Amorphous structures and incommensurate phases

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Amorphous structures and incommensurate phases

J.C.S. Levy

To cite this version:

J.C.S. Levy. Amorphous structures and incommensurate phases. Journal de Physique, 1985, 46 (2),

pp.215-223. �10.1051/jphys:01985004602021500�. �jpa-00209960�

Amorphous structures and incommensurate phases

J. C. S. Levy

Laboratoire de Magnétisme ^des Surfaces, Université Paris 7, 2, ^Place Jussieu, 75251 Paris Cedex 05, ^France (Reçu ^le 16 juin 1983, révisé le 29 juin 1984, accepté ^{le 9 octobre} 1984 )

Résumé.

Les phases incommensurables décrites par le problème classique de Frenkel-Kontorova sont com-

parées ^{ici aux} structures amorphes ^du point ^{de vue du conflit entre} différents groupes de symétrie. ^{Cette similitude}

rend la comparaison fructueuse et introduit un escalier du diable tridimensionnel pour caractériser les ^structures

amorphes. ^{Enfin on met ici en} évidence les propriétés d’invariance ^« self similar » des structures amorphes ^issues

de la symétrie icosaédrique, ^et donc leur caractère d’ordre à longue ^distance.

Abstract.

A comparison ^{is made} between incommensurate phases related with the well known Frendel-Konto-

rova problem, ^and amorphous structures, ^on the basis of a problem of conflict between different symmetries.

This similarity ^{leads to a fruitful} comparison of solutions with the introduction of a 3-dimensional devil’s staircase for the characterization of amorphous structures. Finally ^for amorphous structures which propagate ^an icosahedral symmetry, self similar properties ^are shown with evidence for a long-range propagation.

Classification

Physics ^Abstracts

61.40D - 64.80G - 61. 0D

Introduction.

In the recent years the words solitons, amorphous

structures, low energy excited states, and low frequency

noise have become strongly associated, ^{and there} is strong evidence for common behaviours [1]. ^How-

ever, first most of the exact results on these problems

come from the solution of the one-dimensional Frenkel-Kontorova problem ^{and related} problems [2],

while amorphous structures are three dimensional, and moreover, most of the work on amorphous

structures neglects ^the energy problem, ^{at least in a}

first step. ^The great many ways of building ^dense

random packing, ^D.R.P. [3], ^{more or less} experimental,

which are suggested ^as typical ^{units of} amorphous

structures seem far from the solution of a Frenkel- Kontorova problem. ^{However, there is a way for} minimizing ^the 0 K energy of amorphous ^structures by ^{means of a} variational method [4] ^{which is ana-} logous ^{to the usual} Euler-Lagrange derivation of

Lagrange’s equations, and this method leads to the effective construction of amorphous structures [5].

This method can be compared with the solution of the one dimension Frenkel-Kontorova problem, quite directly, ^{and this is the} aim of this _paper.

First of all, ^{the FK} Frenkel - Kontorova one-

dimensional problem [6] ^{is raised} by ^a conflict between two interactions, ^{i.e. two} potentials. First there is an

elastic interaction V between neighbouring ^atoms,

and second each atom is directly ^{submitted to a}

periodic potential U, ^{which can} represent ^different interactions. Thus, ^the conflict between interactions V and U is external, and is rooted on the different

symmetries ^involved by the interactions. Interaction V alone leads to a periodic solution with an arbitrary

lattice parameter b, while interaction U alone leads to a periodic solution with a fixed lattice parameter ^{2 a.}

Are a and b commensurate ? In other words, ^{are the}

internal symmetries ^{of V and U} compatible ? Many

other structural problems, ^{such as in} magnetism ^for instance, ^{are raised} by the conflict of different poten- tials, ^or of different parts ^{of the} potential, ^{which is} actually ^a conflict between the internal symmetries

which can be associated with these parts of the inter- action. We call these conflicts « external » because they

arise between different explicit parts ^{of the} potential.

For amorphous structures, there is ^no such simple,

direct _way to distinguish ^between parts of the Hamil- tonian, but the conflict of symmetries is clear : systems with a small number of particles ^can be invariant under local point symmetries only, ^while systems with an infinite number of particles ^{can be trans-} lationally invariant. Thus quite ^{different states of}

symmetry can occur. We will speak ^{in this case of an}

« internal » conflict because of its implicit, ^less

apparent character, mathematically speaking. ^These

comments show that this distinction between external and internal conflicts is a weak _one, and that the

comparison is sensible and thus probably ^fruitful.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004602021500

216

Then the comparison ^between amorphous ^struc-

tures and FK problems ^bears upon methods, ^i.e.

variational methods and Poincar6’s mapping ^which

both result in a writing ^{of the} equations ^{of motion}

and lead to a local field concept. Finally, ^Poincar6’s mapping propagates ^{an initial} condition, ^{i.e. a boun-} dary condition, while the equation of structural

propagation obtained for amorphous structures [4]

can be used to propagate ^{a seed} structure. The next

point of view for comparison ^{bears on} symmetries

which are, of _course, different in a one-dimensional

problem ^{and in a 3d one. In} one-dimensional problems

there is commensuration or incommensuration of lattice spacings, ^i.e. translational periods, ^{one of} them, a, is due to the potential ^{U and the other, b,}

is locked by boundary conditions and an elastic interaction described by ^the potential ^{V. In 3d} pro-

blems, point symmetries ^and translational symmetries

are the possible opponents. ^{Here it} may be pointed

out that the _group of symmetry Yh ^{of the} icosahedron, which is important ^for the local point ^{of view, is} isomorphous ^{to a} subgroup ^{of the} group of symmetry Q6 ^{of the} simple ^{cubic structure} in dimension 6 : SC6

which is compatible ^{with a} group of translations.

