A geometrical analysis of limit layers and interfaces of lattices

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A geometrical analysis of limit layers and interfaces of lattices

Michel Duneau

To cite this version:

Michel Duneau. A geometrical analysis of limit layers and interfaces of lattices. Journal de Physique

I, EDP Sciences, 1992, 2 (6), pp.871-886. �10.1051/jp1:1992185�. �jpa-00246608�

Classification

Physics

Abstracts

61.50J 61.50K 61.70N 64.70K

A geometrical analysis of limit layers and interfaces of lattices

Michel Duneau

(*)

Centre de

Physique Thdorique,

Ecole

Polytechnique,

⁹¹¹²⁸ Palaiseau Cedex, France

(Received

3J October I99I, revised 3 December I99I,

accepted

13 December

I99I)

R4sum4. Les couches limites d'un rdseau _au

voisinage

d'un

plan quelconque

peuvent ^dtre d£crites par la mdthode de coupe. Les sites

correspondants

sont

proches

d'un r6seau moyen bi- dimensionnel contenu dans le

plan

limite. La modulation assoc16e _a la structure d'un cisaillement fractionnaire

simple.

Dans les _cas oh deux r£seaux de mdme densit6 sont lids _{par un} double cisaillement nous montrons que pour certaines orientations d'interface les deux couches limites ont le mdme rdseau moyen, favorisant airtsi _une coh6rence

optimale.

Abstract. Limit

layers

of a lattice in the

neighborhood

^of any

plane

can be described

by

the cut

method. The

corresponding

nodes are close to a 2-dirnensional _average lattice contained in the limit

plane.

The associated modulation has the structure of _a

simple

^fractional shear. ln _cases where two lattices of

equal density

_are related

by

_a double shear we show that for

particular

orientations of the interface both limit

layers

share the _same average lattice, thus

favoring

_an

optimal

coherence.

I. Introduction.

The _cut method was

extensively

used in connection with

quasicrystals

in order to

provide

models of

quasiperiodic

structures with

non-crystallographic symmetries (see

^for instance

II).

In such constructions _a

higher

dimensional space is introduced

(for

instance R6 in the _case of icosahedral

structures)

and the cut method allows _one to select nodes of _a

non-physical

lattice

(for

instance

26)

and to

project

these nodes in the 3-dimensional

physical

_space.

Thus,

quasiperiodic

structures such _as the 3-dimensional Penrose

tilings

_are

finally

obtained.

In this paper we show that the _cut method _can be used with benefit in the

ordinary

3-

dimensional space for the

description

of limit

layers,

interfaces and

displacive

transformations

between lattices. The _occurrence and the

importance

of

quasiperiodic ordering

in

grain

boundaries _was

already

noticed in

[2-4].

The limit

layers

of _a half lattice cut

by

a

plane

_are obvious instances of the cut method. The tools

developed

in the

quasicrystal

field _can be

applied

in this _more intuitive situation. As _an

example

these limit

layers

tum out to be close _to 2-dimensional lattices _even when the cut

plane

^is irrational. This result is based _on the construction of average lattices of

quasiperiodic

structures. We shall _see that the bounded transformation from _a

(possibly quasiperiodic) layer

(*) CNRS-UPRA 0014.

of _a lattice into its 2-dimensional average lattice is _a fractional version of _a shear transformation.

The fractional shears _are introduced and discussed. We show that

they provide

the

simplest

bounded transformations between lattices and therefore

give

the best

geometrical

candidates for

displacive

transformations. As mentioned above

they

_are involved in the construction of

average

lattices,

but

they

also prove to be useful in the

analysis

of _more

general displacive

transformations.

Shear transformations _can be used to

analyse

the

relationships

between two lattices of

equal density.

Four successive shears _are

usually required

to map a lattice

L~

onto a lattice

L~

^{in 3} dimensions. However for

particular pairs

of

lattices,

or for

particular

relative orientations this number _can be lowered to two

(or

_even

one).

Such

exceptional

_{cases are}

believed to have _a

physical meaning.

^For instance it _was shown in

[5]

that icosahedral twins of cubic lattices, as observed in

[6]

in the Almnfesi system, are

pair

wise related

by only

two

shears. On the other hand the relative orientations between

grains giving

rise to usual

coincidence lattices _can be shown to

correspond

to one or two shears with rational

parameters

[7].

In section 2 _we recall the definition and

properties

of linear shear transformations and _we introduce fractional shear transformations. Then _we show in section 3 that most

simple

bounded transformations between lattices _are

given by

such fractional shears. In section 4 _we consider limit

layers

of lattices bounded

by

a

plane

^of

typically

irrational orientation. We show that such limit

layers

are close _{to an} 2-dimensional average lattice

lying

in this

plane.

The transformation

mapping

the nodes of the limit

layer

onto the nodes of the average lattice is

proved

to be _a fractional shear. In section 5 _we consider the most

general

relative orientations between lattices of

equal density.

