VOLUME 74, NUMBER 8 PH YSICAL REVIEW LETTERS 20 FEBRUARY 1995 Equation of State and Shear Strength at Multimegabar Pressures: Magnesium Oxide to 227 GPa Thomas S. Duffy, Russell J. Hemley, and Ho-kwang Mao Geophysicai Laboratory and Center for High Pres-sure Research, Carnegie Institution of Washington, 5251 Broad Branch Road, NW, Washington, DC 20015 (Received 9 June 1994) The equation of state, elasticity, and shear strength of magnesium oxide were examined to 227 GPa using synchrotron x-ray diffraction in a diamond anvil cell. Static compression, ultrasonic elasticity, and shock data for MgO from ambient pressure to above 200 GPa can all be described by a single Birch-Murnaghan equation of state when the static shear strength, which is determined to be at least 11 GPa at 227 GPa, is taken into account. Our results show that there are significant changes in the degree and character of the elastic anisotropy of MgO at high pressure. PACS numbers: 62.50.+p, 64.30.+t, 64.70.Kb Understanding the physics of materials at ultrahigh pres- Structural studies of materials at multimegabar pres- sure is an important focus of current work in condensed sures have largely been restricted to high atomic weight matter science. The equation of state (EOS) and phase sta- materials, due to the difficulty of performing x-ray diffrac- bility are the most fundamental properties obtained from tion on the extremely small volumes which can be sub- these investigations. The reliability of equation of state jected to such pressures. We show that with synchrotron determinations at large compressions is uncertain because radiation methods x-ray diffraction can be carried out on the stress state at multimegabar static pressures is not well MgO, a low-Z oxide, to pressures in excess of 200 GPa. understood [1,2]. Shear strength provides a fundamental We also show that MgO remains in the Bl (NaC1) struc- description of a material's mechanical behavior, but little ture to these ultrahigh pressures, unlike other oxides is known about this quantity under high static pressures which undergo pressure-induced phase transitions under [2,3]. Upon compression, the stress state of all solids is moderate pressures [12]. nonhydrostatic when the sample has finite strength. If un- MgO (99.998% purity) was mixed with 5 wt. % corrected, the presence of shear stresses can lead to system- molybdenum and compressed in a Mao-Bell diamond atic errors in the resulting bulk modulus and its pressure anvil cell with beveled diamonds having central Oats derivative [4,5]. Shear strength effects can also introduce of 19.5 and 49.8 p, m. Energy-dispersive synchrotron errors in measured pressures when secondary calibrants x-ray diffraction was carried out using superconducting (e.g. , ruby fluorescence or diffraction standards) are used wiggler radiation [13]. Figure 1 shows a representative [6]. As a result, there are typically discrepancies in equa- x-ray diffraction pattern. For MgO, the (200), (220), and tions of state determined by different techniques. In this occasionally (420) peaks were resolved. Pressures were study, we examine a low-Z prototype material (magnesium determined from Mo diffraction lines using an isotherm oxide) by x-ray diffraction at pressures above 200 GPa derived from shock data [14]. The pressure-volume and provide the first determination of shear strength un- relation is shown in Fig. 2. The range of stability der such conditions. When this strength is taken into ac- count, a single equation of state can describe ultrasonic, 209 GPB shock, and static compression data for this material. We Mo (&&0) show that it is possible to characterize material strength and elastic anisotropy at ultrahigh pressure directly from MgO (200) diffraction data. Magnesium oxide is an ionic solid that has been widely studied at high pressure and is a prototype for developing and applying first-principles theoretical methods to oxides MgO Mo [7,8]. MgO has also been extensively employed in the development of experimental methods for shear strength determination at low pressures (to 40 GPa) 9,10]. A [5, q~e ply' major problem with current data, however, is that equations state from ultrasonic, and static experiments of shock, I I are not mutually consistent [11]. Furthermore, theoretical 30 40 50 60 models for MgO have been complicated by questions Energy (keV} concerning the degree of ionicity and characterization of FIG. 1. Energy dispersive x-ray diffraction pattern of MgO at the charge distribution of the oxygen atoms [7,8]. 209 GPa. The spectrum was taken in 150 min. 0031-9007/95/74(8)/1371(4)$06. 