Benchmarking boron carbide equation of state using computation and experiment
Shuai Zhang, Michelle C. Marshall, Lin H. Yang, Philip A. Sterne, Burkhard Militzer, Markus Daene, James A. Gaffney, Andrew Shamp, Tadashi Ogitsu, Kyle Caspersen, Amy E. Lazicki, David Erskine, Richard A. London, Peter M. Celliers, Joseph Nilsen, Heather D. Whitley
BBenchmarking boron carbide equation of state using computation and experiment
Shuai Zhang,
1, 2, ∗ Michelle C. Marshall, † Lin H. Yang, Philip A. Sterne, Burkhard Militzer,
3, 4, ‡ MarkusD¨ane, James A. Gaffney, Andrew Shamp, Tadashi Ogitsu, Kyle Caspersen, Amy E. Lazicki, David Erskine, Richard A. London, Peter M. Celliers, Joseph Nilsen, and Heather D. Whitley § Lawrence Livermore National Laboratory, Livermore, California 94550, USA Laboratory for Laser Energetics, University of Rochester, Rochester, New York 14623, USA Department of Earth and Planetary Science, University of California, Berkeley, California 94720, USA Department of Astronomy, University of California, Berkeley, California 94720, USA (Dated: August 5, 2020)Boron carbide (B C) is of both fundamental scientific and practical interest due to its structuralcomplexity and how it changes upon compression, as well as its many industrial uses and potentialfor use in inertial confinement fusion (ICF) and high energy density physics experiments. We reportthe results of a comprehensive computational study of the equation of state (EOS) of B C in theliquid, warm dense matter, and plasma phases. Our calculations are cross-validated by comparisonswith Hugoniot measurements up to 61 megabar from planar shock experiments performed at theNational Ignition Facility (NIF). Our computational methods include path integral Monte Carlo,activity expansion, as well as all-electron Green’s function Korringa-Kohn-Rostoker and moleculardynamics that are both based on density functional theory. We calculate the pressure-internalenergy EOS of B C over a broad range of temperatures ( ∼ × –5 × K) and densities (0.025–50g/cm ). We assess that the largest discrepancies between theoretical predictions are (cid:46)
5% near thecompression maximum at 1–2 × K. This is the warm-dense state in which the K shell significantlyionizes and has posed grand challenges to theory and experiment. By comparing with different EOSmodels, we find a Purgatorio model (LEOS 2122) that agrees with our calculations. The maximumdiscrepancies in pressure between our first-principles predictions and LEOS 2122 are ∼
18% andoccur at temperatures between 6 × –2 × K, which we believe originate from differences in theion thermal term and the cold curve that are modeled in LEOS 2122 in comparison with our first-principles calculations. In order to account for potential differences in the ion thermal term, wehave developed three new equation of state models that are consistent with theoretical calculationsand experiment. We apply these new models to 1D hydrodynamic simulations of a polar direct-drive NIF implosion, demonstrating that these new models are now available for future ICF designstudies. (LLNL-JRNL-812984)
I. INTRODUCTION
The design of high energy density and inertial confine-ment fusion experiments requires a good description ofthe ablator equation of state (EOS). Materials that aretypically used as ablators are plastics, such as hydro-carbons (CH) and glow discharge polymers (GDP).
However, formation of condensed phase microstructuresand mixing with the DT fuel during implosion could af-fect the performance of the ignition target . Additionalmaterials with higher density and hardness, such as high-density carbon (HDC), boron-materials, and berylliumalso provide current and future options for ablators. .In comparison to plastics, these high-tensile strength ma-terials typically exhibit ablation pressures that are 15-20% higher . Using these materials as the ablator canhave higher x-ray absorption and use a shorter laser pulsewith a higher ablation rate for a given temperature, andthereby require a thinner ablator shell while maintainingthe same mass and outer diameter . Ablators dopedwith boron have also been the subject of more recent pro-posals to use reactions with γ -rays as a means of quan-tifying ablator mix in inertial confinement fusion (ICF)experiments, and boron carbide is of particular interestfor ignition experiments because a method for producing hollow capsules has already been demonstrated. In recent studies, Zhang et al. combined several com-putational methods to set accurate constraint for theEOS of boron (B) and boron nitride (BN) over a widerange of temperatures ( ∼ and clarifying the dominat-ing physics (cold curve, ion thermal, or electron thermal)at different regions of the temperature-density space.They also performed 1D hydrodynamic simulations ofpolar direct-drive exploding-pusher experiments to ex-plore the performance sensitivity to the EOS.Boron carbide is another important member in thefamily of boron materials. At ambient condition, ithas a high melting point, superior hardness, low specificweight, good resistance to chemical agents, and high neu-tron absorption cross section. These outstanding proper-ties allow it to be widely used for mechanical, electrical,chemical, and nuclear applications. The ambient crys-tal structure of B C has rhombohedral symmetry (space a r X i v : . [ phy s i c s . p l a s m - ph ] A ug group R¯3m), similar to that of α -B, and is characterizedby B-rich icosahedra and C-rich chains. X-ray diffrac-tion experiments reveal this structure to be stable understatic compression at up to 126 GPa. Single-crystal ex-periments show that the icosahedral units are less com-pressible than the unit cell volume and the static com-pression is governed by force transfer between the rigidicosahedra. However, dramatic structural changes havebeen reported under shock compression , scratchingand nanoindentation , or depressurization and at-tributed to amorphization or structural transition that isaccompanied by changes in hardness, compressibility, orelastic modulus. There have also been studies thatshow the shear strength of boron-rich boron carbide canbe lowered due to nanotwins and multi-scale molec-ular dynamics (MD) simulations that relate structuralchanges to hydrostaticity of compression .Over the last few years, knowledge about the EOSof boron carbide has advanced significantly. The EOSand melting curve of B C were constructed by Molodets et al. that agree with available experiments at up tomegabar pressures, featuring melting with a negativeClapeyron slope at pressures below 150 GPa and a pos-itive one above 170 GPa. Jay et al. performed com-prehensive ab initio calculations for boron carbide at upto 80 GPa and 2000 K, and their temperature-pressure-concentration phase diagrams show phase separation ofboron carbides in multiple stages and into B and Cat above 70 GPa. Fratanduono et al. extended theHugoniot, sound velocities, and thermodynamic proper-ties measurements of liquid B C to 700 GPa. Shamp etal. performed MD calculations based on density func-tional theory (DFT) to determine the Hugoniot curveup to 1500 GPa, and predicted discontinuities along theHugoniot at <
100 GPa as results of phase separation andtransformation in solid B C. An equation of state table(LEOS 2122) based on an average atom-in-jellium model(Purgatorio) has thus been developed that fits all avail-able experimental Hugoniot data above 100 GPa . How-ever, accurate EOS at higher pressures and temperatures,in particular those corresponding to the partially ionized,warm dense state, is still unknown.The goal of this work is to benchmark the EOS ofB C in a wide range of temperatures and pressures bycombining theoretical calculations and experiments. Ourtheoretical methods include path integral Monte Carlo(PIMC), pseudopotential DFT-MD approaches realizedin multiple schemes, an activity expansion method (AC-TEX), and an all-electron, Green’s function Korringa-Kohn-Rostoker (KKR) method. Our experiments consistof seven Hugoniot measurements conducted at the NIF.The paper is organized as follows: Sec. II outlines ourcomputational details; Sec. III describes our shock exper-iments; Sec. IV compares our EOS and Hugoniot resultsfrom computation and experiments, constructs new EOSmodels, and explores the role of EOS in hydrodynamicsimulations; Sec. V discusses the microscopic physics ofB C by combining electronic structure and QEOS per- T e m pe r a t u r e ( K ) Density (g/cm )10 FIG. 1. Schematic diagram showing the temperature-densityregions at which different methods are used in this work forcalculating the EOS of B C. The principle Hugoniot fromLEOS 2122 is shown (white curve) for comparison. spectives; finally we conclude in Sec. VI.
