ARTICLE IN PRESS

Physica B 348 (2004) 151–159

*Corresp

27887830.

E-mail a

0921-4526/$

doi:10.1016

Microscopic spin-Hamiltonian parameters and crystal fieldenergy levels for the low C3 symmetry Ni2þ centre in

LiNbO3 crystals

Zi-Yuan Yanga,b, Czeslaw Rudowiczc,*, Yau-Yuen Yeungd

aMicroelectronics Institute, Xidian University, Xi’an 710071, PR Chinab Institute of Chemistry and Physics, Department of Physics, Baoji University of Arts and Science, Baoji 721007, PR China

cDepartment of Physics and Materials Science, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong SAR, ChinadDepartment of Science, The Hong Kong Institute of Education, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong SAR, China

Received 26 November 2003; received in revised form 26 November 2003; accepted 27 November 2003

Abstract

The microscopic spin-Hamiltonian (MSH) parameters and the crystal field (CF) energy levels for Ni2þ ions in

LiNbO3 crystals have been investigated using the crystal field analysis/microscopic spin-Hamiltonian package recently

developed. The investigations considered for the first time the spin–spin (SS) and spin-other-orbit (SOO) interactions.

The low-symmetry effects (LSE) arising from the additional terms ðImðB43Þa0Þ induced at the C3 symmetry sites by the

distortion angle j; which have been omitted in earlier works, have also been dealt with. This study shows that for

LiNbO3 : Ni2þ the contributions arising from SS and SOO interactions to the zero-field splitting parameter D are

appreciable, whereas those to gjj and g> are quite small. Since the distortion angle j ðD0:68�Þ for LiNbO3 : Ni2þ is

rather small, the contributions to the spin-Hamiltonian (SH) parameters arising from LSE are also small. Feasibility of

application of the superposition model is also discussed. A good overall agreement between the theoretical and

experimental results for the SH parameters and the CF energy levels has been obtained.

r 2003 Elsevier B.V. All rights reserved.

PACS: 76.30.F; 71.70.C; 75.10.D

Keywords: Microscopic spin-Hamiltonian (MSH) parameters; Spin–spin (SS) interaction; Spin-other-orbit (SOO) interaction; Low

symmetry effects (LSE); LiNbO3: Ni2þ crystals

1. Introduction

LiNbO3 is an important technological crystalbecause of its applications in optoelectronics as anoptical waveguide substrate [1], non-linear materi-

onding author. Tel.: +852-27887787; fax: 852-

ddress: apceslaw@cityu.edu.hk (C. Rudowicz).

- see front matter r 2003 Elsevier B.V. All rights reserve

/j.physb.2003.11.085

al [2], and as solid-state laser host matrix [3]. Inparticular, LiNbO3 crystals doped with the transi-tion-metal (TM) ions, e.g., Ti3þ [4,5], Cr3þ [6–9],Ni2þ [10], Mn2þ [11,12], and Fe3þ [13,14], andrare-earth (RE) ions, e.g., Er3þ [15], have attractedmuch attention. The TM or RE ions in suchcrystals are responsible for modifications of theoptical properties of the host matrix. Hence theseimpurity ions in LiNbO3 crystals play a major

d.

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Z.-Y. Yang et al. / Physica B 348 (2004) 151–159152

role, e.g. some dopants are used to increase thephotorefractive sensitivity to light. The spin-Hamiltonian (SH) parameters and, to a lesserextent, the crystal field (CF) ones (see, e.g. Refs.[16–18]), are known to reflect very sensitively evensmall variations in the coordination of the TMimpurity ions in such materials. Thus the theore-tical studies of the SH and CF parameters as wellas their experimental studies using electron para-magnetic resonance (EPR) and optical spectro-scopy, respectively, can provide a great deal ofmicroscopic insights concerning the crystal struc-ture, structural disorder, phase transitions, pres-sure behavior as well as the observed magneticand spectroscopic properties of TM ions incrystals [4–18].The SH parameters and CF energy levels have

