Partial pair distribution functions in icosahedral Al-Li-Cu quasicrystals

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Partial pair distribution functions in icosahedral

Al-Li-Cu quasicrystals

M. de Boissieu, Ch. Janot, J.M. Dubois, M. Audier, B. Dubost

To cite this version:

Partial

_pair

distribution functions

in

icosahedral Al-Li-Cu

quasicrystals

M. de Boissieu

_{(1, 2),}

Ch. Janot

_(1),

J. M. Dubois

_(2),

M. Audier

₍₃₎

and B. Dubost

₍₄₎

(1)

Institut

_{Laue-Langevin,}

156X, 38042 Grenoble Cedex, France

(2) LSG2M,

Ecole des Mines, Parc de

_Saurupt,

54042

_Nancy

Cedex, France

(3)

LTPCM-ENSEEG, B.P. 75, 38402 St Martin d’Hères

_Cedex,

France

(4)

Pechiney,

Centre de Recherches de

_Voreppe,

B.P. 27, 38340

Voreppe,

France

(Reçu

le 5 décembre 1988, révisé le I S mars _1989,

_accepté

le 16 mars

₁₉₈₉₎

Résumé. 2014 Des informations structurales concernant la

phase icosaédrique

T2 du

système

AlCuLi sont _{obtenues par diffraction des neutrons,} avec substitution

_isotopique

sur le cuivre et le lithium. Des fonctions de distribution de

_{paires partielles}

sont alors déterminées et

_comparées

à

celles que l’on peut déduire de la structure connue d’une

_phase

_(R)

_cubique

voisine. Cette étude

permet de

_dégager

les

_analogies

et les différences de l’ordre à courte distance des deux

_composés.

Abstract. 2014 Neutron diffraction with

isotopic

substitution on _{copper and lithium is used} to obtain

structure information in the T2-icosahedral AlCuLi

_phase.

Partial

_pair

correlation functions are

then determined and

_compared

to those deduced from the known structure of the related cubic

R-phase.

Similarities and differences in short range order of the two

_compounds

are

_suggested.

Classification

Physics

Abstracts

61.55H - 61.50E - 64.70E - 61.10L

1. Introduction.

The

_discovery

of

_systems

with diffraction

_patterns

_exhibiting

the icosahedral

_symmetry

_[1]

forbidden in classical

_{crystallography}

has

_recently

been a

_challenge

for theoreticians and

experimentalists

who have tried to find models

_explaining

the observed features. In

fact,

what is

_{retrospectively}

_surprising

is that Shechtman

_discovery

startled so much the

crystallog-raphers.

Mathematicians

_[2]

had

introduced,

early

in the

_century,

_{quasiperiodic}

and almost

periodic

functions _{and it is easy} to realize that these families of functions Fourier transform

into dense sets of

_{sharp Bragg-like peaks.}

_Moreover,

incommensurate

_phases

_{[3, 4]}

were at

the time an

_already

known

_example

of non

_periodic

structures

_showing

_strong

diffraction

peaks. MacKay

[5, 6]

had also drawn attention to two- and three-dimensional Penrose

_tilings

(3 DPT)

and their diffraction

_patterns

which, indeed,

were

_{qualitatively}

similar to that

observed

_by

Shechtman et al.

_[1].

_However,

this icosahedral

_phase

has remained somewhat exotic

and, indeed,

considered as the

_archetype

of a new state in condensed matter. Since

then,

numerous

_examples

of metallic

_alloys

have been

_discovered,

the atomic structures of which

_display

« forbidden »

_symmetries

_(see

a

_complete

review in

[7]).

The debate has

resulted in a wealth of

_published

_{papers. A}

_bibliography

of

_{quasicrystals}

which chronicles the

so-called initial

_{period, roughly}

as far as

_mid-1986,

contains about 700 references

[8].

A

collection of

_{significant reprints}

has

_recently

been edited

_by

Steinhardt and Ostlund

_[9]

and or

European

Workshop

[10]

has somewhat

_updated

the

_knowledge

state of the

_subject.

Basically,

successful

_descriptions

of

_{quasiperiodic}

_geometrical

networks have been achieved

using

a

_{surprising variety}

of

_quite

different schemes for

_generating

them : space

_{tiling by}

two

rhombohedral cells with

_matching

rules

_[11-13],

inflation-deflation

_procedure

_{[13, 14],}

multigrid

or dual methods

_[15-17],

_{strip-projection}

or

_{cut-projection approaches}

[3, 18-21].

The

_{cut-projection}

method

_{[3, 21]}

in

_particular

_{shows that any}

_{quasiperiodic}

network has

actually

hidden

_periodic

translations which can be recovered if the structure is

_properly

described in a

_{higher-dimensional}

_{space. This is} a

_simple

_consequence

_of,

for

instance,

the icosahedral

_point

_group

_symmetries

m 3 5

_{being compatible}

_{with space-group in 6-dimensions}

while it is not in 3-dim.

Once

_{generated by}

one of the above

_methods,

a

_long

_{range ordered}

_quasilattice

without

periodicity

is not the end of the

_description

of a

_{quasicrystalline}

structure. We still have to _say

where the atoms are, not

_only

to

_{specify completely}

the structure but also to shed

_light

on the

problems

of

_{understanding}

_{why only}

certain elements and stoichiometries

_pxoduce

icosahedral

phases

and how the icosahedral

_phase

_grows.