II. COMPUTATIONAL METHODS
In this section, we briefly describe the computationalsettings of the theoretical methods that we employ tocompute the internal energies and pressures of B C acrosswide ranges of temperatures and densities. Figure 1 sum-marizes the conditions at which each of the methods hasbeen used. The computations are performed by lever-aging the applicability, accuracy, and efficiency of eachmethod. More theoretical details can be found in ourrecent paper and references therein.We perform PIMC simulations of B C using the
CUPID code . All electrons and nuclei are treated explicitly. Inorder to deal with the Fermionic sign problem, we applythe fixed-node approximation using free-particle nodes torestrict the paths . The pair density matrices areevaluated in steps of Hartree − (Ha − ) and the nodalrestriction is enforced in steps of Ha − . The calcu-lations are performed at densities of 0.25–50.17 g/cm [0.1 to 20 times the ambient density ( ρ ∼ . ) ]and temperatures of 10 –5 × K. Each simulation cellconsists of 30 atoms, which is comparable to our previoussimulations for pure B, BN, and hydrocarbons .The finite cell size effects on the EOS are negligible atsuch high temperature conditions .Our DFT-MD simulations for B C are performed intwo different ways. One way is by using the frozen-1s-core projector augmented wave (PAW) or opti-mized norm-conserving Vanderbilt (ONCV) pseudopo-tentials and plane-wave (pw) basis; the other is aFermi operator expansion (FOE) approach using all-electron ONCV potentials. The PAWpw calculationsare performed using the Vienna Ab initio
SimulationPackage (
VASP ) and employing the hardest availablePAW potentials (core radius equals 1.1 Bohr for bothB and C), Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional, a large cutoff energy (2000 eV)for the plane-wave basis, and the Γ point to sample theBrillouin zone. The PAWpw calculations of the EOS areperformed at 6.7 × –5.05 × K ( ∼ ρ . We conducted ONCVpw simulations at temperatures up to 3 × K, using PBE exchange-correlation functional and a 900 eV energy cutoff (coreradius equals 1.125 Bohr for both B and C) for the plan-wave expansion, in order to cross check the PAWpw re-sults. For both PAWpw and ONCVpw calculations, aNos´e thermostat is used to generate MD trajectories(typically ∼ . × –1 . × K. Note that FOE takes advantage ofthe smooth Fermi-Dirac function at high temperatureby approximating the function with polynomial expan-sion, which provides a very efficient way to conduct theKohn-Sham DFT-MD calculation. We use 30-atom cellsand conduct
N V T simulations that last 3000–6000 steps(0.05–0.1 fs/step) to ensure sufficient statistics to ob-tain the EOS. To be consistent with the plan-wave cal-culations, the FOE calculations employ PBE exchange-correlation functional and much larger energy cutoff(4000 eV) due to smaller core radius (0.8 Bohr) due tothe inclusion of 1s core states in both B and C pseu-dopotentials. We also use the all-electron ONCV poten-tials and pw basis to perform calculations at densities of12.544 g/cm or higher and temperatures of 1 . × Kor lower, in order to reduce the possibility of frozen-coreoverlap in the MD simulations.Over the last ten years, Militzer et al. have devel-oped and employed the approach combining PIMC andDFT-MD to calculate the EOS of a series of elemen-tal materials (He , B , C , N , O , Ne , Na ,Al , Si ) and compounds (H O ,LiF ,CH , BN ,MgO , MgSiO ) over wide ranges of temperatures andpressures. The PIMC data were shown to reproduce pre-dictions by classical plasma theories (such as the Debye-H¨uckel and the ideal Fermi-gas model) in the limit ofinfinitely high temperatures and agree remarkably well(differences up to ∼ ∼ –10 K (or10–100 eV), while the DFT-MD predictions of the Hugo-niot are consistent with dynamic shock experiments thatare available up to multi-megabar (Mbar) pressures. Byfully capturing the ionic interaction effects (DFT-MD),nuclear quantum effects (PIMC), and electronic many-body effects (PIMC), these computations set accurate constraints for the EOS of these materials ( Z up to 14)from condensed matter to hot plasma states (degener-acy parameter ∼ , coupling parameter ∼ , an all-electron, Green’s functionKKR electronic-structure method based on Kohn-ShamDFT and an activity expansion method, in addition toFOE and a spectral quadrature method, were used tocompute the EOS of BN and compare with the PIMCand pw DFT-MD data. The Green’s function methodsimplifies the calculation by using a static lattice and ap-proximating the ion kinetic contribution with an idealgas model, and show good agreement with PIMC andDFT-MD predictions at above 10 K when the ion ther-mal contribution becomes less significant in comparisonto electron thermal or cold curve contributions. The ac-tivity expansion approach is based on an expansion ofthe plasma grand partition function in powers of the con-stituent particle activities (fugacities) , and the EOScalculations include interaction terms beyond the Debye-H¨uckel, electron-ion bound states and ion-core plasmapolarization terms, along with relativistic and quantumcorrections , and therefore produce accurate EOS attemperatures down to ∼ K. It is thus interesting toexplore the ranges of applicability of these approaches forB C.We use the Multiple-scattering Electronic-structureCalculation for Complex Applications (
MECCA ) codefor the all-electron, Green’s function KKR calcula-tions. The KKR spherical-harmonic local basis in-cluded L max = 2 within the multiple-scattering contri-butions, and L up to 200 are included automaticallyuntil the free-electron Bessel functions contribute zeroto the single-site wavefunction normalizations. We uselocal density approximation (LDA) for the exchange-correlation functional, a 12 × ×
12 Monkhorst-Pack k -point mesh for Brillouin zone integrations for energieswith an imaginary part smaller than 0.25 Rydberg, anda 8 × × k -point mesh otherwise. A denser mesh wasused for the physical density of states calculated alongthe real-energy axes when needed. We use a static 5-atom cubic cell for the calculations and approximate theion-kinetic contribution by the ideal gas model. Thisstructure can be viewed as a body-centered cubic carbonlattice that has a simple-cubic boron sublattice inscribedat ( ± / , ± / , ± /
4) and ( ± / , ± / , ± / C at ambient conditions. Therefore, it is not expectedto agree with experiments or other computational meth-ods that do not assume this static structure. However,the structure is space filling and might be a representa-tion for higher temperatures and pressures.Activity expansion calculations are performed usingthe
ACTEX code . We cut off
ACTEX calculations attemperatures below the point where many-body termsbecome comparable to the leading-order Saha term (
T > . × K).
III. EXPERIMENTS
We present Hugoniot data for B C to 61 Mbar, ex-ceeding the shock pressures achieved in previous experi-ments by a factor of eight. The new data were obtainedfrom experiments at the NIF , where the B C Hugo-niot was measured relative to a diamond standard usingthe impedance-matching technique. The planar targetpackage, which was affixed to the side of a laser-drivenhohlraum, had a 200- µ m-thick diamond ablator, 5- µ m-thick gold preheat shield, a 100- µ m- or 125- µ m-thick dia-mond baseplate (the impedance-matching standard), andB C, diamond, and quartz samples as shown in Fig. 2(a).The surfaces opposite the drive of the diamond baseplateand smaller diamond sample were flash coated with 100nm of aluminum to facilitate shock break out time mea-surements. Densities of the polycrystalline diamond, z-cut α -quartz, and B C were 3.515 g/cm , 2.65 g/cm ,and 2.51 g/cm , respectively. The inner walls of thehohlraum were irradiated with 176 laser beams, whichproduced a ∼
200 eV x-ray bath that drove a planar andnearly steady shock through the target package. Thetime-dependent shock velocity history in the quartz, mea-sured using a line-imaging velocity interferometer for anyreflector (VISAR) , showed only ±
3% variation fromthe average over the relevant time period of the experi-ment. The laser pulse duration, either 5 or 7.5 ns, andthe total energy, between 519 and 820 kJ, varied shot-to-shot to produce high-pressure states in the B C spanning27 to 61 Mbar.The shock velocities in the diamond baseplate (stan-dard) and B C sample at the material interface are re-quired to determine the pressure-density state on theB C Hugoniot using the impedance-matching technique.Average shock velocities through the smaller diamondand B C samples were calculated from their thicknesses,measured using a dual confocal microscope, and theshock transit times, measured using VISAR. The in situ shock velocities in the B C and diamond samples were de-termined from the measured shock velocity history in thequartz using an analysis technique to correct for shockunsteadiness . The average and in situ shock velocitiesare shown in Fig. 2(b). The Hugoniot and release datafor the diamond standard were determined using LEOS9061, a multiphase EOS for carbon based on DFT-MDand PIMC calculations . The experimental B C Hugo-niot data are given in Table I. Further details on theexperimental configuration and analysis techniques canbe found in Ref. , which reports on quartz and molyb-denum data that were acquired simultaneously with theB C data presented here.
IV. RESULTS AND DISCUSSIONA. Hugoniot comparison
In this section, we compare our experimental measure-ments of the pressure-density Hugoniot of B C with ourtheoretical predictions. Figure 3 compiles the experimen-tal and theoretical Hugoniot curves in pressure-densityand temperature-density plots.The comparison in Fig. 3 shows very good consistencybetween the measurements and the theoretical predic-tions. Assisted by the theoretical predictions, we esti-mate Hugoniot temperatures for the experimental datato be in the range of 1–5 × K. Our results also showthat the PIMC and DFT-MD predicted Hugoniot are inoverall good consistency with LEOS 2122 (L2122). Ourcalculations and the L2122 model predicts B C to have amaximum compression ratio of 4.55 at 9 × Mbar and2 × K, below which L2122 predicts B C to be slightlysofter. We also note that the pressure-density Hugo-niots predicted by a different Thomas-Fermi based tabu-lar model L2120 is very similar at pressure ranges otherthan that around the compression maximum, at whichthe L2120 prediction is stiffer by ∼ Weexpect future, accurate experiments at Gbar pressures totest our predictions for B C.At 3–400 Mbar and 10 –10 K, the Hugoniot curve ob-tained from MECCA and those from DFT-MD (PAWpw,ONCVpw, and FOE) agree remarkably well with eachother. Because MECCA calculations are based on astatic lattice and the ion thermal contribution to theEOS is added following an ideal gas model, the goodconsistency implies that the ion thermal contribution isdominated by the ion kinetic effect. We note that AC-TEX predictions of the Hugoniot down to 6 × K and140 Mbar also agree very well with the DFT predictions.The computational predictions are consistent with theNIF experimental data at pressures above 27 Mbar, aswell as those conducted at the Omega laser facility upto 5 Mbar. However, at 5–10 Mbar, the experimentalHugoniot seems to be softer than DFT-MD predictions,similar to findings by a previous DFT-MD study that wasperformed up to 15 Mbar , which might be attributedto chemical separation of the B C samples as has beencarefully explored for solid B C at low temperatures inRef. 36.
