Specific-heat measurements in phason-strained quasicrystalline AlFeCu

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Specific-heat measurements in phason-strained quasicrystalline AlFeCu

K. Wang, C. Scheidt, P. Garoche, Y. Calvayrac

To cite this version:

K. Wang, C. Scheidt, P. Garoche, Y. Calvayrac. Specific-heat measurements in phason-strained quasicrystalline AlFeCu. Journal de Physique I, EDP Sciences, 1992, 2 (8), pp.1553-1557.

�10.1051/jp1:1992221�. �jpa-00246639�

Classification Physics Abstracts

65.40 62.20D

Short Communication

Specific-heat measurements in phason-strained quasicrystalJine

Alfecu

K.

Wang(~),

^C.

Scheidt(~),

P.

Garoche(~)

and Y.

Calvayrac(~)

(~)

Laboratoire de Physique des Solides, Universit6 Paris-Sud, 91405 Orsay Cedex, France (~)

CECM/CNRS,

15 _rue G. Urbain, 94407 Vitry Cedex, France

(RecHved

7 April1992, accepted 4 June

1992)

Abstract. We measured the low temperature specific-heat _on phason-strained and phason- free samples of the quasicrystalline A162Cu25.5Fe12.5 alloy. A strong enhancement of the vibra-

tional contribution is observed for the phascn-strained sample. This clearly indicates _a softening

of low-energy phonon modes, which allows _us to deduce _a decrease of the elastic shear modulus induced by phason strains.

Quasiperiodic

structures

display

_a

special

kind of structural

disorder, conventionally

known

as

phason

strain

[I].

Theoretical calculations show that

phason

strains in

quasicrystals (QC)

lead to _an X-ray diffraction

peak-broadening,

which scales with

corresponding perpendicular

wave vectors Ki [2], ^{and that}

phason

strains affect the elastic constants [3,

4].

^The

Ki

related

peak-broadening

has been

observed,

but until _{now, no}

experimental

observation has been _re-

ported concerning

the relation between

phason

strains and the elastic constants.

Furthermore,

the structure

instability

observed for _some

QC'S

is characterized

by

the onset of

phason

strains [5]. ^{Several authors} [6] have

proposed

that this

instability

is related to the

phason elasticity.

In order _to

study

the

phason-strain

influence _on the atomic vibrational

properties

in

QC'S,

we have measured the low temperature

specific-heat

_on

phason-strained

and

phason-free QC

Alcufe

samples.

^{This allows} the

experimental investigation

of the

phason-strain

influence

on the elastic constants, as it is well known that the

thermodynamic properties

of solids _are

intimately

related to the elastic

properties.

At low temperatures,

only long wave-length

vibra- tional modes _are

excited,

and it is

precisely

these modes which _are related _to the

macroscopic

elastic constants in the

Debye

continuum model.

The

samples,

of nominal

composition A162Cu25.5Fe12.5,

have been elaborated at the

CECM/

CNRS in

Vitry (France). They

_are obtained

by planar flow-casting

and submitted _to various thermal treatments. This

composition produces

different structural states _{as a} function of the thermal treatment: _an

annealing

at 600°C leads to _a

single phased

but

phason-strained QC

structure; whereas _an

annealing

_up to 812°C leads to _a

perfect

structure. Two

samples,

obtained

repectively by

the 600°C

(during

I

hour)

and 812°C

(during

two

hours) annealing

and both with _{a mass} of about 1.5 _mg, have been

investigated.

1554 JOURNAL DE PHYSIQUE I N°8

Both

samples

are character12ed

by high

resolution

X-ray

diffraction

experiments (Fig. I).

These two

samples

_are both

single-phased.

For the 600°C-annealed

sample,

the

X-ray Bragg peak- broadening

^scales

roughly

with the

corresponding perpendicular

wave vectors

Ki,

_so this _sam-

ple

_can ^be considered to contain

essentially

frozen-in

phason

strains [2,

ii,

and will be referred to below _as the

"phason-strained" sample.

For the 812°C-annealed

sample,

the

phason

strains

are

entirely

eliminated. The

X-ray

diffraction

peak-width

is of the order of the instrumental resolution

(Aq

~-

10~~ i~~,

q = 2

sin(@/~)),

and the

peak positions

_are

exactly

those of _an ideal icosahedral lattice. Its 6-dimensional lattice parameter is reduced

by

about 0.03il _as

compared

to the

phason

strained

sample (from

6.320 to 6.318

I).

_This

_sample

_{remains stable}

over the whole temperature range from 800°C down to room temperature. ^{It will be referred} to as the

"phason-free" sample.

