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Rhombohedral Phases in AlCuFe Alloys
A. Le Lann, J. Devaud
To cite this version:
A. Le Lann, J. Devaud. Structure and Decoration of the Icosahedral and Rhombohedral Phases in AlCuFe Alloys. Journal de Physique I, EDP Sciences, 1995, 5 (1), pp.129-151. �10.1051/jp1:1995110�.
�jpa-00247043�
Classification
Physics Abstracts
61.16D 61.55H
Structure and Decoration of trie Icosahedral and Rhombohedral Phases in Alcufe Alloys
À. Le Lann and J. Devaud
Centre d'Etudes de Chimie Métallurgique C.N.R.S., là rue'Georges Urbain, 94400 Vitry
sur
Seine, France
(Received 29 March 1994, revised 8 September 1994, accepted 19 September 1994)
Abstract. Structural models for Alcufe icosahedral and rhombohedral phases are obtained
by decomposition, in perpendicular space, of the atomic surfaces into cells of identical local environment. The models are descnbed in 3D real space and compared to the corresponding
H-R-E-M- images obtained on structurally perfect icosahedral and rhombohedral phases. The
specificity of their apenodic structure and chemical decoration is outlined.
l. Introduction
Unexpected electronic properties have been observed in stable F ordered icosahedral Alcufe
iii, AlcuRu [2] and AlPdmn [3] phases, whose aperiodic structure and chemical decoration are still unknown. Fortunately, some of the crystalline approximants (which are periodic arrange-
ments of large volumes of quasicrystalhne structure) also present these particular electronic
properties [4, Si. The determination of the structure of a large approximant cell is an easier
objective, which enables a study of the specificity of a quasicrystalline structure. Presently, the Alcufe alloys, in which both the icosahedral phase [6] and the rhombohedral approximant [7] have been obtained as structurally perfect sarnples (Fig. l), are the best candidates for
this structure determination. A 6D neutron diffraction study of a A162Cu25.5Fe12.5 icosahedral phase prepared with isotopic elements [8] has led to a complex model of decoration of the 6D hypercubic lattice. A simplified model, proposed by diiferent authors [9, loi, is used here
together with a method of "cell decomposition" of the atomic surfaces [9], to obtain a 3D model of the icosahedral structure. Assuming that the A162.8Cu26Feii.2 rhombohedral phase,
whose chemical composition is very close, corresponds to the same decoration of the 6D lattice, Gratias has also obtained a 3D structural model of the rhombohedral approximant structure.
Hereafter:
1) These 3D structural models of the icosahedral and rhombohedral phases obtained by
6D - 3D projections, are descnbed by 2D projections along three mutually perpendicular
twofold axes (a twofold orientation allows to identify the modes belonging to the four F
sublattices). The 2D projections of the rhombohedral model are compared to H-R-E-M- images of the rhombohedral phase observed along trie corresponding directions.
© Les Editions de Physique 1995
a) b) Fig. 1. la) Perfect rhombohedral structure observed in A162.sCu26Feii
2 alloy annealed for 11 days
at 700 °C. b) Corresponding diffraction pattem.
2) The determination of the chemical decoration of the icosahedral lattice is based on the
hypothesis that ail sites of equivalent local environments are occupied, as in crystalline
structures, by the same kind of atoms. It consists of a complex "cell decomposition"
of the atomic surfaces in perpendicular space [9], according to the neutron diffraction experimental results.
The correspondence between the 3D icosahedral and rhombohedral models allows to propose
an internai structure and decoration for the rhombohedral cell.
2. Icosahedral and Rhombohedral Structural Models Obtained by 6D - 3D
Projections
The Alcufe icosahedral structure (F ~l m35) is defined in 6D space (Fig. 2a) by three atomic surfaces located at the N (000000) and N'(100000) nodes of the underlying primitive lattice and at one body-centre BC 1/2(llllll); the other body-centre BC' 1/2(llllll) is vacant [11].
