Exploration of Structure-Property Relationships

CRYSTALLOGRAPHY IN SWITZERLAND 562

CHIMIA 2001,55, No, 6

Chimia 55 (2001) 562-569 © Schweizerische Chemische Gesellschaft ISSN 0009-4293

J(jrgen Schreuer*

Abstract: Electrostriction and elasticity provide highly sensitive probes forthe exploration of structure-property relationships. With the aid of modern high-resolution capacitive dilatometers reliable linear and quadratic electrostrictive constants can be determined. The unusual thickness dependence of the converse piezoelectric effect in a-quartz and the electrostrictive behavior of the alkali halides are presented as examples. The close correlations between elastic properties and crystal structures are reflected by several heuristic and empirical rules. Resonant ultrasound spectroscopy (RUS)belongs to the most promising methods for the determination of elastic constants in the acoustic frequency regime. The potential of RUS for the simultaneous investigation of elastic and electrostrictive properties including electromechanical coupling effects is discussed. Keywords: Crystal physics' Elasticity' Electrostriction . Piezoelectricity' Structure-property relationships

1. Introduction was the advent of modern computers that known crystal structure. In the case of eventually made ab initio calculations higher order effects the situation is even Since the establishment of the first lattice feasible. A detailed overview of compu- more dramatic. Among the different rea- theoretical models about 100 years ago, tational methods in crystallography and sons for the large discrepancy between the interpretation of structure-property mineralogy has been recently given by importance and experimental knowledge relationships ranks among the most im- Winkler [1]. of macroscopic physical properties, the portant tasks in solid-state sciences. Besides the fascination for this devel- lack of high quality crystals of sufficient Models possessing high predictive power opment one should bear in mind that the size is certainly the most important. Ad- can help, for instance, to provide struc- interpretation of properties is not excluded hindrances against a broader investi- tural and physical data of solids under sively the ability to predict properties gation of crystal properties are the high extreme conditions which are not acces- from first principles, but rather to reveal technical effort for development and sible by experiment, and to tailor new the relations between chemical composi- maintenance of the mostly self-made ex- materials with special properties. The tion, structure and properties of a broad perimental set-ups, and the high expendi- different modeling approaches either variety of crystalline materials. In this ture of time on preparation and measure- make use of transferable force fields de- context one has to face the following ment of samples. termined by certain adjustable parame- questions: (i) Is the present amount of As second order derivatives of ther- ters or rely on first principles calcula- available experimental data sufficient for modynamical potentials the elastic stiff- tions. While the advantage of empirical the derivation of general relations and nesses are directly connected with quan- models lies in their high computational rules? (ii) Which important properties tities in which the lattice energy per unit efficiency, the often poor accuracy and will not be easily accessible by calcula- volume is involved. The Christoffel de- the limited availability of transferable po- tions in the near future? (iii) Is there a terminants combine the elasticity tensor tentials make the use of quantum me- great difference between a crystal de- with the propagation velocities of sound chanical methods desirable. Although the scribed by computer simulation and a waves and related properties such as De- basic concepts were developed in the real macroscopic individual? These top- bye temperature, thermal conductivity, twenties and thirties of the last century, it ics have been controversially discussed thermal expansion (Griineisen relation), on two workshops [2] organized by the melting temperature (Lindemann formu- Laboratory of Crystallography of the la), infrared resonance frequencies and ETH Zurich. In order to illustrate the mo- dielectric behavior. Therefore a general tivation for our own research in the field qualitative interpretation of elastic prop- 'Correspondence: Dr. J. Schreuer of crystal physics, I would like to address erties - in some favorable cases also Laboratorium fOr Kristallographie these questions brief!y. nearly quantitative - will open access to ETH Zurich Sonneggstr. 5 To date the complete sets of elastic such other properties. With regard to pos- CH-B092 Zurich constants of about 2000 crystal species sible applications, additionally piezo- Tel.: +41 1 6323752 are known. At a first glance this seems to electric properties and electromechanical Fax: +41 1 6321133 E-Mail: [email protected] be a remarkable number. In fact it is less coupling effects are of considerable inter- http://www.kristalf.ethz.ch/LFK than 1% of the crystalline materials with est. Higher order effects provide access CRYSTALLOGRAPHY IN SWITZERLAND 563 CHIMIA 2001,55, NO.6 to the anharmonicity of lattice potentials. Table 1.Typical values for longitudinal deformations.1l of a sample with thickness J == 1mm caused For small and high symmetry structures by thermal expansion, linear and quadratic electrostriction, respectively. calculations from first principles nowa- Property days yield elastic constants which agree Value Force Induced change in length to within a few percent with experimental data [3], However, these calculations are thermal expansion a' II == 50 . 