J Phys
Pnnted
Chem
Solrdr
Vol.
in Great Britain.
49,
No.
8. PP.
945-956,
1988
0022.3697/88
S3.00 + 0.00
0 1988 Pergamon Press plc
A PRESSURE-VOLUME-TEMPERATURE
EQUATION
OF
STATE FOR Sri(P) BY ENERGY
DISPERSIVE
X-RAY
DIFFRACTION
IN A HEATED
DIAMOND-ANVIL
CELL?
MARK
Department
E. CAVALERI,~THOMAS G. PLYMATES and JAMES H. STOUT
of Geology
and Geophysics,
(Received
University
of Minnesota,
Minneapolis,
16 April 1987; accepted in revised form 3 December
MN 55455, U.S.A.
1987)
total of 36 molar volume determinations
measured by energy-dispersive
X-ray diffraction in
a heated diamond-anvil pressure cell form the basis for a P-V-T equation of state for Sn@). Isothermal
Mumaghan regressions for the 25, 100, 160, and 225°C subsets of these data yield - 1.38 (kO.13) x
Ah&act-A
lo-* GPa degree’
for (dK/aT),.
The slopes of the Sri(b) isochores increase from approx. 235” GPa-’
at room pressure and temperature
to 260” GPa-’ at the Sn triple point to 370” GPaa’ at the room
temperature
Sn(&Sn(bct)
transition
at 9.4GPa,
indicating
that the product
apKr decreases with
decreasing volume by more than 35% of its initial value. The a crystallographic
direction is significantly
less compressible
and slightly less expandable
than c; the extent of this anisotropic
behavior decreases
at simultaneously
elevated pressure and temperature.
A five-parameter
temperature-corrected
Murnaghan
equation fits the entire data set well within the
experimental
error. This explicit V(T, P) equation is used to integrate literature heat capacity data to
elevated pressure, yielding entropies and Gibbs energies for all pressures and temperatures
within the
Sn(j?) stability field.
Keywordr:
Sn(B), molar
volume,
pressure,
temperature,
INTRODUCTION
The behavior of the molar volume of Sn(/?) is well
understood as a function of pressure at room temperature [l-S] and as a function of temperature at
room pressure [6,7]. Pressure-volume-temperature
(P-V-T) equations of state based on such data commonly assume that the isothermal incompressibility,
KT, does not depend on temperature, or, equivalently,
that the isobaric expansivity, ap, does not depend
on pressure. Isochores based on such an assumption
consistently overestimate the molar volume of a
homogeneous solid for all simultaneously elevated
pressures and temperatures.
Direct measurements of molar volume at simultaneously elevated pressure and temperature can now
be. made in situ in a heated diamond-anvil pressure
cell by energy-dispersive X-ray diffraction (EDXRD).
The EDXRD method is ideally suited for this application because of the simple diffraction geometry and
the relatively short data acquisition time. As a
demonstration of the general utility of this technique,
we present here a large body of P-V-T data for
a material with a relatively simple structure and
moderately high compressibility, Sn( /I).
t This is publication
number 1065 of the School of Earth
Sciences, Department
of Geology and Geophysics,
University of Minnesota,
Minneapolis,
MN 55455, U.S.A.
$ Present address: Department
of Geology,
Macalester
College, St. Paul, MN 55105, U.S.A.
§ Present address: Department
of Geosciences, Southwest
Missouri State University,
Springfield,
MO 65804, U.S.A.
X-ray
diffraction,
diamond
anvil.
The P-T stability relationships involving Sn( /3),
Sn(bct) and Sn(melt) are quite simple; the positions
of the single invariant point and the three univariant
curves are all well established [&lo]. This communication is the first of two and is restricted to
the homogeneous
behavior of the low-pressure
phase, Sn( /3). The second paper [l l] deals with the
P-V-T
behavior of the high-pressure polymorph and
the volumeentropy
systematics and heterogeneous
equilibria of the entire Sn system.
EXPERIMENTAL PROCEDURE
The experimental system used in this study is
illustrated in Fig. 1. The design of the diamond-anvil
pressure cell is from Mao and Bell [12, 131 and is
capable of generating static pressures in excess of
100 GPa at room temperature (RT). Mechanical
force is applied to the back of the l-in. diameter
piston through a 5: 1 lever arm by means of a
hydraulic hollow cylinder driven remotely by a handcrank hydraulic pump. This hydraulic system was
designed specifically to avoid the application of any
torque to the diamond-anvil cell which could change
the diffraction angle during the course of a series of
experiments [ 141.
Finely powdered Sn(@ (J. T. Baker Chemical
Company; 99.993% purity) was mixed with finely
powdered KC1 (Baker Reagent Grade) as an internal
pressure standard and loaded into a 250 pm hole
drilled through the center of a pre-indented metal
gasket (hardened T-301 stainless steel). This gasketed
945
MARK E. CAVALERIet al.
946
12 KW rotatmg anode
%---.
-;-
primary collimator
hydraulic
hollow cvlinder
Si (Li)
thermocot
momtor
thermocou
DETAIL OF GASKETED SAMPLE
Fig. 1. Schematic design of the heated diamond-anvil
pressure. cell used for the in srtu measurement
of
molar volume at simultaneously
elevated pressure and temperature
by energy dispersive X-ray diffraction.
was compressed between the two opposing
diamond anvils (l/4 carat; 500 pm diameter pressure
face) as shown in Fig. 1. At the beginning of each
experiment, the sample was approx. 100pm thick.
Compression to 10 GPa typically thinned the sample
by 50% or more and enlarged the diameter of the
sample cavity to 300-4OOpm.
The sample and admixed pressure standard are
heated by two 20-watt resistance heaters which
consist of 36 gauge chrome1 wire wound (on a thin
coating of Rutland
High-Temp
Stove Gasket
Cement) directly around the tool steel cups in which
the diamonds are mounted. Thin shims of zirconium
foil separate the diamonds from these cup heaters in
order to reduce the effects of differential thermal
expansion. An additional 140-watt resistance heater,
wound with 30 gauge nichrome wire on a toroidshaped pyrophyllite core, encircles the inner cup
heaters and provides background heating as needed
(Fig. 1).
