ARTICLES. Optical, elastic, and dielectric studies of the phase transitions in lawsonite

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1 PHYSICAL REVIEW B VOLUME 62, NUMBER 0 ARTICLES SEPTEMBER 2000-II Optical, elastic, and dielectric studies of the phase transitions in lawsonite P. Sondergeld, W. Schranz, and A. Tröster Institut für Experimentalphysik, Universität Wien, Strudlhofgasse 4, A-090 Wien, Austria M. A. Carpenter Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, United Kingdom E. Libowitzky Institut für Mineralogie und Kristallographie, Universität Wien, Althanstraße 4, A-090 Wien, Austria A. V. Kityk Institute of Physical Optics, Dragomanova Street 23, Lviv, Ukraine Received 20 July 999; revised manuscript received 30 March 2000 Results of optical birefringence, ultralow-frequency elastic, dilatometric, and dielectric measurements of lawsonite crystals are presented in a wide temperature range including two successive phase transitions Cmcm(T 273 K) Pmcn(T 2 20 K) P2 cn. Pronounced pretransitional effects are observed in macroscopic properties in a large temperature region (200 K) above T. The low-temperature transition at T 2 20 K is found to be proper ferroelectric. A deviation from the Curie-Weiss law appears in the temperature dependence of the dielectric susceptibility in the region of the ferroelectric transition below T 2. Frequency dependent measurements of the dielectric permittivity yield evidence that ferroelectric domain-wall motion is responsible for this deviation from the standard behavior. I. INTRODUCTION Lawsonite, CaAl 2 Si 2 O 7 (OH) 2 H 2 O, is a mineral which is a common constituent of high-p and low-t metamorphic environments like blueschists. Containing wt % H 2 O, it may be responsible for carrying water down into Earth s mantle. After several subsequent investigations by different authors, Baur has refined the structure at room temperature in space group Cmcm Fig. a. The phase transitions in lawsonite are primarily related with proton ordering. Below T 273 K the structure changes to space group Pmcn, which is accompanied by a loss of the 00-mirror plane Fig. b. 2 The H 2 O molecules rotate within the 00 plane. The phase transition Cmcm Pmcn occurs within the same crystal class. As will be shown below, this transition is related with an instability at the Brillouin zone boundary, i.e., can be classified as antiferrodistortive. Results of polarized Fourier transform infrared FTIR spectroscopy 3 at different temperatures provide strong evidence that this phase transition is not caused by the real rotation of hydroxyl groups and H 2 O molecules to static, symmetric positions, as drawn in Fig. a, but rather by a dynamical and disordered occupation of hydroxyl and H 2 O sites around the highly symmetric positions. In addition, a single-crystal NMR study also indicates a disordered H sublattice in lawsonite, the single H sites being close to those of the low-temperature Pmcn structure. 4 The librational motion of the hydroxyl and H 2 O groups is also indicated by the anisotropic displacement parameters of the H atoms. Refinements using the x-ray data of Libowitzky and Armbruster 2 and a first neutron data set at room temperature 5 revealed that the Hh atoms show an extremely strong anisotropic displacement parameter along 00. This is in agreement with the Hh dynamic disorder around the O4 atoms essentially within the 00 plane. Below T 2 20 K the structure changes to the polar space group P2 cn, which is characterized by a loss of the 00-mirror plane Fig. c. The Hwa atoms of the H 2 O groups deviate downwards from the 00 plane, and those of the hydroxyl groups (Hha) deviate upwards. In this paper we report results of optical birefringence, ultralow-frequency elastic, dilatometric, and dielectric studies together with their phenomenological analysis, which is essential for under- FIG.. Structure of lawsonite projected along 00: a Cmcm at room temperature; b Pmcn at 233 K; c P2 cn at 0 K. Numbers denote the deviation of the atom from the 00 plane (X0) in Å 0 2. Dashed lines indicate H...O bonds shorter than 2.00(2) Å /2000/620/6435/$5.00 PRB The American Physical Society

