Dynamics of the commensurate-incommensurate phase transition in C2O4D ND4, 1/2 D2O: a polarized Raman study under pressure

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Dynamics of the commensurate-incommensurate phase transition in C2O4D ND4, 1/2 D2O: a polarized Raman

study under pressure

M. Krauzman, A. Colline, D. Kirin, R. Pick, N. Toupry

To cite this version:

M. Krauzman, A. Colline, D. Kirin, R. Pick, N. Toupry. Dynamics of the commensurate- incommensurate phase transition in C2O4D ND4, 1/2 D2O: a polarized Raman study under pressure.

Journal de Physique I, EDP Sciences, 1993, 3 (4), pp.1007-1029. �10.1051/jp1:1993180�. �jpa-00246766�

Classification

Physics

Abstracts

33.20F 64,70R 64.60C

Dynamics ^of the commensurate-incommensurate phase

transition in C~04D ND4, ^1/2 D~O

_{: a}

polarized Raman study

under pressure

M, Krauzman

(~),

A, Colline

(~),

D, Kirin

(2),

R, M. Pick

(I)

and N,

Toupry (~)

(~)

D6partement

de Recherches

Physiques (*),

Universit6 P, et M, Curie, 75252 Paris Cedex 05, France

(2) Institut Ruder Boskovic, 41000

Zagreb,

Croatia

(Received 5 June J992,

accepted

in

final

form 28 October

1992)

R6sum6. Les spec~es Rarnan

polaris6s

de basse

frdquence

de AHOD ont 6t6 d6terrr1n6s h 6,4 kbar dons les

phases

norrrale et incommensurable.

L'analyse

des deux

pics

centraux visibles

dans les deux

phases,

et des trois

phonons

acoustiques rendus visibles dons la distorsion incommensurable, corrobore les r£sultats obtenus r6cemment par diffusion de neutrons dons la

phase

norrrale [10], et 6tend ceux-ci h la

phase

incommensurable, En

particulier,

l'dvolution du temps ^de r6sidence des ions

ND(

dons _cette demi~re

phase,

ddterrr1n6e h

pant

de

plusieurs expdriences, indique

_un accroissement notable de la bam~re de

potentiel

_par rapport ^{h la}

phase

norrrale.

Abstract, Polarized low frequency Rarnan spectra ^of AHOD have been measured _at 6,4 kbar both in the normal and in the incommensurate

phases,

The evolutions of _two central

peaks,

visible in the two

phases,

and of the flwee acoustic modes made active

by

^the incommensurate distortion

have been

analyzed.

The results _are in good agreement ^{with the values deduced from} a recent

neutron experiment in the normal

phase [10],

and extend them _to the incommensurate one. In

particular,

the

ND(

residence time is determined

through

various methods in the latter

phase,

and

provides

evidence for _an increase of the barrier

height

with respect to the disordered

phase.

1. Introduction.

Ammonium Oxalate

hemihydrated, NH~C~O~H, H~O (in

short

AHO)

and its

fully

deuterated 2

analog (AHOD)

have been

recently

the

subject

of several

investigations

motivated

by

their remarkable

phase diagram,

This

diagram

has been constructed from Raman

experiments [1-3]

up ^to 8 kbar and down to 80 K, The nature of the

phases

involved has been established

through

(*) U-R-A- 71.

a series of X

Ray [4-5],

Brillouin

[6]

and elastic neutron

[7] scattering

measurements and

figure

I

reproduces

our present

knowledge

of the

phases appearing

ⁱⁿ AHOD, Phase I

belongs

to the

D(((Pmnb ₎

_{space group} with Z

=

8 formula units per cell, The

phase

I-II transition is

equitranslational,

and of the second order type, ^{Phase II}

belongs

to the

Cl

~ PI ~~

l space group and the elastic constant

C55

tends to zero at the

phase

transition.

n

Phase I-III transition is also second order, while

phase

III-II transition is first order. It _was established

[3-7]

that

phase

III is incommensurate with wavevector qo =

3c*,

where 3 is _a function of pressure

only,

3

increasing

from 0.04

[8]

in the

vicinity

of the critical _pressure

P~

to

approximately

0.23 at 8 kbar. Furthermore, from _a

polarization analysis

of the Raman

spectra [3],

it _was concluded that the variable which freezes in

phase

III

transforms,

in

phase I,

as the _v~

representation [9]

of qo.

These

experiments

have

suggested

that the two second-order

phase

transitions have the _same

origin

and _can be understood in the

following

_way. In the

high

temperature

phase

I, the 8

ND(

cations lie _{on a} mirror

plane [4]

and thus form _two distinct families, One of them is disordered

[4],

and the two

symmetrical positions

of each cation with respect to this statistical mirror

plane

_may be

represented by

the two values of _a

pseudospin.

For each lattice

cell,

_one

2

can form four different linear combinations of these

pseudospins,

each of them

transforming

as a different irreducible

representation

of the

crystal point

_group. One of these

combinations,

which will

play

the

major

role

here,

transforms _as the

B~~ representation,

and _we shall, in accordance with

[10],

^call

~r)

this linear combination in the

L~ _cell,

_and

~r~(q)

^{its Fourier}

transform

(up

to

phase

factors

[10]

which _are irrelevant for the

present discussion).

