Study of RbDP phase transition in the paraelectric (upper) phase by phonon echoes

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Study of RbDP phase transition in the paraelectric (upper) phase by phonon echoes

A. Billmann, G. Guillot

To cite this version:

A. Billmann, G. Guillot. Study of RbDP phase transition in the paraelectric (upper) phase by phonon echoes. Journal de Physique, 1978, 39 (12), pp.1317-1321. �10.1051/jphys:0197800390120131700�.

�jpa-00208874�

STUDY OF RbDP PHASE TRANSITION

IN THE PARAELECTRIC (UPPER) ^{PHASE BY PHONON ECHOES}

A. BILLMANN and G. GUILLOT

Laboratoire d’Ultrasons (*), Université Pierre-et-Marie-Curie, Tour 13, 4 place Jussieu, ^{75230 Paris Cedex} 05, ^France

(Reçu ^{le 22 mars} 1978, révisé les 19 mai et 17 juillet 1978, accepté ^{le 23 août} 1978)

Résumé.

Nous avons étudié la transition de phase ^du RbH2PO4, ^{en mesurant le} temps ^{de vie} T2

et l’amplitude A ^du signal ^{d’écho de} phonons ^{à 2 03C4, dit} dynamique, ^{dans des} poudres résonnantes.

L’atténuation suit la loi de variation obtenue par d’autres techniques ultrasonores dans le même type ^de composés. L’allure de A(T) ^{donne de} plus quelques indications sur le comportement critique

des non-linéarités.

Abstract.

A ferroelectric phase transition (RbH2PO4) ^{has been} investigated by measuring

the damping ^constant T-12(T) ^{and the} amplitude A(T) ^{of the} dynamic ² 03C4-phonon ^{echoes in resonant} powders. Our results on the attenuation confirm those obtained with other ultrasonic methods

on the same type ^of compounds. ^{We also} get from the behaviour of A(T) ^more information about non-linear effects near the transition.

Classification Physics ^Abstracts

62.25201364.70D201377.80B

Introduction. - The phonon echo method is a

basic acoustic technique, ^which gives ^results compa- rable to those obtained by usual ultrasonic techniques.

With it, ^the study of surface defects [1] ^{or of acoustic} properties ^{of some breakable} [2] ^{or microsize} crystals [3] is much easier.

We have studied the phase transition of RbH2P04 by ^{means of} phonon echoes. The static properties ^of

this compound ^{are known to be very much the same}

as those of KDP (§ 1), ^{but its} dynamic properties ^had

not been investigated ^till recently [4]. ^{We will} shortly present ^{the 2} t-phonon ^{echo in} powders (§ 2), ^and interpret ^{our results on} the lifetime T2 (§ 3) ^and amplitude (§ 4) considering ^our specific ^{case of reso-}

nant powders.

1. Static properties ^{of RbDP :} analogies ^{with KDP.}

-

The rubidium diacid phosphate (RbDP) ^{has similar}

structural and critical properties ^to KDP, ^{a well}

known ferroelectric crystal. ^{Its inverse} longitudinal

dielectric susceptiblity, ^{measured at constant stress} X, varies linearly above the transition temperature Tc ~ ^{145 K} (122 ^{K for} KDP), ^{in the} upper 4 2 m

paraelectric ^and piezoelectric ^{phase :} X "DIx = (T - Tc)/C [5, 6]. ^{The lower} phase ^{has the 2 mm} symmetry ^and is ferroelectric for these compounds.

The elastic constants of RbDP have been measured

through ^the transition [7], ^{and have the same charac-}

teristic behaviour as the constants of KDP [5, 8] ⁱⁿ

the upper paraelectric phase. C66 (measured ^{with E} kept constant) ^{vanishes at} the transition temperature, whereas CP66 (measured ^{with P} kept constant) ^remains

constant as well as all the other elastic constants.

