PHYSICAL REVIEW B 15 JANUARY 1997-IIVOLUME 55, NUMBER 4
ARTICLES
Structural properties of plutonium from first-principles theory
Per So¨derlindPhysics Department, Lawrence Livermore National Laboratory, Livermore, California 94550
J. M. WillsTheoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
B. Johansson and O. ErikssonCondensed Matter Theory Group, Physics Department, Uppsala University, Box 530, Sweden
~Received 8 May 1996; revised manuscript received 6 September 1996!
First-principles theory is shown to account for the unique low-temperature crystal structure of plutoniummetal (a-Pu!. Also the observed, and debated, upturn of the equilibrium volume between neptunium andplutonium is reproduced and found to be a consequence of the different crystal structures for these two metals.Thus it is shown that density-functional theory is able to accurately describe bonding properties of 5f electronsin an outstandingly complex system, where also relativistic effects are large. The electronic structure fora-Pu and for plutonium in competing close-packed crystal structures are also presented. Moreover, an expla-nation for the occurrence of the highly complexa-Pu structure is given. The mechanism is described in termsof a Peierls distortion in conjunction with a narrow 5f -band width. The energy gained from the splitting of the5f bands outweighs the electrostatic energy which favors the high symmetry structures found for most othermetals. At lower volumes we predict that plutonium should become bcc.@S0163-1829~97!01304-0#
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I. INTRODUCTION
Plutonium (94Pu! metal is one of the heaviest metaknown and it belongs to the series of metals that are refeto as the actinides. Plutonium metal, which is not foundnature, has since the first sample was prepared, been sto have a number of spectacular physical and chemical perties. The phase diagram is particularly interesting with fiallotropes discovered early (a, b, g, d, ande), and a sixthphase that was found later (d8),1 see Fig. 1. The latest discovered solid phase (z) ~Ref. 2! has a yet unknown complecrystal structure. Thus plutonium has as many as sevenferent crystalline forms1 which is more than any other metaThe a, b, andg phases, stable up to about 380 K, 460and 570 K, respectively, are very complex structures. Thiin contrast to the simple structures ofd-Pu ~fcc, face cen-tered cubic! ande-Pu ~bcc, body centered cubic!, which arefrequently found in other metals. Thea phase is especiallyinteresting with a unique open and low symmetry~mono-clinic! crystal structure. In this allotrope there are 16 atoper unit cell and the atomic arrangement seems to recovalent chemical bonding, where the nearest neighbortance between certain atoms is very small and betweeners very large.1,3 The crystal structures of the light actinide(90Th294Pu! display increasingly distorted crystal structurewith plutonium showing the most extreme complexity in threspect. This fact has been of great scientific interestdecades and a good understanding of this behavior hasrecently emerged. For plutonium, however, there hasbeen presented any reliable theory for the crystal struc
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and in this paper we address fromab initio theory the pecu-liar crystallographic properties of this material. We woualso like to give additional background to the complexitythis element and thus mention that other physical and checal properties of plutonium are also very unusual.4 For in-stance, two of the allotropes,d and d8 ~bct, body centeredtetragonal!, contract rather than expand when heatwhereas the low-temperaturea phase shows the largest themal expansion among the elemental transition metals.4 Alsoother properties, like the thermal and electrical conductivare anomalous for this spectacular material.4 Based uponthese remarkable properties, it has been speculated thattonium cannot be related to other metals and that a ra
FIG. 1. The experimental~Ref. 1! phase diagram for plutonium
specific theory for plutonium is needed to describe its mafacets.5
The electronic structure of plutonium also shows a vcomplex behavior. The actinide elements near94Pu have theinteresting property that for the earlier metals (90Th, 91Pa,92U, and 93Np! the 5f electrons are delocalized,6–10whereasthe heavier metals (95Am and beyond! have 5f electronswhich are chemically inert, in analogy to the localizedfelectrons in the lanthanide metals.6–10 Hence plutonium islocated at a position close to the border where, as a funcof atomic number, a transition from delocalized to localiz5f states appears.6–10 This complex and intriguing scenariprovides a great challenge for theory to accurately treatelectronic structure of plutonium. For instance, the appeance of f states in the valence band of plutonium probamakes it hard to use theoretical band structure methodsutilize nonlocal basis functions~plane waves or similar! suchas the pseudopotential methods. In addition to this the plnium atom is very heavy, which means that relativisticfects need to be taken into account in the description ofelectronic structure of plutonium metal.10 In particular, thespin-orbit coupling cannot be neglected when such hemetals are investigated. Because of the mentioned diffities, an accurate total energy treatment of plutonium inground state (a-Pu! has not been performed until now, to oknowledge.
