Shock Response and Phase Transitions of MgO at Planetary Impact Conditions

0.24 MB·English

by  Seth Root

SAND2015-0557 O
Shock Response and Phase Transitions of MgO at Planetary Impact Conditions
∗
Seth Root, Luke Shulenburger, Raymond W. Lemke,
Daniel H. Dolan, Thomas R. Mattsson, and Michael P. Desjarlais
Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
(Dated: January 30, 2015)
Themoon-formingimpactandthesubsequentevolutionoftheproto-Earthisstronglydependent
on the properties of materials at the extreme conditions generated by this violent collision. We
examinethehighpressurebehaviorofMgO,oneofthedominantconstituentsintheearth’smantle,
5
1 using high-precision, plate impact shock compression experiments performed on Sandia National
0 LaboratoriesZ-MachineandextensivequantumsimulationsusingDensityFunctionalTheory(DFT)
2 andquantumMonteCarlo(QMC).Thecombineddataspanfromambientconditionsto1.2TPaand
42,000 K, showing solid-solid and solid-liquid phase boundaries. Furthermore our results indicate
n
under impact that the solid and liquid phases coexist for more than 100 GPa, pushing complete
a
melting to pressures in excess of 600 GPa. The high pressure required for complete shock melting
J
places a lower bound on the relative velocities required for themoon forming impact.
0
3
PACSnumbers: 91.60.Hg,91.60.-x,81.40.Vw,71.15.Pd

]
i
c The leading theory of moon formation is a giant im- ture, which is stable over a wide pressure-temperature
-s pact event occurring approximately 4.5×109 years ago range [14–16]. Dynamic compression experiments start-
rl [1]. Recent large scale simulations show that giant im- ingfromambient[17–21]andfromT0=1900K[22]show
t pacts are capable ofproducing the correctEarth- Moon noindicationsofphasetransitionsupto230GPa. Abini-
m
masses and angular momentum [2, 3]. Complicating the tio studies on the MgO phase diagram agree that three
.
t giantimpacttheory,however,isthattheEarthandmoon phases exist: the B1 solid, the B2 (CsCl) solid, and the
a
m have a nearly identical chemical and isotopic composi- liquid [23–25], but disagree on the location of the phase
tion [4]. This implies that either the impactor was com- boundaries. Along the Hugoniot, which is relevant for
-
d positionally similar to the proto-Earth,which is thought planetary impact scenarios, the location of the B1-B2
n tobeunlikely,orthatextensivemixingofthepostimpact transitionand the melt transitionhas not been precisely
o materials occurred. The simulations needed to test the determined. Recently McWilliams et al. have shown
c
post-impactmixingaredependentonanaccurateunder- that MgO can be dynamically compressed to pressures
[
standingofthepropertiesofmantlematerialsatextreme exceeding 1 TPa using a decaying shock technique [26].

1
pressures and temperatures. For example, post impact The authors also proposed locations for the B1-B2 and
v
mixingforchemicalequilibrationintheproto-Lunardisk B2-liquid transitions along the Hugoniot, but the mea-
9
0 has been shown in simulations [5], but requires melting surements relied heavily on prior MgO Hugoniot data,
0 and vaporization of the mantle in order for material to which was not well known above 200 GPa.
0 diffuse. Testingsuchtheoriesisdifficultbecausethemelt
We present a comprehensive study of the MgO Hugo-
0
lineofthemostcommonmantlematerialsisnotwellcon-
niotusingexperiments,densityfunctionaltheory(DFT),
.
2 strained at these conditions [6].
and quantum Monte Carlo (QMC) methods over a wide
0
Advanced facilities for performing dynamic materials pressure range covering the B1, the B2, and the liquid
5
1 experimentshavegreatlyincreasedthepressureandtem- phases from 0.27 to 1.2 TPa. The high-precision data
: peratureregimesthatcanbeprobedforimportantplane- constraintheHugoniotatmulti-Mbarpressures,andthe
v
i tarymaterials[7–10]. TheRankine-Hugoniot(RH)equa- DFT and QMC results further elucidate information on
X tions[11]relatetheexperimentallymeasuredshockveloc- the phase boundaries. This work provides accurate EOS
r ity to the thermodynamic state, pressure and density, of dataatextremeconditionsandfurthermorerevealslower
a
theshockedmaterial. Theabilitytoperformexperiments limitsoftherelativevelocityrequiredinthegiantimpact
with steady planar shocks and with well-characterized scenario for moon formation.
