JOURNAL OF APPLIED PHYSICS 103, 054908 共2008兲

P-V-T equation of state of platinum to 80 GPa and 1900 K from internal resistive heating/x-ray diffraction measurements Chang-Sheng Zha,1,2,a兲 Kenji Mibe,3,4 William A. Bassett,3 Oliver Tschauner,5 Ho-Kwang Mao,1 and Russell J. Hemley1 1

Geophysical Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Rd. N.W., Washington, DC 20015, USA 2 Cornell High Energy Synchrotron Source, Wilson Laboratory, Cornell University, Ithaca, New York 14853, USA 3 Department of Earth and Atmospheric Sciences, Cornell University, Ithaca, New York 14853, USA 4 Earthquake Research Institute, University of Tokyo, Tokyo 113-0032, Japan 5 High Pressure Science and Engineering Center and Department of Physics, University of Nevada, Las Vegas, Nevada 89154, USA

共Received 17 August 2007; accepted 11 December 2007; published online 12 March 2008兲 The P-V-T equation of state 共EOS兲 of Pt has been determined to 80 GPa and 1900 K by in situ x-ray diffraction of a mixture of Pt and MgO using a modified internal resistive heating technique with a diamond anvil cell. The third-order Birch–Murnaghan EOS of Pt at room temperature can be fitted with K0 = 273.5⫾ 2.0 GPa, K0⬘ = 4.70⫾ 0.06, with V0 = 60.38 Å3. High temperature data have been treated with both thermodynamic and Mie–Grüneisen-Debye methods for the thermal EOS inversion. The results are self-consistent and in excellent agreement with those obtained by the multianvil apparatus where the data overlap. MgO is taken as the standard because its thermal EOS is well established and based on a wealth of experimental and theoretical data, and because the EOS at room temperature has been determined by a primary method that is completely independent of any assumptions or measurements by other methods. Improvements to previous internal resistive heating methods were made by using a Re gasket that replaces the original gasket composed of diamond and MgO powder. We have thereby extended the P-T range to nearly 80 GPa and 1900 K. Use of this method in combination with synchrotron radiation has advantages in the study of EOS, phase diagrams, and materials synthesis for a variety of problems in physics, chemistry, geosciences, and material sciences. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2844358兴 I. INTRODUCTION

The P-V-T equation of state 共EOS兲 is a fundamental physical property crucial for addressing numerous problems in the physical sciences. In modern static high pressure and high temperature experiments, pressure calibrations also rely on the precision P-V-T EOS of calibrants. It is therefore important to obtain precise structural and thermodynamic information on materials at simultaneous high pressures and temperatures. To obtain the precise P-V-T EOS by combining static compression and x-ray diffraction methods, pressure measurement and its relationship to volume and temperature is the most challenging issue. This is because it is not possible to obtain this relationship from the measurements themselves as is done in shock wave experiments. A secondary pressure standard, calibrated against the primary pressure scale,1–3 is typically used. The primary pressure scale can be obtained either by a reduced isothermal EOS from a shock wave experiment,4–6 or by simultaneous volume and elasticity measurements under static compression.7,8 The most widely used primary pressure scales to date are reduced isothermal P-V-T EOS from shock wave experiments. Unfortunately, the accuracy of these reduced isothermal EOSs suffers from either uncertainties in theoretical asa兲

Electronic mail: [email protected].

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sumptions or low precision in measurements.4 Although the experiments can cover a large pressure-temperature range, self-consistency among these EOSs is poor at higher pressures.9 On the other hand, reliable P-V-T data can be obtained at moderate pressures and temperatures using a large volume press 共LVP兲, as most of the pressure scales are reasonably consistent in this P-T range. Particularly important among the LVP studies have been those in which simultaneous measurements of elasticity and volume were made.10,11 A reliable high P-T experimental method for filling the gap between LVP and shock wave compression techniques is urgently needed for overlapping data sets and cross-checking the results that come from them. The generation and measurement of simultaneous high pressures and temperatures in the diamond anvil cell 共DAC兲 in conjunction with synchrotron radiation sources have undergone rapid development in recent years. Laser heating DAC techniques12,13 represent the most widely used approaches for this purpose, but reliable measurements with these methods can be challenging due to large temperature gradients and instabilities, depending on how the methods are implemented.13–15 Internal resistive heating in the DAC has been reported previously.16 Similar to laser heating, this method is capable of producing temperatures of several thousand degrees inside the gasket hole while the temperature of

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© 2008 American Institute of Physics

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054908-2

Zha et al.

J. Appl. Phys. 103, 054908 共2008兲

FIG. 1. The schematic diagram of the new arrangement in the sample-heater assemblage of the internal resistive heating DAC.

the DAC body remains at a much lower temperature. This method is expected to be more reliable than the laser heating technique for EOS studies as a result of the heating homogeneity and time stability. However, the pressure ranges of previous internally heated DAC methods were limited because of the lower strength of the polycrystalline gasket material used. Improvements to the experimental design have made it possible to significantly extend the pressure range. FIG. 2. 共Color兲 Pictures taken before and during heating for this experiment.