In other words, ^{there is a} fibration which projects

this 6d crystalline ^{structure onto a 3d} amorphous

structure, and this fibration is defined by symmetry considerations. Finally, it will be seen that this 3d

amorphous ^{structure is} nearly self-similar, ^{i.e. that}

similar parts appear ^at rather regular ^intervals,

which are given by commensurate ratios. This pseudo periodicity ^can be called a locking oj’ amorphous

structures, in analogy ^{with the} commensurate pro-

blem, and defines localized areas of conflict in the intermediate _range. Thus these ratios define a cohe-

rence length of this 3d structure.

In section 1 different conflicts will be analysed

in terms of the symmetries involved. Section 2 com-

pares the methods of resolution, while section 3 is devoted to remaining symmetry ^{and its} result, the locking ^of structures.

,

1. The model of Frenkel and Kontorova, FK, ^and other models of external and internal conflicts.

The _energy in the FK model [2] ^{is :}

it applies ^{to a chain of atoms} of coordinates _ui sub- mitted to an elastic coupling ^V = 1/2(u1+1 - ^ui)2

of unit string ^{constant and to a} periodic potential U(ui) ^with amplitude À ^and period 2 a, ^U

(,h/2) [1 - ^cos (7rui/a)], ^{J.l is a} tensile force applied ^to

the ends of the chain. The elastic coupling ^{does not}

determine a period; ^{when A}

0, ^{there is a} periodic

arrangement :

and i.e.

where C is an arbitrary ^constant. Thus, the tensile force defines the second period ^b

p. In other

words, ^the external conflict between two different

potentials ^is arbitrated by ^the boundary ^condition.

Here two kinds of symmetries ^are conflicting : ^one

translation of given period ^{and one} translation with

a period ^the length ^{of which can be fitted} according

to the boundary ^condition.

Similar problems ^{are usual in} physics. ^For instance,

in ferromagnetic samples ^{there is a} distribution in

magnetic domains which results from a conflict between exchange interactions and dipolar ^ones.

In this conflict different parts of the interaction are

opposed. ^{Thus, this is an «} external » conflict as the FK model. When the exchange integral ^{is a} rapidly fluctuating variable of _space as in RKKY inter- actions [7], ^{there are numerous} ferromagnetic couplings ^and antiferromagnetic ^{ones. In this case}

there is a conflict between terms issued from the

same kind of interaction. It is convenient to call this

an internal external » conflict since the conflict is exhibited by ^different parts ^{of the} Hamiltonian, which, ^{as a matter of fact} belong ^{to the same class of}

interactions. A similar « internal external » conflict

occurs for spins interacting ^via dipolar interactions

only, ^{or via truncated} dipolar interactions [8]. ^Besides

this more internal series of external conflicts, ^there

are internal conflicts, ^i.e. properly speaking, ^{where the}

source of conflict lies in the geometry, ^{and the} potential

arbitrates the conflict. Such conflicts arise when the

geometrical dimension is larger ^{than 2 and occur}

because the optimal ^local configuration ^cannot freely propagate ^{in the whole} space. This internal conflict leads to the different crystallographic structures and to amorphous structures as well.

These conflicts have been introduced on the basis of interaction terms which _may be in conflict with other interaction terms, ^or similar interaction terms or geometrical requirements. ^{It is useful to note the}

symmetry ^{conflicts as} usual when speaking ^of phase

transitions. In the FK model there is a conflict bet-

ween two subgroups ^of translation, ^{i.e. between}

two periods a ^{and b,} with the main question ^{of com-}

mensuration between a and b. In the case of domains, exchange interactions are invariant under a large

class of symmetry operations ^while dipolar ^inter-

actions are not invariant. Generally, ^{there are} point

symmetry groups which leave invariant the local interactions, ^but they ^{are not the same} everywhere

and the resulting geometrical conflict is often called

« frustration ».