impactorsandtargetsiscriticalfordeterminingtheequa- Toattainplanetaryimpactconditions,weperformeda
tion of state (EOS) and the phase. To fully address seriesofshockcompressionexperimentsusingtheSandia
the physics relevant to planetary science, this thermo- Z-Machine[27]. TheZ-machineisapulsedpowersystem
dynamicinformationmustbe augmentedwithanunder- capable of producing shaped current pulses and induced
standing of the phase transformations. magnetic fields in excess of 20 MA and 10 MG respec-
tively. Thecombinedcurrentandmagneticfielddensities
InthisworkwefocusonMgO,theendmemberofmag-
generate magnetic pressures up to 650 GPa that can ac-
nesiowu¨stite, a major constituent of the earth’s mantle
celerate aluminum flyers up to 40 km/s [28].
[12]andlikely otherterrestrialexoplanets [13]. Atambi-
ent conditions, MgO exists in a NaCl (B1) lattice struc- Figure 1 shows a schematic view of the target geom-
2

Flyer Velocity 1300 LANL Handbook
14
Shock in MgO 1200 Vassiliou, expt.
Quartz Shock Velocity Duffy, expt.
1100
Zhang, expt.
12 1000
Fratanduono, expt.
m/s) a) 900 McWilliams, expt.
y (k10 GP 800 Z, Al flyer, this work
Velocit 8 MgOQuartz T, Transit Time essure ( 670000 ZBB,12 C DDuFF fTTly,, etthhr,ii ssth wwisoo wrrkkork
Pr 500 Liquid DFT, this work
VISAR 400 Linear Fit B1 Data < 230 GPa
6 300
VF 1145 1150 1155 1160 1165 1170 1175 1180 200 overlapped Z and DFT data
Time (ns)
4 100
950 1000 1050 1100 1150 1200 1250
0
Time (ns) 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5
3
FIG. 1. The experimental target and representative VISAR Density (g/cm)
data. TheVISARmeasurestheflyervelocityasitapproaches FIG. 2. The MgO Hugoniot data in ρ−P space from the Z
the MgO (grey line). For this low velocity impact, as the experiments, previous experimental data[17–21, 26], and the
shocktransitstheMgO,theVISARlosessignal(redline). As DFTsimulation resultsfrom thiswork. Thecurrentdataset
the shock transits into the quartz, the VISAR signal returns shows a deviation from theextrapolation of thefit to theB1
and the quartz shock velocity is measured (blue line). The MgO data from < 230 GPa.
inset shows a VISARrawsignal. Thefiducials correspond to
impact and to shock transit into therear window.
24 LANL Handbook
Vassiliou, expt.
etry; a more detailed Z target geometry is found else- 22 Duffy, expt.
where[29]. AnAlorCuflyerplateisshocklesslyacceler- Zhang, expt.
20
atedtowardthetargetstackconsistingofasingle-crystal Fratanduono, expt
McWilliams, expt.
MgO sample ([100], 300-500 µm, Asphera Corp., ρ0 = 18
3.584 g/cm3) backed by a window. Although the back m/s)16 ZZ,, AClu F Flylyeer,r ,t hthisis w woorkrk
sideoftheflyerismeltedbythehighcurrent,thesurface (kS
oftheflyerimpactingthesampletypicallyhasafewhun- U14 e
c
dverleodcmityicirnotnesrfoefrosomliedtedrensysisttyemmafoterraianlyatreiflmepcatocrt[(2V8I]S.ATRhe) 12 Phase xisten
measures the flyer plate velocity (VF) up to impact at 10 2- oe
the target (Fig. 1). Impact produces a steady shock in B1-Phase B C Liquid
8
the MgO sample. At low impact velocities (and conse-
quently low shock pressures), the MgO sample scatters 6
0 2 4 6 8 10 12 14
light from the VISAR preventing direct measurement of
UP (km/s)
the shock velocity. Instead, sharp fiducials are observed
intheVISARsignal(Fig.1inset)thatcorrespondtoim- FIG.3. TheexperimentalUS−UP dataincludingresultsfrom
previous experiments from Refs [17–21, 26]. The optimized
pactandtoshocktransitintothequartzwindow. Inthis
linear fitsare also plotted. The dashed vertical lines indicate
case, we calculated the MgO shock velocity (US) using
theoptimizedphaseboundariesalongtheprincipalHugoniot
the transit time determined from the fiducials and the
and theshaded cyan regions indicate theuncertainty.
