Tao Sun et al- Lattice dynamics and thermal equation of state of platinum

8/3/2019 Tao Sun et al- Lattice dynamics and thermal equation of state of platinum

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Lattice dynamics and thermal equation of state of platinum

Tao Sun,1,* Koichiro Umemoto,2 Zhongqing Wu,2 Jin-Cheng Zheng,3 and Renata M. Wentzcovitch21Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA

2Department of Chemical Engineering and Materials Science, Minnesota Supercomputing Institute, University of Minnesota, Minneapolis,

Minnesota 55455, USA3Condensed Matter Physics & Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA

Received 22 January 2008; revised manuscript received 29 May 2008; published 28 July 2008

Platinum is widely used as a pressure calibration standard. However, the established thermal equation ofstate EOS has uncertainties especially in the high P-Trange. We use density-functional theory to calculate thethermal equation of state of platinum up to 550 GPa and 5000 K. The static lattice energy was computed byusing the linearized augmented plane-wave method with local-density approximation LDA, Perdew-Burke-Ernzerhof, and the recently proposed Wu-Cohen functional. The electronic thermal free energy was evaluatedusing the Mermin functional. The vibrational part was computed within the quasiharmonic approximationusing density-functional perturbation theory and pseudopotentials. Special attention was paid to the influenceof the electronic temperature on the phonon frequencies. We find that, in overall, LDA results agree best withthe experiments. Based on the density-functional theory calculations and the established experimental data, wedevelop a consistent thermal EOS of platinum as a reference for pressure calibration.

DOI: 10.1103/PhysRevB.78.024304 PACS numbers: 71.15.Mb, 64.30.Ef, 05.70.Ce, 63.20.D

I. INTRODUCTION

Platinum Pt is a widely used high-pressure standard. Itsequation of state EOS at room temperature has been estab-lished by reducing shock Hugoniot14 and by ab initio linear-muffin-tin-orbital LMTO calculations4 up to 660 GPa. Maoet al.5 used the EOS developed in Ref. 4 Holmes et al. tocalibrate pressure in their compression experiment on Fe andFe-Ni alloys. The bulk moduli measured at the earths corepressure are substantially higher than those extrapolatedfrom seismological observations.6,7 A large pressure offset is

needed to remove the discrepancy: about 8% at 100 GPa and15% at 300 GPa. The origin of this offset is under investiga-tion. One possibility is the EOS of Holmes et al. seriouslyoverestimates pressure.6 Singh7 raised other possibilities. Henoticed that the one-parameter EOS of platinum agrees withthe EOS of Holmes et al. to 1% at high pressures and con-cluded that large systematic error in pressure scale is un-likely. He further proposed that the discrepancy due to thepressure on the sample is different from the one on the pres-sure standard in the high-pressure x-ray diffraction measure-ments.

There are conflicting reports on the uncertainties ofHolmes et al.s EOS. Dewaele et al.8 measured the EOS of

six metals at ambient temperature to 94 GPa using adiamond-anvil cell DAC. By cross-checking different pres-sure scales, they found Holmes et al.s EOS overestimatespressure by 4 GPa near 100 GPa at room temperature.This conclusion is confirmed by other groups.9,10 While morerecent calculations based on density-functional theory DFTsuggest that Holmes et al.s EOS underestimates, rather thanoverestimates, pressure. Xiang et al.11 computed the thermalequation of state of platinum using LMTO and a mean-fieldpotential method. The pressure they obtained is 5%6%higher than that of Holmes et al. at high pressures.Menndez-Proupin and Singh12 reached similar conclusionusing pseudopotentials. Both calculations employ the local-

density approximation LDA functional13 and the excesspressure is attributed to LDA in Ref. 12. However it can alsobe caused by other factors. In Table 2 of Ref. 11, the equi-librium volume decreases as the temperature increases. Theelectronic thermal pressure is negative according to this cal-culation, which is contrary to expectations. Reference 12uses an ultrasoft Rappe-Rabe-Kaxiras-Joannopoulos pseudo-potential from the PWSCF website,14 which contains only5d, 6s, and 6p valence states. Its large cut-off radius2.6 a.u. may cause error in studying the highly compressedstructure.

Besides the room-temperature isotherm, accurate thermalpressure Pth is needed to calibrate pressure in simultaneoushigh-pressure and high-temperature experiments. Experi-ments cannot easily determine Pth over a wide temperatureand volume range.15 Consequently Pth is often estimated byassuming it is linear in temperature and independent ofvolume.1,4 Theory can in principle do better. In quasihar-monic approximation QHA, DFT calculations give Pth atany particular temperature and volume. It is desirable tocombine the experimental data with DFT calculations, takingthe advantages of both, and construct a more accurate ther-mal EOS for pressure calibration.

In this paper we have three goals: first is to check theaccuracy of the theoretical EOS of platinum predicted by

different exchange-correlation functionals. In contrast withprevious calculations, we find the room-temperature iso-therm computed with LDA lies below and nearly parallel tothe experimental compression data. The Fermi level of plati-num lies in the d band and gives a very large density of stateDOS NEF. Its vibrational frequencies are more sensitiveto the electronic temperature than those of many other met-als. A Kohn anomaly has been observed in platinum at 90K.16 It becomes weaker and finally disappears when the tem-perature increases. Thus our second goal is to discuss theelectronic temperature dependence of vibrations ETDV andits influence on the thermal properties. Our last goal is toprovide an accurate thermal EOS for pressure calibration.

PHYSICAL REVIEW B 78, 024304 2008

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For this purpose we make corrections to the raw DFT results.We correct the room-temperature Gibbs free energyGP ,300 K to ensure that it reproduces the experimentalisotherm and then combine it with the DFT calculated tem-perature dependence to get GP , T. The thermal EOS andthermal properties deduced from the corrected Gibbs freeenergy are in good agreement with the known experimental

data.

II. COMPUTATIONAL METHOD

The EOS of a material is determined by its Helmholtz freeenergy FV, T, which consists of three parts:

FV,T = UV + FvibV,T + FeleV,T, 1

where UV is the static energy of the lattice, FvibV, T is thevibrational free energy, and FeleV, T accounts for the ther-

mal excitation of the electrons. UV is calculated by usingthe linearized augmented plane-wave LAPW method17 andthree different exchange-correlation functionals: LDA,13

Perdew-Burke-Ernzerhof PBE,18 and Wu-Cohen WC.19

The 4f, 5s, 5p, 5d, and 6s are described as valence stateswhile others are treated as core states. The convergence pa-rameter RKmax is 10.0 and the muffin-tin radius R is 2.08a.u.. A 161616 Monkhorst-Pack20 uniform kgrid is usedand the integration over the whole Brillouin zone is donewith the tetrahedron method.21 All the calculations usingLAPW are performed with and without spin-orbit effect.

