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1969-05-10

Studies of phase transitions in order-disorder

ferroelectrics. III. The phase transition in KH(2)PO(4)

and a comparison with KD(2)PO(2)

Reese, W.

American Physical Society

Physical Review, v.181, no. 3, May 10, 1969, pp. 905-919

http://hdl.handle.net/10945/47768

PHYSICAL REVIEW VOLUME 181, NUM BER 2 10 MAY 1969

Studies of Phase Transitions in Order-Disorder Ferroelectrics. III. ThePhase Transition in KH,PO4 and a Comparison with KD,PO,

W. RzzszDepartment of Physics, Naval Postgraduate School, Monterey, California

(Received 15 October 1968)

Calorimetric, dielectric, and electrocaloric measurements have been performed on single-crystal samplesof KH2PO4 in the neighborhood of the ferroelectric phase transition, and show that the transition is of erstorder, with a discontinuous entropy change of (0.0456 0.00M '"')R and a change in the dielectric constantby a factor of 2. Calorimetric measurements in applied electric fields were also carried out, and show thatthe e8ect of an applied field is to shift the transition to higher temperatures and to broaden it. Calorimetricmeasurements on KD2PO4 are reported which can be combined with previously reported results to givethe heat capacity for this material in the range 80—280'K. Comparison of the heat capacity of KH2PO4and its deuterated isomorph allows estimates of the lattice heat capacity to be made. From this estimate,the entropy change associated with the ferroelectric transition is found to be (0.400&0.036)R for KH2PO4and (0.457+0.034)R for KD2PO4. A signilcant feature of the comparison is the lack of evidence for theheat-capacity contribution above the transition predicted by theories of the Slater type. Comparison of thecalorimetric data with the modi6ed Slater theory due to Silsbee, Uehling, and Schmidt shows that thistheory describes KD2PO4 better than it does KH2PO4. The differences between the theory and experimentare of the type predicted by the modi6cations of the theory which include tunneling motion of the protons.

I. INTRODUCTION' PREVIOUS investigations of the heat capacity of

KHsPO4 (KDP) in the neighborhood of the ferro-electric transition seemed to indicate that the transitionwas of second order, '' although hysteresis has beenreported in measurements of the dielectric constant,which indicates a transition of the 6rst order. Recently,it was found that the transition in the deuteratedisomorph KD,PO4 (DKDP) was of first order. ' On thebasis of these observations it seemed of interest toundertake a reinvestigation of the transition in KDPusing a single-crystal sample, since the previous calori-metric investigations employed samples consisting ofmany small crystals with ill-de6ned electrical boundaryconditions.

One result of this reinvestigation was to 6nd that thetransition, when examined with a single-crystal sample,was of first order. In addition, an investigation wasmade into the sects of applied electric fields, whichwere found to broaden the transition and shift it tohigher temperatures. The latent heat appears to go tozero for fields in excess of about 300 V/cm.

A series of dielectric measurements was madesimultaneously with calorimetric measurements. Thetemperature of the dielectric and calorimetric anomalieswas found to be the same, although this fact wassomewhat obscured by a broadening of the transitionwhen plated electrodes were present. The dielectricconstant is found to change discontinuously by a factorof approximately 2 between the paraelectric and ferro-electric state.

Measurements of the electrocaloric eBect were madewhich substantiate the interpretation of the calori-

1C. C. Stephenson and J. G. Hooley, J. Am. Chem. Soc. 66,1397 (1944).

~ W. Reese and L. F. May, Phys. Rev. 162, 510 (1967).' P. P. Craig, Phys. Letters 20, 140 (1966).4 W. Reese and L. F. May, Phys. Rev. I67, 504 (1968).

metric and dielectric measurements. These measure-ments also point out that nonlinearities in the 6elddependence of the polarization manifest themselves atquite low values of the applied field near the transition.

An extensive series of calorimetric measurements onDKDP between 80 and 280'K is also reported. A com-parison of the heat capacities of KDP and DKDP wasmade, which forms the basis for an estimate of thelattice heat capacity. The most striking feature of thiscomparison is that there exists scant evidence for anonlattice-heat-capacity contribution extending morethan about 20' above the transition. This feature is indirect contrast to the predictions of that class of theoriesof the ferroelectric transition in KDP-type materialswhich places main emphasis on the ordering of theprotons in the possible H-bond positions. The entropyassociated with the destruction of the spontaneouspolarization was calculated, and a slightly greaterentropy change was found by DKDP than for KDP.An analysis of the behavior of the heat capacity in theimmediate vicinity of the ferroelectric transition ofKDP was made. It was found that in the neighborhoodof e= (T, T)/T, =2X 10—' the heat capacity goes froma 1/Qe to a loge dependence.

In the previous papers in this series' 4 some discussionof the theoretical interpretation of the ferroelectrictransition in KDP-type ferroelectrics has been given, sosuch discussion will be omitted here. The emphasis ofthis paper will be on the nature and limitations of theempirical results. After a brief discussion of experi-mental details in Sec. II, the nature of the phase transi-tion in KDP and the phenomena occurring at thispoint will be explored in Sec. III, where the evidencewhich shows the transition to be of 6rst order will bepresented, and an investigation of some thermodynamicaspects of the transition will be made. Data concerningthe eO'ects of electric fields on the heat capacity willalso be presented. In Sec. IV, the heat capacity of

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906 W. REESE

DKDP will be discussed, a separation of the heatcapacity into parts associated with the lattice and theferroelectric transition will be attempted, the transitionentropy will be calculated, and behavior near thetransition will be discussed. In Sec. V a comparison ofthe present results with some theoretical models will bemade, together with some concluding remarks concern-ing the possible interpretation of these results.

II. EXPERIMENTAL DETAILS

The calorimetric measurements were made using ap-paratus and techniques previously described morefully. '4 Briefly, the measurements used conventional,discontinuous-heating, adiabatic calorimetry, using anappropriate low-temperature thermistor' as a tempera-ture sensor. Under favorable conditions, temperaturescould be determined with a precision of approximately50 p'K. The thermistors were calibrated against a Ptresistance thermometer, which in turn was calibratedby the National Bureau of Standards according to theInternational Practical Temperature Scale. Because ofpossible calibration shifts in the thermistors, the tem-perature scale has an accuracy far worse than itsprecision; the over-all accuracy is estimated to beabout 0.1'K.

In addition to the conventional measurements, con-tinuous-heating and cooling experiments were con-ducted. In the continuous-cooling experiments, theadiabatic shield was held at a constant temperature,approximately 2'K below the transition, and thesample was allowed to cool by radiation. Continuous-heating experiments were performed by adding heatelectrically to the sample heater, either with the shieldat the same temperature as in the cooling experimentsor at the same temperature as the sample. A systematicerror of about 0.01'K occurred in the temperature scalewhen the shield was cooler than the sample because ofradiational cooling of the thermometer. Thus the mostreliable such data were obtained during heating withtemperature differences smaller than a few m'K be-tween the sample and the shield. Heating rates of10 4'K/sec or less were used. However, good agreementbetween the continuous-heating experiments'and thediscontinuous-heating measurements was observed onlywhen the heating rate was considerable less than50 p'K/sec.

Samples were obtained from Isomet Corp. as single-crystal pieces cut from larger boules. The DKDP samplewas the same crystal (12.65 g) on which the previousmeasurements4 were performed, while the KDP samplehad a mass of 27.4 g. The usual technique involvedwrapping the crystal with a small quantity of copperfoil (less than 1 g) on which the heater and thermometerwere mounted. The foil was then cemented to the

~ YSI model 44002 (Yellow Springs Instruments) for measure-ments above 180'K and Keystone Carbon Co. type RL10X04below 180'K.

sample with GE 7031 varnish. Such samples will bereferred to as "foil-wrapped" samples.