The bilinear interaction energy between _two

pseudospins

can be written in _terms of these linear

combinations, Let _us call

Jd(q)w k~ Td(q), (1)

(where k~

^{is the} Boltzmann

constant)

the Fourier transform of this interaction energy between two ad variables

belonging

to different cells. The model

proposed

ⁱⁿ

[3]

for the two second- order transitions of

figure

I is that the _same variable

~r~(q

^{drives the} two transitions, and that _:

a) Td(q) depends only

_{on pressure} its maximum

simply

_moves from q = 0 below the critical pressure

P~,

to a wave vector

along c*,

above

P~.

b)

For all values of q, and in

particular

for _q in the

vicinity

of the direction of

c*,

there exists

a strong ^linear

coupling

between

~r~(q)

and _a

pseudo

transverse acoustic

phonon

with normal mode coordinate

Q(q),

^the elastic limit of which for q -

0, qflc*

is the e~~ shear

(which

transforms,

in this

limit,

_as the

B~~ representation).

This model

agreed,

inter alia, with the Raman detection

[3],

in

phase

III, of _two low

frequency

excitations which _were identified _{as two} acoustic

phonons

with _wave vectors qo. These _two excitations _were

mostly

^{studied in}

AHO, though

some measurements _were also

performed

_on AHOD

[I],

From the

polarization

ⁱⁿ which

they

were

detected,

and from their

frequencies

which could be related to the known elastic _constants of AHO in

phase

I, one of these excitations _was found to be the LA

phonon,

which

appeared

in the _ac

polarization,

while the second _one, ab

polarized,

was identified _as the b

polarized

TA

phonon,

A weaker

signal,

_aa

polarized,

at the _same

frequency

_as this TA

phonon

_was also

detected,

and attributed to a

polarization leakage

the absence of _a lower

frequency signal

in that

geometry

^{led the auth6rs} of

[3]

to

conjecture

that the

coupling

_{of ~r~}

(qo )

with

Q (qo )

was so strong ^{that the}

frequency

of the _a

polarized

TA

phonon

remained too low to be detected with their

equipment

in the whole

temperature

^{domain accessible in}

phase

^III,

Neutron inelastic

scattering experiments [10] (here

after referred to as

I)

^have been

recently

performed

in

phase

I of

AHOD,

at 0 kbar and 5 kbar _{at some}

temperatures

above

(including

close

to)

the second-order

phase

transition.

They

confirmed the

predictions a)

and

b)

of the

preceding

model but also showed that, at 5

kbar,

the _a

polarized

TA

phonon

branch

essentially

did not

soften, except

^{in the} immediate

vicinity

of q =

0,

_{contrary to} the

hypothesis

made in

[3].

This

implied

that, if w~~ ^{is the}

frequency

of _a

phonon

on this TA branch, ^and _v is the

mean residence time of _an individual

ND(

ion in _one of its _two orientational

wells,

the

relationship

_{w~~ v ~} l is fulfilled for most of the

phonons

of that branch in the whole

phase

I _; values of _v

(T)

consistent with this

inequality

_were in fact deduced from the 5 kbar

experiment.

The purpose of this Raman paper is twofold. First, we wish to confirm the value of this _mean residence

time,

which _can be deduced from the

study

of _a central

peak,

_{as was} shown in

[2],

This central

peak

is related to the individual

pseudospin dynamics,

and appears in _two different

polarizations,

which

probe

the

dynamics

of two different collective

variables, ~r~(q)

and

~r~(q) [10],

in the q =

0 limit. Due _to

improvements

of _our Raman

equipment,

and to the construction of _{a new} pressure cell

[I Ii,

such measurements, which _were

formerly possible only

at zero pressure and for strong

signals [2]

_{are now}

feasible,

under pressure, With _an increased accuracy, whatever the

polarization

_geometry, and _can be made both in

phases

I and III.

They

will

provide

_us with information _{on v}

(T)

in both

phases

on an AHOD

crystal

of the

same

origin

and

quality

_as that used in

[10].

We also wish _to

clarify

the

origin

of the weak low

frequency signal

with _aa

polarization

obtained in

[3]

at the _same

frequency

_as the b

polarized

TA

phonon,

It _was found in

[10]

that the two bare elastic _constants

C44

and

C~5

^have

approximately

the _same value _; this

suggested

that the _aa

polarized signal

could be due _to the _a

polarized

TA

phonon

and _{not to a}

polarization leakage

of the other TA

phonon,

With the

improved quality

of _our

equipment,

it is also

possible

to check this

point,

and this will also represent an

independent

_way of

measuring v(T)

in

phase

III,

In order to illustrate the various aspects ^{of this}

work,

_{our paper} is

organized

_as follows, In section

2,

_we

briefly

discuss the

dynamics

of all the low

frequency

excitations which could be visible in _a Raman

scattering experiment,

first in

phase I,

and second in

phase III,

where the

new

couplings provided by

the

freezing

^of an incommensurate distortion lead to a much more

complex

discussion. Section 3 is devoted to the

description

of the Raman

experiments performed

_at 6.4 kbar and of the

tecllniques

used to extract numerical information _on the

various spectra we are interested in. The

qualitative

thermal behaviour of these spectra ^is discussed in section 4 and found to agree with the

predictions

of section 2. The

quantitative analysis performed

in section 5 and the discussion of these results show that _a coherent

picture

of the

phase

I-III transition and of the

dynamics

of the relevant excitations _can

presently

be

made.