In contrast with the static lattice energy UV, which issensitive to the relaxation of the core states and requires a

full-potential treatment, thermal excitations contribute tomuch smaller energy variations and mostly depend on thevalence states. We use pseudopotentials to compute the ther-mal effects. An ultrasoft Vanderbilt pseudopotential22 is gen-erated from the reference atomic configuration5s25p65d96s16p0, including nonlinear core corrections.23

There are two projectors in the s channel 5s and 6s, two inthe p channel 5p and an unbound p at 0.2 Ry above thevacuum level, and one in the d channel 5d. The localcomponent is set in the f channel at the vacuum level. Thecut-off radii for each channel s, p, d, and local are 1.8, 1.9,1.9, and 1.8 a.u., respectively. We use the scalar relativisticapproximation and the spin-orbit effect is not included. This

pseudopotential reproduces the LAPW electronic band struc-ture both at the most contracted volume and the 0 GPa ex-perimental volume. We find that pseudopotentials with dif-ferent exchange-correlation functionals yield very similarelectronic band structures for platinum and we use LDA tocompute all the thermal effects.

We consider 20 different volumes with lattice constantsfrom 7.8 to 6.2 a.u.17.588.83 3 in volume. For eachvolume Vi, we use LAPW to compute its static energy UViand the LDA pseudopotential to evaluate its thermal freeenergy FvibVi , T and FeleVi , T. FvibVi , T is treated withinQHA with phonon frequencies dependent on electronic tem-perature denoted as eQHA as

FvibeQHAVi,T =

1

2 q,jq,jVi,Tele

+ kBTq,j

ln1 exp q,jVi,TelekBT

,2

where q,jVi , Tele denotes the phonon frequency computedat volume Vi and electronic temperature Tele. In thermal equi-librium the system temperature T, the ionic temperature Tion,and the Tele are equal. We distinguish these three tempera-tures to emphasize that the temperature dependence of pho-non frequencies coming from different sources. Anharmonicphonon-phonon interactions cause phonon frequencies to de-pend on Tion but they are omitted in QHA. Electronic thermalexcitations disturb the charge distribution in the crystal andcause phonon frequencies dependent on Tele. In the normalQHA used for insulators and some metals, this effect is alsoignored and q,j has no temperature dependence except

through VT. Platinum has a larger NEF than many othermetals and ETDV may have noticeable effects on its thermalproperties. To quantitatively measure the influence of ETDV,we compare the vibrational free energies at volume Vi andtemperature TjTj =500,1000,...,5000 K computed with-out and with ETDV. Without ETDV normal QHA, phononfrequencies are computed at Tele= 0 K by usingMethfessel-Paxton24 MP smearing with a smearing param-eter of 0.01 Ry. The corresponding vibrational free energy isdenoted as Fvib

QHAVi , Tj. With ETDV eQHA, phonon fre-quencies have to be computed separately for each Tj. This isachieved by using the Mermin functional25 and Fermi-DiracFD smearing. The corresponding vibrational free energy isdenoted as F

vib

eQHAVi

, Tj

. The difference between these two,FETDVVi , Tj= FvibeQHAVi , Tj FvibQHAVi , Tj, describes thecorrection caused by ETDV. To get FETDV at arbitrary tem-perature between 05000 K, we fit a fourth order polynomialfrom FETDVVi , Tj,

FETDVVi,T = FvibeQHAVi,T Fvib

QHAVi,T

= a1Vi T+ a2Vi T2 + a3Vi T

3

+ a4Vi T4 . 3

The final vibrational free energy is computed as FvibVi , T= Fvib

QHAVi , T+FETDVVi , T we omit the superscripteQHA and denote Fvib

eQHA as Fvib.

Phonon frequencies in the above procedure are deter-mined by density functional perturbation theory DFPTRef. 27, as implemented in the QUANTUM ESPRESSO Ref.28 package. The dynamical matrices are computed on an888 q mesh 29 q points in the irreducible wedge of theBrillouin zone. Force constant interpolation is used to cal-culate phonon frequencies at arbitrary q vectors. The sum-mation in Eq. 2 is evaluated on a 323232 q mesh.

The electronic free energy FeleVi , T is determined byusing the Mermin functional25,26 and Fermi-Dirac smearing.Similar to getting FvibVi , T, we first compute Fele at every50 K from 50 to 5000 K, then we fit them to a fourth orderpolynomial

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FeleVi,T = b1Vi T+ b2Vi T2 + b3Vi T

3 + b4Vi T4.

4

Terms other than b2ViT2 represent deviations from

the lowest-order Sommerfeld expansion Fele=

2

6 NEF, VikBT2, where NEF, Vi is the electronic den-

sity of states at Fermi energy EF

and volume Vi

. We find that,below 1000 K, keeping only the quadratic term does notintroduce much error. The influence of the higher order termsbecomes prominent at high temperatures. At 2000 K, theerror reaches about 15%. The fitted quadratic coefficientb2Vi differs from the Sommerfeld value

2

6 NEF, VikB2 by

5% Vi =8.83 3 to 15% Vi =15.63

3. It seems the Som-merfeld expansion works better at high pressures where theelectronic bands are more dispersive and NEF is smaller.We combine FeleVi , T with the static energy UVi fromLAPW and the vibrational free energy FvibVi , T from thesame pseudopotential to get the total free energy FVi , T atvolume Vi. There are two popular parametrized forms to fitthe total free energy FV, T: fourth order Birch-Murnahan29

BM and Vinet.30 We find BM and Vinet are comparable inaccuracy to fit the static and low-temperature free energiesbut BM yields much lower residual energies than Vinet forthe high-temperature results. Thus we use fourth order BM toget FV, T. Other thermodynamical properties are computedby finite difference.

All the pseudopotential calculations are carried out withthe same plane-wave cutoff of 40 Ry, charge-density cutoffof 480 Ry, and a shifted 161616 Monkhorst-Pack mesh.To determine the convergence uncertainties of our results, wechoose one volume Vi =15.095

3, and recompute its pho-non frequencies at Tele =0 K with a 242424 mesh and ahigher plane-wave cutoff 60 Ry. The two sets of phonon

frequencies differ by 0.5% at most. The corresponding FvibQHAdiffer by 0.07 mRy/atom at 2000 K and 0.18 mRy at 5000 K.The influence of ETDV is much greater than the convergenceuncertainties. For some modes phonon frequencies changeby 10% or more from Tele=500 K to Tele=2000 K. Thefree-energy correction FETDV is about 1 mRy/atom at 2000K.