These variations are fully explained ^{if in the} expres- sion for the Landau free _energy ço(q), ^{the order} para- meter ~ is identified with P 3 (third component ^{of the} spontaneous polarization) ^{and if the} coupling ^between P 3 ^and only ^one elastic mode S6 ^{via the} piezoelectric

modulus _a36 is considered [9]. Other elastic modes

cannot be coupled ^to P3 because of the zero values of the relevant piezoelectric moduli in _{the upper}

phase,

The dynamic ^elastic properties ^{of KDP had been}

well investigated ^{near the} transition [8 and references therein, 3]. Our results on RbDP will confirm the

analogy ^{with KDP in the} paraelectric phase ^{near the}

transition.

(*) Associated with the Centre National de la Recherche Scien-

tifique.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197800390120131700

1318

2. 2 T-phonon ^{écho in} piezoelectric ^média.

^The

echo phenomena ⁱⁿ piezoelectric crystals ^{has been} widely studied for some years, either in single crystals [10, 11] ^{or in} powders [12-14]. ^{It is} essentially ^the

re-radiation of a coherent signal ^{at time 2 r} by ^a sample ^{which was excited} by ^two radio-frequency pulses of electric field at times 0 and -r. T2 is the life- time of this signal when s varies and is directly ^related

to an acoustical lifetime [15].

We measure the 2 T-echo decay ^constant T2, ^and

the echo amplitude A (extrapolation ^{at s}

^{0 of the} amplitude A(,r», ^as functions of the temperature T, from 165 K to 145 K (Fig. 1), ^{in RbDP} powders (size range : 20-40 03BC) ^{at 110 MHz. The} powder ^{is in}

a capacitor, part ^{of a matched} electrical circuit, ^under

a low pressure of _gaseous helium.

FIG. 1.

2-r dynamic echo. Lifetime (T2(+)) ^and amplitude (A (0)) ^{measured in RbDP} powders. ^The continuous curve repre- sents the expected theoretical law for T2(T), ^{where the} parameters

T2P1,

^{and Tc have} been fitted to the values : T2pl ^{= 15.2 ps ;} Tc = ^{143.5 K.}

3. Study ^{of the} damping ^constant T2 ’(T).

3. 1 CONTRIBUTIONS TO T1- l(T). ^{- The} attenuation processes involved in T2- ¹ have three main origins : a) ^The acoustic transmission loss to the external medium must not show _any variation with the tem-

perature [3].

03B2) ^The internal attenuation processes, intrinsic ^to the properties of the solid medium (such ^{as diffusion} by defects, impurities, ^{or thermal} phonons) may be

supposed ^to give ^a contribution constant with the temperature ^{over the} temperature range ; and the

coupling between the order parameter P3 ^{and the}

elastic mode S6 ^{is the} only ^source of critical attenuation

(which ^we will examine in detail further _on, in § 3.2).

y), ^{The third} process is due to mode conversion and is specific ^{to the} experimental condition of microsize

resonant powders : ⁱⁿ every particle, ^{there is a complex} stationary ^vibration mode, ^{which is a} mixture of _pure

resonance modes with different damping ^constants

and different non-linear efficiencies ^to build the 2 T-

echo. Every particle ^has already ^a complex damping

constant related to the _pure mode damping ^constants.

Further, ^{we measure an} average value of T2- ^{1 taken}

on all the particles. Consequently, ^{the critical} part ^we

extract from the measured attenuation should not

coincide numerically with the theoretical critical atte-

nuation of the S6 ^mode.

3.2 THEORETICAL CRITICAL ATTENUATION.

Gar- land [8] ^{made a} thorough study ^{of the critical attenua-}

tion in the _upper phase ^{of KDP. But} he worked with

single crystals ^{of much} larger sizes than the half acoustic wavelength ^{at his} frequency (at ¹⁵ MHz, Â/2 - ¹⁵⁰ 03BC) : ^the crystal ^was clamped by ^{the use of}

a frequency ^much higher ^{than the acoustic resonance} frequency, ^or the surface of the crystal ^{could not} follow ^the excitation.

This is different from our case : we have studied

mechanically ^resonant powders, ^{and we are in the}

case of a free crystal.

