Planes and rows in icosahedral quasilattices

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Planes and rows in icosahedral quasilattices

T. Kupke, H.-R. Trebin

To cite this version:

T. Kupke, H.-R. Trebin. Planes and rows in icosahedral quasilattices. Journal de Physique I, EDP

Sciences, 1993, 3 (7), pp.1629-1642. �10.1051/jp1:1993205�. �jpa-00246821�

Classification

Physic-s

Absfi.acts

61.50E 61.50J 61.80J 63.90

Planes and

_rows

in icosahedral quasilattices

T.

Kupke

^and H.-R. Trebin

Institut far Theoretische und Angewandte Physik, Pfaffenwaldring 57, D-7000

Stuttgart

80,

Germany

(Receii,ed 8 December J99?, accepted in final form 9 March J993)

Abstract. The planar and linear substructures of _a threedimensional icosahedral Ammann- Kramer-Penrose quasilattice have been analysed.

Infinitely

_many families of planes ^and rows are

existing

such that each

family

_covers all vertices of the quasilattice. ^{The members of} a

family

of parallel planes are separated

by

at least three but

a finite number of different distances, whose sequence is quasiperiodic and _can be

uniquely

characterized by _a

strip-projection

method from _a

two-dimensional

periodic

lattice. The _same is valid for the sequence of vertices _on each _row. The

vertex pattem ^{in each} single plane ^{results from} a

strip-projection

from 2 ₊ 2 dimensions. In

addition _to several

specific

patterns we have determined the _vertex occupation ^densities of planes and _rows and their statistics. The families of planes with

highest occupation

densities

correspond

well to minima of

backscattering profiles,

which have been calculated in numerical simulation of

heavy ion-channeling

in the

primitively

decorated

quasilattice.

1. Introduction.

The geometry of

periodic

lattices is rich but its

variety

is

by

far not

comparable

to that of

quasiperiodic

lattices. A computer

model,

which _we constructed for the well-known

icosahedral Ammann-Kramer-Penrose

quasilattice [I]

indicated that the vertices _can be

assigned

to

planes

and _rows

[2].

But contrary to

periodic crystal lattices,

the

planes

^of a

family

neither _are stacked

equidistantly

_{nor are}

geometrically

_congruent.

Nonetheless,

numerical simulations

[2]

Showed that the

planes guide

^{ions in}

channeling-backscattering,

^{and the}

feasibility

of

channeling

has been confirmed in the meantime

experimentally

for

decagonal quasicrystals [3].

A mathematical

proof

for the existence of

planar

and linear substructures in the Ammann-

Kramer-Peniose quasilattices

has been

given by

Katz and Duneau

[41.

From their

theory

follows

immediately,

that in the

quasilattice

there _are

countably infinitely

_many families of

planes

and _rows. The sequence of vertices _on each row and the vertex pattem ⁱⁿ each

plane

_are

quasiperiodic

_as well _as the Sequence of

separations

of

planes

in _a

family

_or the

projection

of _a

family

of _{rows onto an}

orthogonal plane.

All these

quasijeriodic

_{patterns and sequences}

can be constructed with the

strip-projection

method

[5

and _are characterized

uniquely by

_a lattice and the width of the

strip.

We have

applied

the

theory

of Katz and Duneau [41 to construct several of the

patterns.

^A

quantity

most

important

for the

ability

of

planes

and _{rows to} channel ions is the

density

of

vertices, according

to the

theory

of Lindhard [61 for

periodic

lattices. We have determined those families of

planes

and _rows which carry the

largest occupation density.

Their normal or

their direction vectors,

respectively~

are

pointing

^either

parallel

or

perpendicular

to

high

symmetry ^{directions of the} icosahedron. In _one

family countably infinitely

_many

occupation

densities _are found. No two

planes

_or rows are congruent.

An

investigation

of the _structure of rows in twodimensional

quasilattices

has been

published

in reference

[71.

To

clarify

the notations~ _we repeat ^the

strip projection

method in section 2. In section 3 the

correspondence

of

planar

and linear _structures in

parallel

and

orthogonal

_Space is outlined.

Quantitative

data _on

planes

_are

Supplied

in section 4, and _{on rows} in Section 5.

2. The

strip projection

method.

An

important

tool for _our

analysis

is the

strip projection

method

[51.

In its _most

simple

form it enables _us to construct a one-dimensional

quasilattice.

