Structural Analysis of the Decagonal Quasicrystal Al70Ni15Co15 Using Symmetry-Adapted Functions

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Structural Analysis of the Decagonal Quasicrystal Al70Ni15Co15 Using Symmetry-Adapted Functions

L. Elcoro, J. Perez-Mato

To cite this version:

L. Elcoro, J. Perez-Mato. Structural Analysis of the Decagonal Quasicrystal Al70Ni15Co15 Us- ing Symmetry-Adapted Functions. Journal de Physique I, EDP Sciences, 1995, 5 (6), pp.729-745.

�10.1051/jp1:1995163�. �jpa-00247098�

Classification PhysJcs Abstracts

fil.43Bn 61.44+p 61.50Ks

Structural Analysis of the Decagonal Quasicrystal AlmNii5Coi5 Using Symmetry-Adapted ^Functions

L. Elcoro _(~ and J-M- Perez-Mato (~)

(~) Departarnento de Ingenieria Mecànica _y de Materiales, Universidad Pùbfica de Navarra, Campus de Arrosadia,

s/n

31.006 Pamplona, Spain

(~) Departamento de Fisica de la Materia Condelrsada, Facultad de Cielrcias, Ulriversidad del Pais Vasco, Apdo. 644, Bilbao, Spain

(Received

18 Jalruary 1995, revised in final form and accepted 16 February 1995)

Abstract. Trie decagolral quasicrystalline structure of AlmNii5Coi5 is analy2ed within trie superspace formalism. Symmetry-adapted functions have been used for trie parametrization of the limits of the atomic surfaces that represent ^trie atoms in superspace. As diffraction data,

253 independent reflections from

[Steurer

W., Haibach T., Zhang B., Kek S. and Lück R. Acta Cryst. 849

(1993)

661.] bave been considered. Starting from _a circular approximation for the atomic surfaces, their boundaries _were refined with trie program QUASI. A fit comparable to that reported in trie above reference _was attained with 15% fewer adjustable parameters. Trie _main

diflerence _is trie _non- inclusion of Debye-Waller-type factors

in internai space. Trie boundaries of trie resulting atomic surfaces _are rather wavy or circular _in contrast with trie polygonal

forms ofthe previous model. It could be verified that these polygons evolve in _a free refinement towards trie obtained _wavy forms. Trie eflect _on trie diffraction intensities ofDebye-Waller factors

along internai space is rather equivalent to that of wavy boundaries for the atomic surfaces.

An _open question _is, then, whether trie polygonal ^{forms considered} in previous models _are

physically significant _or model-forced. The resulting diflerences in physical _space between trie two models _are subtle, trie _main features being essentially identical; however, both of them present

a significant proportion of non-physical atomic distances that _concem fully occupied atomic

positions. Previous quantitative diffraction analyses of other quasicrystalline structures have also shown this tendency to introduce _a sigmficant number of unphysical interatomic distances;

its _{cause remains} ulrclear, but could be related with trie recently conjectured existence of _non-

dense atomic surfaces _in real quasicrystals.

1. Introduction

Tue structure determinatiolr of

quasicrystals

bas been

greatly improved

witu trie introduction of trie _superspace formalism

[1-3].

In this

framework, quantitative analyses

bave been made of trie structure of icosauedral

quasicrystals [4-11]

and

decagonal quasicrystals [12-15].

Trie first model for tue Al-Ni-Co

phase

_was

presented by

Yamamoto et ai. [13] ^{wuo could refine 41}

independent

reflections _{up to} WR

= ù-ii. A

considerably larger

set of

independent

reflections

© Les Editions de Physique 1995

(253)

_were measured

by

Steurer et ai. [15], ^who

proposed

_a dilferent structural mortel much better

adjusted

to the _new diffraction dataset. It is based _on the _one

previously proposed

for the

decagonal phase

of Al-Cu-Co [12]. ^{The AS} are

polygonal (irregular decagons

and

regular pentagons)

_very dilferent from the _ones of reference [13].

In tuis

work,

we report _a structural model obtained

using

_an alternative

approacu

_[16].

Tue radial functions

describing

tue boundaries of the AS _are

expressed

in terms of symmetry-

adapted functions,

which

satisfy

the

point-group

symmetry of the AS in internai _space. Tue

amplitudes

associated to eacu function _are tuen tue continuous parameters to be determined.

The values of these

amplitudes

_are

optimized using

tue refinement _program

QUASI il?],

wuicu

was first

applied

in the refinement of tue icosahedral

phase

of Al-Li-Cu

iii].

Four AS with

constant

occupation probabilities

have been considered instead of the rive AS of reference

il

5]. ^Another important initial dilference is the non-inclusion of

Debye-Waller "perpendicular"

factors,

_1-e-,

Debye-Waller

factors

along

the internai _space.

The

decagonal phases belong

to the _group that _can be called

"polygonal quasicrystals"

il?].

Tue

quasiperiodicity

is present

only

_{in a}

plane,

tue

stacking

of

planes being periodic.

Following

tue framework

presented

in reference

il?]

