- Email: [email protected]
Semistatistical model of AlMnSi-type icosahedral quasicrystal
] O U R N A L Of"
Journal of Non-Crystalline Solids 143 (1992) 225-231 North-Holland
NON-CRYSTALLISOLIDS NE
Semistatistical model of A1MnSi-type icosahedral quasicrystal M. Mihalkovi~ a n d P. M r a f k o Institute of Physics, Slovak Academy of Sciences, Dubravska cesta 9, CS-842 28 Bratislava, Czechoslovakia
Received 3 October 1991 Revised manuscript 13 January 1992
The i-A1MnSi-typemodel inferred from c~-AIMnSiphase was investigated. The shapes of 3D atomic motifs at nodes and body centers .of 6D hypercubic lattice are optimized by the Monte Carlo method, optimizing the density and average coordination number. The resulting 6D fcc ordering is qualitatively compared with X-ray and neutron diffraction experiments on i-A1CuFe alloy. The model structure simulating the i-Al(Si)Mn quasibinary system with 6D body-center domain occupied by Al,is found to be stable upon molecular dynamics relaxation with Morse pair potentials.
1. Introduction Elser and Henley [1] for the first time recognized that Mn sublattice plus central voids of 54-atom Mackay icosahedra (MI) clusters in the a-A1MnSi F r a n k - K a s p e r phase is a periodically repeated fragment of 3D Penrose tiling. The relation between a- and i-A1MnSi phase thus naturally leads to a more general 6D-hyperspace description of both structures ([2], CP method [3,4]), in which rational approximant is viewed as a commensurately locked quasiperiodic structure with uniform 'phason' strain, U i j = O R [ - / O R / , where R ± and R are the coordinates in 'phason' and 'real' mutually perpendicular 3D-spaces, respectively. Essentially, the two approaches to the deterministic modelling have been developed utilizing the known structure of rational approximant: (i) 3D, in which A m m a n n rhombohedra are decorated by the icosahedral clusters and longrange order is propagated by hierarchical inflation procedure [5] and (ii) 6D, in which the structure is specified by 3D hypersurfaces located at the nodes and body centers of 6D hypercubic lattice. While the models of type (i) have well mastered local ordering but fail to guarantee the quasiperiodicity, the models of type (ii) predeter-
mine the long range order but to achieve reasonable density many parameters must be specified and the shapes of hypersurfaces become complicated. The aim of this paper is to develop the method combining the two approaches. The structures we construct are the tilings of A m m a n n rhombohedra decorated using the non-local Monte Carlo algorithm, optimizing the density and average coordination number. Since the decoration of the tiles depends on their environment, the structures can be most efficiently described through a 6Dscheme. In phason space, our technique can be viewed as the optimization of a 3D hypersurface at non-zero uniform phason strain. A final stage of the technique (not deduced here) should be to infer the shape of 3D hypersurfaces at zero phason strain.
2. Method 2.1. Initial structure
In the AIMgZn structural class, the model of i-phase can be easily constructed by the direct decoration of A m m a n n rhombohedra, since their
0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
M. Mihalkovi?, P. Mrafko / AlMnSi-type icosahedral quasicrystal
226
grouping is not constrained by the matching rules [6]. On the contrary, in the A1MnSi structural class there are no simple matching rules satisfying the geometrical constraints imposed by the facetype decoration of a rhombohedral tiling [1]. Our idea is to decorate the rhombohedral tiling in the fashion seen in a-phase while ignoring many unreasonably short distances that occur, and then to take this initial structure and optimize the occupation of the sites so as to satisfy an appropriate hard-core condition. The vocabulary of the site types seen in costructure [1] can be extended on arbitrary rhombohedral tiling: A l ( a ) sites occupy mid-edge points of the vacant 12-fold vertex, forming small Mackay icosahedron (MI); Mn(c¢) sites occupy end-points of the edges of 12-fold vertex, forming large MI; AI(~) sites form icosidodecahedra around 12-fold vertex and they are assigned to the points dividing long diagonals of rhombohedral faces in the ratio r-~ : r - 2 or (a small fraction of them) to the vertices. The rest of the sites are called 'glue' and occupy remaining vertices (Mn([3)), remaining face-diagonal points (Al(y)) (some Al(y) sites are assigned also to the body diagonals of prolate rhombohedra dividing them in the ratio ~'-3(1 : V~- : 1)), and the pairs of points dividing long diagonals of prolate rhombohedra in the ratio T - - 2 : " / ' - - 3 : ' F - 2 ( A I ( ~ ) ) . While Al(~i) sites are the projections of 6D hypercubic lattice body centers, the rest of sites are the projections of 6D nodes [7]. The initial structure is most easily obtained by the CP technique, using the simplest (spherical) acceptance domains with radii sufficiently large to comprise all possible sites of each type. In order to obtain the system with finite size we impose uniform phason strain
( oo)
U( n ) = - ( F~ - Fn_ar) / ( Fnr + F n _ l )
×
0 0
1 0
0 , 1
(1)
where F~ are Fibonacci F,,=17,_ l + F n _ 2 and golden mean, leading boundary conditions in
numbers F0 = 0, F 1 = 1, z = ( 1 + v ~ - ) / 2 is the to the cubic periodic the 'real' space. Under
this distortion, a new phason coordinate, R ~ , reads (2)
R ~ = R " + UR + Uo,
where u 0 is a translation in the phason space. The structures generated by the projection technique at non-zero uniform phason strain depend non-trivially on the shift u0; for example, the choice u 0 = (000) leads to the Pm3 space symmetry [8] while u0=(lll)
(~+r)/[2(F,~'+Fn_I)
] -u s
(3)
