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 compounds 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

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. ͑2.2͒ and ͑2.3͒ is important if atoms. Each atomic site possesses one 4 f and one 5d elec- we associate the first electron with the f state i f while the tron. The position vector of an electron near a crystal lattice second with the d state id . Then in addition to the vectors site nជ is given by ͑2.2͒ we have to consider the states described by the vectors ͉ ជ ͑ id ;i f ͘n the first electron is in the id state and the second is ͒ Rជ ͑nជ ͒ϭXជ ͑nជ ͒ϩrជ͑nជ ͒. ͑2.1͒ in the i f state . However, from the dynamical equivalence of the electrons we can permute the spin-orbitals to the standard ͑ ͒ Here Xជ (nជ ) is the lattice vector which specifies the centers of order, Eq. 2.2 , by using the atoms ͑or Ce nuclei͒ on a rigid fcc lattice. The radius ͉i ;i ͘ ជ ϭϪ͉i ;i ͘ ជ , ͑2.7͒ ជ ជ ជ ⍀ ជ d f n f d n vector r(n) is given in polar coordinates by „r(n), (n)…, where r is the length and ⍀ϭ(⌰,␾) stands for the polar since it requires the interchange of the two electrons. In order angles. We label the two-electron basis ket vectors at a lattice to describe the same quantum state (i f ;id) we will use the ជ basis vectors ͑2.2͒ and apply Eq. ͑2.7͒ when needed. ͓Alter- site n by a single index I fd or, alternatively, by the pair of natively, one can use the procedure of antisymmetrization of single-electron indices (i f ,id): the basis vectors ͑2.2͒ as described elsewhere.͔ Thus, our ͉ ជ ϭ͉ ជ ͑ ͒ basis ͑2.2͒ consists of 140 different vectors ͉I ͘. I fd͘n i f ;id͘n . 2.2 fd The ground-state energy of the (4 f 5d) electron system, The index i stands for the electron orbital and spin projection E0, can be calculated in the local density approximation quantum numbers. The corresponding basis wave functions ͑LDA͒ with spherically symmetric Coulomb and exchange are potentials. Going beyond this model in atomic cerium, one has to take into account multipolar on-site ͑also called intra- ͒ ͗rជ,rជЈ͉I ͘ ជ ϭ͗rជ͉i ͘ ជ ͗rជЈ͉i ͘ ជ , ͑2.3͒ site Coulomb interactions and spin-orbit coupling. In solid fd n f n• d n cerium, the interactions with conduction electrons and inter- where site Coulomb interactions still have to be added. In the following we will study these effects within a uni- fied formalism based on a multipole expansion of the Cou- ͗rជ͉i ͘ ជ ϭR ͑r͑nជ ͒͒͗nˆ ͉i ͘, ͑2.4a͒ f n f f lomb potential and of the systematic use of site symmetry of the crystal lattice. For the case of on-site Coulomb interac- ជ ͉ ជ ϭR ជ ͒͒ ˆ ͉ ͑ ͒ ͗rЈ id͘n d͑rЈ͑n ͗nЈ id͘. 2.4b tions between two electrons ͑charge eϭϪ1) we have

054103-2 INTRASITE 4 f -5d ELECTRONIC CORRELATIONS IN . . . PHYSICAL REVIEW B 66, 054103 ͑2002͒

18 1 part of these correlations in previous work, and here we ͑ ជ ͑ ជ ͒ ជ Ј͑ ជ ͒͒ϭ ͑ ͒ V R n ,R n ជ ជ ជ ជ . 2.8 study all of them in the framework of the fd model. In fact, ͉r͑n͒ϪrЈ͑n͉͒ these multipole interactions are responsible for the electronic 22 The multipole expansion in terms of site-symmetry-adapted terms of atoms. We will see that their combined effect low- ers the energy of the cerium atom by ϳ1 –2 eV in compari- functions ͑SAF’s͒ S⌳(nˆ ) reads son with the spherically symmetric case, yet usually they are not taken into account in the electronic band structure calcu- ជ ͑ ជ ͒ ជ Ј͑ ជ Ј͒ ϭ ͑ Ј͒ ͑ ˆ ͒ ͑ ˆ Ј͒ V„R n ,R n … ͚ v⌳⌳ r,r S⌳ n S⌳ n , lations in solids. Although our consideration in this section is ⌳ based on the original technique for multipole interactions,17 ͑ ͒ 2.9a it overlaps largely with the method of Condon and where Shortley.22 However, for two reasons we have decided to briefly review it here. First, we consider below a more gen- l rϽ 4␲ eral case which is not limited by the LS ͑Russell-Saunders͒ v⌳⌳ ͑r,rЈ͒ϭͩ ͪ ␦⌳⌳ , ͑2.9b͒ Ј ϩ Ј coupling and consideration of diagonal matrix elements. Sec- r(l 1) 2lϩ1 Ͼ ond, the results of this section are used to describe crystal ϭ ϭ ␦ ϭ␦ ␦ with rϾ max(r,rЈ), rϽ min(r,rЈ), and ⌳⌳Ј ␶␶Ј llЈ . electric field effects and the phase transition to the Pa¯3 Clearly, the last expression is site independent. The SAF’s structure. are linear combinations of spherical harmonics and transform The direct matrix elements for the intrasite Coulomb in- as irreducible representations of the site point group ͑Ref. teractions are obtained if we consider only the f -f transitions 37͒. The index ⌳ stands for (l,␶), with ␶ϭ(⌫,␮,k). Here l for the first electron and the d-d transitions for the second. accounts for the angular dependence of the multipolar expan- We start from Eq. ͑2.9a͒ and obtain sion, ⌫ denotes an irreducible representation ͑in the present case of the group O ), ␮ labels the representations that oc- h Coul ͉ ជ ជ ជ ͒ ជ ជ ͉͒͒ ϭ FD ͒ ͒ cur more than once, and k denotes the rows of a given rep- ͗I fd nV͑R͑n ,RЈ͑n J fd͘ ជ ͚ v⌳⌳c⌳͑i f j f c⌳͑id jd , n ⌳ resentation. ͑ ͒ On the other hand, the Coulomb interaction between two 3.1 ជ Þ ជ Ј ͑ ͒ electrons at different sites n n intersite reads where 1 ជ ͑ ជ ͒ ជ Ј͑ ជ Ј͒ ϭ ͑ ͒ V„R n ,R n … ជ ជ ជ ជ . 2.10 FDϭ ͵ 2 ͵ Ј Ј2R 2͑ ͒R 2͑ Ј͒ ͑ Ј͒ ͉R͑n͒ϪRЈ͑nЈ͉͒ v⌳⌳ dr r dr r f r d r v⌳⌳ r,r ͑ ͒ The multipole expansion is given by 3.2

accounts for the average radial dependence while v⌳⌳(r,rЈ) ជ ͑ ជ ͒ ជ Ј͑ ជ Ј͒ ϭ ͑ ជ ជ Ј Ј͒ ͑ ˆ ͒ ͑ ˆ Ј͒ V„R n ,R n … ͚ v⌳⌳Ј n,n ;r,r S⌳ n S⌳Ј n , is given by Eq. ͑2.9b͒. We use the superscripts F and D in ⌳⌳Ј order to indicate that we have transitions between two 4 f ͑2.11a͒ states ͓Fϵ( f , f )͔ and the transitions between two 5d states where ͓Dϵ(d,d)͔. The elements c⌳ are deﬁned by

