First-principles study of phonon modes in PuCoGa5 superconductor

Przemysław Piekarz, Krzysztof Parlinski, and Paweł T. JochymInstitute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, PL-31342 Kraków, Poland

Andrzej M. OleśInstitute of Nuclear Physics, Polish Academy of Sciences, Radzikowskiego 152, PL-31342 Kraków, Poland

and Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

Jean-Pierre SanchezDépartement de Recherche Fondamentale sur la Matière Condensée, CEA Grenoble, 17 rue des Martyrs,

F-38054 Grenoble Cedex 9, France

Jean RebizantEuropean Commission, Joint Research Centre, Institute for Transuranium Elements, Postfach 2340, D-76125 Karlsruhe, Germany

�Received 11 November 2004; revised manuscript received 22 March 2005; published 14 July 2005�

By minimizing the total energy within the density-functional method we determined lattice parameters, atompositions, interatomic forces, and electronic structure of the new PuCoGa5 superconductor. The on-site Cou-lomb repulsion U and Hund’s exchange J lead to changes in the electronic structure, particularly near the Fermienergy, and to a better agreement between the calculated and experimental geometries. Using this ab initioinput, the phonon dispersion relations are determined and classified by their symmetries. Phonon densities ofstates and the lattice heat capacity are discussed. Using individual contributions due to all optical modes at the� point we estimate electron-phonon coupling ��0.7 which does not suffice to explain the observed value ofTc and suggests that electronic interactions also play a role in the pairing mechanism.

DOI: 10.1103/PhysRevB.72.014521 PACS number�s�: 74.70.�b, 63.20.Dj, 63.20.Kr, 71.28.�d

I. INTRODUCTION

Recently, superconductivity was found in PuCoGa5, thefirst intermetallic superconductor containing plutonium, withthe highest critical temperature Tc=18.5 K among f-electroncompounds.1 The crystal structure of PuCoGa5 is identical tothat of Ce-based heavy-fermion superconductors CeCoIn5and CeIrIn5, with Tc=2.3 and 0.4 K. It is quite remarkablethat the value of Tc scales both in doped CeMIn5 �M=Co,Rh, Ir� and in PuCo1−xRhxGa5 compounds with the ra-tio c /a,2 but the values of Tc in Pu-115 are larger by almostone order of magnitude than in Ce-115. Also the similaritybetween the physical properties of PuCoGa5 and the CeMIn5family might indicate a common mechanism ofsuperconductivity,2 and in fact PuCoGa5 may be treated as ahole analog of CeMIn5.3 The mechanism of pairing inPuCoGa5 is unclear until now, but it has been suggested thatthe spin fluctuation model is most consistent with the experi-mental resistivity.4 Yet the phonon mechanism could not beexcluded so far and it is interesting to ask which phononmodes could possibly be responsible for the pairing.

Plutonium itself is a material with very unusual elec-tronic, structural, and magnetic properties. It is one of themost interesting metals with anomalous behavior due topartly filled 5f orbitals. The 5f electrons are strongly corre-lated, and their dual itinerant and localized nature is difficultto implement in electronic structure calculations. The elec-tronic structure of PuCoGa5 was studied by a tight-bindingmethod,5 within the density-functional theory �DFT�,6–9 inlocal spin density approximation with Hubbard U �LSDA+U� Coulomb repulsion,10 and experimentally by the photo-

emission technique.11 These studies suggest that the 5f elec-trons are fully itinerant and their weight dominates at theFermi energy. Only when 5f electrons are assumed to beitinerant in PuCoGa5 do cohesion and crystallographic pa-rameters, as well as the photoemission spectra, compare fa-vorably with experiment.8 In contrast, for more localized 5felectrons in NpCoGa5,9 an antiferromagnetic �AF� order setsin.12

