Critical Sound Damping: When Does Scaling Hold?

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Critical Sound Damping: When Does Scaling Hold?

L. Benguigui, D. Collin, P. Martinoty

To cite this version:

L. Benguigui, D. Collin, P. Martinoty. Critical Sound Damping: When Does Scaling Hold?. Journal de Physique I, EDP Sciences, 1996, 6 (11), pp.1469-1476. �10.1051/jp1:1996158�. �jpa-00247259�

Critical Sound Damping: When Does Scaling ^Holà?

L. Benguigui (*), D. Collin and P. Martinoty (**)

Laboratoire d'Ultrasons et de Dynamique des Fluides Complexes (***), Université Louis Pasteur, 4 _rue Blaise Pascal, 67070 Strasbourg Cedex, France

(Received 29 December 1995, received in final form 2 July1996, accepted 26 July1996)

PACS.64.60.Ht Dynamic critical phenomena PACS.64.70.Md Transitions in liquid crystals

PACS.43.35.+d Ultrasonics, quantum acoustics, and physical effects of sound

Abstract. Using the dynamic specific-heat theory, _we ^show that exponent y, associated

with high-frequency damping at _a second order phase transition (ôa _~J w~+Y), ^is frequency- dependent. This result indicates that the dynamic scaling-laws predicting that the ôa/à&o~

ratio is _a homogeneous function of variable _{wT are never} rigorously verified (7 is trie critical relaxation time). This calculation, which allows trie degree of departure from these laws _to be

determined, is applied to _various experimental situations.

Résumé. Nous montrons, _en utilisant la théorie de chaleur spécifique dynamique, _que l'expo-

sant y caractérisant l'atténuation haute-fréquence à _une transition de phase du 2~~~ _ordre

(à&o~ ~J w~+Y) dépend de la fréquence. Ce résultat indique _que les lois d'échelle dynamiques

qui prévoient _que le rapport ôa/ôao~ est _une fonction homogène de la variable _{w7, ne} sont _en

toute rigueur jamais vérifiées (7 est le temps ^{de relaxation} critique). Ce calcul, _qui permet de

déterminer l'importance des écarts à _ces lois, est appliqué à différentes situations expérimentales.

Introduction

Ultrasound damping and velocity measurements provide important information _on dynamic be- havior in the vicinity of _a phase transition iii. The anomalies observed in the high-temperature phase _are most often the result of _a quadratic coupling between the ultrasound _wave and order- parameter fluctuations. In this _case, theones [2-5] ^based on the fluctuation-dissipation theorem

for the stress tensor predict that high-frequency ^{cntical darnping} is given by

à£im

+~ ~d~~~ Il)

High-frequency damping corresponds to the ~o7 » regime, where ₇ is the critical relaxation time and _~o the angular measunng frequency. This regime _occurs in the vicinity ^{of transition} temperature Tc because of the critical increase of _7. The exponent y = 0 for the transition

(*) ^{Permanent address: Solid State Institute and Physics} Department, Technion-Israel Institute of

Technology, ^Haifa 32000, ^Israel (**) Author for correspondence (***)URA 851 CNRS

@ Les Éditions _de Physique ¹⁹⁹⁶

1470 JOURNAL DE PHYSIQUE I N°11

in helium and _y

= -o/zu for transitions characterized by _a divergence of the specific-heat. _a

is the specific-heat _exponent and _zu the dynamic _exponent associated witu _7.

Analysis of tue _{~o7 »} 1 regime tuus enables tue scaling law given by Il) to be tested,

and gives _us tue value of exponent _y. Tuese tueories also predict tuat tue critical damping

associated witu tue uigu-temperature phase _may be represented by tue scaling law ôa

i ^{= ~i~°~~ 12)}

wuere F(~OT) is _a uomogeneous function of tue reduced variable _~OT. Equation (2) shows tuat

ail tue measurements must fit _{onto a} single _curve, wuicu determines tue form of tue scaling

function F(~OT).

Tuese cntical damping beuaviors would _{appear to} be verified in _a great number of _cases:

tue liquid-gas critical point [6], binary mixtures ii, _8], and structural transitions [9,loi. On tue otuer uand, departures from scaling laws il) and (2) _were observed in tue _case of tue

transition in uelium il1,12]. Tuey _can be identified by tue following features:

. exponent _y > 0, wuereas it suould be _zero;

. y bas diiferent values, according to wuetuer damping measurements are taken at uigu _or low

frequency;

. ratio ôô/à£ice does not scale _{as a} function of tue reduced frequency _~OT.

