The geometrical nature of disorder and its elementary excitations

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The geometrical nature of disorder and its elementary excitations

M. Kléman

To cite this version:

M. Kléman. The geometrical nature of disorder and its elementary excitations. Journal de Physique,

1982, 43 (9), pp.1389-1396. �10.1051/jphys:019820043090138900�. �jpa-00209519�

The geometrical ^nature of disorder and its elementary excitations

M. Kléman

Laboratoire de Physique ^des Solides, Université Paris-Sud, ^Bât. 510, 91405 Orsay, ^France

(Reçu le 20 avril 1982, accepté ^{le 28 mai} 1982)

Résumé.

²⁰¹⁴

En s’appuyant

^{sur une}

analyse géométrique d’un ordre à courte distance frustrant,

^on

propose

^un

modèle de _corps amorphe composé de deux réseaux de Frank conjugués, l’un ^{formé de} disinclinaisons (dans ^la ligne des références [2] ^et [3]), l’autre de domaines cylindriques. ^{Ce modèle} peut ^s’étendre

^au

système ^de spin ^des amorphes magnétiques, ^{à la} phase ^bleue désordonnée des cholestériques et, ^dans

^sa

^version dynamique,

^aux

liquides.

On discute des conséquences ^{du modèle}

^sur

^le spectre des vibrations atomiques élémentaires :

on

explique ^ainsi

le comportement ^du type Bragg pour k

⁼

03C0/d, les modes de rotons, l’existence de modes localisés de basse énergie

conduisant

aux

lois

connues

relatives à la chaleur spécifique ^{à très basse} température.

Abstract.

²⁰¹⁴

We propose,

^on

the basis of

a

geometrical analysis ^of frustating short-range ^order (in the line of references [2], ^and [3]),

^a

model in which

an

amorphous body ^{is made of two} conjugated ^Frank networks,

^one

^of cylindrical domains, ^{the other}

^one

^of disclinations. A similar model could apply ^{to the} spin system ^of amorphous magnets, ^to the disordered blue phase of cholesterics and, ⁱⁿ

^a

dynamic version, ^to liquids. Consequences concerning

the phonon elementary excitations

are

discussed, ^{like the} Bragg-like ^{behaviour at k}

⁼

03C0/d, ^{the rotons} modes, ^and the low energy localized modes invoked in the heat capacity ^low temperature ^behaviour.

Classification

Physics ^Abstracts

1. Introduction.

^-

It is a well established experi-

mental fact that all amorphous ^materials (glassy silica,

covalent or metallic glasses, polymers, ...) ^{have com-}

mon physical properties; ^{at low} temperatures (near

1 K) ^{there is an extra} specific heat linear with T, ^{a heat} conductibility quadratic ^with T, saturation of ultra- sonic attenuation at high power inputs, ^{... ; at} high temperatures ^the transport coefficients obey ^{a Fulcher-} Vogel law, ... [for ^{a review see} 1]. Spin glasses ^{seem also}

to display ^{the same} properties, ^whose general ^occur-

rence has been found remarkable enough ^to bring

them the qualification of universal properties. ^{If this is} really ^{so, these} properties ^must be rooted in some

general structural model of disorder, ^{in the same} way that ordinary specific ^{heat of} insulators, heat conduc-

tivity, conductivity,

^...

^{have common} features which

originate ^{in the} three-dimensional translational model

of crystals.

A structural model of amorphous materials has

recently ^been proposed [2, 3]. ^It is this model ^we shall

develop in this paper. This model involves the consi- deration of symmetry groups in _spaces of constant

curvature, ^either positive ^curvature (3d sphere S3)

or negative ^curvature (3d Lobatchewskyan

^-

^or hyperbolic

^-

space H3). Briefly : ^{it is} possible ^{to tile}

a space of constant curvature with regular polyhedra

whose repetition obey operations ^of symmetry (rota- tions, ^mirror planes, ...) ^which pertain ^{to one of the}

subgroups ^of the continuous _group under which either

S3 ^or H3 ^are transitive. These subgroups ^{are therefore}

similar in nature to the Schoenflies groups of euclidean space R3 ; they could be named the Schoenflies _groups of S3 ^or H3. ^But they ^{describe a} totally ^{different local}

order. For example ^{the five fold} symmetry ^{which has} long ago been advocated by ^Bernal [4] ^{in his} pioneering

work on the structure of liquids, ^and proposed ^more recently by Sadoc et al. [5] ^for non-crystalline solids,

is a symmetry ^{which is} perfectly tolerable in S3 ^or H3 (it pertains ^{indeed to} many of their Schoenflies groups)

but is precluded in the conventional 230 Schoënflies groups. ^{It seems} therefore _very tempting ^to classify

those amorphous (and liquid) systems in which the local order is anomalous in an euclidean sense by ^the crystallographic groups of S3 ^and H3’