The three atomic surfaces are respectively (Fig. 2b):
on N modes, a triacontahedron
T larger (in linear scale) than the triacontahedron which generates the canonical tiling,
on N' nodes, a triacontahedron of same size but truncated along the fivefold directions,
on BC nodes, a triacontahedron T~ smaller than the others.
They are connected, in perpendicular space, along the fivefold directions.
,
~
~
BC
N N'
a) ~~
Fig. 2. a) Localization of the atomic surfaces on nodes N [000000], N' [100000] and BC 1/2[llllll]
in the 6D rational fivefold plane passing through the body-centres. b) The three atomic surfaces located on N, N' and BC nodes are connected along fivefold axes in perpendicular space.
The size and shape of the atomic surfaces determine the occupation of the nodes of the four F sublattices in parallel space (3D real space):
I = Pi e P~
= (N e N') e (Bc e Bc')
The neighbounng nodes belong to diiferent F sublattices (the shortest interatomic distances
are N-N'
= 0.910 a/T = 0.251 nm and BC-N
= a/T = 0.275 nm). Therefore, the selection of the nodes corresponding to the outermost part of the atomic surfaces depends on the dillicult
determination ofthe exact size and shape ofthe atomic surface corresponding to each sublattice.
Adjacent atomic surfaces must be connected in perpendicular space to avoid inhomogeneities
in the density of the selected nodes, but the boundaries between them may not be exactly
defined. This may correspond to an intrinsic indetermination of the icosahedral structure:
whichever F sublattice they belong to, available nodes, doser than an interatomic distance,
have nearly the same environment and the energetic cost of an atomic jump from one to the other is very low.
Therefore, the nodes of the model located on the boundaries between the atomic surfaces in
perpendicular space, are not taken into account in the projection on parallel space; the nodes
in 3D space are thus selected by default. The zones of the icosahedral structure corresponding
to the boundaries appear m the sections of Figures 3 and 4 as "white" regions where nodes are
Iacking. They could be completed by diiferent sets of modes obtained by diiferent fluctuations of the boundaries of the atomic surfaces. The nodes represented on the sections correspond to the internai volume of the atomic surfaces and are determined unambiguously.
A model of chemical decoration of an icosahedral structure results from bath a selection of the nodes of the 3D icosahedral quasilattice which are occupied by atoms, and the distribution of the chemical atomic species on these nodes. As the environment of an atom depends on its
chemical nature, these operations are net independent. The local chemical order carat be determined by experimental methods; it is assumed presently in ail 3D models proposed for icosahedral structures [12-14, 10j, that there exist clusters of icosahedral symmetry resulting
from interactions of the diiferent atomic species, as in crystalline structures; they are analogous
to either the Mackay icosahedron [15, 16] or the Bergman cluster [17]. The chemical decoration proposed here for Alcufe structures is based on analogies of the interatomic distances mea-
sured by EXAFS in the Alcufe quasicrystalline alloys [18] with those observed in crystalline
structures [19]. The method of decomposition of the atomic surfaces into cells corresponding
to the same local environment [9], enables to identify the diiferent clusters existing in the 3D icosahedral network and provides a description of their spatial distribution on large icosahedral orbits.
Neutron diffraction data on Alcufe perfect icosahedral samples prepared with isotopic ele- ments [8] have provided some information on the occupied modes of the four F sublattices:
The small BC volume is occupied by copper: this corresponds in real space, to Cu atoms
separated by large distances (the T" inflation rule of distances in real space is correlated to a T~" deflation rule in perpendicular space (20)). It appears on the section represented
in Figure 3 that they are centres of clusters.
In the same way, the innermost part of the N and N' volumes, almost equally occupied
by iron, corresponds to Fe atoms distributed on large orbits.
A small volume of aluminum is intermingled with the iron near the centre of the N' volume.
The copper is aise found in intermediate shells of bath N and N' volumes (mostly on N') qhile the aluminum is distributed mostly at the periphery.