We show that the transition matrices _are the

product

of at most 4 shear matrices and _we

give

_a

simple algorithm

to compute ^{them. In} 2 dimensions the _most

general

transition matrix is the

product

of 3 shears. We consider the

example

of _two square lattices with _a arm relative orientation. In section 6 _we focus _on the

exceptional

_cases of the double shears and _we show that

they

_can be identified

by

a

simple

criterion. In section 7 the _same double shear _{cases are}

proved

to

yield

most _« coherent _»

interfaces,

the orientations of which _are

specified by special

lattice

planes

of _an intermediate lattice.

2. Shear transformations and fractional shear transformations.

A shear transformation is _a linear

mapping

which

reads, using

Dirac's bracket notation _:

+ Sl ~l St ~2 Sl ~3

~

"

~ ₊

'S) (~

" 52 ~l + 52 ~2 52 ~3

(2.1)

53 ~l 53 ~2 ⁺ 53

3~

S(x)

= x +

(« [x)

s

where _s is _a vector and _« is _a covector.

Therefore S transforms _a

point

_x

by

_a translation of constant direction _s with _an

amplitude (« ix) ills _ii.

The transformation has _an invariant

plane

defined

by (« ix)

= 0 which is

equivalent

to

S(x)

₌ x.

We shall

only

consider volume

preserving

shear transformations which _are characterized

by

the further condition

(« s)

= 0 which is

equivalent

to det

(S)

=

I. In this _case the invariant

plane

contains the shear direction _s and the inverse of S is

given by

S~ ^'

=

I

[s) (« _[.

Let

L~

=

AZ ^~ and

L~

=

BZ~

_{denote two lattices of}

equal density.

The 3 _x 3 real _structure matrix A is

regular

and its columns

give

_a

particular

basis of

L~, la)

=

jai,

_a~,

_a~)

with

a,

=Ae;,

^where

(e~,e~,e~)

is the standard basis of the 3-dimensional space. ^Thus A

specifies

the metric and orientation parameters ^of

L~. Similarly 16)

=

(bi, _b~, b~)

with

b,

=

Be;

where B is _{a structure} matrix of the lattice

L~.

Structure matrices map the abstract lattice

23,

in the space of _« indices _{», onto} the real lattices in the

physical

_space.

However, _any lattice has

infinitely

_many different basis and

consequently infinitely

_many

structure matrices.

Actually,

for any modular matrix U, I.e. with

integral

entries and

determinant _±

I,

_we also have

L~

= AUZ ³ in such _a way that A'

= AU is _an

equally

valid

structure matrix of

L~ corresponding

to the basis

(a')

=

(a(, a(, a()

with

al

=

A'e,

=

£

_{U~, a~.} Therefore _a

given

lattice

only specifies

_a class of _structure matrices

J

defined up ^to

right multiplication by

_a modular matrix.

We shall say that

L~

and

L~

are related

by

_a

simple

shear transformation S

= I ₊

is) («

if _:

I) L~

^and

L~

^have structure matrices A and B such that B

=

AS,

which is

equivalent

to

A B

= S

2)

the vector s is _an irreducible _vector of the lattice 23 and

(« is)

₌ 0.

This condition _means that the

corresponding

bases

la)

and

16) satisfy

the

following

relation _:

b,

₌

ASe;

=

A

(e,

+ «,

s)

= a, + «, w

(2.2)

where _w

=

As is _a lattice vector of

L~.

Notice that since _s in invariant

by S,

_w

=

As

=

Bs also

belongs

to

L~

so that

L~

and

L~

have _{a common} I-dimensional sublattice.

With _more

appropriate

choices of bases

(such

that a~ =

b~

₌

As)

this relation can take _a

simpler

form where the shear matrix S

simply

reads _:

II

o o

S= o I o

«1 «~ l

in such _a way that _:

b~=a~+«~a~

b~

= a~ + «~ a~

(2.3)

b~

₌ a~.

The above shear transformation S is defined and _acts in the space of indices. In

physical

space, the

corresponding

linear transformation which maps

L~

onto

L~

^is

given by

S~

₌ ^ASA One

easily

checks that _:

S~L~=S~AZ3=ASZ3=BZ3=L~

S~

^{has the}

following expression

_:

s~ ~li i i ^i>j Ii1 (2.4)

with _w

= As and

w =

(A

~)~ «.

Thus _w

gives

the shear direction and

w defines the invariant

plane

in the

physical

_space.

One

easily

checks that

(w [w)

=

0 which is

equivalent

to det

(S~)

=

I. The inverse of

JOURNAL DE PHYSIQUE -T 2, N's, JUNE 1992 34

S~

^{is the shear} transformation

Si~

=I-

[w)(w[.

It follows from the definition of

S~

^that

S~ (a~ )

=

b,

for I

=

1, 2,

3 which _means that

S~

maps the basis

(a )

of

L~

onto the basis

(b)

of

L~.

