Disclination density in atomic structures described in curved spaces

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Disclination density in atomic structures described in curved spaces

J.F. Sadoc, R. Mosseri

To cite this version:

J.F. Sadoc, R. Mosseri. Disclination density in atomic structures described in curved spaces. Journal

de Physique, 1984, 45 (6), pp.1025-1032. �10.1051/jphys:019840045060102500�. �jpa-00209832�

Disclination density ^{in atomic structures}

described in curved spaces

J. F. Sadoc (*) ^{and R. Mosseri} (**)

(*) ^{Lab. de} Physique ^des Solides, ⁹¹⁴⁰⁵ Orsay, ^France

(**) Physique ^des Solides, CNRS, ^{1 Pl. A.} Briand, 92190 Meudon Bellevue, ^France (Reçu ^{le 28} octobre 1983, accepte ^{le 8} février 1984)

Résumé.

La courbure de l’espace ^et la densité de disinclinaisons sont deux grandeurs connectées. Il _y

relation exacte à 2 dimensions. Nous montrons comment utiliser

relation approchée à 3 dimensions. Les appli-

cations à la structure W-03B2 ^et

phases ^{de Laves sont} présentées. ^La coordinance, ^{dans les} structures denses aléatoires, ^est expliquée ^à partir de la notion de densité de disinclinaison.

Abstract.

The curvature of

space and the density ^of disclinations

two related quantities. ^{There is}

^exact

relation in 2-D spaces. We show how

approximate ^{solution can} be used in 3-D space. Applications ^{to the} 03B2-W

structure and the Laves phase

presented ^The coordination number in dense random structures is explained

in terms of disclination density.

Classification

Physics ^Abstracts

61.00

61.70

1. Introduction.

In a 2-D space the curvature of _space and the density

of disclinations are exactly ^{related to} each other by

the Gauss-Bonnet theorem [1]. An attractive example

occurs when disclinations are not infinitesimal,

but correspond ^{to a finite} angular ^deficit (or excess) :

the surface is everywhere ^flat except ^on point ^defects corresponding ^{to a} local concentration of curvature.

In this case the 2-D _space is the surface of a polyhedron (in ^a general ^sense, since it is not necessarily ^{a closed} surface). ^{In the} simple ^{case of} regular polyhedra

defined by ^two Schlafli indices { p, q } ^the angular

deficit 6 of each vertex is 6

2 7C 2013 qap, ^where ap is the vertex angle ^{of a} p-gonal ^face.

So

The number of vertices for regular polyhedra ^is

(from ^{the Euler} relation).

Consequently ^{the total} angular deficit for a poly-

hedron is Lb

4 7L Now consider a polyhedron

to be an approximation ^{of a} sphere ^{with a constant} gaussian ^{curvature K. We can write} ff ^Kda

⁴ⁿ

and conclude to the equality between the total angular

deficit of a surface topologically equivalent ^{to a} sphere, ^{and the sum} of the Gaussian curvature on the surface. In fact this is a simple example of the Gauss- Bonnet theorem, and this relation is applicable ^to

all the surfaces approximated by ^a polyhedron

with flat faces. Consequently ^{in 2d} space, there is

an entire equivalence between the disclination density (with ^a weighting ^factor equal ^{to their} angular deficit)

and the curvature.

Unfortunately in 3-D space ^no such simple ^result [2]

exists.

The aim of this _paper is to relate disclination density

and curvature in 3-D _spaces. (For example polytopes

embedded in 4-D Euclidean space.) Application ^to

some complex periodic ^structure [3] ^{which can be}

described in corrugated 3-D space (with ^{a zero mean} curvature) ^is presented. ^{We show how} positive ^and negative disclinations approximatively annul each

other, ^if positive ^and negative curvatures balance

mutually ^{in order to} give ^{a zero mean curvature.}

Coordination numbers in dense structures are explai-

ned in terms of disclination density.

2. Polytopes ^and disclinations.

A polytope ^{is an} approximation by ^flat spaces (cells)

of a 3-D spherical space. The curvature is concen-

trated on edges acting ^as disclinations.

