REVIEW OF SCIENTIFIC INSTRUMENTS

VOLUME 72, NUMBER 2

FEBRUARY 2001

Synchrotron radiation and laser heating in a diamond anvil cell Denis Andraulta) Laboratoire des Ge´omate´riaux, ESA-7046, Institut de Physique du Globe, Universite´ Paris-7, France

Guillaume Fiquet Laboratoire de Mine´ralogie-Cristallographie, UMR-7590, Universite´ Paris-6, France

共Received 6 July 2000; accepted for publication 26 October 2000兲 The last decade has seen the development of very bright third generation x-ray synchrotron sources that have significantly improved x-ray diffraction experiments at high pressure and high temperature. In the same time, focusing optics as well as detectors have been improved in such a way that x-ray diffraction experiments can be routinely carried out on samples in laser-heated diamond anvil-cell, i.e., under extreme static pressure and temperature conditions. It is now possible to obtain from a laser heated sample in a diamond anvil cell 共DAC兲 very high quality powder patterns, needed for accurate in situ phase boundary and pressure-volume-temperature equation of state determinations, as well as for structural refinements. The setup for in situ x-ray investigation at simultaneous high pressure and temperature is complex, since perfect alignment of x-ray path, infrared-laser hot spot, and optical temperature measurement is required. It provides, however, the most accurate information on the structural behavior with step by step increases of pressure and temperature. In addition, a range of experiments can also be realized at room temperature after laser annealing of DAC samples, which has been shown very efficient in releasing stresses build on compression, and/or overcoming phase transitions kinetic barriers. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1343866兴

I. INTRODUCTION

ters. It is then preferable to use a harder pressure medium like MgO, SiO2 , or Al2O3 , that are intrinsically too hard to be good pressure transmitting medium, but remain good thermal insulator at the highest pressures. In this case, deviatoric stresses have to be relaxed using laser annealing after each sample compression step. Not only does the choice for a particular sample configuration depends on the P – T range to be covered during the experiments, but obviously also on the sample chemistry, since chemical interactions between sample and pressure transmitting medium have to be avoided. A direct consequence of an improved thermal insulation of the sample in a DAC, is that samples can be heated with moderate laser energy density, which will preserve optics and diamond anvils, and optimize infrared 共IR兲 beam profiles at sample location. The DAC laser heating technique could thus evolve from the use of highly focused yttrium– aluminum–garnet 共YAG兲-laser sources, producing on the sample very punctual hot spots of about 5–10 ␮m in diameter, to heating with slightly defocused multimode YAG or TEM00CO2 lasers that can produce hot spots of about 30–50 ␮m in diameter. Such heating on a much larger sample surface introduces much smaller radial temperature gradients, more suitable to in situ x-ray diffraction investigation using synchrotron radiation.

The laser-heated diamond-anvil cell is the only experimental tool able to create extreme static pressures ( P ⬎100 GPa) and temperatures (T⬎3000 K) and it has therefore a major impact in high-pressure research and geophysics. After some reconnaissance studies,1,2 x-ray diffraction experiments on laser-heated diamond-anvil cell samples can now be optimized by considering several parameters, such as sample environment, minimization of temperature gradients and instabilities, accurate pressure measurements, and x-ray diffraction detection configuration. II. SAMPLE ENVIRONMENT

A good control of static P and T conditions is required when using laser-heated diamond anvil cell 共LH–DAC兲. Main improvements come from a better design of the sample configuration, where a thin sample foil 共or pellets兲 is embedded in a pressure transmitting medium that insures minimum P and T gradients.3 The use of a soft pressure medium such as rare gases 共He, Ne, Xe, Ar...兲 or highly compressible materials 共NaCl, KBr, LiF...兲 minimizes deviatoric stresses that can develop during compression. In addition, these pressure transmitting media have to be low-Z elements and their structure has to be simple, so that their contribution to diffraction signal is not important. The high compressibility of these soft materials, however, makes them poor thermal insulator at the highest pressure, since insulation layers between sample and diamond anvil decrease to a few microme-

III. SYNCHROTRON RADIATION

Recent development of very bright third generation x-ray synchrotron sources greatly improved the size and the intensity of the x-ray beams available for experiments on extremely small samples. On the ID-30 beamline of the

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FIG. 1. Setup for the on-line laser heating on the ID-30 x-ray diffraction beamline. The incoming x-ray beam 共from left to right兲 is focussed on a 10⫻15 ␮m spot on the sample, using various stages of slits and mirrors. The membrane-type diamond anvil cell provides high pressure up to more than 100 GPa. All alignments must be monitored on a micron scale, so that the diffraction is done at the exact position where the laser heating and the T measurement are simultaneously performed.

