Approximate icosahedral periodic tilings with pseudo-icosahedral symmetry in reciprocal space

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Approximate icosahedral periodic tilings with pseudo-icosahedral symmetry in reciprocal space

J.-L. Verger-Gaugry

To cite this version:

J.-L. Verger-Gaugry. Approximate icosahedral periodic tilings with pseudo-icosahedral symmetry in reciprocal space. Journal de Physique, 1988, 49 (11), pp.1867-1874.

�10.1051/jphys:0198800490110186700�. �jpa-00210868�

Approximate icosahedral periodic tilings ^with pseudo-icosahedral symmetry ⁱⁿ reciprocal space

J.-L. Verger-Gaugry

Institut National Polytechnique ^de Grenoble, LTPCMIENSEEG (CNRS ^UA N° 29) BP 75, ^Domaine Universitaire, 38402 St Martin d’Hères, ^France

(Requ ^{le 3 mai} 1988, révisé le 6 juillet 1988, accepté ^{le 27} juillet 1988)

Résumé.

Nous caractérisons, par des résultats classiques ^de géométrie ^des nombres, les pavages périodiques

3D de réseau de Bravais quelconque, construits

les gros ^et petits rhomboèdres d’Ammann (les prototuiles

des pavages de Penrose 3D) tels que la transformée de Fourier de ^tous les sommets du pavage présente

presque la symmétrie icosaédrique. ^Une expression analytique ^des déplacements ^des pics intenses par rapport à la symétrie icosaédrique parfaite ^est donnée. On prouve que les positions icosaédriques exactes sont situées

sur

la projection d’un réseau de R6, ^stable

l’action du groupe de l’icosaèdre.

Abstract.

We characterize, by classical results of geometry ^of numbers, 3D periodic tilings of any Bravais lattice built with the prolate and oblate Ammann rhombohedra (the prototiles of 3D Penrose tilings) ^{such that}

the Fourier transform of all the vertices of the tiling ^almost presents the icosahedral symmetry. ^An analytical expression ^{of the} displacements ^of the intense peaks ^{from the exact} icosahedral symmetry ^is given. ^We prove that exact icosahedral positions

^located

^the projection ^of

lattice of R6, stable under the action of the icosahedral group.

Classification

Physics ^Abstracts

61.50E

61.55H - 02.40

1. Introduction.

Since the discovery by ^{Shechtman et al.} [1] ^{of the}

icosahedral phase ^{in a} rapidly solidified AIMn alloy,

it was shown that this phase exists in many other systems, mostly ^{in Al based} alloys. Mathematical models were developed ^{in order to} explain ^the

diffraction patterns ^{and the} high resolution electron images ^{of this} icosahedral quasicrystalline phase [2, 3] : dualization of multigrids [4], projection

methods from a higher dimensional _space [5-8].

They provide distributions of Bragg peaks ^{in the} reciprocal space which possess the perfect ^icosahed-

ral symmetry. However, there exist now several

experimental ^results showing that this perfect

icosahedral symmetry ^{is not} always respected, ^for

instance in the AlLiCu system [9-11]. ^A possible explanation ^of such deviations can be made by considering ^the phason ^strain approach [12-14], ^i.e.

disorder can be introduced in the perfect ^{3D Penrose} tiling. ^Another possibility consists in saying ^{that the}

icosahedral symmetry ^is apparent and arises from

large ^lattice parameters structures [15], ^{for which}

unit cells possess high symmetry ^elements [16]. ^An

intermediate interpretation, suggested by ^the obser-.

vation of enlargements ^of peak widths, ^{can also be}

made by considering the presence of microaggregates

in twinning position ^{of some} crystal structure, ^at random or disposed ^{in a} deterministic _way [11, 17].