^More

generally,

if _x=

£n;a;

is _a lattice node of

L~ (n; integer)

then

S~(x)= £n;b;

is the lattice node of

L~

^{with the} same indices. Notice that since

w is invariant

by S~,w

has the _same coordinates with respect to

la)

and

(b).

Now,

if

L~

and

L~

are related

by

_a

simple

shear transformation

S,

_{we can}

easily

build _a bounded transformation

3,

called hereafter _a fractional

shear,

which satisfies the

following properties

_:

Ii

3 is _a one-to-one

mapping

of

R3

such that

3~

=

A3A _maps

L~

onto

L~.

2)

3 is _« bounded _» in the _sense that the

displacement 3(x)

_-x remains bounded when

x runs over R3.

3 is

given by

^the

following equivalent

definitions

(see Fig, 1)

_:

3(x

= x + Frac

j (« [xi j

_s

(2.5) 3(x)

=

S(x)

-Rnd

I("lxl)I

_S

where

Rnd(t)

is the

(integer valued)

round function

(I)

in such _a way that

Frac

(t)

= t Rnd

(t)

lies in the interval

( 1/2, 1/2].

By

definition of

3~

_we have _:

3~(x)=x+Frac j(«[A~~x)]As=x+Frac _j(w[x)]w 3~ (x)

=

S~ (x)

Rnd

j (w [xi

_w

^~~'~~

Clearly,

3 is bounded since the _norm of the

displacement 3(x)

_x

= Frac

j(« [x)

_s is

bounded

by

i

s11 /2.

Similarly, 3~ (x)

x =

Frac

j(w xi

_w is bounded

by

ⁱ_{w11 /2.}

Fig.

I. The square lattice L~ ^is transformed into L~

by

a shear S of direction _w

= (2, 1). Both lattices

lay

in the bundle of dashed lines A

parallel

to w. The linear shear involves

diverging

translations between

x in L~ and S(x) in L~ whereas the fractional shear 3 translates _x onto the nearest L~ node _on the _same line.

(1) The round function maps a real number _{t onto} the _nearest

integer

(with Rnd (n + 1/2) _n for all

integer

n).

3 is _a one-to-one

mapping

for it has _an inverse transformation

given by

3~

~(x )

₌ x Frac

[ (« xi

_s.

Similarly, 3~

is also _one-to-one with _an inverse

given by Ii ~(x)

= x Frac

(w xi

w. The

property

^that

3~

_maps

_L~

onto

L~

^{then follows from the} remark that

3~

differs from

S~ by multiples

of _w which also

belong

to

L~ (see (2.6)).

Using coordinates,

the

mapping 3~

is

explicitly given by

:

x ₌

£x;

_a,

-

3~ (x)

₌

£

_{x, b~} ^Rnd

[£

_«;

_x;j

_w

(2.7)

where

(x~,

_x~,

x~)

denote the coordinates of _x with respect to the basis

(a).

The coordinates

~y~, y~,

y~)

_{of y}

=

3~ (x)

with respect to the basis

(b)

_are therefore

given by

_:

y~ = x~ Rnd

£

_{«; x~ w;}

(2.8)

where

(w~,

_w~,

w~)

^{denotes the} coordinates of _w with respect to both bases

(a)

and

16 )

As _a

conclusion,

the fractional shear

3~

has the

following specific property

: it translates _a

node of

L~

onto the nearest node of

L~ lying

_on the _same lattice line

parallel

to

w ; this follows from the

amplitude

Frac

(w ix)

of the translation

being

bounded

by 1/2

and from _w

being

irreducible.

3.

Simple

bounded transformations between lattices.

In this section _we show that the

simplest

bounded transformations between lattices _are

provided by

fractional shear transformations _as defined in the above section. The terms

«

simple

_» and _« bounded _»

require

_a clear definition which is

given

below. It _{tums out} that such transformations _are

possible only

between

particular pairs

of

lattices, namely

^those for which the transition matrix is

equivalent

to a shear matrix. More

complex relationships

between lattices will be considered in section 5.

By

a

simple

bounded transformation _{we mean a}

global

one-to-one

mapping 4

of the 3- dimensional space such that

4 (x)

= x + w

(x)

u where _q~ is _a bounded real function and where _u is _a fixed vector. Thus

4

transforms any

point

_x

by

_a translation of _constant direction

u and bounded

amplitude

_q~

(x )

i u ii.

This definition is devised to

correspond

_more or less to

simple displacive

transformations with the difference that in _{our context no} energy consideration is involved.

As will become clear in the

following

^the fact that such _a transformation maps a

given

lattice

L~

onto some lattice

L~

is not obvious. This

actually corresponds

to

strong

restrictions

on

L~

^and

L~.

THEOREM.

Let

L~=AZ~

and

L~ =BZ~

_denote

two lattices and _assume

4

^is a

simple

bounded

transformation,

_as defined

above,

which maps

L~

onto

L~.

