Acoustic absorption by the soft modes of defects in

NH4Cl

F.J. Bartis

To cite this version:

F.J. Bartis. Acoustic absorption by the soft modes of defects in NH4Cl. Journal de PhysiqueLettres, 1980, 41 (7), pp.161-164. <10.1051/jphyslet:01980004107016100>. <jpa-00231749>

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L-161

Acoustic absorption by the soft modes of defects in NH4Cl

F. J. Bartis

Department of Physics, Indiana University, Bloomington, Indiana 47401, U.S.A.

(Re~u le 23 aofit 1979, revise le 7 janvier, accepte le 8 fevrier 1980)

Résumé. 2014 Des défauts intrinsèques tendent à osciller par rapport aux domaines dans la région diphasée de latransformation d’orientation. Traitant le solide comme un continu élastique, on trouve que les fréquences devibrations des lacunes et des dislocations mobiles diminuent comme (To - T)1/3 et | T0 - T |1/2, respectivement,près du point de transition To. De l’excitation ultrasonore du mode mou de dislocations à pulsation 03C9 le frottementintérieur acquiert des maximums déplacés à To ± k03C92 et des queues étendues de la forme 03C9 | T0 - T|-1. Onpropose une étude spectroscopique de l’opalescence de transition pour vérifier le devis de 50 MHz comme unelimite inférieure de la fréquence du mode mou de dislocations dans un bon cristal.

Abstract. 2014 Intrinsic defects tend to oscillate relative to the domains in the two-phase region of the orientationaltransformation. When the solid is treated as a continuous elastic medium, the vibrational frequencies of vacanciesand mobile dislocations are found to decrease as (To - T)1/3 and | T0 - T |1/2, respectively, near the transitionpoint To. From the ultrasonic excitation of the soft mode of dislocations at angular frequency 03C9 the internalfriction acquires displaced maxima at To ± k03C92 and extended tails of the form 03C9 | T0 - T|-1. A spectroscopicstudy of the transitional opalescence is proposed to check the estimate of 50 MHz as a lower bound on the frequencyof the soft mode of dislocations in a good crystal.

J. Physique - LETTRES 41 (1980) L-161-L-164 leI’ AVRIL 1980,

Classification

Physics Abstracts62.65 - 63.20P - 78.35

1. Introduction. - Valuable insight into the quasi-static properties of solids during weakly first-ordertransitions is provided by the continuum theory oflattice defects. By way of illustration consider theopalescence of ammonium chloride during its orien-tational ordering reaction. Assume that the lightundergoes multiple Rayleigh-Gans scattering fromthe domains formed in the two-phase region of thereaction [1]. It can be plausibly argued that the inten-sity of the light emerging at 900 is proportional tothe surface area of the domains [2]. Suppose furtherthat the transition points of the crystallites around avacancy are determined solely by its Alr 3 stress

field [3]. The transition curve of the reaction beingalmost a straight line, the radius of the sphericaldomain that develops around the vacancy should

vary roughly as (To - T) - 1/3 In contrast, the radiusof the cylindrical domains that appear alongside anedge dislocation should go approximately as

I To - T I - 1, if its B/R stress field [4] regulatesdomain development. Accordingly the intensity of theopalescence should diverge as (To - T)- 2/3 and/orI To - T I 1. From the observation of both kinds ofsingular behaviour [5] it is concluded that domainevolution is controlled by vacancies in the lower partof the coexistence region and by dislocations in its

upper portion [6].

The continuum theory of crystal imperfections alsohelps to explain the growing sluggishness of weaklyfirst-order transformations near their nominal tran-sition temperatures. In the spirit of irreversible

thermodynamics let the speed of the domain boundaryunder isothermal conditions be proportional to theunderpressure of the boundary. When the departurefrom equilibrium is minuscule, the underpressurebecomes proportional to the gradient of the stress

field of the vacancy or the dislocation at the domain

boundary. It follows then that the thermal relaxationtime diverges as (To - T) - ’1’ and/or I To - T 1-2.Again both types of divergent behaviour are seen in theequilibration of Cu 3Au during its spatial orderingreaction [7].

2. Defect modes. - The vibrations of crystal imper-fections in single-phase solids have been under activeinvestigation for over thirty years [8]. Yet no thoughtseems to have been given to their vibrations in two-phase solids. Here I want to show that intrinsic defectsacquire special modes of oscillation during passagethrough coherent two-phase fields. Assuming thatdomain growth is regulated by the stresses around thedefects, I determine that the oscillation frequencies ofvacancies and mobile dislocations fall as (To - T) 1/3

and I To - T 11/2 , respectively, in the vicinity of the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:01980004107016100

L-162

transition point. I suggest that these soft modes areprimarily responsible for the dynamic anomalies ofweakly first-order transformations.