10·8K·1 T == 1K ill == 50 nrn still restricted to the athermal limit. linear electrostriction d'lI) == 2 10 12rnV1 E = 106Vm-1 J1 == 2 nm Moreover, the consideration of tempera- 21 ture effects will remain nearly impossible quadratic electrostriction d'lllI= 5· 10· m2y-2 E = 106Vm-' .jJ = 0.005 nm in the foreseeable future. The answer to the third question is definitely 'Yes'. On the one hand, it is can be neglected in first approximation. the lack of reliable experimental data the intrinsically impossible to obtain a real The interest in quadratic electrostriction interpretation of quadratic electrostric- crystal without defects (the surface is the is further stimulated by their fundamental tion has stagnated in the last decades. Just worst possible defect). On the other hand, relation to some non-linear optical effects recently two research groups, one in the certain electrical and optical properties such as stimulated Brillouin scattering Netherlands [7] and the second in Ger- are known to be strongly influenced by and quadratic electrooptic effects (Kerr many/Switzerland [8], studied the sourc- point defects and dislocations. For many effect). es of several disturbing effects and devel- properties no systematic study of their The mechanical strains Eij induced by oped apparative tools which allow for the dependence on different types of defects an electric field E with components Ek successful investigation of quadratic has been performed at all. Deviations are described phenomenologically by the electrostrictive properties. from the periodic structure are currently Taylor series expansion not tractable by computational methods. 2.1. Experimental Approach: High- Even with a supreme effort only a negli- resolution Capacitive Dilatometer gible, probably non-representative part Among the different approaches to of a real macroscopic crystal, which con- where the coefficients dijb dijkl etc. de- the determination of quadratic electro- sists of about 1023 atoms, can be mod- note the corresponding first, second and strictive constants ([8] and references eled. higher order electrostrictive material therein) the direct strain measurements Therefore, our work focuses on the properties. employing dilatometers based on the fre- improvement and development of dy- The components dijb often referred to quency-modulation technique are the namic methods for the experimental in- as converse piezoelectric constants, form most promising. vestigation of electromechanical proper- a polar third-rank tensor. Therefore, line- The heart of our capacitive dilatome- ties of small samples. Employing these ar electro striction can only occur in crys- ter [8] (Fig. 1) consists of an oscillator techniques we study the elastic and elec- tals with a non-centrosymmetric point with basic frequency in the range be- trostrictive properties of crystals as a symmetry group (PSG), with the excep- tween 60 and 110 MHz. The two elec- function of external forces and of their tion of PSG 43. Linear electrostrictive ef- trodes of the frequency-controlling ca- real structure. fects are in the order of 1O-12mY-l(Table 1) pacitor are supported by transducer rods and can be easily measured employing which are connected with the sample and optical or electromechanical dilatometers with a reference crystal, respectively. 2. Electrostrictive Effects [6]. Due to the important technical appli- Thus, a periodic longitudinal deforma- cations of piezoelectric materials (piezo- tion of one of the crystals leads to a mod- The best criterion to test lattice theo- electric actuators, sensors, oscillators, ulation of the high-frequency signal of retical models is proof of their predictive switches, frequency filters etc.) most of the oscillator. The modulation is elec- power. One of the rare examples where a the recent work is devoted to the search tronically transformed in a voltage signal crystal property was predicted without for new non-centrosymmetric crystal that is finally filtered by means of a previous experimental knowledge is pro- species possessing large polar properties. Lockln amplifier. For small deforma- vided by quadratic electro striction [4]. In contrast to linear electrostriction, tions (L11 ~ 10 nm) the linear relationship Such electromechanical effects represent quadratic effects, represented by a polar L11 = SA between Lockln-amplitude A th the second approximation of the mechan- 4 rank tensor {dijkl}, exist in all crystals and deformation L11 holds in sufficient ical response of matter on an applied and even in amorphous materials. How- approximation. The factor S can be deter- electric field. In 1969 Grindlay and ever, in non-ferroelectric compounds mined by applying a low voltage ac-sig- Wong [5] proposed a model of the 'elas- these effects are rather small. Particular- nal, Ure == 10 Y, to the reference crystal tic dielectric' and first calculated the ly, the mechanical response of a centro- (for example an X-cut a-quartz speci- quadratic electrostriction in some alkali symmetric crystal on an applied electric men) with the effective converse piezo- halides of rocksalt structure type. The field is often barely above the lower limit electric effect d!lr The change of length straightforward calculation encouraged of detectable strains (Table 1). It is there- of the sample is now obtained by the authors to suggest quadratic electro- fore not surprising that the early experi- striction as a useful probe for the anhar- mental results for the quadratic electro- monicity of interionic forces. Unlike for striction in alkali halides are contradicto- the coefficients of thermal expansion, the ry, i.e. the tensor components deviate relationship between the higher order significantly in magnitude and in sign. electrostrictive coefficients and the an- For example, the literature values of the Quantities with subscripts sand rc are harmonicity of the lattice potential is rel- component dIll! of sodium chloride vary related to the sample and to the reference atively simple since temperature effects between-6.7 and +34·1O-2Im2y-2.Due to crystal, respectively. The sign of the elec- CRYSTALLOGRAPHY IN SWITZERLAND 564