Finally

a summary of this work, as well _{as some}

concluding

remarks _are done in

section 6.

2. Low

frequency

excitations visible

by

^Raman

spectroscopy.

A review.

This section is devoted to _a brief review of the effects which have been detected in _our Raman

experiments

_on AHOD under pressure,

performed

both in

phase

I and in

phase III,

in the low

frequency region.

For temperatures ^{above the}

transition, light couples only

to excitations _at q =

0, and,

in the

frequency regime studied,

_no

dynamics

other than the

pseudospin

one is accessible, Below the

transition,

_{on top} of the _same

dynamics, light couples primarily

to the excitations at ± qo, and the latter

comprise

both the

spin dynamics

^{and the acoustic}

phonons.

Except

for _one

special scattering

_geometry, in the incommensurate

phase,

these

dynamics

_are

not

linearly coupled,

_so that the situation will tum out to be

relatively simple,

On the other hand,

through

the incommensurate modulation, the

pseudospins

at q = qo are

coupled,

in

phase III,

to other low

frequency excitations,

and _we shall have _to

explain why

these spectra

can nevertheless be understood in terms as

simple

as in the

high temperature phase.

2.I THE _PHASE I SftUATION. As recalled in the

Introduction,

in the

high

temperature

phase,

in each

elementary cell,

4

ND(

ions exhibit orientational disorder,

They

all

belong

to the _same

family,

and, inside _one

primitive cell,

_{one can} form four different linear combinations of the

corresponding pseudospins,

which transform

respectively

_as the

B~~, A~, Big

and

B~~ representations

of the

crystal point

_group

[2]

_; these combinations have been labeled

respectively

~r~, ~r~, ~r~ and ~r~ in I.

Only

the last two are Raman active and _we shall discuss in tum their

spectra.

2,I.I The

B~~ pseudospin.

The free energy related to the ~r~

pseudospins

and to the

transverse acoustic

phonons, Q (q ),

to which

they

_are

linearly coupled

has been discussed in I.

When

specialized

to wave vectors q

parallel

to

c*,

this free energy, limited _to its harmonic

part,

^reads :

Fd

=

z ik~ (T

Td

(q)) ~r~(q) ~r~(-

_q

)

₊

idq iQ (q)

~r

(- q)

_c-c-

i

+

o] _{q2 Q (q) Q (-}

_q)1

~

(2)

where _:

~r~(q)

is the Fourier transform of

~r).

Q(q)

is the normal coordinate of the _a

polarized

TA

phonon

with sound

velocity

ilo.

d is _a constant which

couples

these _two

variables, while,

in the

approximation

used in

I, T~(q) reads,

for

qflc

_:

T~(q)=

^-A

(B-c)cos~j~-Kcosq.c (3)

where

A, B, C,

K _are interaction energy constants related to different ty1Jes ^of

neighbours

of a

given pseudospin. d,

_as well _as the coefficients ofT~

(q ) depend

_{on pressure}

only,

and d is

large enough

to increase the transition

temperature

^from

T~(0 )

= 65 K to

Tf

=

160 K at 0 kbar and from

T~(qo )

= 57 K to

T(

= 137.5 K at 5 kbar for instance.

Writing

for the

equations

of motion of these _two variables

Q

and ~r~.

~

~F

= w

2

Q (q )

~

~~

=

i w

k~

Tv ~r~

(q ) (4)

where _v is _an individual

pseudospin

_mean residence time, one obtains for the

pseudospin dynamics

_:

lad (q,

_w

)

_ad

(q,

W

) *>

₌

(I

+ n

(w ) )

3m

l~

~ ~

(5)

