Supplemental Material
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The Sm:YAG primary fluorescence pressure scale
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Dmytro M. Trots,1 Alexander Kurnosov,1 Tiziana Boffa Ballaran,1 Sergey Tkachev,2 Kirill
Zhuravlev,2 Vitali Prakapenka,2 Marek Berkowski,3 and Daniel J. Frost1
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Germany.
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GeoSoilEnviroCARS, University of Chicago, 9700 South Cass Ave. 434A, Argonne, IL 60439,
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Bayerisches Geoinstitut, Universität Bayreuth, Universitätsstraße 30, D-95447 Bayreuth,
USA.
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Institute of Physics, Polish Academy of Science, Al. Lotnikow 32/46, 02-668 Warsaw, Poland.
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Corresponding author: D. M. Trots, Bayerisches Geoinstitut, Universität Bayreuth,
Universitätsstraße 30, D-95447 Bayreuth, Germany. (d_trots@yahoo.com)
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Pressure (GPa)
20
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
15
10
5
0
618
620
622
624
626
Pressure (GPa)
Wavelength (nm)
50
45
(a)
40
(b)
35
(c)
30
(d)
(e)
25
(f)
20
(g)
(h)
15
(i)
10
(j)
(k)
5
(l)
0
618 620 622 624 626 628 630 632 634
Wavelength (nm)
Supplementary figure S1. Comparison of Y1-Sm:YAG calibrations. Calibration
of fluorescence wavelength of Y1 band of Sm:YAG vs. absolute pressure and its
comparison with other pressure scale dependent calibrations: (a) – Bi et al., [1990]; (b) –
Hess and Schiferl, [1992]; (c)– Liu and Vohra, [1993]; (d) – Yusa et al., [1994],
polynomial; (e) Yusa et al., [1994], linear; (f) – Zhao et al., [1998]; (g)– Sanchez-Valle et
al., [2002]; (h) – Goncharov et al., [2005]; (i) – Raju et al., [2011]; (j)– Wei et al.,
[2011]; (k) – fit to our data; (l) – our data.
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25
Pressure (GPa)
20
15
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
10
5
0
616
618
620
622
624
Pressure (GPa)
Wavelength (nm)
60
55
50
45
40
(a)
35
(b)
30
(c)
25
(d)
(e)
20
(f)
15
(g)
10
(h)
5
(i)
0
616 618 620 622 624 626 628 630 632 634
Wavelength (nm)
Supplementary figure S2. Comparison of Y2-Sm:YAG calibrations. Calibration
of fluorescence wavelength of Y2 band of Sm:YAG vs. absolute pressure and its
comparison with other pressure scale dependent calibrations: (a) – Hess and Schiferl,
[1992]; (b) – Liu and Vohra, [1993]; (c) – Yusa et al., [1994], polynomial; (d) Yusa et al.,
[1994], linear; (e)– Zhao et al., [1998]; (f)– Goncharov et al., [2005]; (g)– Raju et al.,
[2011]; (h) – fit to our data; (i) – our data.
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Supplementary figure S3. Large single crystals of Sm:YAG were grown using
the Czochralski method at the Institute of Physics, Polish Academy of Science. The
crystals were each cut lengthwise and ICP MS measurements were performed on the
sliced phases in 41 points for each crystal: the upper and lower crystals contain 0.5–0.7
and 1.7–2.2 wt % of Sm2O3, respectively. SINGLE CRYSTAL FRAGMENTS CAN
BE PROVIDED ON REQUEST.
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Supplementary methods: high-temperature powder diffraction
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Thermal expansion measurements were performed at the synchrotron facility HASYLAB/DESY
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(Hamburg, Germany) with the powder diffractometer at beam-line B2 [Knapp et al., 2004a]. The sample
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was ground in an agate mortar and sieved through a mesh. A quartz capillary 0.3 mm in diameter was
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filled with powdered Sm:YAG and sealed. Subsequently, the capillary was mounted inside a STOE
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furnace in Debye–Scherrer geometry, equipped with a Eurotherm temperature controller and a capillary
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spinner. The furnace temperature was measured by a TYPE–N thermocouple and calibrated using the
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thermal expansion of NaCl [Pathak and Vasavada, 1970]. In order to avoid the adsorption of the K edge
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from Yttrium around 0.73 Å, a wavelength of 0.74925(1) Å was selected using a Si(111) double flat-
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crystal monochromator from the direct white synchrotron beam. The x-ray wavelength was determined
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from eight reflection positions of LaB6 reference material (NIST SRM 660a) measured with a
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scintillation single counter detector, which has Ge(111) analyser crystal in its front. Eighty-eight
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diffraction patterns were collected for Sm:YAG (5-7 minutes per pattern depending on the synchrotron
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ring current with 1 minute for temperature stabilization) at different temperatures up to 1197 K with a
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temperature step of approximatelly 10 K during heating cycles, using an image-plate detector [Knapp et
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al., 2004b]. The image plate detector was calibrated using the diffraction pattern of LaB6 and the
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wavelength was determined using the tandem scintillation counter/analyser crystal. An additional pattern
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was taken at ambient temperature after the heating cycle. All diffraction patterns were analyzed using
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the software WinPLOTR [Rodriguez–Carvajal, 1993].
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Supplementary note 1: thermal expansion
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The temperature dependence of the volumes the Sm:YAG measured at the powder diffraction
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beamline B2 (see supplementary methods) is summarized in supplementary table S1. The temperature
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evolution of the volume was expressed as [Fei, 1995]:
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T