Time (ns) S ho c k v e l o c it y ( k m / s ) (b) Shock velocities Δt B4C Δt C B CC Quartz (a) Target C a b l a t o r AuQuartzB C C b a s e p l a t e C VISAR
FIG. 2. (a) Target design and (b) shock velocities in the boron carbide (B C), diamond (C), and quartz samples attached tothe diamond baseplate for NIF shot N160414. Dotted lines in (b) show the average shock velocity in the samples determinedfrom the measured thickness and shock transit time (∆t). Solid curves show the time-dependent shock velocity histories,measured using VISAR for quartz (orange) and determined using the nonsteady waves correction for B C (blue) and diamond(gray).TABLE I. B C Hugoniot data using the impedance-matching technique with a diamond standard. Shock velocities ( U s ) at thediamond standard/sample interfaces were measured in situ using VISAR for quartz ( Q ) and determined using the nonsteadywaves correction for B C and diamond ( C ). U Cs and U B Cs were used in the impedance-matching analysis to determine theparticle velocity ( u p ), pressure ( P ), and density ( ρ ) on the B C Hugoniot. The average shock velocities ( (cid:104) U s (cid:105) ) determined fromthe measured thickness and shock transit times are also listed. The uncertainties for (cid:104) U s (cid:105) are the same as those given for U s .Shot U Qs (cid:104) U Cs (cid:105) U Cs (cid:104) U B Cs (cid:105) U B Cs u B Cp P B C ρ B C (km/s) (km/s) (km/s) (km/s) (km/s) (km/s) (Mbar) (g/cm )N160414 50.65 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± We note that, at temperatures of 1–4 × K, ourPIMC data for B C have large errors (up to ∼ –10 K. Thismay be due to the methodological difference between AC-TEX/MECCA and PIMC/Purgatorio.In order to better understand the origin of the dif-ferences at the compression maximum, we compare inFig. 4 the energy term E − E i and the pressure term( P + P i )( V i − V ) / .
0, where (
E, P, V ) and ( E i , P i , V i ) re-spectively denote the internal energy, pressure, and vol- ume of B C under shock and in its initial state (300 Kand 2.51 g/cm ), of the Hugoniot function from PIMC,ACTEX, MECCA, and LEOS 2122 along two isotherms1.3 × K and 2 × K. The cross point between thecurve of the energy term and that of the pressure termgives the Hugoniot density at the corresponding temper-ature. Our comparison shows that the internal energyslowly decreases while the pressure term dramatically in-creases, as the density increases from 9 to 14 g/cm .Due to the high computational expense of PIMC sim-ulations at low-temperature conditions, our PIMC dataat low temperatures exhibit significantly larger error barsand stochastic noise than the higher temperature results.The error bars of the PIMC data lead to estimations ofthe 1 σ uncertainty in Hugoniot density, as is shown withshaded green areas in Fig. 3. L2122 and PIMC agree wellwith each other in both energy and pressure, explainingthe excellent consistency between their predicted Hugo-niots. MECCA pressures are slightly higher than PIMC,whereas energies are similar, therefore the Hugoniot den- FIG. 3. Comparison of the Hugoniot of B C predicted byvarious simulations and the LEOS/SESAME models in (a)pressure-density and (b) temperature-density representations.Also shown in (a) is our experimental data collected at theNIF and those by Fratanduono et al. at Omega laser fa-cility. The shaded areas around the lower end of the PIMCcurve represent 1 σ uncertainty in the corresponding Hugoniotdensity due to EOS errors. All pressures in our MECCA EOStable have been shifted up by 97.1 GPa, so that the value atambient is zero. The deviation between PIMC/L2120 (andMECCA) and ACTEX/L2122 curves above 10 Mbar is dueto the electron relativistic effect, which is considered in AC-TEX and L2122 but not in PIMC/L2120 (and not fully inMECCA). The initial sample density ρ i =2.51 g/cm for allthe Hugoniot except that by Shamp et al. , which is 2.529g/cm . sity is also lower. In comparison to PIMC, ACTEX en-ergies are lower, while pressures are similar at 1.3 × Kbut lower at 2 × K, therefore the Hugoniot densitiesfrom ACTEX are also lower. P t e r m E t e r m T=1.3x10 K P t e r m E t e r m T=2x10 K E ne r g y ( e V / B C ) Density (g/cm )PIMCL2122ACTEXMECCA FIG. 4. Comparison of the energy and pressure terms ofthe Hugoniot function for B C from different theories andLEOS models at two temperatures around the compressionmaximum. The shaded area denote the standard error of thePIMC EOS.
B. EOS comparison
The principle Hugoniot samples a specific pathway inthe phase space from 2.5 to 11.5 g/cm accompanied byincreasing temperatures. These conditions are very im-portant because the corresponding states are reachableusing shock experiments. However, off-Hugoniot states,as those simulated in the present work, also play vital rolein hydrodynamic simulations and the underlying physicscan be different. We therefore make detailed comparisonsof the EOS among various methods in this subsection.The pressure-temperature data along several isochoresfrom our calculations are compared in Fig. 5. At 4 × K and above, all our methods (PIMC, ACTEX, andMECCA) agree and are consistent with the L2122 model.This is understandable because the system is approach-ing the limit of a fully-ionized classical plasma, which isaccurately described by PIMC, ACTEX, and the DFTmethods MECCA and Purgatorio.At lower temperatures, the different ways of employ-ing DFT-MD (PAWpw, ONCVpw, and FOE) give thesame EOS and consistent trend with the PIMC data.Several differences are noteworthy when other methods
FIG. 5. Comparison of the pressure-temperature profiles ofB C along several isochores from PIMC, DFT-MD [PAWpw,frozen 1s; ONCVpw, frozen 1s; FOE or pw, all-electron(ae)],ACTEX, MECCA, and L2122. Also included is a set ofMECCA data that have been shifted up by 97.1 GPa, so thatthe value at ambient is zero. Subplot (b) is a zoom-in versionof (a). (ACTEX, MECCA, and L2122) are considered: (1) AC-TEX pressures being lower than others, more so at higherdensities; (2) MECCA pressures being significantly dif-ferent from L2122 at 5 g/cm and below, in particular at T < K; (3) with a rigid shift-up of 97.1 GPa (so thatthe ambient pressure is zero), MECCA pressures agreebetter with L2122 at ambient density and above, butworse at lower-than-ambient densities; and (4) FOE pres-sures gets slightly lower than L2122 for densities higherthan 25 g/cm .Figure 6 focuses on the differences between the first-principles PIMC/DFT-MD data and L2122 ∆ P =( P FP − P L2122 ) /P L2122 ∗ × K. At lower temperatures, ∆ P varies between ±
17% depending on the density—DFT-MD pressures are in general higher at densities below10 g/cm and lower above. | ∆ P | becomes smaller than 10% and gradually vanishes when temperature increasesto 3.5 × K or above. PAWpw and ONCVpw/FOEpredictions are overall the same. FOE smoothly bridgeswith PIMC predictions at 10 K.We also compare the pressures and energies from ourdifferent computations with those from L2122. The re-sults along two isotherms 1.3 × and 2 × K areshown in Fig.7. We find that PIMC, MECCA, and FOEagree with each other to within 5%, which is compara-ble to what we found about differences between PIMCand DFT-MD in previous work on B , BN , and hy-drocarbon systems . The cross validation of the dif-ferent DFT methods and their consistency with PIMCpredictions confirm that both the PIMC and the DFT-MD approaches, albeit carrying approximations in each,are reliable for studying the EOS of warm dense mat-ter. Our ACTEX data also show remarkable consistency(e.g., <
2% at 2 × K) with L2122 at densities below10 g/cm . However, the ACTEX data get way too low athigher densities, which is due to breakdown of the AC-TEX method when the two-body term at order 2 in theactivity becomes comparable to the Saha term, similarto what has been found for BN . C. Modifications to L2122 and 1D hydrodynamicsimulations
We have shown in Fig. 3 that L2122 predicts slightlysofter behavior for B C at 5–500 Mbar, despite theoverall good consistency, in comparison with our first-principles and experimental Hugoniot. We have thus cre-ated three new models for the B C EOS, with the intentto span the range of Hugoniot behavior that is in betteragreement with the experimental data from both NIFand Omega. Recent advances in ICF design methodolo-gies that leverage Bayesian inference techniques to findmost probable physics models based on a range of ex-perimental outcomes and recent interest in B C as anablator for such experiments motivated us to create thisrange of possible EOS models rather than just a singletable. By considering the range of reasonable EOS mod-els for B C as obtained from our above comparisons oftheoretical methods and experimental uncertainty, we de-veloped these three new tables by making modificationsto the Gr¨uneisen parameter within the QEOS methodol-ogy.The Hugoniot curves corresponding to the new models(L2123, L2124, and L2125) are shown in Figure 8, alongwith the experimental data. The PIMC Hugoniot witherror bars is also shown. The new baseline model (L2123)has a slight modification to the Gr¨uneisen parameter,which determines the ion thermal EOS, to bring it intobetter agreement with both sets of experimental data.L2124 and L2125 have modified forms of the Gr¨uneisenparameter that span the range of the experimental errorbars. Both L2123 and L2124 (the softer model) closelytrack L2122 near peak compression, whereas the L2125
1 10 T e m pe r a t u r e ( K ) Density (g/cm ) -4-2 0 2 4 10 T e m pe r a t u r e ( K ) Density (g/cm ) -15-10-5 0 5 10 15 FIG. 6. Percent difference in pressure of B C between PIMC/PAWpw (in spheres) or ONCV (in diamonds) and L2122.FIG. 7. EOS differences of PIMC (red), FOE (black),MECCA (blue), and ACTEX (yellow) relative to LEOS 2122along two isotherms [1.3 × (dashed curves) and 2.0 × K (solid curves)]. Because of the different references cho-sen in the EOS datasets, all energies have been shifted bythe corresponding values at ambient condition (2.5087 g/cm and 300 K). The pressure differences are normalized by thecorresponding LEOS 2122 values; the energy differences arenormalized by the fully-ionized ideal gas values (46 . k B T perB C). The statistical error bars correspond to the 1 σ uncer-tainty of the FOE and PIMC data. The gray vertical bar at11.54 g/cm denotes the maximum Hugoniot density accord-ing to LEOS 2122 and PIMC. FIG. 8. Comparison of the Hugoniot ( ρ i =2.51 g/cm ) ofB C from newly constructed QEOS models (L2123, L2124,and L2125) and those from experiments, PIMC simulations,and L2122. (the stiffer model) shows significantly modified behaviornear peak compression.We applied these new models to 1D hydrodynamic sim-ulations of a polar direct drive fusion experiment basedon previous studies.