ESRF for example 共Fig. 1兲, two phased undulators provide a very bright monochromatic x-ray beam, in an energy range between 25 to about 60 keV. Two silicon mirrors in a Kirkpatrick–Baez geometry, located at 40 and 60 cm from the sample, are use to focus the primary x-ray beam to a beam size of about 10⫻15 ␮m full width of half maximum 共FWHM兲.4 Further slits collimating is applied to clean the focused beam from diffuse scattering, to finally achieve an optimal probe with all x rays within 20⫻25 ␮m at the sample location 共Fig. 2兲. The gain in intensity is high enough so that not more than 5 min are needed to collect at 33 keV on imaging plates 共IP兲 a high-quality angle dispersive x-ray diffraction spectra on LH–DAC samples5 with the machine

D. Andrault and G. Fiquet

running at 200 mA. Images are read in about 15 s by an on-line IP reader, the so-called Fastscan detector.6 Similar on-line reader using the IP technique as well as charge coupled device 共CCD兲 camera have to be used during these experiments which require short acquisition and readout time. ID-30 is in this sense the most optimized beamline, since it provides a high-quality focused monochromatic microbeam coupled with fast two-dimensional detectors. X-ray focusing optics are of primary importance since a very punctual x-ray probe is required to integrate data for sample at temperature as homogenous as possible. In the same manner, a good quality x-ray microfocused beam is a prerequisite to work with minute samples mounted in gasket pressure chambers less than 60 ␮m in size, typical of experiments carried out above 100 GPa. It should be noted, however, that alignments of the different paths have to be achieved within a few micrometers, so that x rays, IR laser, and temperature measurement are positioned at the same sample location. This is the first requirement for any reliable in situ measurement in a LH–DAC. The current trend indicates that energy-dispersive diffraction 共EDX兲, which has been so far the standard configuration for high-pressure powder diffraction with synchrotron sources, will be progressively replaced by angle-dispersive diffraction. The combined use of monochromatic x-ray beam and two-dimensional detectors appears to be the most appropriated technique for the LH–DAC experiments, as it provides an optimum coverage of the reciprocal space 共Fig. 3兲. Laser heated DAC samples are particularly small, which yields a reduced number of crystallites in the volume probed 共about 20 ␮m in diameter and 5–10 ␮m in thickness兲. This often results in relatively spotty powder diffraction patterns, which is unavoidable for such small samples. This is the main reason why angle-dispersive diffraction is required, since when using EDX with a polychromatic beam, one can clearly miss some important features because the Ge detector, typically a few millimeters in diameter, only intercepts a very small portion of the reciprocal space. EDX has many advantages, such as fast alignment procedure and real-time pattern, but it suffers from an intrinsically lower spectral resolution 共⌬E/E of the order of 1%兲, which makes small structural changes difficult to resolve. On the other hand, this technique is helpful to study kinetics of phase transformation or transition on a few seconds time scale, as long as enough randomly oriented crystallites are present in the sample. However, advances in techniques involving CCD cameras are already very competitive in terms of acquisition and readout time. Two-dimensional detection and time-resolved studies are thus feasible at extreme conditions of pressure and temperature. IV. TEMPERATURE GRADIENTS

FIG. 2. The top curve represents the x-ray flux transmitted through a 60 ␮m diameter hole drilled in a preindented Re gasket, as a function of the gasket position relative to the x-ray beam. The bottom curve represents the size of x-ray beam at the sample location. It is obtained after derivation of the right upward part of the top curve. The FWHM of the x-ray profile is found to be 12 ␮m, and all the x rays are located within 25 ␮m. It provides a spatial resolution in perfect accordance with the hot spots generated with multimode infrared lasers.

