REVIEW OF SCIENTIFIC INSTRUMENTS
VOLUME 75, NUMBER 7
JULY 2004
Rhenium, an in situ pressure calibrant for internally heated diamond
anvil cells
Chang-Sheng Zhaa)
Cornell High Energy Synchrotron Source, Wilson Laboratory, Cornell University, Ithaca, New York 14853
William A. Bassett
Department of Geological Sciences, Cornell University, Ithaca, New York 14853
Sang-Heon Shim
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
共Received 27 August 2003; accepted 26 April 2004; published online 23 June 2004兲
The rheologic, chemical, thermal, and electrical properties of rhenium make it an excellent choice
for containing and heating samples to very high pressures and temperatures in diamond anvil cells
共DACs兲. In many experimental configurations, e.g., the internally heated diamond anvil cell
共IHDAC兲, the rhenium parts are at or close to the pressure and temperature conditions of the sample.
Because the pressure and temperature of the rhenium container are close to those of the specimen,
rhenium offers an attractive means for determining pressure at high temperatures in x-ray diffraction
experiments without the requirement of adding an additional material to the intricate and cluttered
sample assembly. For this reason, we set out to determine an equation of state 共EOS兲 of rhenium.
We combine the isothermal equation of state of rhenium at ambient temperature with volume data
collected at randomly distributed, simultaneous high pressure-temperature conditions. A linear
dependence of thermal pressure on temperature at constant volume has been assumed. Data were
collected using synchrotron radiation x-ray diffraction in conjunction with an IHDAC equipped with
a rhenium internal resistive heater developed recently at the Cornell High Energy Synchrotron
Source. The consistency over a large P – T range between our EOS and shock EOS within the
experimental uncertainty suggests that the thermal pressure is measurable using the method
proposed in the article, and that the rhenium can be used as a convenient pressure calibrant although
the accuracy of it depends on many factors including the reliability of the pressure scale at high
temperature. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1765752兴
I. INTRODUCTION
fundamentally important for extending its scientific and engineering applications including its usefulness as a pressure
calibrant.
In this article, we present a method for developing a
P – V – T equation of state employing one set of volume data
collected randomly throughout an extended pressure–
temperature range. The equation of state of a solid in its most
general form is
Recently, Zha and Bassett1 described a diamond anvil
cell with an internal resistive heater 共IHDAC兲 in which the
sample resides in a hole in a strip of Re foil that serves as the
resistive heater. In this and in other similar devices, Re has
proven to be one of the most satisfactory materials for containing and heating a sample to achieve simultaneous high
pressure and high temperature.2– 4 Although the ductile, but
incompressible, properties of metallic rhenium were reported
by Bridgman in 1955,5 it has only been within the past two
decades that rhenium has been favored for use in diamond
anvil cells 共DACs兲.4,6 In addition to the rheologic properties
described by Bridgman, the very high melting point
共3180 °C兲 and lack of ductile-to-brittle transition enhance its
value for use under simultaneous high pressure and temperature conditions. In recent years, engineering interests in rhenium and its alloys for parts subjected to extreme conditions
have increased dramatically.7,8 There have been many studies
on the physical properties including the P – V equation of
state 共EOS兲 under shock and static pressure conditions.4,9–16
The P – V – T equation of state reported in this article will be
P 共 V,T 兲 ⫽ P 共 V,0兲 ⫹ P th共 V,T 兲 .
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 pressure–volume relationship at absolute zero,
and the second term is the thermal pressure. Our usual concern is the difference in pressure between ambient conditions
and a hot, compressed state,
P 共 V,T 兲 ⫺ P 共 V a ,300兲 ⫽⌬ P 共 V a →V,300兲 ⫹ P th共 V,300→T 兲 .
共2兲
The subscript a refers to ambient conditions. Equation 共2兲
tells us that the pressure change from ambient conditions to
higher P – T conditions is equal to the change in pressure due
a兲
Electronic mail: [email protected]
0034-6748/2004/75(7)/2409/10/$22.00
共1兲
2409
© 2004 American Institute of Physics
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2410
Rev. Sci. Instrum., Vol. 75, No. 7, July 2004
Zha, Bassett, and Shim
to isothermal compression at ambient temperature plus the
thermal pressure change due to isochoric temperature
change.
For most solids, the first term on the right of Eq. 共2兲 can
be determined by the third-order Birch–Murnaghan equation
of state,
冋冉 冊 冉 冊 册
冋冉 冊 册冎
再
7/3
Va
V
3
⌬ P 共 V a →V,300兲 ⫽ K Ta
2
⫺
5/3
Va
V
Va
V
3
⬘ ⫺4 兲
⫻ 1⫹ 共 K Ta
4
2/3
⫺1
,
共3兲
⬘
where K Ta is the isothermal bulk modulus, and K Ta
⫽( ⳵ K T / ⳵ P) T , both at ambient conditions. The units of Eq.
共3兲 are GPa when K Ta is in GPa. Equation 共3兲 is valid for
isothermal compression at any high temperature, provided
⬘ are evaluated at the elevated temperature and
K Ta and K Ta
zero 共omit the-兲 pressure conditions, respectively.
