ab initio studies of electric field gradients and magnetic properties of...

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PAPER

Ab initio studies

aDepartment of Physics, Faculty of Science,

Avenue, Isfahan 81746-73441, Iran. E-m

[email protected]; Tel: +98 9133287908bCenter for Computational Materials Scien

Pakistan. E-mail: [email protected]; TcDepartment of Physics, University of Malak

Cite this: RSC Adv., 2015, 5, 37592

Received 14th February 2015Accepted 17th April 2015

DOI: 10.1039/c5ra02881g

www.rsc.org/advances

37592 | RSC Adv., 2015, 5, 37592–3760

of electric field gradients andmagnetic properties of uranium dipnicties

E. Ghasemikhah,a S. Jalali Asadabadi,*a Iftikhar Ahmad*bc and M. Yazdani-Kacoeia

In this paper, we explore the electric field gradients (EFGs) at 238U sites for antiferromagnetic UX2 (X ¼ P, As,

Sb, Bi) using LDA, LDA + U, GGA, GGA + U, and the exact exchange for correlated electrons schemes by

considering the diagonalization of the spin–orbit coupling Hamiltonian in the space of the scalar

relativistic eigenstates using the second-order variational procedure. The electronic structures and

magnetic properties of the compounds are also investigated. It is found that the density functional

theory approaches except exact exchange for correlated electrons are not successful in reproducing the

experimental zero electric field gradient value in UBi2, even LDA + U and GGA + U within their default 4f

density matrices by varying the U parameter in an energy interval of [0; 4 eV], though these techniques

with no need to manually adopt their initial conditions (elements of the occupation matrix) are effective

in the calculation of the nonzero electric field gradients for the other compounds. The exact exchange

for correlated electrons has efficiently provided a null electric field gradient in UBi2 and nonzero electric

field gradients for the other compounds by adjusting its dimensionless parameter a to 0.4. The physics

of the null electric field gradient in UBi2 is revealed in this article and it is discussed that the source of the

ignorable electric field gradient originates from the antiferromagnetic ordering of [Y as compared to the

long-range antiferromagnetic ordering of [[YY in the other compounds. Furthermore, our calculated

magnetic moments for the uranium atoms in these compounds are consistent with the available

experimentally measured values as compared to the severely underestimated theoretical results.

I. Introduction

Electric eld gradient (EFG) is a powerful tool to investigateatoms in a crystalline environment.1 It reveals the deection ofthe electron cloud around the nucleus from spherical mode,where a larger value of this parameter means more asymmetryfrom the spherical shape. Therefore, in crystals EFG serves as aprevailing criterion for the measurement of the electron'slocalization within an atom.2 The electric quadrupole moment(Q) and EFG are nonzero for all nuclei with a nuclear quantumnumber equal or higher than one.3

The principle component of the electric eld gradient (EFG)tensor, Vzz, is used to reveal the asymmetry of electrons around anucleus. This value sensitively indicates the amount (orienta-tion) of deviated electron charge density distributions (ECDDs)from spherical symmetry.4 The Vzz value is zero for sphericalECDDs and nonzero for asymmetrical ECDDs,5 where it servesas a very sensitive indicator to the asymmetry of semicore,6 andvalence states,7 as well as electron density of states (DOS).2

University of Isfahan (UI), Hezar Gerib

ail: [email protected];

ce, University of Malakand, Chakdara,

el: +92 332 906 7866

and, Chakdara, Pakistan

2

Although Vzz is not a measurable quantity in the laboratory,1 itcan be indirectly evaluated by a variety of accurate experimentaltechniques,8 like Mossbauer spectroscopy, where in this tech-nique the nuclear quadrupole moment is measured and thenthe basic concept that the electric hyperne splitting, DEhf, isproportional to Vzz times nuclear electric quadrupole moment(Q)1 is used to evaluate Vzz. Interestingly this parameterconnects DEhf and Q, where one is a measurable condensedmatter parameter and the other one is a measurable nuclearquantity. Hence, Vzz may be generally considered as a connect-ing bridge between condensed matter physics and nuclearphysics (indebted to the Mossbauer's discovery) and it can beprecisely predicted by modern ab initio density functionaltheory (DFT) calculations. In this work, we are interested in thetheoretical condensed matter aspects of the Vzz for actinidesspecically UX2 (X ¼ Bi, Sb, As, P) compounds.

The electronic and magnetic properties of actinide materialshave always been a topic of interest in condensedmatter physicsbecause of their 5f electrons which display some remarkablephysical properties heavy fermions behavior, Pauli para-magnetism, spin uctuation and unconventional supercon-ductivity.9–15 The properties of a crystal depend on thelocalization of its electrons,16 but the localization of the 5felectrons depends upon the environment and hence the atomicradii for elements with 5f orbitals are not xed.15 The difference

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of the same actinide atom from one material to another mainlyoriginates from the fact that the 5f electrons have a dual naturebetween itinerant and localized states.17 The 5f electrons havean intermediate property, something between the localizationof the 4f electrons in the lanthanide compounds and the itin-erant 3d electrons in the transition metals.18 Sometimes, the deHaas–van Alphen and Shubnikov–de Haas experiments areperformed to clarify the nature of 5f electrons in acompound.18,19 Most of the actinides are extremely rare andtherefore the main concentration of the experimental projectsare focused on uranium compounds and uranium alloys.