A way to reduce radial temperature gradient is to x ray the sample with a beam as small as possible. In a similar manner, it is possible to minimize the axial temperature gradient 共along the x-ray path兲 by making the sample as thin as possible. For a given gap between the two diamonds 共i.e., at a given pressure兲 a very thin sample of less than, or about, 5

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minimize axial T gradients. It should be noted, however, that the double-sided heating yields more complicated laser paths on both sides of the DAC, which sometimes makes the use of two-dimensional detection systems very difficult. In this case, the advantages of the double-sided laser heating are questionable. Another particularity of the LH–DAC on a synchrotron beamline is the necessity for all measurements to be performed simultaneously on a single sample location, where x rays, IR laser, and optical measurements should be coaxial. This clearly results in an increase number of optical devices around the DAC, and thus a lack of space. IR-focusing lenses with longer focal length have to be used instead of those usually used for a standard LH–DAC setup under an optical microscope, resulting in a relatively larger IR spot on the sample, which is a convenient feature to reduce the radial T gradients in the x-rayed zone but this makes it also more difficult to heat samples correctly at very high pressures. Similarly, temperature measurements are not as simple as they could be in a home laboratory experiment. Long working-distances objectives are needed, which sometime preclude large magnification and/or the use of the best true achromatic lenses such as reflective objective. It should be noted that the correction proposed for dispersion of the diamond anvil8 is only valid at a given pressure since diamonds anvils deform in a significant manner.9 FIG. 3. Selected portion of an imaging plate recorded on iron at about 2000 K and 45 GPa. The sample is a 5-␮m-thick iron foil embedded in two pellets of corundum that is used as an insulating material. The diffraction lines of iron are not the continuous circular rings usually found for a classical powder diffraction pattern, because the number of crystallites in the x-ray spot are too reduced. Still, there is a sufficient diffraction information to obtain an acceptable statistic for the diffraction lines intensity, after for the pattern integration. Without the use of the two-dimensional detection system, one can clearly miss important diffraction features from the studied sample.

␮m clearly enables the use of thicker insulating material, thus providing a more efficient thermal insulation. Still, the use of too small samples can yield an insufficient amount of crystallites in the x-ray spot, which significantly reduces the quality of the x-ray diffraction pattern. At this point, it is important to consider the differences between single- or double-sided laser heating of DAC samples on a synchrotron beamline. For a given sample thickness, it is clear that double-sided LH can provide lower axial thermal gradient,7 and this technique should be used as often as possible. However, several studies have shown that the axial T gradient in very thin samples 共5–10 ␮m thick兲 remains moderate, as evidenced by comparable diffraction line FWHM for the cold and LH sample.2 Experiments show for instance that temperature differences between both sides of an iron foil 10 ␮m thick does not exceed 40–80 K when heating only from one side as long as a good thermal insulation is achieved. In fact, axial T gradients can really become crucial when pressure exceeding 100 GPa are reached. Double-sided laser heating is even more necessary when using energy-dispersive instead of angle-dispersive diffraction, because the coverage of the reciprocal space with EDX is intrinsically lower, a diffraction pattern of sufficient quality can only be obtained with a larger sample volume, hence a thicker sample which has to be double-sided heated to

V. TEMPERATURE INSTABILITIES

The recent improvements related to the T measurement and control is discussed in other chapters of this journal issue. From the point of view of diffraction experiments, we shall distinguish two kinds of instabilities. High frequency instabilities are mainly due to the laser and can be corrected with built-in laser-stabilization systems within 1% peak to peak. There are other instabilities such as those created by progressive thermal drift of the laser head, with temperature variations with periods between a few seconds and minutes. These are clearly the most important in our case, because the typical exposure time for obtaining a convenient imaging plate is of the same order. Several feedback systems have been proposed,10 which use the thermal emission of the sample as an input signal. Even if this procedure is very attracting for in situ x-ray diffraction experiments, one notices, however, that the temperature stability is closely related to sample properties such as absorption coefficient and surface state, which can evolve during an experiment. Diffusion of infrared absorbers 共platinum black for YAG lasers, for example兲 intimately mixed with intrinsically nonabsorbing samples is a good illustration of such problems, when the migration of this absorber toward the edge of the cell pressure chamber makes it difficult to keep the temperature at a maximum. Also, when using rare gases as pressuretransmitting medium, motions in the thin molten layer at the contact with the laser-heated sample can affect the quality of the thermal insulation, and thus the sample temperature. In any cases, the procedure is to find a sample location where the measured temperature is the most stable and record the diffraction pattern from this sample location, until a new