measured thickness. At high impact velocities (and high
shockpressures),the shockfrontinthe MgOisreflective
and the VISAR directly measures the shock velocity. As shock velocity, we calculate the MgO Hugoniot state
the shock transits into the quartz, the VISAR measures density (ρ), pressure (P), and particle velocity (UP).
the quartz shock velocity directly. Multiple VISAR sig- The Hugoniot state is determined using a Monte Carlo
nals were recorded for each sample eliminating 2π am- impedance matching analysis [10] to solve the RH equa-
biguities and providing redundant measurements for im- tions [11]. The Monte Carlo method accounts for the
proved precision. For directly measured velocities, the uncertainties in the experimental measurement and the
uncertainty is better than 1% and for transit time mea- Al and Cu Hugoniot standards. Further discussion on
surements the uncertainty is on the order of 1-2%. the impedance matching method along with the list of
Knowingthe initial densities ofthe MgO andthe flyer Hugoniot data are found in the supplement [30].
plate and measuring the flyer velocity and the MgO Figure 2 plots the experimental and DFT principal
3
Hugoniot in ρ-P space. The Z experimental data span ever, as our continuum level experiments do not provide
the rangefrom0.27TPa upto 1.2TPa - the highest,di- microstructure information, we performed ab initio cal-
rectly measured Hugoniot states attained in MgO. Also culations of the Hugoniotand the phase diagramto bet-
included are the DFT simulation results for the B1, B2, ter understand the high pressure states of MgO.
and liquid phases of MgO (discussed later). Although The high precision requirements of this work necessi-
the VISAR diagnostic does not give direct information tated refinements of previous ab initio methods [23–25].
about the MgO phase upon shock compression, we can We performed calculations utilizing DFT and QMC fo-
infer phase transitions given our data. Figure 2 shows cusing on the solid-solid phase transformation from B1
anextrapolationofthe linearfitto the US−UP datafor to B2 and the melting of MgO along the Hugoniot, pre-
B1-phase Hugoniot states < 230 GPa (converted to ρ- sumably fromthe B2 phase. Using DFT to calculate the
PusingtheRHequations)determinedfromtheprevious Hugoniot requires prior knowledge of the phase state, so
experiments[17–21]. Below≈360GPa,theZexperimen- we first calculated the phase diagram. We used a three-
tal data are consistent with the gas-gun data but above part approach to determine the phase boundaries. In
360 GPa they deviate from the extrapolation. This sug- order to determine the melt boundary from both the B1
geststhattheB1phaseisstableupto360GPaandlikely and B2 phases, we performed two-phase calculations of
undergoesaphasetransitionfromtheB1statetoanother melting using VASP 5.2.11[32, 33]; further details are
phase, presumably the B2 state, at that shock pressure. presented in the supplemental material [30]. To deter-
At higher pressures(≈>700 GPa) the shock front in the minethesolid-solidphaseboundarieswedecomposedthe
MgO was reflective. From the reflective shock front, we solid’s Helmholtz free energy into two pieces.
infer that the MgO has melted into a conductive fluid,
Fsol(V,T)=E(V)+Fvib(V,T) (1)
similar to what is observed for quartz [7].
