The Al3Ni2 Structure as Approximant of Quasicrystals

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The Al3Ni2 Structure as Approximant of Quasicrystals

Chuang Dong

To cite this version:

Chuang Dong. The Al3Ni2 Structure as Approximant of Quasicrystals. Journal de Physique I, EDP

Sciences, 1995, 5 (12), pp.1625-1634. �10.1051/jp1:1995109�. �jpa-00247163�

J.

Phys.

I France 5

(1995)

1625-1634 DECEMBER 1995, PAGE1625

Classification

Physics

Abstracts 61.44+p 61.66Dk

The A13N12 ^Structure

_as

Approximant of Quasicrystals

Chuang Dong

Department of Materials

Engineering,

Dalian

University

of

Technology.

Daban 116024, P. R. China

(Received

6 March 1995, received 7 June 1995,

accepted

5

September 1995)

Abstract. The present _paper ^discusses the approximant nature of the A13N12 structure from the

viewpoint

of valence electron concentration

(VEC) correspondence

_as well _as from atomic

structure correlation. Its VEC is close to that of

corresponding quasicrystals

and its atomic

structure can be descnbed

by layers

stacked

along

the direction

corresponding

to the 10-fold

axis of the

decagonal phase.

A

large

orthorhombic

pseudocell

is

proposed

whose

edges

_are

integer multiplication

of those of _a basic Cscl unit

along

directions

corresponding

to the

orthogonal

10- fold and _two 2-fold _axes of the

decagonal phase.

Thus its structure _con be expressed ^{in the} form of

A13N12-(1,3,3)

_or

A13N12-(3,1,3),

^the integers

being

the

multiphcity

of the Cscl unit

along

the three directions. The

generalization

of this expression to several other approximants mdicates that approximants can be related to each other

by packing

the basic Cscl

pseudocells

m different ways.

1. Introduction

The

approximants

^of

quasicrystals

_are defined in the

light

of

hyper

dimensional

crystallography [1-3].

The

approximants

_are

periodic

structures that _arise tram the

projection

of _a

hyper

crystal along

^rational

directions,

whereas the

corresponding quasicrystalhne

structures are

projections along

irrational directions. The term

"approximant"

_was first introduced

by

Elser and

Henley

_[1] to denote the

crystalhne

structure

generated

from _a

hyper-cubic crystal using

a

rational ratio

p/q

to substitute for the irrational

golden

_{mean r}

=

(1+ vÎ) /2.

The

projection

scheme

is, however,

so versatile that _even non-existent structures may be

produced.

In

practice,

it is often difficult _to

distinguish approximants

from _common

crystals,

_{smce a}

quasicrystalline alloy

_may well contain several

crystalhne phases

of

quite

^different structures.

One

example

_is the A13N12

type

^of

phase,

_a well-known

vacancy-ordered

Cscl

superstructure,

which exists in many

quasicrystalline systems

such _as Al-Cu-Co

[4],

^Al-Cu-Fe

[5, 6],

^Al-Pd- Mn

[7],

^Al-Ni-Fe

[8],

Al-Cu-Fe-Cr

[9, loi. Ho~vever,

this kind of

phase

_is

relatively

remote from

quasicrystals

_m

composition

and

apparently

does not contain any structural characteristic of

quasicrystals.

Due to these _reasons, trie

study

_on its

relationship

to

quasicrystals

lias been

largely ignored.

©

Les Editions de

Physique

1995

Fe (atomic fraction)

u

~-Afi~Fe4

C~~= ^0.23

0A~é~

^(ela ₌ ^1.86)

~ ~

/

O a Î'A162.3~~24.9~~12.8

o-i °

*

~-Aliocuiofe

0 0

~

Al 0 o-1 0.2 0.3 0A 0.5 0.6 o.7

CU (atomic fraction)

Fig. 1. Schematic concentration chart of the Al-Cu-Fe system. ^The

ela-constant

hne, with

ela

= 1.86, _passes

through

_or

nearby

four

phases:

À-A1i3Fe4, 1-A162.3Cu24 9Fe12 _8, 4-Aliocuiofe and

(-A13Cu4.