The second term on the right side of Eq. 共2兲 is more
complicated because of its definition, and direct determination by experiment still is challenging. On the basis of data
analysis for a wide range of materials, Anderson and
colleagues17–19 have indicated that, for adequate precision of
approximation, P th often takes the linear form of
P th⫽a⫹bT
共4兲
for most solids when the temperature is higher than the Debye temperature, ␪. From thermodynamic identities assuming ␣ K T (V a ,T) is independent of the temperature, they have
demonstrated18
P th共 V,300→T 兲
⫽ P th共 V,T 兲 ⫺ P th共 V,300兲
冋
⫽ ␣ K T 共 V a ,T 兲 ⫹
冉 冊 冉 冊册
⳵KT
⳵T
Va
V
ln
V
共 T⫺300兲 ,
共5兲
where ␣ ⫽(1/V)( ⳵ V/ ⳵ T) P is the volume thermal expansion
coefficient, and ( ⳵ K T / ⳵ T) V is the temperature derivative of
the isothermal bulk modulus at constant volume. So Eq. 共2兲
finally becomes
P 共 V,T 兲 ⫺ P 共 V a ,300兲
冋冉 冊 冉 冊 册
冋冉 冊 册冎
再
冉 冊 冉 冊册
冋
3
⫽ K Ta
2
Va
V
7/3
⫺
Va
V
3
⬘ ⫺4 兲
⫻ 1⫹ 共 K Ta
4
⫹ ␣ K T 共 V a ,T 兲 ⫹
5/3
Va
V
⳵KT
⳵T
x-ray diffraction are the most commonly used methods. To
develop a P – V – T equation of state, one generally needs a
large number of experimental data in order to adequately
cover the three-dimensional variable space. The ideal situation would be for those data to be obtained in the form of
two-dimensional variable groups with the third variable fixed
for each group 共e.g., isothermal, isobaric, or isochoric兲. Otherwise, data randomly dispersed through variable space can
lead to annoying problems for their reduction. Furthermore,
random dispersion of data throughout variable space is likely
to be one of the sources of uncertainty in the results. Unfortunately, the ideal methods for data collection are difficult
experimentally. As stated above, we will show in this article
how to use limited data points that are spread out in P – V – T
space for the creation of a P – V – T equation of state.
Equation 共6兲 greatly simplifies the experimental procedure for obtaining the P – V – T equation of state. The part
dealing with isothermal compression at ambient temperature
关Eq. 共3兲兴 can be easily and precisely achieved over a very
wide pressure range with modern DAC techniques. The thermal pressure part 关Eq. 共5兲兴 is assumed to have a simple linear
relationship between P th and T, the slope for temperature
dependence of 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 equation of state measured at
some lower temperature condition using the same volume
value as that measured at high P – T. Fortunately, the relationship is essentially linear for most solids if the temperature is higher than the Debye temperature, ␪, based on the
analysis by Anderson et al.17 With the slopes determined in
this way at different volumes, thermal pressures at various
temperatures can be obtained that correspond to these volumes. Hence, only one set of high PT data at random high
PT conditions, corresponding to different volumes 共or compressions兲, combined with a well defined isothermal equation
of state measured at some lower reference temperature can
form several isothermal data sets at multiple temperature
points. Two parameters, ␣ K T (V a ,T) and ( ⳵ K T / ⳵ T) V , can
be determined by fitting each data set to Eq. 共6兲. The slope of
thermal pressure on the right side of Eq. 共5兲 can be obtained,
and isothermal equations of state at different temperatures
can be created. The P – V – T equation of state for rhenium in
this study is obtained using this method.
2/3
⫺1
ln
V
Va
V
II. PREVIOUS DATA
共 T⫺300兲 .
共6兲
By using Eq. 共6兲 and basing their model on a vast
amount of experimental data, they proposed a P – V – T equation of state for gold,18 which has been widely used as pressure standard for static compression experiments.2,20
Obtaining P – V – T equations of state for any other interesting materials has been an active area of research for the
past few decades.21,22 Laser heated DACs or a resistively
heated multianvil apparatus in conjunction with synchrotron
In order to use the method described above, we need to
know two things, the degree of linearity of the thermal pressure of rhenium versus the temperature and the starting temperature for this linearity.
The thermal pressure P th 共between 300 K and T兲 is determined by
P th⫽
冕
T
Ta
共 ␣ K T 兲 V dT,
共7兲
where T a ⫽300 K. The isothermal bulk modulus K T can be
obtained from
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Rev. Sci. Instrum., Vol. 75, No. 7, July 2004
Re calibrant for laser heated DAC’s
2411
TABLE I. Physical properties of rhenium at room pressure and high temperatures. Elasticity measurements were obtained from Ref. 10. The thermal
expansion coefficients ␣ and specific-heat capacity C P at constant pressure are calculated based on the data in the American Institute of Physics Handbook,
pp. 4 –129 and 4 –107, respectively. Densities are calculated using ␣ ⫽(1/V)( ⳵ V/ ⳵ T) P and ␳ 共298 K兲⫽21.024 g/cm3.
T
共K兲
␳
共g/cm3兲
␣
(⫻10⫺6 K⫺1 )
C 11
共GPa兲
C 33
共GPa兲
C 44
共GPa兲
C 13
共GPa兲
C 12
共GPa兲
KS
共GPa兲
CP
共J/g K兲
␥
KT
共GPa兲
P th
共GPa兲
4
23
73
123
173
223
273
298
323
373
423
473
523
573
623
673
723
773
823
873
923
21.12
21.11
21.10
21.08
21.07
21.05
21.03
21.02
21.01
21.00
20.98
20.96
20.94
20.92
20.90
20.88
20.86
20.84
20.81
20.79
20.77
0.027
0.689
9.142
13.808
16.219
17.325
17.629
17.758
17.887
18.144
18.402
18.659
18.916
19.174
19.431
19.688
19.946
20.203
20.461
20.718
20.975
644.6
643.9
641.0
636.4
631.1
625.8
620.6
618.2
615.4
610.3
605.3
600.2
595.4
590.3
586.1
582.2
578.1
574.0
570.0
566.0
561.9
717.0
717.0
713.5
707.2
700.1
693.3
686.7
683.5
680.4
674.8
669.6
664.1
659.0
654.1
649.3
645.0
640.6
636.5
632.1
627.9
623.7
168.5
168.5
167.9
166.3
164.8
163.2
161.5
160.6
159.8
158.2
156.6
154.9
153.2
151.4
149.6
147.9
146.1
144.4
142.5
140.8
139.1
195.9
196.2
197.4
200.0
202.6
204.8
206.8
207.8
208.4
209.8
211.0
211.8
212.8
213.8
214.6
215.4
216.0
216.8
217.4
217.8
218.4
277.0
276.7
275.8
275.1
274.5
274.8
275.0
275.3
275.3
275.2
275.1
275.0
275.0
274.5
274.8
275.6
275.9
276.0
276.4
276.5
276.2
371.5
371.4
370.7
370.0
369.1
368.2
367.2
366.9
366.2
365.0
363.8
362.4
361.2
359.9
358.8
358.0
357.0
356.0
354.9
353.8
352.6
0.0002
0.0054
0.0718
0.1081
0.1265
0.1347
0.1377
0.1383
0.1387
0.1392
0.1400
0.1411
0.1426
0.1441
0.1456
0.1469
0.1482
0.1494
0.1507
0.1522
0.1536
2.21
2.21
2.21
2.22
2.22
2.22
2.23
2.24
2.25
2.27
2.28
2.29
2.29
2.29
2.29
2.30
2.30
2.31
2.31
2.32
2.32
371.5
371.4
370.2
368.6
366.8
365.0
363.3
362.6
361.5
359.6
357.6
355.4
353.4
351.2
349.3
347.6
345.6
343.7
341.8
339.7
337.6
0
0.020
0.117
0.316
0.584
0.887
1.203
1.363
1.525
1.849
2.177
2.507
2.840
3.175
3.513
3.854
4.198
4.544
4.892
5.243
5.596
K T⫽
KS
,
1⫹
␣␥ T 兲
共
共8兲
where the Grüneisen parameter ␥ at constant pressure is defined by
␥⫽
␣KS
.