In this paper, we present an investigation of lattice dy-namics in PuCoGa5 using the ab initio direct method.13 Ourmain objective is to identify the phonon modes which mightcontribute to the pairing in PuCoGa5 and to estimate �i� theelectron-phonon coupling constant and �ii� the transitiontemperature Tc which follows from the phonon mechanismof pairing. So far, theoretical studies of lattice dynamics inactinides are very limited due to notorious difficulties in thetreatment of f electrons in the electronic structure calcula-tions. Nevertheless, an unprecedented success of the ab initioapproach was demonstrated recently in the example of thephonon spectra of the � phase of Pu, determined using thedynamical mean-field theory.14 The phonon dispersions werecorrectly predicted by the theory and were soon after ob-served by inelastic x-ray scattering.15 These studies haveshown that the lattice properties of plutonium are anomalous,with unusually large softening of phonon frequencies at in-creasing temperature, which might indicate a strong tempera-ture dependence of the electronic structure and largeelectron-phonon coupling.15,16 Similar behavior could also beexpected for PuCoGa5.

The paper is organized as follows. In Sec. II we presentthe calculation method for the electronic structure which in-

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cludes electronic interactions within the generalized gradientapproximation �GGA�, leading to the so-called GGA+Umethod.17 This ab initio input is next used for the calculationof force constants and phonon frequencies in Sec. III. Fi-nally, an estimation of the electron-phonon coupling constant� is presented in Sec. IV, and we discuss the implications ofour findings for the mechanism of superconductivity inPuCoGa5.

II. CALCULATION METHOD AND ELECTRONICSTRUCTURE

The density-functional total-energy calculations were per-formed using the VASP program18 with the GGA for theexchange-correlation energy.19 The local electron interac-tions have been included in the multiband Hubbard modelfor the f states of Pu, with intraorbital Coulomb repulsion Uand interorbital Hund’s exchange J.20 These interactionswere treated in the Hartree-Fock approximation, and we sub-tracted the interaction terms for the uniform f-electron dis-tribution to avoid double counting. The electron-electron in-teractions are implemented in a similar way to that used inthe LSDA+U method21 and lead in the present case to theGGA+U approach.17

The Kohn-Sham Hamiltonian obtained within the GGA+U method was diagonalized self-consistently using the it-erative blocked Davidson scheme, which updates all bandssimultaneously.18 Valence electron wave functions—3d and4s states for Co, 4s and 4p states for Ga, and 5f , 6s, 6p, 6d,and 7s states for Pu—were evaluated using the projectoraugmented-wave �PAW� method.22 On average, one findsclose to 9 valence electrons per Co ion, 3 per Ga ion, and 16per Pu ion. The energy cutoff for the wave function expan-sion was 400 eV. For the energy minimization we used twosupercells: 1�1�1 and 2�2�1, with 7 and 28 atoms, re-spectively. The crystal geometry was optimized in the tetrag-onal space-group symmetry P4/mmm with periodic bound-ary conditions. The k-point grids were generated for bothcases by the Monhorst-Pack scheme.23 During optimizationthe Hellman-Feynman �HF� forces and stress tensor werecalculated after each update of ionic positions. The optimi-zation was finished when the residual forces on all atomswere less than 0.03 meV/Å.

We optimized the geometry for the nonmagnetic �NM�state at no Coulomb interactions �U=J=0� and for ferromag-

netic �FM� and AF states, obtained for finite interactions us-ing the GGA+U method with U=3 eV and J=0.7 eV. Thevalue of J follows from the atomic data,24 while the value ofU is lower than U=4 eV suggested for Pu,25 but is in theballpark of the commonly accepted values for PuCoGa5.10 Inthe tetragonal unit cell, there are seven atoms grouped inlayers. The basal plane includes one plutonium at the posi-tion �0,0,0� and one gallium �GaI� at the position �0.5,0.5,0�.The remaining gallium atoms �GaII� are located in two planesat positions: �0.5,0 ,z�, �0,0.5,z�, �0.5,0 ,1−z�, and�0,0.5,1−z�. Co atom is situated above Pu at the high-symmetry position �0,0,0.5�.

The calculated crystallographic parameters for the NMstate at U=0 and for both magnetic states at U=3 eV, J=0.7 eV are presented in Table I and compared with theexperimental data from single-crystal x-ray diffraction atroom temperature.1 We have also included the results of theelectronic structure calculations using the full-potential localorbitals method.6,8 In these papers both the NM and magneticground states were studied, with the energies of FM and AFstates being slightly lower than those of NM ones. As mag-netic long-range order has not been observed experimentally,the magnetic order indicates only the tendency towards theformation of local moments in the paramagnetic phase,which cannot be properly captured within electronic struc-ture calculations. The crystal parameters calculated in thepresent GGA+U study at finite U are closer to those ob-tained for the AF structure8 and to the experimental ones1

than the values found at U=0. Thus, the obtained equilib-rium geometry with realistic electronic structure provides areliable starting point for studying the lattice dynamics inPuCoGa5.

