Theory of G-approximant crystals with G, a non-crystallographic point group. I

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Theory of G-approximant crystals with G, a non-crystallographic point group. I

J.-L. Verger-Gaugry

To cite this version:

J.-L. Verger-Gaugry. Theory of G-approximant crystals with G, a non-crystallographic point group. I.

Journal de Physique I, EDP Sciences, 1991, 1 (9), pp.1303-1320. �10.1051/jp1:1991208�. �jpa-00246413�

J.

Phys.

IFrance1 (1991) 1303-1320 SEPTEMBRE 1991, PAGE t303

Classification

Physics

Abstracts

02.40 61.10D 61.50E

Theory of G-approximant crystals with G,

a

non-crystallographic point group.1

J.-L.

Verger-Gaugry

LTPCM

(*)-ENSEEG-INPG,

BP 75, Domaine Universitaire, 38402 Saint Martin d'Hdres,

France

(Received18 October1990, reviked 2 May 1991,

accepted

21

May1991)

Rksumk. on d6finit le concept ^{de cristal}

G-approximant,

_avec G _un groupe

ponctuel

_non

cristallographique

et on

analyse

son facteur de structure au moyen d'un nouvel

objet

_: le G-cristal virtuel associb _au cristal

G-approximant.

Los variables

qui

contr61ent le

dkveloppement

_en skrie de

Taylor

de la transformke de Fourier de l'ensemble des sites

atomiques

d'une maille

blkmentaire du cristal

G-approximant

sont : l'excds de facteur de diffusion

atomique,

l'excds de

multiplicitb

de site, les fonctions de dkformations totales, dans l'espace rkel et dans l'espace

rkciproque,

dkfinies _par rapport au G-cristal virtuel, et la dbviation

depuis

_une interaction

moyenne onde-G-amas. Ceci permet ^de caract6riser le caractdre

pseudo-G-invariant

du cristal G-

approximant

et

d~analyser

des

rdgles

de

pseudo-inflation.

Abstract. The concept ^of

G-approximant

crystal, with G, _a

non-crystallographic point

_group, is defined and its structure factor is

investigated by

_means of _{a new}

object

the virtual G-crystal

associated with the

G-approximant crystal.

The relevant variables in the

Taylor

series

expansion

of the Fourier transform of the _set of the atomic sites in _a unit cell of the

G-approximant crystal

are : the atomic

scattering

factor _excess, the site

multiplicity

_excess, the overall deformation

functions,

in real and

reciprocal

_spaces, defined with respect to the virtual

G-crystal,

and the

deviation from _an average interaction wave-G-cluster. This allows _us to characterize the

pseudo-

G-invariant character of the structure factor of the

G-approximant

crystal and to analyze

pseudo-

inflation rules.

1. Inhoducfion.

Although

3D

crystals

in condensed matter

physics

_{appear as} 3D

periodic entities, fully

described

by crystallographic

_groups and _some numerical data such _as

parameters positions [I], they

_may exhibit local combinations of atoms which

present

^{almost forbidden}

symmetries,

such _as

pseudo-n-fold symmetries,

with _n

= 5 _or above

7,

_or

pseudo-icosahedral symmetry [2- 4],

and the

study

of such

crystals

_may contribute to

provide

_a valuable

insight

into the laws of

organization

of atoms in

quasicrystals [5].

These

(finite) configurations

of atoms appear

(*)

CNRS UA 29.

almost stable under the action of _a

non-crystallographic point

_group

G,

but

generally,

there is

no such

(fixed)

action _common and

compatible

for all the clusters of _atoms in the

crystal.

However,

it is the _case for _some

phases

like

a-Almnsi,

R-AlLicu for icosahedral symmetry

[2, 6],

and it is _now usual to call these

phases G-approximant crystals

_» to indicate that local

order is

depending

_upon the symmetry group of the

crystal

_as well _as the forbidden

point

group G. The main

goal

of this article is to

analyze

the G

dependence

of the

approximant crystal

in real and

reciprocal

_spaces, and this

approach

^{lies outside} classical

crystallography.

The first part

(I)

of the article aims at

investigating

coherence conditions between the

configurations

of atoms in _a

crystal

with

respect

to the action of _a

non-crystallographic point

group

G,

and at

defining formally

_a

G-approximant crystal

_». In this

respect,

we start

by defining G-clusters,

their deformations and

G-approximant crystals

in _a

general

_way, before

applying numerically

in

part (II)

these notions with G

=

m§5

_{and the four}

approximate

icosahedral

crystals, namely: R-AlscuLi~, a-Almnsi, fl-Almnsi, c-Ti~Ni.

It is often

observed that local order in

crystalline approximant phases predetermines

the orientation

relationships

between the

quasicrystalline phase

and its

crystalline

_« cousin »

phases [7, 8].

Therefore,

it _seems reasonable to define _a G-action in _a

crystal

_on the basis of _an

analysis

of the

geometry

of local

configurations

in the

crystalline approximant phase [9].

Roughly speaking

and

fairly realistically,

_an

approximant crystalline

cousin _»

phase

of _an icosahedral

quasicrystal,

^such as

R-AlscuLi~, a-Almnsi, c-Ti~Ni,

_can be described

I)

ⁱⁿ

reciprocal

_space,

by

a distribution of intense

Bragg peaks

which almost follows icosahedral

symmetry ^{and inflation rules}

(by

the

golden mean),

without

respecting

them since diffraction spots are located _{on a}

lattice, it)

in real space,

by

the existence of _an

important density

of

CN12

sites, empty

or

occupied,

as in many

Frank-Kasper phases,

about which

pseudo-

icosahedral units of _{more or} less

important

diameters lie _; these units

being strongly

imbricated and

packing together coherently,

in the structure.