Of course, this classification raises a number of

questions, ^{the first one} being obviously : a) ^{how to}

map ^a non-euclidean lattice onto ordinary space,

remembering that this operation ^{must lead to a}

disordered state. Another question ^{is :} b) ^{what are the} physical properties which relate to the anomalous

crystallographic properties defined above ?

To question a) ^{we have} given

^a

preliminary ^answer

in [2, 3], where it is claimed that the introduction of disclinations in the lattice is the natural _way of

«

decurving » space, while maintaining ^a fairly ^similar

local order (with fairly ^constant lengths ^and angles)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019820043090138900

1390

except ^on disclinations cores. This mapping ^{« in the}

disclination mode » favours probably ^the H3 ^model

versus

the S3 ^model, since in this last case the finiteness

of S3 permits ^{to build} only

^a

^small piece ^of amorphous

materials (an ^{extension to an infinite} body requires

the introduction of walls ; conversely ^{we note here that}

the Sg ^{model can be} particularly ^{relevant to the small} aggregates ^of Farges et ^al. [6]). ^{In the first} part ^{of this}

paper ^we will proceed further in the investigation ^of

the geometry ^of

^an

amorphous body mapped ^{from an} H3 ^{lattice. We show in} particular ^{that an} amorphous body ^can be described as

a

random network of ^small cylindrical domains of finite length, ^{in contact} by ^their

ends (the nodes of the network being ^{at those} ends).

The axes of these domains are mappings of reticular

axes

which exist in the H3 ^{lattice, and} along ^{which the}

lattice repeats ^itself by ^an operation of translation.

These reticular ^axes (R. ^{A. in} short) ^are fundamental features of the hyperbolic lattice, in which there is

nothing like the usual reticular planes and reticular lines of the euclidean lattice, repeated by the elements of

a

3d translational group. In ^an hyperbolic ^lattice

the translations are strictly ^{limited to the R.A.} (one

characteristic translation _per R.A.). To the best of our

knowledge ^these geometrical properties ^{of the} Hg

lattice were not described before in such a way,

although the notion of localized translation symmetry has long been known (these translations are the transvections of Cartan [7]).

The network formed by ^the cylindrical ^{domains in}

the amorphous body ^cannot be accomodated without

disclinations; they ^run through the interstices left by

the domains and form themselves another network, conjugated ^to the first. These

are

the disclinations of

our mapping process from H3 ^to E3, ^{which was at}

that stage very superficially described in [2, 3]. ^There

is

a

striking analogy between this model which makes

use of cylindrical domains and the model of the cholesteric blue phases of Saupe [8], recently ^revived by

Meiboom et al. [9]. ^We will argue that the cholesteric blue fog [10] is in fact an amorphous ^state correspon-

ding ^{to our model,} and that the spins ⁱⁿ amorphous magnetic ^{materials with local} anisotropy might ^also

have a similar topology.

Question b) ^{offers another} challenge. ^In this paper

we

shall stress the implications ^{of our} geometrical

model for excitations of -the phonon type (including anharmonicity). ^In particular ^{we shall} gain ^some insight ^on the existence of a (Bragg) maximum in the

phonon dispersion ^{curve, the existence of roton-like}

modes at twice the (Bragg) ^wave number, ^{and the}

existence of states of small energy with ^a few number of levels. The two first points ^{relate to} experiments ^and

calculations

on

the phonon dispersion ^{curve which we}

shall recall, the last one to the low temperature speci-

fic heat effect (and ^related effects).

2. The projective representation ^of hyperbolic space.

-

A major difficulty ^with hyperbolic space is that

we Wtally lack intuitive insight ^{of its} geometrical

Fig. 1.

^-

Projective representation ^of H2. ^{The absolute A is}

a

conic representing ^the points ^at infinity ^of H2. Its interior

points represent ^the points ^of H2. ^The right ^lines passing through p (the polar ^of QQ’)

^are

^all perpendicular ^to QQ’

(in

^an

hyperbolic sense). ^The right ^{lines of the} representation

are

right ^{lines of} H2. ^Various hypercycles ^{with axis QQ’}

^are

represented.

properties, because of

our

educational background.