One pronounced experimental evidence for the itinercharacter of the 5f electrons in the light actinide elemencan be found from the trend displayed by their density~orvolume! as one proceeds from90Th to 94Pu, where the vol-ume shows a parabolic decrease of the volume as a funcof nuclear charge~Fig. 2!. In contrast to this, from95Am andonwards the metallic volumes are much larger (; 50%! witha weak linear decrease with atomic number, similar tolanthanide contraction~not shown!. This behavior strongly
FIG. 2. Measured and calculated room temperature atomic elibrium volumes~Ref. 24! for the light actinide metals. The calculations were done both for the observed room temperature stures ~curve labeled ‘‘theory’’! as well as for a hypothetical fccstructure~curve labeled ‘‘theory-fcc’’! and for Np and Pu also fotheb-Np structure~stars labeled ‘‘theory-b-Np’’ !.
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suggests that the 5f states for95Am and the heavier actinideare localized.6 The theoretically predicted6 and observed11
super conductivity of americium metal give much supportthis view. The volume behavior of the early actinid(90Th294Pu! on the other hand, is very similar to whatfound for the nonmagneticd-transition metals, and it can bunderstood from the filling of first bonding and then anbonding orbitals of thed ~or f for the actinides! bands.6 Thisimplies that the lowest volume~highest density! would befound for the actinide metal with the highest numberbonding, and lowest number of antibonding electrons, ifor plutonium. Interestingly, from Fig. 2 we note that plutonium shows an additional anomalous behavior, namelysomewhat larger volume than its preceding metal, neptunium (93Np!.
Intrigued by the complex electronic and crystallographproperties ofa-Pu we have performed first-principles, totenergy calculations of this material over a wide volumrange. By comparing the total energy of thea phase with theother structures found in the actinides as well as the sttures found for the transition metals we aim to demonstrthat the delocalized 5f states drive the peculiar crystal struture of a-Pu. Furthermore we will show that at high presures the bcc crystal structure becomes the most favorfor 94Pu. With the total energy dependence of the atomvolume for the eight studied structures we have been ablexamine how the crystal structure influences the atomic elibrium volumes for plutonium and it is our intention to shothat the anomalous volume increase between93Np and94Pu is actually a consequence of the anomalous crystructure of thea phase. In the following we describe outheoretical method in Sec. II. In Secs. III, IV, and V ouresults for the crystal structure, occupation numbers,atomic volume, respectively, are presented and we concin Sec. VI.
II. COMPUTATIONAL DETAILS
The reported results are obtained from electronic struccalculations for plutonium in eight different crystal strutures:a-Pu, a-Np, b-Np, a-U, bct, bcc, fcc, and hcp. Thetotal energy of these structures was calculated as a funcof volume. The presentab initiomethod solves the Dirac~forthe core electrons! or a ~modified! Schrodinger equation~forthe valence and semicore!. The total energy of the systemwas obtained within density-functional theory. In thdensity-functional approach, it is common to make the lodensity approximation for the exchange and correlationteractions between the electrons. Since the recently presegeneralized gradient approximation~GGA! ~Ref. 12! hasbeen shown to significantly improve the accuracy of thesults for f -electron metals,13 we have chosen to adopt thapproximation for the exchange and correlation energy futional in the present calculations. The relativistic effectsincluded in the Hamiltonian, and the spin-orbit interactiterm is considered according to the recipe proposedAndersen.14 The wave functions are expanded by meanslinear muffin-tin orbitals inside the nonoverlapping muffitin spheres that surrounds each atomic site in the crystal.muffin-tin radius was consistently chosen such thatmuffin-tin spheres occupied 41% of the total volum
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55 1999STRUCTURAL PROPERTIES OF PLUTONIUM FROM . . .