Let me begin with a vacancy surrounded by adomain of the disordered modification of ammoniumchloride. I presume that at the prevailing temperaturethe vacancy makes a large number of jumps per unittime between locally equivalent lattice sites. Thedomain boundary however moves much too slowly tokeep up with the nomadic vacancy. Consequently thevacancy needs more energy in order to wander furtherfrom the domain centre. If its displacement S fromthe centre is very small compared to the domainradius rn, then the vacancy must make up for the extralattice strain energy

where f is the fractional change of volume in passingfrom the disordered to the partially ordered modifi-cation. Differentiating with respect to S gives theforce acting to return the vacancy to the centre of thedomain

Subject to this force the vacancy tends to oscillatewith an angular frequency Wv given by

where M is the effective mass of the vacancy. With theinsertion of the temperature variation of the radius it is

seen that the oscillation frequency of the vacancyfalls as (To - T) 113A similar line of reasoning can be used to treat the

soft mode of edge dislocations. One envisages a shift ofthe whole dislocation line a distances to a locallyequivalent lattice position. One assumes that under thecircumstances the Peierls force on the mobile dislo-cation can be neglected. From the change in thelattice strain energy the restoring force per unit

length of the dislocation is estimated to be [9]

where G is the shear modulus, b is the length of theBurgers vector of the dislocation and Rn is the radiusof the domain decorating the dislocation. Taking theeffective mass per unit length to be npb2, one arrivesat the expression

Here p is the density of ammonium chloride. Remem-bering that the radius goes roughly as I To -- 7"!’~one concludes that We decreases as I To - 7"~.The soft mode of edge dislocations should be

carefully distinguished from their Koehlermode [10, 11]. When the Koehler mode is excited, adislocation pinned at several points along its lengthvibrates like a violin string. On the other hand, whenthe soft mode is excited, the whole length of a mobile

dislocation oscillates back and forth in a slip plane.The frequency of the Koehler mode can be raised byshortening the length of the vibrating segmentsthrough, say, gamma irradiation of the crystal [12].Likewise the frequency of the soft mode can be shiftedby varying the size of the domain, lying alongside thedislocation, through a change in the temperature of thesample. There is in any case a lower bound set on thefrequency of the soft mode by the defect density.It is a direct result of the limit on the domain size.In an ammonium chloride crystal containing 4 x 104dislocations/cm2 the domain radius reaches about10- 2 cm. From eq. (5) it follows that the soft modeof dislocations can only drop to 50 MHz !

3. Ultrasonic losses. - I submit that resonant

absorption by the soft mode of dislocations underliesthe anomalous attenuation of longitudinal ultrasoundin ammonium chloride [13, 14]. Like the Koehlermode, the soft mode should be broadened by a strongradiation damping. In the formal description of thesoft mode a retarding force varying as the speed of thedislocation can be used. The internal friction Q -1of such a damped resonance may be written [15]

where y characterizes the damping force per unit massper unit speed of the dislocation. It is We that causes theinternal friction to vary during the orientationaltransformation. The internal friction passes throughmaxima when We = c~ or when the temperature is

where k is practically a constant. On the other hand,when ~ ~ w, the temperature differs substantiallyfrom To and the internal friction goes inversely as thetemperature displacement

The soft mode of vacancies should also be subjectto a strong radiation damping when it is excited

acoustically. In this case the internal friction shouldexhibit a peak at

and a tail of the form

where h denotes a positive constant and 6 characterizesthe damping force per unit mass per unit speed of thevacancy.

Physically the long tails of the transitional internalfriction indicate that the second modification ofammonium chloride is present, even when the tempe-rature is far from To. The extended tails of its transi-tional opalescence may be viewed in a similar fashion.

L-163ACOUSTIC ABSORPTION BY THE SOFT MODES OF DEFECTS IN NH4Cl

It may be recalled that the amount of light scattered bythe domains lying alongside dislocation is proportionalto their surface area and hence their radius. From