to the Pa3 ( F n + F , _ 1 odd) or I213 ( F ~ + F n _ 1 even) space symmetry [9]. For the reasons discussed in section 3.1, we choose u 0 = u s. 2.2. Monte Carlo optimization Monte Carlo (MC) optimization assigns to each site from the initial structure one of two states: the ith site is occupied (S i = 1) or unoccupied (S i = 0) by the sphere with radius r 0 =0.5aq, w h e r e a q = 4.6 A is the edge length of a rhombohedron (quasilattice constant) and r 0 is chosen so as to allow forming of small Mackay icosahedra. The MC run is initialized setting S i = 0 Vi and consists of two parallel processes: (i) choose randomly the site and set S i = 1 - S i if [ ( e x p ( - a l A e ) < x ~ ) / , ( A E > 0)1 V ( h e _ < 0),
(4) and (ii) choose randomly the pair of sites S i and Sj such that ( S i q- S j ) = 1)/~ (rij < r 0) and s e t S i = 1 - Si, S j = l - Sj if
((1/(1 + exp(-a2
e)))
< (5)
where x I and x 2 are random numbers uniformly distributed on the interval (0, 1), ri] is the pair distance, and AE is the change of the potential energy, E, of the system. We define potential energy as E = -N b
ifrmin > r°'
E = ~
otherwise,
(6)
where N b is a number of bonds in the system with length from the interval (r 0, 1.4r0), repre-
M. Mihalkovi~, P. Mrafko / AlMnSi-type icosahedral quasicrystal Table 1 Rms deviations (in ,&) from idealized sites in a-phase. Numbering of orbits is taken from ref. [2] Model/orbit
AI(6)
Al(7)
Al(6 + 7) - Al(y)
After ref. [1] After ref. [2] Present study
0.302 0.415 0.302
0.913 0.032 0.032
0.680 0.294 0.215
senting a first shell cutoff and rmin is the length of the shortest bond in the system. The definition (6) represents hard-sphere and monoatomic approximation of the pair potentials. Parameter a 1 > 0 governs the annihilation of the spheres, while a 2 > 0 enables the diffusion of vacancies (the vacancy is defined as an unoccupied site with just 1 occupied site in forbidden distance). Our experience is that with a t = 0 (terminated annihilation) and ce2 = 0 (the m6ves of the vacancies are fully random), the system always reaches the state with the maximal density; subsequent slow 'cooling' ( a 2 ~ ~) maximizes the average coordination number. 2.3. Modification: real-space c o m p o n e n t s
The basic test the model has to pass is the comparison of the ' 1 / 1 ' (n = 2 in eq. (1)) approximant structure with the a-phase. However, the described algorithm cannot achieve complete agreement (including an e v e n - o d d modulation in the third shell of Mackay icosahedra [10]) with the set of sites described in the previous section. Therefore we (i) eliminate all sites with pair distance r < r 0 from A l ( a ) sites in order to support forming of small Mackay icosahedra and (ii)
227
assign real-space shifts to some AI(~/) sites ('phason' coordinates Rr~ remain unchanged). A similar philosophy has been exploited by the authors of ref. [11] when attempting to fit a neutron diffraction experiment. Our rule is: let the fourfold vertex of the rhombohedral tiling have the coordinate R 4 and the endpoints of the four edges emanating from it R 4 + el, i = 1 . . . 4. Only one pair combination of the four edges, say ei, ej, satisfies eie j < 0; the remaining pair of edges is denoted e k and e t, respectively. Half of the idealized AI(-~) sites in the a-phase are by the usual CP-type model [2] assigned coordinates R~ = R 4 + e i + r - 1 (e k + el ) or R 4 + e i + "r- 2 (e k + e l) producing short AI(~/)AI(~) distances (0.812r0). We shift these sites to the R'r = R 4 + e i + 2(ek + e t ) / 3 and R 4 + e i + (e k + et)/3. The minimum energy 1 / 1 structure obtained by MC optimization is then a compromise between the descriptions of E l s e r - H e n l e y [1] and Cahn et al. [2]; root mean square deviations from the real atomic positions are reduced (table 1) and e v e n - o d d modulation is correctly reproduced. The modification of idealized AI(~) site positions is visualized in fig. 1.
3. Properties of optimized structures 3.1. Density
The structural characteristics of the sequence of optimized models with increasing n (see eq. (1)) are gathered in table 2. The models have densities about 3.5% larger than comparable 5 / 3
Table 2 Structural characteristics of optimized approximants compared with a-phase
Fn/Fn
-
1
1/1 (a-ph.) 2/1 3/2 5/3 5/3*
Mn (%)
AI(~) (%)
MI (%)
CNav
Atoms per cubic cell
p ( a t o m s / A 3)
17.4 17.9 18.2 18.0 18.2
4.4 4.8 4.7 4.3 2.7
78.3 74.5 70.4 68.0 68.8
12.652 12.593 12.616 12.581 12.368
138 580 2456 10 388 10364
0.0680 0.0674 0.0674 0.0673 0.0666
AI(~) are the sites projected from BC motif MI (%) is a fraction of sites contained in 54-atom MI clusters. The last column, p, is density of the structure ( a t o m / A 3) calculated for aq = 4.6 A. Model 5 / 3 * has been optimized without modification of AI(~/) sites.
228
M. Mihalko~i~, P. Mrafko / AlMnSi-type icosahedral quasicrystal
Another aspect is that 6D-type models fail to guarantee the conservation of the matter under the translation u 0. However, a detailed discussion of this topic is beyond the scope of the paper; we note that the reasons are not purely technical (how to distort the shapes of complicated 3D polyhedra in phason space in accordance with eq. (2)). Again, the simplest way to avoid this difficulty is to set u 0 = u S.
3.2. 6D fcc ordering Fig. 1. Fourfold vertex (e) constitutes assembly of two prolate (PR) and two oblate (OR) rhombohedra (the latter are not drawn in figure except for their c o m m o n face) - rhombic dodecahedron (RD), linking a pair of MI in twofold direction. Al(y) sites (after ref. [1]) are Al(6) plus AI(7) (after ref. [2]); on the figure A1(6)=-@ and AI(7)-= ©. T h e arrows indicate real-space components assigned to the Al('y) sites by the rule described in section 2.3. E v e n / o d d modulation in a-phase is then naturally described as follows: even R D are decorated by eight Al(y) sites on P R face diagonals (four of them are shown in figure), while odd R D are decorated by the two Al(6) sites (body-center motif) on the triad axes of both PR.