S⌳͑nˆ ͒S⌳ ͑nˆ Ј͒ ͑ ជ ជ Ј Ј͒ϭ ⍀͑ ជ ͒ ⍀Ј͑ ជ Ј͒ Ј v⌳⌳Ј n,n ;r,r ͵ d n ͵ d n . ͑ ͒ϭ ͵ ⍀͗ ͉ ˆ ͘ ͑ ˆ ͒͗ ˆ ͉ ͘ ͑ ͒ ͉Rជ ͑nជ ͒ϪRជ Ј͑nជ Ј͉͒ c⌳ i f j f d i f n S⌳ n n j f , 3.3a ͑2.11b͒ The intersite multipole expansion ͑2.11a͒ is anisotropic and 38 ͑ ͒ϭ ͵ ⍀͗ ͉ ˆ ͘ ͑ ˆ ͒͗ ˆ ͉ ͘ ͑ ͒ converges fast since c⌳ id jd d id n S⌳ n n jd . 3.3b

͑ ͒l͑ Ј͒lЈ ជ ជ r r v⌳⌳ ͑n,nЈ;r,rЈ͒ϳ . ͑2.12͒ The other possibility is to consider the transitions 4 f Ј ជ ជ ជ ជ lϩlЈϩ1 ͉X͑n͒ϪX͑nЈ͉͒ →5d for the first electron and the transitions 5d→4 f for the second. This gives the exchange interactions and then we Therefore, it is sufficient to consider it only for nearest should use Eq. ͑2.7͒ in order to return to the standard order neighbors. From the practical point of view, one can calcu- ជ ជ Ј Ј Ј of the spin-orbitals. We find late v⌳⌳Ј(n,n ;r0 ,r0) only for two fixed radii r0 and r0 . ជ ជ Ј Ј Ј Then one obtains v⌳⌳Ј(n,n ;r,r ) as a function of r and r ជ ជ ជ ជ exch ͗I ͉ ជ V R͑n͒,RЈ͑n͒ ͉J ͘ ជ by employing the dependence ͑2.12͒. fd n „ … fd n

ϭϪ ( fd)(df) ͒ ͒ ͑ ͒ III. INTRASITE INTERACTIONS ͚ v⌳⌳ c⌳͑i f jd c⌳͑id j f , 3.4 ⌳ The interactions which we analyze in this section are present already in atomic cerium.22,34 We have considered a where

054103-3 A. V. NIKOLAEV AND K. H. MICHEL PHYSICAL REVIEW B 66, 054103 ͑2002͒ v( fd) (df) TABLE I. Term energies of the fd configuration in the absence ⌳ ⌳ ⌬ of spin-orbit coupling. Em and E stand for the singlet-triplet energy means and the singlet-triplet energy differences, respectively. L ϭ ͵ 2 ͵ Ј Ј2R ͑ ͒R ͑ ͒R ͑ Ј͒R ͑ Ј͒ ͑ Ј͒ drr dr r f r d r d r f r v⌳⌳ r,r is the orbital quantum number of the two electrons. The energy corresponding to the spherically symmetric description is taken as ͑3.5͒ zero. All energies are in units K. and ⌬ L Singlets Triplets Em E ͑ ͒ϭ ͵ ⍀͗ ͉ ˆ ͘ ͑ ˆ ͒͗ ˆ ͉ ͘ ͑ ͒ P 13 541.8 5384.5 9463.1 8157.3 c⌳ i f jd d i f n S⌳ n n jd , 3.6a D Ϫ3011.8 1951.8 Ϫ530.0 Ϫ4963.5 F 2767.8 Ϫ6616.1 Ϫ1924.1 9383.9 df͑ ͒ϭ ͵ ⍀͗ ͉ ˆ ͘ ͑ ˆ ͒͗ ˆ ͉ ͘ ͑ ͒ G Ϫ9378.4 Ϫ1485.3 Ϫ5431.8 Ϫ7893.1 c⌳ id j f d id n S⌳ n n j f . 3.6b H 12 310.8 Ϫ5653.4 3328.7 17 964.2 We observe that in the basis with real orbitals, Eq. ͑2.6a͒ and ͑2.6b͒, and with real functions S⌳ the coefficients c⌳ are real 4␲ and we get v( fd)ϭ G5͑4 f ,5d͒. ͑3.9͒ 5 11 ͒ϭ ͒ ͑ ͒ c⌳͑id j f c⌳͑ j f id . 3.7 ͑Notice, however, that our coefficients c⌳ are different from We start with the description of spherically symmetric those of Condon and Shortley.22͒ We have estimated the in- ϭ ͑ ͒ R R terms (l 0) corresponding to the trivial function S0 tegrals in Eqs. 3.9 from the radial dependences f and d ϭ1/ͱ4␲. The coefficients c⌳ in Eqs. ͑3.3a͒ and ͑3.3b͒ be- obtained from calculations on a cerium atom in the LDA. We FDϭ FDϭ ( fd)ϭ come diagonal, have found that v2 85 858, v4 24 260, v1 83 128, ( fd)ϭ ( fd)ϭ v3 26 563, and v5 12 584, in kelvin. 1 1 Finally, we rewrite expressions ͑3.1͒ and ͑3.4͒ in matrix ͑ ͒ϭ ␦ ͑ ͒ϭ ␦ clϭ0 id jd i j , clϭ0 i f j f i j , form as ͱ4␲ d d ͱ4␲ f f ͑ ͒ 3.8 ͗ ͉ ͑ ͉͒ ͘Coulϭ FD FD͑ ͉ ͒ϩ FD FD͑ ͉ ͒ I fd V intra J fd v2 c2 I fd J fd v4 c4 I fd J fd ϭ ϭ ͑ ͒ while clϭ0(i f jd) clϭ0(id j f ) 0. Hence, we obtain a contri- 3.10a bution to ͗I͉V͉J͘Coul which is proportional to the unit ma- and trix. Since it corresponds only to a shift of the ground-state energy, it is irrelevant. ͗I ͉V͑intra͉͒J ͘exch In considering the other contributions ͑with lϾ0) in Eqs. fd fd ͑ ͒ ͑ ͒ ϭϪ͓ ( fd) ( fd)͑ ͉ ͒ϩ ( fd) ( fd)͑ ͉ ͒ 3.1 and 3.4 we will take advantage of the selection rules v1 c1 I fd J fd v3 c3 I fd J fd imposed by the coefficients c⌳ , Eqs. ͑3.3a͒, ͑3.3b͒ and ϩ ( fd) ( fd)͑ ͉ ͔͒ ͑ ͒ ͑3.6a͒, ͑3.6b͒. First of all, we notice that the coefficients c⌳ v5 c5 I fd J fd , 3.10b are diagonal in terms of spin components us . From the FD ͉ theory of addition of angular momenta ͑see, for example, Here the direct Coulomb matrices cl (I fd J fd) are defined ͒ as Ref. 36 we know that nonzero coefficients c⌳(i f j f ) can occur if l ͑in ⌳) equals to 0,1,2,...,6,Eq.͑3.3a͒. Furthermore, FD the odd values of l are excluded due to the parity of the c ͑I ͉J ͒ϭͩ ͚ c ␶ ͑i j ͒c ␶ ͑i j ͒ͪ , ͑3.11a͒ l fd fd (l, ) f f (l, ) d d integrand in Eq. ͑3.3a͒ and finally we obtain that lϭ0, 2, 4, ␶ and 6. Analogously, for the d-d transitions the allowed coef- where lϭ2 and 4, and the three ‘‘exchange’’ matrices ficients are with lϭ0, 2, and 4, Eq. ͑3.3b͒. For the f -d tran- c( fd)(I ͉J )(lϭ1, 3, and 5͒ are given by sitions, Eq. ͑3.6a͒, ͑3.6b͒, we find that lϭ1, 3, and 5. Next l fd fd R R we notice that if the radial parts f , d , are the same for all ( fd) spin-orbitals of 4 f and 5d states, correspondingly, then the c ͑I ͉J ͒ϭͩ ͚ c ␶ ͑i j ͒c ␶ ͑i j ͒ͪ . l fd fd (l, ) f d (l, ) d f integrals ͑3.2͒ and ͑3.5͒ depend only on l. We can condense ␶ FDϭ FD ϭ ( fd) ͑3.11b͒ the notation vl v⌳⌳ for l 0, 2, and 4 and vlЈ ( fd) (df) ϭv⌳ ⌳ for lЈϭ1, 3, and 5. In fact, these integrals are We solve the secular problem for the 140ϫ140 matrix of proportional to Fk and Gk in the notation of Condon and intrasite interactions Shortley,22 ͉ ͉͒ ϭ ͉ ͉͒ Coul ͗I fd V͑intra J fd͘ ͗I fd V͑intra J fd͘ 4␲ 4␲ vFDϭ F2͑4 f ,5d͒, vFDϭ F4͑4 f ,5d͒, ϩ͗I ͉V͑intra͉͒J ͘exch, ͑3.12͒ 2 5 4 9 fd fd and obtain the ten energy levels E quoted in columns 2 and ␲ ␲ fd 4 4 3 of Table I. This term spectrum corresponds to the usual LS v( fd)ϭ G1͑4 f ,5d͒, v( fd)ϭ G3͑4 f ,5d͒, 1 3 3 7 ͑Russell-Saunders͒ coupling without spin-orbit interactions.