The present ^{work is devoted to the} study ^{of 3D} periodic tilings ^{which can} be built with the prolate

and oblate rhombohedra, ^{for which we fix once for}

all the size (edge length equal ^to 1) (the prototiles ^of

3D Penrose tilings), ^such that the Fourier transform of the whole set of the vertices of the tiling ^almost

exhibits the perfect icosahedral symmetry. They

come from particular rational 3D cuts in the cut and

projection technique applied ^to R6, ^as it will be demonstrated by proposition ^{2. This} approach ^is purely geometrical ^{and consists in} analyzing ^which

deviations of the intense peaks ^{can be} expected ^{if the}

icosahedral quasicrystal ^is interpreted by ^{such a 3D} periodic tiling, ^{whatever its} Bravais lattice. This leads us to define the notion of approximation ^with

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198800490110186700

1868

respect ^to intensities and positions ^{in the} reciprocal

space, and ^we show that some classical results of

geometry ^{of numbers} (Diophantine approximation)

allow us to treat this double approximation fully analytically. ^The present approach provides ^for-

mulae for the deviations of the intense peaks ^from

the exact icosahedral symmetry (distortions), ^and

does not take into account any type ^{of defects. The} corresponding ^exact icosahedral positions ^{are shown}

to be located on the projection ^{onto our} physical

space of ^a 6D lattice in R6 which can be easily computed.

2. Some definitions.

First we recall the principles ^{of the cut and} projection

method applied ^{to R6 in order to fix} the notations used in the following. ^E designates ^our physical

space embedded in R6 as the image ^{of the}

«

irrational » rank 3 projector PI expressed by ^the

matrix :

in an orthogonal ^basis {ei / i ^{= 1 to} 61 ^of R6, ^where

each ei ^{is of norm} wi. ^Several possibilities ^{can of}

course be considered [6, 8]. ^{We note as usual}

e!l

Pll (ei), and e--’ 1

P_L (ei) ^with P_L = Id 6 - Pi

This projector Pj ^{induces a} Q-isomorphism ^{from the} Q-vector space generated by ^the e,’s ^{onto the} Q-

vector space in E generated by ^the el’s. ^In particular,

the lattice, ^called B/2 Z6, generated by ^the ei’s, ^{is in}

one to one correspondance ^{with its} image by Pj , ^{which is dense and} uniformly ^{dense in E} (i.e. ^it

satisfies the modulo 1 uniform distribution criteria).

This lattice B/2 ^{Z6 comes} naturally ^{from an} integral

linear representation of the icosahedral group G,

I G I ^{= 120, in R6.} The elements of G are written ye in the following, ^with 1 -- t -- 120 and yl

I d.

We call _{p :} G -+ GL (R6 ) ^{the linear} representation ^of

G such ,that the lattice B/2 ^{Z6 be} G-invariant, ^and PO : G -+ GL (E) ^the subrepresentation ^{of p, which} provides ^{a set of} orthogonal transformations of E

compatible ^{with the} permutation operations ^{of the}

lattice h Z6 [6]. ^{For each t} and each element x in

h ^{Z6 we have :}

A generic point ^{x of} J2 Z6 ^is given by ^its coordinates ni (i ^{= 1 to} 6) ^{in the basis of the} ei’s. ^{We call}

and N2 ^{the sum} of the squares nf of the coordinates of _x, which is half the square of the Euclidean ^norm of x in R6. _xj or xl designates ^often PI (x), ^and

xl ^or xl represents P_L (x). ^{We call} El ^{the space of} R6 orthogonal ^to E, image ^{of the} « irrational »

projector P, . We call TR the « irrational » cut

volume, image b ^P_L of the open elementary ^6-cube

of the lattice 2 Z6 associated with the origin. ^{TR is}

therefore a triacontahedron ; it is the open ^convex set generated by ^{the vectors} et . By Pi , the 3-faces of the 6-cubes of .J2 ^{Z6 are} projected ^{onto both} types of rhombohedra, ^the prolate ^one and the oblate one.

A selection of the points ^of .J2 Z6 by ^{TR leads to a}

sequence ^of 3-faces which, ^when projected ^{onto E} by Pj , constitutes a tiling, without hole nor overlap

of the prototiles, ^{of our} physical space E.

