High Pressure Elastic Neutron Scattering Study of the Incommensurability in ND4DC2O4-1/2 D2O (AHOD)

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High Pressure Elastic Neutron Scattering Study of the Incommensurability in ND4DC2O4-1/2 D2O (AHOD)

J. Toulouse, R. Pick, F. Moussa

To cite this version:

J. Toulouse, R. Pick, F. Moussa. High Pressure Elastic Neutron Scattering Study of the Incom-

mensurability in ND4DC2O4-1/2 D2O (AHOD). Journal de Physique I, EDP Sciences, 1995, 5 (2),

pp.235-244. �10.1051/jp1:1995125�. �jpa-00247054�

J.

Phys.

I France 5

(1995)

235-244 _FEBRUARY

1995,

PAGE 235

Classification

Physics

Abstracts

61.12 64.70K 64.60C

High Pressure Elastic Neutron Scattering Study of the

Incouunensurability in ND4DC204-1/2 D20 (AHOD)

J. Toulouse (~>*), R-M- Pick

(~)

and F. Moussa

(~)

(~)

Département

de Recherches

Physiques,

Université Pierre & Marie Curie 4,

place

Jussieu, 75230 Paris

Cedex,

France

(~)

Laboratoire Léon Brillouin

(CEA-CNRS),

CE Saclay 91191 Gif-sur-Yvette

Cédex,

France

(Received

20

Aprjl 1994,

received in final form 20 October

1994, accepted

27 October

1994)

Abstract. Several

experimental

studies have

identified,

in

ND4DC204-1/2

D20

(AHOD),

the existence of _an incornmensurate phase above _a critical pressure, Pc m 2.6 kbar. This phase, and the transition

leading

to it, has been well described

by

_a

compressible

ANNNI model _up to

approximately 8 kbar. In order to further test the

model,

_we have

performed

elastic _neutron scattering measurernents frorn 7 kbar _up to 13 kbar and have followed _a

Bragg

peak satellite

as a function of temperature. ^{The results show} that, for intermediate pressures, the system refrains incornmensurate _{over a} broad range of ternperatures below trie transition and that the c*

/4

and

c*/5

lock-in

phases

_appear

only

ht low temperatures. ^At

higher

_pressures, the c*

/3

phase

_{appears to} be stable

even at 0

K;

the

reasons for these dilferences with trie

rigid

^ANNNI

rnodel _are discussed.

l. Introduction

Several

experimental

studies of _ammonium oxalate

(NH4HC204.I/2H20)

in its

hydrogenated (AHO)

_or deuterated

(AHOD)

form have shown this substance _to be _an

original example

of

trie ANNNI model. The

high temperature phase

I is orthorhombic

(Pmnb)

with Z

=

8,

and

possesses ^two

equivalent

mirror

planes perpendicular

to the _a axis at _{x =}

1/4

and _x

=

3/4

[1](~

).

^The

eight NH(

ions have their N atom _on these _mirror

planes,

and

they

form two distinct families: for _one of these

familles,

these

planes

_are true _mirror

planes,

but the second

farnily displays

onentational disorder: each ion may take two different

orientations, symmetrical

_one

to the other with

respect

to those

planes

which thus _are mirror

planes only

from _a statistical

point

^of view. The almost

planar

oxalate ions lie

approximately

normal to the _c axis with the

center of their C-C bond at _z

=

1/4

and _{z =}

3/4.

The disordered

NH(

ions have also their

(*)

Permanent address: Physics

Department,

16 Memorial Drive

East, Lehigh

University, Bethlehem,

PA 18015

(USA).

(~)

In this _{paper, we} make _use of the non-standard notation Pmnb for

phase

I, introduced by Keller et ai. [2], and

used, thereafter,

to describe ail the lower syrnrnetry

phases

derived frorn this parent

phase.

©

Les Editions de

Physique

1995

N atom

approximately

located _on two

planes perpendicular

to c, at z = 0 and _z

=

1/2.

The

cogwheel type

motion of the

NH(

ions from _one orientation to the other induces _a

ghde

of the oxalate

planes,

_a motion which is

linearly coupled

to the TA

phonon propagating along

c and

polarized

in the _a

direction,

_or, in the _q

= o

limit,

to the _e5 deformation.

All the

phase

transitions which take

place

in AHOD

(or AHO)

_are driven

by

the orientational

ordenng

of the disordered

NH( family,

the _two

possible

orientations of which _can be descnbed

by

the two values of _{a one} half

pseudospin.