Such _a definition with another

non-crystallographic point

_group G' _can be

given similarly

for _a

G'-approximant phase.

We are now

dealing

with the simultaneous double

approximation,

in

«positions

and intensities» in

reciprocal

_space. For

this,

_we

analyze

in

reciprocal

_space the

positions

of intense

Bragg peaks

and the

quasi-invariance

of the structure factor of the

G-approximant crystal

under

G,

at the first

order,

_{as a} function of the four

quantities (defined

in the

text)

which appear to be relevant in the formal

expressions

_: the overall

deformation,

the atomic

scattering

factor _excess, the site

multiplicity

_excess and the deviation with respect to an

average response per cluster to an incident _wave. The type ^of wave is

implicitly X-rays

or

electrons. The main result of this work is contained in

proposition

2.

2. Geomehic

approach

in real space.

2, I COLORED G-CLUSTERS _{AND THEIR} DEFORMATIONS. Assume G is _a

non-crystallogra-

phic point

_group and that _we have identified G with its

image

under _a real faithful linear

representation

_{pi :} G

- O

(3,

M

),

_an

origin

and _an

orthogonal

basis

being

chosen _once for all in 3D Euclidean _space M~, so that _{we are}

dealing

with _a finite _group of 3 _x 3

orthogonal

matrices with coefficients in

M [10].

A G-cluster

X(n

is defined

by

_a finite collection of

points

in M~, ^which is not included in _a

plane,

such that _:

I)

It is

G-invariant,

it)

it is _a union of _n shells

(layers),

each of them

being

constituted

by

the vertices of

perfect regular

G-invariant

polyhedra,

it)

is of _{course a} consequence of

I).

It may include the

origin

or not. The definition of _a

layer depends

_upon the realization of G in 3 dimensional

(real)

Euclidean _space. For

bt 9 THEORY oF G-APPROXIMANT CRYSTALS 1305

G

=

m35,

there _are two

unequivalent representations and,

in both _{cases, a}

layer

amounts to _a G-invariant collection of

points

_{on a}

sphere

centered at the

origin.

For

cyclic

and dihedral

non-crystallographic

_{groups, C~} and

D~~,

^with n #

1, 2, 3, 4, 6,

there _are several

possibilities

of realizations in 3 dimensions

[10, 11], but,

^{for each of them}

(adding possibly

the trivial _one- dimensional

representation

to fill up the

dimension),

_a

layer

will be _a set of

points

_on the surface of _a

cylinder,

whose

height

is

equal

to the diameter of the disc

constituting

its basis.

The

origin

is chosen _at the _center of the

cylinder

and is

equally

distant of its two faces and its

cylindrical

wall.

Each

layer

is _a finite union of orbits of

points

under the action of G. The coordination number of the

origin

is

conventionally

taken

equal

to the number of atoms in the first

layer.

For G

=

m§5,

it is the usual notion. In

figure I,

_we illustrate this notion in 2

dimensions, by taking

the _group

Cs

^of

plane

rotation matrices and collections of dilated

pentagons

^all

centered at the

origin.

We _now introduce

(atomic)

colorations and deformations of G-clusters.

Coloration _{: we} consider that each

point

in _a G-cluster has _a color which makes it _a

ball,

centered at this

point,

with _a radius

depending only

_upon the color. The introduction of colors

(Fig. lb)

to the whole set of

points

of _a G-cluster makes _some

configurations impossible,

when

points

_are too close to each other.

Indeed,

_as in hard

sphere packing problems,

_we allow

no intersection between the

balls,

_or

possibly, tangential

connections. We do _not

impose

that the coloration is invariant under G _a colored G-cluster is then _not

necessarily G-invariant,

whereas its

underlying (not colored)

G-cluster is G-invariant

by

definition. When the

origin belongs

to the

G-cluster,

_we obtain _a ball at the

origin by

atomic coloration.

Deformation _{: a} deformation of _a colored G-cluster

X(n

=

(xi,

color of _x~ is _a map

fl

of

X(n)

into M~ which _preserves coloration and does not

change

the

position

of the _center of the G-cluster. In other terms, it is _a collection of

displacements (fl(x~)

_x~, at the site

J~)

of all the atoms of the

G-cluster,

_one

displacement

for each atom, such that _no

interpenetration

of the balls _occurs at the final stage after

moving,

whatever the

path (Fig. lc).

The sites of _a G-cluster _are therefore in one-to-one

correspondence

with those of the related deformed G-cluster. The _center

position

does not

change

it remains _a fixed

point.

@ @

,'

© ~

/ _~ ~

" '

'

o Ii

'

z

^'

~

ii

'

jj

^~

'

~

' ~'

~ _~

t

«

Fig.

I.

a) X(2)

=

Cs-cluster

composed of two shells _; the coordination number of the

origin

is

equal

to 5. The second shell is the union of three orbits under C5. The center is empty. b) X(2)

=

colored

Cs-cluster composed

of _two

layers

_; coloration is made

by

three types of discs of difserent radius ; the

coordination number of the

origin

is

equal

to 10. The center is

occupied.

Color is distributed

independently

of the C5-action. c) Small deformation of the previous colored

Cs-cluster X(2)

_; the

intersection of two discs remains empty or is reduced to one

point. Conventionally,

the coordination number of the

origin

remains

equal

to 10, as before the deformation. The center does not _move.