The physical properties of uranium compounds are relatedto their lattice constants, the distance between U atoms andhybridization of 5f electrons with other valence electrons.According to the Hill criteria,20 uranium based compounds withthe U–U interatomic distances smaller than 3.4–3.6 A are typi-cally nonmagnetic with itinerant 5f states.21 Metallic uranium isparamagnetic but shows magnetic ordering when it combineswith group 5 (N, P, As, Sb, Bi) or group 6 (S, Se, Te) elements. AllUX2 (X ¼ Bi, Sb, As, P) actinide compounds areantiferromagnetic with relatively high Neel temperature. Themagnetic ordering in these compounds is the consequence ofthe partially lled 5f shell.22,23 The antiferromagnetic structureof these compounds is determined by neutron diffraction,which shows that these compounds consist of ferromagneticsheets of uranium stacked perpendicularly to the c axis.24,25 Themagnetic ordering for UBi2 is [Y, while for USb2, UAs2 and UP2is [[YY. Therefore, the magnetic and chemical unit cells ofUBi2 are identical but in the other compounds the magneticunit cell is double with respect to the chemical unit cell alongthe [001] direction.17,26,27 The double magnetic unit cell in USb2,UAs2 and UP2 brings about a at magnetic Brillouin zone.18,28,29

The main focus of the present studies is to theoreticallyreproduce the experimental EFG values30 at 238U sites for anti-ferromagnetic UX2 (X ¼ P, As, Sb, Bi) dipnicties. These experi-ments which were performed at 5.1 K,30 reveal zero EFG valuefor UBi2 and nonzero values for the other dipnicties. Therefore,along with the theoretical understanding of the EFGs for thesecompounds, the second main motivation of this study is toexplore the cause of the zero EFG of the UBi2. The electronicstructures and magnetic properties of the compounds are alsoinvestigated and discussed in details. The calculations arecarried out with the different avors of the density functionaltheory i.e., local density approximation (LDA), generalizedgradient approximation (GGA), LDA plus Hubbard potential(LDA + U), GGA + U, and exact exchange for correlated electrons(EECE) to reveal a clear theoretical description of thecompounds. As these compounds have 5f electrons which mayhave strong relativistic effects, so those expected relativisticeffects are included by using the spin–orbit coupling (SO) in ourcalculations.

II. Computational details

The calculations presented in this paper have been performedwithin the framework of the density functional theory (DFT)using the augmented plane waves plus local orbitals (APW + lo)


method as implemented in the WIEN2k code.31 In the APW + lomethod the wave functions, charge density and potential areexpanded in spherical harmonics within non-overlappingMuffin-tin spheres and in plane waves in the remaining inter-stitial region of the unit cell. The exchange correlation func-tional is treated with the LDA, GGA, LDA + U and GGA + U. It isworth to mention that in these calculations we use LDA ofPerdew–Wang 92 and GGA of Perdew–Bourke–Ernzerhof 96.The relativistic effects are included by diagonalizing the spin–orbit (SO) coupling Hamiltonian in the space of scalar relativ-istic eigenstates using the second-order variational procedure32

in our calculations for the proper treatment of the 5f electronsof uranium.

It is well understood that the most popular approximationsfor the exchange correlation functional in the DFT33 are thelocal density approximation (LDA)34–36 and the generalizedgradient approximation (GGA).37–40 These approximations inmost regular cases give appropriate results, consistent with theexperimental values. The main reason for using these localschemes is that they lead to calculations which are computa-tionally cheap in comparison to the more sophisticatedschemes. These functionals have some shortcomings, whichhave been revealed over time. The main shortcoming of theseapproximations is that they have problems to describe theelectronic structure of strongly correlated compounds, whereasstrongly correlated systems have attracted enormous attentionin physics research in the recent years. The importance of thestrongly correlated systems and the shortcoming of LDA or GGAin describing these systems encourage researchers to thinkabout new solution to this problem. The DFT approach withHubbard parameter such as LDA + U and GGA + U is one of thesolutions to the problems of the strongly correlated systems.41–44

The number of calculations using this scheme, since the 1990decade, has shown that this approach has a specic role in theelectron correlation of crystals. Nevertheless, this approach hasalso some disadvantages, e.g., for applying this method, oneneeds to know the Ueff (U–J) parameter whereas its determina-tion is time-consuming theoretically and costly in experiments.Furthermore, perchance in some cases by using these func-tionals too we cannot obtain the acceptable results that arecompatible with the experimental results due to the fact thattheir predicted self-consistent results in some cases can dependon the initial conditions, i.e., the so called multiple solutionsproblem.45 In orbital dependent schemes, the initial conditions,i.e., the elements of the occupation matrices can be manuallyadapted (by performing several self-consistent calculations for avariety of occupation matrices and eventually selecting the onewhich leads to the minimum energy) to achieve consistentresults with experiment. But, this task can be also time-consuming for the 4f-electron systems with angular momentumquantum number l ¼ 3 having two real and complex (2l + 1) �(2l + 1)¼ 7 � 7 density matrices. Although the hybrid method isalso orbital dependent, fortunately for the cases under studycan give excellent results by adjusting only its alpha parameter.We show that the exact exchange for correlated electrons (EECE)method can be considered as a successful approach to study oursystems which was presented in 2006 by Novak et al.46,47 The

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RSC Advances Paper

EECE scheme in many cases leads to the acceptable results forstrongly correlated systems. Therefore, in order to achievelogical theoretical results we are also using EECE besides theother functionals mentioned earlier.