From different

experiments,

^{it has} been deduced [3] that the interactions between the

pseudospins

of the nearest

neighbor planes perpendicular

to the c-axis _are of the ferro

type (1.e. they

favor

parallel

orientations of the

NH()

while the interactions between

pseudospins

located in next nearest

neighbor planes (1.e. separated by c)

are of the antiferro

type(~).

This is the

typical

ANNNI situation

[4],

^{but for} one

important

difference and additional

ingredient

which will

play

_a fundamental role in the

explanation

of the

expenmental

results to be

given

below: it is the

strong

hnear

coupling

^of the

NH(

ordenng

to the TA

phonon.

The

complete

ANNNI model

taking

into account the

coupling

^of

the

pseudospins

to the TA

phonons

_as well _as the

phonon-phonon

interactions will be referred to _as the

"compressible

ANNNI model".

At

atmospheric

_pressure, the ferro type interaction dominates and

leads, through

a second order transition which takes

place

at 146 K in AHO [5] and 160 K in AHOD

[6],

to

phase

II in which the

pseudospins

have the _same value in ail the cells of the

crystal.

The

coupling

with the e5 deformation leads to _a ferroelastic transition with _a considerable

softening

of the

C55

elastic

constant in the

vicinity

of the transition

temperature

_[7]. Above

P~

_m 2.6

kbar,

_a Raman

[8] and _an inelastic _neutron

scattering

_[9]

study

have shown the existence of _an intermediate incommensurate

phase

III. The transition between this

phase

III and the

high temperature phase

I is of second

order,

while the

phase III-phase

II transition is first order.

The incommensurate

phase

III _was identified

by

the observation of _a satellite of the

[4, o,

o]

Bragg peak

of

phase

I at qo " ôc*

[9],

^à

being temperature independent

and

monotonically mcreasing

with _pressure [8]. The above mentioned Raman and _neutron

scattenng

^{studies have}

m fact shown

that,

at least between 5 and 8

kbar,

à is _a linear function of pressure with

dô/dp

₌ 0.o17

kbar~~

These measurements have also shown

that,

below 8

kbar,

the domain of

stability

of

phase

III increases with

increasing

_pressure,

indicating

that _pressure increases the antiferro interaction relative to the ferro

type

interaction.

Nevertheless,

_contrary to the usual ANNNI

model,

no

sign

of _a lock-in transition at commensurate values of à close _to the incommensurate _ones could be detected. This absence of _a lock-in transition _is

likely

due to the

strong coupling

^of

the

pseudospins

to the TA

phonon

_or to the _e5 deformation. This

couphng

is at the _core of the

compressible

ANNNI model used to describe _[3] the above

measurements;

it will be

briefly

summarized in Section _4.

The

phase diagram

of the normal ANNNI

model, reproduced

from reference

[4],

^{and reduced}

to its _most stable

phases,

is

given

in

Figure

1. The coordinates _are the

ratio,

_r, of the antiferro

type

interaction parameter,

(J2(,

to the

ferro-type

interaction parameter,

Ji,

and the reduced

temperature kBT/Ji

It is however

important

to note that the lattice parameter used in

Figure

1 is the distance between next nearest

neighbor planes,

which

actually corresponds

to c in AHOD since there _are two

NDt _{planes through}

each unit cell. Wider lock-in

phases,

corresponding

to the smaller à

=

1/n,

_are found with

increasing

_{n up} to _a

singular point

at T

= o.

(~) ^In fact, _as there _are four such

pseudospins

_m the unit

cell,

which transforrn into _one another

by

the sym~netry

operations

of the

D()

space group, this description _is

only

vabd _{once a} specific

syrnmetrized

linear cornbination of the four

pseudospins

_is considered.

N°2 HIGH PRESSURE INCOMMENSURATE PHASES IN AHOD ₂₃₇

J~i/J~

Ù-S ?

à

= 1/2 _'Î~

.:.j

Î

0.6

6=1/3

o1

~j~

Ii

o.2

FERRO _II

~0 2 ( 6

k~T/J~

Fig.

l. Most stable

phases

of the 3d ANNNI model

(frorn _[4]).

Note that the lattice pararneter _c

bas been chosen _as the distance between nearest

neighbor planes,

_in order to allow _easy cornparison with AHOD.