III. SUMMARY OF PREVIOUS WORKS

Besides the two calculations11,12 mentioned in the intro-duction, which focus on the thermal EOS of platinum, thereare some other papers related to this subject. Cohen et al.31

computed the static EOS of platinum using LAPW and PBE,and treated it as an example to discuss the accuracy of dif-ferent EOS formations. They found that Vinet fitted betterthan third order BM. The accuracy of fourth order BM andVinet were comparable. Tsuchiya and Kawamura32 computedthe electronic thermal pressure Pele of Au and Pt usingLMTO and LDA. At 2200 K, Pele is 1.01 GPa for Pt whileonly 0.06 GPa for Au. This is caused by the different NEFof the two metals. The small ETDV effect 1% change inphonon frequency from Tele=0 K to 3000 K observed ongold33 is consistent with this picture. Wang et al.34 usedLAPW and an average potential method to determine thethermal contributions. Then they reduced the experimental

shock Hugoniot and got the room-temperature isotherm ofPt. This isotherm is very similar to that of Holmes et al., inspite of the fact that in the latter case, thermal pressure isestimated semiempirically. Reference 35 computed the staticEOS of platinum using pseudopotentials with and withoutspin-orbit effects up to 150 GPa. In the following section, wecompare our calculations with these previous ones whenever

appropriate.On the experimental side, The reduced isothermal P-V-T

EOS from shock wave experiments are widely used as pri-mary pressure scales. At present they are also the only ex-perimental sources for P-V-T data at very high pressures.The shock Hugoniot of platinum was first obtained by usingchemical explosives.1 The reduced room-temperature iso-therm was up to 270 GPa. Holmes et al.4 went to highercompression ratio using a two-stage light-gas gun. The finalshock Hugoniot is a combination of these two sets of data. Inspite of the crucial role of the reduced shock EOS, its accu-racy suffers from low precision in measurements and theo-retical simplifications made in the reducing process.10,36 With

the development of DAC and third-generation synchrotronlight source, cross-checking different pressure scales becamefeasible. More accurate thermal EOS were obtained by usingthis method.8,15

Recently, Dorogokupets and Oganov37 constructed asemiempirical model to describe the thermal properties of Al,Au, Cu, Pt, Ta, and W. The model contains about 20 param-eters, which are fitted to the available experimental data onthe heat capacity, enthalpy, volume, thermal expansivity,bulk modulus, and shock Hugoniot. Based on this model,they reanalyzed the data in Ref. 8 up to 100 GPa. The result-ing EOS, which are consistent with the measured thermalproperties, are believed to be more accurate than the original

in the corresponding pressure range.38

A simplified versionof the model39 yields similar EOS at low pressures. Howevertheir high-pressure extrapolations differ by 2.5% near 240GPa. It will be interesting to use DFT to explore the EOS atvery high pressures, which are still out of reach for the cur-rent DAC experiments.

IV. RESULTS AND DISCUSSIONS

A. Static equation of state

Before studying the EOS at finite temperature, we exam-ine the static EOS computed by using different exchange-correlation functionals and compare them with previous cal-

culations. Excluding the thermal effects which amount to2 GPa at room temperature helps to identify the origin oftheir differences. Figure 1 shows the static pressure vs vol-ume relations using different exchange-correlation function-als. The corresponding EOS parameters are listed in Table I.The experimental data at room temperature are also includedin the figure to give a rough estimate of the difference. Com-paring to the experiments in the entire volume range, LDAunderestimates pressure while PBE overestimates. WC im-proves on PBE but still overestimates. A detailed comparisonbetween the calculated room-temperature isotherms includ-ing the thermal effects and the experimental data will begiven in Sec. IV C. DFT has many different implementations

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such as LAPW, LMTO, and various pseudopotentials. If the

calculations are good, they should yield similar results. Wecompare our LDA calculations with previous ones in Fig. 2.Two of our own pseudopotential calculations are included forcomparison. One is the Vanderbilt pseudopotential that weuse to compute the thermal effects, denoted as pseudo-1. Theother is a Rappe-Rabe-Kaxiras-Joannopoulos pseudopoten-tial from the PWSCF website Pt.pz-nd-rrkjus.UPF, denotedas pseudo-2. The static EOS predicted by pseudo-1 is similarto that of LAPW. Their EOS parameters differ by no morethan 0.5%. The previous overestimations of pressure areprobably caused by the large cut-off radius and insufficientnumber of valence electrons Ref. 12, or another issue re-lated to the negative electronic thermal pressure Ref. 11.

Platinum is a heavy element and its electronic band struc-ture is sensitive to spin-orbit effect.41 We find inclusion ofthe spin-orbit effect increases the equilibrium volume nomatter which exchange-correlation functional is used. Thistendency has also been observed by Bercegeay andBernard35 in their pseudopotential calculations. However, theEOS parameters are not independent of each other. Thevariation of the equilibrium volume largely compensates thatof the bulk modulus and the actual pressure difference iswithin 0.7% at high pressures.

Using pseudopotentials instead of the all electron LAPWmay introduce error in computing phonon frequencies, espe-cially at high pressures. Since lattice vibrations are closely

related to the force stress on the atoms, we estimate theerror in phonon frequencies by analyzing the error in staticpressure. At high pressures 150550 GPa, the pressure dif-ference between LAPW with LDA functional and pseudo-1is about 1.4%. The error in phonon frequencies caused byusing pseudo-1 is likely to be of the same order. Since theinfluence of spin-orbit effect is half of the pseudopotential

uncertainty, it is ignored completely in the following calcu-lations.

B. Phonon dispersion and its electronic temperature

dependence

Figure 3 shows the phonon dispersions at the experimen-tal ambient condition lattice constant a =7.4136 a.u. 8 One iscomputed at Tele=0 K while the other at Tele=2000 K, closeto platinums melting point at ambient pressure 2041.3 K.42

The Kohn anomaly near q = 0,0.35,0.35 disappears whenthe electronic temperature is high and the vibrational DOSvaries noticeably. The corresponding corrections to the vibra-

tional free energy,

FETDVVi , T, are shown in Fig. 4a.FETDV is always positive. As volume decreases, it dimin-ishes and finally becomes negligible. ETDV originates in thethermal excitations of the electrons near the Fermi surfaceand the number of thermal excited electrons is proportionalto NEF in the lowest-order Sommerfeld expansion. Forsmaller volumes, the electronic bands are more dispersiveand NEF decreases, as shown in Fig. 4b. ETDV dimin-ishes accordingly.