054103-4 INTRASITE 4 f -5d ELECTRONIC CORRELATIONS IN . . . PHYSICAL REVIEW B 66, 054103 ͑2002͒

TABLE II. Calculated lowest term energies of the fd configu- Comparison with the experimental data34 on a Ce atom ͑ 1 3 ration with the spin-orbit coupling and the Lande g factors columns shows that the order of the three lowest levels G4 , F2, and 2–4͒, ⌬EϭE ϪE (1G ). Last two columns are experimental 3 1 fd fd 4 H4 with the multiplet G4 as the ground state is correct. On data ͑Ref. 34͒. All energies are in units of K. the other hand, the experimental data show that in atomic Ce different levels ͑such as 1G, 3F, 3H) are considerably ⌬ E fd g E CeI LaII(fd) mixed up. Due to the strong spin-orbit coupling, there is an 1 3 3 1G Ϫ10 570.3 0.9323 0 0 0 appreciable mixing between G4 and H4, and between F2 4 1 1 3F Ϫ9226.5 0.7030 1343.8 329.2 881.7 and D2. The ground state of a Ce atom has 55% of G4 and 2 3 34 3H Ϫ8155.1 0.8958 2415.2 1840.9 1764.7 29% of H4. The next level which lies only 329 K above 4 3 1 34 3 Ϫ the ground state has 66% of F2 and 24% of D2. In F3 7048.8 1.0825 3521.5 2393.0 2354.5 3 Ϫ conclusion, although the actual atomic spectra of cerium dif- H5 6449.4 1.0334 4120.9 3177.9 2850.7 3G Ϫ4247.3 0.7659 6323.0 1998.5 5472.8 fer somewhat from our results obtained for the fd configu- 3 ration, Table II, our approach captures the main properties and we will use it for Ce in the solid state. In the following We have also checked our results by working with the eigen- we will extend the calculations of the energy level scheme by vectors of Eq. ͑3.12͒ which can be obtained independently including intersite interactions. We will treat separately the ␥ by exploiting the formulas ͑Table 43 of Ref. 22͒ of vector and the ␣ phase of solid Ce. ϭ ϭ addition of angular momenta ( j1 3 and j2 2). The levels of Table I correspond to the following parameters in the no- IV. INTERSITE INTERACTIONS 22 ϭ ϭ tations of Condon and Shortley: F2 325.4, F4 25.1, ϭ ϭ ϭ ͑ ͒ ͓ In this section we will first consider the matrix of intersite G1 567, G3 47, and G5 7.2 in K . These parameters should not be confused with those in Eqs. ͑3.9͒.͔ interactions for the ( fd) system on a crystal in general. Next We now consider the effect of the spin-orbit coupling. we will show that the interaction is largely simplified by Starting with the spherically symmetric LDA calculation of a crystal symmetry and derive the crystal electric field ͑CEF͒. cerium atom we obtain that in the one-electron approxima- We calculate the energy spectrum of the ( fd) system in pres- ⌬ ϭ Ϫ ϭ ⌬ ence of the crystal electric field in the disordered cubic tion so( f ) E f (7/2) E f (5/2) 4003.4 K and so(d) ϭ Ϫ ϭ phase. Ed(5/2) Ed(3/2) 2344.0 K. This gives for the spin- orbit coupling constants ␨ ϭ1143.8 K and ␨ ϭ937.6 K. We start from expression ͑2.11a͒ and write it in the space f d ͉ ͘ Therefore, a typical value of spin-orbit splitting is ϳ1000 K of two-electron state vectors I fd . Carrying out the angular which shows that it cannot be treated as a small perturbation integrations d⍀(nជ ), d⍀Ј(nជ ), d⍀(nជ Ј), and d⍀Ј(nជ Ј), we to the LS term scheme, Table I. In order to take into account obtain the spin-orbit coupling exactly we have to consider the operator ͗ ͉ ជ͗ Ј ͉ ជ ͑ ជ ͑ ជ ͒ ជ Ј͑ ជ Ј͉͒͒ Ј ͘ ជ ͉ ͘ ជ I fd n I fd nЈV R n ,R n J fd nЈ J fd n ϭ ͒ϩ ͒ ͑ ͒ Vso Vso͑ f Vso͑d , 3.13 ϭ ␣␣ ␣Ј␣Ј͑ ជ Ϫ ជ Ј͒ ͚ ͚ ͚ v⌳ ⌳Ј n n where ␣␤ ␣Ј␤Ј ⌳⌳Ј

͑ ͒ϭ␨ ជ ͑ ͒ ជ͑ ͒ ͑ ͒ ϫ͕c⌳͑i␣ j␣͒␦͑i␤ j␤͖͕͒c⌳Ј͑i␣Јj␣Ј͒␦͑i␤Јj␤Ј͖͒. Vso f f L f •S f , 3.14a ͑4.1͒ ͑ ͒ϭ␨ ជ ͑ ͒ ជ͑ ͒ ͑ ͒ Vso d dL d •S d . 3.14b Here each of the indices ␣, ␣Ј, ␤, ␤Ј runs over the labels f, The matrix of interactions reads d. The coefficients c⌳ are defined by Eqs. ͑3.6a͒ and ͑3.6b͒, ␦ ͗I ͉V ͉J ͘ϭ ͗i ͉V ͑ f ͉͒j ͘␦ ϩ͗i ͉V ͑d͉͒j ͘␦ . while (i␤ j␤) stands for the Kronecker delta symbol. For fd so fd „ f so f id jd d so d i f j f … ␣ϭ f , we have ␤ϭd and for ␣ϭd, ␤ϭ f , with a similar ͑3.15͒ correspondence between ␣Ј and ␤Ј. The intersite interaction ͗ ͉ ͉ ͘ ␣␣ ␣Ј␣Ј Since we know the matrix elements i f Vso( f ) j f and element v is given by ͉ ͉ ͓ ͑ ͒ ⌳ ⌳Ј ͗id Vso(d) jd͘ see, for example, the explicit Eq. A.2 of Ref. 17͔, we can calculate the matrix elements for the fd ␣␣ ␣Ј␣Ј͑ ជ Ϫ ជ Ј͒ configuration. We now solve the secular problem for the sum v⌳ ⌳Ј n n of the intrasite and spin-orbit interactions, starting from the matrix of ϭ 2 Ј Ј2R 2 ͑ ͒R2 ͑ Ј͒ ͑ ជ ជ Ј Ј͒ ͵ drr ͵ dr r ␣ r ␣Ј r v⌳⌳Ј n,n ;r,r . ͉ ϭ ͑ ͒ϩ ͑ ͒ U intra V intra Vso , 3.16 ͑4.2͒ and obtain 20 energy levels ͕E fd͖. Since we are interested only in the lowest levels, we quote in Table II the first six out Notice that only direct Coulomb interactions are present of the 20 levels. in Eq. ͑4.1͒, and with the help of the selection rules for the ͑The procedure of calculation of magnetic moments and coefficients c⌳ we conclude that only the interactions with the Lande g factors is given in Appendix A.͒ lϭ0, 2, 4, and 6 have to be considered.