In the specific internal conflict which leads ^to

amorphous structures among other crystallographic

structures, it is necessary ^to define a simple ^model

with crude assumptions, and later more details will be introduced as corrective terms. Thus, ^{a set}

« A » of basic assumptions ^{to be used a} first time is

i) ^a Hamiltonian with only pair potentials, ii) only

one kind of particles, ^{which are} spherical, i.e. with a

radial pair potential, iii) ^a short-ranged interaction

occurring ^{for nearest} neighbours only, ^the only para- meter being the width w of the potential ^{well of} depth unity ^{for a} unit distance which is the optimal ^nearest neighbour distance. With such an interaction, obviously ^{the most stable structure is the one with}

the highest connectivity ^where only neighbouring

atoms are connected. Thus, ^{we define the} optimal connectivity Zo ^as the maximum number of identical atoms which can surround by ^{close contact a} given

atom, i.e. the ratio between the total solid angle ^Q

and the solid angle ^{Q’ subtended} by ^{atom 0’ seen}

from the neighbouring ^{centre 0.}

In the Euclidean d-dimensional space ^{of centre 0} and axis Oz, ^a running point ^{M on} the unit-modulus

hypersphere ^{centred on 0 defines an axis OM.}

Let 0 ^{be the} angle between Oz and OM. Because of similarity, the element of area on a hypersphere

around M is dO

Ad-1 (sin l/J )d- 2 do, ^where Ad

measures the area of the unit-modulus hypersphere

of dimension d. The total ^area of the hypersphere ^of

unit radius, ^i.e. the total solid angle Sl, ^{is obtained} when 0 ^runs through [0, n],

Let 0’ be the centre of a neighbouring hypersphere

which is tangential ^to that centred on 0, ^{and P the} running point of this unit-modulus hypersphere

centred on 0’. In the plane 00’P, ^{line 00’ and OP} make an angle 0 which is less than or equal ^to n/6.

The solid angle (1’ in which is seen the atom (0’, 1)

from 0 is

Thus, ^the optimal connectivity Zo ^{reads :}

Equation (6), ^valid for d > 2 gives :

For d

2, ^the optimal connectivity is reached by ^the honeycomb ^{structure for all} particles. This remark

explains ^the exceptional physical abundance and

stability ^{of this structure} among the _{17 space} groups of two-dimensional paving [9]. ^{For d =} 3, ^{such a} Zo(3) ^cannot be reached. A high ^{value of} connectivity

for the central atom, ^{Z =} 14, ^is reached for a b.c.c.

cluster B of 15 atoms, ^{where two} competing ^nearest neighbour distances of ratio 2//3 ^{= 1.155 have to be}

admitted simultaneously, ^{and where true hard} spheres overlap. ^{In other words, such a structure can be stable} only for smooth potentials ^{with w} larger ^{than 0.155.}

The pseudopotentials ^to be considered when dealing

with metals become smoother and smoother when the temperature ^{is increased} [10], typically ^because

energy fluctuations of about kT are allowed at the atomic level. Thus, the smoothness of these pseudo- potentials ^{is the reason for the} stability ^at high ^tem- perature ^of many b.c.c. metals [11]. Then different ways of putting Zo

^{12 atoms around a central one}

can be found. The most symmetrical ^{ones are a f.c.c.}

unit F, ^a hexagonal ^unit H, ^an icosahedral unit I and

a pseudoicosahedral ^{unit P with a mirror} symmetry with a plane perpendicular ^{to the} pentagonal axis, instead of the central symmetry ^{of I. Without} dealing

with the problem ^{of clusters,} which has been studied

by many people [12], ^{we must notice that I is the most}

stable cluster _among them if an isotropic contraction of the internal atom of 0.036 and anisotropic ^dilation

of the peripheral ^{atoms of 0.04 are admitted. In other} words, ^{this is a} requirement ^{on w to be in the} vicinity

of 0.076. If such contraction is not possible, ^{i.e. w = 0,}

then F is the most stable configuration, ^{while H is}

favoured by ^some anisotropy.

In the light ^{of the} previous ^{comments on FK models}

and conflicts, ^a particular interest lies in the com-

parison ^between symmetries. The interaction described in assumptions ^{A is} rotationally invariant. The best local solutions proposed have different symmetries.

The b.c.c. clusters B and f.c.c. F are invariant under the symmetry group Oh ^{of 48} elements. The hexagonal

unit H is invariant under the symmetry group ^T of 24 elements, ^{while I is} invariant under Yh, ^the sym- metry group of the icosahedron which contains 120 elements. It can be noticed that all these _sym- metry groups contain C3 ^{axis, and thus common} subgroups. Moreover, ^{some of these} symmetry groups,

namely Oh ^and Td ^are compatible ^with translations,

while Yh is not; in other words, ^the crystalline repeti-

tion of b.c.c. B, ^{f.c.c. F or} hexagonal ^H clusters is

possible and, ^{as a} matter of fact, ^well known, ^while that of icosahedral units is not possible, ⁱⁿ spite ^{of its} being ^{for some} potentials ^a stable local configuration.

Quite obviously ^this symmetry conflict leads to

amorphous structures.

218

2. Comparison ^of resolutions.

2.1 BULK EQUATIONS.

The principle ^of Aubry’s mapping ^is simply ^to write the classical equation ^of

motion [2], where with Hamiltonian (1). ^One gets :

and then writing ^this equation ^{in a 2 x 2} matrix form

which can also be read as :

Equation (10) ^{is a} discrete translation of the sine- Gordon equation ^for stationary displacement ^{u. This}

is a relation between FK problems and soliton ^ones.