Reading Coxeter’s books several times [11, 12] pencil

in hand, ^{and in} particular drawing ^{all the}

^«

strange »

figures ^{where two} parallels ^to

^a

given ^line

^run

^from

^a

single point, proved ^the best exercise for me to become

acquainted ^{with this} geometry. ^{I see no better} way of

summing up here this practical knowledge ^I acquired

than describing

^on

^the projective representation of H2

and H3 ^{some of the} geometrical properties ^{of these}

spaces.

In figure 1, ^the points ^at infinity ^{in the} hyperbolic plane H2 ^are represented by

^a

^{conic A} (the absolute),

the points inside the absolute being ^the image ^of points ⁱⁿ H2. Any straight ^line

^yc

^{is the} image ^of

^a

straight ^{line in} H2. ^Point p, which is the polar ^of

with respect ^to A, ^{is a} point ^{common to all the} straight

lines perpendicular ^to

^7r

ⁱⁿ H2.

Two lines which meet on the absolute (i.e. ^at infinity

in H2) ^{are said to be} parallel ^{in a} given ^direction.

There are clearly ^two parallels ^to

^a

given ^line passing through ^a given point. Two lines like _pp and _pv which meet outside the absolute ^are said to be ultra-

parallel. Clearly, they ^have one common perpendi-

cular QQ’ and only ^one.

There are three families of

^«

cycles )). I) ^{The circle}

is the locus of points ^{at a} given distance of a fixed

point. It is also the locus of the mirror images ^{of a} given point through all the lines passing through ^the

centre; 2) ^The hypercycle ^{is a locus of} points equi-

distant to a fixed line n, the axis. A few hypercycles ^with

7r

as axis are drawn in figure ^{1. Note that an} hypercycle

is also the locus of the mirror images ^{of a} given point through ^{all the lines} passing by p. p is in this sense the

«

centre » of the hypercycle. The definition of the

hypercycle ^{in term of} symmetries ^will prove ^to be

important ^to

^us.

3) ^The horocycle ^is

^a

cycle ^whose

Fig. ^2.

^-

Projective representation ^of H3. The absolute A is

a

quadric, ^{whose interior} points represent ^the points ^of H3.

The two planes tangent ^{to A in} QQ’ ^and containing p define

a

right ^line QQ’ ^{which is} perpendicular (in ^the hyperbolic sense) ^to any plane containing p. p and QQ’

^are

reciprocal polars. ^The planes ^{of the} representation

^are

(hyperbolic) planes ⁱⁿ H3.

centre is at infinity, ^on the absolute (or whose axis is tangent ^{to the} absolute). ^{It is not}

^a

straight ^line.

All these properties generalize ^{to 3 dimensions} (H3). ^The absolute is now a quadric. Duality by pola- rity couples planes

ⁿ

^to points p (outside A), points

inside to planes ^M outside, ^{lines A to} lines 1. Two

parallel planes intersect in a line which is tangent ^{to A.}

Two hyperparallel planes (which ^{intersect 1 outside} A)

have

a

common perpendicular ^A which is the polar ^of

I. The three types ^of cycles generalize ^{to three} types ^of spheres (Fig. 2).

The projective representation preserves projective properties, ^and particularly the cross-ratio of four

points ^{on a} line, and of four lines with common point.

The hyperbolic distance d between 2 points p ^and

^v

^is

defined in function of a cross-ratio (see Fig. 1) :

where R - 1 is

a

positive quantity measuring ^{the cur-}

vature of _space, and the cross-ratio of four points ^a, p,

y, 6 is

Similary, ^the angle ^{between two lines} (in H2) ^{or the}

dihedral angle ^{between two} planes (in H3) ^{is defined} simply ^{as a} function of the cross-ratio of these two lines (resp. planes) ^with respect ^{to the lines} (resp.

planes) tangent ^{to A and} having ^{the same} intersection.

Note the important fact that the hyperbolic ^dis-

tances and angles ^{we are} discussing ^are also euclidean distances and angles : ^{it is} always possible, by ^defini-

tion of a Riemannian _space (like Hn), ^to develop ^it (to

roll it without gliding) ^{on an euclidean} space R.

along

^a

line, ^{with no} distorsions of the lengths along ^the line,

^as

^{well as no relative} angular variations of the directions attached to it. (However this isometric

mapping along

^a

^{line cannot} be extended continuously

outside the line, and it is indeed this difficulty ^which

led

us

to the idea of

a

mapping in the disclination mode [2]).