II. EXPERIMENT

Recently, a new type of sample assemblage for this method has been developed. Similar to the previous design,16 two supporting gaskets serve as the electrical leads and the holding frame 共Fig. 1兲. A Re gasket 共thickness of 0.05 mm兲 replaces the original gasket composed of a diamond-MgO powder mixture. The new Re gasket has a slot that is filled with insulating material 共silica glass powder兲 to electrically insulate the heater strip from the gasket. Electric current flows from the upper supporting gasket through the Re heater strip, passes the sample portion, then reaches the Re gasket, which electrically contacts the lower supporting gasket. Because the Re gasket is insulated from the upper supporting gasket, this design has simplified the previous zigzag style of heater, making it easier to pack the sample and the heater into the gasket hole. The sample, a Pt and MgO powder mixture 共1:5 volume ratio兲, is packed into the center hole of the rhenium heater strip with an initial diameter of 0.02 mm and a thickness of 0.012 mm. This portion of the heater is surrounded by silica glass and fills the Re gasket hole. The hot zone is formed at the area surrounding the sample not only because of its smaller cross section 共and therefore higher resistance兲 but also because of the good thermal insulation offered by the silica glass separating it from the diamond anvil 共which is a heat sink due to its high thermal conductivity兲. Since the sample is surrounded by the heating zone, the temperature gradients in the sample are small.16 With this new design, the pressure range has been extended to more than 100 GPa, while up to 2000 K in temperature can be reached. Figure 2 shows the sample before and during heating.

A programmable direct current power supply 共HP 6552A, 20 V, 25 A兲 was used for the power source. The ripple of voltage and current are 0.3 mV and 5 mA, respectively, and the corresponding temperature variation from this power fluctuation is negligible. Power control can be achieved by either constant voltage or constant current control modes, with 2 mV or 6 mA minimum variations. The voltage control mode was used in this experiment. In this mode, the current changes in response to the change in heater resistance because of the temperature change while the voltage is held constant. The increased or decreased resistance results in a decrease or increase in current, which in turn leads to a decrease or increase in power, respectively, thus compensating for the cumulative temperature change. The very small adjustable step in voltage results in a measurable, stable temperature change smaller than 10 K. We have tried to keep the sample temperature as close as possible to the target temperature for the data collection. The unavoidable small differences 共艋10 K兲 between measured and target temperature, which is within the uncertainty of temperature measurement, can be ignored. We consider the data collected this way to be isothermal. The controllable, stable temperature offered by the internal resistive heating method makes isothermal measurements the best method for the high P-T diffraction experiment. During a run, we heat the sample to the desired temperature and hold the temperature constant until the data are collected. The temperature is then increased to the next desired value and held constant for the data collection. The temperature was lowered to 300 K after data collection at the highest

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FIG. 3. Representative x-ray diffraction pattern at high P-T. The unmarked peaks are either from the heater or from the pressure medium.

temperature for a run. Data for the room temperature isotherm are always collected at this time in order to avoid large deviatoric stresses. This assumption is based on our previous experience16 that this preheating cycling procedure was necessary for relaxing the stress-strain condition induced by the random stacking and mechanical deformation in the pressure chamber after each pressure loading. The pressure of the cooled sample then is increased by changing the loading at this temperature. A similar heating and data collection procedure is used again for this different loading. Temperature is measured using the spectroradiometry13,17 method. Since the temperature gradients are quite small between the heater edge and the sample center,16 temperature was measured by sampling the radiation of 8 ␮m area on the heater edge near the sample in order to use the gray body assumption for the temperature calculation. The temperature uncertainty is within ⫾20 K. Energy dispersive x-ray diffraction measurements were carried out at the Cornell High Energy Synchrotron Source 共CHESS兲. An x-ray beam of 0.015⫻ 0.02 mm2 in area was aligned to the sample, the 2␪ angle was set at 15°, and the average data collecting time was 5 min. One of the key advantages of internal resistive heating for x-ray diffraction experiments is the well-defined sample position, which is defined by the heater and fixed by the pressure medium. X-ray transmission scans are often used for precisely locating the sample position, taking advantage of the high contrast in x-ray absorption between heater and sample. This procedure eliminates the possibility of position mismatch in x-ray and temperature measurements. Figure 3 shows the x-ray diffraction pattern taken at the highest pressure and temperature. Normally, two to three diffraction peaks for each phase are clearly visible in most of the patterns as shown. However, some patterns show only one peak for MgO. This is probably due to the spotty nature of the diffraction rings resulting from recrystallization and preferred crystal orientation. Consequently, some diffraction rings may fail to register in the energy dispersive x-ray diffraction system because of the limited range of detection

FIG. 4. P-V-T EOS for platinum.

angles along each ring. The unmarked peaks are either from the Re heater or the crystallized pressure medium. The Re of the heater was calibrated and used as a pressure marker in previous experiments.18 However, a heterogeneous pressure distribution between the sample and the surrounding heater materials was observed in this experiment because the thickness of the gasket assembly is much thinner than in previous experiments. Therefore, Re diffraction lines were not used for in situ pressure calibration in this experiment, and the role of the heater was simply as a secondary hot gasket. As a result, it offers a good pressure environment for the sample in the small hole at the heater center.