_we cuoose _a set of _wave vectors

indexing

tue diffraction pattern:

ki

₌

a,jaj

i

=

1,.. ,4

k5

_"

ai (1)

wuere

(a,1>

_a12>

a13)

"

(cos ~",

sin

~",

0)

and tue basis

(a] )

is

orthogonal

witu

[ai

=

[ai #

5 5

[ai

_[. In tue AlNico _case

[ai

₌ 0.2636

À~~

_and

[a(

₌ 0.24506

À~~

_{[15]. Tue first four vectors}

are on the _same

plane, pointing

to four vertices of _a

regular

pentagon. ^{The fifth} vector of tue pentagon ^is not included _as it is not

rationally independent

of the other four. The

k5

wavevector is

orthogonal

to tuis

plane

and is

parallel

to tue

periodic

direction wuicu coincides witu tue

decagonal

axis.

Tue A matrix (Ù =

Ax),

tuat relates the _Ù; coordinates in wuicu tue superstructure is

periodic

and tue _{xi, xii}

coordinates,

tuat

correspond

to

parallel

and internai _or

perpendicular subspaces

_is

il?]:

T-1

@É

T

(T-1)@É

Ù

₂ ^~ ₂

~

T

(T-1)@t

T-1 2+T

~2

₂ ^~ ₂

/~

A= _T

(T-1)@0

_T-1 2+T

(2)

T

~l /~

^{~ T}

(T

1)~É

Ù

₂ ^~ _{2 2}

0 0 1 o o

where _T

=

~

~,

trie first three columns _are trie aij parameters

previously

defined

(Eq. (1))

2

and tue two last columns _are

conveniently

cuosen

il?].

Tue rive parameters _Ùi are tue

global phases

of the modulation associated _to tue wavevectors ki. Tue turee variables _xi

correspond

to the

parallel subspace

and _are also tue relative coordinates in the

(ai)basis (reciprocal

to the

(a] )previously defined)

of _{a generic vector r} in

physical

_space, while the _two last variables xi; generate ^{the internai}

subspace.

From tue

systematic

extinctions observed in tue dioEraction pattern [15] ^{tue structure must} be

assigned

to _one of tuese _groups:

Plosmc

_or

Plos /mmc.

As in tue

previous

reference _we will

assume tuat tue _group is tue

centrosymmetric

_one. As generators of tue _group, tue elements

(Ciol

o, o, o, o,

1/2), (m[

o,

o,

o, o,

o)

and

(I[o,

o, o, o,

o)

_can be cuosen. Tue form of tueir rotational parts ^{in tue basis is:}

o o o -1 o o o o 1

1 1 o o o 1 o o

Cio" -1 o o o o _m= o 1 o o o

(3)

o -1 o o o o o o o

o o o o o o o o 1

wuile I is tue inversion matrix.

2. Trie Structural Model of Steurer et al.

A diffraction dataset of 253

independent X-ray

reflections

(MoKa,

₌ o.70926

À)

corrected of Lorentz-Polarization elfects and

spuerical absorption,

ail reflections

satisfying I(H

>

20e(1(H)

and o _{< <}

45°,

bas been worked out

by

Steurer et ai. [15]. ^Tue rest of

reflections, including

some "satellites" _{were not} used in the refinement.

They proposed

_a structural model which could be fitted _{up to a} value of

wR(RF)

of o.078

(o.o91).

Tue structure is similar _to tue _one

proposed

for tue Alcuco

alloy

[12]; ^tuere exist rive symmetry

independent

AS in tue unit cell centered at tue points

(j/5, j/5, j/5, j/5, 1/4) (j:

_even

integer)

witu 5m symmetry ^and

multiplicity

2. Tue first two AS bave _{a common} center

(j

=

2)

and _are

irregular decagons,

defined

by

two parameters. Tue internai AS

(AS

1

internai)

is

fully occupied by Nilco

and tue externat AS

(AS

1

externat)

bas _a

composition:

47%

Al,

3%

Nilco. Hence,

tuis latter bas total

occupation probability

o.5. Tuis makes

puysically acceptable

the "too-short" interatomic

distances of1.78

À

_that

are

generated by

this AS

together

with its

symmetry-related

_{ones, as}

these

pairs

of atomic positions _can be considered not

simultaneously occupied [15].

^The

point

for

j=4

is also the center of _two AS

(AS

2 external and AS 2

internai),

but in this _case

they

are concentric

regular

pentagons ^defined

by only

_one parameter. ^Both are

entirely occupied by Al,

their

only

dilference

being

dilferent

Debye-Waller

factors. In

addition,

_a

perpendicular

Debye-Waller

^factor

,

1-e- of the type

exp(-Bp~rh])

with

hi being

the internai component ^of tue diffraction vector, is associated _to AS 2 external.

Finally,

AS 3

(j

=

o)

is

occupied

at 50%

by Nilco.

Tuis surface and its

symmetry-related

located _at

(0,

0,

0, 0, 3/4) give

rise to

pairs

of

Nilco positions only

2.04 apart, ^but

again,

each _one

only occupied

^with

probability

0.5.

The form of AS 3 is also

pentagonal.

A scheme of tuese rive

polygonal

AS

proposed

in reference [15] ^is

presented

in

Figure

1, wuere

tuey

_are

approximately represented using

^{uarmonic radial}

functions,

_as

explained

in tue next section.

Tue number of

adjustable

parameters of tue model _was 21

including

_an extinction factor.

Tue atomic

composition

coincides with tue

experimental

_one, but trie

density

is

forger,

_{pc =}

4.48

g/cm3 against

tue

experimental

4.17

g/cm3

_[15]. Trie authors

pointed

out that tue

sarnple presented

internai

cavities,

_so tuât tue nominal value could be taken _{as a} Iower Iimit of tue real _one. Once tue superspace structure _was

refined,

tue autuors used tue calculated

structure factors _{as a}

starting

point in tue

application

^of tue Maximum

Entropy

Metuod

[18,19].