␳CP
共9兲
To obtain the thermal pressure P th , we need to know the
temperature dependences of the coefficient of thermal expansion ␣, the specific heat at constant pressure C P , and the
elastic properties. Shepard and Smith16 and Fisher and
Dever10 have studied the elastic moduli of rhenium over a
large temperature range. Table I lists the fundamental thermodynamic parameters given by Fisher and Dever and other
sources we used for estimation of the temperature dependence of thermal pressure. T and K S are the temperature and
adiabatic bulk modulus calculated from elastic constants of
Fisher and Dever, ␣ and C P are from American Institute of
Physics Handbook. The density, ␳, is calculated by using
thermal expansion coefficients ␣ at higher temperatures than
298 K and ␳ 共at 298 K兲⫽21.024 g/cm3.23 The ␣ values below
room temperature are not available, but can be calculated
from Eq. 共9兲 using the extrapolations of ␥ and ␳. The Grüneisen ␥ at room pressure was found to be 2.24, which is close
to 2.39 reported by Manghnani et al.14
Figure 1 is a plot of ␣ K T and thermal pressure P th versus
the temperature for rhenium at zero compression. The Debye
temperature of this material is reported to be higher than
room temperature 共406.2 K兲.10 But Fig. 1 clearly shows that
both ␣ K T and thermal pressure, P th , are nearly linear with
the temperature starting even from lower than room temperature. Figure 1 also shows that the ␣ K T is not completely
independent of the temperature at its linear portion because
of the electronic contributions in metals at high
temperature,24 a feature that would lead to a small quadratic
term for the thermal pressure although it is not obvious in
Fig. 1. However, the least-square fitting result of a very small
value for the quadratic term 共see discussion below兲 suggests
that linear approximation for the temperature dependence of
thermal pressure is adequate for experimental precision. On
the other hand, the linear treatment means that the
␣ K T (V a ,T) in Eq. 共6兲 either should be temperature
independent18 or have an average value over the entire temperature range of study.
Because the results are derived from zero compression
data, there is a question as to the behavior expected at higher
compressions. The thermodynamic identity
冉
⳵共 ␣KT兲
⳵V
冊
⫽⫺
T
冉 冊
1 ⳵KT
V ⳵T
共10兲
V
FIG. 1. Temperature dependence of ␣ K T and thermal pressure P th at ambient pressure for rhenium. The starting point of linearity of thermal pressure
is even lower than 300 K.
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2412
Rev. Sci. Instrum., Vol. 75, No. 7, July 2004
Zha, Bassett, and Shim
FIG. 2. Temperature dependence of adiabatic 共subscript S兲 and isothermal
共subscript T兲 bulk moduli of rhenium at ambient pressure and volume.
has been used to evaluate how ␣ K T depends upon volume
compression. If ␣ K T is independent of a change of volume,
the term ( ⳵ K T / ⳵ T) V on the right side of Eq. 共10兲 should be
zero. Because the measurements were taken at constant P,
not constant V, another thermodynamic identity,
冉 冊 冉 冊
⳵KT
⳵T
⫽
V
⳵KT
⳵T
⫹␣KT
P
冉 冊
⳵KT
⳵P
共11兲
,
T
FIG. 3. Energy dispersive x-ray diffraction spectrum taken at simultaneous
high pressure and high temperature during the experiment. The unmarked
small peaks are from crystallized pressure medium of SiO2 glass at high
P – T conditions.
has been used for the calculation of ( ⳵ K T / ⳵ T) V , 19 which is
⫺0.0094 GPa/K for the ambient condition here, and is plotted in Fig. 2. Obviously, ( ⳵ K T / ⳵ T) V0 is nonzero, but a small,
negative value indicates that the slope of the thermal pressure curve will be changed slightly at different compressions.
The second term in the square bracket on the right side of
TABLE II. 共a兲 Measured lattice parameters and volumes of gold at different temperatures and pressures. P T are
calculated from Anderson’s EOS for gold with V a ⫽67.847 Å3 共04-0784, 1997 JCPDS-ICDD兲. The average
lattice parameters for each PT point, ā, are obtained by arithmetic averaging the four diffraction lines. 共b兲
Measured volumes of rhenium at different temperatures and pressures. The pressures at different T are obtained
from a gold sensor based on Anderson’s EOS for gold.