The heat capacity of the addendum for the foil-wrapped samples was measured at the conclusion of themain measurements. In the addendum measurements,the foil was compressed so that as little surface area aspossible was exposed. In one series of measurements,carried out between 80 and 140'K, the KDP andDKDP crystals were measured using the same ad-dendum and thermometer calibration, so that, for thisone series, a comparison of the heat capacity of the twoisomorphs could be made with a minimum of adverseeGects caused by systematic errors in the temperaturescale and/or addendum heat capacity.

In addition to the series of measurements on DKDPreferred to above, a separate series, employing separateaddendum measurements and thermometer calibra-tions, was made between 80 and 180'K. A third seriesof measurements was made in the range 200—235'K,using the crystal with a minimum of addenda: no foil,only the thermometer, heater, diGerential thermocouple,mounting strings, and the minimum quantity ofGE 7031 varnish needed for mechanical stability. Forthis series, the heat capacity of the addendum wascalculated, using the results of previous measurementsof the heat capacity of these materials made in thislaboratory. This series of measurements of the heatcapacity of DKDP are not reported in detail here, butwere incorporated with the other "series in Table II.Finally, several measurements, using foil-wrappedsamples, were made at about 10'K intervals in therange 240—280'K, using temperature increments ofabout 0.2'K.

Additional calorimetric measurements were made,and are reported in detail here, on a KDP sample withelectrodes attached. The sample was a rectangularplate, 2.30)&2.12X0.382 cm, with the short dimensionparallel to the ferroelectric axis, which was cut fromthe original KDP crystal using a wet string saw.Following water polishing, gold electrodes were evapo-rated on this crystal, which will be referred to as the"plated" sample. The heater and thermometer weremounted on opposite faces, and electrical connectionswere made to the electrodes. Simultaneous measure-ments of the dielectric and thermal properties weremade using this sample, as were measurements of theheat capacity in applied electric ields and measure-ments of the electrocaloric eGect.

Capacitance measurements of the plated sample weremade using a General Radio type 716 capacitancebridge operating at 1 kHz with a peak field of 15mV/cm. Such small fields give a negligible electrocaloriceffect, even in the immediate vicinity of the transition,so that there is no practical distinction to be madebetween the isothermal and adiabatic susceptibilities.Narrow-band amplification of the output signal fromthe bridge was employed so that such small measuringfields could be used. During the course of the experiment

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FIG. 1.Measurement of the electro-caloric eGect made about 0.015'Kabove the transition in KDP. Thesecond pair of curves duplicates theconditions of the first. This figure,which is a tracing from the recorderoutput of temperature versus time,demonstrates that the measurementswere made under nearly adiabaticconditions.

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it was verified that no observable difference in calori-metric measurements resulted if the capacitance bridgewas operating, if the sample leads were shorted, or if thesample leads were open-circuited during measurements.

Measurements of the electrocaloric effect—the tem-perature change accompanying an adiabatic fieldchang" were made by switching the sample betweenan appropriate battery and a short circuit. One set ofmeasurements, which was taken about 0.015'K abovethe transition, is shown in Fig. j.. The slight base-linedrift is caused by radiational cooling from the sample,due to the inability of the adiabatic shield to follow therapid temperature variations of the sample. Except forthe base-line drift, the measurement in Fig. I demon-strates the reversibility of the effect observed.

rapid temperature variation, a plot of logjo C as afunction of T is shown. Even in this compressed pre-sentation, the temperature dependence of C increasesmarkedly in the neighborhood of the dashed line, whichis taken to represent the transition temperature. Thepoints for which the initial and/or final temperaturesfall above this temperature are thought to representmeasurements on a superheated sample.

If the transition is of first order, as implied by theabove interpretation, the most revealing variable

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III. NATURE OF THE FERROELECTRICTRANSITION IN KDP

A. Calorimetric Measurements in Zero Field P 2000—Lal

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The present calorimetric data agree to within 0.1'Po

with the earlier data of Stephenson and Booley' morethan 15'K away from the transition, so that, other thannoting this excellent agreement, such data will not bediscussed. In addition, the differences between thecurrent results and previous data'' seem adequatelyexplained by differences in transition temperature' asclose as a few hundredths of a degree to the transition.

The present heat-capacity data, obtained fromthe discontinuous-heating measurements on the foil-wrapped sample in the immediate vicinity of thetransition, are presented in Fig. 2. Because of the very

6 The transition temperature given in Ref. 2 was in terms of aprovisional temperature scale which the current scale replaces.Stevenson and Hooley find the transition on their scale to occur0.26'K higher than is found in the present work. Whether thisdifference is real or merely reQects different temperature scalesdoes not seem germane at this point.

500

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Fzo. 2. Heat capacity of the "foil-wrapped" sample of KDP,deduced by discontinuous-heating experiments, shown in theneighborhood of the ferroelectric transition. The horizontal barsgive the temperature range, initial to final, over which the pointwas taken and are typical of all points in vicinity of the points somarked. The dashed vertical line is thought to represent thetransition temperature. The solid curve has no significance otherthan that of fitting the data.

ORDER —D ISORDER FERROELECTRI CS. I I I

908 W. REESE 181

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40

FIG. 3. Enthalpy-temperature relationship deduced for the"foil-wrapped" sample of KDP. The discontinuous-heating datawere taken from calorimetric data such as those shown in Fig. 2.The continuous-cooling curve was obtained under conditions dis-cussed in the text. The dashed line shows the assumed thermalequilibrium enthalpy-temperature relationship. The evidence ofsuperheating and supercooling as well as the isothermal absorptionof heat mark the transition as one of first order.

against which to plot measured quantities is theenthalpy (which, to within an additive constant, is theheat supplied), since in the two-phase region the portionof the sample which has undergone the phase transitionwill be linear in enthalpy. In Fig. 3, the heat-capacitydata have been recast in the form of an enthalpy-temperature plot. The curve is reproducible, since thepoints represent the results of many independent passesthrough the transition. This presentation of the datashows an isothermal absorption of heat and bothsuperheating and supercooling. These features establishthe transition as being of first order. Because of thesuperheating, it is difficult to give an unequivocal valuefor the latent heat, but on the basis of the assumedtransition temperature, indicated by the dashed linesin Figs. 2 and 3, the latent heat is found to be 46.2+0.2J/mole. The uncertainty indicates the reproducibilityover seven independent determinations. A much largeruncertainty is due to the problem of locating the transi-tion, so that, if this uncertainty is included, the finalresult for the latent heat is 46.2 s.o~' J/mole.

Also shown in Fig. 3 are the results of a continuous-cooling experiment. In this experiment heat was re-moved from the sample at the rate of 0.79 mJ/sec,producing a cooling rate of less than 60 p'K/sec. Thecooling rate was determined by matching the latentheat obtained from the continuous-cooling experiment

r In a preliminary report of this work I W. Reese, Bull. Am.Phys. Soc. 13, 504 (1968)g, a different value, 50.5 J/mole, wasreported. This value was obtairied with diferent assumptionsconcerning the nature of the phenomena at the transition, andassumed that the lowest temperature reached after the isothermalabsorption of heat represented the true transition temperature.

with that obtained from the discontinuous-heating data.The cooling rate so obtained was consistent with valuesobtained by matching the heat capacity determined bythe two techniques about 0.2'K above the transition.Two small shifts have been made in the temperaturescale in the cooling experiment. The first shift, about1 m'K to higher temperatures, was made to correct forsmall temperature gradients in the sample which re-sulted because cooling took place from the surface.Thus, the average sample temperature is actuallyslightly higher than the surface temperature. Thiscorrection was found by a comparison of cooling curvestaken at different cooling rates with extrapolation tozero cooling rate. The second correction, about 12 m'Kto higher temperatures, accounts for the imperfectthermal contact between the sample and thermometer.Because of radiation to the shield, the thermometer wascooler than the sample. This correction was determinedby comparing cooling experiments with continuous-heating experiments in which the shield was held atthe same temperature as during the cooling experiments. .