been reported for Ni2þ ions doped into LiNbO3

crystals [19,20]. Ni2þ ions exhibit the ground state3A2 at the C3 symmetry sites in LiNbO3 and showanisotropic EPR spectra [19,20] with the zero-fieldsplitting (ZFS) parameter D ¼ �5:31 cm�1 andthe Zeeman factor Dg ¼ gjj � g> ¼ 0:04 [19].These values are remarkably larger in magnitudethan those of Cr3þ (D ¼ �0:39 cm�1; DgE0 [21]),Mn2þ (D ¼ 0:07245 cm�1; Dg ¼ �0:012 [11]), andFe3þ (D ¼ 0:1640ð5Þ cm�1; Dg ¼ �0:008 [14]) ionsin LiNbO3: In order to investigate the SHparameters and the CF energy levels of Ni2þ ionsin LiNbO3 crystals, Zhou et al. [22] developed acomplete diagonalization method (CDM) for 3d8

ion at C3v symmetry using the strong CF scheme.The CDM [22] results were criticized as incorrectby Li [23], who independently developed a similarCDM. However, more recent studies [24,25] revealthat the results [22,23] are incorrect, most prob-ably due to errors in the matrix elements assuggested by Zhang et al. [25] concerning Li’sresults [23]. So far, no satisfactory theoreticalinterpretation for the experimental findings [19,20]has been proposed.Recently, in order to provide a better under-

standing of the electronic structure of 3d2 ð3d8Þions in crystals, an extended crystal-field analysis/microscopic spin-Hamiltonian (CFA/MSH) com-puter package has been developed by us. Thispackage is based on the CFA package for 3dN ionsat arbitrary symmetry sites [26–28] and its recent

Windows version [24], whereas the MSH modulesat present are applicable for 3d2 ð3d8Þ ions attrigonal type I (C3v; D3; D3d) and type II (C3; C3i)symmetry sites including the ‘imaginary’ CF terms[29]. The Hamiltonian adopted in the CFA/MSHpackage includes all terms considered earlier[26–28] and, additionally, the spin–spin (SS) andspin–other-orbit (SOO) [29,30] interactions, whichhave not been considered in the previous micro-scopic studies [22–25]. The CFA/MSH pack-age enables to study not only the CF energylevels and wave functions but also the SHparameters as a function of the CF parameters(B20;B40;ReB43; ImðB43Þ) for 3d2 ð3d8Þ ions attrigonal types I and II symmetry sites. In thepresent work, the CFA/MSH package is used toinvestigate the SH and the CF energy levels takinginto account the actual C3 symmetry of the Ni

2þ–O6 complex in LiNbO3: Ni

2þ crystals.

2. Theoretical background of the CFA/MSH

package for 3d2 and 3d8 ions

A large amount of the recent work has beendevoted to the microscopic studies of the SHparameters for the transition-metal 3dN ions atvarious symmetry sites in crystals. The incentivecomes from the need to explain the experimentalresults accumulated due to extensive applicationsof the electron magnetic resonance (EMR) techni-ques. However, the theories published so farappear not to be fully satisfactory. In particular,the SS interaction, SOO interaction, and C3 lowsymmetry effects have as yet not been consideredmainly due to computational difficulties. Hence, itis worth investigating the role of these interactionsin explaining the MSH parameters. The CFA/MSH package enables such comprehensive theo-retical studies, since it includes these interactionson top of those considered earlier [19,22–25] aswell as takes into account all 45 states in the 3d2

ð3d8Þ configuration.

2.1. Energy matrices

The Ni2þ ions doped into LiNbO3 experience adistorted octahedral CF with the local site

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Z.-Y. Yang et al. / Physica B 348 (2004) 151–159 153

symmetry given by C3 point group. In the CFframework, the total Hamiltonian is written as[26–29]

H ¼HeeðB;CÞ þ HTreesðaÞ þ HCFðBkqÞ

þ Hmðz;M0;M2Þ; ð1Þ

where the respective terms represent the Coulombinteractions, the Trees correction, CF interactions,and magnetic interactions. The latter include theSO, SOO, and SS interactions [29,30]:

Hm ¼ HsoðzdÞ þ HsooðM0;M2Þ þ HssðM0;M2Þ; ð2Þ

the explicit form of which has been given inEqs. (2)–(4) of Ref. [29]. In Eq. (2), zd is the spin–orbit interaction parameter whereas M0 and M2

are the Marvin’s radial integrals used for repre-senting the SS and SOO interactions. The CFHamiltonian for trigonal symmetry in the Wy-bourne notation [17,31] is given as [32]

HCF ¼B20Cð2Þ0 þ B40C

ð4Þ0 þ B43C

ð4Þ3

þ B4�3Cð4Þ�3; ð3Þ

where Bkq are the CF parameters. In general, theB20 and B40 are always real, whereas for thetrigonal symmetry B43 and B4�3 are real for type I,while complex for type II. Since in the Wybournenotation the relation

Bk�q ¼ ð�1ÞqB�kq ð4Þ

holds [32], the two components Re and Im ofeither B43 or B4�3 are enough for full parameter-ization; we chose here B43 ¼ ReðB43Þ þ i ImðB43Þ:The CFA/MSH package constructs the com-

plete 45� 45 energy matrix for 3d2 ð3d8Þ ion attrigonal type II symmetry and, for the first time,incorporates the SS and SOO interactions omittedin previous studies [19,22–25]. The completeenergy matrix can be partitioned into three smallermatrices, each of dimension 15� 15: Since theð10� NÞ-electron system can be regarded as theN-hole system (see, e.g. Ref. [33]), one can obtainthe d8 matrix by changing the signs of only the CFand SO matrix elements within the completeenergy matrix for the d2 configuration, whereasno such change is required for the SS and SOOmatrix elements. The methods of calculation of thematrix elements for Hes; Hso; and HCF have been

described in Ref. [26], whereas those for HSS andHSOO in Ref. [29].The Hamiltonian matrices obtained in this way

are the functions of the Racah parameters B andC; CF parameters Bkq; SO interactions constantzd; and SS (or SOO) interactions parameters M0;M2: Provided the values of these microscopicparameters are available, diagonalization of thefull Hamiltonian matrices yields the energy levelsand eigenvectors, including the ground stateeigenvectors to be used in the calculations of theMSH parameters.

2.2. MSH parameters for d8 ions at C3 symmetry

The cubic orbital singlet ground state 3A2ð3d8Þ

of Ni2þ ion is not spilt by the C3 symmetry CF,whereas the magnetic interactions, Eq. (2), splitthis state into the spin states jE71ð3Fk3A2k

3AÞSand jAð3Fk3A2k

3AÞS: We adopt here thelabeling of the final states [24]:jG

C�3ð2Sþ1Lk2Sþ1GOh

k2Sþ1GC3ÞS; which indicates

explicitly the parentage of the states. Using theCFA/MSH package, the ground spin ðS ¼ 1Þstates of the d8 configuration are obtained bycomplete diagonalization of the three 15� 15matrices in the form of linear combinations ofthe basis LS states as [26,29]

jcþ1S jEþ1ð3Fk3A2k3AÞS

¼X15

j¼1

aþ1;j jjjS; ð5aÞ

jc�1S jE�1ð3Fk3A2k3AÞS

¼X15

j¼1

a�1;j jjjS; ð5bÞ

jc0S jAð3Fk3A2k3AÞS ¼

X15

j¼1

a0;j jjjS: ð5cÞ

The states jcþ1S; jc�1S; and jc0S are representedby real functions for the trigonal type I symmetryused in [24], whereas complex ones for the trigonaltype II symmetry used here.For Ni2þ ions at C3 symmetry, the effective spin

Hamiltonian, taking into account the ZFS and

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-5.28

-5.24

-5.20

D (

cm-1)

Z.-Y. Yang et al. / Physica B 348 (2004) 151–159154

Zeeman terms [17,18], can be written as [24]

