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Charge (electrostatic) formation of the non-magnetic ground state in the Kramers system
Solid State Communications, Vol. 89, No. 6, pp. 553-557, 1994 Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/94 $6.00 + .00
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0038-1098(93)E0093-D CHARGE (ELECTROSTATIC) FORMATION OF THE NON-MAGNETIC G R O U N D STATE IN THE KRAMERS SYSTEM R.J. Radwafiski Centre for Solid State Physics, Sw. Filipa 5/5, 31-150 Kraktw, Poland
(Received 12 July 1993; in revised form 25 September 1993 by P. Wachter)
Electrostatic interactions can produce the localized non-magnetic state also in case of the Kramers system. The full suppression of the local moment is attained by highly anisotropic charge distribution at the vicinity of the f-shell electrons. This charge-formation mechanism for the non-magnetic state of the f magnetic ion is compared with the Kondo-compensation mechanism and the hybridization, of f and conduction electrons, mechanism. THE PROBLEM of properties of a magnetic impurity in a metallic host has a very long lasting discussion; here I mention a mile-stone work in 1961 by Anderson [1]. The formation of the non-magnetic (N-M) or very weakly-magnetic (VW-M) state of the magnetic impurity got tremendous impetus with the discovery of heavy-fermion (h-f) compounds, some compounds containing the Ce and U ions; namely CeAI 3 in 1975 [2]. The heavy-fermion compounds show very spectacular phenomena like the very large specific heat at low temperatures, discussed in terms of the large effective-electron mass, and a non-magnetic ground state at low temperatures contrasting with the "normal" sizefmagnetic moment observed at ambient temperatures. The N-M ground state appears from the finite, though remarkably large, susceptibility and hardly any magnetic-field effect on the h-f state. An extensive review of heavy-fermion compounds and their properties can be found in [3-8]. It is commonly accepted that the ground state of the h-f state has a collective nature. Based on this first-glance general impression, localized-based approaches have been, say, not in a fashion. The present main stream of theoretical works underlines the importance of mutual electronic correlations in which the local f moment is somehow dissolved by means of the hybridization of the f electrons with conduction electrons [3, 4, 7, 8] (the hybridization model). In Kondo-type explanations [9] for the formation of the non-magnetic state it is assumed that below a certain temperature a complex bound state of the localized spin and conduction
electrons is formed in which the local moment of the magnetic impurity is fully compensated by antiparallel oriented spins of conduction electrons. This singleimpurity Kondo model when extended to the lattice meets serious objections as has been pointed out by Nozitres [10] that in the Kondo lattice there is not a sufficient number of conduction electrons to compensate all localized moment by means of the spin-compensation mechanism. It is worth noting that both models, i.e. the hybridization model and the Kondo-lattice model are dealing with low-temperature properties and both of them have difficulties to describe experimentally observed the smooth transformation with temperature of the non-magnetic h-f state to the "normal localized" state. From an experimental point of view Franse et al. [11] pointed out significant similarities of the heavy-fermion UPt3 and the paramagnet PrNi 5 though further localizedelectron predictions have been found to fail in more detailed description of UPt3. Rietschel et al. [12] argued that anomalies above 20 K in the specific heat of four uranium h-f compounds, UBel3, UPt3, URu2Si2 and U2ZnlT, are of the Schottky type and are explainable as crystalline-electric-field (CEF) effects indicating strong localization of 5f electrons. The valency of the U atom is, however, difficult to determine from specific-heat studies. The thermal variation of the susceptibility at high temperatures also does not give unambiguous results about the uranium valency as the U 3+ and U 4+ ions have almost the same value for the effective moment. Above
553
554
N O N - M A G N E T I C G R O U N D STATE IN THE K R A M E R S SYSTEM
mentioned evidences illustrate the "normal localized"f behaviour above certain temperature and a mysterious non-magnetic state formed at lowest temperatures. From the CEF point of view for U compounds the configurations f2 (U 4+) a n d f 3(U 3+) can be concerned. The configuration f ' , corresponding to the nonKramers system, has been theoretically favoured [1315] owing to the possible formation by CEF interactions the singlet ground state for thefelectrons. Here it is worth mentioning that the quadrupolar Kondo effect proposed by Cox [16] for UBeI3 employs the nonmagnetic doublet F3 of the non-Kramers 5f 2 configuration. The configurationf3 has not been exploited in CEF approaches owing to the general conviction about the magnetic character of localized states for the Kramers system. Recently, the non-magnetic Kramers state has been surprisingly proven to exist and to be realized as the ground state within the ordinary CEF approach for the case of the f3 system with the hexagonal symmetry [17]. In this paper, I would like to discuss the charge mechanisms for the formation of the non-magnetic local f state in the case of the Kramers system f3 and to compare it with other existing mechanisms. The crystalline-electric-field approach is an elegant and handle form for the description of the localized states resulting from the charge (electrostatic) interactions of the f-shell electrons with surrounding charges. In CEF approach the local symmetry is directly incorporated from the starting point. The CEF approach has been found applicable [18-20] to 4f "normal" rare-earth ions (i.e. Nd, Tb, Dy, Ho, Er and partly Pr, Sm and Tm ions; Yb and mainly Ce often exhibits an "abnormal" behaviour). In the