CHIMIA 2001, 55, No.6

halides, the longitudinal component dIll I varies between 2·IO-2]m2y-2 and 6.10-21 m2y-2, whereas the transversal component dl ]22always exhibits a negative sign II FM oscillator and dl212 is one order of magnitude smaller than dill]' The volume-stricti on 1 ()lInV/(oE)2 = dllll + 2d1122, a scalar invariant in PSG m3m, is positive in all in- vestigated alkali halides and increases with decreasing elastic stiffness. The close correlation between volume-striction and the quantity alK = yCvl3 V (a linear coefficient of thermal expan- 2 sion, K compressibility, V molar volume, Cv specific heat capacity at constant strain, y Grtineisen parameter) shows the 3 anharmonic nature of quadratic electrostriction. The longitudinal effect d'llll = U1iUljUlkUI/dijkl (Uli direction cosine) pos- 4 sesses a strong anisotropy with distinct maxima and minima along (100) and • 5 (Ill), respectively (Fig. 2). At first ~---6 glance the coincidence of the maxima of d'l] II and of the longitudinal elastic stiffness in most alkali halides is surprising. 7 The different behavior of lithium fluoride (Fig. 2), however, indicates that the origin of the electrostrictive anisotropy is different from the origin of the elastic anisotropy. The relative sizes of the cations and anions dictate the directions with 8 close contacts between neighboring ions. Correspondingly, the maxima of the longitudinal elastic stiffness appear in lithium halides along (Ill) whereas in other alkali halides the maxima are observed in (100) directions. In contrast to the elastic Fig. 1. Sketch of our capacitive dilatometer. 1: electrically shielded oscillator chamber, 2: behavior the anisotropy of the quadratic reference crystal holder, 3: reference crystal, 4: plate capacitor, 5: electrical shielding, 6: sample, electrostriction is nearly independent of 7: sample holder, 8: adjustable support. the radius ratio. The dominant contributions to the macroscopic electrostriction trostrictive deformation of the sample and pOSItIon of the conducting wires. stem from the polarizability of the ions can be derived from the phase difference Fortunately, in most cases large intervals and from dispersive interactions between ,1f/J = f/Js - f/Jrc of the LockIn signals pro- without such spurious effects can be opposite charged ions. This picture is vided that the polarity of the reference tracked down by frequency dependent confirmed by lattice energy calculations crystal is known. measurements. based on empirical force fields [5][10]. In order to achieve high resolution of Sufficient agreement between theory and about ,11 '" 2.10-13m thorough measures, 2.2. Example: Quadratic experiment is only achieved by model as for example good electrical and ther- Electrostriction in Alkali Halides calculations which adequately take into mal shielding and excellent vibration iso- Currently, the quadratic electrostric- account short-range interactions. lation, have to be taken to minimize ex- tive coefficients of alkali halides deter- Although it is not yet a routine exami- ternal disturbances. The most critical mined with the aid of the new capacitive nation, our capacitive dilatometer as well source of errors are Coulomb-interac- dilatometer are likely to be regarded as as the sophisticated, but slightly less ac- tions between certain parts of the the best available. Earlier experimental curate double-beam Michelson-interfer- dilatometer. Particularly, the high-volt- data, the spread of which was interpreted ometer recently constructed by Sterken- age of up to 3 kV required for the investi- by a strong dependence of quadratic elec- burg et al. [7] allow for the determination gation of very small quadratic effects trostriction on impurities and defects [9], of reliable quadratic electrostrictive coef- leads to strong electromechanical vibra- should be considered with some reserva- ficients. tions which superimpose the strain signal tion. It cannot be excluded that these data It is expected that a larger amount of caused by the electro strictive effect of the suffer from spurious effects as outlined in experimental data will become available sample. It is practically impossible to the previous section. soon and thus will give new impetus to avoid electromechanical resonances, be- According to the results obtained by the study of quadratic electro-striction in cause they depend on shape, dimension Schreuer and Haussiihl [8] on eight alkali centrosymmetric crystals. CRYSTALLOGRAPHY IN SWITZERLAND 565