This

diagram

_is dra».n

according

to the data reported in

Bradley

and Goldschmidt [5] ^and Faudot et ai. [6].

In _a recent paper

[10],

^the

discovery

of _an

ela-constant

fine _was

reported

in temary Al-

Cu-Fe and

pseudo-temary Al-Cu-(Fe,Cr) phase diagrams, ela representing

valence electron concentration

(VEC)

_per atom.

Figure

1 illustrates

schematically

the

composition

chart for the Al-Cu-Fe

system.

^{The chemical}

composition

of this fine is

CFe

" 0.228

0.4Ccu (e la

=

1.86)

when the _outer electron number of Fe is taken _as -2

Ill].

This _means that there _is

only

_one

degree

of freedom in

composition

variation if _e

la

is _constant. The

composition

zone of the Al- Cu-Fe icosahedral

phase (I-Alcufe)

is aise

elongated along

this fine

[1ii.

Similar _e

la-constant

hnes _can aise be observed in the

phase diagrams

of other

systems

^such as Al-Cu-Mn

[12],

^Al- Cu-Co

[4j,

^Al-Pd-Mn I?i and Al-Cu-Ru

[13]. Therefore,

the existence of the

ela-constant

fine should be _a

general phenomenon

for

quasicrystalhne phase diagrams.

It is noted that in

Figure

the

ela-constant

fine _passes

by

net

only

the

composition

_zones of I-Alcufe and its well-known

approximants À-A1i3Fe4

but aise _some Cscl-based structures

such as

#-Al~ocuiofe

with A13N12 structure and

(-A13Cu4

with _a modified

~f-brass

structure.

Therefore,

_we tend to consider ail these

crystalline phases

_as

approximants.

As

suggested

_ear-

lier,

the

apprommants

are those

Hi~me-Rothery crystaiiine phases

that possess

apprommateiy

the _same valence eiectron concentration _as that

of

the

corresponding qi~asicrystai _[14].

This introduced _{a ne~v} criterion to the usual definition of Elser and

Henley iii

which is

entirely

based upon

crj,stallographic

correlation. The

Hume-Rothery

charactenstic of

quasicrystals

has been established _since 1987

[15],

^{and the} new definition of

approximants

stresses bath the

crystal- lographic

and electronic structure

correspondence

between

quasicrystal

and

approximants.

The Cscl

superstructures, having

the _same VEC _as their

corresponding quasicrystals,

_rep-

resent _an

important

_group ^of

approximants,

as contrasted ta those

having complex

structures

(e.g. A1~3Fe4

and

A14Mn). Ho~vever,

their structural

relationship

to

quasicrystals

is

usually

unknown. As _an

example,

this _paper will deal with the A13N12 structure. As _a matter of fact this

phase

_{is a}

decomposition product

of metastable Al-Ni

decagonal phase prepared by rapta

quenching _[16].

The

j-brass

structure Will be discussed in _a seperate paper

[17].

N°12 A13N12 APPROXIMANT 1627

Table I. A

comparison

between

phases of

the

A13Ni2 type

and two reiated

qi~asicrystais

in the

Ai-Ci~-(Fe,

Go

) systems.

Atomic

density

_{p is} in

À~~.

_N

_represents

_VEC

pet i~nit uoii~me.

The iast coii~mn shows

do,

the interatomic distance that

corresponds

_to the

edge of

the basic Csci

psei~doceii.

For the

A13N12-type phases, do

"

(Vcejj/3)~/~

is the auerage size

of

the

Csci

psei~doceii.

The icosahedrai and

decagonai phases

houe

respectiueiy do

"

ar/ cas(18°) /T

and

do

" ar x 2 _x

cas(18°)/T

that _are

edge iengths of

the basic

pentagons.

Since the iattice

parameters

^{and atomic densities aboi~t}

#-Aiio Ci~iofe

and

H-A13 (Ci~,

Go

)2

are Rot

known,

those

of A13Cu2 f2$j

_are used instead.

phase composition parameters ela

N

a = 2.903

c = 5.094

a = 1.81

c = 5.094

1-Alcufe

A162.3Cu24.9Fe12.8

_{ar =} 4.465 0.0660 1.86 0.123 2.902

Al6oCu3oCoio

_a

= 4.106 0.0672 2.0 0.134 2.903

c = 5.094

D-Alcuco

Al6sCuii Co21

_{ar =} 2.437 0.0724 1.94 0.140 2.865

2.