Polytopes ^are very well described in the books

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019840045060102500

1026

by ^Coxeter [4, 5]. They ^are characterized by ^three Schlafli indices I p, q, r } : I p, q } is characteristic of the polyhedral cell and r is the number of cells sharing

a common edge. ^{In the} following paper the word

polytope is used with two different meanings : ^some

times it is an ensemble of cells with straight edges,

and some times it is a spherical ^space containing

vertices connected by geodesic edges; ^{in this case}

we call it a spherical polytope.

3. Orthoscheme tetrahedron and symmetry sphere.

The orthoscheme tetrahedron is a very efficient _way to describe a polytope. The faces of this tetrahedron

are mirrors. A given ^{vertex is} reproduced ^{in these}

mirrors like in a kaleidoscope. ^{All these} images ^are

the vertices of the polytope. The faces of the ortho- scheme tetrahedron are rectangular triangles ^defined

in the spherical space. An Euclidean tetrahedron

can also be defined by analogy.

A M6bius spherical triangle (which ^is equivalent

to the orthoscheme tetrahedron in 2-D spherical space) ^is presented ⁱⁿ figure ^1. All identical triangles

obtained by reflection in their sides form a pattern

covering ^the sphere just ^{once. If a vertex is} placed ^on

one of the vertices of the M6bius triangle, ^its images

form a polyhedron.

Fig. ^1.

^{A Mobius} spherical triangle VCE, ^{and other} identical triangles ^obtained by reflection in their sides to form

pattern covering ^the sphere just ^{once. The} angles

of this triangle ^{are V}

nlp, ^C

nlq, ^E = n/2 ^{if the} poly-

hedron is { p, q }.

The orthoscheme tetrahedron of {4, 3, 4 } (the

Euclidean cubic tessellation) ^is represented ^on figure ^2.

The relation between a polytope cell and the ortho- scheme tetrahedron is clearly ^{defined on this} figure.

In order to define angles of the orthoscheme tetra-

hedron faces, it is drawn unfolded on figure ^{3. The}

vertex of the polytope { p, q, r } ^{is called} Po, ^{the centre}

of cell is P3, ^{the centre of face is} P2 ^{and the centre of} edge ^is Pl. ^{In the} appendix ^we give the different ele- ments of the Mobius triangle and of the orthoscheme tetrahedron following the Coxeter notation.

Looking ^at figure ^{1 we} observe that sides of the Mobius triangle ^are parts ^of great circles of the

Fig. ^2.

The orthoscheme tetrahedron for the {4, 3, 4 } tessellation (the euclidean cubic tessellation). ^Point Po correspond ^to

^vertex, P1 ^{is the centre of} edge, P2 ^{is the}

centre of face, P3 ^{the centre of cell.}

Fig. ^3.

The unfolded orthoscheme. The value of all

angles

^functions of p, q,

^are given ^{in the} appendix.

2-sphere. ^These geodesic ^{lines are the} reflection lines of the polyhedron. ^{In 3-D} spherical space ^a face of the

spherical orthoscheme is part ^{of a} 2-sphere ^{which is}

called ^« symmetry sphere ^{of the} polytope ^».

4. Disclinations and symmetry spheres.

Consider a spherical 3-D space with ^an inscribed

spherical polytope. ^This polytope is characterized by

its symmetry spheres. ^{If the curvature of the} space is then concentrated on the edges ^{of the} polytope, ^a symmetry sphere ^is distorted and becomes a poly-

hedron (but ^we always ^{call it «} symmetry sphere »).

The basic idea of this _paper is to study the effect of disclinations going through ^a symmetry sphere ⁱⁿ

order to understand the combination law of these disclinations. We take advantage of the existence of an

exact solution for ^a symmetry sphere (which ^{is a 2-D} space) ^{and we} extend results to the spherical ^3-D

space characterized by ^the symmetry spheres.