D u n e a u - O g u e y model [12] studied in ref. [13]. The density depends mainly on the density of Mackay icosahedra centered at 12-fold vertices, since they concentrate about 70% of the mass and on the ordering of 'glue' atoms. As the frequency of 12-fold vertices for small n in eq. (1) exhibit the strong dependence on the shift, u0, the same holds also for the density of low-order approximant structures generated by 6D technique. For example, in the 2 / 1 (n = 3) case, we find the number of 12-fold vertices per cubic cell ranging from 4 to 8. How drastically this effects the density can be demonstrated on the comparison of the u 0 = 0 Pm3 structure [7] (seven 12-fold vertices, seven MI, 507 atoms) and u 0 = u s Pa3 structure constructed here (eight MI, 580 atoms). The choice u o = u s defines the sequence of the most ordered approximant structures [9] reminiscent of perfect self-similar quasiperiodic tiling [3,14] with the number of 12-fold vertices equal to 4F3(n_2). Careful choice of u 0 becomes crucial in any extensive calculations (electronic structure) requiring relatively small size of investigated system.
The manifest feature of our optimized structures is a 6D fcc modulation. Each atomic position in the model can be indexed by six-integer indices ni: 6D-node sites are at ~2nie i, while 6D-body-center sites are at ½~(2n i + 1)e i. In fig. 2, we show the cuts through the optimized motifs perpendicularly to the two-fold direction, in which the crosses represent the sites with E n i even projected onto the phason space (N 1 node motif, BC a body center motif) while the full circles represent the sites with ~,n i odd (N 2 and BC 2 motifs). We find, that about 56.3% of the atoms belong to the N 1 motif, 39% to the N 2, 4.7% to the BC~ while none of the atoms sit at BC 2 position. Assigning normalized scattering powers to each motif, F ( N 1) = 38.4%, F ( N 2) = 55.9%, F ( B C 1) = 5.7% and F ( B C 2) = 0%, we find the best agreement of simulated diffraction pattern with X-ray diffraction data of i-A163Cu25Fe12 alloy [15] (fig. 3). The difference in scattering powers of even and odd motifs is most pronouncedly reflected by the intensity of (7/11) peak, which vanishes if the two motifs are identical. Recent neutron diffraction measurements on the same i-A1CuFe system [16] determine the scattering powers as F ( N 1 ) = 4 0 . 2 5 % , F ( N 2 ) = 5 1 . 2 5 % , F ( B C 1) = 8.5% and F ( B C 2) = 0%, while the N 1 domain is larger than N 2, which is also in qualitative agreement with our result. Note, that i-A1CuFe belongs to the i-A1MnSi structural class [17] and the tendency to an 6D fcc ordering has been observed also in i-A1MnSi system [18]. To bring to light the origin of a 6D fcc ordering, we have, at first, eliminated parallel shifts of AI(-,/) sites. The result is shown on the example of 5 / 3 model (denoted 5 / 3 * in table 2). The modu-
M. Mihalkovi{, P. Mrafko / AlMnSi-type icosahedral quasicrystal
,,1~,,,I,,,~1,,,,
BC domains disappear and even-odd modulation vanishes. Surprisingly, the density is slightly improved. The network of occupied sites we have obtained has been recognized as the the tiling of four canonical cells [9], introduced recently by Henley [19]. This network can be considered as an interesting alternative to the filings of Ammann rhombohedra. The vertices of the 'canonical' tiling can be decorated by Mackay icosahedra mutually linked only through 'b' (twofold) and 'c' (threefold) linkages [19], known from s-phase. In the current deterministic 6D-type models [7,12], Mackay icosahedra are centered at the 12-fold vertices of the 3D Penrose tiling. For example, in 8 / 5 approximation (n = 6), the CP technique gives maximum of 4F3(n_2)= 576 12-fold vertices and yet few of them have to be removed due to the short fivefold linkages. On the vertices of 8 / 5
I ,
II,,LI,,,,I,~I,II,II
~,I~,,,I,~I,I,,JILJ
229
IIIIIIkZXlIXlIIIII
' " ' I I ........ L x z x x z z J IIIIII Jill
xxxxxxxxxxxxxxzzxx~x ixzxxzx~:--~xxxzxxxx~ ~xxxx~zx xxxx~xx~x~
>,
ri{iii{{iii{[[!rr (c)
~*+H+*-'444+"
~1 ' ' ~
. . . .
(d)
'1
I''''1''''1'
. ÷iiiiiiiiiil![! l l I I I l l l l I I I I I I I I I i i I I I I I I I X l I
Uc oJ
....
I ....
I ....
I ....
o)
E
. I I X l I I I Z I I I X l I I I I I I I III1 I I I I I I
" + ~ + ~ '
'
b)
Fig. 2. 2D cuts of node (left) and B C (right) motifs. Crosses are even parity points and full circles are odd parity points. Scale is in A. (a) 2 / 1 , (b) 3 / 2 , (c) 5/3, (d) 5 / 3 * model - see text and table 2. Neighboring crosses are connected so that they constitute square lattice. Edge center Al(c0 sites (lst MI
shell) are omitted. 40
lation is only weakly reduced, the BC motifs have a smaller relative weight and the density is slightly decreased. However, the elimination of Al(a) mid-edge sites during the optimization (setting r0 = 0.562aq) affects the structure drastically. The
6O
80
100 °26)(XCoKs)
120
Fig. 3. X-ray diffraction pattern of i-A163Cu25Fe12 alloy: (a) experiment [15] and (b) 3 / 2 model. The best agreement is achieved when the 6D motifs are assigned normalized scattering powers F ( N 1) = 38.4%, F ( N 2) = 55.9%, F ( B C 1) = 5.7% and F(BC 2) = 0% and temperature factor B = 0.6.
230
M. Mihalkovi(, P. Mrafko / AlMnSi-type icosahedral quasicrystal
canonical tiling obtained after optimization [9] can be placed 592 MI; all of them are correctly linked and remaining free space can be easily filled in a deterministic way because the network is the tiling of four canonical cells. Only about 65% of the 'canonical' vertices are identified with 12-fold vertices of Penrose tiling (the rest are 7, 8, 9 and 10-fold vertices), hence the Fourier transform of both models (as it mostly depends on the positions of MI, concentrating about 70% of the matter) should significantly differ. The work on the more detailed comparison between the two decorated networks is in progress.