III. RESULTS

Figure 4 shows the experimental results of this study. Detailed data are also listed in Table I. Four pressure loadings and five isotherms are measured. The MgO scale of Speziale et al.3 is used in the plot of pressure against the volume of platinum. Our data obtained at 300 K are very consistent with the measurement of Dewaele et al.,9 which was conducted under hydrostatic pressure conditions and plotted with the ruby pressure scale.2 The room temperature EOS data from Fei et al.19 obtained up to 28 GPa with a multianvil press are also consistent with this study. The volumes of Pt at higher temperatures are located very close to each other on each curve of the EOS of Fei et al.19 except for the higher pressure range; at higher pressure, the present study shows a slight higher pressure existing at each isotherm.

A. P-V-T equation of state

The P-V-T EOS of a solid normally has the following form if the electronic contribution to its free energy can be neglected:20,21 P共V,T兲 = Pa共V,Ta兲 + Pth共V,T兲.

共1兲

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TABLE I. The measured lattice parameters and volumes of MgO and Pt at different temperatures and pressures. The PT are calculated from the EOS of Speziale et al.3 for MgO with Va 共MgO兲 = 74.71 Å3, V / V0 for Pt are calculated with Va 共Pt兲 = 60.38 Å3. The average lattice parameters for each P-T point ¯a are obtained by arithmetic average of multiple diffraction lines. MgO ¯a 共Å兲

⌬a / ¯a

V 共Å3兲

PT 共GPa兲

¯a 共Å兲

⌬a / ¯a

V 共Å3兲

V / V0

4.1802共26兲 4.0034共35兲 3.9374共11兲 3.8373共17兲 4.2096共45兲 4.0973共0兲 3.9116共46兲 3.8683共21兲 4.2128共17兲 4.0819共38兲 3.9130共27兲 3.8763共52兲 4.2154共0兲 3.9935共0兲 3.9114共36兲 3.8770共64兲 4.2139共17兲 4.0227共22兲 3.9168共39兲 3.8764共37兲

0.0006 0.0009 0.0003 0.0004 0.0011 0.0000 0.0012 0.0005 0.0004 0.0009 0.0007 0.0013 0.0000 0.0000 0.0009 0.0017 0.0004 0.0005 0.0010 0.0010

73.05共13兲 64.16共17兲 61.04共5兲 56.50共7兲 74.60共25兲 68.79共0兲 59.85共22兲 57.88共9兲 74.76共9兲 68.01共18兲 59.91共13兲 58.24共23兲 74.91共0兲 63.69共0兲 59.84共16兲 58.27共30兲 74.83共9兲 65.10共10兲 60.09共18兲 58.25共17兲

3.8 33.2 48.5 78.4 6.6 21.7 61.9 74.1 7.5 25.7 62.9 73.7 8.5 44.4 64.8 75.0 10.1 39.4 64.8 76.7

3.9082共14兲 3.7982共15兲 3.7561共10兲 3.6878共8兲 3.9230共15兲 3.8599共72兲 3.7421共9兲 3.7181共57兲 3.9274共22兲 3.8487共20兲 3.7460共110兲 3.7222共32兲 3.9286共72兲 3.7973共24兲 3.7462共86兲 3.7227共70兲 3.9266共19兲 3.8250共33兲 3.7551共67兲 3.7228共79兲

0.0004 0.0004 0.0003 0.0002 0.0004 0.0019 0.0002 0.0015 0.0006 0.0005 0.0029 0.0009 0.0018 0.0006 0.0023 0.0019 0.0005 0.0009 0.0018 0.0021

59.69共7兲 54.79共7兲 52.99共5兲 50.15共3兲 60.38共7兲 57.51共33兲 52.40共3兲 51.40共23兲 60.58共11兲 57.01共9兲 52.57共46兲 51.57共14兲 60.63共33兲 54.76共10兲 52.57共36兲 51.59共29兲 60.54共9兲 55.96共15兲 52.95共29兲 51.60共33兲

0.9886 0.9075 0.8776 0.8306 0.9999 0.9524 0.8679 0.8512 1.0033 0.9442 0.8706 0.8541 1.0042 0.9068 0.8707 0.8544 1.0027 0.9268 0.8769 0.8545

T 共K兲 300 300 300 300 1300 1300 1300 1300 1500 1500 1500 1500 1700 1700 1700 1700 1900 1900 1900 1900

Pt

The subscript a refers to ambient conditions. The left side of this equation represents the total pressure P at volume V and temperature T, the first term on the right side is the isotherm at ambient temperature, and the second term is the thermal pressure due to isochoric temperature change. For most solids, the first term on the right of Eq. 共1兲 can be well determined by a third-order Birch–Murnaghan EOS:

冋冉 冊 冉 冊 册再 冋冉 冊 册冎

3 Pa共V,Ta兲 = KTa 2 ⫻

Va V

Va V

7/3

−

Va V

5/3

3 1 + 共KT ⬘ − 4兲 a 4

2/3

−1

,

共2兲

where KTa and KT ⬘ = 共⳵KT / ⳵ P兲T are the isothermal bulk a modulus and its pressure derivative at ambient conditions respectively. The units of Eq. 共2兲 are in gigapascals 共when KTa is in gigapascals兲 and T in kelvins for the temperature. Equation 共2兲 is also valid for isothermal compression at any temperature. When it is used this way, the subscript a in KTa and KT ⬘ = 共⳵KT / ⳵ P兲T needs to be changed to 0. These two a parameters are the isothermal bulk modulus and its pressure derivative at zero pressure conditions, respectively. Normally, two approaches were used for calculating the thermal pressure Pth共V , T兲 from static compression experimental data.20,21 In the thermodynamic approach, the thermal pressure can be determined by Pth =

