Dislocations and disvections in aperiodic crystals

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Dislocations and disvections in aperiodic crystals

Maurice Kléman

To cite this version:

Maurice Kléman. Dislocations and disvections in aperiodic crystals. Journal de Physique I, EDP

Sciences, 1992, 2 (1), pp.69-87. �10.1051/jp1:1992124�. �jpa-00246464�

Classification

Physics

Abstracts

61.42 61.70

Dislocations and disvections in aperiodic crystals

Maudce Kl£man

Laboratoke de

Physique

des Solides (Associ£ au C-N-R-S-), Universit£ de Paris-Sud, Bit. 510, 91405

Orsay

Cedex, France

(Received 25

July

J99J,

accepted

4 October J99J)

Abstract. The search for

topological

invariants of dislocations in _a

periodic

crystal

employs mappings

of the

Burgers

circuit

surrounding

the defect _on 2 types ^{of manifolds} : the

perfect

lattice (this

mapping yields

the

Burgers

vector) and the order parameter space

(which

classifies the defects _as elements of the

homotopy

_groups of this op

space).

Both classifications _are

equivalent

because the

perfect

lattice is _a

covering

(in _a

topological

sense) of the op space. ^We deime _an

aperiodic crystal

_as the dj -dimensional boundary of _a d-dimensional crystal and get two similar manifolds _{: an} op space U which is the acceptance ^{domain in}

Pi

with suitable identifications of the faces, and _a

perfect

lattice H~ which is _a curved

periodic

lattice in _a space of negative curvature and _a covering of the former. H~ _{is invariant under the action of}

a non-abelian group of translations H~ _{which classifies} all the

(di 2)-dimensional singularities

of the

aperiodic crystal,

viz. al complete dislocations and b) novel

topological

defects which _we call disvections. The latter

are classified _as the elements of _some normal

subgroup

3~ of H~ _which includes the commutator

subgroup.

The former appear as the elements of the quotient group of H~ _mod.

3~

_or else _as the elements of _a

subgroup

of the group of dislocations Z~ _{of the}

hypercubic

lattice the other elements of Z~ _which do _{not enter} in H~

correspond

to partial dislocations. We argue that disvections _are the usual

phason

defects (mismatches, etc.). The above

theory

of defects

applies,

mutatis mutandis, to

approximants,

for which _we

provide

_a

topological

classification.

1. Introduction.

Defects in

aperiodic crystals (commonly

called

quasicrystals

and abbreviated hereunder _as

qc)

show up a number of features which make them

essentially

different from defects in

periodic crystals [1-3].

These differences and their

peculiarities

_appear best when _one defines the qc as

some

projection

from _a

crystal

of

higher

dimension.

Let _us recall that the _structure of _a

quasicrystal

obtains

by

the

cut-and-project

method

[4]

performed

in _a d-dimensional Euclidean

hypercrystal ^E~:

a

djj-dimensional planar

cut

Pjj ^is introduced _{at some} irrational

angle

with

respect

to the d-dimensional

lattice,

and _a restricted _set of vertices and

edges

of E~,

belonging

to _a

cylindrical strip parallel

to Pjj ^and

spanned by

_a unit cell of

E~,

is

projected

_{on Pjj ;}

they

constitute the

« Bravais _» lattice of the

quasicrystal

in

question.

_{Pjj can} be

thought

of _as the

physical

_space. The

hypercrystal

is

usually

a

hypercubic lattice,

with

d=6, dj

=3 for the icosahedral

crystal, d=5,

djj = 2 for the

pentagonal crystal (the

Penrose

pattem).

Pjj ^is a

subspace

of E~ _which is

globally

invariant under the action of the icosahedral group

Y,

which is _a

subgroup

of the

hyperoctahedral

_group in d

= 6

(resp.

the

pentagonal

_group

D~~,

which is _a

subgroup

of the

hyperoctahedral

_group in

d=5).

The Euclidean space which is the substratum of E~ is the

product

Pjj x

Pi,

where

Pi

is the

orthogonal complement

^of

Pi

in E~. In order _to go

farther than the Bravais

lattice,

a more

general

method consists in

attaching

_a

di

(=

d djj

)-dimensional

motif to each cell of the

hypercrystal

and

constructing

the

quasicrystal

as the set of intersections of the ensemble of motifs

(the

so-called atomic

su~fiace [5] E)

with

Pi.

Each intersection is

generically

_a

point

and

represents

an atom. The qc appears then _{as a}

dj -boundary Pi

of _a

d-hypercrystal.

This

picture

of the qc has been

already

used with _some

success

[6]

for the purpose of

analyzing

random

tilings.