For this study, we kept the cap-sule diameter constant at 3000 µ m and set the gas pres-sure to 8 atm of D at room temperature. We used aflux limiter=0.0398 and a square pulse shape with peakpower set to 280 TW. The pulse duration was chosensuch that 476 kJ of energy would be available from thelaser. Due to geometric losses, we assumed that the max-imum absorption of energy would correspond to 75 % ofthe total energy available. Similar to our previous workon boron , we found that the EOS variations we consid-ered here did not produce significant differences in thefuel areal density, peak ion temperature, or ablator arealdensity in these direct drive simulations.In order to expand this sensitivity study to situationsthat might be more relevant to future neutron source de- r (Å) g ( r ) C-C, (c) r (Å) C-C, 5 (f) g ( r ) B-C, (b) B-C, 5 (e) g ( r ) B-B, (a) B-B, 5 (d) K120-atom, 10 K 30-atom, 10 K30-atom, 10 K FIG. 9. Comparison of the nuclear pair correlation functionobtained from DFT-MD (PAWpw) for B C using 30-atom(red) and 120-atom (dark) cells at two different densities andtwo temperatures. The reference density ρ is 2.5087 g/cm . velopment studies, we also examined the neutron yieldvs. ablator thickness for each of the three EOS mod-els. Interestingly, all four EOS models (L2122–L2125)predict similar profiles for the neutron yield with a peakyield that occurs around an ablator thickness of 7.5 µ m.Differences between the models are all within 1% for thin( < µ m) ablators. For ablator thickness between 10–25 µ m, we found the neutron yield from L2123 remains simi-lar ( < V. DISCUSSION
For the sake of benefiting future EOS development,high energy density physics, and warm dense matterstudies, we hereafter discuss the physical origins of theEOS differences shown above from electronic-structureand QEOS points of view.
A. Finite size effects
Our first-principles calculations PIMC, PAWpw, ON-CVpw, and FOE implement the standard way of simulat-ing liquids , which considers a finite number of atoms ina cubic box and under the periodic boundary condition.The finite-cell size effects have been carefully addressed in FIG. 10. Comparison of pressures from single-snapshot cal-culations using various cells [1 formula unit (fu): 5-atom cell;2 fu: 10-atom cell; 3 fu: 15-atom cell] at the ONCV Hugoniotdensities and temperatures. The Hugoniot from three EOSmodels are also shown for comparison. our DFT-MD simulations by choosing large-enough cellswith 120 atoms for all temperatures up to 2.5 × K( ∼
20 eV). This is much higher than the chemical bond-ing (typically about a few eV) is allowed, which justifiesthe usage of 30-atom cells for all simulations at highertemperatures. In order to show this, Fig. 9 compares thenuclear pair correlation function at two different temper-atures (10 and 10 K) and two different densities (2.5and 12.5 g/cm ) using two different cells sizes (30 and120 atoms), from our PAWpw calculations. The resultsshow remarkably good agreement in the features of g ( r )using 30-atom cells with those using the much larger 120-atom cells even at the relatively low-temperature (10 K), high-density (12.5 g/cm ) condition. This is differentfrom our recent findings for BN, which show stronger sizedependence at similar conditions, and is probably due tolarger polarization effects in BN than in B C. Moreover,structures can be clearly seen in the pair correlation plotat 10 K, which are signatures of chemical bonding. At10 K, these structures smooth out and the g ( r ) becomesmore ideal-gas like, which validates the ideal mixing ap-proximation in multi-component average-atom EOS ap-proaches .At temperatures below 10 K, chemical bonding hasto be described using reasonably big simulation cells sothat the EOS can be accurately obtained. In Sec. IV B,we show that MECCA calculations using a 5-atom cellproduce a pressure (-97.1 GPa) that is significantly dif-ferent from 1 bar at ambient condition, and therefore arigid shift in pressure for the MECCA EOS table has tobe applied to improve the agreement between MECCAand DFT-MD Hugoniots. It is worthwhile to investigatethe effect of using such small sizes in more depth by mak-ing comparisons with slightly larger ones.0
FIG. 11. Fractional decomposition of pressure (right axes) inthe LEOS 2122 model and ONCV calculations along their re-spective Hugoniots (black/grey curves, left axes). Also shownare the K-shell occupancy (right axes) of boron and carbonatoms and the average values as obtained from ONCV calcu-lations using different cell sizes (shown in the legend). TheONCV pressures calculated at the Hugoniot temperature anddensity conditions and using smaller cells [short dashed: 1formula unit (fu) (5-atom cell); dotted: 2 fu (10-atom cell);dashed: 3 fu (15-atom cell)] are shown for comparison.
We constructed three structures consisting of 5, 10,and 15 atoms respectively , and performed additionalpw-based single-snapshot calculations using all-electronONCV potentials along the density-temperature Hugo-niot predicted using the ONCVpw/FOE EOS. The pres-sure data as a function of density from the new ONCVcalculations are compared in Fig. 10). The results showthat using 10-atom cells brings the pressure down rela-tive to that using 5-atom cells. However, using larger,15-atom cells leads pressure to the opposite direction,instead of approaching the converged values. The differ-ences as signatures of ion thermal and cold-curve effectson the EOS of B C are observable along the Hugoniotat densities up to 9 g/cm , which is ∼
100 Mbar and ∼ × K. B. Roles of kinetic and interaction effects fromions and electrons
In order to clarify the roles of kinetic and interac-tion effects and those from the ions and from the elec-trons, we performed additional analysis of our pw-basedall-electron ONCV calculations. The calculations allowdecomposing the total pressure into an ion-kinetic (IK)term, which is calculated using the ideal gas model, anda remaining term (P-IK)(Fig. 11). In comparison to theQEOS way of decomposing the L2122 Hugoniot pressureinto ion-thermal, electron-thermal, and cold curve com-ponents, we find that the IK contribution is overlapping with the ion-thermal term in L2122.In addition, we find that the temperatures at whichfinite cell size effects are significant, as characterized bythe differences between solid and dashed curves, over-lap with those at which the cold-curve surpasses the ion-thermal contributions. The turn-over point T t , ∼ × K for B C, may be interpreted as a conservative estima-tion of the uppermost temperature at which finite-sizeeffect remains significant in a theoretical computation,or the lowermost temperature at which an average-atomapproach is feasible. Below T t , interactions are so signif-icant that the ideal mixing approximation becomes lessreliable and a large simulation cell is required for the ac-curacy of computations.As temperature increases to a critical value T c whereK-shell ionization starts, the electron-thermal contribu-tion becomes dominant. This leads to a saddle point inthe IK and the P-IK curves in Fig. 11. Our present cal-culations show T c = 3 × K for B C, which is closeto what we previously obtained for pure boron andslightly below that for carbon. This is not unexpectedbecause the K level is deeper for elements with higher Z . At temperatures above ∼ × K, B C is fully ion-ized and the EOS is dominated by ideal-gas contributionsfrom the nuclei and the electrons.In order to further elucidate the roles of interaction andkinetics in the EOS, we calculate their respective con-tributions to the heat capacity C V along the Hugoniotusing the all-electron ONCV potential, and the resultsare shown in Fig. 12(a). The ion kinetic term ( K ion )contributes 7.5 k B /B C to C V independent of temper-ature, where k B is the Boltzmann constant. Electronkinetic contributions ( K ele ) are generally higher (above15 k B /B C) and show two bumps, one at 10 K and theother at 10 K, which can be attributed to the L- andthe K-shell ionization, respectively. In contrast to K ion which follows an ideal gas model at all temperatures, K ele is dependent on both the electronic orbitals and their oc-cupancy, instead of purely on ionization, and is not idealgas-like until the system is fully ionized. This can be seenfrom its asymptotically approaching the ideal gas valueof 39 k B /B C at above 4 × K.The interaction effects on the EOS are morecomplicated and consist of contributions by ion-ion(“Ewald”), electron-ion (“external”), and electron-electron (“exchange-correlation” and “Hartree”) interac-tions. For simplicity of EOS discussions, it might be eas-ier to group them together than to present individually.This is clearly shown by the difference between the red( K ele ) and the blue ( E − K ion , meaning all except ion ki-netic contributions) line-points in Fig. 12(a). The net ef-fect of interactions can be categorized into two regions: I(gray shaded) is below ∼ K with K ele > E − K ion andimplying negative net contribution of interaction to C V ;II (yellow shaded) is above ∼ K with K ele < E − K ion implying positive contributions of interaction to C V .At ∼ × K, K ele contributions are largely off-set by electron-electron and ion-ion repulsion, therefore1 FIG. 12. (a) Decomposition of the heat capacity from LEOS2122 along two isochores [2.51 g/cm (purple) and 10.03g/cm (green)] and ONCV along the Hugoniot. (b) Heat ca-pacity comparison between LEOS 2122 and PIMC/PAWpw ina broader range of temperatures. In (a), LEOS 2122 resultsare decomposed into electron-thermal (short dashed curves)and ion-thermal (thick dotted curves) terms. ONCV data(dark line-points) are decomposed into ion kinetic (yellow),electron kinetic (red), and interaction (i.e., all-except-ion ki-netic, in blue) terms. C V is dominated by K ion . As temperature increases, therepulsive contribution is gradually offset by the electron-ion attraction, therefore the net interaction contributiongradually increases to zero at ∼ . × K and becomespositive at higher temperatures where K-shell ionizationoccurs. At above 4 × K, K ele and K ion contributionsdominate because the system is fully ionized.Figure 12(a) also compares C V along several isochoresfrom L2122. As a QEOS model, L2122 decomposes thefree energy into three terms: cold-curve, ion-thermal, andelectron-thermal. The ion-thermal term (dotted lines) in-cludes both kinetic and interaction effects such as thosefrom vibration. This explains their differences relative tothe K ion curves, as well as the consistency between theelectron thermal (dashed lines) and E − K ion (blue line- points), because the cold curve does not contribute to C V . We also note [Fig. 12(b)] that the C V curves fromour PIMC/PAWpw calculations in the broad tempera-ture range are consistent with L2122 predictions, exceptfor temperatures above 2 × K, because the electronrelativistic effect that is included in the L2122 modelraises the internal energy and heat capacity and shiftsthe Hugoniot toward the limit of 7 times compression atinfinitely high temperature.