共a兲
T
共K兲
1380.3
1480.2
1506.5
1548.6
1628.5
1716.8
1801.0
1914.5
(d hkl ) (Å)/a hkl (Å)
111
200
220
311
共2.3568兲
4.0821
共2.3540兲
4.0773
共2.3574兲
4.0831
共2.3620兲
4.0911
共2.3654兲
4.0970
共2.3762兲
4.1157
共2.3806兲
4.1233
共2.3826兲
4.1268
共2.0408兲
4.0816
共2.0400兲
4.0800
共2.0428兲
4.0856
共2.0448兲
4.0896
共2.0486兲
4.0972
共2.0550兲
4.1100
共2.0549兲
4.1098
共2.0613兲
4.1226
共1.4452兲
4.0876
共1.4404兲
4.0741
共1.4430兲
4.0814
共1.4457兲
4.0891
共1.4476兲
4.0944
共1.4528兲
4.1091
共1.4588兲
4.1261
共1.4590兲
4.1267
共1.2294兲
4.0775
共1.2305兲
4.0811
共1.2315兲
4.0844
共1.2338兲
4.0921
共1.2359兲
4.0990
共1.2418兲
4.1186
共1.2409兲
4.1156
共1.2438兲
4.1252
ā
共Å兲
⌬a/ā
V
共Å3兲
PT
共GPa兲
4.0823⫾0.0016
0.0004
68.0322
7.29
4.0781⫾0.0013
0.0003
67.8225
8.47
4.0838⫾0.0008
0.0002
68.1073
8.00
4.0904⫾0.0006
0.0001
68.4380
7.58
4.0969⫾0.0007
0.0002
68.7648
7.46
4.1132⫾0.0020
0.0005
69.5888
6.41
4.1186⫾0.0038
0.0009
69.8633
6.48
4.1253⫾0.0011
0.0003
70.2048
6.66
共b兲
T
共K兲
a
共Å兲
c
共Å兲
c/a
V
共Å3兲
PT
共GPa兲
1380.3
1480.2
1506.5
1548.6
1628.5
1716.8
1801.0
1914.5
2.7622
2.7659
2.7640
2.7639
2.7592
2.7656
2.7654
2.7685
4.4604
4.4501
4.4654
4.4717
4.5083
4.4907
4.5092
4.5088
1.615
1.609
1.616
1.618
1.634
1.624
1.631
1.629
29.471
29.482
29.543
29.583
29.723
29.745
29.863
29.927
7.29
8.47
8.00
7.58
7.46
6.41
6.48
6.66
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Rev. Sci. Instrum., Vol. 75, No. 7, July 2004
Re calibrant for laser heated DAC’s
2413
Eq. 共5兲 has included this effect as derived in Ref. 18. Equation 共11兲 shows that ( ⳵ K T / ⳵ T) V is pressure–temperature dependent. This means that ( ⳵ K T / ⳵ T) V in Eq. 共6兲 should be
taken as an average value over the pressure–temperature
range of study.25
III. EXPERIMENT
The internal resistive heating IHDAC technique has
been used here.1 A rhenium ribbon with cross section dimensions of 70⫻20 ␮m was used for the resistive heater and also
for the sample. A small hole of 25 ␮m diameter was drilled
using a Q-switched infrared YAG laser in the center of the
heater, and gold powder was packed into this small hole to
provide a pressure standard at high P – T conditions. This
heater assemblage was carefully packed into a nonmetallic
gasket hole with SiO2 glass as the pressure medium, which
thermally isolated the heater from the diamonds. Energy dispersive synchrotron x-ray diffraction was used for in situ
volume measurements of rhenium and gold at simultaneous
high P – T conditions. Experiment was carried out at the national high pressure facility at the Cornell High Energy Synchrotron Source 共CHESS兲. The x-ray beam size was 20⫻30
␮m and had been aligned to the position of the small hole in
the heater loaded with gold powder. We believe that the pressure and temperature experienced by the gold and the surrounding rhenium were the same because of the x-ray beam
passed through a very small portion of heated material. We
observed a very homogeneous temperature profile in the
sample area, and excellent time stability.
An optical radiometric spectroscopy system was used for
in situ temperature measurement. An optical cable with a
single fiber was used for transferring the radiation signals
from microscope to spectrometer through an optical adaptor.
A multiple channel charge coupled device 共CCD兲 detector
was used to receive signals covering a 500 nm spectral
range. To sample and spectroscopically analyze a portion of
the incandescent light from the sample, we simply place the
cross hairs of the microscope on the desired part of the
sample. The system allows sampling of an area of 10 ␮m in
diameter. The measuring statistical error is ⫾3° in the temperature range of the present experiment according to the
temperature fitting program, and the uncertainty for the temperature was estimated to be ⫾10°. We chose varying temperatures at a constant loading force as the data collection
mode. Experimental details with this technique are published
elsewhere.1
Figure 3 shows one of the x-ray diffraction spectra collected during this experiment. Four gold and six rhenium
lines are present. The volumes for gold at each P – T condition were calculated using the average lattice parameters determined on the basis of reflections having various hkl indices 共Table IIa兲. The volumes for rhenium were calculated by
100 and 101 reflections as only these two reflections are
available for all pressures 共Table IIb兲. But the six rhenium
lines obtained at 1480 K 共Fig. 3兲 offer a good opportunity to
test uniaxial elastic strain at high temperatures 共see the Appendix兲. The volume of gold along with the temperature
measured can be used to estimate the pressure at this P – T
FIG. 4. Pressure–volume relationship of rhenium at different temperatures.
The dashed line shows the isochoric treatment based on linear estimation of
the temperature dependence of thermal pressure at constant volume for the
eight points measured. The solid lines are the fitted isothermal pressures at
different temperatures for those measured points when Eq. 共6兲 is used. See
the text for the details.
condition. The volume of rhenium, therefore, is related to
this P – T condition. On the other hand, the pressure for the
same volume of rhenium at ambient temperature can be
evaluated from the isothermal equation of state at ambient
temperature, which has been measured by many
investigators.3,4,12,14,26 The linear temperature dependence of
pressure for this isochore can then be drawn using just one
measurement at a simultaneous high P – T condition in conjunction with the room temperature isotherm. Because
changing the temperature produces new pressure and volume, x-ray diffraction measurements can be conducted at a
new stable P – T condition, and a new slope of temperature
dependence of the pressure for this new isochore can be obtained in the same way. Linear dependence between the pressure and temperature of each isochore makes it easy to obtain
the pressure–volume data at selected temperatures by linear
extrapolations. Therefore, the pressure–volume data for rhenium at selected isotherms can be obtained by grouping the
P – V data for each temperature selected.