JOURNAL DE PHYSIQUE I T I,M 9, SEPTEMBRE [WI 51

After

deformation,

the

underlying

G-cluster of _a colored G-cluster _can loose _some

(or all,

except

identity) _symmetry

elements of

G,

_{or possess} another

symmetry

group assume H is

now its

symmetry

_group in O

(3, M ).

H may be trivial

(Fig. 2a), conjugated

to _a

subgroup

of G in

O(3, M) (Fig. 2b),

_may have trivial intersection with G

(Fig. 2c),

contain G

(Fig. 2d)...

since the deformation _map, defined _on

X(n ),

is not an

isometry.

In this extent, we see that _a small deformation may

provide

_a colored cluster

having

_as

crystallographic point

_{group a} symmetry group

H,

from _a colored G-cluster

having

the

non-crystallographic point

_group G

as

symmetry

group. In

figure 2c,

for

instance,

_a colored

C4-cluster X'(2), having

two orbits

on the first

layer,

^{and three orbits} on the second _one, is obtained from _a colored

Cs-cluster X(4),

made of four shells of

regular pentagons.

@ ~

i

II

~

i

@

/

~, ~l

Fig. 2. a) Trivial symmetry _group of the deformed colored C5-cluster. b) Deformation of _a colored

Cjo-cluster X(I)

into _a colored

C5-cluster X'(2). c)

The deformation of _a colored

C5-duster X(4)

leads to _a colored C4-cluster X'(2), for which the coordination number of the

origin, initially

5, is

equal

to 8.

We define the coordination number of the

origin

of _a deformed colored G-cluster _as the coordination number of the

origin

in the colored G-cluster before deformation. For small deformations, ^which _preserve ^{the succession order} of the colored shells in the

G-cluster,

it is

as usual the number of atoms which _are the closest _to the

origin,

that is in the first

(distorted)

shell.

2.2 DEFINITION oF G-APPROXIMANT CRYSTALS. We want now to

interpret

local order in

approximant crystals

in terms of

packings

of

slightly

distorted colored

G-clusters,

which

interpenetrate

each other.

bt 9 THEORY oF G-APPROXIMANT CRYSTALS 1307

Assume that F is the space group of _a

given crystal,

in 3 dimensional Euclidean space M~. Let _us denote

by

T the translation lattice of the

crystal

_or its Bravais

lattice,

and

by

R its

point

_group. We say that this

crystal

is _a

G-approximant crystal

when the

following

local conditions _are satisfied :

I)

^In the unit cell

(of

the If-reduced

basis)

of the translation lattice

T,

there exists _a finite number of sites

(occupied by

_an atom _or

not)

CC

=

(si,

_s~,

,

sj),

t m

I, (with

CC _» for clusters

centers)

such

that,

for each I, I _w I _« t, there exists _a

perfect,

not colored

G-cluster, X~(n,),

centered at s~, and _a finite _set of _atoms around _s~ which is obtained

(formally) by

a

small deformation 8~ of this G-cluster

X;(n;)

₌

8~(X~(n~));

and _we

finally

add the

corresponding

colors to the

points

of

8~(X~ (n~))

_so that this cluster becomes identical to the local colored _» environment at s;,

it) (maximality)

CC ₊ T is invariant under the action of

F,

I-e- _no other center of colored

deformed G-cluster in the

crystal

_can be reached

by

the action of elements of F _on

CC

+T,

iii) (orientational compatibility)

there exists _{a common} N _m

I,

with NW n,, for all I, I _« I _w t, such that the intersection n

[X; (n;

_s~

],

where each G-cluster is translated to the

origin (centers

at the

origin),

is _a

(not colored)

_non

_empty

G-cluster

X(N),

(iv) (covering)

_every atom

(=

colored

site)

in the

crystal belongs

at least to one distorted colored

8;(X;(n~)),

centered at s~, for a certain I.

This definition calls for _some remarks. At

first,

it is clear that

iii) requires

_some choices of the s; s. If every deformed colored G-cluster in the

G-approximant crystal

has _a center

occupied by

_an atom, we take

s~ to be the

point

at which this _atom is located. If R does _not contain inversion and if _some colored deformed G-clusters have _no atom in their center, such

as

Mackay icosahedra,

we take _as natural _center of

symmetry

_s~ ^{of such} a colored deformed G- cluster the

highest symmetrical position

inside the cluster

by intersecting

the affine invariant

subspaces

attached _to the symmetry elements of the cluster.

Second,

if

iii)

is satisfied

(see Fig. 3),

_we obtain _a G-action in M~,

expressed

in _a basis of T

by

3 _x 3 matrices. For G

=

m35,

coefficients

belong

to the real

quadratic

field

it(/).

_We

cannot obtain rational matrices since G is _{assumed to be}

non-crystallographic.

We then obtain

physical

space as a 3-dimensional

subspace

of _a

(real) hyperspace

of

higher dimension,

_{say m,} for which all the characters of G _are

rationally-valued [10, 11]

: the

representation

space

M"'

_{is endowed} with _an

integral

action of

G,

for which there may exist G-invariant lattices

(of

rank

m),

_m is chosen minimal to obtain such _an

integral representation

^of

G,

and

physical

space E is _a rank 3

sub-ll [G ]-module

of

M"'.

For G

=

m§5,

_{we as} usual take _m

=

6,

and E is

irreducible.