3.3. Stability against molecular dynamics annealing
Gtxt3 ( r )
GAA
0
i
t
4
GAB
2
0
GBB 4
In the case of i-A1MnSi alloy, neutron diffraction experiments do not fully support even the basical Penrose tiling scheme-Mackay icosahedra at (some) 12-fold vertices of the tiling and well ordered 'glue' decoration [11,20]. The fit of the diffraction data can be improved by assigning the atomic sites components in 'real' space [11]. Such real-space displacement field can be produced in a natural way by the simulated relaxation. In fact, Lancon and Billard have shown [13] that the agreement of 6D-type D u n e a u - O g u e y model [12] with experiment is improved after the relaxation with Morse pair potentials. The relaxation preserved an overall icosahedral symmetry and long-range order but has introduced a large displacement field in real space. A similar, but more reliable result is obtained for different i-A1MgZn model [21] with realistic interatomic pair potentials. However Morse pair potentials are considered as a crude and unrealistic approximation of the reality when applied to AI(Si)Mn quasibinary system, molecular dynamics annealing still can supply us a valuable information about the quality of the packing. We have relaxed 2 / 1 model with composition A1476Mn]04 (the composition is given in atoms per cubic cell). The algorithm we have used is decribed in ref. [21]. Our set of Morse potential parameters is r0AA=0.56aq, r0AB= 0 . 5 6 a q , F0BB = 0 . 6 1 a q , a A A = aAB = O~BB = 4 and eAA = CAB = eBB = 10 - 2 ° J. The parameters r0AB
2
0
r 0
i
j
A/~ r
2
A
! 4
_,~_
i 8
,V-V v ~
i
/3
i
~,. . . . i
10
i
i
12
i 14
r [A) Fig. 4. Partial pair distribution functions gu(r) of the relaxed 2 / 1 (Al(Si)476Mnlo4) model.
and r0BB (B = A1/Si, A = Mn) are inferred from the ~x-phase. The relaxation confirmed the stability of the model. Partial pair distribution functions (PPDF) are shown on fig. 4. The long range fluctuations of PPDFs indicate that the displacement field induced by the relaxation has icosahedral symmetry. Its magnitude, characterizing the measure in which original tiling of rhombohedra is distorted, is tabulated in table 3 and compared with a-phase.
Table 3 Root mean square deviations of the site positions (in A) introduced by the molecular dynamics annealing of the idealized model and averaged through the sites of given type (2/1 model) are compared with the rms deviations from the idealized sites in 1/1 a-phase.
Fn/F~_ ~
Al(a)
AI([3)
Al(3,)
AI(~)
Mn(a)
1/1 2/1
0.15 0.33
0.15 0.23
0.22 0.22
0.24 0.32
0.21 0.33
M. Mihalkouid, P. Mrafko / AIMnSi-type icosahedral quasicrystal
A similar result is obtained also for the 3 / 2 model with composition A12016Mn448. The model we have investigated differs from the Lancon and Billard's model in the two aspects: (i) in the ordering and density of 'glue' atoms and (ii) in the composition. While our body-center motif is occupied by AI in agreement with previous experiments [22,23], in ref. [13] it is occupied by Mn as it is plausible for i-A1MnSi with higher Mn content [20]. We conclude that we have demonstrated the stability of i-A1MnSi model with BC motif occupied by AI.
4. Conclusion
We have used the Monte Carlo technique to optimize the density and the average coordination number of the i-A1MnSi type model. The optimized structures exhibit 6D fcc ordering. It is argued, that the ordering is induced by the forced occupation of 12-fold vertices in rhombohedral tiling by small Mackay icosahedra. If these are removed, body-center motif disappears and the optimization produces the dense packings (with hard-core condition rmin = 0.563aq), recognized as the tilings of four canonical cells introduced by Henley [19]. The 6D modulation is qualitatively compared with X-ray [15] and neutron [16] diffraction data from i-A1CuFe alloy. Finally, the structures simulating i-Al(Si)Mn alloy with 6D body-center motif occupied by A1 are found to be stable against the action of Morse pair forces.
231
References [1] V. Elser and C.L. Henley, Phys. Rev. Lett. 55 (1985) 2883. [2] J.W. Cahn, D. Gratias and B. Mozer, J. Phys. (Paris) 49 (1988) 1225. [3] M. Duneau and A. Katz, Phys. Rev. Lett. 54 (1985) 2688. [4] P. Bak, Phys. Rev. Lett. 56 (1986) 861. [5] M. Audier and P. Guyot, Philos. Mag. 58 (1988) L17. [6] C.L. Henley and V. Elser, Philos. Mag. 53 (1986) L59. [7] A. Yamamoto, in: Quasicrystals, ed. T. Fujiwara and T. Ogawa (Springer, Berlin, 1990) p. 57. [8] Shi-Ye Qiu and M.V. Jaric, in: Quasicrystals, ed. M.V. Jaric and S. Lundqvist (World Scientific, Singapore, 1989) p. 19. [9] M. Mihalkovic and P. Mrafko, submitted to Europhys. Lett. [10] H.A. Fowler, B. Mozer and J. Sims, Phys. Rev. B37 (1988) 3906. [11] M. de Boissieu, C. Janot and J.M. Dubois, J. Phys. Cond. Matt. 2 (1990) 2499. [12] M. Duneau and C. Oguey, J. Phys. (Paris) 50 (1989) 135. [13] F. Lancon and L. Billard, J. Non-Cryst. Solids 117&118 (1990) 836. [14] C.L. Henley, Phys. Rev. B34 (1986) 797. [15] Y. Calvayrac, A. Quivy, M. Bessiere, S. Lefebvre, M. Cornier-Quiquandon and D. Gratias, J. Phys. (Paris) 51 (1990) 417. [16] M. Cornier-Quiquandon, A. Quivy, S. Lefebvre, E. Elkaim, G. Heger, A. Katz and D. Gratias, Phys. Rev. B44 (1991) 2071. [17] C.L. Henley, in: Quasicrystals, ed. T. Fujiwara and T. Ogawa (Springer, Berlin, 1990) p. 38. [18] N.K. Mukhopadhyay, S. Ranganathan and K. Chattopadhyay, Philos. Mag. Lett. 60 (1989) 207. [19] C.L. Henley, Phys. Rev. B43 (1991) 993. [20] J.M. Dubois, C. Janot and M. de Boissieu, in: Quasierystalline Materials, ed. C. Janot and J.M. Dubois (World Scientific, Singapore, 1988). p. 97. [21] J. Hafner and M. Krajci, Europhys. Lett. 13 (1990) 335. [22] C. Janot, M. De Boissieu, J.M. Dubois and J. Pannetier, J. Phys. Cond. Mater. 1 (1988) 1029. [23] J.M. Dubois and Ch. Janot, J. Phys. (Paris) 48 (1987) 1981.