A162 Cu25.5 Fe12,5

2

2h812°C

]1.5

~

~~ ^l c

(m

0.5

0

10 30 50 70 90 110

2

lh 600°C

]1

~

~~ l

c

(m

0 30 5 0 7 0 90 10

2 theta (lKa Co)

Fig. I. Comparison of the X-ray diffraction patterns ^{of the} quasicrystalline A162Cu25.5Fe12.5 after

annealing for 1 hour at 600° C and for 2 hours at 812°C.

The

samples

are

glued

onto _one face of _a

sample

holder which consists of _a thin

sapphire

slab

(2.5

_x 6 _x 2

mm).

On the other face _a

heater,

_a thermometer and thermal links _are

deposited by

thin film

technology.

The

experiments

_are

performed using

_{an a-c-} calorimetric method _at

a

working frequency ranging

from 2 to 6 Hz.

Taking

into account the weak heat

capacity

of the

samples (~-

15

nJ/K

at I Kelvin

(K)),

the

sample

holders _are measured in separate runs

and their contribution is subtracted from the total heat

capacity.

Figure

² presents the molar

specific-heat

of the _two

samples

in _a

C/T

_versus T~

diagram,

in the temperature _range between I to 3 K. This

figure

shows that in this temperature _range, the

specific-heat

for both

samples

_can be described

by

the classical law C

=

~T

+

flT~.

This allows the

separation

of the vibrational contribution

flT~

from the electronic _one

~T.

The _~ and

fl

values for these

samples

_are obtained

by

_a best fit of this customary

plot:

_~

= 0.33 ml

/K~

mole and

fl

= 0.ll ml

/K~

mole for the

phason-strained sample;

_{~ =} 0.3

mJ/K~

mole and

fl

₌ 0.05 ml

/K~

mole for the

phason-free

_one. The

fl

value for the

phason

free

sample

^is

somewhat lower than that obtained

by Biggs

et al. [8], ^{but is in} agreement ^{with the value}

given by

^Klein et al. [9].

.4

~ ^. phason.strained sample

" _o phason.free sample

o

)

£

0.8

E

~' 0.6

~

~

0.2

0 2 3 4 5 6 7 8

T~ (K~l

Fig. 2. Molar specific-heat C for the phason-strained

(e)

and the phason-free

(o)

samples of the quasicrystalline A162Cu25.5Fe12.5 alloy.

The T

= 0

Debye

temperatures _@Do can be calculated with these

fl values,

_we obtain 273 K and 350 K for the

phason-strained

and the

phason-free sample respectively.

Thin strong _@Do evolution indicates that the

low-energy

vibrational

density

of states is enhanced in the

phason-strained sample

as

compared

to the

phason-free

one. This enhancement is

clearly

associated with the

phason strains,

which _are eliminated

by

further

annealing

at 800°C. Low- energy vibrational

density

of _states enhancement due to the

softening

of the

phonon

transverse mode is often observed for metallic

glasses.

This leads _{to a} T ₌ 0

Debye

temperature decrease in

specific-heat

measurements [10].

Low-energy

mode enhancements have also been observed

by

inelastic neutron

scattering

in metastable

QC

PdSiU _as

compared

to the

crystalline phase [11]. However,

our observation allows the first direct

comparison

between _a

phason-strained QC phase

^and _a

perfect

_one. In the

measuring temperature

_range ^this enhancement _cannot be due to the

phason "hopping",

_as

phason

relaxations _occur

only

at

high

temperatures

(~-

^650°C

[7]).

Moreover, the T~ variation does not support resonance-mode contributions. We will show below that this

anomaly

_can be attributed to _a

softening

of the

phonon

transverse

mode, implying

a

softening

of the shear elastic modulus induced

by phason

strains. In

fact,

for _an

isotropic substance,

the T

= 0

Debye

temperature can be related

through

the sound velocities to the

1556 JOURNAL DE PHYSIQUE I N°8

elastic moduli

by [12]:

~ j

@Do =

Ap~+ 2»~'

₊

(B

⁺

(»)

^~

⁽ⁱ⁾

where _{p, p} and B _are the

sample density,

the shear and the bulk elastic

modulus, respectively.

A is _a constant of unit Kelvin-second. The

application

of this

isotropic

model is

justified by

the

high symmetry-degree

of the icosahedral _structure.

For normal metal

alloys,

the relative

change

of the bulk modulus B due to disorder

can be

expressed

_as

[13]:

AB

~

_A~

^~'

^dF(~')

B _a

F(a')

da' ~~~

where _a is the _mean interatomic distance and a'the local interatomic distance related to the defects.