3. Reticular _axes, hypercylindrical domains and model of the amorphous ^state.

^-

^Let

^us

^{consider some} tiling ^{of the} hyperbolic space; ^to simplify ^{matters, we}

start with

a

2-dimensional _space H2 ^and take, ^{for the} building blocks of the lattice, regular polygons ^with p

edges, ^such that q polygons ^share

^a

^{common vertex.}

We note { p, q } the lattice built in this _way (Schlafli symbol). ^The requirement ^of obtaining ^an hyperbolic tiling ^reads [2]

If p ^{is even,} it is evident by ^{reason of} symmetry ^that any line passing through ^the midpoints ^{of two} opposite edges ^is perpendicular ^to both, and is therefore their

unique ^common perpendicular QQ’ (Fig. 3a). Opposite

Fig. ^3.

^-

Reticular axis QQ’ ⁱⁿ

^an

hyperbolic ^lattic. a) p

even

(we have assumed _p

⁼

6); b) p odd, q ^even; c) p odd,

q odd. _q, the number of polygons ^{at each} vertex, ^{is not defined}

in this figure.

1392

edges

^are

^therefore ultraparallel. This line is also the

common perpendicular ^to

^an

^{infinite set of} ultrapa-

rallel edges, ^of length do ^{each, which} repeat ^with

^a

periodicity ^d obeying ^the equation (see ^the appendix

for formulae in hyperbolic geometry)

Note that R (the ^{radius of curvature of} H2) and do

are

related (since ^{there is a natural} length ^{in such a} space). The relation is [13] :

Such common perpendiculars ^{to infinite sets of} edges also exist for _p odd, ^{for reasons of} symmetry which easily ^vizualize (Fig. ^3b for q ^even; Fig. ^3c for q odd). ^{There too the line} QQ’ ^{is an axis of} sym- metry ^of translation, ^{with some} periodicity d ^There

is also no doubt that such lines QQ’ also exist in the

general ^{case, when the} building block of the lattice is not a regular polygon.

We call such lines QQ’ ^{reticular axes} (R.A.) ^because

of their analogy ^{with the} corresponding ^euclidean objects. ^{But an} essential difference is of course their

uniqueness, ^{for each set of} ultraparallel edges.

A number of interesting consequences ^can be drawn from the existence of R.A. We derive them starting

from the remark that the symmetry group of the

hyperbolic tessellation is generated by reflexions on

the edges ^of

^a

characteristic triangle (Fig. 4) ^of angles nlp, nlq, n/2.

We restrict here to a tessellation of regular polygons,

for simplicity. Therefore :

-

any vertex, when reflected in _any lattice edge perpendicular ^to QQ’, generates ^{another vertex of the} tessellation. All the vertices of a family generated ^from

a

given ^{vertex lie on an} hypercycle ^of QQ’. ^Therefore QQ’ ^{is a line of} glide-reflexion ^{for the} crystal lattice ;

-

any other element of the symmetry group

acting ^on QQ’ transforms QQ’ ^to another R.A.

Therefore _p R.A. _pass through ^{the centre of} any

polygon. ^{The whole} family of R.A. obtained in this way forms ^an hyperbolic lattice which can be studied

easily ;

-

since the space is hyperbolic, ^{there is an} expo-

nentially growing ^{number of} polygons, ^measured per unit length ^of QQ’, ^{when one} goes away from QQ’

Fig. ^4.

^-

^The characteristic triangle OPQ ^{in the} hyperbolic

lattice { p, q I.

along ^{one of its} perpendiculars. ^{The area} spanned

between the hypercycle, ^at

^a

^distance

^x

^of QQ’, ^and QQ’, is, ^for

^an

^unit length along QQ’

and the area of a { p, q }.polygon ^is

Therefore the density ^{6 of} polygons ^is growing ^{with x}

like

The mapping ^{of the} hyperbolic ^{lattice on} R2 ^can

now be better understood. If

we

choose for d some

repeat distance favoured in the amorphous body, ^the region ^{in the near} vicinity ^{of an R.A.} constitute clearly

a

domain in which the mapping introduces little dis- torsion, ^if part ^{of the} mapping consists in rolling H2 ^on R2 along ^{the R.A. itself (on} which there is