V (T )  VR exp    (T ) dT 
T

 R

(S1)
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where VR is the volume at a chosen reference temperature TR and (T) is the thermal expansion
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coefficient. We fitted equation (S1) to the experimental V vs. T data at fixed experimental values of VR
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and TR (taken at ambient temperature) with a simple 2nd order polynomial, i.e.:
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(T)=a0+a1T+a2T2
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This approach was found to give a suitable description of the experimental data and
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determination of the volumetric thermal expansion coefficient 0 at room pressure and room
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temperature. The fitted values a0 = 8.41(50)*10-6 K–1, a1= 4.31(15)*10-8 K–2 and a2= –2.49(11)*10-11 K–
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3
(S2)
yield an estimate for the thermal expansion coefficient at ambient conditions of 0=19.0(7)*10-6 K–1.
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Supplementary table S1. Unit-cell volumes of Sm:YAG at 88 temperaturesa.
T (K)
V (Å3)
T (K)
V (Å3)
T (K)
V (Å3)
T (K)
V (Å3)
298
1735.15
519.3356
1743.473
734.3692
1753.031
963.7077
1764.158
314.9491
1735.692
528.9673
1743.999
744.3724
1753.734
974.8285
1764.733
324.9886
1736.078
538.6059
1744.212
754.403
1754.066
986.0417
1765.15
334.9712
1736.386
548.2527
1744.626
764.4622
1754.481
997.3526
1765.746
344.9022
1736.785
557.909
1745.126
774.5515
1755.075
1008.767
1766.04
354.7865
1737.046
567.5757
1745.431
784.6724
1755.587
1020.292
1766.576
364.6288
1737.406
577.254
1746.076
794.8264
1756.024
1031.932
1767.107
374.4337
1737.74
586.9449
1746.332
805.0154
1756.452
1043.695
1767.669
384.2052
1738.24
596.6491
1747.016
815.2411
1756.929
1055.589
1768.161
393.9473
1738.561
606.3677
1747.395
825.5056
1757.515
1067.619
1768.657
403.6638
1738.856
616.1014
1747.509
835.8111
1758.053
1079.795
1769.272
413.3581
1739.295
625.851
1748.245
846.1598
1758.525
1086.055
1769.711
423.0335
1739.655
635.6173
1748.703
856.5543
1758.845
1097.09
1770.12
432.6932
1739.92
645.401
1749.052
866.9973
1759.466
1108.15
1770.45
442.3399
1740.407
655.2029
1749.453
877.4917
1759.974
1119.233
1771.236
451.9764
1740.872
665.0238
1749.933
888.0404
1760.504
1130.341
1771.755
461.6052
1741.098
674.8644
1750.412
898.6468
1761.069
1141.473
1772.353
471.2285
1741.576
684.7255
1750.783
909.3144
1761.687
1152.629
1772.626
480.8487
1741.933
694.608
1751.429
920.0468
1762.006
1163.81
1773.343
490.4677
1742.255
704.5127
1751.647
930.848
1762.58
1175.014
1773.929
500.0873
1742.663
714.4404
1752.332
941.7223
1763.102
1186.243
1774.426
509.7094
1743.29
724.3923
1752.817
952.6739
1763.422
1197.495
1775.069
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a
Standard deviations of volumes are of the order of 0.1 Å3 as obtained from a whole–profile fitting
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procedure.
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Supplementary note 2: EoS formalism and determination of absolute pressure
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Physical properties at ambient conditions and estimation of the magnitude of the
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adiabatic–to–isothermal correction
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The room temperature volume V0 was measured precisely three times from three different pieces
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of Sm:YAG by single crystal diffraction using the 8–positions centring procedure and refined using the
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vector least square algorithm implemented in the code Single [Angel and Finger, 2011]. The values for
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V0 displayed a maximum difference of only 0.03%. Therefore, we have set the mean value
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V0=1735.15(0.26) Å3 for the ambient pressure volume. Using compositional constraints from laser
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ablation ICP MS, the calculated density, 0, is equal to 4678.7(7) kg/m3.
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The acoustic sound velocities of two different single crystal fragments of Sm:YAG were
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measured by Brillouin spectroscopy at ambient conditions. The values for adiabatic bulk moduli for both
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fragments were derived from the sound velocity data. They lie within the estimated standard deviations
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(Table 1). The value for adiabatic bulk modulus KS0 at ambient conditions is, thus, set to the mean value
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of 186.5(1.5) GPa as measured by Brillouin spectroscopy. By analogy, the room pressure Debye
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temperature 0= 730(10) K was averaged from two determinations, which were calculated from
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experimental sound velocities using the equation of Robie and Edwards [1966]:
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1
1
 