Because _our

previous

measurements mdicated that

hydrostatic pressure,in

our

system, plays

a role

analogous

to r, the

present experiment

_was

designed

to

explore

the

phase diagram

of AHOD _{up to} 16 kbar and down to 1.7 K. As _we shall _see, the most obvious difference between

Figure

and _our results _{is a}

partial suppression

of the lock-in transitions at à

=

1/5

and

à

=

1/4,

these

phases being apparently

unstable with

respect

to the

incommensuraie phase

in

some

temperature

^{domain below}

phase I, regardless

ofthe _pressure. A discussion ofthe _pressure set up is

given

in Section 2 and the diffraction results and their discussion _are

presented

in

Sections 3 and 4

respectively.

2.

Experimental

The present

study

_was

performed

with the 4Fl

spectrometer

on the cold _source of the

Orphée

reactor ai

Saclay.

The

crystal

_was mounted in _a pressure cell which has been described else-

where

[loi.

The pressure

transmitting

fluid used _was deuterated

isopropanol

and the _pressure

was set at _room

temperature

^before

mstalling

the cell in _a

cryostat.

A critical aspect ^{of the}

experiment

was the accurate measurement of the pressure mside the cell. It is well known

that,

because of the

progressive freezing

^of the pressure

transmitting

fluid and of the

higher

^thermal

expansion

^{of fluids} with respect to

solids,

the pressure in the cell decreases with

decreasing temperature;

a nominal _pressure,

Po,

set up at _room

temperature yields

_{a pressure}

P(Po, T)

_<

Po

for _a lower

temperature

^T.

Unfortunately,

little

quantitative

information _was available _on the pressure loss _m this cell _{as a} function of

temperature.

^Con-

sequently,

_we had to

perform

an

auxiliary investigation

of the

temperature dependence

of the

pressure inside the cell _as measured with _a

manganin

resistor. As shown in reference

[11],

^the

change

of resistance of

manganin

is related to pressure in _a

quadratic

form

P

=

ARR/Ro

+

B(AR/Ro)~ ₍₁₎

where

Ro designates

the resistance at _zero pressure.

However,

the resistance of

manganin

at constant pressure also

changes

with

temperature. Assuming

that the pressure and

temperature dependencies

_are

unrelated,

_{we con} write

AR/Ro

"

f(P,

_{TO) +}

_g(T> Po) (2)

in which

To

^and

Po respectively designate

_room

temperature

^and

atmospheric

_pressure.

f(P, To

was determined

using equation (1)

and then

g(T, Po) using equation (2),

from measurements

of AR between _room temperature ^{and 4 K} for

atmospheric

_pressure and 7 kbar.

Having

thus determined

AR/Ro

^for

manganin

as a function of

temperature

and pressure, we were

able, using equation (1) again,

to calculate the

change

in pressure with

temperature

^{inside the} cell. In the _{pressure range} of

interest,

^and

making

_a linear

aproximation,

a pressure loss of

o.o19

kbar/K

_was estimated with _a

precision

of

+la%.

In order to

verify

this

result,

_we

measured the

temperature

^{evolution of the lattice}

parameter

of sodium chloride in the _same pressure

cell, setting

_up the initial _pressure at 7

kbar,

and

compared

it with the

published

data

[12].

^This measurement

yielded

the _same value for the loss of pressure, within the

precision

^of

the measurement, a value which is also close _to that

reported

in reference

[13].

^This pressure loss _was then assumed to be valid in the whole _{pressure range} covered in this

study (7-16 kbar).

In the text

below,

_as each

experimental

_run

corresponds

to a given pressure

Pn

set up ^at

room

temperature,

we use this nominal _pressure

Pn

to

designate

_a

given experiment,

in

spite

of the

temperature dependence

^of the _pressure.

However,

when used _{as a}

coordinate,

the pressure mdicated in the

figures

_are the true pressures,

taking

into account this

temperature dependence.

3. Results

Measurements _were

perforrned

at nominal _pressures

5, 7, la.7, 11, 11.7,

13.5 and 16 kbar. At each pressure, scans

along _[4

o

i~] ^were made for

decreasmg

temperatures and allowed for the determination of the satellite

position.

The value of the lattice

parameter

c was

accurately

obtained from [o ^o 4]

Bragg peak

measurements; the value of à _was then deduced. The values of the _c

parameter

are

presented

in

Figure

2 _as a function of temperature and for the nominal

pressures

applied.

