Magnetic-field-induced effects in the electronic structure of itinerant d- and f-metal systems

A paramagnetic response of transition metals and itinerant d- and f -metal compounds in an external magnetic field is studied by employing ab initio full-potential LMTO method in the framework of the local spin density approximation. Within this method the anisotropy of magnetic susceptibility in he...

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Veröffentlicht in:	Физика низких температур
Datum:	2009
1. Verfasser:	Grechnev, G.E.
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Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2009
Schlagworte:	Электронные свойства проводящих систем
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Zitieren:	Magnetic-field-induced effects in the electronic structure of itinerant d- and f-metal systems / G.E. Grechnev // Физика низких температур. — 2009. — Т. 35, № 8-9. — С. 812-828. — Бібліогр.: 63 назв. — англ.

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spelling	Grechnev, G.E. 2017-05-22T15:20:35Z 2017-05-22T15:20:35Z 2009 Magnetic-field-induced effects in the electronic structure of itinerant d- and f-metal systems / G.E. Grechnev // Физика низких температур. — 2009. — Т. 35, № 8-9. — С. 812-828. — Бібліогр.: 63 назв. — англ. 0132-6414 PACS: 71.20.–b, 71.20.Be, 75.10.Lp https://nasplib.isofts.kiev.ua/handle/123456789/117352 A paramagnetic response of transition metals and itinerant d- and f -metal compounds in an external magnetic field is studied by employing ab initio full-potential LMTO method in the framework of the local spin density approximation. Within this method the anisotropy of magnetic susceptibility in hexagonal close-packed transition metals is evaluated for the first time. This anisotropy is owing to the orbital Van Vleck-like paramagnetic susceptibility, which is revealed to be substantial in transition metal systems due to hybridization effects in electronic structure. It is demonstrated, that compounds TiCo, Ni₃Al, YCo₂, CeCo₂, YNi₅, LaNi₅ and CeNi₅ are strong paramagnets close to the quantum critical point. For these systems the Stoner approximation underestimates the spin susceptibility, whereas the calculated field-induced spin moments provided a good description of the large paramagnetic susceptibilities and magnetovolume effects. It is revealed, that itinerant description of hybridized f electrons produces magnetic properties of CeCo₂, CeNi₅, UAl₃, UGa₃, USi₃ and UGe₃ compounds in close agreement with experiment. In the uranium UX₃ compounds the strong spin–orbit coupling together with hybridization effects give rise to peculiar magnetic states, where the field-induced spin moments are antiparallel to the external field and the magnetic response is dominated by the orbital contribution. The author dedicates this work to the 90th anniversary of B.I. Verkin, who was one of pioneers in the field of magnetic properties studies in transition metals and compounds. The author is grateful to O. Eriksson, P.A. Korzhavyi, A.S. Panfilov, I.V. Svechkarev, A. Grechnev and J.M. Wills for fruitful scientific discussions. This work has been supported by the Russian-Ukrainian RFBR-NASU project 8-2009. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Электронные свойства проводящих систем Magnetic-field-induced effects in the electronic structure of itinerant d- and f-metal systems Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Magnetic-field-induced effects in the electronic structure of itinerant d- and f-metal systems
spellingShingle	Magnetic-field-induced effects in the electronic structure of itinerant d- and f-metal systems Grechnev, G.E. Электронные свойства проводящих систем
title_short	Magnetic-field-induced effects in the electronic structure of itinerant d- and f-metal systems
title_full	Magnetic-field-induced effects in the electronic structure of itinerant d- and f-metal systems
title_fullStr	Magnetic-field-induced effects in the electronic structure of itinerant d- and f-metal systems
title_full_unstemmed	Magnetic-field-induced effects in the electronic structure of itinerant d- and f-metal systems
title_sort	magnetic-field-induced effects in the electronic structure of itinerant d- and f-metal systems
author	Grechnev, G.E.
author_facet	Grechnev, G.E.
topic	Электронные свойства проводящих систем
topic_facet	Электронные свойства проводящих систем
publishDate	2009
language	English
container_title	Физика низких температур
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
format	Article
description	A paramagnetic response of transition metals and itinerant d- and f -metal compounds in an external magnetic field is studied by employing ab initio full-potential LMTO method in the framework of the local spin density approximation. Within this method the anisotropy of magnetic susceptibility in hexagonal close-packed transition metals is evaluated for the first time. This anisotropy is owing to the orbital Van Vleck-like paramagnetic susceptibility, which is revealed to be substantial in transition metal systems due to hybridization effects in electronic structure. It is demonstrated, that compounds TiCo, Ni₃Al, YCo₂, CeCo₂, YNi₅, LaNi₅ and CeNi₅ are strong paramagnets close to the quantum critical point. For these systems the Stoner approximation underestimates the spin susceptibility, whereas the calculated field-induced spin moments provided a good description of the large paramagnetic susceptibilities and magnetovolume effects. It is revealed, that itinerant description of hybridized f electrons produces magnetic properties of CeCo₂, CeNi₅, UAl₃, UGa₃, USi₃ and UGe₃ compounds in close agreement with experiment. In the uranium UX₃ compounds the strong spin–orbit coupling together with hybridization effects give rise to peculiar magnetic states, where the field-induced spin moments are antiparallel to the external field and the magnetic response is dominated by the orbital contribution.
issn	0132-6414
url	https://nasplib.isofts.kiev.ua/handle/123456789/117352
citation_txt	Magnetic-field-induced effects in the electronic structure of itinerant d- and f-metal systems / G.E. Grechnev // Физика низких температур. — 2009. — Т. 35, № 8-9. — С. 812-828. — Бібліогр.: 63 назв. — англ.
work_keys_str_mv	AT grechnevge magneticfieldinducedeffectsintheelectronicstructureofitinerantdandfmetalsystems
first_indexed	2025-11-27T09:23:09Z
last_indexed	2025-11-27T09:23:09Z
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fulltext	Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9, p. 812–828 Magnetic-field-induced effects in the electronic structure of itinerant d- and f-metal systems G.E. Grechnev B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkov 61103, Ukraine E-mail: grechnev@ilt.kharkov.ua Received February 17, 2009 A paramagnetic response of transition metals and itinerant d- and f -metal compounds in an external magnetic field is studied by employing ab initio full-potential LMTO method in the framework of the local spin density approximation. Within this method the anisotropy of magnetic susceptibility in hexagonal close-packed transition metals is evaluated for the first time. This anisotropy is owing to the orbital Van Vleck-like paramagnetic susceptibility, which is revealed to be substantial in transition metal systems due to hybridization effects in electronic structure. It is demonstrated, that compounds TiCo, Ni 3Al, YCo 2, CeCo 2, YNi 5, LaNi 5 and CeNi 5 are strong paramagnets close to the quantum critical point. For these systems the Stoner approximation underestimates the spin susceptibility, whereas the calculated field-induced spin mo- ments provided a good description of the large paramagnetic susceptibilities and magnetovolume effects. It is revealed, that itinerant description of hybridized f electrons produces magnetic properties of CeCo2, CeNi 5, UAl 3, UGa 3, USi 3 and UGe 3 compounds in close agreement with experiment. In the uranium UX3 compounds the strong spin–orbit coupling together with hybridization effects give rise to peculiar magnetic states, where the field-induced spin moments are antiparallel to the external field and the magnetic response is dominated by the orbital contribution. PACS: 71.20.