The results which _are

presented

below _are in fact

independent

of the _nature of the atomic surface

(they depend only

_on the

symmetries

of the construction

process)

hence we shall for the sake of

simplicity

invoke the

cut-and~project

method

(and

the Bravais

lattice)

each time _a

reference _to the construction of the qc appears necessary or convenient.

This paper deals with the classification of defects of _a

quasicrystal.

For _reasons of

simplicity,

_we shall discuss

only

^{the defects of} the qc which break the

symmetries

of translation of

E~,

I,e. the dislocations and _{a new} class of line defects

(in 3D)

which _we call disvections and which _are

singularities

of the

phase. Disclinations,

which break rotational

symmetries,

will therefore not be considered. Defects of other dimensionalities will _not be considered either _:

they

are far less

interesting

in the

proposed

frame of

study.

The paper is

organized

_as follows.

In the _next

section,

_we comment at a

general

level _{on some} not very well-known aspects ^of the

general theory

of defects which _are of

particular

_use for

qc's

_{; one} of them refers _to the

utility

of the

algebraic theory

of

coverings,

which

provides

a

deep insight

on the links between the

Volterra-Burgers approach

to dislocations and the

topological

methods _{; a} second _one

concems the

specific approaches

which have been

developed

for the

study

of

crystal

surface

defects. The

impatient

reader will first

glance

over those sections, and _come back to them afterwards if necessary. Sections 3 and 4

give

a short _account of the nature of the

phase singularities

in

qc's

and of the type ^{of order} parameter

(abbreviated

as

op)

_space we expect.

For _reasons to be

discussed,

the acceptance domain with faces identified

U, despite

its

usefulness,

is _{not a}

perfect

_{op space,} and _must be

supplemented by

_a second

manifold, H~,

the universal

covering

of

U,

which in _{a sense} is the

equivalent

of the

perfect crystal

of the

Volterra-Burgers mapping.

The

topologies

of these _two

manifolds,

and how

they

relate _to the classification of dislocations and

disvections,

_are discussed in details in section 4. In section

5,

we introduce _a third manifold

E~,

whose

topological properties

_{separate very}

nicely

the

dislocations and the

disvections,

and allow for _a

study

of their

topological

interaction _; it is indeed well-known that each dislocation is escorted

by

a

phason

strain

[I] analyzable

_{as a}

cloud of disvections which cannot

mutually

anneal. A somewhat

unexpected

result of _our

analysis

is that there _{are two}

types

^of dislocations in _a qc,

partial

dislocations and

complete dislocations, according

to the value of their

Burger's

vector. The word _«

partial

_» refers _to the

same concept ^of

partial

_as in usual

crystals.

^The «

complete

_» dislocations _are not the

analogs

of the usual _«

perfect

_» dislocations. Both

types

^of dislocations _are indeed escorted

by phasons strains,

but with

qualitative

differences which _are

currently

under

study.

2. The

theory

of defects revisited.

2. I VOLTERRA _PROCESS AND TOPOLOGICAL THEORY. All the results which follow in this

work _concem line defects in _a 3D medium

(or topological point

defects in _a 2D

medium).

We

shall have to invoke the two well-tried methods of

approach

of the classification of defects in

an ordered

medium,

I-e- the usual method known _as the Volterra process

[7, 8],

and the

topological

method

[9].

Both will appear necessary, and

they

will

play

_a

complementary

role.

This is in _{a sense}

quite surprising,

since it is

precisely

for the line defects that the two methods

give equivalent results,

while it is well-known that the Volterra process is ineffective to

explain

^{defects of other} dimensionalities. Let _us make _some comments on the relation between the two methods in _a usual

periodic crystal.

The traditional method

(derived

from the Volterra

process)

to

recognize

the presence of _a

singularity

of the

displacement

(«

phonon »)

field _a dislocation in _a

periodic crystal

makes _use of _a

mapping

of _a

loop

₇₀

(the Burgers circuit), surrounding

the

dislocation,

onto the _«

perfect crystal

» E~

[7, 8]

_; this

mapping yields directly

the

Burgers'

_vector b _as the

closure failure of the

image

r of To ^{in the} considered

mapping.

The

topological

method makes _use of _a

mapping

_{of yo on} the order parameter space V

=

T~ _(the d-torus),

the

image

is _a

loop

_y and the class of the defect is then _an element of the

homotopy

_group

arj

(T~)

which is the group of oriented

loops

with base

point

_on V. Of course the two _answers are

isomorphic,

a result which

originates

in the fact that the _«

perfect crystal

_» considered here is

nothing

else than _a

tiling

of _a d-dimensional _space with

equivalent hypercubes (unit cells),

I.e. is the universal

covering

^{of the order} parameter space V

=

T~.