Figures 5 and 6 show the volume thermal-expansion co-efficient , heat capacity at constant pressure CP, entropy S,and the temperature-dependent part of the Gibbs free energyGP , T = GP , T-GP , T=300 K. Including ETDV re-moves about half of the discrepancies between experimentsand calculations based on normal QHA. The remaining smalldifferences between theory and experiment are attributed toanharmonic phonon-phonon interactions43 and electron-phonon interactions.44 These two effects are of the same or-der of magnitude44 as Fele for metals. But explicit perturba-tive calculations to determine their magnitudes arecomputationally demanding and beyond the scope of the cur-rent paper. We notice DFT calculations based on QHA de-scribe well the thermal properties of other metals, such asgold,33 silver,45 and copper,46 up to melting point. This is incontrast with ionic crystals such as MgO where there arelarge deviations from QHA at high temperatures. It is pos-sible that the effects of anharmonic phonon-phonon and

electron-phonon interactions tend to cancel each other inthese metals. Further work is needed to clarify this issue.

C. Room-temperature isotherms

By fitting the total Helmholtz free energy at 300 K, we getthe theoretical 300 K isotherms, as shown in Fig. 7. Theirparameters are listed in Table II. In the low-pressure range,the LDA isotherm and the experimental data are almost par-allel. As pressure increases, they start to merge. It seemsLDA works better at high pressures. Regarding to EOS pa-rameters, LDA gives equilibrium volumes closest to the ex-periments while WC yields closest bulk modulus K0 and

0

50

100

150

200

250

300

10 11 12 13 14 15

Pressure(GPa)

Volume (3)

McQueenDewaele

HolmesLDA StaticPBE StaticWC Static

0

5

10

15

20

25

30

35

10 11 12 13 14 15

PPLDA,s

tatic

(GPa)

Volume (3)

McQueenDewaele

HolmesPBEWC

LDA+SO

(b)

(a)

FIG. 1. Static EOS computed by the LAPW method using vari-ous exchange-correlation functionals. a Pressure vs volumecurves. Spin-orbit effect is too small to be identified on this scale.b Pressure difference with respect to the static LDA EOS. In spiteof the relative large change in EOS parameters, the actual pressuredifference with and without spin-orbit effect is very small. Experi-mental data labeled as McQueen are from Ref. 1, Dewaelefrom Ref. 8, and Holmes from Ref. 4.


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the derivative of the bulk modulus K0. Some people8,35prefer to compare pressures from two EOS labeled as EOS-Iand EOS-II at the same compression, i.e., the same value ofV/V0. V0 is the corresponding equilibrium volume, V0,I forEOS-I, and V0,II for EOS-II. Such comparisons can give fa-vorable agreement when K0 and K0 of EOS-I are close tothose of EOS-II even when V0,I and V0,II are quite different.8

As mentioned before, the EOS parameters are not indepen-dent of each other. It can be fortuitous that K0 and K0 agreewell. Judged from pressure vs volume relation, LDA is theoptimal functional for platinum. It is worth noting that the

LDA Hartwigsen, Goedecker and Hutter HGH pressure vsvolume relation reported in Ref. 35 is similar to those ob-tained in this study. However, Ref. 35 presents data in vol-ume vs compression and concludes that LDA overestimatespressure by 8 GPa near 100 GPa. In fact, although K0,LDA291 GPa from this study is much larger than K0,expe 273.6GPa from Ref. 8, the bulk modulus computed at the experi-mental equilibrium volume V0,expe15.095

3 is 270 GPa,quite close to K0,expe. Thus when plotted in pressure vs vol-ume, the isotherm computed by LDA is nearly parallel withthe experimental data in the low-pressure range.

D. Thermal equation of state of platinum for pressure

calibration

In the previous sections the thermal properties of platinumis discussed from a pure theoretical point of view. We havecomputed the static lattice energy UV using LAPW andfound that spin-orbit interactions are not important in deter-mining the EOS of platinum. We have used QHA to calculatethe vibrational free energy FvibV, T and found that includ-ing ETDV improves the agreement on the thermal properties.We have calculated the electronic free energy FeleV, T us-ing Mermin functional. The resulting thermal properties, e.g.,the temperature-dependent part of the Gibbs energyGP , T, are close to the experimental data at 0 GPa. The

TABLE I. Static EOS parameters obtained from LAPW calculations and compared with those in litera-ture. Parameters from the pseudopotential calculations pseudo-1 and pseudo-2 are also listed. For conve-nience, both Vinet and fourth order BM parameters are shown. V0 denotes the equilibrium volume, and K0,K0, and K0 are the isothermal bulk modulus, the first and second derivatives of the bulk modulus at V0,respectively. Note their different fitting ranges: 0550 GPa this study, 01000 GPa Ref. 11, 0660 GPaRef. 12, 0150 GPa Ref. 35, and 0350 GPa Ref. 31. Reference 35 uses third order BM EOS so thecorresponding K0 are not listed.

Vinet B-M

V0 3 K0 GPa K0 V0

3 K0 GPa K0 K0 GPa1

LDA 14.752 308.02 5.446 14.761 309.29 5.295 0.02666

LDASO 14.784 301.17 5.533 14.785 301.13 5.510 0.03214

LDA pseudo-1 14.719 308.69 5.423 14.726 309.61 5.295 0.02681

LDA pseudo-2 15.055 297.48 5.515 15.060 299.28 5.375 0.02873

LDAa 14.90 300.9 5.814

LDAb 15.073 293 5.56

LDAc HGH 14.82 305.99 5.32

LDASOc,d TM 15.2 291.18 5.35

PBE 15.679 242.50 5.639 15.678 245.88 5.464 0.03620

PBESO 15.751 231.97 5.762 15.754 229.96 5.850 0.04932

PBEa 15.77 243.3 5.866

PBEc HGH 15.59 250.85 5.65

PBEe 15.69 248.9 5.43

WC 15.171 280.63 5.500 15.177 283.49 5.306 0.02889

WCSO 15.223 269.97 5.630 15.223 269.00 5.670 0.03893

aReference 11.bReference 12.cReference 35.dReference 40.eReference 31.