Electronic

density

_{maps in}

puysical

_{space were} derived

and,

from

tuem,

tue atomic

positions.

Tue structure consists of the

periodic stacking

of two atomic

planes,

one rotated

27r/10

with

respect to tue otuer and

separated by

_a distance of 2.04

À.

_{Tue main motif of tue structure}

are

decagonal

clusters

forming

_a set of

antiprismatic decagonal

cuannels [15]. ^Tue

pairs

of too-close atomic

positions resulting

from AS 1 externat and AS 3 and mentioned above bave _an

appealing

a) b)

C)

Fig. 1. Approximate forms of AS 1

(a),

AS 2

(b)

and AS 3 (c) according to the model of reference [15]. ^Trie contours of trie AS _are

approximated

by _means of trie three first terms _{in an} expansion of

the type

(4).

Note trie diflerent scale used for each AS.

Fig. 2. Projection _on internai _space of the AS whose superposition _gives

rise to unphysical inter- atomic distances in trie model of refrence [15]. ^{Trie central} decagonal ^AS is placed at (2

là,

2

là,

2

là,

2

là, 1/4)

and trie pentagonal AS at

(-1là,

4

là, -1là, 4/5, 1/4), (-1là,

4

là,

4

là, -1là, 1/4),

(4

là, -1là, 4/5, -1là, 1/4), (4/5, -1là, 4/5, 4/5, 1/4),

and

(4/5, 4/5, -1là, 4/5, 1/4).

interpretation: they

_con be tue signature m the _{average structure} of

phason-type

disorder which

produce flippings

^of atomic

positions

in real _{space: one} AS is translated

along

internai _{space so} tuat it _starts

crossing

tue real _space section wuile another

neigubouring

AS does tue contrary.

The

model, however,

also mcludes _a

significant

amount of

unphysical

distances between

pairs

of Al

positions,

_one of tuem

being fully occupied,

that have _no dear

interpretation

Tuis set

of close atom

pairs

result from tue closeness in

puysical

_space

(only

0.94

À apart)

of AS 1 and surfaces tuât _are symmetry related witu AS 2. In

Figure

2 _we represent ^tueir

projection along

internai _space. As it _con be _seen in the

figure,

the

superposition

between these

projections

is

significant.

The Al _atoms

represented by

the _zones of tue AS 2 surfaces that _superpose in the

projection

with AS 1 _are situated at an

unacceptable

distance of tue order of1 of 0.5

occupied AI/Nilco positions corresponding

to AS externat. As tuis

superposition

_zone is quite

large,

it

gives

rise to a considerable number of

unpuysical

atomic

pairs.

We bave calculated tueir relative number

by comparing

tue

superposition

_area witu tue total _areas of tue AS. 54% of 0.5

occupied Al/Nilco positions

due to AS externat bave anotuer

fully occupied

AI atomic

position

at 0.94

À, resulting

^{from tue} AS 2 externat.

Analogously

^20% of Al atoms from AS 2 externat hâve another 0.5

occupied Al/Nilco

atomic position at 0.94

À.

_This

_important

unrealistic feature of the model does not

disappear

in tue Maximum

Entropy

_maps.

3. Refinement

by

Means of

Symmetry-Adapted

Harmonics

As usual _{we assume} tuat trie AS _are

parallel

to internai _space.

Following

tue method introduced in reference

il?],

tue internai and externat limits of tue AS _are described

by

radial

functions,

wuicu in this _case must bave 5m

symmetry:

~,in ce

a(,~~

~ 4

~j,ex f aÎ'~~

~~s(5j4), rf(4)

=

fi~

+

( _W

_{~°~~ ~~~}

rj~(ç$)

"

$

^~

~

Qi

~

Tue

ability

of tuese expansions truncated up ^to a lower order for

describing

quite

complex,

_even

polygonal forms,

is evidenced in

Figure

wuere the

polygons proposed

in reference [15] ^{for tue} AS'S _are

approximated

witu sucu expansions with

only

three terms

(including

tue

uomogeneous one).

Tue r matrices

il?]

for aII AS _are the

identity,

because the orientation of the symmetry-

eIements of tue AS in internai _space coincides witu tue _one cuosen for tue construction of tue functions _in

equation (4). Hence, expression (28)

^{of reference}

il?]

^takes tue form:

F(H)

₌

@ £ _{pm(~) fm(H)} £

e

Î~~~Î~~e~"~'(~~P+~)

'

»,m R

2r r~~(#)

x

/ _d4 /

^' _dr

r e2~iÉ<h<.x<

j5)

T'n(~)

p

According

to

equation (2)

and tue definition of tue _ai

basis, [A[

₌

5(2T -1)/4, V(az)

=

[ai[.[azl.Îa31

_" 58.73

À3.

_Tue

sum in /J extends to tue

independent

AS in tue unit

cell,

and tue _sum in R _to the two symmetry ^elements

(

E

0,

0,

0,

0, 0

)

and

(1[ 0, 0, 0,

0, 0

).

pm(~)

is the

occupation probability

of element

m in AS _~.

fm(H)

is the

scattering

factor of the element _m for the reflection H.

lî

=

(hi

_h2>

,

hn)

_are the indices of H in the

(kj) basis,

while hi stands for trie internai coordinates of

lî.