The electronic densities of states were obtained by per-forming the summation over the Brillouin zone on the 14�14�14 grid with 288 irreducible k points, using the tetra-hedron method. First we analyze the total electronic densityof states for the optimized unit cell in the reference NM stateat U=J=0, shown in Fig. 1. A narrow peak with large inten-sity at the Fermi energy EF=0 is due to Pu 5f states for bothspins. The density of states at the Fermi energy EF, N�EF�=13.94 eV−1, and the characteristic maximum at E�−1.2 eV are in good agreement with earlier electronicstructure calculations performed in NM states, using eitherthe tight-binding5 or full-potential8 in muffin-tin orbitalsmethod and the full-potential local orbitals band-structuremethod.6

TABLE I. Lattice constants a and c �in � and internal Ga z coordinate as obtained in the present �PAW� method using a 2�2�1supercell, for the NM state at U=0 and the FM and AF states at U=3 eV, full-potential electronic structure nonmagnetic �Ref. 6�, andmagnetic �Ref. 8� calculations, compared with experimental values of Ref. 1.

Present GGA+U

U=0 U=3 eV Full potential Expt.

NM FM AF Ref. 6 Ref. 8 Ref. 1

a 4.197 4.246 4.249 4.120 4.259 4.232

c 6.693 6.917 6.883 6.600 6.870 6.786

z 0.304 0.311 0.312 0.304 0.310 0.312

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The electronic structure changes in a drastic way whenlocal potentials due to finite U and J are present �see Fig. 1�.The overall density of states N�E� redistributes, and its valueat the Fermi energy drops to N�EF�=4.32 eV−1, showing asimilarly strong renormalization to that found by Shick, Ja-niš, and Oppeneer.10 To understand better these changes weanalyzed the projected densities of states for s, p, d, and forbitals in Fig. 2 and compared them with those at U=J=0�not shown�. When U and J increase, the f-electron densityof states moves away from the Fermi energy and consists ofcharacteristic maxima, both below and above EF. Below EFone finds also two broader features with predominantly d andp character at ��−2 eV—we have verified that these fea-tures are similar in the NM state at U=J=0. The s statesextend over a broad energy range, but show two main fea-tures with large intensity around −5 and −7 eV. The posi-tions and intensities of all non-f bands are in good agreementwith the tight-binding muffin-tin orbital method.5

The decreased spectral weight of f states near the Fermienergy found at U=3 eV and J=0.7 eV follows from themagnetic splitting of f states which are fully polarized, asshown in Fig. 3. This splitting is somewhat larger than U andmoves the spectral weight for the minority-spin f statesabove the Fermi energy. At the same time, the broad maxi-mum for the majority band moves from above to below EFand is found at E�−0.7 eV in Fig. 3. As a result, the fdensity of states at EF is strongly reduced as compared withthe NM state at U=J=0 and is only due to the majority-spinband.

III. PHONON DISPERSIONS

The phonon dispersion curves were calculated using thedirect method,13 which consists of following steps. First, theHF forces are obtained for the crystal with atoms displacedfrom their equilibrium positions. The number of necessarydisplacements is determined by the group symmetry and bythe number of nonequivalent atoms. A complete set of HFforces for PuCoGa5 was obtained from nine independentatomic displacements �equal to 0.02 �: two for Pu, Co, andGaI �along x and z� and three for GaII �along x, y, and z�, inthe 2�2�1 supercell. Next, the matrices of force constantsare found using the singular value decomposition method.Finally, the dynamical matrix is constructed and diagonalizedfor each wave vector k.

TABLE II. Maximal values of the force constants �in eV Å−1�,as obtained for local distortions �Rij =0� and for two representativevalues of distance Rij �in Å�, within the 2�2�1 supercell ofPuCoGa5.