We also remark that the function

8~,

in

point I),

is defined

locally, only

_on the G-cluster

X~(ni),

that is in the

neighbourhood

_{of s;.}

However,

the real number 8 _measures the overall deformation in the

following

_{sense : we} define 1

3;

ii

by

: 1

3,

ⁱ

=

Sup

^[_8~

(, )

_x~^[

,

for

J~

running

_over all the sites in

X(N)

_{+ s;,} and call it the _« deformation

implitude

» of the G-

cluster

X;(n~)

at s;.

By

small _» in

I),

_{we can} understand atomic

displacements

all less than the value of the smallest atomic

radius,

for instance 0. I _nm. The

G-approximant

character of

a

crystal

is then measured

by

the deformation 8

=

Sup11 8,

ⁱ

taking

into account the

;

deformation

amplitudes

of all the deformed G-clusters centered at the

s~'s

in the unit cell of

T,

when the G-action is made

optimal by

the suitable

position

of

X(N),

_as

explained

below.

Condition

iv)

_means that _{we can} describe the whole

crystal knowing

the centers

s~, the sequences of atomic

layers

in each cluster

X~(n;)

and the deformations

8~.

^{It is clear}

.A~

T

Fig.

3. Schematic

representation

of

interpenetrating

distorted colored G-clusters _{m a}

G-appro»mant crystal.

The site ~

belongs

to 2 G-clusters and its

multiplicity

_A~ is 2. Layers of atomic sites in G-clusters

are represented by circles. Atomic sites _{are not} indicated. Only the _common orientation is

schematically

indicated

by

the pentagon

(condition iii)

in sect. 2.2).

that the distorted G-clusters in the

crystal interpenetrate

to cover the entire set of atoms and that any atomic site is

likely

to

belong

_to several distorted G-clusters

simultaneously.

For

quantizing

this

dependence

_on the structure

factor,

_we will define below the

quantities

of average site

multiplicity

and site

multiplicity

_{excess ».}

A collection of G-clusters is

schematically represented

in

figure

3

by

concentric

circles,

which

interpenetrate,

centered at some atomic sites in _a unit cell of the translation lattice T of

a

G-approximant crystal.

The _common orientation is

symbolically represented by

_a

regular

pentagon over the G-clusters.

2.3 EXISTENCE oF A UNIQUE OPTIMAL coMMoN G-CLUSTER

X(N).

In

practice,

_qe take

the _most

interesting

value of

N,

that

is,

the G-cluster centered at the

origin

with the greatest

possible

number of

shells, N,

which

fits,

_up to the translations _s~, the local environments

around the

s~ ^s with _a

good approximation.

There exists _a

priori

an

infinity

of _ways to

position

a G-cluster

X(N)

at the

origin

in order to

render,

from

it,

for all I, all the deformations 18~ⁱ small at each site s~.

However,

_one

(unique) position

is

privileged

since it

corresponds

to the

position

of the G-cluster

X(N)

obtained

by

the least square method _over the collection of distorted clusters

8,(X~(n~))

_s,. Part 2 is

dealing

with the

equations

and the numerical resolution of this

problem

of

finding

the

optimal

G-cluster

X(N). Here,

_we

just

need the existence of this

optimal positioning

to

analyze

further the structure factor of the

G-approximant crystal.

The function

Sup

_8~

ii,

^{for I} « I _« t, is assumed to be _non-zero for G

=

m§5,

it _comes

i

from the fact that _{we cannot} tile 3D space

by perfect

icosahedra. We have that

X(N)

is

positioned optimally

when this

function,

calculated with

X(N ),

is minimal

(variables

are Euler

angles

to

parametrize rotations,

_see

part

^{2 for}

details).

This

provides

_an

optimal

G- action in

physical

_space in «direction». This

optimal

G-action is the

key

_to make _a modelization of the icosahedral

phase, by

_means of _a rational cut and the construction of _a 6D discrete

crystal,

which respects ^{the local} atomic orientations

[9].

bt 9 THEORY OF G-APPROXIMANT CRYSTALS 1309

2.4 VIRTUAL _{CRYSTAL ASSOCIATED WITH A} G-APPROXIMANT _CRYSTAL. The virtual

crystal V(£)

associated with the

G-approximant crystal

His defined

formally by

the

following

_: it is the

periodic assembly

of the not deformed colored G-clusters

X; (n~),

^centered

at the _{s~ s,} where the

periods

lattice is the _{same as} that of the

crystal E,

^{that is the lattice T.}

This

assembly

of colored

points

is

generally

not _a

~physical) crystal

since many ^atoms

overlap.

We _can also view

V(£)

_as the

crystal

obtained from £

by

the collection of transformations

(8/~), respecting colors,

each of them

applied locally

to the distorted G-clusters

8~(X~(n~)).

We call

V(£)

the _« virtual _»

crystal

obtained from £

by

this «inverse»

transformation,

in the _sense that _a color at _a

point

does not

correspond

to a

volume,

that

is,

to a

spatial

extension of the atom at this

point.

Many

colored sites in the

G-approximant crystal

£

belong

to several deformed colored G-

clusters, and,

transformed

by

the collection of_« inverse _» deformation

functions, 8j

_~,

applied

to all the G-clusters at all the sites _s~ of the

crystal,

the

crystal

becomes

V(£),

_a

periodic

collection of colored

sites,

_many of them

being

_{DEDOUBLED » as many} times _as

they belong

to G-clusters. This DEDOUBLING, and _more

generally,

this

«multiplying»

of colored sites

occurs over very small

distances, depending

_upon the values of the deformation

amplitudes

of

the

family

of G-clusters.