We make use of a so-called double basis set since we atwo tails with different kinetic energy for each muffin-tiorbital with a given l -quantum number. The calculationwere done for one, fully hybridizing, energy panel in whithe values forEn’s related to the valence orbitals 7s, 7p,6d, and 5f , and to the semicore orbitals 6s and 6p weredefined. Within the muffin-tin spheres, the basis set, chadensity, and potential were expanded in spherical harmowith a cutoff lmax56. Outside the muffin-tin spheres, in thinterstitial region, the wave functions are Hankel or Nemann functions which are represented by a Fourier seusing reciprocal lattice vectors. The same expansion is uto represent the charge density and the potential. This trment of the wave function, charge density, and potential dnot rely upon any geometrical approximations and thescribed type of computational method is usually referredas a full potential linear muffin-tin orbital method~FP-LMTO!. This so-called full potential method15 has previ-ously been successfully applied to many systems, includalso some of the actinides,16,17 proving its reliability.
In the calculation of the one-electron band structuresspecialk-point method has been used with various sampldensities of thek points. In thea-Pu anda-Np structures 16k points of the irreducible part of the Brillouin zone~IBZ!were used whereas forb-Np the corresponding number wa18. This may seem to be relatively small numbers ofk pointsin the IBZ for these structures where the IBZ is 1/2, 1/2, a1/4 of the full Brillouin zone~FBZ! for a-Pu, a-Np, andb-Np, respectively. However, an increase of the numbek points to 32 (a-Np! and 40 (b-Np! only lowered the totalenergy with about 0.1 mRy/atom at the theoretical equirium volume. The electronic structure for plutonium in tha-U crystal structure~orthorhombic with 2 atoms/cell! wasobtained using 100k points in the IBZ~1/8 of the FBZ!. Inthe case of the more symmetric bcc, fcc, and bct structuthe symmetry of the bct unit cell was consistently appland a total number of 150k points were used in the IBZ fothose ~1/16 of the FBZ!. For the hexagonal close-packelattice, which we assumed to have an idealc/a ratio, we used162k points in the IBZ~1/12 of the FBZ!. To further inves-tigate the convergence of thek-point sampling for the vari-ous crystal structures we chose the crystallographic pareters (c/a, b/a, and positional parameters! for a-Np,b-Np, and bct in such a way that the structures describedbcc geometry. Close to the theoretical equilibrium volumthese test calculations showed that the total energy differebetween the structures was converged to about 1 mRy/a
III. CRYSTAL STRUCTURES
This section focus on the crystal structures of plutoniat low temperatures and pressures, their origin and influeon the anomalously low equilibrium density exhibited by tplutonium metal. For that reason we have performed toenergy calculations for94Pu in the above mentioned eighcrystal structures. As stated above we have chosen tstructures since they represent typical structures found ind-transition metals~bcc, fcc, and hcp! or in the actinides@bct(a-Pa!, a-U, a-Np, b-Np, anda-Pu#. In order to find thetheoretical equilibrium density for plutonium we have minmized the total energy with respect to volume for the m
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tioned eight crystal structures. The results from these calations are shown in Fig. 3, where we plot the total enefor the different structures as a function of the atomic volurelative to the bcc total energy, which defines the zero enelevel. In agreement with experiment our results confirm tthe a-Pu structure is the ground-state structure for plunium. However, it is also interesting to notice that the enefor 94Pu in thea-Np structure is very close. Notice in Fig.that thea-Pu,a-Np, b-Np, a-U, bct, and the bcc structureare all considerably lower in energy than the hcp andstructures.
