clusters are energetically more stable than nickel clusters with cubo-octahedral geometry [5]. On the basis of such calculations, it has often been assumed that for metals in the solid environment local icosahedral packing is nearly universally preferred, but is not always observed because it cannot be propagated indefinitely. It is therefore essential to establish the stability of local icosahedral packing relative to other types, without the complications due to, for example, the spurious inclusion of surface effects in small cluster calculations. Nelson and Widom [9, 10] and Sadoc and coworkers [11] have argued that curved-space models provide a useful way of modelling local icosahedral packing. One of the attractive features of such models is that they allow us to circumvent the frustration effects associated with attempting to propagate icosahedral packing in fiat space [2]. In fact, recent Landau-type theories have assumed that in the absence of such frustration effects, icosahedral configurations could be states of minimum free energy [12-14]. In the present work, we examine the electronic structure, moments of the density of states, and bonding energies of a model elemental transition metal on polytope {3, 3, 5} as well as a 147 atom extension of the Mackay icosahedron [ 15]. Although, by far, most structures containing icosahedral sites are obtained only in alloy systems, a great deal about the relative magnitude of angle dependent forces and the universality (or lack of) of the energetic stability of local icosahedral packing can be learned from these calculations. Following earlier work on bulk solids [ 16], we calculate the low-order moments of the densities of states, and use the recently implemented principle of maximum entropy [ 17-19] to reconstruct the site projected density of states (DOS). From the DOS, we then compute the bonding energy per site as a function of d-band filling, and compare this with the bonding energy per site in the f.c.c, structure. We adopt this procedure instead of using the exact eigenvalues associated with the clusters because it utilizes only information concerning the local environment, in keeping with the aim of our calculations. For the same reason, we consider only the DOS on the central atom © Elsevier Sequoia/Printed in The Netherlands
350 of the Mackay icosahedron, which is the only one having exact icosahedral symmetry (all sites of the polytope are equivalent). 2. Method Polytope {3, 3, 5} is a 120 atom cluster on S3 in which every atom has perfect icosahedral coordination. The mathematical structure of the polytope was analyzed in detail by Coxeter [20]. By contrast, the Mackay icosahedron is a geometric structure in R 3 with 55 atoms. We extended the Mackay icosahedron so that atoms in the second, third and fourth coordination shells would have 12-fold coordination. The primary difference in geometry between the polytope and the extended Mackay icosahedron first shows up in the second-neighbor environment, which in the polytope is built up by packing atoms above the faces of the icosahedron, while the extended Mackay icosahedron is built by packing atoms above the edges of the icosahedron (see Fig. 1). In the A15 structure Frank-Kasper phases [21], the second-neighbor shell around the icosahedral sites is built up above the faces of the icosahedron, as in the polytope. The electronic structure of these models is calculated in a one-electron, d-band, tight-binding model where for the flat-space calculations the Slater-Koster table [22-23] is used to compute angular dependence of the couplings. We include only nearest-neighbor couplings at a fixed interatomic spacing, so it is not necessary to obtain the radial dependence of the couplings. The ratio of the couplings were similar to those obtained from canonical band theory [24] and the overall magnitude was chosen to reproduce a typical d-band width of 7-8 eV. The numerical values are Vdd~= -- 1.230 eV, V d d l t = 0.664 eV, and Vdd6 ------0.049 eV. The proper definition of the couplings and their angular dependence in curved space is a subtle ~aestion, and the answer is not unique. Our
(b)
(o) /
/
\~.. \ "-.
POLYTOPE
O t \ I \
MACKAY ICOSAHEDRON
Fig. 1. Packing schemes for second neighbor environment in polytope and extended Mackay icosahedron.
approach is to consider the polytope as embedded in R 4 and to employ the nine d-hyperspherical harmonics [25], which form a basis for an irreducible representation of SO(4), as the angular part of the orbital wave functions. In this case, these nine harmonic functions ~buL~t(r~)are defined at each site with respect to a local set of axes. L labels the magnitude of the angular momentum with respect to the local axis of quantization. Since we are considering d-electrons, we consider only the five L = 2 orbitals. The matrix elements to be evaluated then have the form
n,(i, L = 2, MIHIj, L' = 2, M')nj
(1)
where i a n d j are labels describing the particular atoms being coupled, and hi and fij refer to the direction of the axis of quantization at sites i and j respectively. The angular dependence of the couplings is contained in the difference vector r -- r~ - r~, and also in the orientation of the quantization axes (~i, hi). For any given pair of nearest neighbor atoms, the angle between the quantization axes h~ and hj defines a one parameter family of quantization axes which preserve the icosahedral symmetry. Our quantization axes were chosen to obtain a value of the third moment of the DOS, reasonably close to that of the icosahedron in flat space. Here the moments on site i are defined [ 16] by
(~,),-
Y. (i,~lH"li,~)=
p,(E)E"dE
(2)
~t=l --oo
where p,(E) is the density of states projected on site i, and ct is an orbital index. Our requirement on the quantization axes is made on the grounds that the third moment is the first that reflects the angles in the first neighbor environment, and at this level we want close agreement between the polytope and the flatspace icosahedron. This is achieved by choosing hi and hj such that h,(fij) is given by the direction of the position vector pointing from the origin to atomj(i). If one chooses h~ and hj to be parallel to one another and perpendicular to the difference vector r = rj - ri, one obtains results equivalent to earlier work by Widom [ 10], but in this case the third moment on the polytope differs both in magnitude and sign from the third moment on the flat-space icosahedron. The interatomic coupling strengths were obtained by first choosing the difference vector r = r j - r i to point along the ( l, 0, 0, 0) axis. In this case the interatomic coupling matrix is diagonal and has three distinct elements, denoted by iTdd,, lYdd. and frdda. Using representation matrices [25] for SO(4), and performing rotations which bring the difference vector into the (1, 0, 0, 0) axis the matrix elements were obtained as
