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ICOSAHEDRAL PHASE
J. Cahn, D. Gratias
To cite this version:
J. Cahn, D. Gratias. A STRUCTURAL DETERMINATION OF THE Al-Mn ICOSA- HEDRAL PHASE. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-415-C3-424.
�10.1051/jphyscol:1986342�. �jpa-00225754�
JOURNAL DE PHYSIQUE
Colloque C3, supplbment au n o 7, Tome 47, juillet 1986
A STRUCTURAL DETERMINATION OF THE A1-Mn ICOSAHEDRAL PHASE J. W. CAHN' and D. GRATIAS"
' ~ n s t i t u t e for Materials Science and Engineering,
N B S ,Gaithersburg, MD-20899, U.S.A. . . C . E . C.M.
/ C . N . R. S. , 15, Rue G. Urbain, F-94400 Vitry, France
Introduction
The icosahedral phases C1,23 have generated extensive theoretical and experimental studies. A most puzzling problem is the determination of the actual atomic positions of these phases. Tiling theories may be used to obtain a set of cells, or a set of quasilattice nodes, but the actual location of the atoms need not be confined to the nodes or corners of the tiles. Like most crystal structures which do not reduce to a single atom per unit cell, the icosahedral structures require a description by a nontrivial motif.
We propose herein to discuss the model of structure for the (A1-Si)-Mn alloy which is a packing of large icosahedral motifs. This model has been first suggested independently by P. Guyot and M.
Audier C33 and C. Henley and V. Elser C4,53. The model is most simply described by ignoring the aluminum atoms. Parallel icosahedra of manganese are linked along their three-fold axes by octahedra flattened along this axis. Aluminium atoms. are in two concentric shells, one is a smaller icosahedron within the basic one defined by the manganese atoms, the second one is a icosidodecahedron with almost the same circumscribed radius as the manganese icosahedron. The common center of these three orbits is vacant F i g 1 This 54-atoms cluster is called the Mackay icosahedron (MI) and was proposed by A.L.
Mackay C63 in the early 60's as a possible icosahedral unit for small icoaahedral aggregates.
This model is suggested by crystallographic data of the alpha (A1,Si)-Mn and (Al-Si)-Fe crystalline phases C7,83 which in many respects, approximates, with rational indexes, the icosahedral diffraction patterns. In fact, Henley and Elser C53 have shown how remarkably well these crystalline structures fit with the scheme of the rational approximants of the quasi-crystalline phase. Instead of obtaining the icosahedral quasilattice by projecting a slice of 26 unto a 3-d plane with irrational indexes involving the golden mean, the 3-dim plane is rotated into a orientation with rational indexes, in which the golden mean is replace by an approximant p/q. For the second rational approximant (p/q=l/l) of the cutting slice, a body centered cubic Im3 structure is expected with a lattice parameter a=1.266nm from a 26 hypercubic lattice with parameter A=.65nm. The actual structure of (Al-Si)-Mn is simple cubic almost body-centered with a lattice parameter a=1.268 nm, whereas that of (Al-Si)-Fe is body-centered cubic with a lattice parameter az1.255 nm. The structure of these crystalline phases is essentially a bcc arrangement
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986342
of MI with few additional Aluminium atoms filling the spaces between MI'S. It must be clearly emphasized that manganese (or Fe) atoms are in special positions which differ significantly from the corners of the underlying fundamental tiling of rhombohedra generated by the rational approximant of the quasi-lattice. Actually, manganese atoms are located at .485nm from the central vacancy whereas the elementary quasi-lattice constant in R3 is a=.46nm. As a result, manganese atoms do not sit on the quasi-lattice nodes in either the 3- or 6-dim spaces: the structure is unlikely to be a decoration of the initial 26-lattice, although the systematic extinction rules observed in either diffraction techniques correspond to the primitive 26-lattice.
n
Figure 1: The basic unit of the alpha (Al-Si)-Fe structure is a Mackay icosahedron (MI). It has 12 A1 atoms (small circles) on an icosahedron surrounding a central vacancy, 12 Fe atoms (large circles) on an icosahedron approximately twice larger and 30 A1 atoas on the radial bisector of the edges of the Fe icosahedron. The actual alpha phase contains a few more A1 atoms which are not included in the MI.