Let us now consider rational cuts : {Mi, ^U2, U3) designates ^{a set of} R-linearly independent ^{vectors of}

E such that each of them can be written as a Z-linear combination of e1 ’s. ^{These vectors ui} generate ^a lattice in E which comes from a rational cut, ^called E’, ⁱⁿ R6. _Indeed, these vectors can be lifted _up on

the lattice .J2 Z6, uniluely, ^{since they are Z-linear}

combinations of the ei’s. We call Wl, w2, w3 these elements and we have; ^{for i = 1 to} 3, Pjj (wi) = Ui . E’ is the R-vector space generated by

the w;’s, ^{in R6. We denote} by P I’ ^the orthogonal

«

rational

projector ^{of rank 3} having ^{E’ as} image,

and P’ .L = Id 6 - Pi ^{the « rational}

projector ^whose image is denoted by Ei . the space orthogonal ^to

E’ in R6. Since wl, w2, W3 have integral coordinates in the basis of the ei’s

^{it is} possible ^{to choose 3}

vectors w4, w5, w6 ⁱⁿ E’ L ^such that (wili ^{= 1 to} 6} ^is

a basis of R6 and _{that W4}

w5, w6 ^{have also} integral

coordinates in the basis of the ei’s. ^{This shows that}

the matrix of Pl’ ^{in the basis of the} wi’s ^is composed

of 0 and 1, ^and that, by simple ^basis change formulae, ^{the matrix of} P,í (as ^{well as the matrix of} P’ ) ^{in the basis of the} ei’s ^has systematically ^rational

coefficients. This justifies ^the terminology

«

rational

for the projectors P’ I ^and P’ L

We know how to decorate the unit cell of the

periodic ^{structure in E with both} prolate ^{and oblate}

rhombohedra : by selecting ^the points ^x of the lattice

vf2 Z6 such that P’ (x) belongs ^{to TR’ and} by projecting ^these selected x by the « irrational »

projector PI, ^{we obtain all} the vertices of the

periodic tiling ^{in E. TR’} designates ^{here the convex}

set generated by ^{the vectors} P’ (ei) ⁱⁿ El , slightly

translated in E’ ^if necessary in order to have no

point ^of P N/2 Z6) ^{on the} boundary. ^{TR’ is a} polyhedron, which looks like a slightly ^distorted

triacontahedron when E’ is

close

to E.

Fig. ^1.

^Schematic representation ^{of the} subspaces E, E’, F’, El , El and of the lattices H, P’ (H),

JC in R6 involved in the characterization of pseudo-icosahedral symmetry.

-L -L

We now associate with a rational space E’ a set of 3 rational numbers which tend to the golden ^mean

T

when E’ tends to E. In the next paragraph, ^these

numbers are very useful to characterize the rational spaces E’ which exhibit the pseudo-icosahedral sym- metry ^{in the} reciprocal space. These rational ^num- bers depend only upon the 3 ^x 6

18 coordinates nij (i ^{= 1 to} 6 ; j ^{= 1 to} 3) ^{of the vectors} wl, w2, W3 ^{in the} basis of the ei’s.

These rational numbers associated with E’ arise

naturally ^{from the}

position

^{of E’ with} respect ^to E in R6 (Fig. 1). Since ui and wi correspond solely

one to the other by ^the « irrational » projector Pp ^{and since E and E’ are} both 3D spaces, ^we

«

measure

how far E’ is from E by considering ^the

squares ^{of the} distances 11 wi - ui 112 ^{for i = 1 to 3.}

We have, ^{for i = 1 to 3} [18] :

where N2i, Nji i represent ^the quantities N2 ^and N 1 associated with the vector wi - We then define the three rational numbers pi lqi (i ^{= 1 to} 3) by :

and this definition allows us to write (i ^{= 1 to} 3) :

The relations between N2 i and Pi/qi ^are given ^for

instance in Cassels’s book [19]. ^{However, we} clearly

see that pilqi ^is always greater ^than

^{or smaller} than 1/2, ^{and tends to the} golden ^mean

^when

E’ tends to E, i.e. when 11 wi - ui II ^{tends to zero.}

This expression ^{shows the} contribution of the lattice parameter ^itself by ^{the term} N2 and ^the contribution of the approximation term T - pi lqi which contains the information « how close is E’ to E along ^the

direction ui

It indicates that the lattice parameter does not need to be very large ^to have very small values of 11 wi - ui 11, i.e. to have a good approxi-

mation of E by ^a certain E’. It is known that a

quasilattice ^can be considered as the limit of a

sequence ^{of 3D} periodic tilings characterized by larger ^and larger ^lattice parameters [20-22]. Here,

we insist _upon the fact that it is not necessary ^to have

large ^lattice parameters ^{to obtain a final} good approximation ^{of E} i. e. , anticipating ^on the result of

proposition 2, ^a good approximation ^{of the icosahed-}

ral symmetry ⁱⁿ reciprocal space.