After correcting for temperature gradients in thesample, as discussed above, the cooling and heatingexperiments gave the same transition temperature towithin 0.1 m'K, but 12 m'K lower than in the dis-continuous-heating experiments. Considering all of thecorrections which have been made- to the continuous-cooling data, the agreement with the discontinuous-heating data is perhaps less impressive than mightappear at erst glance. However, the data show anisothermal absorption of heat and supercooling, whichagain mark the transition as being of erst order.

The enthalpy curves display several other noteworthyfeatures. One feature is the unusual supercooling effectobserved at the paraelectric side of the heating transi-tion, i.e., a region of apparent negative heat capacityfollowing the region in which heat is absorbed iso-thermally. This feature is thought to be associated withthe presence of ferroelectric domains. During the heat-ing transition, the domains must be reduced in numberand/or size until they vanish. Since electrostatic andother considerations require a reasonably large numberof domains, the domains must become quite small nearthe end of the heating transition. Since small domainsare unfavorable in terms of the domain-wall energy, itis hypothesized that at soIne point it is less costly interms of the total free energy {including the effects ofexternal fields and domain walls) for the material tomake the paraelectric transition suddenly, and toobtain the required heat by supercooling the resultingparaelectric, rather than to make the remainder of thetransition at equilibrium. Conversely, there is a slight"superheating" effect on the ferroelectric side of thecooling transition, which seems to be related. This effectis thought to represent a transition by the last vestigesof the paraelectric material, before sufficient heat hasbeen removed to allow the entire transition to occur atequilibrium. It is hypothesized that this "superheating

ORDER —D ISORDER FERROELECTRI CS. I I I 909

ICAL PEAK TO PEAK NOISE

FIG. 4. Shown here is a reconstruc-tion of the thermometer response fol-lowing a large heat input while the"foil-wrapped" sample was at the tem-perature of the ferroelectric transition.The heat input caused the outer por-tion of the sample, to which the ther-mometer is most closely coupled, tomake the transition entirely into theparaelectric state. The subsequent be-havior demonstrates supercooling. The6nal equilibrium temperature is nearthe dashed line in Fig. 3.

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on cooling" occurs when a further reduction in size ofthe paraelectric regions would cause the surface terms(probably of the same order of magnitude as domain-wall energies) to outweigh the volume terms, whichtend to keep the sample partly in the paraelectric state.The heat liberated in the sudden transition of theparaelectric vestiges superheats the ferroelectric. Weassume that the effect on heating is larger than theeffect on cooling both because the heat capacity of theparaelectric is smaller than that of the ferroelectric andbecause the domain structure requires larger volumes tobe involved in the heating effect. Quantitative deduc-tions about the size of the domain-wall energy could bemade from this data, provided that the volumes in-volved in the processes were known.

An additional supercooling effect is shown in Fig. 4,where the equilibration of the sample following a large(about 80%%u~ of the latent heat) heat input is plottedThe plot has been reconstructed from the originalthermometer output versus time recording. Followingan initial recovery from an overshoot, which resultsbecause the thermometer was more closely coupled tothe heater than to the sample, the temperature of thethermometer (and probably the outer portion of thesample) stabilizes at a temperature characteristic of thesupercooled paraelectric in the enthalpy plot of Fig. 3.The succeeding warming is interpreted as occurringwhen the outer portion of the sample has lost sufhcientheat to the interior so that the supercooled paraelectricis no longer stable. One or more ferroelectric domainsare then formed in the outer portion of the sample,

liberating sufficient heat to cause the observed tempera-ture rise. The final equilibrium temperature is approxi-mately on the dashed line of Fig. 3. Although it isattractive to associate the supercooling with the domainstructure, this association is not necessary, since aneffect such as this only requires that the transition beof first order.

Figure 5 shows a enthalpy-temperature plot obtainedin a continuous-heating experiment using the "plated"sample and a heat input of 0.45 mJ/sec. During thisexperiment the shield was held at the sample tempera-ture, so that no major corrections to the temperaturescale were made. The heat input was assumed to be onlythe measured electrical heat input. A different transitiontemperature than that found for the foil-wrappedsample was indicated. Since it was later verified byrecalibration that a shift of the calibration of thethermistor had occurred, there is insufficient evidenceto support the shift as an important effect. If the shiftis real, it might well indicate the effects of partialclamping caused by the electrodes or of strains de-veloped in preparing and/or cooling the sample.

Comparison of Fig. 5 to Fig. 3 shows a much smallerisothermal region and no superheating. Thus thetransition shown in Fig. 5 has the appearance of beingnearly of second order. However, the data are inter-preted as indicating the appearance to be illusory, forthe data are consistent with a first-order transition,broadened over a temperature interval of about 0.04'K.The broadening may be due to effects either of samplepreparation or of the evaporated electrodes (perhaps

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I ro. 5. Sample temperature andcapacitance for the KDP platedsample shown as functions of enthalpy.The solid curve shows the tempera-ture-enthalpy behavior as deducedfrom a continuous-heating curve. Thedashed line shows the assumed ther-mal equilibrium transition deducedfrom the "foil-wrapped" sample. TheCurie law was determined from equi-librium measurements made at tem-peratures higher than those shownhere. It is thought that the region be-tween the maximum capacitance andthe Curie-law behavior represents aerst-order transition broadened bysample effects.

partial clamping). ' This interpretation is con6rmed bycapacitance measurements, also shown in Fig. 5, whichwere made during the heating measurements. Thecapacitance measurements will be discussed Sec. III S.The assumed equilibrium transition from Fig. 3 isindicated in Fig. 5 by the dashed line. This assumedtransition encompasses the transition indicated by thecapacitance measurements, thus showing that the samevalue of latent heat as obtained before is consistentwith the data shown in Fig. 5.

A final remark, which indicates the first-order natureof the transition and the coexistance of two phases, canbe made concerning the time required to obtain equi-librium following an input of heat. Figure 4 is typicalof both the foil-wrapped sample and the plated sample,in that thermal equilibrium following an input of heattook approximately 1 h when the sample was in whathas here been interpreted as a two-phase condition.When the sample was definitely in a single-phasecondition —for example, by being several tenths of adegree from the transition —equilibration following theinput of heat typically required 5 min. Thus, in thephase transition region, the thermal equilibrium timewas not limited by the diffusion of heat, unless a verylow thermal conductivity pertains in a very narrowtemperature region, but rather by a longer time,characteristic of the establishment and/or rearrange-

Subsidiary continuous-heating experiments on the platedsample, before and after the electrodes were mechanically re-moved, showed superheating only after the electrodes wereremoved. This strongly indicates that the major cause of theapparent change in the form of the enthalpy-temperature plotwas the presence of the electrodes rather than the cutting of thesample. The most likely effect of the electrodes was in partiallyclamping the sample, as evidenced by small surface cracks whichwere discovered in the sample when the electrodes were removed.

ment of interphase boundaries. Such long equilibrationtimes in the neighborhood of the transition were firstnoted by Stevenson and Booley. ' As a final remark onthis point, some unusual equilibration curves were

observed at the paraelectric end of a pass through thephase transition. Typically, the equilibration would

begin as though with a long time constant and thensuddenly achieve~equilibrium, as though the phasetransition process'&ad come to an end and rapid equi-

libration, limited by the diffusion of heat, ensued.