_In

_brief,

_{at the end of this}

analysis,

_we have _tmJo manifolds at our

disposal,

V and E~. In _one of

them, V,

the defects _are

represented by loops

in the other _one, E~,

they

_are

represented by

_open

paths joining equivalent points.

V is the

analog

of _a reduced Brillouin Zone

(rBZ); similarly,

E~ is _an extended BZ

(eBZ).

A natural

question

to ask is whether there _are

intermediary

manifolds which enable _a finer classification of

defects,

_some

being represented by loops,

other

by

_open

paths

_on the _same manifold which would be then _a iBZ. It appears necessary ^at this stage to

deepen

the

concept

of

covering

_space

[10]

_we have

just

alluded _to.

As _a illustration of the

concept (Fig. I),

take the 2-torus

T~

_{which is the}

op space V for the square lattice E~. V

=

T~

_{is the square Q with}

opposite edges

a and b identified _; its universal

cover is E~

itself,

which is obtained

by glueing together copies

of Q in all

possible

_manners

along

the identified

edges.

This construction maps any

loopy belonging

to

T~

_{onto a}

path

r in E~

and, conservely,

_«

projects

_» onto

T~ along

_{y any}

path

r

joining equivalent points

in

~2

p:E~-T~~p(r)

_=y.

₍₁₎

-1 a

b_~ ~

A

a

a) b) c)

Fig.

I. Illustration of the concept ^of

covering

_{space :}a) C, _a

cylindrical covering

of Q b) its universal

cover E2_; c) il

(Q

)

=

T~ represented

_{as a square} Q with

opposite edges

a and b identified. The inverse

projections

of _a

loop belonging

to il is shown stretched in both _covers.

As _a consequence, we see that the group of translations

Z~

of E~ is

isomorphic

to the

homotopy

_group of

T~ Z~

_{is indeed the}

group with two generators a and

b,

with the relation

r _w aba~ b~

= l. We have _:

"i(T~)

m

Z~

m

(a,

b _; aba~ b~

=

l) (2j

This relation _r

=

I expresses the way a rotation about _a vertex of the square lattice is

performed

with the

help

^{of successive} translations which respect ^the identifications.

But,

while the square lattice is the universal _cover of

Q,

there _are other _covers of

Q,

in-between lli and the square lattice _: for

example

any

cylindrical

lattice C tiled with squares whose

edges

are

along

the

generatrices

^{of the}

cylinder.

Observe that the first group of

homotopy

^of the

cylinder

_ar

j

(C

is _a

subgroup

of arj

(T~),

_since

some

loops belonging

to V _are still

loops

on

C,

while other _ones open up in the inverse

projection

_{p~ :} C

-

E~.

More

generally [10],

l'is _a

covering

_space of X if there is _a continuous

mapping p:v-x

which maps

topologically

each arcwise-connected open

neighborhood

of _a

point

_y e l' _{onto a}

neighborhood

of _xe X. The

mapping

_p is often called _a

projection.

The fundamental theorem of

covering theory

is that the first

homotopy

group

arj(l')

is _a

subgroup

of the first

homotopy

_group

arj(X ).

For the universal _{cover, we} have

arj(l')

=

l. This theorem _states in fact in _a

precise

form the

relationship

between

loops

y ^~ in X and their

inverse-projections p~ ~(f~ )

in the

covering

_{space :} the

loops

whose class of

homotopy belongs

to some

subgroup

of

arj(X)

lift to

loops p~~(y~)=y~

in

l',

and the other _ones lift _to open

paths

P

(i~

=

r ^~

We shall find

illuminating

in the qc case to _use three different

manifolds,

viz.

I)

_a

particular

order parameter space

U, it)

its universal

covering H~,

which

plays

the _same role _{as a «}

perfect crystal

_», but lives in _a space of

negative

_curvature, and

it) in-between,

its Euclidean

covering E~

^{in E~}

(E~

is _a

special

_type of atomic

surface). Unfortunately,

_none of those manifolds

yields

a

mapping

_{as easy} to use as the universal

covering

of

T~

_for dislocations

(there

is _no

simple equivalent

of the Volterra process for

disvections)

_; and _none

provides

either _a

mapping

of the _same type as

T~

_{itself for both} dislocations and disvections. But in _{a sense} these difficulties _are not

artificial,

and

they

shed _{a new}

light

on the _nature of the

singularities,

in

particular

the

singularities

of the

phase

in

qc's.

2.2 SINGULARITIES _OF A djj ^{-BOUNDARY IN A} d-DIMENSIONAL _CRYSTAL. We have all

advantage

to

keep

the

picture

of the qc as a djj

-boundary

_Pjj of _a

d-crystal

_: it tells _us that the defects divide into _two classes _:

a)

those which

belong

to the

hypercrystal

E~ and _cut the

physical plane Pjj,

and

b)

^{those which} are

specific

of the

boundary

of the

hypercrystal.