Disclination lines are edges ^{of the} polytope, ^conse-

quently they go through ^a symmetry sphere ^with

respect ^to the mirror symmetry. A disclination is

perpendicular ^{to this} surface, ^{if there is no vertex} of the

polytope corresponding ^to its intersection with the symmetry sphere. If there is a vertex on the symmetry

sphere ^{it is a} node for disclinations. Some of these disclination segments ^{are in the} surface, ^{others are} disposed symmetrically ^with respect ^to the mirror. In this case they ^can be considered as a continuous discli- nation line with a cusp where it _goes through ^the symmetry sphere.

4.1 EXAMPLE OF THE { 5, ^3, 3} ^POLYTOPE.

^The { 5, 3, 3 } ^{is a} packing of dodecahedra in which an

edge joins three cells. The angular ^deficit, leading ^to disclinations, ^{is due to the} deviation from 2 7r/3 ^{of the} dodecahedron dihedral angle. ^More formally ^this

deficit is related to the deviation of the 7r/r (r

3) angle ^{in the} spherical orthoscheme, ^{from the same} angle ^{in the} Euclidean orthoscheme (2013 2013 ^arcos

cos nlq The disclination angle ^{is 6}

^{2 n - 6 x}

sIn n p ))

1.01722 or 6 - 0.1798 rd

Disclination lines (edges ^{of the} polytope) ^{form a}

tetracoordinated network since there ^are four dode- cahedra sharing ^{a vertex.}

The sphere ^of symmetry ^{of the} { 5, 3, 3 } ^{can be}

obtained from an icosidodecahedron. This polyhedron

contains all equatorial ^vertices of the dual ’polytope { 3, 3, ⁵ }. Consequently ^the icosidodecahedron is the set of all the P3 points ^{of the} { 5, 3, 3 } symmetry sphere (Fig. 4a). ^On figure ^{4b the} positions ^{of other} points (Po, I P I, P2) ^{are shown in a} pentagonal ^{and a} triangular

face of the icosidodecahedron.

Fig. ^4a.

^The symmetry sphere (polyhedral approximation)

for the { ^{5, 3,} 3 } polytope ^can be deduced from

icosido- decahedron.

Fig. ^4b.

^Positions of (Po, P1, P2, P3) points ^{in icosido-}

decahedron triangular ^and pentagonal ^faces.

There are 20 disclinations perpendicular ^{to the}

symmetry sphere ^{in the centre of} triangular ^faces (P 1 points).

The arrangement ^{of other} disclinations close to the symmetry sphere ^{is described on} figure ^{5. There are}

five disclination segments ^{in each} pentagonal ^{face and}

Fig. ^5.

Positive disclinations relative to

pentagonal

face of the symmetry sphere ^{of the} { 5, ^3, 3 } polytope.

there are five disclinations going through ^a pentagonal

face with ^a bend at the crossing point (Po ^point).

We can estimate the contribution to the curvature of the different disclinations crossing ^the symmetry

sphere. (Keeping ⁱⁿ mind that the total curvature is

4 n).

Disclinations perpendicular ^{to the} symmetry ^surface contribute with their exact angular deficit 6 to the curvature.

Contribution bb ^of disclinations bent at the crossing point Po ^can be obtained from the sum of the vertex

angles ^of triangular faces of the orthoscheme which

cover the symmetry sphere.

Disclinations through P 1 point ^{have a} contribution b = 2 ropq - n ^and through Po point (vertex ^{of the} polytope)

using ^the appendix ^notation (see ^also Fig. 3). Applied with p = 5, q = 3, r = 3

6

0.1798 rd (10.30°) and bb

^{0.1494 rd} (8.56°) .

It is obvious that 4 7r

20 6 + 12 x 5 bb*

4.2 EXAMPLE OF THE SIMPLEX POLYTOPE { 3, ^3, 3 }.-

The frame of the symmetry sphere ^{is a} spherical ^tetra-

hedron with non-equilateral triangular faces; angles ^of

these triangles appear ^on figure ^{6. A} disclination line goes through this surface on P 1 point ^{with a contribu-}

tion equal to ð = 2 n - 3 C(T = ^{2.5903 rd} (otr ^{is the}

Fig. ^6.