5

0

5

10

15

20

2530

35

10 11 12 13 14 15

PPLDA,LAPW

(GPa)

Volume (3)

McQueen

DewaeleHolmesXiang

pseudo1pseudo2

FIG. 2. Different LDA static EOS compared with the one com-puted by using LAPW. Xiang denotes the EOS from Ref. 11.


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room-temperature isotherm computed by LDA merges to thereduced shock data at high pressures, indicating LDA worksbetter at high pressures.

Based on these DFT results and all the available experi-mental data, we try to construct a consistent P-V-T EOS ofplatinum up to 550 GPa and 5000 K. To reach this goal, first

we need to include the physical effects that are missing inour original model. A phenomenological term39

FcorrV, T =32 kBaV/V0

mT2 is added to the total Helm-holtz energy to account for the anharmonic phonon-phononand electron-phonon interactions where V0 is the volume of aprimitive cell at ambient condition V0 =15.095

3. Thequadratic temperature dependence comes from the lowest-order perturbation at high temperatures. a and m are twoparameters to be fitted. We found that setting a as equal to105K1 and m as equal to 7 yields good agreement betweentheory and experiments on , CP, and GP , T at 0 GPa, asillustrated in Figs. 5a and 6a. The contribution to thermalpressure can be estimated by differentiating FcorrV, T with

respect to volume. At 2000 K, Pth is 0.38 GPa when V isequal to V0 and 0.2 GPa when V is equal to 0.9V0.Having obtained accurate GP , T, the next step is to get

reliable GP ,300 K. We choose the room-temperature EOSdeveloped by Dorogokupets and Oganov37 as our referencebelow 100 GPa. It has been cross-checked with other pres-sure scales and is likely to be more accurate than the reducedshock data of Ref. 1 in this pressure range. On the otherhand, extrapolating an EOS fitted at low pressures to higherrange can be dangerous. We assume LDA works better athigh pressures, and the difference between the exact ob-tained in an ideal, very accurate experiment and LDA iso-therms approaches to zero as pressure increases.

0

0.01

0.02

0.03

0.04

0.05

0 50 100 150 200

VibrationalDOS

Frequency (cm

1

)

0 K2000 K

0

50

100

150

200

Frequency(cm-1)

[000]

[001]

Dutton0 K

2000 K

[011] [000]

L

[ ]111---222

(a)

(b)

FIG. 3. a Phonon dispersion and b vibrational DOS at a=7.4136 a.u. Solid line corresponds to Tele =0 K and dashed linecorresponds to Tele =2000 K. They are calculated by using the LDApseudopotential. Experimental data labeled as Dutton are fromRef. 16.

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1000 2000 3000 4000 5000

FETDV

(mRy)

Temperature (K)

7.6 a.u.7.1 a.u.6.2 a.u.

0

0.5

1

1.5

2

2.5

3

-20 -10 0 10 20 30

ElectronicDOS

Energy (eV)

6.2 a.u.7.4136 a.u.

(b)

(a)

FIG. 4. a Corrections to the vibrational free energy at variouslattice constants. b Volume dependence of the electronic density ofstates.

0

1

2

3

4

5

6

0 500 1000 1500 2000

(10-5K)

Temperature (K)(a)

Expe0 GPa

0 GPa (Tele=0)0 GPa corrected

20 GPa40 GPa

0

1

2

3

4

5

0 100 200 300 400 500

(10-5K)

Pressure (GPa)(b)

300 K1000 K2000 K

FIG. 5. Color online Thermal expansivity as a function of atemperature and b pressure. Curves with label Tele =0 representproperties computed without ETDV, i.e., computed from Fvib

QHA.Curves without this specification are the default ones computedwith ETDV. 0 GPa corrected denotes the results obtained by add-ing a phenomenological correction to account for anharmonicphonon-phonon and electron-phonon interactions. These correctedthermal data are used to construct the final thermal EOS in TablesIII and IV. The experimental data are from Ref. 47.


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We compare Dorogokupetss EOS Ref. 37 V0=15.095 3, K0 =276.07 GPa, and K0=5.30 in Vinet formwith our room-temperature isotherm computed by LDA V0=14.884 3, K0 =291.25 GPa, and K0=5.547 in Vinetform. The volume difference between these two at eachpressure VP = Vexpe,300 KP VLDA,300 KP is shown inFig. 8. Since VP decreases rapidly as pressure increases,we use exponentially decaying functions to fit and extrapo-late. We correct the calculated room-temperature Gibbs en-ergy GLDAP ,300 K by setting GcorrP ,300 K= GLDAP ,300 K + 0

PVPdP. The isotherm derived from

GcorrP ,300 K coincides with Dorogokupets EOS below100 GPa, which is the upper limit of their fitting. Above 250

GPa, VP is almost zero and the isotherm derived fromGcorrP ,300 K is the same as the uncorrected one. The un-certainty due to volume extrapolation in the intermediate re-

gion 100250 GPa is estimated from bulk modulus to beless than 2 GPa. It is worth noting that the established EOSof platinum are quite similar to each other below 100 GPa, asshown in Fig. 7b. Choosing a different reference such asthe one in Ref. 8 will only change the results near 100 GPaby 1 GPa.

We combine GcorrP ,300 K with the temperature-dependent part of the Gibbs energy GP , T and get thecorrected Gibbs energy GcorrP , T at temperature T. FromGcorrP , T, we derive all the other thermodynamical proper-ties. Thermal properties such as , CP, and S, which dependon the temperature derivatives of the Gibbs energy, are notaffected by changing GP ,300 K. In contrast, the isother-

mal bulk modulus KT and adiabatic bulk modulus KS will beinfluenced, as shown in Fig. 9.After corrections, both thermal expansivity and bulk

modulus agree with the experiments well. We expect theproduct KT to be accurate. Integrating KT, we get the ther-mal pressure, PthV, T = PV, T PV, T0 = T0

TKTdT. The

calculated KT and Pth, both before and after corrections, areshown in Fig. 10. PthV, T is often assumed to be indepen-dent of volume and linear in temperature, i.e., KT is a con-stant. Reference 1 assumes the thermal energy ET =3kBT,the thermal Grneisen parameter =0V/V0, where 0 =2.4,and V0 =15.123

3. The thermal pressure is obtained fromMie-Grneisen relation,

0

10

20

30

40

50

0 500 1000 1500 2000

HeatC

apacityCp(J/molK)

Temperature (K)(a)

Expe0 GPa

0 GPa (Tele=0)0 GPa corrected

20 GPa40 GPa

0

20

40

60

80

100

120

0 500 1000 1500 2000

EntropyS(J/molK)

Temperature (K)(b)

Expe0 GPa

0 GPa (Tele=0)20 GPa40 GPa

0

20

40

60

80

100

120

140

0 500 1000 1500 2000

G

(T=300K)-G(T)(kJ/mol)

Temperature (K)(c)

Expe0 GPa

0 GPa (Tele=0)20 GPa40 GPa

FIG. 6. Color online Thermal properties of platinum. a Heatcapacity at constant pressure, b entropy, and c temperature de-pendence of the Gibbs free energy at constant pressure. The mean-ings of the labels are the same as above. On this scale it is difficultto discern the improvements on entropy and Gibbs free energy,caused by including the phenomenological correction on anhar-monic phonon-phonon and electron-phonon interactions. Thus thosedata are not shown. The experimental data are from Ref. 48.