kB(T T~(q)) iwkB

TV

~~~~

*o

_{q W} ~

For q -

0,

_as the

spectrum

is studied

only

in _a

frequency

_range for which _w »

do

_q, the last term in the denominator of

equation (5)

is

negligible,

and the latter reduces _to its first _two terms, which leads to _a Lorentzian central

peak

for the response function.

2.1.2 The

Bi

~

pseudospin.

Let _us call

~r~(q)

the Fourier transform of the

corresponding pseudospin

variable. For

qfc*,

_{them exists no acoustic}

_phonon

_{which transforms}

as

~r~(q)

^and the

only

^relevant

part

^{of the free} energy is

simply

_:

F~

=

z ik~(T Tc (q)) ~rc(q) ~rc(- q)1 (6)

q

where

k~ T~(q

^{is the}

corresponding

interaction energy.

Calculating

this interaction _energy

by taking

into account the same

pseudospin

interactions _as those used to obtain

T~(q),

_one

finds,

for

qflc*

T~(q)=

-A+

(B-C)cos~'~-Kcosq.c, ₍₇₎

2

Making

_use of _an

equation

of motion similar to

equation (4) yields,

for this

pseudospin

response function _:

(«c(q, W) «c(q, W)*)

=

(i

+

n(w ))

am

~

~~

^~

~

~ ~

(8)

I,e, to another central

peak,

with _an

expression

identical to the

simplified equation (5) _except

for the

change

of

T~(q)

into

T~(q),

2.2 THE PHASE III INCOMMENSURATE CASE. The situation in the incommensurate

phase

is

more

complex,

_{as one can} detect excitations both at q = qo and at q =

0, Let _{us start} with the first _ones, As _was shown e,g, in

[12],

the Raman tensors which

couple

_an excitation at q = qo with the incident

light

_are

proportional

to the order

parameter

~ =

(~r~(q~))

^and the Raman

intensities,

at any

frequency,

of the excitations detected

through

this mechanism will be

proportional

to

~~,

Let _us

discuss,

in tum, the various

spectra

^which can be detected in

AHOD.

2,2,1 The LA

phonon

and the b

polarized

TA

phonon. Along

_c

*,

the

pseudospins

transform

respectively

_as the v~

representation

of the group of q~ for the two

pseudospin

variables

~r~(q )

^and

~r~(q )

and _as the r~

representation

for

~r~(q

_{and ~r~}

(q ),

The LA

phonon

transforms

as v~ and the b

polarized

TA

phonon

_{as v~}

[9],

These two

phonons

_are thus _not

coupled

to any

pseudospin

:

then,

_one

expects

^their

frequency

to be

temperature independent,

and their linewidth

quite

_{narrow as} for any acoustical

phonon, Furthermore,

_as shown in

[3],

the LA

phonon

must appear in the _ac

polaTization,

and the b

polarized

TA

phonon

in the ab

polarization,

2,2,2 The _a

polarized

TA

phonon

and the _~r~

(q ) dynamics.

In order to obtain the

dynamics

of these two variables _{at ±} q~, the usual Landau type treatment

[13]

of the statics and of the

dynamics

of soft variables close to an incommensurate

phase

transition must be

performed, Stating

that the

driving

mechanism of this

phase

transition is the ad

(q)

variable and

adding

to the free energy

(Eq, (2))

_a fourth order _term in this variable, the latter _can be written _{as ;}

F~

=

z

kB

(T Td(q)) "d(q) "d(- q)

₊

q

+

Z ( _{"d(qi ) "d(q2) "d(q3 ) "d(q4)}

₃

_(qi

+ q2 + q3 +

q4)

qt.q~.q3.q4

+

j _{z q(Q (q)}

«

(- q)

_c.c.

)

+

z ^*# q~ Q (q) Q (I _{q) (9)}

where b is _a

coupling

constant which

depends

_on the four q~ ^vectors and

which,

in this subsection, will be used

only

for qi, q~, q~ and q~ in the

vicirdty

of _± qo,

Writing

that the transition which takes

place

at qo leads to an

equilibrium

state

yields

_:

'7^~ =

("d(qo))

^~

=

~

kB _{lTd(qo) Tj}

+

~~

m

~/ _{[Tt Tj (ioa)}

I

*o

iQ(qo)j

=

j i"d(qo)j (lob)

with :

bi

« b

(qo,

_{qo, qo,}

qo). (i i)

In the _mean field

approximation,

below this

transition,

_two of the four

spin

variables

appearing

in the second term of

equation (9)

must be taken _at their

equilibrium value,

while the other

spin

or

phonon

variables _are taken _as the fluctuations around their

equilibrium

values. The

bi

terms, then,

couple

_q

= qo + h to q = qo +

h,

whatever h,

through

_a term

proportional

to

~ ~, while

«~(q)

and

Q (q)

_are still

coupled by

^{the d} term. There _are thus four variables which

are

linearly coupled,

the

dynamics

of which _are

degenerate

in

pair

for h

=

0,

in the absence of the _~^~

coupling,

To understand the solutions of the

dynamical equations

_{for q}

= qo, it is useful to

study them,

first for h x

0,

and then to let h tend to _zero.

Let _us make _use of this classical

method, choosing

for

simplicity

h to be

parallel

to

c*,

and let _us take the

pseudospin amplitude

and

phase

mode variables _as

dynamical variables,

Q~(h)

=

fi j«d(q0

₊

h)

e~'~

«d(~

_q0 +

h) e~~j (12b)

2 where _:

(~r~(qo))

_{= ~} ^e~~

(13)

with similar

expressions

for the

phonon amplitude

and

phase

mode variables,

Using

equation (4)

^and

postulating

^that a part ^{of the} response function is located in the

vicinity

of

w =

0,

while the other part appears in the

vicinity

of _too

=

do

_{qo, one}

obtains, through

_a

calculation which is

briefly

sketched in the

Appendix,

the

following

results.

Vicinity of

_{too =}

do

_qo, Two excitations have their spectrum ⁱⁿ the

vicinity

of the

unperturbed

TA

phonon frequency

in the h

- 0 limit. One is of the

amplitudon

_type, and its spectrum ^is

proportional

to _:

~~~~~°

~~ ~

~~~°

_~~ ~~

l~

~

2

~°

~°

~

2

k~

(T~

T)

+

k~

_(T~

T~(qo))

I

wk~

Tv

~°

(14)

The

intensity

of this spectrum ^is

proportional

to ~~. It represents ^the

dynamics

of _an acoustic

phonon coupled

to a diffusive

pseudospin,

this

coupling becoming

smaller _as T

decreases,

because the real part of the

pseudospin dynamics

increases with _T~ T, while its

imaginary

part ^increases

exponentially through

the temperature

dependence

of _v. It is Raman active in any of the three

diagonal polarizations.

The second excitation is of the

phason type,

ⁱⁿ the h

- 0

limit,

both

through

^its

eigenvector,

which is _a linear combination of the two

phase

mode

variables,

and

through

the

algebraic expression

of its spectrum ^{which reads}

~~~~~~

_~~ ~

~~~~~

~~

~j(

~j_

~

~2

~~~~

k~(T(qo)

_T~) I

wk~

Tv

Indeed,

_as for any

phase mode,

the real part ^{of the} term in

d~

does _not

depend

_on

temperature (compare Eqs. (14)

and

(15)).