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loading of sample is required. In a more general manner, a DAC loading can often be used for several heating steps at a given pressure, but it is common that the sample must be reloaded after very few pressure increases. VI. MELTING

We also found it difficult to measure melting curves using LH–DAC coupled with x-ray diffraction, even if diffraction is normally the best tool to identify the melting. At the melting point, liquid and solid have a different absorption coefficient for the IR-light, which often results in a ‘‘yoyo’’ effect for the sample temperature. This particular behavior has been used in the passed to determine the onset of melting by optical observations.11 Using x-ray diffraction, if this effect occurs, part of the sample can reach the solid state for a short amount of time and make visible the sample diffraction lines, even if the time-averaged sample temperature is well above that of the melting. VII. THERMAL PRESSURE

A range of x-ray diffraction measurements carried out on laser-heated samples have shown the difficulty to properly characterize pressure conditions. Heinz12 suggested that thermal pressure and relaxation could play an important role in such experiments, which was confirmed by in situ x-ray diffraction measurements using several internal pressure standards2,13,14 as well by calculations.15 Experiments in the well-known phase diagrams of Mg2SiO4 and SiO2 have been for instance conducted to control the P – T path during in the laser-heated DAC.13 As shown in Fig. 4, Mg2SiO4 wadsleyite could for example be synthesized during the laser heating at 2200 K of forsterite previously compressed at a nominal pressure of 11 GPa. Because the Clapeyron slope between these two polymorphs is positive, the occurrence of wadsleyite clearly evidenced an increase of pressure in LH sample. This issue remains controversial, but such effect for instance quantitatively explains the discrepancy between diamondanvil cell and large volume press experiments on the position of the coesite-stishovite transition in the P – T phase diagram.13,15 This could also explain why so few pressurevolume-temperature 共PVT兲 equation of state determination are published, even though an increasing number of hightemperature high-pressure data are collected on material of geophysical relevance. It is clear that the amount of the theoretical thermal pressure ( ␣ K⌬T) converted into a real increase of pressure, or in a volume relaxation, can be different from a sample configuration to another 共with a dominant parameter being the hardness of the pressure medium兲. In any cases, it is clearly recommended to measure sample pressure within the laser hot spot, especially if the goal of an experiments is to determine the PVT equation of state of a given material. For this purpose, internal pressurestandard are available, such as metals 共Pt, Au,...兲, oxides 共MgO,...兲 or salts 共NaCl,...兲. Note that the pressure standard must be intimately mixed with the sample, before the mixture is embedded in the pressure-transmitting medium within the DAC gasket hole. This is the only configuration where one can be sure that both sample and P-standard encounter

FIG. 4. Calculated and experimental phase boundary 共Ref. 20兲 for Mg2SiO4 . OL, ␤, SP, Per and Pv stand for olivine, wadsleyite, ringwoodite, and the periclase and MgSiO3 -perovskite mixture, respectively. The P – T path during the laser heating in the DAC is estimated from the in situ x-ray diffraction at various temperatures. The ␤-wadsleyite diffraction lines appeared after the heating of forsterite at 2200 and 2600 K, for a nominal pressure of 11 GPa. The pressure increase in the laser hot spot is due to a difficult sample expansion in such a highly constrained volume between the two diamonds.

the same P and T conditions during experiment. Temperature is then determined by spectrometry and pressure calculated from the PVT equation of state of the pressure standard, given by the presence of a suitable diffraction pattern for this calibrant. VIII. CHEMICAL REACTION