Approximant

Nature of trie

A13N12

Phase

2.1. CRYSTALLOGRAPHIC STRUCTURE. The _structure of this

type

^of

phase

_was known

long

time _ago

[18].

It has _a rhombohedral structure with _space group P-3m1. It is characterized

by vacancy-ordering

in _a Cscl

superlattice.

One unit cell contains three Cscl

pseudocells,

with cell formula

Al3Ni2ai,

o

representing

_{one vacancy} site. The vacancy ^sites are ordered into

hexagonal

network _{on a}

( ii1) plane

of the Cscl

pseudocell

with

three-layer period, determining

the

diagonal

of the rhombohedral cell _{or in} the

hexagonal coordination,

the

length

of the axis

c. See Pearson

[19j

^{for its detailed structural}

description.

This

phase

is the prototype ^of a series of

vacancy-ordered

"T"

phases,

uncovered

by

Lu and

Tang [20j

and studied in detail

by

Van

Sande,

De

Ridder,

Van

Landuyt

and Amelinckx

[21],

whose structure _can be described in terms of _a basic Cscl type of cell with

ordenng along

the

[111]

^direction.

Chattopadhyay, Lele, Thangaraj

and

Ranganathan _[22j

first

pointed

Dut the

possible

link between the

vancancy-ordering

and 1-dimensional

quasipenodicity. They

noted that the vancancy

layers

in the _T

phases

follow _a Fibonacci _sequence. The

A13N12

structure,

designated

_{as T3,} has _a

vacancy-layer period

of three

layers.

2.2. VEC CORRESPONDENCE. In order ta illustrate the VEC

correspondence,

_we have

calculated the VEC per ^atom

(ela

and _per unit volume

(N)

for this kind of

phase

and two related

quasicrystals

_in the

Al-Cu-(Fe, Ca) _systems.

The results _are shown in Table I. For trie Al-Cu-Fe

system,

^the

agreement

bath in VEC and interatomic distances between the

#- Ahocuiofe phase

with

A13N12

structure and the I-Alcufe

phase

is excellent indeed.

By using

the free electron

mortel,

_we obtain the

corresponding

Fermi

sphere

diameters of the _two

phases

ta be

2kf

= 3.066

À~~ (d

= 2.050

À)

and

2kf

= 3.07î

À~~, (d

= 2.042

À), respectively.

The intense reflections

(110) (K

= 3.060

À~~,

d

= 2.053

À)

and

(102) (K

= 3.034

À~~,

d = 2.071

À)

of the

# phase

^define a Brillouin _zone

(BZ)

similar ta the second-order

(110)

BZ of the Cscl

phase

but with lower

syiumetry.

^Note that these

planes correspond

ta the

(110) planes

of the Cscl

phase.

For the Al-Cu-Co

system,

the N values of

H-A13(Cu,Co)2

and

D-Alcuco _are

quite

^close ta each

other,

toc.

However,

these values _are

slightly larger

than in the _case of the Al-Cu-Fe

system.

^This can be attributed _ta smaller

negative ela

contribution

(ela

=

-1)

and smaller atomic radii

il.25 À)

^{of Ca than} those of Fe

le la

=

-2,

atomic radii

= 1.27

À).

These _two factors increase overall _e

la

value _as well _as atomic

density.

Therefore,

the

approximant identity

^the A13N12 structure _can be established from the VEC

correspondence

ta

quasicrystals.

2.3. AToNiic STRUCTURE CORRESPONDENCE.

According

ta the orientation

relationship

between _an Alcufecr

approximant

with

complex

orthorhombic _structure

(O1AicuFecr)

and

a Cscl surface

layer

induced

by

ion-beam

milhng,

it has been

suggested

that _a deformed

pentagonal network, resembling

the Penrose

Tiling,

can be defined _on

(110) planes

of the Cscl

phase [23j.