3. Dielectric Measurements

The results of the dielectric measurements can besummarized by stating that the same general results asreported by many others'9 were observed, namely, thesusceptibility obeyed the Curie-Weiss law above thetransition and decreased very slowly with temperaturebelow the transition. The Curie-Weiss temperatureneeded to fit the present results was approximately0.11'K less than the transition temperature, in remark-

ably good agreement with Craig, ' who found a Curie-Weiss temperature 0.115 K less than the transitiontemperature observed on heating. (Craig observed ahysteresis of about 0.050'K which was not observed inthe present work. )

The most interesting dielectric results are thoseshown as a function of enthalpy in Fig. 5. Superficially,the maximum in the dielectric constant occurs about0.04'K below the point of maximum heat capacity.There is then a region in which the dielectric constant

The earlier work is summarized in such references as W.Kanzig, in Solid State I'hysics, edited by F. Seitz and D. Turnbull(Academic Press Inc., New York, 1957), Vol. 4, p. 1.

ORDE&R —DISORDER FERROELECTRI CS. I I I 9ii

FIG. 0. Summary of some of theelectrocaloric effect data showing theratio of temperature change to Geldchange as a function of electric Geld.The temperatures refer to the tem-perature of the sample in zero Geld. T,is the temperature of the heat-capacitymaximum in zero Geld. The lines showthe behavior expected for the para-electric phase if nonlinear effects areneglected as calculated by Eq. (3).The error bars given for some of theexperimental points are typical valuesfor all of the points given. The errorsassociated with the calculated curvesshow the uncertainty in the Curie-Weiss temperature.

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decreases almost linearly with increasing enthalpy,followed by a Curie-Weiss law dependence on tempera-ture. This last feature is obscured in an enthalpy plot.These observations have a rather natural explanationif it is assumed, as in the previous section, that theregion between the maximum in the dielectric constantand the Curie-gneiss law represents a sample in whichtwo phases, ferroelectric and paraelectric, coexist. Interms of this model the ferroelectric phase has adielectric constant approximately twice that of theparaelectric phase at the transition temperature. Thedependence of the sample capacitance, and henceapparent dielectric constant, on enthalpy in the transi-tion region then shows the low-dielectric-constant para-electric phase growing at the expense of the high-dielectric-constant ferroelectric phase. Since the volumein the paraelectric phase should be linearly related tothe enthalpy, the Inost simple, but not the only, inter-pretation of the linear enthalpy dependence of thecapacitance through the bulk of the transition region isthat, in this region, the paraelectric and ferroelectricphases extend in patches through the entire thicknessof the sample. The total capacitance is then calculatedas the parallel combination of two capacitors whosearea is a linear function of the enthalpy. The behaviorat either side of the linear region represents a morecomplicated two-phase pattern, perhaps reAecting theconditions at the edges of the sample.

G. Electrocaloric Measurements

In addition to the calorimetric and dielectric measure-ments, studies of the electrocaloric eGect were made on

the plated sample. These measurements support theconclusion reached in the previous section, that thesample was still at least partly ferroelectric at tem-peratures above those at which the dielectric constantand the heat capacity were maximum. This supportsthe hypothesis of a broadened transition in the platedsample.

Thermodynamics shows that

.where Cz is the heat capacity at constant field and E isthe spontaneous polarization. In the ferroelectric phase(8P/BT)~ is nonzero, even in zero field, while in theparaelectric phase it is zero in zero Geld. This leads to auseful experimental test for spontaneous polarizationin the sample. In the ferroelectric phase, if the Geldchanges are sufhciently small, one can approximate allof the terms in Eq. (1) by constants to And that thetemperature change associated with the adiabatic ap-plication of a field is given by

where the subscript refers to zero Geld. Thus the tem-perature change is linear in the applied Geld, for smallenough Gelds, when spontaneous polarization is present.For purely paraelectric material, if the fields are sufE-ciently small so that nonlinear effects can be neglected,the polarization is given by EE/(T —Te), where Eis'

912 W. REESE 181

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FIG. 7. Heat capacity of the KDP"plated" sample in various appliedelectric fields. The continuous curvesare primarily intended to organize thedata presented rather than to depictthe true form of the transition. As dis-cussed in the text, the transition inzero field is thought to be a broadenedfirst-order transition, so that the ap-pearance of a second-order transitiongiven by the solid curve is largelyillusory. There was some evidence ofa small latent heat associated with thetransition with E=294 V/crn, butthe available evidence concerning thispoint is highly equivocal. No evidenceof a latent heat was seen for the tran-sition with 8=785 V/cm. T.o is thetemperature of the heat-capacitymaximum in zero 6eld.

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the Curie constant and To is the Curie-Weiss tempera-ture. Under these assumptions, one expects a quadraticelectrocaloric effect

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2Cp (T Tp)(3)

The results of the electrocaloric measurements aresummarized by Fig. 6, in which the results of measure-ments at Gve temperatures are shown. T, is taken to bethe temperature of the heat-capacity maximum. Thetemperature of a series of measurements is taken to bethe temperature of the sample in zero field. The electro-caloric effect is linear in Geld for a small range of tem-peratures below the temperature of the heat-capacitymaximum. Those measurements which indicate anelectrocaloric effect approximately quadratic in field arecompared with calculations using Eq. (3), where theCurie-Weiss temperature is taken to be T,—0.11'K, asfound from the dielectric measurements, and the Curieconstant was taken from Craig's results.

The latter data show departures from the quadraticfield dependence predicted by Eq. (3), which indicatesthe effects of nonlinearities. Although it is possible toextend the treatment of the electrocaloric effect toinclude nonlinear effects, the estimates of the nonlinearterms in the susceptibility which could be made by ananalysis of the electrocaloric measurements would notbe useful, since much more precise measurements canbe obtained directly by determining the Geld depen-dence of the dielectric constant. The present results dopoint out that nonlinear effects occur at very low valuesof applied electric Geld near T,. Thus, one must use

care in interpreting the results of Baumgartner" on thefield dependence of the, dielectric~constant and relatedmes, surements, since theafields~'employed in/his workwere considerably in[excess ot&500„V/cm, which~wouldcause considerable nonlinearities nearIT, .

.OI5-

.OIO-

005

~ ~ ~

II

a 4

/,~ ~

]y FROM HEAT CAPACITY

0 FROM HEATING CURVE

I

200 400

E(Vl cm)

1

600t

800

FIG. 8. Shift in transition temperature as measured by theposition of the heat-capacity maximum shown as a function ofapplied field. The dashed line is the assumed linear shift at lowfields. Data taken from continuous-heating experiments tend togive too large a value for the shift, so that the points taken fromthe heat-capacity data shouM be given most weight in assessingthe true shape of the curve. Particularly noteworthy is the factthat little, if any, additional shift is caused by fields in excess of300 V/cm.

"H. Baumgartner, Helv. Phys. Acta 23, 651 (1950).

ORDER —D ISORD ER FERROELECTRI CS. I I I 913

TABLE I. Thermodynamic parameters characterizing the ferroelectric phase transition in KDP and DKDP in zero 6eld and pressure.L is the latent heat, dT, /dP and dT, /dE are the shifts of the transition temperature with pressure and electric field, and AP and A Vare the discontinuous jumps in spontaneous polarization and value.