The unfolded symmetry sphere (polyhedral

approximation) ^{for the} simplex polytope f 3, 3, ³ }.

1028

dihedral angle ^{of a} regular tetrahedron). Three other disclination lines _go through Po points making ^an angle n/3 ^{at these} points. Contribution to the symmetry sphere ^{curvature is} 6b

^{3.325 rd.}

In this example ^{we observe an} increase of the disclination contribution on bending. ^So any approxi-

mation of 6b by bb ^{= 6 sin a,} in which a is the angle

between the disclination with the symmetry surface, ^is completely ^false. (In ^{the case of the} { 5, 3, 3} ^with

a

108°/2 bb/b

^0.809, compared ^{to the exact value} 6bl6 = 0.83119.)

4. 3 APPLICATION ^TO #-TUNGSTEN STRUCTURE. - The

fl-W (A 15) ^{structure can} be described [3] by ^disclina-

tions in a tetrahedral packing. The tetrahedral packing

is built _up in a first step by ^atoms put ^on the vertices of a { 3, 3, 5 } polytope ^{or in a dual} description by

atoms in the center of cells of a { 5, 3, 3 } polytope (these ^{cells are} Voronoi cells of the structure). ^This

tetrahedral packing ^{is in a} spherical ^curved space. In ^a second step it is transformed into the fl-W ^structure by

disclinations along ^{the A 15} linear chains (Fig. 7). ^This

Fig. ^7.

Cubic cell of

f3- W ^structure.

structure can be considered as a mixing ^of positive

disclinations on the edges ^on the Voronoi cells, ^and negative disclinations on the straight ^lines going through ¹⁴ coordinated atoms (A ¹⁵ chains). ^The

Voronoi froth is a packing ^of regular dodecahedra with polyhedra characterized by ¹² pentagonal ^faces

and 2 hexagonal ^faces (a disclinated dodecahedron).

On figure ^{8 is} presented ^an orthogonal projection ^of

the fl-W structure. On the right part ^{of the} figure ^some

Voronoi cells _appear. The projection plane ^{is a mirror}

of the structure. On the left part ^{of the} figure ^some points ^are quoted with the notation used to characte- rize orthoscheme vertices.

This mirror of the structure can be considered as a

symmetry sphere ^distorted by negative disclinations, ^as

the A 15 structure is obtained from a { 3, 3, 5 } polytope

distorted to an Euclidean structure by ^a disclination

procedure.

Notice that in this description ^the symmetry ^surface is not exactly ^{flat, but is a} corrugated surface tiled by

Fig. ^8.

Projection

[100] plane ^{of the} fl-W ^structure.

Heavy ^circles

^for

family ^{of atoms} just ^{in the} projection plane. Some Voronoi Cells

shown. Points Po, P1, P2, P3

defined

symmetry ^surface

^noted 0, 1, 2, 3.

geodesic triangles. This results from the supposition

that all tetrahedral interstices are regular ^tetrahedra

with equal edges. ^{In this} description ^{A 15 structural}

space is Euclidean only ⁱⁿ average. In paragraph ^{4 the} { 3, 3, 5 } symmetry sphere is described using ^{an icosi-}

dodecahedron which is a tiling by pentagons ^and triangles. ^The symmetry surface of the A 15 structure can be deduced in a similar _way from a tiling ^of triangles ^and hexagons (see ^{in the} figure P3 points).

This tiling correspond ^{to an} icosidodecahedron trans- formed by negative disclinations, ^which change penta- gons into hexagons.

Account of the positive ^and negative disclinations in

fl-W ^structure

We describe fl-W ^structure using ^a mixing ^of positive

and negative disclinations. The positive disclinations

(edge ^{of the Voronoi} froth) ^{are identical to} { 5, 3, 3 } edges. Their contribution is determined in paragraph

4.1. Disclinations going through symmetry sphere orthogonally ^on P 1 points contribute with 6

0.1798 rd; disclinations going through point Po (with ^{a bend on Voronoi} vertices) contribute to the curvature with 6 b

0.1494 rd. The part ^{of the} sym- metry plane in the cubic crystallographic ^{cell is a}

square which contains two P 1 points ^{and six} Po points.