_Furthermore,

the detailed atomic structure is

certainly

necessary to understand the

_properties

of these

_alloys,

in

_particular

electronic and

magnetic

properties. Experimental approaches

and

_modelling

have both contributed to the

point

which has been

_extensively

reviewed

_{[7, 22].}

The 6-dim

_approaches

of decoration models is

_probably

the most

_generic

and

_global

one and

_is,

_actually,

a direct

_{extrapolation}

of

conventional

_{crystallography}

methods

_{[23, 24].}

On the other

hand,

sophisticated techniques,

such as contrast variation effects with neutron

_diffraction,

have been shown to be a

_requisite

for

_deciphering

raw data in the framework of a _{N-dimensional space ; this has}

_successfully

resulted into a

_{experimentally}

_{derived structure, with occupancy modulation of} one Mn and

two Al different sites

_{[25, 26]}

for the AIMnSi icosahedral

_{quasiperiodic}

_crystal.

The second

_{major quasicrystalline}

_system,

_namely

AlLiCu

_{alloys, although extensively}

investigated,

does not seem to have

_kept

its

_promises

so far.

Indeed,

it was

_{enthutiastically}

thought

of a tremendous

_leap

forward into

_{investigations}

when Dubost et al.

_{[27, 28]}

and others

_[29-32]

_reported

the

_growing

of AlLiCu

_{quasicrystal grain approaching}

a millimeter across and

_{consequently raising}

the

_prospect

for

_{single crystal X-ray}

and neutron diffraction

pictures.

Single-crystal-like

approaches

[32-35]

have

_mostly

illustrated that the

_major

recorded reflections

_only

exhibits

_2-fold,

3-fold or 5-fold

_{point symmetries.}

Four circle

_single

crystal

diffraction scans

(Fig. 1)

have measured

_peaks

which are

_mostly

indexable with six

indices that

_belong

to

_primitive

icosahedral Bravais lattice. This was in

_{good consistency}

with electron diffraction

_patterns

and

_high

resolution electron

_{microscopy images}

_[36-41]

which

also lead to the derivation of

_{relationships}

between structures of the icosahedral

_phases

and the

_{approximant crystalline}

structures of the same

_system.

_However,

the

_prospect

of

_getting

experimentally

into the proper atomic decoration of this

quasicrystal by using

modified direct

crystallography

methods seems to be

_hopeless

and on the other hand there are so far very few

available

_comparisons

between models and

_experimental

data

_[33,

34,

42-46].

Such a

_difficulty

to

_{specify completely}

the structure _{may be} not

_surprising,

_considering

the structural

complexity

of this

_{ternary system}

and the tedious work to

_analyse

diffraction intensities to

which one of the atoms,

namely

Li,

does not contribute at all

_(X-rays)

or with a

_weight

of less than 4 %

_(neutrons)

to the total.

_{Clearly enough, only}

contrast variation

_{techniques might}

have a

_(little)

chance to reach the

_goal,

a

_point

which will be further elaborated later in the

paper.

Moreover,

several

_peculiar

observations were made which indeed did not contribute to

Fig.

1. -

Four-circle neutron diffraction scans obtained from a centimeter size oriented dendrite of

T2-AL6Li3Cu ; scattering

vectors are

_along

two different 3-fold axes.

_Indexing

has been made within a

frame of 2-fold axes.

diffraction

_patterns

_[35]

which remain somewhat

_mysterious

even if Levitov

_[47]

has

_proposed

an

_explanation

within a modified

_{strip-projection}

model.

_Despite

the appearance of well-defined

facets,

the AlLiCu

_quasicrystal

_samples

seem to have a

_{high degree}

of atomic disorder

[48, 49]

and the

_single

_(quasi)

_{crystal Bragg peaks}

have

_complex

and

_{symmetry-dependent}

shapes. Attempts

to describe some of these observed

_{peak shapes}

[50, 51]

were made

_by

postulating

a

_{superposition}

of uniform

_phason

strains,

or

spatially

varying phason

strains

either thermalized or

_quenched

_[52],

which is for instance a _way to introduce local violation of the

_matching

rules into a 3-dim Penrose

_tiling.

Detailed

_analysis

of electron diffraction

patterns

[53]

and

_{convergent-beam}

electron diffraction

_[54]

have shown some

_asymmetries

which have been taken as indices of

_{non-icosahedrality}

and have

_brought

_support

to twin-models

_{[55, 56].}

_But,

the twin model has been

_strongly

dismissed

_by

Field Ion

_Microscopy

large single grains,

as-cast or heat treated

_{powder samples}

have also shed doubts on the

perfect

quasicrystallinity

of the

_system

_{[58, 59].}

Conclusively,

there is still

_place

for further

_{experimental approaches.}

Neutron diffraction

measurements,

along

with contrast variation

methods,

which have

_proved

to be so successful

into the derivation of Al-Mn

_quasicrystal

structure

_[25,

_26,

_60]

have

_obviously

to be

_applied

to

the AlLiCu case. The purpose of this paper is to

_report

on the first

_part

of such an

_approach,

namely

derivation of

_{partial pair}

distribution functions.

2. The

_samples

and some of their

_properties.

The

_knowledge

of the

_{equilibrium phase diagram}

of the

_{aluminium-lithium-copper}

_system

has

long

been limited to the isothermal sections at 500 °C and 350 °C established some _{30 years}

ago

by Hardy

and Silcock

_[61].

The identified

_phases

were the

_{tetragonal 0-Al2CU,}

fcc

&-AILi,

fcc

_{TB-AI7.sCu4Li,}

bcc

_R-AlSCuLi3,

hex

_Tl-Al2CuLi

and

_{(yes !) a non-identified}

structure

T2-Al6CuLi3.

Within the frame of researches for

_light

AlLi-base

_alloys

to _{be used for aerospace purposes,}

the AlLiCu

_{phase diagram}

has been revisited

_{carefully along}

with

_{thermodynamic properties}

of the

_phases

of interest

_[62-66].

In

_particular,

it has been shown that

_only

slow

_cooling

rates

are

_required

to form the icosahedral

_T2-phase

which behaves

_actually

like an

_equilibrium

phase going,

in

_{particular, directly}

to the

_liquid

state _upon

_heating.