In a clamped crystal, Garland found a critical atte- nuation of the mode S6 at constant strain (S) :

where u is the ultrasonic velocity ^{of the} mode S6 (pu’

CE66) ^and rs is ^the polarization ^relaxation

time at constant strain. The Landau model gives ^this

relaxation time for the order parameter ^{above the} transition température ; it is related ^to the suscepti- bility ^{and to} the kinetic coefficient L describing ^the

order parameter fluctuations by Ts

1/2 LX331s. ^Some

thermodynamic considerations [9, 3] give :

with

and

But in our case of the free crystal, ^{we must} suppose

we measure the critical attenuation of the mode S6

at constant stress (X) :

where

So :

3.3 EXPERIMENTAL RESULT.

We extract from

T2- ’(T) ^the part ^{which is constant with} temperature

over the temperature range T2-1plateau, ^{and keep only}

the temperature dependent part of the attenuation

8i l(T) :

As we expect the variation of 021(T) ^{to result} only

from the coupling between the mode S6 and the order parameter P3, ^we try ^{to describe} 02(T) ^{with a law :}

The experimental ^{data fit} quite well with such ^a law (Fig. 1). ^{We find :}

3.4 DISCUSSION.

Thus we estimate Tc ^{from our}

measurement of T2(T) ; ^we find 143.5 K. From the

literature, the value of T, ^should vary from one

sample ^{to another.}

For acoustic studies of RbDP, ^{Pierre et al.} [7] ^found Tc ~ ¹⁴⁷ K, ^{and in a recent} paper, Singh ^{and Basu} [4]

found in two different samples Tc = (145.2 ^± 0.1) ^K

and (143.5 ± 0.1) K, studying ^then particularly ^from

the lower phase. ^A numerical evaluation of Mlh. ^can

be made with the results on the elastic and piezo-

electric constants of RbDP, ^obtained by ^{Pierre et}

al. [7] ^{and on the order} parameter relaxation rate

made in KDP by ^Garland [8].

With

we find

The theoretical temperature dependence ^is experi- mentally ^well found but the measured temperature

dependent relaxation time is higher than the theore- tical one of the mode S6 [8] by ^{about two orders of} magnitude. ^We attribute this discrepancy ^{to the}

effect of mode conversion.

Comparable ^{results were obtained in the} study ^of

the K.D.P. crystal by phonon ^{echoes in the same} phase : ^the temperature dependence ^{found was}

correct, but the measured temperature dependent

relaxation time was too high by ^{two orders of} magni-

tude [3].

4. Study ^{of the écho} amplitude A(T).

^{The echo} amplitude ^{can also} give information on some thermo-

dynamic coefficients playing ^{a part in its formation.}

But this amplitude ^is only ^an average value on all

the modes, ^the anisotropy ^{effects are not measurable} by ^{such a method.}

4.1 TWO MODELS, TWO INTERPRETATIONS OF A(T).

- We will first summarize the features of two main models for the 2 T-echo with which we could interpret

the variation of A(T).

a) ^{First model.}

The first model was elaborated for phonon ^{echoes in} single crystals [10] ^{and is} easily

also applied ^{to the case of} powders ^{where the} particle

sizes are much larger than the acoustic half wave-

length. ^{A first} pulse ^of frequency ^v generates phonons

at the surfaces of the crystals ^{at time 0.} They propagate freely ^{in the} crystals till the instant r of application

of a second pulse ^{at the same} frequency. They ^are

then reversed by ^the double harmonic of the electric field, during ^the application ^{of the second} pulse : ^the

free evolution of their phases afterwards is the _oppo- site to their evolution between 0 and T. At time 2 T has been achieved a complete phase reversal and the initial coherent state of phase is recovered. As T increases the echo amplitude should decrease as

e

Such a model gives generally ^a good ^{account of}

the experimental observations in the out of resonance cases.

foi) Second model.

