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Intrasite 4F-5D Electronic Correlations in the Quadrupolar Model of the ␥-␣ Phase Transition in Ce

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Intrasite 4F-5D Electronic Correlations in the Quadrupolar Model of the ␥-␣ Phase Transition in Ce PHYSICAL REVIEW B 66, 054103 ͑2002͒ Intrasite 4f-5d electronic correlations in the quadrupolar model of the ␥-␣ phase transition in Ce A. V. Nikolaev1,2 and K. H. Michel1 1Department of Physics, University of Antwerp, UIA, 2610 Antwerpen, Belgium 2Institute of Physical Chemistry of RAS, Leninskii prospect 31, 117915, Moscow, Russia ͑Received 5 December 2001; revised manuscript received 15 March 2002; published 2 August 2002͒ As a possible mechanism of the ␥-␣ phase transition in pristine cerium a change of the electronic density from a disordered state with symmetry Fm¯3m to an ordered state Pa¯3 has been proposed. Here we include on-site and intersite electron correlations involving one localized 4 f electron and one conduction 5d electron per atom. The model is used to calculate the crystal field of ␥-Ce and the temperature evolution of the mean field of ␣-Ce. The formalism can be applied to crystals where quadrupolar ordering involves several electrons on the same site. DOI: 10.1103/PhysRevB.66.054103 PACS number͑s͒: 64.70.Kb, 71.10.Ϫw, 71.27.ϩa, 71.45.Ϫd I. INTRODUCTION 20 theory of the orientational phase transitions in solid C60 Elemental solid cerium is known to undergo structural where a similar space symmetry change (Fm¯3m!Pa¯3) oc- 1,2 phase transitions. In the pressure-temperature phase dia- curs at 255 K at room pressure.21 gram of Ce the puzzling long-standing problem is to under- The present article is a continuation of our approach to the stand the apparently isostructural transition between the cu- problem of the ␥-␣ transition in Ce based on the technique bic ␥ and ␣ phases.1,3 An isostructural phase transformation of multipolar interactions between electronic densities of cannot be ascribed to the condensation of an order parameter conduction and localized electrons in a crystal.17,18 Our sec- and therefore cannot be explained by the Landau theory of ond motivation is to extend our initial method ͑Ref. 17͒ for phase transitions. In the past, several models and theories the case when two electrons ( f and d) are at the same site of have been suggested to address this problem.4–12 Among cerium and all intrasite interactions ͑including the on-site them, a Mott-like transition for 4 f electrons4,5 and a Kondo- exchange͒ between them are taken into account. Our treat- effect-based approach9–12 are the competing ones. Also, new ment of intrasite correlations is closely related with the computational schemes have been applied to the problem method used by Condon and Shortley for many electron 22 using dynamical mean-field theory combined with the local states of atoms. Provided that the average number of elec- density approximation.13 trons per site is conserved this method is exact for intrasite ␣ correlations and goes beyond the usual self-consistent-field Under pressure above 5 GPa -Ce becomes unstable and 23–25 transforms first to a crystal with C2/m or ␣-U space approach employed by band structure calculations. 2 ␣Ј Besides the problem of the ␥-␣ phase transition in Ce, the symmetry ( -Ce) and then to a body-centered-tetragonal 26 ͑bct͒ structure (␣Љ-Ce) above 12 GPa.14 This series of trans- microscopic method can be applied further to describe qua- formations cannot be explained by invoking the concepts of drupolar ordering and to perform crystal field calculations of the 4 f localization-delocalization transition or Kondo vol- many f electrons on the same site. There are numerous com- pounds exhibiting quadrupolar ordering at low ume collapse models and indicates that there are anisotropic 15 ␣ temperatures and there is a sustained interest in understand- interactions present in the phase of cerium. Such electron ͑ ͒ ing their properties. Thus, recently DyB2C2 Ref. 27 , DyB6 interactions can be of quadrupolar origin, which are known ͑ ͒ ͑ ͒ ͑ ͒ ͑ Ref. 28 , UCu2Sn Ref. 29 , PrPb3 Ref. 30 , YbAs, Ref. to drive symmetry lowering phase transitions in many lan- ͒ ͑ ͒ thanide and actinide compounds.15 31 , YbSb Ref. 32 were reported to undergo a quadrupolar Recently, the isostructural character of the ␥-␣ phase ordering. transition has been questioned by Eliashberg and Capellmann.16 They suggest that ␣-Ce should have a dis- II. MODEL torted fcc structure. Independently, the present authors have 17,18 put forward a theory of quadrupolar ordering in cerium. As follows