For cubic and rhombohedral tilings, ^it is easy ^to check that these three rational numbers are always equal.

3. Mathematical approach ^{of the} pseudo-icosahedral symmetry.

3.1 FOURIER SPACE AND RECIPROCAL LATTICE AS-

SOCIATED WITH E’.

As usual, ^{we define the}

1870

vectors ut ^{of E} by setting Ui*. Uj

5ij’ ^for

1, i , j , ^3. They generate ^the reciprocal ^{lattice of}

the tiling in E. It is easy ^to observe that the vectors

u*,1, i ^=$ 3, ^are R-linearly independent ^{and can be}

written as Q-linear combinations of the ej ’s, ^what-

ever the Bravais lattice. Since P 11

restricted to the

Q-vector space generated by ^the ei’s, ^is injective ^and surjective ^{onto the} Q-vector space generated by ^the

ei s, ^{we can} lift up the three ^vectors u?l ^{in a} unique

way in R6 and we obtain the vectors Wi*. They ^are

such that Pp (Wi*)

u.*, ^{1, i , 3,} with wi. wJ

2 5 ij, ^{for 1 -- i, j -- 3, and they} have rational coordi- nates in the basis of the ei’s. ^{We define} the Fourier space F’ in R6, associated with the rational cut E’ given by ^a primitive basis {Ul, ^u2, U3), ^{by the 3D} R-subspace ^of R6 generated by {Wl*’ wi, W3*}. ^We

call H the rank 3 Z-module generated by ^these

vectors. H is a lattice in F’ which can be considered

as the reciprocal lattice of the tiling given by E’, ^{but viewed in} R6 (Fig. 1). The space F’ has the

following important properties, ^{which can be} easily

demonstrated in the case of cubic tilings (and, ^{in the}

case of tilings of other Bravais lattice, ^{lead to} heavy algebraic expressions) :

Proposition ¹

Consequence

E[.

2) ^{The lattice} generated by (w/, wi, W3*’

We will see below that this 6-dimensional lattice almost gives ^the key ^to calculate the exact icosahed- ral positions ^{in the} diffraction patterns but it is

generally ^{not stable under G for most} of the rational cuts E’.

Expression of ^{the structure} factor

In order to calculate the intensities, ^{we make the}

Fourier transform of the local potential ^{of the} periodic tiling determined by E’, ^{which is} given by

zero everywhere except ^at the vertices of the tiling

where it is equal ^{to 1. The} intensity ^I (gj ) ^calculated

at a point gll in E is the square ^{of the} module of the

complex ^{number :}

where the sum is taken over all the vertices « ir- radiated by ^the beam ». The rj 11 ’s ^{represent the} projections by Pp ^{of the} vertices rj of the lattice

J2 ^{Z6 which have the} property ^that P’ (rj) ^belongs

to TR’ .

We transform this « infinite

sum into a finite

sum by considering ^as usual the contribution of the lattice and the contribution of the unit cell. We call

B = {j ^E N/riH ^{belongs to the unit cell), and n the}

number of cells

under the beam

For all g ^E H,

we have:

3.2 PSEUDO-ICOSAHEDRAL SYMMETRY - CON- STRUCTION OF THE LATTICE 3fl. - We now investi- gate pseudo-icosahedral symmetry. ^The problem

amounts to prove the existence of an element g belonging ^{to H} such that the intensities I ( p (,ye) gl )

vary only slightly ^{when f} varies from 1 to 120, ^with

all the _p ( ye ) g ^{in H.} Unfortunately, ^{it is} incorrectly

formulated like this since H is 3-dimensional and G operates

^{in 6 dimension}

ⁱⁿ R6, ^{and even} though

we extend H by P1 (H) to obtain 6-dimensional lattice, ^{as was} considered in the last paragraph, ^we generally ^{do not obtain a} G-invariant lattice of

R6. This means that we have a good ^{chance to loose}

intense peaks ^{in the} orbit (pj (ye) gl /1 -- t -- 120)

in the reciprocal space. We shall pass round the

difficulty ^below by constructing ^a G-invariant lattice je from H and P’ (H) which will allow us to consider simultaneously ^the « intensity and pos- ition » approximation.

