Elastic constants and volume changes associated with two high-pressure rhombohedral phase transformations in vanadium

0.18 MB·English

by  B. Lee

Elastic constants and volume changes associated with two high-pressure rhombohedral
phase transformations in vanadium
∗
Byeongchan Lee, Robert E. Rudd, John E. Klepeis, and Richard Becker
Lawrence Livermore National Laboratory, Livermore, California 94551
(Dated: February 2, 2008)
We present results from ab initio calculations of the mechanical properties of the rhombohedral
phase (β) of vanadium metal reported in recent experiments, and other predicted high-pressure
phases(γandbcc),focusingonpropertiesrelevanttodynamicexperiments. Wefindthatthevolume
8 change associated with these transitions is small: no more than 0.15% (for β – γ). Calculations of
0 thesinglecrystalandpolycrystalelasticmoduli(stress-straincoefficients)revealaremarkablysmall
0 discontinuityintheshearmodulusandotherelastic properties across thephasetransitions evenat
2 zero temperature where thetransitions are first order.

n
a PACSnumbers: 62.50.+p,61.50.Ks,31.15.Ar,62.20.Dc
J

8
I. INTRODUCTION to be weakened by increased temperature. Recent ramp
1
wavetechniquesbasedonZ,8laser9,10andgraded-density
] impactor11drivesareabletogeneratehighpressurewith-
i The existence of a high-pressure rhombohedral phase
c outthe entropygenerationofshockwavetechniques,and
s of pure crystalline vanadium has been the focus of an are therefore preferable in the present context. An-
- intense research effort recently. The first indication of
l other challenge is that the subtle rhombohedral distor-
r a phase transition came from the theoretical observa- ◦
t tion (< 1 ) detected by x-ray diffraction in the DAC is
m tion that the C44 shear modulus of bcc vanadium de- probablytoosmallforin-situx-raydiffractionindynamic
. creases and becomes negative at pressures greater than experiments.12 Indirect techniques are an alternative to
t ∼1.3 Mbar,1,2 pressures that are experimentally acces-
a detect the transition. For example, VISAR free-surface
m sible. A negative shear modulus means the material is velocity measurements can detect changesin the density
mechanically unstable under trigonal (prismatic) shear,
- due to a volume change, and they can be used to infer
d suggesting a phase transition. At that time the ex-
the longitudinalstress,andhence the changein strength
n perimental evidence showed no phase transition up to if the equation of state is known.10,13 Rayleigh-Taylor
o 1.54 Mbar.3 Then recently, Mao and coworkers4 con- growth rate is another way to probe strength.9
c
ductedx-raydiffractionexperimentsinthediamondanvil
[ In this article we use DFT to make predictions about
cell (DAC) up to 1.5 Mbar and found features in the
the properties of high-pressurevanadium relevant to dy-
2 diffraction peaks that were consistent with a second-
namic experiments. We compute the magnitude of the
v order phase transformation to a rhombohedral struc-
volume change associated with the three phase tran-
3 ture with an R¯3m point group symmetry at pressures
8 sitions related to the rhombohedral structure in Sec-
above 0.69 Mbar. It was soon confirmed that density
3 tion III. We also compute the elastic properties and cal-
4 functional theory (DFT) finds the rhombohedral phase culate bounds on, and an explicit estimate of, the poly-
1. to be the ground state at zero temperature and pres- crystalline shear modulus in Section IV and Section V
suresabove0.8Mbar,inreasonablygoodagreementwith
1 respectively. Since the strength is typically assumed to
7 experiment.5 Infact,itwasshownthatDFTpredictsad- vary with the shear modulus,14 any anomalies in the
0 ditional phase transformations that had not been found shear modulus are likely to provide a signature in the
v: in experiment, i.e. a first-order transformation to a dif- VISAR trace. Indeed, an important motivation for the
ferent rhombohedral structure at 1.2 Mbar and a third
i present work is to assess whether the bcc shear modulus
X transformation back to the bcc structure at 2.8 Mbar.5
C going to zero is likely to produce a strong signa-
44
r The prediction of the existence of the two high pressure ture. The shear modulus also affects defect energetics,
a
phase transformations has been subsequently confirmed
and may have a measurable effect on transition kinetics.
with DFT phonon calculations.6
We consider the implications of our results for dynamic
Alternative techniques may provide the pressures experiments to detect the high-pressure phases.