–b Electron density of states and band structure of crystalline solids; 71.20.Be Transition metals and alloys; 75.10.Lp Band and itinerant models. Keywords: electronic structure, itinerant systems, spin susceptibility, orbital paramagnetism. Introduction The interaction between the external magnetic field and the electronic spin moment is usually the only inter- action which is taken into account when one is interested in the magnetic susceptibility χ of transition metal (TM, or T hereafter) or itinerant f -metal (IFM) systems. How- ever, the orbital moments in these systems are by no means negligible, and they are expected to contribute es- sentially to magnetic susceptibility. First principles calcu- lations proved to be a useful tool in modeling of various physical properties, including magnetic ones, of transi- tion metal systems. Nonetheless, it is still controversial whether the one-electron band theory within the local spin density approximation (LSDA) [1] or the genera- lized gradient approximation (GGA) [2] describes consis- tently the magnetic properties of TM and, particularly, IFM compounds, which are alternatively discussed in a number of papers within the models based on a localized character of d and f electrons (see, e.g., Refs. 3, 4 and ref- erences therein). We have carried out ab initio calculations of the elec- tronic structure and magnetic properties of transition met- als and representative series of TM and IFM compounds. On the basis of a comparison of the experimental data on χ and the results of calculations, we analyzed the elec- tronic states and interactions responsible for the magnetic properties of these systems. This comparison allowed to identify domains of applicability of our method for calcu- lating the paramagnetic susceptibility of TM and IFM compounds. Also, in this paper the atomic volume effect on magnetic susceptibility was studied with the aim to shed more light on the nature of paramagnetism and elec- tron–electron interactions in TM and IFM systems. It is expected that corresponding volume (or pressure) deriva- tives of χ are especially sensitive to the mechanism of these interactions. © G.E. Grechnev, 2009 Computational techniques The calculation of magnetic susceptibility of metallic systems is a rather difficult problem in solid state physics [5–8]. From a relativistic treatment, based on the Dirac equation, the total susceptibility in the absence of sponta- neous magnetic moment can be expressed as the sum ([5,6]): χ χ χ χ χ χtot spin orb so dia= + + + + L , (1) where these terms correspond to the Pauli spin suscepti- bility, a generalization of the Van Vleck orbital paramag- netism, a contribution due to the spin–orbit coupling, the Langevin diamagnetism of closed shells (core electrons), and generalization of Landau conduction electrons dia- magnetism, respectively. The χso term is a relativistic correction to the suscepti- bility, and it has been shown [6,7], that χso is a higher or- der term as compared with χspin and χorb , and its value was estimated to be much smaller than the other two terms for transition metals (about few percents). Hence χso has been usually neglected in theoretical studies of magnetic susceptibility of transition metals [5,9–13]. In order to evaluate the various terms in Eq. (1), it was proposed in Refs. 5–7 to calculate the corresponding wave-vector dependent susceptibilities χ( )q by using a realistic band structure of some transition metals, and then taking the limit q → 0 either analytically [5] or by numerical extrapolation [6,7]. Also the linear response formalism based on a Green's function technique was em- ployed to calculate the spin [10,11,13] and orbital [13] magnetic susceptibilities in some transition metals. In these calculations the exchange enhancement of the Pauli spin susceptibility was taken into account within the Sto- ner model. In the framework of the Stoner model the ex- change-enhanced Pauli paramagnetic spin susceptibility χ P can be written in the form: χ χ μ 2 ston = ≡ − −S N E IN EP B F F( ) [ ( ) ]1 1 , (2) where χ P = μ 2 B FN E( ), S is the Stoner enhancement fac- tor. The Stoner integral I, describing the exchange-corre- lation interaction of the conduction electrons, can be ex- pressed in terms of the calculated parameters of the electronic structure [14,15]: I N E N E J N EF ql qll F qll ql F= ′ ′ ′∑1 2/ ( ) ( ) ( ) . (3) Here N EF( ) is the total density of electronic states at the Fermi level EF , N Eql F( ) is the partial density of states for atom q in the unit cell, J qll′ are the local exchange integrals: J g r r r rll l l′ = φ φ∫ ( ( )) ( ) ( ) ,'ρ 2 2d (4) where φl r( ) are the partial wave functions, and g r( ( ))ρ is a function of the electronic density [16]. More elaborated spin-polarized approach was proposed in Ref. 17 for the calculation of χspin , however the orbital contributions to χ tot have not been evaluated in that work. In the present paper we apply another approach, which has been successfully used in Refs. 8, 18 to calculate mag- netic susceptibility of some TM and IFM systems, includ- ing even anisotropy of χ. This approach is based upon the ab initio full-potential linear muffin-tin orbital method (FP-LMTO) [18,19] within the local spin density approx- imation for the exchange-correlation effects [16]. The un- derlying Kohn–Sham equations are solved for a generic potential without any shape approximation, and the vol- ume and crystal structure are the only input parameters to calculations of this kind. The details of our method are given elsewhere [8,18,19], and here we mention only main features of the present implementation, which are different from other used FP-LMTO techniques. In the present calculations, the FP-LMTO basis set included the s p d f, , , orbitals for f metals, and s p d, , orbitals for all other elements within a single, fully hybridizing, energy panel [19]. All relativistic effects, including spin–orbit coupling, were incorporated, and the effect of an external magnetic field B was taken into account self-consistently at each iteration by means of the Zeeman term: � ( � �H Z = ⋅B s + l2 ) , (5) where �s is the spin operator and �l is the orbital angular mo- mentum operator. By this way, the present approach al- lows to calculate ab initio the field-induced spin and or- bital magnetic moments. When the field induced spin and orbital magnetic mo- ments are calculated, the corresponding volume magneti- zation can be evaluated, and the ratio between the magne- tization and the field strength provides a susceptibility, which basically corresponds to χspin and χorb terms in Eq. (1). We should note, that within these self-consistent calculations the effect of the spin–orbit coupling upon the calculated field-induced magnetic moments is actually in- cluded implicitly, i.e., without calculations of a separate χso term in Eq. (1). In the case of hexagonal or tetragonal crystal structure the components of magnetic susceptibility, χ \|\| and χ ⊥ , can be derived from the magnetic moments obtained in an external field, applied parallel and perpendicular to the c axis, respectively. We have tested the linearity of the M B( ) dependence, where M is the magnetization, and found it quite linear for the fields 0.5–10 T. For external fields about 1 T, however, a huge number of k points Magnetic-field-induced effects in the electronic structure of itinerant d- and f -metal systems Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 813 (about 100 K) were necessary to get the desired accuracy, and we actually used the field of 10 T for most calcula- tions in order to reduce the number of k points used. The calculated total energies were well converged (∼ −10 6 Ry) with respect to all parameters involved, such as k-space sampling and basis set truncation. In the present work we did not calculate the diamag- netic contributions to the susceptibility coming from core and conduction electrons, which correspond to the χdia and χ L terms in Eq. (1). The Langevin contributions χdia can be estimated based on results of Refs. 5, 20, 21 and appeared to be between free-atom and free-ionic diamag- netic susceptibilities. To calculate the Landau diamag- netic contribution χ L is a considerably more difficult problem (see [5,22–24] and references therein). The free-electron Landau limit is often used for estimations, giving a χ 0 L that equals − /1 3 of the Pauli spin susceptibil- ity, though for many systems this crude approximation was found not to provide even the correct order of magnitude of the diamagnetic susceptibility. Beyond the free-electron limit, χ L can be expressed qualitatively as inversely proportional to an average ef- fective mass of conduction electrons, m. Therefore this contribution appears to be large for some nontransition metals and semimetals (graphite, beryllium, cadmium, bismuth) with small values of m [22,23,25,26]. For tran- sition metal systems χ L is usually assumed to be negligi- ble in comparison to considerable paramagnetic contribu- tions, due to the predominantly large effective masses m. On the other hand, it was shown [23,24] that the anoma- lous diamagnetism can originate from a tiny group of quasi-degenerate electronic states with small m, situated in the vicinity of the Fermi energy EF and this contribu- tion can be many times larger than the free-electron Landau estimation χ 0 L. Magnetic susceptibility of cubic transition metals We first inspect the accuracy of the present method by comparing experimental and theoretical data for the mag- netic susceptibility of TMs. The calculated densities of electronic