As this mode is _not Raman

active,

it will be of _no interest here.

Vicinity of

_w

= 0. It _was shown in I that the

pseudospin dynamics

becomes

critical,

I-e-

gives

rise _{to a} central

peak

with HWHM

tending

to zero, in the

vicinity

of the normal- incommensurate

phase transition,

for q in the

vicinity

_{of qo.} Below T~, in the _{same manner as} for the excitations in the

vicinity

of the bare

phonon frequency

discussed above, the _two

corresponding

diffusive modes at ± qo

couple

into _an

amplitudon

_type central

peak,

and a

phason

_type central

peak

_; ^this is shown in the

Appendix,

and _was

briefly discussed, recently,

in

[14].

The

amplitudon

_type central

peak

has _an

intensity proportional

_{to :}

'l~SA«(W)

=

'l~il

₊

_n(w)13m

~

hi

_'l

iWkBTT (16)

Its

integrated intensity

is thus

proportional

to

k~ T,

^I-e- non critical close _to T

= T~ while its linewidth is

proportional

to T _T~

similarly

to the

phonon

_type

amplitudon (Eq. (14)),

this

spectrum

^has a

diagonal polarization,

and could

thus,

be detected _a

priori,

ⁱⁿ any of these

polarizations.

In

fact,

_we have not detected

it, and,

as will be shown in section

6,

this is due to its _too small

HWHM,

in the whole temperature range studied.

As for the

phason

type central mode

(Eq. (A7),

second

line)

it has, in _a

light scattering experiment,

_a linewidth much _narrower than the central

peak

of

equation (16).

This excitation is thus _not detectable

by

Raman spectroscopy even if its Raman

activity

is not

identically equal

to zero, as is

presumably

the _case if _one makes _use of arguments ^similar to those used for the

displacive phason [12].

2.2.3 The

Bi~ ^pseudospin

^variable spectrum. ^The

spectrum

^{of the}

~r~(q)

variable in the

vicinity

_{of q =} 0 _was

already discussed,

above the

phase

transition, in subsection 2.1.2. We must reconsider the situation below the

phase transition,

because the

freezing

of the

pseudospin

at ± qo

couples

this variable _to

pseudospins

^{in the}

^vicinity

^of ± 2 qo.

Indeed, taking

into account

only

the terms which lead to such _a

coupling,

and

limiting

the

pseudospin

wave

vectors to be

along

the c*

axis,

the relevant part ^{of the free} energy reads _:

F~

=

zkB(T- T~(q)) ~r~(q) «~(- q)

+ q

~c

+

Z

~ ^{"c(qi "c(q2) "d(q3) "d(q4)}

³

^(qi

^{+ q2 + q3 +}

^{q4) (1?)}

q; q2. q3. q4

where b~ is _a function of the four _wave vectors. In the _mean field harmonic

approximation,

below T~,

~r~(q~)

and

~r~(q~)

^will be

replaced by

their thermal _mean value, with

amplitude

~. If _one takes q~ = q~ = ± qo this _term

simply

renormalizes the

k~ (T T~(q ))

term,

adding

to it _{a term}

equal

to

b[

_{~ ~,} which in view of

equations (6)

to

(8),

modifies the thermal evolution of the

corresponding

central

peak.

If,

_on the other

hand,

_one

considers,

for

instance,

_{q~ = q~ = qo,} ^the

corresponding

term

couples

the

dynamics

of

~r~(q)

with that of

~r~(q

2

qo) through

a term

proportional

to

b[

_~^~_{; a} similar

coupling

exists with

~r~(q

+ 2

qo)

_{for q~ = q4 "} qo. Nevertheless, ^if _q ^{is in} the

vicinity

of the Brillouin _zone center, T~

(q )

x

T~(q

+ 2

qo)

m

T~(q

2

qo).

This _means that

the bare

dynamics

of

~r~(q)

^is not

degenerate

with that of

~r~(q

± 2

qo).

In _a

perturbative approach,

this

gives only

a correction in

~~

_to the

dynamics

^of

~r~(q),

_an effect which _can be

neglected

with

respect

to the

b[

_~ ^~ correction described above.

Thus

equation (7) yields

:

(«c (q,

_W

)

_«c

(q,

_W

)*)

=

Ii

+ n

(w )j

am

~ ~

(18)

with _:

bi

m bc

«c(q) «c(- q) «d(qo) «d(- qo)

_{q -} o.