Journal of Non-Crystalline Solids 143 (1992) 225-231 North-Holland
NON-CRYSTALLISOLIDS NE
Semistatistical model of A1MnSi-type icosahedral quasicrystal M. Mihalkovi~ a n d P. M r a f k o Institute of Physics, Slovak Academy of Sciences, Dubravska cesta 9, CS-842 28 Bratislava, Czechoslovakia
Received 3 October 1991 Revised manuscript 13 January 1992
The i-A1MnSi-typemodel inferred from c~-AIMnSiphase was investigated. The shapes of 3D atomic motifs at nodes and body centers .of 6D hypercubic lattice are optimized by the Monte Carlo method, optimizing the density and average coordination number. The resulting 6D fcc ordering is qualitatively compared with X-ray and neutron diffraction experiments on i-A1CuFe alloy. The model structure simulating the i-Al(Si)Mn quasibinary system with 6D body-center domain occupied by Al,is found to be stable upon molecular dynamics relaxation with Morse pair potentials.
1. Introduction Elser and Henley [1] for the first time recognized that Mn sublattice plus central voids of 54-atom Mackay icosahedra (MI) clusters in the a-A1MnSi F r a n k - K a s p e r phase is a periodically repeated fragment of 3D Penrose tiling. The relation between a- and i-A1MnSi phase thus naturally leads to a more general 6D-hyperspace description of both structures ([2], CP method [3,4]), in which rational approximant is viewed as a commensurately locked quasiperiodic structure with uniform 'phason' strain, U i j = O R [ - / O R / , where R ± and R are the coordinates in 'phason' and 'real' mutually perpendicular 3D-spaces, respectively. Essentially, the two approaches to the deterministic modelling have been developed utilizing the known structure of rational approximant: (i) 3D, in which A m m a n n rhombohedra are decorated by the icosahedral clusters and longrange order is propagated by hierarchical inflation procedure [5] and (ii) 6D, in which the structure is specified by 3D hypersurfaces located at the nodes and body centers of 6D hypercubic lattice. While the models of type (i) have well mastered local ordering but fail to guarantee the quasiperiodicity, the models of type (ii) predeter-
mine the long range order but to achieve reasonable density many parameters must be specified and the shapes of hypersurfaces become complicated. The aim of this paper is to develop the method combining the two approaches. The structures we construct are the tilings of A m m a n n rhombohedra decorated using the non-local Monte Carlo algorithm, optimizing the density and average coordination number. Since the decoration of the tiles depends on their environment, the structures can be most efficiently described through a 6Dscheme. In phason space, our technique can be viewed as the optimization of a 3D hypersurface at non-zero uniform phason strain. A final stage of the technique (not deduced here) should be to infer the shape of 3D hypersurfaces at zero phason strain.
2. Method 2.1. Initial structure
In the AIMgZn structural class, the model of i-phase can be easily constructed by the direct decoration of A m m a n n rhombohedra, since their
0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
M. Mihalkovi?, P. Mrafko / AlMnSi-type icosahedral quasicrystal
226
grouping is not constrained by the matching rules [6]. On the contrary, in the A1MnSi structural class there are no simple matching rules satisfying the geometrical constraints imposed by the facetype decoration of a rhombohedral tiling [1]. Our idea is to decorate the rhombohedral tiling in the fashion seen in a-phase while ignoring many unreasonably short distances that occur, and then to take this initial structure and optimize the occupation of the sites so as to satisfy an appropriate hard-core condition. The vocabulary of the site types seen in costructure [1] can be extended on arbitrary rhombohedral tiling: A l ( a ) sites occupy mid-edge points of the vacant 12-fold vertex, forming small Mackay icosahedron (MI); Mn(c¢) sites occupy end-points of the edges of 12-fold vertex, forming large MI; AI(~) sites form icosidodecahedra around 12-fold vertex and they are assigned to the points dividing long diagonals of rhombohedral faces in the ratio r-~ : r - 2 or (a small fraction of them) to the vertices. The rest of the sites are called 'glue' and occupy remaining vertices (Mn([3)), remaining face-diagonal points (Al(y)) (some Al(y) sites are assigned also to the body diagonals of prolate rhombohedra dividing them in the ratio ~'-3(1 : V~- : 1)), and the pairs of points dividing long diagonals of prolate rhombohedra in the ratio T - - 2 : " / ' - - 3 : ' F - 2 ( A I ( ~ ) ) . While Al(~i) sites are the projections of 6D hypercubic lattice body centers, the rest of sites are the projections of 6D nodes [7]. The initial structure is most easily obtained by the CP technique, using the simplest (spherical) acceptance domains with radii sufficiently large to comprise all possible sites of each type. In order to obtain the system with finite size we impose uniform phason strain
( oo)
U( n ) = - ( F~ - Fn_ar) / ( Fnr + F n _ l )
×
0 0
1 0
0 , 1
(1)
where F~ are Fibonacci F,,=17,_ l + F n _ 2 and golden mean, leading boundary conditions in
numbers F0 = 0, F 1 = 1, z = ( 1 + v ~ - ) / 2 is the to the cubic periodic the 'real' space. Under
this distortion, a new phason coordinate, R ~ , reads (2)
R ~ = R " + UR + Uo,
where u 0 is a translation in the phason space. The structures generated by the projection technique at non-zero uniform phason strain depend non-trivially on the shift u0; for example, the choice u 0 = (000) leads to the Pm3 space symmetry [8] while u0=(lll)
(~+r)/[2(F,~'+Fn_I)
] -u s
(3)
to the Pa3 ( F n + F , _ 1 odd) or I213 ( F ~ + F n _ 1 even) space symmetry [9]. For the reasons discussed in section 3.1, we choose u 0 = u s. 2.2. Monte Carlo optimization Monte Carlo (MC) optimization assigns to each site from the initial structure one of two states: the ith site is occupied (S i = 1) or unoccupied (S i = 0) by the sphere with radius r 0 =0.5aq, w h e r e a q = 4.6 A is the edge length of a rhombohedron (quasilattice constant) and r 0 is chosen so as to allow forming of small Mackay icosahedra. The MC run is initialized setting S i = 0 Vi and consists of two parallel processes: (i) choose randomly the site and set S i = 1 - S i if [ ( e x p ( - a l A e ) < x ~ ) / , ( A E > 0)1 V ( h e _ < 0),