The lattice JC can be defined now. It is simple cubic, 6-dimensional, ^and characterized by ^{the two} following properties : i) it contains the 6 dimensional lattice generated by ^{H and its} image P’ (H) ^and ii) it is the smallest lattice (i. e. possessing ^the largest

lattice parameter) obtained from vI2 Z6 by ^scalar multiplication having ^this property. ^In other terms, JC is generated by the icosahedral orbit of H, up ^{to a}

scaling ^{factor À} which is determined by ^{the icosahed-}

ral orbit of P’ L (H). Indeed, ^{H is} « bearing ^the

information » about Bragg peaks positions, ^while P’ L (H)

contains » the approximation ^term (measured ^here by ^the proximity ^of E ^and E_L ). The lattice JC is then generated by ^{the basis} I ei /Ali ^{=1 to} 6} ^{for a certain non zero number}

,k. The parameter A, solely determined is a global

invariant of the tiling given by ^{E’. It has the} important property ^{that it} belongs ^to Q : ^{since the}

vectors wl* have rational coordinates in the basis of the ei’s ^{and since, in this basis, the matrix of} P’ has rational coefficients, ^{the vectors} P’ (w.-7’c) (i ^{= 1 to} 3) also have rational coordinates in the basis of the ej’s. ^{The fact that A} is rational implies

that the lattice Joe is stable under G [26], and that the action of G on the Z-module Pl (JC) ^{in E is} compatible with the action of G on the lattice .

Another property ^{of JC is the} following ^{one : for}

each element z belonging ^{to JC,} there exists a strictly positive integer ^s, depending ^{on z, such} that, ^for all f

from 1 to 120, ^{the 120} projections p (yf)(SZ)H ^{of the}

elements _p (yf ) (sz) ^{onto F’} parallelly ^to El belong

simultaneously ^{to H.} Indeed, all the elements

p (yi) z belong ^to JC, ^{and can be} expressed ^as Q- linear combinations of the ei’s, ^{since we have shown}

above that the lattice generated by ^{H and} P’ L (H) ^is

6-dimensional and has a basis ({ w 1 *, W2 * w3 , pi (w*),P’ (w*),P’ (w*) 1) composed ^{of vectors} having rational coordinates in the ea’s. Thus, ^there

exists a strictly positive integers ^such that, ^for

1, Q _ 120, s p ( y e ) (z )H belongs actually ^{to the} reciprocal ^lattice H, in F’. We take in the following

the smallest value of s realizing ^these conditions (for

each node z in Je).

Although ^the terminology ^{is not} yet defined, ^we

have the following fundamental result:

Proposition ^2: pseudo-icosahedral symmetry ⁱⁿ reciprocal space ^can be obtained if and only ^{if the}

rational space E’ is close ^{to our} physical space E. In this _case, the exact icosahedral positions ^{of the}

intense peaks ^are projections by Pl ^{of some vertices}

of a G-invariant lattice of R6.

Proposition ^{2 can be} examplified by ^{two well-}

known situations. At first, ^{if we consider the} rhombohedral tiling ^obtained by piling up period- ically ^the elementary prolate rhombohedron only, ^it

is clear that we cannot obtain dffraction patterns exhibiting almost the icosahedral symmetry. ^This

comes from the fact that the corresponding ^rational

E’ space is ^not close to E in the sense we have defined above. Secondly, Gratias, Cahn et al.

[21, 23] ^and similarly ^{Elser et al.} [20, 22] ^{have con-}

sidered certain subfamilies of cubic tilings ^{for which}

it is easy ^to show that the rational cuts E’ are in each

case close to E. In the case of rhombohedral tilings, only ^one tiling ^{has been studied in the cut and} projection ^{method framework} up till ^now [17, 24].