CEF approach the electronic states of the fn system, where n is an integer number, are described by the quantum number of the total angular momentum J that results from the consideration of the LS scheme, i.e. by assuming the spin-orbit coupling constant As-o to be 0o. Within the CEF approach for t h e f 3 configuration, that is relevant for the Nd 3+, U 3+ and Np 4+ ions, Hund's rules provide the multiplet 4•9/2 with J = 9/2 as the ground multiplet. Under the CEF interactions of the hexagonal symmetry the 10-fold degenerate 419/2 multiplet is split into 5 charge-formed (CF) doublets. The sequence of the doublets of the energy level scheme and the energy separations among them depend on the mutual strength of the CEF interactions of the different order. The charge-formed doublet I'~: cos/3] + 3 / 2 ) + sin/3] q: 9/2) (notation with cos/3 and sin/3 assures the automatic normalization) has only the component z of the total angular momentum J given by (Jz) = 4-(1.5-6 sin2/3). The dependence of the parameter/3
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on the strength of the CEF interactions is tg2/3 = 4v/-2-1k/(18a + 15b - 3g),
(1)
where k = 360B 6, a = 3B °, b = 84B ° and g = 5040B6°. Within the states F~ a non-magnetic doublet occurs if/3 equals 30 ° . The state I '1 = x/3/21 :F 3/2) + 1/21 + 9/2).
(2)
written in the ]J, Jz) representation with the z-axis taken along the hexagonal axis, has fully quenched the f-system moment as (Jx) = (Jy) = (Jz) = 0. This N-M Kramers state can be realized as the ground state by a set of CEF parameters: B ° = +6.0K, B°=-123mK, B°=-2mK and B 6 6 = - 5 2 . 3 m K [determined from equation (1)]. Some important outcomings concerning magnetic properties resulting from the CF N-M Kramers doublet ground state I'91 are (i) the vanishing magnetic moment at 0 K, (ii) the finite susceptibility at lowest temperatures, (iii) very anisotropic magnetic properties with the easy magnetic direction perpendicular to the hexagonal axis, (iv) almost linear increase of the reversal susceptibility X-1 with temperature marking good applicability of the Curie-Weiss law above 200 K though for the given CEF parameters the overall CEF splitting ACEv amounts to 846 K, (v) the effective moment derived from the high-temperature susceptibility is close to that expected for the configurationf3. The points (i) and (v) evidence the fact that the effective moment does not carry any information about the lowtemperature moment but rather about the ground multiplet [14]. The N-M Kramers doublet ground state is largely insensitive to the external/internal field. It makes difficult for spin-dependent interactions to establish a long-range magnetic state. According to numerical experiments an external field of 5 T separates the twodoublet levels by 0.13 K for the x-axis and by 0.004 K only for the c-axis revealing extremely weak influence of the magnetic field on these levels. The separation by 6.7 K one could expect for an effective spin S of 1/2. This remarkable disagreement means that any approximation of the behaviour of the N-M Kramers doublet in magnetic fields by any fictitious spin S with 1/2, as is a common practice for the doublet state in the spin Hamiltonian approach, is completely invalid. The field dependence of the N-M Kramers doublet is quite unusual as in principal the levels of the Kramers doublet should go up and down symmetrically what is related with opposite values for the associated magnetic moments. It should be noted, however, that the N-M behaviour appears only in the first approximation. Above 10T the levels of the N-M
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Kramers doublet are more and more split revealing is inestimable due to their large penetration of the that there is no violation of the time reversal symmetry intraionic space the effect o f p and d electrons seems to behind the discussed N-M Kramers doublet. be much larger due to the very anisotropic shape of The N-M Kramers doublet ground state results their orbitals [23]. Other atoms constituting the from higher-order CEF interactions. From equation intermetallic compound are also important for the (1), it shows that ~ depends critically on the anisotropy of the charge distribution at the vicinity of parameter B66 that describes in-plane six-order the f-shell electrons as the spatial allocation of the interactions. This parameter is indispensable for the anisotropic shape of the own p and d orbitals in the formation of the N-M doublet allowing the mixing of lattice depends on the charge distribution of the the [ + 3/2) and t 4- 9/2) states. The CEF interactions neighbouring atoms. In principle, it is possible to of the second-order produce the ground state with the transform the CEF parameters into the charge dominant contribution associated with the extremal distribution over the crystal. In this point experimenvalue of Jz, i.e. 0 (for B ° > 0) or + J (for B ° < 0) in tal evaluation for the CEF parameters will meet results case o f a non-Kramers' ion and + 1 / 2 (for B ° > 0) or of fully band-structure calculations if these calculations + J (for B ° < 0) in case of a Kramers' ion. The shown reach required accuracy of, say, 1 meV. set of the CEF parameters has been chosen in order Next, a question is if the N-M Kramers state can (i) to have the N-M state F91 as the ground state and be formed by electrostatic interactions also for other (ii) to have the first excited level at an arbitrarily symmetries and for other electronic configurations chosen value of 50 K. In fact, the value of 50 K has l i k e f I o r f 13. As far as the ordinary CEF approach is been chosen in [17] in order to account for two considered the answer is negative. However, going experimental results found for UPt 3. Namely, (i) the beyond the pure CEF approach by taking into Schottky-type of maximum in the extra specific heat account, for instance, the finite value for the spin[12, 21] at 23K points to the energy separation of orbit interaction, the