351 linear combinations of l?dd,,, Pdd,~and ITda6.The values of these parameters we used are lYdd~= -- 1.566 eV, ITad~ = 0.730 eV, and Pda6 = --0.049 eV. These were selected so that the dimer in curved space would have the same eigenvalues as the flat-space dimer. To calculate the moments of the density of states, we employed the recursion method [26], as well as path counting [27] and explicit diagonalization of the Hamiltonian matrix. From the moments with n ~< 6, the DOS was obtained using the maximum entropy method [16-19]. This was employed because it is the optimally "unbiased" scheme for reproducing a nonnegative function from its moments (see ref. 28 for a discussion). 3. Results The calculated structural energy differences per atom between the polytope and f.c.c., and the central atom on the extended Mackay icosahedron and f.c.c., are shown in Fig. 2. For both the polytope and the extended Mackay icosahedron, we see that they are significantly more stable than f.c.c, only in the region of d-band filling between about 2 and 5 d-electrons per atom (Nd). This range of Nd does not correlate well with the observed range of average d-band occupation in real alloys exhibiting local icosahedral packing [29], though it does correlate with earlier tight-binding calculations on the phase stability of A15 compounds [30]. The disagreement with observed trends is undoubtedly partly due to the fact that in most alloys having icosahedral sites, many of the other sites have 14-, 15-, and 16-fold coordination [21]. Furthermore, we have not treated the distortions of the icosahedra found in real alloy systems. However, our calculations show that, when angle-dependent forces are included, local icosahedral packing is not necessarily the minimum energy configuration, even when this packing is done in the artificially frustration-free environment of curved space. The magnitude of the structural energy differences is of order 0.5 eV. For convenience we define a scaled energy difference
0.6 0,4-
O.2 .: /
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,
,
/X
"\
/
0.2
/ ',
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0.4 0.6
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Nd Fig. 2. Calculated structural energy differences between polytope and f.c.c. ( ), and extended Mackay icosabedron and f.c.c. ( ) vs. number of d-electrons per atom. AE is energy per atom for polytope and energy projected on central atom for Mackay icosahedron. icosahedral packing and f.c.c, packing, yields (using a typical transition metal bulk modulus B of 2 Mbar) Ae3 = B ( A V ) 2 / ( 2 V x bandwidth) ~ 0.03 The energy scale of the elastic energy is that which would be expected from pair potential calculations without surface effects. It is seen to be smaller than that associated with the angle-dependent forces in our bonding energy calculations. Furthermore the energy scale associated with icosahedral vs. conventional packing schemes is of the same order of magnitude as the f.c.c.-b.c.c, energy difference.
fcc
I >q9 1.5
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Mackay/
/
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/ I
A~2 = [AE(b.c.c.-f.c.c.)[/(bandwidth) ~ 0.07.
/
0
Ae~ = [AE(polytope-f.c.c.)l/(bandwidth ) ~ 0.06 Earlier calculations [31] of the f.c.c.-b.c.c, structural energy difference, and the bandwidth of a standard transition metal such as ruthenium (bandwidth, approximately 7.5 eV) [32], yield the result
/
-6
/
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/
/
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,I
-4
/
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4
6
E (eV) However, an estimate of the elastic energy difference resulting from the atomic size mismatch between
Fig. 3. Densities of states for polytope, extended Mackay icosahedron, and f.c.c.
352
Acknowledgments 4,
0.5
The authors are grateful to Nick Papanicolaou and Lawrence Mead for the use of their maximum entropy code M A X E N T . This work was supported by the Department of Energy under G r a n t Number DE-FG0284ER45130.
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4
6
Fig. 4. Difference in local densities of states between icosahedral environments and f.c.c. ( , polytope-f.c.c.; , Mackay-f.c.c.). Figure 3 shows the site projected density of states for the polytope, the extended Mackay icosahedron and the f.c.c, structure based on the sixth-moment calculation. The differences between the DOS are shown in Fig. 4. Both the polytope and the Mackay icosahedron difference DOS display a peak near the lower band edge, which appears to be due to a degeneracy associated with the high symmetry of the icosahedral environment. This peak is followed by a dip and a smaller peak. In a moment analysis, this type of difference between the icosahedral DOS and the f.c.c. DOS first appears in #4 (the second moments are identical for all the structures). This is significantly smaller in the icosahedral structures even though these have a larger number of paths contributing to #4. The region of relative stability for the icosahedral structures (compare Fig. 2) corresponds closely to Fermi level positions residing in the dip in the difference DOS. In summary, this work shows that even in a curved-space background, where frustration effects are artificially turned off, local icosahedral packing is not energetically favored for all d-band fillings. Though the calculated range (2-5 d-electrons per atom) of energetic favorability is not in good agreement with the observed range of average d-band occupation in icosahedral alloys (4.5-7.5 d-electrons per atom), we believe that our calculations demonstrate the important role played by angle-dependent forces in such packing schemes, in comparison with atomic size mismatch effects. Future work will take into account higher and/or lower coordination polyhedra and distortions of the ideal polyhedral environments, and will treat alloying effects explicitly.
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