Such _a

partition

is _{not new} in the

theory

of defects in condensed _{matter ;} it has been used first

by

Volovik

[I

I for the purpose of

classifying

surface

singularities

with constrained boundaries in usual ordered

media,

with the

help

of the

topological

method of classification of defects

[9].

Consider,

_{as a}

simple (and helpful

for what _we have in mind later

on)

illustration of Volovik's

method,

the _case of _a nematic

phase

with

parallel anchoring

conditions of the director at the

boundary

:

I)

^{in the bulk the order} parameter _space ^V is the

projective plane ^P~

^and the defects of class

a)

are classified

by

the classes of

equivalence

of the various groups of

homotopy ar~(P~), I.e.,

for _n

=

I say,

arj(P~)

=

Z~,

the Abelian group with two elements which tells _us that the line defects

belong

_{to two} classes,

experimentally recognized

_as the

thins

(which

are

topologically singular

lines,

represented by

the _non trivial element in

Z~)

^{and the} thicks

(these

lines _are

represented by

the trivial element in

Z~,

I-e-

they

are

topologically

unstable lines of

defects) [12]

_;

it)

_on the

boundary,

the order

parameter

space is

restricted _to U

=

§~

_; each

point

of the standard circle

§~

_represents

a

possible

direction of the director at the

boundary. arj(§~ ₎

=

Z,

the group of

integers,

classifies both types ^{of line} defects

a)

and

b) belonging

to the

boundary

;

a)

defects _are

represented by

the odd

integers

S

= 2 p ⁺ I in

Z,

and

b)

^defects

by

^the even

integers

S

= 2 p. The way U is included in V tells

us

something

about the

relationship

between the surface and the bulk

defects;

here the solution is rather easy to grasp

(Fig. 2)

_: the odd

integer,

class

a),

defects _{map on} the _non- trivial element of

Z~

and _are intersections of thins with the

boundary

_;

conversely,

the class

b)

surface defects of _even

integer strength

ⁱⁿ

Z,

which _are

singular

on the

boundary,

_are the terminations of the thicks and map on the trivial element of

Z~. they

are isolated

point

defects _on the

boundary,

since the thicks are

usually non-singular (Friedel's

nuclei _are most

probably

surface defects of this type ^{in small molecules}

nematics).

These considerations

point

towards _an

homomorphism

between the _two groups

ii arj(§~

-

art(P~) ₍₃₎

whose kemel ker

ii

^{is the invariant}

subgroup

^which consists of the _even

integer

elements S

= 2 p in art

(§~ _).

_{All the results of some} interest

concerning

the surface defects in a nematic with

parallel anchoring

conditions _are included in

equation (3).

This is _a rather

simple

_case. In

more

complex instances,

the

analysis

of the surface defects would

require

_{some more}

algebraic-topology

theoretical

results, involving higher homotopy

_groups. Notice that in the

example

under

study

the

singular points

of the

boundary

_can also be defined 0

directly

_as the elements of the relative

homotopy

_group

ar~(V, U),

since this group classifies indeed the classes of

equivalence

of the defects surrounded

by loops

_To

belonging

to the

boundary

and

capped

in the bulk

by

a 2D surface

ahomotopic

to _a

half~sphere.

We find indeed

ar~(V, U)

= ker I

trim j~ (see below).

Results of _a similar _nature extend _to any situation in which _two order parameter spaces V and U _are related

by

_an

operation

of inclusion _{; more}

precisely,

it exists in each such _{case an} exact sequence

of homomorphisms

between

homotopy

groups, which reads

a~ ,~ j~ a~ ,,

-

ar~(u)

_-

ar~(il)

-

ar~(il, u)

-

arj(u)