intermediate coupling instead of 50 K and (ii) a transition peak in the optical reflection the pure LS coupling, a possible lowering of the local observed by Marabelli et al. [22] at about 4meV is symmetry and a possible cancellation of spin and interpreted as associated with the excitation to the first orbital moments open some possibilities for the excited CEF doublet. So, it is supposed that the formation of a VW-M ground state or may be even discussed CEF interactions could be a prototype for the fully non-magnetic [24]. the charge interactions in UPt3, a compound with the According to my knowledge, the charge mechanism hexagonal structure. As a further argument for these for the formation of the non-magnetic state in case of charge interactions in UPt3 it could be added that the Kramers system has not been discussed yet in these CEF parameters yield the inelastic-neutron- literature. This charge mechanism differs completely scattering (INS) transition with the lowest energy of from the hybridization and the Kondo mechanism. 50 K that has been correlated with an INS transition The hybridization model [1, 3] requires thefstates to lie observed by Felten [21]. Until now little attention has at the Fermi level resulting in the high density of (quasibeen given to this experimental observation. particles) states at E F necessary for explanation of the In the presented model the full suppression of the large specific heat at low temperatures. The hybridlocal moment is attained by the electrostatical ization theory, though seems to be quite successful in (approximated by the CEF) interactions of the f description of low-temperature properties in terms of shell with the surrounding charge distribution. This the Fermi liquid, cannot explain the smooth transforcharge distribution is highly anisotropic as is revealed mation of the heavy-fermion state to the localizedby CEF parameters with significant values for higher- moment state with the increase of temperature. It is order terms. The term B °, for instance, is associated highly intriguing how the increase of temperature with the charge-density variation with the polar angle sometimes by 10-20 K, what corresponds to the energy 0 given by the Legendre function p0 (cos 0) that has of about 0.002 eV, can drive the shift of thefelectronic [18] extrema at 0 = 30 ° and 65 °. The anisotropy of states down by 2-5 eV. In the Kondo mechanism, a the charge distribution is the direct consequence compensation of thef(significant) local moment occurs of the existence of the crystallographic lattice. The by conduction-electron spins and a non-magnetic biggest part in the suppression of the f-ion moment singlet ground state refers to an object containing the likely comes from own conduction electrons as f moment and the cloud of conduction electrons with the electrostatic interactions strongly depend on the opposite spins as it has been numerically shown by distance. The CEF terms B ° and B 6, for instance, Wilson [25] for the one-impurity Kondo problem. In depend on the distance in power up to - 7 . Although discussions of the Kondo mechanism, the attention is the role of the s electrons, among conduction electrons, not put to the spatial distribution of the charge cloud
556
NON-MAGNETIC G R O U N D STATE IN THE KRAMERS SYSTEM
associated with the spin cloud. In the presented view this charge-cloud distribution, if it is not spherically isotropic, modifies substantially the internal electric field experienced by the f-shell electrons that, in a consequence, produces a localized state with largely reduced or even completely suppressed local magnetic moment. The anisotropic charge distribution in the neighbourhood of the f-shell electrons seems to be much more effective in production the VW-magnetic state of the originally magnetic f impurity than the direct spin compensation. The priority of the charge mechanism over the Kondo-spin mechanism seems to lie in the fact that the charge mechanism does not suffer from the Nozirres' argument. In this point it is worth stressing that, in contrary to the hybridization model, both models, i.e. the presented model and the Kondotype model, have in common to treat thefelectrons as localized ones. However, according to the presented model the local moment associated with thefelectrons is rather small whereas a substantial local moment, close to that seen in the paramagnetic Curie-Weiss law, is thought to be in Kondo-type considerations. In brief, the message of this letter is that charge interactions can produce a very weakly magnetic or a non-magnetic state for the localfmoment also for the Kramers systems. This charge mechanism associated with the anisotropic charge distribution at the neighbourhood of the f-shell electrons is very effective in the suppression of the local magnetic moment and overcomes the Nozirres objection for the Kondocompensation mechanism in the Kondo-lattice model. The present mechanism underlines the strong correlation of the ground state of thefion and its magnetic state with the charge distribution around the ion, i.e. the CEF potential at the f shell. Moreover, this correlation is a good illustration of the fundamental interplay of electrostatic and magnetic phenomena. Further studies on the physics of the non-magnetic or very-weakly magnetic f ground state and its physical realization are in progress. In this point heavy-fermion compounds with nearly non-magnetic ground state could be mentioned as being suspicious to have the CEF very-weakly magnetic Kramers doublet state as the ground state of their f electronic subsystem. Then the large specific heat is associated with many-electron excitations of the f-electronic subsystem within the charge-formed Kramers doublet ground state that is slightly split by spin-dependent interactions [26]. This low-energy excitation to the Kramers-twin state causes the reversal of the local moment and subsequently the polarization of the spin density of conduction electrons.