Equation (8) is also the result of ^a variational method when applying ^{the variation}

where c is an arbitrary ^{low constant and} bm,i ^the

Kronecker symbols. ^As equations ^{of motion are}

derived from a variational treatment of Lagrangian,

i.e. the least action principle, ^{there is an ab initio} equivalence ^between variational method and Poin- car6’s mapping. Precisely, equation (8) expresses that the resultant of forces applied ^on point ^{i is nil, i.e.}

there is an exact balance which is a «local field»

concept : the local tensile forces _{Pi + 1} balance the local field (Àn/2 a) ^sin (7rui/a).

The mapping equations (9) ^define (Ui+1, Pi+1) ^{as a}

function of (ui, pi) by ^{means of a} generalized ^transfer

matrix T of which coefficients depend ^on ui. By interaction, (ui, pi) ^{is a} function of (uo, po) ^{and this}

defines a discrete trajectory ^{in the} phase space (u, p).

A question ^{must be raised as to what} happens ⁱⁿ

the problem of internal conflict described by ^the

Hamiltonian :

with ni ^{the local} density of atoms, and V(r) ^the pair potential which defines the set of assumptions ^A.

The results of ^a variational method using local varia- tions similar to these of equation (11) ^is [4] :

where nk ^and Vk are ^the respective ^{Fourier transforms}

of the density n ^{and the} pair potential ^{V. The calcula-}

tion has been done in detail from a finite element

approach ^{of the} density ^{and from a continuous} approach also in reference [4]. ^The previous arguments

suggest ^that equation (13) ^{is a «} local field » equation,

valid when taking ^{into account} all anharmonic contri- butions of the pair potential ^V. Equation (16) ^is easily

solved in real _{space :}

with

The kj’s ^are the nodes of the Fourier transforms of the

pair potential, ^{and the} c/s ^{are arbitrary constants.}

Thus, equation (4) expresses the propagation ^{in the}

whole _space of a structure defined by ^the c/s, ^while equation (10) in their matrix form _expresses the _pro-

pagation ^of (uo, po) in the whole _space. The structure has an exponential, wave-like, propagation ^defined by equation (14) ^while Aubry’s trajectories propagate according ^{to the} powers of the transfer matrix T.

Another comparison ^can be done. The variational method for pair potentials may be used in the FK

problem when there are only pair potentials, ^i.e.

for A

0. Then the Fourier transform of the elastic

potential ^{reads :}

if V is not restricted to a short _range, and

if V(x) ^{is cut at x}

^{± a’. In both cases the} only

is ki

^{0 and} equation (14) ^{reads :}

This means any Id lattice spacing ^{b is} admissible,

i.e. b can be continuously varied, according ^{to boun-} dary conditions for instance. In other words, ^this introduces the first remarks on FK problem.

2.2 GLOBAL CONDITION

STABILITY.

For the FK

problem, the recursive equation (14) ^{is a bulk} equation

and there are boundary equations ^at the ends of the linear chain. At the first end, ^the boundary equation

links _uo and _{po ;} thus, ^{one can obtain} _po ^{as a function} of uo. Then the bulk equations ^{enable us to calculate}

uN and _PN as a function of _uo. Finally, the second

boundary condition links _uN and _PN, and thus, ^{can be}

written in the following ^{form of a} global condition :

where F is a characteristic function which determines the allowed initial values _uo, and can be written

explicitly with Hamiltonian (1). ^{There is a countable}

set of uo which fulfill equation (17) { ^{u x} p }. Equation

(17) ^{involves a} cascade of N imbricated sines :

as typical ⁱⁿ non-linear problem, ^{it is a} transcendental

equation which admits _many solutions which define

a countable ^set since F(uo) ^{is not} identically ^zero.

As a matter of fact, ^another meaning ^{can be} given

to the functional F(uo). ^The (N ⁺ 1) particular equa- tions of motion 00IOui

⁰ involved in equation (8)

are resolved in N relations _ui ⁼ fi(uo) (17a) ^with

i ⁼ 1, ^{2... and one} equation ^on uo : 0010uo

⁰ (17b).

When using equation (17a), ^the problem ^{reduces to}

a 1-body problem characterized by 0(uo) = 0(fi(uo)),

with the simple ^result

where equations (17a) ^and (17b) have been taken into account. Quite obviously, characteristic equations (17)

and (18) ^{are identical since} they ^{result from the same}

set of equations (8) ^{resolved in two different} ways.

We can take advantage ^{of the} potential ^nature of 0 by saying ^{that for a stable} configuration 0 ^{is minimum}

and thus

This is closely ^connected with the results of Lyapounov [13] ^on conditions for stability. ^{As a matter of} fact, the global equations (17) ^or (18) result from the

compatibility ^{of the} (2 ^{N +} 2) equations ^{of motion} (9)

over the (2 ^{N +} 2) ^variables u,,, pn. Thus 2 N + 2 glo-

bal equations ^{can be} written, namely /J(un)

^{0 or} t/J(Pn)

^{0. In other words,} any particle ^can be selected

as the relevant body of the effective 1-body problem.