For 7

kbar,

AHOD indeed goes

successively through phases I,

III and

II,

with the two transition

temperatures being

134 K and 125 K

respectively.

For all

higher

pressures

investigated,

the parameter c decreases

smoothly

with _no

sign

of _an accident at the

I-III incommensurate

phase

transition

(T).

As _we show

below,

for these

higher

_pressures, the

crystal

_no

longer

ends in

phase

II but first _{assumes an} mcommensurate structure and then goes

through

_{one or more} lock-in transitions. The

parameter

_c, ^measured in

phase I,

at 160

K,

is

plotted

in

Figure

3 _{as a} function of _{true pressure.} All _c values fall _{on a}

straight

fine:

c=co-aP

with _ca

= 6.765

À

_and

a = o.025

À kbar~~.

The measurements of the satellite _near the

Bragg point (400)

_are shown in

Figures 4,

5 and

6, corresponding

to the nominal _pressures of 11.o

kbar,

11.7 kbar and 16 kbar

respectively.

Although

measurements of the satellites have also been made _at the other pressures rnen-

tioned

previously,

these three are

representative

^of the incornrnensurate behavior of AHOD.

N°2 HIGH PRESSURE INCOMMENSURATE PHASES IN AHOD ₂₃₉

6.9

6.8

_/1

^bar

£~

6.7

~~~

Î

^~

fi-fi ^~~~

6.5 C-

135 kbar 6.4

16kb

6.3

0 50 100 150 200 250 300

Temperature

(K)

Fig.

2. Lattice parameter c versus ternperature for dilferent _pressures. Given _pressure values _are rneasured at _roorn ternperature

(nominal values).

Lines _are

guide

for the eye.

6.9

6.8 T=160K

t~

^6.7

6.6 E

Î

_6.5

6.4

6.3

0 2 4 6 8 10 12 14

Pressure (kbar)

Fig.

3. Lattice para~neter c versus pressure ^at 160 K. At this temperature the loss of pressure relative _to the nominal values _is

nearly

2A kbar.

For 7

kbar,

trie satellite is visible at qo " 0.167c* between 134 K and 125

K,

below which trie

system

_is in

phase

II. Il.0 kbar falls _{m an} intermediate and very

interesting

_range of _pressures;

for this pressure, the satellite _{appears at} T

= 124 K with à

= 0.230 and

slowly

drifts to

higher

à values with

decreasing temperatures.

It stabilizes _near 0.255 at the lowest

temperatures

with- out any further transition. Because of the

proximity

of this value of à from

1/4,

_we

carefully

checked the

alignment

of the

spectrometer

and the

shape

^of the

Bragg peaks.

Neither _was found

capable

of

explaining

the 0.005

departure

^from the

1/4

value. As _we show

below,

this

is not

surpnsing

in _a

system

in which weaker intermediate lock-ins should be

expected;

trie results for 11.7 kbar

provide

trie _necessary confirmation of this statement. For this _pressure,

the incommensurate modulation appears ^at T

= 125 K with à

= 0.238. As T

decreases,

the modulation _wavevector drifts _up

continuously

towards 0.25 _over a12 Kelvin interval _approx-

imately

centered _on 100 K. At 45

K,

^à = o.25 but at 39 K the satellite has

abruptly

shifted to à

= 0.33. This last shift

dearly

marks _a lock-in transition which is much

sharper

thon

what _{occurs near} à

=

0.25, suggesting, by comparison,

^{that the à} ₌ 0.25

phase

is

only weakly

25

= il k bar

P~ = 11.7 kbar

60 20

~ 15 ^~ w

u1 ^ù'

c C

W W

é 10 C

20 5

o o

-0 3( -0 30 -0 26 -0 22 0,18 ~°'° °36 ~° 32 ~°28 -0 2( -020

11,0,~i

1,

0,~i

Fig.

4

Fig.

5

Fig.

4. Neutron _scans of the

satellite(s) along

the [4, o, _~] direction for

decreasmg

ternperatures at 11 kbar

(nominal value).

For

clarity

each spectrum is

vertically

shifted

by

2250. Lines _are best lits with Lorentzian laws.

(Ô)~124.3

_K;

(.)

122.I K; (Zh): IIB.6 K;

(+):

II3.8 K;

ID):

Io5.2

K; IA):

69.3

K;

(q):

44,o