New characteristics of echoes in resonant powders ^cannot be understood with such

an interpretation. ^We restrict here ourselves to the so-called dynamic ^{2 T-echo} [12, 14]. ^Its amplitude

varies no longer simply exponentially ^{with T, as} expected ^{from a} signal ^{due to a} parametric interaction of the type ^above suggested. ^The signal ^{observed in}

resonant powders ^first increases with T instead of

being ^{maximum for small -r; then it reaches a maximum} at Tmax ^{and decreases as} e - 2t/T2 for large T(r ~ ² !max).

This behaviour had been previously ^{seen in} single crystals ^{of CdS at 9 GHz and 4} K, ^for very high amplitudes ^of the excitation fields [10].

Therefore, ^{when the} excitation fields are stronger

as in CdS single crystals ^{or more efficient} (as ^{in reso-}

nant powders) ^{a new} type ^of non-linearity ^{must be}

invoked. This process should permit ^a build-up

period ^{of the echo} amplitude for small ^T. Any type of anharmonic interaction between the excited modes

can reverse the phase conditions of the modes _pro- duced by ^{the first} pulse, mixing ^{them with the modes}

issued of the second pulse ^{from the time T.} So, ^a coherent signal ^can appear ^at time 2 r ; ^its amplitude

is roughly ^described [12] by ^{a law :} e - 2t/T2( 1 - e - 2t/T 2),

which shows a build-up period ^{for small r.}

y) ^{RbDP case : short} T2.

^When T2, measuring

the average mode lifetime, ^{is short} enough, the initial increase of the echo amplitude ^for small T is not

visible. In the limit T > Irmax (Tmax ~ ^0.2 T2), a para- metric process ^{or an} anharmonic interaction would both give ^{an echo} amplitude proportional ^{to e- 2t/T2.}

For our experiments, T2 ^{is too short to choose bet-}

ween the two models. But the relevant non-linear

1320

coefficients would not be the same for these two types of echoes. We shall present ^{both cases with the res-}

pective relevant coefficients.

4.2 INVOLVED LINEAR AND NON-LINEAR COEFFI- CIENTS.

a) ^The electromechanical coupling ^factor k2(T)

e2(T)/CE(T) XI(T) [5] (where ^{e is the} piezo-

electric coefficient) ^{measures the} conversion, ^either

from electric to elastic, ^or from elastic to electric energy. It ^must be taken into account at least twice, for the elastic modes generated by the first electric

pulse, and for the echo detection at 2 r. Therefore,

the detected echo amplitude ^{varies as} A(T) ^oc k2(T).

This factor also measured the amplitude ^{of the} signal

radiated after a single pulse, signal ^{which is a noise}

in our measurement.

The electromechanical coupling ^factor k 36 2 ^between

the critical mode P3 ^{and its} unique coupled mode S6

is the only ^one varying ^{with the} temperature ^as k2 ^{T -} Te - To

from an ex erimental estima-

k36Crit(T) = T - T ; crit ^{from an} experimental ^estima-

T -T ₀

tion of (Tc - TO) - ^{4.2 K [7],} k36crit(T) ^{would not}

be doubled over our experiment range. The other

coupling ^{factors are} quite ^{constant with the} tempe-

rature near the transition [7].

We also find through ^the attenuation study ^that

the critical experimental lifetime is much greater ^than the S6 ^{mode lifetime, and we} attribute this to mode conversion effects (§ 3.4); it shows that the S6 ^mode

excitation is not prevailing ^as compared ^{to those of}

the other pure elastic modes.

03B2) ^{We call} ~(T) the factor measuring ^the efficiency

of the non-linearity building ^{the echo.}

-

First model. - If the echo is due to a term as

03B2ESS in the Hamiltonian (E ^{is the electric} field, S ^is

the strain) [10], ^the electric field acts via its second

XSN B

.

XNL 2

harmonie E2W (E2w=

XNL E2w), ^XL ) and the echo amplitude

should finally vary ^{as :}

-

Second model.