Our principal concern in this paper will be a critical discussion of the simplest model that can be inferred from the crystalline phase, i-e., a quasi-periodic stacking of MI'S which seems a priori to be an appealing possible solution for the icosahedral (Al-Si)-Mn/Fe structures.
In the first part of the paper, we will discuss which kind of convex acceptance volume may be used for generating the quasi-lattice nodes corresponding to the central vacancies of the basic MI'S. In the second part, we will construct the MI'S in the 6-dimensional space and compare the computed diffraction intensities of the model with the experimental results.
Quasi-periodic framework of MI'S.
The shortest allowed distance between two adjacent MI'S is the long diagonal of the prolate rhombohedron; it corresponds to the centering of the bcc crystalline phase. Shorter distances, the edge length, the two face diagonals, the other body diagonals of the rhombhedra are all not allowed. This shortest allowed length occurs as ten vectors along 3-fold axes, and the simplest description would be using Z10 with the ten vectors forming the basis. In this paper we will continue to use 26. Z10 projects with rational indexes unto 26.
Each of the basic vectors becomes a basic vector of type (111000) in 26. Not all of the (111000) type vectors in 26 are allowed, e.g.
(110100), and (110001) project to give shorter diagonals of the
rhombohedra. The acceptance volume in the orthogonal 3-dim space must
therefore be small enough to exclude the (1,0,0,0,0,0) and
(1,1,0,0,0,0) nodes as well as all the unallowed other nodes.
Assuming unit vectors of 26 to project in R3 and R'3 according to the following matrix (where t=1.618034 is the golden mean):
we obtain a simple classification of the projected lengths based on two integers t91: let X 1 =(nl,n2,n3,n4,n5,n6) be a 6-dim lattice vector, X and Xperp its projections onto respectively R3 and R'3, and
(u/u',v/v',w/w'~ the integer coordinates of X defined by:
we obtain the following lenghts (in 26-lattice parameter unit):
a
2X 6 =N/2; X =(Nt+M) /2(2+t); ~:erp =t(Nt-M) /2(2+t) /3/
where
2 2 2 . 2 2
N= 2Ln;= u +v +w +ul +vta+wL and M= u" +v? +wl +2(uu'+w'+w' 141 From these relations it is easily shown that the vectors
<1,0,0,0,0,0> project on the vertices of a icosahedron onto R3 and R'3 with a radius of 1/ 2 and the vectors <1,1,0,0,0,0> on the vertices
of two different icosidodecahedra in R3 and R'3, one with radius r= J(4+4t)/ J2(2+t) in R3 and r 1 = 2 / J2(2+t) in R'3, the other conversely with radius r'in R3 and r in R'3. The simplest convex acceptance volume which will avoid the simultaneous presence of the forbidden distances is therefore the intersection of the duals of an icosahedron of radius 1/ 2 $2 (a dodecahedron) and an icosidodecahedron of radius r/2 (a triacontahedron). This acceptance volume is a truncated dodecahedron as shown on figure 2 . It has an inscribed sphere of radius r=.354 and a circumscribed sphere of radius r=.391.
The sphere of same volume V has a radius of r=.375. As pointed out by A. Katz C133 and C. L. Henley C141, this volume is not exactly the optimum acceptance volume; a nonconvex volume can be obtained which captures a few
%more nodes than the one presented here. Many properties are not sensitive to the shape of the acceptance volume.
For these we approximate it with a sphere.
The density d of surviving quasi-lattice nodes (number of central vacancies/nm3) is (A is the 26 lattice parameter in nm):
which is 82% that of the crystalline alpha phase (d=.981 MI/nm3). The expected mass density of the material would be approximately 2.97gIcm3 which is rather far below the experimental one. Hence, the model which can be built with a quasi-periodic stacking of MI'S would have to be augmented by additional atoms to fit with the observed density.
Of great interest is the comparison of the different distances between MI'S in the quasi-periodic stacking and those in the crystalline phase. The calculation is performed in the following way:
let X; =(XO , X : e r p be a 26-lattice node which projects in R'3 inside
the acceptance volume. We first construct the complete star of the NO
Figure
2 :Acceptance polyhedron (truncated pentagonal dodecahedron).