states (DOS) for a series of TMs belonging to successive groups of the Periodic Table are presented in Fig. 1. One can see, that due to the strong s p, –d hybri- dization the conduction bands of TMs are rather wide, and the valent d states can be considered as itinerant ones. This justifies the use of the proposed technique to evalu- ate magnetic susceptibility for TMs. The calculated para- magnetic contributions to χ of the cubic TMs are given in Table 1 together with the estimated Langevin diamagne- tism of closed shells and the experimental data on mag- netic susceptibility. Table 1. Magnetic susceptibility of cubic transition metals M χston χspin χorb χdia [20,21] χspin+χorb+χdia χexp [27] d ln χ / d lnV 10 6− emu/mol theor exp V 180 175 113 –10 278 300 2.2 1.9 [28] Nb 90 120 67 –10 177 212 1.9 1.7 [29] Ta 82 90 45 –14 121 162 1.5 1.1 [29] Cr 30 30 178 –15 193 180 2.1 2.0 [30] Mo 21 23 92 –23 92 81 1.5 1.2 [31] W 13 15 68 –36 47 53 0.5 0.5 [31] Rh 71 75 65 –22 118 110 2.5 — Ir 40 46 52 –50 48 30 2 — Pd 481 587 119 –25 681 720 6 5.2 [28] Pt 110 189 79 –40 228 220 2.8 — 814 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 G.E. Grechnev –0.2 –0.1 0 0.1 0.2 0.3 E, Ry 0 20 20 20 (hcp) 20 20 N (E ), st at es /( R y ·a to m ) 20 20 20 Y Zr Nb Cr Re Fe Pd Rh Fig. 1. Density of electronic states of transition metals: Y, Zr, Nb, Cr, Re, Fe (hcp), Rh, Pd. The Fermi level is marked with a vertical dashed line. The TMs of group VA — V, Nb, and Ta — have the bcc crystal lattice, and the Fermi level is situated within the descending part of a broad DOS N(E) peak (see Fig. 1). As can be seen in Table 1, for these metals the orbital paramagnetic contributions χorb are very important since the spin contributions χspin alone are not sufficient to de- scribe the experimental values of susceptibility. We note that the sum of calculated paramagnetic contributions ap- pears to be lower than the experimental χ. This should be considered as rather expected underestimation for the LSDA ground state of group VA TMs. Due to the known over-bonding tendency of LSDA [19] the theoretical equilibrium volumes are about ∼5% smaller than the ex- perimental ones, and this resulted in slightly suppressed values of ab initio calculated χspin and χorb . The TMs of the next group VIA — Cr, Mo and W — also have the bcc crystal lattice, and the Fermi level is po- sitioned at the local DOS N E( ) minimum (see Fig. 1). In accordance with the low N EF( ) values in these metals, the orbital paramagnetic contributions χorb are found to be substantially higher than the spin contributions. As is seen in Table 1, the calculated χspin and χorb appear to be in agreement with the experimental data on susceptibility. Obviously, the diamagnetic contributions χdia and χ L have to be taken into account. For the fcc transition metals of group VIIIA the Fermi level lies at a sharp DOS N E( ) peak, correspondingly be- low (Rh and Ir) and just after the peak (Pd and Pt). In Rh and Ir the calculated spin and orbital paramagnetic contri- butions are about the same size, and with additional dia- magnetic contributions (χdia and χ L) the good agree- ment with the experimental susceptibilities can be seen in Table 1. On the other hand, in Pd and Pt the high N EF( ) values provide the substantially enhanced spin suscepti- bilities χspin . However, it should be noted that the orbital contribution χorb is needed to achieve a good agreement with the experiment. One can see from Table 1 that the Stoner approach un- derestimates substantially the spin susceptibility in Pd and Pt. There is actually the general trend found also for other TMs that the Stoner model provides smaller values of the spin susceptibility χston , comparatively to the field-induced calculated χspin . We should emphasize that ab initio LSDA calculations take into account the nonuni- form induced magnetization density in the unit cell and presumably provide more accurate values of χspin . Based on the calculated χspin ( )V and χorb ( )V , the vol- ume dependence of the magnetic susceptibility (VDMS) d dln / lnχ V was evaluated and compared with the ex- perimental data for the cubic TMs (see Table 1). The calculated large value of VDMS in Pd is in agreement with the experimental values of the pressure effect on χ [28] and also with the magnetostriction measurements for Pd [31]. For other cubic TMs the VDMS is smaller, d ln χ /d ln V � 1–2, and also is in agreement with the available experimental data. For the isovalent metals be- longing to the same group of the Periodic Table one can notice from Table 1 that d ln χ /d ln V appears to decrease with increasing atomic number of the element, i.e., in the 3 4 5d d d→ → -metal series. Magnetic susceptibility of hcp transition metals In the present work the field-induced spin and orbital magnetic moments were also calculated for TMs which possess the hcp crystal structure at ambient pressure. The calculations were performed for varying atomic volume at the corresponding experimental lattice parameters ratios c/a. The averaged values of the calculated and experimental susceptibilities χ χ χ= + ⊥( )\|\| 2 3/ and the anisotropies Δχ are listed in Table 2. Also the Langevin diamagnetic terms χdia , estimated according to Refs. 20, 21, and the calculated exchange-enhanced Pauli suscepti- bilities χston are presented in the Table 2. Table 2. Magnetic susceptibility of hcp transition metals M χ s to n χ s p in χ o rb χ d ia [2 0 ,2 1 ] χ s p in + χ o rb + χ d ia χ e x p [2 7 ] Δχ th eo r Δχ ex p [2 7 ] d ln χ/ d ln V 10 6− emu/mol theor Sc 207 267 38 –10 295 300 –5 –10 4 Y 163 191 26 –15 202 178 –8 –22 2 Lu 146 199 25 –20 204 193 –10 –25 2 Ti 42 61 73 –9 125 140 8 21 1.2 Zr 46.5 59 51 –10 100 120 10 51 0.2 Hf 31 37 39 –16 60 70 5 32 0.9 Re 25 34 77 –36 75 68 –4 –5 1.0 Ru 35 40 68 –23 85 50 –12 –9 1.9 Os 23.5 30 56 –44 42 20 –5 –5 1.7 As can be seen from Table 2, for trivalent TMs Sc, Y, and Lu the spin contributions to susceptibility χspin are substantially larger than Van Vleck orbital paramagnet- ism χorb . Also χspin is notably larger than the χston sus- ceptibilities which were calculated within the Stoner ap- proximation (Eq. (2)). For these IIIA group TMs the sums of χspin , χorb and χdia appear to be in agreement with the corresponding experimental susceptibilities. Regarding the anisotropy of susceptibility, our calculations repro- duce the sign and order of magnitude of Δχ, and also a rise of \| \|Δχ in a the series Sc → Y → Lu (see Table 2). Magnetic-field-induced effects in the electronic structure of itinerant d- and f -metal systems Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 815 For the IVA group TMs — Ti, Zr, and Hf — the calcu- lated values of χ are also in a qualitative agreement with experimental data, taking into account somewhat reduced χspin and χorb for the theoretical LSDA atomic volumes, as discussed above. The contributions χspin and χorb are about the same size indicating again the importance of the Van Vleck paramagnetism in TM. Also, in the case of Ti, Zr, and Hf the exchange enhanced Stoner susceptibility χston is about 20% lower than the field-induced calcu- lated χspin . It is seen in Table 2 that our calculations reproduce the positive sign of the anisotropy Δχ for the IVA group TMs, however, the calculated anisotropy appeared to be sub- stantially lower than experimentally observed Δχ values. One should note that in these metals the Fermi level is si- tuated close to the DOS N E( ) minimum, as can be seen in Fig. 1 for Zr. This implies the presence of the sp–d hybri- dized states with low effective masses, which can provide a notable anisotropic diamagnetic term χ L, giving addi- tional contribution to Δχ. In the case of a complicated multiband structure of hcp transition metals it is not feasi- ble to calculate correctly or even estimate the χ L contri- bution, which can be responsible for the noted discrep- ancy with the experimental data on Δχ. In connection with this we can also assume that the dif- ference between the calculated and experimental suscep- tibilities in the case of group VIIIA hcp transition metals Ru and Os (see Table 2) could arise from a substantial dia- magnetic contribution χ L. For these metals the calculated Van Vleck susceptibility χorb is about twice as large as the χspin term, and presumably this dominant χorb contribu- tion is the main source of the observed Δχ, which is repro- duced quite well by the calculations. This is also the case of the hcp Re, where a good agreement with the respective experimental data was obtained for the calculated χ and Δχ. Also, for Re, Ru and Os the field-induced calculated contribution χspin is notably higher than the correspon- ding Stoner estimation χston , calculated according to Eq. (2). At ambient conditions all hcp transition metals have a c a/ ratio close to the ideal value of 8 3/ ≈ 1.633, but, as can be seen from Table 2, Δχ > 0 for