If the space group, translation group and

point

_{group are}

F,

T and

R, respectively

for the

G-approximant crystal E, considering

colors instead of

balls,

_{we can} still attribute _{a space} group, say

F',

and _a

(crystallographic) point

_{group, say}

R',

to the

periodic assembly

of

atoms _» V

(£ ).

The lattice T remains _common to the

crystals

Hand V

(£ ). Although locally

atom environments become G-invariant

by

this _process, the

point

_group R' is different from

G since there is _no accumulation of colored

points

in any bounded

region

of space

by

construction. The effect of this _« inverse _» transformation

(8

j _~) ^is to

keep

_constant, not

only

the lattice but also the sites which _are centers of G-clusters.

Consequently,

the diffraction

peaks

obtained from £ _or from

V(£)

will appear on the _same lattice

(= T*)

at the _same

positions

in

reciprocal

_space, which is the

reciprocal

lattice T* of T.

This modification of the

G-approximant crystal

£ into _a virtual

crystal

is made

possible by privileging

the translation elements of the space group

F,

that is T is

preserved,

_over the group of the

orthogonal operations

deduced from the elements of

F,

and R becomes another

point

group R'. We have

already

_use this remark to lift the atomic decoration in _a

hyperspace

endowed with _a host lattice for atomic sites

[9].

3. The structure factor of _a

G-approximant crystal.

We want to

analyze

the structure factor of the

crystal

£ with respect to the action of G _as defined

optimally above,

and look at

pseudo-inflation

rules. In the _case of G

=

m§5,

inflation rules _are

given by

a

quasidilatation

in the

projection

framework

[I1-13],

which amounts to the

invariance

by multiplication by

_T

=

(1+ /)/2,

_T~

or

r~

_{of the}

peak positions

in the diffraction

pattern

ⁱⁿ

reciprocal

_space.

3,I STRUCTURE FACTOR TAYLOR _sER1Es DEVELOPMENT. Before

entering

the

formalism,

we would like to

give

_some definitions and show

intuitively

that the variables which _are chosen below to

perform

the series

development, namely

deformations

8;

in real space, distortions dis

(q )

of

Bragg peaks positions

in

reciprocal

_space, atomic

scattering

factor _excess

q

^{and site}

multiplicity

_excess A

j, ^are

physically

relevant.

Let _us call K the _average atomic

scattering

factor of the

crystal

S If

C~

denotes the atomic concentration of the

species

_m, of atomic

scattering

factor

K~,

in the

crystal £,

_we have _: K

=

IC~ K~,

where the _sum is _over the set of the atomic

species

in £. We

implicitly

consider

interactions with

high

_energy ^electrons _or

X-rays.

Define _now

z~ ⁼

Jj K,

the atomic

scattering

factor _excess for the

j-th

atom. In the

expression

of each

K~,

the

temperature

^factor correction is

included,

_so that

K~

_can be

approximated by

a

polynomial

of

degree,

_{say w,}

greater

^than or

equal

to

2, independent

of _m, of the variable sin

0/v,

with the

angle

9

corresponding

to the

scattering

vector q in the

reciprocal

lattice T* and _v the

wavelength

of the incident beam _:

Km( iiq )

=

zm(«om

₊

«im(sin 9/v)

+

«~m(sin o/v)~

_{+ +}

«

wm(sin o/v)w) (i)

ao ~

is

equal

to I for

X-rays

and to 4

ar

~

m * e

~(r~) /3 h~,

for

electrons,

where _m * is the

moving

mass of the

electron,

_e the

charge

of the

electron, (r~)

the _mean square radius of the atom m, h the Planck constant

[I].

For iq ⁱ close to 0 and

X-rays interaction,

_z~ is reduced to the

atomic number _excess

Z~ -Z,

with Z

=

I~ C~Z~. Choosing

_w

sufficiently large,

the a,~ ^{s can be}

easily

calculated from the tabulation in

[I] by

the least squares method.

Similarly,

let _us define A _as the _average site

multiplicity

of the

crystal

E. If

A~ denotes the site

multiplicity

of the atomic site

j,

that

is,

the number of distorted G-clusters which contain this atomic site in _one of their

layers,

we have A

=

I(1/34~

A~, where the _sum is taken _over the M atomic sites in _a unit cell of the lattice T. We call

A~ ⁼ A~ A the site

multiplicity

excess for the atom

j (Fig. 3).

Both functions

K~

and

A~

_are invariant

along

_any

crystallographic

orbit of

F,

but the collection of

(vector)

deformations

(8~(xj) x~/8~(xi)

₌

x~)

attached to the colored

point

J~~,

joining

_x~ to the very close

positions

_x~, in the

perfect

colored G-clusters which contain J~~, is not invariant

along F(x~).

Now,

consider

plane

waves

propagating

to the

crystal

S The response in Fourier space will be

roughly

G-invariant with

peaks

_on T*. The main

goal

is _now to describe the simultaneous

double

approximation

in

positions

and intensities for the intense

Bragg peaks

Peaks

positions

were

already analyzed

in

[14]

for

approximate

icosahedral

periodic tilings

with

expressions

of distortions

depending

_{on a} rational cut in M~ and the number

theoretical characteristics of the

reciprocal

lattice T* lifted _up in M~. It will be shown later _on that the geometry ^{of the}

perfect

G-clusters in _a

G-approximant crystal

allows to calculate

directly

the exact

peak positions

without

using

the N-dimensional

formalism,

with N ~ 3 _;

intensities

of peaks,

located _{on a}

slightly

distorted

sphere,

deviate from _an average value

given ideally by

the G-action. What _can be the variables which govem the average

intensity

value and the

slight

deviations from this value at each intense

peak position

?