This

pointed

Dut _a

possible

mechanism that the two

distinctively

dilferent

structures _can be

mutually

transformed. Structural

relationship

of the A13N12 type ^of

phases

to that of

quasicrystals

_can be resolved in _a similar

fashion,

since this is _a

vacancy-ordered superstructure

^based _on trie Cscl

pseudocell.

Figure

2 sho~vs the atomic structure of the

A13Cu2 Phase

with A13N12 structure

[24].

^Its

hexagonal

lattice parameters are a = 4.106

À,

_c

= 5.094

À.

_The

upper

plot

is _a stereo view of its

hexagonal

unit cell

together

~vith _one

rhombohedrally

distorted Cscl

pseudocell,

and the lower

plot

illustrates the

[010j projection. Special

attention is

para

to the

(102)

and

-120) planes

that

correspond

to the

Cscl-(110) planes

and hence

quasi-penodic planes

of the

decagonal phase (or

the

plane

normal to _a 5-fold axis _in the _case of the icosahedral

phase).

For this

ieason the

(102)

and

(-120) planes

_are marked eut in

Figure

2 The atomic

configurations

on these t~vo

planes

_{are very} similar ta eacli other.

There may be _two

possible

orientation

relationships

which _are based

respectively

_on these two

planes. Wang [16j

^{has observed that after}

annealing

the metastable

decagonal

AlNi

phase

is transformed into the A13N12

Phase. Though

the orientational

relationship

between them _are net

yet

^known

completely,

the

experimental

results support at least the second

(-120)

scheme but do net exclude the first

(102)

scheme.

The two orientation

relationships

ai-e best illustrated in the

[010j-projected

structure of

Fig-

ure 2

(the

lower

plot).

In trie first

(102)-based _scheme,

the

[212j direction, being perpendicular

to the

(102) plane,

_is

supposed

to be

parallel

to the 10-fold axis of the

decagonal phase

_{or a}

s~fold _axis of trie icosahedral

phase.

The 3-fold

[001]

direction of the

A13Cu2 Phase

_is

placed parallel

to _a 3-fold _axis of

quasicystals.

^The

angle

between

[212j

^and

[001j

^{is 35.61}

,

close _ta that between 5- and 3-fold _axes. In the second

(-120)-based scheme,

the

[-2 iii, [212j, _[010j

directions _are

respectively parallel

ta 2P-fold, 2D-fold and 10-fold of the

decagonal phase.

The

relationships

_among

Cscl, A13N12,

icosahedral and

decagonal phases

_can be summanzed

as follows:

Cscl

T3/(102) T3/(-120) decagonal

icosahedral

[110j [212j [010j

^{10-fold 5-fold}

[-1loi _{[010j [212j}

2-fold D 2-fold

[001j [-2 iii [-2 iii

2-fold P 2-fold

Figure

3

presents

^the atomic

arrangement

in the

(102)

and

(-120) planes

of the

A13Cu2 phase together

with _one similar atomic

configuration

of trie

decagonal phase

_in Al-Mn

system (abbreviated

_as

D-Almn). Exactly speaking,

the

(102) plane

_is net a flat

layer:

_some atoms

fluctuate above and below the

plane,

however the

displacements

_{are very} small _as

compared

ta the interatomic distances

(about 2-3%).

The unit cell _on

(102) plane

consists of three deformed

N°12 A13N12 APPROXIMANT 1629

(102)

~

_,---. (-120)

Q

Al

. eu

o.149 11 Vacancy Site

, z = o.352

, 1,

a b

3-fold

. (102)