The sample temperature
is measured by a
chromel-alumel microthermocouple (25 pm diameter
bead) placed directly within the sample cavity. A
second, larger chromel-alumel thermocouple (250 pm
diameter bead) is mounted on the edge of the cylinder
diamond as a monitor (Fig. 1). This monitor thermocouple was calibrated against the sample thermocouple at low pressure (co.5 GPa) at the start of
each set of experiments so that very accurate estimates of the sample temperature could be made for
experiments in which the sample thermocouple failed
at elevated pressure. The thermal gradient over the
500pm distance between the sample and monitor
thermocouples is directly proportional to the monitor
thermocouple temperature and is typically no more
than 20” at 250°C. In all cases, the temperatures
reported in this paper are those at the location of the
sample thermocouple.
No corrections have been made for the effects of
pressure on the e.m.f. of the sample thermocouple.
The sample temperatures measured at elevated pressure agree well with those predicted by the monitor
thermocouple calibrated at low pressure, consistent
with the observation of Getting and Kennedy [ 151
that the effect of pressure on the e.m.f. of chromelalumel thermocouples is small. Furthermore,
as
Bassett [16] points out, no significant correction
should be necessary in this type of application
because the points at which the thermocouple leads
enter the high-pressure zone are at very nearly the
same temperature as the thermocouple junction. We
are confident that the uncertainty in our reported
sample temperatures, caused by the combination of
the imprecision of the thermocouples and the temperature gradient across the irradiated portion of the
sample, does not exceed f 2% of T - T,.
The energy-dispersive X-ray diffraction spectrum
of the sample and admixed pressure standard is
collected in situ by a Si(Li) solid state X-ray energy
detector. This detector is mounted on a horizontal
goniometer table (vertical-axis of rotation) so that it
can be positioned at the optimum diffraction angle 20
at the start of each set of experiments. The X-ray
source is the unfiltered bremsstrahlung from the gold
A pressure-volume-temperature
equation of state for %I@)
941
both the sample material and the admixed pressure
standard are then determined from these d-values.
Pressure calibration
15
20
25
30
;5
Energy (KeV) Fig. 2. Representative energy-dispersive X-ray diffraction
spectrum of Sn(& (mixed with KC1 as the internal pressure
standard; 1: 3 by volume) collected at 15” 20 for 60 min in
the heated diamond-anvil pressure cell.
target of a rotating anode X-ray generator operated
at 750 W (50 kV, 15 mA) with a micro-focus filament
(0.1 x 1.0 mm electron beam cross-section). The Xray flux produced by this type of source is approx. 10
times that from a conventional fixed-target X-ray
generator. A gold target was chosen in order to avoid
any interference from characteristic fluorescence
peaks of the target material and to maximize the
intensity of the bremsstrahlung in the 15-30 keV
range, the energy range of maximum efficiency of the
detector. In order to minimize the effect of pressure
gradients across the compressed sample, the primary
X-ray beam is collimated through a 125 pm gold
pinhole centered in the piston, directly beneath the
piston diamond (Fig. 1).
The spectra are recorded and processed in 4096
channels of a multi-channel analyzer. Typically a
usable spectrum can be collected in 60-90min,
although data collection was occasionally extended
to as much as 3 h per spectrum when the sample had
become thinned significantly. A typical spectrum is
reproduced in Fig. 2. The principal Sn( /I) diffraction
peaks which were used to calculate the molar volumes,
as well as the diffraction peaks of the internal pressure
standard [KCl(I) in this example] and the Sn Ka and
K/l fluorescence peaks, are all easily distinguished.
With the aid of a Gaussian peak-fitting routine the
centroid of a typical strong diffraction peak can be
determined to within f5 eV. The d-value (interplanar spacing) of each diffraction is determined from
the energy (E) of the peak centroid by using the
energy-dispersive form of the Bragg relationship
d.E
= (h.c)/2.sinfI),
where h is Planck’s constant and c is the speed of
light. The lattice parameters and molar volumes of
The criteria for choosing internal pressure standards for diamond-anvil X-ray diffraction studies are
discussed at length elsewhere [17-191. The basic
requirements of a good pressure standard for our
application are (1) a simple, stable structure, characterized by a simple, strong X-ray diffraction pattern
which does not interfere with the diffraction pattern
of the sample material, (2) a well-established P-V-T
equation of state from which the pressure may be
calculated given the unit cell volume and temperature, and (3) a relatively high compressibility so that
the pressure can be transmitted uniformly throughout the sample cavity and small changes in pressure
can easily be detected by measurable changes in the
unit cell volume. Unfortunately, there is no single
material which, when used as the pressure standard
for Sn(/I), would simultaneously satisfy all three
of these criteria. NaCl, the most commonly used
internal pressure standard, is not appropriate for this
study because its two most intense X-ray diffraction
peaks [(200) and (22011 interfere with the four most
intense peaks [(200) + (101) and (220) + (21111 of
Sn( /I). KC1 has no such interference with the Sn( /I)
diffraction pattern and was therefore used as the
pressure standard in this study.
KC1 is one of the more compressible alkali halides
(K, = 17 GPa) and thus should be a very good
pressure transmitting medium. However, no elevatedtemperature P-V data for KC1 are currently available in the literature and, because of the significant
solubility of NaCl and KC1 at elevated temperature,
none are likely to be forthcoming in the near future
[20]. Furthermore,
KC1 transforms from the Bl
(fee-NaCl) structure to the B2 (bcc-CsCl) structure
at approx. 1.9 GPa [21], and thus two different
equations of state are needed to span the pressure
range of interest in this study. For our Sn + KC1
spectra collected within the stability field of KCl(I),
we estimated the pressure from the theoretical
P-V-T
equation of state derived by Decker [22]. For
our Sn + KC1 spectra collected within the stability
field of KCl(II), the pressures were estimated as
follows: first, a spectrum was collected at RT and the
pressure was estimated from a Murnaghan regression
of the RT compressibility data for KCl(II) from
Yagi [20]. While holding the gauge pressure on the
hydraulic system constant, the sample was heated
and additional spectra were collected at the various
elevated temperatures of interest. The pressure for
these elevated-temperature
spectra was assumed to
be equal to that of the RT spectrum collected at the
same gauge pressure. Although Plymate et al. [ll]
show that an assumption of this type is not justified
for spectra collected at temperatures above approx.