Ti, Zr, Hf, whereas for Sc, Y, Lu, Ru and Os we find that Δχ < 0. This means that the c a/ ratio is not the main factor determining the sign of Δχ. In fact, the behavior of Δχ in hcp TMs is deter- mined by a delicate balance between the Zeeman energy, exchange effects and the spin–orbit coupling. We empha- size that the sign and order of magnitude of Δχ are always reproduced for the hcp transition metals in the framework of our field-induced band structure calculations. As seen from Table 2, the sign of the experimental anisotropy of the magnetic susceptibility [27,32] is positive in Ti, Zr, and Hf (group IVA), but Δχ is negative in Sc, Y, Lu (group IIIA), Re (group VIIA), Ru and Os (group VIIIA). The absolute values of the calculated anisotropy are also in a qualitative agreement with the experiment, with allow- ance made for the observed [27,32] strong temperature dependence of Δχ. Magnetic properties of hcp iron Recent ab initio total energy calculations for different crystal structures of iron, performed over a wide volume range, as well as the recent experimental studies with dia- mond anvil cells, have confirmed the stability of the hcp ground state at pressures above 15 GPa (see [33–35] and references therein). The hcp phase of iron is also expected for temperatures and pressures corresponding to the Earth's inner core conditions [35]. These calculations, however, have not provided a clear picture of magnetic properties of the hcp iron and their anisotropy. Our spin polarized electronic structure calculations have not revealed a spontaneous magnetic moment for the hcp iron, and the absence of macroscopic magne- tization is in agreement with the recent observation of the onset of superconductivity in iron at pressures 15 GPa < <P 30 GPa and below 2 K. On the other hand, the presence of magnetic fluctuations in the hcp Fe has been put forward as a possible origin of the superconduc- tivity [34]. It has been demonstrated in previous sections that our calculations of magnetic susceptibility for TMs reproduce experiments at ambient conditions accurately enough, including the measured anisotropy of hcp transi- tion metals. This provides the basis on which magnetic properties of the hcp iron can be studied ab initio over a wide volume range, from the superconducting state to the Earth's inner core conditions. The field-induced magnetic moments of the hcp iron were calculated for a number of atomic volumes, corre- sponding to pressures from 20 to 350 GPa where the hcp phase is stable. The c/a axial ratio was taken equal to 1.59, which minimizes the total energy of the hcp iron [33] and conforms with recent experimental data. At these conditions Fe does not spontaneously order magnetically, and a small magnetic moment develops only in the pres- ence of a magnetic field. The corresponding density of electronic states in hcp Fe, presented in Fig. 1, is very similar to the DOS of the isoelectronic metals Ru and Os, calculated at ambient conditions. Our calculations demonstrate that for the hcp iron the induced spin moments are parallel to the orbital moments, in agreement with Hund's third rule. It is particularly remarkable that the spin contribution to χ is even smaller than the orbital contribution (χ χspin orb� 0 7. ) at pres- sures about 300 GPa. The averaged value of the sus- ceptibility of hcp iron χ is found to be ranging from 170 10 6⋅ − emu/mol (P = 350 GPa) to 350 10 6⋅ − emu/mol (P = 50 GPa). The evaluated VDMS d dln / lnχ V is pre- sented in Fig. 2 and appears to be consistent with the 816 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 G.E. Grechnev calculated and experimental paramagnetostriction data for nonmagnetic transition metals. Based upon the calcu- lated VDMS, the hcp iron occupies some intermediate po- sition between Pd, which has strongly exchange-en- hanced paramagnetism, and other TMs. For comparison, the Pauli spin contribution to the magnetic susceptibility was also calculated within the Stoner model (2) and ap- pears to be in agreement with the field induced spin sus- ceptibility, evaluated by using the full Zeeman term (5). The Stoner enhancement factor S was found to be about 3 for the hcp Fe at pressures of 20 GPa, which indicates on a possible role of spin-fluctuations as mediators of the su- perconductivity. As expected, the anisotropy of χ comes almost exclu- sively from the orbital contribution χorb , which indicates that a relativistic treatment is necessary to explain mag- netic properties even of comparatively «light» elements, like Fe. The calculated pressure and temperature depen- dencies of the magnetic susceptibility anisotropy of the hcp Fe are presented in Fig. 3. The thermal effects were taken into account through the Fermi–Dirac distribution function. It has been demonstrated [8] that the observed seismic anisotropy of the Earth's inner core could have its origin in the anisotropy of the magnetic susceptibility of the hcp iron. A suggested mechanism of the seismic anisotropy of the Earth's inner core involves the anisotropy of the mag- netic susceptibility of the hcp iron, and it is argued [8] that if χ is sufficiently anisotropic, a preferential orienta- tion of the hcp crystals may occur. A validity of this mech- anism is crucially dependent on the elastic anisotropy of the hcp iron, and with the calculated anisotropy of the elastic constants of the hcp Fe for c/a = 1.59 [33], the compressional velocity is faster along the c axis than along the a axis. Therefore, the present theoretical calcu- lations demonstrate that the anisotropy of magnetic sus- ceptibility of the hcp Fe can explain the seismological ex- periments. Transition metal compounds with enhanced para- magnetic susceptibility In the recent years the properties of the exchange-en- hanced paramagnets and weak itinerant ferromagnets near magnetic instabilities has attracted substantial inter- est in connection with the so-called quantum critical points (QCP) [36,37]. The phase transitions in these QCP are due to nonthermal quantum fluctuations and take place at T = 0, though their influence can extend even at finite temperatures. The TM systems such as Pd, Sc, the hcp phase of Fe, TiCo, Ni 3Al, Ni 3Ga, ZrZn 2, TiBe 2, and MnSi compounds, a number of TCo 2 and TNi 5 alloys, and also some cerium and uranium based systems are close to QCP. Remarkably nonconventional properties were recently found in these materials, such as non-Fermi liquid behavior, metamagnetic transitions, unconven- tional superconductivity, co-existing with ferromagne- tism in some cases. Though a number of theoretical mo- dels have been put forward [37], there is still no sufficient Magnetic-field-induced effects in the electronic structure of itinerant d- and f -metal systems Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 817 8 16 V, � 3 0 1 2 3 4 5 6 d ln χ/ d ln V Pd Sc Cr V Nb Rh Pt Fe (hcp) Fig. 2. Calculated volume dependence of the magnetic suscep- tibility d dln / lnχ V for the hcp phase of iron (�, in a logarith- mic scale) in the volume range corresponding to pressures from 350 to 50 GPa. The dashed line is a guide for the eye. The calculated for other transition metals d dln / lnχ V at ambi- ent pressure are presented for comparison (�, see also Tables 1 and 2). 100 200 300 P, GPa –25 –20 –15 –10 –5 0 , 1 0 Δχ – 6 em u /m o l 3000 K 0 K Fig. 3. Pressure dependence of the magnetic susceptibility ani- sotropy of the hcp iron. The solid and dashed lines correspond to the temperatures 0 and 3000 K, respectively. understanding of these phenomena. Also the validity lim- its of band theory approaches and the LSDA approxima- tion to the analysis of QCP phenomena have not been es- tablished. In this section the fine details of electronic spectra and magnetic properties of systems close to QCP were studied by means of the filed-induced FP-LMTO calculations. Also, a detailed comparison with available experimental data has allowed to validate and tune up the quality of the band theory methods employed. Particular attention has been given to investigations of the magnetovolume effect in these systems in proximity of QCP. TiCo, Ni3Al and YCo2 systems The titanium compounds TiFe and TiNi were exten- sively studied for years in connection with a possibility to store hydrogen and the observation of the shape memory effect, respectively. TiCo, a compound with the interme- diate band filling, has the same simple B2 cubic crystal structure (CsCl type) and exhibits a strong paramagne- tism [36,38]. Also a transition to ferromagnetic (FM) state has been observed upon substitution of Ti with nontransition metals — Al and Ga [36,39]. This indicates itinerant character of the observed ferromagnetism in these alloys and proximity of TiCo to QCP. Earlier band structure calculations have given contro- versial results, including a possibility of the FM phase in TiCo with a small magnetic moment M B� 0 02. μ ([36,38] and