_{R and Ri}

are trie matrices in

parallel

and internai _space associated to

É _by

_{the relation:}

A~~ÉA

=

( l(6)

The thermal tensors, ^assumed constant for the whole of each

AS,

_are forced

by

symmetry to be

diagonal

with

flii

" fl22

#

fl33. For trie refinement _process trie _same dataset of reference [15]

JOURNAL DEPHYSIQIJE -T. 5,N°6,JUNE 1995 29

bas been used. An extinction pararneter ^{and anomalous}

scattering

eflects bave been considered.

Tue

expression

used for tue extinction correction _is [15]:

~~

ext.Fj(H)

~~~~

" ~ ~

sinÙ/À

^~~~

wuere

Fo(H)

is tue observed

intensity.

In tue refinement _{process two}

penalty

functions bave been included: _one for trie concentration and another for tue

density.

Tue relative

composition

of _a cuemical

species

m is related witu tue volumes of AS [4]:

£pm(~)N~V~

~~

(

cte(v)Nvvv

^~~~

v

where V~ ^is trie volume of AS _~,

N~

^{is its}

multiplicity, pm(~)

is trie

occupation probability

of specimen m in trie AS _~, and

cte(v)

is the overall occupation

probability

of AS _v. Tue _sums

extends to ail

independent

AS in tue unit cell. Tue volume takes _a

simple

form in tuis _case, owing ^to tue

orthonormality

of tue functions of

expansion (4):

V» =

j~~ ^dl /~~)~

^{dr r =}

^£(af'~~)~ £(af"")~ ₍₉₎

, , ,

Tue _mass

density

is:

~

NA~(a,) ~ ~ ~~~~~~~~~"

_~~~~

wuere

Mm

is tue atomic _mass of atom _m and

NA

is the

Avogadro's

number. The

QUASI

program used minimizes tue

quantity:

~ _{~°H(Fo(H) k[F (H)[}

_)~

_{~j(cm ~m~2}

2 H

~ 0 c

~

[

_~°H

l~° ~~))~

^{~ ~°~}

~

ll(cm)2

^{~ ~°"}

~~°

i~~~~

~~~)

m

Tue

integrals

in

equation (5),

in tue internai _space, bave been calculated with 40 points of

integration by

the Gauss method [20], ^whicu

gives

_{an accuracy} of turee

digits

in tue WR factor.

In

general,

the

weigut required along

tue refinement for the

penalty

^functions _was

quite

small.

As _a

starting

point in the refinement process, a

simplified

version of the model in reference [15] was considered. Tue AS _were

given

circular

forms,

centered _at tue

points

of tue model of reference [15], ^{but tue weak}

proportion

^of

Nilco

in AS 1 externat _was

disregarded,

_so tuât AS 1 externat _was

assigned

_a 0.5

occupation probability

of Al. A

single Debye-Waller

factor

was associated to tue wuole AS 2, so no dilference _was doue between tue externat and internai part of tuis

surface, reducing

tue effective number of

independent

AS to four. In

addition,

no

puason-type Debye-Waller

factors

(along

internai

space)

_were included in tue refinement.

Occupation probabilities

_were fixed. As

weighting scheme,

tue usual inverse of the standard deviations of tue moduli of the structure factors _was used. AS 1 externat has _an internai

Iimit,

wuich coincides with the externat limit of AS 1

internai,

therefore the paranleters ^{a~~ of this} latter and tue parameters ^{a"' of tue former} are constrained to be

equal

in tue refinement. Tue

least-squares refinement

quickly converged

_up to WR

= 0.103.

Successively, uigher-order

terms for tue radial functions

(4) describing

tue Iimits of tue AS _were included in tue refinement.

Table I. Structural parameters

of

the

final

model.

Debge-Waller factors

_are

giuen

in

À~.

AS externat AS internai AS 2 AS 3

~',lÎl,fl3>~4,~5

^{~ ~ ~}

l~

^{~ ~ ~}

~) (~

^{~ ~ ~}

_~)

fl~ ^{8.1(6) 1.18(2) 1.90(6) 2.2(7)}

0.4(1)

0.51(2) 3.9(1) 0.6(6)

o.5 o i o

0 0.5 0 0.25

0 0.5 0 o.25

~ex 0.996(7) 0.666(2) 0.895(3) 0.148(5)

1

~ex -0.031(3) 0.074(5)

2

~ex 0.051(7) 0.066(6)

3

Tue second and third harmonics _was

significant only

for AS 1 internai and AS 2. Tueir standard deviation in the other surfaces _was much

forger

tuan their fitted

values;

_in

subsequent

refinements tueir values _were fixed _to

0,

_so that AS 1 externat and AS 3 _were maintained circular

in trie final model that attained WR

= 0.080 and RF _" 0.092, witu _a Goodness-of-Fit of 2.66.

a) b)

C)

Fig. 3. AS 1

(a),

AS 2

(b)

and AS 3 (c) ^{of the final model of Table I. The hori20ntal} axis is parallel

to trie _xn direction (a fine of symmetry), and the vertical _{one is} parallel to trie _x12 direction.

The introduction of _a fourth harmonic in the radial functions did not _cause any

significant improvement.

Trie final structural

parameters

^and

Debye-Waller

factors _are listed in Table I.