^Then

L~

^and

L~

have the _same

density,

_u is _a lattice direction of both lattices and after _a convenient

scaling

_(q~

- rq~ and

u - r~

u)

^which does not

change 4,

_u is _an irreducible _vector of

L~

^and

L~.

PROOF.

One

easily

checks that if

L~

and

L~

had not the _same

density,

a bounded _one to one

mapping

from

L~

to

L~

^{would be}

impossible.

Now for any x and _y in

L~

we have _:

w(x)- w(y)- w(x-y)

=

iq~(x) q~(y) q~(x-y)i

u.

First,

_we can assume this

identity

does not reduce

always

to 0

=

0 otherwise the

amplitude

q~ would be _a linear

(and bounded)

function in this _{case q~} would vanish and

consequently L~

₌

Lb.

If the above

equation

is _not trivial

then,

since the left hand side

belongs

to

L~,

u is _a lattice direction of

L~.

Since

4

is assumed _to be _one-to-one from

L~

to

L~,

^for any x and y in

L~

one can find

z in

L~

such that 4

(x) 4 (y)

=

4 (z).

This

implies

_:

0=w(x)-w(y)-w(z)

_=x-y-z+

iq~(x)-q~(y)-q~(z)iu

which in _tum

implies

that u is also _a lattice direction of

L~.

Consequently

^both lattices have _{a common} bundle of lattices lines A

parallel

to u. The

mapping

4 thus transforms the

L~

^nodes on such _a line into the

L~

nodes

lying

on the same line. Since the

displacements

_are assumed to remain

bounded,

the I -dimensional lattices

L~

ⁿ A and

L~

n A must have the _same

density,

I,e, the _same lattice parameter. ^{From this}

remark we conclude that

q~ and _{u can} be rescaled

by

_some real number

r#0

(q~ -rq~ and _u

-r~~u)

in such _a way that _u is _a

generator

^{of both lattices}

L~

n A and

L~

n A. Thus _u is _an irreducible vector of

L~

^and

L~.

At this

point

_{we can} raise _{a new}

question

_:

assuming _L~

and

L~

are related

by

_some

simple

bounded

transformation,

is there _a

particular

transformation of this type ^{which minimizes all}

displacements?

The _answer is

positive

and the solution is

given by

_a fractional shear transformation.

Actually,

_{on any} line A the I-dimensional lattices

L~

n A and

L~

ⁿ A are

simply

translated from each other

(they

have the _same basis _vector

u).

So there is

only

_one

way ^to minimize the

displacements

_{: map any} node of

L~

ⁿ A onto the nearest node of

L~

n A. But this solution is

precisely

what is obtained

by

the fractional shears _as shown _at the end of the

previous

section.

Finally,

if _two lattices are related

by

a

simple

bounded

transformation, they

_are also related

by

a fractional shear which minimizes the

displacements.

This fractional shear is therefore _a

simple

and natural candidate for _a

possible displacive

transformation between these lattices.

4. Limit

layers

and their average lattices.

In this section _we show that limit

layers

of _a lattice in the

vicinity

of _a

plane

_are close _{to a} 2- dimensional lattice.

Furthermore, simple displacements transforming

the limit

layers

into their average lattices _are

given by

fractional shears _as described in section 2.

Let

L~

=

AZ ^~ denote _a 3-dimensional lattice with structure matrix A. Assume this lattice is cut

by

_a

plane

lI of

equation (v ix)

_{= c} and _assume for

simplicity

that _c

= 0. lI may be

considered _{as an} ideal free surface of the lattice _{or as an} interface

plane

with respect to an

other structure. In either _case different

phenomena

will contribute to

change

the structure in the

neighborhood

of lI _so that _a definite limit

layer, supposed

to contain all

deformations,

is

hard to

specify.

We shall

only

consider the _case where lI is

completely

irrational with

respect

to

L~. Rational,

_or

partially

rational orientations of lI

give

rise _{to a}

simpler

situation because the cut

plane

contains _at least _a I-dimensional sublattice.

For this _reason, and for

geometrical

_reasons

explained below,

_we shall consider different but

particular

^limit

layers satisfying

the condition that their width

correspond

to some lattice

vector. In other

words,

we consider limit

layers

bounded

by

two

planes

lI-w/2 and

lI+ w/2 where w is an irreducible _vector of

L~ (which

does not

belong

to

10.

Once

w is

chosen,

_{we can} scale the normal vector v of lI in order that

(

_v

w)

= I. The limit

layer

thus contains all nodes _x such that 1/2 _<

(

_v

ix)

_w 1/2 and each node _x has _a

decomposition

_:

x =

(

_{v x}

)

_{w + y}

(4. 1)

where y = x

(v ix)

w lies in lI for the above

equation implies ( vi y)

₌ 0.