IV. RESULTS
Tables IIa and IIb list detailed data for gold and rhenium
from all eight measurements. The pressure–volume data for
rhenium at three isotherms, which are calculated based on
linear isochoric treatment of each measured volume at high
temperatures, are plotted in Fig. 4. Seven temperatures of
300, 500, 1000, 1500, 2000, 2500, and 3000 K are chosen as
temperatures in this study, but only three isotherms are plotted for clarification. The negative pressures of isotherm 300
K for the experimental volumes, which are larger than the
volume at the ambient condition, are results of the mathematical treatment only. Equation 共6兲 was used for the leastsquares fit to each of seven isotherms to find two parameters:
␣ K T (V a ,T) and ( ⳵ K T / ⳵ T) V with V a ⫽29.4087 Å3 共data
⬘
from 05-0702, 1997 JCPDS-ICDD兲 K Ta and K Ta
⫽( ⳵ K T / ⳵ P) T fixed. It is interesting but not surprising that
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2414
Rev. Sci. Instrum., Vol. 75, No. 7, July 2004
Zha, Bassett, and Shim
TABLE III. Fitting parameters and results for Eq. 共6兲.
Parameter
Va
K Ta
⬘ ⫽( ⳵ K T / ⳵ P) T
K Ta
␣ K T (V a ,T)
( ⳵ K T / ⳵ T) V
Reference
3
29.4087 Å
360 GPa
4.5
0.00776 GPa/K
⫺0.00815 GPa/K
05-0702, 1997 JCPDS-ICDD
3
3
This study
This study
for a certain isothermal EOS at 300 K, all six isotherms gave
the same results of ␣ K T (V a ,T) and ( ⳵ K T / ⳵ T) V from the
fitting, which reflects the self-consistency of the measurements. A number of equations of state for rhenium at ambient
temperature have been published in the past.3,4,11 The ambient pressure isothermal bulk modulus K Ta ⫽360 GPa and its
⬘ ⫽( ⳵ K T / ⳵ P) T ⫽4.53 for the thirdpressure derivative K Ta
order Birch–Murnaghan equation of state were chosen,
which are consistent with shock wave data.27 Table III lists
the fitting results. As mentioned above, both ␣ K T (V a ,T) and
( ⳵ K T / ⳵ T) V in Eq. 共6兲 are average values over the pressure–
temperature range studied. Unlike the data shown in Fig. 1,
the fitted ␣ K T (V a ,T) does not change with the temperature,
and the value is close to, but higher than, the average value
of that listed in Table I. This is because the temperature
range in which the fitted ␣ K T (V a ,T) being averaged was
much larger than that in Table I. For the same reason, the
fitted ( ⳵ K T / ⳵ T) V is slightly different from the corresponding
value shown in Fig. 2, which was obtained based on the data
at the ambient condition shown in Table I.
When ␣ K T (V a ,T) and ( ⳵ K T / ⳵ T) V are known, pressure
versus volume at any temperature of interest can be calculated from Eq. 共6兲. Six isothermal PV data at 500, 1000,
1500, 2000, 2500, and 3000 K were calculated, and fitted to
the third-order Birch–Murnaghan equation of state 关Eq. 共3兲兴
⬘ for each of these isotherms as
to obtain V 0 , K 0T , and K 0T
listed in Table IV, where the subscript 0 indicates zero pressure.
We performed two fitting inversions: one inversion without fixing any of the three parameters, the other inversion
with only V 0 fixed. V 0 was calculated based on published
thermal expansion coefficients 共American Institute of Physics
Handbook, pp. 4 –129兲 at ambient pressure. Since V 0 has a
significant effect on the equation of state, and the thermal
expansion coefficients at room pressure measured by several
investigators are in reasonable agreement,13,23,28 –30 we prefer
the results of the latter inversion. Figure 5 shows the equation of state of rhenium at PT conditions inverted with these
FIG. 5. P – V – T equation of state comparison between shock wave 共Ref.
27兲 共dashed lines兲 and this study 共solid lines兲. The discrepancies at the lower
pressure range in 共a兲 were improved as shown in 共b兲 when zero pressure
volume V 0 was fixed during fitting.
two fitting methods. Figure 5 also shows the inverted equation of state from shock wave data,27 which are the only
available high PT data for rhenium to date. It is interesting
that our static compression data are quite consistent with
shock data except for the very high PT range. The discrepancy between shock and this study in the lower pressure
range shown in Fig. 5共a兲 indicates that fitting of the equation
of state with fixed V 0 calculated based on published thermal
expansion coefficients is necessary and adequate. Table V
lists the pressure at selected compressions and temperatures
based on the equation of state fitted with fixed V 0 in this
study. We would like to point out an interesting comparison
here for demonstration of the data reliability of this study.
Table I gave the thermal pressure P th⫽4.23 GPa from 298 to
923 K at 1⫺V/V a ⫽0 by direct integration of ␣ K T . The
results of this study, which are obtained from linear treatment
for the temperature dependence of thermal pressure listed in
TABLE IV. Equation of state of rhenium along six different isotherms. The numbers in parentheses are the
fitting results without fixing V 0 ; see the text for the details.
Fitting with fixed V 0 共Fitting without fixed parameters兲
Parameter
3
V 0 (Å )
K 0T (GPa)
⬘ ⫽( ⳵ K T / ⳵ P) T
K 0T
500 K
1000 K
1500 K
2000 K
2500 K
3000 K
29.517
共29.540兲
354.84
共350.53兲
4.49
共4.55兲
29.814
共29.880兲
340.87
共329.61兲
4.46
共4.62兲
30.154
共30.250兲
322.98
共308.17兲
4.47
共4.69兲
30.537
共30.659兲
302.93
共286.25兲
4.52
共4.77兲
30.965
共31.111兲
281.44
共263.80兲
4.59
共4.87兲
31.438
共31.616兲
259.82
共241.01兲
4.66
共4.97兲
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Rev. Sci. Instrum., Vol. 75, No. 7, July 2004
Re calibrant for laser heated DAC’s
2415
TABLE V. Equation of state of rhenium. Pressure 共in GPa兲 at selected compressions and temperatures. V and
V a are volume at PT and ambient conditions, respectively.