Tue number of

adjustable

parameters,

taking

into account tue scale and extinction factor

(ext

₌

5.2)

_was 18. Tue calculated

composition

coincides

practically

witu tue

experimental

one and tue calculated _mass

density

_{is pc =}

4.44g/cm3.

In

Figure

3, _a

grapuical representation

of tue AS in tue final model is suown.

Once trie best model with 4 AS _was

obtained,

_{a new} refinement

dividing

AS 2 in two parts with dilferent

Debye-Waller factors,

_as done in reference [15], was

attempted.

Tue

improvement

in tue R-factors _{was very} small and did not

justify

^tue

increasing

in tue number of parameters.

Also,

_a refinement witu

adjustable

cuemical disorder _{in some} of tue AS _was

tried,

but did Dot introduce

significant

dilferences.

Ù _Q

ù

~~

~~

$ Ô'O

~Q ~Ù

_~

Ù°'

ù

Q $

'2 -' 0 -O8 -06 -O4 -<2 -' 0 O 8 -O6 04

a) b)

Fig. 4. Fourier

(a)

and Fourier-dilference (Fo

Fc) (b)

_maps in internai space around AS 1, _in

absolute ulrits

(eIectro1rs/À~)

Referelrce fraIne _as in Figure 3.

In order to

analyze

tue

goodness

of tue final structural

model,

Fourier and Fourier-dilference maps in tue internai _space around tue centers of tue AS bave been derived

(Figs. 4-6).

As

con be _seen in tue

figures,

trie Fourier-differences _are not

significant

in trie _two first

AS;

tue

-2 O -<8 -' 6 -<4 -2O -'8 _-' 6 -<4 -' 2

a) b)

Fig. 5. Fourier

(a)

and Fourier-diflerence (Fo Fc) (b) _{maps in} internai space around AS 2, _in

absolute units

(electrons/À~)

Reference fraIne _{as in} Figure 3.

->o

~ ~

jÎ ~

@)

~'~'~~~

~

~

~

m

~

^'~

a)

04 -02 OO 02 04 -04 02

'~

OO 02 04

b)

Fig. 6. Fourier

(a)

and Fourier-dilference (Fo

Fc) (b)

_maps in internai _space around AS 3, in

absolute units

(electrons/À~).

Reference frame

as m Figure 3.

uighest peaks

_are about 1% and 3% of tue observed

density

in AS 1 and AS 2,

respectively.

AS 3, on trie contrary, presents a maximum in trie Fourier-dilference _map of the order of 33% of the observed

density,

which is

also,

in absolute

value,

two or three times

Iarger

than the

peaks

observed in trie other two dilference _{maps. The model} exhibits,

therefore,

_a

significant

deficit of electron

density

at AS 3. It should be noted that this AS is

assigned

to

Nilco

with

occupation probability of1/2,

and this

occupation

cannot be increased since this

surface, together

with its symmmetry ^related one in the

upper-plane (x3

_"

0.75), gives

rise to pairs of close atomic

positions

wuich cannot be

occupied simultaneously. Tuerefore,

this

shortcoming

^{of tue} model, also present ⁱⁿ the _one of

Steurer,

cannot be avoided and its _{cause is} unknown.

Nevertheless,

tuis AS

scarcely

^{affects the}

global refinement,

its _size

being

20 to 40 times smaller thon tue other three AS. From the Fourier maps, one coula

naively

take for

granted

that AS 2 and AS 3 must have _a

perfect pentagonal form,

in contrast with the wavy Iimits and the circular form

respectively assigned by

the mortel. The approximate

pentagonal

form of the maps may be,

however,

_an elfect of the truncation of the Fourier _serres. This type ^{of elfect} is demonstrated

in

Figure

7, where _a Fourier map of AS 3 obtained with the 253 calculated structure factors _is shown.

According

to the model

employed,

the

represented

AS should be _a circle; the _map of

O 02 02 ₀₄

Fig. 7. _-

Fourier

map in

AS 3,

however,

rather resembles _a

polygon.

This deviation from the circular

suape

is

clearly,

in tuis _{case, a} truncation elfect. On tue otuer

uand,

in tue _case of AS 2, _a ^surface _as the

one

depicted

in

Figure

3,

given

its ratuer small deviation from _a

pentagonal form, produces (at

tue available truncation

Ievel) quasi-perfect pentagonal

Fourier _{maps as} tue _one suown in

Figure

5. Tuis _was cuecked

directly

and is confirmed

by

tue small values in the

corresponding

Fourier-dilference _maps.

Hence,

in

general,

_a

straightforward

evidence _on the actual form of the AS boundaries cannot be derived from the Fourier maps.

The

possibility

of two dilferent

Debye-Waller

factors within the surface AS 2 _was further

investigated by

_means of Fourier and Fourier-dilference _maps in sections tuat include _a

puysical

and _an internai direction.

Figure

8 shows tue Fourier maps for AS and AS 2. Tue horizontal axis is

parallel

to tue

decagonal

direction in

puysical

_space and tue vertical _one is in the internai space. The width of the AS

along

the

decagonal

direction is

Iarger

for AS 2, in agreement with

-<

O'O 020 OOO 040 OW OOO O'O 020 OOO 040 OW

a) _~)

Fig. 8. Fourier _maps of AS 1

(a)

and AS 2

(b).