KDP

DKDP 219.62e

L(J/mole)

46.2

440'

dT./dp('K/kbar)

—4.52+0.06'—5.56&0.3c—2.63+0.05'

3.9 ~0.2g

—0.017+0.0010.021+0.001—0.053+0.001—0.079%0.005

0.022&0.02b0.03 &0.02d0 057~0 006c

aV(cm'/mole)

Kq (5) Measuredd T,/dE

(10 4'K cm/V)

0.47+0.05

1.9 ~0.3~

b,P(pC/cm')Eq. (4)

0.31+0.03

6.5 +1.0

a Reference 15.b Reference 20.

e Reference 16.d Reference 19.

e Reference 4.f Reference 18.

& Reference 17,

D. Measurements in Electric Fields

In addition to the calorimetric measurements in zeroGeld, a number of calorimetric measurements were madein applied electric fields. The results of some of thesemeasurements are presented in Fig. 7, where the resultsof measurements at three different fields are shown. Theeffect of the Geld is to shift the transition to highertemperatures, to lower the heat capacity at tempera-tures below the transition, and to inciease it at tem-peratures above the transition. At a field of 785 V/cm,the pronounced X-like heat-capacity curve has becomea smooth curve which is almost symmetric about thetemperature of the heat-capacity maximum. Moredramatic alterations of the shape of the heat-capacitycurve by fields have been reported for triglycine sul-

phate, "in which the transition in zero field is apparentlyof second order.

Figure 8 shows the shift of the temperature of theheat-capacity peak as a function of Geld derived fromboth discontinuous-heating calorimetric studies andcontinuous-heating experiments. At low fields, theshift of the transition seems linear in field, with dT, /dE= (4.7+0.5)X10 "K cm/V. Clapeyron's equation, inthe form

(4)dE hS

where V is the molar volume and AI' is the discon-tinuous change in spontaneous polarization at thetransition, gives DP= (3.1&0.3))&10 'C/cm', when theresults of the present work are used.

For fields much in excess of about 300 V/cm theapplication of electric fields causes very little, if any,additional shift in the temperature of the transition.Additionally, there is little, if any, evidence from thecalorimetric studies of any latent heat for such fields.This indicates that the transition (if indeed it is ap-propriate to dedne a transition under such circum-stances) loses its first-order character above 300 V/cm.

E. Comyarison with Previous Results

If, as has been argued above, the ferroelectric transi-tion in zero field in KDP is of first order, this result

' 3.A. Strukov, S. A. Turaskin, T. L. Skomorokhova, and K. A.Minaeva, Izv. Akad. Nauk SSSR, Ser. Fiz. 29, 982 (1965).

should be reconciled with the previous results. Each ofthe previous experiments employed suSciently smallheat increments so that an isothermal absorption ofheat would not have been overlooked. There seem twopossible explanations, both based on the fact thatprevious measurements have been made on polycry-stalline samples with ill-defined electrical boundaryconditions. The simplest explanation is that difIerentcrystals, even when grown from the same batch, havedifferent transition temperatures, which obscures thenature of the transition in a polycrystalline sample. Asan example, di6ering samples of the antiferromagnetMnF2 were found to have transition temperatures asmuch as 0,1'K apart, "and even larger shifts of transi-tion temperature with impurity content have recentlybeen investigated in the ferromagnet Gd." A secondpossibility is that, because of the pyroelectric effect,some of the crystals in a container of small crystals Gndthemselves under electric fields during the course of thecalorimetric experiment. This would shift and broadenthe transition. This latter interpretation of the heatcapacity of polycrystalline ferroelectric samples haspreviously been discussed by Strukov. " A third, butmuch less likely, explanation could be that large andsmall crystal samples exhibit diGering behaviors. Sincecooling curve measurements using large fragments andpulverized crystals show that the transition temperatureis independent of crystal size with a precision of 0.1'K,»

and since the samples used in previous experiments hada volume of several mm' while it has been observed thatisolated crystals 10 ' as large showed "normal ferro-electric behavior, '"' this suggestion seems unlikely tobe correct.

The previous' observation of a difference between thetemperature of the calorimetric and dielectric anomaliesmay be related to the inAuence of electrical boundaryconditions imposed by differing surface treatments ofcrystals with and without electrical contacts or topartial clamping efIects of electrodes. In any case, in

"D T. Teany, Ph. ys. Rev. Letters 14, 898 (1965)."F.J. Cadieu and D. H. Douglas, Jr., Phys. Rev. Letters 21,680 (1968).

'4 3. A. Strukov, in I'roceedings of the International Meeting onFerroelectri city (Czechoslovakia Academy of Sciences, Prague,1966), Vol. 1, p. 191."C. Jaccard, W. Kanzig, and M. Peter, Helv, Phys, Pq&g g6,521 (1953).

W. REESE

T('K)

8090

100110120130140150160170

C(J/mole 'K)

46.0351.2555.8960.2064.4068.6872.9977.2581.1185.19

r C('K) (J/mole 'K)

180190200

Region of ferroelectric230240250260270280

89.3593.1698.10

transition106.8108.6111.0113.4115.7118.2

TABLE IZ. Heat capacity of DKDP, except in the immediatevicinity of the ferroelectric transition. Data in the vicinity of thetransition can be found in Table I of Ref. 4, but those data mustbe corrected as discussed in the text by adding 2.40+0.0306TJ/mole 'K to obtain agreement with the present data.

which Hill and Ichiki' made their measurements had alower transition temperature than the one on which thepresent calorimetric measurements were made; it there-fore presumably had a lower deuterium content andperhaps a smaller discontinuous entropy change as well.If both of the values employed in the calculation of thepolarization jump are assumed to be correct, a problemarises, since the calculated value of the jump at thetransition may be significantly larger than the mostrecent experimental values for the spontaneous polariza-tion of O'K."There are, in the opinion of the presentauthor, no really reliable measurements of the polariza-tion in the immediate neighborhood of T, with whichthe values calculated using Eq. (4) can be compared.

the present work the same transition temperature isdetermined calorimetrically and dielectrically when themeasurements are made simultaneously on the samecrystal.

F. Further Thermodynamic Relationships

The determinations of the latent heat in zero field forKDP and DKDP can be combined with recent deter-rninations of the shift of the critical temperature withpressure" " in Clapeyron's equation to predict thediscontinuous change of volume at the ferroelectrictransition,

These results, together with the results of using Eq. (4)to relate the shift of critical temperature with appliedGeld" to the polarization jump, are given in Table I.

While the reported values of the pressure shift of thetransition temperature do not agree within the statederrors, the predicted volume changes are reasonablyconsistent with the best estimates which can be madefrom available x-ray data.""In preparing Table I, areanalysis of Cook's data has been made. The volumechanges originally reported refer to the total change inexcess of that expected by normal thermal expansionover an interval of about 10'K, rather than the dis-continuous volume change at T„which is the valueobtained from Clapeyron's equation. The uncertaintiesin the x-ray data are rather large because of the largeuncertainty in extrapolating the lattice constants to T.,which in turn is due to the very rapid temperaturedependence of the lattice constant expected near T,.

The value calculated for the spontaneous polarizationjump for DKDP may be too large, since the crystal on

'6 H. Umebayashi, B.C. Frazer, G. Shirane, and W. B.DanielsSolid State Commun. 5, 591 (1967).