To avoid the overlap of the Muffin-tin spheres during thecalculations; the Muffin-tin sphere of radius 2.8 a.u. is chosenfor U in UBi2, USb2, UAs2 but 2.6 a.u. in UP2 and 2.4 a.u. ischosen for X in UX2 compounds. The experimental latticeparameters measured by X-ray diffraction technique,24,48 shownin Table 1, are used as input parameters. For sampling of theBrillouin zone, a mesh of 45 special k-points which correspondsto the grids of 10 � 10 � 5 is used for UBi2, while 11 � 11 � 2grids with 21 special k-points in the irreducible part of theBrillouin is used for the other compounds. The maximum ‘ forthe waves inside the atomic spheres is conned to ‘max ¼ 10.The wave function in the interstitial region is expanded in planewaves with a cutoff of RMTKmax ¼ 7. The charge density isFourier expanded up to Gmax ¼ 14

ffiffiffiffiffiRy

p. The full relaxation is

performed with the criterion of 1 Ry/Bohr on the exerted forces.In the LDA + U, GGA + U and EECE schemes, U and a parametersare gradually increased from about zero to values havingconverged calculations with enough accuracy.

III. Results and discussionsA. Electric eld gradients

The electric eld gradient is a second order symmetric tensorwith zero trace and ve independent components. The experi-mental measurement and theoretical calculations of the electriceld gradient is a tricky job. The EFG cannot be measureddirectly from experiments, therefore rst one has to performMossbauer spectroscopy to measure nuclear quadrupole inter-action and then, calculate electric eld gradient (EFG) by usingrelation Vzz ¼ DEhf/e

2Q, where Q, e and DEhf are nucleus quad-rupole moment, electron charge and electric hyperne splitting,respectively.30

As a matter of fact f(I, mI, h) is a well known function of thenuclear spin angular momentum quantum number I, and themagnetic quantum number mI, as well as the axial asymmetry

parameter h ¼ Vxx � Vyy

Vzz, where Vzz $ Vyy $ Vxx in the principle

axes system (PAS).49 Q(h) is zero for spherical nuclear charge

distribution with I ¼ 0 or12

(axially symmetric crystalline

environments with Vxx ¼ Vyy) and nonzero for nonsphericalnuclear charge distribution with I $ 1 (axially asymmetric

Table 1 Lattice parameters (a and c) and lattice parameter ratio forUX2 compounds

Compound a (A) c (A) c/a

UBi2 4.445 8.908 2.004USb2 4.270 8.884 2.048UAs2 3.954 8.116 2.053UP2 3.810 7.764 2.038

37594 | RSC Adv., 2015, 5, 37592–37602

crystalline environments with Vxx s Vyy).8,49 h vanishes due tothe axially symmetric environments5 for the systems underquestion in this paper, viz. f(I,mI, h)¼ f(I,mI), and can serve as apowerful gauge for measuring such a degree of localization.

Aer the above discussion it is obvious that the onlyparameter which can be determined at the uranium site in UX2

(X ¼ P, As, Sb, Bi) dipnicties is the principle component ofEFG:3,50

Vzz ¼ limr/0

�5

p

�ð1=2Þ�V20ðrÞ

�r2�

(1)

where, V20 is the radial potential and can be calculated as:3

V20ðr ¼ 0Þ ¼ 4p

5

ðRMT

0

r20

rdr� 4p

5

ðRMT

0

r20

r

�r

RMT

5

dr

þ 4pXk

VðkÞJ2ðkRMTÞY20ðkÞ (2)

where, r20 is the L ¼ 2 and M ¼ 0 of density expansion, Y20(k) isspherical Bessel function and RMT is the Muffin-tin (MT) sphereradius. The rst term in eqn (2) is related to valence electrons(Quasi-core) in MT sphere. It is called valence contribution ofEFG and indicated with Vvalzz . The sum of the second and thirdterm is related to outMT sphere electrons and interstitial regionand therefore is known as lattice contribution of EFG and isindicated by Vlatzz .3

The calculated EFGs and their valence and lattice parts forUX2 (X ¼ Bi, Sb, As, P) compounds using LDA, GGA, LDA + SO,GGA + SO, LDA + U and GGA + U, are presented in Table 2. Thecalculated values of the EFGs are also compared with the EFGvalues evaluated from the Mossbauer spectroscopy.30 Thetable clearly indicates nonzero calculated EFG values forUSb2, UAs2, and UP2, through GGA, LDA + U and GGA + Uapproximations. Though, these results are underestimatedfrom the experimental values but still provide a clue that thecharge distribution is not symmetric. Unlike these delibera-tions, these approximations are not successful in attainingthe zero EFG value for UBi2. In the expectation of furtherelaboration, here, we have not tried to adjust the elements ofoccupation matrix to force the LDA + U or GGA + U to repro-duce the experimental values. Instead, in order to reach zerovalue for the EFG of UBi2, only the Hubbard parameter (U) inLDA + U and GGA + U approximations are varied from 0.01 to0.40 Ry. Fig. 1 shows the effect of U on the EFG value of UBi2. Itis clear from the gure that the EFG increases with U,consequently the difference of the calculated value increasesfrom the experimental results and hence none of the used “U”value provides correct EFG for UBi2. On the other hand for Uhigher than 0.40 Ry, calculations are not well converged andantiferromagnetic property of UBi2 vanishes. So, results ofcalculations for U higher than 0.40 Ry are not presented here.The results presented in Table 2 and Fig. 1 conrms that theseapproximations are not suitable to calculate EFG for UBi2 andwe may use some appropriate methods and approximation forthis compound, although these approximations are not badchoices for the other UX2 compounds, as they provide resultswhich are in somewhat manner comparable with the