2 644 P. SONDERGELD et al. PRB 62 FIG. 2. Temperature dependences of the optical birefringence along the principal crystallographic directions of lawsonite. In the inset the temperature dependence of the derivative d(n xy )/dt is shown. standing the origin of both transitions. There is not much information about the phase transition at T 2. Up to now, even its transition temperature has not been precisely determined. Here we present clear evidence that its origin is proper ferroelectric. II. EXPERIMENTAL SETUP For the present investigation we used crystals of lawsonite from Tiburon Peninsula, Marin County, California. The sample no. G4555 of the South Australian Museum, Adelaide, Australia was kindly given to us by Dr. Allan Pring. The crystals were oriented according to the natural faces morphology and checked optically for macroscopic defects, inclusions, etc. The sample dimensions were: mm 3 for birefringence, mm 3 for dilatometric and elastic measurements and mm 3 for dielectric measurements. A polarizing microscope Zeiss Axiophot equipped with a Linkam hot stage was used for the optical experiments. The birefringence was measured fully automatically using the Senarmont compensation method. 6 A variable rectangular aperture allowed selection of a part of the crystal sample free from macrodefects and natural inclusions. Oriented crystal plates along 00 direction with silver painted electrodes were used for measurements of the dielectric constant. The dielectric measurements were performed using an impedance analyzer manufactured by Hewlett-Packard HP492A and controlled by a personal computer. The temperature of the sample chamber was regulated by a commercial temperature FIG. 3. Temperature dependence of the lattice parameters of lawsonite. controller Eurotherm 88 between 80 and 750 K. The lowfrequency elastic measurements were performed by a parallel plate stress PPS method using a commercial dynamic mechanical analyzer DMA-7, Perkin-Elmer. In the PPS mode this device also provides simultaneous automatical determination of the thermal expansion with an accuracy better than 2 nm. 7 III. EXPERIMENTAL RESULTS Figure 2 shows the temperature dependences of the optical birefringence along the principal crystallographic directions. The baselines dashed for the birefringence and the thermal expansion data were obtained from fits of equation y(t)y 0 y s coth( s /T) ( s 26.6 K; quantum saturation has been taken into account to the experimental curves in the temperature range 450T630 K. However, since the precise theoretical description of the birefringence data is yet unknown, they should be regarded as a guide to the eye. The variations of birefringence above T are nonlinear along all directions. The corresponding deviations, which appear to be especially pronounced along the 00 direction, indicate strong pretransitional effects. These fluctuation effects occur in an extremly wide temperature interval (200 K above T ). A kink, which usually occurs more or less pronounced in the temperature dependence of the birefringence at T c in many other crystals, does not appear here at all. Instead, a smeared anomaly is observed and the precise phase transition temperature cannot be determined directly from the data. To estimate the phase transition temperature we investigate the temperature derivative of the optical birefringence. From its maximum value see Fig. 2, inset one obtains T

3 PRB 62 OPTICAL, ELASTIC, AND DIELECTRIC STUDIES OF FIG. 4. Temperature dependences of the Young s moduli of lawsonite crystals measured along the principal crystallographic directions. The applied force F.3 N, f Hz. The resolution is about %. 27 K. In the low-temperature region additional anomalies of the optical birefringence are observed at T 2 20 K along all crystallographic directions, but the most pronounced one occurs along the 00 direction. These anomalies are attributed to the low-temperature transformation Pmcn P2 cn. Figure 3 shows the temperature dependence of the lattice parameters of lawsonite calculated from our thermal expansion data, using the room-temperature values of the lattice parameters as reference points. 2 They show anomalous changes at the same temperature as the optical birefringence data. The most pronounced anomalies occur for the a parameter, where the behavior of thermal expansion clearly shows a kink at T 2 and a rounded smeared anomaly near T in agreement with the optical birefringence data. It is emphasized that the thermal-expansion coefficient along this direction changes the sign in the vicinity of the transition point. This indicates a strong coupling between the ordering process order parameter and the strain tensor component U xx U. Such a strong coupling should also influence the temperature dependence of the Young s modulus. Indeed, Fig. 4 shows that the most pronounced softening near T occurs again along the 00 direction. In Fig. 5 the temperature dependence of the dielectric constant a measured along the 00 direction is presented. In particular, it shows an anomalous peak at about 20 K. In addition we observed hysteresis loops below 20 K, implying that the P2 cn phase is ferroelectric. The inverse dielectric constant a behaves linearly with temperature in a large temperature range above and below T 2 except in the