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REFERENCES 1. 2. 3. 4. 5.
6.
7. 8. 9.
10. 11. 12. 13. 14. 15.
16. 17. 18. 19. 20.
21. 22.
P.W. Anderson, Phys. Rev. 124, 41 (1961). K. Andres, J.E. Graebner & H.R. Ott, Phys. Rev. Lett. 35, 1779 (1975). P. Fulde, J. Keller & G. Zwicknagl, Solid State Phys. 41, 1 (1988). N. Andrei, K. Furuya & J.K. Loewenstein, Rev. Mod. Phys. 55, 331 (1983); G.R. Stewart, Rev. Mod. Phys. 56(4), 755 (1984). A. de Visser, A. Menovsky & J.J.M. Franse, Physica B147, 81 (1987); A. de Visser, J.J.M. Franse & J. Flouquet, J. Magn. Magn. Mater. 108, 15 (1992). N. Grewe & F. Steglich, Handbook on the Physics and Chemistry of Rare Earths (Edited by K.A. Gschneidner, Jr. & L. Eyring), Vol. 14, p. 343, North-Holland (1991). G. Zwicknagl, Adv. Phys. 41, 203 (1992). T. Kasuya, Prog. Theor. Phys. Suppl. 108, 1 (1992); T. Saso, Prog. Theor. Phys. Suppl. 108, 89 (1992). It is a suggestion to reserve the name of the "Kondo effect" for the experimental fact associated with the increase of the resistivity with lowering temperature observed in some intermetallic compounds and diluted metallic systems as this fact has been exploited in the original paper by J. Kondo, Rep. Theor. Phys. 32, 37 (1964). P. Nozirres, Ann. de Phys. 10, 19 (1985). J.J.M. Franse, A. de Visser, A. Menovsky & P.H. Frings, J. Magn. Magn. Mater. 52, 61 (1985). H. Rietschel, B. Renker, R. Felten, F. Steglich & G. Weber, J. Magn. Magn. Mater. 76 & 77, 105 (1988). G.J. Nieuwenhuys, Phys. Rev. B35, 5260 (1987). R.J. Radwafiski, J. Magn. Magn. Mater. 103, L1 (1992). C. Geibel, A. Brhm, R. Caspary, K. Gloos, A Grauel, P. Hellmann, R. Modler, C. Schank, G. Weber & F. Steglich, Physica B186-188, 188 (1993). D.L. Cox, Phys. Rev. Lett. 59, 1240 (1987); J. Magn. Magn. Mater. 76 & 77, 53 (1988). R.J. Radwafiski, Physica B182, 209 (1992). R.J. Radwafiski & J.J.M. Franse, Physica B154, 181 (1989). R.J. Radwafiski, J.J.M. Franse, D. Gignoux, F.E. Kayzel, C. Marquina & A. Szewczyk, Physica B177, 291 (1992). R.J. Radwafiski, N.H. Kim-Ngan, F.E. Kayzel, J.J.M. Franse, D. Gignoux, D. Schmitt & F.Y. Zhang, J. Phys.." Condens. Matter 4, 8853 (1992). R. Felten, Thesis, Technical University Darmstadt (1988); for INS results see p. 103. F. Marabelli, G. Travaglini, P. Wachter & J.J.M. Franse, Solid State Commun. 59, 381 (1986).
Vo1.'89, No. 6 23. 24.
NON-MAGNETIC G R O U N D STATE IN THE KRAMERS SYSTEM
R. Coehoorn, K.H.J. Buschow & M.W. Dirken, Phys. Rev. B42, 4645 (1990). In [14] similarities of CEF interactions of the Pr 3+ ion in Pr2FelaB and the U 4÷ ion in URu2Si 2 (the f 2 configuration) have been discussed. Here, I would like to comment that though many features for URu2Si2 can be explained by the f2 configuration but the large specific heat of this h-f compound in my present view can hardly be explained within the f2 configuration; consult, however, this view with the quadrupolar Kondo effect proposed by Cox
25. 26.
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[16] that operates on the non-magnetic doublet of the f2 configuration. In view of the present paper, considerations of the charge interactions within the f3 configuration are more appropriate for uranium compounds exhibiting the h-f behaviour, i.e. having the large specific heat at lowest temperatures. K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975). R.J. Radwafiski, Conf. Strongly Correlated Electron Systems, Poster 8P-92, Sendai, September (1992).