300°C repeated independent runs of this type using
pure NaCl as the sample material confirmed that the
MARK E. CAVALEFUet al.
948
Table I. Molar volume determinations for Snfj?) using
KC@) as the internal pressure standard and pressure
transmitting medium
Temperature
(“C)
2s
64
90
90
90
130
140
210
225
225
KC](I)
alawS
0.9774
+
1.0006
0.9991
0.9703
A
Pressuret
@Pa)
Sn(@)
alawS
Sn(8)
VlVo!J
1.4
:9
0.9923
1.0005
0.999 I
0.9997
0.9854
1.0030
I.0023
1.0041
1.0038
1.0056
0.9760
1.0019
1.0007
1.0016
0.9665
1.0090
1.0081
1.0157
1.0163
1.0183
0.2
2.1
0.0
0.0
0.0
0.0
0.0
Uncertainties: temperature, *2% of T-T,; pressure,
rf:3%; lattice parameters, & 0.06%; volume, kO.I8%.
t Estimated from the theoretical P--V-Tequation of state
for KCl(1) from Decker [22].
$ The subscript “00” indicates evaluation at P,, and r,,
i.e. at atmospheric pressure and RT. KCl(I) a, = 0.62931
nm; Sri(p) a,=O.S8316nm;
Sn(@) V,=0.10819nm3 unit
cell-’ (1.6288 x 10-5rn’ mot-‘).
0 Data collected at atmospheric pressure.
KC1 phases are isometric, the a lattice parameter
could be determined independently from each of
these peaks, and these values were also checked for
internal consistency before the molar volume of the
pressure standard was calculated.
The uncertainty associated with the resolution of
our detector and related electronics, coupled with the
uncertainty associated with our Gaussian peak-fitting
routine, yields a total uncertainty of approx. rtO.O6%
for each lattice parameter determination.
The
calculated molar volumes of both the Sn(/?) and the
pressure standard phases therefore have an un~~ainty
of approx. &0.18% (50.0018 V/V,,), consistent with
the estimated uncertainty reported by Manghnani
er al. [19] for a similar diamond-anvil-EDXRD
system. This uncertainty in the molar volumes of the
pressure standards, coupled with the uncertainties in
the P-V-T relations used to estimate the pressure,
produce a total uncertainty of approx. +3% in the
pressure determinations reported in this paper.
RESULTS
sample pressure, as dete~ined
by the Decker 1221
NaCl equation of state, does indeed remain very
nearly isobaric (+0.2 GPa at 5 GPa) up to the 250°C
temperature limit required for the present study.
Volume measurements
Cavaleri [23] discusses in detail the procedure for
optimizing the precision and accuracy of lattice
parameter measurements made by energy-dispersive
X-ray diffraction through a diamond-anvil pressure
ceil. Two of the more critical factors are (I) the
diffraction angle 28, and (2) the relative amounts of
sample and pressure standard loaded into the sample
cavity. To eliminate interference with the fluorescence
peaks of both the Sn and the Au X-ray target, and
to maximize the intensity of the strongest diffraction
peaks of both the Sn(j3) and the stable KC1 phase,
we collected our Sn + KC1 spectra at 15” 28. At
this angle, and with typical sample thicknesses of
50-100 pm, the optimum ratio of pressure standard
to sample was found to be approx. 3: 1 by volume
(approx. 0.8 KCl/Sn by mass).
Sn(fl) is tetragonal (14,/amd) and therefore a
minimum of two linearly independent diffraction
peaks is needed to uniquely determine the lattice
parameters and molar volume. At 15” 28 the (200),
(IOI), (220) and (211) peaks all provide reliable
centroid values. For each spectrum, separate a and c
lattice parameters were computed from each independent pair of these peaks and the resulting values were
checked for internal consistency before the molar
volume of Sn(fi) was calculated. At this angle,
usually either two or three peaks of the stable KC1
phase [(200), (220), k(222) for KCl(1) and (llO),
(200), +(21 I) for KCl(II)] were also well enough
developed to yield reliable centroids. Because both
Tables 1 and 2 list our P-V-T data for Sri(B)
derived respectively using KU(I) and KCl(I1) as the
internal pressure standard and pressure transmitting
medium. Of our 36 volume measurements, 17 were
determined at simultaneously elevated temperature
and pressure and are the first such data for Sn( fl) in
the literature.
Room temperature compressibility
Figure 3 compares the RT subset of our data with
the available literature RT compressibility studies
[l-5]. Our volume determinations
along this RT
isotherm are internally consistent and are also quite
consistent with the data of Bridgman fl, 2] and
Vaiyda and Kennedy [4]. On the other hand, some
of the volumes reported by Barnett et al. [3] appear
to be a bit low while some of those reported by Liu
and Liu [S], especially those determined at the highest
pressures, appear to be a bit high.
Table 3 compares the room pressure values of
the isothermal bulk modulus, KT, for the various RT
static compression studies with the ultrasonically
determined K, values for Sn(fi) [24-26). The values
of K, and its pressure derivative, K&, tabulated on
the first six lines of Table 3 were determined by ieastsquares regression using an isothermal Murnaghan
equation [27] of the form
where the superscript “ “’ indicates differentiation
with respect to the independent variable (in this case
P). From these K, values it is clear that our RT
volume determinations are consistent not only with
the static compression studies of Bridgman [I, 21 and
Vaiyda and Kennedy [4] but also with all of the
available ultrasonic studies.