(4) and (ii) choose randomly the pair of sites S i and Sj such that ( S i q- S j ) = 1)/~ (rij < r 0) and s e t S i = 1 - Si, S j = l - Sj if
((1/(1 + exp(-a2
e)))
< (5)
where x I and x 2 are random numbers uniformly distributed on the interval (0, 1), ri] is the pair distance, and AE is the change of the potential energy, E, of the system. We define potential energy as E = -N b
ifrmin > r°'
E = ~
otherwise,
(6)
where N b is a number of bonds in the system with length from the interval (r 0, 1.4r0), repre-
M. Mihalkovi~, P. Mrafko / AlMnSi-type icosahedral quasicrystal Table 1 Rms deviations (in ,&) from idealized sites in a-phase. Numbering of orbits is taken from ref. [2] Model/orbit
AI(6)
Al(7)
Al(6 + 7) - Al(y)
After ref. [1] After ref. [2] Present study
0.302 0.415 0.302
0.913 0.032 0.032
0.680 0.294 0.215
senting a first shell cutoff and rmin is the length of the shortest bond in the system. The definition (6) represents hard-sphere and monoatomic approximation of the pair potentials. Parameter a 1 > 0 governs the annihilation of the spheres, while a 2 > 0 enables the diffusion of vacancies (the vacancy is defined as an unoccupied site with just 1 occupied site in forbidden distance). Our experience is that with a t = 0 (terminated annihilation) and ce2 = 0 (the m6ves of the vacancies are fully random), the system always reaches the state with the maximal density; subsequent slow 'cooling' ( a 2 ~ ~) maximizes the average coordination number. 2.3. Modification: real-space c o m p o n e n t s
The basic test the model has to pass is the comparison of the ' 1 / 1 ' (n = 2 in eq. (1)) approximant structure with the a-phase. However, the described algorithm cannot achieve complete agreement (including an e v e n - o d d modulation in the third shell of Mackay icosahedra [10]) with the set of sites described in the previous section. Therefore we (i) eliminate all sites with pair distance r < r 0 from A l ( a ) sites in order to support forming of small Mackay icosahedra and (ii)
227
assign real-space shifts to some AI(~/) sites ('phason' coordinates Rr~ remain unchanged). A similar philosophy has been exploited by the authors of ref. [11] when attempting to fit a neutron diffraction experiment. Our rule is: let the fourfold vertex of the rhombohedral tiling have the coordinate R 4 and the endpoints of the four edges emanating from it R 4 + el, i = 1 . . . 4. Only one pair combination of the four edges, say ei, ej, satisfies eie j < 0; the remaining pair of edges is denoted e k and e t, respectively. Half of the idealized AI(-~) sites in the a-phase are by the usual CP-type model [2] assigned coordinates R~ = R 4 + e i + r - 1 (e k + el ) or R 4 + e i + "r- 2 (e k + e l) producing short AI(~/)AI(~) distances (0.812r0). We shift these sites to the R'r = R 4 + e i + 2(ek + e t ) / 3 and R 4 + e i + (e k + et)/3. The minimum energy 1 / 1 structure obtained by MC optimization is then a compromise between the descriptions of E l s e r - H e n l e y [1] and Cahn et al. [2]; root mean square deviations from the real atomic positions are reduced (table 1) and e v e n - o d d modulation is correctly reproduced. The modification of idealized AI(~) site positions is visualized in fig. 1.
3. Properties of optimized structures 3.1. Density
The structural characteristics of the sequence of optimized models with increasing n (see eq. (1)) are gathered in table 2. The models have densities about 3.5% larger than comparable 5 / 3
Table 2 Structural characteristics of optimized approximants compared with a-phase
Fn/Fn
-
1
1/1 (a-ph.) 2/1 3/2 5/3 5/3*
Mn (%)
AI(~) (%)
MI (%)
CNav
Atoms per cubic cell
p ( a t o m s / A 3)
17.4 17.9 18.2 18.0 18.2
4.4 4.8 4.7 4.3 2.7
78.3 74.5 70.4 68.0 68.8
12.652 12.593 12.616 12.581 12.368
138 580 2456 10 388 10364
0.0680 0.0674 0.0674 0.0673 0.0666
AI(~) are the sites projected from BC motif MI (%) is a fraction of sites contained in 54-atom MI clusters. The last column, p, is density of the structure ( a t o m / A 3) calculated for aq = 4.6 A. Model 5 / 3 * has been optimized without modification of AI(~/) sites.
228
M. Mihalko~i~, P. Mrafko / AlMnSi-type icosahedral quasicrystal
Another aspect is that 6D-type models fail to guarantee the conservation of the matter under the translation u 0. However, a detailed discussion of this topic is beyond the scope of the paper; we note that the reasons are not purely technical (how to distort the shapes of complicated 3D polyhedra in phason space in accordance with eq. (2)). Again, the simplest way to avoid this difficulty is to set u 0 = u S.
3.2. 6D fcc ordering Fig. 1. Fourfold vertex (e) constitutes assembly of two prolate (PR) and two oblate (OR) rhombohedra (the latter are not drawn in figure except for their c o m m o n face) - rhombic dodecahedron (RD), linking a pair of MI in twofold direction. Al(y) sites (after ref. [1]) are Al(6) plus AI(7) (after ref. [2]); on the figure A1(6)=-@ and AI(7)-= ©. T h e arrows indicate real-space components assigned to the Al('y) sites by the rule described in section 2.3. E v e n / o d d modulation in a-phase is then naturally described as follows: even R D are decorated by eight Al(y) sites on P R face diagonals (four of them are shown in figure), while odd R D are decorated by the two Al(6) sites (body-center motif) on the triad axes of both PR.
D u n e a u - O g u e y model [12] studied in ref. [13]. The density depends mainly on the density of Mackay icosahedra centered at 12-fold vertices, since they concentrate about 70% of the mass and on the ordering of 'glue' atoms. As the frequency of 12-fold vertices for small n in eq. (1) exhibit the strong dependence on the shift, u0, the same holds also for the density of low-order approximant structures generated by 6D technique. For example, in the 2 / 1 (n = 3) case, we find the number of 12-fold vertices per cubic cell ranging from 4 to 8. How drastically this effects the density can be demonstrated on the comparison of the u 0 = 0 Pm3 structure [7] (seven 12-fold vertices, seven MI, 507 atoms) and u 0 = u s Pa3 structure constructed here (eight MI, 580 atoms). The choice u o = u s defines the sequence of the most ordered approximant structures [9] reminiscent of perfect self-similar quasiperiodic tiling [3,14] with the number of 12-fold vertices equal to 4F3(n_2). Careful choice of u 0 becomes crucial in any extensive calculations (electronic structure) requiring relatively small size of investigated system.