references therein). According to the present calcula- tions, TiCo exhibits a paramagnetic (PM) ground state and the corresponding density of electronic states is given in Fig. 4. The two groups of bands in the figure are formed by strong hybridization of d states of Ti and Co, and the Fermi level is situated above the pseudo-gap, in contrast to the TiFe compound. The calculated contributions to magnetic susceptibil- ity χspin and χorb are presented in Table 3. As seen in the table, the spin contribution χspin is large and dominant in TiCo, whereas the Stoner approach (2) provides substan- tially lower corresponding value of χston . Taking also into account the Van Vleck contribution χorb we obtain very good agreement with the experimental susceptibility (see Table 3). Table 3. Magnetic susceptibility of cubic compounds TiCo, Ni3Al, YCo2, CeCo2 M χston χspin χorb χspin+χorb χexp d ln χ /d lnV 10 4− emu/mol theor exp TiCo 7 10.5 2.7 13.2 13 6.4 5.4 [38] YCo2 12 15.5 2.5 18 20 12 15 [40] CeCo2 10 12 3 15 13 6 5 [41] Ni3Al 23 29.3 0.6 30 20* 7 3–7.7 † [38] T = 293 K; †T = 100–300 K. Note that the values of χspin and χorb in Table 3 are calculated at the theoretical lattice parameter a. As a in- creases and the system approaches QCP, a sharp rise of the contribution χspin determines the behavior of mag- netic susceptibility, presented in Fig. 5. The correspond- ing calculated value of VDMS d dln / lnχ V is listed in Table 3 and appears to be higher than in elemental TMs, including Pd and Sc (compare with the data in Tables 1 and 2). As seen in Table 3, the theoretical ground state 818 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 G.E. Grechnev 0.6 0.8 E Ry, 0 50 100 N , st at es R y (E ) / TiCo E F Fig. 4. Density of electronic states of TiCo. The Fermi level is marked by a vertical dashed line. 5.54 5.56 a, a.u. 1.3 1.4 TiCo , 1 0 em u /m o l χ – 3 Fig. 5. The calculated magnetic susceptibility of TiCo as a function of the lattice parameter. The theoretical lattice param- eter is marked by a vertical dashed line. value of d dln / lnχ V is in agreement with the experimen- tal result of Ref. 38, obtained for TiCo at T = 78 K. The intermetallic compound Ni3Al has the cubic Cu3Au-type structure, and it is widely used in air- and spacecraft technologies. At low temperatures Ni3Al is an itinerant FM with the ordering temperature TC = 41K and the small magnetic moment about 0.2 μ B [36,38]. Above TC the magnetic susceptibility sharply decreases with temperature until at T � 300 K it gradually falls into the region of χ typical for exchange-enhanced paramagnets (see Table 3). In addition to observed non-Fermi liquid transport properties under pressure and a transition to the PM phase at P = 8.1 GPa [42], strong pressure effects on magnetic susceptibility have been also detected in Ni 3Al [36,38]. The band structure of Ni 3Al calculated in the present work is found to be in good agreement with previous cal- culations (see Refs. 36, 38, 43). The calculated at the ex- perimental lattice parameter total energy differences be- tween the PM and FM phases are barely detectable, whereas in the close vicinity of the theoretical (LSDA) lattice parameter we get the PM ground state. The density of electronic states for the PM phase of Ni3Al is given in Fig. 6. One can see the similarity of the N E( ) structure in Fig. 6 to that of the fcc palladium in Fig. 1. In fact this re- semblance can be attributed to the moderate hybridization of d states of Ni with p states of Al in Ni3Al, where the Fermi level is situated within the descending part of al- most d character DOS peak with a high value of N EF( ). The paramagnetic contributions to susceptibility of Ni3Al, χspin and χorb were calculated within FP-LMTO scheme for the PM phase in an external magnetic field and presented in Table 3. According to our calculations, at the equilibrium ground state volume of Ni 3Al the total energy of the FM phase is larger, but close to E total of the PM phase. This makes the calculations of field-induced magnetic moments in Ni 3Al rather difficult, with a num- ber of precautions exercised to get convergency, like in- creasing the number of k points used, and decreasing the external field value to 5 T. Moreover, the convergency was actually obtained for slightly reduced lattice parame- ters of Ni 3Al, a a≤ 0 98. theor , and we can consider the cal- culated χspin and χorb as approximations. As seen in Table 3, the χspin contribution is definitely dominant in Ni 3Al, and it is substantially larger than the Stoner susceptibility value χston , obtained according to Eq. (2). The agreement with the experimental data on χ and d dln / lnχ V can be considered as satisfactory, taking into account the proximity of total energies for the FM and PM phases. This forced us to reduce the lattice param- eters to get convergency of the field-induced calculations. Also it should be noted, that the experimental data for Ni3Al in Table 3 were obtained for the most part at room temperatures, whereas the FP-LMTO calculations are done at T = 0 and apply to the ground state only. The compound YCo 2 with the cubic Laves phase C15 structure is known as strongly enhanced Pauli paramag- net, which exhibits a distinct maximum in the temperature dependence of the magnetic susceptibility at T � 230 K, and the itinerant metamagnetic transition to the FM phase in fields of about 70 T [36,44,45]. Also, a strong pressure effect on χ has been observed in YCo 2 and some Y(Co 1 2−x xM ) alloys [40,45,46]. In order to find the origin of pronounced volume and temperature effects on magnetic susceptibility in YCo 2 compound, the volume dependent electronic structure was calculated ab initio by employing the FP-LMTO Magnetic-field-induced effects in the electronic structure of itinerant d- and f -metal systems Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 819 –0.1 –0.08 –0.06 E, Ry 0 40 80 120 N (E ), st at es /R y E F Ni 3 Al Fig. 6. Density of electronic states of Ni3Al (PM phase). The Fermi level is marked by a vertical dashed line. 0.6 0.8 E, Ry 0 100 200 N (E ), st at es /R y YCo 2 E F Fig. 7. Density of electronic states of YCo2. The Fermi level is marked by a vertical dashed line. method, within LSDA. According to our calculations, the PM phase is the ground state of the system. There is a pro- nounced hybridization of 3d states of Co with 4d states of Y, which provides a peculiar N E( ) structure at EF (Fig. 7). The Fermi level is situated close to a local mini- mum of the DOS, just after a high and sharp DOS peak with the dominant d states of Co. The spin and orbital contributions to the magnetic sus- ceptibility were derived from the corresponding field- induced moments, which have been calculated at T = 0 in an external magnetic field of 10 T. These χspin and χorb contributions and the total paramagnetic susceptibility are presented in Fig. 8 and in Table 3. The main part of the strongly volume dependent spin contribution originates from the 3d states of Co, and these states are also respon- sible for the orbital Van Vleck contribution to χ. As seen in Fig. 8, as a increases and the system approaches QCP, a sharp rise of χspin determines the behavior of the magnetic susceptibility. The obtained very large value of magnetovolume effect in YCo 2, d ln χ /d ln V = 12, ap- peared to be in agreement with the experimental data of Ref. 40 (15 2± ). YNi 5, LaNi 5 and YNi 5−xCu x systems Intermetallic compounds between nickel and trivalent TM (or rare earth R) elements of the RNi 5 type are distin- guished by a great diversity of magnetic structures and properties [44]. For example, compounds with Y, La, Ce, and Lu are considered as the exchange-enhanced Pauli paramagnets, while PrNi 5 is a Van Vleck paramagnet. The compounds RNi 5 have a number of interesting physi- cal properties, and they are used for making portable bat- teries. The compound LaNi 5 is promising for hydrogen storage, since the unit cell of this compound can store up to 7 hydrogen atoms. There are indications that substitu- tion of some amount of nickel in RNi 5 compounds with other metals (Al, Co, Cr, Cu, Mn) alters their structural, electronic, and magnetic properties and can also improve their electrochemical characteristics [47]. In this work a detailed investigation of magnetic properties and elec- tronic structure of RNi 5 (R = Y, La, Ce) compounds and YNi 5−xCu x alloys was carried out by means of the first-principles calculations of their electronic structure and magnetic properties. Ab initio calculations of the electronic structure were carried out for the RNi 5 compounds by employing the FP-LMTO method. These compounds crystallize in a hexagonal structure of the CaCu 5 type with six atoms per unit cell and two inequivalent types of transition metal atoms occupying positions of different symmetry (2c and 3g positions). The corresponding contributions to mag- netic susceptibility were