Let _us review

briefly

In order to tile _a

straight

line, called the

parallel

_{space Ej(,} with two types ^{of intervals in} a

quasiperiodic

_way (see

Fig. 1)

consider the square lattice

Z~

_{in the}

plane

^E~. The

slope

of Ej( with respect to the canonical basis of E~ is irrational. Parallel to Ej( a

strip S(

is obtained

by shifting

the unit square of the lattice

along

the line _Ej) (the

subscript

U indicates that the

strip

is

produced by

^the

shifting

of the unit cell of the lattice). The one-dimensional

quasilattice consisting

of the vertices,t. is obtained

by

the

orthogonal projection

of the lattice

points

n ~

Z~

_{within the}

strip S[,

_onto

_El

,r =

ar~j

^(n), fi ~

S(

_n

Z~ _II

Projection

onto

perpendicular

_space

E[ _{yields points}

.<'

= ar~i in ), n ~

S(

_n

Z~, ₍₂₎

which _cover a finite section of

E[

of the width of the

strip S[.

Since the

slope

of

Ej( is irrational the distribution of the

points

x' is dense and

homogeneous.

In a first step ^{the method is}

generalized

_as follows. Consider the n-dimensional lattice Z" in the Space

E~

=

Et

e

E[~~

₍₃₎

Et

is the d-dimensional

parallel

space. The

strip St

is defined

by shifting

the n-dimensional unit cell of Z~

along Et.

The

quasiperiodic tiling

in

Et

results from the

orthogonal projection

of the

lattice

points

_n of the

strip St.

The

special

_{case n}

= 5, d

=

2

yields

the Penrose pattern. ^{For the} three-dimensional Ammann-Kramer-Penrose

quasilattice

the data _{are n}

= 6, d

= 3. The pattern ^{in the}

physical, parallel

_space

El

consist of _two types ^of rhombohedra, _a flat _one and _a thick _one, with identical

facets and

edges [8, Il.

The

projection

of the six-dimensional unit

hypercube ^W~

_onto

E(

_{generates a} rhombic triacontahedron

ar~j (W~).

The

points

x'

=

gr~j

^{(n), n ~}

S[

n

Z~

_fall

within this _convex

polyedron.

Their distribution is dense and

homogeneous.

In the

following

we

frequently

_use a

generalized

version of the

strip projection

method, where the

hyperlattice

must not be cubic and where the

strip

it _not

generated by shifting

the unit cell rather has

arbitrary

width.

y

. o ^{. . . .} . . . .

. o o . . . o . e .

. °

Sl

. . . . .

Any

^{direction D in} E(~ ^defined

by

two

points

_xj~ x~ ~ E(~ ^has a

unique

partner ^direction

D' in

E(,

defined

by.i(

and

xi ~E(.

_The

_planes perpendicular

to D _are denoted

P~j~

(I

~Z numbers the

infinitely

_many members of the

family,

the

subscript

ID _or

ID' indicates

quantities,

which

depend

on the directions D and D'

respectively).

Each

plane P,j~

includes at least _{one vertex x.} The

position

of _a

plane P,j~

I D is

given by

the

orthogonal projection ar~(P,j~)

onto the direction D. Each

plane

has _a

unique

_partner

P,'jo,

^{ID' in} E

(,

which is

orthogonal

to D' and

intersecting

^D' at

ar~,(P,'jo,).

The natural

isomorphism

is valid also if restricted to D _or

P,j~.

Hence every

point.i

of D _or of _a

plane P,,~

ID has its counterpart on D' and P,'jj~, I D'

respectively.

Remember that all

points

x' _are contained in the

triacontahedron ar~i

(W~).

4. Planes.

Choose _an

arbitrary

direction D in the

quasilattice.

We _are interested in the

following questions

about the

family

of

planes P,j~.

. How can the

position ar~(P~jD)

^{be determined and what distances} _appear between

neighbouring planes

^?

. How _{can we} calculate the

occupation density

of _a

Single plane

and its average over the

family

?

. Which families of

planes

Show the

highest

_average

occupation

densities ? This

point

is very

important

for

channeling experiments.

. How _can the pattern ^{of vertices in} a

single plane

^be constructed ?

In order to determine the

positions

of the

planes

P~/~ i D note that E~ _can be

decomposed

into _a direct _sum of

subspaces

:

E~=E(%E( =D%P,/~%D'%P,'~D>. ⁽⁴⁾

Therefore the lift of the

projections

_no

(P,/~)

and

ar~.(P,'/~,)

^is contained in _a space

W=D%D', (5)

which itself embraces _{a two} dimensional

plane

of the

hyperlattice.