The

_T2-phase

can be

obtained as

_{grain boundary precipitates}

_upon

_annealing

in an aluminum rich AlCuLi

_alloy,

or

by

direct solidification of

_large

dendrites embedded into an Al- rich

matrix,

or else as

_already

mentioned,

by

free solidification into

_{single grain quasicrystal}

_[28].

The

_phase

_diagram

as

reported

in

_[62]

shows that the bcc

_R-phase

and the icosahedral

_T2-phase

_{have very similar} features. Their densities are almost the same

_(2.46

and 2.47

g/cm3, respectively)

and

_they

form within a _very narrow

_composition

_{range :}

_{Als.60CU1.20Li3.2o}

for the

_R-phase

and

AIs.7oCu1.ogLi3.22

for the

_T2-phase

_(within

3 % error

bars).

The

_R-phase

is

_likely

to melt

congruently

at 638 ± 2 °C while the

_T2-phase

_undergoes

a

_{non-congruent}

_melting

at 622 ± 2°C.

_{A very}

_{unfortunate consequence is that} a

_completely

_pure

_T2-phase

cannot be

obtained

_easily

and one has to

_accept

contamination

_by

residual a-Al or

_(and)

_Tl-phase

except

perhaps

for the small triacontahedral

_{single grains}

which result from free solidification with

_separation

of the dendrites from the

_{residual liquid}

in internal

_shrinkage

cavities. This has of course to be

_kept

in mind when

_analysing

diffraction data from bulk

_samples,

even if

the

_point

is somewhat dedramatized

_by

the

_relatively

small distance in the

_{phase diagram}

between the true

_liquid-solid

transition and the virtual

_congruent

_melting

_temperature

of the T2

_compound.

The maximum volume fraction of T2 forms when

_keeping

_{long enough}

the

right compositional

alloy

within the 500-620 °C

_temperature

_range, while the

_R-phase

forms

more

_easily

between 625 °C and 635 °C. At this

_stage

it is

_interesting

to

_point

out that R and

T2-phases

differ,

slightly,

only

in their Al/Cu relative concentrations.

The use of neutrons is a

_priori

_particularly

attractive in

_studying

the atomic structure of

AlCuLi

_compounds.

First of

all,

the Li atoms are

_hardly

« visible » with

_X-ray

since their

atomic

_scattering

factor is

_only

2/10 that of

_Al,

which combined with concentration

effects,

leads to a mere contribution of less than 5 x

10- 3

of the total diffracted

_intensity.

With

neutrons, the ratio of the

_scattering

_lengths

of Li to Al _{rises up} to more than 0.5.

_Moreover,

Li

has two stable

_{isotopes, 6Li}

and

_7Li,

whose

_{respective scattering lengths}

are + 0.20 x

10 -12

and - 0.222 x

10-12

_cm,

_with

natural abundances of 7.5 and 92.5 %. This allows

_significant

changes

into the contrast on the Li sites of the structure when

_alloys

are

_prepared

with different

6Li/7Li

mixtures,

of course _{without any} disturbance of the

_crystal

_chemistry

of the

of the two

_{isotopes. Copper}

has also two stable

_{isotopes : 63Cu}

with a

_{scattering length}

+ 0.643 x

10-12

cm and natural abundance

69.17 %,

65Cu

with a

_{scattering length}

+ 1.061 x

10- 12

cm ;

unfortunately

both Cu

_isotopes

have

_{positive scattering length}

which forbids the desirable zero-scatterer mixture and somewhat restricts the available contrast variation.

The

6Li/7 Li

mixtures,

at different

_composition

were chill cast from 250 °C into boron nitride

coated steel crucibles under argon controlled

atmosphere.

These lithium mixtures were then

added to _{proper Al-Cu}

_{liquid alloys}

at 730 °C. The

_resulting

AlCuLi

_liquid

was

_finally

child

cast within five minutes into

_preheated

_graphite

coated steel molds and maintained a 500 °C

for about 80 hours in

_dry

air. The solidified

_ingots

of

_T2-phase

₍₀

18 x 60

mm)

were

_ground

into fine

_powder

and

put

into thin walled vanadium _{containers for the purpose of} neutron diffraction measurements. Five

_samples

of the icosahedral

_phase

were

_produced

with natural

copper and different

617 Li isotopic compositions corresponding

to

(Li)

scattering length

b (Li) = - 0. 222 (pure

7Li

isotope), - 0.190 (natural Li),

0, + 0.102,

+ 0.20

_(pure

6Li

_{isotope) (in}

10-12

_cm),

and two more

_samples

with Li-zero scatterer

_{(b (Li) = 0)}

and either

63Cu

or

65Cu.

A

_{sample of the}

bcc

_R-phase

with natural

isotopic compositions

was also

prepared following

the same

procedure

but for the solidification

_temperature.

Parts of the

samples

were characterized

_{by X-ray}

and electron diffraction. The neutron

_scattering

measurements were carried out at the

_High

Flux Reactor of the Institut

_{Laue-Langevin}

(Grenoble),

on the D4B diffractometer which is set on a hot source beamline. A

_fairly

short

wavelength

À =

0.5 A

was used in an

_attempt

to minimize the drastic

_absorption

effects of the

6Li

_isotope

and to obtain data at

_{large Q}

values. The

Q

resolution of the D4B

diffractometer,

currently

used for structural

_{investigation}

in

_liquids

and

_amorphous

materials,

depends

upon the

_{scattering angle}

but increases

_roughly

from about 0.06 to 0.20

Â- 1 over

the measured

_Q

range

(FWHM).