054103-5 A. V. NIKOLAEV AND K. H. MICHEL PHYSICAL REVIEW B 66, 054103 ͑2002͒

A. crystal electric field ␥ phase „ … ␣␣ ᭹ϭ ͵ 2R 2 ͑ ͒ ͑ ជ ជ Ј Ј͒ ͑ ͒ v⌳ dr r ␣ r v⌳ 0 n,n ;r,r 4.6b In the ␥ phase the electronic density is compatible with 1 0 1 ” the crystal structure Fm¯3m. At each atomic site the CEF has and the point group symmetry Oh . In lowest approximation, the CEF corresponds to the potential experienced by a charge at 2 2 Q␣ ϭ ͵ drЈ rЈ R ͑rЈ͒. ͑4.6c͒ a central site nជ , when spherically symmetric (lЈϭ0) contri- Ј ␣Ј butions from charge densities at the 12 neighboring sites nជ Ј ͑ Notice that Q␣Ј stands for Q f or Qd which are charges in on the fcc lattice and similar terms from the electronic den- units e) of the 4 f or 5d electron. As before17,18 the integra- sity in the interstitial regions are taken into account. tion is taken over 0ϽrЈϽR , where R is the radius of 17,18 MT MT Previously these effects were studied for a single 4 f the muffin-tin sphere. Besides 4 f and 5d electrons we can electron per Ce atom. Here we present an extension to the also consider similar contributions from 6s electrons and nu- (4f 5d) system. clei belonging to nearest neighbors. Notice that the interac- ជ Ј ff ᭹ dd ᭹ ͑ ͒ In the crystal field approximation the functions S⌳Ј(n )at tion parameters v⌳ 0 and v⌳ 0 , Eq. 4.6b , remain the same ជ Ј 0ϭ ͱ ␲ 1 1 any of the 12 sites n reduce to Y 0 1/ 4 . We will write an for all these contributions and all we have to do is to collect ⌳ ϭ ͑ ͒ index 0 for Ј (lЈ 0,A1g). The coefficients c⌳Ј in Eq. 4.1 the charges together. Finally, after summation over 12 near- now reduce to est neighbors and simplifications, we obtain

͗ ͉ ជ ͑ ជ ͑ ជ ͉͒͒ ͘ ជ 1 I fd nVCF R n J fd n ͒ϭ ␦ ͒ ͑ ͒ c0͑i␣Јj␣Ј ͑i␣Ј , j␣Ј . 4.3 ͱ4␲ ϭ f ͒␦ ͒ϩ d ͒␦ ͒ ͚ ͓B⌳ c⌳ ͑i f j f ͑id jd B⌳ c⌳ ͑id jd ͑i f j f ͔, ⌳ 1 1 1 1 At the central site nជ , the electronic density has full cubic 1 ជ ͑4.7͒ symmetry. We denote the corresponding SAF’s by S⌳ (n), 1 ⌳ ϵ where 1 (l,A1g), where A1g stands for the unit representation of ϭ the cubic site group Oh . We retain the functions for l 4 and ϭ 12 l 6, which correspond to the cubic harmonics K4 and K6. f ϭ F ᭹ ͑ ͒ B⌳ Qeffev⌳ 0 , 4.8a The selection rules imply that the d-d transitions are per- 1 ͱ4␲ 1 turbed by K4 only, while for the f -f transitions both K4 and ͑ ͒ K6 are relevant. Expression 2.11a reduces to 12 d ϭ D ᭹ ͑ ͒ B⌳ Qeffev⌳ 0 . 4.8b 1 ͱ4␲ 1 1 ជ ជ ͒ ជ ជ ͒͒ϭ ជ ជ ͒ ជ ͒ V͑R͑n ,RЈ͑nЈ ͚ v⌳ 0͑n,nЈ;r,rЈ S⌳ ͑n . ͱ4␲ ⌳ 1 1 Here again we write D for (dd) and F for ( ff). We take as 1 ϭ ͑4.4a͒ an effective charge Qeff QMT the total charge inside a MT sphere. In contradistinction to our previous work17,18 here we The elements have neglected the effect of interstitial charges. Previously it was found that a homogeneous distribution of negative ជ ជ charge in the interstices increases the effective charge by an v⌳ ͑n,nЈ;r,rЈ͒ 10 amount 2.85QMT . This fact is due to the angular dependence of the leading cubic harmonic S⌳ , ⌳ ϭ(lϭ4,A ), in Eq. S⌳ ͑nˆ ͒ 1 1 1g 1 ជ ជ 1 ϭ ͵ d⍀͑n͒ ͵ d⍀Ј͑nЈ͒ ͑4.4b͒ ͑4.4b͒. Indeed K (nˆ ) is positive and maximum along the ͱ ␲ ͉ ជ ͑ ជ ͒Ϫ ជ Ј͑ ជ Ј͉͒ 4 4 R n R n cubic direction ͓100͔ ͑the centers of the interstices͒ and negative and small along ͓110͔ ͑the sites nជ Ј). On the other have the same value for all 12 neighbors nជ Ј on the fcc lattice. hand, if we consider an inhomogeneous charge distribution In addition they are independent of rЈ, as follows from ex- where most of the electronic density in the interstices is lo- pression ͑2.12͒ for lЈϭ0. We then define the crystal field cated close to ͓110͔, then the contribution to the crystal field operator by from interstitial charges can be assumed to be negligibly small. We observe that previously the inclusion of contribu- 12 ជ ជ ͒͒ϭ ជ ជ ͒ ជ ͒ ͑ ͒ tions from a homogeneous charge distribution in the inter- VCF͑R͑n ͚ v⌳ 0͑n,nЈ;r,rЈ S⌳ ͑n . 4.5 ͱ ⌳ 1 ” 1 stices has led to an overestimation of calculated crystal field 4␲ 1 splitting in comparison with the experimental values.18 ␣␣ ᭹ Returning to expression ͑4.2͒ we write within the crystal In practice, it is convenient to calculate v⌳ from 1 0 field approximation ជ ជ ͒ v⌳ 0͑n,nЈ;RMT ,rЈ ␣␣ ␣Ј␣Ј ␣␣ ᭹ ␣␣ ᭹ 1 ” ␣ ͑ ជ Ϫ ជ Ј͒ϭ ͑ ͒ v⌳ ϭ q , ͑4.9a͒ v⌳ 0 n n v⌳ 0 Q␣Ј , 4.6a 1 0 l l 1 1 • RMT where where