A pressure-volumhemperature
Table 2. Molar volume determinations for Sn(/I) using
KCl(I1) as the internal pressure standard and pressure
transmitting medium
Temperature
(“C)
25
160
225
25
25
25
25
225
256
25
25
100
160
225
25
100
160
25
130
25
100
160
25
160
25
100
KCI(II)
alaooS
Pressuret
(GPa)
Sn(fi)
alo&
Sri(B)
VIVCJ
0.9296
-t
2.0
2.0
2.0
2.7
2.9
3.3
3.6
3.6
3.6
4.1
4.1
4.1
4. I
4.1
4.8
4.8
4.8
5.9
5.9
6.1
6.1
6.1
6.4
6.4
8.5
8.5
0.9886
0.9906
0.9922
0.9862
0.9845
0.9836
0.9823
0.9838
0.9831
0.9796
0.9797
0.9810
0.9797
0.9819
0.9767
0.9779
0.9796
0.9728
0.9713
0.9712
0.9708
0.9734
0.9685
0.9708
0.9645
0.9647
0.9641
0.9745
0.9800
0.9521
0.9496
0.9507
0.9437
0.9520
0.9500
0.9379
0.9334
0.9370
0.9406
0.9478
0.9262
0.9289
0.9292
0.9122
0.9155
0.9125
0.9096
0.9149
0.9040
0.9144
0.8767
0.8853
-t
0.9228
0.9209
0.9173
0.9149
1;
0.9111
0.9108
-t
z;
0.9058
1:
0.8983
-t
0.8972
1:
0.8953
-t
0.8839
-t
Uncertainties: temperature, +2% of T-T,; pressure,
+3%; lattice parameters, *0.06%; volume, &0.18%.
t For the RT data the pressure was estimated from a
Mumaghan regression of the KCl(I1) compressibility data
from Yagi [20]. Because no elevated-temperature equation
of state for KCl(I1) is available, the elevated-temperature
data points were measured at the same gauge pressure as,
and are therefore assumed to be isobaric with, the RT data
point under which they are tabulated.
$ The subscript “00” indicates evaluation at PO and T,,
i.e. at atmospheric pressure and RT. The tabulated values
for KCl(II) a are normalized to KCl(1) a, (0.62931 nm);
Sri(g) a,.,, = 0.58316nm; Sn(/l) VW = 0.10819 nm3 unit cell-’
(1.6288 x 10-5m3m01-‘).
equation of state for Sri(B)
949
This suggests that, at least in some applications,
a relatively large amount of a very soft solid can
provide a more-nearly hydrostatic pressure in a
gasketed diamond-anvil sample than can a smaller
proportion of a liquid pressure transmitting medium.
Volume at simultaneously elevated pressure and
temperature
The effect of temperature on the isothermal
compressibility of Sn(j?) can be seen by comparing
the room pressure values of KT and K; derived by
Murnaghan
regressions along each of our four
isotherms (Table 4). If we allow both K,,% and KG,,,
to vary for each Mumaghan regression (top half of
Table 4), linear regressions of the resulting parameter
values yield -3.97( + 1.97) x 10m2GPa degree-’ for
(dK/lJT), and +6.45 (k9.71) x 1O-3 degreee’ for
(aK’/i?T),.
Although the signs of both these parameters are correct (i.e. consistent with theoretical
expectations), the standard deviation for the (dK’/
JT), determination clearly indicates that our data
are not sufficiently precise to yield a reliable value for
this parameter. Therefore, the second half of Table 4
lists the results of Murnaghan regressions for each
of our isotherms derived with K;,, constrained to
equal 4.0. The resulting linear regression yields
- 1.38(f0.13) x 10m2GPa degree-i for (c?K/JT),
(Fig. 4). This value is consistent with the average of
the three available ultrasonic determinations of this
parameter (Table 5).
Figure 5 shows our entire Sri(B) data set plotted
on a P-T phase diagram for the Sn system. The
solid lines are isochores (contours of constant molar
volume) and represent the smoothed average of the
contours produced by three topographic contouring
programs, each based on a slightly different linear
interpolation algorithm. These isochores therefore
provide an unbiased representation of the P-V-T
variation in the Sri(B) data set, completely indepen-
Because the heater configuration
in our experimental setup does not allow for the loading of a
liquid or gas pressure transmitting
medium, we have
relied upon our solid pressure standard
(KC]) to
serve this function as well. Liu and Liu [5], on the
other hand, used both H,O and a methanol-ethanol
mixture as pressure transmitting
media in their RT
experiments.
The effect of non-hydrostaticity
in a
diamond-anvil cell such as ours (and such as the one
used by Liu and Liu [5]) would be to produce X-ray
lattice parameter measurements which are anomalously high. Because our high-pressure, RT Sn(/I)
molar volume measurements are slightly lower than
those of Liu and Liu [5], and because our measurements virtually coincide with those of Bridgman [ 1,2]
and Vaiyda and Kennedy [4] made with largervolume devices’ in which non-hydrostaticity
is less
of a problem, we conclude that any non-hydrostatic
effects in our measurements are essentially negligible.
Fig. 3. Compressibility of Sri(b) at RT. The curve represents
an isothermal Mumaghan eqn (1) using the parameter
values derived by regression of our RT data (Table 3, line 5;
K, = 56.82GPa and K& = 2.30).
MARK
950
E. CAVALERIet al.
Table 3. Comparison of determinations
of the RT isothermal bulk modulus of St@)
&a)
Static compression studies
1 Bridgman [ 1,2]
55.59
(0.65)
44.31
(2.91)
54.34
(0.17)
50.09t
(1.66)
56.82.