The finding that among the symmetric structures thephase is energetically similar to the more open structurequite interesting and deserves attention. First of all howewe note that the energy differences between the fcc, hcp,bcc structures of the light actinides can be explained insame way as was done for the transition metals, wherework of Skriver18 and Duthie and Pettifor19 showed that theoccupation of thed band is the important parameter for thtransition metals. By evaluating the band energies fortransition metals, one was able to conclude that the shapthe d density of states (d-DOS! in conjunction with thed-band occupation determined the crystal structures for thmetals. With the same technique as for thed-transition met-als ~using canonicalf bands20! it has been shown that fo5f populations between 3 and 6~Ref. 17! the bcc structure islower in energy than the fcc and hcp structures. The reafor this is that for 5f occupations in this interval the Fermlevel is situated between the two pronounced peaks incanonical f -DOS for the bcc structure. Consequently thstructure is very favorable for these 5f populations. The lightactinides with 5f occupations in this interval (93Np294Pu,92U has somewhat less than three 5f electrons! could beviewed as counterparts to the nonmagnetic bccd-transitionmetals Nb, Mo, Ta, and W which all haved occupationssuch thatEF lies in between the two peaks of the bcd-DOS. Thus, for the light actinides,92U294Pu, canonicalband theory20 can be used to show that the bcc structure
FIG. 3. Total energy for plutonium, calculated in thea-Pu,a-Np, b-Np, a-U, bct (c/a50.85), hcp~idealc/a), and fcc crystalstructures, relative to the bcc structure, as a function of volume
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2000 55SODERLIND, WILLS, JOHANSSON, AND ERIKSSON
lower in energy than the fcc and hcp energies, in agreemwith Fig. 3 as well as with the calculations of Skriver,18
although for uranium the bcc and hcp energies are vclose. However, this theoretical model is only useful whthe comparison is restricted to close-packed~fcc or hcp! ornearly close-packed~bcc! structures.18 This is because theelectrostatic energies are very similar for these structurescan be essentially neglected when the comparison isstricted to these three phases. For other phases this differin Madelung energy has to be considered explicitly togetwith overlap repulsion. The Peierls–Jahn-Teller distortmechanism, favorizes low symmetry structures over hsymmetry structures for narrow band systems, and musconsidered in the present context. Taking into account bof these arguments it is natural to find crystal structuresuranium, neptunium, and plutonium at low temperaturespressures that have low symmetry, and can be obtathrough a distortion of the bcc parent lattice. For92U ~Ref.21! and 93Np ~Ref. 17! it has been shown theoretically ththeir equilibrium structures, which both for uranium and netunium can be viewed upon as distorted bcc structures, trform to the bcc structure at sufficiently high pressure. Opresent work shows that this also happens for pluton~Fig. 3! but this prediction has not yet been observed. Tbasic explanation for the occurrence of the bcc structurecompressed plutonium metal is that a broadening of thefbands will eventually make the Peierls–Jahn-Teller distion less important~since less energy can be gained by tmechanism for broader bands! which will favor more sym-metric structures.21
From the above discussion, that bcc is the most stablthe high symmetry structures of plutonium, it is not surpring to find that this phase becomes the stable one atpressure. Fig. 3 confirms this picture and the transition tophase occurs at about 20% compression (V/V0; 0.8!. Thisresult is also partially supported by very recent experimefindings for plutonium22 which show that the metal transforms to a structure with higher symmetry atV/V0; 0.8,which was interpreted as a distorted hcp structure. We nhere that the hcp and bcc structures can be viewed as ditions of each other, along the Burgers path.23 The fact thatthe low-temperature structures of plutonium sometimescompared to the hcp structure22,24 has inspired us to alsoconsider this structure in our calculations. In Fig. 4 we sh
FIG. 4. Relative total energies for the hcp crystal structureplutonium calculated as a function of the axial ratioc/a at a volumeof 18.2 Å3 ~i.e., a volume close to the theoretical equilibrium voume!.