Rij 0.0 4.25 6.0

Pu 11.78 −0.21 −0.03

Co 8.13 −0.71 0.03

GaI 5.83 0.17 0.07

GaII 7.40 0.07 0.01

FIG. 1. The total electronic density of states N�E�, as obtainedfor PuCoGa5 in the PAW method with U=J=0 �NM state, top� andin the GGA+U �Ref. 17� with U=3 eV and J=0.7 eV �FM state,bottom�.

FIG. 2. Partial �s, p, d, f� electron densities of states, as obtainedfor the FM state of PuCoGa5 in the GGA+U method with U=3 eV and J=0.7 eV.

FIG. 3. Spin-dependent f densities of states for up-spin �top�and down-spin �bottom� electrons, as obtained for the FM state ofPuCoGa5 in the GGA+U method with U=3 eV and J=0.7 eV.

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The direct method allows one to determine the exact pho-non frequencies for a few k points in the Brillouin zone�usually at high-symmetry points�. The number of thesepoints depends on the group symmetry and on the supercell

size; here phonon frequencies were calculated at all high-symmetry points. If the interaction range is limited to thesupercell, phonon frequencies in the entire Brillouin zonehave correct values. This is the case of PuCoGa5, as we showin Table II. Indeed, force constants for Pu and Co atomsdecrease by three orders of magnitude and for Ga atoms bytwo orders of magnitude inside the supercell, so all phononfrequencies should be computed with good accuracy withinthe present approximation.

Eighteen optical modes of Fig. 4 �E modes are doublydegenerate� are classified in the zone center �at the � point�according to the following irreducible representations:

� = A1g + B1g + 2Eg + 3A2u + B2u + 4Eu. �1�

The obtained phonon frequencies are given in Table III to-gether with their optical activities—R and I denote the Ra-man and infrared active modes. The frequencies ��U� of allthe modes depend on Coulomb interaction U, and we quan-tified this dependence by a parameter

��U� =��0� − ��U�

��U�. �2�

The largest frequency changes were found for two low-energy modes A2u and Eu at frequencies 2.90 and 2.93 THz�Table III�. These changes follow both from the modifiedgeometry at finite U and from the modified force constantswhen the f-electron distribution becomes more rigid due tofinite Coulomb repulsion.

Let us analyze different optical modes in more detail. AllRaman modes involve only movements of GaII atoms. B1gand A1g are out-of-plane �along the z direction� and two Egmodes are in-plane vibrations of GaII atoms �Fig. 4�. In thelowest infrared A2u mode, Pu and GaI oscillate along the zdirection with opposite phases. The infrared Eu mode with��2.93 THz ��141 K� includes mainly in-plane vibrations

TABLE III. Phonon frequencies � �in THz� of different symmetry as obtained at the � point at U=3 eV and U=0, the relative shift of phonon frequency ��U� �in percent; see Eq. �2��, and their opticalactivities in Raman �R� and infrared �I� spectroscopies. Four infrared-active modes with Pu character aremarked by boldface.

�

�

��U� ActiveU=3 eV U=0

Eg 2.733 3.192 17 R

A2u 2.896 3.815 32 I

Eu 2.929 3.691 26 I

B2u 3.589 4.097 14 —

Eu 4.002 4.297 7 I

A2u 4.216 4.623 10 I

Eg 5.273 5.688 8 R

Eu 5.330 5.585 5 I

B1g 5.835 6.151 5 R

A2u 5.972 6.406 7 I

A1g 6.409 6.619 3 R

Eu 6.650 7.365 11 I

FIG. 4. �Color online� Schematic representation of optical pho-non modes in PuCoGa5, with atomic displacements at the � point�arrows� and their phonon frequencies in THz given in parantheses,as obtained with U=3 eV and J=0.7 eV. Pu ions participate in twoEu �2.93 and 5.33 THz� and two A2u �2.90 and 4.22 THz� modes.

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of Pu atoms. Two higher infrared Eu modes, with frequencies��4.00 and 5.33 THz, consist of in-plane vibrations of Coand GaI atoms. The highest infrared A2u and Eu modes rep-resent out-of-plane and in-plane vibrations of Co atoms, re-spectively �Fig. 4�.