(19)

In the _mean field

approximation

used

here,

_~ ^~ is

proportional

_{to T~} T

but,

since the value of

b[

is

unknown,

_{one can}

only predict

the appearance of _a second _term linear in T in the real

part

of the denominator of

equation (18).

As will be _seen in section

4,

in the

present

case, I-e- for AHOD at 6.4

kbar,

^it will _reverse the

sign

of the coefficient of T. Let _us

finally

note

that,

_as this

spectrum

was

already

visible

above the

phase transition,

ⁱⁿ the ab

polarization,

^{it will} remain visible

through

the _same mechanism below T~.

2.2.4 The

B~~ pseudospin

^{variable at} q = 0. In

principle,

the situation is

quite complex

_as

one must combine the difficulties met in subsections 2.2. I and 2.2.3

(I.e.

the linear

coupling

between

phonons

and

pseudospins)

on the _one

hand,

the

coupling

between the excitations _at q, q + 2 qo and q 2 qo ^{and the} renormalization of the coefficient

k~(T T~(q)),

_on the other

hand. The

analysis

is rather cumbersome but shows that the

only significant

effect is the last one, as was

already

the _case for the

«~(q) dynamics,

both the

coupling

with the acoustic

phonons (as

in subsection 2.

I-I)

and the

coupling

with the

pseudospins

at q ± 2 qo

(as

in subsection

2.2.3) being negligible.

One thus

obtains,

in

analogy

with

equation (18)

_:

("d(q, W) "d(q,

_W

)*)

=

Ii

₊

n(w)]

am

~ ~

(20) kB(T Td(q))

₊

b2

_'l i

wkB

Tr

with _:

b(

m b

(q,

_{q, qo,}

qo)

_q

- ° ~~~~

b(qi,

_{q~, q~,}

q~) being

defined

by equations (9)

and

(

II

).

Comparing equations (5)

and

(20),

_as well _as

equations (8)

and

(18),

_{one sees} that the effect of the

phase

transition _on the spectra ^related to the

dynamics

of the

«~(q)

and

«~(q)

pseudospins

at q =

0 is

absolutely

similar. It

simply changes

^the

slope

of the coefficient in Tin the real

part

^of a

denominator,

thus

changes

the thermal behaviour of the HWHM of the

corresponding

central

peak.

It does _not

change

its

polarization properties

_so that the

«~(q) dynamics

will still appear in the _ac

polarization.

3.

Experiments.

3.I EXPERIMENTAL _DEVICE. The present

experiments

have been

performed

with _a

8 _x 7 _x 7

mm~ crystal

of AHOD obtained

by

the

evaporation technique

in the _same bath _as for

the _neutron

experiment

of I. The

sample

was oriented with its _a axis

perpendicular

_to the

scattering plane,

the

incoming

beam

being parallel

to _c and the scattered beam

parallel

to b. A helium pressure ^{cell with three}

saphire

windows

containing

the

crystal

_was

placed

in _{a vacuum} cryostat ^{with silica windows. Pressure} was measured within _± 5 bar

by

a

manganin

resistor in _a

secondary

cell, at room temperature, ^{and the} temperature ^of the

crystal

was

probed by

a

Chromel Alumel

thermo-couple

at room pressure inside _a 3 _mm in diameter

pipe penetrating

the

sample cavity.

Raman

spectra,

excited

by

the 514.5 _nanometer line emitted with _a power of 0.I _to 0.2 W

by

_a

Spectra Physics Argon

ion

laser,

_were recorded with _a

Coderg

T800

triple spectrometer

modified in order _to be driven

by

_an IBM AT3 Personal

Computer

which collected and stored the

signal

from _a cooled RCA 31034-06

photomultiplier.

Data _were taken

during

_one

second,

_every 0.125 cm~ ~, and the

origin

of the

frequency

scale _was checked and

occasionally

shifted within half _a step ^after

comparing

the Stokes and the antistokes part ^{of the}

spectra.

All the

experiments

discussed below _were

performed

at 6.4

kbar,

between _room

temperature

and 78 K

although

_some

preliminary

tests _were

performed

at a lower pressure ^to ascertain the

position

of the

phase

II-HI transition for

fully

deuterated AHOD.

Taking

this information into account, the

general

form of the

phase diagram,

_as well _as the other transition

temperatures

shown in

figure I,

the 6.4 kbar pressure was chosen _{as a}

compromise

between two

opposite points

of view _:

The pressure ^must be _as close _as

possible

to 5

kbar,

in order to allow for _a fruitful

comparison

with the neutron

experiment

of I.

The first order

phase

II-m transition must take

place

below 78

K,

_so that the

phase

III

dynamics

_can be studied in _as

large

a temperature ^interval as

possible.

iii

i

Z

r0

f

^Ii

T

140

~

T K ^~

Fig.

1. Phase diagram ^of fully deuterated AHOD. The transition points ^{have been} obtained from

neutron data (round [8] and square [10] dots) and Rarnan data obtained

during

the present set of

experiments

(crosses). Phase III exists only ^above

P~

= 2,4 kbar and below T~ = 147 K [10].