Katz and Duneau [41 have demonstrated~ that the

positions

of the

planes

P~/~ ^and

P)j~,, respectively,

are determined

by

a

strip projection technique.

The

corresponding

lattice is the

projection arw(Z~)

of the

hyperlattice,

the

strip

SW ^is the

projection arw(S[) _(Fig.

_{3). This fact is}

_{expressed by}

_the

equations

~j "D(P'/D)

"

"D("W(z~)

n

sW), (6)

~J ~rD(Pi/D.)

=

~rD(~rw(Z~)

n

Sw>, (7)

which _are based _on the identities

ar~(Z~

n

S[)

=

ar~(arw(Z~)

n

Sw), (8)

"D'(z~

n

s[)

"

"D'("Wlz~)

n

sW), 19)

derived from

equation (4).

We have

applied

this

strip projection

method for many directions D and have

found,

that _at least three different

lengths separate

^the

planes

in each

family. Examples

for five families _are

displayed

in

figure

4, the

separations being

tabulated in table I. The

edge length

^of the

rhombohedra is assumed to be

I/,I

~i

_~

~ W

=

De

D'

SW

~

o

o

o D

o o

o

o <

«D(fl/D)

o o

Fig.

3.

Strip projection

method in W D e D' _to obtain the

positions

_7rD(P,~D) and 7rD.(P,[DI of the planes P,,D ^and P,',D.

respectively.

The smallest basis cell of the lattice

7rw(Z~)

is marked

by

the thick

lines.

---_---_---_-- -_---_---_---_---

-5.loo 5.~oo

^~

5.j~~~~~~~~~~~~~~'~~~~~~~~~~~~~~~~~'~'~'ooo

_~

5.

jjj~~~~~~~~~~~~~~~~~~~~~~~~~~~~~)~oo

_o ^~

_-___-___-_-_=-___-_-__=-_=-__,-_-___-__-__-___-,-___-___-_-_=-_-___-__--,-=-,-=-

-5.~oo 5.~oo

^~

Fig. 4.

Quasiperiodic

_sequence of the

separations

(indicated by the horizontal lines) for five different families. Every family (labeled by _a direction D

perpendicular

to the planes) _is named by _a capital letter.

Table I.

Sepa;ations

w>ithin the

families of planes

whose notation is

gii>en

in

figure

4.

Direction H R I A C

0.142 0.075 0.133 0.062 0.075

0.230 0.242 0.164 0.100 0.121

0.372 0.316 0.297 0.162 0.195

Let _{us now} be concerned with the average

occupation density fij~

in _a certain

family

of

planes. fil~,

which is the average number of vertices per unit _area, is of

particular importance

for the

interpretation

^of

planar channeling experiments

because it determines the

angular

^half

width ~ ' of the

backscattering profile [91.

i

JOURNAL DE PHYS<QUE T 3 N'7. JULY <99~ iy

The

quantity fi,~

and the average

separation d/~

of the

planes

are related

by

the number of vertices per unit volume _p

BID =

PdiD ( lo)

p has been derived from the statistics of vertex environments for the Ammann-Kramer- Penrose pattem ^{from the}

probability

of _occurrence for the local patterns ^around a

point [101

and the

length

II

/

_of

an

edge

^of the rhombohedrons _as

p =

4.353

(ll)

The average

separation d,~

is derived

by

the

strip projection

method _as follows denote

by F$,(/~

the _area of the smallest basis cell of the lattice

arw(Z~)

in W

=

D % D'.

Exactly

_one

point

of

arw(Z~) corresponds

to each cell. The average

separation

of the vertices

ar~(arw(Z~)

_n

_Sw)

_{on D} is

j~j2)

d/D ₌ ^~'~~~

(12)

b(Sw)/D

where

b(Sw)/D

denotes the width of the

strip Sw.

With

equation j10)

the average

occupation density

of the

planes

in _a certain

family

is _:

~ F$,(/D

~~ ~

b(Sw)/~

^~~~~

Because the width of the

strip

_SW

equals

the extension of the triacontahedron ar~

j (W~

₎ in the direction D,

b(Sw)~~

results from

b(Sw)/o

=

~~'~

(14)

F

(P)%'

_Vi

_j (W~)1

V~~,~ = 4.353 denotes the volume of the triacontahedron _ar~

j (W~

₎ and

f (P,'/~.

n _ar

~

j (W~))

is the size of the average section of _a

plane P,'jo, through

_ar~i

(W~ ).

