Ultrasonic study of the normal — incommensurate — commensurate phase transitions in [N(CH3)4]2ZnCl 4

HAL Id: jpa-00210228

https://hal.archives-ouvertes.fr/jpa-00210228

Submitted on 1 Jan 1986

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Ultrasonic study of the normal - incommensurate - commensurate phase transitions in [N(CH3)4]2ZnCl 4

J. Berger, J.P. Benoit, C.W. Garland, P.W. Wallace

To cite this version:

J. Berger, J.P. Benoit, C.W. Garland, P.W. Wallace. Ultrasonic study of the normal - incommensurate - commensurate phase transitions in [N(CH3)4]2ZnCl 4. Journal de Physique, 1986, 47 (3), pp.483-489.

�10.1051/jphys:01986004703048300�. �jpa-00210228�

Ultrasonic study ^{of the} normal 2014 incommensurate 2014

commensurate

phase

transitions in [N(CH3)4]2ZnCl4

J.

Berger (*),

J. P. Benoit

(**),

^{C. W. Garland} and P. W. Wallace

(***)

(*) ^DRP (LA71), Université Pierre et Marie Curie, ⁷⁵²³⁰ Paris Cedex 05, ^France (**) LURE, CNRS, Université Paris-Sud, ⁹¹⁴⁰⁵ Orsay Cedex, ^France

(***) Department ^of Chemistry and Center for Materials Science and Engineering,

Massachusetts Institute of Technology, Cambridge 02139, Massachusetts, ^U.S.A.

(Refu ^{le 12} septembre ^1985, accepti le 23 octobre 1985)

Résumé. 2014 Les constantes élastiques c44, c55, ^et c66 d’un crystal ^de

[N(CH3)4]2ZnCl4

^ont été mesurées à 10 MHz

près de la transition de la phase prototype ^{vers la} phase incommensurable à Ti ~ ^23°C puis près de la transition de lock-in (incommensurable-ferroélectrique) ^à Tc ~ ^{8 °C et au} voisinage ^{de la} transition du 1er ordre ferroélectrique- ferroélastique ^à T1 ~ ^{4°C. Une variation} importante de la vitesse et de l’atténuation du mode de cisaillement c66 dans la phase incommensurable peut ^être interprétée qualitativement par ^un couplage solitons-phonons. ^L’atte-

nuation 03B1 associée au mode longitudinal ^{c11 a été étudiée en fonction de la} fréquence (10 ^{à 70} MHz) près ^de Ti.

Dans la phase incommensurable l’atténuation critique ^{est bien} représentée par ^une loi de la forme : 03B1

~ f2(Ti -

T)-0,9.

Abstract ²⁰¹⁴ The shear elastic constants c44, c55, and c66 of single-crystal

[N(CH3)4]2ZnCl4

have been measured at 10 MHz near the normal-incommensurate transition at Ti ~ 23°C, the incommensurate-ferroelectric commen- surate lock-in transition at Tc ~ ^{8°C, and the} first-order ferroelectric-ferroelastic transition at T1 ~ ^{4°C. A}

substantial variation in the velocity and attenuation of the c66-mode ultrasonic shear wave in the incommensurate phase ^{can be} understood qualitatively ^{in terms of} soliton-phonon coupling. ^The longitudinal attenuation 03B1 associat- ed with the c11 mode ^was also studied ^{as a} function of frequency (10-70 MHz) ^near Ti. The critical attenuation in the incommensurate phase ^{could be well} characterized by

03B1 ~ f2(Ti - T)-0.9.

Classification

Physics ^Abstracts

64.70K201362.65201362.80

1. Introduction.

Tetramethylammonium

tetrachlorozincate

is a member of the

isomorphous

^{class of}

compounds [N(CH3)4J2MX4,

^{where M can be} Zn, Cu, Co, Fe, Mn,... and X is Cl or Br. All of these

compounds belong

^to the space group

Dih (Pnam

^or

Pmcn)

ⁱⁿ

their

high-temperature

^normal

(prototype) phase [1].

On

cooling, they

exhibit first an incommensurate

(INC) phase

and then ferroelectric

and/or

ferroelastic

commensurate

phases [2-7].

^The p-T

phase diagrams

have been

extensively

^{studied for these}

compounds [8-13],

^{and a master}

phase diagram showing

^the

sequence of various transitions is

given

ⁱⁿ

figure

^1.