冕

T

共␣KT兲VdT,

共3兲

Ta

where ␣ = 共1 / V兲共⳵V / ⳵T兲 P is the volume thermal expansion coefficient and KT is the isothermal bulk modulus. A more general form of Eq. 共3兲 has been derived:21

Pth =

冕

再

T

关␣KT兴共Va,T兲dT + 共⳵KT/⳵T兲V − ln共V/Va兲共T

Ta

− T a兲 +

冕冕 T

T

Ta

Ta

冎

␣dTdT ,

共4兲

where 共⳵KT / ⳵T兲V is an average value. Normally it is close to zero for the P-T range of this study. The second term in the parentheses is actually very small compared with the first term and can be considered negligible without any significant effect on the results. Also, a large amount of experimental data indicates that the product 关␣KT兴共Va , T兲 is less sensitive and linearly depends on temperature when the temperature is higher than the Debye temperature, ⌰.22–24 Based on the above analysis, a simplified formula

冋

Pth = ␣KT共Va,T兲 +

冉 冊 冉 冊册 ⳵KT ⳵T

ln

V

Va V

共T − 300兲

共5兲

has been proposed for the equality of Eq. 共4兲.24 Equation 共1兲 becomes

冋冉 冊 冉 冊 册再 冋冉 冊 册冎 冋 冉 冊 冉 冊册

3 P共V,T兲 = KTa 2 ⫻

Va V

+

⳵KT ⳵T

Va V

7/3

−

2/3

−1

ln

V

Va V

Va V

5/3

3 1 + 共KT ⬘ − 4兲 a 4

+ ␣KT共Va,T兲 共T − 300兲.

共6兲

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Equation 共6兲 forms an isotherm at high temperature. Zha et al.18 have shown that because the isochoric thermal pressure linearly changes with temperature, the slope for the temperature dependence of the thermal pressure at each constant volume can be easily deduced from a straight line between two points. One point is the pressure measured at simultaneously high pressure and temperature conditions, and the other point can be obtained from an isothermal EOS measured at ambient conditions using the same volume value as that measured at high P-T. With the slopes determined in this way at different volumes, extra thermal pressure points for each volume, which are not obtained by direct measurement at desired temperatures, can be calculated corresponding to these volumes. Several isothermal data sets 共including measured and calculated兲 grouped at each desired temperature can be used to fit Eq. 共6兲. Two parameters, ␣KT共Va , T兲 and 共⳵KT / ⳵T兲V, can be determined in this fitting, and isothermal EOSs at different desired temperatures can be created without a vast amount of experimental data. In the Mie–Grüneisen–Debye20,21 共MGD兲 approach, the thermal pressure is determined using the Mie–Grüneisen equation Pth = Pth共V,T兲 − Pth共V,Ta兲 = 关␥共V兲/V兴关Eth共V,T兲 − Eth共V,Ta兲兴.

共7兲

The Grüneisen parameter ␥, assumed to be independent of temperature but volume dependent, is

␥ = ␥0

冉 冊 V V0

q

共8兲

.

The parameter q is usually taken to be 1 implying ␥ / V = const. This commonly accepted formulation has been questioned recently.25–27 Speziale et al.3 have proposed a model for the volume dependence of the Grüneisen parameter with q = q0

冉 冊 V V0

q1

共9兲

,

where q0 and q1 are constants. The internal energy term Eth can be evaluated in terms of the Debye function ⌬Eth =

Pth =

9nRT4 ⌰3

冕

xD

0

9n␥共V兲RT4 V⌰3

x3 dx, ex − 1

冕

xD

0

x3 dx, ex − 1

共10兲

共11兲

where n is the number of atoms per formula unit, R is the gas constant, and xD = ⌰ / T. The Debye temperature with form ⌰ = ⌰0 exp关共␥0 − ␥兲/q兴

共12兲

is related to the volume change. Equation 共1兲 indicates that a correct P-V-T EOS strongly depends not only on the data collected at simultaneous high P-T conditions but also on the accurate isothermal EOS at the ambient temperature. Dewaele et al.9 have obtained a precise data set for platinum to 94 GPa at quasihydrostatic pressure and room temperature conditions in the DAC with