no

distorsion at all). ^{Consider now}

^a

ring ^{made of R.A.} segments ⁱⁿ H2, ^{and roll} H2 ^on R2 along ^this ring. ^This process maps the ring ^{on an open,} self-cutting, circuit in R2,

but it is feasible to close this circuit on R2 by removing

the extra R.A. segments, building ^{therefore a} ring ^of

R.A. in R2. ^{The centre of the} rings ^is occupied by ^{a dis-}

clination point (a positive disclination : ^we have removed segments ^{in order to} decrease the density ^of

vertices). Continuing ^this process,the ^entire mapping

consists in connected rings ^{of R.A.} segments, ^these segments ^{are axes} of domains in which the density

increases _away from the axis, enclosing disclination

points ^whose strength ^is evidently ^a multiple ^of p-1 or q-l (see Fig. 5).

Let us now investigate how there results extend to three dimensions. We restrict too our discussion to a lattice built of regular polyhedra.

If the polyhedron ^{is made of} polygons containing ^an

even number of edges, ^{we can} consider the section of

adjacent polyhedra by ^a plane passing through ^the midpoints ^{of two ultra} parallel edges ^and perpendi-

cular to them. The intersection obtained ^{in this} _way is similar to that one of figure 3a, ^the ultraparallel edges being perpendicular ^{to the} figure. ^{Therefore there}

exist R.A. passing through ^{the centres of} polygons belonging ^{to two} adjacent polyhedra ^and perpendi-

cular to the common polygon ^{of these} polyhedra. ^All

the planes passing through ^{such a R.A. are perpen-}

dicular to

a

set of ultraparallel ^{faces ; some of them}

are

also perpendicular ^{to the} edges ^{of these faces,}

which are therefore ultraparallel. The consequences drawn before for H2 ^extend easily; a) ^{all the vertices}

lie on hypercycles equidistant ^to QQ’, which is there- fore so-to-speak ^{an axis of} cylindrical symmetry. ^In fact QQ’ ^{is an axis of} glide-reflexion ^{if the constitu-}

tive polygon ^is

^a

square, and possibly ^{an axis of}

Fig. ^5.

^-

^{A model for}

^a

^two dimensional amorphous body.

The atoms have translational symmetry along

^axes

AB, which

are

local

axes

of glide-reflexion. The atomic density

is minimum

on

AB and increases towards the centre of each polygon ^{of the network, which is} occupied by

^a

^discli-

nation point ^{L. The} polygonal array is random.

helicity ^{if the} symmetry of the constitutive polygon ^is larger; b) ^the density ^of polyhedra, ^measured per unit

length of R.A. increases exponentially with the dis- tance to QQ’ ; c) ^{a whole set} of R.A. is obtained by applying all the reflexions generated by the faces of the characteristic tetrahedron; ^{this set forms an} hyperbolic lattice; d) ^the mapping ^on R3 ^{in the}

disclination mode introduces a random network of

cylindrical domains; positive disclinations allowed by

the lattice symmetries pass through the interstices of this network.

If the polyhedron ^has ultraparallel ^faces (this ^{is the}

case for the dodecahedron 15, ³ }, ^the icosahedron

{ 3, 5 } and the octahedron { 3, ³ }), ^{the R.A. are lines} joining ^{the centres of these} faces; they

^are

^{lines of}

rotation and transvection for the lattice.

The regular honeycombs of H3 ^{fall into one of these}

two categories, ^{or in both} (for ^a description ^{of these} honeycombs, ^see [13]). ^More generally, the search for the R.A. lattices necessitates first to recognize ^the

families of ultraparallel faces of the hyperbolic crystal.

They ^{should not fail to exist, even if it} might ^{be neces-}

sary ^to consider clusters containing ^{more than one} building ^{block to find} ultraparallel ^faces.

4. Elementary excitations ; ^the Bragg ^case.

^-

^{Is it} possible ^to recognize ^{the nature of the} H3 crystallo- graphic arrangement by ^{some sort} of diffraction method in the resulting amorphous body ? ^{It is this}

kind of question ^{which led}

^me

^{to consider the} Bragg problem ^{in an} amorphous material, ^{i.e. to find under}

which conditions (on ^{the wave} length A and the dif- fraction angle) ^an horospherical ^wave (issued ^from

Fig. ^6.