(2.19)
56.61
(0.79)
2 Bamett et al. [3]
3 Vaivda and Kennedv 141
4 Liu and Liu [5]
5 This study
(RT subset)
6 Combined
RT subsetS
Ultrasonic studies
7 Rayne and Chandrasekhar
Single crystal
8 Kammer et al. [25]
Single crystal
9 Kamioka [26]
Polycrystalline
K&
[24]
2.45
(0.19)
5.48
(1.03)
3.83
(0.11)
5.167
(0.52)
‘2.30
(0.81)
2.25
(0.27)
Standard
error
(V/V,)
7.788 x 1o-4
2.926 x IO-’
9.199 x 10-S
9.848 x 10-l
2.626 x lo-’
1.695 x 1O-3
55.0
55.4
-
54.6
The subscript “00” indicates evaluation at P,, and T,, i.e. at atmospheric pressure and
room temperature. The values tabulated for the static compression studies were
determined by regression to an isothermal Murnaghan eqn (1); the numbers in
parentheses are the estimated standard deviations.
t Liu and Liu [5] report K, = 50.2 GPa and K& = 4.9 determined by regression of
their data to a Birch equation.
$ The “Combined RT subset” includes the data from Bridgman [I, 21and Vaiyda and
Kennedy [4] as well as the RT subset of our data.
dent of any assumptions
about the Sn(/?) equation
of state. The resulting Sn( /I) P-V-i” surface displays
all of the general features of P-V-T
surfaces
expected from theoretical considerations. The surface
is clearly concave upward throughout the Sn(j?)
stability field, consistent with the expectation that a
normal, homogeneous material should become less
compressible with decreasing volume. Each Sn( /I)
isochore in Fig. 5 is very nearly straight, and
therefore the isochore slope at any given volume
[(CJT/i?P),] is nearly constant. However, the slope of
the Sn( 8) isochores increases markedly with decreasing volume, from approx. 235 degrees GPa-’ at RT
and pressure (V/VW = 1.0) to approx. 260 degrees
GPa-’ at the Sn triple point (308°C; 2.9 GPa; V/V,
= 0.97) to approx. 370 degrees GPa-’ at the RT
Sn(&Sn(bct)
transition (9.4 GPa; V/V, = 0.87).
The dashed lines on Fig. 5 are contours of
constant (a/~~)~ drawn in the same manner as the
isochores. The extent to which these contours deviate
from the isochores is a measure of the anisotropic
behavior of Sn(/3). Along the RT isotherm the contours of constant (a/~~)~ are clearly offset to higher
pressure relative to the isochores. This behavior
indicates that Sri(p) is less compressible along the a
crystallographic direction than along c, an observation consistent with that of Barnett et al. [3].
Furthermore, the data also suggests that, along any
low-pressure isobar, the (a/~,,,,)~ contours are offset
to higher temperature relative to the isochores; thus
the a crystallographic direction also appears to be less
expandable than c. The convergence of the (~/a~))
contours with the isochores indicates that Sn(/I)
becomes more nearly isotropic at simultaneously
elevated temperature and pressure.
DISCUSSION
P-V-T equation of state for Sn(/?)
One of the most common ways to estimate the
P-V-T behavior of a material for which no highP-T volume data exist is to assume that the
isothermal incompressibility KT is independent of T
(or, equivalently, that the isobaric expansivity ap
is independent of P) [28,29]. Isochores based on
this assumption have strongly negative curvature
((a2 T/c?P~)~ + 0). Our experimentally-determined
isochores (Fig. 5) reveal that this type of P-V-T
equation would significantly overestimate the molar
volume of Sn( fl) for all simultaneously-elevated
temperatures and pressures within its stability field.
Despite the limited temperature stability range of
Sn( b), we can clearly detect the effect of temperature
on the isothermal incompressibility (Fig. 4). Because
(c?K/aT), is proportional to (a2 VIaTaP), we should
expect that in order to adequately model the variation of the molar volume of Sn(& over its entire
stability field we will need a P-V-T equation with
A pressure-volume-temperature
951
equation of state for Sn(/?)
Table 4. Mumaghan regression parameters for St@) molar
volume isotherms
Temperature
(“C)
Standard error
Kr. P,
PT.&t
(GPa)
Kr. r0
56.61
(0.79)
49.72
(2.38)
47.19
(3.70)
48.93
(4.85)
2.25
(0.27)
4.34
(0.85)
5.24
(1.62)
3.29
(2.93)
unconstrained
K;, PO
2X
I .OOOo
100
1.0055
160
1.0107
225
1.0170
(V/VT,,)
1.695 x IO-”
1.557 x 10-j
1.589 x lO-5
KG.po= 4.0 (constrained)
25$
1.OoOo
100
1.0055
160
1.0107
225
1.0170
52.03
(0.39)
50.64
(0.47)
50.02
(0.72)
49.23
(0.78)
2.434 x 1O-3
= V,exp[a,(T-To)
Fig. 4. Temperature variation of the room pressure isothermal bulk modulus of Sn(b). The K7 po values were
derived by isothermal Mumaghan regressions with K;.Po
constrained to equal 4.0 (bottom half of Table 4).
1.322 x lo-’
2.053 x IO-’
1.377 x 10-r
The numbers in parentheses are the estimated standard
deviations.
t For each Mumaghan regression, VT,h was constrained
to take the appropriate value determined from a thermal
expansion curve of the form
V,,(T)
0
2.153 x 1O-3
+ a&(T-To)2/2]
using a, = 6.645 x 1O-J deg-’ and a& = 18.03 x lo-* dege2
derived by simultaneous regression of the atmosphericpressure subset of our data along with the expansivity data
of Deshpande and Sirdeshmukh [6,7].
$ The 25°C isotherm includes the literature data from
Vaiyda and Kennedy [4] and Bridgman [l, 21 as well as the
25°C subset of our data.
at least five refinable parameters, one associated with
each of the two first-order partial derivatives of
volume [(aV/M), and (aV/kW),] and one associated
with each of the three independent second-order
partial derivatives [(a* V/aP*)r, (a’ V/8T2)p and
(8 v/aTaP)].