-

arj(il) (4)

-

~~~/

a

/~~

Fig. 2. Defects of classes a) and b) at the boundary of _a nematic with

parallel anchoring

conditions (see text).

The property ^of exactness means that the

image

of any

homomorphism belonging

to the sequence is the kemel of the

homomorphism

^which follows in the sequence ; for

example

:

im a~ = ker

ii-

Note that

j~

is another

operation

of inclusion which consists in

considering

_any

oriented 2D

cycle

in U _as

belonging

to

V,

and a is the

operation

which consists in

restricting

« to its

boundary

am ₌ y. The last two

homomorphisms

of the sequence suffice to discuss the

relationships

between defects of different classes in the _case of

aperiodic crystals.

But it is clear that the

knowledge

of _arj

(U

is

enough

to

classify

all the defects in

Pjj.

Observe that in the nematic _{case we} have

ar~(V)

₌ Z _;

hence,

because of the property ^of exactness of the

sequence,

ar~(V, U)

is _as stated above.

3. General remarks _on the

topology

of defects in

aperiodic crystals.

Since the defects

belonging

to the

hypercrystal

^E~ can be

readily

classified

by

the standard methods relevant to ordered media

by simply generalizing

them to

high dimensionalities,

it is rather easy ^to

classify

the defects of the qc which

belong

to class

a).

The dislocations and the

disdinations of the qc are indeed

nothing

else than the intersections with

Pi

of the defects of the _same

topological

nature in E~, I-e- classified

by

_arj

(V ).

If _one

forgets

the

disclinations,

the order parameter space V for the defects in E~ _{is the unit cell with}

opposite

faces

identified,

I-e- the d-dimensional torus

T~.

_{Defects of class}

_a)

_{of other} dimensionalities than dislocations

originate

in

non-generic

intersections and _are

probably

of small

physical

^{interest. We} refer the reader to the relevant papers for _more details

[13~15].

Defects of class

b)

_are much _more subtle and _more novel.

Let _us define the nature of the constraint _on the

boundary

_Pjj of the

hypercrystal

: all the

parallel positions

of Pjj ^{which intersect}

Pi

inside the

projection /ti

of the unit cell _on

P~ provide equivalent ground

states of the qc,

differing

_one from the other either

by

_a

global

translation and/or _a

phase

shift

(Fig. 3).

We call

/ti

the acceptance ^{domain. It is} a

tdacontahedron TR in the icosahedral _case and _a set of 4

pentagonal

sections of _a rhombic icosahedron al in the

(slightly

_more

involved)

Penrose _case.

By conveniently identifying opposite

faces and

edges

^of

/ti,

_we obtain _a closed manifold U which is the op space of the

hypercrystal boundary

_Pjj constrained _as defined

above, although

some difficulties _are

immediately

apparent

I)

_a

given phase

state of the qc is

represented by

an

infinity

of

points

in U

(and

not

by

a

single point) differing by

translations _e

belonging

to the group of symmetry ^of E~

(Fig. 3)

note that the _set of _{vectors e} do not form _a group ;

it)

the set of such translations _e which _connect

equal phase

states in

/ti (hence

in

U)

is _not the set of all the translations of the symmetry group, but

only

those which _{connect two}

points

of the

strip,

I-e- it excludes _a number of translations

(forming

_a

group) belonging

to the set

£

_{m, y,}

(see Fig. 3).

In other

words _we have in U _too many values of the same

phase

^and not

enough

^of the different

translations. These

peculiar

characters _can be rationalized _so that U

obeys

the usual definition of _an op space, as follows.

The translations _e and _y

together

form _a group

Z~ again, they

are

represented

^{in il}

by

different

loops

which are not

homotopic

to zero

(except

the zero element of the

group),

or

equivalently, by

_segments which connect

equivalent points

_on the

boundary

of the unfolded version

T~

of

V,

I-e- of the unit cell in E~

(Fig. 4).

But observe that _a segment ^of type e

projects orthogonally

in

/ti

as a segment ^{which does connect two}

points

which _are both in

/t~,

i.e. which does not

yield

a

loop

in

U,

whereas _a

segment

^of

type

y generates a

loop

in U.

Therefore _{y connects}

equivalent points

in

/t~,

while _{e connects}

points

which

qualify

as

«

inequivalent

_»,

although they

generate

qc's differing by

a «

phonon

_» ^shift

only,

I-e- _a pure

displacement.

Henceforth the

only

dislocations of E~ _which

are classified _as defects in

arj(U)

are those of type y

(they

form _a group

isomorphic

to

Z~).

We call them

complete

3 P

I

I

i

Fig.

3. The acceptance ^domain

Jti

of

equivalent perfect crystalline

states for the d= 2 _case.

PI

