Extracting structural information from low symmetry crystal field parameters-case study: Er3+ and Nd3+ ions in YAlO3

JOURNAL OF RARE EARTHS, Vol. 27, No. 4, Aug. 2009, p. 619

Extracting structural information from low symmetry crystal field parameterscase study: Er3+ and Nd3+ ions in YAlO3 P. Gnutek, C. Rudowicz (Institute of Physics, West Pomeranian University of Technology, Al. Piastów 17, 70–310 Szczecin, Poland) Received 1 October 2008; revised 1 February 2009

Abstract: The experimental monoclinic CF parameter (CFP) sets obtained by Duan et al. (Phys. Rev. B 75 (2007) 195130) for Er3+ and Nd3+ ions in YAlO3 were reanalyzed. These CFPs fitted using R-approach, i.e. with the monoclinic second-rank CFP set to zero, and additionally with one six-rank CFP fixed to zero, turned out to be non-standard. In order to understand better the low symmetry aspects involved in the fitted CFPs and extract useful structural information inherent in monoclinic CFPs, an approach comprising four methods was utilized. First, superposition model (SPM) was applied to calculate CFPs in the crystallographic axis system. Second, the principal values for the SPM determined CFPs and the orientation of the principal axis system w.r.t. the crystallographic axis system were obtained using the procedure 3DD for diagonalization of the 2nd-rank CFPs. Third, analysis of higher symmetry approximations, i.e. orthorhombic and tetragonal, was carried out using the pseudosymmetry axes method. Fourth, the closeness factors and norm ratios were employed for quantitative comparisons of various CFP sets. Partial results for Er3+ ions in YAlO3 were presented here, whereas detailed results would be given in a follow-up paper. Keywords: Er3+; Nd3+; YAlO3; crystal-field; energy levels; low symmetry; rare earths

In this study a comprehensive approach to analysis of the low symmetry crystal field parameters (CFPs) was utilized to reanalyze the experimental CFP sets obtained by Duan et al.[1] for Er3+ and Nd3+ ions at monoclinic symmetry sites in YAlO3. This approach comprised four methods: (1) superposition model (SPM) of CFPs[2,3], (2) the procedure 3DD for diagonalization of the 2nd-rank CFPs[4], (3) the pseudosymmetry axes method (PAM)[5] extended to lower symmetry cases[6], and (4) quantitative comparison of CFP sets and other quantities using the closeness factors and norm ratios[7,8]. Our aim was to understand better the low symmetry aspects involved in the fitted and theoretical CFPs for Er3+ and Nd3+ ions in YAlO3 and extract useful structural information inherent in the monoclinic CFPs. Due to space limitations, only partial results of our reanalysis for Er3+ ions in YAlO3 were provided here, whereas detailed results would be presented in a follow-up paper[9]. For details of the methods and CFP notations, the readers may refer to Refs.[3,10,11]. Section 1 provided brief discussion of the CFPs of Duan et al [1]. Section 2 presented the results of our modeling of CFPs using SPM and preliminary analysis of the low symmetry aspects inherent in the CFPs for Er3+ ion in YAlO3.

1 Experimental Duan et al.[1] presented and interpreted own optical spectral investigations extended into vacuum ultraviolet range and the synchrotron radiation excited emission and excitation spectra of Er3+ and Nd3+ ions in YAlO3. For CF analysis they used, however, the energy levels deduced from emission and absorption spectra for Er3+ in YAlO3 at 4.2 K by Donlan and Santiago[12], whereas the luminescence transitions for Nd3+ in YAlO3 reported by Kaminskii[13]. The observed transitions were employed[1] to assign about 100 energy levels used in CF analysis. The general form of CF Hamiltonian in the Wybourne k operators Cq [3,10,14 ] used[1]: k

H CF

¦ ¦ B C k q

k

q

reduces for monoclinic symmetry to one of the three possible forms depending on the choice of the monoclinic direction[15]. The computational procedure[1] was based on fitting 20 real “atomic” parameters following the notation[16,17] and 15 real CFPs in Eq.(1). The only symmetry axis, i.e. the monoclinic direction for Cs site symmetry, was chosen[1]

Foundation item: Project partially supported by the research grant from the Polish Ministry of Science and Tertiary Education in the years 2006-2009 Corresponding author: C. Rudowicz (E-mail: [email protected]) DOI: 10.1016/S1002-0721(08)60301-4