It is convenient to express the first four of these
derivatives in terms of k&,, h, K& and a&,. (As
before, the superscript ““’ indicates differentiation
with respect to the independent variable; therefore, tl’
denotes @/aT),.) Because volume is a state variable,
its value is path independent and hence the secondorder cross partial derivatives (a2 V/%?P) and (a* V/
aPaT)are equal. Thus we can describe this crossterm dependence with a single variable and, if we
assume that it is constant, we can express it as either
Table 5. Comparison of determinations of the temperature dependence of the
isothermal bulk modulus of Sn(&
(=/a %I
(GPa degree-‘)
Static compression isotherms
(this study)
1 Unconstrained isothermal
Mumaghan regressions
(top half of Table 4)
2 Constrained isothermal
Mumaghan regressions (Kr, p0= 4.0)
Ultrasonic studies
3 Rayne and Chandrasekhar [24]
Single crystal, - 19627°C
4 Kammer et al. 1251.
Single crystal, 28-232°C
5 Kamioka [26]
Polycrystalline, 25-139°C
(aK/aT),
(degree-‘)
-3.97 x 10-z
(1.97 x 10-Z)
6.45 x 1O-3
(9.71 x 10-3)
- 1.38 x lo-*
(0.13 x 10-Z)
-
- 1.50 x 10-Z
-
- 1.83 x lo-*
-
-0.47 x 10-Z
-
The subscript “00” indicates evaluation at PO and To, i.e. at atmospheric
pressure and*RT. The values tabulated for our static compression study were
determined by linear regression of the isothermal Mumaghan parameters
from Table 4; the numbers in parentheses are the estimated standard
deviations for those linear regressions.
952
PRESSURE
(GPR)
Pig. 5, Phase diagram for the Sn system showing P-Tlocation of maiar volume determinations for Sn(jJ):
a = our data; A = literature RT data [I, 2,4]; 0 = iiterature room pressure data [6,7J_ The solid
contours are isocbores (lines of ~ns~~~ molar voIume; in units of the d~~~nsio~~e~ quantity v/Y,>
drawn throu& &e experimental data by a h.w interpolation procedure which is independentof any
P-Y-T
equation of state. The dash& contours are lines of constant (~/a)~.
~aK~aT~~ or ~a~laP~* Because there is a timch
greater change in vofume along the Sn@) isotherms
than along the isobars, it is more convenient to use
(r3K,@r3, in this study.
Plymate and Stout [30,31] derive an explicit,
empirical V(T, P) equation incorporating the five
refmable parameters K,, K;, , cl,, or&and @K/dT), .
They demonstrate that their fo~ula~ion
has far
more power to accurately model the P-V-T behavior
of a typic& homogeneous solid than does Amyother
five-parameter V(T, Pf equation currently in use in
the literature. Their equation,
x (T - WP),
is derived by integrating
the identity
after assuming that it and K each vary linearly with
both T and P. (For the details of the derivation, see
Plymate and Stout [31].) The resulting expression is
basically a Murnaghan equation multiplied by an
exponential temperature correctian. However, eqn
(2) differs from other eve-pa~me~er tern~~~~corrected Mumaghan equations in the literature f32]
in that it was derived in such a manner as to satisfy
the Maxwell relationship
Regression of our entire Sn(/?) data set ~inc~nd~ng
the literature RT data from Bridgman [I, 2] and
Vaiyda and Kennedy [4] and the room pressure data
from Deshpande and Sirdeshmukh [6,7]) to eqn (2)
yields the values for the five refinable parameters
listed on line 4 of Table 6. The isochores predicted
by eqn (2) using these parameter values match the
expe~meneally-determined isochores nearfy perfectiy.
Both the ma~~mnrn misfit between the p&&ted and
ex~~men~aUy=dete~~n~
isochores (0.001 V/VW)
and the standard error of the regression (0.0017 ff/
V,) are fess than the total experimental un~r~a~n~y
in our Sn(@f Y/VW measu~ments. Furthermore, the
regression values for the parameters arc alt quite
reasonable (Table 6, line 4 compared to lines l-3).
The parameter which matches its independently
determined counterpart least well is a&,, as would be
expected considering the limited temperature range of
the Sn(@) $~ab~~~~y
field.
By assuming that KG, a;, and (%~,@T),
are
constants in the derivation of eqn f2), Plymate and
Stout [30,31] have irnpl~c~~~yassumed that all thirdorder and higher derivatives of vohzme are zero.
En generaf, one would not expect this to h the case.
In fact, comparison of the X& values for the unconstrained Yucatan
regressions for our four ST@)
isotherms suggests a positive value for ~~~~/~~)~
(Table 5). Zfowever, as noted above, the standard
error for the (aK’/H),
value derived in this manner
is too large ta allow any confidence in this determination, Furthermore, an anaiogous comparison
of the expansivity afong isobars failed to reveal any
consistent variation of a; with P. Moreover, the
standard errors for the RI” Mumaghan regressions
(Table 3, tines 5 and 6) and for the room pressure
953
A pressure-volume-temperature equation of state for Sn(/?)
Table 6. Regression parameters for St@) equations of state
Equation
Data set
1 V/V, Combined
RT subset7
2 V/V, Combined
room pressure
sub&%$
3 V/V,
Isotherms
4 Entire V/V,
data set?
5 (U/U&~ Data??
Isothermal
Murnaghan
(1)
Isobaric
expansivity$
Linear/l
(Fig. 4)
T-corrected
Murnaghan
(2)
T-corrected
Murnaghan
(2)
(&)
56.61
(0.79)
K;,
(degz-t)
2.25
(0.27)
-
-
-
-
6.645 x 10-s
(0.352 x lo-‘)
-
-
-
1.695 x 1O-3
-
4.947 x 10-d
18.03 x lO-s
(4.29 x 1O-B)
-
55.45
(0.72)
2.70
(0.23)
7.616 x 1O-5
(0.820 x 10-s)
53.70
(1.66)
6.28
(0.71)
3.967 x 1o-5
(1.24 x 10-r)
- 1.38 x
(0.13 x
-2.07 x
(0.39 x
3.79 x 10-s
(9.63 x 10-8)
26.36 x lo-*
(14.85 x lo-*)
Standard
error
(V/Vim)
(aKid n,
(GPa degree’)
-
lo-’
10-Z)
1O-2 1.684 x 1O-3
10-Z)
-3.40 x lo-’
2.507 x 1O-3
(0.72 x IO-*)
Tbe subscript “00” indicates evaluation at POand r,. i.e. at atmospheric pressure and RT. The numbers in parentheses
are the estimated standard deviations.