0

50

100

150

200

250

300

10 11 12 13 14 15

Pressure(GPa)

Volume (3)

McQueenDewaele

HolmesLDA 300 KWC 300 K

2

2

6

10

14

18

22

10 11 12 13 14 15

PPLDA,3

00K

(GPa)

Volume (3)

McQueenDewaele

HolmesDorogokupets

WC 300 K

(b)

(a)

FIG. 7. 300 K isotherms. a Pressure vs volume. b Pressuredifference. Dorogokupets denotes EOS from Ref. 37, which isvery close to the reduced shock EOS McQueen near 100 GPa.PBE is not plotted as its EOS is way off.


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PthV,T =ETV

V=

3kB0V0

T= 6.57 103 TGPa .

5

In Ref. 4, KT is estimated to be 6.94103 GPa/K. Both

values lie within the variation of the calculated KT, asshown in Fig. 10a. We find that KTPth has noticeable

volume dependence. At fixed temperature, it first decreases,reaches a minimum at about V/V0 =0.8, and then increases.Such behavior originates in the pressure dependence of thethermal expansivity Fig. 5b and bulk moduli Fig. 9b.This feature has also been observed in Ref. 37, as shown inFig. 10b. However it is missing in the previous ab initiocalculation.11

TABLE II. EOS parameters of the theoretical 300 K isotherms, compared with the experiments. V0,expe is15.095 3 . Pressure range: 0550 GPa this study, 0660 GPa Refs. 4 and 12, 094 GPa Refs. 8 and 37,and 0270 GPa Ref. 1.

Vinet B-M

V0 3 K0 GPa K0 V0

3 K0 GPa K0 K0 GPa1

LDA 14.884 291.25 5.547 14.886 291.65 5.496 0.03232LDA V0,expe 269.96 5.640 269.91 5.626 0.03730

LDAa 15.188 281 5.61

PBE 15.864 225.55 5.751 15.866 225.34 5.741 0.04709

WC 15.322 263.93 5.601 15.325 264.72 5.530 0.03580

Holmesb 15.10 266 5.81

Dewaelec 15.095 273.6 5.23

Dorogokupetsd 15.095 276.07 5.30

McQueene 15.123 277.715 4.821 0.01379

aReference 12.bReference 4.cReference 8. When K0 is set to 277 GPa, the value measured by ultrasonic experiments, K0 is equal to 5.08

GPa.dReference 37, improved analysis using data from Ref. 8.eReference 1. Fitted from the tabulated shock reduced isotherm at 293 K.

0.05

0

0.05

0.1

0.15

0.2

0.25

0 50 100 150 200 250

VVLDA

,300K

(3)

Pressure (GPa)

DorogokupetsExtrapolation

DewaeleMcQueen

FIG. 8. Volume correction to the theoretical isothermat 300 K. We use VP=0.1215expP /34.0846+0.0885expP /109.9892 to fit the volume difference in therange P100 GPa. The exponential functional form guaranteesthat it approaches zero at high pressures. It happens that this ex-trapolation agrees well with McQueens reduced shock data.

150

200

250

300

350

400

0 200 400 600 800 1000 1200 1400

BulkModuli(GPa)

Temperature (K)

KSKS uncorrected

KTCollard

Dorogokupets

0

500

1000

1500

2000

2500

0 100 200 300 400 500

B

ulkModuli(GPa)

Pressure (GPa)

KS 300 KKS 2000 KKT 2000 K

(b)

(a)

FIG. 9. a Bulk moduli at 0 GPa. Uncorrected denotes theraw DFT results without any correction. As noted in Ref. 37, theexperimental data in Ref. 49 is inconsistent. The data points weshow here are digitized from the graphs in Refs. 49 denoted asCollard and 37 denoted as Dorogokupets, respectively. KS de-duced from the corrected Gibbs free energy agrees well with theone computed from empirical models in Ref. 37. b Bulk moduli asa function of pressure.


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Thermal Grneisen parameter =KTV

CVis an important

quantity. Empirically it is often assumed to be independent oftemperature. Its volume dependence is described by a param-

eter q = ln ln V and can be represented in q as =0

V

V0 q

.From Mie-Grneisen relation, it is obvious that q is related tothe volume dependence of KT. If q is equal to 1, KT isindependent of volume. If q is greater than 1, KT getssmaller as volume decreases. In Ref. 1 q is assumed to beequal to 1. Fei et al.9 determined by fitting the measuredP V T data to the Mie-Grneisen relation up to 27 GPa.They gave 0 =2.72 and q =0.5. Zha et al.10 extended

measurements to 80 GPa and 1900 K. Their fit gave 0=2.75 and q =0.25. Our calculation indicates that the tem-perature dependence of is small. The volume dependenceof is shown in Fig. 11. The uncorrected DFT calculationtends to overestimate . At ambient condition 0 is equal to2.87. After corrections, 0 =2.70. The corresponding q isequal to 2.35, much larger than the value obtained in Refs. 9

and 10. We notice previous DFT calculation on gold33

alsogives much larger q than the value in Ref. 9. This is probablydue to the small pressure range explored in Ref. 9 and lim-ited number of data points measured in Ref. 10.