(1)

k 2,4,6 q  k

620

JOURNAL OF RARE EARTHS, Vol. 27, No. 4, Aug. 2009

as z-axis. Then the authors[1] used a suitable rotation in the x-y plane to transform one complex CFP Bqk to be real without changing the eigenvalues. Thus the number of independent real CFPs was reduced to 14, whereas the atomic parameters were further constrained in the fittings. This approach has resulted in the free-ion parameters (not reproduced here) and the CFP sets (listed in Table 1) for Er3+ and Nd3+ ions in YAlO3 [1]. We used the notation for CFPs with the real and imaginary parts denoted as Bqk and Sqk, respectively. In fact, the R(reduced) approach with the monoclinic 2ndrank CFP S22 set to zero was employed, and additionally the monoclinic six-rank CFP S46 was fixed to zero for computational convenience, since in the process of calculation this CFP was found not well defined for both ions. The authors[1] asserted that such truncation of HCF did not affect the residual error. Importantly, the CFP sets for both ions[1] turn out to be non-standard[18,15]. Hence, to enable meaningful comparison with other CFP sets for similar ion-host systems available in literature, such sets should be first standardized (see Section 2), using e.g. the package CST[19] or the recently developed module 3DD[4]. B

B

2

Modeling and analysis of CFPs for Er3+ and Nd3+ ions in YAlO3

The unit cell of YAlO3 contains four magnetically inequivalent Y sites which may be occupied by rare-earth ions (Fig. 1)[20]. The Y sites with the monoclinic local symmetry Cs are 12-fold coordinated, so may be also approximated as 8-fold coordinated sites by neglecting four more distant 16 oxygen ligands. For the orthorhombic space group D2 h (Pbnm) the crystallographic axes (a, b, c) form a Cartesian axis system and coincide with the symmetry axes. Hence, the axes (a, b, c) can be adopted as the symmetry adapted axis system. For SPM calculations the following axis system Table 1 Fitted experimental monoclinic CFPs[1] (R-approach, in Wybourne notation) for Er3+ and Nd3+ ions in YAlO3/ cm–1 CFPs

Er

CFPs

Nd

B02

–178.5

B02

–154

B22

578

B04

–541

B

B22 B

489.6

B

S22

S22

[0]

967

S24

24

B44

–309

S44

608

B06

–671

-62

B26

512

S26

–18

1611

S46

[0]

0

S66

132

[0]

B

B04

–134.0

B24

464

S24

–183

B24

B44

–9

S44

627

B

B

B

B

B

B

B06

–453

B62

199

S26

B46

808

S46

[0]

B46

B66

–74

S66

24

B66

B

B

B

B

B

B

B

Fig. 1 Unit cell of YAlO3 for the crystallographic data[20]

is adopted: the monoclinic direction being perpendicular to the reflection ab-plane (ıh) is taken as the z-axis (z||c), whereas the remaining axes are taken as x||a and y||b. In ‘usual’ approach to SPM[2,3], expressed in the extended q Stevens (ES) operators Ok [21,22], the model parameters, i.e. the intrinsic parameters B  R and the power law exponents tk, are obtained by matching the SPM calculated CFPs with the experimental CFPs. In this study a novel approach is proposed based on the ‘geometrical’ rotational invariants s  ES defined for given ion-host system as: k

0

cc

k

S k  ES

Bk  R0 ˜ R0 ˜ skcc  ES tk

(2)

where Sk(ES) are by definition the rotational invariants for CFPs (experimental or theoretical) and R0 is reference distance[2,3]. Thus the intrinsic parameters are obtained in terms of ‘geometrical’ rotational invariants as: t cc Bk  R0 S k sk ˜ R0 (3) The major advantages of expressing the intrinsic parameters via the ‘geometrical’ rotational invariants are as follows: (1) intrinsic parameters remain positive, unlike negative intrinsic parameters arising occasionally from the ‘matching’ procedures in ‘usual’ approach to SPM[2,3], (2) intrinsic parameters for a given ion-ligand system rotated by arbitrary Euler angles (Į, ȕ, J) are invariant, (3) intrinsic parameters are independent of the axes chosen in SPM calculations, hence more consistent intrinsic parameters are obtained, and (4) the relation (due to electrostatic interactions) tk=k+1 is preserved for all transition ions and this agrees with the corq   k 1 responding equation Bk ~ R arising from other models, i.e. angular overlap model (AOM), exchange charge model (ECM), and simple overlap model (SOM); see, e.g. Ref. [23]. Using the ionic positions of ligands[20] expressed in the chosen axis system, the reference distance R0=0.25 nm (being a middle value between the average for 12- and 8-fold