Occurrence of any new diffraction line can result from a chemical reaction or a phase transformation in sample probed by x rays. Oxidation of Fe can for example be tracked by the appearance of the FeO diffraction lines, which will be more and more intense as iron progressively oxidizes. This is a simple procedure that provides strong arguments for or against the simplest chemical reactions. Chemical alloying, however, is difficult to detect using x rays, since a few percent alloying elements is not always detectable in lattice parameters. The partial oxidation of a Fe foil in a MgO pressure medium, for example, can yield a formation of some 共MgFe兲O magnesiowustite whose diffraction lines can easily superimpose with those of the periclase. X-ray diffraction has to be coupled with optical observations as often as possible. For example, the FeO diffusion in a MgO matrix is easy detectable. To prevent undesirable chemical reactions and sample oxidation, one must choose a convenient P medium for a given sample, and, as often as possible, use various P media to confirm observations. Another interest of using various P medium is to probe the sample diffraction

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FIG. 6. Typical Rietveld refinement performed on angle dispersive x-ray diffraction pattern recorded for a San-Carlos olivine sample at 94 GPa, after the YAG-laser heating at 2200 共⫾200兲 K for several minutes. Lower, middle and upper ticks correspond to d hkl lines of (Mg,Fe兲SiO3 Pv, 共Mg,Fe兲O Mw, and the gold P standard, respectively. Differences between experimental and calculated profiles are much reduced, and no extraline are observed, which clearly evidences the reliability of the diffraction lines intensities.

created, several out-of-equilibrium transformations can be induced, and in particular unexpected phase breakdown can occur as a consequence of a strong chemical diffusion or thermal stresses in a T gradient. IX. RIETVELD REFINEMENTS

FIG. 5. Chemical profiles of Fe and Si after YAG laser heating of a SanCarlos olivine in a DAC, determined using an ionic microprobe 共see Ref. 16兲. It appears a lower iron content on the center of the heated strip relative to the starting material 共top photo兲. It results a higher Si content in the center of the strips 共bottom photo兲. This chemical segregation occurs in strong thermal gradient created by this type of laser-heating experiment, when the sample is not thermally insulated from the diamonds. It should be avoided to probe an homogeneous sample with the x rays.

lines in a complete angle 共or energy兲 range, because the different P media do not show annoying diffraction lines at the same d(hkl) position. One should also not neglect any possible interdiffusion processes that can occur in the T gradient of the laser hot spot. For example, it is very difficult to measure any accurate PVT-equation of equation of state for the 共Mg,Fe兲O magnesiowu¨stite using the LH–DAC, since iron content at the x-ray location can vary with time, as iron clearly diffuses away from the hottest zone. Similar behavior has been observed in (Mg,Fe兲SiO3 systems where iron migrates at the edge of laser tracks. A zone clearly enriched in silicon is clearly visible on Fig. 5.6 To prevent such type of problems, a solution is to work with very small samples about the size of the central zone of laser hot spot. Lack of control of the P and T gradients during the laserheating in a DAC can produce severe chemical problems on a micron scale. When T gradient as large as 100 K ␮m⫺1 are

The quality of the x-ray diffraction patterns is sometimes sufficient to extract more information than only cell parameters from even a mixture of phases 共Fig. 6兲. Furthermore, the use of a monochromatic x-ray beam and imaging-plates 共on ID-30–ESRF for example兲 yields diffraction patterns with reliable intensity profile. It is thus tempting to try to extract crystallographic information from the patterns recorded in a LH–DAC. The main problem arises then from the small number of crystallites contributing to the pattern, resulting in a real spottiness of diffraction ring, often pronounced at high pressure and temperature when recrystallization is important. One could argue that there are too few crystallites in the x-ray spot to perform reliable structural refinement, but we clearly demonstrated that there is sufficient information to infer some of the atomic parameters: The diffraction pattern recorded for iron at high P and T, for example, confirmed the Pbcm structure model for the ␤ form of iron.17 For the MgSiO3 perovskite, we could quantify the effect of the T on the Si–O–Si angles at about 90 GPa.18 We also described the evolution of Si–O and O–O bonds along the Landau-type phase transformation occurring in stishovite at about 50 GPa.19 These results opened a wide range of structural investigations in an extended P – T field. X. FUTURE MEASUREMENTS

Synchrotron radiation offers many other techniques for the analysis of structural and physical properties of solids and liquids at high pressure. For example absorption spectroscopy, inelastic scattering, x-ray emission spectroscopy and nuclear forward scattering are all techniques which are already used for studies at high-pressure in a DAC. How-