Adding the thermal pressure to the room-temperature iso-therm, we get the thermal EOS of platinum, as shown inTable III. We compare our results with two DAC measure-ments in Fig. 12. Within the error of the experiments, the

TABLE III. Pressure in GPa as a function of compression 1-V/V0,V0 is the experimental volume at ambient condition; 15.095 3 and

temperature in K, deduced from the corrected Gibbs free energy. Pressures above melting point is not shown. The melting point isdetermined using TmP=2057+27.2P 0.1497P

2 K up to 70 GPa from Ref. 42

1-V/V0 300 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0.00 0.00 1.51 5.31 9.14 12.97

0.05 16.22 17.62 21.20 24.83 28.48 32.16

0.10 38.32 39.64 43.05 46.52 50.03 53.57 57.14

0.15 68.41 69.67 72.97 76.34 79.77 83.24 86.73 90.25

0.20 109.46 110.71 113.99 117.37 120.82 124.31 127.83 131.39 134.98

0.25 166.45 167.74 171.15 174.69 178.29 181.94 185.63 189.36 193.13 196.94 200.81

0.30 247.37 248.72 252.34 256.07 259.88 263.73 267.64 271.58 275.57 279.58 283.63

0.35 362.30 363.65 367.31 371.10 374.98 378.91 382.90 386.94 391.01 395.12 399.25

0.40 525.86 526.93 530.04 533.39 536.86 540.40 543.99 547.62 551.28 554.99 558.76

5

6

7

8

9

0.6 0.7 0.8 0.9 1

K(103GPaK1)

V/V0

300 K300 K uncorrected

1000 K3000 K

0

5

10

15

20

25

0 500 1000 1500 2000 2500

ThermalPressure(GPa)

Temperature (K)

V/V0=1.0V/V0=1.0, uncorrected

V/V0=0.8V/V0=0.7V/V0=1.0V/V0=0.8V/V0=0.7

(b)

(a)

FIG. 10. a Temperature derivative ofPth, KT=Pth

T, before and

after corrections. b Thermal pressure Pth at different V/V0 whereV0 is the experimental volume at ambient condition 15.095

3.Points are the thermal pressures from Ref. 37.

0

0.5

1

1.5

2

2.5

3

3.5

0.6 0.7 0.8 0.9 1

Gr

neisenParameter

V/V0

300 K

300 K uncorrected

2000 K

0

1

2

3

4

0.6 0.7 0.8 0.9 1

q

V/V0

300 K

300 K uncorrected

2000 K

(b)

(a)

FIG. 11. Thermal Grneisen parameter a volume dependenceat fixed temperature. b The corresponding q.


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agreement is reasonably good. For convenience of interpola-tion, parametric forms of the thermal EOS are listed in TableIV.

The P-V-T thermal EOS we obtained are very similar tothe one in Ref. 37 below 100 GPa. This is to be expected aswe used the 300 K isotherm in Ref. 37 as the reference tocorrect the room-temperature Gibbs energy and the thermalproperties calculated from both approaches agree well withthe experiments. In this P-T range, the uncertainty of ourEOS is comparable to the one in Ref. 37. Above 100 GPa,the uncertainty is about 1.4%, which is the difference be-tween the LAPW LDA and pseudo-1 static EOS. Othersources of error, e.g., convergence uncertainty 0.5% andignoring spin-orbit effect 0.7% are smaller effects. Tocheck the accuracy of our thermal EOS at high pressures, westart from the corrected Gibbs energy and compute the the-

oretical Hugoniot by solving the Rankine-Hugoniot equa-tion:

EHV,T EiV0,Ti = PHV,T + P0V0,TiV0 V

2, 6

where EHV, T and PHV, T are the internal energy, pres-sure at volume V, and temperature T. EiV0 , Ti andP0V0 , Ti are the internal energy, pressure at the initial vol-ume V0, and temperature Ti. The results are shown in Fig. 13.The predicted Hugoniot pressure is in good agreement withmeasurements. The temperature predicted by DFT is lowerthan the empirically deduced value in Ref. 1. The reduction

in Ref. 1 neglects the electronic thermal pressure and thismay cause the overestimation of Hugoniot temperature.4

We end this section by comparing our room-temperatureisotherm with that of Holmes et al. Below 70 GPa, they arealmost identical. At high pressures 200550 GPa, the pres-

sure from our EOS is about 3% lower than the one fromHolmes et al. Holmes et al. used LMTO with the atomic-sphere approximation to get the static EOS. In principle, thefull potential LAPW method used in this study is more ac-curate. It seems the EOS of Holmes et al. overestimates pres-sure systematically at high compression ratio. However, themagnitude is much smaller than the pressure offset needed tocompensate the discrepancy between Mao et al.s experimentand seismological extrapolation. The real cause of the dis-crepancy might be a combination of several factors.

V. CONCLUSIONS

In this paper, we report our calculations on the static andthermal EOSs of platinum using DFT with differentexchange-correlation functionals. Contrary to previous re-ports, we found that the room-temperature isotherm com-puted with LDA lies below and nearly parallel to the experi-mental compression data. We studied the lattice dynamics ofplatinum within QHA and found that the electronic tempera-ture dependence of vibrations plays a noticeable role in de-termining the thermal properties of platinum. Combining theexperimental data with DFT calculations, we propose a con-sistent thermal EOS of platinum up to 550 GPa and 5000 K,which can be used as a reference for pressure calibration.

13.5

14

14.5

15

15.5

16

16.5

0 5 10 15 20 25 30

Volume(3)

Pressure (GPa)

300 K1473 K1673 K1873 K

300 K, Expe1473 K, Expe1673 K, Expe1873 K, Expe

12.5

13

13.5

14

14.5

15

15.5

16

0 10 20 30 40 50 60 70 80

Volume(3)

Pressure (GPa)

300 K1300 K1900 K

300 K, Expe1300 K, Expe1900 K, Expe

(b)

(a)

FIG. 12. High temperature isotherms after corrections. Linescorrespond to the calculated isotherms. The experimental data in aare taken from Ref. 15 except those at P =0 GPa, which are ob-tained by integrating the thermal expansivity listed in Ref. 47. Datain b are from Ref. 10.

0

50

100

150

200

250

300

10.5 11.5 12.5 13.5 14.5 15.5

Pressure(GPa)

Volume (3)

TheoryMcQueen

0

1000

2000

3000

4000

5000

10.5 11.5 12.5 13.5 14.5 15.5

Temperature(K)

Volume (3)

TheoryMcQueen

(b)

(a)

FIG. 13. a Theoretical shock Hugoniot compared with the ex-perimental data from McQueen et al. Ref. 1. b Temperaturealong the Hugoniot.


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ACKNOWLEDGMENTS

We thank P. B. Allen, P. I. Dorogokupets, A. Floris, and B.B. Karki for discussions and help. We also thank A. Dewaelefor suggestions and taking Ref. 39 to our attention, and Y. W.