1;The “V/VW combined RT subset” includes the data from Bridgman [1,2] and Vaiyda and Kennedy [4] as well as the
RT subset of our data (Fig. 3 and line 6 of Table 3).
$ The “V/V, combined room pressure subset” includes the data from Deshpande and Srrdeshmukh [6,7] as well as the
room pressure subset of our data.
$ V,,(T) = V~exp[~~~~-~~) + a&&“-&)z/2] derived by assuming that ap (defined as V-’ dV/aT) is linear tn 7’.
]ILinear regresston of the Kr pnvalues determined by isothermal Murnaghan regresstons with KG., constramed to equal
4.0 (bottom half of Table 4).
7 The “entire V/V, data set” includes the RT data from Bridgman [l, 21 and Vaiyda and Kennedy [4] and the room
pressure data from Deshpande and Sirdeshmukh [6,7] as well as all of our data.
tt The “(a/~~)~ data” includes RT data from Bridgman [l] and room pressure data from Deshpande and Sirdeshmukh
[6,7] as well as all of our a/aoa measurements.
expansivity regression (Table 6, line 2) are sufficiently
below the experimental
uncertainty
inherent in
the V/V, measurements that there is no need to
assume a non-zero value for either K”( =aZC/aP) or
tl”(=&‘/~W). Therefore, we conclude that the fiveparameter temperature-corrected Murnaghan eqn (2)
with the parameter values determined by regression
from the entire Sn(/?) data set (line 4 of Table 6) is
k’i
f
__-
Y’ 00
PRESSURE
_ _[ib.o
-
1
7
6’ 00
-
8’ 00
10’ 00
(GPA)
Fig. 6. Molar volume and entropy of Sn(B). The solid contours are isochores (in units of J MPa-’ mol-‘)
predicted by our five-parameter temperature-corrected Murnaghan V(T, P) eqn (2) using the regression
parameter values from line 4 of Table 6. The dashed contours are isentropes (contours of constant entropy;
in units of J deg-’ mol-‘) determined by numerical integration of eqn (3) using the same V(T, P) equation
and the room pressure heat capacity data from Robie et al. [33]. Also shown for reference are the Sri(B)
phase boundaries from Fig. 5. Tbe Sn(fi) isochores and isentropes are extended into the Sn(melt) and
Sn(bct) stability fields only for the purpose of labeling; although both the molar volume and entropy of
metastable Sn(@) should be expected to extend into these fields as smooth, continuous functions of P and
T, we make no claim that our data accurately predict the P-T locations of these contours for metastable
St@) in these fields.
MAZXE.
954
CAVALERI et al.
the “correct” P-Y-T equation of state for Sn(#?)
over its entire stability field. The isochores shown on
Fig. 5 are based on this equation (contoured in units
of J MPa-’ mol-‘) and we use this equation in the
subsequent calculations of various other thermodynamic properties of St@).
Gibbs
free
energy of Sn( /I)
We calculate the Gibbs free energy, G, from the
expression
T
G(T, P) = Goo-
SW, PO)dT
s TO
P
+
Entropy
W,
s PO
of Sn( /?)
Given an explicit equation for the molar volume
of a homogeneous solid as a function of all P and
T, all we need is an expression for heat capacity, Cr,
as a function of T along a single isobar in order
to calculate all of the remaining thermodynamic
properties of that solid at any P and T. Robie et af.
[33] tabulate C, as a function of T at atmospheric
pressure for Sri(B) from RT to its melting point at
232°C.
We calculate the entropy of Sn(fi) from the
expression
Substitution of an explicit expression for the temperature variation of the room pressure heat capacity
(from Robie et al. [33]) into the first integral and
substitution of the temperature derivative of our
explicit V(T, P) expression [eqn (2)] into the second
integral then allows us to determine S at any
specified T and P.
We have determined the entropy of Sn(fi) at
various elevated temperatures and pressures by
numerical integration of eqn (3). Figure 6 shows the
resulting isentropes (contours of constant entropy),
contoured in units of J deg-’ mol-’ . The entropy
increases
clearly
with increasing
temperature
[(aS/aT), > O] and decreases with increasing pressure
[(&S/a& < 01, consistent with our intuitive expectations. The fact that the temperature and pressure
contributions
to the entropy are opposite in sign
requires that the isentropes, like the isochores, all
have positive slope [(dT/aP), > O]. In contrast to the
isochores, the isentropes are more nearly perpendicular to the temperature axis, reflecting the fact
that entropy is more strongly dependent on temperature whereas volume is more strongly dependent on
pressure. The spacing between the isentropes clearly
increases both with increasing temperature and with
increasing pressure. This behavior indicates that the
entropy surface is concave downward along isobaric
sections [(CJ2S/aT2), < 0] but concave upward along
isothermal sections [(a*S/aP*~ > 01, consistent with
theoretical expectations [34]. The curvature of the
isentropes is negative throughout the Sn( /I) stability
field [(a2T,k?P2), < 01, and the magnitude of this
curvature increases with increasing entropy.
Evaluation
P)h
dP.
(4)
of eqn (3) at PO yields
f
C,/T
S(T, PO) = IS, +
dT.
s ra
Substitution of this expression into the first integral
of eqn (4) and substitution of our volume expression
[eqn (211 into the second integral then allows us to
determine G at any specified T and P.
We have determined the Gibbs free energy of
Sn(/J) at various
elevated temperatures
and
pressures, relative to G(T,, P,,) = 0.0 as the standard
state, by numerical integration of eqn (4). Figure 7
shows the resulting contours of constant Gibbs
energy for Sn(/J), contoured in units of kJ mol-' .