These tilings, ^which approximate ^the quasilattice,

exhibit the pseudo-icosahedral symmetry ^{in their} diffraction patterns, the Fourier transforms being

calculated by ^the computer [25], ^with systematically

the approximity of E’ and E. The present proposition generalizes these results.

We try ^to give ^{at first a} definition of pseudo-

icosahedral symmetry by taking ^{into account simul-} taneously ^the intensity ^and position approximation : Definition : ^a periodic tiling ^defined by {Ul’ ^U2, u3} ^{(or by E’, the} w;’s,...), ^{is said to} possess the pseudo-icosahedral symmetry ^of type (s, q ) -

for intensities and n ^for positions

^{if there exists a}

non-zero vector qeE ^{such that :}

with

and

Experimentally, ^{we are} mainly concerned with in- tense peaks, ^for they give ^the appearance ^of icosahedral symmetry ; this definition is fitted to the

experimental approach. ^In addition, ^{this definition}

becomes interesting ^when

^and q ^are simul-

taneously ^small. Assume -qo ^to be the maximal bound for _-q we want to tolerate ; ^{it can be} quantified

on diffraction patterns and reflects the perception

we have of the

quasi »-regularity of the distribution of intense peaks. Assume also Eo ^to be the maximal admissible value of _{8 ,} for instance Eo = n2 x IB 12/10. ^{We can now define the} pseudo-

icosahedral symmetry ^{of the} tiling given by ^E’ by saying ^{that it occurs when the} tiling possesses the

pseudo-icosahedral symmetry ^of type (8, q ) ^with

£ , Eo, q ^-- q 0 simultaneously. It is clear that, ^{if we}

can find rational cuts E’ of not too large ^lattice parameter, such that

is rendered small and

q lies below the spatial resolution, ^{we can} interpret

the quasicrystalline phase ^{as a} crystal arising ^from

the tiling given by ^one of these spaces E’.

We now demonstrate proposition ^{2. We assume}

that E’ is close to E and show the existence of a

vector q in E, which satisfies the conditions (7). ^Let

us assume that the ge’s ^and q exist and satisfy ^the

conditions (7), and introduce the notations for the

phases corresponding ^{to the intense} peaks ^{located at}

the positions gt jj in reciprocal space. For

1 -- f -- 120, ^we define of ^{as follows :}

We are now dealing ^{with the existence of an element}

q. This ^means that we fix arbitrary real values for the

CPt’S ^{and that we have to} show, whatever the distribution of the CPt’S, the existence of a q satisfying (7) ^and such that the values :

are almost all equal and maximal. We now prove that q ^can always be found in Pj (JC). ^{There are two}

cases : 1) ^{All the} phases CPt ^are equal; 2) ^the 4)f’s ^{are not all} equal. ^We fully ^{treat the first case}

with all the t>t’S equal ^{to zero :} 1) ^{We use the}

property ^of G-invariance of the lattice JC to build linear forms on R6 which globally characterize the action of G on the intensities and we apply ^the following ^{result of} homogeneous Diophantine ap-

proximation [19, 27] :

Proposition 3 : ^let L;(I ^{= 1 to} m) ^{m linear forms}

defined on Rn. For all A:::> 0, ^{there exists}

a

(ai) ^{EZm and b}

(b j) ^{eZn _ {0} such that :}

1872

Vj:1 ^to n, IbilA ^{and Vi:1 to m,} I Li (b) - ai I ^{A - nlm.}

We take m = 120 x IBI, n

^{6 ; Z6 is} considered as

the lattice of R6 generated by ^{the vectors} ei/ B/2. ^In

order to obtain the existence of oints ⁱⁿ Je, ^we multiply Z6 by ^a scaling ^factor B/2/A. This scalar

multiplication ^{is called d. We now} define the linear forms Lej ^{by :}

for

By application ^{of this} proposition, ^{we obtain the}

existence of a non zero element g ^{in JC and} integers atj ^such that 11 g 11 . ^{-- A} /2/,k ^{[18] and that}

for all e, j

We take of course A sufficiently large ^{to have these}

absolute values arbitrarily ^small.

We now claim that the q ^{we are} looking ^{for is} Pli (so), ^{for a certain} strictly positive integers, ^{as it}

was considered at the beginning of section 3.2.