054103-6 INTRASITE 4 f -5d ELECTRONIC CORRELATIONS IN . . . PHYSICAL REVIEW B 66, 054103 ͑2002͒

␥ ␣ TABLE III. Lowest levels of the energy spectrum of H B. Quadrupolar ordering „ phase… ϭ ͉ ϩ ␥ U intra VCF , -Ce. Numbers in parentheses stand for degen- ⌬␧ ϭ ⌬␧ ϭ eracy; 1 1409.6 K, 2 2499.4 K; the site group is Oh . In the cubic phase (␥-Ce) with the Fm¯3m space symme- ⌫ ␮ ␧ ͑ ͒ ␧ Ϫ␧ ͑ ͒ M ␮ try the nontrivial electron density distribution is given by , i K ( i 1) K z ( B) ⍀ ⍀ ϭ cubic harmonics K4( ) and K6( ) with l 4 and 6. All ͑ ͒ Ϫ ϭ Јϭ A1,1 1 10 661.1 0.0 0 quadrupole densities with l l 2 average to zero. Only ͑ ͒ Ϫ Ϯ T1,1 3 10 613.7 47.4 0.4596; 0 ﬂuctuations of electric quadrupoles are allowed in the inter- E,1 ͑2͒ Ϫ10 580.9 80.2 0; 0 action, Eq. ͑4.1͒, and those lead to an effective attractive ͑ ͒ Ϫ Ϯ 17,18 T2,1 3 10 505.6 155.5 2.3112; 0 interaction at the X point of the Brillouin zone. This in- ͑ ͒ Ϫ ⌬␧ Ϯ T2,2 3 9251.5 1 0.6634; 0 teraction drives a transition to a new phase which is charac- ͑ ͒ Ϫ ⌬␧ ϩ E,2 2 9224.4 1 27.1 0; 0 terized by an ordering of electric quadrupoles such that the ͑ ͒ Ϫ ⌬␧ Ϯ T1,2 3 8161.7 2 2.2447; 0 space group symmetry is Pa¯3. This order-disorder transition ͑ ͒ Ϫ ⌬␧ ϩ Ϯ T2,3 3 8154.2 2 7.5 0.4503; 0 is accompanied by a contraction of the crystal lattice which ͑ ͒ Ϫ ⌬␧ ϩ A1,2 1 8145.9 2 15.8 0 stays cubic. We have associated this phase transition with the ͑ ͒ Ϫ ⌬␧ ϩ E,3 2 8137.5 2 24.2 0; 0 ␥→␣ transition of Ce. In real space the Pa¯3 ordering implies the appearance of four distinct sublattices of simple cubic structure ͑see Fig. 3 of Ref. 17͒. We label these sub- ␣ϭ ͵ Ј Ј(lϩ2)R 2 ͑ Ј͒ ͑ ͒ lattices which contain the sites ͑0,0,0͒ (a/2)(0,1,1), (a/2) ql dr r ␣ r . 4.9b ϫ ជ ϭ (1,0,1), and (a/2)(1,1,0) by ͕n p͖, p 1 –4, respectively. ͑ ͒ Then the CEF operator 4.5 , with an effective charge Qeff , In principle, one can proceed as in Ref. 17 and derive an can be rewritten as effective mean-ﬁeld Hamiltonian. Here we will start from the crystal in real space and consider the following four quadru- ជ ជ ͒͒ϭ ˆ ͒ l ͑ ͒ ¯ VCF͑R͑n ͚ B⌳ S⌳ ͑n r , 4.10a polar SAF’s corresponding to the four sublattices of Pa3: ⌳ 1 1 1 where39 1 S ⍀͒ϭ ⍀͒ϩ ⍀͒ϩ ⍀͒ ͑ ͒ ជ ជ ͕n ͖͑ ͓S1͑ S2͑ S3͑ ͔, 4.13a v⌳ ͑n,nЈ;R ,rЈ͒ 1 ͱ 12 10 MT ” 3 B⌳ ϭ Q e . ͑4.10b͒ 1 ͱ ␲ eff l 4 RMT ␥ ϭ Our calculations for -Ce (a 9.753 a.u.) yield QMT 1 ϭϩ ͉ ͉ 40 d 4 ϭ f 4 ϭ S͕ ͖͑⍀͒ϭ ͓ϪS ͑⍀͒ϪS ͑⍀͒ϩS ͑⍀͔͒, ͑4.13b͒ 0.9136 e , q4/RMT 0.222 71, q4/RMT 0.036 04, n2 ͱ 1 2 3 f 6 ϭ ͑ ͒ dϭ f ϭ 3 q6/RMT 0.017 39 in a.u. , and B4 1198.4, B4 193.9, f ϭ ͑ ͒ 41 and B6 74.4 all in K . We then consider the Hamiltonian

␥ ជ ͒ϭ ͉ ϩ ជ ͒ ͑ ͒ 1 H ͑n U intra VCF͑n , 4.11 S͕ ͖͑⍀͒ϭ ͓S ͑⍀͒ϪS ͑⍀͒ϪS ͑⍀͔͒, ͑4.13c͒ n3 ͱ 1 2 3 which we associate with the ␥ phase of Ce. By diagonalizing 3 H␥ we have found that in the cubic CEF the 20 atomiclike levels of cerium are split into 58 distinct sublevels which can be labeled by single-valued irreducible representations 1 S ͑⍀͒ϭ ͓Ϫ ͑⍀͒ϩ ͑⍀͒Ϫ ͑⍀͔͒ A (⌫ ), E(⌫ ), T (⌫ ), and T (⌫ )ofO . In particular, ͕n ͖ S1 S2 S3 . 1 1 3 1 4 2 5 h 4 ͱ3 three lowest levels of cerium are split according to the fol- ͑4.13d͒ lowing scheme:42,36

1 → ϩ ϩ ϩ ͑ ͒ G4 A1 T1 E T2 , 4.12a Here we use the short notation S ϵS ϭ ϭ for real k (l 2,T2g ,k 1–3) 3 → ϩ ͑ ͒ spherical harmonics Y 1s , Y 1c , and Y 2s which belong to a F2 T2 E, 4.12b 2 2 2 three-dimensional irreducible representation T2g of Oh . 3 → ϩ ϩ ϩ ͑ ͒ ͑ H4 T2 A1 T1 E. 4.12c These spherical harmonics are proportional to the Cartesian components yz, zx, and xy for kϭ1–3.͒ The calculated splittings of these levels are quoted in Table Below we consider the intersite quadrupole interactions III. In presence of a magnetic ﬁeld, there occurs an additional VQQ(nជ ,nជ Ј) which involve only the functions ͑4.13a͒– splitting of the triplets. The corresponding magnetic mo- ͑ ͒ ͓ ϭ ments are given in the last column of Table III. ͑Details of 4.13d . There are also SAF’s with l 4 and 6 allowed by 37 the calculations can be found in Appendix A.͒ In the second the Pa¯3 symmetry but those lead to weaker multipole in- part of the present section we will study the energy levels in teractions, Eq. ͑2.12͒.͔ We then rewrite Eq. ͑4.1͒ for a case ជ ෈ ជ ෈ ϭ ͒ the ordered phase. when n ͕n1͖ and nЈ ͕n pЈ͖ (pЈ 2, 3, 4 :