Fei for sending us Ref. 10. We are indebted to the anony-mous referees for careful reviews. The pseudopotential cal-culations were performed at the Minnesota Supercomputing

Institute MSI with the QUANTUM ESPRESSO package http://www.pwscf.org. The LAPW calculations were performed atBrookhaven National Laboratory BNL with the WIEN2Kpackage http://www.wien2k.at . T.S. was supported by NSF/ITR Grant No. ATM0426757. R.M.W., K.U., and Z.W. weresupported by NSF/EAR Grants No. 0230319 and No.0635990, and NSF/ITR Grant No. 0428774 VLab.

*[email protected] R. G. McQueen, S. P. Marsh, J. W. Taylor, J. M. Fritz, and W. J.

Carter, in High Velocity Impact Phenomena, edited by R. Kin-slow Academic, New York, 1970, Chap. 7.

2 J. A. Morgan, High Temp. - High Press. 6, 195 1974.3 J. C. Jamieson, J. N. Fritz, and M. H. Manghnani, in High-

Pressure Research in Geophysics, edited by S. Akimoto and M.H. Manghnani Center for Academic, Tokyo, 1982.

4 N. C. Holmes, J. A. Moriarty, G. R. Gathers, and W. J. Nellis, J.Appl. Phys. 66, 2962 1989.

5 H. K. Mao, Y. Wu, L. C. Chen, and J. F. Shu, J. Geophys. Res.95, 21737 1990.

6 F. D. Stacey and P. M. Davis, Phys. Earth Planet. Inter. 142, 1372004.

7 A. K. Singh, Phys. Earth Planet. Inter. 164, 75 2007.8 A. Dewaele, P. Loubeyre, and M. Mezouar, Phys. Rev. B 70,

094112 2004.9 Y. W. Fei, A. Ricolleau, M. Frank, K. Mibe, G. Shen, and V.

Prakapenka, Proc. Natl. Acad. Sci. U.S.A. 104, 9182 2007.

10 C. S. Zha, K. Mibe, W. A. Bassett, O. Tschauner, H. K. Mao, andR. J. Hemley, J. Appl. Phys. 103, 054908 2008.

11S. K. Xiang, L. C. Cai, Y. Bi, F. Q. Jing, and S. J. Wang, Phys.Rev. B 72, 184102 2005.

12 E. Menndez-Proupin and A. K. Singh, Phys. Rev. B 76, 0541172007.

13 J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 1981.14 There are two LDA ultrasoft Rappe-Rabe-Kaxiras-Joannopoulospseudopotentials on the PWSCF www.pwscf.org website.Pt.pz-rrkjus.UPF is the one without nonlinear core correction.Pt.pz-nd-rrkjus.UPF includes this correction. Both have a cut-off radius of 2.6 a.u. and both yield static EOS stiffer than thatof N. C. Holmes et al. Ref. 4. The results reported in Ref. 12are very close to our calculations using Pt.pz-nd-rrkjus.UPF.

15 Y. W. Fei, J. Li, K. Hirose, W. Minarik, J. V. Orman, C. Sanloup,W. V. Westrenen, T. Komabayashi, and K. Funakoshi, Phys.Earth Planet. Inter. 143-144, 515 2004.

16 D. H. Dutton, B. N. Brockhouse, and A. P. Miller, Can. J. Phys.50, 2915 1972.

TABLE IV. Parametric form of the thermal EOS, PV = P0 +32 K0V/V0

73 V/V0

53 1+

34 K0

4V/V0 2

3 1+38 K0K0+ K03K04+

359 V/V0

23 12. At high temperatures, the equilibrium vol-

ume may exceed the largest volume we calculated. For better accuracy we fit the P-V-T data in three differentpressure-temperature intervals: 1 0100 GPa and 02000 K, 2 50250 GPa and 03000 K, and 3150550 GPa and 05000 K. P0 denotes the starting pressure of the corresponding interval. V0 , K0, K0, andK0 are temperature-dependent parameters, and are fitted to a fourth order polynomial a0 + a1t+ a2t

2 + a3t3

+ a4t4, where t= T/1000. They have a formal correspondence to the usual fourth order BM EOS parameters,

which are defined at P0 =0 GPa.

1 a0 a0 a1 a2 a3

V0 3 14.9924 0.295837 0.194441 0.0917141 0.024365

K0 GPa 290.539 45.4082 9.38792 5.09573 1.40266

K0 5.11956 0.52903 0.0733263 0.0195011 0.0229666

K0 GPa1 0.0275729 0.0120014 0.0114928 0.00672243 0.00359317

2 a0 a1 a2 a3 a4V0

3 13.2246 0.128227 0.049052 0.0160359 0.00241857

K0 GPa 523.48 30.3849 3.86087 1.31313 0.222027

K0 4.24183 0.217262 0.0235333 0.00944835 0.000371746

K0

GPa1

0.00125873 0.00268918 2.13874e 05 3.57657e 05 1.75847e 053 a0 a1 a2 a3 a4V0

3 11.4929 0.0672156 0.0119585 0.00243269 0.000219022

K0 GPa 951.004 21.0874 2.84254 0.654708 0.0639296

K0 4.31383 0.05775 0.00505386 0.00245414 0.000167453

K0 GPa1 0.00588145 0.00130468 0.000221904 6.51359e 05 4.99978e 06


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17 P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J.Luitz, in WIEN2k: An Augmented Plane Wave and Local Orbit-als Program for Calculating Crystal Properties, edited by K.Schwarz Vienna University of Technology, Vienna, Austria,2001.

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19

Z. Wu and R. E. Cohen, Phys. Rev. B 73, 235116 2006.20 H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 1976.21 P. E. Blchl, O. Jepsen, and O. K. Andersen, Phys. Rev. B 49,

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1982.24 M. Methfessel and A. T. Paxton, Phys. Rev. B 40, 3616 1989.25 N. D. Mermin, Phys. Rev. 137, A1441 1965.26 R. M. Wentzcovitch, J. L. Martins, and P. B. Allen, Phys. Rev. B

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2007.40 This Troullier-Martins TM pseudopotential is constructed from

5d96s0.956p0.05 atomic configuration. The cut-off radius of 6s is2.6 a.u., which is probably too large. The HGH pseudopotentialis constructed from 5s25d65d10 and does not have this problem.

41 A. Dal Corso and A. Mosca Conte, Phys. Rev. B 71, 1151062005.

42 A. Kavner and R. Jeanloz, J. Appl. Phys. 83, 7553 1998.43 R. A. Cowley, Rep. Prog. Phys. 31, 123 1968.44

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1992.


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