needed to observe the second rhombohedral phase and
the reentrant bcc phase. Dynamic experiments do not
rely on the mechanical integrity of anvils and are able II. THEORETICAL BACKGROUND
to reach multi-megabar pressures. They have been used
to study similar transformations such as the diffusion- The rhombohedral crystal structure of vanadium at
less α-ε transition in iron.7 There are several challenges high pressure results from a slight distortion of the bcc
specific to vanadium, however. The softening of the structure. Specifically the distortion is a uniaxial strain
shear modulus and the rhombohedral phase transition alongh111i,whichremainsathree-foldsymmetryaxisof
are related to subtle electronic effects,2 which are likely thecrystal. Thisstructureisknownastheβ-Postructure
2
(Strukturbericht A, Pearson hR1). It still has a single
i
TABLEI:Equationsofstateforthebcc,β andγ phasesand
atom per unit cell, so the rhombohedral transition may
metastable structures. Pressures are in Mbar, and volumes
be expected to be diffusionless (martensitic) and likely are in unitsof the ambient volume Vo=13.518 ˚A3.
rapid despite the small energy difference. There are four
independent three-foldaxes,so there arefour variantsof Volume Pressure
the rhombohedral crystal that are degenerate in energy. bcc β γ
The ground state of the single-crystal rhombohedral
1.000 0.000 - -
phase has been determined from first principles using a
0.831 0.479 - -
volume-conserving rhombohedral shear path,5
0.804 0.596 - -
k δ δ 0.779 0.726 0.724 -
T(δ)= δ k δ  (1) 0.759 0.840 0.838 -
 δ δ k  0.754 0.870 0.869 -
0.729 1.031 1.030 -
intheusualbcccrystalframewherekisdeterminedfrom
0.717 1.118 1.117 1.110
therealpositivesolutionofdet(T)=1toinsureconstant
0.707 1.191 1.190 1.183
volume. The approach is to use DFT in combination
0.705 1.210 1.209 1.202
with a gradient-corrected exchange and correlation en-
ergyfunctional15 asimplementedintheViennaAb-initio 0.681 1.408 1.407 1.399
Simulation Package (VASP) code along with the projec- 0.659 1.627 1.627 1.620
tor augmented-wave (PAW) method.16 Specifically, the 0.636 1.869 1.870 1.866
PAW potentials with 13 valence electrons (3s, 3p, 3d, 0.614 2.136 2.138 2.140
and 4s states) are used. The planewave cutoff energy is 0.593 2.430 2.433 2.445
66.15 Ry and an unshifted 60×60×60 uniform mesh is 0.588 2.494 - 2.510
used for the k-point sampling: this results in 5216 and
0.572 2.769 - 2.782
18941kpointsintheirreducibleBrillouinzoneoftheun-
0.568 2.841 - 2.852
strainedbcc andrhombohedrallattices,andup to 18941
0.551 3.149 - -
and 54932 k points for the strained bcc and rhombohe-
dral lattices respectively. For all of the calculations we
use a primitive cell. 0.2
Stable β
)
% Metastable β
III. VOLUME CHANGE DUE TO e ( 0.1 Stable γ
TRANSFORMATION ng Metastable γ
a
h
c
e
In Ref. 5, we calculated the enthalpy and pressure m 0
u
as functions of strain along the rhombohedral deforma- ol
v
tion path, and used the enthalpy to find any stable or e
metastable crystal structures. We noted that the equa- ativ-0.1
tionsofstate(EOS)forthebccandrhombohedralstruc- el
R
turesarenearlyidentical,sotheirbulkmoduliareessen- bcc β γ bcc
tiallyequal(differingbynomorethan3%),andreported -0.2
0 1 2 3
the EOS of the ground state up to 2 Mbar. We now use Pressure (Mbar)
those data together with additional data on the EOS of
themetastablestructurestocalculatethevolumechange FIG. 1: (Color online) The volumetric strain ∆V/V of the
associatedwiththephasetransformationsinareadilyac- ground state of single-crystal vanadium at zero temperature
cessible form. with respect to that of the bcc structure. Dashed lines cor-
Using the EOS P (V) for the stable and metastable respondtothreevolumechangesassociated with threephase
i
structures (i = bcc,β,γ), we have calculated the associ- transitions. Thelight(dark)grayarearepresentsthepressure
rangeinwhichtheβ(γ)phaseisthegroundstaterespectively.
ated volume change ∆V according to
The bccphase is stable in the otherregions.