Intuitively,

the average

intensity

value will be _a function of the number _t of clusters per unit cell and of the _average site

multiplicity

A.

Indeed,

these two

quantities

characterize the

spatial

extension of each cluster _as well _as how clusters

interpenetrate

to _cover all the atomic sites of the

G-approximant crystal

E. Another relevant variable is the

quantity

S defined below which

reflects the average interaction behaviour per G-cluster with _an incident

plane

_wave. In

fact,

it is conceivable that the ideal interaction between _a

plane

_wave and _one colored distorted

cluster

8; (X, (n;))

is dictated

by

the

underlying

colored _non distorted cluster

X;(n; present

ⁱⁿ the associated virtual

crystal V( E),

its deformation

8;

^{and its} distribution of atomic numbers

which is not G-invariant

(as

it _was said

above,

coloration is not a

priori

^invariant under G in _a

G-cluster).

The average atomic

scattering

factor K should therefore

naturally

_appear in the

expression

of _an average

intensity value,

with its

dependence

in sin

of

_v.

We _{can now} define the

scattering

function _: for _{any q e}

T*,

H(q,

_y

j, y~,

y~)

=

3k 3~(K+

_y

j

z~)(A

+ y~

Aj)~ exp[2 iarq. _(xy

₊

y~[8k(~)

_x~]

)] (2)

N 9 THEORY OF G-APPROXIMANT CRYSTALS 1311

with yj, y~, y~ in the interval

[0

_;

1],

where the first _sum is taken _over the t

G-clusters,

that

is,

I Sk _w t, ^where

j

runs over all the atomic sites of the k-th distorted G-cluster

3k(Xk(nk)).

The

Bragg peak

intensities for the

crystal

£

given by

_:

I~ (q

₌

H(q, I, I,

I

1~ for all q in T^*

(3)

3.I.I Contributions in real space. We _assume in the

following

that the

scattering

function

His,

within _a

good approximation, given by

its first order

development

at

(q, 0, 0, 0)

the

partial

derivatives _{are :}

~~

=

3k _3~

_z~ A~ ^' exp

[2 iarq x~] (4)

Y1 _{(q, o,o,}o)

~~

=

3k 3j _KA~

A~ ^~ exp

[2 iarq

xy]

(5)

~Y2

(q,0,0,0)

~~jqooo~~~k~J~~ ~(21"q. [3k(~)-X~])eXp[2igrq.X]. (6)

This

gives,

at the first

order,

with second order terms assumed to be

negligible,

H(~,

_{Y1> Y2,}

Y3)

₌

H(~, °, °,

°

)

₊

t

j~,~,~,~

Y.

(7)

and therefore

H(q, I, I,

I

) H(q, 0, 0,

0

)1

«

w3~z~A~A~'+3~KA~~A~A~+3k3j2grKA~'iiqii[[3k(~)-Xj[[

w

3~ C~

_{z~ +} KA ^'

I~

~

+ 2 arKA iq ⁱ

jj

1

3~

ⁱ Card

(Xk (nk) (8)

k i

where _{r runs over} the atomic sites in _a unit cell of Rand _{m over} the different

species

in £. Let

us fix the

wavelength

_v of the incident beam and let _us have ⁱ qⁱ

varying

between 0 and _a maximal value

iiq¹

~~

to

analyze just

what

happens

for the G-action in _a

fairly

_{narrow cone} in

reciprocal

_space

involving

the first

layer (in

the _sense of Sect. 2.

I)

of intense

Bragg peaks.

The variable sin 0

Iv

lies in the interval

[0

sin

00/v],

with

00 strictly

less than

ar/2.

Let _us call

z =

Sup (z~(

ⁱ _q

ii

)i

the maximal value of the

z~'s

associated with this _cone, m,o« i qj ~ ii qj

~~

A the maximal value of the

A~'s

and M the number of atoms of the

crystal

Sin _a unit cell of the lattice _T

(note

that the number of colored

points

in the unit cell of T for the virtual

crystal

V(E)

is

equal

to MA at the first

order).

We obtain _:

H(q, I, I,

I

) H(q, 0, 0,

0

)

_«

Mz

₊ KA ^' MA ₊ 2 «KM i

q11 8

(9)

which _says that the three

contributions, namely

the atomic number _{excess z,} the site

multiplicity

_excess and the overall deformation 8 in real _space, control the distance of the structure factor of the

crystal

£ with

respect

to the

exponential

_sum

H(q, 0, 0,

0

).

3.1.2 Contributions in

reciprocal

_space. We _now

analyze

the

exponential

_sum

H(q, o, o,

o

)

= KA

3k Ij

_exp

[2 _iarq

_x

j] (lo)

dis(q)

w-t

q

q~.

«

n

,

/

T~

^/

.'

~,~,

Fig. 4. Irrational 2D section of reciprocal _space exhibiting the distribution of the exact icosahedral

positions q"

and their

respective

inflated values

rq", together

with the

positions

_{q of} the intense Bragg

peaks

located in their

respective neighbourhood,

_on the

reciprocal

lattice T*. Here, _we

implicitly

assume that all the distortions dis

(q) belong

to the

figure,

to

simplify

the

representation.

when _q is

decomposed

_as follows

(Fig. 4)

q =

q~~

+ dis

(q) (ii)

to take into account the distortions dis

(q)

in

reciprocal

_space with

respect

to the exact

[14]

peak positions

_q~~.