CHIMIA 2001, 55, No, 6

2.3. Thickness Dependence of Converse Piezoelectric Effects While testing the high-resolution capacitive dilatometer an unexpected thickness dependence of the longitudinal converse piezoelectric effect was discovered along the polar directions of certain piezoelectric crystals. The well reproducible thickness effect (Fig. 3) was first observed on thin plates of a-quartz (X-cuts of synthetic and natural crystals with diameters and thicknesses in the ranges 12-40mm and O.08-13mm, respectively). For I > O.5mm (a) (b) dIll increases in proportion to the recip- rocal thickness l/l of the specimens. With thinner plates dill gradually reaches a constant value. Crystals of LiNb03, NaBr03, Li2S04·H20 and benzil show a similar behavior whereas in CS2S206 no influence of the sample thickness on d333 is observed. The electrostrictive effect along the six-fold axis in lithium iodate even decreases rapidly with decreasing thickness. Due to the specific behavior of the different crystal species, systematic errors originating from the experimental set-up cannot account for the strange thickness dependence. Further distur- (e) (d) bances from electrode effects, excitation of vibrational modes of the samples, boundary conditions and sample mount- Fig.2. Representation surfaces of longitudinal elastic stiffness C'IIII = UliUljUlkulluijkl (Uli direction cosine, i= 1,2,3) and of the longitudinal effect d'llll = UliUIPIkUlld'ijklOf quadratic electrostriction ing have been excluded by experiments. in LiF. (a),(b):surface of c'llll viewed along [100] and [111], respectively. (c),(d):surface of d'lill A simple sandwich model, which is viewed along [100] and [111]. based on the assumption that the electrostrictive properties of the surface zone differ from those of the bulk material, al- lows for a qualitative interpretation of the observed thickness effects. In the case of quartz the model yields a thickness of about 100-150 ~ for the surface zone. () ace -lo- The electrostrictive effect d1f{'if = 8.5· 30 1O-12mV-1is about a factor 3.7 enlarged -- u1k ~ 019 rr compared to the value dlfI of the bulk en 25 -1.876mm crystal (l ~ 00). The reason for the varia- - tion of electrostrictive propelties in sur- «-- 20 c: 0 :;:; ctI 15 E al0- 10 Q.) -"0 5

-Q.) a: 0 0 10 20 30 40 50 60 70 Fig. 3. Relative deformation tJ - As I Arc of two synthetic X-cut a-quartz plates with different thicknesses but equal diameters ofabout 20 mm Applied voltage [V] as a function of the applied voltage. Reference crystal is a disk-likespecimen ofX-cuta-quartz with diameter of 20 mm and thickness of 1.872 mm. CRYSTALLOGRAPHY IN SWITZERLAND 566 CHIMIA2001, 55, No.6