Tuese diiferences bave been explained by Ferrell and Buattacuarjee _[12], wuo used _a dynamic specific-ueat tueory to show tuat scalings il) ^and (2) _are only respected wuen tue specific-ueat anomaly ôcp is low witu respect to tue regular _{part Cr~g.} Tue problem of critical dynamics

bas also been addressed by Gurovicu _{et ai.} [13], wuo bave suown tuat ultrasound damping does not, ⁱⁿ general, exuibit scaling beuavior.

In tuis article, _{we propose a} calculation of tue exponent _y wuicu shows tuat scaling laws il

and (2) are never ngorously verified. Tuis calculation bas tue advantage of providing _a simple

cnterion for determining tue conditions for wuicu departure from tuese laws is important (no scaling), _or too sligut to be expenmentally observed (apparent scaling). Tue calculation will be applied in particular to tue Nematic-Smectic-A transition, for wuicu tue ratio ôcp/Cr~g varies

greatly from _one compound to tue next, and will enable _a certain number of predictions to be made conceming ^ultrasound beuavior _at tuis transition.

Tue article is divided into _two parts. Firstly, _we calculate exponent y, and tuen tue result of tuis calculation is applied to various situations:

a) tue transition _m uelium

b) transitions based _on tue Ising mortel

c) tue N-SmA transition.

1. Calculation of Exponent _y

Ferrell and Buattacuarjee _{[12] presented a very} elegant metuod of calculating sound absorption

at tue superfluid transition in liquid uelium, _usmg tue dynamic specific-ueat _{approacu. Tue} starting-point is tue relationsuip between sauna velocity U and specific ueat Cp _[14]

1 g

@ ^{= K} z ¹³⁾

wuere _g and K _are functions of _serve tuermodynamic quantities wuicu _are supposed to be

weakly dependent on temperature ^{and volume.} Cp _can be written _as tue _sum of _{two terms:}

Cp _{= Cr~~} + ôcp Ii)

In (4), ôcp is the critical part of the specific ueat, whereas Cr~g is tue regular part. Cr~g con

be taken _{as a} constant because only temperatures near Tc _are considered. For ôcp we use the renormalisation-group expression ils]

ôcp ₌ cc + At~" là)

in which t is the reduced temperature (t

= (T Tc)/Tc), and Cc _a constant. It is important

to note that A _> 0 if _{a >} 0 (divergence behavior) and A < 0 if _{o <} 0 (cusp behavior).

Expression (4) _can be written _as

Cp ₌ Go + At~" (6)

where Go is _a positive constant equal to Cc + Cr~g.

Equation (3) shows that the velocity Uc at Tc is given by:

j

= K for _a divergence Ii)

~

and ~

= K ~

for _{a cusp.} (8)

U~ Go

Since U is not too diiferent from Uc, it is possible to rewrite equation (3) in ternis of the

expenmentally ^{measured quantity} )

1 c

~ Go +~At-" ^{~°~ ~ ~~~~~~~~~~ ~~~}

Î~ _j

Go +~At-" ^{o ~°~ ~ ~~~~ ~~~~}

In order to calculate ôoe, ^{it is} necessary:

1) to write Cp as a function of the critical relaxation frequency r

= rot~~

r -n/zv

Cp = Go + A (-) _Ill)

~o

2) to use the dynamic scaling hypothesis, ^which consists in replacing ^r by -i~o when _{~o >} r _j~ -OE/zv

c = c~ ₊ A (12)

~ ro

Specific heat is _{now a} complex quantity. Velocity, related to Cp by equations (9, 10), ^{is also} a

complex quantity, and its imaginary part is related _to the damping ~°",

which is wntten as

~o

~

-OE/ZV

~ ~

~°"