For TMATC-Zn in its normal

phase,

^{we have}

labelled the

crystallographic

^{axes in} agreement ^with international standards

(c

^a

b)

^{so that a =}

12.268

A,

^{b = 15.515}

A,

^{c = 8.964 A and} the space group is Pnam. The normal-incommensurate ^tran-

sition occurs at

Ti --

^{23 OC. Below}

T;,

^{a modulation}

develops along

^the

pseudohexagonal a

^{direction with}

an incommensurate wave vector q rr ^{0.42 a*. At the} transition temperature

Tc

^{8 OC, the}

crystal

^under-

goes ^a lock-in transition into ^a commensurate ferro- electric structure

(Pna21 symmetry) with q

^{= 2}

a*/5.

On further

cooling,

^the

crystal undergoes

^a first-order transition at

T1

^{4 OC into a} commensurate ferro- elastic

phase (space

group

P21/nii) with q = a*/3.

These three transitions have been

extensively

^studied

using

dielectric measurements [2, 3,

14], X-ray

^dif-

fraction

[5,

6,

13],

^neutron diffraction

[7, 15-17],

^heat

capacity

measurements

[ 18],

and theoretical

modelling [ 15,19, 20].

^{There is also a second set of} transitions at

low temperatures

(- 95°C, - 102°C, - 114°C) [ 18].

These involve

ordering

^{among the}

N(CH3)4

^{ions and}

will not concern us in the present ^work.

It should be noted that

[N(CH3)4]2MX4

^com-

pounds

^are

isomorphous

^{with the}

simpler A2MX4 compounds

^like

K2Seo4, Rb2ZnC’4

^and

(NH4)2BeF 4 (AFB),

^{all of} which exhibit incommensurate

phases.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703048300

484

Fig. ^{1. - Universal} phase diagram for TMATC-M com-

pounds (M ⁼ Co, Zn, Mn) using appropriately ^scaled p and T variables; ^{h =} hydrogen compounds ^{and d = deuterated} compound. Zero pressure for each compound is indicated

by ^the position of the labelled vertical line.

However, ^{there are}

significant

differences _among these

compounds,

^and

they

^can be classified into three distinct _groups

[17].

^{The first} group contains com-

pounds that behave like

K2Seo4l

^{for which the}

incommensurate wave vector qinc = 0.7 a* and lock-in

occurs at qcomm ⁼ 2

a*/3

^{in the extended Brillouin}

zone. The second _group

(AFB)

^has qinc close to 0.5 a*

and locks in at qcomm ⁼

a*/2.

^TMATC-Zn

belongs

^to

the third _group, all of which have _{qin, -} 0.42 a* and

a stable commensurate ferroelastic

phase

^with

qcomm ⁼

a* /3.

The

dynamical

^{behaviour of}

A2MX4 crystals

^has

been

extensively

studied. A soft-mode

instability

^has

been found in the

first-group compounds K2SeO4

and

Rb2ZnCl4 [21],

and this has stimulated sub- sequent ^Raman

[22,

23], ultrasonic

[24-26],

^and

Brillouin

[27-29]

studies. The acoustic

investigations

reveal that several elastic constants exhibit anomalies

near Ti ^{with a} strong attenuation of the elastic ^waves at 10 MHz and 20 GHz [24, 25,

29]

^{and that the shear}

constant c55 softens in the INC

phase

^on

approaching Tc [26].

^Thus,

first-group compounds

^{show a} strong

coupling

between the order parameter and elastic

deformations, and the relaxation of the order _para- meter is fast

enough

^to produce critical attenuation at Brillouin

frequencies.

AFB (a second-group

compound)

exhibits order-

disorder type

dynamical

behaviour with no soft mode

instability.

Acoustic studies

[30]

^indicate

velocity

and attenuation anomalies for the _{cl i}

longitudinal

mode similar to those observed in

first-group

^com- pounds. ^A

softening

^{also occurs in the} c55 shear mode,

but in this case c55 softens in the normal

phase

^on

approaching

Ti and remains

essentially

^{constant in}

the INC

phase.

The

dynamical

behaviour of TMATC-Zn, ^which

seems to be a

typical

member of the third group, has been studied

by

^{neutron and Raman}

scattering

[16,17, 31, 32]. ^These

investigations

show that the normal- INC transition at

Ti

^{is not caused}

by

^the

instability

of a

phonon

branch, ^{and no soft mode was observed.}

We have

previously

studied the

longitudinal

^elastic

behaviour in TMATC-Zn from Brillouin

scattering [33]

^{and found no} anomalies in the

hypersonic

^elastic

constants and

only

^a small increase in the

longitudinal

attenuation at - 20 GHz on

cooling

^below

T;.