TheinterpretationfortheMgOphasealongtheHugo- The first piece is the density dependent energy of either
niotisnotasobviousbetweenthe lowpressureB1phase theB1orB2phase. ThisiscalculatedviadiffusionQMC
andthehighpressureliquidphase. Priorab initio calcu- using qmcpack [34] following methodology detailed in
lations suggest that MgO transitions from B1 to B2 and Ref. 35 with particular concern paid to the construction
then from B2 to liquid with a small B1-B2 coexistence of pseudopotentials. The second piece of the free en-
regionandapotentiallylargecoexistenceregionbetween ergy is due to the finite temperature motion of the ions
the B2 and liquid phases [25]. To investigate the phase and electrons and is calculated in two parts. First the
regions,weanalyzetheUS−UP datausingaMonteCarlo harmonic part of the free energy is calculated using the
optimization(MCO)methodsimilar to the methodused finite displacement method as implemented in the phon
in previous work on carbon[31]. code [36]. The quasiharmonic approximation (QHA) is
Slope changes in the US − UP data often indicate known to break down as temperatures increase and this
phasetransitionsandphaseboundaries. UsingtheMCO isparticularlytrueforMgO[37]. Forthisreasonandbe-
method, we fit four lines to the experimental US −UP causetheHugoniotisexpectedtocrossthephasebound-
data. The US −UP data were converted to a “cloud” of ary relatively close to the melt line, we have augmented
points (as described in Ref. 31), allowing region bound- ourQHAcalculationsoffreeenergywiththermodynamic
aries to move smoothly during optimization. For a par- integration. This is performed by using
ticular set of data clouds, the eleven parameters (four T
f 1 ∂E
slopes, four intercepts, and three region boundaries) of ∆S = dT (2)
a four-segment curve were obtained by minimizing the ZTi T (cid:18)∂T(cid:19)V
square minimum distances to each cloud point. Revised thatallowsthechangeinentropyalonganisochoretobe
clouds were generated by randomly drawing a new cen- calculated directly in terms of the internal energy. The
ter for each cloud. Optimization was repeated (≈ 10000 energyiscalculatedusingDFTbasedquantummolecular
times) using the revised clouds to characterize the dis- dynamics (QMD) at points spaced by 250 K along sev-
tributions of the parameters. The parameters are listed eralisochoresintheregionofthephasetransition. Using
in the supplement [30]. It is important to note that this entropy from the QHA calculation at low temperatures
analysis is only possible because of the high precision asareference,wecalculatetheGibbsfreeenergyofboth
data produced from the steady shocks. phases and determine the phase transition pressure di-
Figure 3 shows the compiled experimental US − UP rectly. Thismethodalsodeterminestherangeofvalidity
Hugoniotdata,thefourlinearfits,andthephaseregions for the QHA. We find the range to be smaller than pre-
determined from the MCO method. Following the lit- viously estimated[23] with significant deviations in the
erature [25] and our DFT results, we propose the four free energy occurring by 5000 K and 400 GPa. The pos-
regionsbeclassifiedasfollows: 1. TheB1solidfromam- itive effect of the anharmonic entropy was significantly
bient to 363 GPa; 2. The B2 solid from 363 to 462 GPa; largerin the B1 phase than in the B2 phase, moving the
3. The B2-liquid coexistence region between 462 and phaseboundarytohigherpressuresathightemperature.
620 GPa; and 4. The liquid state above 620 GPa. How- Specificcomputationaldetailsareinthesupplement[30].
4
TABLE I. Phase boundaries along theprincipal Hugoniot. TABLEII.Impactorvelocitiesforcommonplanetarymateri-
Method B1-B2 B2-Coexist. Coexist.-Liquid alsrequiredtocompletelymeltMgOwithT0 atambientand
(GPa) (GPa) (GPa) at1900Kinashockevent(assumingplanarnormalimpact).
Expt. (MCO) 363±6 462±20 620±17 Initial MgO Temp. Impactor Impact Velocity
DFT 330 475 600 [K] [300K] [km/s]
300 MgO 18.6
300 Dunite 19.4
300 Iron 15.3
Figure 4 shows the DFT determined phase diagram 300 Quartz 20.1
andthecalculatedP-TstatesontheMgOHugoniot. Us- 1900 MgO 16.0
ingtheDFTphasediagram,wecaninferthemicrostruc- 1900 Dunite 16.3
ture during our plate impact experiments. However, the 1900 Iron 12.9
1900 Quartz 17.7
experimental Hugoniot data is an incomplete thermody-
namic description due to the lack of temperature mea-

surements. To assign the data to a given phase within
thesinglephaseregions,weuseQMDcalculationstopro- 24000 AlfŁ B1 Melt
DFT B1 Melt
vide the missing temperature by calculating the Hugo- 22000
DFT B2 Melt
niot. For each of the three phases at given densities, 20000 B2 Melt Line Fit
we perform several QMD calculations at various tem- K)18000 TQIH BA1 -BB12- BB2o Bunoduanrdyary
peratures. We then find the temperature for which the ure (16000 Svendsen & Ahrens
RH energy equation is satisfied to determine the shock at14000 Fat’yanov et al.