00 1995 The American Physical Society 1371 VOLUME 74, NUMBER 8 PHYS ICAL REVIEW LETTERS 20 FEBRUARY 1995 250 under nearly hydrostatic conditions are also consistent with the ultrasonic and reduced shock data. The vol- 200— umes predicted by theoretical methods [7,8] differ from the combined shock and ultrasonic equation of state by only 2% or less at pressures near 200 GPa. A third-order &50— Birch-Murnaghan fit to the present static compression data U yields KpT = 177 ~ 4 GPa and KoT = 40 —0-1. CD CD differs markedly from the ultrasonic, shock, and quasihy- CD 100— CD drostatic compression data (Fig. 2). The ambient-pressure CL bulk modulus is 10% larger than the ultrasonic value, and the volumes are 1.5% larger at 150 GPa. Other static compression data for MgO under nonhydrostatic stress [20] also yield high bulk moduli (-180 GPa) and P V- states above the inferred hydrostatic compression curve. 10 9 The role of nonhydrostatic stresses in opposed anvil Volume (cm /mol. ) devices has been discussed in detail [1]. The principal FIG. 2. Pressure-volume data for MgO. The filled symbols stress in the load direction o-3 is greater than the stress in are static compression data (circles —this study, triangles— the radial direction o ]. In the diamond cell geometry, the [20]), and the open symbols are from shock compression (cir- incident x-ray beam travels along the load direction and cles —single-crystal [16,17], squares polycrystalline [18]). the diffracting plane normals are nearly at right angles to The solid line is a third-order Birch-Murnaghan fit to the the load present data. The dashed line is a third-order Birch-Murnaghan axis. Thus diffraction measures strain near the extrapolation of ultrasonic data. The dash-dotted line is a direction of minimum principal stress. The pressure is fourth-order Birch-Murnaghan fit to the combined ultrasonic given by and reduced shock data. The inset shows pressure versus the difference in the lattice parameters determined from the (220) 03 + 267& and (200) diffraction lines. 3 and, assuming a von Mises yield condition, the shear predicted by theory for MgO in the 81 structure has strength r is [1] — varied widely (100 1000 GPa) (e.g., [7,8]). Our results ]. r = —, (~3 — show the Bl structure is stable to at least 227 GPa ~i). (2) (V/Vp = 0.612) at room temperature, demonstrating the A quantitative determination of the shear stresses in remarkable stability range of MgO in this structure. our sample may be made in two ways. The first method Ultrasonic sound velocity measurements [15] on MgO utilizes the dependence of the lattice parameter on crys- single crystals to 3 GPa yield high-precision values of the tallographic direction. The difference in lattice parame- isothermal bulk modulus (Ivor = 160.3 +. 0.3 GPa) and ters determined from the (220) and (200) MgO lines are its first pressure derivative (Kpr = 4.15 ~ 0.10). Using shown as a function of pressure in Fig. 2 (inset). With these parameters in a third-order Birch-Murnaghan equa- increasing pressure, the lattice parameter from the (220) tion yields the equation of state shown in Fig. 2. Shock diffraction line is systematically larger than that from the compression data for MgO have been reported to 199 GPa (200) line by an amount that increases from —0.1% at [16—18]. At low pressures, where thermal effects are 100 GPa to 0.9% above 200 GPa. negligible, the single-crystal Hugoniot results are con- From linear elasticity theory, the following expres- sistent with extrapolated ultrasonic data (Fig. 2). Mie- sions describing the lattice strain in a nonhydrostati- Griineisen theory [16] was employed to reduce the shock cally compressed cubic sample in the diamond cell can data, and a Birch-Murnaghan equation fit to the result- be obtained [1]: ing isotherm yields = 157 ~ 7 GPa and = 4.3 ~ K» KpT 2~ 0.2, in agreement with ultrasonic values. A combined fit E (hkl) = ep — (1 —3 sin 2 0) 3 to the reduced shock and ultrasonic data, in which KpT and KpT are fixed, constrains the second pressure deriva- 1 A' = — x a(S„—5„—3SI ) (3) tive of the bulk modulus, Kp'T 0.022 0.004 GPa +,2G~ This yields a fourth-order Birch-Murnaghan EOS which is a„,(hkl) ao slightly less compressible than the third-order fit (Fig. 2). I (4) Recently, static compression data for MgO have been Qp obtained to 23 GPa under quasihydrostatic conditions in the diamond cell using neon as a pressure medium S = (W —1), 2 and NaC1 as an internal standard [19]. A third-order fit to these data yields KoT = 4.2 ~ 0.1 (««oz. = h2g2 + g2)2 + I2h2 = 160.3 GPa and Vp 11.253 cm3/mole). Thus static data (h2 + k~ + l2)~ 1372 VOLUME 74, NUMBER 8 PHYS ICAI REVIEW LETTERS 20 FEBRUARY 1995 Here e (hkl) is the lattice strain (which along with line can only be explained by a negative value of 5. This the stress is taken to be negative in compression) and, provides direct experimental evidence for a fundamental in general, is a function of the Miller indices hkl of change in the nature of the elastic anisotropy of MgO at the diffraction plane.