Table 2 Summary of EFG (Vzz) in units 104 V nm�2, also the lattice (Vlatzz) and valence (Vval

zz ) contribution of EFGs from LDA, GGA, LDA + U and GGA+ U calculations

Compound LDA GGA LDA + SO GGA + SO LDA + U GGA + U EXP.a

UBi2 Vtotzz 0.523 0.556 0.545 0.558 0.532 0.645 0 � 0.2Vvalzz 0.524 0.560 0.543 0.558 0.528 0.641 —Vlatzz �0.001 �0.004 0.002 0.000 0.004 0.004 —

USb2 Vtotzz 1.237 1.227 1.287 1.275 1.182 1.195 1.6 � 0.4Vvalzz 1.242 0.241 1.289 1.286 1.179 1.190 —Vlatzz �0.005 �0.014 �0.002 �0.011 0.003 0.005 —

UAs2 Vtotzz 1.409 1.353 1.412 1.365 1.306 1.303 1.5 � 0.3Vvalzz 1.414 1.357 1.417 1.368 1.305 1.303 —Vlatzz �0.005 �0.004 �0.005 �0.003 0.001 0.000 —

UP2 Vtotzz 1.544 1.423 1.501 1.391 1.260 1.244 2.1 � 0.5Vlatzz 1.549 1.428 1.505 1.395 1.260 1.244 —Vvalzz �0.005 �0.005 �0.004 �0.004 0.000 0.000 —

a Ref. 30.

Fig. 1 The electric field gradients of UBi2 using different Hubbardparameters in LDA + U.

Paper RSC Advances

experimental values. The strong inuence of U on the elec-tronic properties of the UBi2 conrms that this compound isstrongly correlated electron system. So, to resolve the issue weneed to be more careful in the choice of the exchange-correlation energy for the treatment of this compound. Inorder to resolve this issue we proceed with another exchange-correlation energy which is more suitable for the stronglycorrelated compounds, called exact exchange for correlatedelectrons (EECE) scheme, introduced by Novak et al.46 in 2006.The method of performing the EECE calculations is like LDA +U. Therefore, they are called U-like functionals. The calculatedresults by using these functionals are presented in Table 3. Itis clear from the table that all these functionals lead toacceptable values of EFGs for UBi2, which are in agreementwith the experimental results. A general form of the EECEfunctional can be written as follows:46

E[r, {Bm}] ¼ ESL[r] + EHFx [nss

0(r), {Bm}] � Edc[n

ss0(r), {Bm}],

(3)

where,Bm represents the correlated orbitals, ESL is the energy ofthe semilocal functional such as LDA or GGA, EHF

x is theexchange energy which is calculated using the Hartree–Fock


approach, and the last term is the double counting correctionwhich can be written as:

Edc[nss0

(r), {Bm}] ¼ ESLxc [rcorr]. (4)

According to these equations, the EECE approach improvesthe exchange-correlation energy of semilocal functionals usingthe exact exchange energy of Hartree–Fock method and as aresult improves the semilocal functional results with respect tothe experiments. This originates from the localized character ofU-5f electrons which is well described by the EECE approach.

The EECE functionals reproduce the experimental zero EFGof UBi2, which is at least time-consuming if not impossible forthe other DFT schemes. The calculated EFG for the other UX2

compounds are also close to the experimental values, except theresults for UP2, which has larger difference than the experi-mental results as compared to those of the LDA and GGA. Infact, in UP2 case, the approximations with Hubbard parameterssuch as LDA + U and GGA + U are also not successful to calculateEFG. The lattice and valence parts of EFG for UX2 compoundsare also shown in Tables 2 and 3. One can see from these valuesthat almost all of EFG is related to the valence electrons in theMT sphere and the electrons which are out of MT sphere do notsignicantly affect EFG. Therefore, in the EFG calculations, it isessential to choose appropriate MT sphere radii otherwise onecan obtain wrong EFG value. On the other hand lattice param-eters do not substantially affect EFG, since out of MT sphereelectrons have no signicant effect on it, and therefore we cancautiously use experimental lattice parameters in the EFGcalculations.