1⫺V/V a
300 K
500 K
1000 K
1500 K
2000 K
2500 K
3000 K
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
0
3.70
7.61
11.74
16.11
20.73
25.61
30.77
36.23
42.00
48.11
54.58
61.43
68.68
76.36
84.49
93.12
102.27
111.97
122.26
133.19
1.31
5.02
8.94
13.07
17.45
22.07
26.95
32.11
37.57
43.35
49.46
55.92
62.76
70.00
77.68
85.80
94.41
103.54
113.23
123.50
134.40
4.81
8.53
12.46
16.60
20.98
25.61
30.49
35.65
41.11
46.87
52.96
59.41
66.23
73.44
81.07
89.16
97.72
106.79
116.41
126.61
137.42
8.54
12.25
16.16
20.29
24.64
29.24
34.10
39.23
44.64
50.37
56.42
62.82
69.58
76.74
84.32
92.33
100.82
109.82
119.35
129.45
140.17
12.42
16.09
19.97
24.07
28.39
32.96
37.78
42.87
48.24
53.93
59.93
66.28
73.00
80.11
87.63
95.59
104.02
112.96
122.43
132.47
143.12
16.34
19.98
23.82
27.88
32.16
36.68
41.46
46.50
51.83
57.47
63.42
69.72
76.39
83.44
90.90
98.81
107.18
116.06
125.46
135.44
146.03
20.26
23.86
27.67
31.69
35.93
40.41
45.14
50.14
55.42
61.00
66.91
73.15
79.76
86.75
94.16
102.00
110.31
119.12
128.46
138.37
148.89
Table V, give P th⫽4.33 GPa for the same V – T range. The
difference between two independent experiments is 2.3%,
which is also reasonably within the experimental uncertainty
of this study and will be discussed later.
When the P – V – T EOS is known, the temperature dependence of volume at constant pressure can be obtained.
The volume coefficients of thermal expansion are evaluated from ␣ ⫽(1/V)( ⳵ V/ ⳵ T) P , and are shown in Fig. 6.
Other thermoelastic parameters can be derived as shown in
Table VI.
V. DISCUSSION
Equation 共6兲 tells us that the total pressure experienced
by a solid at high P – T conditions is constrained by the
isothermal portion at a reference temperature and the isochoric portion from the reference temperature to higher temperatures, i.e., Mie–Grüneisen assumption. A correct
P – V – T equation of state strongly depends not only on the
data collected at a simultaneous high P – T condition, but
also on the accurate isothermal equation of state at the ref-
erence temperature. To demonstrate this, volume data collected at high P – T conditions in this study have been combined with a different isothermal equation of state of
rhenium at ambient temperature based on ultrasonic measurement at the low pressure range14 to create the P – V – T
equation of state. It also is compared to the same converted
shock equation of state mentioned above in Fig. 7. The large
discrepancies between this equation of state and the shock
equation of state are very clear and become severe at higher
P – T conditions. It seems that to obtain the most accurate
isothermal equation of state at the reference temperature 共for
many materials, it is at ambient temperature兲 is critical for
creating the correct P – V – T equation of state. Fortunately,
this has become more practical because quasihydrostatic
pressure media, such as helium,31–33 are often used, and the
well characterized pressure scales, such as the ruby pressure
scale34 at ambient temperature, has been proved to be quite
reliable.35
Figure 5共b兲 shows a comparison of our equation of state
with the P – V – T data converted from the shock Hugoniot.27
Generally, the consistency between these two studies is quite
good. With increases of both the pressure and temperature,
the discrepancies become larger. At compression of 1
⫺V/V a ⫽0.2 and 3000 K for rhenium 共the total pressure is
⬃149 GPa; see Table V兲, the thermal pressure of this study is
TABLE VI. Thermoelastic parameters evaluated for ambient conditions
from a combination of previous and this study. Subscript a denotes ambient
condition.
FIG. 6. Volume coefficients of thermal expansion at different temperatures
and pressures.
( ⳵ ␣ / ⳵ T) Pa
( ⳵ ␣ / ⳵ P) Ta
( ⳵ ␣ / ⳵ T) Va
( ⳵ K T / ⳵ T) V
( ⳵ K T / ⳵ P) Ta
( ⳵ K T / ⳵ T) Pa
0.5148⫻10⫺8 K⫺2
⫺0.171⫻10⫺6 K⫺1 GPa⫺1
0.4072⫻10⫺8 K⫺2
⫺0.008 15 GPa K⫺1
4.5
⫺0.036 52 GPa K⫺1
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2416
Rev. Sci. Instrum., Vol. 75, No. 7, July 2004
Zha, Bassett, and Shim
deviation of 3.9 GPa 共or 2.6%兲 between this study and the
shock equation of state at ⬃150 GPa and 3000 K actually is
within the measurement uncertainties as discussed in the Appendix. The fact that the systematic deviation is within the
measurement uncertainty in this study does not necessarily
mean this measurement completely agrees with the shock
equation of state. The accuracy of both the gold standard and
shock equation of state may need to be revised when a more
reliable, primary pressure standard is available.
ACKNOWLEDGMENTS
FIG. 7. P – V – T equations of state produced from the high P – T data of this
study and a different isothermal equation of state at 300 K 共Ref. 14兲. The
large discrepancies between these equations of state and the shock equations
of state indicate that an accurate isothermal equation of state at ambient
temperature is critically important for the method we used in this study.