'r E. Higenbarth and S. Ullwer, Cryogenics 7, 306 (1967).» G. A. Samara, Phys. Letters 25A, 664 (1967).'s R. M. Hill and S. K. Ichiki, Phys. Rev. 132, 1603 (1963)."M. DeQuervain, Helv. Phys. Acta 17, 509 (1944)."W. R. Cook, J. Appl. Phys. 38, 1637 (1967).

IV. HEAT CAPACITY OF DKDP ANDCOMPARISON WITH KDP

A. Heat Capacity of DKDP

In addition to the studies of KDP in the vicinity ofthe ferroelectric transition, an extensive study of theheat capacity of DKDP in the temperature range80-280'K was carried out. Five independent experi-ments involving separate thermometer calibrations andaddenda heat-capacity determinations were conductedover various portions of this temperature range. One ofthese experiments, covering the temperature range185—235'K, which encompasses the ferroelectric transi-tion, has been previously reported in some detail. 4

At the conclusion of these experiments, it was dis-covered that the previously reported results disagreedin the region of overlap with the lower-temperature dataand with a second experiment conducted between 205and 240'K using the crystal with minimum addenda. Itwas concluded that the addendum heat-capacity deter-minations employed in the analysis of the previous datawere seriously in error because of an incorrect procedureused to account for radiation heat loss during equilibra-tion. To check this conclusion, the addenda from theprevious experiment were compressed so as to present assmall a radiating surface area as possible, and were re-measured. The new addendum measurements, whenapplied to the previous data, gave results which agreedwith all other sets of measurements. The most seriousresult of this error is that all the entries in Table I ofRef. 4 should be increased by 2.40+(0.0306'K ')2'J/mole'K. This systematic error had little effect on theanalysis presented since a corresponding incorrectestimate of the background heat capacity was alsomade. After correction of this error in the one set ofdata, all data sets agreed in regions of overlap to betterthan 1%; in most cases the agreement was of the orderof 0.3%.The data are summarized in Table II.

B. Comyarison of KDP and DKDP

If one is to obtain the maximum useful informationconcerning a phase transition from calorimetric mea-

"J.L. Bjorkstam and R. E. Qettel (private communication)

ORDER-DISORDER FERROELECTRICS. lI I 915

suremenis, one must separate the heat capacity into abackground part, here called CG or lattice heat capacity,and an excess part, which is taken to be connected withthe phase transition. A variety of techniques can beadopted to eGect the required separation, ranging froma simple guess made by drawing a "reasonable" smoothcurve consistent with the data to more elaborateguesses based on fitting the heat capacity away fromthe transition to functions with theoretical significance,such as one or more Debye functions. In the currentinstance, any separation of the second type is fraughtwith dangers, since the transitions occur at elevatedtemperatures where the lattice-heat-capacity contribu-tion is expected to be large, a fact complicated by thecomplicated unit cell which will lead to a rich spectrumof possible lattice vibrations. In order to circumvent atleast some of these pitfalls, a technique previously em-

ployed in the analysis of the heat capacity of DKDP4was adopted, namely, estimating C6. from a comparisonof the heat capacities of the normal and deuteratedmaterials. This technique is based on the hope that thelattice heat capacity of the two materials is not toodiQ'erent, even when one is in the paraelectric, tetragonalphase and the other is in the ferroelectric, orthorhombicphase. This seems reasonable since the lattice distortionbetween phases is quite small. A secondary assumptionis that, in the vicinity of a transition in one salt, theheat capacity of the other is entirely, or almost entirely,background heat capacity. This seems reasonablebecause the transitions are widely spaced in tempera-ture. Regardless of the validity of these assumptions,such a procedure is well defined operationally.

The results of the comparison, concentrating on theregions not too close to the transitions, are presented inFig. 9. The present measurements of both KDP andDKDP were used below 140'K. The results of theexperiment in which heat-capacity measurements werecarried out on both isomorphs with the same addendumand temperature scale were weighted most heavily,since the possible effects of systematic errors are thenminimized so that the accuracy of the comparison is ofthe order of the combined precision of the measurements,about 0.25%.At higher temperatures the present DKDPmeasurements are compared with the KDP measure-ments of Stevenson and Booley. ' Because of the goodagreement between the Stevenson-Booley results andthose of the present work, which was cited at thebeginning of Sec. III, it is felt that the accuracy of thecomparison is better than the combined accuracies ofthe two measurements. An over-all estimate of accuracyof the comparison shown in Fig. 9 is perhaps &0.1E.The heat capacity of DKDP is found to be larger thanthat of KDP above about 130'K. This seems a remark-able conclusion, with important implications concerningthe theoretical analysis of the underlying mechanismsof the phase transition in KDP-type ferroelectrics.

One expects the vibrational frequencies in DKDP tobe smaller than in KDP because of the larger mass of

l.2

0,9

0.6

0.3K

O

OI

ILCIVC5

o -0,3

0 t

«0.6

l00I

!50T{'x}

1

2001

2 50.

P FIG. 9. Difference in heat capacity of KDP and DKDP asobtained from the present results and those of Stephenson andBooley. The arrows mark the location of the ferroelectric transi-tions. The dash-dot line gives the result of the simple calculationof the residual difference based on the infrared spectrum outlinedin the text. The dashed curve gives the estimate of the residualdifference which was used for further analysis. Perhaps the mostsignificant feature of this figure is the scant evidence for a con-tribution of about 0.5E to the heat capacity above T, due to theresidual disordering of protons within the possible H-bondpositions.

the deuteron as compared to that of:.the proton. Thusit is expected that the heat capacity of KDP will besmaller than the heat capacity of DKDP. An estimateof this contribution is shown as the dash-dot line inFig. 9.This estimate was based on the observed infraredabsorption spectrum of the two isomorphs. "" Thedata of Imry et al. have been given preference, becausethese data give absorption spectra for both isomorphsin both phases. It was assumed that the heat capacityassociated with optical modes could be calculated usingEinstein functions with temperature-independent fre-quencies. The largest part of the diBerence in heatcapacity in such a calculation comes from the p(OH)and B(OH) modes, but the modes associated with PO4vibrations also give non-negligible contributions. Ac-count of acoustic modes was taken by employing Debyefunctions with Debye temperatures scaled by the squareroot of the unit-cell mass. This last contribution isalmost negligible in the temperature range considered,never amounting to as much as 0.018..

Although the above calculation has the virtue ofsimplicity, it does not work particularly well. Forexample, at 280'K the above calculation underestimatesthe total heat capacity of both isomorphs by about

"R.Blinc and D. Hadzi, Mol. Phys. 1, 391 (1958).24A. S. Barker and M. Tinkham, J. Chem. Phys. 3'8, 2257

(1963).25 Y. Imry, I. Pelah, and E. 'Wiener, J. Chem. Phys. 43, 2332

(1965).

55

35C3

Cl

25

l5

previous, much simpler, estimate of the lattice heatcapacity of KDP.' Here the conclusion seems almostinescapable, unless one assumes that C—Cg for DKDPis non-negligible at much lower temperatures thanmight be expected from the temperature dependence ofthe spontaneous polarization, or that there are largedifferences in Cg between the tetragonal and ortho-rhombic phases. This conclusion is surprising, since themost widely accepted theories of the ferroelectric transi-tion in KDP are elaborations of an idea presented longago by Slater, "that the transition could be understoodin terms of the ordering of protons in the availableH-bond positions. Any such theory predicts a sizeablecontribution to the heat capacity due to disorderingprocess (about 0.5E) for a sizeable temperature region(about 100K') above the ferroelectric transition. Sincethe estimates of lattice heat capacity made here arenot expected to be wrong by such a substantial amount,little evidence for the existence of such a heat-capacitycontribution is found.