The BC' volume is empty.
The rhombohedral structure is denved from the icosahedral structure by a shear of the 6D lattice. It is assumed that this shear has no component in parallel space. A deformation of the atomic surfaces in perpendicular space is associated with this shear. The selection of nodes in parallel space is thus diiferent from the icosahedral selection but the "cell decomposition" in
perpendicular space is the same and leads to the same clusters.
The rhombohedral cell constructed on the BC nodes is assumed to be centrosymmetric (this
has net been refuted by HREM images) the centre of the cell is a BC' mode (unoccupied).
3. Icosahedral Structure
3.1. DESCRIPTION OF THE CLUSTERS. Figure 3a is a section of the AÎCUFe icosahedral structure perpendicular to a twofold axis and passing through the origin of coordinates placed
here on a BC node. The nodes of the N, N' and BC sublattices are represented by white, shaded, and black circles respectively. The BC' nodes are absent.
The clusters, constituted of concentric orbits, are represented as volumes in [19] and by their section in Figure 3. They can be classified into two types: those, hereafter labelled A clusters,
in which the centre and the orbits belong to the same P lattice and those, called B clusters, in which the centre belongs to a third F sublattice.
The A cluster (Orly N and N' modes, Fig. 3b) is formed of a dodecahedron of radius
r = 0.910 a/T
= 0.251 nm (Orly seven of the twenty modes con be simultaneously occu- pied by atoms), an icosahedron of radius a
= 0.446 nm, an icosidodecahedron of radius
r = I.oàa
= 0.469 nm, and a rhombicosidodecahedron of radius r = 0.648 nm. This type of cluster is the most dense that cari be constructed in Alcufe alloys if the centre of the cluster is occupied by an atom [19] it is labelled pseudo Mackay icosahedron in [14]. The A
A1
Az
A1
' NM
" a)
O
f~~
~
A b) B c)
o
o
O°
Fig. 3.
model. The ircles are odes elonging to the ection;
the smaller ores are near the section.
The "white" zones of
the section where
modes are lacking orrespond to zones of
"undetermined" local
structure: BC (.) odes, N' (O) modes, N (o) modes, unoccupied BC' (.) mode. b) Concentric
in a A cluster centred by N or N' mode): dodecahedron of
radius r = 0.910a/T =
of the
odes cari be r = a = 0.446
nm;
r = I.oàa = 0.469 nm;
rhombicosidodecahedron r =
0.648 nm. c) Concentric orbits in
B cluster (centred by a BC or
BC' mode): icosahedron r = a/T =
0.275 nm;
r = 0.910a = .406 nm. d) istances between neighbou~ing
along
ivefold
and threefold
A1
' NM
A6 ~)
o
NM b)
Fig. 4. Orbits formed by the centres of clusters: they are represented by their section by the twofold plane passing through the engin: a) Concentric orbits around a BC (.) centre of cluster:
on
N' modes (O), icosahedron of radius r
= aT; on N modes (o), icosahedron r
= aT~;
on BC (.) modes,
icosidodecahedron r
= 1.05 aT~; on N' (o) modes, dodecahedron r
= 0.910aT~; on BC' (.) modes,
icosahedron r = aT~ (trot drawn); on BC (.) modes, icosidodecahedron r
=
1.o5aT~. Around
a N'
centre of cluster distant by aT~ along a fivefold axis, the same orbits belong to different sublattices
(reciprocal description around the BC and N' origins). b) Orbits centred around BC' (.) centres of cluster (scheme of icosahedra corresponding to a section of Fig. 7): on N modes (o), icosahedron
r = aT; on BC (.) modes, dodecahedron r = 0.910 aT~; on N' modes (O)> lcosahedron r
=
aT~; on N
(O) modes, dodecahedron r = 0.910aT~; on BC (.) modes, icosahedron r = aT~
clusters can be centred by either a N or a N' node; in the A clusters centred by a N node the rhombicosidodecahedron is not occupied.