HS ¼HZFS þ HZe ¼ D S2z �

13

SðS þ 1Þ� �

þ mBgjjBzSz þ mBg>ðBxSx þ BySyÞ ð6Þ

with the z-axis along a [1 1 1] direction. For 3d8

ions at trigonal symmetry sites the CFA/MSHpackage computes the MSH parameters in Eq. (6)in the following way. The ZFS parameter D isdirectly related to the ZFS between the spin singletðMS ¼ 0Þ and doublet ðMS ¼ 71Þ and is com-puted using the built-in MSH expression [29]

D ¼ eðjE71ð3Fk3A2k3AÞSÞ

� eðjAð3Fk3A2k3AÞSÞ ð7Þ

with the ground state energies taken as obtainedwithin the CFA module by diagonalization of thecomplete energy matrices. In the external magneticfield B the energy levels are further split by theactual Zeeman interaction

HZe ¼ mBðkLþ geSÞ B; ð8Þ

where ge ¼ 2:0023; and k is the orbital reductionfactor, and the angular momenta: orbital L andelectronic spin S; represent the summation oversingle electron variables. The CFA/MSH packagecomputes the axial components of the Zeeman g-factors gjj and g> in Eq. (6) using the built-in MSHexpressions [24]

gjj ¼ k/cþ1jLð1Þ0 jcþ1Sþ ge/cþ1jS

ð1Þ0 jcþ1S; ð9aÞ

g> ¼ kð/cþ1jLð1Þ�1jc0S�/cþ1jL

ð1Þþ1jc0SÞ

þ geð/cþ1jSð1Þ�1jc0S�/cþ1jS

ð1Þþ1jc0SÞ: ð9bÞ

The matrix elements of the irreducible tensoroperators in Eq. (9) are obtained using theWigner–Eckart Theorem [24,34].

260 280 300 320 340 360 380-5.40

-5.36

-5.32

ZF

S p

aram

eter

ImB43

(cm-1)

Fig. 1. The variation of SFS parameter D with CF parameter

IMðB43Þ:

3. Applications to Ni2+ ions in LiNbO3 crystals

Using the MSH expressions derived in Section 2and the energy levels and wave functions obtainedfrom the CFA module, the MSH parameters canbe estimated for any set of the input values of B &C; Bkq; zd; k; and M0 & M2: For Ni

2þ in LiNbO3:Ni2þ crystals, B ¼ 816 cm�1; C ¼ 3224 cm�1; zd ¼540 cm�1; and k ¼ 0:83 have been obtained [24]

using the optical absorption data [19]. The Treescorrection a is chosen as 43:48 cm�1 for free Ni2þ

ion [36]. The SS or SOO parameter values M0 ¼0:3382 cm�1 and M2 ¼ 0:0264 cm�1 are availablefor Ni2þ ions from atomic data [35]. Having fixedthe values of these input parameters, the MSHparameters D; gjj; and g>; which can be experi-mentally measured by EPR, as well as the energylevels, which can be derived from optical spectro-scopy measurements, become functions of the CFparameters B20; B40; and B43 only.

3.1. Crystal structure and CF parameters

The LiNbO3 structure has been determined byX-ray and neutron diffraction [37–39]. The centersof the oxygen octahedra are occupied by cations,whereas the structural vacancy (SV) forms analternating sequence Liþ–Nb5þ–SV–Liþ–Nb5þ–SV [8,38–41]—see Fig. 1 in Ref. [8]. It has beenreported [42] that in LiNbO3: Ni

2þ crystals thedivalent Ni2þ ions occupy Nb5þ sites with C3 pointgroup symmetry. However, the previous studiesfor LiNbO3 : Ni

2þ were based on the approxi-mated C3v [22,23,25] or Oh [20] symmetry ratherthan on the actual C3 symmetry. Such approxima-tions (C3v or Oh) may not provide a reliable insightinto the energy level structure, and the MSHparameters for Ni2þ ions in LiNbO3 crystals. Inthe present work, we have considered the con-tributions to the MSH parameters and the CF