Local Coulomb interactions on Pu atoms �U and J�modify phonon frequencies in the entire Brillouin zone. Inparticular, one finds significant changes in the modes belong-ing to the �-X-M plane. For U=0, the lowest acousticphonons with polarization along the z direction are separatedfrom other. modes �Fig. 5�. For finite U, the transverse opticmode of the lowest energy softens and mixes with the acous-tic phonon at point X �Fig. 6�. Both modes include primarilyout-of-plane movements of Pu and GaI atoms. The interac-tion between acoustic and optic modes is also visible alongthe X-M and �-M directions.

All these effects can be explained by the dependence offorce constants on U. As we observe, the force constantsrelated to displacements of gallium atoms are very sensitiveto changes in U. For GaI atoms, the force constant in the zdirection is strongly reduced �more than 50%� when Uchanges from U=0 to U=3 eV. There is also a significantdecrease in force constants related to the in-plane displace-ments of GaII atoms. It explains the large softening of thelowest optic mode along the �-M direction and near point A.

Total and partial phonon densities of states ����, shown inFig. 7, were obtained by summation over randomly selectedwave vectors in the Brillouin zone. The heaviest Pu atoms

vibrate with the lowest frequencies below �=4 THz, as in-dicated by a distinct maximum in �Pu��� at ��2.5 THz, andone finds an average frequency ��Pu�=3.26 THz�156 K, avalue being much lower than the estimated Debye tempera-ture 240 K,1 or the average phonon frequency ���=4.42 THz �212 K�. As expected, vibrations of Ga atoms,which participate in several modes, span a broad spectrum offrequencies in �Ga��� and provide a dominating contributionto the total density of states ����, which we used to calculatethe basic thermodynamic functions. Finally, Co atoms con-tribute mainly to the highest and almost dispersionless Euoptical mode at ��6.65 THz and give large phonon densi-ties �Co��� at high frequensies �6 THz. We emphasizethat although two modes with appreciable Pu character arestrongly renormalized when U increases, the largest changein the overall partial density of states is found for Ga atoms.

FIG. 5. Phonon dispersion relations as obtained for PuCoGa5

along the high-symmetry directions in the Brillouin zone, as ob-tained for U=J=0. The special points are �= �0,0 ,0�, X= � 1

2 ,0 ,0�,M = � 1

2 , 12 ,0�, Z= �0,0 , 1

2�, R= � 1

2 ,0 , 12

�, and A= � 12 , 1

2 , 12

�.

FIG. 6. The same as in Fig. 5, but for U=3 eV andJ=0.7 eV.

FIG. 7. Total ���� and partial �A��� �A=Pu,Ga,Co� phonondensities of states for PuCoGa5, as obtained at U=3 eV, J=0.7 eV �solid lines� and at U=J=0 �dashed lines�.

FIG. 8. Temperature dependence of the lattice heat capacity atconstant volume Cph�T� for PuCoGa5 and sublattice contributions:Pu �dash-dotted line�, Ga �dashed line�, and Co �dotted line�. Theinset shows the phonon Cph�T� /T �solid line� and electron Ce�T� /T�dashed line� terms in the total heat capacity in the low-temperatureregime.

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The phonon heat capacity Cph�T� and its contributions dueto sublattices are presented in Fig. 8. Note that at low tem-perature only Pu and Ga atoms contribute to Cph�T�. Thephonon heat capacity exceeds the electronic contributionCe�T�=T, with =77 mJ/mol K2 �Ref. 1�, for T11 K.

IV. DISCUSSION AND SUMMARY

The mechanism of superconductivity in PuCoGa5 remainsan open question. The critical temperature Tc�18.5 K is stillin the range of standard phonon-mediated mechanisms. Onthe one hand, in several systems, like in A-15 compounds orin MgB2,26 the phonon-mediated superconductivity leadseven to higher transition temperatures �23–40 K�. On theother hand, a remarkable similarity between PuCoGa5 andf-electron heavy-fermion systems or, to some extent, high-Tccopper-oxide superconductors might suggest a commonmechanism. In PuCoGa5, the main role in superconductingtransport is played by PuGaI planes, in analogy to CuO2planes in the cuprates. A two-dimensional character of theFermi surface in PuCoGa5 was demonstrated by the DFTcalculations.6,9 The origin of superconductivity and the roleof phonons are still under debate in the cuprates,27 and nu-merous experiments point out the importance of the high-frequency breathing mode in the CuO2 planes.28