The B duster (Fig. 3c) is formed of an icosahedron (r
= a/T
= 0.275 nm) and a dodecahe- dron (r
= 0.910a
= 0.406 nm) as in the Bergman duster il?]. These orbits are the internai orbits of a triacontahedron corresponding to the 64 nodes of the 6D cubic unit cell. Here, the
centre of the cluster is a BC node; the distance from the centre to the first orbit is langer than
in the A cluster. In bath types of clusters A and B, the successive orbits belong alternatively
to the N and N' sublattices and the distance between neighbouring nodes on successive orbits
is the shortest one compatible with interatomic distances in Alcufe alloys [19].
This selection of clusters is in agreement with the results obtained by neutron diffraction [8]
(the N and N' nodes are the most numerous, there
are few BC occupied nodes and no BC'
nodes). The distance between BC nodes is at least 0.759 nm. Although there are no occupied BC' modes in the model, there exist on fivefold axes, at the distance aT~ from the origin, N and N' orbits of a B cluster, surrounding an unoccupied BC' node (Fig. 3a).
The distance from trie BC origm of Figure 3d to a centre of a cluster belonging to a diiferent sublattice is, along a fivefold axis: aT to the N' centre, aT~ to the N centre, aT3 to the BC' centre, aT~ to the N' centre; there aise exist intersecting clusters separated by a (N' - N and BC' -+ BC); on a threefold axis, the distances are 0.910aT and 0.910aT~ from the BC to the N centres, 0.910ar3 from the BC to the N' centre, and 0.910ar~ from the N to the N' centre.
3.2. DESCRIPTION OF THE ORBITS FORMED BY THE CLUSTERS. The centres of the clusters form large orbits that reproduce at an inflated scale the A and B families descnbed above.
However, at those inflated scales, bath A and B families coexist around the centre of the cluster (Fig. 4a); the separation of the small orbits into A or B clusters is a consequence of the finite size of the atoms which imposes the choice of convenient spacings between the orbits; in
another icosahedral alloy the structure of the clusters can be diiferent.
The description of the orbits formed by the centres of clusters belonging to each sublattice
depends on the node (N, N', BC or BC') chosen as the origin. This can be seen in Figure
4a where orbits are drawn around tue BC engin and around a N' centre of cluster distant by ar~ along a fivefold axis; the description of the orbits is reciprocal: icosahedra of N' centred
clusters around a BC origin and icosahedra of BC centred clusters around a N' engin, etc...
Figure 4b corresponds to another description of the same structure: icosahedra (radius
r = aT~) and dodecahedra (r
= 0.910aT~) drawn on BC nodes are centred by an unoccupied BC' node. These orbits are used hereafter to establish the 3D correspondence between the
icosahedral and rhombohedral structures. The problem to be studied determines the convement
description.
4. Correspondence between Alcufe Icosahedral ar~d Rhombohedral Structures
4.1. RHOMBOHEDRAL CELL. The Alcufe rhombohedral phase (a = 3.216 nm; a = 36°),
first identified in 1989 [21]
,
has been recognized as the so-called 3/2 rhombohedral approxi- mant: three ei, e2, e3, vectors of the 6D lattice
: [21111(, [121111], and il l2lÎl], project with the same size and orientation m the icosahedral and rhombohedral structures; they correspond
to the edges, (of length aT3 along AS axes), of an inflated prolate rhombohedron.