VI. CONCLUSIONS
In this work, we present a comprehensive study of theEOS of B C over a wide range of pressures and tempera-tures by implementing several computational methods,including PIMC, DFT-MD using standard plane-wavebasis and PAW or ONCV pseudopotentials, ACTEX, and
MECCA .Our EOS data by PIMC, FOE, ACTEX, and
MECCA show good consistency at 10 K where 1s electrons areionized. Our detailed EOS comparison provides strongevidences that cross validate both the PIMC and theDFT-MD approaches for EOS studies of the partiallyionized, warm-dense plasmas.At 2.5–3.2 × K and 1.0–1.3 × Mbar, our PIMC,
ACTEX , and
MECCA calculations uniformly predict amaximum compression of ∼ C ( ρ i =2.51 g/cm ), which originates from Kshell ionization. This compression is underestimated byTF models by ∼ ρ i =2.26 g/cm ) andslightly smaller than pure boron ( ρ i =2.31 g/cm ) .We also report Hugoniot data up to ∼
61 Mbarfrom experiments at the NIF. The measured data showgood agreement with our theoretical predictions basedon DFT-MD.By comparing QEOS models with the electron thermalterm constructed in different ways (Purgatorio in LEOS2122 or TF in LEOS 2120/SESAME 7082), we find thatthe Purgatorio-based EOS models provide excellent over-all agreement with our numerical simulations, similar toour previous studies on pure boron and BN. Because thelargest differences in the Hugoniot response of the modelsoccurs near peak compression, performing experimentsfor materials near peak compression would providea rigorous experimental test of our understanding of elec-tronic structure in high energy density plasmas. It wouldalso be worthwhile to pursue experiments that providemeasurements of the temperature and the pressure in ei-ther Hugoniot or off-Hugoniot experiments, which wouldprovide data to test the first principle calculations.Based on the experimental data, we have developedthree new EOS models (L2123, L2124, and L2125) byvariations of the ion thermal EOS model to span therange of experimental error bars. These models were de-veloped to span the range of EOS models that are consis-tent with the experimental error bars. 1D hydrodynamic2simulations of direct drive implosions with a B C ablatordemonstrate that the nominal polar direct drive explod-ing pusher design is not sensitive to the equation of statemodel. Our work should motivate similar studies for fu-ture ICF designs using B C ablators.
VII. SUPPLEMENTARY MATERIAL
See the supplementary material (Table II) for the EOSdata table of B C from this study.
ACKNOWLEDGMENTS
This work was in part performed under the auspices ofthe U.S. Department of Energy by Lawrence LivermoreNational Laboratory under Contract No. DE-AC52-07NA27344. Computational support was provided byLLNL high-performance computing facility (Quartz) and the Blue Waters sustained-petascale computing project(NSF ACI 1640776). B.M. is supported by the U. S.Department of Energy (grant DE-SC0016248) and theUniversity of California. S.Z. is partially supported bythe PLS-Postdoctoral Grant of LLNL.This document was prepared as an account of worksponsored by an agency of the United States govern-ment. Neither the United States government nor anyagency thereof, nor any of their employees, makes anywarranty, express or implied, or assumes any legal li-ability or responsibility for the accuracy, completeness,or usefulness of any information, apparatus, product, orprocess disclosed, or represents that its use would notinfringe privately owned rights. Reference herein to anyspecific commercial product, process, or service by tradename, trademark, manufacturer, or otherwise does notnecessarily constitute or imply its endorsement, recom-mendation, or favoring by the U.S. Government or anyagency thereof. The views and opinions of authors ex-pressed herein do not necessarily state or reflect those ofthe U.S. Government or any agency thereof, and shall notbe used for advertising or product endorsement purposes.
TABLE II:
Supplementary Material : first-principles equation ofstate data for B C based on PIMC and DFT-MD [PAWpw, ON-CVpw(1.07), FOE/ae-pw(2.06)] simulations by Burkhard Militzer, ShuaiZhang and Lin H. Yang. Energies are relative to the corresponding val-ues of B C at ambient condition (300 K, 2.51 g/cc). ρ T P P error
E E error note(g/cm ) (K) (GPa) (GPa) (Ha/B C) (Ha/B C)0.2509 1010479 978.3 9.7 206.906 1.219 PIMC A33680.2509 1347305 1402.8 9.4 283.310 1.184 PIMC A33670.2509 2020958 2254.4 10.9 404.807 1.372 PIMC A33660.2509 4041916 4641.0 15.3 711.485 1.926 PIMC A33650.2509 8083831 9400.4 18.0 1312.607 2.266 PIMC A33640.2509 16167663 18839.9 20.1 2501.695 2.528 PIMC A33630.2509 32335325 37744.2 28.3 4881.135 3.559 PIMC A33620.2509 64670651 75667.1 38.4 9653.671 4.832 PIMC A33610.2509 129341301 151298.9 50.4 19170.798 6.348 PIMC A33600.2509 258682602 302675.0 70.6 38219.473 8.871 PIMC A33590.2509 517365204 605430.7 65.9 76316.150 8.280 PIMC A33580.6272 1347305 3476.4 68.1 266.271 3.426 PIMC A35650.6272 2020958 5653.7 69.0 395.583 3.473 PIMC A35640.6272 4041916 11487.2 59.9 700.416 3.015 PIMC A35630.6272 8083831 23286.6 79.8 1298.108 4.016 PIMC A35620.6272 16167663 47143.0 85.3 2500.912 4.296 PIMC A35610.6272 32335325 94249.1 112.8 4873.197 5.674 PIMC A35600.6272 64670651 188925.7 154.8 9639.451 7.786 PIMC A35590.6272 129341301 378160.5 174.2 19164.543 8.762 PIMC A35580.6272 258682602 756073.9 270.9 38186.621 13.628 PIMC A35570.6272 517365204 1513078.5 253.9 76289.408 12.782 PIMC A35561.2544 1347305 6449.7 134.8 238.128 3.390 PIMC A35761.2544 2020958 11170.0 140.0 381.263 3.522 PIMC A35751.2544 4041916 22672.4 119.1 686.744 3.000 PIMC A35741.2544 8083831 46886.6 158.6 1301.944 3.988 PIMC A35731.2544 16167663 94533.2 157.3 2504.194 3.960 PIMC A35721.2544 32335325 188899.6 214.1 4880.955 5.392 PIMC A35711.2544 64670651 378262.2 264.7 9647.721 6.666 PIMC A35701.2544 129341301 757312.8 382.3 19187.998 9.627 PIMC A35691.2544 258682602 1514056.9 533.5 38232.909 13.454 PIMC A35681.2544 517365204 3026294.1 690.7 76291.346 17.394 PIMC A3567continued . . . . . . continued ρ T P P error E E error note(g/cm ) (K) (GPa) (GPa) (Ha/B C) (Ha/B C)2.5087 2000 8.8 0.3 0.178 0.001 PBE B4C120 01372.5087 6736 34.2 0.2 0.451 0.000 PBE B4C120 