^-

^The geometry for the calculation of the Bragg

diffraction conditions; ^the figure displays

^a

planar ^{section of} H,, containing the incident and diffracted directions (Q

¹

and O2), ^{and a} diffracting ^R.A.

some point Q 1 ^at infinity) ^is diffracted by ^{an R.A.}

towards some other point Q2 at infinity.

The geometry is illustrated figure ^6, which shows

some planar section of H3 by ^a plane containing 921, Q2’ and the R.A. here QQ’. ^{Points a and b}

^are

^at

^a

transvection distanced. hi and h2 ^are the feet of the

perpendiculars ^to QQ’ erected from 92, ^and Q2’ ^{Call h}

the length hl h2. ^It is easy ^to prove, by using ^the right- angled asymptotic (Ql ^and Q2 at infinity !) triangles Q I hl a, Q2 h2 a,..., ^{that :}

a

the + or - sign depending ^on the relative values of a 1

^versus

a2, fl,

^versus

P2’

Formula (8) ^{are obtained} by using the fundamental

equation ^for

^a

right-angled asymptotic triangle, ^which

here read (i

⁼

1, 2)

As soon

as

the lattice node considered (a, b) ^{is a few}

intersites distances d from hi, it appears clearly ^on equation (9) ^{that the} angle cxi is _very small (remember

that d and R are comparable, ^from equations ^{(3) and} (4)). ^Therefore

^we

^can safely replace ^sin x,, tg cxig ^etc...

by cxi for most of the nodes, ^the exception being ^{a very}

few nodes in the vicinity of hl and h2 (which ^{are fixed} points).

Now consider the asymptotic triangles Qi ^ab, 02 ab. The differences in length ð1 = 01 a - Q2 b, 62 = Q2 a - Q2 b, ^can be calculated using hyper-

bolic trigonometry. ^{We find}

1394

(we have assumed that, ^{like on} figure 6, (XI

1

PI’

(X2 P2)’

Hence the total path difference for the two rays issued from Q 1 ^{reads :}

Taking ^now equation (9) ^{into account with the same} approximation, ^{we find}

We are now able to state the Bragg ^{condition in an} hyperbolic ^lattice.

From equation (12) it appears that ^we have to

satisfy

a condition which is similar to the well-known Bragg

condition in crystals (2 ^{d sin 0}

⁼

nÀ.) ^{for 0}

⁼

7r/2.

This is in fact what happens ^here, approximately,

since _{ai and} fli ^{are so small} 0 - 7E - ^oci, p) ^{so that,}

when considering ^the corresponding amorphous body,

the corresponding rays ^are practically parallel ^{to the}

R.A. The hyperbolic

^«

limit » of the Bragg law is then obtained for 0

⁼

7r/2.

However equation (13) ^states also that the Bragg

condition is _very isotropic ^{in an} hyperbolic ^{lattice :}

this is not very surprising, since there is a natural

length ^{in such a lattice} (the ^{radius of} curvature), ^{and we}

expect ^{nA to be} of the same order of magnitude.

Note that our demonstration does not require ^that ol Q2 ^{and QQ’ be in the same} plane. ^For 0, ^{and the}

R.A. given, 02 is ^at infinity ^{on a cone} which is well defined by equations (8) ^and (13) and whose axis is

along ^{the R.A.}

A natural extension of these considerations con-

cerning ^the building ^{of coherent waves} (but ^not only

the horospherical ^{wave have discussed} here) ^would

lead to an investigation ^{of the} question ^{of the}

^«

^reci- procal lattice » of ^a lattice ^on H3 ^We leave this ques- tion for a further study ^{and limit our} discussion to the consequences ^on the Bragg ^{law we have found} (Eq. 13) ^{on the} phonon dispersion ^curve.

In our model of an amorphous body consisting ^of cylindrical ^domains centred round R.A., ^we _expect that

planar ^{acoustic waves with wave number k}

⁼

n/4

whatever their direction _may be, ^are Bragg diffracted, because there are many cylindrical ^{domains at} Bragg position (0~ 7r/2). If there is some correlation in the orientations of the cylindrical domains, ^{some direc-}

tions of diffraction might be favoured.