_and

Pi

_{have the same}

_phase

_state, but differ

by

_a translation of the type e =

£

_{n, e,,} where _e is _a free

vector

belonging

to the

strip

whose cross~section is

Jti

;

PI

and

PI

_differ

_by

a

phase

shift (the translation

which

brings PI

_upon

PI

is _{not a} lattice translation)

PI

and

PI

do _not

belong

both to

Jti,

have the _same

phase

state, but differ

by

a translation t

= e + y, where _y

=

£

_{m, y,} is not a free vector of the strip.

e

Ai

^~

Fig.

4. Relation between

Jti

and the d-dimensional _torus T~. Dislocations of type e are

partials,

dislocations of type _y are

complete.

d

= 2. See text for _more details.

dislocations. On the

contrary,

dislocations of

type

e, which _are of course true defects in

Pi,

since

they

can be constructed as intersections of the relevant defects in

E~,

should be considered _as

partial

dislocations of the qc, since their

representation

in

/ti

connects two

points

which _are

inequivalent. They

_are true

partials

in the _{exact sense} of usual

crystals,

where the _set of

partial Burgers'

vectors

emerging

from a common

origin

form _{a star} in the unit cell

(this

_star is

easily

related _to the

Thompson representation

of

partial Burgers'

vectors, which is

a tetrahedron in fcc lattices

[7])

and _{a set} of open

paths

^{in the} op space

T~.

_{In the qc, the}

relevant _star is the

projection

^of the vertices and

edges

of the Bravais lattice of the qc in

/t~. They

_are also

partials

in the _sense of

approximants

the _{y vectors} transform _to the _true lattice parameters ^{of the}

approximant

when the

boundary

Pjj ^is

rational,

and the _{e vectors tum} to the

partials (in

the usual _sense of this

term)

of the

approximant crystal.

We _can summarize this discussion

by stating

that the

homomorphism

ii, gr~(u)

_-

grj(v) (5)

is _{not onto} im

ij

does _not contain all the elements of

arj(V).

The first

homotopy

_group

arj(U) yields

all the types ^of

complete

defects

a)

and

b)

which break the _« translational _»

symmetries

of

Pjj.

Some of those defects

(we

have

just

characterized them

by

their

Burgers' vectors)

_are

dislocations,

I-e-

singularities of

the _«

phonon

_»

field,

_we call the other _ones

disvections

;

they

are

clearly singularities of

the

«

phason

_»

field

_: and _we shall argue that

they correspond precisely

to _« mismatches _» of the Penrose

lattice,

in the d

= 5 case

(Fig. 5),

and to their

generalizations.

Both fields _are indeed modified when Pjj ^is

displaced parallel

_to itself

and spans U. The

partial

^defects _appear as the elements of

arj(V)

which _are not in

arj(U). They

are all of the dislocation type. ^The rest of this paper deals

essentially

with disvections.

Again

U is _not the

complete

order

parameter

_space

(although

it is _a valid op

space)

because it does not include rotational

symmetries.

,,

,' ₁ "

""#*'

^'

'

1 , ^Al

i , 15

1 ,,

1 , ,

1 , ,

' 35

' '

54

1 , ³

1 , ,

, , , 42

, , ,

',

1

~~

23

' '

',

,

$~ _{43 25}

34 ~~

, ,

, , ^,

, ⁵¹

,

' ' ' "

' ~,

' '

' ' '

24 j 4

' '

' $

, ^' ,

' '

*

, 2

'

13

a)

~~

b)

Fig.

5. Triacontahedron TM a) bird's eye view b) its Schlegel diagram with faces identified by the symmetries ^of translations of the parent

hypercubic

cell.

4. Defects classified _{as a} whole.

In this

section,

_we define U and

H~

and indicate how these manifolds _are used. We

delay

a

similar discussion for

E~

to the next section. As

already indicated,

U and

H~

are

respectively

the

analogs

of the order

parameter

space and of the _«

perfect crystal

_»

(the

universal

covering

of the op

space).

But

E~

is _a manifold much _more novel in the

theory

of defects

by

_many

aspects.

4.I THE ORDER PARAMETER SPACE U _{AND ITS COVERING SPACE}

H~(d= _6).

_{In the}

icosahedral _case the op space U is _a TR with

equivalent

faces identified. We do _not enter into the details of the geometry ^of this manifold and of the relevant identifications this is

published

elsewhere

[16],

_as well _as the _more delicate _case of the Penrose

tiling.

The identifications in

question

_are the identifications induced _on

/ti by

the

orthogonal projection

on

Pi

of the identifications of the

opposite

faces and

edges

of the unit cell in

E~.

Figure

5

gives

_{as an}

example

the identifications of the faces of

/ti

=

TR. These

identifications

provide

the group of translations of the universal _cover

H~

_of U.

P:H~-U. ₍₆₎

For

TR,

the

corresponding

group of translations has 15

generators

_{y, =}

A~~ (I

=

1,

,

15 p, q = 1,

,

6)

and there _are 12 relations between generators, ^{of the form}