The raw data were then corrected for

_{absorption, multiple scattering,}

inelastic

_scattering

_{(Placzek correction),}

incoherent

_scattering

and other

_background

such as

the

_scattering

from the container

_[67],

to obtain the total interference function

_S(Q)

_up to

Qmax

= 23

Â- 1

(space

resolution thus limited to about AT? = 0.27

Â

FWHM).

Some of these

corrections have to be treated with

_special

cares. Incoherent

_scattering

is very

_large

for the

7Li

_isotope

and is

_always

_present,

even in the

_particular

case of a

6/7Li isotopic

mixture with zero coherent

_{scattering length.}

The correct subtraction of these incoherent contribution is of basic consequence for

getting

proper renormalization of the different

S ( Q )

functions. The correction for inelastic effects is also rather

_large

due to Li atoms

_being

not so

heavy. Finally

the enormous

_{absorption by}

the

6Li

_isotope

_(see

Tab.

_I)

makes the whole

_procedure

tedious

Table I. - Measured

neutron

_(A

= 0.5

Â)

transmissions

of

the

_samples

_of

AILiCu

and delicate.

_{Consequently,}

a direct

_validity

check,

using

the

_{R-phase sample,}

has been carried out as

_explained

later in the paper. The contamination

_by

diffraction contribution

from residual a-A1 was subtracted from the raw data

_patterns

as

_explained

in detail

elsewhere

_[60].

Some of the

_resulting

_S(Q)

functions are shown in

_figure

2. Obvious

_intensity

changes

due to contrast variation are

_easily

observed. Diffraction features become

_vanishingly

small

_beyond

15

A -1

and the

_{S ( Q )}

_conveniently

_converge to

_unity

_{which will make easy and} accurate the next

_step

of _treatment,

_namely

the calculation of

_pair

distribution functions

(PDF)

by

direct Fourier transform of

_{S (Q )}

into

_physical

_space.

Fig.

2.

_{- Typical}

_{S ( Q )}

function measured with the

_T2-phase

in neutron diffraction

_(A

=

0.5 À).

From

top to bottom the

_{scattering length}

of the Li

_isotopic

mixture is

_{b (Li ) _ -}

0.190, 0, + 0.102, + 0.200

_(in

10- 12

_cm).

3. General formalism of the

_{non-crystallographic approach}

to structural studies.

A conventional

_{crystallography approach}

of the AlLiCu

_{quasicrystals, using possibly}

the

contrast variation methods to

_simplify

data

_deciphering,

would mean first

_working

with much

better

Q

resolution as

_compared

to the one available on

_D4B,

then

_measuring

the

_integrated

intensities of the

_pseudo-Bragg

reflexions

_correctly

indexed and

_{finally trying}

to work out

some sort of

_phase

reconstruction

_procedure

to determine the

_partial

structure factors as

already

done for the Al-Mn

system

[25, 26].

Such a

_{crystallography approach}

is

_actually

in progress,

using

both

single crystal

and

powder

data,

but does not seem to be

_{easily completed}

very soon because of the

_large

amount of neutron beam time which is

_required

and some

peculiar

intricacies into the calculations of atomic

_density

in real space.

The calculation of

PDF,

using

methods

_{commonly employed}

to

_study

_{non-crystalline}

solids

(liquids,

glasses, amorphous

alloys)

is,

in

_principle,

more

_{straightforward}

and

_requires

more

reasonable allocation of beam time

_(less

than one week on

D4B).

Moreover information

gained

in such an

_approach

are

_indeed,

to the

_best,

limited to the

_PDF,

or

_{partial pair}

distribution functions

_(PPDF),

but

_might

be

_{complementary}

of the

_{crystallography approach}

in as much as the whole diffracted

_intensity,

both

_{Bragg peaks}

and « diffuse »

_scattering,

is

accounted for. This is an

_interesting

_aspect

for

quasicrystals

whose

peculiarity

is

_precisely

to

produce

scattering

intensity

everywhere

in the

_reciprocal

_space.

This,

to some extent,

The

_{non-crystalline}

material

_approach

to the structure of

_{quasicrystals}

has been

_extensively

used to determine the total

_(average)

PDF in different families of icosahedral

_{phases :}

Pd58.8U20.6Si2O.6 [68, 69], Al-Mn-Cr-Si [70]

and

_{Al-Li-Cu [71-73].}

Contrast variation effects have been

_applied

to measure the PPDF in icosahedral AIMnSi and AIMnFeCrSi

phases

[60]

and also for the

_decagonal

AIMn/AIMnFeCr

_system

_[74].

The basic

_principles

of the method are _very

_simple.

Let consider for instance the

_elementary

formula for

_X-ray

or

neutron diffraction

_by

an

_assembly

of N identical atoms whether

_{crystalline, amorphous}

or

liquid, namely

the

_expression

of the interference function :

The definition of

_{S (Q),}

which is related to the Fourier transform of the structure Patterson

function,

requires

a summation over all sites in a

_{macroscopic specimen}

and can be

_replaced

with an _{ensemble average} over site

_positions

measured relative to some standard site at

R = 0.

Introducing

further the

oscillatory

_part

of the

pair

distribution function

g (R )

allows

S (Q)

to be rewritten as :

in which the

_{brackets (...)}

_{denote the ensemble average and n stands for the number atomic}

density.

Now let transform

_equation

₍₃₎

_by

_introducing

_spherical

coordinates

_{R, 0, ~}

with

respect

to a « vertical » axis

_{lying along}

the

_Q

vector, i.e. :

To

_proceed

further we need to assume that our

_sample

is

_{macroscopically}

_isotropic,

i.e.

g (R )

does not

_depend

on 0 or

_¢

and can be

_expressed

as a

_simple

function

_{g (R )}

of the

_pair

distance

_R,

inside _{the ensemble average}

_pool.