The manifest feature of our optimized structures is a 6D fcc modulation. Each atomic position in the model can be indexed by six-integer indices ni: 6D-node sites are at ~2nie i, while 6D-body-center sites are at ½~(2n i + 1)e i. In fig. 2, we show the cuts through the optimized motifs perpendicularly to the two-fold direction, in which the crosses represent the sites with E n i even projected onto the phason space (N 1 node motif, BC a body center motif) while the full circles represent the sites with ~,n i odd (N 2 and BC 2 motifs). We find, that about 56.3% of the atoms belong to the N 1 motif, 39% to the N 2, 4.7% to the BC~ while none of the atoms sit at BC 2 position. Assigning normalized scattering powers to each motif, F ( N 1) = 38.4%, F ( N 2) = 55.9%, F ( B C 1) = 5.7% and F ( B C 2) = 0%, we find the best agreement of simulated diffraction pattern with X-ray diffraction data of i-A163Cu25Fe12 alloy [15] (fig. 3). The difference in scattering powers of even and odd motifs is most pronouncedly reflected by the intensity of (7/11) peak, which vanishes if the two motifs are identical. Recent neutron diffraction measurements on the same i-A1CuFe system [16] determine the scattering powers as F ( N 1 ) = 4 0 . 2 5 % , F ( N 2 ) = 5 1 . 2 5 % , F ( B C 1) = 8.5% and F ( B C 2) = 0%, while the N 1 domain is larger than N 2, which is also in qualitative agreement with our result. Note, that i-A1CuFe belongs to the i-A1MnSi structural class [17] and the tendency to an 6D fcc ordering has been observed also in i-A1MnSi system [18]. To bring to light the origin of a 6D fcc ordering, we have, at first, eliminated parallel shifts of AI(-,/) sites. The result is shown on the example of 5 / 3 model (denoted 5 / 3 * in table 2). The modu-
M. Mihalkovi{, P. Mrafko / AlMnSi-type icosahedral quasicrystal
,,1~,,,I,,,~1,,,,
BC domains disappear and even-odd modulation vanishes. Surprisingly, the density is slightly improved. The network of occupied sites we have obtained has been recognized as the the tiling of four canonical cells [9], introduced recently by Henley [19]. This network can be considered as an interesting alternative to the filings of Ammann rhombohedra. The vertices of the 'canonical' tiling can be decorated by Mackay icosahedra mutually linked only through 'b' (twofold) and 'c' (threefold) linkages [19], known from s-phase. In the current deterministic 6D-type models [7,12], Mackay icosahedra are centered at the 12-fold vertices of the 3D Penrose tiling. For example, in 8 / 5 approximation (n = 6), the CP technique gives maximum of 4F3(n_2)= 576 12-fold vertices and yet few of them have to be removed due to the short fivefold linkages. On the vertices of 8 / 5
I ,
II,,LI,,,,I,~I,II,II
~,I~,,,I,~I,I,,JILJ
229
IIIIIIkZXlIXlIIIII
' " ' I I ........ L x z x x z z J IIIIII Jill
xxxxxxxxxxxxxxzzxx~x ixzxxzx~:--~xxxzxxxx~ ~xxxx~zx xxxx~xx~x~
>,
ri{iii{{iii{[[!rr (c)
~*+H+*-'444+"
~1 ' ' ~
. . . .
(d)
'1
I''''1''''1'
. ÷iiiiiiiiiil![! l l I I I l l l l I I I I I I I I I i i I I I I I I I X l I
Uc oJ
....
I ....
I ....
I ....
o)
E
. I I X l I I I Z I I I X l I I I I I I I III1 I I I I I I
" + ~ + ~ '
'
b)
Fig. 2. 2D cuts of node (left) and B C (right) motifs. Crosses are even parity points and full circles are odd parity points. Scale is in A. (a) 2 / 1 , (b) 3 / 2 , (c) 5/3, (d) 5 / 3 * model - see text and table 2. Neighboring crosses are connected so that they constitute square lattice. Edge center Al(c0 sites (lst MI
shell) are omitted. 40
lation is only weakly reduced, the BC motifs have a smaller relative weight and the density is slightly decreased. However, the elimination of Al(a) mid-edge sites during the optimization (setting r0 = 0.562aq) affects the structure drastically. The
6O
80
100 °26)(XCoKs)
120
Fig. 3. X-ray diffraction pattern of i-A163Cu25Fe12 alloy: (a) experiment [15] and (b) 3 / 2 model. The best agreement is achieved when the 6D motifs are assigned normalized scattering powers F ( N 1) = 38.4%, F ( N 2) = 55.9%, F ( B C 1) = 5.7% and F(BC 2) = 0% and temperature factor B = 0.6.
230
M. Mihalkovi(, P. Mrafko / AlMnSi-type icosahedral quasicrystal
canonical tiling obtained after optimization [9] can be placed 592 MI; all of them are correctly linked and remaining free space can be easily filled in a deterministic way because the network is the tiling of four canonical cells. Only about 65% of the 'canonical' vertices are identified with 12-fold vertices of Penrose tiling (the rest are 7, 8, 9 and 10-fold vertices), hence the Fourier transform of both models (as it mostly depends on the positions of MI, concentrating about 70% of the matter) should significantly differ. The work on the more detailed comparison between the two decorated networks is in progress.