K; ("):

29.4 K;

(X):

29.2

K; (+):

20.4

K; (Q):

1.7 K.

Fig.

5. Neutron _scans of the

satellite(s) along

the [4, 0, ~] direction for

decreasing

temperatures

at 11.7 kbar

(nominal value).

For

clarity

each spectrum _is

vertically

shifted

by

6000. Lines _are best Lorentzian lits.

(0):

121.2 K;

(.)

106.5 K; (Zh): 98.5 K;

(+):

90 K;

(D):

80 K;

(A):

70 K;

(q):

59,1

K;

(")

48 K;

(X

): 39 K;

(+):

35.5

K; (Q):

11 K.

$=

16 k bar

60

~

~ _À0 c

?

w

20

o

-036 -03( -032 -030 -028 -026 -02(

(1,0,~i)

Fig.

6. Neutron _scans of the

satellite(s) along

the [4, o, ~] ^{direction for}

decreasmg

temperatures

at 16 kbar

(nominal value).

For

clanty

each spectrum _is

vertically

shifted

by

6000. Lines _are best Lorentzian lits.

(~):

l13.7

K; (0):

Ill.2 K;

(.)

lo4.3 K; (Zh): loo

K; (+):

94.8

K; (D):

90.3 K;

(A):

76.6 K;

(v):

74.3 K;

("):

63.2 K;

(X

): 47.6 K;

(+):

35.5

K; (Q):

1.7 K.

stable. For 13.5 kbar and 16

kbar,

the satellites first appear ^at 0.259 and 0.269

respectively and,

in both _cases, the modulation locks _{m at} à

= 0.33. No satellite _is observed at 0.25. The results for 16 kbar _are shown in

Figure

6. All numerical results _are summanzed in Table I.

In

Figure 7,

_we have

plotted

^the

temperature dependence

of the satellite

position

for _some of

the _pressure studied and in

Figure

8 the initial and maximum values within

phase

III of the

reduced wavevector à _as a function of pressure.

N°2 HIGH PRESSURE INCOMMENSURATE PHASES IN AHOD 241

Table I.

Summary of

transition

temperat~lres (in Kelvin)

and main mod~llation waueuectors

(ezpressed

in red~lced

ua1~les) for

the

dijferent

_pressures studied

(in kbar). Pn

_means nominal value

of

_pressure ^measured at _room

temperature. ô;n;t designates

the waueuector

of

the

jirst appearing

satellite at

high temperature T

except

for

intermedtate values

of

_pressure 10.7 and 11 kbar where seueral satellites

successiuely

_appear at low

temperatures.

ôfin and

Tfin correspond

to the

final

values

of

euolution

of

the

incommensurability. T;

is the teck-in

phase

transition

temperature

and ôj;

is1/3. fl;

_is the estimated _{pressure at} the lock-in transition.

134 0.167 125

129 0.219 86 0.226

86 0.205 50 0.200

50 0.256 +0 0.248

11 124 0.230 80

80 0.253 +0

II.7 125 0.238 +91

13.5 120 0.259 60 0.287 60

16 l17 0.269 71 12 0.333

O.40

0.35

XX xX x ~***

c O.30 ~

é ^*.~Éd%+

^{l 6} kbar

~

0.25 ^{" "" " "} 13.5 kbar

( ^"~~~

^{l kbar}

~

_0.20

" 7.4 kbar

O.15

~

o.io

O 45 90 135 180

Temperature (K)

Fig.

7. Satellite reduced wavevector à _versus temperature for dilferent _pressures

(nominal values).

4. Discussion

The above results _on the

position

of the satellites

along

_some

specific

lines _m the P-T

plane

allow _us to extend the earlier

phase diagram

_[GI of AHOD towards

higher

_pressures.

The _new

phase diagrarn

is

presented

in

Figure

9. The low pressure

phase

boundaries

(broken hnes)

_are

reproduced

from reference _[GI and show the mcommensurate

phase

III _as intermediate

between the disordered

phase

I and the ferroelastic

phase

II. At

higher

_pressures, the _same incommensurate

phase

is intermediate between

phase

I and _some lock-in

phase

characterized

by

à =

1In (n

₌

3,

4 _or

5).

^{The dashed fines}

represent

^the

paths

^followed

by

the measurements in the P-T

plane.

Some

paths (e.g.

the 11.7 kbar

one)

_cross two

transitions,

the first _one

10

IV

( _jj

_{8 à}

= 1/3

JJ r~

Îi 1

w

6 "1"1~

Î ÎÎ

Î

,

II

~++ 2

~+')~

~,+~

₀

~+/ 0 (0 80 120 160

~~~+'?'~

01 0 2 0 3

Fig.