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ever, the simultaneous use of DAC and LH makes the optical access to sample more complicated. In the same manner, laser heating require techniques with a spatial resolution of a few micrometers, as well as with a short exposure time. Only x-ray diffraction is therefore currently performed on LH– DAC samples, but advances in x-ray focusing optics and insertion devices might soon make these new investigation tools compatible with laser heating in a diamond-anvil cell. For any further developments, it is clear that the control of P and T conditions in such experiments will remain a key parameter. ACKNOWLEDGMENTS

The authors thank all past and present members of the ESRF high-pressure group for their hard work that made all these experiments possible. This work is supported by ITCNRS. 1

R. Boehler, N. Von Bargen, and A. Chopelas, J. Geophys. Res. B 95, 21731 共1990兲; G. Fiquet, D. Andrault, J. P. Itie´, P. Gillet, and P. Richet, Ann. Geophys. 共Germany兲 12, 24 共1994兲; C. Meade, H. K. Mao, and J. Hu, Science 268, 1743 共1995兲; S. K. Saxena, L. S. Dubrowinski, P. Ha¨ggkvist, Y. Cerenius, G. Shen, and H. K. Mao, ibid. 269, 1703 共1995兲; C. S. Yoo, J. Akella, A. J. Campbell, H. K. Mao, and R. J. Hemley, ibid. 270, 1473 共1995兲. 2 G. Fiquet, D. Andrault, J. P. Itie´, P. Gillet, and P. Richet, Phys. Earth Planet. Inter. 95, 1 共1996兲.

D. Andrault and G. Fiquet R. Boehler and A. Chopelas, Geophys. Res. Lett. 18, 1147 共1991兲. D. Ha¨usermann and M. Hanfland, High Press. Res. 14, 223 共1996兲. 5 G. Fiquet and D. Andrault, J. Synchrotron Radiat. 6, 81 共1999兲. 6 R. Thoms, S. Bauchau, M. Kunz, T. le Bihan, M. Mezouar, D. Ha¨usermann, and D. Strawbridge, Nucl. Instrum. Methods Phys. Res. A 413, 175 共1998兲. 7 G. Shen, H. K. Mao, R. J. Hemley, T. S. Duffy, and M. L. Rivers, Geophys. Res. Lett. 25, 373 共1998兲. 8 R. Boehler, Nature 共London兲 363, 534 共1993兲. 9 R. J. Hemley, H. K. Mao, G. Shen, J. Badro, P. Gillet, M. Hanfland, and D. Ha¨usermann, Science 276, 1242 共1997兲. 10 D. Heinz, J. S. Sweeney, and P. Miller, Rev. Sci. Instrum. 62, 1568 共1991兲. 11 S. K. Saxena, G. Shen, and P. Lazor, Science 260, 1312 共1993兲. 12 D. L. Heinz, Geophys. Res. Lett. 17, 1161 共1990兲. 13 D. Andrault, G. Fiquet, J. P. Itie´, P. Richet, P. Gillet, D. Ha¨usermann, and M. Hanfland, Eur. J. Mineral. 10, 931 共1998兲. 14 A. Kavner et al., EoS, transaction, AGU, supplement, 1998. 15 A. Dewaele, G. Fiquet, and P. Gillet, Rev. Sci. Instrum. 69, 2421 共1998兲. 16 M. Tarrida, M. Madon, J. Peyronneau, G. Catillon, and F. Degreve, in Secondary Ion Mass Spectrometry, edited by A. Benninghoven, B. Hagenhoff, and H. W. Werner 共1977兲, p. 925; A. J. Campbell, D. L. Heinz, and A. M. Davis, J. Geophys. Res. B 19, 1061 共1992兲. 17 D. Andrault, G. Fiquet, M. Kunz, F. Visocekas, and D. Ha¨usermann, Science 278, 831 共1997兲. 18 G. Fiquet, A. Dewaele, D. Andrault, M. Kunz, and T. le Bihan, Geophys. Res. Lett. 27, 21 共2000兲. 19 D. Andrault, G. Fiquet, F. Guyot, and M. Hanfland, Science 23, 720 共1998兲. 20 Y. Fei, S. K. Saxena, and A. Navrotsky, J. Geophys. Res. B 95, 6915 共1990兲. 3 4