1/2

~~~

l/2 (~120)

*

~

. _~~~~

~~~~~

Q

^~'

~~~

1/2 p

Q1/2

1/2

~~

l/2 *

j010)projection

Fig.

2. Structure of the A13Cu2

phase

of A13N12 type. ^The

hexagonal

system is chosen to show this rhombohedral

phase. Laige

circles represent Al atoms and the small

ones Cu atoms. The upper

plot ^is _a stereo _view of

a

hexagonal

unit cell, inside _is drawn _a

rhombohedrally

deformed Cscl basic unit. The lower plot is the [010] projection, where its

crystallographic relationship

with

quasicrystals

is indicated. The

(102)

plane _is marked out which corresponds to the

quasi-periodic planes

^{of the}

decagonal phase.

Cscl

rectangles.

One of the

rectangles

contains _a vacancy ^site so that trie

composition

on the

layer

_is

A13Cu2a,

_or

A150Cu33a17

in atomic

percentage,

the _{same as} the cell formula. The unit cell _on

(-120) plane

aise consists of three deformed Cscl

"rectangles".

The

stacking

of the

(102) plane along [212j

^{direction and}

(-120) plane along [010j

^{direction has} a

six-layer penodicity.,

or 12.426

À.

The D-Almn structure

(originally proposed b»

Steurer

[25j

^and

recently

modified

by

Beeli and Honuchi

[26]) presents

^{similar local} atomic

configurations (Figs. 3, 4).

It is formed with two kinds of

layers,

_one flat B

(and

its 10-fold _rotation

image b)

and the other

puckered

A

A13Cu2-(010) A13Cu2-(102) D-Almn (flat layer)

4 299

~

+0

~

o

~

Q

~

~ R

"

4 064 ^~

-0 077

ce ^u~ ce

É

11 ~ ~

_~Î

~

oe ce ce

2.977

~ _~

^{Î -0.0?3}

~

0

4 106 ⁴ 817

©

_Al

_~

Cu

~J

Vacancy

Fig.

3. Atomic arrangement ^{in the}

(102)

and

(-120)

planes of trie A13Cu2 phase

(left)

and _a similar

atomic

configuration

from D-Almn

(nght).

Some interatomic distances and the

displacements

above

and below the

plane

_ai-e marked.

Large

circles represent ^Al atoms and the small

ores Cu

(Mn)

atoms.

Al

u

Cu _.

u

Cu O _.

~~ O

n o

Cu _.

u

u Al

u

Cu

Rat layer

Fig.

4. Part of the flot

quasi-periodic plane

of D-Almn where _{one can see} traces of the A13N12

structure feature

(trie

shaded

areas).

Trie atoms _are relinked to resemble trie

Cscl-(110)

atomic

gnd.

Trie

original

structure _is taken from Beeli et ai. [26].

(and

its 10-fold rotation

image a),

which _are stacked

along

the 10-fold direction in the

stacking

sequence of ABAaba. Part of the flat

quasi-periodic plane

_is shown in

Figure

^4. The atoms are relinked to resemble the

Cscl-(110)

atomic

grid.

Such _a

reorganized hnkage

reveals the

N°12 A13N12 APPROXIMANT 1631

following important points:

1 The shaded _{areas on} the flat

plane _present

_A13N12 characteristic: three deformed Cscl

"rectangles"

with the central site vacant. This

implies

that trie

A13N12-type

of structure retains indeed certain structural

properties

of

quasicrystals.

2 The A13N12

Phase

is formed

by stacking

six

(102)

_or

(-120) la»em,

while D-Almn aise has _a

six-layer stacking

_sequence

along

the 10-fold _axis.

3

Quasicrystals

_are

vacancy-ordered

based _on Cscl basic

pseudocell.

The vacancy sites

are net distributed in random. Such _an

ordering

should

play

_an

_important

rote in the stablization of

quasicrystals.

4 The _vacancy concentration VC of the A13N12 structure is

comparable

to that of _qua-

sicrystals.

On the flat and

puckered layers

of

D-Almn,

VC values _are

respectively 21%

and

11%(~).

The average value _is about

14%.

Trie A13N12 structure has _a

slightly higher

VC of about

17%.

3. Unified

Expression

for Cscl-based

Approximants

A

periodicity

of12.426

À along

the

[212]

^{direction is achieved}

by stacking

_six

(102) layers (refer

to the

[010] projection

_m

Fig. 2). Adding

the

periodicity along

the

[010]

^{direction of} 4.1off

Àand

_that

along [-2 -11]

of 8.748

À,

_a

larger

orthorhombic

pseudocell

_can be constructed which shows _a

simple crystallographic

correlation to the

decagonal quasicrystal:

AAI~N;e " 4.106

À, [010]hexagonal II

2-fold

D,

BAI~N;~ " 12.426

À, [212]hexagonal 11

^10-fold.

CAI~Ni~ " 8.748

À, [-2 11]hexagonai II

2-fold P when using the orientation

relationship

based _on the

(102) plane,

or:

AAI~Ni~ " 12.426

À, [212]hexagonai II

2-fold

D,

BA13N12 " 4.ÎÙÙ

À, (ÙlÙjhexagonal II ÎÙ-fOÎd,

CAI~Ni~ " 8.748

À, [-2 11]hexagonai II

2-fold P

when

using

the orientation

relationship

based _on the

(-120) plane.

This orthorhombic cell _is half that of the

Al4Cug phase

with

~i-brass

structure. As discussed in another _paper, the

Al4Cug phase

is _an approximant of

quasicrystals

and it _can aise be

described

by stacking

two kinds of

layers,

_one flat and _one

puckered, along

a < l10 > direction in _a

stacking

_sequence similar to that of D-Almn

il?].

The

large

"orthorhombic"

pseudocell developed

^from it follows the _same orientation

relationship

to

quasicrystals

_as the

above,

_1-e-,

AAi~cug

" 12.31

À, [-110]cubic Il

2-fold

D, BAi~cug

"

12.31À, _[110]cubic II 10-fold, CAi~cug

" 8.7068

À, _[001]cubic II

2-fold P.

(~) ^These values _{may vary} in different part ^of the

quasi-penodic planes

_as well

as m different alloy systems. ^We bave counted _{an area} of 33 _x 39 À~ of trie D-AIÀ/In mortel of Steurer-Beeh [25,

26].

2-fold P

,

~°~lAfi~~~~

Î010]A1i3Fe4

(

,

1 ,

, ,

1 ,

,_, /

[010101 _Alcufecr ,~ ,

, i

',aAl13Fe4 ',

, ,

, ,

, ,

,

, i

, i

1 1

Îl_00]A13Mn ,/

ÎÙIù]A13Mn /

r i

i t

r /

t i i r

[212)A13N~

t

-/