3.3. Stability against molecular dynamics annealing
Gtxt3 ( r )
GAA
0
i
t
4
GAB
2
0
GBB 4
In the case of i-A1MnSi alloy, neutron diffraction experiments do not fully support even the basical Penrose tiling scheme-Mackay icosahedra at (some) 12-fold vertices of the tiling and well ordered 'glue' decoration [11,20]. The fit of the diffraction data can be improved by assigning the atomic sites components in 'real' space [11]. Such real-space displacement field can be produced in a natural way by the simulated relaxation. In fact, Lancon and Billard have shown [13] that the agreement of 6D-type D u n e a u - O g u e y model [12] with experiment is improved after the relaxation with Morse pair potentials. The relaxation preserved an overall icosahedral symmetry and long-range order but has introduced a large displacement field in real space. A similar, but more reliable result is obtained for different i-A1MgZn model [21] with realistic interatomic pair potentials. However Morse pair potentials are considered as a crude and unrealistic approximation of the reality when applied to AI(Si)Mn quasibinary system, molecular dynamics annealing still can supply us a valuable information about the quality of the packing. We have relaxed 2 / 1 model with composition A1476Mn]04 (the composition is given in atoms per cubic cell). The algorithm we have used is decribed in ref. [21]. Our set of Morse potential parameters is r0AA=0.56aq, r0AB= 0 . 5 6 a q , F0BB = 0 . 6 1 a q , a A A = aAB = O~BB = 4 and eAA = CAB = eBB = 10 - 2 ° J. The parameters r0AB
2
0
r 0
i
j
A/~ r
2
A
! 4
_,~_
i 8
,V-V v ~
i
/3
i
~,. . . . i
10
i
i
12
i 14
r [A) Fig. 4. Partial pair distribution functions gu(r) of the relaxed 2 / 1 (Al(Si)476Mnlo4) model.
and r0BB (B = A1/Si, A = Mn) are inferred from the ~x-phase. The relaxation confirmed the stability of the model. Partial pair distribution functions (PPDF) are shown on fig. 4. The long range fluctuations of PPDFs indicate that the displacement field induced by the relaxation has icosahedral symmetry. Its magnitude, characterizing the measure in which original tiling of rhombohedra is distorted, is tabulated in table 3 and compared with a-phase.
Table 3 Root mean square deviations of the site positions (in A) introduced by the molecular dynamics annealing of the idealized model and averaged through the sites of given type (2/1 model) are compared with the rms deviations from the idealized sites in 1/1 a-phase.
Fn/F~_ ~
Al(a)
AI([3)
Al(3,)
AI(~)
Mn(a)
1/1 2/1
0.15 0.33
0.15 0.23
0.22 0.22
0.24 0.32
0.21 0.33
M. Mihalkouid, P. Mrafko / AIMnSi-type icosahedral quasicrystal
A similar result is obtained also for the 3 / 2 model with composition A12016Mn448. The model we have investigated differs from the Lancon and Billard's model in the two aspects: (i) in the ordering and density of 'glue' atoms and (ii) in the composition. While our body-center motif is occupied by AI in agreement with previous experiments [22,23], in ref. [13] it is occupied by Mn as it is plausible for i-A1MnSi with higher Mn content [20]. We conclude that we have demonstrated the stability of i-A1MnSi model with BC motif occupied by AI.
4. Conclusion
We have used the Monte Carlo technique to optimize the density and the average coordination number of the i-A1MnSi type model. The optimized structures exhibit 6D fcc ordering. It is argued, that the ordering is induced by the forced occupation of 12-fold vertices in rhombohedral tiling by small Mackay icosahedra. If these are removed, body-center motif disappears and the optimization produces the dense packings (with hard-core condition rmin = 0.563aq), recognized as the tilings of four canonical cells introduced by Henley [19]. The 6D modulation is qualitatively compared with X-ray [15] and neutron [16] diffraction data from i-A1CuFe alloy. Finally, the structures simulating i-Al(Si)Mn alloy with 6D body-center motif occupied by A1 are found to be stable against the action of Morse pair forces.
231
References [1] V. Elser and C.L. Henley, Phys. Rev. Lett. 55 (1985) 2883. [2] J.W. Cahn, D. Gratias and B. Mozer, J. Phys. (Paris) 49 (1988) 1225. [3] M. Duneau and A. Katz, Phys. Rev. Lett. 54 (1985) 2688. [4] P. Bak, Phys. Rev. Lett. 56 (1986) 861. [5] M. Audier and P. Guyot, Philos. Mag. 58 (1988) L17. [6] C.L. Henley and V. Elser, Philos. Mag. 53 (1986) L59. [7] A. Yamamoto, in: Quasicrystals, ed. T. Fujiwara and T. Ogawa (Springer, Berlin, 1990) p. 57. [8] Shi-Ye Qiu and M.V. Jaric, in: Quasicrystals, ed. M.V. Jaric and S. Lundqvist (World Scientific, Singapore, 1989) p. 19. [9] M. Mihalkovic and P. Mrafko, submitted to Europhys. Lett. [10] H.A. Fowler, B. Mozer and J. Sims, Phys. Rev. B37 (1988) 3906. [11] M. de Boissieu, C. Janot and J.M. Dubois, J. Phys. Cond. Matt. 2 (1990) 2499. [12] M. Duneau and C. Oguey, J. Phys. (Paris) 50 (1989) 135. [13] F. Lancon and L. Billard, J. Non-Cryst. Solids 117&118 (1990) 836. [14] C.L. Henley, Phys. Rev. B34 (1986) 797. [15] Y. Calvayrac, A. Quivy, M. Bessiere, S. Lefebvre, M. Cornier-Quiquandon and D. Gratias, J. Phys. (Paris) 51 (1990) 417. [16] M. Cornier-Quiquandon, A. Quivy, S. Lefebvre, E. Elkaim, G. Heger, A. Katz and D. Gratias, Phys. Rev. B44 (1991) 2071. [17] C.L. Henley, in: Quasicrystals, ed. T. Fujiwara and T. Ogawa (Springer, Berlin, 1990) p. 38. [18] N.K. Mukhopadhyay, S. Ranganathan and K. Chattopadhyay, Philos. Mag. Lett. 60 (1989) 207. [19] C.L. Henley, Phys. Rev. B43 (1991) 993. [20] J.M. Dubois, C. Janot and M. de Boissieu, in: Quasierystalline Materials, ed. C. Janot and J.M. Dubois (World Scientific, Singapore, 1988). p. 97. [21] J. Hafner and M. Krajci, Europhys. Lett. 13 (1990) 335. [22] C. Janot, M. De Boissieu, J.M. Dubois and J. Pannetier, J. Phys. Cond. Mater. 1 (1988) 1029. [23] J.M. Dubois and Ch. Janot, J. Phys. (Paris) 48 (1987) 1981.