It is clear from Fig. 2(a) that the EECE functionals providegood results in case of UBi2 for a¼ 0.40. The calculations do notconverge for a larger than 0.40, and in case if the self-consistentcalculations are elaborated to be converged, then the antifer-romagnetic character of the crystal would be destroyed, becausein this case the up and down density of states curves of thecrystal are not symmetric and thereby cannot cancel each otherto form zero antiferromagnetic feature. Keeping all these

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Table 3 Summary of EFG (Vzz) in units 104 V nm�2, also the lattice (Vlatzz) and valence (Vval

zz ) contribution of EFGs from EECE calculations

Compd LDAFock B3PW91 WCFock PBEsol PBEFock EXP.a

UBi2 Vtotzz 0.108 0.118 0.065 0.050 �0.057 0 � 0.2Vvalzz 0.103 0.099 0.062 0.056 �0.066 —Vlatzz 0.005 0.019 0.003 �0.006 0.009 —

USb2 Vtotzz 1.277 1.293 1.261 1.280 1.283 1.6 � 0.4Vvalzz 1.275 1.289 1.258 1.278 1.280 —Vlatzz 0.002 0.004 0.003 0.002 0.003 —

UAs2 Vtotzz 1.346 1.377 1.311 1.317 1.295 1.5 � 0.3Vvalzz 1.346 1.378 1.311 1.317 1.295 —Vlatzz 0.000 �0.001 0.000 0.000 0.000 —

UP2 Vtotzz 1.269 1.202 1.246 1.262 1.215 2.1 � 0.5Vlatzz 1.269 1.201 1.245 1.261 1.215 —Vvalzz 0.000 0.001 0.001 0.001 0.000 —

a Ref. 30.

Fig. 2 (a) Vzz values using different a in EECE functionals. Orbitalcontributions in Vzz for all uranium atoms using LDAFock approxi-mation in (b) UBi2 and (c) UAs2 compounds. (d) Vzz's for all uraniumatoms in UAs2 compound.

Fig. 3 (a) Chemical structure for UX2 (X ¼ Bi, Sb, As, P), (b) magneticstructure of UBi2 called as (I) and (c) magnetic structure of USb2, UAs2

RSC Advances Paper

aspects in considerations, the EFG values presented in Table 3of UBi2 are calculated for a ¼ 0.40. The same problems areobserved during the EFG calculations of USb2, UAs2 and UP2,but in these compounds the appropriate value for a is opti-mized to be 0.20. Therefore, this value is used in the EECEfunctionals results of USb2, UAs2 and UP2, shown in Table 3. Itis clear from the table that these values are closer to theexperimental values. If the convergence and antiferromag-netism problems for a higher than 0.20 were not there, then wecould have reproduced the experimental EFG values for thesecompounds.

The interesting point that needs to be addressed here is thereason for the difference between EFG of UBi2 and that of theother compounds. It is necessary to nd the physics behind thequestion, why nuclear quadrupole interaction and EFG is zerofor UBi2 and nonzero for the other compounds? To answerthese questions, the rst logical step is to know factors that

37596 | RSC Adv., 2015, 5, 37592–37602

affect EFG value. Hence, we obtained p–p, d–d and f–f orbitalcontributions of EFG for UBi2 and UAs2. Uranium atoms in theunit cell of UBi2 are in the same position as UAs2; but withspecial magnetic structure its ferromagnetic sheets order anti-ferromagnetically [Y. In other words the distribution of themagnetic or spin of atoms in a unit cell is different for UBi2 thanUAs2, though both have the same crystal structure. Fig. 2(b)shows orbital contributions of U(1) and U(2) atoms in the EFGof UBi2. It is clear that different contributions with up and downspin states are different for each atom. Also, different up anddown contributions of both uranium atoms are reversely veryclose; that is, p–p orbital contribution of U(1) with up spin isclose to that of U(2) with down spin.

and UP2 compounds called as (II).


Paper RSC Advances

The magnetic structure of UAs2 is shown in Fig. 3(c), where itis obvious that for this compound U(1) and U(3) are at the sameposition but have opposite spins and similarly U(2) and U(4) areat the same positions but magnetically opposite to each other.However, U(1) and U(4) atoms, as well as, U(2) and U(3) atomsare magnetically the same, though their positions are different.Fig. 2(c) shows the p–p, d–d and f–f contributions of EFG for upand down spin states for all uranium atoms of UAs2 compound.This gure demonstrates that the magnetic structure for U(1) issimilar to U(4) and for U(2) is similar to U(3), whereas the U(2) isarranged antiferromagnetically with the U(3) in EFG. The totalEFG at all uranium atoms sites of UAs2 for different approxi-mations is shown in Fig. 2(d). This gure shows that for allapproximations, the total EFG of U(1) and U(4) together, andtotal EFG of U(2) and U(3) are consistent. Aer the analysis ofthe EFG of UBi2 and UAs2 we can infer that the magneticstructure and spin arrangement of atoms have a signicanteffect on the EFG in uranium atoms positions. Therefore,orbitals which have more magnetic properties have greatereffect on EFG. Therefore, it seems that the main reason of thedifference between the EFG of UBi2 and the other UX2

compounds is their difference in magnetic structures, becausetheir spatial structure and point group are the same.