Since _w is _a lattice vector of

L~,

^the

mapping

defined

by

x _- y = x

(v [x)

_{w maps} all nodes of the limit

layer

_{on a} 2-dimensional lattice contained in lI. This lattice is

simply

the

intersection of lI with the bundle of lattice lines

parallel

to w

(or

the

image

of

L~ by

the

oblique projector parallel

to w onto

10. Moreover,

since the width of the

layer

is

specified by

the irreducible _{vector w,} each lattice line

parallel

to w contains _a

unique

node inside the

layer

and

consequently

the

mapping

is one-to-one

(see Fig. 2). Finally,

the 2-

dimensional lattice thus obtained _can be considered _{as an average} lattice of the limit

layer

and the

displacements

are bounded

by

ⁱ_{w11 /2} and

parallel

to w.

The

mapping

x _- y defined above is _{not a} shear transformation since

( vi w)

is different from 0.

However,

for _x inside the limit

layer,

this

mapping

_proves to be

equivalent

to _a

fractional shear transformation of the form _x

- y = x + Frac

(w [x)

_w which is defined _as

follows _:

Since _w

belongs

to

L~

^{there exists} a

reciprocal

vector p in

Lfl (different

from

v),

^such that

(p [w)

=

I. Now _we define

w = p v and _we observe that _:

I) (w[w)

=0.

2) (w[x)

=

(p[x) -(v[x)

_so that for any lattice node _x of

L~

we have Frac

[(w ix)]

=

Frac

[- (v[x) ].

Therefore for any lattice node _x inside the limit

layer

_we have Frac

(w [x) ]

=

(v [x)

and

consequently

the

mapping

reads _:

y=x-

(v[x)

w=x+Frac

[(w[x)]w. (4.2)

This

equation

shows that the

mapping

of the limit

layer

_onto its average lattice is

given by

_a

fractional shear _as defined in section 2. The shear direction is

given by

the lattice vector w and the invariant

plane

is

specified by

_w with the condition

(w [w)

=

0 satisfied.

The definition of _w

= p v is _not

unique

because there _are

infinitely

_many

reciprocal

vectors such that

(p[w)

=

I. The

corresponding

linear shears

S~

=I ₊

[w)(w[

_are

different but the associated fractional shears

3~

_are

equal

because Frac

[(w ix)

] is

independent

of _p for _x in

L~.

^{Notice also that all} shears

S~

transform the lattice

L~

^into a

unique

lattice A

spanned by

w and

by

the 2-dimensional _average lattice of the limit

layer

_so that lI is _a lattice

plane

of A.

fl+w/2

fl-w/2

w

Fig. 2. A layer is specified by the two planes 1I w/2 and 1I ₊ w/2 where _w is _an irreducible lattice

vector and His _an

arbitrary

direction. The nodes (black dots) lying inside the layer ^have an average

lattice (white dots) in 1I _on which

they

_map

by

translation bounded

by

ii w [[/2.

Conversely,

_assume A is _a lattice such that A

=

SL~

for _some shear S

= I ₊

w) (w

where

w is _an irreducible _vector of both lattices. If lI is _a lattice

plane

of A such that

w and the 2-dimensional lattice A n lI span

A,

then A n lI is the average lattice of the

layer

of

L~

^bounded

by

lI w/2 and lI ₊ w/2.

5. Shear factorization of _a transition matrix and bounded transformations between lattices.

If

L~

^is a

given

lattice and

L~

^is a different lattice of

equal density,

the transition matrix T

=

A B _cannot in

general

reduce to a

simple

shear matrix _even with the free choice of the bases of the lattices.

Actually,

_a shear matrix with rational direction

only depends

_on 2

independent

real

parameters

^{whereas the} set of lattices of

given density

is _an 8-dimensional manifold. For this _{reason we} guess that _a

general

transition matrix will be

equivalent

_to the

product

of 4 different shears. A

rigorous proof

is

given

below in 3 dimensions

(see [8]

for _a

general proof

in _n dimensions where _{n +} I shears prove ^to be

necessary).

The _cases where

only

_one or two shears _can transform

L~

into

L~

are therefore

exceptional

and

correspond

to

particular pairs

of lattices _as will be _seen in _more details in section 6. If

L~

^{is defined}

only

_up to _a rotation we see that these

exceptional

_cases could

possibly point

out

particular

orientational

relationships.

If

L~

^is

equal

to

L~,

_up to _a

rotation,

the _same conclusion holds and it _can be shown that the relative orientations associated with the existence of coincidence lattices

correspond

to such

exceptional

_cases

[7].

In

[5]

it _was shown that the relative orientations

giving

rise to icosahedral twins of cubic lattices _as observed in

[6], namely

2 ar/5 rotations around

pseudo

5-fold

axis,

also

correspond

to transitions matrices

which _are

equivalent

to a double shear.

Let T denote _a real 3 _x 3 matrix with determinant I. For instance T

=

A B is _a transition

matrix between two lattices

L~=

^{AZ ~} and

L~ =BZ~

_of

equal density.

Then T _can be

decomposed

_as a

product

of 4

simple

shear transformations

So, St,

_S~ and

S~.