01382.5087 10000 49.7 0.2 0.593 0.001 PBE B4C120 01392.5087 20000 98.2 0.2 1.060 0.001 PBE B4C120 01402.5087 50523 264.2 0.4 2.906 0.001 PBE B4C120 01412.5087 67364 360.7 0.5 4.102 0.002 PBE B4C120 01422.5087 101047 568.4 0.7 6.747 0.003 PBE B4C120 01432.5087 202095 1241.3 1.0 15.738 0.010 PBE B4C120 01452.5087 252619 1590.9 1.5 20.516 0.012 PBE B4C120 01462.5087 505239 3447.0 1.4 45.647 0.020 PBE B4C30 00192.5087 1347305 12396.6 265.9 215.509 3.347 PIMC A35872.5087 2020958 21582.0 288.2 357.506 3.625 PIMC A35862.5087 4041916 45205.4 248.0 676.904 3.119 PIMC A35852.5087 8083831 92769.4 310.1 1283.561 3.902 PIMC A35842.5087 16167663 187865.4 337.2 2484.609 4.243 PIMC A35832.5087 32335325 378004.1 412.5 4880.292 5.206 PIMC A35822.5087 64670651 756470.6 612.0 9644.225 7.704 PIMC A35812.5087 129341301 1510913.9 833.1 19138.679 10.495 PIMC A35802.5087 258682602 3024944.4 1072.2 38190.873 13.480 PIMC A35792.5087 517365204 6054012.4 1131.2 76307.047 14.213 PIMC A35785.0174 1347305 25101.2 503.4 200.936 3.166 PIMC A35985.0174 2020958 38909.9 550.8 313.556 3.472 PIMC A35975.0174 4041916 88884.2 486.5 655.812 3.062 PIMC A35965.0174 8083831 185650.7 597.0 1276.586 3.755 PIMC A35955.0174 16167663 374783.7 638.2 2472.912 4.010 PIMC A35945.0174 32335325 753018.7 890.7 4856.619 5.603 PIMC A35935.0174 64670651 1511237.5 1126.5 9629.535 7.068 PIMC A35925.0174 129341301 3027710.6 1432.9 19172.825 8.963 PIMC A35915.0174 258682602 6056069.4 2235.2 38226.650 14.063 PIMC A35905.0174 517365204 12103991.4 2778.3 76278.717 17.526 PIMC A35895.0175 6736 443.7 0.3 0.892 0.001 PBE B4C120 01485.0175 10000 478.5 0.4 1.042 0.001 PBE B4C120 01495.0175 20000 580.7 0.8 1.504 0.002 PBE B4C120 01505.0175 50523 908.6 1.0 3.216 0.003 PBE B4C120 01515.0175 67364 1106.9 1.1 4.344 0.004 PBE B4C120 01525.0175 101047 1516.6 1.5 6.821 0.005 PBE B4C120 01535.0175 126309 1838.0 0.9 8.832 0.003 PBE B4C120 01545.0175 202095 2844.7 0.9 15.336 0.004 PBE B4C120 01555.0175 252619 3535.9 2.1 19.938 0.008 PBE B4C120 01565.0175 505239 7122.1 3.9 44.159 0.029 PBE B4C30 00357.5261 1010479 24092.6 676.4 119.081 2.836 PIMC A36107.5261 1347305 34984.9 616.5 180.443 2.586 PIMC A36097.5261 2020958 58336.2 713.7 303.149 2.991 PIMC A36087.5261 4041916 132474.5 725.5 643.917 3.038 PIMC A36077.5261 8083831 278394.0 933.9 1270.287 3.920 PIMC A36067.5261 16167663 561169.8 988.3 2464.149 4.144 PIMC A36057.5261 32335325 1132212.6 1304.3 4864.353 5.483 PIMC A36047.5261 64670651 2266120.7 1450.6 9624.077 6.105 PIMC A36037.5261 129341301 4537038.6 2389.5 19151.236 9.991 PIMC A36027.5261 258682602 9075883.8 3734.0 38189.980 15.688 PIMC A36017.5261 517365204 18160616.9 3508.9 76296.115 14.737 PIMC A36007.5262 6736 1336.5 0.5 1.866 0.001 PBE B4C120 01587.5262 10000 1391.2 0.5 2.027 0.001 PBE B4C120 01597.5262 20000 1545.6 0.8 2.501 0.001 PBE B4C120 01607.5262 50523 2017.5 0.8 4.148 0.002 PBE B4C120 01617.5262 67364 2297.1 1.4 5.208 0.003 PBE B4C120 01627.5262 101047 2894.1 2.0 7.583 0.006 PBE B4C120 01637.5262 126309 3372.8 2.5 9.535 0.007 PBE B4C120 01647.5262 202095 4850.5 1.8 15.857 0.006 PBE B4C120 01657.5262 252619 5870.5 1.9 20.376 0.006 PBE B4C120 0166continued . . . . . . continued ρ T P P error E E error note(g/cm ) (K) (GPa) (GPa) (Ha/B C) (Ha/B C)7.5262 505239 11135.2 7.6 44.177 0.036 PBE B4C30 005110.0349 6736 2699.3 0.6 3.105 0.001 PBE B4C120 016810.0349 10000 2778.3 0.7 3.276 0.001 PBE B4C120 016910.0349 20000 2989.2 1.1 3.770 0.002 PBE B4C120 017010.0349 50523 3597.4 2.7 5.382 0.005 PBE B4C120 017110.0349 67364 3954.4 3.0 6.403 0.006 PBE B4C120 017210.0349 101047 4723.7 3.1 8.694 0.007 PBE B4C120 017310.0349 126309 5332.8 2.9 10.572 0.006 PBE B4C120 017410.0349 202095 7267.9 3.5 16.787 0.009 PBE B4C120 017510.0349 252619 8592.0 4.0 21.239 0.010 PBE B4C120 017610.0349 505239 15437.1 9.6 44.666 0.036 PBE B4C30 006710.0349 1010479 34986.5 1044.5 123.038 3.285 PIMC A362110.0349 1347305 46667.0 1012.2 173.850 3.185 PIMC A362010.0349 2020958 77656.4 1082.6 294.180 3.405 PIMC A361910.0349 4041916 177948.6 977.7 641.285 3.079 PIMC A361810.0349 8083831 367611.3 1269.2 1254.130 3.992 PIMC A361710.0349 16167663 746834.2 1262.5 2456.294 3.969 PIMC A361610.0349 32335325 1508100.1 1689.9 4857.032 5.321 PIMC A361510.0349 64670651 3023617.9 2169.2 9628.178 6.870 PIMC A361410.0349 129341301 6043779.5 2638.7 19132.188 8.317 PIMC A361310.0349 258682602 12096112.7 4951.6 38172.125 15.585 PIMC A361210.0349 517365204 24216797.6 5382.9 76302.638 16.961 PIMC A361111.2892 1010479 41709.7 1143.1 127.812 3.197 PIMC A363211.2892 1347305 54706.0 1098.7 177.545 3.072 PIMC A363111.2892 2020958 84970.2 1195.4 284.316 3.347 PIMC A363011.2892 4041916 194649.0 1080.3 622.685 3.019 PIMC A362911.2892 8083831 414955.6 1405.5 1255.831 3.931 PIMC A362811.2892 16167663 838384.2 1474.3 2449.395 4.124 PIMC A362711.2892 32335325 1692733.1 2027.9 4844.705 5.676 PIMC A362611.2892 64670651 3407184.7 2353.1 9643.450 6.580 PIMC A362511.2892 129341301 6802116.0 4041.0 19138.514 11.329 PIMC A362411.2892 258682602 13620367.2 4405.1 38206.034 12.316 PIMC A362311.2892 517365204 27241435.2 5301.2 76295.358 14.821 PIMC A362212.5436 10000 4628.5 0.7 4.676 0.001 PBE B4C120 017912.5436 20000 4901.3 1.2 5.196 0.002 PBE B4C120 018012.5436 50523 5647.4 1.8 6.797 0.003 PBE B4C120 018112.5436 67364 6070.7 1.0 7.780 0.002 PBE B4C120 018212.5436 101047 6998.4 2.2 10.002 0.005 PBE B4C120 018312.5436 126309 7737.4 5.0 11.839 0.010 PBE B4C120 018412.5436 202095 10085.9 4.5 17.947 0.011 PBE B4C120 018512.5436 252619 11697.0 4.5 22.338 0.011 PBE B4C120 018612.5436 505239 20042.1 7.9 45.469 0.029 PBE B4C30 008312.5436 1347305 58420.8 1183.6 169.416 2.978 PIMC A364212.5436 2020958 98531.3 1292.6 291.848 3.249 PIMC A364112.5436 4041916 216565.9 1198.0 620.836 3.019 PIMC A364012.5436 8083831 457590.9 1556.2 1244.895 3.923 PIMC A363912.5436 16167663 930693.5 1648.7 2445.848 4.151 PIMC A363812.5436 32335325 1879207.6 2326.5 4839.516 5.857 PIMC A363712.5436 64670651 3774359.1 3080.0 9613.114 7.730 PIMC A363612.5436 129341301 7564430.1 3763.7 19153.993 9.483 PIMC A363512.5436 258682602 15135191.9 5628.8 38208.688 14.165 PIMC A363412.5436 517365204 30257769.9 4928.2 76268.123 12.396 PIMC A363315.0523 1010479 53347.1 1465.9 117.897 3.077 PIMC