These results are in

fairly good

agreement ^{with those} of

Karajamaki et

al. for this

compound [34]

^{and for}

TMATC-Cu

[35].

In contrast to the Brillouin measurements, ^an ultrasonic

study

^{at 8 MHz of the}

longitudinal

^modes

in TMATC-Zn

[36]

shows that ci i, c22 and _C33 all exhibit clear anomalies near

Ti

^but

only

very small

changes

^near

Tc

^or T1.

Unusually

^narrow attenuation

peaks

^were also observed at

Ti

for all three modes.

It was not clear whether these attenuation

peaks

^were

due to some type ^of

domain-scattering

^mechanism

or were the result of critical relaxation related to the normal-INC

phase

transition. If the latter mechanism is dominant, the relaxation rate must be

fairly

^slow

in view of the virtual absence of

hypersonic

( ~ ²⁰ GHz)

anomalies at Ti.

No acoustic

investigations

of shear elastic waves

have been carried out in TMATC-Zn. However, ^the

c55 shear mode has been studied in TMATC-Cu for a

sequence of

phase

transitions which differ from those

occurring

in TMATC-Zn

[37].

^{In this} copper ^com-

pound,

c55 softens

dramatically

in the INC

phase

^and

approaches

^{zero at the} INC-ferroelastic transition.

Thus, ^{it seems that} many

A2MX4 compounds

^exhibit

some kind of shear

anomaly [26,

^{30, 37,}

38].

^{In those}

cases where shear

softening

^occurs in the INC

phase,

this can be related to

coupling

between the shear strain and

phasons

^near

T;

^{or solitons near}

Tc [26].

The present ^{work was} undertaken to

explore

^shear-

wave behaviour in TMATC-Zn and to

clarify

^the

longitudinal

attenuation near Ti. The shear elastic constants c44, css and C66 have been measured at 10 MHz over a temperature range that _spans the three transitions at

Ti,

T, ^and

T;.

The directions of the

wave vector k and the

polarization

^{vector e for these}

shear waves were k //c and e //b for _c44, k //c and

e // a for C55, k // a and ^{e // b} for _c66. The attenuation of the _Cll

longitudinal

mode associated with the normal-INC transition was measured as a function of temperature ^{at several}

frequencies

^{in the} range 10-70 MHz.

2.

Experimental

procedure.

Large single crystals

of TMATC-Zn were obtained

by

^slow

evaporation

^{of a} saturated solution at 40 OC.

The specimens ^were colourless, transparent, ^{and free} from _any visible defects. The ^two

samples

^{used for}

acoustic measurements were

parallelepipeds

^with

edges parallel

^{to the} a, b, ^c axes, and their

respective

sizes were 14.22 x 10.51 ^x 14.93 mm’ and 17.28 ^x 15.66 x 16.31 mm’ at room temperature. ^The

density

of TMATC-Zn is 1.387 _g cm-3.

Ultrasonic

velocity

^{values at room} temperature

were determined with the

pulse-superposition

^tech-

nique [39]. Velocity changes

^{as a} function of tempera-

ture and attenuation values were obtained

using

^a

MATEC MBS 8000 system, ^which

provides

^{for the}

coherent

phase-detection

^{of two echo}

pulses [40].

The

signal

^{from a CW} oscillator is divided into two channels : one

provides

the reference

signal

^{and the}

other contains a

gated amplifier

^to

produce

^a

high- voltage

^RF

pulse

which excites the transducer affixed to the

sample.

^Two

phase-detected

output

signals (in-phase

^and

out-of-phase components)

^{are obtained}

for each of the two echo

pulses.

^Four

sample-and-hold

devices are used to measure the

phase-detected

outputs, and a

multiplexed

^14-bit

A/D

^converter

provides digital

output ^{to a} Hewlett-Packard 9845 micro- computer, ^which averages 100

readings

^{and then}

calculates the attenuation and

velocity. Fairly large samples

^{must be} used in order to obtain

well-separated

echo

pulses.

^{As a} result, ^the

longitudinal

attenuation is

quite high

^near

Ti. Fortunately,

^the

phase-detection

method

provides good signal-to-noise

ratios for weak

signals ( ~ 55

^dB

dynamic range).

Longitudinal

^and transverse quartz transducers

were used at their fundamental resonance

frequency

of 10 MHz and at odd harmonics. These transducers

were bonded to the

sample

^{with a} very thin

layer

of

Dow-Coming

276 VA resin. The temperature ^of the

sample

^{holder was}

electronically

controlled, ^and

the

sample

temperature ^{was measured to within 2 mK}

using

^a calibrated Rosemount

platinum

resistor and

a Leeds and

Northrup

^K5

potentiometer.