~~~~]A14Cu9

r

r t

~ t

'

t t

l r

'

l' Î001]A1i3Fe4

l

10]csCÎ

ÎÎ~Î~~~~~~

[00110,,Alcufeci

1-11°lcsci 2-fold D

/'°~°lA>3N<2

projection

direction

corresponds

to 10-fold

Fig.

5. Construction in trie

Cscl-(110)

atomic

grid

of trie

(pseudo)

orthorhombic cells

having edges parallel

to three

orthogonal

_axes of the

decagonal

phase, 1-e-, 2-fold D and P, 10-fold

(trie projection

direction). For trie Cscl, A13N12 and Al4Cug phases, ^their

original

cells must be

reorgamzed

to form the _ne~v orthorhombic

pseudocells.

The A1i3Fe4 monoclmic cell _is twmned to give the doubled

pseudo

orthorhombic cell. The orthorhombic approximants 01-Aict,Fecr and 03-Aicufecr

(A13à/In type)

directly bave their edges parallel to trie three _axes. Each orthorhombic

(pseudo cell)

_is labelled

by

_a

shaded

triangle

and the

projection

direction _on its

upnght

_corner.

In this _lvay~ the Cscl-based

approximants A13N12, Al4Cug

and Cscl _con ail be

expressed

in

larger

~'orthorhombic" cells ~vith three

edges parallel

to the

orthogonal

2-fold D, 10-fold and 2-fold P directions of the

decagonal phase.

A schematic illustration _on the construction of the

pseudocells

_is

presented

in

Figure

.5.

If the orthorhombic

pseudocell

of the basic Cscl _{structure is} taken _as the unit with

edges ,40

= ac~o x

vi,

_BO

= acsci x

vi, _Go

= acsci, trie other _{two cari} be

regarded

as

superstructures

~vith

integral multiplication

of the basic

edges.

For the A13N12

structure,

_{AAI~Ni~ "} 1 _x

Ao.

BAI~Ni~ " 3 x

Bo,

_CAI~NI~

" 3 x

Go

For the

Al4Cug phase, _AAi~c'ug

" 3 _x

Ao, BAi~cug

" 3 _x

Bo.