In order to estimate the electron-phonon coupling con-stant �, we computed partial contributions �� due to indi-vidual phonon modes at k=0 �at the � point�. For each pho-non mode of frequency ��, �� can be calculated using theformula29

�� =N�EF��g�,0

2 �EF

��2 . �3�

The electron-phonon matrix elements are given by30

g�,k,n = A�

eA��

MA

�k,n��A�V�k,n� , �4�

where n labels different electron bands, A runs over all atomsin the unit cell, �=x ,y ,z labels the directions of local dis-placements, eA�

� are the orthonormalized polarization vectors,and �A�V is the change of the potential associated with thedisplacement uA� of atom A. Diagonal matrix elements inEq. �4� can be obtained from the deformation potential forthe state �k ,n�,31

�k,n��A�V�k,n� =� k,n

�uA�

, �5�

where k,n are electron energies near the Fermi level.It may be expected that the phonon modes which interact

strongly with high 5f density of states at the Fermi energy inPuCoGa5 involve rather low-frequency vibrations of Pu at-oms. These are predominantly the acoustic modes and thelow-energy optical A2u and Eu modes �at 2.90 and2.93 THz�, which directly modify the Pu–Pu distance. Forthe actual estimation of � we used the calculations performedin the NM state at U=J=0, as problems with convergencehindered good accuracy at finite U=3 eV. Following the

above approach, we estimated the electron-phonon couplingconstant � using the phonon modes at the � point. Here wefound the largest contributions due to the B2u and Eg modeswhich involve Ga atoms, at frequencies 4.10 and 5.69 THz atU=0. Averaging over all optical modes one finds an esti-mated value of ��0.7. For a rough estimate of Tc, one maythen employ the Allen-Dynes formula32

Tc =���1.20

exp�−1.04�1 + ��

� − �*�1 + 0.62�� , �6�

for a non-s-wave pairing channel. Using the above value of arelatively strong electron-phonon coupling ��0.7, ���=212 K, and a representative value of the effective Coulombrepulsion, �*=0.1, one finds Tc�7.4 K ��14.1 K is an up-per limit at �*=0�. Although further increase of Tc couldfollow from large effective mass, these values are still toosmall to explain the observed transition temperature Tc=18.5 K. A better estimation of the transition temperaturecould be achieved by determining the electron-phonon cou-pling constant by averaging over the entire Brillouin zoneand will be the subject of future study.

In summary, we identified four optical phonon modes �atfrequencies 2.90, 2.93, 4.22, 5.33 THz for U=3 eV� withdominating Pu character. By considering all optical modeswe determined a relatively strong electron-phonon couplingconstant ��0.7, and a similar value of 0.8 was deducedfrom the present calculations using only the modes withdominating Pu character. We found that the estimated valuesof Tc are too low to explain the superconductivity inPuCoGa5 by the phonon mechanism alone. Thus, we con-clude that another electronic mechanism is necessary andprobably even a driving force for the pairing. However, sig-nificant changes in phonon frequencies due to local Coulombinteractions indicate possible interplay of electron correla-tions and phonons in the mechanism enhancing Tc.

After this paper was completed, we learned about the re-cent measurements33 of the nuclear spin-lattice relaxationrate and Knight shift, which suggest that PuCoGa5 is an un-conventional superconductor with antiferromagnetic fluctua-tions as likely pairing mechanism. This provides additionalsupport to our main conclusion that the phonon mechanismalone does not suffice to explain the superconducting prop-erties of PuCoGa5.

ACKNOWLEDGMENTS

We thank R. Zeyher for valuable discussions, as well asA. Szajek and J. A. Morkowski for providing the details oftheir electronic structure calculations published in Ref. 5.The computations were partly performed at the Interdiscipli-nary Center for Mathematical and Computational Modellingat Warsaw University. A.M.O. acknowledges partial supportby the Polish State Committee of Scientific Research �KBN�under Project No. 1 P03B 068 26.

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