054103-7 A. V. NIKOLAEV AND K. H. MICHEL PHYSICAL REVIEW B 66, 054103 ͑2002͒

␣␣␣Ј␣Ј ␣␣␣Ј␣Ј ͗ ͉ ជ͗ Ј ͉ ជ QQ͑ ជ ជ Ј͉͒ Ј ͘ ជ ͉ ͘ ជ ␭ ϭ ␥ I fd n I fd nЈV n,n J fd nЈ J fd n where we have defined 4 and used F and D for (␣␣), with ␣ϭ f or d, respectively. The values of ␣␣␣Ј␣Ј ␥ ␭␣␣␣Ј␣Ј 18 ϭϪ ͕ ͑ ͒␦͑ ͖͒ have been calculated before. Here we use the val- ͚ c͕n ͖ i␣ j␣ i␤ j␤ ␭FFϭ ␭DFϭ␭FDϭ ␭DD ␣␣Ј 3 1 ues 2241 K, 6489 K, and ϭ18 793 K, calculated for ␣-Ce. Including the intrasite part ϫ͕ ͑ ͒␦͑ ͖͒ ͑ ͒ c͕n ͖ i␣Јj␣Ј i␤Јj␤Ј , 4.14 ͉ ជ ͑ ͒ pЈ U intra and the crystal field VCF(n1), Eq. 4.1 , which are also present in the ordered phase, we obtain the full mean- where, as before, ␣,␣Ј,␤,␤Ј run over f and d, and where the field Hamiltonian same exclusion rules ͑if ␣ϭ f then ␤ϭd and vice versa͒ hold between ␣ and ␤ as well as between ␣Ј and ␤Ј. Here we HMF͑nជ ͒ϭUMF͑nជ ͒ϩV ͑nជ ͒ϩU͉ . ͑4.21͒ have 1 1 CF 1 intra Finally, the expectation values ͑the order parameter ampli- Q,e Q,e ␣␣␣Ј␣Ј 2 2 2 2 ជ ជ tudes͒ ␳ and ␳ , Eq. ͑4.19͒, are found by solving the ␥ ϭ ͵ drr ͵ drЈrЈ R ͑r͒R ͑rЈ͒v⌳⌳͑n,nЈ;r,rЈ͒, F D ␣ ␣Ј following mean-field equations ͑4.15͒ ͕␳Q͑ ជ ͒ ͓Ϫ MF͑ ជ ͒ ͔͖ ជ ជ ជ ជ Tr F n1 exp H n1 /T with v⌳⌳(n,nЈ;r,rЈ) where nϭ(0,0,0), nЈϭ(a/2)(0,1,1) ␳Q,eϭ , ͑4.22a͒ F Ϫ MF ជ ͒ and ⌳ϭ(lϭ2,T ,kϭ1). The coefficients c͕ ͖ are defined Tr͕exp͓ H ͑n1 /T͔͖ 2g np as Tr͕␳Q͑nជ ͒exp͓ϪHMF͑nជ ͒/T͔͖ ␳Q,eϭ D 1 1 ͑ ͒ c͕ ͖͑i␣ j␣͒ϭ͗i␣͉S͕ ͖͉j␣͘. ͑4.16͒ D . 4.22b np np Ϫ MF ជ ͒ Tr͕exp͓ H ͑n1 /T͔͖ We introduce the quadrupolar density operators for the ( fd) It is convenient to rewrite these equations in the basis electron system on each sublattice: ͉ ϭ͉ MF K fd͘ k f kd͘ where H is diagonal,

Q ជ 1 Ϫ⑀ ␳␣␣͑n ͒ϭ͚ ͉I ͘c͕ ͖͑i␣ j␣͒␦ ͗J ͉, ͑4.17͒ ␳Q,eϭ F K /T ͑ ͒ p fd np i␤ j␤ fd ͚ c͕ ͖͑k k ͒e fd , 4.23a I,J F n1 f f Z K fd ជ ෈ ϭ ␣ϭ␤ ␤ϭ ␣ where n p ͕n p͖, p 1 –4. Here again , d or 1 Ϫ⑀ ϭ ␤ϭ ␳Q,eϭ D ͒ K /T ͑ ͒ d, f . In terms of quadrupolar density operators, the ͚ c͕ ͖͑kdkd e fd , 4.23b D Z n1 quadrupolar interaction operator between two ( fd) systems K fd ជ ෈ ជ ෈ at site n1 ͕n1͖ and n pЈ ͕n pЈ͖ reads with

␥␣␣␣Ј␣Ј Q Ϫ⑀ /T ͑ ជ ជ ͒ϭϪ ␳Q ͑ ជ ͒␳ ͑ ជ ͒ Zϭ͚ e K fd . ͑4.23c͒ V n1 ,n pЈ ͚ ␣␣ n1 ␣Ј␣Ј n pЈ . ␣␣Ј 3 K fd ͑ ͒ 4.18 Equations ͑4.20b͒–͑4.23c͒ are solved self-consistently. First, ␳Q,e ␳Q,e The mean-field potential at site nជ is obtained by summing we introduce nonzero expectation values F and D in the 1 mean-field Hamiltonian HMF. After this we diagonalize HMF V(nជ ,nជ ) over the 12 nearest neighbors nជ of nជ on the fcc 1 pЈ pЈ 1 and calculate new values of ␳Q,e and ␳Q,e at a given tem- lattice and by approximating the quadrupolar densities at F D perature T according to Eqs. ͑4.22a͒ and ͑4.22b͒. Then we these nearest-neighbor sites by their thermal expectation val- use these values to improve the mean-field Hamiltonian ␳Q ជ ues. The thermal expectation value of ␣␣(n p) does not de- ͑ ͒ ␳Q,e ␳Q,e ͑ ͒ 4.21 , etc., until the input and output values of F and D pend i.e., is the same on any site of a given sublattice and converge. The results of the numerical calculation are shown from the equivalence of the four sublattices it follows that it in Fig. 1. is the same on all sites of the fcc lattice. We then write The procedure outlined above converges very slowly in the vicinity of 100 K, i.e., at the phase transition point. We ␳Q ͑ ជ ͒ ϭ␳Q,e ͑ ͒ ͗ ␣␣ n p ͘ ␣␣ , 4.19 ϭ have found the transition temperature T1 97 K and the or- ␳Qe ϭϪ where the superscript e stand for thermal expectation. The der parameter discontinuities F (T1) 0.069 56 and ␳Qe ϭϪ ͑ mean-field potential is then given by D (T1) 0.038 75. From symmetry considerations it follows that the phase transition is of first order.17͒ At Tϭ0 the averages in Eq. ͑4.19͒ are taken over the ground-state dou- MF͑ ជ ͒ϭϪ ␥␣␣␣Ј␣Ј␳Q ͑ ជ ͒␳Q,e ͑ ͒ U n1 4 ͚ ␣␣ n1 ␣Ј␣Ј 4.20a blet and we obtain ␳Qe(Tϭ0)ϭϪ0.154 62, ␳Qe(Tϭ0)ϭ ␣␣Ј F D Ϫ0.084 85. The lowest five levels of HMF for this case are or, explicitly, given in Table IV. Notice, however, that unlike the crystal field VCF the MF͑ ជ ͒ϭϪ͑␭FF␳Q,eϩ␭DF␳Q,e͒␳Q͑ ជ ͒ MF ͑ ͒ U n1 F D F n1 mean-field potential U , Eq. 4.20b , and the Hamiltonian HMF depend implicitly on temperature T since the order pa- Ϫ͑␭DD␳Q,eϩ␭FD␳Q,e͒␳Q͑nជ ͒, ͑4.20b͒ ␳Qe ␳Qe D F D 1 rameter amplitudes F and D change with temperature,