CAi~cug

" 3 _x

Go. Then,

these Cscl

superstructures

_con be

expressed

into the

following

simple

forms:

Cscl-(1,1,1), A13N12-(1,3,3)/(3.1,3), Al4Cug-(3,3,3).

The three

figures

_represent

N°12 A13N12 APPROXIàIANT 1633

respectively

trie

multiplication

of the basic Cscl unit

along

(lie directions 2-fold

D,

10-fold and 2-fold P of the

decagonal phase.

Furthermore,

this

type

^{of formula is} net confined to these

commonly accepted

Cscl _su-

perstructures.

^The

approximants

are divided into tvTo groups

according

to their

positions

in

phase diagrams [10.14],

one consists of

complex approximants (lying

to the left of I-Alcufe _m

Fig. 1)

and the other the Cscl superstructures

(lying

to the

right

of I-Alcufe in

Fig. 1).

Trie

phases

in the former _{group can} aise be

expressed

in _a similar _way with

help

of trie basic Cscl

pseudocell.

In the Al-CU-Fe

system, only

_one

phase

is included in this _group, 1-e-, trie

A1~3Fe4 phase.

It has

C2/m

_as its space group and trie

follovTing

lattice

parameters:

a = 15.489

À,

b = 8.0831

À,

_c

= 12.476

À, fl

= 10î.72°

[27].

^Its

approximant

nature lias been

extensively

discussed

[28].

^{Its monoclmic cell} can be transformed

by

a

twinning operation

^into an or-

thorhombic

pseudocell containing

two

original

monochnic unit cells. Such _a

t~vinning

mode is often observed

experimentally.

New the

edges

_are

_AAii~fe~

" c = 3 x

Ao, BAii~fe~

_" ^b ₌ 2 _x

Bo,

CAii~fe~

-S 2 _x

cas(18°)

_{x a}

= 10 _x

Go,

_{or in} the

simplified

form

A1i3Fe4-(3,2,10).

In the Al-Cu-Fe-Cr

system,

there _{are a} series of

complex approximants [9. 29].

Two

examples

_are

OlAicufecr (B-face

centered

orthorhombic,

_a

=

32.54,

b

= 12.27 and _c

= 23.64

À [9,23])

and

O3AicuFecr (a

= 14.582, b

= 12.320, _c

= 12.363 -~

[30],

isostructural with

A13Mn [31])

The

edges being parallel

to trie three directions mentioned

above, they

can be

expressed

_as

OlAicufecr-(8,3,8)

and

03Ai~Nin-(3,3,5),

_as shown in

Figure

5.

4.

Summary

We have

analyzed

the valence electron concentration and atomic structure

correspondence

between the

A13N12

type ^of

phase

^and two relevant

quasicrystals.

The results support ^trie conclusion that this type of Cscl superstructure ^is an

appi~oximant

of

quasicrjrstals,

in

spite

of its

relatively simple

atomic structure. Its structure cari be described

by

_a

six-layer stacking

of

(102) planes

that

corresponds

to a

Cscl-(l10) plane

_or to the

quasi-periodic planes

of the

decagonal phase.

Such _a

description

leads to _a

langer

orthorhombic

pseudocell

lvith

edges being parallel

to the

orthogonal

10-fold and _two 2-fold directions of

quasicrystals.

Trie

pseudocell edges

_are

integer multiplication

of those of the basic Cscl unit _so that this

phase

_can be

simplified

into the expression

A13N12-(1,3,3)

or

A13N12-(3,1,3),

the three

integers being

the

multiplicity

of the basic Cscl

pseudocell aloiig

the three _axes of the

decagonal phase.

This form of

expression

con aise be

applied

to other

approximants

_so that it _can be treated _{as a} unified _way to relate

approximants

to the basic Cscl _units.

Acknowledgments

Part of the financial

support

^for this work _was

provided by

Dalian

à~unicipal

Foundation for

Young

Scientists. The author _is

grateful

for fruitful discussions with Piof. R. l&~ang from ^Sci.

& Techn. Univ. of

Beijmg.

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