The positive contribution to the curvature of this sur-

face in this _square is C+

^{1.257 rd}

The negative ^{curvature is due to} negative ^disclina-

tions orthogonal ^{to the} symmetry surface in P2 points.

Negative curvature at these points ^{is 6}

^{27r -}

12 5 n

1.257 rd The negative ^curvature exactly

annuls the positive ^curvature.

4.5 LAVES PHASE EXAMPLE.

A similar description

can be done with the Laves phase ^structure (Cu2Mg)

[3]. ^{This structure can} also be described using ^a

{ 5, 3, 3 } polytope ^{with atoms} inside dodecahedral

cells, ^and inserting negative disclinations in order to

reduce this curvature to zero. But in this case the

negative disclination network is a diamond network.

A symmetry plane ^{of the structure} appears in figure ^9.

It is a [ 110] crystallographic plane (Fig.10). ^This plane

can be obtained from an icosidodecahedron by changing ^some triangles ^into pentagons ^{in order to} have the same number of pentagons ^and triangles (this ^{is needed to} follow the Euler relation for the

plane). Disclinations are represented ^{on the left} part of the figure. ^{In a} unit cell of this plane ^{there are}

10 positive disclinations of type bb through Po points,

2 positive disclinations of ty-pe 6 through P 1 points ^and

2 negative disclinations through P3 points occupied by large ^atoms.

Fig. ^9.

Projection ^on

[110] plane of the Laves phase

structure. Positive disclinations go through ^this plane

on

points Po ^and P,, negative disclinations go through

this plane ^on points P3 occuped by large ^atoms.

Fig. ^10.

Projection plane in the cubic cell and negative disclinations.

We deduce the positive contribution in the cell

The negative contribution of negative disclinations is

usirig ^the appendix ^{notation. We observe the exact} compensation ^of positive ^and negative ^curvature (there ^{are 2} negative disclinations in the cell).

5. Disclination density ^{and the} Regge ^calculus [6].

In paragraph ^{4 we} have described exactly how discli- nations contribute to the curvature of particular geodesic surfaces which are mirrors of the structure.

Intuitively, ^{it seems} that the disclination density depends ^{on the} length ^of geodesic ^lines going ^{from one}

of these surfaces to another close surface. This can be

explained ^more precisely.

A disclination segment goes from a Po point ^to

another Po point (vertices ^{of the} polytope) following

an edge ^{of the} polytope. This disclination is ortho-

gonal ^{to a face of a cell at a} P, point. ^The disclination concentrates all the curvature of the bipyramidal

volume built with this face and the two Po points (Fig. 11). (Called ^an E-Cell).

Fig. ^11.

^An edge Po Po ^{between two connected vertices}

of

polytope, ^and

^face P3, P3, P3 ^define

bipyramid

If the curvature of the space is concentrated ^on

Po Po edge, ^this given edge ^{contained all the curvature of the} bipyramid

The disclination contribution to the curvature

depends ^{on the} length ^{of the} Po Po disclination _seg- ment and on the volume of the pyramid ^The difficulty

comes from the evaluation of this volume in a non-

Euclidean _space. Nevertheless in a space of constant

curvature the area of the face ^can be evaluated : it is 2 ^r times the area of the P, P3 P2 triangle (Fig. ^{3). In}

non-Euclidean space the ^area of this triangle ^is

(Î + ; ⁺ ’" pq - 1t). ^{(The radius of curvature is}

used as unit).

The area of the face is S

r 2 r 7r - 2 qlpq - 7r

it is _very important ^{to note that this area} is also the

angular ^deficit corresponding ^to the disclination.