054103-8 INTRASITE 4 f -5d ELECTRONIC CORRELATIONS IN . . . PHYSICAL REVIEW B 66, 054103 ͑2002͒

leads to a uniform lattice contraction conserving the cubic symmetry of the lattice. The change of the energy spectrum at the transition implies a change of the magnetic susceptibility. Indeed, we have found that the calculated magnetic moments are different in ␥- and ␣-Ce. Moreover, in the ground state of ␣-Ce the magnetic moment of the 4 f electron is bound to the magnetic moment of the 5d electron ͑a qualitative origin of this correlation is given in Appendix B͒. The lowest magnetic excited state (E, 2 in Table IV͒ is separated from the ground state by an energy gap ⌬⑀ϳ350 K which is much larger than a typical crystal field excitation in the ␥ phase, Table III. However, here our treatment is incomplete. Al- though the present model carefully takes into account the FIG. 1. Calculated evolution of the order parameter amplitudes intrasite interactions, it does not describe properly the metal- ␳Q,e ␳Q,e F and D with temperature. lic bonding in Ce. The question of correspondence between our approach ⑀ Ϫ⑀ Fig. 1, and the energy splittings ( i 1) decrease with in- and electron band structure calculations deserves a special creasing T up to T1. Above the phase transition point the attention. As we have discussed in Sec. II, in the ‘‘muffin- ␥ spectrum transforms discontinuously to that of -Ce, Table tin’’ approximation a localized 4 f electron experiences only ͑ III. The double degenerate states the representations E in a field of spherical symmetry and occupies a 14-fold degen- Table IV͒ are due to time-reversal symmetry.36 For a further MF erate level. The localized states of the 4 f electron then are discussion of the energy spectrum of H we refer to Ap- uncorrelated with the states of conduction electrons, because pendix A, where we study the magnetic moments of the two the spherical component of the 4 f density is independent of electrons. its spin (sz) and orbital (m f ) projections. In our study we show that this simple picture is not correct and there exist V. DISCUSSION AND CONCLUSIONS strong local correlations between localized 4 f and delocalized 5d electrons omitted in a conventional band structure This work is an extension of our previous model of the calculation. These correlations arise due to the Coulomb on- ␥-␣ phase transition in Ce based on the idea of quadrupole site repulsion and reflect the electronic term structure of ordering.17,18 In addition to the intersite quadrupolar cou- atomic cerium, Sec. III. We show that the excitations of the plings here we consider the multipolar intrasite ͑direct Cou- 4 f electron are combined with those of 5d electron in a lomb and exchange͒ interactions between one localized 4 f single spectrum, which is sensitive to crystal site symmetry electron and one delocalized 5d electron taken instanta- because of intersite interactions, Sec. IV. neously at a same cerium site. The intrasite interactions are In principle, band structure calculations with the full po- treated exactly in the adopted 4 f 5d model. In ␥-Ce we have tential ͑FP͒ extension ͓so-called FP-linear muffin-tin calculated and analyzed the crystal electric field excitations, orbital25 ͑FLMTO͒ and FP-linear augmented plane-wave23 Table III. In ␣-Ce the Pa¯3 quadrupolar ordering sets in and ͑FLAPW͒ methods͔ are capable of dealing with nonspherical drives the ␥-␣ phase transition. The quadrupolar order has contributions of density and potential. Provided that the site been studied in the mean-field approximation, Eqs. ͑4.20b͒– symmetry is introduced explicitly, calculations with the full ͑4.23c͒. We have calculated the phase transition temperature potential option can describe some, but not all, structural ϭ ¯ → ¯ (T1 97 K) and the evolution of the order parameter ampli- properties associated with the Fm3m Pa3 transformation. ␳Qe ␳Qe ͒ tudes ( F and D ; see Fig. 1 by solving self-consistently The reason is that the band structure calculations are based the mean-field equations ͑4.20b͒–͑4.23c͒. We have shown on the single-determinant Hartree-Fock method. In our treat- ͑ ͒ before17,18 that quadrupolar ordering in the Pa¯3 structure ment each local two-electron basis function, Eqs. 2.2 – ͑2.6͒, corresponds to a Slater determinant ͓with the permuta- ͑ ͔͒ MF tion property 2.7 . The solutions are expressed as linear TABLE IV. Lowest levels of the energy spectrum of H at T combinations of all these functions ͑determinants͒. As such, ϭ0 and magnetic moments M for ␣-Ce. Numbers in parentheses z our method corresponds to a many-determinant treatment or stand for degeneracy; the site group is S ϭC ϫi. 6 3 configurational interaction ͑CI͒. Clearly, it is not the case ⌫,␮⑀͑K͒ (⑀ Ϫ⑀ ) ͑K͒ M (␮ ) with the band structure approach. This explains why the con- i i 1 z B ventional band structure treatment is missing some intrasite E,1 ͑2͒ Ϫ10 888.5 0.0 Ϯ2.0683 and intersite correlations which are taken into account in our A,1 ͑1͒ Ϫ10 721.4 167.1 0 approach. A,2 ͑1͒ Ϫ10 699.7 188.8 0 The other drawback of the FP electronic band structure E,2 ͑2͒ Ϫ10 542.2 346.3 Ϯ1.0116 calculations is connected with the LDA. In the LDA the ex- E,3 ͑2͒ Ϫ10 427.1 461.4 Ϯ0.4882 change potential is a function of density and has the full ͑unit͒ symmetry of the crystal. This implies that the density