^{The rate of} temperature

change

^was very slow

(typically -

^{2 K h-1}

far from _any transition and - 0.04 Kh-1 near a

transition). ^When

taking

^{data near a} transition, ^the temperature controller was

adjusted

ⁱⁿ steps ^of

~ 20 mK with a wait of 30 to 60 min between steps

to allow for

equilibration prior

to measurement.

Data obtained on

heating

^and

cooling

^{were in} very excellent agreement for all the elastic waves studied.

3.

Experimental

^results.

The temperature

dependence

of the shear velocities

corresponding

^to the c44, C55 and c66 modes are shown in

figures

^{2 and 3. Both the} c44 and _{cs s} modes show

only

^{a small}

discontinuity

ⁱⁿ

slope

^at

Ti,

whereas the C66 mode

undergoes

^{a marked}

change

ⁱⁿ temperature

dependence

^at

T;.

^On

cooling

^{the INC}

phase

^between

Fig. ^{2. -} Temperature dependence ^{of two} ultrasonic shear velocities in TMATC-Zn at 10 MHz : (a) Vi =

(C44/P)1/2

and (b) V2 ⁼

(CSS/p)1/2.

These data were obtained with a

conventional pulse-echo technique.

Fig. ^{3. -} Temperature dependence of the ultrasonic velo-

city V3 ⁼

(C66/P)1/2

in TMATC-Zn at 10 MHz.

T; and the lock-in transition at Tc, C66 decreases

monotonically by -

³⁵

%

^while c44 and _C55 show much smaller

changes. Figure

⁴ shows that there is a

significant

increase in the attenuation a of the _c66 shear wave as the INC

phase

^{is cooled, and then a}

dramatic

drop

^{in a occurs}

when q

^{locks-in at} Tr.

In the ferroelectric commensurate

phase

^between T 1

and

Tc,

^{both the} C66 and _c44 waves show substantial attenuation which is

perhaps

^{due to} the appearance of ferroelectric domains.

486

Fig. ^{4. -} Attenuation of 10-MHz c66 shear ^{waves versus} temperature.

The variation near

T;

^{of the} c11 mode

longitudinal

wave

velocity

^{and the} associated attenuation at 10 MHz are shown in

figure

^5. Similar but smaller variations occur for the C22 and c33 modes

[36],

^but

these modes were not

investigated

in this work. It should be noted that the attenuation maximum occurs at a

higher

temperature ^{than the}

velocity

minimum,

a feature that has been observed

previously

^for

longitudinal

^{modes near the} commensurate-incom-

mensurate

phase

transition in

K2Seo4 [23]

^and

SC(NH2)2 [41].

^The attenuation a of the _{cl 1}

longi-

tudinal mode was also measured at 30 MHz, ^{50 MHz} and 70 MHz as a function of temperature ^{both above} and below

T ;. Unfortunately,

^{it was not}

possible

^to

measure attenuation values _very close to

T;

^{at these}

frequencies

^{since a became so}

large

^{that the echo}

signals

^were lost in the noise.

The

longitudinal

attenuation in the normal

phase

exhibits rather erratic behaviour as a function of tem-

perature ^and

frequency.

^{As indicated in}

figure

5,

appreciable

attenuation at 10 MHz is observed

only

over a very ^narrow temperature range. The

a(IO)

values are very small

(~

^0.25

dB/cm)

^{above 23.4 °C.}

At

higher frequencies, significant

attenuation values could be measured over a wide temperature range.

Between - 24.5 OC and 40 OC, ^{these values are}

independent

^of temperature and exhibit no

systematic frequency dependence : a(30) =

^2.0

dB/cm, a(50) =

1.6

dB/cm,

^and

a(70) =

^3.0

dB/cm

in this range. On

cooling

below - 24.5 OC, ^the

high-frequency

^atte-

nuation increases

rapidly

^{to about 14}

dB/cm (at

- 23.3 °C for 30 MHz and 50 MHz and - 23.5 °C for 70

MHz),

^{the echo} pattern then becomes nonexpo-

nential, ^{and the echo}

signals subsequently disappear.

Fig. ^5. - Temperature dependence ^{of the} velocity V. =

(Clllp)112

^{of 10-MHz} longitudinal ^waves propagating along [100] ^{and the} corresponding attenuation a.