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Z.-Y. Yang et al. / Physica B 348 (2004) 151–159 155

energy levels arising from the low symmetry effectsdue to the actual C3 symmetry as well as the SSand SOO interactions omitted in the previousstudies [20,22–25]. With the z-axis for HCF chosenalong the [1 1 1] axis, we adopt the definition of thex- and y-axis as given in Ref. [29]. Then from theX-ray data [37] we obtain the structural para-meters for the Nb centers in the undistortedLiNbO3 crystals: R1 ¼ 0:1889 nm; R2 ¼ 0:2112 nm;a ¼ 61:65�; b ¼ 47:99�; and j ¼ 0:68� (see, Fig. 1in Refs. [8,29]).In principle, these values of the structural

parameters could be used as starting values withinthe superposition model (SPM) [43–47], whichprovides quantitative relationships between thestructural parameters and CF ones, for modelingthe lattice distortions as well as the CF parametersfor various structural distortion configurations(see, e.g. Refs. [7,8,11,14,24,29]). Since the applica-tion of SPM for the present ion/host system istechnically feasible, we have initially considered‘‘fitting’’ or actually matching the structuralparameters for the distorted LiNbO3 : Ni

2þ hostvia the SPM relationships. Various sets of the CFparameters were calculated by successive SPMcalculations and subsequently used in the CFA/MSH package to obtain the theoretical CF energylevels and states as well as the SH parameters,which were then compared with the respectiveexperimental data. In fact, we have obtained aperfect agreement between the theory and experi-ment using this SPM-based ‘‘fitting’’ approach.However, in view of the inherent uncertainties inthe input data, we have finally limited usage of thisapproach to consideration of the CF parameterImðB43Þ (see below). Note that the value of theintrinsic SPM parameter %A2 is unknown for Ni2þ

in LiNbO3 and can only be approximated from theratio %A2= %A4: Hence the SPM-calculated values ofthe CF parameters could hardly have uncertaintiesless than 10%. This would induce much greateruncertainties in the final ‘‘fitted’’ quantities,rendering this approach unreliable. Thus werefrain from providing the SPM results in full here.In order to consider the C3 low symmetry effect

(LSE), the knowledge of the CF parameterImðB43Þ is necessary. The differences in crystalstructure between the C3v and C3 symmetry can be

described by the distortion angle j of the rotationof the upper and lower oxygen triangles in theoctahedron away from the sv plane (see Fig. 1 ofRef. [29]). For j ¼ 0; ImðB43Þ ¼ 0 and C3 sym-metry reduces to C3v: In view of the small jðD0:68�Þ; the magnitude of ImðB43Þ may beexpected to be quite small and thus its effect onthe CF energy levels and states as well as the SHparameters may be insignificant. To verify thisexpectation, we may employ SPM, since ImðB43Þdoes not depend on the intrinsic parameter %A2 andsuch estimates may be more reliable than in thecase of other CF parameters. Using the SPMexpression [29]

ImðB43Þ ¼ 6ffiffiffiffiffi35

p%A4Q

t4 sin3b cos b sin 6j; ð10Þ

where Q is the radial ratio ðR0=R2Þ (see below), t4is the power-law exponent, %A4 is the intrinsicparameter [44,47], and b is the bond angle withinthe oxygen octahedron defined in Fig. 1 of Ref. [8].Note that in Eq. (18) of Ref. [29] the sign at theimaginary term inside the bracket (+i) is mis-printed and should be replaced by ð�iÞ; conse-quently the RHS in Eq. (10) above is positive. Weadopt the values of R1; R2; b; and j [35,37]obtained above for the undistorted LiNbO3

crystals. The reference distance R0 is approxi-mated as R0EðR1 þ R2Þ=2: %A4 can be found fromthe relation for the cubic CF parameter Dq [48]:%A4E3Dq=4; whereas we take t4 ¼ 5 [44,49]. ThenEq. (10) yields ImðB43Þ ¼ 314 cm�1; i.e. a smallvalue indeed. In order to check the influence ofImðB43Þ on the SH parameters, we vary ImðB43Þ by720%: The dependence of the SH parameters onImðB43Þ is plotted in Figs. 1 and 2 for D and g-factors, respectively. It turns out that the effect ofabout 20% uncertainty in the value ImðB43Þ on D;gjj; and g> is indeed very small.