different pressure scales, including the ruby pressure scale.2 Fei et al.19 conducted a high P-T measurement in a multianvil press for Pt with MgO as the pressure marker, in which the P-V-T EOS of MgO obtained by Speziale et al.3 was used as the pressure scale. The results at ambient temperature of Fei et al.19 are very well consistent with those of Dewaele et al.9 when the ruby scale is used for the data of Dewaele et al.9 Figure 4 includes the data point obtained at 300 K in this experiment; data from Dewaele et al.9 and Fei et al.19 are also plotted for comparison. The consistency between previous work and this study is very good. Least squares fitting to Eq. 共2兲 using only the data points of this study yields KTa = 274.6⫾ 5.3 GPa, KT ⬘ = 4.65⫾ 0.24 with Va = 60.38 Å3 a fixed, and using data from all three studies yields KTa = 273.5⫾ 1.0 GPa, KT ⬘ = 4.70⫾ 0.06 with Va = 60.38 Å3 a fixed. We choose the latter result as our 300 K EOS for the P-V-T EOS inversion. Both thermodynamic and MGD approaches were used for the P-V-T EOS inversion of this study. A total of 16 data points at 1300, 1500, 1700, and 1900 K with different volumes is obtained in this experiment 共Table I兲. In the thermodynamic approach, the linear temperature dependence of thermal pressure at constant volume is assumed, and the isothermal EOS represented as Eq. 共6兲 is used for the least squares fitting. Only four data points are directly measured at each temperature; however, a total of 16 data points is available for the inversion of the EOS at each temperature, as shown in Fig. 5. The remaining 12 points can be calculated by linear fitting of the isochors between room and measured temperatures at different volumes. The fitting yields ␣KT共Va , T兲 = 0.0067⫾ 0.0003 GPa/ K and 共⳵KT / ⳵T兲V = 0.013⫾ 0.003 GPa/ K for all the isotherms, which indicate self-consistency in the measurements. Using the MGD approach, Fei et al.19 found that constant q = 0.5 can fit their data very well up to 28 GPa. This study shows a systematic, slightly higher thermal pressure than the extrapolation of their EOS in the higher pressure region, as shown in Fig. 4. This deviation cannot be fitted with constant q = 0.5 when the other parameters cannot be changed significantly for the consistency with many other independent measurements. The final parameters for the best fit to Eqs. 共8兲–共12兲 with only data points obtained in this study are listed in Table II and plotted in Fig. 4. The isothermal P-V data at any desired temperature can be calculated either from Eq. 共6兲 by using the fitted ␣KT共Va , T兲 and 共⳵KT / ⳵T兲V or Eqs. 共8兲–共12兲 by using the parameters of Table II. It is important to compare the isotherms obtained by thermodynamic and MGD approaches, which are shown in Fig. 6. As indicated by the theory, the results from these two approaches should be consistent with each other. Our results show that they are in remarkably good agreement. The standard deviations between fitting value and experimental data points for both EOS approaches are ⫾1.0 GPa, but the maximum pressure deviation between these two EOS is only 0.4 GPa. Table III shows the parameters determined in the high temperature isothermal EOS fitting to Eq. 共2兲 using the data obtained from both the thermodynamic and MGD approaches. Even though the results given by these two methods are very comparable, the argu-

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TABLE II. Fitting parameters for the P-V-T EOS of platinum with MGD equation. Parameters Va 共Å3兲 KTa 共GPa兲 KT ⬘ a ⌰0 共K兲 ␥0 q0 q1

Fei et al.

This study

60.38共1兲 273共3兲 4.8共3兲 230 2.69共3兲 0.5共5兲 0

60.38 273.5共10兲 4.70共6兲 230 2.75共3兲 0.25 15共10兲

冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 ⳵ 共 ␣ K T兲 ⳵V ⳵KT ⳵T

=

V

=−

T

1 ⳵KT V ⳵T

⳵KT ⳵T

+ ␣KT

P

共13兲

,

V

⳵KT ⳵P

.

共14兲

T

If ␣KT is independent of a change in volume, the term 共⳵KT / ⳵T兲V on the right side of Eq. 共13兲 should be zero. 共⳵KT / ⳵T兲V can be evaluated using Eq. 共14兲 as all the terms on the right side of this equation are available from the fitted P-V-T EOS 共MGD兲 and have been listed in Table IV. The value of 共⳵KT / ⳵T兲V for the EOS obtained by the MGD method is 0.012 GPa/ K, which is very close to the value obtained by the thermodynamic method. Existing thermodynamic data for a large number of materials have demonstrated that 共⳵KT / ⳵T兲V ⬇ 0, the so-called “Swenson law.”22 Realistic values for different materials may vary within ⫾0.2 around zero.23 The small, positive value obtained here implies that the thermal pressure increases with compression, as shown in Fig. 8. FIG. 5. Isothermal data fitting to Eq. 共6兲 with data points obtained from measured 共solid circles兲 and isochoric calculation 共open circles兲.

ment has been made that the MGD method is preferable because it better represents the temperature dependence of ␣ and other thermoelastic properties. This, in turn, provides a more secure basis for interpolating or extrapolating the results beyond the studied P-T ranges.21

IV. DISCUSSION

Platinum has been used widely for pressure calibration in high P-T x-ray diffraction experiments due to its simple structure, phase stability over a large pressure range, strong signal, and inert chemical properties. For a long time, the

B. Thermoelastic properties

The volume thermal expansion coefficient ␣ = 共1 / V兲 ⫻共⳵V / ⳵T兲 P at different pressure-temperature conditions can be obtained from the P-V-T EOS 共MGD method兲 and is plotted in Fig. 7. The value for ambient condition is 2.26 ⫻ 10−5 K−1, which agrees well with the value of 2.24 ⫻ 10−5 K−1 obtained by Edwards et al..28 The P-V-T EOS 共MGD兲 can also be used for calculating higher-order thermoelastic parameters. Among these parameters, 共⳵KT / ⳵T兲V and 共⳵KT / ⳵T兲 P are of interest because of their relation to the volume dependence of thermal pressure by

FIG. 6. Comparison of the EOSs obtained with thermodynamic and MGD approach for this study.