Before proceeding ^{further we have now to} argue that s depends only upon g and ^not upon A, otherwise the following calculations would be false.

Let us call (p ( y t ) (sg ) )’ ^{= g t these 120} (non-zero) projections ^{in H} obtained from sg eJe. ^{We now}

check the intensity ^and position approximation :

Intensities : we have :

A is sufficiently large ^{in order to} have all the terms

exp (2 i,7r p (y ) (sg ) . rj i ) ^{almost equal to 1 owing to}

equation (11). On the other hand, ^{for all} Q, ^the

terms - p (1’e )(sg) + gQ belong ^to E’ , ^{and since we}

have assumed E’ close to E, ^{we have} El ^{close to} E 1. ; therefore, ^{since the vectors} rj ¹ belong ^to E, ^all the scalar products (- ^p ( yQ ) (sg ) ⁺ ge ) . _ri I ^are al- most equal ^{to zero. The sum} (12) is then almost

equal ^to I B the conditions (9) ^are satisfied and the action of G modifies only slightly ^the intensities calculated at the points Pi (gf).

Positions : in order ^to check that the intense peaks

located at the positions Pi (ge) ^{are almost} distributed

according ^to the icosahedral symmetry, ^{we first} write :

Since sg - (sg )H belongs ^to El , ^{we see that the term} Pj (sg - (sg )H ) ^{is almost} equal ^{to zero} (E’ ^{is close to} E). Similarly, ^{we have for all I :}

a term almost equal ^{to zero. We therefore} prove by

these equalities ^{that the exact} icosahedral positions

are given by :

but, ^since they ^are generally ^{not elements of the} reciprocal ^lattice (in E), ^{the intense} peaks appear

just besides, ^{at the} positions Pj (gi). ^{In this extent,}

we have just ^{derived a} general expression ^{for the}

distortions for each peak ^indexed by ^the subscript t :

Since the lattices H, P’ L (H) ^{and can be}

determined once the 18 coordinates nij ^are given ^and

since the positions sg ^can be found by ^the computer

fairly easily ^{in JC} by scanning ^and sorting ^{the nodes}

of JC in the neighbourhood ^{of the} origin, ^a complete

calculation of the distortions can be made once

E’ is given. ^The expression (14) ^shows that, ^the closer E’ to E, the smaller the distortions, ^{as we} expect ^it.

When all the phases Of ^{are taken} equal ^{to a non-}

zero (modulo 1) ^number, we can use the following

case 2).

2) ^{When an} arbitrary ^{set of} phases cp t ^{is con-} sidered, ^{we use} similarly Kronecker’s theorems of

nonhomogeneous Diophantine approximation ^{to ob-}

tain the existence of an element sg ^{in JC} [19, 27].

To summarize, pseudo-icosahedral symmetry ap- pears when E’ is close ^to E, ^{whatever the} distribution of phases in the intense peaks.

To finish up the demonstration of proposition 2, ^it is _easy to check that, when E’ is not close to E, distortions are systematically important ^{and there-}

fore the intensity ^and position approximation ^cannot

be fulfilled.

Finally, ^{let us consider a rational cut} E’ of R6 of

arbitrary Bravais lattice exhibiting ^the pseudo-

icosahedral symmetry. We have the following ^im- portant ^{result :}

Proposition 4 : (given E’) ^the distortions of the

intense peaks ^{in the} reciprocal space decrease to

zero at large distances from the origin.

It is just ^an application ^{of the inflation} operator ^T of R6 defined by, in the basis of the ei’s :

on the exact icosahedral position PI (p (’Yt)(sg»

and the distortions dise. Indeed, ^{T has the} properties

that Pi T = TPI, P-L T = - 1ITP, ^and T2

T + 1 d6. ^{When a shell} of intense peaks

^times

farther from the origin ^is considered in the reciprocal

space, the associated distortions are given by, ^with

(sg )’ - sg EE’ :

(with ^{a suitable value of} s), ^and correspond ^{to the}

exact icosahedral positions :

Since T contracts distances by

ⁱⁿ E I

^and

E’ is close ^to E, ^{T contracts} distances in El « by

rational numbers » close to

and consequently

divides distortions by ^{the same} quantity, ^{what in-}

duces the result.