, lO-4 2 IO" 5 2

/Tc

Fro. 10. Behavior of the heat capacity of the "foil-wrapped"sample just below the temperature of the ferroelectric transition.hC is the difference between the measured heat capacity and Cg.The two curves are empirical its to the data.

10—12%. This indicates that the spectrum of latticevibrations is not nearly so simple as has been assumed.Such assumptions as temperature-independent fre-quencies for the optical modes are not strictly correcteither. The hope on which the calculation is based isthat, however the true spectrum of lattice vibrationsdiffers from the simple model, the variation is roughlythe same for both isomorphs. Fortunately, the majorityof the region of interest is off the scale of Fig. 9, showingthat the uncertainties discussed above are not as

important as the discussion may make them seem.The results of the calculation outlined above seems

unsatisfactory because, when the calculated curve istaken as the residual difference between the lattice heatcapacities, there is little symmetry in C—Cg betweenthe two isomorphs, particularly above the transitiontemperature. The dashed curve in Fig. 9, which repre-sents a compromise between the simple calculation out-lined above and a desire for a degree of similarity in thetransitions of the toro isomorphs, has been adopted forfurther analysis. Uncertainty in the difference betweenCg for the two substances is of the order of a few tenthsof E unless the assumptions on which the estimate wasmade are invalid.

The analysis shows that C'—Cg is small for KDP atany temperature above 140'K, seemingly with anyreasonable assumption as to the residual difference inlattice heat capacity. This result was anticipated in a

C. Transition Entroyy and Behavior near T,

Using the measured heat capacity of DKDP andKDP and the estimate of the residual difference betweenC6 for the two isornorphs given by the dashed line in

Fig. 9, an estimate of C|.- for both materials is available.Using this estimate of CG, the entropy change associatedwith the ferroelectric transition to T, is (0.457+0.034)8for DKDP and (0.400&0.036)R for KDP. The quoteduncertainties take into account bo]h the uncertainties inthe experimental data and the much larger uncer taintiesin the lattice-heat-capacity estimate. These values differsomewhat from previous determinations of the transi-tion entropy primarily because a different estimate ofthe lattice heat capacity was made. No attempt wasmade to calculate the entropy difference above T,(which is presumably associated with the destructionof short-range order), since the heat-capacity excess is

small, and the value obtained in such a calculationdepends critically on the estimate of Cz.

Using the present estimate of Ct.-, the behavior of theheat capacity in the neighborhood of the phase transi-tion can be examined. Such an examination for KDP,using data from the "foil-wrapped" sample, is presentedin Fig. 10. Only the results below T, are examined,since any uncertainty in Cg is much more importantabove T, than below and can seriously prejudice theanalysis above T,. In the region 10 '& c)3&(10 ', theheat capacity seems proportional to 1/e'~s, which is thewell-known Ornstein-Zernike result. However, in theregion inside about &=2&10 4, there seems a decidedpreference for a logarithmic fit. In Fig. 10 any pointswhich may have been affected by superheating havebeen excluded by casting out any points for which theinitial and/or final temperature fell above the dashedline in Fig. 2. Very similar behavior was observed for

"J.C. Sister, J. Chem. Phys. 9, 16 (1941).

OR DER —D ISORD ER I' E RROELECTRI CS. I I I 917

DKDP, namely, a 1/ge dependence starting at about&=10 ' which went over to a logarithmic dependencecloser to the transition. The behavior in KDP is muchmore pronounced than in DKDP, as one might expectin terms of the usual interpretation of "critical-point"phenomena, since the transition in KDP is much morenearly of second order than in DKDP.

Because the transition is of first order and may besomewhat broadened, it is dificult to decide if one orthe other of these temperature dependencies representsthe "true" behavior in the vicinity of the phase transi-tion or if the apparent rapid transition from one formto the other is a real effect rather than an artifact of theexperiment. However, the author does view the resultsas oGering quite interesting suggestions concerning thebehavior of the heat capacity near the transition.

V. COMPARISON WITH THEORETICAL MODELS

The simplest model with which one can compare ourresults is the phenomenological model, widely identifiedwith Devonshire" in the fieM of ferroelectricity, whichin the current instance is equivalent to the Landau"expansion of the free energy. In this model one expandsthe free energy per unit volume as

F=Fp+ (1/2K) (T Tp)P'+ 'y—P'+ pbP' -(6-)where the coefficients are assumed, in the absence ofcontrary evidence, to be temperature-independent andare chosen to fit experiment. In the above, Fo is thefree energy with zero polarization, E is the Curieconstant, To is the Curie-Weiss temperature, and P isthe polarization. The thermodynamic results of thisassumed equation of state have been thoroughly ex-

plored, " so that we will use the results withoutderivation.

The difference between T, and To and the latent heatcould be used together with the Curie constant toevaluate the parameters in Eq. (6). From this theexpected behavior in the electrocaloric measurementsand heat-capacity measurements in fields, etc., could becomputed. Such a procedure has little merit, since thetheory does not seem to work particularly well in theimmediate neighborhood of the ferroelectric transitionwhere the bulk of the current work is concentrated.However, before making an examination of this point,it may be well to point out a success of the theory. Thetransition entropy is predicted to be

65= (1/2K)Pp'

where Po is the spontaneous polarization at O'K. Usingthe most recent values for Po" and E," the model

'~A. F. Devonshire, Phil. Mag. 40, 1040 (1949); 42, 1065(1951);Advan. Phys. 3, 851 (1954).

'SL. D. Landau, Zh. Eksperim. i Teor. Fiz. 7, 1232 (193'l)."Perhaps the most detailed exposition is that of E. Fatuzzo

and W. J. Merz, Ferroelectricity (North-Holland Publishing Co.,Amsterdam, 196i), Chap. 3.' R. J. Mayer and J. L. Bjorkstam, J. Phys. Chem. Solids 23,619 (1962).

predicts 65/R=0. 452 for DKDP and 0.352 for KDP.Since there is about a 10—20% uncertainty in the calcu-lated value, rejecting the uncertainty in Po, rather goodagreement between this calculation and the presentexperimental results is found. This does not contradictthe claim concerning the failure of the theory to describeKDP near T„since the bulk of the transition entropycomes from more than 1'K below T,.

Ginzburg" pointed out some time ago that one doesnot expect the Landau theory to work well near a phasetransition because of the effects of Ructuations, whichbecome particularly serious if the transition is ofsecond order or is nearly of second order (as in the caseof KDP). Since it is not surprising that the theory wouldencounter difficulties near T„only the most obviousfailure will be noted. Based on the present latent heatmeasurement one would predict that the spontaneouspolarization jump would be about 34% of Pp, which isto be contrasted with a jump of about 6% of P p deducedfrom the present measurements through the Clayperonequation.