In the INC

phase,

^the

longitudinal

attenuation varies

quadratically

^with

frequency

^{and exhibits a} strong temperature

dependence

^{over the entire inves-}

tigated

range. A

plot

^{of a}

versus f at

^{various constant}

temperatures ^below

Ti

^is

given

ⁱⁿ

figure

^{6. These a}

values represent smooth-curve values based on a

large

number of attenuation measurements made as a

Fig. ^{6. - Plot of the} ultrasonic attenuation a versus

f2

^for longitudinal ^waves propagating along [100] ^{at various cons-}

tant temperatures ^{in the} incommensurate phase.

function of temperature ^{at each}

frequency.

^{All the}

isotherms conform _very well with an

f dependence

except ^{for some} scatter at 21.5 °C. At temperatures above 21.5 OC, reliable attenuation data could be obtained

only

^{at 10 MHz}

(up

^{to 23.03} OC) ^{and 30 MHz}

(up

^{to 22.82} OC). ^Note also that all the lines in

figure

⁶

have a common

intercept

^{at zero;} thus there is no

frequency-independent

^«

background »

attenuation _ao in the INC

phase.

4. Discussion.

The elastic behaviour of TMATC-Zn shows some

analogies

^to that observed

previously

^{for TMATC-}

Cu

[4,

35]. The rounded step ^{observed near}

T;

^{for the}

longitudinal

^constants cl l, c22 and _c33 can be described in Landau

theory

^{as due to a}

coupling

^{between elastic}

strain S and an

appropriate

^order parameter

Q

^that

gives

^{rise to a}

SQ2

^term in the free energy. It is well known that this type ^of

coupling

^{term is}

responsible

^for

a

steplike

variation in the related elastic constant

[41-43].

No

hysteresis

^was observed in our measurements in contrast to the substantial

hysteresis reported

^for

ultrasonic measurements on TMATC-Cu

[4].

^How-

ever,

Sugiyama et

^al.

[4]

^{used a}

heating

^and

cooling

^rate

of about 0.1

K/min,

^{and their}

hysteresis

^{may very well}

be due to remnant domain structure associated with

rapid

temperature ^{scans. The} very slow

changes

ⁱⁿ

temperature used in the present work have eliminated this

problem.

It should be noted that in order to discuss and com-

pare the elastic behaviour of various isomorphous

A2MX4 compounds

^{one must be alert to the way}

in which the

crystallographic

^{axes are labelled. We}

have used the international convention c a b ;

thus the space group is Pnam and a is the

pseudohexa- gonal

^axis

along

which the incommensurate wave

develops.

^{Some other} authors have chosen a c b

so that c will be the

pseudohexagonal

axis; ^{in this case}

the space group is Pmcn. Indeed, ^{this latter convention}

is used for the acoustic

study

of TMATC-Zn in references

[2

^and 36]. ^{As a} result, ^{our elastic constant}

cl 1 becomes _C33 in the Pmcn notation. In the same

way, ^our shear constants c44 and _c66 become _c66 and _C44,

respectively,

^{in the Pmcn} notation. The elastic constant C55 is unaf’ected

by

^a

change

^from

Pnam to Pmcn notation.

We shall turn now to a discussion and

analysis

^of

the

longitudinal

attenuation near

T;

^{for the} Cn mode.

There can be two contributions to the critical atte- nuation of this wave

[38].

^{One is} associated with

energy-density

fluctuations and can contribute both above and below T; ; the other is associated with

order-parameter

relaxation

(the

Landau-Khalatnikov

mechanism)

^and contributes

only

^below

Ti.

^{Our data}

above T; ^{are not} suitable for a

quantitative analysis

but

they

do indicate that _any fluctuation contribution is

significant

only ^at

higher frequencies

^{when T -}

Ti

is small

(AT

^{1.5 K} for 50 and 70 MHz). ^{In any}

event, all the attenuation values below

Ti

^{shown in}

figure

^{6 are}

appreciably larger

than those at the same

I

AT above T ;.

^{Thus, we shall}

neglect

^the fluctuation contribution and assume that the a values in the INC

phase

^can be described

by

^the

single

relaxation expres- sion :

The relaxation

strength A

⁼

(V2 _ Yo)/2 V30

^will

be a

slowly varying

function of temperature, ^{and the} relaxation time r

characterizing

^order parameter relaxation is

expected

^to show conventional critical behaviour

[44] :

Since a is

proportional

^{to úJ2} for all of our data in the INC

phase (with

^the

possible exception

^{of 10 MHz}

values _very close to

Ti),

the condition úJ2

r2

¹

must hold and

equation (1)

^{can be}

simplified

^to

We shall write this in the

general

^form

where B is a constant and _p is

expected

^{to be close to 1.}

A

log-log plot

^of

rxl12 versus I AT

I ⁼ Ti - ^{T is}

show in

figure

^{7. We} have chosen

Ti

⁼ 23.07 °C,

which is the temperature ^{at which a} (10

MHz)

^attains

its maximum value.