T

=

so s~ s~ si _(5.1)

' So2 So3 o o

So

= o I o S~ ₌ ^{o I o}

°

° '

31

532 with

0 0 S12 S13

S~

= s~j s23 S~ = 0 0

(5.2)

o o 0 0

This statement holds for almost all matrices T.

Actually

it

requires

the

following

conditions _:

T~i

# 0 and T~~

T~i T~i

_T~~ # 0. If these conditions _were not met, one

easily

checks that _a

permutation

matrix P _can be found such that

they

_are satisfied for

T'

=

TP

(taking advantage

of the fact that det

(T~

#

0).

The

proof

of the factorization

(5,I)

is

given by

the

following algorithm.

. Find _a shear matrix

So,

^{of the form}

given

in

(5.2),

such that

(So)~~

T

= U satisfies

Uii

= I and

Uii

U~~

Ui~ U~i

=

I. This is

easily

achieved since _:

1) lfll

_"

l~ll

_{S021~21 S031~31}

~

l

~) ~fll ~22 ~12 ~21

_"

l~ll

_{1~22 1~121~21} 503(1~221~31 1~21 1~32) ~ l.

The second

equation gives

_so~ and the rust _one afterwards

gives

_so~,

provided

the condition mentioned above is satisfied _:

l~ll1~22

+ 1~121~21

~~~

1~221~31 1~21 1~32 1-

Tii

₊

so~T~i

SW "

~m 21

. Find _a shear matrix

Si

such that

U(Si )~

=

V satisfies

Vi

j ⁼

V~~

= I and

Vi~

₌

Vi~

= 0.

These

equations

read _:

1) V22

_"

~22

_S12

_~21

"

~) l~12

_"

~12

_S12

" °

3) ~'13

_~

~13

_S13

" °

They

_are satisfied

by Uking

_si~

=

Ui~

and _si~

=

Ui~.

^{Now V reads} :

0

1

~'"

~'21 ~'23

'31

~'32 ~'33

. Find _a shear matrix S~ ^{such that}

V(S~)~

=

W satisfies

W~i

= W~~ = 0 and W~~ = I.

These conditions read _:

1)

_f~'21

"

~'21

_{S21 ~} ^°

~)

_f~'23

"

~'23

523 " °

3)

f~'33 "

"33

523~'32 _"

They

_are satisfied

by taking

_s~i

=

V~i

and s~~ =

V~~.

^{Now W reads} :

Ii

^{o o}

w= o i o.

W31

W32

. The last

step yields

_a shear matrix S~ ^{such that}

W(S~)~~

=

I,

I,e. S~ =

W,

_so that

S31 ~ f~'31 ~tld 532 ~

~'32.

Finally

_we have obtained

(So)~ T(Si)~ (S~)~ (S~)~

=

I which reads _:

T

=

SOS~S~SI.

The above factorization of the transition matrix T=A

~~B

_{shows that}

any ^two lattices

L~

and

L~

with the _same

density

_are

usually

related

by

the

product

of four shear

transformations _:

Lb

~ $A,_O~A,₃ ~A,₂~A, _I

~a

with

S~,

_{I =} AS

; A

By replacing

each linear shear transformation

S~,, by

^the

corresponding

fractional shear

3~,

_{; =} ^AZ _;^A _~, as defined in section

2,

_{we see} that _a bounded transform- ation from

L~

to

L~

^is

given by

the

composition

of four fractional shears _:

3~,

_o

3~,

~

3~,

~

3~,

1.

As _an

example

_we shall consider here the 2-dimensional _case of _two square lattices with _a arm relative orientation.

First the factorization theorem in 2 dimensions reads _as follows _:

If T is _a 2x2 matrix with determinant

I,

there exist 3 shear transformations

So, Si

and S~ ^{such that}

T=

SOS~SI (5.3)

where

I

oy

i

oj

^I

_sjj

~°

o 1

~~

s2

~~ o

~~'~~

The

product

of these shear matrices reads

So S~ Si

=

~

~ ~°~~ ~° ~ ~~ ~ ~° ~~~

(5.5)

s~ I ₊ s~ si

The three

parameters

s~ si and s~ are

easily

derived from the entries of T

(notice

however that if

T~i

=

0,

T _must be

replaced

with TP where P is _a convenient

permutation matrix).

If

L~

^and

L~

^{denote the} two square lattices related

by

_a arm

rotation,

the transition matrix T is

simply

the arm rotation and the above factorization reads _:

/jl -lj _[I I-/j[

0j[1 1-/j _~~6)

~ _~~

l 0 II

fi

0

According

to the definitions of section

2,

the

corresponding

fractional shears _are

given by

30(x)

= x + Frac

[(I /) _x~]

ei

~2(Y

" Y + Frac ~yl~

II

e2

(5.7)

Ii (z

= z + Frac

[(I /) _z~]

ei

% / j~ ^I /~m ^~ _~

j

' ~ *-w ~ *+ ~ ~a

/+m

^{~ ~}

^{~ d} _~~m

f / ~ j

w~ , ~ ~ ~

GO

W~G*~

. OW C~ ~B

+~ ~ +~ ~ +~

~ ~ ~ /

~*Q~

~R

'~

~ ~

*9e~

~

_wv

j~ ^, j--~, ^~

~, ~/, ~i /,

Fig.

3. The _one-to-one

mapping

between the two square lattices at arm is

given by

the

composition

of