A365415.0523 1347305 73340.1 1428.0 171.417 2.997 PIMC A365315.0523 2020958 119085.1 1610.1 288.524 3.371 PIMC A365215.0523 4041916 262569.8 1486.4 621.390 3.115 PIMC A365115.0523 8083831 552139.3 1882.7 1247.669 3.949 PIMC A365015.0523 16167663 1117688.5 1965.7 2444.803 4.127 PIMC A364915.0523 32335325 2260439.8 2810.9 4848.438 5.904 PIMC A364815.0523 64670651 4537668.1 3241.8 9629.701 6.808 PIMC A3647continued . . . . . . continued ρ T P P error E E error note(g/cm ) (K) (GPa) (GPa) (Ha/B C) (Ha/B C)15.0523 129341301 9069639.2 5110.8 19136.901 10.681 PIMC A364615.0523 258682602 18144450.8 7854.1 38170.075 16.462 PIMC A364515.0523 517365204 36320053.4 6770.2 76289.249 14.184 PIMC A364415.0524 6736 6775.1 12.0 5.933 0.017 PBE B4C120 018815.0524 20000 7261.8 1.1 6.704 0.001 PBE B4C120 019015.0524 50523 8149.9 1.9 8.315 0.002 PBE B4C120 019115.0524 67364 8648.5 2.2 9.287 0.004 PBE B4C120 019215.0524 101047 9713.2 3.5 11.446 0.005 PBE B4C120 019315.0524 126309 10560.4 3.8 13.222 0.007 PBE B4C120 019415.0524 202095 13290.7 5.6 19.243 0.011 PBE B4C120 019515.0524 252619 15162.4 6.5 23.602 0.016 PBE B4C120 019615.0524 505239 24961.7 9.4 46.532 0.025 PBE B4C30 009917.5610 1010479 62336.2 1580.2 115.857 2.836 PIMC A366517.5610 1347305 85805.3 1608.8 168.247 2.891 PIMC A366417.5610 2020958 136741.9 1818.2 279.581 3.273 PIMC A366317.5610 4041916 300880.5 1708.3 607.206 3.071 PIMC A366217.5610 8083831 641212.4 2159.3 1238.969 3.886 PIMC A366117.5610 16167663 1305330.3 2254.3 2444.524 4.053 PIMC A366017.5610 32335325 2628791.0 3092.4 4831.496 5.553 PIMC A365917.5610 64670651 5290724.2 3841.7 9621.624 6.923 PIMC A365817.5610 129341301 10568812.8 5871.5 19112.975 10.583 PIMC A365717.5610 258682602 21175001.0 8164.4 38180.974 14.699 PIMC A365617.5610 517365204 42363812.8 8271.6 76271.245 14.867 PIMC A365517.5611 10000 9598.7 13.4 7.613 0.019 PBE B4C120 019917.5611 20000 10070.4 1.8 8.279 0.002 PBE B4C120 020017.5611 50523 11094.5 1.8 9.900 0.002 PBE B4C120 020117.5611 67364 11661.7 2.9 10.860 0.004 PBE B4C120 020217.5611 101047 12849.0 3.1 12.955 0.005 PBE B4C120 020317.5611 126309 13818.2 4.7 14.720 0.007 PBE B4C120 020417.5611 202095 16870.0 6.6 20.642 0.011 PBE B4C120 020517.5611 252619 18989.2 6.6 24.999 0.016 PBE B4C120 020617.5611 505239 30145.1 15.9 47.653 0.046 PBE B4C30 011520.0697 1010479 70289.2 1815.5 111.781 2.854 PIMC A367620.0697 1347305 93427.7 1819.3 158.288 2.859 PIMC A367520.0697 2020958 159835.5 2151.1 281.486 3.384 PIMC A367420.0697 4041916 341929.4 2005.1 599.969 3.152 PIMC A367320.0697 8083831 725349.4 2370.9 1223.976 3.732 PIMC A367220.0697 16167663 1489538.2 2722.7 2438.785 4.282 PIMC A367120.0697 32335325 3008323.1 3484.4 4835.791 5.471 PIMC A367020.0697 64670651 6033813.2 4696.1 9600.594 7.390 PIMC A366920.0697 129341301 12085003.6 6140.8 19121.899 9.673 PIMC A366820.0697 258682602 24204304.1 9811.3 38186.636 15.483 PIMC A366720.0697 517365204 48408536.7 9345.6 76258.850 14.675 PIMC A366620.0698 10000 12762.8 0.9 9.181 0.001 PBE B4C120 020920.0698 20000 13320.4 1.4 9.905 0.001 PBE B4C120 021020.0698 50523 14477.6 2.4 11.543 0.003 PBE B4C120 021120.0698 67364 15100.8 4.8 12.485 0.006 PBE B4C120 021220.0698 101047 16425.3 3.8 14.561 0.006 PBE B4C120 021320.0698 126309 17466.7 4.6 16.264 0.008 PBE B4C120 021420.0698 202095 20828.9 5.1 22.165 0.009 PBE B4C120 021520.0698 252619 23147.8 6.6 26.468 0.013 PBE B4C120 021620.0698 505239 35571.3 14.3 48.863 0.030 PBE B4C30 013125.0872 1010479 87214.1 2216.6 107.373 2.787 PIMC A368725.0872 1347305 123235.7 2239.0 160.989 2.820 PIMC A368625.0872 2020958 196124.9 2664.1 269.727 3.354 PIMC A368525.0872 4041916 428037.6 2444.2 593.804 3.077 PIMC A368425.0872 8083831 908550.6 2946.7 1220.664 3.706 PIMC A368325.0872 16167663 1864015.2 3317.5 2436.957 4.176 PIMC A368225.0872 32335325 3747985.1 4716.4 4816.846 5.943 PIMC A368125.0872 64670651 7543547.7 5674.7 9599.259 7.151 PIMC A3680continued . . . . . . continued ρ T P P error E E error note(g/cm ) (K) (GPa) (GPa) (Ha/B C) (Ha/B C)25.0872 129341301 15119588.9 6727.7 19136.352 8.486 PIMC A367925.0872 258682602 30258390.7 11453.7 38187.745 14.433 PIMC A367825.0872 517365204 60534837.3 10922.3 76287.187 13.773 PIMC A367730.1046 1010479 107601.8 2604.2 107.318 2.730 PIMC A369830.1046 1347305 140893.9 2693.8 149.817 2.828 PIMC A369730.1046 2020958 232091.0 3088.5 261.337 3.240 PIMC A369630.1046 4041916 512012.4 2940.2 586.090 3.084 PIMC A369530.1046 8083831 1078105.1 3821.0 1202.934 4.006 PIMC A369430.1046 16167663 2228993.3 3817.0 2424.901 4.009 PIMC A369330.1046 32335325 4520030.4 5136.8 4837.597 5.372 PIMC A369230.1046 64670651 9058273.3 7317.4 9603.443 7.667 PIMC A369130.1046 129341301 18123765.5 9637.7 19113.151 10.120 PIMC A369030.1046 258682602 36320568.3 12420.2 38198.164 13.042 PIMC A368930.1046 517365204 72622107.5 13340.5 76265.622 14.002 PIMC A368837.6307 1010479 147119.9 3181.3 113.815 2.671 PIMC A370937.6307 1347305 193405.1 3225.9 159.107 2.707 PIMC A370837.6307 2020958 298300.7 3899.0 261.283 3.272 PIMC A370737.6307 4041916 636287.4 3753.1 574.957 3.146 PIMC A370637.6307 8083831 1348937.5 4607.3 1197.395 3.863 PIMC A370537.6307 16167663 2777583.7 4929.8 2412.462 4.135 PIMC A370437.6307 32335325 5640862.8 6172.6 4825.974 5.175 PIMC A370337.6307 64670651 11304155.2 8029.8 9584.400 6.776 PIMC A370237.6307 129341301 22665528.6 10246.9 19119.821 8.547 PIMC A370137.6307 258682602 45377914.9 14656.3 38176.243 12.356 PIMC A370037.6307 517365204 90789135.2 18077.2 76272.864 15.166 PIMC A369950.1743 1010479 228810.2 3608.3 129.142 2.269 PIMC A337950.1743 1347305 268222.6 3711.4 158.940 2.338 PIMC A337850.1743 2020958 416612.9 4532.8 264.081 2.850 PIMC A337750.1743 4041916 856514.5 3610.6 569.466 2.272 PIMC A337650.1743 8083831 1808604.3 4427.3 1194.311 2.786 PIMC A337550.1743 16167663 3717607.4 4748.5 2413.995 2.991 PIMC A337450.1743 32335325 7494789.2 7111.9 4803.281 4.481 PIMC A337350.1743 64670651 15097727.3 9275.7 9596.045 5.833 PIMC A337250.1743 129341301 30227397.3 12043.7 19120.537 7.599 PIMC A337150.1743 258682602 60498356.0 19034.1 38169.777 11.969 PIMC A337050.1743 517365204 121037580.3 17009.5 76260.744 10.735 PIMC A33692.5090 2000 9.3 0.8 0.231 0.001 PBE ONCV 1.072.5090 6736 34.8 2.7 0.517 0.002 