In a _{very fine}

_powder

the whole

_sample

is

isotropic

in as much as

_grains

are

_randomly

oriented and there is no further difficulties to

work out

_{completely equation}

₍₄₎

which

_{gives :}

and

In these

_expressions,

it has been accounted for the radial distribution function

_tending

to

_unity

for

_{large R.}

There is also a delta-function

_singularity

_{8 (Q)}

as

_{Q ---> 0}

as

_easily

seen in

equation

(1).

Subtracting

this

_singularity

as the Fourier transform of

_unity

in

g(R)

and

_tacitly

ignoring

it in the

_S(Q)

factor result in the final form of

_{equation (5).}

It is clear that

S ( Q )

tends to

_unity

for

_large

Q

so that

_{g (R )}

is

_convergent

_provided

that the

_singularity

8

_(Q)

has been

_{dropped. Actually, Rmax}

and

_6max

should have been written as oo. The finite

present

case are

_likely

to be

_completely

screened out

_by

the instrumental resolution. The

Qmax

limit has a direct effect on _{space resolution of the PDF}

(OR

= 2

’TT /Qmax).

The limitation of the

_Q

resolution

_{produces spurious broadening}

of the diffraction

_peak

as a

result of their convolution with the instrument resolution function.

_Assuming

a Gaussian

shape

for this resolution

_function,

with the maximum FWHM of about 0.2

Â- 1,

which

corresponds

to the worse resolution in the measured

Q

range, would result in a

_damping

of

the calculated PDF or PPDF

_according

to a Gaussian

_profile

of about 30

Â

FWHM which

somewhat deteriorates _{the accuracy into calculation of the coordination numbers for}

_pair

distances

_{beyond 10 Â}

about.

To illustrate the kind of information that can be

_gained

from

_{non-crystallography}

method

for the

_study

of ordered structure,

figure

3 shows a

_comparison

between the measured PDF of

the bcc R-AlLiCu

_phase

and a simulated PDF as calculated from the

_crystal

structure

determined

_by

four-circle

_X-ray

and neutron

_experiments

_[75].

The simulated function has

genuinely

been obtained

_{by isotropic regrouping}

of the

_pair

distances in this

_compound,

convoluted with a Gaussian

_broadening

function to best

_reproduce

the

_experimental

_pair

distributions. The used

_{variance (T2(Â2) =}

0.02 + 0.006

_{(R -}

2

_)°.45

_corresponds

to a FWHM on the first

_{g (R )}

_peak

_equal

to about 0.30

À

which compares

quite

well with the limitations

arising only

from termination at

_6max

Peak

_positions

are

_fairly

well

_reproduced

and

« intensities »

(or

rather

pair

coordination

numbers)

show

_{only reasonably}

weak

discrepancies

especially

at short distances. An

_{interesting point}

is also that the method can be a valuable tool to

_investigate

relative

_changes

in a structure. As an

_{example, figures}

4 and 5 show

simulations of the

_R-phase

PDF and of its lithium-lithium

_PPDF,

_{respectively,}

either in the actual structure or in the fake

_perturbed

modification obtained

_{by shifting}

the Li atoms of the

24 g

sites

_[75]

_{by 0.2 Â parallel}

to the

_{(001 )}

direction

_(from

0.3047 to 0.3191 in reduced

coordinates).

Curiously,

the result on the total PDF

(Fig. 4)

_appears as enhancements of the

first

_(positive)

coordination shell

_{(Al-Al mainly)}

and of the second

_(negative)

one

(Al-Li,

due

to

_{negative scattering length}

of natural

_Li).

But

_looking

at the lithium-lithium PPDF

_{(Fig. 5)}

shows

_clearly

the

_splitted

first Li-Li

distance,

with two

_components

_{having weights}

in

proportion

to the number of concerned atomic

_pairs.

Fig.

3. - The measured total

pair

distribution function of the cubic

_R-phase

_{(full line)}

_compared

to

calculated

_{g (R )}

as deduced from the

_{crystallography}

structure

_{(dashed line).}

The Cu and Li elements

Fig.

4. -

Simulations of the PDF of the cubic

phase

either in its actual structure

_{(full line)}

or in a

structure modified

_{by displacements}

of Li atoms

_{(see text) (dashed line).}

Fig.

5. - Same

representation

as in

_figure

4 but for PPDF of the Li-Li correlations instead of the total

PDF.

Finally,

the

_{good consistency}

between simulated and measured data for this

_R-phase

(Fig. 3)

is also

_{quite reassuring}

with

_respect

to the correction

_procedure.

Indeed,

as

_previously

stated lithium

_(especially

_6Li)

is

_{simutaneously}

a

_strong

neutron absorber and a

_light

_{element ;}

a combination of

_{large absorption}

effects and inelastic correlations is a rather tedious situation

to be corrected for

_properly

and it is better to have a final check of a successful

_procedure.

4. The

_{partial pair}

distribution functions of the T2-AILiCu icosahedral

_phase.

Within the

_approach

described in section

_3,

_{S (Q )}

_data,

as obtained

_{by dividing}

the corrected

intensities

_{by (b2),}

can be Fourier transformed

into g (R )

functions

_according

to

_equation

_(5).

different elements

_{( a , f3, ...)}

in concentrations _{ca ,}

_{c,6, ...}

and

_having

neutron coherent

scattering lengths

b a’ b 3 ’

..,

they

are linear combinations of PPDF defined for each atomic

pairs,

i.e.

_{g al3}

_{(R ),}

_according

o :

In _{the presence of}

_completely

unknown structures, there are as _many

_{gaB (R)}

to be determined as number of different

_pairs

₍₃

in a

_{binary alloy,}

6 in a

_{ternary system,}

_etc...).