+~

~ ~o

~~~ 2 _zu

~ cj _~ ~~~~ ^1°

"~~~

~

qr a ~ ^~J

~2"/zv (13)

~~ ^{~~ ~} ~~

~ ~ ~0

1472 JOURNAL DE PHYSIQUE I N°11

This form is valid whatever tue specific ueat beuavior (cusp _or divergence). Since A and _{a are}

always of tue _{saine sign,} tue product A (sin gra/2zu) is positive. If _a is small, cos(gra/2zu)

r~ 1

and equation (13) becomes:

~

~"/zv

ôdoe (ro~

Ù ^~

~

-o/zv ^~ ~~~~

~~ ~ ~ ~0~

It is possible to define _an effective exponent _{y, as} ^follows:

d

(ôdoe)

Y" ) _à/j

(là)

àJ àJ

Equation (là) gives:

~

-n/zv

Go A (-)

~ ~ ^~~

-OE/ZV ~~~~

Go + A ~o^~°

Expression (16) _can be mfde _more convenient by separating tue _cases wuere _{a >} 0 and _{o <} 0.

Taking B

= (A(/Co, _we obtain for _{a >} 0:

n/zv

1 B (Î)

~

~ ~~

n/zv ~~~~

~~

l ₊ B (Î)

~o and for _{a <} o:

-n/zv

+ B ^~°

~ Î

-"/zv ~~~~

l B -)

ro

Formulae (17, 18) show tuat tue dynamic scahng laws Il) and (2) cannot be rigorously verified,

smce tue exponent _{y is} frequency-dependent. Tuey also show tuat it is possible for systems belonging to tue _same universality dass not to show tue _{same y} values, _smce tuis exponent depends, via terni B, on tue regular _part of tue specific ueat, wuicu is _a non-universal quantity.

Formula (17), corresponding to tue _{a >} 0 _case, shows tuat y varies from -a/zu for B

= 0

to +a/zu for B _- cc, and cancels out for B

= (~o/ro)"/~~. Tuis beuavior is illustrated by Figure 1, obtained for _a

= o-II (tue Ising mortel) and _zu

= 2. Tue values of _uJTo correspond

to tue uigu-frequency _(~oTo "10~~) _and low-frequency _(~oTo

= 10~~) regimes.

Formula (18), corresponding to tue _{o <} 0 case, shows tuat _y presents a discontinuity wuen B = (~o/ro)"/~~ ^Tuis beuavior _cornes from formula (13), wuicu shows tliat à£ice/~o tends towards infinity for tuis particular value of B. Tuis beuavior of ôôoe /uJ _is not puysical, because it would correspond to _a vanisuing specific ueat, and tuerefore tue discontinuity of _y bas _no

0.04

j GJTO=10~~

a =0.Il

0.00 ~

~ GJTO"10~~

-o.oz

-0.04

-0.06

0.00 0.20 0.40 0.60 0.80 1.00 1.Zo

B

Fig. 1. Variation of exponent _{y as a} function of B for _o

= 0. ii (Ising model) and _zv

= 2. _y presents different values depending _on whether dampmg measurements _are taken _in the high (w% _" 10~~) _or

low (w% _" 10~~) frequency regime (see text for details).

significance. It suould also be noted tuat tue v>lue ^{of B} cannot exceed _a limit which _can be deduced from tue ratio

~jj~ jOE/~i

~ ~

i

Cc

tue tueoretical value of wuicu is -1.057 [16]. Dl represents tue first scaling correction and Ai tue associated exponent. Witu Di

" o.241 iii], _{a =} -7 _x lo~3 and Ai _" o-à, _we obtain

(Dl("/~i ₌ 1.02. Since Go _" Cc ₊ Cr~g, ^{it is deduced tuat B <} (R(/(Di("/~i

= 1.03.

Tue beuavior of _y wuen B is less tuan 1 is given in Figure 2 in wuicu _a

= -o.oo7 _as in tue 3DXY mortel. Tuis figure ^{shows tuat} exponent _{y is} very small (o.02) and virtually independent

of frequency for values of B ranging from o to o-G- For tuese values, tue scaling ^laws given by Il and (2) are more or less respected, ^and tue beuavior observed uere coula be referred to as

quasi-scaling.

Tue results tuus obtained will _now be applied to various systems.

2. Comparison with Experimental ^Results

2.1. THE TRANSITION IN HELIUM. Exponent _y was rneasured _{over a} wide set offre-

quencies ranging ^{from lo~} to lo~ Hz iii,18,19]. Tue results obtained show tuat _{y ~J} o-là for frequencies below 35 MHz and

~J

o.33 for frequencies over 40 MHz [20]. ^{Tue fact tuat}

y ^is tuus frequency-dependent dearly shows tuat scaling dqes not apply in tuis _case. Ferrell and Buattacuarjee [12] ^{bave used tue} dynamic specific-ueat approacu to explain tuis beuavior of _y, assuming tuat tue specific ueat diverges logantumically. ^{In wuat follows,} we suait _use