If the anharmonic elastic interaction of a term 41 ! ^{Ce S4 in the} Hamiltonian

produces ^{the echo} [ 12], the electromechanical coupling

factor must still be considered twice, ^{as the elastic}

modes from the second electric pulse ^act quadratically,

to reverse the phases of the elastic modes from the first pulse. ^{In this case, the echo} amplitude ^should

vary ^{as : :}

4.3 EXPERIMENTAL RESULT. - Our experimental

data are well aligned ^{on a} straight ^{line of} slope ^{n = - 1}

when plotted ^{with 20} Logo (T - Tc) ^{taken as abscis-}

sa, for Tc

^{143.5 K.} The relative variation of n does

not exceed 10 % ^if Tc ^{varies of ± 0.5 K} (Fig. 2).

We obtain :

FIG. 2.

2,r echo amplitude (dB) plotted ^{with 20} Log10 (T - Tc)

takén as abscissa. When Tc

^143.5 K, the experimental points

are well aligned ^{on a} straight ^{line of} slope - ^1.

We also notice that near the transition, ^the signal (echo) ^to noise ratio is greatly ^enhanced (the ^noise

in our measurement is the ringing consequent ^{on a}

single pulse). ^{From this} observation, ^we attribute the

divergence ^of A(T) ^above Tc uniquely ^to ~(T), ^the efficiency ^{of the} non-linearity, ^{and we} neglect ^the

influence of k2(T). Anyway, ^{as we} discuss it in sec-

tion 4.2 _a, only ^one electromechanical coupling ^factor

.

2 Tc - To

varies above ^{varies above} T Tc as K36 crit = ^as k2

Tx - TO T - T° ^{; and the and the atte- atte-}

nuation study ^{indicates that the} S6-mode ^{is not} mainly excited. We thus conclude that only q(T) diverges ^as ( T - Te)-l.

4.4 DISCUSSION.

1) ^{In the case of a} parametric

effect with the electric field at the doubled frequency,

~(T) . P(T) XSNL(T)

~(T) ^is proportional ^to A

-CE ^{(T) XSL(} XSL(T) . A ^precise

knowledge ^of X’NL(T)IXI(T) (efficiency ^{of the electric}

field double harmonie conversion) ^{at our} frequencies

should give ^some insight ^{about an} eventual critical behaviour of ’1(T).

2) ^{In the case of an elastic} anharmonicity, ~(T) ^is

.

2 CNL(T )

proportional ^to k (T) CE (T ). ^{As we deduce from}

propotional ^to k2(T)

CE(T)

°

As we deduce from the signal ^to noise ratio behaviour that ~(T) diverges

near the transition as (T - Tc)-l (the electromecha-

nical coupling ^factor k2(T) ^varies slowly ^near T,,

anyway), ^we should conclude :

Some examples of critical divergences of the third order elastic constant are known, ^even if the second order elastic constant vanishes at the transition tem-

perature ⁱⁿ perovskite structures [16]. ^{But no} compa- rable result on the fourth order elastic constant is known to us.

As a conclusive remark, ^it may be noticed that in both models, ^the divergence ^of C 6(T) ^could explain

the divergence ^of q(T) ^{we observe.}

So, complementary studies of dielectric and elastic non-linearities of this type ^of compound ^{in our fre-}

quency range would be _necessary for a complete interpretation ^{of our result on the echo} amplitude.

But it gives already ^some information on non-linear coefficient products.

Concerning the model one could finally retain,

Laïkhtmann [17] has shown that acoustic non-

linearities should be expected ^{to have a} greater impor-

tance than other non-linearities.

5.

Conclusion.

Our experiment gives ^{a correct} temperature dependence of the critical ultrasonic

lifetime, though its order of magnitude ^is largely

overestimated by mode conversion effects. We can

also attribute ~(T), ^the efficiency ^{of the} non-linearity building ^the echo, ^the divergence ⁱⁿ (T - Tr ^{1 we}

observe on the echo amplitude ; this could be a pre- cious piece ^of information to study non-linear effects.

This confirms the value of the phonon echo method for investigation ^of phase transitions with _very light experimental ^devices.

Acknowledgments.

^{We are} very grateful ^to

M. Remoissenet who provided ^{us with the} samples

we studied and had previously investigated ^their

elastic constants [7].

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