Due to the non-linearity ^of 0, ^which appears in the cascade of sines for instance, ^{there are at least N} solutions for equations (17) ^and (18). ^{When N becomes} larger ^and larger, ^the average distance between the solution _uo decreases as N -1, thus dense ^sets of solutions _appear either on isolated points ^{of accumu-}

lation or stretched on bands, and isolated points

occur. The number of isolated solutions compared

with the number of accumulated solutions is of order of a/N, ^{where a is a} minimum distance between isolated solutions. As a matter of fact, ^this spectrum of solutions is parametrized by ^{one continuous}

parameter, the tensile force _{p. For} a given tensile force J.1a let us assume that (Uo)a is the stable solution. When

is slightly varied from _ga, the stable solution (uo) ^must

remain as close to (UO)a ^as possible, thus, ^if (uo)a belongs ^{to a series} converging ^{towards an accumu-}

lation point, (uo) belongs quasi every time to the same

series, ^{and if} (uo)a belongs ^to the interior of a band, (uo) ^{does too. Thus, the} variation of the tensile force ,u

enables us to explore ^{the spectrum of} (uo), ^{and the} previous description ^of bands, accumulation points

and isolated points ^{defines a} devil’s staircase for the

curve MoM [2]. ^This description ^{of the} spectrum of (uo) corresponding ^to metastable configurations, ^{into bands} D, accumulation points ^C, and isolated points P,

leads to the existence of excited states of negligible

energy in the cases C and D and not negligible ^for

isolated points ^P.

When going ^{to the 3d} case, ^a first technique ^{is to}

consider by analogy ^three independent ^FK problems

on (ui, via, Wi) with three independent potentials ^and boundary conditions. The resulting characteristic

equation ^{reads :}

where G(vo) ^and H(vo) ^are functions similar to F(uo).

The previous result for the solution of equation (20)

means for the solution Uo(uo, vo, wo) ^of equation (21)

that when N and N 3 are large ^{there are} many dense sets of solution Uo ^and possibly only ^a few isolated

points. ^For given ^8, which defines the minimum distance between isolated points, ^and N,,, ^{the number}

or particles, the relative number of isolated points compared with the number in dense series is weak, and weaker in 3d than in Id. This remark does not

really depend ^{on the} shape ^of potentials, if anharmonic, and on the boundary conditions. Thus, ^it may be

applied ^{to our} problem ^defined by Hamiltonian (15),

where it justifies ^{the use of a} variational method with some approximations, because this leads to an

approximate definition of Uo, ^{i.e. the} trajectory ^{or the}

3d structure which remains close to a dense series of

{ Uo }. ^Later realistic relaxations with convenient

boundary conditions will _converge rapidly ^because

the distances between these { Uo } ^{are small.} Changes

in boundary conditions can be used, ^as previously changes of tensile force p ^{for FK} problems, ^to explore

these series of configurations. These variations can be done either on initial boundary conditions, ^{i.e. on the}

variational solution, ^{or on} boundary ^conditions

of the relaxation. Other variations can be brought ^on

the solutions of the variational method by adding

interstitial atoms for instance, ^or displacing ^some

atoms, and then relaxing ^the system. During ^this analysis, many metastable configurations, ^with nearly

the same energy per particle, ^{are obtained} [5]. ^The quasi absence of a gap of excitation _energy means

that the considered { Uo } ^{in reference} [5] ^{are not}

isolated points.

On a practical ^{level, the} question ^{is how to choose}

the cj’s ^of equation (14) ^{in order to obtain a 3d structure}

rather close to metastable configurations, ^{and as there}

are many possible ^such choices of { cj }, ^{how to choose}

a simple one, i.e. tractable. Such a solution has already

been published [5]. ^{Thus, we} just ^{want here to define}

the basic practical rules for such a choice. They ^{are :} i) Neglecting ^the cj’s ^{of large} k. As the goal ^{is to}

obtain a full 3d structure, which has in any ^case spatial

Fourier transforms with amplitudes decreasing ^accor- ding ^{to a k- 2} law, ^the _cj ^{decrease as} kj ^2, contributions of high k ^{can be} neglected ^{in this} approximation.

This restricts the practical kj’s ^{to a few} concentrical

spheres. Basically ^{it is an} hologrammatic approxima-

tion [14].

220

ii) ^Choice oj’the c/ s ^{according to a boundary condition.}

iii) ^{Return to a discrete} density nd(r) by selecting ^the

maxima of n(r) ^defined by equation (14) for the sites of the atoms. These maxima are discrete, but too close

together, thus, ^{in order to} obtain a convenient density,

we shall choose some threshold value _no to be deter- mined self-consistently, allowing ^a possibility ^of overlapping ^atoms. Then in such ^cases of conflict,

the highest n(r) will define the site retained. This’

process defines conflicts which in this first step ^are solved in a deterministic _way, but which can be later solved in different _ways in order to obtain some series of metastable configurations ^as previously suggested.

As noticed before, ^this process defines a structure and its possible corrections.