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J. Appl. Phys. 103, 054908 共2008兲

Zha et al. TABLE III. The Birch–Murnaghan EOS of platinum along five different isotherms. Parameter

Approach

Va 共Å 兲

Thermodynamic MGD

KTa 共GPa兲

Thermodynamic MGD

3

KT ⬘ = a

共 兲 ⳵KT ⳵P

T

Thermodynamic MGD

300 K

1300 K

1500 K

1700 K

1900 K

60.38 60.38

61.91 61.92

62.23 62.28

62.55 62.65

62.88 63.02

273.5 273.5 4.70 4.70

commonly used P-V-T EOSs for platinum have been determined from shock wave data.5,6 Recently, the EOS at 300 K has been studied by the static compression technique.9 A simultaneous high P-T study using a multianvil apparatus led to the first P-V-T EOS of platinum at a static pressure up to 28 GPa and 1900 K.19 It is highly desirable to validate the EOS of Pt proposed by Fei et al.19 at higher P-T condition with different experimental methods. This is necessary for making it a practical pressure scale for the in situ high P-T experiment. This work demonstrates that the DAC with internal resistive heating described above has extended the P-V-T EOS of platinum proposed by Fei et al.19 to near megabar pressures with good agreement in the P-T range overlapping that of the multianvil press. In addition, since temperature was measured by the thermocouple in the multianvil apparatus, this agreement also demonstrates that the spectroradiometric temperature measurement used in internal resistive heating is consistent with the temperature measured by thermocouple to 1900 K. Figure 4 shows some data scatter from the fitting curves. Since the temperature uncertainty is small and should vary with pressure in the same direction 共higher temperature corresponds to higher thermal pressure兲, its effect on the data scatter is negligible. The volume uncertainty will be the major source of this scatter. For cubic materials, each diffraction peak provides an independent lattice parameter and therefore an independent volume. The sample volumes in this study were calculated from the arithmetic averages of lattice pa-

253.1 253.2

249.0 248.0

4.77 4.65

4.79 4.65

244.8 242.7 4.80 4.66

240.8 237.4 4.81 4.67

rameters provided by the diffraction peaks, as listed in Table I. Differences between the volumes calculated from individual diffraction peaks may offer an opportunity for evaluating the deviatoric stress-strain condition at any given P-T condition. The overall volume uncertainties including the effect of stress-strain for the MgO and Pt in this study are ⬃0.3% and ⬃0.4%, respectively. The uncertainty for MgO corresponds to ⬃1.5 GPa pressure difference at the highest P-T condition according to the EOS of Speziale et al.,3 while the uncertainty for Pt corresponds to ⬃3 GPa pressure difference in the same P-T region according to the EOS obtained in this study. The maximum pressure uncertainty from the combination of these two volume uncertainties, therefore, is 3.4 GPa, which is ⬃4% of the maximum pressure of this study. This is consistent with a previous study18 although the pressure ranges are quite different. The largest data scatter shown in Fig. 4 is near 2.3 GPa, which is within the maximum pressure uncertainty estimated above. It is important to note that the ⬃0.3– 0.4% in volume uncertainty is comparable to the intrinsic uncertainty of the synchrotron radiation x-ray diffraction technique,29 which is due mainly to instrumentation effects. From this point of view, there is no measurable deviatoric stress-strain condition in this experiment. It is also important to compare the present study to the reduced shock wave EOSs which have been used for pressure calibration in modern in situ high P-T x-ray diffraction experiments in both the DAC and multianvil apparatus. Figure 9 shows the comparison of this study and those of Holmes et al.5 and Jamieson et al.6. The data of Jamieson et al. were extrapolated based on the assumption of a linear function in the temperature dependence of thermal pressure at constant volume. The room temperature isotherms in both reduced shock wave EOSs give higher pressures than that of the present study, but the pressure discrepancies at higher TABLE IV. The fitting parameters and results for thermoelastic properties. Parameters

FIG. 7. Volume thermal expansion coefficient of platinum at high pressure and temperatures.

Va K Ta KT ⬘ = 共⳵KT / ⳵ P兲Ta a 共⳵␣ / ⳵T兲 Pa 共⳵␣ / ⳵ P兲Ta 共⳵␣ / ⳵T兲Va ␣KT 共Va , T兲 共⳵KT / ⳵T兲V 共⳵KT / ⳵T兲 Pa

Thermodynamic

MGD

60.38 Å3 273.5GPa 4.70

60.38 Å3 273.5 GPa 4.70 0.543⫻ 10−8 K−2 −0.164⫻ 10−6 K−1 GPa−1 0.442⫻ 10−8 K−2 0.0062 GPa/ K 0.012 GPa/ K −0.0169 GPa K−1

0.0067 GPa/ K 0.013 GPa/ K

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FIG. 8. Volume dependence of thermal pressure.

temperature are different. The result of Holmes et al. has higher pressures than the present study 关Fig. 9共a兲兴 and that of Jamieson et al. shows lower pressures 关Fig. 9共b兲兴. For the latter, at 1900 K, the difference is about 1.3– 1.5 GPa, close to the thermal pressure difference from 1700 to 1900 K. The measurements of Holmes et al.5 show progressively higher pressures at each isotherm than those of the present study. The room temperature data reported by Dewaele et al.9

FIG. 9. Comparison of reduced shock wave EOSs and EOS from this study. 共a兲. Comparison between the EOS reported by Holmes et al. and this study. 共b兲 Comparison between the EOS reported by Jamieson et al. and this study.