Figure

7 combines information obtained from the

slopes

ⁱⁿ

figure

^{6 and} direct values of

Lx/f’

^{measured at} 10 and 30 MHz near

Ti.

^{The latter}

Fig. ^{7. - Plot of} a/f2, in units of 10-14 dB cm-1 Hz-2, versus AT ) I = Ti - ^{T for} [100] longitudinal ^{waves. The} points ^denoted by ⁺ represent ^the slopes ^{of the} straight ^lines

shown in figure ^6.

a/f2

points obtained from direct measure- ments at a single frequency ^are also shown close to T; ^for

10 MHz (0) and 30 MHz (·). The horizontal error bars indicate the effect of changing T; by ^{0.01 OC.}

488

can be used without _any correction since the « back-

ground

attenuation » _a° has been shown to be

negli- gible.

The best-fit line drawn in

figure

⁷ corresponds

to

equation

(4) ^with p ⁼ 0.9 and B ⁼ 0.56 ^x 10-14 dB cm-1 Hz-2 KO.9. None of the

points

^near

Ti

^deviate

systematically

from the line in a way that suggests ^{a breakdown in the} W2

T2

¹ condition.

Such deviations could

only

be created

by choosing

^a T; value below 23.07 °C, and that would have the undesirable effect of

putting

the maximum in

a (10

MHz)

^above

T;.

^The significance ^{of a}

dynamical

exponent p ⁼ 0.9 less than 1 is not clear, but compa- rable values have also been observed below the order- disorder critical

point

in ammonium chloride

[45].

Finally,

^{let us comment on} the dramatic behaviour of the _c66 shear mode. The shear

velocity

^decreases

substantially

^on

cooling

^between

Ti

^and

Tc

^and

then become almost constant below Tc

(see Fig.

3).

The associated attenuation increases to a very

high

value in the INC

phase

^before

dropping

^almost

discontinuously

^{to a low value at the} lock-in transition

Tc

(see Fig.

4). There is also a second attenuation

peak

in the middle of the commensurate ferroelectric

phase.

The latter attenuation _may be related to the _presence of

antiphase

domains, which have been observed

recently

ⁱⁿ

high-quality

^TMATC-Zn

crystals by

X-ray

topography [46].

Since domains ^seem to form very

easily

in this material

they

may have a

major

influence on the elastic

properties,

^{as has} been observ- ed in other materials

[47].

An

interesting comparison

^can be made between the behaviour of c66 in the INC

phase

of TMATC-Zn and that of _c55 in

K2SeO4 [26].

^{In both cases the}

incommensurate wavevector locks-in at

Tc

^to

give

^a

ferroelectric commensurate phase

(although

^the

qcomm values are

different).

Each material exhibits a

comparable

decrease in shear stiffness between

T;

and

Te

^{and a}

large asymmetric

attenuation

peak

^near T c. ^{In the INC}

phase

^{close to}

T;,

the modulation of the order parameter is sinusoidal. This

plane-wave regime

is characterized in terms of

amplitudons

^and

phasons,

and anomalous elastic behaviour can be described

as a result of

phason-phonon

interactions

[25].

^How-

ever, ^as the

crystal

^{is cooled toward}

T c

the modulation

changes

^{into a} sequence of commensurate

regions

^with

alternating positive

^and

negative

^strains

separated by

discommensurations. In this soliton

regime,

the elastic shear anomalies

of K2SeO4

^{arise due to a} fourth-order

S5Q’ coupling

^term in the free _energy

[26]. Approa- ching

the lock-in transition from above, the distance between solitons _goes to

infinity,

the shear stiffness decreases to a minimum value at

Tp,

and the associated attenuation increases. Thus it seems that the _c66

velocity

and attenuation behaviour we have observed in TMATC-Zn can be

qualitatively

^described

by

Rehwald’s

phonon-soliton

interaction model

[26].

However, it should be stressed that the

higher

^order

elastic ^-

order-parameter coupling

^{terms in TMATC-}

Zn must differ from those in the

isomorphous K2Se04

since the critical mode

softening

^{occurs in} C66 rather than in c5 s.