The next point is the choice of a boundary condition, which can be defined from mathematical or physical

reasons. The mathematical argument ^{used when} establishing equation (20) ^{for the FK} problem ^{is to}

introduce a rather arbitrary initial condition _uo and then to derive all _ui from the propagation equation,

in order to obtain at the other end an equation ^{on the}

tensile force. This means propagating ^{an initial}

structure, and then checking ^the external forces on the

resulting structure. For initial structure, either ^a local one, i.e. a germ ^{or a} two-dimensional _one, much as

in epitaxial problems, ^or why ^{not a} one-dimensional germ, ^seems rather interesting ^for constructing ^amor- phous structures, ^or crystalline ^ones. By ^the way, in the last sentence mathematical conditions have found their physical interpretation : boundary ^conditions

are read as germination conditions. Moreover, ^the principle ^of introducing ^one boundary ^condition

in the bulk equation ^{and then} checking ^{the self} consistency with other boundary conditions, ^which

was already ^{used in} establishing equation (20) ^{for the}

FK problem, ^is quite natural for frustration problems

where there is a global balance between local contra- dictions.

Now _among the great many clusters to be consi- dered, ^the previous ^{comments on solid} angle problems

suggest ^the study of the b.c.c. unit B of 15 atoms, the f.c.c. unit F, ^the hexagonal ^unit H, the icosahedral unit I and the pseudoicosahedral ^{one P.} B, ^{F and H} give ^{rise to extended} crystalline fragments, ^when using ^the previously developed algorithm ^for n(r) ^and nd(r). ^{The most} important cluster is obviously ^the

centred icosahedron I which is the stable local struc- ture for a large category ^of pair potentials, ^as already analysed, ^{and has been a central} object ^{of interest} [15],

with a 3d Fourier transform which shows well defined

peaks ^{on the first zones} of ^the reciprocal space. The

simplest ^{choice of the} c/s according ^{to this} boundary

condition consists in selecting ^the peaks ^{of one or}

several ^zones of the Fourier transform of I for k, ^of

non-zero cj ^and using ^the amplitudes of these Fourier transforms ^on these peaks ^as the values of cj. ^There

are two direct proofs ^{of the} quality ^{of such a} process :

firstly trying ^{to use other} zones, secondly relaxing

these structures. Other proofs ^are given ^{in the next} chapter.

The dual of the icosahedron is a dodecahedron which _appears in the Fourier transform of I, ^{but on}

some central zones of the Fourier transform of I, there are only ¹² peaks ^{located at} the summits of an

icosahedron of central distance k. Let us define the reference system ^{of I} by ^its origin ^{0 at the} centre, ^the axis Oz which links 0 and one of the summits and an

other summit which by definition of x belongs ^{to the} plane Oxz. Then the practical ni (r) ^{reads :}

with

and

z is the classical golden ^number [9, 16], ^{which is} strongly connected with the icosahedron. In the

practical determination of the ML structure [5], ^the

threshold value useful for _n was ^9.

The ML structure [5] ^was constructed from the set 8 of maxima of n(r) which have a maximum of nl(r) higher ^{than 9. 8 contains} points ^{too close} together

which occur because of conflicts to be discussed later in this _paper, thus additive rules ^were used to define

unambiguously ^{the ML} structure. Such rules _were,

i) ^to classify 8 according ^{to a} spatial ^{order of} scanning, ii) ^{if two or more} points ^{of 8 are} closer than some

value, ^{i.e. 0.9 atomic} diameter, ^to select the point

which gives ^{rise to the} highest ^{maximum. In reference} [5], ^some variations of this ML structure were also studied and shown to be stable under different relaxa- tion processes.

3. Symmetries, similarities of the amorphous ^structure

and lock-in effects.

As already ^stated, nl (r) written in equation (21) ^{is the} simplest n(r) satisfying ^{both the} general ^{form of} equation (14) ^{and the} boundary condition of germi-

nation with I for the _germ. Practically, nl (r) is invariant under the symmetries ^{of the} group Yh of the icosa- hedron which has :

-

6 axes of fivefold symmetry,

-

10 axes of threefold symmetry.

-

15 axes of twofold symmetry

and a centre of symmetry, ^{i.e. 6} x 4 + 10 x 2 + 15 + 1

60 direct isometries, and inasmuch inverse iso- metries, i.e. 5 ! : 120 distinct symmetry operations.

Of _course, this symmetry group is well known by

mathematicians [16, 17]. ^The stability of the icosa- hedron I is due to this high ^{level of} symmetry

^for

comparison Oh ^has only 48 elements. Yh ^{has the same}

number of elements as the _group of permutation ^{II. 5}

of 5 elements which is the quotient group of the _group of permutation ^{of 6 elements} P6 by ^the cyclic operation.