face zones is not yet clear. It is probable to the coefficients dijk of the piezoelectric the S-values holds within 10% for a large that defects which are induced by the tensor according to emij = dmk,cSkl' Super- variety of compounds, including spinels, preparation procedure play an important scripts Sand E indicate adiabatic condi- perovskites, garnets and feldspars. Even role. Further investigations are in progress. tions and constant electric field, respec- the influence of water of crystallization The most surprising aspect of the tively. can be separated. On the other hand devi- thickness effect is not the existence of a Due to their symmetry properties, the ations from the rule yield hints on anoma- surface layer possessing different proper- elastic stiffnesses Cijk{ are usually written lous bonding contributions. ties, but its extension. A cube with edge in an abbreviated form employing One approach to the qualitative inter- lengths of 0.5 mm, 1 mm and 5 mm and Voigt's notation, i.e. the index pairs ij pretation of the anisotropy of the longitu- surface zones of about 100 J..lmthickness and kl are replaced by simple indices ac- dinal elastic stiffness C'llll=UliUlPlkUlICijkl consists of 78%, 49% and 12% surface cording to ij ~ i for i =j and ij ~ 9-i-j for is provided by the system of principal zone, respectively. With regard to the on- i:;t:j. Four types of elastic constants Cij can bond chains (PBC-vectors according to going miniaturization of experimental be distinguished: Hartman and Perdok [16]). Often the set-ups careful investigations of potential CII>C22, C33 longitudinal elastic maximum of C'llll corresponds with a surface zones and their influence on the stiffnesses, minimum of the longitudinal thermal ex- experimental determination of various C12,C13, C23 transverse interaction pansion a'ij=uI iUljay and both usually oc- physical properties are essential. coefficients, cur along the direction of the strongest C44, C55, C66 elastic shear stiffnesses, PBC-vectors (Fig. 4). Instructive exam- C14,c15,C16,etc. coupling coefficients. ples can be found among the phthalates 3. Elastic Properties of Crystals [18], oxalates [19] and tetroxalates [20]. The number of independent elastic However, the description of the bonding Elasticity provides one of the most constants of a crystal depends on its point system in terms of PBC-vectors is not useful probes for the exploration of struc- symmetry group. For example, 21 con- fully compatible with the interpretation ture-property relationships. As spring stants Cij have to be determined in order of the elastic stiffnesses as spring con- constants elastic stiffnesses are directly to describe the elastic behavior of a tri- stants, because it neglects the spatial ar- related to the three-dimensional network clinic crystal whereas in cubic crystals rangement of the bonds within a periodic of interatomic interactions in the crystal the knowledge of three constants (e.g. bond chain. The alkali feldspar sanidine, and, consequently, to many other of its CII>Clb C44) is sufficient. Details about KAISi30g, represents a prominent excep- thermodynamical properties. Hence, the symmetry reduction of property ten- tion from the rule. Due to the zigzag-like elastic constants permit subtle tests of the sors can be found in many text books character of the strongest bond chains reliability of atomistic model calcula- dealing with crystal physics and applica- running parallel to the aI-axis a longitu- tions. Further, elastic constants are highly tions of group theory [12-14]. dinal deformation along [100] is mainly sensitive to structural instabilities con- caused by small changes of the O-Si-O nected with phase transitions. Character- 3.1. Empirical Rules bond angles instead by compression or istic anomalies of the elastic behavior al- For new compounds simple heuristic dilatation of the rather stiff Si-O bonds. low not only the detection of phase tran- and empirical rules allow quick estima- Consequently, in sanidine the maximum sitions but offer valuable hints on the tion of the order of magnitude of certain of C'1I11is observed along [010], i.e. it is driving mechanisms [11]. physical properties (e.g. dielectric behav- located in the plane perpendicular to the In the linear approximation of ior, refractive indices, certain non-linear main bond chains. Hooke's law the components eJij of the optical properties and mean elastic stiff- A hint on the nature of the bonding mechanical stress tensor and the compo- ness). These rules are based on the as- interactions in crystals can be derived nents £ij of the strain tensor are related by sumption that the constituents of the from the deviations from Cauchy rela- eJij = Cijkl£kl where {cijd is the elasticity crystal yield quasi-persistent additive tions [21] represented by the second rank tensor [12]. In the case of electrostrictive contributions to the various physical tensor invariant {gmn} with the compo- crystals Hooke's law must be completed properties. Once the crystal structure is nents gmn = emikenj{cijd2, where eijk de- by additional terms which take into known, one can further deduce a qualita- note the components of the Levi-Civita account electro-mechanical coupling tive picture of the anisotropy of the prop- symbol. In metallic and ionic crystals, effects. For small deviations from equi- erties. particularly in those built up from aspher- librium the mechanical and electrical var- Haussiihl [15] discovered that the ic ions and constituents with large polar- iables of the corresponding thermody- quantity S = c· Mv, defined as the prod- isability, the transverse interaction coef- namic potential are related by uct of the mean value of the principal ficients dominate considerably over the elastic constants C = (CII+ C22 + C33 + CI2 corresponding shear stiffnesses resulting _ l! I: ,f £:a a¢ + cn + c23 + C44+ c55 + C66)/9 and the in positive deviations from Cauchy rela- (T -c~lJ-e H '. I,;'Jll'" +e/qj "- ('x, "". molecular volume Mv, opens the possi- tions. Strong covalent or other bonds . a~ ~ flf/J bility to estimate the mean elastic stiff- with preferential directions cause oppo- D=e-~ +~ E =e .:..:a-E - I .u-lJ 'p I 101 ex I u ax I ness of ionic crystals provided that chem- site effects. Extreme examples are gold ical composition and density are known. and diamond with gIl"" +l2·1010Nm-2 10 where E is the electric field vector and D The S-value varies only slightly within and gIl "" -24·10 Nm-2, respectively. the electric flux density. ~k denote the isotypic crystals and chemically related Crystals possessing a layer-like structure components of the displacement vector, compounds. Furthermore, in ionic crys- such as gypsum and micas show often a is the electric potential and {E ij} repre- tals the S-value can be decomposed in distinct minimum of the deviations from sents the tensor of the dielectric con- additive contributions S(X;) of the con- Cauchy relations along the direction per- stants. eijk are the components of the pi- stituents Xi of the compound X according pendicular to the plane of the layers indi- ezoelectric stress tensor which are related to S(X) = IS(X;). The quasi-additivity of cating covalent bonding contributions CRYSTALLOGRAPHY IN SWITZERLAND 567

CHIMIA2001, 55, No.6 , t~3--

"~---'It'n .. \

.-~ '.' ~, ,~ i. ~~' ...~

".I~.': ...:;•.. .- \ "

I Fig. 4. Correlation between longitudinal elastic stiffness (a), longitudinal effect of thermal ex- (a) (b) (c) pansion (c) and crystal structure (b) of trigonal ((CH3bNCH2COOb-2MnCI2 [17].