state. Finally, the pressure and temperature of these per12000
m
shock states are compared to the phase boundaries to Te10000 B1 Hugoniot, DFT
B2 Hugoniot, DFT
determine whether that state is thermodynamically sta- 8000 Liq. Hugoniot, DFT
ble. ThisapproachalsoprovidesavalidationoftheQMD 6000 B1 Hugoniot, T0=1900K DFT
bycomparingthe calculatedHugoniottothe experimen- 4000 B2 Hugoniot, T0=1900K DFT
tal results. The calculations and the experiments are in 2000 Liq. Hugoniot, T0=1900K DFT
goodagreementinρ−P space(Fig.2)andinP−T space
0 100 200 300 400 500 600 700 800 900
(Fig.4). TheDFTcalculatedphaseboundariesalongthe
Pressure (GPa)
HugoniotcorroboratetheMCOfittingmethodresultsfor
the experimental data suggesting the Hugoniot has four FIG.4. P-TphasediagramofMgOwithcalculatedHugoniots
starting at ambient and elevated initial temperature condi-
major regions: B1, B2, coexistence, and liquid. Table I
tions. Experimental P-T data [22, 40] and the low pressure
lists the phase boundaries along the principal Hugoniot
B1-B2 melt line from Ref. 41 are included.
from the MCO method and the DFT simulations.
Both the experimental and DFT results show that
a minimum shock pressure of 620 GPa is required to Hugoniot crosses the B2 region with a large solid-liquid
achievecompletemeltinginMgOthatisinitiallyatambi- coexistence region lasting over 100 GPa before complete
enttemperature. Inthegiantimpactscenario,theproto- melt. The DFT simulation results corroborate the find-
Earthhas alsobeen assumedto have anelevatedsurface ings inferred from the experimental data and also show
temperature prior to the moon-forming event [2]. We thatatelevatedT0 theHugoniotpersiststhroughalarge
have performed additional DFT simulations to calculate coexistence region before melt. The results help place a
the Hugoniot of MgO starting from an initial tempera- lower bound on impact velocities for the moon-forming
ture of 1900 K. From T0 = 1900K, a minimum shock giantimpactscenario. The dataandphase diagrampro-
pressureof445GPaisrequiredtoachievecomplete melt videasolidbasisforthedevelopmentofanMgOequation
in the MgO. Assuming planar normal impact, we can of state that can be used in planetary collision studies.
determine a minimum impact velocity required to melt The authors thank the Z-Operations and Fabrication
MgO.TableII liststhe requiredimpactvelocitiesforim- team for assembling targets and fielding the Z experi-
pactorsofcommonplanetarymaterials. Inarealimpact ments. The authors also thank K. Cochrane for insight-
event,obliqueimpact[38]andshockattenuationthrough ful discussions. QMC calculationsby LS were supported
the mantle [39] affect the amount of energy transferred throughthe PredictiveTheoryandModeling forMateri-
and imply even higher velocities are likely required for als and Chemical Science programby the Office of Basic
significant melting of MgO in the mantle. Energy Science (BES), Department of Energy (DOE).
We have performed an extensive experimental and Sandia National Laboratories is a multiprogram labora-
computational study of the high pressure - high temper- tory managed and operated by Sandia Corporation, a
ature behavior of MgO. The Hugoniot is experimentally wholly owned subsidiary of Lockheed Martin Corpora-
determinedupto1.2TPaandthedatasuggeststhatthe tion, for the U.S. Department of Energy’s National Nu-
5
clear Security Administration under Contract No. DE- and N.C. Holmes, J. Appl.Phys.114, 043518 (2013).
AC04-94AL85000. [22] O. V. Fat’yanov, P. D. Asimow, and T. J. Ahrens, in
ShockCompressionofCondensedMatter,editedbyM.L.
Elert, W. T. B. M. D. Furnish, W. W. Anderson, and
W. G. Proud (AIP,2009) p. 855.
[23] A. B. Belonoshko, S. Arapan, R. Martonak, and
∗ [email protected] A. Rosengren, Phys. Rev.B 81, 054110 (2010).
[1] W.K.HartmannandD.R.Davis,Icarus24,504(1975). [24] B.BoatesandS.A.Bonev,Phys.Rev.Lett.110,135504
[2] R.M. Canup, Science 338, 1052 (2012). (2013).
[3] M. Cuk and S.T. Stewart, Science 338, 1047 (2012). [25] D. Cebulla and R. Redmer, Phys. Rev. B 89, 134107
(2014).
[4] U. Wiechert, A. N. Halliday, D.-C. Lee, G. A. Snyder,
L. A.Taylor, and D.Rumble,Science 294, 345 (2001). [26] R. S. McWilliams, D. K. Spaulding, J. H. Eggert, P. M.
Celliers, D. G. Hicks, R. F. Smith, G. W. Collins, and
[5] K.PahlevanandD.J.Stevenson,EarthPlanet.Sci.Lett.
262, 438 (2007). R. Jeanloz, Science 338, 1330 (2012).