3.2. Results and discussions

For the reasons discussed above, instead of thefull-scale SPM approach, we utilize the CFparameters obtained from the optical data, i.e. v ¼�950 cm�1; v0 ¼ 600 cm�1; and Dq ¼ 792 cm�1

[19,24]. This approach should be more reliable asit involves no a priori assumed values. Substitutingthe parameter values selected above for B; C; zd; k;

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Z.-Y. Yang et al. / Physica B 348 (2004) 151–159156

a; M0; and M2 together with the values of Bkq

(obtained using the relationships [24] between Bkq inEq. (3) and v; v0; and Dq—as listed in Table 1) intothe CFA/MSH package, we obtain for Ni2þ ions inLiNbO3 crystals the SH parameters listed in Table 1and the CF energy levels in Table 2. Note thatcalculations including the SOO interactions requireusing instead zd the effective value [29] of the SOcoupling constant z0 ð¼ zd�7ð2N �3ÞM0þ42M2 ¼zd � 7M0 þ 42M2 ¼ 538:7414 cm�1 for Ni2þ).This is done internally in the CFA/MSH package.Our previous calculations [24] were aimed atcomparative analysis of earlier results, which didnot include the Trees correction, and hence useda ¼ 0: The present calculations have been carriedout for both a ¼ 0 and 43:48 cm�1 however, only

260 280 300 320 340 360 3802.00

2.05

2.10

2.15

2.20

2.25

2.30

Zee

man

g-f

acto

rs: g

// and

g ⊥

ImB43

(cm-1)

g// g ⊥

Fig. 2. The variation of gjj and g> with CF parameter ImðB43Þ:

Table 1

The values of the ZFS parameter D (in cm�1) and g-factor

Dq ¼ 792 cm�1; zd ¼ 540 cm�1; M0 ¼ 0:3382 cm�1; M2 ¼ 0:026�10657:6 cm�1; ReðB43Þ ¼ �13432:7 cm�1 as well as ImðB43Þ ¼ 314 c

for Ni2þ ions in LiNbO3 crystal

Symmetry Interactions included

(1) C3 SO SOO

(2) C3 SO

(3) C3 SO

(4) C3 SO SOO

(5) C3v SO SOO

Expt. at T ¼ 300 K (no uncertainties given) [19]:

the latter results are provided. We note thatneglecting the Trees correction induces about lessthan 1% difference in the calculated values of D;whereas negligible differences in the g-factors.Table 1 reveals that taking into account the SO,

SS, and SOO interactions yields a good agreementbetween the theoretical values of D and g-factors(row (1) or (5)) and the experimental ones. On theother hand, using the same input parameters as inrow (1) of Table 1, a good agreement betweentheory and experiment is also obtained for the CFenergy levels of Ni2þ in LiNbO3 crystals (see Eð1Þin Table 2). The assignment of the opticaltransitions in Table 2 has been based on theenergy-level calculations and the selection rules. InTable 2, we also present the energy levels shifts dueto the SO, SS, and SOO interactions, as well asLSE. The average values of the energy levels shiftsinduced by a given contribution are 1.81, 5.36, and2:06 cm�1 for the SS, SOO interactions, and LSE,respectively. It can be seen from Table 2 that thecombined contributions arising from the SS andSOO interactions to various multiplet energy levelsfor Ni2þ ions in LiNbO3 crystal were estimated tobe in the range of several cm�1; respectively. Thisrange is comparable with that for V3þ ions inAl2O3 [29].Concerning the magnetic interactions, our study

reveals that the contributions to D from theSO, SS, and SOO interaction are 90.4%, 3.9%,and 5.7%, respectively (see Tables 1 and 3).These results show that the ZFS parameter D ismostly induced by the SO interaction, whereasthe combined (SS+SOO) contributions are also

s (unitless), calculated with B ¼ 816 cm�1; C ¼ 3224 cm�1;4 cm�1; k ¼ 0:83; a ¼ 43:48 cm�1; B20 ¼ �2647:1 cm�1; B40 ¼m�1 for C3 symmetry and ImðB43Þ ¼ 0 cm�1 for C3v symmetry