Now, comparing ^to experimental ^{results, it appears}

that it is a very general ^{feature of the} phonon disper-

sion curve of liquid ^{materials that it bends down from}

the linear behaviour

a) ⁼

ck (expected ^{in a model of}

disorder at all scales), ^and displays ^{a maximum at}

some value of k which is always comparable ^to

^a

reciprocal ^lattice parameter. This has been observed and computed ⁱⁿ liquid ^Rb [14, 15] ^{and Pb} [16], ^for longitudinal ^modes. Experiments ⁱⁿ amorphous ^mate-

rials are rather _scarce; the maximum itself has never

been observed, ^to the best of _my knowledge, ^{but it has}

been computed ⁱⁿ a-Mg70Zn30 [17] ^for longitudinal

waves; also various computed ^{results on transverse}

waves have been obtained for k n/d [17] ^and

k > n/d [18], ^which clearly ^{let us} expect ^{a maximum} for transverse waves, inasmuch as the descending part (k > 7r/d) ^{of the} dispersion ^{curve has been}

observed [19]. ^{There are all reasons to believe that this}

behaviour is related to our model of cylindrical domains, which favors Bragg diffraction along ^R.A.

5. Elementary excitations : the limit of long ^wave- lengths ; ^{the rotons.}

^-

By extrapolating ^{the short} wavelength behaviour of these observed transverse waves (in a-Mg7oZn3O), it appears that ^a soft mode near k

⁼

2 nld, ^{or a roton of} very small energy, should exist. We want to argue that our model predicts ^such

a roton, ^{on a} physical ^basis very similar to that which

explains ^the

^«

^{solidon »} recently proposed ^{for 4He} [20].

The small cylindrical ^domain is indeed a small ordered region ^{which can} trap phonons ^{whose wave-} length compares with d ^{or some} multiple of d. These

are the Bragg phonons ^{we discussed} above. Such

phonons ^can be transferred to the

«

crystallon » by Umklapp processes involving waves k1 and k2 ^near Bragg diffraction, ^which certainly ^{can interact} strongly

in the cylindrical ^domain. They would decrease their energy by creating ^an excitation of small _energy and small k (we discuss afterwards of their nature), ^and

a localized phonon ^{of momentum}

with Kc

⁼

² nld. ^The corresponding energy of the

crystallon ^{would be}

where V is the volume of the cylindrical ^{domain. If} E c = nw c wither - ¹⁰¹² S-1 ^one get V - 1O - 2 2 cm3 (this ^{value of} We has been measured in liquid argon and seems typical ^{of a small roton} energy).

To the difference with H6ritier et al. solidon, ^the

«

crystallon » ^{described here is not a} dynamical object,

but this does not really ^{make a} difference. The excita- tion of small energy and small k appearing ^{in the pro-}

cess would simply replace ^the hydrodynamic part which is essential in the solidon process.

These excitations of small _energy ^are presumably

related to the modes of vibration of the cylindrical

domain. Vibrations of a cylinder ^{of small section} compared ^to length, ^with boundaries free of traction,

have been studied in details (for ^{ex., see Love} [21]

and Rayleigh [22]). ^The present situation is different : the cylindrical domain has

a

density ^which grows

exponentially ^{from axis to} exterior : this condition,

which is difficult to treat exactly, ^is probably correctly

fitted by taking

^a

cylinder with fixed boundaries. The calculation is _easy for torsional and longitudinal vibrations; ^the frequency spectrum contains in both

cases an acoustic branch

co

= ck, ^where

^c

^{is some}

sound velocity, ^and

^«

optical » branches, starting ^at

w ⁼

cla, where a is the radius of the cylinder ^and

^c

some (other) ^sound velocity. ^{There are} vibrations of

possibly ^smaller frequencies : those related to flexural

(transverse) deformations of the cylinder, propagating along ^the cylinder ^or helically. They might be favoured because they correspond truly ^{to the} crystallography

of the cylindrical ^{domain. But I have not made a}

calculation of these modes. In all these cases, ^one

expects that the smallest excited k would be related to the finite length ^{of the} cylinder, corresponding ^{to a}

localized phonon ^of very small energy trapped ^{in the} cylindrical domain. This could be a model for the soft

zones so often invoked to explain ^{the low} temperature behaviour of the specific ^heat [1].

Another possibility for this low temperature ^beha- viour, ^{not exclusive of that one} just described, ^{is with}

the regions ^where cylindrical domains meet; ^{the atoms}

sitting there have practically equal energetic ^choices

between the various positions ^related

^«

crystallogra- phically » ^to the various domains. These atoms can

therefore be easily ^{excited on a} finite number of levels,

either by ^the phonons generated ^{in the} cylindrical domains, ^{which cannot} propagate very far without

interchanging energy with the lattice, ^or by ^the Bragg

diffracted beams.