Isotopic

substitutions on one, or

_better,

several elements allow to

_modify

the

_ba

_bp

_products

in

_equation

₍₆₎

_and,

_thus,

to measure as _many

_independent

total

_{g (R )}

as _necessary to

calculate the different

_partial

_{gaB (R)}

_by

inversion of the linear

_system

of

_equation

_(6).

The T2-AlLiCu

_compound

is

_obviously

a

_ternary

mixture with six different

_partial

_{gaB (R )}

to be determined. But the

_possibility

of

_{preparing samples}

with a « zero scatterer » lithium

(Li(O))

yields

some

_{simplifications}

in as much as _any

T2-AlCuLi(O)

_{sample actually}

behaves like a

binary compound

from the

_point

of view of neutron diffraction. Then three such

Li(O)

bearing samples prepared

with natural copper

(Cunat),

,

63Cu

and

65Cu

should lead to the

determination of all

_Al-Al,

Al-Cu and Cu-Cu

correlations,

through

the inversion of the

corresponding

system

of

_equation

_(6).

The normalized coefficients

_{Waf3 =}

Ca

c

ba

b aca

ba ] 2are

gathered

in table II and the determinant

Il Wa,6 Il

is

equal

to

0.006.

Thus,

despite

a reasonable efficient contrast on the Cu atoms

_(see

_{Wcucu in}

Tab.

_II)

the calculation of the Al/Cu correlation functions had been

_expected

to be rather difficult and inaccurate. But

_again

simulation with the

_R-phase

structure has shown that it was not as

hopeless

as it looked like. In the

_R-phase

[75]

there is a certain Al-Cu chemical order but it is so weak that it would

_certainly

_escape to a

63Cu/65Cu

contrast

_experiment

as illustrated in

figure

6

_showing

simulated

_{g (R )}

curves for

_{R-phases containing}

the two

_isotopes.

_However,

assuming

either a total Cu/Al chemical disorder or the maximum chemical order

correspond-ing

to all Cu atoms located on one of thé four Al/Cu sites available

_{gives g}

_{(R )}

functions which

are different

_{(Fig. 7)}

_especially

around 5 and

8-9 A

distances and would have been

differentiated in a

63Cu/65Cu

contrast.

_{Comparatively,}

the two

_S(Q)

and

_{g (R )}

functions

corresponding

to

T2-AI63 CuLi(O)

and

T2-AI65CuLi(O)

are shown in

_{figures 8}

and

_9,

respectively. They

look very much the same for both

_alloys

which means that Al/Cu

correlations,

if any, have little chance to be extracted from these data. As a

_qualitative

important

result,

we _{may conclude-that Al/Cu chemical order in the}

_T2-phase

_is,

to the

least,

as weak and somewhat similar as the one in the cubic

_R-phase

(Fig. 10).

The

_positive

consequence is that

(Al, Cu)

atoms can now be treated as a

_single

_element,

_{say A} atoms, and the

_T2-phase

as a

_{pseudo-binary}

A-Li

_compound

whose

_{partial pair}

distribution functions

Fig.

6. -

Simulation of the copper

isotopic

effect on the

_R-phase

PDF

_(full

line :

65Cu ;

dashed line :

63Cu).

In both cases Li is

supposed

to be a zero-scatterer.

Fig.

7. - The simulated

g (R )

functions of the cubic

_phase

with Cunat and

Li(’) assuming

either a

perfect

chemical order

_(-)

or a

_perfect

chemical disorder for Al/Cu

(...)

(see text).

Fig.

8. -

The measured

_S(Q)

functions of the

T2-phase

when

_prepared

with either the

Fig.

9. -

The total PDF’s calculated from the two

_S(Q)

functions show in

_figure

8.

Fig.

10.

_{Fig. 11.}

Fig.

10. - Calculated

AI-AI, Al-Cu, and Cu-Cu

partial pair

distribution functions of the cubic

_R-phase.

Fig.

11. -

Typical

PDF of the

_T2-phases

as deduced from the

_{S (Q )}

functions shown in

_figure

2.

_Copper

is Cunat and

_isotopic

mixtures of lithium

_correspond

_(from

_top to

_bottom)

to

_{b(Li) = - 0.190,}

0,

+ 0.102 and + 0.200

_(in

10-12

_cm).

corrélations _{may be} in

_principle

measured with three

_{samples having}

different lithium

_isotopic

compositions

after calculations

_{using equation (6).}

The _{gAA function} is

_merely

a

_weighted

average of the true Al-Cu correlations.

Contrast effects on the Li sites are well visible in

figure

11,

which shows different PDF

measured with values of b

_(Li)

_ranging

from - 0.190 to + 0.200

_(in

10- 12

_cm).

In

_particular,

in

the

_region

of about 3

_Â,

_{9 (R)}

has the same

sign

as

_{b (Li)}

which

_obviously

_may come from the contribution of Al-Li

_{pairs. Comparisons}

with the

_{corresponding}

measured or simulated

9 (R)

for the

_R-phase

_(Figs.

_12,

₁₃₎

_suggest

_qualitative

similarities and differences between

Fig.

12. -

Large

pair

distances behaviour of the

_T2-phase

PDF

_{(full line)}

_compared

to that of the

R-phase

(dashed line).

Both

_correspond

to _Linat, Cunat

_compounds

and are measured functions.

Fig.

13. - Detailed

comparison

between the PDF of the

_T2-phase

_(-)

and of the

_R-phase

_{(... )}

for the

extreme contrast situations :

_{a) b (Li) = -}

0.190 ;

b)

b (Li) =

+ 0.200. The additional curves

(-)

are

large

values of the

_pair

distances.