To examine this idea, we considered magnetic structure I forUSb2, UAs2, UP2, as shown in Fig. 3(b), andmagnetic structure IIfor UBi2, as shown in Fig. 3(c), and then calculate EFG aturanium site in these new structures. The calculated resultspresented in Table 4 show that the EFG of UBi2 using structureII is 1.24 � 104 V nm�2, while for structure I it is 0.11 � 104 Vnm�2, whereas the EFG of USb2 is 0.09 � 104 V nm�2 forstructure I, and 1.28 � 104 V nm�2 for structure II. The quantityfor UP2 using structure I is as small as 0.12 � 104 V nm�2 andvery large for structure II. The EFG of UBi2 with structure I isconsistent with that of the USb2. Regarding to EFG of UX2

compounds with different magnetic structures and previousdiscussion about the relationship between EFG and spin ofatoms, it is concluded that the different EFG value of UBi2originates from the difference in the magnetic structure.

The EFG of an atom can be different in different compoundscontaining this atom, as there are many factors affecting theEFG value. This means that for a specic atom we can have zeroEFG in one compound and nonzero for another one. This is dueto the fact that the point symmetry and magnetic structure of acell affect EFG. Furthermore, external factors are also affectingEFG, whereas one of those factors is temperature. There are twoequations for EFG at zero and ‘T’ temperature, regarding thepartially lled orbitals and type of electrons. For s, p and dorbitals it is given below:51

Table 4 The EFGs values for UX2 compound in 104 V nm�2 units atdifferent structures

UBi2 USb2 UAs2 UP2

Structure I 0.11 0.09 — 0.12Structure II 1.24 1.28 1.35 1.27


Vzz(T) ¼ Vzz(0)(1 � BT1.5) (5)

This is a nonlinear relation. For f orbitals we have:51

Vzz(T) ¼ Vzz(0)(1 � BT) (6)

where Vzz(0), Vzz(T), B and T are EFG at zero temperature, EFG atT temperature, a constant (which is in the order of 10�5 to 10�4)and temperature, respectively. These equations show that EFGdecreases as temperature increases. We used density functionalcalculations which are performed at zero temperature, butexperimental results are obtained at room temperature; there-fore eqn (6) is used to obtain EFG at room temperature. In orderto consider temperature effect in EFG, we supposed that B is10�4. The EFG values of all UX2 compounds at zero and 300 Kare shown in Table 5. The results clearly indicate that the EFGdecreases as temperature increases but this decrease is notprofound for 300 K, though these results at zero temperatureare reasonable as they can be condently compared to theexperimental data. It is obvious that with the increase intemperature electrons are excited and matter is expanded. It isalso known that the EFG represents the asymmetry of electronsaround a nucleus, therefore with the increases in temperaturethe asymmetry in electron decreases which consequentlydecrease the EFG.

B. Density of states

The density of states (DOS) curve shows the distribution ofelectronic state in terms of energy. The area under DOS curve ineach energy interval is equal to the number of allowed elec-tronic states in that interval. Fig. 4 shows total, U and X atomsDOSs for all UX2 compounds. In this gure, plots (a) to (d) showthe total DOS curves for UX2 compounds. All curves coincidewith the Fermi level at nonzero value, so these compounds areconductors. The symmetry in the up and down states for all thecompounds shows that they are antiferromagnetic materials innature. A valley in the DOS of USb2, UAs2 and UP2 can be seen inthe energy interval 0 to 0.5 eV but this valley goes far away fromthe Fermi level for UBi2 and occurs around �1 eV. The DOS ofUBi2 at the Fermi level and around it and especially far belowthe Fermi level states are more split and have less degeneracy.Furthermore, the DOS of UBi2 coincides with the Fermi level atsmaller values than that of the other compounds, and theheight of peaks in this energy interval is less than that of theothers. Therefore, the contribution of the Fermi level electronsof UBi2 in conduction is easier. It is further noted that at lowenergies �6.0 to �4.5 eV the tail of curve of UBi2 reaches to zero

Table 5 The EFGs values for UX2 compound in 104 V nm�2 unit atdifferent temperatures

UBi2 USb2 UAs2 UP2

T ¼ 0 K 0.050 1.280 1.317 1.262T ¼ 300 K 0.049 1.242 1.277 1.224

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Fig. 4 The DOS curves for: (a)–(d) the total UX2 compounds, (e)–(h) the U(1) and U(2) atoms in all UX2 compounds, (i)–(l) the X atoms in all UX2compounds.

Fig. 5 PDOS curves for all orbitals of uranium in UBi compound.

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sooner, and therefore the band width of UBi2 is lesser than theother compounds.

The plots (e) to (h) of the gure show DOS curves for U(1) andU(2) atoms of UX2 compounds. As seen, in all compounds, DOSfor each uranium atom for up and down spin states are asym-metric. It demonstrates that uranium atoms are magnetic in allcompounds and have a local magnetic moment. The DOS of thesecond uranium atom with a reverse spin arrangement ascompared to the rst atom is shown in these plots. The DOS ofthis atom in up and down spin states is reversely the same as theDOS of the rst atom. That is, up states curves for U(1) is thesame as the down spin state curve for U(2) and vice versa.Therefore, the magnetic moment which is caused by U(2) isequal but in opposite direction to that of U(1) and hence the netmagnetic moment of the compound is zero. This means that themagnetic moment of U(2) atom is canceled by that of U(1) atomand as result no net magnetic moment is observed in the unitcell. As a result, although uranium atoms are magnetic, but theyhave no contribution in the total magnetic moment of thesecrystals. As discussed earlier about the magnetic structure thatin all UX2 compounds the number of uranium atoms in oppo-site spin directions is equal.