Pergamon
0038-1098(93)E0093-D CHARGE (ELECTROSTATIC) FORMATION OF THE NON-MAGNETIC G R O U N D STATE IN THE KRAMERS SYSTEM R.J. Radwafiski Centre for Solid State Physics, Sw. Filipa 5/5, 31-150 Kraktw, Poland
(Received 12 July 1993; in revised form 25 September 1993 by P. Wachter)
Electrostatic interactions can produce the localized non-magnetic state also in case of the Kramers system. The full suppression of the local moment is attained by highly anisotropic charge distribution at the vicinity of the f-shell electrons. This charge-formation mechanism for the non-magnetic state of the f magnetic ion is compared with the Kondo-compensation mechanism and the hybridization, of f and conduction electrons, mechanism. THE PROBLEM of properties of a magnetic impurity in a metallic host has a very long lasting discussion; here I mention a mile-stone work in 1961 by Anderson [1]. The formation of the non-magnetic (N-M) or very weakly-magnetic (VW-M) state of the magnetic impurity got tremendous impetus with the discovery of heavy-fermion (h-f) compounds, some compounds containing the Ce and U ions; namely CeAI 3 in 1975 [2]. The heavy-fermion compounds show very spectacular phenomena like the very large specific heat at low temperatures, discussed in terms of the large effective-electron mass, and a non-magnetic ground state at low temperatures contrasting with the "normal" sizefmagnetic moment observed at ambient temperatures. The N-M ground state appears from the finite, though remarkably large, susceptibility and hardly any magnetic-field effect on the h-f state. An extensive review of heavy-fermion compounds and their properties can be found in [3-8]. It is commonly accepted that the ground state of the h-f state has a collective nature. Based on this first-glance general impression, localized-based approaches have been, say, not in a fashion. The present main stream of theoretical works underlines the importance of mutual electronic correlations in which the local f moment is somehow dissolved by means of the hybridization of the f electrons with conduction electrons [3, 4, 7, 8] (the hybridization model). In Kondo-type explanations [9] for the formation of the non-magnetic state it is assumed that below a certain temperature a complex bound state of the localized spin and conduction
electrons is formed in which the local moment of the magnetic impurity is fully compensated by antiparallel oriented spins of conduction electrons. This singleimpurity Kondo model when extended to the lattice meets serious objections as has been pointed out by Nozitres [10] that in the Kondo lattice there is not a sufficient number of conduction electrons to compensate all localized moment by means of the spin-compensation mechanism. It is worth noting that both models, i.e. the hybridization model and the Kondo-lattice model are dealing with low-temperature properties and both of them have difficulties to describe experimentally observed the smooth transformation with temperature of the non-magnetic h-f state to the "normal localized" state. From an experimental point of view Franse et al. [11] pointed out significant similarities of the heavy-fermion UPt3 and the paramagnet PrNi 5 though further localizedelectron predictions have been found to fail in more detailed description of UPt3. Rietschel et al. [12] argued that anomalies above 20 K in the specific heat of four uranium h-f compounds, UBel3, UPt3, URu2Si2 and U2ZnlT, are of the Schottky type and are explainable as crystalline-electric-field (CEF) effects indicating strong localization of 5f electrons. The valency of the U atom is, however, difficult to determine from specific-heat studies. The thermal variation of the susceptibility at high temperatures also does not give unambiguous results about the uranium valency as the U 3+ and U 4+ ions have almost the same value for the effective moment. Above
553
554
N O N - M A G N E T I C G R O U N D STATE IN THE K R A M E R S SYSTEM
mentioned evidences illustrate the "normal localized"f behaviour above certain temperature and a mysterious non-magnetic state formed at lowest temperatures. From the CEF point of view for U compounds the configurations f2 (U 4+) a n d f 3(U 3+) can be concerned. The configuration f ' , corresponding to the nonKramers system, has been theoretically favoured [1315] owing to the possible formation by CEF interactions the singlet ground state for thefelectrons. Here it is worth mentioning that the quadrupolar Kondo effect proposed by Cox [16] for UBeI3 employs the nonmagnetic doublet F3 of the non-Kramers 5f 2 configuration. The configurationf3 has not been exploited in CEF approaches owing to the general conviction about the magnetic character of localized states for the Kramers system. Recently, the non-magnetic Kramers state has been surprisingly proven to exist and to be realized as the ground state within the ordinary CEF approach for the case of the f3 system with the hexagonal symmetry [17]. In this paper, I would like to discuss the charge mechanisms for the formation of the non-magnetic local f state in the case of the Kramers system f3 and to compare it with other existing mechanisms. The crystalline-electric-field approach is an elegant and handle form for the description of the localized states resulting from the charge (electrostatic) interactions of the f-shell electrons with surrounding charges. In CEF approach the local symmetry is directly incorporated from the starting point. The CEF approach has been found applicable [18-20] to 4f "normal" rare-earth ions (i.e. Nd, Tb, Dy, Ho, Er and partly Pr, Sm and Tm ions; Yb and mainly Ce often exhibits an "abnormal" behaviour). In the CEF approach the electronic states of the fn system, where n is an integer number, are described by the quantum number of the total angular momentum J that results from the consideration of the LS scheme, i.e. by assuming the spin-orbit coupling constant As-o to be 0o. Within the CEF approach for t h e f 3 configuration, that is relevant for the Nd 3+, U 3+ and Np 4+ ions, Hund's rules provide the multiplet 4•9/2 with J = 9/2 as the ground multiplet. Under the CEF interactions of the hexagonal symmetry the 10-fold degenerate 419/2 multiplet is split into 5 charge-formed (CF) doublets. The sequence of the doublets of the energy level scheme and the energy separations among them depend on the mutual strength of the CEF interactions of the different order. The charge-formed doublet I'~: cos/3] + 3 / 2 ) + sin/3] q: 9/2) (notation with cos/3 and sin/3 assures the automatic normalization) has only the component z of the total angular momentum J given by (Jz) = 4-(1.5-6 sin2/3). The dependence of the parameter/3