within and weaker Coulomb or van-der- echo techniques, acoustic resonance limited by a number of problems includ- Waals interactions between the layers. methods. ing transducer ringing, transducer-sam- A more quantitative interpretation of • Scattering methods: scattering of pho- ple coupling, beam diffraction, the sam- elastic behavior can be obtained from lat- tons or neutrons on thermal phonons, ple size, and the necessity of measuring tice energy calculations [1]. Within the e.g. Brillouin spectroscopy (scatter- samples with different orientations in or- framework of thermodynamical poten- ing of light on acoustic phonons), Ra- der to estimate all independent elastic tials elastic constants are defined as sec- man spectroscopy (light scattering on constants. ond order deri vati ves of potential func- optical phonons), thermal diffuse x- tions, e.g. ray scattering (Compton scattering of 3.3. Resonant Ultrasound X-ray photons), inelastic neutron Spectroscopy u scattering (phonon spectroscopy). In the last three decades a very power- ful alternative experimental approach has Other techniques such as the investi- become available due to significant ad- gation of surface acoustic waves or the vances in computer technology: resonant Here V = U( {Ejj} ,D,S) is the free ener- application of atomic force microscopy ultrasound spectroscopy (RUS). Here the gy, and Eij, D and S are the components of are still in an early stage of development. elastic constants are derived from an ex- strain tensor, the electric flux density and The potential of laser-induced phonon perimentally measured ultrasonic reso- entropy, respectively. spectroscopy, LIPS [23], and ultrasonic nance spectrum of a freely vibrating crys- interferometry in the GHz frequency re- tal with well-defined shape (Fig. 5). The 3.2. Experimental Determination of gime [24] for use with diamond-anvil main advantages offered by RUS are (i) Elastic Constants cells under high pressures is currently ex- the elastic, piezoelectric and dielectric Many different techniques covering plored. Other methods even allow for im- properties of the crystal can be in princi- broad frequency-, temperature- and pres- aging of phonon propagation in crystals ple simultaneously studied on one sam- sure-ranges are available for the experi- [25]. ple, (ii) the small sample size, (iii) no mental investigation of elastic properties The most accurate determination of medium is required for transducer-sam- [22]: elastic constants in the important acoustic ple coupling and (iv) the very short data frequency regime is facilitated by ultra- acquisition time. Detailed reviews of this • Static methods including stretching, sonic techniques. Conventional ultrason- very promising technique have been re- bending and torsion experiments. ic methods actually determine the veloci- cently published [26]. The principles of • Dynamic methods such as periodic ties of ultrasonic plane-waves injected RUS and its advantages and challenges stretching, bending and torsion exper- into the sample. The velocities are (if not to say disadvantages) are briefly iments at frequencies below the first derived either from the traveling time of described in this section. mechanical resonance frequency of short ultrasonic pulses (pulse-echo tech- The natural vibrational modes of a the sample. niques) or from resonance frequencies of sample correspond to stationary solutions • Ultrasonic methods based on excita- thick plane-parallel plates (improved of its Lagrangian tion of ultrasonic waves in the sam- Schaefer-Bergmann method, impedance ple, e.g. normal and improved method). Although these techniques are L = J ( I.Jncuc - pol nil.! )dV Schaefer-Bergmann method, pulse- highly sophisticated, their application is CRYSTALLOGRAPHY IN SWITZERLAND 568

CHIMIA 2001, 55, No.6

Following Hamilton and using the functions for the approximation of the usually do not restrict the application of Ritz prescription, i.e. expanding the dis- electric potential and of each component RUS. placement vector ~ = ~iei and the electric of the displacement vector. While in cu- Different experimental set-ups suited potential in sets of basis functions {1m}, bic crystals about 1000 basis functions for samples of about] 00 ~ up to several {gm} according to ~i = aim!m and = bm give a good compromise between accura- meters in size and covering the temper- 8m' respectively, one obtains stationary cy and computing time sets of about 4000 ature range from about 4K to I800K have solutions by solving the general eigen- basis functions or more are required in been reported (see [26] and references value problem the case of monoclinic or triclinic crys- therein). RUS-apparatus for routine tals in order to avoid significant trunca- measurements are already commercially (G.+·p-lt{t).a :p~Ma (1) tion errors. available [29]. l;).'!All:'-ll(ta Unfortunately, no analytical solution For crystals with piezoelectric effects exists for the inverse problem. Thus the larger than about 30% of dIll of a-quartz where v = m/2T[ are the corresponding sample parameters have to be evaluated and moderate dielectric constants the pi- resonance frequencies and p is the mass by an iterative non-linear least-squares ezoelectric contribution to eigenfrequen- density of the sample. The coefficients of procedure that matches observed and cal- cies, as included by the term KKIKr in the matrices G, K, E and M depend on culated resonance frequencies by adjust- (1), can be forced to be considerably larg- the shape and dimensions of the sample ing values for the independent parame- er than the experimental error (the repro- as well as on the coefficients of the elec- ters. The convergence of a refinement de- ducibility of experimental resonance fre- tromechanical property tensors of the pends critically on the proper choice of quencies is usually ±O.l KHz or better) crystal: the initial values. Statistical [27] and ex- by choosing an appropriate sample size. perimental [28] procedures were pro- Due to the strong anisotropy of piezo- posed for mode identification in samples electric properties, fortunately, in most with high symmetry from which the deri- cases no starting value problem exists for vation of starting values c8 is expected. the computation of piezoelectric con- However, the use of conventional tech- stants from resonance frequencies [30]. niques for the determination of appropri- Employing the following procedure ate c8 values is still necessary for crystals with low symmetry. • Step I: Prerequisites are the precise The progress of the refinement proce- knowledge of dielectric constants and dure is further affected by a number of of proper starting values for the elas- errors originating from mechanical load tic constants. Oij denotes the Kronecker symbol. To on the sample, misorientation of the sam- • Step 2: Refinement of cij neglecting keep the computational effort within rea- ple, deviations from the ideal shape and piezoelectric coupling effects. sonable limits, the size of the matrices crystal defects. In particular, two-dimen- • Step 3: Simultaneous refinement of must be restricted by the use of a truncat- sional defects such as twin boundaries elastic and piezoelectric constants used set of appropriate basis functions. A and cracks can make the quantitative in- ing any set of initial e;Jk values. suitable choice are powers of Cartesian terpretation of a resonance spectrum im- coordinates !m=x]p] xf'2xl3 ,m= (P],P2>P3), possible because the sample has to be and using elastic and dielectric constants because these functions can be easily ap- considered as a system of two or more listed in Landolt-Bornstein [31] we were plied to samples with simple shape resonators of unknown shape and dimen- able to derive piezoelectric stress con- (spheres, ellipoids, rectangular parallele- sions coupled by springs of unknown stants from ultrasonic resonance spectra pipeds, rods, cones etc.). The truncation force constants. More homogeneously in excellent agreement with literature condition T ~ PI + P2 + P3 leads to N = distributed defects with dimensions data (Table 2). Larger deviations of up to (T + 1)(T + 2)(T + 3)/6 independent basis much smaller than the elastic wavelength 30% from literature values are only ob-