To the best of our knowledge, this study is the first
to present an experimentally-dete~ined
G(T, P)
relationship for a solid phase over its entire stability
field. A similarly determined G(T, P) relationship for
all the other phases in any particular system would
allow calculation of the P-T location of all heterogeneous (multi-phase) reactions within that system,
regardless of their stability or kinetics and therefore
regardless of whether those reactions can be traced
experimentally.
Such an analysis of the heterogeneous equilibria in the Sn system is the subject of
the second paper in this series [II].
Our ex~~mentally-dete~ned
Snf 8) free energy
surface displays all of the general characteristics
predicted by theory ]35,36]. The Gibbs energy clearly
decreases with increasing temperature [(aG/D’), < 0]
and increases with increasing pressure [(aClaP),
> 01, consistent with the Second Law requirements
that (aG/aT), = -S and (aclaP),=
V. Like the
isochores and isentropes, the Gibbs energy contours
all have positive slope [(aTlaP), > 01, reflecting the
fact that the temperature and pressure contributions
to the Gibbs energy are opposite in sign. The spacing
between the Gibbs energy contours clearly decreases
with increasing tem~rature
and increases with increasing pressure. This behavior indicates that the
free energy surface is concave downward for all T
and P, consistent with the Second Law requirements
that
(a*G/dT*),
= -
(a*c/a~*),
=
(aslaT),
<0
and
(avjap),
< 0.
A pressure-volum&emperature
equation of state for Sn(/Q
7’ 00
6’ 00
PRESSURE
6’ 00
955
10’ 00
(GPA)
Fig. 7. Gibbs free energy of Sn(&. The solid contours represent the Gibbs energy (in units of kJ mot-’
relative to G(T,,, PO) = 0.0 as the standard state) as determined by numerical integration of eqn (4) using
the room pressure heat capacity data from Robie el al. [33] and our five-parameter temperature-corrected
Murnaghan V(T, P) eqn (2) with the regression parameter values from line 4 of Table 6. Also shown for
comparison (dashed contours) is the Gibbs energy predicted by the assumption that molar volume is
independent of both T and P (i.e. that V(T, P) = V,). As before, the contours are extended into the
Sn(melt) and Sn(bct) stability fields only for the purpose of labeling.
curvature
of the Gibbs Energy contours is
negative throughout the Sn(/?) stability field [(a2T/
dP2), < 01, and the magnitude of this curvature
increases slightly with increasing G.
To illustrate the effect on the Gibbs energy of the
pressure and temperature variation of the molar
volume, we also show on Fig. 7 contours of predicted
Gibbs energy based on the assumption that molar
volume is independent of both T and P (i.e. that
V(T, P) = V,). Far-fetched as this assumption may
seem, many studies that predict the equilibria among
solids of geologic interest jncorporate precisely this
The
assumption [37,38]. As Fig. 7 shows, the error in
the prediction of the Gibbs energy caused by this
assumption
exceeds
10 kJmol_’
at the RT
Sn( fi)-Sn(bct) transition.
The product apK, for Sn( 8)
Anderson [39] (and numerous earlier papers)
has suggested that for any solid above its Debye
temperature, in general, the product of tlr and KT
is nearly constant for all T and P. Based on this
assumption he has proposed a “universal thermal
equatton of state” for solids under P-T conditions
Y 00
PRESSURE
6 00
a 00
10 00
(GPR)
Fig. 8. apKT for Sn(&. The product of the isobaric thermal expansion coefficient ar and the isothermal
bulk modulus KT, normalized to the T,, PO value aooKoo = 4.223 x 10e3 GPa deg-' , as determined by eqn
(5) using our five-parameter temperature-corrected Mumaghan V(T, P) eqn (2) with the regression
parameter values from line 4 of Table 6. As before, the contours are extended into the Sn(melt) and Sn(bct)
stability fields only for the purpose of labeling.
PCS 49,8-o
956
MARK E. CAVALERIet al.
equivalent to planetary interiors. Our Sri(p) P-V-T
data can provide a convenient test of this assumption
because the Debye temperature for Sri(B) is well
below RT (approx. -75°C [24]).
From the definitions of CQ,and Kr,
a&
= -(av/aT),/(av/aP),
= (aP/aT),.
(5)
Therefore, V(i”, P) equations based on the assumption that ark’r is constant would predict isochores
which are straight and parallel. As noted above, our
ex~~mentally-dete~ined
Sn( /3) isochores are very
nearly straight, but they are distinctly not parallel;
their slopes [(U/a&]
increase significantly with
decreasing volume.
Substitution
of the temperature and pressure
derivatives of our explicit V(7; P) eqn (2) into
expression (5) allows us to evaluate the product ar KT
for Sn( /?) at simultaneously elevated temperatures
and pressures. Figure 8 shows contours of arKr
for Sn(& derived in this manner, normalized to the
To, P,, value a,&
= 4.223 x 10e3 GPa deg’.
The
value of this product decreases steadily with decreasing volume; the magnitude of this decrease exceeds
35% of the initial value at the RT Sn(~~Sn(~t)
transition.
One could argue, of course, that the behavior of
a moderately compressible solid such as Sn(/?) is of
little relevance for evaluating models for the behavior
of the highly incompressible materials in planetary
interiors; the bulk modulus of Sri(b) is a factor of
3 or 4 lower than the bulk moduli of the silicates and
oxides of geophysical interest. On the other hand,
the pressure and temperature ranges of geophysical
interest are far more than 3 or 4 times greater than
those represented by the P-T stability field of Sn( 8).
Therefore, our data seem to suggest that thermodynamic models which assume a,& to be constant
for planetary interiors may not be completely
justified.
Acknowledgements-We
wish to acknowledge the support
of the National Science Foundation throuah research grants
EAR 7812947 and EAR 8518158 to JHS Additiona? support was provided by the University of Minnesota Graduate
School and Computer Center. Financial support for TGP
was provided by the Exxon Corporation and Southwest
Missouri State University.
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