[17]

:

~l"~12~13~16~14~15~ ~12"~46~26~56~36~16

"1

(7)

where

A~~

^{is the} translation which

brings

face pq over face _qp

(Fig. 5).

^{The reader will}

recognize

that each of them involves _a series of faces

belonging

to the _{same zone on} TR. In these

expressions,

the

operators

r; act to the

right, I-e-,

in rj, say,

Aj~

acts

first,

then

Ai~,

etc. The group

H~

=

lY,

; r; =

11 (8)

which is defined

by

generators ^{and relations is the} group of translations of the universal _cover

H~

of U

(TR ) and, by

definition of _an universal

covering,

it is also the first

homotopy

_group of U.

gri

(u )

-

H6 (9)

I.e. it is the group which classifies

complete

line defects in _a qc

(partials

are excluded from

H~,

_as it has

already

been

emphasized

above _; but these

points

^{will be} taken _over in _more details in the

sequel). H~

_{is a}

non-abelian,

infinite group, since the number of

independent

generators ^is

larger

than the number of relations. Therefore

H~

is _{a «}

crystal

_» tiled with TR'S in _a space of

negative

curvature

[16, 17]

of dimension

di

₌ 3.

4.2 MAPPINGS OF LOOPS OF THE QUASICRYSTAL ON U _AND

H~.

The

precise

definition of

the order parameter ^of a deformed

quasicrystal

presents some fundamental

difficulties,

whose nature can be

already

^{realized when}

trying

to

give

a measure of this order

parameter

^{in the}

simple

_case of _a

perfect _qc.

Of _course, if this qc is defined

by

_{some djj}

-plane Pjj,

^it

might

_seem

sensible _to

give

to the order parameter a

unique

value

w(Pjj

in

Pi,

as follows _: lift any vertex r = Mjj

belonging

to the Bravais lattice of the qc to the

point

JL _e

E~

_from which it has been

projected orthogonally,

then

project orthogonally

JL _on

Mi

in

Pi,

take the

envelop

of all these

projections,

which is _a window

congruent

to the acceptance ^domain

/ti (we

denote it also

by

^the same

symbol /ti,

since there is _no ask of confusion, and since this

property

^of congruence allows _us to use the window _{as an}

acceptance domain,

I-e- _{as a} copy of the rBZ of the

op)

^{and define}

w(Pjj

_as the value of

w at the center of

/ti,

for all r's. However this

unicity

of

w(Pj ) depends strongly

_on the fact that the infinite qc is known. If this is _not the

case, there is _some

indeterminacy

in the choice of the window in which _to locate the

projection

of the finite qc. A solution to this

difficulty

is to

assign

a variable order parameter to _a

perfect

_qc, viz. the value

w(Mi

to each vertex Mjj ^{of the} qc. Let _us show that this does

not lead _to

topological

inconsistencies. Note indeed

that, although

all the

physical points

Afj ^do not

project

at once on the _same

point

ⁱⁿ

Pi,

all the

loops

To in the qc lift _{to a}

image loop 7~

in E~ which

belongs entirely

to the

strip

^§ of the

cut-and-project

method, since the qc is

perfect.

Hence its

orthogonal projection

_yi in

Pi

is

entirely

inside the window

/ti

and _can be

smoothly

reduced to _a

point

inside

/ti

without ever

meeting

the boundaries of

/ti Consequently,

this

property

^{of smooth}

reductibility

to _a

point

in

/ti

is not modified

by

an identification of the

edges

and faces of

/ti

which transforms

/ti

to U and

yi ^to 7. The

image yof

_To in U is

homotopic

to zero, And the

image

r of To in

H~

_is

obviously

_a

loop,

I.e. is also

homotopic

to _zero in

H~,

since any

loop

is reducible to _a

point

in the universal _cover. In that _sense, the infinite

perfect

_qc is

truly homotopic

to zero.

This is true also of any finite

perfect _crystal,

for which the above method of the

envelop

does not

yield

_a

unique w(Pi ).

The above discussion extends _to any

loop

_yo in the qc

(not only loops joining vertices)

it suffices for the sake of the argument to lift any

point Mj

of the

loop

_yo to the

corresponding point

JL e E~

belonging

to the _djj -face in E~ which is the lift of the tile in

Pi

to which M~

belongs.

Has this

mapping

_{a sense} for _a deformed qc ? The _answer is

straightforwardly positive

for _a

«

phonon

_» strained _qc, and for dislocations. For in this _case the

image y~

in E~ of _a

loop

To

surrounding

_a dislocation is _an open

path,

and its closure failure _measures

precisely

the

Burgers'

vector b in d dimensions. The

projection y~

^of

y~

in

/t~

is also _an open

path,

but its

image yin

U is either _an open

path

if To ^surrounds a

partial dislocation,

_{or a}

loop

^if it surrounds

a

complete

dislocation. In this latter _case, this

loop

is

represented by

_some class of

arj(U)

which _we shall _soon

display.

The _use of the _same