3.2 EXPERIMENTAL TECHNIQUE. At each temperature, ^{the four different}

spectra

^with

respective polarizations

aa,

ab,

_ac and bc _were

recorded,

from 150 to 150 cm~ as well _as

(he

aa

polarized

_spectrum from 840 _to 910

cm~~

which contains _a

sharp

line _near

8~ficm~

_~, ^the

integrated intensity

of which _was used _{as an} intemal

intensity

standard. In this

high frequency

_range, there also

exists,

in the bc

polarization

a line whose

frequency (851

cm~ ~) is

temperature independent

in

phase I,

and which

splits

into _two lines in

phase

^III.

The transition

temperature

was

accurately

determined

by recording

^{the 851} cm

intensity

as a

function of temperature : due _to this

splitting,

the 851 cm~

intensity drops rapidly

^{below the}

transition and the _same

result,

_T~ =135.5 K _± 0.3

K,

_was found either

by cooling

or

by

warming

the

sample,

in _a

good agreement

with the

extrapolation

of the neutron results

(Fig.

I

).

Careful

setting

of the

polarizers

resulted in correct

polarizations

_{as can} be checked in

figure

2 which represents, at the temperature ^T ₌ ^78.3 K, far below T~, the Stokes part ^{of the} four recorded spectra up to I IO cm~ _:

indeed, leakages

^from strong ^{lines do not} appear in

unallowed

polarizations.

As

expected,

Lorentzian

peaks

_appear

only

in the _ac and ab

polarizations.

Nevertheless, ^{extemal modes} of the

crystal

_are located above 50 cm~ ~, and

they

are broad

enough

to

produce

_a substantial

(and frequency dependent)

contribution 15 _to 20 cm~ below their maximum,

especially

in

phase

I. In order to obtain the pure

pseudospin

dynamics,

it is necessary ^to subtract this

frequency dependent

«

background

_». This _was

performed by representing

the

corresponding

extemal modes

by damped oscillators,

charac- terized

by

their

frequency,

linewidth and

intensity

evaluated

by

_a fit of the

spectra

^above 50 cm~

78.3K 6.Skbar

[

~ ab

~

-

bc

aa

50 ioo

w

(cm-') Fig.

2. Rarnan spectra at T

=

78.3 K and P

=

6.4 kbar in the _{aa, ac,} bc and ab

polarizations.

3.3 AcousTic _PHONONS.

IySTRUMENTAL

LINEWIDTH DEcoNvoLuTioN. In order to

study

in

detail the thermal evolution of the acoustic

phonon

lines, it _was necessary to deconvolute the latter from the instrumental linewidth _; this deconvolution

plays

_no role for the continuous spectra ^discussed above. It

is,

of _course, the deconvoluted values of the linewidths and the

frequencies

of these acoustic

phonons

which _are

reported

in

figures

5 _to 7.

4.

Qualitative

discussion~

4,I AcousTlc _PHONONS.

Figure

3 represents, ^{for the} same temperature as in

figure 2,

T

=

78.3

K~T~,

_a

portion

of the _same spectra,

represented

_{on an}

expanded

scale and

arbitrarily

normalized _at the _same 885 cm~

peak integrated intensity.

Three

spectra

^{show the}

LA,

the _a

polarized

and the b

polarized

TA

phonons respectively

at q = qo. Due to the very

good polarization quality

^of our

experimental setting,

there is _no

polarization leakage,

while our detection accuracy allows for _a resolution of about 1.7 _{cm ~.} In

figure 3,

it _appears that the

aa and ab spectra ^{have maxima} at different

frequencies. Furthermore,

_{as can} be _seen in

figure

⁴ which shows similar spectra recorded at a

higher

temperature, ^the two lines have different

linewidths, indicating clearly

that these _two spectra ^do not

correspond

to the _same

excitation,

and _are thus the two TA modes.

~

~

i

" ~

§

_~

d ^~

ac

6.Skbar

l17K

20 50 20 30

w

(cm-')

^W

^(cm-<

F~g. 3. Fig. 4.

Fig.

3. Enlargement of the low frequency part ^of figure 2 (presented in _a different order). Note the excellent

polarization

of the four spectra.

Fig.

4.

Comparison

of the b and _a

polarized

TA

phonons

at 117 K. The latter is broadened by ^the interaction with the «~

pseudospin.

The

meaningful

results of fits of each line with _a

simple damped

oscillator above _a linear

background,

convoluted with the

spectrometer

^function are

represented

in

figures

5 and 6. In

figure

5, the thermal variation of the three

frequencies

in

phase

III, from 78K up to

T~, is

represented

as

~

for each of the three

phonons.

One _sees

that,

while the

frequencies

w

(78 )

of the LA and the b

polarized

TA

phonons practically

^do not

change

^with

temperature,

the

frequency

of the _a

polarized

TA

phonon

^{decreases when}

approaching

_T~ from below. A

parallel

type ^of behaviour _can be _seen in

figure 6,

where the

corresponding

deconvoluted linewidths

are

plotted

_: for the first two

phonons,

the linewidths remain, in the whole

phase III,

smaller than the instrument

resolution,

while this linewidth is

larger

for the _a

polarized

TA

phonon,

and increases

notably

when

approaching

the transition temperature.