15.7 GPa, whereas the shock wave data gives 19.6 GPa,
leaving 3.9 GPa difference. There has been a long debate
over the accuracy of the gold pressure scale.2,36 Pressures in
the scale proposed by Anderson et al.18 are normally lower
than those proposed by Heinz and Jeanloz.37 Assuming the
shock wave data are correct, our data would suggest a pressure correction for the gold scale somewhere in the middle of
the difference in pressure between these two scales.
Shim et al.36 have proposed a new equation of state for
gold based on the quasihydrostatic EOS measurement conducted recently by Takemura.33 According to this new scale,
the pressure corresponding to the compression and temperature mentioned above is slightly higher than in Anderson’s
scale but not in the middle between Anderson’s and Heinz’s
scale. Assuming the pressure scale proposed by Shim et al. is
closer to the truth, there must be another source of error for
the systematic difference in pressure between that in this
study and the shock wave data. Could it be neglect of a
quadratic term of thermal pressure integration?
Thermal pressure integration in Fig. 1 has been fitted to
a third-order polynomial equation in order to estimate the
difference in pressure introduced by linear treatment. At 1
⫺V/V a ⫽0 and 3000 K, nonlinear fitting gives 21.22 GPa
thermal pressure, while linear treatment of this study gives
20.26 GPa. The difference is 0.96 GPa. That is about 4.7% of
the rise in pressure from 300 to 3000 K. Assuming the same
difference ratio happens at high compressions, at 1⫺V/V a
⫽0.2, this difference would be 0.74 GPa. It seems that this
error is too small to cause the difference. It also demonstrates
that linear treatment for the thermal pressure does not introduce significant error to this study. Because this P – V – T
equation of state is built on large extrapolations from a low
P – T range in which the measurements were conducted to
very high P – T conditions, the extrapolation errors should be
the reasonable source responsible for the systematic deviations.
We would like to point out that the systematic pressure
The authors thank Dr. Thomas Dufffy, Dr. Donald G.
Isaak, and Dr. Sol Gruner for their valuable comments and
suggestions. This work is based upon research conducted at
the Cornell High Energy Synchrotron Source 共CHESS兲,
which is supported by the National Science Foundation and
the National Institutes of Health/National Institute of General
Medical Sciences under Award No. DMR0225180.
APPENDIX: UNCERTAINTIES OF THIS STUDY
Apparently, the uncertainties of the P – V – T relationship
will come from pressure, volume, and temperature measurements. In the experiment, temperature and volume can be
measured independently. But pressure cannot be measured
independently, it depends on the other two measurements as
well as on the pressure scale used.
1. Effect of temperature uncertainty
As pointed in Ref. 1, the temperature uncertainty at the
temperature range of this study was estimated to be less than
20°. The thermal pressure of rhenium was evaluated by linear extrapolation of the isochore P – T line, one end of which
is an experimental P – T point from the gold measurement,
and the other end is the pressure of the isothermal EOS of
rhenium at 300 K. We also know that the thermal pressure is
linearly proportional to the temperature for both gold and
rhenium, which means that P and T will vary in the same
direction, with similar temperature slopes for gold and rhenium in the compression range of this study 共see both Table
V of this study and Ref. 18兲. So the temperature uncertainty
will have negligible effects on the results.
2. Effects of volume uncertainties
Because the pressure is calibrated by using volume data
of gold, the main uncertainties in this study come from both
volume measurements of gold and rhenium. The uncertainties of the lattice parameter of gold and rhenium could
strongly be affected by the stress–strain condition in the
sample chamber and lead to inconsistent lattice parameters
represented by different diffraction lines. As Singh38 and
many other investigators3,39,40 have pointed out, pressure determined using a sensor’s volume measured under uniaxial
compression with its axis parallel to an incident x-ray beam
is always lower than the real pressure because of the deviatoric stress–strain condition. Accurate deviatoric stress estimation requires knowledge of the hydrostatic pressure component or strain, which is not available in this experiment.
On the other hand, numerous investigators have found that
deviatoric stress dramatically decreases with an increase in
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Rev. Sci. Instrum., Vol. 75, No. 7, July 2004
Re calibrant for laser heated DAC’s
TABLE VII. Six diffraction lines of hexagonal close packed rhenium at
1480.2 K and 8.47 GPa.
Formulas
for d, a, c
hkl
1
100
2
d
4
⫽
3a2
c (Å)
共when using a
of hkl)
d
共Å兲
a
共Å兲
2.3953
2.7659
4.4501 共100兲
1
101
d2
⫽
4
3a2
1
110
d2
⫽
⫹
1
4.4833 共110兲
4.4473 共200兲
2.1092
c2
4
1.3800
a2
2.7600
4.4570 共110兲
1
103
d2
⫽
1
200
2
d
112
1
d2
⫽4
4
3a2
⫽
冉
⫹
9
1.2618
c2
16
1.1979
3a 2
1
a2
⫹
4.4534 共100兲
4.4531 共200兲
1
c2
冊
2.7664
4.4390 共200兲
4.4411 共100兲
4.4658 共110兲
1.1739
temperature.39– 41 Theoretical analysis as well as some
experiments39,40 indicate that the elastic strain of gold under
uniaxial stress manifests itself as the largest deviatoric strain
in the 200 diffraction peak and smallest in the 111 diffraction
peak. As shown in Table IIa, gold diffraction peaks obtained
in this study show random strain change at each P – T condition. The same appears to be true of rhenium. Table VII
2417
shows six diffraction peaks of rhenium obtained at 1480.2 K.
Three peaks determine lattice parameter a independently, but
parameter c cannot be determined independently. Table VIII
shows nine possible a, c, and volume data from different
combinations of diffraction peaks. With those different volumes, the corresponding pressures at 300 K will be different,
and the linear isochore fit between 300 and 1480.2 K for
obtaining the slope of thermal pressure will be different.
Table VIII also shows calculated thermal pressures corresponding to each of those volumes at selected temperatures.
A combination of 100 and 101 共combination 1 in Table VIII兲,
was used in this study for all volume calculations. Singh and
Balasingh proposed a ratio of anisotropic lattice strain to
isotropic bulk strain, R, to describe the error introduced by
assuming that an elastically anisotropic sample is isotropic.