However,

the

_{crystalline R-phase}

exhibits measured g

_{(R )}

functions which still oscillate around

unity

far

_beyond

the

_point

at which correlations in the

_T2-phase

are

_completely

flat

_{(Fig. 12).}

This is consistant with the icosahedral

_phase

_having

a coherence

_length

which is shorter than the one in the bcc

_periodic

structure. _{The sequences}

of

_pair

distances look also

_fairly

similar but with a

_significant

shift of the distribution

_peaks

towards

_larger

distances for the

_T2-phase.

Now,

the calculation of the PPDF

_using

these total

_{g (R )}

and the

_{corresponding}

_system

of

equation

(6)

should be

_readily

achievable. The normalized coefficients

_WAA

_(remember

A is for Al and Cu

_altogether)

_WALi

and

_WLiLi

are

_given

in table III. Their determinant

Il Wa/311

is

_equal

to 0.05 if

NatLi,

_{Li (bcoh}

=

0 )

and

6Li

containing samples

_are selected.

Accordingly,

the PPDF were

_anticipated

to be

_{« easily »}

obtained for A-A and A-Li correlations but difficulties should accompany gLiLi calculations. It is indeed

straightforward

for the A-A and A-Li correlations whose PPDF are

_pictured

in

_figure

14

_along

with their

counterparts

calculated for the cubic

_phase.

But

attempts

to extract the PPDF for the Li-Li correlations have resulted in

_unphysical

_profiles,

with in

_particular

_strong

_negative

contri-butions. In

_fact,

_despite

the

_quite

_large

range of contrast variation on the lithium

_sites,

the

combined effects of

_relatively

low Li concentration with

_respect

to

_{(Al-Cu) (32}

% vs 68

_%)

and of the smaller b value for the same atoms

_{( 1 bmax (Li) |}

= 0.20 vs

bA

=

0.41)

reduce the

gLiLi contribution to a mere 4 % on its maximum value. The

_point

is

_{spectacularly}

illustrated in

figures

13a and 13b which show little differences between measured total

_{g (R )}

and their reconstructed

counterparts

using

the extracted _{gA_Ll and gA_A and}

_assuming

_gLiLi = 0. This is _a

fact we have to live with. In

_figure

_13,

the Li-Li function of the

_R-phase,

without the

_missing

function of the

_T2-phase,

is

_pictured

for the sake of

_{completeness.}

Table III. - Normalized

coefficient

Wa/3

of

the PPDF

_for

AILiCu

_samples

with

_isotopic

contrast on the lithium atoms. A is

_for

Al + Cu atoms

_altogether.

We have checked that the _gAA and _{gALi calculated} for the

_T2-phase

were not influenced at

all

_by

their ill _{determined gLiLi}

_counterpart.

Either

_ignoring

its

_unphysical

contribution while

inversing equation

(6)

or

_fixing

it

_arbitrarily

_((i)

_gLiLi = 0 or 1 for all R

values,

(ii)

gLiLi identical to the one calculated for the

_{bcc-phase)_}

_give

the same result or _{gAA and}

gALi within

experimental

accuracy.

Pair distances and coordination numbers as obtained from these PPDF are listed in tables IV and V. General trends of the icosahedral PPDF with

respect

to those of the

crystalline

cubic

_phase

are :

-

a

_slight

increase of

_pair

distances

_except

for the shortest one ;

-

an

asymmetrical broadening

of the distribution

peaks

on the side of the

_large

_{distances ;}

-

an overall

similarity

_except

_perhaps

around 4.5

Â.

Fig.

14. - Measured

_{partial pair}

distribution function of the

_T2-phase

_{(full line)}

_compared

to those simulated for the

R-phase

(dashed line)

for the A-A, A-Li and Li-Li

_(R-phase

_only)

from _top to bottom.

5. Discussion and

conclusion.’

Some

_insight

into the structure of the icosahedral T2-AlLiCu

_phase

_{may be}

_gained

_by

analysing

further,

and in a

_comparative

_{way, this PPDF} and their modifications in the

R-phase.

The structure of the

_R-phase

as determined in reference

[75]

is schematized in

figure

15 and the refined atomic coordinates with site

_occupancies

are

_given

in table VI. The structure

_belongs

to the

Im3

_(bcc)

_{space group with} a lattice

parameter

of 13.9056

Â.

The

_Al,

Cu,

Li atoms are distributed over shells around the

_origin.

The first shell

(Fig. 15a)

is an

icosahedron of

_{(Al, Cu),}

with

_unoccupied

centre.

_Twenty

Li atoms are then

_placed

above the

centres of the

_twenty

_triangular

faces of the

icosahedron,

forming

the dual

_polyhedron,

a

Table IV. - Pair distances and coordination numbers

_ZAA

for the A-A

_[A

_for

_{(Al, Cu)]}

correlations in the T2 and

_R-phases.

_R,

_Rm

and

_RM

stand

_for

_position

_{o f the}

maximum,

smallest

and

_largest

_pair

distances within the distributions

_peaks.

Table V. - Same as in table IV but

_for

A-Li correlations.

pentagonal

faces of the dodecahedron in an icosahedral

_arrangement

to

_give

a « small »

rhombic triacontahedron. The next

_layer

of

_{(Al, Cu)}

atoms

_{(Fig. 15d)}

forms a distorted

truncated icosahedron. The twelve

_pentagonal

faces of this

_polyhedron

are centred on the 12

(Al, Cu)

atoms of the « small » rhombic

triacontahedron,

whereas the

_twenty

hexagonal

faces

match the 20 Li atoms of the

_{underlying pentagonal}

dodecahedron. The set of successive

Table VI.