3 successive fractional shears.

The

composition

of these three

mappings

therefore maps

L~

onto

L~ (see Fig. 3)

and writes _:

30 3~ 30(x)

= x + Frac

[(I /) _x~]

ei + Frac

[(xi

+ Frac

[(

I

/) _x~])/ ~fi]

e~

+ Frac

[(

I

/)(x~

+ Frac

[(xi

+ Frac

[(

I

/) _{x~] )/} II _)]

ei

(5.8)

This formula shows that the

displacements

_are

always

bounded

by

ⁱ_{et +}

e~/2[[

=

l12.

This transformation _can also be

computed explicitly using

coordinates associated to the successive

lattices,

like in section 2. As _a

result,

_x

= xi at +x~a~ ⁱⁿ

L~

is

mapped

onto

y ₌ yj

bi

+ y~

b~

ⁱⁿ

L~

^with :

yj = xi Rnd

[(I /)x~]

Rnd

i(

i

/) _(x~

_Rnd

(/(x~

_Rnd

_{( (i} /) _{x~))/2))1 (5.9)}

y~ = x~ Rnd

[/ _(xi

_Rnd

_{( (I} /) _{x~) )/2]}

Such _a bounded one-to-one

mapping

is not obvious. At first

glance

_a natural candidate for

such _a map would consist in

mapping

_a

point

_x of

L~

onto the

origin

_y of the square of

L~

^{which contains} x.

However,

_squares of

L~

may contain

0,

_or 2

points

of

L~

so that the _one- to-one

property

^could not be satisfied.

6. The double shear _case.

The _cases where the transition matrix

T=A~~B

between _two lattices

L~ =AZ~

_and

L~

₌

BZ~

_{is the}

product

of

only

two shear transformations _seem to be of

physical

interest.

This

algebraic property

^{of the transition matrices has} a more intuitive

meaning

_{as we} shall _see below.

Moreover,

a

geometrical

characterization is welcome since the transition matrices

between _two lattices _are defined up ^to

right

and left

multiplication by

^{modular matrices}

(owing

to

changes

of

bases).

Let _us first consider the _case where T

= A B ₌ S is _a

single

shear transformation for an

appropriate

choice of the structure matrices A and B. Then, _as

explained

^{in section}

2,

both lattices have _{a common} lattice _{vector w}

=

As

=

Bs and

they lay

_on a common bundle of lattice lines.

In Fourier space, ^{the dual lattices}

Lfl

and

Lf

have structure matrices A*

=

(A~~)~

and B*

=

(B~~)~.

If the bases

la)

and

16)

_are ^{such that} _{a~ = b~ = w,} ^{like in}

(2.3),

the

corresponding

dual bases

la *)

=

jai, at, at

and

16 *)

=

16(,

_b~*,

b?)

_are related

by

the

following equations

:

at

=

b? (6.1)

~~"~~+~l~l+~2~~.

All lattice

planes corresponding

to

reciprocal

vectors nj

al

+ n~

at belong

to both lattices.

These

planes

are those which _are built

by

_means of any

pair

of lattice lines

parallel

_to

w. The dual lattices

lay

on a stack of

planes parallel

to the

plane spanned by

at

and

at

and their intersection

generically

reduces to the 2-dimensional lattice

spanned by

the _same basis.

The double shear _case is characterized

by L~

and

L~ having

structure matrices

A and B such that T

= A B is the

product

of _two shear matrices _:

~

~2 ~l (~

+

'52) (~2' )(~

+

'Sl) (~l )

(~ ~)

Tx

= x +

(«i ix)

_{si +}

(«~[x)

_{s~ +}

(«~[sj) («~ ix)

_s~

where si and s~ are two different irreducible

integral

vectors

defining

the directions of the shears in the space of indices. The invariant

planes

_are defined

by

_«~ and «~ and

satisfy

(«~[s~)

=

(«~[s~)

=

0.

Since A B

=

S~ St,

we may define the structure matrix r

= AS

~ ⁼ BS

p

of _{a new} lattice A

=

rZ~.

_{A is}

a

simple

shear deformation of both

L~

and

L~

^and can be considered _{as an}

intermediate lattice between them: if

S~=AS2A~'

and

S~=BSi~B~'

_{we have}

r

=

S~A

₌

S~

B _so that A

=

S~ L~

₌

S~ L~.

^{These shears} are

explicitly given by

_:

s~

₌ ^I +

jw~j jw~j

~ ~

s~

= i +

iw~i iw~i

with w~ = As

z, w~ =

(A

_{~)~ «~, w~}

=

Bsi

and w~ =

(B~

_{~)~ «~.}

Now, using

the above relations _{we see} that _:

rsi

= AS

~ si =

As1

+

(«~[si) As~

=

BSj

_si

=

Bsi

= w~

(6.4) rs~

₌ AS

~s~ =

As~

₌ BS

j

s~ =

Bs~ (« ii _{s~) Bsi}

= w~.

Consequently