Therefore _we

finally

arrive at

~lD

_~

(

~

i'~/D

~ ~~~/D' ^~ ~~~

~~~

Da

=

F$,(/~

F

(P,'j~,

n ar~3

(W~

(15)

For

explicit

calculations of BID ^relation

(13)

^is more

appropriate

because

fl

_{(P,'jD, n}

ar~3

(W~))

has _to be determined

numerically.

Relation

(15)

will become useful later.

For ion

channeling

it is of

particular

interest which families of

planes

have the

highest

average

occupation

densities. These should exhibit strong

channeling properties.

All directions D

(characterized by

unit vectors) ^which are not

equivalent

under the

icosahedral _{group are} contained in _a solid

angle spanned by

three fivefold _axes

(R)

of the icosahedron

(see Fig. 5a)

which

projects

_{on a}

plane perpendicular

_to I _as in

figure

5b.

Those ten directions which carry the

highest

_average

occupation density

_BID

(according

to

Eq. (13))

are marked in

figure

^{5b. The numerical} values of BID are

displayed

in table II, the

angles

between the directions D in table III.

1

R

H

i

d

~

[.

.

~

.

~

~B°

f~~

_.

c 9 1

~

~

° .

~

h

o o ~ o °

Fig. 5. al Space sector of

non-equivalent

directions D b)

projection

of the space ^sector on a

plane perpendicular

_to the threefold axis (I) and presentation of the _ten families of

planes

(indicated

by

_a capital letter) which show the highest _average

occupation density fi,o.

The radius of the point~ is

proportional

to the

respective

fi,D. The small letters denote the

angles

between the directions D.

Now _we turn to the

occupation density B,j~

of _a

single plane P,/~

in _a fixed

family.

Each vertex x on a

plane _P,jD corresponds

to a

point.i'

on the section

P,'%,

n

ar~j (W~).

Since the

occupation

of the triacontahedron

ar~j (W~)

is dense and

homogeneous

the

following

relation holds

B,jD

^F

(P,'/~,

n gr~i

(W~)), (16)

the

proportionality

factor

being

determined

by comparison

^with

equation (15)

B>/D ~

F I>(/D ^F

(Pl/D'

n _" El

(w~ _ii (17)

The

occupation

densities vary

strongly

within _a

family

because the sections F

(P,'/~.

n ar~

j (W~))

_range from _{zero to a} maximum value _{as a} rule. Due _to the geometry ^{of the}

triacontahedron each value

F(P,'j~,

n ar~3

(W~))

_occurs

_exactly

twice which _means that in _an

infinite

family only

two

planes P,jD

show the _same

occupation density.

The situation is

completely

different from

periodic

lattices with a

single occupation density

_per

family.

The construction of the _vertex

pattern

ⁱⁿ a fixed

plane P,jD

^{is based} on two arguments

first,

the six-dimensional lifts _{n ~}

Z~

_{of the vertices}

x ~ P~/~ ^{and their}

partners

^x' ~ P~'jD, lie within _a

Table II.

Ai>erage occupation

densities

of

^the

families of planes fi.om fi~qure

5b.

Direction

Description

_fi~~

H twofold axis I. 13

R fivefold axis 0.850

threefold axis 0.694

A 0.488

C axis in the mirror

plane

0.477

J 0.324

G 0.322

L 0.321

E 0.320

B 0.310

Table III.

-Angles

betw,een the djrections D

of figure

5b.

Angle

_a b _c d _e

f

[°]

15.451 5.455 5.451 13.283 9.l10 4.172

Angle

_g h I

j

^k

[°]

^{20.905 9.732 3.882}

four-dimensional

hyperplane V,/~

=

P,jD

e

P,'/~,.

Second, the

points

~~' are contained in the triacontahedron. It follows that the six-dimensional lifts of _x and x' _are

placed

within the