P (V +∆V)=P (V ) (2)
j i i i
forpairsofstructuresiandj. Inpractice,wehavecalcu- changewithrespecttothebccphase,∆V/Vbcc,isplotted
latedthepressureatasetofvolumesandusedpiece-wise in Fig. 1.
quadraticinterpolatationtosolvetheequalpressurecon- Inprinciple,thereisavolumechangeduringthetrans-
dition (2) between those points, equivalent to the com- formation from bcc (α) to the first rhombohedral phase
mon tangent construction at the phase boundaries. The (β), and a second volume change associated with the
EOS data are tabulated in Table I. The relative volume transformation to the second rhombohedral phase (γ),
3
and a third volume change associated with the transfor- B β γ
12 11 1.2
mation back to bcc (α) at high pressure. There is no B
12
B
psptfohoaereatmthrhkaetictnyooieonbmtnniecuestccsototoinnbfbngeetehβcdoetivesatetdnrirnaddancrtγtsivfifpotenhhrnimaatetssaoetptseir;tomehtnshapemteervrtataewhytseoutcrhabieneuc.i2cstpeiIaornetlinhgpptierohgantarcrssotaeiauncppseis--,, moduli (Mbar) 69 BBB12343434 00..69moduli (TPa)
ruenttaiilntehdeinneawmphetaassetahbalseastcahtaenpcaesttothneucplehaatseeabnodungdroawry. Elastic 3 0.3Elastic
For this reason it is interesting to examine the entire
curve in Fig. 1, and not just the values in between the
equilibrium phase boundaries. 0 0
(a)0 1 2 3
In each case the initial ∆V is a volume change, so the Pressure (Mbar)
vchoalunmgee aisssroecdiuacteedd fwoiltlohwtihneg tbhcecttroanβsittiroann.sfTorhmeavtoiolunmies ar)3 β γ 100
b
M
small, 0.03% or less in magnitude. It would not be easy
to detect such a small change in a dynamic experiment. ues ( Anisotropy ratio -1A)
Themagnitudeofthevolumechangeassociatedwiththe val2 o (
secondtransformationislarger: about0.15%fortheβ to en 10 ati
γbetraabnosufotrmthaetisoanmaeti1f.2bcMcbwaerr.eTrheteavinoeludmteocahapnregseswuroeulodf ulus eig otropy r
∼1.2 Mbar and then transformed directly to γ. This mod1 Anis
However, if the bcc or β phase persists to higher pres- c
sures, the volume change becomes progressively smaller asti 1
El
and eventually changes sign, becoming a volume expan-
0
sion near 2 Mbar. The final transitionback to bcc again (b)0 1 2 3
Pressure (Mbar)
has a change of over 0.1% in magnitude. So the β to
γ transformation has the strongest signature in terms of FIG. 2: (Color online) The elastic moduli (specifically, the
volumechange,butitmaynotbelargeenoughtodetect. stress-strain coefficients) of the ground state single-crystal
structureas afunction ofpressure; (a) B written in Voigt
ijkl
notationintheframeoftheprimitiverhombohedralcell,and
(b)thecorrespondingeigenvaluesofthe9×9stress-strainco-
IV. SINGLE CRYSTAL ELASTIC MODULI
efficient matrix B (see text). In the rhombohedral phase
ijkl
there are six independent elastic moduli (stress-strain coef-
We next considerhow the single crystalelastic moduli ficients) (vs. three for bcc),17 and three independent shear
changewithpressure. Specifically,wecalculateBijkl(P), eigenvaluesforvanadiummetal(dashedcurvesrepresentdou-
the elastic moduli with respect to a shear-stress-freeref- blydegenerateeigenvalues). TheanisotropyratioA−1 isalso
erence state at pressure P (either the bcc or rhombohe- plotted,showinganextremevaluejustbelowtheγ–bcctran-
dral structure, as specified). The B are often called sition.
ijkl
the stress-straincoefficients, as we do below.17 They are
directlyrelatedtosoundvelocitiesathighpressure. B
ijkl
can be obtained fromthe deformation paths used for or- is not readily apparent from the elements of Bijkl. The
thorhombiclattice18 orfortrigonallattice,19 andthe de- bulk modulus of the rhombohedral phases is within 3%
tails are given in the appendix. of that of the bcc structure, as we already discussed.