All these results agree with the

predictions

of subsections 2.2,I and 2.2.2 _: the LA

phonon

and the b

polarized

TA

phonon, decoupled

from the

pseudospins,

must be very narrow and have

frequencies essentially

temperature

independent. Conversely,

the

coupling

of the _a

polarized

TA

phonon

with

«~(q)

was

expected

to

slightly

decrease the

phonon frequency

and to increase

notably

its linewidth. This increase of the linewidth

is,

in fact, so

large

that it _was

already

noticed in I : in Phase

I, along

the whole c*

direction,

the b

polarized

TA

phonon

_was

found _to be much _{narrower at} 0 kbar and 300

K,

than the _a

polarized

TA

phonon

which _was measured at various temperatures ^both at 0 kbar and 5 kbar.

On the contrary, at 5

kbar,

the

frequency change

^of the _a

polarized

TA

phonon

with temperature was too small _to have been detected

by

neutron

scattering

in

phase

I _{: one} needs the accuracy of Raman spectroscopy to evidence it in

phase

III.

~

~x>

~

~ TAb

-

LA

~

-

T4

3

d

80 loo

Tt

T(K)

Fig.

5. Thermal variation of the

frequencies

of the three acoustic

phonons

^relative to their

respective frequency

at 78 K.

TAb

80 loo 120 Tt

T _K

Fig.

6. Deconvoluted linewidth r(T) for the three acoustic

plionons.

The small linewidths of the LA and

TA~

phonons are evaluated with _a poor

precision

due to the instrumental resolution ( 1.5 cm- ~).

4.2 THE PSEUDOSPIN DYNAMICS. The evolution of the

pseudospin dynamics

with

temperature

^{is shown in}

figures

7 and 8. In

phase I,

the linewidth of these spectra ^decreases with

temperature

down to T~, while the

intensity

at _{w =} 0 increases when

approaching

the

phase

I-III transition. Below this

transition,

_on both spectra,

opposite

effects _{are seen} for the

ac

loo Tt

T(K)

Fig.

7. Thermal variation of the width (left scale) and

intensity (fight

scale) of the «~

pseudospin dynamics

as fitted to the ac spectrum.

ab

i-

~

-

5

~

E d

U ^~j

~'

-

~

fi

~

loo

T

(

^{K )}

Fig.

8. Same _as

figure

7 for the «~

pseudospin dynamics

_as fitted to ab spectrum.

intensities _: when still

decreasing

the

temperature,

^the

intensity

decreases

again.

But the linewidth still decreases below T~, and saturates, or

slightly

increases

again

_near 78 K.

The

high

temperature ^behaviour was

anticipated

from subsection 2,I. For

instance,

the

spectral

_response of the _ac spectrum was

predicted,

from

equation (5),

to be

equal

to

~~~

~~

_~~~~~~

j~(0 ₎

₂

~

~ 2

~~~~

T ~

j~~(o)

T where R~~ is the

corresponding

Raman tensor.

The individual relaxation time increases when the

temperature

^{decreases while}

~~/~

decreases for

T~T~ >T~(0).

This results in _an increase of

I~(0)

and _a decrease of the

HWHM,

and the _same

is,

of _course, true for the ab spectrum.

On the contrary, ^below T~, the thermal evolution of the HWHM and

of1(0)

could not be

anticipated

from the sole

analysis

of the discussion

performed

in subsections 2.2.3 and 2.2.4.

For

instance,

from

equations (10a), (20)

and

(21),

the

spectral

_response of the _ac spectrum ^is

given

below _T~

by

:

~~~

~~

_~~~~

Mac (T)j~

~ ^W

T ~

~ ^~

c

~~~~

(T)

with _:

2

b(

_j 2

b(

fa~(T)

=

I _{+ T~}

T~(0) (24a)

hi

T

bi

It is the thermal evolution both of

fa~(T)

and of _r which fixes the thermal evolution of

Ia~(0)

and of the HWHM. As

b(

is

expected

to be of the _same order of

magnitude

as

bi, fa~ (T

^is

expected

to increase with

decreasing

temperature, as well _{as r.} The present ^results indicate that _r

increases,

close to T~, faster than

f~~(T)

but slower than

lf~~ (T)]~,

and similar

considerations must hold for the

Bi

~

where

f~~(T)

is

replaced by

12b[

^j

^2b[

fab(T)

~

i

~

⁺

y ~

^Tt

^{Tc(o) (24b)}

One may

simply

_note, at this

point,

that _we find here for AHOD at 6.4

kbar,

_an evolution of

1(0)

and of the

HWHM,

below _as well _as above T~, very similar _to the results _we

reported

in

[2]

at normal pressure for AHO. In

particular,

in these

early experiments,

below the

phase

I-II transition temperature,

1(0)

decreased with

decreasing

temperature ^{while the HWHM still} decreased in _some temperature range before

increasing again.

In section 5, _we shall

show,

that

we find for _v

(T),

_a thermal evolution similar to the evolution of this residence time in AHO _at normal pressure, an evolution which

explains

the behaviour

of1(0)

and of the HWHM.

5.

Quantitative analysis

and discussion.

All _our

experimental

results

being

in

qualitative

_agreement with the theoretical

predictions,

_we

have

proceeded

to their

quantitative analysis.

In the next

subsection,

we shall consider first the

integrated intensity

of the acoustic

phonons

in

phase

III, second the

consistency

of the individual relaxation time measurements in both

phases.