The horizontal axis _is parallel to the decagonal axis, _x3 ⁱⁿ parallel _space, and the vertical _one corresponds to internai space

(xn direction).

the values of the component fl33 of the thermal _tensors.

However,

there does _{not seem to} be any

significant inhomogeneity

of the AS widths

along

the internai direction. It should be noted that AS 1 is divided in two parts,

Nilco

in the internai and Al in the externat _one, but the two

Debye-Waller

factors fl33 are very similar

(see

Tab.

I)

in accordance with the maps. In AS 2 the

quite

uniform widtu of the AS supports a model with _a

single

thermal _tensor for tue

wuole AS. In

addition,

tue Fourier-dilference _maps did not show

significant peaks.

As the model of Steurer et ai. [15], ^tue one

presented

above introduces

unphysical

interatomic distances:

they

_are pairs ^{of Al}

positions

_one

fully

and tue other half

occupied separated

0.94

À,

which arise from AS 2 and AS

externat, respectively.

This fact is illustrated in

Figure

9,

analogous

to

Figure

2. In the present model the

proportion

of 0.5

occupied

Al positions from AS 1 that have another too-close Al atom from AS 2 is

48%,

while the inverse relation is 16%.

These values _are 20% lower thon the _ones in the model [15], ^{but still} quite

high.

Moreover, the

superposition

between the dilferent AS 2 shown _in

Figure

9 also

produces

_a small number of

unphysical

interatomic distances of _a dilferent type. A last refinement _was made with _an

occupation factor 1 for the AS 1 externat. The number of

unphysical

interatomic distances _is reduced but the WR factor _increases

significantly.

Fig. 9. Superposition _on internai _space of the _same AS _as in Figure 2, accordilrg to trie final1nodel of Table 1.

4.

Comparison

^of trie Two Structural Models in

Superspace

The AS

proposed

in referelrce [15] ^{and those}

reported

above dilfer

siglrificantly.

The refinement methods

employed

in each _case make _use of dilferent "basic units" to construct the AS. While in reference [15] ^{the AS} are

approximated by polygons

with _{a maximum} of _ten

vertices,

in the

present ^{work their limits} are described

by

smooth radial functions. This latter

description

is

necessarily

less constrained and includes _{up to an}

acceptable degree

of

approximation polygonal forms,

_as shown in

Figure

1.

Nevertheless,

_our refinement did not converge ^to the

polygonal

model of reference [15],

although

the WR factor attained _a

comparable

value. An essential dilference is the inclusion in reference [15] of a

"phason-type"

_or

perpendicular Debye-Waller

factor of the form

exp(-Bp~rh))

for AS 2 externat. In order to

investigate

its eoEect, _we

performed

_an alternative refinement

starting

from the solution of reference [15] expressing tueir AS

by

_means of

expansions

of type

(4)

witu turee terms, as suown in

Figure

1, ^but not

including

the

phason-type Debye-Waller

factor. For this

starting

model the

weighted

WR factor

is 0.086, the o-où? increase witu respect to tue solution of reference [15]

being essentially

due to tue absence of tue mentioned

perpendicular Debye-Waller

factor. After trie

least-squares

process WR attained _again 0.079, and trie AS took tue forms shown in

Figure

10. It is clear from tue

figure

tuat tue

polygonal

AS evolve towards tue AS in _our mortel.

Except

for tue internai limit of tue second

AS, separating

two dilferent

Debye-Waller factors,

whicu _{was not} considered in _our

description,

and _a small dilference in AS 3, doser to a

decagon

tuan to a cirde, tue rest of tue surfaces _are

practically

tue _same. However, tue AS 3, as stressed

above,

bas _a minimal importance in tue refined model because of its small size.

Moreover,

tue standard deviations of tue second and tuird terms tuat

modify

^tue circular form

proposed

in Table I for tuis

surface,

_are

larger

tuan tueir actual values.

Hence,

tue elfect _on tue dilfracted

intensities of tue

perpendicular Debye-Waller

factor _is

apparently

quite

equivalent

to tuat of _a wavy form of _some of tue

AS,

instead of _a

polygonal

one

(Dot only

for AS 2 wuose scattering is

directly

corrected

by

tue

perpendicular Debye-Waller

factor, but also for AS 1

internai).

Tuis fact _was confirmed

by

furtuer refinements done witu tue Ameba minimization mode of

QUASI il?]

^wuere _an

adjustable perpendicular Debye-

Waller factor for AS 2 _was induded. Its introduction did not

improve

_or

change significantly

the model of Table I and

Figure

3. On the

a) b)

C)

Fig. 10. AS obtained in _a refinement without Debye-Waller perpendicular factors taking _as starting point trie model of reference [15]

(Fig.

l).

other

hand,

_a refinement

starting

from the

polygonal

model in reference [15]

(Fig. l) yielded

wavy AS with its main features

basically equal

to those of

Figure

3, while

WR(RF) improved

up ^to 0.077

(0.088)

and the

perpendicular Debye-Waller

^{factor sulfered} _a 30% decrease. This last refinement has not,

however,

much

significance

since the number of

adjustable

parameters

was increased with respect to the _one of reference [15] ^{from 21 to 27.}

In any case, there is _a clear correlation between corrections to a

pentagonal

form for trie AS and _a

perpendicular Debye-~valler

factor.