PBE ONCV 1.072.5090 10000 59.7 3.9 0.658 0.004 PBE ONCV 1.072.5090 35001 176.0 5.1 1.921 0.006 PBE ONCV 1.072.5090 126313 714.1 8.2 8.732 0.018 PBE ONCV 1.072.5090 673653 4948.3 34.5 78.542 0.149 PBE ONCV 2.062.5090 842066 6686.7 22.2 111.475 0.101 PBE ONCV 2.062.5090 1010479 8472.6 24.3 146.350 0.157 PBE ONCV 2.062.5090 1347305 12374.3 15.2 214.048 0.098 PBE ONCV 2.065.0170 6736 438.2 4.8 0.932 0.002 PBE ONCV 1.075.0170 10000 488.2 6.3 1.113 0.004 PBE ONCV 1.075.0170 35001 728.7 13.3 2.308 0.011 PBE ONCV 1.075.0170 126313 1821.7 20.6 8.701 0.026 PBE ONCV 1.075.0170 350013 4898.4 12.6 28.724 0.043 PBE ONCV 2.065.0170 673653 9984.3 10.1 71.185 0.055 PBE ONCV 2.065.0170 842066 12831.4 12.2 100.502 0.057 PBE ONCV 2.065.0170 1010479 15990.7 33.0 130.330 0.106 PBE ONCV 2.065.0170 1347305 23939.0 6.7 193.193 0.102 PBE ONCV 2.067.5260 6736 1316.7 7.0 1.894 0.004 PBE ONCV 1.077.5260 10000 1376.0 9.2 2.084 0.005 PBE ONCV 1.077.5260 35001 1770.7 14.1 3.298 0.009 PBE ONCV 1.077.5260 126313 3330.8 47.2 9.306 0.039 PBE ONCV 1.077.5260 350013 7905.6 28.0 28.870 0.066 PBE ONCV 2.067.5260 673653 15539.5 23.1 67.735 0.060 PBE ONCV 2.06continued . . . . . . continued ρ T P P error E E error note(g/cm ) (K) (GPa) (GPa) (Ha/B C) (Ha/B C)7.5260 842066 20140.4 23.1 93.497 0.079 PBE ONCV 2.067.5260 1010479 24590.6 16.9 119.636 0.104 PBE ONCV 2.067.5260 1347305 34532.3 22.9 180.619 0.133 PBE ONCV 2.0610.0350 6736 2690.7 10.4 3.140 0.003 PBE ONCV 1.0710.0350 10000 2751.2 4.1 3.308 0.003 PBE ONCV 1.0710.0350 35001 3297.3 27.5 4.589 0.015 PBE ONCV 1.0710.0350 126313 5327.5 46.0 10.417 0.034 PBE ONCV 1.0710.0350 350013 11301.0 49.0 29.423 0.068 PBE ONCV 2.0610.0350 673653 21476.6 91.1 66.968 0.216 PBE ONCV 2.0610.0350 842066 27203.8 42.3 90.131 0.157 PBE ONCV 2.0610.0350 1010479 33457.0 74.1 117.018 0.187 PBE ONCV 2.0610.0350 1347305 46468.1 63.8 174.733 0.148 PBE ONCV 2.0611.2890 6736 3543.6 8.2 3.832 0.004 PBE ONCV 1.0711.2890 10000 3634.3 10.9 4.017 0.005 PBE ONCV 1.0711.2890 35001 4206.0 31.0 5.246 0.016 PBE ONCV 1.0711.2890 126313 6486.6 56.3 11.048 0.043 PBE ONCV 1.0711.2890 350013 13071.0 35.4 29.683 0.052 PBE ONCV 2.0611.2890 673653 24818.9 141.9 67.657 0.034 PBE ONCV 2.0611.2890 842066 30471.3 16.0 88.406 0.011 PBE ONCV 2.0611.2890 1010479 38382.2 67.2 116.519 0.023 PBE ONCV 2.0611.2890 1347305 52575.8 46.7 172.974 0.015 PBE ONCV 2.0612.5440 6736 4519.2 2.9 4.543 0.002 PBE ONCV 2.0612.5440 10000 4663.0 1.8 4.767 0.001 PBE ONCV 2.0612.5440 35001 5302.7 16.2 6.002 0.012 PBE ONCV 2.0612.5440 126313 7840.6 17.6 11.940 0.025 PBE ONCV 2.0612.5440 350013 15317.2 77.9 30.819 0.105 PBE ONCV 2.0612.5440 673653 26264.0 19.6 64.244 0.071 PBE ONCV 2.0612.5440 842066 34878.4 81.5 89.036 0.069 PBE ONCV 2.0612.5440 1010479 42918.5 22.6 115.451 0.023 PBE ONCV 2.0612.5440 1347305 58855.8 43.6 167.049 0.067 PBE ONCV 2.0615.0520 2000 6618.1 1.1 5.769 0.002 PBE ONCV 2.0615.0520 6736 6809.5 2.6 6.074 0.003 PBE ONCV 2.0615.0520 10000 6925.4 6.0 6.269 0.004 PBE ONCV 2.0615.0520 35001 7623.3 21.8 7.432 0.016 PBE ONCV 2.0615.0520 126313 10488.1 57.0 12.961 0.056 PBE ONCV 2.0615.0520 350013 19377.3 101.4 31.455 0.103 PBE ONCV 2.0615.0520 673653 33914.0 30.5 65.918 0.090 PBE ONCV 2.0615.0520 842066 41941.1 79.4 92.213 0.132 PBE ONCV 2.0615.0520 1010479 51904.3 97.5 113.040 0.125 PBE ONCV 2.0615.0520 1347305 72318.8 342.6 169.570 0.169 PBE ONCV 2.0617.5610 10000 9638.9 6.7 7.734 0.003 PBE ONCV 2.0617.5610 35001 10408.7 6.4 8.958 0.008 PBE ONCV 2.0617.5610 126313 13972.8 24.0 14.757 0.032 PBE ONCV 2.0617.5610 350013 24066.7 124.8 33.087 0.124 PBE ONCV 2.0617.5610 673653 40863.3 152.1 66.182 0.149 PBE ONCV 2.0617.5610 842066 50853.8 154.1 88.308 0.156 PBE ONCV 2.0617.5610 1010479 62119.5 333.4 116.191 0.141 PBE ONCV 2.0617.5610 1347305 84179.8 44.1 163.040 0.124 PBE ONCV 2.0620.0700 10000 12811.8 11.2 9.369 0.007 PBE ONCV 2.0620.0700 35001 13915.9 8.6 10.773 0.005 PBE ONCV 2.0620.0700 126313 17586.3 65.4 16.123 0.051 PBE ONCV 2.0620.0700 350013 28877.3 115.3 34.082 0.093 PBE ONCV 2.0620.0700 673653 48008.1 224.8 67.112 0.030 PBE ONCV 2.0620.0700 842066 59253.3 301.9 91.678 0.024 PBE ONCV 2.0620.0700 1010479 70396.8 89.0 111.236 0.066 PBE ONCV 2.0620.0700 1347305 97864.6 171.5 161.321 0.097 PBE ONCV 2.0625.0870 10000 20383.0 5.2 12.629 0.003 PBE ONCV 2.0625.0870 35001 21740.1 28.1 14.131 0.014 PBE ONCV 2.0625.0870 126313 26153.1 50.1 19.250 0.036 PBE ONCV 2.0625.0870 350013 40240.6 173.9 37.224 0.116 PBE ONCV 2.06continued . . . . . . continued ρ T P P error E E error note(g/cm ) (K) (GPa) (GPa) (Ha/B C) (Ha/B C)25.0870 673653 62748.5 133.9 67.642 0.119 PBE ONCV 2.0625.0870 842066 76860.4 451.1 91.864 0.114 PBE ONCV 2.0625.0870 1010479 92273.8 551.9 110.436 0.141 PBE ONCV 2.0625.0870 1347305 124901.1 564.6 161.060 0.127 PBE ONCV 2.0630.1050 10000 29636.4 12.5 15.982 0.004 PBE ONCV 2.0630.1050 35001 31097.4 41.0 17.386 0.014 PBE ONCV 2.0630.1050 126313 36480.6 81.2 22.816 0.043 PBE ONCV 2.0630.1050 350013 52548.5 157.1 39.666 0.099 PBE ONCV 2.0630.1050 673653 79783.9 529.2 70.934 0.106 PBE ONCV 2.0630.1050 842066 98505.0 204.4 90.862 0.108 PBE ONCV 2.0630.1050 1010479 114099.0 653.8 113.600 0.100 PBE ONCV 2.0630.1050 1347305 151123.5 518.9 159.254 0.082 PBE ONCV 2.0637.6310 10000 46591.7 14.1 21.192 0.005 PBE ONCV 2.0637.6310 35001 48478.9 56.0 22.750 0.020 PBE ONCV 2.0637.6310 126313 54509.5 135.1 27.698 0.071 PBE ONCV 2.0637.6310 350013 74028.7 839.7 43.372 0.081 PBE ONCV 2.0637.6310 673653 108684.7 484.0 74.384 0.042 PBE ONCV 2.0637.6310 842066 127839.8 630.1 93.539 0.050 PBE ONCV 2.0637.6310 1010479 152358.0 600.1 113.126 0.070 PBE ONCV 2.0637.6310 1347305 200300.2 1428.8 160.424 0.047 PBE ONCV 2.0650.1750 10000 82106.4 25.8 29.732 0.008 PBE ONCV 2.0650.1750 35001 84498.8 67.5 31.221 0.026 PBE ONCV 2.0650.1750 126313 91888.0 151.8 35.933 0.055 PBE ONCV 2.0650.1750 350013 117311.6 756.2 59.691 0.298 PBE ONCV 2.0650.1750 673653 160348.0 612.3 83.482 0.161 PBE ONCV 2.0650.1750 842066 188902.7 1262.9 99.851 0.415 PBE ONCV 2.0650.1750 1010479 213986.7 1638.1 117.712 0.518 PBE ONCV 2.0650.1750 1347305 273743.7 1461.5 159.957 0.224 PBE ONCV 2.06 ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] M. 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