(The optimum acceptance polyhedron is slighhy larger and shows more facets C13,143)
X c equivalent vectors in R6 which project in R3 at the same distance (XI froom
X O. These vectors project in R'3 at the distance (Xpurpl from Xperp. The only surviving ones in R3 will be those which project inside the acceptance volume (see Figure 3). Hence the average number N of MI'S projecting at a distance (XI from a given central MI is obtained by integrating all possible projections of the central node.
Within the sphere approximation (acceptance sphere of radius R) this integration can be eaaiiy performed analytically:
n
a
N= ~ N O / ~ * R ~ J
2TC( ( 1 - c o s 0 ) d ~ /6/
0
where the variables are explicitely defined in figure 3.
Figure 3: Geometric variables used in equation /6/ for the calculation of the average number of MI'S surrounding
a central MI at distance (XI.
The resulting integral is:
where ') = IXperpl/R. This function is shown on Figure 4. The
discrete radial distribution of HI'S is seen on figure 5 together with
the one corresponding to the crystalline phase. Each MI of the
quasi-crystalline phase is surrounded by 5.58 MI instead of 8 in the
crystalline phase; also the number of "second" neighboors is slightly
lower (5.88 instead of 6). On the other hand, there are many new
distances which do not appear in the crystalline phase that show up in the quasi-periodic stacking. An interesting feature of this stacking is that there are never three colinear MI'S equally spaced as it is the case in the crystalline structure; colinear MI'S are always at distances which correspond to some powers of the golden mean. In particular the well-known t scaling of the intrinsic icosahedral stacking is fully observed in this model.
m 0 -
O m
2 0 -
'.
Zr
0 -
7"
0 -
0 0
I I I , , , , , , , I
0 0 0 2 0 4 0 6 0 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6
X p e r p / R O
Figure 4: Occupancy average N/NO as a function of IXperpi/R in the sphere approximation
X
( i n nm)Figure
5 :Discrete radial distribution function of the MI'S in the sphere approximation Rs.375.
(a) quasi-periodic structure;(b) crystalline structure.
In fact, as shown by V. Elser El03 there is a 6x6 matrix with
integer coefficients (which therefore transforms a Z6-lattice node
into nother 26-lattice node) which has a 3-dim degenerate eigenvalue of tg in R3 and a 3-dim degenerate eigenvalue of -t-' in R13: any vector projecting inside the acceptance volume generates -by successive application of this scaling matrix- infinitely many vectors which all project inside the acceptance volume.
The Parallel Motif Algorithm (PMA)
Once the framework of MI'S has been achieved, there are two parts to the calculation of the Fourier spectrum of the model. We start with the hypercubic lattice in R6, and place a 3-dim hyperplane parallel to R3 through each lattice point. Each MI is then a "Bravais motif" in R6, lying on an hyperplane parallel to R3. This placement allows for the slice and project method to be an efficient way for recognizing which actual Wyckoff positions of the alpha (A1,Si)-Mn/Fe crystalline phases do correspond to a rational projection of a possible "Bravais motif" of the 6-dim space. This "Bravais motif" has to be located in such a
waythat it projefts completely in R3 if the corresponding 26-lattice node does. Let e be the 6-dim Wyckoff position of an atom attached to the X; 26-lattice node; the
I,
i
projection onto R'3 of X 6 + e must be such that:
/ b
i
n (x6 + e
) =R' tx 6 1 i. e. IT.'( ei
) =o / a /
The resulting structure consists of a quasi-periodic stacking of a single kind of motif. Let
x,y,zbe the coordinates of a given atom in R3; the coordinates in R6 of the corresponding parallel motif are given by:
Choosing the X* ,Ys ,Za coordinates of the %toms of the alpha phase defined of the basic vectors of type A=(2t /\J2(2+t), 0, 0) we obtain the following relations between the R3 and R6 coordinates:
Xq =k( X a + tY= 1; X 2 =kt tX4 + 2-
);X s =k( Yw + t201
X', =k( tYo, - X ? .
);X t =k( tx, - 2, 1; X g =k( tz* - Ye.
)/lo/
with k= (t+l)/(t+2)
Table 1 shows the R6 coordinates of the alpha phase calculated in the PMA.