eqs. (5) and (8) it can be inferred that the contributionof these domains to the internal friction is also pro-portional to their radius. For ten degrees Kelvin or soabove To the data on both parameters [5,16,17]bear out such a relationship, if in line with the conti-nuum description of lattice imperfections the radiusis taken to be inversely proportional to the temperaturedisplacement. It would be unrealistic to expectagreement over a larger temperature range, if aneutron diffraction study of the disordered state ofmanganous oxide is any guide. The weakly first-ordernature of its antiferromagnetic transition near 122 K isclearly established by Bloch and Maury [18]. Ren-ninger et al. [19] detect split peaks near the super-lattice positions at 133 and 298 K. They attribute thepeaks to neutrons scattered by the antiferromagneticmodification of MnO. However, they deduce fromtheir data domain sizes of 46 ± 5 A and 61 ± 10 Aat 133 and 298 K, respectively. The nearly constant sizeprobably means that in this temperature interval thesecond modification is almost all located near inclu-sions of Mn304 [20]. Returning to the orientationaltransformation of ammonium chloride, one notes thatthe growth of domains of its second modificationnear pinned dislocations would alter the frequency ofthe Koehler mode. Since the Koehler mode does notmimic the soft mode, the contribution of the pinneddislocations to the transitional internal friction wouldnot follow eq. (8).Blocked from the use of eqs. (8) and (10), one turns

to eqs. (7) and (9) as a means of testing the presentapproach to ultrasonic absorption. Their predictionsclearly differ from the implication of the Landautheory that the displacement of the peak varies as thefirst power of the frequency [21]. It should be empha-sized that the displaced maxima of the internal frictionare observable only if the acoustic frequency exceedsthe lower bounds of the soft modes. These boundscan be estimated theoretically but it seems preferable todetermine them experimentally. The experimentalresults can then serve as a check on the earlier calcu-lation of 50 MHz for a crystal having 4 x 104 dislo-cations/cm2.For this purpose I propose a measurement of the

spectral width of the light quasielastically scatteredby ammonium chloride during its orientational order-ing reaction. Near To the dynamic width is expected toremain finite, reflecting the boundedness of the defectmodes. Lazay [5] finds that the opalescence of anammonium chloride crystal can rise an order of

magnitude when it is cycled through the reaction.In contrast, Yagi et al. [22] and Courtens [23] reportthat long anneals can greatly reduce the opalescenceof potassium trihydrogen selenite and potassiumdihydrogen phosphate. Extending the ideas of Cour-

tens, I would argue that the passage of an ammoniumchloride crystal through the reaction increases thedensity of intrinsic defects and so should raise thelimiting frequencies of their soft modes. Thus I predictthat an increase in the intensity of the Rayleigh orelastic component should be accompanied by a

broadening of the dynamic component of the quasi-elastically scattered light.

4. Some questions. - The present letter is addressedto ammonium chloride during its orientational order-ing reaction. Yet its findings are expected to apply toany solid undergoing a weakly first-order transition.This includes quartz during its a-~i inversion. As thetemperature of a quartz crystal is lowered toward theinversion point To, a normal mode of the lattice is

observed to soften. According to Axe and Shirane [24]the frequency of this lattice mode decreases roughlyas (T - T~) 1~ 2, where T~ is some ten degrees Kelvinbelow To. This lattice mode is therefore not the softmode of mobile dislocations whose frequency is

predicted to drop as (T - 7~)~. Still the two modesmay be related. Both seem to be governed by the samepower-law exponent. On the other hand, a power-lawexponent of 1/3 appears to characterize the soft modeof the quartz lattice below the inversion point [25]as well as the soft mode of vacancies. Is there somefundamental reason for such an identity ?A lower bound of 50 MHz is given here for the

oscillation frequency of mobile dislocations in ammo-nium chloride during its orientational transformation.A similar limit should exist for the soft mode ofdislocations in quartz. Such a limit would explain thefailure of Shigenari et al. [26] to observe a dynamiccomponent in the transitional opalescence of quartzbetween 1 Hz and 50 MHz. Studies like that of

Shigenari et al. should be especially useful in decidingwhether thermodynamic fluctuations play a major rolein weakly first-order transitions. In the case of ammo-nium chloride a careful search for a dynamic compo-nent below 50 MHz in its transitional opalescencecould settle once and for all the question of whethercritical fluctuations are responsible for its anomalousabsorption of sound [27] and its strong scattering oflight [5]. According to Ginzburg [28] and Benedek [29]the hallmark of critical opalescence is a spectralnarrowing near the transition point. If the dynamiccomponent does not go below 50 MHz, then by thiscriterion the phenomenon is not critical opalescence.Prevailing opinion seems ready to accept such ajudgment for the li.-f3 inversion of quartz [26] and theferroelectric transformation of potassium dihydrogenphosphate [30] but not for the ferroelectric transitionof lead germanate [31]. Is a two-phase region in thelatter really ruled out by the experimental informationat hand ? Could not its anomalous ultrasonic absorp-tion [32] result from the excitation of the soft mode ofdislocations ?

L-164 JOURNAL DE PHYSIQUE - LETTRES

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