In summary, the shear elastic constants c44, cs s, and C66 of TMATC-Zn have been measured ^near the normal-incommensurate transition at

T;,

^{the incom-}

mensurate-ferroelectric commensurate lock-in tran- sition at Tc, ^{and the} commensurate ferroelectric- ferroelastic first-order transition at T1. ^A substantial variation in the

velocity

and attenuation of the _c66 mode in the INC

phase

^near

Tc

^can be ascribed to

soliton-phonon

interaction. The

[100] longitudinal

attenuation associated with _cll was studied as a

function of temperature ^and

frequency.

^{In the INC}

phase

^near

Ti,

^critical relaxation behaviour was

observed with

cx - f(Ti - T)- 0.9 Acknowledgments.

This work was

supported

ⁱⁿ part

by

the National Science Foundation under grant CHE 84-00982 and in part

by

^{a NATO} grant ^{to one} of the authors

(J.

B.)

during

^{a visit at MIT. We wish to} thank J. P.

Chapelle,

E. Guiot, and N. Lenain for

providing

^{us with}

high- quality

^oriented

single crystals

of TMATC-Zn. The current address of P. Wallace is E. I. Du Pont de Nemours and Co., Inc., P. 0. Box 13999, Research

Triangle

Park, N. C. 27709.

References

[1] MOROSIN, ^{B. and} LINGAFELTER, ^E. C., ^Acta Cryst. ¹² (1959) 611.

[2] SAWADA, S., SHIROISHI, Y., YAMAMOTO, A., TAKASHIGE,

M. and MATSUO, M., ^J. Phys. ^Soc. Japan ⁴⁴ (1978)

687.

[3] SAWADA, S., SHIROISHI, Y., YAMAMOTO, A., TAKASHIGE,

M. and MATSUO, M., Phys. ^{Lett. 67A} (1978) ^56.

[4] SUGIYAMA, J., WADA, M., SAWADA, ^{A. and} ISHIBASHI, Y., ^J. Phys. ^Soc. Japan ⁴⁹ (1980) ^1405.

[5] TANISAKI, ^{S. and} MASHIYAMA, H., ^J. Phys. ^Soc. Japan

48 (1980) ^339.

[6] MASHIYAMA, ^{H. and} TANISAKI, S., Phys. ^{Lett. 76A} (1980)

347.

[7] GESI, ^{K. and} IIZUMI, M., ^J. Phys. ^Soc. Japan ⁴⁸ (1980)

337.

[8] SHIMIZU, H., OGURI, A., ABE, N., YASUDA, N., FUJI- MOTO, S., SAWADA, ^{S. and} TAKASHIGE, M., ^Solid

State Commun. 29 (1979) ^125.

[9] SHIMIZU, H., KOKUBO, N., YASUDA, ^{N. and} FUJIMOTO, S., ^J. Phys. ^Soc. Japan ⁴⁹ (1980) ^223.

[10] SHIMIZU, H., ABE, N., KOKUBO, N., YASUDA, N., FUJIMOTO, S., YAMAGUCHI, ^{T. and} SAWADA, S.,

Solid State Commun. 34 (1980) ^363.

[11] MASHIYAMA, H., TANISAKI, ^{S. and} GESI, K., ^J. Phys.

Soc. Japan ⁵⁰ (1981) ^1415.

[12] GESI, K., ^J. Phys. ^Soc. Japan ⁵¹ (1982) ^1043.

[13] HASERE, K., MASHIYAMA, H., TANISAKI, ^{S. and} GESI, K., J. Phys. ^Soc. Japan ⁵³ (1984) ^1863.

[14] GESI, ^{K., J.} Phys. ^Soc. Japan 51, ²⁵³² (1982).

[15] PARLINSKI, ^{K. and} DENOYER, F., ^{to be} published.

[16] MARION, G., ALMAIRAC, R., LEFEBVRE, ^{J. and} RIBET, M., J. Phys. ^{C :} Solid State 14 (1981) ^3177.

[17] IIZUMI, ^{M. and} GESI, K., Physica ^120B (1983) ^291.

[18] ^RUIZ LARREA, I., LOPEZ-ECHARRI, ^{A. and} TELLO, ^M. J., J. Phys. C : Solid State 14 (1981) ^3171.

[19] MASHIYAMA, H., ^J. Phys. ^Soc. Japan ⁴⁹ (1980) ^2270.

[20] ISHIBASHI, Y., Ferroelectrics 35 (1980) ^111.