close vicinity of T 2. This behavior is typical for proper FIG. 5. Temperature dependences of the dielectric constant a measured a and inverse value a calculated from the data b of lawsonite measured along 00 direction for different frequencies. The electric measuring field E ac V/cm, E dc 0. The dashed line indicates Curie-Weiss behavior. ferroelectrics. However, more detailed considerations also indicate a nontypical T dependence of the dielectric anomaly: The ratio of slopes a (TT 2 )/ a (TT 2 ), which according to the Curie-Weiss law should be :2, essentially deviates from the value predicted by Landau- Ginzburg theory. As will be shown below, this disagreement may be due to the influence of ferroelectric domain-wall motion. IV. DISCUSSION A. Free energy Let us denote by 2 x,2 y,2 z the length of the conventional Cmcm base-centered unit cell b o. According to Ref. 8 the primitive unit cell vectors a, a 2, and a 3 are a x, y,0, a 2 x, y,0, a 3 0,0,2 z and the reciprocal-lattice vectors are b, x y,0 ; b 2, x y,0 ; b 3 0,0, z. The intermediate phase Pmcn belongs to the primitive orthorhombic Bravais-lattice-type o, the unit cell of which coincides with the conventional Cmcm unit cell. The transition Cmcm Pmcn is associated with an instability at the wave vector k 5 (b b 2 )/2 here and below in Kovalev s notation 8. The star of the wave vector consists of a single

4 646 P. SONDERGELD et al. PRB 62 vector and all space-group irreducible representations in this point of the Brillouin zone are real and one dimensional. The order parameter Q for this transition transforms according to the irreducible representation T 3. The low-temperature phase transition Pmcn P2 cn can be described in the simplest case as a distortion P (k0) where P is the x component of the polarization vector of the intermediate phase, where the soft mode instability appears in the center of the Brillouin zone (k 9 0) associated with the one-dimensional representation T 4 of the space group Pmcn. Following the standard prescription of, e.g., Ref. 9, the free-energy expansion for the two phase transitions can be written as FF Q,U,DF P,U,DF U F D, F Q,U,D A 2 Q 2 B 4 Q 4 C 6 Q 6 a U a 2 U 2 a 3 U 3 Q 2 2 b 4U 4 2 b 5 U 5 2 b 6 U 6 2 Q 2 FIG. 6. Temperature dependence of the squared spontaneous strain. n i n o 3 2 i o i n 3 o 2 3 F 2 2 D n o i 2 c iq 2 o, 2 2 c D 2 c 2 D 2 2 c 3 D 3 2 Q 2, F P,U,D A 2 2 P 2 B 2 4 P 4 C 2 6 P 6 a U a 2 U 2 a 3 U 3 P 2 2 b 4U 4 2 b 5 U 5 2 b 6 U 6 2 P 2 2 c D 2 c 2 D 2 2 c 3 D 3 2 P 2, F U 2 C o U U, F D 2 o i D i 2. Here,,...,6Voigt notation and i,2,3. A A o (TT ) and A 2 A 2o (TT 2 ); A o,a 2o, o B,B 2,C,C 2 are assumed to be positive; C and o i are the bare values at TT ) of the elastic constants and optical polarizabilities, respectively. The coupling parts of the free energy responsible for the interactions between the order parameters and static deformations U or electrical displacements D j at high optical frequencies are also included. It should be noted that only the basic interactions are included in these parts of the above free-energy expansion Eq.. B. Optical birefringence, thermal expansion, and elastic anomalies In the mean-field approximation, the anomalies observed in the optical, 0 thermal, and elastic properties are easily obtained using the free-energy expansion Eq.. In particular, for the refractive indices and thermal expansion one obtains a qualitatively similar behavior: T 2 TT : U S o a Q 2 o,,,2,3, 3 where Q o is the equilibrium order parameter. TT 2 : n i n o 3 2 c iq 2 o c i P 2 o, 4 U S o a Q 2 o S o ap 2 o,,,2,3, 5 where P o is the equilibrium order parameter in the ferroelectric phase; n i and U are the anomalous changes of the refractive index and spontaneous deformation, respectively; n o is the average refractive index; S o C o is the bare elastic compliance tensor. One realizes that the anomalous changes of birefringence and the deformation in the mean-field approximation are both proportional to the square of the order parameters. Indeed, in a plot of n i versus n j or U versus U, straight lines are obtained in accordance with Eqs. 2 and 3. As an example, Fig. 6 illustrates the temperature dependences of U 2 2 Q 4 o. The straight lines suggest that the phase transition at T 273 K in lawsonite is at least very near to a tricritical point, i.e., Q o (T T) /4. However, due to the large contribution of order parameter fluctuations no definite conclusion can yet be drawn about this point. 2,3 C. Dielectric susceptibility The dielectric susceptibility is readily calculated using the free-energy expansion and the relation ij 2 F P i P j. 6 One obtains for T TT 2 while for TT 2 A 2o TT 2 7