This strong connection of Yh and P6 ^{leads one to}

compare Yh ^{to the} point symmetry Q6 ^{of a} generalized simple hyper ^{cubic structure} SC6 ^{in a Euclidean} space of dimension 6. Quite obviously, each axis of SC6

is a fivefold axis of symmetry, because the 5 other

axes are equivalent, ^{and there are 6 such axes, i.e.}

6 axes of fivefold symmetry. They ^are C’

20 ways of taking ^{3 axes among the 6 axes of} SC6, ^{and thus}

10 ways of selecting ^two conjugated couples ^{of 3 axes,}

which will define threefold symmetry, ^{and there are}

C2

15 ways of taking ^{2 axes} among the 6’s of SC6,

and to define twofold symmetry. Thus, Yh ^is simply

related to a subgroup of Q6 ^{and this defines a} mapping

of R6 and SC6 ^{on the 3d} space which conserves most of the symmetry properties. By analogy ^with equa- tion (21), ^{the 6d} density n6(r) ^is introduced with _xi, i

1 to 6 as coordinates :

n6 oscillates between 12 and - 12. Maximum values 12, ^are obtained when xi

² np%k, ^with pi integer, ^for all i, and minimum values - 12 are obtained when xi

(2 qi ⁺ 1) n/k ^with qi integer, ^{for all i. Other}

maxima derived from V.n6

⁰ satisfy ^to xi = nrJk

with _ri integer ^{for all i.} Thus, ^the process of selecting

the maxima of n, or I n defines unambiguously simple hypercubic structures SC6 ^{of R6.} The maxima of n ¹

are issued from those of _n6, i.e. from one SC6. ^The

threshold level for _nl prevents ^{us from} taking ^{too dense}

networks in 3d _space.

4. Symmetries.

The ML structure is invariant under the symmetry operations ^of Yh, ^since nl (r) is invariant under these

operations, and thus the places where its maximum values are reached form a set, ^a « dust _», invariant under Yh ^{with the same maximum value.}

5. Similarities and approximate similarities.

All previous symmetries ^{have the same centre located}

at the origin. Thus, ^the question ^{is : are} they points ^of

the ML structure which are nearly ^{in the same condi-}

tions as the origin ? ^A glance ^at the numerical results of reference [5] ^{convinces one that} probably ^such points

do exist, and that some of them belong ^to the axis Ox.

If _so, their _n, is nearly equal ^to 12, ^{as at the} origin,

and their only non-nil coordinate x is submitted to the conditions from equation (21) :

with _v, v’ and v" integers. These three conditions,

which are necessary and sufficient for having ^a point (x, 0, 0) ^{similar to the} origin, ^{lead to two distinct}

conditions only : ^{v and Vt both even} integer. ^These

conditions are approximately ^fulfilled by ^{the double}

of the Fibonacci series [9, 26] ^{with for v} the values : 0, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, ..., where the _pro-

gression ^{is obvious} Vn+1 = vn ⁺ Vn-1.

By ^difference

between (23a) ^and (23b) ^{v is even, and from} (23b),

v appears in the development in continued fraction ofr,

as a Fibonacci number fn ^{does : lim} fn + 1 = r [16].

n-+ 00 fn

The occurrence of such Fibonacci numbers in physical growth ^of germs ^as well as in phyllotaxis ^{is due to the} aggregative character of the equation : fn + 1 ’-fn +fn - 1.

In table I we plotted Av’, ^the non-integer part ^of v’, which is equal ^to Av", ^{and cos} (2 n Av’), ^{which measu-}

res the difference between this point ^{and the} origin. ^It

appears that already for v" ^{= 16, the} phase ^{shift Av’}

is weak and cos (2 ⁿ Av’) ^{too. These} phase ^shifts

decrease when n is increased. Thus, equations (24)

define an infinite series of points Ai fully equivalent

to the origin. ^{Near these} points Ai ^the phases

nearly ^{with the} coordinates, thus, ^a translation with vector OAi ^{leads to a} change ⁱⁿ phases ^of nearly ^{2 n.}

The approximation, nearly ^{2 1t,} is valid in a domain of centre Ai and radius OAi. This remark confirms the

progression of the Fibonacci series, and proves the

quasi-self-similarity ^{of the ML} structure, when taking

into account at the same time the symmetry group Yh,

these quasi-translations ^{and the} products ^of symmetry operations ^of Yh by quasi-translations.

Table I.

v, Ov’ and cos (2 ⁿ A v’) ^{when v =} f,, ^with jor j" the Fibonacci series.

222

The term ^« quasi-translation » emphasizes ^{the local}

character of these translations, ^{i.e. the} occurrence of conflicts between these operations which leads to bad interferences for _nl, and low values of the local maximum of n (r). ^{Thus we found an other} description

of the conflict : when such bad interferences occur, there can be sites of maximum values of nl (r) ^{too close} together, and that is really ^{what occurs} [5]. ^{It defines} places for «localized conflicts. At points ^{0 and} Ai,

there is no conflict, but between them, ^and especially

in the mediatory planes ^between Ai ^and Ai, ^{the phases}

can differ from an integer ^{number of 2 n.} This defines

polyhedra of conflicts. These first quasi-translations

occur with A1 (10 ^{x 2} n/k1, 0, 0) A2 (16 ^{x 2} n/k 1,0,0), thus, ^one may expect ^for the radius of these polyhedra

of conflicts : 5 ^x 2 7r/k, ^{or 8 x 2} n/ki and ^{so on. The}

first value is consistent with that obtained for the location of conflicts from the calculation of local elastic coefficients in this ML structure [18] , ^{6.8 x}

2 n/k ~ ^{5.4 x 2} n/k1. ^{Of course,} another value can

be obtained from Ao (6 ^{x 2} n/k1, ^{0, 0),} but in this case

the phases of Ao ^{are not} exactly integer ^{numbers oaf 2 n}

and the corresponding ^{conflict at} Ao/2 ^{is not a hard}

conflict but a smooth one, ^as observed from the calculation of local elastic coefficients [18].