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on the strength of the CEF interactions is tg2/3 = 4v/-2-1k/(18a + 15b - 3g),
(1)
where k = 360B 6, a = 3B °, b = 84B ° and g = 5040B6°. Within the states F~ a non-magnetic doublet occurs if/3 equals 30 ° . The state I '1 = x/3/21 :F 3/2) + 1/21 + 9/2).
(2)
written in the ]J, Jz) representation with the z-axis taken along the hexagonal axis, has fully quenched the f-system moment as (Jx) = (Jy) = (Jz) = 0. This N-M Kramers state can be realized as the ground state by a set of CEF parameters: B ° = +6.0K, B°=-123mK, B°=-2mK and B 6 6 = - 5 2 . 3 m K [determined from equation (1)]. Some important outcomings concerning magnetic properties resulting from the CF N-M Kramers doublet ground state I'91 are (i) the vanishing magnetic moment at 0 K, (ii) the finite susceptibility at lowest temperatures, (iii) very anisotropic magnetic properties with the easy magnetic direction perpendicular to the hexagonal axis, (iv) almost linear increase of the reversal susceptibility X-1 with temperature marking good applicability of the Curie-Weiss law above 200 K though for the given CEF parameters the overall CEF splitting ACEv amounts to 846 K, (v) the effective moment derived from the high-temperature susceptibility is close to that expected for the configurationf3. The points (i) and (v) evidence the fact that the effective moment does not carry any information about the lowtemperature moment but rather about the ground multiplet [14]. The N-M Kramers doublet ground state is largely insensitive to the external/internal field. It makes difficult for spin-dependent interactions to establish a long-range magnetic state. According to numerical experiments an external field of 5 T separates the twodoublet levels by 0.13 K for the x-axis and by 0.004 K only for the c-axis revealing extremely weak influence of the magnetic field on these levels. The separation by 6.7 K one could expect for an effective spin S of 1/2. This remarkable disagreement means that any approximation of the behaviour of the N-M Kramers doublet in magnetic fields by any fictitious spin S with 1/2, as is a common practice for the doublet state in the spin Hamiltonian approach, is completely invalid. The field dependence of the N-M Kramers doublet is quite unusual as in principal the levels of the Kramers doublet should go up and down symmetrically what is related with opposite values for the associated magnetic moments. It should be noted, however, that the N-M behaviour appears only in the first approximation. Above 10T the levels of the N-M
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555
Kramers doublet are more and more split revealing is inestimable due to their large penetration of the that there is no violation of the time reversal symmetry intraionic space the effect o f p and d electrons seems to behind the discussed N-M Kramers doublet. be much larger due to the very anisotropic shape of The N-M Kramers doublet ground state results their orbitals [23]. Other atoms constituting the from higher-order CEF interactions. From equation intermetallic compound are also important for the (1), it shows that ~ depends critically on the anisotropy of the charge distribution at the vicinity of parameter B66 that describes in-plane six-order the f-shell electrons as the spatial allocation of the interactions. This parameter is indispensable for the anisotropic shape of the own p and d orbitals in the formation of the N-M doublet allowing the mixing of lattice depends on the charge distribution of the the [ + 3/2) and t 4- 9/2) states. The CEF interactions neighbouring atoms. In principle, it is possible to of the second-order produce the ground state with the transform the CEF parameters into the charge dominant contribution associated with the extremal distribution over the crystal. In this point experimenvalue of Jz, i.e. 0 (for B ° > 0) or + J (for B ° < 0) in tal evaluation for the CEF parameters will meet results case o f a non-Kramers' ion and + 1 / 2 (for B ° > 0) or of fully band-structure calculations if these calculations + J (for B ° < 0) in case of a Kramers' ion. The shown reach required accuracy of, say, 1 meV. set of the CEF parameters has been chosen in order Next, a question is if the N-M Kramers state can (i) to have the N-M state F91 as the ground state and be formed by electrostatic interactions also for other (ii) to have the first excited level at an arbitrarily symmetries and for other electronic configurations chosen value of 50 K. In fact, the value of 50 K has l i k e f I o r f 13. As far as the ordinary CEF approach is been chosen in [17] in order to account for two considered the answer is negative. However, going experimental results found for UPt 3. Namely, (i) the beyond the pure CEF approach by taking into Schottky-type of maximum in the extra specific heat account, for instance, the finite value for the spin[12, 21] at 23K points to the energy separation of orbit interaction, the intermediate coupling instead of 50 K and (ii) a transition peak in the optical reflection the