F(a')

has the

asymptotic

form

~- sin

(2kFa') la'~

with kF ^{the Fermi} vector.

According

to the

X-ray

diffraction

experiments,

the relative

change

of

Aala

for the

phason-strained sample

_can be estimated at about 0.03il relative to

phason-free

one. This

implies

_a finite

F(a')

value

(a

_zero

F(a')

leads _to ha

=

0,

_so AB ₌

0) [13]. Taking

for a' the _mean interatomic distance _a

~- 3

I

_{and for kF the} free-electron value kF _{free *} I-S

i~~,

we estimate that the

corresponding change

^{in the bulk modulus}

AB/B

does not exceed lit. So the _@Do decrease in the

phason

^strained

sample

is

essentially

due to the decrease of the elastic shear modulus p. In fact,

neglecting

the

vanishingly

small

density

variation between the two

samples,

_{we get}

from formula

(I):

_{A@Do/@Do *}

Ap/2p,

and _{we can} estimate for the

phason-strained sample

_a strong ^{variation of the elastic shear modulus}

Ap/p

_m -40it _as

compared

to the

phason-free

one.

Similar estimation

concerning

the bulk modulus _can be made

using

an

isotropic

elastic contin-

uum model and

taking

into account the lattice variation

[14].

in

fact,

the elastic bulk modulus of solids is

essentially

related to the _mean atomic volume. Then the bulk modulus is

hardly

altered

by

structure disorder if _no strong ^{atomic volume variation is}

implied.

This is indeed the

case for the

phason-free

and the

phason-strained samples,

for which the lattice parameter is

only changed by

about 0.03il. The elastic shear modulus is far _more structure-sensitive. Shear modulus decreases _up to 30il relative to the

crystals

have been observed for metallic

glasses [10].

^{Phason strains}

correspond

to rearrangements ^{of local} structures in _a

perfect QC

and

thus

destroy

the

long-range

order. So it is not

surprising

that the

long wave-length

transverse

phonon modes,

as well _as the shear elastic

modulus,

_are affected.

To _our

knowledge,

this result

gives

the first

experimental

evidence that

phason-strain

disorder

drastically

reduces the

QC

elastic shear modulus. This _can be

compared

with Jaric and

Mohanty's

calculation of the elastic moduli for icosahedral

quasicrystals

_[4]

according

to which the elastic bulk modulus for _a

phason-strained

icosahedral

QC

remains

unchanged,

whereas the shear modulus _p is reduced _as

compared

to the

phason-free

value _po

c2

~ ~°

3C314C4

^~~~

where

C3

and

C4

_are the

phason

elastic constants and

C5

results from the

phason-phonon coupling (These

constants _are related to the parameters _c; of reference [4]

by C3

_"

c3/3,

C4 _"

-c411i

and

C5

"

cslli).

The shear elastic modulus _p is reduced

by

the

phason-

strain field

through

the

phason-phonon coupling

constant C5

squared.

So the drastic shear modulus decrease deduced for the

phason-strained sample implies

_a strong

phason-phonon

coupling.

Moreover, ^the onset of

phason-strains

is characteristic of

phase

transitions between unstable

QC'S

and low temperature

approximant crystalline phases

_[5]. Our results support ^the _propo- sition that such transitions _are due _{to an} elastic

instability

_[6]

corresponding

to _a

softening

of the sbear elastic modulus _p

~- 0.

However,

^{this elastic}

instability

_may not be

merely

related to

phason

elastic modes

(through C3

and

C4). Phason-phonon coupling

_can

play

_an

important

role

through

the

coupling

constant

C5.

If the

QC instability

is driven

by

the

phason-phonon coupling,

the low temperature

approximant phases'

order parameter _may be

closely

related to the nature of this

coupling.

In

conclusion,

we studied the atomic vibrational

properties

of _a

phason-strained

and _a

phason-

free

sample

of the

quasicrystalline

A16~Cu~5.5Fe12.5

alloy by

low temperature

specific

heat measurement. The _excess vibrational contribution in the

phason-strained sample

indicates _a

strong

enhancement of the low energy vibrational

density

of states. This

implies

_a

softening

of the

phonon

transverse modes that is caused

by

_a decrease of the elastic shear

modulus,

which is related to

phason

strains

through phason-phonon coupling.

This

phason-phonon coupling

may

play

_a

key

role in the

phase

transition between the

QC'S

and the

approximant phases.

References

[ii

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AUDIER M. et al., Philos. Mag. B 63

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