Although having

similar elfects _on the

diffraction,

their structural

meaning

^is

quite

dilferent. The

polygonal

AS

produce tiling-type

structures [2

ii,

which _con be considered

highly

ordered

However,

_a

perpendicular Debye-Waller

factor for

a

polygonal

AS indicates _a disorder of the structure _as it describes _some diffuse

spreading

of the AS effective boundaries in internal space that reflects its variation from _one

supercell

to another

one. The disorder _on the actual border of the AS

imphes

random atom

"phason jumps"

_in real space with respect to the

perfectly

ordered

quasiperiodic

_average structure [22]. ^In contrast,

a structure with wavy limits for the AS describes _a

perfect (average) quasiperiodic

atomic

ordering,

but in

general,

^of

higher complexity.

For instance, if _we start from _a

polygonal

AS and

we modulate its

frontiers,

_some of the AS which crossed the real _space section _in the

polygonal

model will not do it _now and vice _versa,

giving place

to

changes

_in the atomic

positions

with respect to the

perfect simple tiling corresponding

to the structure with

polygonal

AS. However, tuese atomic

"jumps"

_are located in concrete

points

of the structure. Tuese "defects" witu respect to the

tiling configuration

do not break tue

quasiperiodic ordering.

From the results above, it _seems

impossible

to conclude what of the two alternative pictures is doser to the real

system. ^{It has been}

conjectured

that the AS must be

polygonal (or polyhedral

in icosahedral

quasicrystals)

in order to have "local

matching

rules" [23]. But it should be stressed that there

is no fundamental _reason that

prouibits

_a

quasicrystal

to have AS with "smootu" boundaries.

Moreover, the

necessity

^{of local} rules is also _a

conjecture.

The first coordination

spheres

of _an

important

proportion

of _atoms in the _structure

only depend

_on the

rough

volume features of the

AS,

_so tuat in tuis broad _sense the details of the borders of the AS _{we are}

discussing

do not make any dilference

(see, uowever,

Section

6).

It suould also be noted tuat in reference [15]

only

AS 2 external is considered to have _{a non-}

zero

perpendicular Debye-Waller

factor. As stressed

above,

this

implies

_an effective diffuse

broadening

of the AS limits in internai _space

(together

with _a

corresponding

decrease of the

occupation probability

in this diffuse

bordering region). However,

the mortel also includes _a concentric internai AS with

coinciding

^external

limits,

but with _no

perpendicular Debye-Waller

factor. This _means that the model is

depicting

a certain

superposition

of the atomic densities associated _to AS 2 internal and AS 2 externat in their border

region.

^As both AS _are

fully occupied

and _one of the AS has not effective

spreading

of its atomic

density

at this

border,

the

model is, in

fact, introducing

ⁱⁿ this border

region

_{an excess} of atomic

(electronic) density

with respect to that associated to Aluminium atomic

positions.

The elfect _{we are}

talking

about is

by

_{no means} small. The

magnitude

^of

Bp~r

^{for AS 2 externat}

proposed

in reference [15] ^would

correspond

to a mean-squareroot

"displacement"

of tue AS

along

internai _space of about o-1?- One _{can see} in

Figure

1 tuat tuis is tue _same order of

magnitude

_as tue linear size of AS 2 internal.

Tuerefore,

AS 2 externat witu its _mean "fluctuations" _covers almost

completely

the space where AS 2 internal is. This situation has _no clear

physical

interpretation except

an undefined chemical disorder between Al and the transition metals. One could _say that,

in

general,

if

perpendicular Debye-Waller

factors _are

introduced, fully occupied

concentric AS surfaces _must have

equal perpendicular Debye-Waller

factors to avoid inconsistencies in the

corresponding real-space

structure, or uncontrolled

implicit changes

in the chemical order of the model.

5. Structure in

Physical Space

In

Figure

1la _a two-dimensional section _x3

"

1/4

in

physical

_space

corresponding

to the model

in Table I is

depicted.

The

complete

structure is formed

by

the

periodic stacking along

_x3 ^of

two

planes

like tuis _one

separated

2.04

À

_{and rotated}

an

angle 7r/5.

The two

planes

form _a

period

of tue structure

along

the

decagonal

axis.

Nilco

atoms _are tue black cirdes and trie wuite _ones represent ^Al atoms from tue AS 1

external,

whose

occupation probability

is

1/2.

Finally,

white

triangles

_are

fully occupied

Al

positions

wuicu _come from tue AS 2. Tue

figure

is similar to tue _ones obtained in reference [15]

(Figs.

7a and 9 of this

reference).

A

rephca

of the

same

layer

for the model of reference [15] ^{with the}

symbols

used here is shown _in

Figure

^116.

Tue

similarity

between tue _two atomic

configurations

_is obvious. In the center of trie

figures

a

regular

pentagon ^with transition metals in its vertices is present. ^{Trie successive}

stacking

^of

these pentagons ⁱⁿ the two types of

planes produces

the

pentagonal

antiprismatic channels that

are considered _as basic units of _{tue structure in} reference [15]. ^{The too-close Al-Al}

pairs (one

of them

being fully occupied)

mentioned in the

preceding

sections _con be

clearly

_seen in the two

figures (pairs

^of _a ^{white circles and}

triangles),

their

density

_is however

larger

in

Figure 116,

where the dilferences of the AS limits _are suflicient for

making

these too-close atom

pairs

to appear ^at much smaller distances from the

origin.