pure LS coupling, a possible lowering of the local observed by Marabelli et al. [22] at about 4meV is symmetry and a possible cancellation of spin and interpreted as associated with the excitation to the first orbital moments open some possibilities for the excited CEF doublet. So, it is supposed that the formation of a VW-M ground state or may be even discussed CEF interactions could be a prototype for the fully non-magnetic [24]. the charge interactions in UPt3, a compound with the According to my knowledge, the charge mechanism hexagonal structure. As a further argument for these for the formation of the non-magnetic state in case of charge interactions in UPt3 it could be added that the Kramers system has not been discussed yet in these CEF parameters yield the inelastic-neutron- literature. This charge mechanism differs completely scattering (INS) transition with the lowest energy of from the hybridization and the Kondo mechanism. 50 K that has been correlated with an INS transition The hybridization model [1, 3] requires thefstates to lie observed by Felten [21]. Until now little attention has at the Fermi level resulting in the high density of (quasibeen given to this experimental observation. particles) states at E F necessary for explanation of the In the presented model the full suppression of the large specific heat at low temperatures. The hybridlocal moment is attained by the electrostatical ization theory, though seems to be quite successful in (approximated by the CEF) interactions of the f description of low-temperature properties in terms of shell with the surrounding charge distribution. This the Fermi liquid, cannot explain the smooth transforcharge distribution is highly anisotropic as is revealed mation of the heavy-fermion state to the localizedby CEF parameters with significant values for higher- moment state with the increase of temperature. It is order terms. The term B °, for instance, is associated highly intriguing how the increase of temperature with the charge-density variation with the polar angle sometimes by 10-20 K, what corresponds to the energy 0 given by the Legendre function p0 (cos 0) that has of about 0.002 eV, can drive the shift of thefelectronic [18] extrema at 0 = 30 ° and 65 °. The anisotropy of states down by 2-5 eV. In the Kondo mechanism, a the charge distribution is the direct consequence compensation of thef(significant) local moment occurs of the existence of the crystallographic lattice. The by conduction-electron spins and a non-magnetic biggest part in the suppression of the f-ion moment singlet ground state refers to an object containing the likely comes from own conduction electrons as f moment and the cloud of conduction electrons with the electrostatic interactions strongly depend on the opposite spins as it has been numerically shown by distance. The CEF terms B ° and B 6, for instance, Wilson [25] for the one-impurity Kondo problem. In depend on the distance in power up to - 7 . Although discussions of the Kondo mechanism, the attention is the role of the s electrons, among conduction electrons, not put to the spatial distribution of the charge cloud
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associated with the spin cloud. In the presented view this charge-cloud distribution, if it is not spherically isotropic, modifies substantially the internal electric field experienced by the f-shell electrons that, in a consequence, produces a localized state with largely reduced or even completely suppressed local magnetic moment. The anisotropic charge distribution in the neighbourhood of the f-shell electrons seems to be much more effective in production the VW-magnetic state of the originally magnetic f impurity than the direct spin compensation. The priority of the charge mechanism over the Kondo-spin mechanism seems to lie in the fact that the charge mechanism does not suffer from the Nozirres' argument. In this point it is worth stressing that, in contrary to the hybridization model, both models, i.e. the presented model and the Kondotype model, have in common to treat thefelectrons as localized ones. However, according to the presented model the local moment associated with thefelectrons is rather small whereas a substantial local moment, close to that seen in the paramagnetic Curie-Weiss law, is thought to be in Kondo-type considerations. In brief, the message of this letter is that charge interactions can produce a very weakly magnetic or a non-magnetic state for the localfmoment also for the Kramers systems. This charge mechanism associated with the anisotropic charge distribution at the neighbourhood of the f-shell electrons is very effective in the suppression of the local magnetic moment and overcomes the Nozirres objection for the Kondocompensation mechanism in the Kondo-lattice model. The present mechanism underlines the strong correlation of the ground state of thefion and its magnetic state with the charge distribution around the ion, i.e. the CEF potential at the f shell. Moreover, this correlation is a good illustration of the fundamental interplay of electrostatic and magnetic phenomena. Further studies on the physics of the non-magnetic or very-weakly magnetic f ground state and its physical realization are in progress. In this point heavy-fermion compounds with nearly non-magnetic ground state could be mentioned as being suspicious to have the CEF very-weakly magnetic Kramers doublet state as the ground state of their f electronic subsystem. Then the large specific heat is associated with many-electron excitations of the f-electronic subsystem within the charge-formed Kramers doublet ground state that is slightly split by spin-dependent interactions [26]. This low-energy excitation to the Kramers-twin state causes the reversal of the local moment and subsequently the polarization of the spin density of conduction electrons.