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the calculated total energy of the hcp structure as a funcof c/a ratio. Interestingly, a nonidealc/a value (c/a;1.85) optimizes the hcp structure. This optimized hstructure has an energy close to the fcc energy at this vol~18.2 Å 3, i.e., close to the theoretical equilibrium volum!and this is in agreement with the hcp calculations by vanet al.25 However, this phase is still much higher in enerthan, for instance, the bcc energy. At elevated pressuresoptimized hcp structure is also much higher in energy ththe bcc structure~data not shown!. The reason for the largec/a ratio in the hcp structure is interesting, since normaone encountersc/a ratios close to 1.63~ideal packing!. Thedistortion of thec/a ratio of the hcp structure away fromideal packing is in line with the fact that the hcp and bstructures are related via a so-called Burgers transformapath.23 In the Burgers transformation,~1 1 0! planes of thebcc structure are shifted and followed by a Bain strain ofbcc lattice.26 These displacements lead to a crystal structwith a hcp geometry, but thec/a axial ratio of this hcpstructure is larger than the ideal value and therefore, alonBurger transformation, a hcp structure with ac/a.1.63 iscloser to the bcc structure. This gives additional informatfor why a nonideal hcp structure has lower energy thanideally packed hcp structure since the former structurecloser to the bcc structure which in turn is even lowerenergy. We mentioned above that using a canonical bmodel for thef bands one could argue that forf -band occu-pations between 326, the bcc structure should be favoreover the hcp and fcc structure.17 Hence for metals with 5fpopulations within 326 one should expect a nonidealc/aaxial ratio to optimize the hcp structure. Both neptunium aplutonium have 5f occupations within this interval, whereauranium has somewhat less than three 5f electrons at ambi-ent conditions. For neptunium the optimizedc/a axial ratiowas calculated~FP-LMTO! to be larger than the ideal valu(c/a; 1.75! whereas for uranium thec/a ratio was rela-tively close to the ideal value (c/a; 1.70!.27
In view of the structural properties of plutonium, discussed above, it is interesting to note~Fig. 5! that the 5fDOS for a-Pu ~upper panel! and bcc Pu~middle panel! arerather similar, with two broad features, one above andbelow the Fermi level, whereas the hcp 5f -DOS ~lowerpanel! and fcc 5f -DOS ~not shown! are somewhat differentFora-Pu ~upper panel! we show the 5f -DOS only for one ofthe 16 atoms of the cell since the difference betweenvarious atoms is not significant for this discussion. The simlarity in the 5f -DOS for the bcc anda-Pu structures indi-cates thata-Pu is closer to the bcc structure than, for istance, the hcp or the fcc structures and this supportsarguments outlined above as well as the ones of eastudies.17 This observation further explains why many of thstructures found in the light actinides (92U294Pu in particu-lar! may be viewed upon as distortions of the bcc structuOur calculated DOS curve fora-Pu in Fig. 5 may be com-pared with x-ray photoemission spectra of the valence bfor this material.28 The calculated DOS curve has a; 2 eV~0.15 Ry! broad feature centered at; 0.5 eV ~0.04 Ry!binding energy. This is in acceptable agreement withmeasured spectrum, showing a; 2.5 eV broad feature centered at 0.5 eV binding energy.
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55 2001STRUCTURAL PROPERTIES OF PLUTONIUM FROM . . .