formula (18), _m wuicu _a

= -o.oo7 (3DXY mortel) to determine y, and show tuat variation of y witu frequency is _in good agreement ^{witu tue} experimental results, tuereby validating _our

approacu.

Tue values of A and Go, wuicu _are necessary ^to determine y, show tuat (A(/Co _~ l [21], and comparison witu Figure 2 indicates tuat y ^{is a} function of _~o. Tue exact calculation using

formula (18), and the value of ro (ro

~

3 _x la+~~ Hz) _{[18] grues y =} ^{o-là for f} = lo~ Hz, and

y = o.32 for f

=

lo~ Hz, ^which is _m good agreement ^{with the} experiments.

1474 JOURNAL DE PHYSIQUE I N°11

°'~°

a =-0.007

0.30

GJTO"ÎÙ ^~

~ ~

Ù.2Ù

o.io

~GJT

~ ~~.50 _{0.60 0.70}

0.80 0.90 1.00

B

Fig. 2. Same

as for Figure 1 but for _o

= -7 _x 10~~ (3DXY model) and _zv

= 1.

2.2. ISING-LIKE TRANSITIONS. We shall consider 2 systerns (binary mixtures and normal- incommensurable transition) for which the specific-heat _{exponent is of} the Ising type la o-il). =

Ultrasound damping in the 3-methylpentane-nitroetuane mixture _was measured by Garland and Sancuez _{[8] and} scaling laws Il and (2) _were cuecked witu _an imposed value of _y

= -o.06 deduced from tue formula _y

= -a/zu, in wuicu _a

= o.Il and _zu

= 1.93.

Tue specific-ueat measurements [22] ^{show that A} = 2.96 and Go ₌ 17.72 which gives B

=

o.16. Use of this value for calculating _y from formula (là) shows that _y

= -o.03 for tue ultrasound frequencies u8ed (~oTo +~

lo~3 lo~~) Tuis value _{of y} shows tuat scaling laws Il)

and (2) suould not be applied. Tuey _are apparently respected, since tue value _{of y} Iv

= -o.03)

is not far enougu from tue imposed value Iv ₌ -o.06).

In tue _case of tue normal-incommensurable transitions in NaN02 _[9] and Rb2ZnC14 [loi, _an

apparent scaling _was obtained witu _a value of _y

= -o.l, corresponding to values of o.Il and 1 for _a and _zu respectively. According _{to our} calculation, tuis implies tuat B suould be _very

small.

2.3. THE N-SMA TRANSITION. _{Like the transition}

m uelium, tuis transition belongs

to tue 3DXY universality dass. Tue 3DXY feature _{is not} always _apparent, uowever, _as it is sometimes very close _{to a} tricritical point. Tuis _is wuy, depending _on tue compound under

study, _a 18 found _{to vary} between _-o.oo7 (3DXY) and o-à (tricritical point) _{[23, 24].}

If _we consider only tuose compounds for wuicu _a

r~ -7 _x lo~~, ratio B

= (A(/Co _is found _to vary from o.99 _to o.40 [17,24,25]. It is dear from Figure 2 tuat _y suould be strongly dependent

on frequency wuen B is close to 1, and suould not show any marked variation witu frequency

wuen B is smaller tuan o-G- Consequently, ultrasound damping suould bave quasi-scaling

beuavior for compounds ^for wuicu B is less tuan o.6.

Recent experiments carried out at tue N-SmA transition _m TBBA [26] ^show tuat scaling laws Il and (2) _are venfied, witu _{y m} tue vicinity of _zero and _zu of tue order of1. Tue value of _y, wuicu implies tuat _a

r~ o, _suggests tuat tue transition is of the 3DXY type, in which

case Figure 2 shows tuat B suould be less tuan o.6. Recent prelimmary measurements of tue specific ueat show tuat tue cntical part is small in companson witu tue background [2î]. Tuis

result is consistent witu tue small expected value of B.

0.35

°.~~

a =0.424

o.15 ^' ~

GJTO"10_~

o.05

~ ~o.05 ^'

-o.15

jG~To"io~~

-0.35

-0.45

°°°°°~ ~°°°~

B ^°'°~

°'~

Fig. 3. Sartre _as for Figure 1 but for _{a =} 0.424 (mixture with X

= o-à of Ref. [24]) ^and zv = 1.

Trie B value

is deduced from trie specific-heat data analysis of reference [24] ^{and is} represented by trie vertical dashed line (see text).

Finally, we suait examine compounds wuicu _are under trie influence of _a tricritical point, sucu _as tue mixtures given ^{in reference} [24], ^wuicu are cuaracterized by _{an a} exponent ^of o.3