the three

pairs

of _vectors

(rsj, rs~)

in

A, (As1, As2)

in

L~

^and

(Bsj, Bs2)

in

L~

span the _same lattice

plane

W which therefore

belongs

to the three lattices. W is invariant

with

respect

to both shears S~ ^and

S~

^{since the} directions of the shears w~ and w~ are in W. Therefore

L~, L~

and A

lay

_{on a} common stack of lattice

planes parallel

to W.

Finally,

the double shear

Sp

^'S~ ^which maps

L~

onto

L~

^{transforms the}

points by

translations

parallel

to W.

In Fourier space, the existence of _a common sequence of lattice

planes implies

^{that the}

reciprocal

lattices of

L~

^and

L~

^have a I-dimensional intersection

spanned by

a

reciprocal

vector _w

orthogonal

to W. The Miller indices of

w are

proportional

to sj x s~

(and equal

if this vector is

irreducible). Actually,

the dual lattices

Lfl

and

Lt

have

respective

structure matrices A*

=

(A

_~)~ and B*

=

(B~

_~)~ which

satisfy B*~~A*

=

T~=

(I

+

i«i) isii)(I +1«2)iS21),

therefore si x s~ is invariant

by

^T~ and A* _{si x s~}

=

B* si x s~

belongs

to both

reciprocal

lattices.

7. Interfaces and smooth transitions in the double shear _case.

An ideal

planar

interface between two lattices

L~

and

L~

is defined

by

5

parameters

: two of them

specify

the orientation of the interface

plane

and the others

give

the relative translation

between the lattices. The double shear _case has

interesting

consequences with

regards

to the

description

of

possible

interfaces

planes

between

L~

and

L~.

More

precisely

_we show _now that in this situation

pa~icular

interface orientations _are

pointed

out for

simple geometric

reasons.

Assume His _a lattice

plane

of the intermediate lattice A defined in the above section which satisfies the

following

conditions _: w~ and A n lI span

A,

and

similarly

_w~ and A n lI span A

(see below).

We consider the

layer

of

L~

^bounded

by

lI

wJ2

and lI ₊

wJ2

and the

layer

of

L~

bounded

by

lI-

w/2

and lI+

x/2.

NOW these two

layers

have the _same average

lattice, namely

the 2-dimensional lattice

A n lI. This

actually

follows from the construction of average lattices of

layers explained

at the end of section 4.

More

precisely,

the fractional shear

3~

which maps the

L~

^nodes

lying

between lI-

wJ2

and lI+

wJ2

_on the

plane

lI is

given by

:

3~(x)

= x + Frac

[(w~[x)

_w~

(7,1)

3~

_maps the nodes of limit

layer

of

L~

onto the 2-dimensional average lattice A n lI.

Similarly,

the fractional shear associated to

S~

^is

given by

_:

3~(x)

= x + Frac

[(w~[x)

_w~

(7.2)

3~

_maps the nodes of limit

layer

of

L~

onto the _same 2-dimensional lattice A n lI

(see Fig. 4).

The condition

bearing

_on the

planes lI,

mentioned at the

beginning

of this

section, requires

that the 2-dimensional lattice An lI spans A with both irreducible vectors w~ or

w~. Therefore if

v is the

reciprocal

vector of A* associated _to lI _{we must} have

~~' ~a)

~ ~ ~'~~

~~' ~b)

" ~ ~.

~ o _o

~ o

~ o ^°

~ o

~ o

~ o

~ o

~ o

~ o o

~ o ^°

~ o °

a

~ o o

~ o

~ o °

~ ^o _~ a a

~ o o

o o °

~ o °

a D a

~ o o

o o °

~ o °

a a a a

~ o

~ o °

~ o D a a a D D

~ o

~ o

o ^o D D a D a a a a

o o

o o °

a a a a a a a a a

o ^o

'I

a D a D D D a a a a

a D D a D D a D a a a a

a D D D a D D a D D D D a

Fig.

4. The limit

layers

of L~ ^and L~ arc

parallel

to a lattice

plane

of the intermediate lattice A.

They

have the _same average lattice _on which

they

_map

by simple displacements

of _constant direction

and bounded

amplitude.

Finally

the double shear _case is characterized

by

the

following prope~ies

_:

I) _L~

and

L~

have _{a common} lattice

plane

W

spanned by respective

bases

(As

i,

As~)

and

(Bsi, Bs~)

such that

Bsj

=

As i +

r~AS~

and

As~

=

Bs~

_{+ ri}

Asi

with si and s~ in

Z~,

and

ri, r~ real numbers

(see Eq. (6.4)).