IV. OCCUPATION NUMBERS
It was recently argued that the reason for the tendetowards open crystal structures in the actinides~and pluto-nium in particular! is primarily due to an increased popultion of the 6d band, which has a strongly itinerancharacter.25 Furthermore it was suggested that the bondingthese structures wasnot dominated by the 5f band althoughthe increased 5f bonding also encountered in these structuallowed for considerable lowering of the total energy. Heand in our previous studies,21 we argue that it is indeed thoccupation of the 5f bands that is responsible for the opand complex structures in the actinide series. In Table Ipresent our calculated occupation numbers for plutoniumthe eight investigated crystal structures at a volume closthe (a-Pu! volume~19.4 Å3). These numbers depend on th
FIG. 5. 5f ~full line! and 6d ~dashed line! electron density ofstates~states/eV/atom! for a-Pu ~upper panel!, bcc Pu ~middlepanel!, and hcp Pu~lower panel! calculated at the experimentaroom temperature equilibrium volume~Ref. 24!. The calculateddensity of states has here been convoluted with a Gaussian funof width 0.05 eV.
TABLE I. Calculated occupation numbers for plutoniumeight crystal structures at an atomic volume 19.4 Å3 and a 1.24 Åmuffin-tin radius.
choice of muffin-tin radius in our computational method acould therefore not be compared directly with the resupresented in the paper by van Eket al.25 However, a com-parison of the population between the different structu~they all have the same muffin-tin radius! is still possible. Inagreement with van Eket al.we find that thed occupation issomewhat larger fora-Pu compared to the other structurealthough the difference between thea-Np and thea-Pustructures is quite small. Our calculateds occupation, how-ever, is very much independent of the actual crystal strture, in disagreement with the results obtained by vanet al. Table I further reveals that thef population is slightlyenhanced for the most complex structures (b-Np, a-Np, anda-Pu!. Considering the featureless shape of thed-DOS inFig. 5 we conclude that the small increase in thed popula-tion of thea-Pu phase compared to the other structures isthe reason that this phase is the most stable. Note alsothe bcc and hcp phases have very similard occupation~Table I! but the bcc total energy is much lower~Fig. 3!.This thus illustrates that the occupation number of thedorbital is not the most important factor determining tchemical bonding and crystal structure ofa-Pu. Instead weargue, as has been done before,6,9,10 that it is the 5f stateswhich dominate the bonding and as discussed in the prevsection it is the shape of the 5f -DOS, the position of theFermi level, and finally the electrostatic~Madelung! energythat are the most important factors for the determinationthe crystal structure in the actinide materials, including ptonium.
V. ATOMIC VOLUMES
Next we analyze the equilibrium volume for plutoniumAs mentioned in the introduction~Sec. I!, plutonium showsan anomalous volume behavior since it has alarger atomicvolume than its preceding metal,93Np. This anomaly is im-portant since it gives rise to appreciable effects on otproperties ofa-Pu. For instance, it is the main reason for textremely large thermal expansion ofa-Pu.29 Suggested ex-planations for the upturn of the atomic volume for94Pu,relative to the 93Np volume, are that the 5f electrons inplutonium show a tendency towards localization25,30~i.e., the5f localization that dramatically lowers the density in amecium has to some degree already begun in94Pu! or that therelativistic spin-orbit splitting of the 5f band gives rise tothis behavior. Fully relativistic calculations~based on theDirac equation and with a spherical approximation for tcharge density and potential! for fcc plutonium show that therelativistic spin-orbit coupling leads to a slightly decreas5f bonding for 94Pu and consequently an increased equilrium volume.10 One purpose of the present section iselaborate on this long-standing issue.