D gjj g> Dg

SS �5:293 2.2427 2.2035 0.0392

�4:785 2.2373 2.1989 0.0384

SS �4:993 2.2374 2.1990 0.0384

�5:086 2.2426 2.2034 0.0392

SS �5:296 2.2428 2.2035 0.0393

�5:31 2.24 2.20 0.04

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Table 2

The calculated and observed CF energy levels for Ni2þ in LiNbO3 at C3 symmetry sites (cm�1)

Calculated in this work (see Table 1 for values of input parameters) Observed

C3ð2Sþ1LÞ C�3 E(1)b E(2)c E(3)d E(4)e E(5)f DEðLSEÞg DEðSSÞh DEðSOOÞi Oh s[20] p[20] s [22,54] p[22,54]

3Að3FÞ E 0 0 0 0 0 0 0 0 3A2g 0 03Að3FÞ A 5.293 4.785 4.993 5.086 5.296 0.003 0.208 0.301 3A2g 5.31a

3Að3FÞ E 7573 7576 7575 7574 7571 1.9 1.1 1.33Að3FÞ A 7626 7625 7627 7624 7624 1.9 1.8 1.3 3T2g 7812 78103Eð3FÞ E 7947 7949 7950 7946 7945 1.6 0.7 2.53Eð3FÞ E 8190 8185 8184 8191 8188 1.6 1.2 3.6 3T2g 7968 79703Eð3FÞ A 8314 8308 8306 8316 8313 1.5 1.5 2.13Eð3FÞ A 8381 8368 8372 8377 8379 1.5 3.9 1.41Eð1DÞ E 12153 12145 12145 12153 12152 0.3 0.3 7.7 1E 12120 121203Eð3FÞ A 12409 12419 12416 12412 12407 2.7 2.8 6.83Eð3FÞ A 12695 12692 12694 12692 12693 2.0 2.3 0.4 3T1g 12990 129903Eð3FÞ E 12931 12928 12930 12929 12929 2.2 1.9 1.13Eð3FÞ E 13465 13455 13456 13464 13463 2.0 0.7 8.9 3T1g 13333 133333Að3FÞ A 13806 13796 13797 13805 13803 3.2 0.4 9.2 3T1g 13900 137733Að3FÞ E 14037 14030 14029 14038 14034 2.4 1.7 8.01Að1DÞ A 19145 19139 19138 19145 19143 1.7 0.2 6.7 1T2g 19417 194201Eð1DÞ E 20022 20015 20015 20022 20020 1.9 0.1 6.7 1T2g 20408 204501Að1GÞ A 21025 21024 21024 21025 21025 0.8 0.1 1.5 1A1 206203Að3PÞ E 21436 21433 21436 21433 21433 2.5 2.6 0.23Að3PÞ A 21564 21572 21566 21569 21561 2.5 5.2 3.1 3T1g 21978 222203Eð3PÞ E 23289 23303 23303 23289 23286 2.5 0.1 13.3 3T1g 23364 232603Eð3PÞ E 23416 23413 23416 23413 23414 2.5 2.7 0.13Eð3PÞ A 23559 23525 23533 23551 23557 2.5 8.1 25.83Eð3PÞ A 23612 23594 23582 23625 23610 2.5 12.2 30.91Að1GÞ A 24368 24366 24366 24368 24366 2.0 0.3 1.91Eð1GÞ E 24750 24748 24749 24750 24749 1.5 0.2 1.61Eð1GÞ E 29912 29909 29910 29912 29908 3.6 0.1 2.11Eð1GÞ E 31175 31173 31173 31175 31172 3.0 0.1 2.31Að1GÞ A 31718 31716 31716 31718 31716 2.8 0.1 2.41Að1SÞ A 51849 51847 51847 51849 51846 2.7 0.1 2.1

Average 2.1 1.8 5.4

aRef. [19].bC3 symmetry with SO, SS, and SOO.cC3 symmetry with SO and without SS and SOO.dC3 symmetry with SO and SS without SOO.eC3 symmetry with SO and SOO without SS.fC3v approximation with SO, SS, and SOO.gDEðLSEÞ ¼ jEð1Þ � Eð5Þj;hDEðSSÞ ¼ jEð3Þ � Eð2Þj;iDEðSOOÞ ¼ jEð4Þ � Eð2Þj:

Z.-Y. Yang et al. / Physica B 348 (2004) 151–159 157

appreciable reaching 9.6%. Hence the lattercontributions to D shall not be neglected inthe detailed investigations of the SH parameters,which are very sensitive to the lattice dis-tortions [8,50–53]. However, the SS and SOOcontributions to gjj; g>; and Dg are very small(see Table 3). Our study indicates that the

low symmetry effects induced by the anglej contribute negligibly to the SH parametersfor Ni2þ in LiNbO3 crystals and hencethe calculations based on C3v and C3 symmetryyield nearly the same results. However, theseeffects may be appreciable for V3þ in Al2O3

crystal, especially for D and Dg [29], due to much

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Table 3

The percentage contributions to the SH parameters for Ni2þ

ions at C3 symmetry sites in LiNbO3 arising from the SO, SS,

and SOO interactions

Parameters gSO (%) gSS (%) gSOO (%)

D 90.4 3.9 5.7

gjj 99.76 0.004 0.24

g> 99.79 0.005 0.20

Dg 97.96 0 2.04

Calculated using the formula: (a) gSO ¼ fSO=fSOþSSþSOO; (b)

gSS ¼ ðfSOþSS � fSOÞ=fSOþSSþSOO; (c) gSOO ¼ ðfSOþSOO � fSOÞ=fSOþSSþSOO:

Z.-Y. Yang et al. / Physica B 348 (2004) 151–159158

larger jD3:0� than 0:68� for the undistortedLiNbO3 lattice.

4. Summary

The CFA/MSH package recently developed byus [29] enables to study not only the CF energylevels and wave functions but also the MSHparameters as functions of the CF parameters(B20; B40; and B43) for 3d

2 and 3d8 ions at trigonaltype I ðC3v; D3;D3dÞ and type II ðC3; C3hÞsymmetry sites. In this paper we utilized theCFA/MSH package to study the spectroscopicproperties of Ni2þ ions in LiNbO3 crystals. Wehave taken into account for the first time the SSand SOO interactions and the low symmetryeffects arising from the additional CF termsImðB43Þa0 induced by the angle j for C3

symmetry, which have been omitted in earlierworks [19,22–25]. A good overall agreementbetween the theoretical and experimental valuesof the CF energy levels as well as the ZFSparameter D and the g-factors has been obtained.The general conclusions, which can be drawn fromthe present results, may be summarized as follows:(1) The ZFS parameter D for LiNbO3:Ni

2þ ismostly induced by the SO interaction, whereas thecombined (SS+SOO) contributions are also ap-preciable and shall be considered in detailedinvestigations involving the lattice distortionsand structural disorder.(2) The contributions to gjj; g>; and Dg from

the SS and SOO interactions are very small.

(3) The CF energy levels have been calculated,taking into account for the first time the SS andSOO interactions as well as the C3 LSE induced bythe angle j; and assigned according to theobserved optical spectra (see Table 2). Althoughthe average contributions to the energy levels fromthe SS and SOO interactions as well as the LSE forNi2þ: LiNbO3 crystals are only 1.81, 5.36, and2:06 cm�1; respectively, they become appreciablefor certain terms, e.g., for some of the 3Eð3PÞstates the energy level shifts due to the SOOinteraction reach the value of about 31 cm�1:(4) Since the distortion angle jðD0:68�Þ for

LiNbO3: Ni2þ is rather small, the contributions to

the SH parameters arising from the LSE inducedby the C3 symmetry are also small.

Acknowledgements

The authors would like to thank the Editor,Prof. Frank de Boer, for his help in resolvingtechnical issues concerning the early version of thispaper. This work has been partially supported bythe Education Committee Natural Science Foun-dation of Shaanxi Province (Project No. 02JK045)and by Baoji University of Arts and Science KeyResearch Grant as well as the City University ofHong Kong Research Grant (Project No.7001099).

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