If we suppose the volume calculated by ^V = ’3 ^{S. h}

(as ^{it is in Euclidean} space) the disclination density

3 M .

p = S. h ^is equal ^{to 3.}

This result can be compared ^{with a} relation intro-

duced by Regge [7] ^{in order to} study ^an approximation

1030

of space-time by tetrahedral Euclidean cells connected

by their faces and edges ^{as in a} honeycomb :

where (3)R is the scalar curvature and where £ ^extends

i

to all disclinations segments bi ^of length li in the volume of summation. Applied ^{to a} 3-sphere ^{with (3)R}

6/R2

(R is the radius of curvature) gives :

we deduce

or with a sphere of unit radius :

all disclinations volume of the 3-sphere

which is equivalent ^{to the 2-D relation}

all disclinations ^area of the 2-sphere

But the 3-D result is an approximation. ^This approxi-

mation is (in 3-D) ^{similar to the} approximation (in 2-D) ^{of the area of a} 2-sphere by ^{the area of a} poly-

hedron. But in the present description ^{it is the} geodesic length ^{and the} spherical surface of the bipyramid ^which

are used to approximate volume, ^{and not} their Eucli- dean counterparts. This increases the _accuracy of the

approximation. ^{Table I shows} application ^{of this} approximation ^{for the} regular polytopes.

Table I.

Approximative ^{volume V} for {p, q, r}

polytope ^{in relation to the} angular deficit ^{6 on} edge ^and

to the edge length ^{I. N is} the number of edges. ^{s is the}

ratio between V and the volume of ^the 3-sphere ^{2 n2.}

6. Coordination number in dense packing ^with only

tetrahedral sites.

The { 3, 3, 5 } polytope ^{is an} example of a dense packing

in a spherical space [8]. ^{In this case} the coordination number is 12. This polytope ^{can be used to build}

different models of Euclidean dense structures. Nega-

tive disclinations are introduced in order to change ^the

curvature. Numerous structures are obtained if various networks of negative disclinations are used. With this

description positive disclinations ^{run on} {3, 5, 5 } edges (1), ^and negative disclinations are superposed ^to

some of the edges. They appear ^{on a} tetrahedron edge

if 6 tetrahedra share this edge (in ^the following ^dis-

cussion we suppose that all tetrahedra are regular, consequently ^the space is corrugated). The coordina- tion number of these structures can be determined if the relative length ^of positive disclinations and _nega- tive disclinations is known.

We have defined _p

161V ^{with I} equal ^{to the} geo- desic length ^of the E-Cell height ^{and V} equal ^{to the}

volume of the E-Cell in curved space. This volume ^can be calculated if the _space is spherical. But in the _pre- sent case the _{space is} corrugated ^with positive ^and negative ^{curvature which cannot be taken as constant}

in a whole E-Cell. An exact calculation will be, ^{if it is} possible, extremely ^{difficult We} only give ^{an estima-}

tion of the density ^of disclinations.

The density ^of positive disclinations _p+

L+ 6+

is _p+

31R 2if ^we suppose infinitesimal disclinations.

In the following discussion the radius of curvature R of the spherical space is the unit length, ^then p +

3.

In a spherical space approximated by ^a polytope

p +

N16,12 ^{n 2. The} quantities N, ^I and 6 + ^are

defined in table I. For the particular ^{case of the} { 3, 3, ⁵ } polytope p +

2.942.

The density ^of negative disclinations (with ^{the same}

unit length) ^{in the} approximation ^of infinitesimal disclinations is _{p_}

L - 6_ = - 3 in order to ba- lance positive disclinations. If we suppose the E-Cell space ^to have a negative ^constant curvature, ^then I p - I :> ^{3. The exact} calculation of the volume of a

tetrahedron in hyperbolic ^{space is not an easy pro-}

blem [9]. ^{In the} present ^case the reference hyperbolic

structure is { 3, 3, 6 } ^{which is a} very special honey-

comb with all vertices at infinity ^and consequently

the hyperbolic approach ^{does not offer} any advantage.

But the reader ^can easily convince himself that the

approximation of the tetrahedron volume by V

-116

is an under estimate in spherical space ( p + 3) ^and

an over-estimate in hyperbolic space. (For example : ⁱⁿ hyperbolic space the tetrahedron with point ^at infinity

has a finite volume.)