Figure 2 shows the calculated atomic volumes for theand thea-Pu structure and we compare them with the callated equilibrium volumes for the other light actinide~evaluated both in the low-temperature structures as wein a hypothetical fcc structure!. The zero-temperature calculations are corrected for thermal expansion since the expmental data are measured at room temperature.24 We havedone this by means of the experimental thermal expanscoefficient~a correction using the theoretical thermal expa
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2002 55SODERLIND, WILLS, JOHANSSON, AND ERIKSSON
sion coefficient29 gave a very similar result!. Notice that
94Pu in the fcc structure has a considerably lower voluthan 93Np, and hence these isostructural calculations doreproduce the observed behavior of the atomic volume mtioned in the introduction. This result is in agreement wearlier full potential calculations for plutonium in the fccrystal structure.16 The same conclusion can be drawn frocalculations based on theb-Np structure. As is clear fromFig. 2, minimizing the total energy of Np and Pu in thb-Np structure with respect to volume leads to a larger eqlibrium volume of Pu than for Np. However, when the trulow temperature structures are considered (a-Np anda-Pu!, the calculations indeed reproduce the experimeobservation with an upturn of the atomic volume betwethese two metals. The reason for this anomaly of thea-Puvolume is thus mostly due to its unique crystallograpproperties. It has been argued from so-called full charge dsity calculations~albeit with a spherical potential!30 that thestructural properties of thea phase are not responsible fothe rise in volume between93Np and 94Pu. Our present calculations show that the crystal structures, in fact, are decifor the equilibrium volume and that a consideration of tcorrect structures indeed reproduce the experimental trTo exemplify the important influence that the structures hon the equilibrium volumes we observe that the differencethe equilibrium volume between bcc and fcc neptuniumabout 7%~Ref. 17! and for plutonium this difference is almost as much as 10%~not shown!. In the description of theparabolic trend displayed by the equilibrium volume of tlight actinides it seems sufficient to use almost any crystructure in combination with spherical potentials. Howevin order to resolve the smaller features, such as the risatomic volume between neptunium and plutonium, the tcrystal structure obviously needs to be considered.
We now put our results in context with previous theorattempting to explain the upturn of thea-Pu volume. Firstwe note that since we are able to describe accuratelycrystal structure and equilibrium volume of plutonium, telectron correlation effects seem to be equally well describy the GGA ina-Pu as they are in the other light actinideThe suggestion that the anomalous atomic equilibrium vume is a sign of an onset of localization, or a fingerprintstrong electron correlations,25,30 not being accurately described by density-functional theory, thus seems less likThe other argument, that the large volume of94Pu is relatedto the relativistic spin-orbit effect,10 is more plausible. Infact, we find that by neglecting this interaction in our calclations we obtain a slightly different volume~about 1%, notshown! for a-Pu. However, the inclusion of the spin-orbcoupling did not affect the trend of the calculated atomvolumes for the light actinides. The mechanism shownRef. 10, that relativistic effects reducing the chemical boing from the 5f shell, are also present in our calculationthereby expanding the equilibrium volume. However, retivistic effects also influence the semicore states such thasum of all the effects cause the equilibrium volumea-Pu to be rather insensitive to the treatment of the sporbit coupling. However, we find that the detailed electrostructure of for instance the 5f band is heavily influenced bythe spin-orbit coupling, in agreement with the conclusimade by other researchers.10,31
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VI. CONCLUSION