from electronic band structure calculations of There, it was proposed that the ␥-␣ transformation is not ␥-Ce there exist three conduction electrons per atom which really isostructural. Rather, it was associated with hidden form the (6s6p5d)3 metallic band and one localized 4 f 19 electronic degrees of freedom. In our previous work ͑Refs. electron.33,6–8 In our previous work ͑Ref. 18͒, we have al- 17 and 18͒, we have suggested that the symmetry change is ready considered electric multipole interactions between from Fm¯3m to Pa¯3. This symmetry lowering is a special conduction electrons and the localized 4 f electrons. Below one. Although accompanied by a lattice contraction, it con- we focus on the on-site and intersite correlations in the sys- serves the fcc structure of the atomic center-of-mass points tem and will simplify the model. We consider the instanta- ͑cerium nuclei͒ and is solely due to the orientational order of neous configuration 6s25d4 f as having the largest statistical electronic densities. Such a scenario reconciles the ␥-␣ weight on a cerium site ͑in comparison with other possibili- transformation with the Landau theory of phase transitions. ties such as 6s6p5d4 f ,6s25d2,6s24 f 2, etc.͒. The two 6s Our considerations for cerium have been inspired by the electrons give only a spherically symmetrical density on the 0163-1829/2002/66͑5͒/054103͑12͒/$20.0066 054103-1 ©2002 The American Physical Society A. V. NIKOLAEV AND K. H. MICHEL PHYSICAL REVIEW B 66, 054103 ͑2002͒ R R cerium site and their lowest-energy state corresponds to a Here f and d are radial components of the 4 f and 5d singlet. Therefore, as a first approximation we discard them electrons, respectively; nˆ stands for ⍀(nជ ). There are 14 ori- ͑ as giving a closed shell. Indeed, the level structure of entational vectors ͑or spin-orbitals͒ ͗nˆ ͉i ͘ fora4f electron 6s25d4 f corresponding to atomic Ce I is similar to that of f ϭ ͗ ˆ ͉ ͘ 34͒ (i f 1 –14) and 10 orientational vectors n id fora5d elec- 5d4 f of La II. In fact, in doing so we omit the s-d elec- ϭ ͒ tron transitions which can contribute to the quadrupolar tron (id 1-10 . These spin-orbitals can be written as density.18 We are left then with one 5d conduction electron ͗ ˆ ͉ ͘ϭ͗ ˆ ͉ ͘ ͑ ͒ ͑ ͒ and one localized 4 f electron. In the electronic band struc- n i f n m f us„sz f …, 2.5a ture calculations the charge density of the 5d electron on the ͗ ˆ ͉ ͘ϭ͗ ˆ ͉ ͘ ͑ ͒ ͑ ͒ cerium site is considered as an average over all occupied kជ n id n md us„sz d …. 2.5b ជ ␣ р ␣ ϭϮ states ͓E(k, ) EF , where is the band index and EF is Here us is the spin function (s ) for the spin projections the Fermi level͔. We have shown in Appendix B of Ref. 17 ϭϮ ˆ ͉ sz 1/2 on the z axis. The orbital parts ͗n m f ͘ (m f that the electron density is mainly spherical which corre- ϭ1 –7) and ͗nˆ ͉m ͘ (m ϭ1 –5) are expressed in terms of sponds to the standard ‘‘muffin-tin’’ ͑MT͒ treatment in elec- d d m ⍀ ϭ͗ ˆ ͉ ͘ tronic band structure calculations. The spherical density of spherical harmonics Y l ( ) n l,m . We find it convenient 5d and 4 f electrons will be the starting point in this work. to work with real spherical harmonics. We consider We consider the 5d electron on a cerium center being instan- ͕ 0 1c 1s 2c 2s 3c 3s͖ϭ͗ ˆ ͉ ͘ ͑ ͒ taneously coupled with the 4 f electron and include in the Y 3 ,Y 3 ,Y 3 ,Y 3 ,Y 3 ,Y 3 ,Y 3 n m f 2.6a model all corresponding intrasite interactions, crystal electric for 4 f electrons ͑corresponding to m ϭ1 –7) and field effects, and intersite multipolar electric interactions. f From the technical point of view, this fd model is a many ͕Y 0 ,Y 1c ,Y 1s ,Y 2c ,Y 2s͖ϭ͗nˆ ͉m ͘ ͑2.6b͒ electron generalization of the concepts of Ref. 17. We are 2 2 2 2 2 d ͑ ϭ aware that the model based on the fd configuration is incom- for 5d electrons corresponding to md 1 –5). We use the plete, but it has an advantage of taking into account all in- definition of real spherical harmonics of Ref. 37 ͓see also trasite interactions ͑often referred to as Hund’s rules͒ which explicit expressions ͑2.1͒ in Ref. 17͔, which is different from are usually omitted in the electron band structure the definition of Condon and Shortley.22 The advantage of calculations.23,24 Later we will briefly discuss a possibility to using the basis with real spherical harmonics is that the ma- refine our model with the help of the valence bond ͑or trix elements of Coulomb and exchange interactions stay Heitler-London͒ theory of chemical bonding.35,36 real. We consider a face-centered-cubic ͑fcc͒ crystal of N Ce The order of indices in Eqs.
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