The six independent stress-strain coefficients, B11, The eigenvalues of the 9×9 matrix B(ij)(kl) provide a
B , B , B , B , and B (here given in Voigt nota- description of the elasticity that is less coordinate de-
33 12 13 44 24
tion in the rhombohedral frame with directions [¯110] , pendent, but there is a technical issue. In the rhombo-
bcc
[¯1¯12] and[111] as1,2and3,respectively),areplot- hedral phase, shear and compression are mixed in the
bcc bcc
ted in Fig. 2(a). They are discontinuous at the first- sense that a non-equiaxed strain is required to produce
order phase transitions; however, within the domain of purely hydrostatic pressure, and the tetragonal strains
each stable phase, most of the stress-strain coefficients to produce hydrostatic pressure and pure shear are not
increase monotonically with pressure. The exception is orthogonal. To eliminate any ambiguity, we restrict to
B neartheγ toreentrantbccphaseboundary(roughly the space of constant volume strains using a projection
33
2.8 Mbar). Since B33 is associated with uniaxial strain matrix Π(ij)(kl) =δ(ij)(kl) − 31δijδkl. Then the 9×9 ma-
along the three-fold axis, its anomalous behavior is sug- trix ΠBΠ/2 has 5 nontrivial eigenvalues, corresponding
gestive, but a better presentation is needed to separate to different shear moduli. This matrix is closely related
the effects of shear and compression. We turn to it now. to von Mises stresses.
Indeed, there is a remarkable approximate continuity TheeigenvaluesareplottedinFig.2(b). Thetopcurve
ofthe elasticpropertiesacrossthe phasetransitionsthat represents two degenerate eigenvalues that are equal to
4
′
B = (B −B )/2 in the bcc phase, the usual shear
11 12 1.5
modulus for tetragonal shear in the cubic crystal. It is
quite smooth. The remaining three eigenvalues are de-
generate in the bcc phase and equal to B44, the shear bar) 1
modulusfortrigonalshearinthecubiccrystal(nottobe M
confused with the B44 in the rhombohedral frame). In uli ( 0.5
d
therhombohedralphasestwooftheseeigenvaluesremain o
degeneratebutonesplits off. Thatsingleeigenvaluerep- ar m 0 bcc β γ bcc
resents a pure shear corresponding to the rhombohedral She-0.5 bcc Voigt
deformation. Its value is (B +2B +B −4B )/6 bcc Reuss
11 33 12 13 Rhombohedral Voigt
whichdecreasestowardzeroasthepressureintherhom- Rhombohedral Reuss
-1 Virtual Test Sample (VTS)
bohedral phases approaches the bcc phase boundary.
This decrease is most pronounced approaching the high- 0 1 2 3
Pressure (Mbar)
pressurereentrantbccphase(2.8Mbar),butitispresent
at both. In the energy curves, it is clear that the width
FIG. 3: (Color online) The polycrystalline shear modulus
of the rhombohedral well is broadening with the change
as a function of pressure, along with the Voigt and Reuss
in pressure as it rises above the bcc well and quickly be- boundsforbcc,β andγ. Thisshearmodulusisbasedonthe
comesunstable. Bythesametoken,thesingleeigenvalue stress-strain coefficients B of the single-crystal structure
ijkl
reachesitsmaximumat1.87Mbar,thepressurethatthe (see Fig. 2 (a)), and calculated from the virtual test sample
rhombohedralwellisdeepestandmoststableagainstthe with random grain orientation shown in color (inset).
bcc phase. The eigenvalues can also be used to study
the elastic anisotropy of the crystal. The anisotropy ra-
′
tio (A = B /B in bcc) has been calculated for all four
44
ically. Both of these effects have been neglected. The
phasesfromtheratioofthesmallestandlargesteigenval-
homogenized shear modulus in the rhombohedral phase
ues. For an isotropic material A = 1; for vanadium 1/A
ispositive,indicatingmechanicalstability. Thevariation
is never less than its ambient value, fluctuates through-
in the Voigt-Reuss difference results from the changing
out the entire pressure range studied, and becomes ex-
crystalline anisotropy.