6. Discussion.

^-

In this paper, I have proposed,

on the basis of a geometrical analysis ^{of the conse-}

quences of frustrating short-range order, ^{a model}

in which the amorphous body is divided in cylindrical

domains forming themselves a sort of Frank network of domains, through which disclinations (all ^{of the}

same sign) ^are finding way. I have discussed some

effects of this model on elementary excitations. Other consequences should of ^course be discussed, among them the simplest ^{would concern} plastic behaviour;

since this model predicts ^some underlying network,

this should facilitates the understanding of processes which have, by ^many aspects, ^much resemblance with those observed in ordinary crystals, ^like slip (shear)

bands and slip lines, brittleness, ^etc...

Let us also notice that the present ^model reconcile,

in some way, the supporters of a model of an amorphous body ^{made of} microcrystallites, ^and others : the

microcrystallites would be the cylindrical domains,

but they ^{are not} separated ^one from the other by grain boundaries.

Also, ^a number of experiments ^are required ^to

check the model :

more

measurements of the phonon dispersion ^curve, especially ^near n/ d ^{and 2} nld, firstly; also careful microscopic examination of frac-

ture areas (by scanning ^electron microscopy) ^and

attention to density ^contrast in transmission electron

microscopy ^{would be} rewarding. Finally, ^{other mate-}

rials than conventional amorphous bodies should

display similar structural properties,

^as

already stated;

the blue fog [10] ^{is an obvious} candidate, ^{on which} light scattering experiments ^{on one} hand, ^electron microscopy observation of freeze-etched specimens

on

the other, ^{should be} tempted. Also it is most possi-

ble that the spin ^{structure of} amorphous magnets with local anisotropy present ^{the same} structure;

there is indeed a striking analogy, stressed in refe-

rence [23], between the cylindrical domains of the cholesteric blue phase (in which the molecules rotate from a direction along the axis towards

a

direction inclined at rc/4 ^{from the} axis), and the static curling

mode of the magnetization ^{vector, discussed} long ago

by Frei et ad. [24] apropos the modes of small _energy in elongated ferromagnetic particles, ^{and it is} tempting

to adopt ^{the model to} amorphous magnets.

It would be interesting ^to look for one-dimensional

stackings of polyhedra in numerical experiments. ^Note

also that it would be interesting ^to compare the results of numerical experiments ^{with fixed} (or periodic) boundary conditions and free boundaries. In this last case, and for a sufficiently small number of atoms, ^the result should be a spherical ^{model of an} amorphous

material (S3), ^{or a model} mixing ^different regions ^with 83 ^and Hg ^character.

Finally, ^we want to stress the importance ^of

^a

pro- blem that we have completely neglected up ^to now;

in modelizing ^our amorphous body,

^we

have intro- duced a random network of cylindrical domains,

without discussing ^{this new} process of randomization.

Clearly, ^{it should} be treated in the same way

^we

treated the randomization of atoms, ^{but at a} larger spatial ^{case. A new} process of randomization will

necessarily appear ^{at a} still larger scale, ^{that one} being performed, ^{and so on...} up ^{to some} cut-off length

determined either by the size of the system, ^{or the} method of preparation ^{of the} amorphous body, ^etc...

Therefore we suspect ^{that a} complete ^model of disorder

implies multiscale physics, extending between the smallest scale (atomic size) ^{towards a} highest ^scale.

This is reminiscent of turbulence, and of the model of chaos proposed ^first by Kolmogorov [25] ^and

Obukhov [26], which has been so fruitful and to which the numerous experiments ^done to-day ^{on the struc-}

ture of turbulent flows are not irrelevant.

Acknowledgments.

^-

^I thank Professor J. Friedel for illuminating ^{comments and a critical} reading ^{of the} manuscript, and Dr. G. Bellessa for _many fruitful discussions.

Appendix.

^-

^{In a} right-angled triangle ^{with two}

sides parallel (Fig. 7) ^{the so-called} parallel-angle

depends only ^{on the} length d of the finite side

1396

Fig. ^7.

^-

Asymptotic right-angled triangle (see appendix).

In an ordinary right-angled triangle, the essential

trigonometric ^{formulae are} (notation ^of Fig. 8) :

Fig. ^8.

^-

Right-angled triangle (see appendix).

The area of

a

triangle ^with angles

^c

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