Having pointed out that the Landau theory isinadequate to describe KDP in the immediate vicinityof T„ the question of interest is how near one must getto T, before the inadequacy becomes apparent. Sincethere are no sufficiently precise measurements of thespontaneous polarization available to test the prediction

hC= (T/2K) (d/dT) (P') (g)

only a few general comments on the question can bemade. Taken strictly, the Landau theory does notpredict the 1/ge dependence of AC which is observedfor t.&10 '. From this point of view one might suspectthat the &=10—' marks the region of validity of thefree-energy expansion. If true, this may be consideredrather surprising, since, in his consideration of theregion of validity of- the Landau theory to ferro-electrics, Ginzburg estimated the region of validity tobe outside &=10 4, because of the long range of theinteractions involved. If &=10 ' marks the range ofvalidity of the Landau theory, when applied to KDP,this is an indication that the relevant interactions havea short range in KDP, as envisaged by theories of theSlater type. On the other hand, it seems possible toargue that the Ornstein-Zernike 1/ge dependence of5C represents the same type of account of short-rangecorrelations (in the mean) as is implied by the Curie-Weiss law, and hence that, while not predicted by theLandau theory, this dependence of hC is compatiblewith microscopic models which lead to the same rela-tionship between P and E. In this way it seems possibleto argue that the Landau theory should be essentiallyvalid, at least as far as the relationship between P andE is concerned, until the Ornstein-Zernike dependenceof the heat capacity is seriously violated, which isaround 2)&10 4 for KDP. (See note added in proof. )

3'V. L. Ginzburg, Fiz. Tverd. Tela 2, 2031 (1960) tEnglishtransl. : Soviet Phys. —Solid State 2, 1824 (1960)).

ISI

~ KDPR DKDP

Ie -20

S.U. S.

FIG. 11.Heat-capacity excess C—Cgfor KDP and DKDP as deduced fromour measurements and the latticeestimate as shown in Fig. 9. Note thechanged vertical and horizontal scalesin the region 0.975(T/T, (1.0. Thecurves are calculated using the SUStheory with parameters chosen to fitthe data near T/T, =0.96.

e~ ~

e

~ RR«R~

~ Re e e1 I

.70 .80 s90 1.00 l.OT5

A completely different approach from that embodiedin Eq. (6) consists of detailed calculations based onmicroscopic models. For KDP the best developedamong such models are elaborations of the originalSlater model which consider that the ordering ofprotons in the possible H-bond positions plays anessential role in the phase transition. ""However, as

emphasized by Tokunaga and Matsubara, " it isessential to include a coupling of the proton ordering tothe motions of the K+ and PO4 ions in order to explainthe spontaneous polarization. Because of its greatersimplicity, the model of Silsbee, Uehling, and Schmidt"(SUS) will be emphasized. This model will not beexpected to work as well for KDP as for DKDP becauseof the neglect of tunneling.

In the SUS model, one introduces three parameters:eo and e~, which give the energy of various proton con-figurations, and P, which is a mean-field term. Theparameter e& has been measured for DKDP by theanalysis of the temperature dependence of the NMRrelaxation time. The transition temperature is taken asan input parameter, reducing the number of free

parameters to one in the case of DKDP and two forKDP. One expects the energy parameters eo and ~~ tobe related to the proton off-center distance in theH-bond, because the conigurations would not physi-cally differ if the off-center distance were zero. Thus,one can scale from one isomorph to the other using thesquare of the oB-center distance and greatly reduce thetotal number of free parameters.

ss Y. Takagi, J. Phys. Soc. Japan 3, 271 (1948); 3, 273 (1948).'3H. B. Silsbee, E. A. Uehling, and V. H. Schmidt, Phys. Rev.

133, A165 (1964)."R.Blinc and S. Svetina, Phys. Rev. 147, 430 (1966).35M. Tokunaga and T. Matsurbara, Progr. Theoret. Phys.

(Kyoto) 35, 581 (1966).ss K. K. Kobayashi, J. Phys. Soc. Japan 24, 497 (1968).

In Fig. 11 the heat-capacity data are compared withthe calculations of the SUS model. The calculation forDKDP has previously been presented using the sameparameters. Parameters were chosen which Gt thecalorimetric data for DKDP in the region of T/T, near0.96, since in this region the value of Ac is little affectedby uncertainties in the choice of C&. This calculation isgiven by the dashed line. The same parameters alsogive a good 6t to the measured spontaneous polariza-tion. The energy parameters were then scaled by thesquare of the off-center distance and adjusted slightlyto better fit the calorimetric data for KDP in the samereduced temperature region. The resulting curve is thesolid line in Fig. 11.

The calculation for DKDP is a much better fit to thedata than is the calculation of KDP, presumablybecause the neglect of tunneling is a better approxima-tion for DKDP. For both substances a transitionentropy of 0.522 is predicted, which is too large to fitthe experimental data. The predicted values for thelatent heat are 30% too large for DKDP and 580%%uo toolarge in the case of KDP. In the first case, this mayreQect just the difficulties of a mean-Geld model insatisfactorially accounting for phenomena at phasetransitions. In the second case, one may also haveserious corrections due to the neglect of protontunneling.

The calculations of Blinc and Svetina'4 are the mostcomprehensive of any Slater-type theory which incor-porates proton tunneling. Because of the complexity oftheir equations and the multitude of free parameters,no attempt to carry out computations based on theirtheory was made. The original computations'4 showthat the deviations between the data and the SUStheory are of the type which can be explained by theinclusion of tunneling. In particular, these results show

ORDER —DISORDER FERROELECTRI CS. I I I 919

that tunneling reduces the value of the heat capacity inthe neighborhood of the transition, thus qualitativelyexplaining the fact that the region of experimentallysignificant DC extends to significantly lower reducedtemperatures than is predicted by the SUS theoryfitted to the data near T,. Since tunneling is moreimportant in KDP than in DKDP, one can qualitativelyunderstand why the Gt is better for DKDP than forKDP and why the transition entropy is greater inDKDP than for KDP. Further, tunneling calculationsof the transition entropy greatly reduce the discrepancywith experiment.

There is, however, one significant discrepancy be-tween the computations and experiment which cannotbe resolved simply by the inclusion of tunneling intothe SVS theory. This is the observation that there islittle, if any, heat-capacity excess over background atreduced temperatures greater than about 1.10. Assum-

ing that this e8ect is real, and not a result of an in-correct estimate of C|.-, presents a distinct problem toany theory based on a Slater-type model. Two possibleresolutions of this problem seem worthy of mention.First, perhaps the assumption of constant energy forthe proton configurations is incorrect. Physically, thismight be due to the volume changes in the lattice, whichare emphasized by the values of the discontinuousvolume change given in Table I. The volume changesmight change the energies sufficiently to give theobserved results. A second, and perhaps more interest-ing, suggestion might be constructed along the followinglines. Recently Kobayashi" has suggested that thetransition in KDP is of a mixed type, in which there isa lattice instability arising because of a coupling of the

proton ordering with the K+-P04 optical vibration.Evidence for the existence of the resulting "soft mode"has been recently found in Raman scattering data. "Although the calculation has not been performed, it ispossible that, because of the coupling between theproton ordering and the optical mode, the behavior ofthe heat capacity in the Kobayashi theory is qualita-tively diGerent above T, than is predicted by moreconventional theories, which emphasize;the role ofproton ordering the KDP phase transition. It'is to behoped that further theoretical work will be devoted tothis question.

)Vote added ws proof. The observed Qe dependencefollows from Eq. (6) if y'K/hTO(&1. Baumgartner'sdielectric data imply that this is the case for KDP. Thusthe Qe dependence does rot imply a failure of theLandau theory.

ACKNOWLEDGMEN'TS

The author would like to acknowledge the contribu-tions of L. F. May for his expert assistance in theproblems of an experimental nature and for taking someof the data reported here. The cooperation of Dr. W. R.Cook in making his data on thermal expansion of KDPand. BKDP available proved most"helpful. The authoralso wishes to extend his thanks+to~&Professor J. L.Bjorkstom for making the recent data concerning thespontaneous polarization available to him. Finally, theauthor would like to acknowledge Iuseful communica-tions from Dr. G. A. Samara and Professor V. H.Schmidt.

3~I. P. Kaminow and T. C. Damen, Phys. Rev. Letters 20,1105 (1968).