Following Verger-Gaugry [14],

_we know that there exists _a

unique

q~~ associated with each _q. The distribution of the

g(q~~l's,

where _g describes

G,

is G- invariant and is located _{on a}

sphere

of radius iq~~^{i in}

reciprocal

_space for

m§5 (resp.

_{on a}

cylinder

for

C~

^and

D~~,

_see Sect.

2.I),

whereas the distribution of the

g(al's giving

the appearance of the forbidden

symmetry G,

is located _on

T*,

and also in _a small

neighbourhood

of this

sphere (resp. cylinder).

In order to make the situation

symmetrical

in

reciprocal

_space,

we continue the calculation _as if all the _non distorted colored G-clusters _were moved

by

translation to the

origin (centers

at the

origin).

We

have,

still

neglecting

second-order terms

H(q,0,0,0)=KA~'3k3~(1+2iardis (q).xj)exp[2iarq~~ _x~]

=

" ~i~

~k ~j

_eXp

[21gr _q~~

(X~

Sk)

_~XP

[21"

_q ~~

Skj

+ KA ^'

3~ 3j (2

iar dis

(q xj)

exp

[2

_{iar q} ^~~ _x~

(12)

bt 9 THEORY OF G-APPROXIMANT CRYSTALS 1313

For any element _{g in}

G,

_any k and any colored

point _,

in the G-cluster

Xk(nk),

we have

[14]

:

g

(q~~) (J~ sk)

= q~~ g~ ^'

(J~ sk) (13)

Therefore,

since

Xk(nk)

is

G-invariant,

^the sums

3y

exp

[2

iar

q~~ ~ Sk)] (14)

are G-invariant and

depend only

_on the variable k. Let _us call

i

S

= t ^'

£ 3j

eXp

[2

_{1gr q} ^~~ _{~ Sk}

)j (l 5)

k

the _{average response per} cluster of _a scattered

plane

_wave characterized

by

the

scattering

vector q~~, very close to q. Since all G-clusters _are

roughly

^oriented in the _same way in the structure, the _average value S will not be _very different from each

scattering

term

[3~ exp[2

iar

q~~ (, Sk)] [,

^{when k varies from I to t. Let} us call also _e the maximum

e = maxk S

(

exp

[2

I_ar q ^~~

(,

_Sk

(16)

corresponding

to the

scattering

deviations for the _t G-clusters.

We _now

develop

the second term of the

expression (12).

We have it

equal

to

= 2 iar KA ^'

Ik I~ (dis (q) (, s~))

_exp

[2

iar q ^~~

x~]

+

+ 2 iar KA ^'

I~(dis (q ) s~) I~

exp

[2

iar q^~~

xj] (17)

Its modulus is then less than

2 _ar KA ^'

I~

dis

(q) _[3~ (fi ^s~)]

+ 2 _ar KA ^' dis

(3~

ⁱ_s~i Card

(X~(n~) (18)

where dis denotes

max~

^{i dis}

(q) ii,

^when q describes the

Bragg peaks positions

_on the

slightly

deformed

sphere

of radius ⁱq~~ⁱ

(resp. cylinder), forming

almost _a G-orbit. When the G- clusters

X~(n~)

_{s~ are}

inversion-invariant,

that is

centrosymmetric,

the first term in

(17)

is

equal

to _zero. When it is not the _case, this first term in

(17)

is bounded

by

2 _ar KA ^' t dis _max~

3~

§

s~)

= KA C

j dis

(19)

and

Cj

reflects the very ^nature of the

geometry

^{of the} common G-cluster

X(N)

with respect to inversion. We then obtain

iH(q,0,0,0)-KA~'StI wKA~'tj+KA~~cdis ₍₂₀₎

where C is the constant

C

=

2 ar

(Ikjjsk

i Card

(Xk(nk) ))

₊

Cj (21)

We have then

proved

the

following result, putting together equations (9)

and

(20)

_:

Proposition

I.

For any

G-approximant crystal

_as defined above and with the variables

given

in the text, the

scattering

function H

obeys

the

pseudo-G-invariance

_property

iH(q,I,I,I )-KA~~StI wmg+KA~~MA +2arKMiiqii 8+KA~~tj+KA~~cdis.

(22)

The

corresponding

intensities of the

Bragg peaks

_can therefore be deduced

owing

to

equation (3),

at the first order.

Proposition

2.

For any q in T* such that the associated q~~ lie on a

unique G-orbit,

the

pseudo-G-

invariance character of the intensities

along

this orbit is

given by

:

(Ia(q) ^(KA-~ St)~(

«

w2KA~~St[Mg+KA~'MA+2arKMiiqii 8+KA~~tj+KA~~cdis] ₍₂₃₎

where S is evaluated at q~~ ^{and is}

G-invariant,

K is i qⁱ

-dependant through

the factor sin

of

_v. The _average

intensity

value

Ifl(q) along

this G-orbit is

equal

to

(KA~ St)~.

By

this formula

(23),

_{we see} that the intensities _are controlled

by

the wave-G-cluster interaction via the modulation term S

(Fig. 5).

The

density

of clusters is

expressed by

_t, and the term A~ reflects the

magnitude

of cluster

interpenetration.