tremely highnear the γ–bcc boundary (A∼1/130). For
comparison, the most anisotropic cubic transition metal In the case of a microstructure with more equiaxed
at ambient conditions is copper20 with A = 3.21, and grains,wecalculatethepolycrystallineshearmodulusby
among all cubic elements recent calculations found for homogenizingthe single-crystalB ofthegroundstate
ijkl
polonium A=1/6 to 1/18 at T=0K.21 structure at each pressure using a virtual test sample
(VTS).23 The procedure in Ref. 23 has been repeated:
the VTS is strained in six pure shear modes (ε , ε ,
12 23
V. POLYCRYSTALLINE SHEAR MODULI ε31, ε11−ε22, ε22−ε33, and ε33−ε11), and the results
fromsixcalculationsareaveragedtoreducetheeffectsof
anisotropy. The isotropic average shear modulus(G )
Polycrystallinevanadiumwithouttexturehasisotropic VTS
plotted in Fig. 3 has been obtained by equating the cal-
mechanical behavior, described by just two independent
culatedelasticenergypervolumetoanidealelasticsolid
elastic moduli: the bulk modulus K and the shear mod-
at the same strain:
ulus G. Regardless of phase, K is within 3% of that of
the bcc structure as mentioned earlier. Using the single-
crystalBijkl,Gmaybe boundedbythe VoigtandReuss u=G (ε2 +ε2 +ε2 +ε2 +ε2 +ε2 ). (3)
VTS 11 22 33 12 23 31
approximations of constant strain and constant stress,
respectively. We have calculated these approximations
using expressions equivalent to those in the literature.22 Onlyoneortwoofthesixstrainsarenonzeroateachrun
Since dynamic experiments conducted at Z-pinch and depending on which of the six modes has been applied,
laserfacilitiesoftenusethin-filmtargetswithmicrostruc- and the resulting variance in six independent runs is in-
turesthatcanvaryfromcolumnartoequiaxeddepending dicative of the anisotropy. The overall VTS prediction
on how they are processed, the Voigt and Reuss bounds lies between the Voigt and Reuss bounds, and the VTS
arehelpfulinassessingtherangeofpossibleresponses. In valuesarewithin5%oftheVoigt-Reuss-Hillaverage24ex-
calculating the Voigt and Reuss bounds shown in Fig. 3 ceptatthereentrantbccphaseboundary,wheretheVTS
(as well as the explicit polycrystalline calculations be- value is 31% greater. The Voigt-Reuss difference at this
low), we assume that the deformations are infinitessi- phaseboundaryisquitelarge: 1.34Mbar. Theconstant-
mal. With the low energy barriers, switching between stress Reuss average is sensitive to the most compliant
variants of the rhombohedral phase may contribute to orientation, whereas the Voigt average is fairly insensi-
the strain with no cost in stored elastic energy, leading tive; the VTS shearmodulus is closerto the Voigtvalue.
to a reduction in the shear modulus. At larger strains Itsignificantlydecreasesatthispointofhighanisotropy,
the response to rhombohedralstrains stiffens anharmon- and may lead to an anomalous dynamic response.
5
VI. CONCLUSION Acknowledgments
We have investigated the properties relevant to dy- We would like to thank G. Collins, A. Landa, D. Or-
namic experiments for two high-pressure rhombohedral likowski, B. Remington, and P. So¨derlind for useful dis-
phases in vanadium metal. It will be challenging for dy- cussions. This work was performed under the auspices
namicexperimentstodetecttherhombohedralphaseun- of the U.S. Dept. of Energy by Lawrence Livermore Na-
ambiguously. Thedistortionisprobablytoosmallforin- tionalLaboratoryunderContractDE-AC52-07NA27344.