In summary we have demonstrated that the highly anomalous crystal structure ofa-Pu is reproduced by accuratedensity-functional calculations. We conclude that the peculiar crystallographic properties of this material are not a fingerprint of strong electron correlations or quasilocalizationof the 5f states. We further conclude that the 5f band isdominating the bonding in these systems and that the 6dband is of less importance for the crystal structure. The spetacular crystallographic properties of this material are insteaunderstood from delocalized 5f states, in the same way asfor the earlier actinides, with the key ingredient being aPeierls distortion caused by the 5f bands. Let us investigatethese arguments in some detail by comparing two similabut yet different crystal structures, namely, the bcc and thb-Np structure. Both these structures yield low total energiefor plutonium, see Fig. 3, but bcc has a very high symmetras opposed to theb-Np structure. Nevertheless, they are verysimilar and an approximate one-parameter path can be dfined between these structures.17 When the parameteru ~anatomic position! is equal to 0.5 and 0.375 the structure be-comes bcc and approximatelyb-Np, respectively. We havecalculated the total energy for the light actinides~at theirexperimental equilibrium volume! for this path and the re-sults are shown in Fig. 6. Notice that for uranium, nep-tunium, and plutonium the approximateb-Np structure islower than the bcc structure which indicates that the Peierdistortion is effective in lowering the energy for these met-als. Neptunium and plutonium, in particular, are more stablin the distortedb-Np type of structure and this follows fromthe fact that lowering the crystal symmetry results in fewedegenerate 5f bands in the vicinity of the highest occupiedlevel. This can be illustrated by comparing the 5f -DOS forthe bcc and ‘‘b-Np’’ structures. In the upper panel of Fig. 7the 5f -DOS for the ‘‘b-Np’’ structure is shown and in the
FIG. 6. Calculated total energy as a function of atomic positionu, for a crystal structure which is approximately the observedb-Np structure foru50.375 and the bcc structure foru50.50~Ref.17!. The calculations are done for the light actinides90Th294Puclose to their respective experimental equilibrium volumes and thenergies are shifted so that the bcc total energy is equal to zero feach metal.
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55 2003STRUCTURAL PROPERTIES OF PLUTONIUM FROM . . .
lower panel the corresponding DOS for the bcc structurean atomic volume of 20.4 Å3. Notice that the ‘‘b-Np’’ 5f -DOS is relatively low at the Fermi level. For the bcc cain the lower panel, the 5f -DOS atEF is larger and moreweight of the DOS is found at higher energies. Furthermthe bcc 5f -DOS has a peak at about 0.5 eV below theEF ,which is much less pronounced in th‘‘ b-Np’’ 5 f -DOS in the upper panel of Fig. 7. The loweresymmetry for the ‘‘b-Np’’ structure thus shifts electronic
FIG. 7. Calculated 5f ~full line! and 6d ~dashed line! electronicdensity of states for plutonium in an approximateb-Np structure~upper panel! and in the bcc structure~lower panel!. The density ofstates has here been convoluted with a Gaussian function of w0.03 eV. Energies are in eV and the vertical line indicates the Felevel.
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energy states away fromEF to lower energies and consequently the band energy decreases. The 6d-DOS, on theother hand, is very similar for these two crystal structurand does not seem to be of any importance for the crystructure. The results shown in Fig. 6 illustrate that ancreased 5f occupation drives low symmetry structures. Wcompare Fig. 6 with Fig. 6 of Ref. 17, where a very similstudy was performed for Np at ambient and compressedumes. Curves of exactly the same shape as in Fig. 6found in Fig. 6 of Ref. 17. The reason for this is that compressing neptunium broadens the 5f band, reducing the ef-fect of the Peierls–Jahn-Teller distortion. The same redtion is found when depopulating the 5f band by going from93Np to 92U, 91Pa, and90Th.We have also predicted that the high pressure phas
plutonium will attain the bcc crystal structure and this trasition is calculated to occur at about a 20% compressionthis metal.
Finally, we have shown that the unique crystal structuof plutonium results in an anomalously large equilibriuvolume at room temperature, which is larger than for tpreceding metal, neptunium. Hence we demonstrate thathough many of the ground-state properties of plutoniumanomalous the description of itsa phase does not requirespecial theoretical approach, and thata plutonium is the lastactinide metal with a delocalized 5f band.
ACKNOWLEDGMENTS
We thank C. Slaughter for a critical reading of our manscript. A.K. McMahan, J. Akella, and L. Fast are acknowedged for valuable discussions. Two of us~B.J. and O.E.! arethankful for financial support from the Swedish Natural Sence Research Council and for support from the materconsortium No. 9. Part of work performed under the auspiof the U.S. Department of Energy by the Lawrence Livemore National Laboratory, Contract No. W-7405-ENG-48
thi
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