[6] R.Boehler, Rev.Geophysics 38, 221 (2000). [27] M.E.Savageet al.,in2007 IEEEPulsed Power Confer-
ence, Vol. 1-4 (2007) p. 979.
[7] M. D. Knudson and M. P. Desjarlais, Phys. Rev. Lett.
103, 225501 (2009). [28] R.W.Lemke,M.D.Knudson,D.E.Bliss, K.Cochrane,
J.-P.Davis,A.A.Giunta,H.C.Harjes, andS.A.Slutz,
[8] R. G. Kraus, S. T. Stewart, D. C. Swift, C. A. Bolme,
R.F.Smith,S.Hamel, B.D.Hammel,D.K.Spaulding, J. Appl.Phys. 98, 073530 (2005).
[29] S. Root, T. A. Haill, J. M. D. Lane, A. P. Thompson,
D.G.Hicks,J.H.Eggert, andG.W.Collins,J.Geophys.
Res. - Planets 117, E09009 (2012). G. S. Grest, D. G. Schroen, and T. R. Mattsson, J.
Appl. Phys.114, 103502 (2013).
[9] D. K. Spaulding, R. S. McWilliams, R. Jeanloz, J. H.
[30] See Supplemental Material at [URL will be inserted by
Eggert, P. M. Celliers, D. G. Hicks, G. W. Collins, and
R.F. Smith, Phys.Rev.Lett. 108, 065701 (2012). publisher].
[10] S. Root, K. R. Cochrane, J. H. Carpenter, and T. R. [31] M. D. Knudson, M. P. Desjarlais, and D. H. Dolan,
Mattsson, Phys.Rev.B 87, 224102 (2013). Science 322, 1822 (2008).
[11] Y. B. Zel’Dovich and Y. P. Raizer, Physics of Shock [32] G. Kresseand J. Hafner,Phys.Rev.B47, R558(1993),
Waves and High Temperature Phenomena (DoverPubli- Phys. Rev. B 49, 14251 (1994); G. Kresse and J.
Furthmu¨ller, Phys.Rev. B 54, 11169 (1996).
cations, Inc.,Mineola, NY,2002).
[12] W.F.McDonoughandS.Sun,Chem.Geology 120,223 [33] P. E. Bl¨ochl, Phys. Rev. B 50, 17953 (1994); G. Kresse
and D.Joubert, Phys.Rev.B 59, 1758 (1999).
(1995).
[13] N.M. Batalha et al.,Astrophys.J. 27, 729 (2011). [34] J. Kim, K. P. Esler, J. McMinis, M. A. Morales, B. K.
[14] A.Zerr and R. Boehler, Nature371, 506 (1994). Clark, L. Shulenburger, and D. M. Ceperley, J. Phys.
Conf. Series 402, 012008 (2012).
[15] T.S.Duffy,R.J.Hemley, andH.Mao,Phys.Rev.Lett.
74, 1371 (1995). [35] L. Shulenburger and T. R. Mattsson, Phys. Rev. B 88,
245117 (2013).
[16] F. Coppari, R. F. Smith, J. H. Eggert, J. Wang, J. R.
[36] D. Alf`e, Computer Physics Communications 180, 2622
Rygg, A. Lazicki, J. A. Hawreliak, G. W. Collins, and
T. S. Duffy,NatureGeo. 6, 926 (2013). (2009).
[17] S.P.Marsh,LASLShockHugoniot Data,Vol.5(Univof [37] Z. Wu,Phys. Rev.B 81, 172301 (2010).
[38] R.G.Kraus,S.Root,R.W.Lemke,S.T.Stewart,S.B.
California Press, 1980).
[18] M.S.Vassiliou andT.J.Ahrens,Geophys.Res.Lett.8, Jacobsen, andT.R.Mattsson,Accepted forPublication,
729 (1981). Nature Geo. (2014).
[39] S. K. Croft, Geol. Soc. America, Spec. Pubs. 190, 143
[19] T. S. Duffy and T. J. Ahrens, AIP Conference Proceed-
ings 309, 1107 (1994). (1982).
[40] B.SvendsenandT.J.Ahrens,Geophys.J.R.Astr.Soc.
[20] L. Zhang, Z. Gong, and Y. Fei, J. Phys. Chem. Solids
69, 2344 (2008). 91, 667 (1987).
[41] D. Alfe, Phys.Rev.Lett. 94, 235701 (2005).
[21] D. E. Fratanduono, J. H. Eggert, M. C. Akin, R. Chau,