The DOS curves for X atoms of the UX2 compounds areshown in Fig. 4(i)–(l). As shown, in all compounds the up anddown states curves for X atoms are symmetric, which reveal thatX atoms are non-magnetic. So, in all these compounds themagnetic sheets of uranium are separated with non-magneticsheets of X atoms. It causes a quasi-two dimensional propertyof them. Since both DOSs of U and X atoms coincide the Fermilevel at nonzero value, it reveals that all electrons of thesecompounds contribute in the conduction process. Thecomparison of the DOSs of U and X atoms in Fig. 4 shows thatfor all UX2 compounds the main contribution of DOS at the

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Fermi level is related to U atoms, whereas X atoms have domi-nant contribution at higher energies.

Fig. 5 shows the DOS curve for uranium orbitals in UBi2. Allcurves have nonzero values at the Fermi level, so all orbitals inuranium atom contribute in conduction. The DOS curve for fand d orbitals coincide the Fermi level at much higher energythan s and p orbitals, which shows that the electrons of theseorbitals (f and d) are magnetic electrons of U and have moreeffect on the magnetic properties of the uranium atom. TheDOS curves for uranium atom orbitals of the other UX2

compounds are almost similar to Fig. 5, and therefore we havenot shown them to avoid repetition.

The DOS curves for f and d orbitals of uranium are shown inFig. 6, where the plots from (a) to (d) in this gure are related tothe f orbital of uranium and (e) to (h) show d orbital. In all of thefour compounds under study, the DOSs of the up and downstates are asymmetric, which shows magnetism for the elec-trons of this orbital. It is clear from the gure that the f orbital

2


Fig. 6 The PDOS curves of uranium atoms in UX2 compounds. (a)–(d) The f orbitals and (e)–(h) the d orbitals.

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touches the Fermi level at nonzero values for all compounds,which conrms that the electrons of this orbital contributes inthe conduction process. The high effective mass24,28 of theconduction carriers of these compounds is due to these elec-trons, though f is an internal orbital. These f orbital electronsare not completely localized, where the amount of localizationis different for different compounds. Therefore, to investigatethe localization of these orbitals we need case by case study ofall the compounds. In all the compounds under study, the spindown electrons of the f orbital are easily participating in theconduction process as these electrons coincide the Fermi levelat lower values, however the spin up electrons have a largervalue at the Fermi level which shows higher effective mass ofconduction carriers than that of the spin down electrons. Theshape of DOS for f orbitals is the same as that of U atom.

The DOSs of the d orbitals of uranium for up and down statesfor all the compounds shown in the gure display that theseorbitals are symmetric for all compounds, and from area underthe DOS curves we can say that the magnetic properties of forbital is more than that of the d orbital, or in other words d isalmost non-magnetic. As a result, the magnetic properties ofuranium and consequently UX2 compounds are due to the forbital electrons. Therefore, the study of the f orbital electronsis important because they are strongly affecting the magneticproperties as well as high effective mass of conduction carriersof UX2 compounds.

The partial DOSs (PDOSs) of f orbital of uranium in UBi2 andUSb2 for various ‘,m and j(‘� s) are shown in Fig. 7. There are 7values for m in f orbital with ‘ ¼ 3. The plots in the gure from(a) to (c) show partial curves for m ¼ �3 to m ¼ �1. These plotsshow that the curves for UBi2 and USb2 are almost the same andtheir difference is only in the height of peaks, where for someenergies the height of UBi2 curve is higher while for otherenergies the height of USb2 curve is a higher. The contributionof down spin states for energies lower than Fermi energy, i.e.occupied states, is very low and ignorable, where almost all ofDOS curves in this energy interval are related to up spin states.

The PDOSs of f orbital for m ¼ 1, 2 and 3 are shown in theplots (d) to (f) of Fig. 7. The plots show that the contribution ofthese partial parts in occupied states is very low and most part


of these curves are in the higher energies than the Fermi leveland unoccupied states. It is also shown that the curves areshied to the le towards the Fermi energy whenm is decreasedfrom 3 to 1. The curve for m ¼ 1 exists at lower energy than theFermi energy, but even in these parts, the contribution of thedown states electrons in DOS of occupied states is very small.The plot (g) of the gure shows PDOS of f orbital for ‘¼ 3 andm¼ 0. This curve is also shied to the le and has more contri-bution in occupied states relative to that of m ¼ 1, 2 and 3.Furthermore, form¼ 0 the contribution of the down spin statesat lower than the Fermi energies is more than the other partialparts of the f orbital. However, this contribution is still lesserthan the up spin states. In general, by comparing the partialcurves of f orbitals (Fig. 7(a)–(g)) we can see that most part ofDOS of f orbital at lower than Fermi energies is related tonegative and zero m. As the curves are expanded to the Fermilevel with the change of m from m ¼ 3 to m ¼ 1, then the bandwidth also increases in this region. Regarding the asymmetry ofthe up and down states of the partial parts, we can say that allparts with variousm contribute in the magnetism of the orbital.Fig. 7 reveals that the up and down PDOSs are asymmetric withrespect to each other for all m values. Thus, the PDOSs withdifferent m values contribute in magnetism of f orbital. Thecurves for m ¼ 1, 2 and 3 coincide the Fermi level at highervalues, so the effective mass of conduction carriers is higher inthese partial parts. Plots (h) and (i) of the gure show the partialcurves of f orbitals of UBi2 and USb2 for j ¼ 5/2 and 7/2,respectively. It is clear from the plots that the behavior of DOSof these partial parts is almost the same for both crystals. For j¼5/2, the spin up states coincide Fermi level at higher value but incontrast for j¼ 7/2 the spin down states coincide the Fermi levelat higher value. Therefore, the spin up states in j ¼ 5/2 and spindown states in j ¼ 7/2 have higher effective mass of conductioncarriers. This reveals, easier conduction of f orbital electrons inthe spin down states for j ¼ 5/2 and spin up states for j ¼ 7/2.