derived from the calculated field-induced moments in an external field of 10 T. In the FP-LMTO basis set the maximum value of the orbital quantum number l was taken as lmax = 2 for Y, Ni, and Cu and by lmax = 3 for La. The electronic structure calcula- tions were performed for a number of lattice parameters close to the experimental one and the ratio c/a was fixed at its experimental value (c/a ≈ 0.8 for RNi 5). The equi- librium lattice spacings a theor and corresponding theoreti- cal bulk moduli B theor were determined from dependence of the total energy on the unit cell volume E V( ) by using the well known Murnaghan equation. The differences about 10% between the theoretical B theor and experimen- tal bulk moduli of RNi 5 are presumably related to the over-bonding tendency of the LSDA approach [19]. The electronic densities of states of RNi 5 calculated over a wide energy interval are presented in Fig. 9. The exchange-enhanced Pauli susceptibilities (2) for RNi 5 compounds were calculated in the framework of the Stoner model on the basis of the calculated densities of states at the Fermi level and the Stoner integrals (3). The results are presented in Table 4 and compared to the experiment. It is seen from the table that for YNi5 and LaNi5 the principal contribution to χ comes from the ex- change-enhanced spin susceptibility. As a consequence of the large value of the Stoner enhancement factor for these compounds (S � 6), small variations in N EF( ) calculated by different methods will lead to noticeable differences in the calculated susceptibility χston . For YNi5 and LaNi5 the FP-LMTO method was used to calculate the induced spin and orbital magnetic mo- ments in an external magnetic field, and the correspond- ing spin and orbital contributions to the susceptibility are listed in Table 4. As seen from the table, for these com- 820 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 G.E. Grechnev 13.2 13.4 a, a.u. 0 1 2 3 total spin orbital χ , em u /m o l 1 0 – 3 YCo2 Fig. 8. The calculated contributions and total magnetic suscep- tibility of YCo2 as a function of the lattice parameter. The the- oretical lattice parameter is marked by a vertical dashed line. pounds the spin contribution χspin is definitely dominant, but taking the Van Vleck contribution χorb into account provides almost complete agreement between the calcu- lated and measured values of the susceptibility. Note that according to Refs. 20, 21, for RNi 5 compounds the abso- lute value of diamagnetic contribution to susceptibility from the ion core χdia amounts approximately to 3–4% of the sum of χspin and χorb . Table 4. Magnetic susceptibility of compounds YNi5, LaNi5 and CeNi5 M χston χspin χorb χspin+χorb χexp d ln χ / d lnV 10 3− emu/mol theor exp YNi5 1.5 1.78 0.23 2.0 1.9 [51,55] 7 6.6 [54] LaNi5 1.6 1.85 0.24 2.1 2.0 [52,53] 7.5 — CeNi5 1.6 2.39 0.62 3.0 3.0 [52,56] 4 3.9 [56] It should be noted that earlier calculations [48,49] have given a spin-polarized ground state of the LaNi 5 system at the experimental values of the lattice parame- ters. In our FP-LMTO calculations in an external field of 10 T, the theoretical values of the lattice parameters of YNi 5, LaNi 5 and CeNi 5 were determined from the total energy considerations and appeared to be approximately 1% lower than the experimental values. A paramagnetic response was obtained for these compounds at the theo- retical lattice parameters, with induced moments repre- sented by the susceptibilities χspin and χorb in Table 4. At the experimental values of the lattice parameters we were unable to obtain stable induced moments in our calcula- tions for these compounds. When the external field was «turned off», however, spontaneous spin polarization did not arise in YNi5 and LaNi5 systems during the self-con- sistent calculations of the electronic structure, in agree- ment with the experimental data [47,50–53]. The sponta- neous spin polarization found in the calculations of Refs. 48, 49 for LaNi5 is most likely due to insufficient- ly detailed calculations of the electronic structure under conditions where the system is close to magnetic insta- bility. The calculated volume derivatives of susceptibility d ln χ /d ln V are also listed in Table 4 and appear to be in agreement with the corresponding derivative resulted from the experimental studies of the pressure effect on χ for YNi5 [54] at T = 77.3 K. According to these calcula- tions the strongly volume dependent spin contribution to χ originates predominantly from the 3d states of Ni. In Table 4 the values of induced magnetic moments M, calculated with the FP-LMTO method in the field of 4.8 T are listed together with the experimental moments, re- sulted from the polarized neutrons diffraction in YNi5 at the same value of the external magnetic field [57]. The emphasis can be put on spin moments M spin on the Y atom and in the interstitial area, which have opposite di- rection to the dominant spin moments of Ni. This is due to the inhomogeneous distribution of the spin density in the unit cell of YNi 5 because of hybridization of the elec- tronic states of nickel and yttrium. The values of induced orbital moments Morb are approximately the same on the atoms of Y and Ni, thus for yttrium Morb is directed op- positely to M spin . Here, presumably, one can see the cer- tain analogy with the third Hund's rule for f -systems moments, as Y is situated at the beginning of the 4d period. Thus M spin and Morb are partly compensated, and resulting small moment on the atom of yttrium M Btotal � 0 4 10 3. ⋅ − μ appears to be below the precision level of the neutron experiments (± ⋅ −0 6 10 3. μ B [57]). For the Ni atoms there is a good agreement of the calculated M total with the experimental data [57]. In particular, the moments on the 3g sites appear to be larger than the mo- ments on 2c sites. As it follows from Table 5, for YNi 5 one can observe a good agreement between the experi- mental magnetic susceptibility, the calculated value of χ, and also the neutron data on χ( )n , obtained in external magnetic field [57]. It has been observed [47] that with increasing copper concentration x in YNi 5−xCu x alloys, their magnetic sus- ceptibility varies nonmonotonically and reaches a maxi- mum at x � 0 3. (see Fig. 10). In general, the nonmag- netic copper impurity enhances the paramagnetism of RNi 5−xCu x alloys in the region 0 1< <x . This effect is the most pronounced in alloys with cerium and less pro- nounced in alloys with lanthanum [47,55,56]. Magnetic-field-induced effects in the electronic structure of itinerant d- and f -metal systems Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 821 –0.1 0 0.1 E, Ry 0 200 400 0 200 400 N (E ), R y – 1 N (E ), R y – 1 N (E ), R y – 1 0 200 400 E F YNi 5 LaNi 5 CeNi 5 Fig. 9. Density of electronic states of RNi5 compounds. The partial contributions of the f states are indicated by the dashed lines. The Fermi level is marked with a vertical dashed line. Table 5. YNi5: Calculated and experimental [57] magnetic mo- ments (in 10 3− μB), induced by magnetic field of 4.8 T, and corre- sponding magnetic susceptibilities Atom Mspin Morb M total χ, 10 3− emu/mol Y (1a) –0.79 0.36 –0.43 –0.05 Ni (2c) 2.93 0.42 3.35 0.78 Ni (3g) 4.07 0.32 4.39 1.53 Interstitial –1.05 — –1.05 –0.12 YNi5 16.23 2.16 18.39 2.14 Neutron data Ni (2c) 24 06. .± 057 009. .± Ni (3g) 41 08. .± 143 020. .± Ni (tot) 171 26. .± 20 03. .± [57] 20 01. .± The presence of a peak in N E( ) slightly above the Fermi level in the RNi 5 compounds (see Fig. 9) suggests, that as nickel is substituted by copper in these compounds and the 3d band is filled, the Fermi level will shift into the region of the local maximum of N E( ), and one will there- fore expect an increase of the Pauli spin susceptibility. However, in a quantitative analysis of this effect one must take into account that in alloys the fine structure of the DOS N E( ) should be smeared on account of impurity scattering of conduction electrons. In Ref. 55 the beha- vior of χston (2) in YNi 5−xCu x alloys has been calculated by taking into account the smearing on N E( ). However, the use of Lorentz function with the damping parameter Γ and, alternatively, a so-called «effective temperature» T within the Fermi–Dirac distribution function have not provided reasonable description of the experimental χ( )x . For more rigorous calculations of the density of states and spin susceptibility of the alloys we have used the KKR-ASA Green-function method in the coherent poten- tial approximation (CPA) (details of the KKR-ASA-CPA method employed are presented in Ref. 58). In the self-consistent LSDA calculations by the CPA method a random distribution of Ni and Cu atoms in the 2c and 3g positions of the unit cell of the CaCu 5 crystal lattice was assumed. The CPA calculations give maxima of the local DOS N EFNi ( ) for the nickel atoms in the 2c and 3g sites at x � 0 5. (Fig. 11), leading to growth of the Stoner pa- rameter I according to Eq. (3). Here, as seen in Fig. 11, the total DOS N E xF( , ) turns out to be practically constant in the interval 0 ≤ ≤x 0.5 owing to the competition between the contributions N EFNi ( ) and N EFCu ( ). The magnetic susceptibility χ( )x of YNi 5−xCu x alloys was evaluated according to Eqs. (2) and (3) with the use of the partial and total DOSs at the Fermi level, calculated with the KKR-ASA-CPA method (Fig. 11), and also the Stoner integrals I. As seen in Fig. 10, the results of this calculation are found in a qualitative agreement with the experimental data on χ. We note that the Van Vleck or- bital contribution to the susceptibility χorb was not calcu- lated in the framework of the KKR-ASA-CPA computa- 822 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 G.E. Grechnev 0 0.5 1.0 1.5 x 1 2 3 , 1 0 χ – 3 em u /m o l Fig. 10. Concentration dependence of the magnetic suscepti- bility of YNi5−xCux alloys. � — experimental data of Ref. 47. The solid curve corresponds to the KKR-ASA-CPA calculations. 