1e-3

1e-4 Fig. 5. (a) High-temperature RUS: A sample of ~ monoclinic sanidine, KO.S9Nao.11AISi30S'pre- Q) pared as rectangular parallelepiped with edge '0 ~ lengths of about 5 mm is weakly clamped on :t: 1e-5 a. two opposite corners by alumina buffer rods E which act simultaneously as sample holder « 1e-6 and as ultrasonic wave guides. Two piezoelec- r'r' tric transducers operating as ultrasonic gener- ator and detector, respectively, are glued to the 1e-7 ends of the buffer rods outside the heated area. 600 610 620 630 640 650 The central alumina tube supports a thermo- Frequency [KHz] couple. (b) Part of the resonance spectrum of a spodumene (LiAISi206) sample showing four (a) (b) sharp eigenmodes. Note the logarithmic scale of the signal amplitude. CRYSTALLOGRAPHY IN SWITZERLAND 569 CHIMIA 2001.55. No.6

Table 2. Electromechanical coupling effects in selected piezoelectric crystals. N shift of man, W.G. Perdok, Acta Crystallogr. resonance frequencies of the respective samples induced by electromechanical coupling;fo.Jc 1955, 8, 525. observed and calculated resonance frequencies; L1eijk= I (e~~- eijk)leijkI relative difference [17] J. Schreuer, S. Haussiihl, Z. Kristallogr. between ejjkcalculated from resonance spectra and literature data ej~Btaken from [31]. Resonance 1993, 205, 309; S. Haussiihl, Z. Kristallo- spectra have been collected on samples which are prepared as rectangular parallelepipeds with gr. 1989,188,311. edge lengths in the range 4-9 mm. [18] S. Haussiihl, Z. Kristallagr. 1991,196,47. [19] S. Haussiihl, Z. Kristallogr. 1991,194,57. [20] H. Kiippers, H. Siegert, Acta Cryst. 1970, Compound PSG l/ Ifdol IJk data/parameter A26,401. (maximum) (average) (average) ratio [2l] S. Haussiihl, Phys. kondens. Materie 1967,6, l8I. (KHz] [KHz] [%] [22] Handbook of elastic properties of solids, liquids, and gases, vol. I, 'Dynamic methods for measuring the elastic properties of NaBr03 23 0.2 0.28 0.5 12.5 solids', Eds. A.G. Every, W. Sachse, Aca- CS2S206 6m 4.3 0.27 1.0 8.0 demic Press, San Diego, 2001. [23] J.M. Brown, L.I. Slutsky, K.A. Nelson, tourmaline (elbaite) 3m 1.0 0.24 2.8 8.1 L.T. Cheng, J. Geophys. Res. 1989, 894, 9485; E.H. Abrahamson, J.M. Brown, L.I. LINb03 3m 50.9 0.57 3.2 7.8 Slutsky, J. Zaug, J. Geophys. Res. 1997, Si02 (Ct.-quartz) 32 0.3 0.19 0.5 8.6 102, 12253. [24] H. Spetzler, A. Shen, G. Chen, G. Her- La3Ga5Si014 32 4.0 0.23 0.2 9.1 mannsdorfer, H. Schulze, R. Weigel, Phys. Earth Planet Int. 1996,98,93; H.-J. L1 S0 H2O 2 4.4 0.21 6.7 6.3 2 4 Reichmann, R.I. Angle, H. Spetzler, W.A. Bassett, Am. Mineral. 1998,83, 1357. [25] J.P. Wolfe, 'Imaging phonons', Cam- served for some of the small constants of quadratic electromechanical effects. How- bridge University Press, Cambridge, monoclinic lithium sulfate monohydrate. ever, in this paper 'electrostriction' is used 1998. In summary, the RUS-technique is al- in a broader sense as the strain response of [26] R.G. Leisure, F.A. Willis, J. Phys. Can- matter on an inducing electrical force. ready a standard tool in non-destructive dens. Matter 1997,9, 600l; A. Migliori, This includes the linear electrostriction or material testing. Other promising appli- J.L. Sarrao, 'Resonant Ultrasound Spec- converse piezoelectric effect, respective- troscopy', Wiley Sons, New York, 1997; cations are the investigation of piezoelec- ly, and all higher order terms. J. Schreuer, Z. Kristallagr. 