The detailed discussion on the magnetic properties of UX2 (X¼ P, As, Sb, Bi) dipnicties revels that these compounds areantiferromagnetic with short-range ordering of [Y for UBi2 andlong-range ordering of [[YY for UP2, UAs2 and USb2compounds. We also know from this study that the magnetic

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Fig. 7 The PDOS curves of f orbital of uranium in UBi2 and USb2 compounds for various ‘, m and j.

RSC Advances Paper

properties of these compounds are mostly originated from theuranium atoms especially from its f orbitals, whereas the dorbitals of uranium and X atoms are approximately nonmag-netic compared to their f orbitals. Aer all these considerations,it will not be appropriate if the magnetic moment of theuranium atom in these compounds is not discussed. Therefore,the magnetic moment of uranium atom in these compounds iscalculated withWCFock approach and the calculated results arepresented in Table 6. The calculated results are also comparedwith the experimental and other theoretical results. Thecomparison of the data presented in the table clearly indicatesthat the theoretical values of the magnetic moment of uraniumatom in these compounds by Lebegue et al.17 are severelyunderestimated from the experimental results,24 however ourcalculated values for all the compounds except UP2 are consis-tent with the experimental results. The table also presents thecontribution of the spin and orbital in the total magneticmoment of uranium for each compound. These results reveal

Table 6 Orbital (ORB), spin (SPI) and total (TOT) magnetic momentsper uranium atom in mB using WCFock together with theoretical andexperimental results of the others

UBi2 USb2 UAs2 UP2

ORB �4.243 �3.904 �2.790 �2.221SPI 1.887 1.681 1.209 0.947TOT 2.365 2.223 1.645 1.274TOTa 1.1 � 0.1 0.90 � 0.01 0.7 � 0.1EXP.b 2.1 � 0.1 1.88 � 0.03 1.61 � 0.11 2.1 � 0.1

a Ref. 17. b Ref. 24.

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that the orbital motion of the f-electrons has a profound effecton the magnetic properties of uranium atoms.

IV. Conclusions

In this paper, the electric eld gradients (EFGs) and their latticeand valence contributions are calculated for UX2 (X ¼ Bi, Sb, As,P) compounds. The calculated results show that the EFG is zerofor UBi2 and nonzero with fairly large values for the other UX2

compounds, which are in agreement with the experimentalresults. It is concluded that the zero EFG cannot be obtained byusing LDA, GGA, regular LDA + U and GGA + U functionals, butit is possible with EECE functional. The origin of the zero EFGfor UBi2 and nonzero for the other compounds is investigatedand it is concluded that the antiferromagnetic ordering of [Yfor UBi2 andmore long-range ordering of [[YY for UX2 (X¼ Sb,As, P) are responsible for the difference in the EFG values. Theanalyses of EFGs show that the magnetic structure and spinarrangement of atoms affect EFG; whereas in the mentionedfactors, the magnetic orbitals of uranium have the highestinuence on EFGs. Furthermore, valence electrons (Quasi-core)in the MT sphere play an important role in EFG and thecontribution of electrons which are out of the MT sphere is verysmall. The calculated EFG data for different temperatures showthat EFG decreases as temperature increases but this increase inEFG is signicant only at high temperatures, whereas at roomtemperature it is ignorable.

The DOS curves show that uranium atoms are magnetic andX atoms are non-magnetic for all UX2 compounds. Therefore, inthese compounds, magnetic planes of uranium are separatedwith non-magnetic X planes. So, there is a quasi-two dimen-sional property in these compounds. As DOS curve of f orbital


Paper RSC Advances

has nonzero value on the Fermi surface, therefore f orbitalelectrons contribute in conduction. These f orbital electrons arenon-localized and show different localization in differentcompounds. We also conclude that uranium 5f electrons areresponsible for the large effective mass of the conductioncarriers of UX2 and hence conduction process in thesecompounds. The f orbital is the magnetic orbital of uraniumand its electrons have the highest effect on the magnetization ofthe atom. Therefore, these f orbital electrons have a signicanteffect on the EFG of uranium. Furthermore, our calculatedmagnetic moments for the uranium atom in differentcompounds are consistent with the experimental results24 ascompared to the previously calculated values,17 which areseverely underestimated.

Acknowledgements

We are thankful to Farhad Jalali-Asadabadi for his very friendlycontribution. This work was supported by University of Isfahan(UI), Isfahan, Iran.

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