0.5 1.0 1.5 2.0 2.5 x 0 10 20 30 Ni(3g) Ni(2c) total Y(1a) Cu(2c) Cu(3g)N ( ), at o m ) i μ – 1 (R y · Fig. 11. Concentration dependence of the local densities of states at the chemical potential level in YNi5−xCux alloys: for Ni atoms in the 2c (�) and 3g (�) positions; for Cu in the 2c (�) and 3g (�) positions; for Y in the 1a position (�), calcu- lated by the KKR-ASA-CPA method. The corresponding total density of states (per atom) is represented by the solid curve (�). tional scheme, and the value of χorb obtained for the compound YNi 5 (0.23⋅ −10 3 emu/mol; see Table 4) only improves the agreement with the experiment. On the whole, the results of KKR-ASA-CPA calculations of χ( )x are in agreement with the experiment for YNi 5−xCu x alloys. Itinerant f -metal compounds Magnetism of the itinerant f -electron systems, inclu- ding cerium and uranium compounds, the strength of related electronic correlations and the problem of loca- lization–delocalization are matters of a long standing considerable interest. It is believed that band theory ap- proaches, like LSDA and GGA, are not appropriate for analysis of magnetism in f systems due to strong electron–electron correlations. In this section we check whether the ab initio tech- nique introduced above (the FP-LMTO method) can be used to describe magnetic properties of presumably iti- nerant f -metal compounds. The field-induced spin and orbital moments and the paramagnetic susceptibilities χspin and χorb were evaluated for a number of itinerant cerium and uranium systems and compared with the experimental data. A pronounced pressure effect on χ has been recently observed in UX 3 compounds [18,59], and the corre- sponding VDMS derivatives d ln χ / d ln V were found to be almost temperature independent, except for UAl 3. We expect that the pressure derivative of magnetic suscepti- bility is especially sensitive to the nature of 5 f electrons in these compounds. Therefore, a particular attention was given to investigations of the VDMS in f systems with competing spin and orbital magnetic moments. CeCo2 and CeNi5 systems Among the rare earth RCo 2 compounds with the cubic Laves phase lattice (C15), possessing rather complex magnetic structures and properties, CeCo 2 is regarded as an enhanced Pauli paramagnet. This compound shows su- perconductivity below TC � 1.5 K, often attributed to the Fulde–Ferrel–Larkin–Ovchinnikov mechanism, where the large Pauli susceptibility was proposed to play a lead- ing role [41,44]. It is also believed that this compound shows evidence of the so-called intermediate valence be- havior, which is presumably revealed in the anomalous equilibrium volumes trend in the series of CeFe 2, CeCo 2, and CeNi 2 [44,60]. In order to describe various proper- ties of these CeM 2 compounds a number of theoretical models have been put forward, including Kondo-like models and mixed-valence models, which assume the ce- rium atom being in a state which is a mixture of the local- ized 4 1f and 4 0f configurations, resulting in a non-inte- ger 4 f occupation number (see Ref. 41 and references therein). Ab initio calculations of the electronic structure were performed for CeCo 2 by employing the full-potential LMTO method within LSDA, and the PM ground state with the 4 0f configuration of Ce was established. The DOS at the Fermi level comes mainly from the f electrons of Ce and the d electrons of Co (see Fig. 12), and the par- tial contributions of other states are substantially smaller. As a whole, the electronic structure at EF is governed by a strong hybridization of the 4 f (Ce) and 3d (Co) states, which leads to a filling of the bonding states, with the Fermi level located at the steep downward slope of a broad N E( ) peak, as distinct from YCo 2 (Fig. 7). The main part of the 4 f and 5d states of Ce are situated higher in energy, and these hybridized states can be considered as the antibonding states. The calculated spin and orbital contributions to the magnetic susceptibility of CeCo 2 are listed in Table 3. As seen in the table, the spin contribution χspin is the domi- nant one in CeCo 2, whereas the Stoner model (2) gives somewhat lower value of χston . With the calculated Van Vleck contribution χorb , which amounts to about 25% of χspin , we have obtained a good agreement with the exper- imental susceptibility (see Table 3). The evaluated value of VDMS d dln / lnχ V is also given in Table 3 and ap- pears to be in a nice agreement with the experimental data of Ref. 41, obtained for CeCo 2 at T = 78 K. Basically, the CeCo 2 compound with potentially strong electron correlations can be described with the Magnetic-field-induced effects in the electronic structure of itinerant d- and f -metal systems Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 823 0.72 0.74 0.76 E, Ry 0 100 200 300 N (E ), st at es /R y EF CeCo 2 Fig. 12. Density of electronic states of CeCo2. The total DOS and partial contributions of Ce f states, and Co d states are in- dicated by solid line, dashed line, and dashed-dotted line, re- spectively. The Fermi level is marked by a vertical dashed line. model of itinerant 4 f states, which comes up naturally in the framework of electronic structure calculations per- formed within LSDA. It is shown that magnetic proper- ties of CeCo 2, including magnetovolume effect, can be explained within the theory of itinerant magnetism, as- suming the hybridized 4 f electrons and employing the FP-LMTO calculations of field-induced magnetic mo- ments. For the CeNi 5 compound ab initio calculations of the electronic structure were carried out by employing the FP-LMTO method exactly like in the previous sections. The maximum value of the orbital quantum number l was taken as lmax = 2 for Ni and by lmax = 3 for Ce. The elec- tronic density of states of CeNi 5 calculated over a wide energy interval is presented in Fig. 9, where the total DOS N E( ) is shown together with the partial contribution to DOS from the f states (dashed line). It is seen in Fig. 9 that in CeNi 5 the f states lie in the immediate vicinity of EF and play a substantial role in the formation of the fine structure of DOS N E( ). The experimentally observed magnetic susceptibility of CeNi 5 is noticeably higher than experimental χ of YNi 5 and LaNi 5 (see Table 4). One can note that for CeNi 5 the exchange-enhanced Stoner susceptibility χston , obtained according to Eq. (2), is much lower than χspin and it can not explain the experimental χ. The FP-LMTO method was also employed to calculate the induced spin and orbital magnetic moments in an ex- ternal field, and the corresponding spin and orbital contri- butions to susceptibility of CeNi 5 are listed in Table 4. As seen from the table, for this compound the spin contribu- tion χspin is obviously dominant, but the Van Vleck term χorb , which comes mainly from the electrons in the atomic sphere of Ce, amounts to about 20% of the total susceptibility (to be compared with 11% in YNi 5 and LaNi 5). The sum of the self-consistently calculated at the theoretical lattice parameter χspin and χorb contributions provides a perfect agreement with the experiment for CeNi 5 [47,52,55,56]. The calculated volume derivative of susceptibility d ln χ /d ln V is also listed in Table 4 and appears to be in agreement with the corresponding deriva- tive from the experimental studies of the pressure effect on χ for CeNi 5 [56] at T = 77.3 K. Therefore, it is shown that the magnetic properties of CeNi 5 (which one might erroneously assume to be strongly correlated) can in fact be described with the itin- erant 4 f states within LSDA. Specifically, the magnetic susceptibility of CeNi 5 and the magnetovolume effect can be well reproduced with the ab initio calculations of the field-induced magnetic moments. For CeNi 5−xCu x alloys, unlike the systems YNi 5−xCu x and LaNi 5−xCu