(1) ^In paragraph ⁴

^{have described A15 structure}

with positive disclinations on the { S, ^3, 3} edges. ^The

present ^{treatment offers}

^dual description ^{to that} given

previously ^and gives completely equivalent results for

the coordination number.

If the _space is corrugated ^{there are} necessarily departures ^{from a} constant curvature in each E-Cell.

The effect is an increase of geodesic length ^which

decreases _p (defined by ^p

611V where I and V are

height and volume of the E-Cell in the _space with

spread curvature, ^{and not in the} space approximated by ^Euclidean cells).

Consequently p+ is smaller than 3, ^but p ^{can be} greater ^or smaller than 3 depending ^{on the exact} configuration ^{of the structure.}

In order to determine the coordination number we

needs the length ^of positive ^and negative L+ ^{and L_}

disclinations _per unit volume. They ^{are related to} p + and _p- by p+

6+ L+ ^{and p-}

^{6- L-} (using 6+ =0.1283 rd and 6- = 2 n - 6 C(t (6_ ^{= 1.1026} rd)

for negative disclinations corresponding ^{to six tetra-}

hedra sharing ^an edge).

The mean number of tetrahedra sharing ^an edge ^is

The coordination number is related (9) to nt by

z

12

.

With

3 we et

Z

12 _ ^With I p I

P+

3 we get

( b-n ^{P P+}

which are the values obtained by ^Coxeter [10] using trigonometric arguments.

With p_J I

^{3 and} p +

2.942 (spherical space) ^it

comes nt = 5.106, ^z

^13.42.

nt and ^z values for some crystalline ^{structure which}

can be described with this approach ^are presented ⁱⁿ

table II. We observe variation of z from 13.33 to 14.

Larger value for z are observed with completely

random structures [11] (z

15.54 for random _gas of

points).

These values can be explained ^{in terms of disclina-}

tion density.

z values smaller than 13. 39 are related to I p - [ ^3.

This is the consequence of large variations of the curvature in the corresponding E-Cells. It is the case if

positive ^and negative ^{E-Cells are} strongly ^{mixed This} corresponds ^{to a} highly ^connected negative ^network

of disclinations as is the case for Laves phase. Oppo- sitely ^if negative disclinations are not connected (fl-W) ^{z increases} up ^to 13.50. The fl-uranium

structure is an intermediate case. But I p- I ^fluctua-

tions around 3 cannot explain large ^{a z value, because}

it is unrealistic to supposes p- ! value very different from 3.

The body centred cubic structure is helpful ^to

understand this problem. ^{In this case all} positive

disclinations correspond ^to 6+ = ^{2 n -} 4 at

(5+

^{1.359 rd).} Using ^the polytope { 3, 3, 4} ^then L = 1.909 and _{p +}

2.59. With _{p_ = -} 3 we

deduced nt

^{5.17 and z}

14.51. This is a too large

z value compared ^{to that of 14} in the b.c.c. structure.

p- have ^to be reduced to a value lower than 3 in relation to the great connectivity of disclination networks. From this example ^we conclude that high

coordination numbers are related with strong positive

disclinations. ^{In random gas of points,} disclinations

corresponding ^to 3 tetrahedra sharing ^an edge ^can

appear and increase z. These results have to be com-

pared ^{to N. Rivier’s} [12] ^{results. In the N.R.} paper,

large ^{z values are related to Voronoi} cells with a low symmetry. ^{In the} present paper this is related to the strong positive disclinations. Both analyses ^{are close}

to each other if we consider that strong positive

disclinations completely distort the icosahedral order which is one of the best polyhedral approximations ^of

the spherical symmetry. ^{In the} N.R. paper, small z

value are related to large volume fluctuations of Voronoi cells (or ^to large coordination number

fluctuations). ^{In the} present paper it is related to high connectivity ^{of the} negative disclination network.

Both conclusions are equivalent ^{if we} suppose the second effect is not being ^screened by ^{the first one} (strong positive disclinations).

7. Conclusion.

Glassy structures, but also some crystalline ^structures [1-13] ^can be described from curved structures dis- torted by disclination lines in order to change ^the

Table II.