situ x-ray diffraction, although it might be large enough
in γ.5 We have predicted that the volume change associ-
ated with any phase transformation up to 3.15 Mbar is APPENDIX A: CALCULATION OF
small, and may not have a clear signature in the VISAR STRESS-STRAIN COEFFICIENTS IN THE
trace. Wehavealsopredictedvaluesforthesinglecrystal RHOMBOHEDRAL LATTICE
andpolycrystallinestress-straincoefficients inthe rhom-
bohedralphasesatzerotemperature. Theβandγphases The high-pressure stress-strain coefficients B can
ijkl
smoothly cut offthe negative values ofthe bcc B44. The be obtained in many different ways, but in Table II,
first order transitions between the bcc and rhombohe- we summarize the deformation gradients and the corre-
dral phases give remarkably small changes in the stress- sponding strain energy relations that we used to calcu-
strain coefficients, as evident from the plots of the shear late B here. The pressure term is involved in some
ijkl
modulus and the stress-strain matrix eigenvalues, apart of the strain energy relations, for which the deformation
fromneartheγ–bcctransitionwherethecrystalishighly gradients are not volume-conserving. The stress-strain
anisotropic. coefficientsB (P)areequaltoC whenthepressure
ijkl ijkl
TheresultsherewereobtainedusingDFTatzerotem- vanishes, as explained in detail in Chapter 2 of Ref. 17.
perature for pure vanadium. Since the phase transition For a recent discussion of the stress-strain coefficients
isdrivenbyrathersubtle electronicstructureeffects,the B , see Ref. 26. For P 6=0, the relation is17
ijkl
elasticconstantsmaybesubstantiallyaffectedbychanges
intemperatureorimpurities.2,5,25 Itwouldbeinteresting B =−P(δ δ +δ δ −δ δ )+C . (A1)
ijkl jl ik il jk ij kl ijkl
to see whether the remarkable continuity of the moduli Theresultingstress-straincoefficientsalongwiththecal-
persistsasthephaseboundariesandtherelativestiffness culatedpolycrystallineshearmoduliaretabulatedinTa-
of the bcc and rhombohedral structures change. ble III.
∗ Electronic address: [email protected] 12 D. Kalantar et al., Phys. Rev.Lett. 95, 075502 (2005).
1 N. Suzuki and M. Otani, J. Phys.: Condens. Matter 14, 13 J. B. Aidun and Y. M. Gupta, J. Appl. Phys. 69, 6998
10869 (2002). (1991).
2 A. Landa, J. Klepeis, P. S¨oderlind, I. Naumov, L. Vitos, 14 D. J. Steinberg, S. G. Cochran, M. W. Guinan, J. Appl.
andA.Ruban,J.Phys.: Condens.Matter18,5079(2006); Phys. 51, 1498 (1980).
A. Landa, J. Klepeis, P. S¨oderlind, I. Naumov, O. Ve- 15 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.
likokhatnyi,L.Vitos,andA.Ruban,J.Phys.Chem.Solids Lett. 77, 3865 (1996).
67, 2056 (2006). 16 P. E. Bl¨ochl, Phys. Rev. B 50, 17953 (1994); G. Kresse
3 K. Takemura, in Science and Technology of High Pres- and D. Joubert, Phys. Rev.B 59, 1758 (1999).
sure, Proceedings of AIRAPT-17, Hawaii, edited by M. 17 D. C. Wallace, Thermodynamics of Crystals, (Dover, Mi-
H. Manghnani, W. J. Nellis, and M. F. Nicol (University neola, 1972).
Press, Hyderabad,India, 2000), p. 443. 18 P. Ravindran, L. Fast, P. A. Korzhavyi, B. Johansson, J.
4 Y. Ding, R. Ahuja, J. Shu, P. Chow, W. Luo, and H.-K. Wills, and O. Eriksson, J. Appl.Phys. 84, 4891 (1998).
Mao, Phys. Rev.Lett. 98, 085502 (2007). 19 J.Zhao,J.M.Winey,andY.M.Gupta,Phys.Rev.B75,
5 B. Lee, R. E. Rudd, J. E. Klepeis, P. S¨oderlind, and A. 094105 (2007).
Landa, Phys. Rev.B 75, 180101(R) (2007). 20 J. P. Hirth and J. Lothe, Theory of Dislocations, 2nd ed.
6 W. Luo, R. Ahuja, Y. Ding, and H.-K. Mao, Proc. Nat. (Krieger, Malabar, Florida, 1982).
Acad. Sci. 104, 16428 (2007). 21 D. Legut, M. Fri´ak, and M. Sˇob, Phys. Rev. Lett. 99,
7 G.E.DuvallandR.A.Graham,Rev.Mod.Phys.49,523 016402 (2007).
(1977). 22 J.P.WattandL.Peselnick,J.Appl.Phys.51,1525(1980).