We require now the multiplicities of the R6 Wyckoff positions to
fit with the ones in R3. For that purpose we generate the icosahedral
group m35 with the inversion plus the two following operators:
from which the supports of the different strata are easily obtained (Table 2) by identifying which subspaces are invariant by the induced subgroups of m35.
A Comparison of Table 1 and 2 shows that Mn(l) and Al(1) positions, as obtained from the actual alpha phase, fit almost exactly with the 5m orbit in R6 within a distortion less than 5 % Also, within an accuracy of less than 1%, the Al(2) and Al(3) positions in R3 turn out to belong to an single orbit of R6 (2mm). As already discussed by V. Elser C53, these positions define the MIS. The remaining positions do not fit in any corresponding orbits of same multiplicity in R6 and cannot be described in the PMA. They have necessarily to be outside the plane parallel to R3 and will be ignored in the present section.
Table 1: Values of the different Wyckoff positions of the alpha phase in the 6-dim space in the PMA (Input data are from C73 and C83).
-
Atoms I Mult
I
I
Mn(l)l 12 Al(l)( 12 A1(2)1 6 A1(3)1 24 A1(4)1 6 A1(5)( 12 A1(6)( 12
Table 2: The strata of m35 in the 6-dimensional space.
The mass density of the present model (excluding the remaining A1 atoms) ia obtained by multiplying the density of quasi-lattice nodes by the atomic weight of the MI motif leading to a low value of 2.4 g/cm3 (as compared with 2.93 g/cm3 for the corresponding alpha phase built with the sole MI motif). The stoichiometry of the model is that of the basic MI, i.e., A142Mn12.
R3
.201, -327, .OOO .loo, -164, .000 .000, -364, -000 .189, .119, .298 -500, -210, -000 .096, -168, -500 .117, ,120, .500
The calculation of the diffraction powder patterns for X-rays and neutrons are performed in the standard way; let &((Q() be the atomic scattering factor for species oc , the structure factor F ( Q 6 )is obtained by:
i
F ( Q ~ 1- &. 6 fa( I Q I lexp(2inQIPs. e~ )oc 1 ~ p . 4 /11/
R6
.530, -235, -236, .237, .235,-.236
,260, -117, .118, .119, -117,--118
,426, .000, .263, .426, .000,-.263
.275, .437, ,437, -002, -005, -263
.608, .585, .152,--115, .585,-.152
.266, .474, -707, -127,-.249, ,464
.226, .499, .673, -056,-.224, .498
where G(lQpupl)is the Fourier transform of the acceptance function. In the sphere approximation this acceptance function is:
with (6 =2rl:lQpTlR
The structure factor of the motif in /11/ can be obviously calculated directly in R3 as well. The diffuse intensity maps due to a single MI are seen on figure 6 for both X-rays and neutrons. As expected there are strong differences between X-rays and neutrons due to the negative value of the cross section of Mn in neutron experiments. The powder diffraction intensities result in a sampling of these diffuse maps by Dirac peaks located on the reciprocal quasi-lattice multiplied by the cut function /12/. It is therefore clear that intense X-ray peaks will disappear in neutrons; for example, the (18,29) 2 1 1 , l l - 1 reflection which falls on a maximum of the diffuse map in X-rays should be almost absent in neutrons (compare the 2-fold maps).
Figure 6: Diffuse intensity maps of a single MI motif;
(top) for X-rays; (bottom) for Neutrons. The scale is in l/Ansgtrom.
The resulting intensities are shown on Table 3 and exhibit a
quite good agreement with the actual available data (see, for instance
t11,123) for both X-rays and neutron spectra. The model could
probably be improved by an addition of the atoms in the space between
the MIS, and a partial chemical substitutional disorder betweem
manganese and aluminum orbits. However, these refinements would not
change the intensities drastically. Although the model is in
reasonable agreement with all the scattering data, it predicts too low
a mass density. Since no better stacking may be obtained within the
cut and project method as demonstrated recently by C. L. Henley, it
i s possible that the actual structure contains partial MI'S and a certain degree of random packing which could improve the density as proposed by C.L. Henley.
N H C o o r d I r ~ n L c l i n Z G nult. Q p c r p . Q paral. X - r a y Neutron