strip

~V,,~ "

~~

~

~</D.

To find the

positions

_x ~

P,j~

and x'~

P,'jo,

_we

again perform

a

strip projection

method

~ ~ ~~»D~ ~V,~D~~~ ^~ ~V,>D~~ _' ~~~

,i' _~ grp,,~

(grv ~~(zb

ⁿ

sv,~)), (19)

In

figure

6 for three families the vertex pattems of the

planes

with

highest occupation density

are

plotted.

The patterns ^{consist of} characteristic motives. For the _case of _a fivefold direction there _are

regular

five- _or ten-Sided

figures

which Show up in all

planes though

these may differ in

occupation density.

5. Rows.

Given any lattice direction D all vertices _{x can} be

assigned

to a

family

of _rows

K,j~ parallel

to D.

Analogous

to section 3 the

following questions

are of interest

. How _can the _rows

K,/D,

_r ~

Z,

of _a certain

family

be characterized ?

_. What _{can we} state about the

occupation density L,j~

of _a

single

_row and the average

occupation density L%

^of all the

strings

in _a

particular family

?

. Which families have the

highest

_average

occupation density £/~

?

. How can the vertices _{~ on a}

Single

row

K,/~

be constructed ?

~.~~

~

~.~~

~ ~ ~ ~ ~ ~ ~ ~ ~

~ @ ~ ~ ~ ~ ~ ~

~.~~ ~ ~ ~

~ ~ ~ ~ ~

l.00

~

° *

. ~ ,_ao

.

* AK,

~

@ f' ~ O

fl . o _,

~ ~ ~

~ O ~

~ ~ ~

~

~

~ ~ ~

~ ~ ~

. .

~ ~ ~

~ ~

~ ~

3.00 3_00

3.°0

?.00

a

~ .

o

. ~

i

~

00 ~

~

o ~

o

o

o _~

C)

Fig.

6.

(I)

~

O . .

«

-

0,0 3,0

3,0 0,0 3,0

Fig. 7. Examples for vertex pattems ^{in three}

planes

with different

occupation density perpendicular

to the fivefold axis (RI. One _can identify fragments of the two motives of figure fib.

Examples

for intersections of _rows

K,~~

^with a

plane

_P~~ are

given

in

figure

8.

The average

occupation density

L~~ of _a

family

is

i~~

=

~

(23)

f~

where

f~,~

^{denotes the} average number of

strings

_{per area} of P~~ ^{and results from}

F Wpj~ (WE

) (~~

_II

~K»D j~j41 ~~

min/D

F(wpj~ (w~j(W~)))

and

Ff,(/~

describe the size of

wp,.~

(w~j (W~))

^{and the} area of the smallest basis cell of the lattice

wv(Z~)

in V,

respectively.

Therefore the average

occupation density Lj~

is

j~14^j LID # P

~'~~~

(25)

F Wpj~(WE

) (W~

)

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_~z,z

]..f

. . ]

f

0.00

.~ .~. *k. ..

. .~ .

.z.. .

.Z.. Ml

. . :

:

:

Z.: Z. $

«

i. _{oo . .}

]1z4

:.

z2

cl

z.? ]lzoi

2.°° . ^Z-j . ^.Z. . -

-

*

.

3.0j

~ ~/,l.00 0.00 ' _.°°

~

~~ ~ ~~

C)

The denominator is obtained

by

F

(arp,~ (ar~ _j (w~»>

=

~'~"~

,

(26>

I (Kj~~

n _w

~

j (w6

_>1

where

f(K,'/~,

n w~i

(W~))

denotes the average

length

^of a section

K,'j~,

n w~i

(W~).

Now

equation (25)

transforms into

"

~fi~/D (l~llD'

~ _" El

(~~) ₍₂₈₎

To determine the families of _rows of the

highest

_average

_occupation density

_we have

examined many different directions D

according

to relation

(25).

The _ten families of _rows of

highest density

_are listed in table IV. The sites of the

respective

directions D _are

again

obtained from

figure

5b because

they

_are identical to those of

highest

values B~~.

Table IV. Ten

families of

row,s whit-h show the

highest

_ai,erage

occupation density i~~.

Direction D

L~

H 0.651

R 0.632

0.401

A 0.309

C 0.290

G 0.211

B 0.193

E 0.189

L 0.187

J 0.186

We nevertheless must

point

_out that the fraction