9

à(r

_u)

Fig.

8

Fig.

8. Satellite reduced _wavector à _versus pressure. Trie rate of loss of pressure is

roughly

estirnated

to be o.o19

kbar/K.

This work:

(.)

initial values of à;

(O)

maximum value of à

(see Table); (+)

reference [9]. ^Lines are

guides

for the _eye.

Fig.

9. Pressure-temperature

phase diagram:

Tc:

(+) I-III; (.) III-II;

(Zh)

III-IV;

I:

high

ternpera-

ture phase; II: low ternperature

phase;

III incornmensurate

phase;

IV: à ₌

1/3

lock-in

phase.

Broken fines represent

phase

boundaries deterrnined with _a hehurn

high

_pressure cell

(Ref. [3]).

Solid fines

represent an

interpolation

between

experimental

points. ^Dashed fines represent actual

paths

followed

in the P-T

plane

for _a

given

nominal pressure. Dotted fines represent _a tentative

extrapolation

from the

experimental

data.

between

phase

^III and _a lock-in

phase

and the second between the two lock-in

phases,

but do not

eventually

end in

phase

II at the lowest ternperature. The dotted fine which appears in the middle

part

of the

diagram, represents only

tentative

phase

boundaries which

incorporate

the

present

results but do not

give

definite values of the transition

temperatures

at ail

points.

Our measurements

suggest

^{that AHOD differs from} the

ngid

ANNNI model in _two

respects.

First,

its mcommensurate

phase

forms _a

single

domain in the P-T

plane,

the lock-in

phases appearing only

at lower

temperatures

_in contradiction with the multidomam

topology

of this

phase

^{in the}

^phase diagram

of

Figure

1.

Second,

its lock-in

phases

with _n

=

5, 4,

3

persist

^down

to T

=

0,

while there is

only

_one

smgular point

ⁱⁿ

Figure

1 at which the _{n =} 3 field reaches T = 0. We _now

present qualitative _arguments

which

explam

_our

results,

and also

suggest

^that the first

aspect

is _an intrinsic

property

^of the

compressible

ANNNI

model,

while the second

is, possibly,

_an ^artefact related to

metastability

^effects.

The first

aspect

^{is related} to the

importance,

m

AHOD,

of the

couphng

between

pseudospins

and

phonons

_or, otherwise

stated,

between

pseudospins

and the deformable lattice.

Indeed,

N°2 HIGH PRESSURE INCOMMENSURATE PHASES IN AHOD 243

given

the Fourier transform of trie direct

coupling interaction, Jd(q),

between

pseudospins,

trie transition

temperature

to trie incommensurate

phase

for the _pure

pseudospin _system

should be

given

_[3)

by Jd(qa)

"

kBTd(qo),

where _qa is trie _wavevector for which

Jd(q)

is maximum. It

was shown in reference _[3)

that,

at

atmospheric

_{pressure, qo}

= 0 and

Td (0)

₌ 65 K.

However, expenmentally,

the transition

temperature

^is found to be T~ ₌ 160 K.

Similarly,

at P

= 5

kbar,

qo is

equal

to 0.17c* and

Td(qo

_" 60

K,

while

experimentally,

T~ ₌ 138 K.

As

explained

in reference

[3],

^the

temperature

difference AT

= T~

Td(qo)

is due to the

coupling

between

pseudospms

and

phonons

and is calculated to be:

~$jn

~f2/~

_BUO²

where d the

pseudospin-phonon coupling

constant and _vo is the

velocity

of the transverse acoustic

phonon

involved in the

coupling.

The

large AT'S, respectively

95 K and ₈₀ K for bar and 5

kbar,

_are therefore _a dear indication of the

strength

of this second

coupling,

1-e- of the

importance

^{of the}

compressibility

of the lattice.