or o.4, and _very low values of B(+~ ⁴ x lo~3). _{Figure 3,} corresponding to the mixture with

an o exponent of o.424, shows that, ⁱⁿ spite ^of the very low value of B, _{y varies} strongly with

frequency [28]. ^{Tuese mixtures should not,} tuerefore, have scaling behaviors Il) and (2).

Conclusion

We bave determmed tue exponent _y ^{wuicu descnbes ultrasound} dampmg beuavior _m tue

~OT » 1 regime. This calculation shows that _y is dependent _on frequency, ^{which in} tum mdicates tuat tue simple scaling laws deduced from morte-coupling calculations suould not be

applied.

In tue _case of binary ^{mixtures and structural} transitions, departure from tuese laws is _so

sligut tuat it cannot be observed. Nor suould it be observed in lue

case of tue N-SmA transition for compounds witu _a 3DXY-type _a exponent and _a (A(/Co ratio smaller tuan o.6. On tue

otuer uand, tuis departure suould _{occur m} tue _case of tue N-SmA transition for compounds wuicu, like tue À transition in uelium, bave _an (A(/Go ratio of tue order of1. Tuus compounds belonging to tue _same universality dass _{may or may} not bave tue simple scaling ^beuavior given by equations Il) and (2). A departure ^from tuese laws suould also _occur for compounds witu

a N-SmA transition in tue vicinity ^of a tricritical point. It would be interesting to test tuese

predictions by taking ultrasound measurements on ail the8e _various compounds.

Acknowledgments

It is a pleasure to tuank Professor H. Brand for bis critical reading of tue manuscnpt.

1476 JOURNAL DE PHYSIQUE I N°11

References

iii See for example Garland C.W., _m "Puysical Acoustics", vol. VII (Academic Press, N-Y-, 1970).

[2] Kawasaki K., Phys. Reu. A1 (1970)1750; Phys. Lett. A 31(1970) 165.

[3] Kroll D.M. and Ruuland J-M-, Phys. Lett. A 80 (1980) _45; Phys. Reu. A 23 (1981) 3îl.

[4] Swift J. and Mulvaney B-J-, J. Phys. Lett. IFance 40 (1979) 28î.

[Si Andereck B-S- and Swift J., Phys. Reu. A 25 (1982) lo84.

[6] Eden D., Garland C.W. and Tuoen J., Phys. Reu. Lett. 28 (1972) 726; Roe D.B. and

Meyer H., J. Low Temp. Phys. 30 (1978) 91.

[î] Sancuez G. and Garland C.W., J. Chem. Phys. 79 (1983) 3100.

[8] Garland C.W. and Sanchez G., J. Chem. Phys. 79 (1983) 3090.

[9] Hu J., Fossum J-O-, Garland C.W. and Wallace P-W-, Phys. Reu. B 33 (1986) 6331.

[loi Hu Z.. Garland C.W. and Hirotsu S., Phys. Reu. B 42 (1990) 8305.

[Il] Tozaki K. and Ikusuima A., J. Low. Temp. Phys. 32 (1978) 379.

[12] ^{Ferrell R-A-} and Buattacuarjee J-K-, Phys. Reu. Lett. 44 (1980) 403.

[13] Gurovicu E-V-, Kats E-I- and Lebedev V.V., JETP Lett. 52 (1990) _611, ibid. Sou. Phys.

JETP 73 (1991) 473.

[14] ^{Ferrell R-A- and} Buattacuarjee J-K-, Phys. Reu. B 25 (1982) 3168.

[15] Bagnuls C. and Bervillier C., Phys. Reu. B 32 (1985) 7209.

[16] Bagnuls C. and Bervillier C., Phys. Lett. A 107 (1985) 299; A ils (1986) 84.

[17] ^Garland C.W., Nounesis G. and Siine _K-J-, Phys. Reu. A 39 (1989) 4919.

[18] ^{Williams D. and Rudnick} I., Phys. Reu. Lett. 25 (1970) 276.

[19] Carey R., Bucual Ch. and Pobell F., Phys. Reu. B16 (19îî) 3133.

[20] Ikusuima A., Jpn J. Appi. Phys. 19 (1980) 2315.

[21] Singsaas A. and Aulers G., Phys. Reu. B 30 (1984) 5103.

[22] Sancuez G., Meicule M. and Garland C.W., Phys. Reu. A 28 (1983) 1647.

[23] ^Thoen J., Marynissen H. and Van Dael W., Phys. Reu. Lett. 52 (1984) 204.

[24] Nounesis G., Garland C.W. and Suasuidar R., Phys. Reu. A 43 (1991) 1849.

[25] Garland C.W., Nounesis G., Young M.J. and Birgeneau R-J-, Phys. Reu. E 47 (1993)

1918.

[26] Sonntag P., Collin D. and Martinoty P., _to be publisued.

[2î] Sonntag P., Private Communication.

[28] Formula Iii) _cannot apply rigorously, _smce it is only valid if exponent o is in the vicinity

of _zero. When tue calculation is made witu uigu values of _o la

= 0.3 _or 0.4), it only bas

a very sligut eifect _on formula Iii), wuicu tuus remains valid in practice.