Coordination in metal structures. Notice that the two numbers n and fit ^{are related} by n(6 - nt) ⁼¹²

(from ^Euler relation). If regular tetrahedra are packed ^{with a common} edge ^{there is} place for 5.1043 tetrahedra.

1032

curvature. We have determined the defect density.

There is not an exact mathematical solution in 3-D space, but there are approximate solutions. Using symmetry surface the relation of curvature and disclinations is reduced to a 2-D problem easy ^to solve. A disclination perpendicular ^{to a surface acts}

on this surface as a point disclination with the same

angular deficit If disclinations ^cross together ^{on a}

surface the intuitive law leading ^{to a} contribution

depending ^{on the sine of the} angle with the surface (as will be the case if the effect of disclinations depended

on the flux of a disclination vector through ^the surface) ^{is false.}

Using ^{the exact} calculation in 2-D, ^{we extend it to} 3-D. The density of disclinations ^can be estimated if the whole volume is divided into elementary ^volumes

associated with disclination segments. ^This density

p

Lb where L is the length of disclination in a unit volume is close to p

3 1 ^{R is the radius of curva-}

ture). ^{But in 3-D} space this result is an approximation;

the exact value depends ^{on the} precise geometry ^{of the} disclination network. It is used to explain the fluctua- tion of the coordination number in dense packing.

Appendix.

^The orthoscheme tetrahedron (Figs. 2-3),

This figure ^{defined in a curved} space works like a

kaleidoscope : by reflection into its mirror faces, ^for given ^values of p, q, r, ^some very regular figures appear.

These figures ^are polytopes ^{if the} space ^curvature is

positive, ^or honeycomb ^{if the curvature is} negative.

Following ^Coxeter [5] ^we give the value of angles

defined on the figure ^{3 in the} spherical ^case.

opql qlpql Xpq ^(resp. Xqr, t’lqr, qqr) ^are the sides of the Mobius triangle ^{of the cell} f p, q } (resp. { qr })(Fig. 1).

They ^are obtained from the relations :

and respectively with q ^{and r indices.}

0, t/J, X ^are edge-lengths ^{of the} orthoscheme tetra- hedra

I

where hpq ^{is given by}

with these relations all elements of the orthoscheme tetrahedron can be obtained in terms of p, q, ^r.

References

[1] LELONG-FERRAND, ^Géométrie Différentielle (Masson Paris), ^1960.

[2] GASPARD, ^J. P., MOSSERI, ^{R. and} SADOC, ^J. F., ⁱⁿ Proc. of

Structure of Non-Crystalline ^Mate-

rials » (Edited by Gaskell, ^{P. H. et} al.) 1982,

p. 550.

GASPARD, J. P., MOSSERI, R. and SADOC, J. F., à

paraitre dans Philos. Mag. (1984).

[3] SADOC, ^J. F., ^J. Physique ^{Lett. 44} (1983) ^L-707.

[4] COXETER, ^{H. S.} M., Regular Polytopes (Dover ^Publ.

New York) ^1973.

[5] COXETER, ^{H. S.} M., Regular Complex Polytope (Cam- bridge University Press) ^1974.

[6] REGGE, T., ^{Il Nuovo} Cimento, ^XIX (1961) ^558.

[7] MISNER, ^C. W., THORNE, ^K. S., WHEELER, J. A. Gra- vitation (W. H. Freeman and Company, ^San Francisco).

[8] SADOC, ^J. F., ^J. Non-Cryst. ^{Solids 44} (1981) ^1.

[9] THURSTON, ^W. P., Geometry ^and topology of 3-Manifold

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[10] COXETER, ^{H. S.} M., Illinois J. Math. 2 (1958) ^746.

[11] HANSON, ^H. G., ^{J. Stat.} Phys. ³⁰ (1983) ^591.

[12] RIVIER, N., ^J. Physique Colloq. ⁴³ (1982) ^C9.

[13] NELSON, D., Phys. Rev. Lett. 13 (1983) ^983.