Tuese pairs _were still visible in tue Maximum

Entropy

_maps of reference [15], ^{but tue}

half-occupied

positions

(circles)

_were

diregarded

in the

geometrical

discussion of the

model; they

_were

taken, despite

its dilferent _occupancy

degree,

_as

fully equivalent positions

_in a

phason-type

disorder of the structure [15]. ^The

half-occupied

Al

positions (cirdes)

_con be _seen in both _maps either

forming

also

pairs

1.78

À

_apart

or staying alone. The

pairs

disappear

completely (both

positions are

eliminated)

from the structural

scheme _in reference

[15],

as

typically

these

half-occupied

positions _are also accompanied

by

a 0.94

À-distant full-occupied

Al

position (triangle)

which is, as mentioned

above,

the

only

one

kept

in the schematic picture ^{of reference} [15]. ^{The half Al}

positions

that stand

quite

isolated _on the _{map are}

kept,

but their

half-occupancy

has _no clear

interpretation. Hence,

tue

geometrical interpretation

_in real _space of tue model

proposed

in reference [15] ^contains an

implicit significant change

of its AS in superspace: AS1 externat is reduced to

only

trie fraction that does not sulfer tue superposition

depicted

in

Figure

2. Tuis _{means a} 57% reduction of its

surface,

wuicu

implies

_a reduction to 67% of Al in tue composition, ⁱⁿ comparison witu tue

nominal 70%. On tue otuer

uand,

tue model

density

decreases to 4.15

g/cm3,

_very close to tue

experimental

_one,

wuich, however, according

to reference [15], must be ratuer considered _a lower limit

quite

deviated from tue real

value,

because of the presence of voids in the

samples

used for

density

measurements.

It should be noted that

Figures

1la and 116

correspond

to a very

special

section, because

they

contain tue

origin

chosen in superspace,

througu

wuicu passes

(by construction)

tue

only

exact

decagonal

_axis of tue

real-space

structure. For instance, tue presence of

pentagonal

colums

occupied by only

Ni

/Co

_{atoms, as} tue _one in tue center of

Figures

1la and

116,

is _a ratuer _scarce

puenomenon

_in tue structure. In

fact,

in tue model of reference [15] ^is even a ratuer fortuitous

unrepeated configuration.

Tuese columns _are formed wuen rive close

symmetry-equivalent

AS 1 surfaces _cross the real space section. A

typical

set is shown in

Figure

12

projected

_on the internai space. In order to

clarify

the

figure

the rive

replica

of AS 1 internai _are

depicted separately (and

witu dilferent

scale)

from the external _ores. The

region

wuich

produces

tue

pentagonal

cuannels is tue central

"pentagon"

of

Figure

12a.

Moreover,

if tue relevant internai space

point

is inside tue small "circle" of

Figure

12b the 5 atoms of the pentagon in

puysical

space will be of

Nilco,

_{as in}

Figure

ii. But tuis is Dot a

frequent

_case: if tue

point

^{falls inside}

one of tue rive small

"triangles" surrounding

tue central "circle" in

Figure 12b,

foui- atoms in the pentagon will be of

Nilco

and tue other _one of Al.

Analogously,

if the points,

provided they belong

to the

"pentagonal" region

in

Figure

12a, _are also inside the

regions

indexed

by

1,

turee _atoms in tue real _space pentagon will be transition metals and tue otuer two

Al,

wuile

for tue _areas marked witu 2 tue _reverse will

uappen. Sometuing

similar _can be said from tue

model of reference [15], ^{wuere the dilferent domains}

acquire polygonal forms,

wuile tue central domain

corresponding

to pentagons of

only Nilco

is

essentially

reduced to _a

single point.

Tue cuannels formed

only by Nilco

atoms become tuen not

only

_scarce, but _a fortuitous event.

Therefore,

_in

general,

tue

frequent pentagonal

cuannels _in the _structure have

varying

cuemical

2Ù

,A) . * (Al .

*

a~ no . 4. 40 .

'~ ~~ ~Aj~~a°~~~ ~~ _~~O~ °~j.

~~~fi~~~O

^{à ~~~}

fl~j

O

'O

.~~~

~. ô

_fl~a~

~.

.~~

. O

_fI~A~

à.

à. . ô Oô à. ô. Oh

~ ~Oa ~ .~~&O ~~ô

O

~ .~~ôÉ ~a.

~ô~~~~~~~Q ~&~&~°~~~

^O

. ô .&. 4 & .ô.

ô~ ~&

O

~.~ &~ °~A

Oô~.~

~ ~~

/~. Q

~

° ~.^~

]

~ [ô ~ ~~

/

.

~j

~~ Î ~.^~

(

°

[â

oa

~~~

^°A O

~~

^~a.

lO ~&ÔA. ^~ O .~ A. lO . O A.

&

~ ~ôQ ~

.

~

&

~

. & ô. ôQ .

~AO

~~ ~~ô.~~ôÉ~~.~~ '~ ~~&~~~ôO~.~

. .

20

~

20

-20 15 'O 5 O 5 10 15 2Ù 20 '5 lO 5 O 5 10 15 2Ù

IA) (A)

a) b)

Fig, ii. Atomic positions _m trie plane _x3

=

1/4

around _xi

= x2 = 0 for the model of Table I

(a)

and for trie

one of reference [15]

(b).

Black circles indicate Ni

/Co

atoms. White circles represent ^Al

atoms which _come from AS 1, while white triangles mdicate Al atoms from AS 2.