5 PRB 62 OPTICAL, ELASTIC, AND DIELECTRIC STUDIES OF A 2o TT 2 3B 2 P 2 o 5C 2 P, 8 4 o where B 2 results from a renormalization of the coefficient B 2 due to strain coupling. In case of a second-order phase transition, assuming B 2 C 2 P 2 o, one can neglect the sixth-order term, leading for TT 2 to 2A 2o T 2 T. Indeed, Fig. 5 clearly displays a linear temperature dependence of also below T 2. But the ratio of slopes (T T 2 )/ (TT 2 ):.25 at f 20 khz) is quite different from the expected behavior :2. One possibile explanation of this discrepancy is to assume a contribution of ferroelectric domain walls to the dielectric response. In that case the dielectric permittivity would be enhanced below T 2. In the simplest form, the dielectric permittivity in presence of domains can be written as ij,t 2 F P i P j D. 0 i P i D The last part corresponds to the contribution of domain-wall motion, which we have written as a Debye response for simplicity, while usually a distribution of relaxation times is more appropriate. D can in principle be calculated for a simple arrangements of ferroelectric domains, but for the present discussion we use only the fact that D (TT 2 ) 0, while, of course, D (TT 2 )0. For D one obtains a domain wall contribution to below T 2 in agreement with experiment, whereas for D this contribution vanishes. Figure 5 shows measurements of at three different frequencies. With increasing frequency one observes a clear tendency towards monodomain response. Nevertheless, a significant domain-wall contribution still exists at f 2 MHz, implying D 0 7 s. It should be stressed that no dispersion is found above T 2, i.e., P D 0 7 s, where P is the order parameter relaxation time. 9 V. SUMMARY We have presented results of optical birefringence, ultralow-frequency elastic, dilatometric, and dielectric measurements of lawsonite single crystals performed in a wide temperature range K. We found anomalies in agreement with two continuous phase transitions Cmcm T 273 K PmcnT 2 20 K P2 cn. The phase transition at T is accompanied by enormous birefringence tails starting about 200 K above T. The tail is most pronounced in n xy. A theory for the description of the large fluctuation tails in lawsonite is in preparation. Below T 2 20 K we have detected D-E hysteresis loops, implying that the low-temperature phase is proper ferroelectric. Consistently, the dielectric permittivity increases by a factor of 7 when approaching T 2 from both sides. The inverse dielectric permittivity yields a linear temperature dependence above and below T 2 with a ratio of slopes :.25 instead of :2 as expected from a simple Curie- Weiss law. This deviation can be explained in terms of a contribution of ferroelectric domain-wall motion to the permittivity below T 2. The characteristic relaxation time of the domain-wall motion is D 0 7 s. This value may be compared to the one in KD 2 AsO 4, where domain-wall contributions to have been identified. In this material, just below the transition temperature T c 62.5 K, one finds a relaxation time D 0 8 s, shifting to higher values on lowering the temperature. Around T f 43 K, domain freezing occurs in KD 2 AsO 4. In lawsonite the domain walls are mobile down to the lowest measured temperatures T80 K. Further work to study the domain-wall motion in lawsonite at low temperatures and high frequencies GHz region is in progress. ACKNOWLEDGMENTS The present work is performed in the frame of the EU Network on Mineral Transformations Contract No. ERB- FMRX-CT Support from the Österreichischen Fonds zur Förderung der wissenschaftlichen Forschung, Project No. P2226-PHY is gratefully acknowledged. W.H. Baur, Am. Mineral. 63, E. Libowitzky and T. Ambruster, Am. Mineral. 80, E. Libowitzky and G.R. Rossman, Am. Mineral. 8, S.P. Gabuda and S.G. Kozlova private communication. 5 G. Lager private communication. 6 M. Zimmermann and W. Schranz, Meas. Sci. Technol. 4, W. Schranz, Phase Transit. 64, O.V. Kovalev, Representation of the Crystallographic Space Groups, edited by H.T. Stokes and D.M. Hatch Gordon and Breach Science Publishers, New York, 993, p Yu.A. Izyumov and V.N. Syromyatnikow, Phase Transitions and Crystal Symmetry Kluwer Academic Publishers, Dordrecht, The Netherlands, 990, p J. Fousek and J. Petzelt, Phys. Status Solidi A 55, 979. V. Novotna, H. Kabelka, J. Fousek, M. Havrankova, and H. Warhanek, Phys. Rev. B 47, Below T, both bulk and fluctuation contributions of the equilibrium order parameter enter into any theoretical description of measured birefringence and thermal expansion. Of course, individual isolation of these contributions depends in a crucial way on the underlying model used. Work on this problem is in progress. 3 M. Fally, P. Kubinec, A. Fuith, H. Warhanek, and C. Filipic, J. Phys.: Condens. Matter 7,