_{- Refined}

atomic coordinates

_{( x}

_{104 )}

and site

occupancies (neutron

and

_X-ray

data)

for

the

_bcc-phase

R-AILiCu.

complex »

which contains 104 atoms. At this

_stage,

the structure of

_R-AlSCuLi3

can then be

described as a

_{CsCI-type packing}

of distorted Samson

polyhedra

linked in two _{ways :}

-

along edges

of the cubic cell

_{by sharing}

two aluminium atoms

_{(site 12e) ;}

-

along

the

_{eight body diagonals}

of the cube

_{by sharing}

a common

_hexagonal

face of the

polyhedra

(site 48h).

The

_remaining

lithium atoms

_{(site 12e)}

are found in the interstices formed within the

Samson

_{polyhedron packing. They}

cap the

pentagonal

face of the truncated icosahedron. An alternative and

_perhaps

more fruitful

description

of the structure can be related to the site 12e

_{(Li atoms)}

_{being actually}

located at 24 of the 32 vertices of a

_{« large »}

rhombic

Fig.

15. - Illustration of the

R-AlSCuLi3

bcc structure, from

[75] : a)

icosahedra centred at the

_origin

000 and 1/2 1/2 1/2 ;

b)

dodecahedra,

c)

the addition of icosahedra to the

_previous

dodecahedra form the small

triacontahedra ;

d)

distorted truncated isosahedra, and

_e)

_large

triacontahedra which are

connected either

_by

faces

_along

_{the 100 >}

directions or

_through

and

_overlapping

volume

_defining

a

triacontahedron of radius r = 7.48

À

(Fig. 15e).

The 8

remaining

vertices coincide with the Li

in 16f sites

_already

considered in the formation of the

_underlying

dodecahedral shell

_(15b).

In this

_description

of the

_R-phase,

the «

large »

rhombic triacontahedra are also linked in two

ways :

- in

_{the { 100 }}

_{planes by}

_sharing

rhombus

faces ;

there are six connexions of this

_{type ;}

-

along

the 111 >

directions

_{by overlapping}

with

_neighbour

triacontahedra and thus

defining

small oblate rhombohedra at the

_intersecting

volumes ;

there are

_eight

connexions of this

_type.

Fig.

16. -

Density

function as obtained from data

_(calculated

with the measured 9A-A and 9A-Li functions of the

T2-phase, conveniently weighted by

concentrations)

compared

to

_predictions

of the

Henley-Elser

model

_{[42, 43].}

The distorted truncated icosahedra have not a

_perfect

icosahedral

_symmetry

which would

have forced the atoms in 48 h sites to _emerge at the surface of the outer triacontahedral

atomic shell. The two triacontahedral shells

_(so-called

« small » and

_{« large »}

_heretofore)

have diameters in a ratio

_practically

_equal

to the

_golden

mean T.

Considering

that the shortest

pair

distances of the

_R-phase,

_{with their proper occupancy}

numbers,

are well

_reproduced

in the

_T2-phase,

one _may assume that at least the

_shapes

of the inner

shells,

namely

the small icosahedron

_{(Fig. 15a),}

the

_pentagonal

dodecahedron

_(15b)

and

_perhaps

the next

_{(Al, Cu)}

icosahedron

_(15c),

are

_kept

almost

unchanged.

To check the

point,

_up to some extent, we

have made simulations of the A-A

_partial,

in

_particular,

for several modified versions of the

correlation

_peak

at

5 Â

and the reversion of the doublet around 8-9

À

are indeed

_reproduced

in 9A-A

by introducing

an additional A atom at the

_origin

and

_by

_reducing

the contributions

coming

from the 48 h and 12e

_{(Al, Cu)}

sités

_{(Fig. 15d).}

Thus,

the structural unit of the T2-icosahedral

_{phase might keep}

the 5

À

« radius » rhombic triacontahedron of the

_R-phase

as

shown in

_figure

_15c,

made

_actually

of an icosahedron of A atoms,

containing

a dodecahedron

of Li atoms and a smaller icosahedron of A atoms centred on A atoms. The truncated icosahedron

_{(Fig. 15d)}

of radius about

7 Â

and whose distorsion in the

_R-phase

allows to reconcile local

_{pseudo-five-fold}

_symmetries

and

_{periodic long}

_range

_order,

would then relax

to full

_{icosahedrality}

and be linked

_together

at too

_{large pair}

distances to be

_properly

investigated

in the

_present

work. It is

_interesting

to remark in

_figures

13 and 14 that

_pair

distances in the

_T2-phase

increase with

_respect

to those in the

_R-phase

more

_{significantly}

beyond

the

7 A

distance. Such a

_description

in terms of connected truncated icosahedra is in

good

agreement

with and

_{complementary}

to EXAFS measurements

_[76]

at the Cu K

_edge,

but does not

_correspond

to _{any of the} current atomic models

_proposed

so far to describe the structure of the icosahedral T2-AlLiCu

_phase.

For instance a

_density

distribution function

n (R )/n

calculated

_[43]

with the most

_{popular Henley-Elser}

model

_{(HEM) [42]}

_obviously

fails

to

_reproduce

the measured function

_{(Fig. 16).}

The

_{(Al, Cu)}

chemical order over this truncated icosahedral unit is

_probably

_very

weak,

and in any case _{very difficult} to be

determined

_{accurately by}

diffraction methods. The Li-Li correlations also

_escaped

this

_study

but should be more

_easily

measured in a classical

_{crystallography approach.}

This work is at

present

under way

along

with the construction of a structural model based on the conclusion of the

_present

_study.

Acknowledgements.

The authors wish to

_acknowledge

the Centre de Recherche of the

_{Pechiney Group, Voreppe}

(France)

for

_providing

facilities in

_{sample preparation}

and the Institut

_{Laue-Langevin}

(Grenoble, France)

for allocation of neutron beam time. Fruitful discussions with R. Fruchart and his

_help

in

_preparing

the

_samples

have also been essential to the success of the

present

work.

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