According to their analysis42 103 has smaller lattice strain
than 101, so the volume calculated from combination 110
and 103 共combination 2 in Table VIII兲 should have smaller
volume deviation, or closer to isotropic strain condition, than
that of combination 1. The same analysis also predicts that
combination 1 should have the same volume deviation as the
combination of 110 and 101 共combination 6 in Table VIII兲. A
comparison of 1 and 2 does shows a difference in volume of
0.08 共Å3兲, corresponding to a difference in thermal pressure
of 1.25 GPa at 3000 K. A comparison of 1 and 6 shows a
volume difference of 0.094 共Å3兲 but in an opposite way, corresponding to a difference in thermal pressure of ⫺1.44 GPa
at 3000 K. The presence of random strain for both gold and
rhenium demonstrates that the difference in cell volume,
hence in the pressure, may not originate from deviatoric
stress. The high temperatures probably release the deviatoric
stress as evidenced by numerous other experiments. Figure 8
shows the temperature dependence of the relative difference
TABLE VIII. Different thermal pressures obtained from different volumes of rhenium based on different
combinations of diffraction lines observed at 1480.2 K and 8.47 GPa. ⌬V and ⌬ P th are differences between
individual V, P th , and their average values.
Combination
of hkl
a
共Å兲
c
共Å兲
100 and 101
2.7659
4.4501
110 and 103
2.7600
4.4570
200 and 112
2.7664
4.4390
100 and 103
2.7659
4.4534
100 and 112
2.7659
4.4411
110 and 101
2.7600
4.4833
110 and 112
2.7600
4.4658
200 and 101
2.7664
4.4473
200 and 103
2.7664
4.4531
Average of all
2.7641
4.4545
V
(⌬V)
共Å3兲
29.482
共0.009兲
29.402
共⫺0.071兲
29.419
共⫺0.054兲
29.504
共0.031兲
29.423
共⫺0.050兲
29.576
共0.103兲
29.460
共⫺0.013兲
29.474
共0.001兲
29.513
共0.040兲
29.473
兩 ⌬V/V̄ 兩
0.0003
0.0024
0.0018
0.0011
0.0017
0.0035
0.0004
0.000 03
0.0014
0.0006
共stand.
dev.兲
P 500
th
(⌬ P 500
th )
共GPa兲
P 1000
th
(⌬ P 1000
th )
共GPa兲
P 2000
th
(⌬ P 2000
th )
共GPa兲
P 3000
th
(⌬ P 3000
th )
共GPa兲
0.69
共⫺0.10兲
1.50
共0.71兲
1.33
共0.54兲
0.48
共⫺0.31兲
1.29
共0.50兲
⫺0.24
共⫺1.03兲
0.91
共0.12兲
0.77
共⫺0.02兲
0.39
共⫺0.40兲
0.79
4.66
共⫺0.05兲
5.06
共0.35兲
4.97
共0.26兲
4.55
共⫺0.16兲
4.95
共0.24兲
4.20
共⫺0.51兲
4.77
共0.06兲
4.70
共⫺0.01兲
4.51
共⫺0.20兲
4.71
12.59
共0.04兲
12.17
共⫺0.38兲
12.26
共⫺0.29兲
12.71
共0.17兲
12.28
共⫺0.27兲
13.09
共0.54兲
12.48
共⫺0.07兲
12.55
共0.00兲
12.76
共0.21兲
12.55
20.53
共0.15兲
19.28
共⫺1.10兲
19.55
共⫺0.83兲
20.87
共0.49兲
19.60
共⫺0.78兲
21.97
共1.59兲
20.18
共⫺0.20兲
20.41
共0.03兲
21.00
共0.62兲
20.38
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2418
Rev. Sci. Instrum., Vol. 75, No. 7, July 2004
Zha, Bassett, and Shim
deviatoric stress at high temperatures, most of the other hightemperature patterns for rhenium had only two peaks on
which we were able to base our measurements. Because
there is no reliable method for calculating the uncertainty
resulting from the use of fewer peaks, we can only estimate
the uncertainty on the basis of our judgment. We believe the
smaller number of peaks increases the pressure uncertainty to
⬃5% in our measured P – T range.
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1
2
FIG. 8. Temperature dependence of the relative difference in thermal pressure observed for the 100 and 101 combination of diffraction lines on rhenium at 1480.2 K and 8.47 GPa. ⌬ P th are the differences between individual
P th and their average values shown in Table VIII.
in absolute thermal pressure between combination 1 and
the ‘‘average of all HKL,’’ as shown in Table VIII. The
ratio drops fast with an increase in temperature below 1000
K, and becomes less than 1% when temperature is higher
than 1000 K.
We attribute the random strain condition to an error in
reading of the peaks.
The average uncertainty for gold lattice parameters obtained from different reflection lines is
冉 冊
⌬a
a
⬇0.000 36,
共A1兲
Au
so the volume uncertainty would be
冉 冊
⌬V
V
⫽3⫻
Au
冉 冊
⌬a
a
⬇0.0011.
共A2兲
Au
The statistical volume uncertainties of rhenium at each
P – T condition are not available because only two diffraction
lines were used and only one volume value was available.
But the data at 1480.2 K offer an opportunity by which to
examine the possible statistical volume uncertainty. The standard deviation for the volume of rhenium obtained from different diffraction lines at 1480.2 K is 共see Table VIII兲
冉 冊
⌬V
V
⬇0.0006.
共A3兲
Re
In the pressure–temperature range of this study, the pressure
uncertainties from the volume uncertainties of gold and rhenium would be ⬃0.17 and ⬃0.22 GPa, according to Anderson et al.’s EOS18 for gold and the EOS of rhenium we obtained and shown in Table V, respectively, so the overall
pressure uncertainty of this study can be estimated as
共 0.172 ⫹0.222 兲 1/2⬇0.28 GPa,
共A4兲
which is about ⬃3.8% of the measured pressure.
Although the six peaks in the diffraction pattern of rhenium at 1480.2 K offer an opportunity to test the effects of
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