J. Appl. Phys. 103, 054908 共2008兲

FIG. 10. Comparison between the EOS of Jamieson et al. at 300 K plus the thermal EOS of Fei et al. at high temperature and this study. The EOS of Jamieson et al. of Pt at 300 K is consistent with the data of Dewaele et al. for Pt at 300 K if plotted with the new ruby scale 共Ref. 9兲.

plotted with their proposed new ruby scale,9 as shown in Fig. 10, are consistent with the result of Jamieson et al. at 300 K. Based on these 300 K data, plus the thermal pressure calculated by Fei et al.,19 the total pressures at each high temperature are quite close to the fitted EOS of this study up to 55 GPa. Above that pressure, the total pressure is higher than in this study. Dewaele et al.30 have performed a synchrotron x-ray diffraction study of MgO and Pt using a CO2 laser heating technique, specifically to determine the P-V-T EOS of MgO. Pt was used for the pressure marker and the EOS of Jamieson et al.6 for Pt was used for the pressure calculation at high temperature. The 61 data points they obtained are located from 300 to 2500 K, while 28 data points are within the range of 1900⫾ 200 K. Figure 11 shows the comparison of their MgO data collected at 1900⫾ 200 K 共⫾200 K is the claimed uncertainty in temperature measurement兲 and the

FIG. 11. A cross-check for MgO at 1900 K in this study and a measurement by Dewaele et al. 共Ref. 30兲 using in situ laser heating and synchrotron x-ray diffraction.

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MgO pressure scale of Speziale et al.3 at 1900 K, which we used for the pressure calculation of the isotherm at 1900 K in this study. Their MgO data points are consistent with those of the present study. But as shown in Fig. 9共b兲, their pressure scale 共the EOS of Jamieson et al. for Pt兲 should give about 1.3– 1.5 GPa lower pressures than that of the scale of Speziale et al.3 according to the present study. The most plausible reason for this discrepancy may be the temperature measurement. Spectroradiametric temperature measurement is sensitive to the system calibration and the radiance measurement. The thermal emission spectra they measured are emitted from a MgO–Pt mixture which may have a different wavelength profile of brightness from their calibration spectrum emitted from the hot forsterite surface as a result of larger emissivities for metals than for silicates. Brighter emission leads to overestimation of the temperature, which contributes to the thermal pressure estimation. In the present study, the thermal emission is measured from the portion of the Re heater near the heater-sample interface; this is similar to the calibration spectra of the standard lamp because the emissivity of the materials is similar in both cases. This may also explain the consistency between this study and that using multianvil devices in which temperatures are measured by thermocouple. In the P-V-T EOS for Pt proposed by Fei et al.19 and this study, the MgO scale of Speziale et al.3 was used for the pressure calibration. The MgO scale of Speziale et al.3 was determined by taking all available experimental data into account; it also generally agrees well with molecular dynamics simulations.31 In addition, MgO has been studied as a primary pressure scale at 300 K to 55 GPa by the combination of Brillouin scattering and x-ray diffraction.8 This calibration has shown the consistency with ruby scale2 and the error is within 2% of the new ruby scale of Dewaele et al. at 55 GPa.9 MgO is an ideal candidate for a primary pressure scale at simultaneous high pressure and high temperature. A practical pressure scale calibrated with MgO directly benefits from this effort. Obviously, as a practical pressure scale for in situ x-ray diffraction at high P-T conditions, Pt has the advantage of a stronger signal and more inert chemistry relative to MgO. V. CONCLUSION

In summary, we have performed x-ray diffraction measurements on MgO and Pt mixtures under simultaneous high pressure and high temperature conditions to 80 GPa and 1900 K with an improved internal resistive heating technique in the DAC. Data analysis using both thermodynamic and MGD approaches leads to a consistent P-V-T EOS inversion. The measured P-V-T EOS for platinum is consistent with a previous static compression study conducted in the multianvil apparatus19 in the overlapping P-T range, and the pressure range has been extended threefold in this study. The resulting P-V-T EOS of platinum based on the P-V-T EOS of MgO obtained by Speziale et al.3 could be better as an in situ practical pressure scale in static compression at high temperature than MgO itself when x-ray diffraction is used. For a long time, the P-V-T EOS of static compression

above 1000 K could be obtained only from LVP experiments; however, the pressure range in such experiments is limited. P-V-T EOS data at similar temperature but in a wider pressure range could be obtained up to now only from dynamic compression experiments. The pressure and temperature range of the present technique promisingly fills the gap between dynamic and LVP techniques. This offers the opportunity for cross-checking and reconciling isothermal EOS data derived from these two methods. The wide P-T range and reasonable precision of this technique may also lead to various applications other than P-V-T EOS measurements. Phase diagram mapping for many materials as well as material synthesis at megabar pressures and thousands of degrees may be possible in the future. ACKNOWLEDGMENTS