 1
h  n 3  3  1
2  3

 3 
 0   3

k  4V0    VP 0 3 VS 0 3  


(S3)
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where h is the Plank constant, k is the Boltzmann constant, n = 160 – the number of atoms in the unit
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cell, and VP0 and VS0 are the experimental longitudinal and transverse velocities at ambient conditions,
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respectively. The applicability of this equation is also supported by the fact that Konigs et al. [1998]
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have shown that the dependency of the heat capacity vs. temperature obeys the Debye theory.
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The value of the room pressure Grüneisen parameter 0 is determined via:
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0 
0KS 0
 0C p 0
(S4)
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and it is equal to 1.32(0.13), where Cp0 = 571.72 J*kg-1*K-1 is the experimental isobaric heat capacity of
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yttrium aluminum garnet at room temperature determined by [Konigs et al., 1998].
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The isothermal bulk modulus is calculated via equation (7), so that the value for the isothermal
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bulk modulus KT0 at ambient conditions is equal to 185.1(1.5) GPa. The difference between the adiabatic
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and isothermal values of the bulk modulus is only 0.7 %, whereas, according to Sinogeikin and Bass
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[2000], the experimental accuracy in the determination of the elastic moduli using Brillouin scattering in
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symmetric platelet geometry is approximately 1 %. Moreover, the adiabatic to isothermal correction will
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be even smaller at higher pressures, because the thermal expansivity and Grüneisen parameter decrease
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with pressure. We, therefore, expect that the results from fits of EoS before and after the adiabatic to
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isothermal correction will be essentially the same within the experimental accuracy. In the following we
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will perform fits of EoS parameters using both approaches, i.e. ignoring and accounting for the adiabatic
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to isothermal correction to show that the correction is negligible for Sm:YAG.
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EoS formalism and determination of absolute pressure
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The third–order Eulerian finite strain EoS was used for the analysis of the volume data and
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“cold” variation of the elastic moduli upon compression [Stixrude and Lithgow–Bertelloni, 2005].
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Initially, the adiabatic to isothermal correction was neglected when fitting equations (4) and (5) to the
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experimental KS vs. V and G vs. V dependencies: the parameters V0, KT0`, G0, G0` were fitted, whereas,
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according to recommendations of Kono et al. [2010] and Bass et al. [1981], the value for KT0 was kept
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fixed to the value determined from the Brillouin data.
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In order to check whether the adiabatic to isothermal correction is negligible, we used equation
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(4), where, according to the thermodynamically self–consistent approach [Stixrude and Lithgow–
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Bertelloni, 2005], the volume dependence of the Grüneisen parameter for isotropic solid is represented
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as:
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



1
1
2 f  1 6 0   12 0  36 02  18 0 q0 f
1
6

2
2
1  6 0 f  2  12 0  36 0  18 0 q0 f 


(S5)
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with 0 determined from equation S4 and q0 being the logarithmic volume derivative of the effective
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Debye temperature. The isochoric heat capacity was evaluated within the framework of the Debye
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theory, with the volume dependence incorporated into the volume dependence of the Debye
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temperature:
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 
CV  9nN A k B  
T 
3  / T
x 4e x
 e
x
0