2001, in prepa- tric coupling phenomena, the study of the [5] J. Grindlay, H.C. Wong, Canad. J. Phys. ration. temperature dependence of elastic prop- 1969,47, 1563. [27] H. Yasuda, M. Koiwa, J. Phys. Chern. Sol- erties and the detection of phase transi- [6] L. Bohaty, Z. Kristallogr. 1982,158,233; ids 1991, 52, 723. tions. The focus of current research is on K. Uchino, L.E. Cross, Ferroelectrics [28] A. Stekel, J.L. Sarrao, T.M. Bell, Ming the most challenging problem connected 1980,27,35. Lei, R.G. Leisure, W.M. Visscher, A. [7] S.W.P. van Sterkenburg, T. Kwaaitaal, with the RUS-method: the derivation of Migliori, J. Acoust. Soc. Am. 1992, 92, W.M. van den Eijnden, Rev. Sci. lnstrum. 663. proper initial values for the elastic con- 1990, 61, 2318; S.W.P. van Sterkenburg, [29] DRS, Inc., 225 Lane 13, Powell, Wyo- stants from resonance spectra. Due to the J. Phys. 1991, D24, 1853. ming, USA, www.ndtest.com. ongoing rapid advances in computer [8] J. Schreuer, 'Kritische Untersuchungen [30] J. Schreuer, E. Haussiihl, W. Steurer, Z. technology the application of statistical zur Messung elektrostriktiver Effekte in Kristallogr. Suppl. 1999, 16, 110; J. and simulation methods will become fea- Kristallen', Dissertation Universitat KOln, Schreuer, E. Haussiihl, L. Bohaty, W. sible soon. An alternative approach is the 1994; J. Schreuer, S. Haussiihl, J. Phys. Steurer, Z. Kristallogr. Suppl. 2000, 17, estimation of starting values by lattice 1999, D32, 1263. 53. [9] H. Burkard, W. Kanzig, M. Rossinelli, [31] Landolt-Bomstein, vol. Ill/29b, 'Low fre- energy calculations [3]. Helvetica Physica Acta 1976, 49, 13; G. quency properties of dielectric crystals: Balakrishnan, K. Srinivasan, R. Srinivas- Elastic constants', Ed. D.F. Nelson, Acknowledgement an, J. Appl. Phys. 1983, 54, 2875. Springer- Verlag, Berlin, Heidelberg, New I would like to acknowledge financial sup- [10] R. Srinivasan, K. Srinivasan, J. Phys. York, 1992; Landolt-Bornstein, vol. IlI/ port from the Swiss National Science Founda- Chern. Solids 1972, 33, 1079; R. Srinivas- 29a, 'Low frequency properties of dielec- tion under grant No. 20-53630.98 and 2000- an, K. Srinivasan, phys. stat. sol. 1973,57, tric crystals: Piezoelectric, pyroelectric 06 I482.00/1. 757; J. Shanker, T.S. Verma, A. Cox, J.L. and related constants', Ed. D.F. Nelson, Sangster, Can. J. Phys. 1981, 59, 1359; Springer-Verlag, Berlin, Heidelberg, New Received: April 18,2001 S.W.P. van Sterkenburg, T. Kwaaitaal, J. York,1993. Phys. 1992, D25, 843. [11] M.A. Carpenter, E.K.H. Salje, Eur. J. [I] B. Winkler, Z. Kristallogr. 1999,214,506 Mineral. 1998,10,693. [2] Berichte aus Arbeitskreisen der DGK, vol. [12] J.F. Nye, 'Physical properties of crystals', 2, 'Predictability of physical properties of Oxford University Press, Oxford, 1957. crystals', Ed. J. Schreuer, Kiel, 1998. Be- [13] S. Haussiihl, 'Kristal1physik', Verlag richte aus Arbeitskreisen der DGK, vol. 6, Chemie, Verlag Physik, Weinheim, 1983. 'Measurement and interpretation of me- [14] A.S. N owick, 'Crystal properties via chanical properties of crystals', Ed. J. group theory', Cambridge University Schreuer, Kiel, 2000. Press, Cambridge, 1995. [3] B. Winkler, M. Hytha, M.C. Warren, V. [15] S. Haussiihl, Z. Kristallogr. 1993, 205, Milman, J.D. Gale, J. Schreuer, Z. Kristal- 215. logr. 2001,216,67. [16] P. Hartman, W.G. Perdok,Acta Crystallo- [4] Many authors use the term 'electrostric- gr. 1955,8,49; P. Hartman, W.G. Perdok, tion' exclusively for the description of Acta Crystallagr. 1955, 8, 52 I; P. Hart-