Figure 5 represents two rhombohedra which are preserved in the rhombohedral structure, eut of the twenty that form a stellated polyhedron in the icosahedral structure. The rhombohedral cell is constructed on nodes of only one F sublattice. The radius of concentric orbits formed by the nodes of a P lattice is expressed by r
= a[(N + MT)/(2 + T)]1/~, where N and M are two integers; they are bath even in one F sublattice of a P lattice, and even and odd in the
A5°XIS ~
/~5°~~~
~ ~'
~3°~'~ ~+ê2+~3
/
@lflrhombohedrcl
Il d'nec"°n
52.84 /
- A2axis
/ [00j rhombahedrcl
'Al oxis ~~~~~~~~~
[ll# rhombohedrcl dwection
~' 102-165
/ ~
Fig. 5. Correspondence between the rhombohedral cell (dashed fines) and two prolate rhombohedra opposite by a vertex belonging to a T~ inflated stellated polyhedron (twenty rhombohedra). The cell
is drawn on BC (. o) modes and centred by a BC' (.) mode. The solid (. .) symbols correspond
to modes located in the bisector plane of the cell perpendicular to [f10]; the open (o D) symbols correspond to modes in bisector planes translated by +1/2 [f10].
other one [22]. The triangular sections of the cell correspond to faces of the 18-29 icosahedron
(of radius aT3) and the tips on the A3 axis to nodes of the 102-165 dodecahedron [23]. The edges of the cell and the sides of the triangular sections are parallel to A2 axes; the faces of
trie cell correspond to fivefold planes perpendicular to the e4, e5, e6 vectors of the icosahedral structure.
Une can derme in the rhombohedral cell a set of three mutually perpendicular pseudo- twofold axes, one of which ([Î10] in Fig. 5) is perpendicular to a bisector plane of the cell.
This axis provides the simplest projection of the structure: only two diiferent sections of the cell are superposed, one passing through the ongin, the other through nodes of coordinates
+1/2 [110]. Moreover, as in the icosahedral case, the projection along a twofold axis allows to
distinguish between the nodes of the four F sublattices of the I lattice [19].
The 3D description that follows has been determined from the synthesis of various projec-
tions and sections of the model, perpendicular to this set of three pseudo-twofold axes. We verified on the convenient sections the existence of ail the nodes of the orbits, even outside the
rhombohedral cell considered.
Figures 6a, 6b and 6c are projections along the axis defined by (e2 e3) (the )10] rhom- bohedral direction). Ail the nodes represented here are occupied by atoms; the BC nodes are the largest cirdesj the rhombohedral cell is drawn on BC nodes. The two diiferent sections of the rhombohedral structure are drawn respectively in Figures 6a and 6b. It appears that the volume of structure which is common to both the icosahedral and rhombohedral models
is not limited to the rhombohedral cell. The central part of each cell (antiprism between the
triangular sections) con be completed with BC nodes belonging to the neighbouring cells, in order to form an icosahedron ofradius aT3 which corresponds exactly to the icosahedron drawn in Figure 5; it is centred by a dodecahedron of BC nodes surrounding a BC' vacant node.
The superposition of Figures 6a and 6b provides, in Figure 6c, the spatial configuration
of these icosahedra: the centres of six icosahedra form the two triangular sections of a cell constructed on BC' modes.
Figure 6d corresponds to a projection of the model along the axis defined by (e2 + e3) (the [001] rhombohedral direction in Fig. 5).
o o o
o ce°'
o
~ ~ ~ PC
o o
o o
o o
~~ a)
o
o o
o
~ o°°
o o
o o
m~©O
o o o
°o
o o
o o
oo~ o
NM b)
Fig. 6. a,b,c) Projection along the [i10] rhombohedral direction, of ail the nodes occupied by atoms in the rhombohedral approximant model. The BC (, o) modes are the largest circles. Figures fia and 6b represent respectively the two different sections of the rhombohedral structure drawn on BC modes. The central part of the cell is inscnbed in
an icosahedron of radius aT~ and centred by a dodecahedron of radius 0.910aT~. Figure 6c, obtained by superposition of 6a and 6b, shows the spatial configuration of six icosahedra drawn on BC modes; their centres belong to a rhombohedral cell drawn
on BC' (. D) modes and centred by a BC(.) mode. The
same set of icosahedra cari be drawn around
each BC vertex of the cell. d) Projection of the same six icosahedra along the perpendicular [001]
rhombohedral direction.