Vol. 89, No. 6
REFERENCES 1. 2. 3. 4. 5.
6.
7. 8. 9.
10. 11. 12. 13. 14. 15.
16. 17. 18. 19. 20.
21. 22.
P.W. Anderson, Phys. Rev. 124, 41 (1961). K. Andres, J.E. Graebner & H.R. Ott, Phys. Rev. Lett. 35, 1779 (1975). P. Fulde, J. Keller & G. Zwicknagl, Solid State Phys. 41, 1 (1988). N. Andrei, K. Furuya & J.K. Loewenstein, Rev. Mod. Phys. 55, 331 (1983); G.R. Stewart, Rev. Mod. Phys. 56(4), 755 (1984). A. de Visser, A. Menovsky & J.J.M. Franse, Physica B147, 81 (1987); A. de Visser, J.J.M. Franse & J. Flouquet, J. Magn. Magn. Mater. 108, 15 (1992). N. Grewe & F. Steglich, Handbook on the Physics and Chemistry of Rare Earths (Edited by K.A. Gschneidner, Jr. & L. Eyring), Vol. 14, p. 343, North-Holland (1991). G. Zwicknagl, Adv. Phys. 41, 203 (1992). T. Kasuya, Prog. Theor. Phys. Suppl. 108, 1 (1992); T. Saso, Prog. Theor. Phys. Suppl. 108, 89 (1992). It is a suggestion to reserve the name of the "Kondo effect" for the experimental fact associated with the increase of the resistivity with lowering temperature observed in some intermetallic compounds and diluted metallic systems as this fact has been exploited in the original paper by J. Kondo, Rep. Theor. Phys. 32, 37 (1964). P. Nozirres, Ann. de Phys. 10, 19 (1985). J.J.M. Franse, A. de Visser, A. Menovsky & P.H. Frings, J. Magn. Magn. Mater. 52, 61 (1985). H. Rietschel, B. Renker, R. Felten, F. Steglich & G. Weber, J. Magn. Magn. Mater. 76 & 77, 105 (1988). G.J. Nieuwenhuys, Phys. Rev. B35, 5260 (1987). R.J. Radwafiski, J. Magn. Magn. Mater. 103, L1 (1992). C. Geibel, A. Brhm, R. Caspary, K. Gloos, A Grauel, P. Hellmann, R. Modler, C. Schank, G. Weber & F. Steglich, Physica B186-188, 188 (1993). D.L. Cox, Phys. Rev. Lett. 59, 1240 (1987); J. Magn. Magn. Mater. 76 & 77, 53 (1988). R.J. Radwafiski, Physica B182, 209 (1992). R.J. Radwafiski & J.J.M. Franse, Physica B154, 181 (1989). R.J. Radwafiski, J.J.M. Franse, D. Gignoux, F.E. Kayzel, C. Marquina & A. Szewczyk, Physica B177, 291 (1992). R.J. Radwafiski, N.H. Kim-Ngan, F.E. Kayzel, J.J.M. Franse, D. Gignoux, D. Schmitt & F.Y. Zhang, J. Phys.." Condens. Matter 4, 8853 (1992). R. Felten, Thesis, Technical University Darmstadt (1988); for INS results see p. 103. F. Marabelli, G. Travaglini, P. Wachter & J.J.M. Franse, Solid State Commun. 59, 381 (1986).
Vo1.'89, No. 6 23. 24.
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R. Coehoorn, K.H.J. Buschow & M.W. Dirken, Phys. Rev. B42, 4645 (1990). In [14] similarities of CEF interactions of the Pr 3+ ion in Pr2FelaB and the U 4÷ ion in URu2Si 2 (the f 2 configuration) have been discussed. Here, I would like to comment that though many features for URu2Si2 can be explained by the f2 configuration but the large specific heat of this h-f compound in my present view can hardly be explained within the f2 configuration; consult, however, this view with the quadrupolar Kondo effect proposed by Cox
25. 26.
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[16] that operates on the non-magnetic doublet of the f2 configuration. In view of the present paper, considerations of the charge interactions within the f3 configuration are more appropriate for uranium compounds exhibiting the h-f behaviour, i.e. having the large specific heat at lowest temperatures. K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975). R.J. Radwafiski, Conf. Strongly Correlated Electron Systems, Poster 8P-92, Sendai, September (1992).