mapping

to the case of _a

phason

strained

crystal

does not

display

at once the

topological

nat~Jre of the

phasons, although

it _can be shown

[18]

that the union

~J M~

of the

images

in

P~

of _a

phason

strained qc, finite _or

infinite,

_cannot be included in any window

/t~.

We then first

modify

_our

perception

of this

mapping by noticing

that any

point

JL e E~ and

belonging

_to the

loop y~

_{can be} attached

arbitrarily

to a

particular

unit cell of the

hypercubic lattice,

for

example

_any of the

C(d, di )

=

2~~

hypercells containing

the djj -dimensional face to which JL

belongs.

We prove below that, this first choice

being

made for _some

starting point

_{on 7~} the other

hypercells

which follow when

traversing

yo are no

longer arbitrary.

In the _case of _a

dislocation,

the

starting

unit cell and the final unit cell _are

equivalent by

_a translation

precisely equal

to the

Burgers'

vector measured in E~. _Observe that this

perception

of the

mapping

is reminiscent of the de

Bruijn's pentagrid [19],

because in the

pentagrid

each mesh

(which

is the dual of _a vertex of the Penrose

pattem)

is the intersection of _some

hypercell

in E~ with

Pi.

Therefore the

pentagrid method,

which _can

be extended _to the icosahedral _case,

provides

without any arbitrariness _an

hypercube

specifically

attached _to each vertex of the qc, while the direct method suffers from the arbitrariness in the choice of the first cube _at the

starting point

of the circuit. The

unicity

of the result

provided by

^{the dual} method proves in _a way the result _{we were}

stating

above. But

it _can be proven in another way we schematize the argument.

Assume that the

starting point ^JL°

_on

^7~

is attached to _some

hypercell H°

which contains the

djj-dimensional

face

h°

_{to which it}

belongs.

This

dj"face belongs

itself to a set of

(d

I

)-dimensional

^faces of the

hypercell,

which _are

C(d I, dj )

=

d~

in number. When

JL°

moves to another

position JL~

_in

some other

djj-face h~,

two cases arise _; either h~

belongs

to the _same

hypercell H°,

and _we then make H~

=

H°,

_or it

belongs

to _some other

hypercells.

We look for the condition that there is _no choice _on

H~.

Observe that JL

belongs

to a set of

(d

I

)-dimensional

faces which _are also C

(d 1, ~ )

₌

d~

in number

but _are different from the former. These two sets

belong

to the _same

hypercell

H~ _{if and}

only

if 2 _x C

(d I, dj )

is smaller _or

equal

to the maximum number of faces of _a d-

hypercell,

viz. 2xd. This condition reads

d~

w2xd and is satisfied for

d~ =3,

d

=

2 _or 3.

5. Dislocations and disvections discriminated.

We used above _a

mapping

of To ⁱⁿ E~

(yo

_-

7~)

which

played

_an

intermediary

role.

Although

it presents ^{in itself} some interest

(it

allowed _us, if _some

precautions

are

taken,

to define

unambiguously

the

mapping

on

U),

it has _a

disadvantage

which limits its _{use :} E~ does not

have the _same local

topology

as U.

Hence,

there is _no relation of

covering

between

yo and

y~.

It is the

identity

of the local

topologies

of U and

H~

_{which allows} for _an

unambiguous mapping

from _{one to} the

other,

since

H~

covers U

(infinitely

_many

times).

We

show _now that it is

possible

to embed

7~

in _a manifold which is _a

covering

of

U,

and which has the _same

topology

than

P~,

in _{a sense.}

5.I

i~,

AN ABELIAN COVERING OF THE op SPACE OF AN APERIODIC CRYSTAL.

H~

is the universal _cover of

U,

but there _are

midway

other _covers of U which _are of

interest. We define _{now a cover}

E~

of U where the

Burgers'

circuits of dislocations

(partial

and

complete ones)

in the qc are

represented by

_open

paths

and in which other circuits in the qc map on non-trivial

loops

:

they

surround

topological

defects which _we call disvections.

Ea

consists of the set of all the

orthogonal projections /ti

of all the

hypercells

of the

hyperlattice

in

Pi,

with rules of incidence inherited from the

projection

in

Pi

of the rules of

incidence of the

hypercells

in

E~.

In the d=6 _case,

/ti

₌ TM. A better view of

Ea

is obtained

by lifting

^each

di

-dimensional

/ti

inside the d-dimensional

hypercell

from

which it is

projected,

in such _a way that each face of the lift /t is

glued along

the

«

silhouetting

_» face of the

hypercell

from which it is

projected

onto

/ti (Fig. 6).

These _two

tx

I

I

Y ,

fl

Fig.

E

~ in the_unit of E~.

The

edges a,

a', p, p', _y, y'

are

in « silhouetting » _ositions

PI [I I Ii- Tlie

projection Jti of _the

unit

_cell (or of Jt) on _Pi is

an

hexagon.

Pi

is a ii I Ii plane.