[21] IIZUMI, M., AxE, J. D. and SHIRANE, G., Phys. ^Rev.

B 15 (1977) ^4392.

[22] WADA, M., UWE, H., SAWADA, A., ISHIBASHI, Y., TAKAGI, ^{Y. and} SAKUDO, T., ^J. Phys. ^Soc. Japan 43 (1977) 544.

[23] UNRUH, ^H. G., ELLER, ^{W. and} KIRF, G., Phys. ^Status

Solidi 55a (1979) ^173.

[24] REHWALD, W., VONLANTHEN, A., KRUGER, ^J. K., WALLERIUS, ^{R. and} UNRUH, H.-G., ^J. Phys. ^C:

Solid State 13 (1981) ^3823.

[25] REHWALD, ^{W. and} VONLANTHEN, A., Solid State Commun. 38 (1981) ^209.

[26] REHWALD, W., in Phonon scattering ^{in condensed}

matter (Proceedings of the fourth international conference at Stuttgart), Solid State Sci. 51 (1984)

295.

[27] HAURET, ^{G. and} BENOIT, J. P., Ferroelectrics 40 (1982)

1.

[28] LUSPIN, Y., HAURET, G., ROBINET, A. M. and BENOIT,

J. P., J. Phys. ^{C :} Solid State 17 (1984) ^2203.

[29] BENOIT, ^J. P., HAURET, G., LUSPIN, Y., GILLET, ^{A. M.}

and BERGER, J., Solid State Commun. (in press).

[30] ALEKSANDROV, ^K. S., ANISTRATOV, ^A. J., KRUP- NYI, A. I., MARTINOV, ^O. G., POPKOV, ^{Y. A. and} FUMIN, ^V. I., ^Sov. Phys. Crystal. ²¹ (1976) 296 ; HIKITA, T., TSUKAHARA, ^{T. and} IKEDA, T., ^J. Phys.

Soc. Japan ⁵¹ (1982) ^2900.

[31] ^MARION, G., ALMAIRAC, R., RIBET, M., STEIGENBERGER, U., VETTIER, C. and SAINT GREGOIRE, P., ^Ferro-

electrics 53 (1984) ^277.

[32] SRIMIVASAN, V., SUBRAMANIAN, ^{C. K. and} NARAYANAN,

P. S., Indian J. Pure Appl. Phys. ²¹ (1983) ^271.

[33] BERGER, ^{J. and} BENOIT, ^J. P., Solid State Commun.

49 (1984) ^541.

[34] KARAJAMAKI, E., LAIHO, ^{R. and} LEVOLA, T., ^J. Phys.

C : Solid State 16 (1983) ^6531.

[35] KARAJAMAKI, E., LAIHO, R., LAKKISTO, ^{M. and} LEVOLA, T., Phys. Scripta ²⁵ (1982) ^697.

[36] HOSHIZAKI, H., SAWADA, ^{A. and} ISHIBASHI, Y., ^J. Phys.

Soc. Japan ⁴⁷ (1979) ^341.

[37] SAWADA, A., SUGIYAMA, J., WADA, ^{M. and} ISHIBASHI, Y,

J. Phys. ^Soc. Japan ⁴⁹ (1980) ^89.

[38] HIROTSU, S., TOYOTA, ^{K. and} HAMANO, K., ^Ferroelec-

trics 36 (1981) ^319.

[39] PAPADAKIS, ^E. P., Physical ^Acoustics XII, eds. W. Ma-

son and R. Thurston (Academic, ^New York) 1976,

p. 277.

[40] WALLACE, ^{P. and} GARLAND, ^C. W., unpublished.

[41] BENOIT, ^J. P., THOMAS, ^{F. and} BERGER, J., ^J. Physique ⁴⁴ (1983) 841.

[42] REHWALD, Z., ^Adv. Phys. ²² (1973) ^721.

[43] BRUCE, ^{A. D. and} COWLEY, ^R. A., ^Adv. Phys. ²⁹ (1980)

111.

[44] For conventional van Hove theory, 03C4~03BEz~ | 0394T|- zv

with z ⁼ 2 and 03BD ⁼ 1 /2.

[45] LEUNG, ^R. C., ZAHRADNIK, ^{C. and} GARLAND, ^C. W., Phys. ^{Rev. B 19} (1979) ^2612.

[46] LEFAUCHEUX, ^{F. et} al., ^{to be} published.

[47] JANOVEC, V., Ferroelectrics 35 (1981) 105;

AULD, ^{B. A. and} FEDER, ^M. M., Ferroelectrics 38

(1981) 931.