The average

scattering

factor

term K _comes from the fact that G-clusters exhibit colorations which _{are not}

generally

G-

invariant. Refinements of this _average value

I$(q)

=

(KA~

^'

St)~

_can be made if _we know

that _some subsets of the

X~(n~)

have _a coloration invariant under the action of _some

subgroups

of G.

Similarly,

the coefficients of the variables _z, A, 8, _j, ^{dis in}

(23)

_can be

8

~ _i

i~s _~

n

' '

iqi

e~.j23j

~

Fig. 5. Schematic

representation

of the behaviour a) of the modulation tern S along one axis in

reciprocal

_space and,

b)

of the _average atoInic

scattering

term K

(isotropic behaviour),

_{as a} function of sin 0

Iv,

_or

equivalently

((q((. Vfhen the _average

intensity

value decreases below the

visibility

threshold

as in

c),

the

pseudo-inflation

^rules can still be studied _on the term S _as in

a).

bt 9 THEORY OF G-APPROXIMANT CRYSTALS 1315

optimized,

that is

minimized,

when the coloration of each G-cluster is

coherently

invariant under the action of _some

subgroup

of G.

This formalization of

G-approximant crystals

becomes

interesting

when G-clusters _are

sufficiently large

to exhibit hundreds of atoms, as in a-Almnsi _or

R-Al~cuLi~

in the _case where G is the forbidden icosahedral

point

_group. Let _us

take,

for

instance,

200 atoms per

cluster,

M

=

20 atoms per

cell,

t

=

2 and A

= 4 with _an average atomic number Z

=

20. We obtain

directly,

at the first

order,

for _an interaction with

X-rays,

the

intensity

of the central

spot

as

20~

_x

20~

=

160 000. The intensifies in the first

neighbours G4ayer

of intense

peaks

in

reciprocal

_space will be

roughly equal

to

20~

x

S~/4,

and _{we assume} then to be

equal

to _one fourth of 160 000. For reasonable values of the five variables _z,

A, 8,

_e,

dis,

the first order corrections described

by equation (23)

_can be _seen to be

just

_a few

percents

^{of the} average

intensity

value.

However,

the

~perturbation)

_error term in

(23)

becomes

important

with

respect

to the average value when G-clusters _are

small,

that is the

G~approximant

character of the distribution of intense

Bragg peaks

in

reciprocal

_space has

good

chances to become not _so well

defined,

what is

intuitively

understandable.

3.2 PSEUDO~INFLATION _RULES. The succession of

layers,

their relative

distances,

^the

number of atoms per

layer

in the

perfect

G~clusters

X~(n~)

associated with the colored distorted G~clusters

8~(X~(n~) )

in _a

G-approximant crystal

£ vary within _a

large

extent from

one structure to another _one.

Compare,

for

instance,

the

a~phase

in Almnsi in which

m§s~clusters

_{extend to 16th}

neighbours

and the

crystalline phase c~Ti~Ni

where it becomes difficult to define

realistically m§s~clusters

_{after the 4th}

layer

around

Ni(e)

sites

[3, 15, 16].

We want to

analyze

how the intensities

I~(q),

with q~~

describing

_one

G~orbit,

_vary in _a

general

_way for _a

G~approximant crystal

when _an inflation factor is

applied

to the G~set of exact

peaks positions (g (q~~)/g

_e

G),

in

reciprocal

_space

(Fig. 4).

In this extent, we want to

correlate inflation rules in

reciprocal

_space to sequences of concentric

polyhedra

in real space.

In the

following,

_we will restrict ourselves _to G

=

m§5,

_to the inflation factor _T,

arising naturally

from the 6 dimensional formalism

[I I]

and to G-clusters of _a certain structure, ^that

is,

_we will make _some

assumptions conceming

the sequences of

layers

in the

perfect

G-

clusters.

We start

by decomposing

the

perfect

_common G-cluster

X(N)

into

layers

_up to the N-th

layer

and _we continue from the

(N

₊ I

)-th layer

to the

n~-th

one for each G-cluster in the

following

_{way :} for _any

k,

I Sk _{St, we} call

lay~ (r,

_p~,

m~)

the r-th

layer

of the k-th _non distorted G-cluster in the

crystal £,

that

is, equivalently,

the set of sites in the r-th

layer.

The number _m~ is the number of atoms in this r-th

layer.

We have

nk

Xk(nk)

~

~j~~Yk ^(~,

P

r,

Illr) ~j (Sk)

~~~~

r=1

separating,

_as

usual,

the center of the G-cluster _s~,

occupied by

_an atom _or not, to the _rest of the G-cluster.

3.2.I

Unique inflated layers

series.

Here,

the

assumptions

_are :

I)

for lw km t,

n~ is

independent

of k and

equals N, 2)

for I wrwn~, m~ is

independent

of _r,

3)

for

I _{w r w} n~

I,

_we ^have _p~

~ i ⁼ Tp

~,

4)

for I _{w r w} n~ I and each

§

=

,~~

e

lay~ (r,

_p

~,

m~),

^{there exists} _yy e

lay~ (r

₊

I,

_p

~~ i, m~~ _j

)

such that

yj

s~ = T

(x~

sk). Thus,

_we have _a series of G-orbits of

points,

of radii inflated

by

_{a power} of the

golden

_mean

T from the radius of the first

layer.

When the first

layer

is _an

icosahedron,

because of the existence of _a certain

density

of CN12

sites,

_we obtain _a sequence of concentric