4. Comments.

Proposition ^{4 can} explain ^some experimental ^results [17, ^25, 28]. ^{It is not} the purpose ^to develop ^{here the} algebraic expressions ^{of the} distortions according ^to

the rational space ^E’ and to the Bravais lattice of the structure it determines. Examples ^{will be} give ^else-

where in more details, ^with suggestions ^for indexing coherently ^the distorsions with respect ^{to the} present framework [29].

Since the present approach ^{of the} icosahedral symmetry is considered via rational spaces E’ of

R6 close to the physical space E, ^a complete ^reformu-

lation of the results obtained here can be made in terms of defects of the perfect 3D Penrose patterns

as in the work of Entin-Woklman et al. [30], ^{as will}

be demonstrated later on [29].

The geometric description ^of pseudo-icosahedral

symmetry developed ^{in this} study ^{can be} generalized

in several ways : ^at first, ^{it is} possible ^{to rotate} continuously ^{in R6 the} projection strip, i.e. the 3D rational cut E’, away from the orientation giving

icosahedral symmetry, ^without restricting ^ourselves

to E’ subspaces ^{such that E’ n} .J2 ^{Z6 should remain}

a 3D lattice. Indeed, during ^{such an} arbitrary rotation, ^{E’ n} B/2 ^{Z6 is} generally ^reduced to ( 0 ) , ^or

sometimes it consists in a 1D, ^{2D or 3D lattice} according ^{to the field} (extension ^of Q ⁱⁿ R) ^{which the}

coefficients of Pl( belong to, ^{in the} basis of the

ei’s. ^The corresponding tilings ^{are then} fully aperiodic, ^or partially quasiperiodic ^and partially periodic along 1, ^{2 or 3} independent directions. We

can consider that, ^{each time a} dimension of the affine subspace generated by ^{E’ n} Ji ^{Z6 is} lost, each time the corresponding ^lattice parameter goes to infinity. ^{This leads to a} slightly ^different algebraic description ^{of the} proximity ^{of such a} subspace

E’ (i.e. ^{such that rank} (E’ ⁿ .J2 Z6)

^{0, 1 or} 2)

with respect ^to E, ^{but the} pseudo-icosahedral ^charac-

ter given by E’, though expressed differently ^from

this study, ^{remains once E’ lies in a narrow 6D solid} angle around E in R6 [29] (correspondingly ^inten-

sities and positions ^of Bragg peaks ^{exhibit a continu-}

ous evolution in reciprocal space). Actually, ^{we are} dealing ^{in this} study ^with « fully » ^{rational cuts} E’, i.e. such that E’ n .J2 ^{Z6 is} 3-dimensional, ^since

we would like to investigate ^{how classical} crystal- lography ^{can take into account the a} priori surprising

icosahedral symmetry.

A second generalization ^{can be made} by ^embed- ding ^our physical space E in RN with N greater ^than 6. Whereas there is only ^{one local} isomorphism (LI)

class in the case N

6, several LI classes _may appear (in general) ⁱⁿ higher dimension. This de-

pends only ^{on the structure of the} projected ^Z-

modulus P, (LN) ^of the G-invariant lattice LN in

RN onto E (with ^RN

^{E 0} E1 ; ^and LN being ^the equivalent ^of /2- Z6 ⁱⁿ R6) [27]. ^{We know from}

Gahler [31] ^{that the} Fourier transforms of two

quasiperiodic patterns agree in intensity ^{if the two}

patterns ^are locally isomorphic. Therefore, ^3D

rational approximate ^{cuts E’ in} RN can be classified

according ^{to the LI class to which} (say) ^E belongs. ^In

the case N

6, ^{there is} only ^{one class of rational}

cuts since there is only ^one LI class of quasiperiodic patterns. ^An interesting application ^{of these} concepts would be to investigate ^{4D rational cuts in} R8 and their 3D cross sections in view of approximating ^the

4D and 3D quasiperiodic patterns given by ^{Elser and}

Sloane’s model [32].

Acknowledgments.

Dr M. Sloim is acknowledged for very helpful

discussions.

1874

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