x , a satisfactory description of the mag- netic properties cannot be obtained in the LSDA approxi- mation of band theory used in this study. This may be due to the appearance of the valence fluctuations of cerium in the CeNi 5−xCu x alloys (see Ref. 56), the magnetic pro- perties of which merit a separate detailed examination be- yond the scope of this study. USi 3, UGe 3, UAl 3 and UGa 3 systems The uranium intermetallic compounds UX 3, where X is a nontransition element from the group-III or group-IV series (except for boron and carbon), crystallize in the cu- bic AuCu 3-type structure. The delocalization of f elec- trons in f systems and the related quenching of f -mag- netic moment are usually attributed to either direct f – f overlap, or to the f –spd hybridization [61,62]. Since the U–U spacing in UX 3 compounds is far above the critical Hill limit [61,63], the direct 5 f –5 f interactions are weak, and these systems provide an exceptional opportunity to study the role of the f –spd hybridization in magne- tic properties, ranging from Pauli-like paramagnetism (UAl 3, USi 3, UGe 3) and presumably itinerant antiferro- magnetism (AFM) (UGa 3) to spin-fluctuation behavior (USn 3) and local-moment ordering (UIn 3, UTl 3, UPb 3 [18,61,62]). The electronic structures of UX 3 compounds (X = Al, Ga, Si, Ge) were calculated by using the relativistic FP-LMTO method. The orbital polarization correction [64], corresponding to Hund's second rule, was also taken into account in the calculations. It is found that the main contributions to DOS at the Fermi level come from ura- nium 5 f states and p states of ligand X, and a strong hy- bridization between these states provides narrow bands in the vicinity of EF (see Figs. 13, 14 and 15). Although the Fermi level cuts the U 5 5 2f / peak yielding a relatively 824 Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 G.E. Grechnev –0.1 –0.05 0 0.05 0.1 E, Ry E, Ry 0 100 200 300 N (E ), R y – 1 –0.01 0 0.01 200 UAl 3 Fig. 13. Density of electronic states of UAl3. Inset: The fine structure of N E( ) at EF for UAl3. The Fermi level positions (at 0 Ry) are marked by vertical lines. high value of DOS at EF , a pseudogap opens in the vicinity of EF . The calculated field-induced spin and orbital magnetic moments are found to be antiparallel in each studied UX3 compound, in agreement with Hund's third rule. Also in these compounds the orbital Van Vleck contributions to magnetic susceptibility are substantially larger than the spin ones (see Table 6). Remarkably, these hybridization effects together with the spin–orbit coupling provide the peculiar magnetic states in the UX3 compounds, where the spin moments are antiparallel to the applied field and the magnetic response is dominated by the orbital contri- bution. As can be seen in Table 6, the calculated induced moments appear to be in a good agreement with the expe- rimental data on χ in UX 3. Table 6. Magnetic susceptibility of UX3 compounds M χspin χorb χspin+χorb χexp d lnχ /d lnV 10−4 emu/mol theor exp UAl3 –16 32 16 17 6 5.5 [59] UGa3 –25.5 46.5 21 20 6.3 5.3 [18] USi3 –1.5 7.4 5.9 5.8 3.9 2.5 [59] UGe3 –5.5 18.2 12.7 11 6.1 6.9 [59] The calculated volume dependence of the magnetic susceptibility was found to be more pronounced for the orbital contribution to χ than for the spin contribution. Therefore, the large VDMS in UX 3, observed in Refs. 18, 59, is apparently related to the rapid quenching of the in- duced orbital moment with increasing width of the hy- bridized 5 f band under applied pressure. As can be seen from Table 6, the volume derivatives of the field-induced moments are in a fair agreement with the corresponding experimental volume derivatives of the magnetic suscep- tibility. It should be noted that the significant temperature dependence of d ln χ /d ln V in UAl 3 [59] can be related to the fine structure of DOS at EF (inset in Fig. 13) and a large volume effect on χspin . Magnetic-field-induced effects in the electronic structure of itinerant d- and f -metal systems Fizika Nizkikh Temperatur, 2009, v. 35, Nos. 8/9 825 X M G R –0.3 –0.2 –0.1 0 0.1 0.2 E , R y Γ Fig. 14. Band structures of the PM phase of UGa3. The Fermi level is marked by a horizontal dashed line. –0.1 0 0.1 E, Ry 0 100 200 300 N (E ), R y – 1 UGe 3 Fig. 15. Density of electronic states of UGe3. The Fermi level is marked by a vertical dashed line. –4 –5 –6 –7 –8 ln a 1.35 1.40 1.45 1.50 1.55 1.60 ln χ USi3 UGe3 UAl3 UGa3 UIn3 USn3 UTl3 UPb3 Fig. 16. Magnetic susceptibility of UX3 systems versus lattice spacing (in a logarithmic scale), obtained by extrapolation of χ( )T from the paramagnetic phase (Refs. 18, 59, 61) to zero temperature. The dashed lines represent the calculated deriva- tives d ln χ /d ln V for UAl3, UGa3, USi3, and UGe3. The so- lid line is a guide for the eye. As seen in Fig. 16, the values of d ln χ /d ln V for UX 3 follow a general trend in the dependence of magnetic sus- ceptibility on the lattice parameter a. In particular, the be- havior of the ln χ versus ln a in UX 3 compounds is close to linear with the slope corresponding to d ln χ /d ln V � 6 (except USi 3). This trend allows to conclude that mag- netic susceptibility of UX 3 is predominantly governed by the interatomic spacing variations. For the itinerant AFM phase of UGa 3, observed below TN = 67 K, the calculated spin and orbital magnetic mo- ments appear to be antiparallel, very sensitive to atomic volume, and in a qualitative agreement with previous the- oretical and neutron studies [18,65–67]. In general, the calculated AFM and field-induced magnetic moments are in a fair agreement with the experimental data, indicating the validity of the employed model of hybridized itinerant 5 f states for the studied UX 3 compounds. Conclusions The novel technique for ab initio calculations of the electronic structure in external magnetic field was imple- mented within the density functional theory and the FP-LMTO method. In this way, the field-induced spin and orbital moments and the paramagnetic susceptibilities χspin and χorb were evaluated for a number of TM and IFM systems. For transition metals the calculated susceptibilities and their volume dependencies d ln χ/d ln V appear to be in agreement with experiment. These metals are shown to possess substantial orbital contributions to the induced magnetization due to the hybridization of s p, and d states. It is demonstrated that the corresponding Van Vleck or- bital contributions to magnetic susceptibility χorb are comparable or even higher than χspin in TM. By means of the field-induced calculations for hcp transition metals, the anisotropy of magnetic susceptibil- ity Δχ has been calculated for the first time. The sign and values of the calculated Δχ are in accordance with the ex- perimental data at ambient conditions. The magnetic properties, including Δχ, were evaluated for the high pressure hcp phase of Fe, which is expected to be the dominating element in the Earth's core. For the hcp iron the PM ground state with the substantial Stoner enhance- ment (S � 3) is found in the region of the reported super- conducting state (15 GPa < P < 30 GPa). For the exchange-enhanced paramagnetic compounds TiCo, Ni 3Al, YCo 2, CeCo 2, YNi 5, LaNi 5 and CeNi 5, it is shown that the spin paramagnetism is dominant, while the Van Vleck orbital contribution to susceptibility amo- unts from about 10 up to 20% (TiCo, CeCo 2, CeNi 5) for these compounds. The Stoner approximation (2) is found to underestimate substantially the spin susceptibility, and within this approach, by using ab initio calculated N EF( ) and integrals I, we were not able to explain the experimental susceptibilities and their volume dependen- cies in the studied TM compounds. On the other hand, the LSDA field-induced calculations take into account non- uniform induced magnetization density in the unit cell and provide more accurate values of χspin . Our method was able to describe the susceptibilities and VDMS in the strong paramagnetic compounds, which are close to quantum critical point. Our calculations have revealed that the enhanced spin magnetic susceptibility in YNi 5−xCu x alloys is extremely sensitive to the behavior of the partial contributions of Ni and Cu to the DOS at the Fermi level and also to nonuni- form induced magnetization density at the Ni sites. It is shown that itinerant picture of hybridized 4 f elec- trons produces bulk and magnetic properties of CeCo 2 and CeNi 5 in close agreement with experiment. In general, the LSDA provides an adequate description of peculiar magnetic properties of the studied cerium compounds, in- cluding magnetovolume effects. For the compounds UAl 3, UGa 3, USi 3 and UGe 3 the spin–orbit coupling appears sufficiently strong, and the field-induced spin and orbital moments are found to cou- ple antiparallel, in accord with the Hund's third rule. 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