~~~

is _not constant. It

depends

_on the

L/~

geometry ^{of the} triacontahedron w~~

(W~) _along

the

respective

direction D

(recall Eqs. (15), (28»

h~~ Ff,~D

F

(P),o,

n w~~

(W~))

LID Ff>[/D f

(KllD'

n _"

j (W~

₎₎

< (pj~~,

n w~i

(w6»

=

~

,

(291

f

(K,'/~.

n

ar~j (w~»

where _we

applied

the relation

Fj,(~~

=

Ff,(~~ (the proof

is _too

lengthy

to be

presented here).

By analogy

^with

equation (28)

_{we can} determine the

occupation density L,,o

of _a

single

_row

K,/D

~,/D

_"

~~~/D ~(l~llD'~ _{"E( (W~)}

_).

₍₃₀₎

f(K,'~o.

n w~i

(W~))

is the

length

of the section of the

corresponding string

K,'~~, ^{in the}

triacontahedron.

According

_to

equation (30)

_{one can} calculate the

occupation density

L,~o ^of any

given

row

K,~o.

Within _a certain

family

there exist

infinitely

_many different

occupation

densities because the

length

of the sections K,'~~, ⁿ

w~j (W~)

i D'

again

_range from _{zero to a} maximum value _{as a} rule. In contrast to the

planes

an infinite number of _rows of _a certain

family

show the _same

occupation density

^{because the}

starting

or end

points

of the section K,'~~, ⁿ _w~

j (W~)

with _a

definite

length

form _a closed _{curve on} the surface of the triacontahedron. Because the

distribution of the

points

x' _{~ w~}

j (W~

is dense and

homogeneous,

the

length

of the _curve is _a

measure for the

frequency

H of such _a section, this _means of _a certain

occupation density.

Figure

9 presents ^the distribution for _rows

parallel

to the fivefold direction R.

The sequence of the

points

_x on a certain _row

K,~~

^is also obtained

by

a

strip projection

method in the _space

W,~~=K,~~%K,'~~,

with the lattice

W,~~nZ~.

The

strip

^is

SW,,~ ^" ~'~,/D ^~

~~'

0.316 1.023

Fig. 9. Frequency H of _occurrence of _rows K,,D

Parallel

to the fivefold axis (R) plotted _i,eisus the

occupation

density L,,D (arbitrary units). Rows with _a low occupation density _appear

frequently.

6. Conclusion.

By

an

analysis

^of cross-sections of the acceptance ^domain we have constructed the _vertex

positions

_on each

single plane

or row of _a

quasilattice

as well _as statistical data about the families. Prior _to this work _we had calculated the ion

backscattering profiles

for _a

primitively

decorated Amman-Kramer-Penrose

quasilattice by

numerical simulation of the ion

trajectories

in _a computer ^model

[2]. According

to the Lindhard

theory [6],

which describes ion

channeling

in

periodic crystals,

best

guidance

and hence lowest

backscattering

rates occur

by

^the most

densely occupied planes.

In

figure10

we show _an

angle

scan of the ion beam

along

the directions indicated in

figure

5. The

sharpest backscattering

minima indeed _can be associated with the densest

planes.

Hence, Lindhard

theory

is at least

qualitatively applicable

and

channeling

is _a valuable tool for the structural

analysis

of

quasicrystals.

~~

H L .I C'IL H L I C'.I L H

1.10

oo

o.90

0080

j

^0. 70

(

₀₀₆₀

0050

j

00q0

j

₀₀₃₀

~ 0020

o. lo

o,oo

-sooo ,3000 -1.oo loco 3.oo s.oo

t~[°)

Fig. 10. Backscattering profile in heavy ion-channeling ^for an angular scan through the series of planes, whose normal _vectors follow the sequence HLIC of points in figure 5. Densely occupied plane~

cause strong

channeling

and

yield

low

backscattering

rates. Details of the calculation _are given _in

reference [2].

References

[ii KRAMER P., NERI R., Acia Ciystallogr. A 40 (1984) 580.

[2] KUPKE T., PESCHKE U., CARSTANJEN H. D., TREBIN H.-R., Phys. Rev. B 43 (1991) 13758.

[3] CARSTANJEN H. D., EMRICK R. M., KUPKE T., PLACHKE D., WITTMANN R., TREBIN H.-R., Nat-I.

Instrum. Methods B 67 (1992) 173.

[4] KATz A., DUNEAU M., J. Phys. Franc-e 47 (1986) 181.

[5] DUNEAU M., KATz A., Phys. Ret'. Lent. 54 (1985) 2688.

[6] LINDHARD J., Kgl. Danske Videnskab Selskab, Mater. Phys. Medd. 34 (14) (1965).

[7] HOFFMANN S., TREBIN H.-R., Phys. Status Solidi (hi 174 (1992j 309.

[8] MACKAY A. L., Physic-a l14A (1982) 609.

[9] BARRETT J. H., 1971,

Phys.

Ret'. B 3 (197 Ii 1527.

[10] BAAKE M., KRAMER P., SCHLOTTMANN M., ZEIDLER D.,

Quasicrystals

and Incommensurate

Structures in Condensed Matter, M. J. Yacaman, D. Romeu, V. Castailo, A. G6mez (World Scientific, Singapore, 1990) _p. 85.