054103-9 A. V. NIKOLAEV AND K. H. MICHEL PHYSICAL REVIEW B 66, 054103 ͑2002͒

and the LDA exchange potential are invariant under inver- with sion in both phases. Expanding the exchange potential in terms of spherical harmonics on a cerium site we find that Mជ ͑ f ͒ϭ␮ ͓Lជ ͑ f ͒ϩ2Sជ͑ f ͔͒, ͑A2a͒ only harmonics with even l are allowed by the inversion B symmetry, i.e., lϭ2,4, etc. However, as we observe from Eq. ͑ ͒ Mជ ͒ϭ␮ ជ ͒ϩ ជ ͒ ͑ ͒ 3.10b the contributions with odd values of l are relevant for ͑d B͓L͑d 2S͑d ͔, A2b the exchange between 4 f and 5d electrons, namely, lϭ1, 3, ␮ and 5. Thus, these terms are washed away by the LDA treat- B being the Bohr magneton. The matrix elements of Vmag ment. read On the other hand, the present method should be extended to include the metallic bonding in Ce explicitly. In our opin- ͉ ͉ ϭ ͉M ͉͒ ␦ ͗I fd Vmag J fd͘ ͑͗i f z͑ f j f ͘ i j ion, a combination of the local correlations with the band d d structure approach constitutes a challenge for further studies. ϩ͗i ͉M ͑d͉͒j ͘␦ ͒ H, ͑A3͒ d z d i f j f • We are presently working on the problem and hope to achieve this by employing the valence bond ͑VB͒ or Heitler- ͉M ͉ ͉M ͉ where ͗i f z( f ) j f ͘ and ͗id z(d) jd͘ are one-particle London approach. The VB method is more difficult to imple- matrix elements which can be easily computed. If we diag- ment, but usually it gives a better description of chemical Hϩ 36,43 onalize the matrix of ( Vmag), then the degeneracies of bonding than the method of molecular orbitals. The al- the energy terms are lifted and the magnetic moment of each ternative approach is a merger with one of the existing band sublevel ␯ is given by structure methods and introducing in some way a CI treatment. M ͑⑀ ͒ϭ͗⑀ ͉M ͉⑀ ͘ ͑ ͒ Finally, we would like to mention again that we are aware z ␯ ␯ z ␯ , A4 of the fact that our approach is not complete, but it certainly ͕⑀ ͖ ͕ ͖ ͕⑀ ͖ underlines the importance of the structural factors and the where ␯ stands for the energy levels E fd or i given in ͑ ͒ ␥ local correlations for this long-standing problem. We have Tables II or III and IV, for the atomic case intrasite , the ␣ M shown that the local electronic interactions can trigger the phase, and the phase, respectively. The results for z for ␥→␣ phase transition in Ce, but other aspects of the prob- the solid-state phases are quoted in Tables III and IV. The lem ͑in particular, chemical bonding, etc.͒ should also be Lande g factors are obtained from taken into account. As before,17,18 we suggest synchrotron M ⑀ ͒ϭ␮ ⑀ ͒ ͑ ͒ radiation experiments in order to check the appearance of z͑ ␯ , j Bg͑ ␯ j, A5 weak superstructure reflections in the Pa¯3 structure (␣-Ce) and to study diffuse scattering in ␥- and ␣-Ce. where j is the z projection of a total angular moment: j The present model generalizes our initial approach ͑Ref. ϭϪJ,ϪJϩ1,...,ϩJ. The calculated g factors for the 34 17͒ for a many-electron case. The method can be applied ͑as atomic case, Table II, are close to the experimental values. has been done for TmTe in Ref. 26͒ to study quadrupole In case of ␥-Ce and ␣-Ce the z axis is the ͓001͔ axis of 15 the cubic crystal. For ␥-Ce only triply degenerate levels of orderings in lanthanides and their compounds ͓DyB2C2 ͑ ͒ ͑ ͒ ͑ ͒ ͑ T or T symmetry have nonzero magnetic moments which Ref. 27 , DyB6 Ref. 28 , UCu2Sn Ref. 29 , PrPb3 Ref. 1 2 30͒ YbAs ͑Ref. 31͒, YbSb ͑Ref. 32͔͒ where a few f electrons are ϪM(t), 0 and ϩM(t), where M(t)Ͼ0 and t ϭ ␮ ϭ ␮ ␣ at each site are involved in the process of ordering. (T1 , )ort (T2 , ), Table III. For -Ce only doubly degenerate levels of E symmetry have nonzero magnetic mo- ACKNOWLEDGMENTS ments which are ϪM(e) and ϩM(e), where M(e)Ͼ0 and eϭ(E,␮), Table IV. The doubly degenerate states are We thank D.V. Lopaev for references on atomic spectra due to time-reversal symmetry.36 It is worth noting that in the and A.T. Boothroyd for discussions. This work has been fi- ground state (E,1͒ the magnetic moment of the localized 4 f nancially supported by the Fonds voor Wetenschappelijk electron is not independent. It is attached to the magnetic Onderzoek, Vlaanderen. moment of the 5d conduction electron. The change of sign of the magnetic moment of the 5d electron under time- APPENDIX A reversal symmetry now requires the concomitant change of In order to calculate the effective magnetic moments and the sign of the magnetic moment of the localized 4 f electron. the Lande g factors we study the polarization of electronic We discuss the origin of this correlation in Appendix B. states in a small external magnetic field H. In such case we Since the 5d electron belongs to the conduction band, the H ͓ ͉ ͑ ͒ ground state (E,1͒ gives rise to a Pauli paramagnetism. The add to the Hamiltonian that is, U intra , Eq. 3.16 , for the ͉ ϩ ͑ ͒ ␥ other levels given in Table IV represent excited states of atomic case, U intra VCEF , Eq. 4.11 , for -Ce, and ͉ ϩ ϩ MF ͑ ͒ ␣ ␣-Ce. However, we observe that the first ͑representation A,1 U intra VCEF U , Eq. 4.21 , for -Ce] a magnetic term of Table IV͒ and the second state (A,2͒ are nonmagnetic. The ϭϪM ͑ ͒ ͒ Vmag z•H. A1a first magnetic excited state (E,2 is separated from the ⌬⑀ϳ Here ground state by an energy gap of 350 K. This observa- tion could explain the absence of a Curie-Weiss contribution M ϭ M ͒ϩM ͒ ͑ ͒ ␹ Ͻ z ͓ z͑ f z͑d ͔, A1b to the magnetic susceptibility ␣ at temperatures T T1.

054103-10 INTRASITE 4 f -5d ELECTRONIC CORRELATIONS IN . . . PHYSICAL REVIEW B 66, 054103 ͑2002͒

⍀Ј ⍀Ј where f and d refer to the polar coordinates of 4 f and 5d electrons, respectively. YM L ϭϪ Indeed, there are 11 functions Lϭ5 (M L 5, Ϫ4,...,5),which form a basis of the two-particle irreducible representation H of the group SO͑3͒ of three- dimensional rotations. ͓In Eq. ͑B3͒ we quote only two functions; the others can be obtained from Table 43 of Ref. 22.͔ In the LS ͑Russel-Saunders͒ coupling the triplet 3H lies lower in energy than the singlet 1H, Table I. If now we include the diagonal mean-field coupling UMF, we obtain Y 5 Y Ϫ5 that the six states with the orbital components 5 and 5 ϭϪ ͑ and with the spin components M S 1, 0, 1 originating from the 3H triplet͒ go down in energy. These six states are further split by the spin-orbit coupling in three magnetic dou- ϭ ϫ blets of E symmetry of the site group S6 C3 i. Finally, in FIG. 2. The transformation ͕x,y,z͖→͕xЈ,yЈ,zЈ͖ ͓the Euler ͑ ͒ ͱ the full treatment Sec. IV B the middle doublet is split by angles are ␣ϭ␲/4, ␤ϭarccos(1/ 3), ␥ϭ0 for passive and the crystal field in two nonmagnetic components of A sym- (␥,␤,␣) for active rotations͔ and the function S͕ ͖ , Eq. ͑4.13a͒. n1 metry. This explains qualitatively the origin of the four low- ¯ ␣ APPENDIX B est levels of the Pa3 structure of -Ce, Table IV. The real situation is more complex since there is a mixing of different S ⍀ ͑ ͒ We consider the function ͕n ͖( ), Eq. 4.13a ,inanew configurations in atomic cerium. Nevertheless, a consider- 1 3 coordinate system of axes ͕xЈ,yЈ,zЈ͖ where the zЈ axis cor- able admixture of the H configuration ͑29%͒ is found in the 34 responds to the cubic ͓111͔ axis, Fig. 2. ground state of atomic cerium and it is very likely to occur Then in ␥-Ce. In ␣-Ce orbital functions of 5d electrons on neighboring mϭ0 S͕ ͖͑⍀͒ϭY ϭ ͑⍀Ј͒, ͑B1͒ n1 l 2 sites overlap, giving rise to band structure effects. Here we consider these effects only as a perturbation to the ground where ⍀Ј stands for the polar angles (⌰Ј,␾Ј) in the new F D state E,1, Table IV. Then in the tight-binding approximation system of axes. The coefficients c͕ ͖(i f j f ) and c͕ ͖(id jd), ជ n1 n1 a band state with wave vector k is a mixture of the d func- Eq. ͑4.16͒, become diagonal in the new basis, 2 ⍀Ј Ϫ2 ⍀Ј tions Y 2( d) and Y 2 ( d). However, as we have seen ear- 0 lier, these electronic 5d states are bound to the two- c͕ ͖͑i␣ j␣͒ϭ͗i␣͉Y ͉i␣͘␦ , ͑B2͒ Ϫ n1 2 i␣ j␣ 3 ⍀Ј 3 ⍀Ј 4 f -electron states Y 3( f ) and Y 3 ( f ) and form the two- ϭ ϭϪ YM L 5 YM L 5 ͑ ͒ which facilitates the calculation of the mean-field interaction electron functions Lϭ5 and Lϭ5 , Eqs. B3a and MF ជ ͑ ͒ MF ͑ ͒ U (n1), Eq. 4.20b . In this appendix we show that U is B3b . Therefore, the resulting doubly degenerate states are ជ ជ minimized for the following two functions: in fact two-electron band states with energy Eϩ(k)ϭEϪ(k) ϭ ͑the sign ϩ or Ϫ here stands for two-electron time-reversed YM L 5ϭY 3͑⍀Ј͒ Y 2͑⍀Ј͒, ͑B3a͒ Lϭ5 3 f • 2 d band states͒. In the magnetic field these two bands become ជ ជ ϭϪ Ϫ Ϫ split, Eϩ(k)ÞEϪ(k), and this leads to a temperature- YM L 5ϭY 3͑⍀Ј͒ Y 2͑⍀Ј͒, ͑B3b͒ Lϭ5 3 f • 2 d independent Pauli paramagnetism.

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