8 D. B. Reisman, A. Toor, R. C. Cauble, C. A. Hall, J. R. 23 R.Becker,J.Comput.-AidedMater.Design9,145(2002);
Asay, M. D. Knudson, and M. D. Furnish, J. Appl. Phys. R.Becker, LLNL Report,UCRL-TR-234680 (2007).
89, 1625 (2001). 24 R. Hill, Proc. Phys. Soc. London A 65, 349 (1952); D. H.
9 B. A.Remington et al., Mater. Sci. Tech. 22, 474 (2006). Chung, Philos. Mag. 8, 833 (1963).
10 R. F. Smith et al., Phys. Rev.Lett. 98, 065701 (2007). 25 X. Sha and R. E. Cohen, Mater. Res. Soc. Symp. Proc.
11 J.H.Nguyen,D.Orlikowski,F.H.Streitz,J.A.Moriarty, 987, 0987-PP05-08 (2007).
and N.Holmes, J. Appl.Phys. 100, 023508 (2006) 26 E. Gregoryanz, R. J. Hemley, H.-K. Mao, R. E. Cohen,
6
TABLE II: Deformation gradients for the six independent stress-strain coefficients in the rhombohedral lattice and the corre-
sponding strain energy relations perunit volume at pressure P.
Stress-strain coefficient Deformation gradient Strain energy
1+δ 0 0
0 1
B11 T(δ)= 0 1 0 u(δ,P)=−Pδ+ 12B11δ2
B C
@ 0 0 1A
1 0 0
0 1
B33 T(δ)= 0 1 0 u(δ,P)=−Pδ+ 21B33δ2
B C
@0 0 1+δ A
1+δ 0 0
0 1
B12 (and B66) T(δ)= 0 1−δ 0 u(δ)=(B11−B12)δ2 =2B66δ2
B@ 0 0 1−1δ2 CA
1+δ 0 0
0 1
B13 T(δ)= 0 1+δ 0 u(δ,P)=−3P(δ+δ2)+ 12(2B11+B33+2B12+4B13)δ2
B C
@ 0 0 1+δ A
1 0 0
0 1−δ2 1
B44 T(δ)= 0 1 δ u(δ)=2B44δ2
B C
@ 0 δ 1A
1+δ 0 0
0 1
B24 (−B14) T(δ)= 0 1−δ δ u(δ)= 21(2B11−2B12+B44−4B24)δ2
B@ 0 0 1−1δ2 CA
TABLEIII:Calculated stress-straincoefficientsandvariouspolycrystallineshearmoduluspredictionsforthestablephase. All
quantities are in units of Mbar except that volumehas been scaled by theambient volume Vo=13.518 ˚A3.
Stable Volume Pressure Bulk Single crystal stress-strain coefficients Polycrystalline
phase (V/Vo) (P) modulus(K) B11 B33 B12 B13 B44 B24 VTS Voigt Reuss
bcc 1 0.00 1.82 2.30 2.18 1.52 1.64 0.51 0.18 0.39 0.41 0.35
bcc 0.779 0.73 4.21 4.79 4.44 3.74 4.09 0.87 0.49 0.43 0.59 0.27
β 0.754 0.87 4.63 5.52 5.00 3.96 4.43 0.75 0.57 0.51 0.67 0.33
β 0.729 1.03 5.09 6.00 5.63 4.42 4.84 0.77 0.65 0.50 0.70 0.25
γ 0.705 1.20 5.59 6.17 6.53 5.16 5.27 1.29 0.46 0.69 0.83 0.50
γ 0.659 1.62 6.77 7.87 7.89 6.19 6.23 1.61 0.52 1.03 1.14 0.87
γ 0.636 1.87 7.45 8.79 8.58 6.73 6.85 1.81 0.57 1.19 1.31 1.04
γ 0.593 2.45 9.02 10.81 9.85 8.11 8.38 2.21 0.70 1.43 1.60 1.24
γ 0.572 2.78 10.09 11.56 9.72 9.03 9.97 2.29 0.91 0.99 1.43 0.09
bcc 0.568 2.84 10.15 11.48 10.64 9.07 9.91 2.04 1.19 0.92 1.37 0.56
bcc 0.551 3.15 10.99 12.71 11.84 9.70 10.56 2.37 1.22 1.29 1.68 0.94
and P.Gillet, Phys.Rev.Lett. 90, 079702 (2003).