The role of the

pseudospin-phonon coupling

_on the

phase diagram

_can be understood _as follows. Let _us consider _a

given

ratio _{r =}

-J2 /Ji

If there would be no direct

coupling

between the

pseudospins

and the

phonons,

1-e- if this

coupling

would reduce to _a

simple dependence

of

r on a uniform deformation of the

lattice,

the

phase I-phase

III transition would take

place

at

T~(r)

and the incommensurate lock-in transition _at

Tj; (r),

where the

ordering

of the

pseudospins

decreases their internal _energy

by AUi

This

ordering corresponds

to _a decrease of

AST

in the

pseudospin _entropy

with

AUI

_"

T;(r)ASi

In the presence of the

pseudospin-phonon coupling,

this transition will

produce

^the _same

AUI change

in the

pseudospin

internai energy,

but,

_as the

phonons

_are

linearly coupled

to the

pseudospins,

there will also be _a

change AU2

in the lattice elastic energy. Since the lock-in

phase

_can be viewed _{as a} succession of ferroelastic

domains of

opposite signs,

there

is,

at the domain

boundary,

_an elastic strain which _was much lower when the deformation had _an incommensurate

character; AU2

_is therefore

positive

^and the net internal energy decrease

(AUI

₊

AU2)

^{smaller for} the _same

Asii

the transition thus takes

place

at a lower

temperature.

^The

preceding argument

^{shows that the ratio between} the actual

T;(r) (phase

III lock-in transition

temperature)

and the actual

T~(r) phase

I-III

transition

temperature)

must decrease with the

increasing phonon-pseudospin coupling.

This effect _can

totally

_suppress, at least

experimentally,

the lock-in

phases just

below

T~(r)

for ail

values of _r

=

-J2/Ji

The second aspect ^{is the}

persistence

of the

phases

at à

=

1/3,

and

possibly

also at à

=

1/4,

1/5,

down to the lowest

temperature.

It is

likely

due to the

partial

order-disorder

aspect

of the

phase

transition in AHOD.

While,

in

purely displacive

incommensurate

phases,

the

displacement

of atoms may be

continuous, ND(

_{must overcome an energy} barrier to go from

one orientation to

another, making

this reorientation process more and _more diilicult _as the

temperature

^{decreases. The absence of} a transition from the

1/4

and

1/5

lock-in

phases

to

phase

II is

presumably

due to this fact _as well _as the

partially

reentrant character of

phase

II below 80 K in

Figure

9

(see

dotted

line).

On the

contrary,

the existence of the

c*/3 phase

at the

highest

_pressure is

unlikely

to be the result of the _same

metastability.

An

extrapolation

of the III-II

phase boundary (broken fine)

below 80 K would

only

indude in the

phase

II domain

a hmited

portion

of the

c*/3 phase

field observed

experimentally.

^At

higher

_pressures, the existence of _a stable

c*/3

lock-in

phase

at T

= 0 is therefore _a second

important

^difference

with the

predictions

of the

ngid

ANNNI mode.

Finally, although

the absence of intermediate lock-in

phases (multidomam topology

^discussed

earlier)

_may be _an intrinsic

property

of the

compressible

ANNNI model of

AHOD,

_we cannot

presently

rule out the

possibility

that it may also be _a consequence of the _energy barrer for

reorientations. Those

phases

have _{a very} small

stability

domain

and,

their free _energy

being always

close to that of _more stable

phases, they

should _appear at much lower

temperatures

than in the

rigid

ANNNI model.

Consequently,

at lower

temperatures,

^{the local} _energy barriers may be suflicient to

prevent

their formation.

Conclusion

Measurements of the satellite

position

in AHOD have been

performed

at

high

_pressure, between 7 and 13 kbar. The results have allowed _us to map ^out a

larger portion

of the P-T

phase

diagrarn

than

previously. Significant

differences with the

phase diagram

of _a

rigid

ANNNI

system

have been found which _can be attributed to the

strong coupling

of the AHOD lattice deformation with the orientational

ordering

of the ammonium ions. At intermediate _pressures, AHOD remains incommensurate _{over a} wide range of

temperatures

^and the two lock-in

phases

observed

(c* /5

and c*

/4)

_{are more}

weakly

stable. We

suggest that,

at very low

temperature,

the

phases

at c*

/5

and c*

/4

_are metastable

against

the ferroelastic

phase

II _{as a} consequence of the order-disorder nature of the

phase

transitions in this

system,

^{but that the c*}

/3 phase

is

stable _{over a} wide pressure range down to T

= 0.

Acknowledgments

It is _a

pleasure

to thank René Millet for technical assistance.

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