Thus we have described an ideal structure where the local order of the symmetry group Yh propagates through ^the sample far from its original germ and is locked-in at places Ai. ^This process extends this local order into a long range order, by ^{means of}

« quasi-translations ^{». This ideal structure has well}

known places of conflicts which _appear either as holes of rather large radius, ^{or as} interstitial atoms. Thus, such a structure is stabilized by the introduction of a

non-negligible quantity of smaller atoms, ^as is the case

in practical amorphous ^metallic alloys [19].

The ML structure has a long range order defined

by ^the points { Ai } ^{which as a set are invariant under} Yh. Yet the location of conflict, approximately ^defined by {Ai/2}, ^{is not} precisely defined. As a matter of fact, ^we defined ^an operational algorithm which leads

one to select one structure with some holes, ^{i.e. some}

solution for local conflicts [5]. However, ^{the shift} of these conflicts or holes by ^one atomic diameter

causes negligible differences in _energy [15]. ^This

process defines a large number of low _energy excited states [5], ^{which can be} associated with low frequency

noise.

A question arises : does this structure depend ^on

the choice of (kj, cj) ^{made in equation (21), where these} kj, Cj ^{act as} the initial point uo of configuration ?

The previous ^{comments on} the different solutions of local conflicts show that _uo is not an isolated point

and that there are many metastable structures involved

by small discrete variations of _uo, i.e. (kj, cj) ^which

can be called a « packet » ^of structures in analogy

with the « wavepacket ^». Examples ^{of this ML} packet

have been already ^studied in reference [5] : filling ^of large holes with atoms, relaxation with different pressures and ^stresses, and can be easily ^extended.

Thus, ^the question ^{reads :}

i) ^{are other} structures of the same « packet » ^less ideal, i.e. without infinite long range order, ii) ^{are there}

other packets ^of structures disconnected from this one

and derived from the same boundary conditions.

The answer to the second question ^{can be} given ^{in a}

naive _{way :} since all this packet ^of structures is « oscu-

latory » ^{to a same} initial cluster I, there will be at least local connections between such structures. To the first question ^{there is a first answer} derived from the

projection problem Q6 --> Yhl ^{which means there is}

some universality ^to this structure, but there ^are many such local projections and this remark is more con-

nected with question ii) ^{than to} question i). ^Direct

answers to question i) involve other values for k’s in the same zone and of other k’s in different zones.

If the ki s ^{are in the same zone, they can} be classified into _groups of icosahedral symmetry ^{with a small} deviation dk of k. This will give ^{rise to other sets} { A’i }

of { Ai}, ^{the «} coincidence » points. ^{For small i these}

sets { A’i} ^and { Ai} ^are nearly ^{the same, but for} larger

ones the disconnection becomes clear, typically ^for

values of such that i dk/k ^{be an} integer. ^{This remark}

shows the coherence of such an ideal structure to be function of the admitted dk. On the other hand, ^when considering ^k’s from other _zones, similar Fibonacci sets { A"i} ^will appear because of the same link with

Yh, but their difference with the sets { Ai} lies in the

phase shifts which occur between the k’s, ^{and these} differences lead to disconnections of the Fibonacci sets for larger ^{g values of i such that} 7r ^{n be an integer. g}

As before, this demonstrates that in the same « packet »

of structures there are more or less ideal, ^{i.e. more or}

less partially long-ranged, amorphous structures, where this « idealization » takes place ^{in locked-in}

sites while other parts ^{are more} disordered.

6. Conclusion.

The comparison ^of commensurate phases ^and amorphous structures of icosahedral symmetry ^is fruitful since it deals with similar conflicts, ^{more or} less internal or external, of different dimensionalities, and operates with similar methods : Aubry’s mapping

or equation of structural propagation. ^{It enables us to}

demonstrate the fractal character of amorphous

structures of icosahedral symmetry. The intermittences of this fractal structure define the places ^for stabilizing

smaller atoms. The spatial homogeneity ^{of the}

distribution of these defects is responsible ^{for a} power law of correlation parameters ^{as a function} of distance.

Acknowledgments.

The author wishes to thank _many colleagues ^for helpful discussions, ^and _among ^them especially ^Prof.

P. J. Steinhardt, ^from University ^of Pennsylvania,

Dr S. Aubry ^{and Dr R.} Visocekas, ^from University

Paris VI, ^{and Dr J. M.} Arnaudies from University ^of

Toulouse, and the referees of the previous ^versions.

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