This work was conducted in part at the Cornell High Energy Synchrotron Source 共CHESS兲, which is supported by the NSF and NIH/NIGMS under Award No. DMR 0225180. Help during feasibility experiments performed at sector16, HPCAT, Advanced Photon Source 共APS兲, Argonne National Laboratory, is greatly appreciated. HPCAT is supported by DOE-BES, DOE-NNSA, NSF, and the W.M. Keck Foundation. APS is supported by DOE-BES under Contract No. DE-AC02-06CH11357. Support from NSF-DMR, NASA, and the Carnegie-DOE Alliance Center 共CDAC, DOE/NNSA Grant No. DE-FC03-03NA00144 is also gratefully acknowledged. One of the Authors 共O.T.兲 is grateful for the support from NSF-CSEDI Award No. 0552010 and NNSA Cooperative Agreement No. DE-FC88-01NV14049. The authors thank Y. Fei, A. Goncharov, S. Gramsch, T. Komabayashi, H.-P. Liermann, Y. Meng, and M. Somayszulu for their help, discussions, and comments. 1

A. D. Chijioke, W. J. Nellis, A. Soldatov, and I. F. Silvera, J. Appl. Phys. 98, 114905 共2005兲. 2 H. K. Mao, J. Xu, and P. M. Bell, J. Geophys. Res. 91, 4673 共1986兲. 3 S. Speziale, C.-S. Zha, T. S. Duffy, R. J. Hemley, and H.-K. Mao, J. Geophys. Res. 106, 515 共2001兲. 4 A. D. Chijioke, W. J. Nellis, and I. F. Silvera, J. Appl. Phys. 98, 073526 共2005兲. 5 N. C. Holmes, J. A. Moriarty, G. R. Gathers, and W. J. Nellis, J. Appl. Phys. 66, 2962 共1989兲. 6 J. C. Jamieson, J. N. Fritz, and M. H. Manghnani, in High Pressure Research in Geophysics, edited by S. Akimoto and M. H. Manghnani 共Center for Academic Publications, Tokyo, 1982兲, p. 27. 7 C. S. Zha, T. S. Duffy, R. T. Downs, H. K. Mao, and R. J. Hemley, Earth Planet. Sci. Lett. 159, 25 共1998兲. 8 C.-S. Zha, H.-K. Mao, and R. J. Hemley, Proc. Natl. Acad. Sci. U.S.A. 97, 13494 共2000兲. 9 A. Dewaele, P. Loubeyre, and M. Mezouar, Phys. Rev. B 70, 094112 共2004兲. 10 B. S. Li, J. Kung, T. Uchida, and Y. B. Wang, J. Appl. Phys. 98, 013521 共2005兲. 11 B. S. Li, K. Woody, and J. Kung, J. Geophys. Res. 111, B11206 共2006兲. 12 W. A. Bassett and L. C. Ming, Phys. Earth Planet. Inter. 6, 154 共1972兲. 13 G. Y. Shen, M. L. Rivers, Y. B. Wang, and S. R. Sutton, Rev. Sci. Instrum. 72, 1273 共2001兲. 14 D. Andrault and G. Fiquet, Rev. Sci. Instrum. 72, 1283 共2001兲. 15 T. Yagi, T. Kondo, T. Watanuki, O. Shimomura, and T. Kikegawa, Rev. Sci. Instrum. 72, 1293 共2001兲. 16 C.-S. Zha and W. A. Bassett, Rev. Sci. Instrum. 74, 1255 共2003兲. 17 A. P. Jephcoat and S. P. Besedin, Philos. Trans. R. Soc. London, Ser. A 354, 1333 共1996兲. 18 C.-S. Zha, W. A. Bassett, and S.-H. Shim, Rev. Sci. Instrum. 75, 2409

Downloaded 01 May 2008 to 160.103.2.224. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

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共2004兲. Y. Fei, J. Li, K. Hirose, W. Minarik, J. Van Orman, C. Sanloup, W. van Westrenen, T. Komabayashi, and K. Funakoshi, Phys. Earth Planet. Inter. 143–144, 515 共2004兲. 20 T. S. Duffy and Y. Wang, in Ultrahigh-Pressure Mineralogy: Physics and Chemistry of the Earth’s Deep Interior, Reviews in Mineralogy Vol. 37, edited by R. J. Hemley 共Mineralogical Society of America, Washington DC, 1998兲, p. 425. 21 I. Jackson and S. M. Rigden, Phys. Earth Planet. Inter. 96, 85 共1996兲. 22 O. L. Anderson, Crit. Rev. Anal. Chem. 1, 185 共1984兲. 23 O. L. Anderson, Phys. Earth Planet. Inter. 112, 267 共1999兲. 24 O. L. Anderson, D. G. Isaak, and S. Yamamoto, J. Appl. Phys. 65, 1534 19

J. Appl. Phys. 103, 054908 共2008兲

Zha et al.

共1989兲. O. L. Anderson, H. Oda, A. Chopelas, and D. G. Isaak, Phys. Chem. Miner. 19, 369 共1993兲. 26 R. Boehler, Phys. Rev. B 27, 6754 共1983兲. 27 I. Inbar and R. E. Cohen, Geophys. Res. Lett. 22, 1533 共1995兲. 28 J. W. Edwards, R. Speiser, and H. L. Johnston, J. Appl. Phys. 22, 424 共1951兲. 29 B. Buras and L. Gerward, Prog. Cryst. Growth Charact. 18, 93 共1989兲. 30 A. Dewaele, G. Fiquet, D. Andrault, and D. Hausermann, J. Geophys. Res. 105, 2869 共2000兲. 31 M. Matsui, S. C. Parker, and M. Leslie, Am. Mineral. 85, 312 共2000兲. 25

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