1
2
(S6)
dx

1
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1

2
   0 1  6 0 f   12 0  36 02  18 0 q0 f 2 
2


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where NA is the Avogadro number and 0 is the room pressure value for the Debye temperature
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determined by equation (S3). The fits of the equations (4),(5) and (S5)–(S7) to the experimental KS vs. V
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and G vs. V dependencies were performed with fixed values of KT0, 0 and 0 calculated via equations
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(7), (S4) and (S3), respectively. The parameters V0, KT0`, G0, G0` were fitted. By analogy with Kono et al.
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[2010], q0 was kept fixed during the fitting procedure and was varied manually in the range from 1 to 8.
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As can be seen from Table A1 in Stixrude and Lithgow–Bertelloni [2005], this range is typical for
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numerous oxide silicates and aluminates. Only tiny changes of the V0, KT0`, G0, G0` parameters, lying
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within the uncertainties, were found using the different q0 values, thus demonstrating the insensitivity of
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EoS parameters to changes in q0. The analysis of Table A1 of reference [Stixrude and Lithgow–
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Bertelloni, 2005] shows that q0=1.4 is typical for a number of garnets, therefore q0 was fixed to this
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value during the final fit.
(S7)
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The results of weighted fits are summarized in Table 2 and illustrated in Fig. 2 together with the
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experimental plots of KS vs. V and G vs. V. The weights in the fitting procedure were assigned according
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to the effective variance method [Orear, 1982]. The standard errors for the fitted parameters were
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derived from the variance–covariance matrix of the parameters estimates, which was found by
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multiplying a matrix of partial derivatives, Jacobian matrix, by the mean squared errors. The values of
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the EoS parameters obtained accounting for and ignoring the adiabatic-to-isothermal correction are very
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similar, lying, typically, within 1% of the standard errors (Table 2). The only exception is the difference
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in the value obtained for V0, which falls into the 1.5% standard errors. Thin lines in Fig. 2 represent the
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confidence intervals of any KS and G values predicted by the EoS at any point along the range of V.
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These intervals were obtained from the standard errors for the parameter estimates using Student’s t-
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Distribution at the 99% confidence level, i.e. the 99% probability of the confidence range to capture the
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true values for KS and G parameters.
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The absolute pressure was determined using equation (6). The difference in absolute pressure
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between the two sets of EoS parameters obtained with and without the adiabatic to isothermal correction
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is 0.6 %, or 0.36 GPa, at the highest pressure. The overall uncertainty in absolute pressure is ±1.6–2.7%,
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or ±2.1 % on average over the whole pressure range investigated, calculated using (8). Therefore,
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neglecting the adiabatic to isothermal correction does not significantly influence the determination of the
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high–pressure elastic properties of Sm:YAG by simultaneous XRD/Brillouin scattering technique, as
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also noted for other materials [Goncharov et al., 2007; Sinogeikin and Bass, 2000].
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Supplementary references
147
Bass, J. D., R.C. Liebermann, D.J. Weidner, and S.J. Finch (1981), Elastic properties from
148
acoustic
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dx.doi.org/10.1016/0031-9201(81)90147-3.
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Fei, Y. (1995), Thermal expansion, in Mineral physics and crystallography – a handbook of
physical constants, edited by T.J. Ahrens, AGU, Washington.
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Knapp, M., C. Baehtz, H. Ehrenberg, and H. Fuess (2004), The synchrotron powder
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diffractometer at beamline B2 at HASYLAB/DESY: status and capabilities, J. Synchrotron Radiat., 11,
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328–334, dx.doi.org/10.1107/S0909049504009367.
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H. Fuess (2004), Position-sensitive detector system OBI for high resolution X-ray powder diffraction
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using
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dx.doi.org/10.1016/j.nima.2003.10.100.
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on-site
readable
image
plates,
heat
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dx.doi.org/10.1016/S0040-6031(98)00261-5.
165
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capacity
of
Y3Al5O12
from
A,
521(2–3),
565–570,
0
to
900
K,
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201–206,
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Orear, J. (1982), Least squares when both variables have uncertainties, Am. J. Phys., 50, 912–
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