k

P. Gnutek et al., Extracting structural information from low symmetry crystal field parameters-case study:…

coordination: Rav~0.265 nm and Rav~0.24 nm, respectively), and the relation: tk=k+1, our approach yields intrinsic parameters (in cm–1): B  R0 , B  R , B  R for Er: 1913, 51, 14 and Nd: 2225, 78, 28, respectively. These values for k=4 and 6 agree well with literature data[2,3]. Subsequently, SPM calculations (after conversion to Wybourne notation in Eq.(1)) yield the CFP listed in Table 2 set (a). The axis system used for the experimental CFPs in Table 1, i.e. ‘nominal’ axis system[11] corresponding to the principal axis system of the 2nd-rank CFPs, differs from that for the SPM determined CFPs in Table 2 set (a), i.e. the symmetry adopted axis system coinciding with the crystallographic axis system. Hence, the two CFP sets are not directly comparable. For comparison these CFP sets must be transformed to the same axis system. Hence, the CFPs determined using SPM are transformed to the principal axis system of the 2nd-rank CFPs using the monoclinic standardization option within the package CST[19] or the module 3DD[4]. Analogously to the method applied to the conventional nd 2 -rank ZFS terms in the EMR area[6], the module 3DD[4] diagonalizes the 2nd-rank CFPs and provides the principal values of the 2nd-rank CFPs and the orientation of their principal axis systems w.r.t. the original axis system for monoclinic and triclinic symmetry. Note that the module 3DD[4] enables, for the first time, full triclinic standardization, unlike the package CST[19], which includes only the monoclinic standardization[15]. Results for Er3+:YAlO3 are given in Table 2 set (b). The SPM determined CFPs in Table 2 set (b) and the experimental ones in Table 1 are now expressed in the principal axis system of the 2nd-rank CFPs. However, it turns out that the two sets are non-standard and still cannot be directly compared, since they differ in the value of the ‘rhombicity’ ratio[15,18]: N = B B . It turns out that for orthorhombic[18] 2

2

2

2

0

4

0

6

0

621

and lower symmetry[11,15] , one must first verify if the sets to be compared belong to the same region in the multiparameter space[11,24]. The value of N=–0.756 indicates that CFP set (b) in Table 2 belongs to the area in the multi-parameter space denoted[11,24] as [-II], whereas CFP set Er in Table 1 with N=–2.743 belongs to the area [-III]. Hence, there is a need for monoclinic standardization[11,18]. A problem arises since for monoclinic CFPs expressed in distinct regions in the multi-parameter space[11,24] (total 6 standardization transformations exist) involve different forms of monoclinic HCF[11,15]. Comparison of the transformed CFPs yielding |N|d0.408 listed in Table 3 indicates that the standardized 2nd-rank and orthorhombic 6th-rank CFPs are similar, but differences remain between all 4th-rank CFPs as well as the monoclinic 6th-rank CFPs. Detailed analysis of the possible reasons for these discrepancies will be presented in our further study[9]. As a next step, calculations using the pseudosymmetry axes method (PAM)[5,6] were carried out in order to reconsider the role of 4th and 6th-rank CFPs. The program PAM[6] based on an extension of the cubic/axial pseudosymmetry axes method[6] to monoclinic, orthorhombic, tetragonal I and trigonal I approximations provides the pseudosymmetry axes obtained by minimizing an appropriate CFP combination w.r.t. the Euler angles (D, E, J). The program PAM calculates the 3-D surface of the function Hsym w.r.t. the angles D and E and the contours of Hsym representing its minima. So obtained CFP sets exhibit the maximal and minimal values of the given higher symmetry CFPs and the out-of-given symmetry ones, respectively. A novel approach to PAM proposed here, i.e. simultaneous minimalization of the 4th and 6th-rank CFPs to obtain higher symmetry approximation, enables to obtain more accurate results. The PAM orth-

Table 2 Monoclinic CFPs (in Wybourne notation) for Er : YAlO3/cm–1 (determined using SPM: set (a)-CFPs in the axis system (x||a, y||b, z||c) and set (b)-CFPs after diagonalization of the 2nd-rank CFPs; the rotation Į/Oz=137.4º yields S22= 0)

Table 3 Monoclinic CFPs (in Wybourne notation) for Er3+: YAlO3/cm–1 (determined using the package CST in the principal axis system of the standardized 2nd-rank CFPs: set (a)-experimental CFPs after transformation C2||zoC2||x and set (b)-theoretical CFPs after transformation C2||zo C2||y)

CFPs

3+

Set (a)

CFPs

Set (b)

CFPs

Set (a)

CFPs

Set (b)

B02

–487.0

B02

–487.0

B02

688.9

B02

694.3

B22

29.9

B22

368.1

B22

135.5

B22

114.2

B04

–148.0

–651.2

B24

–50.5

B44 B06

–553.3

B

B

B04

–440.8

B24

B

B

S22

–113.1

S24

215.2

S44

B06

–405.9

B26

93.1

S26

–814.3

S46

93.1

S66

B

B

B

B46 B

B66 B

B

B

S22

B04

–440.8

434.8

B24

–424.0

S24

–558.3

B44

–304.0

S44

B06

–405.9

195.7

B26

–202.8

S26

–22.2

B46

799.5

S46

–12.0

B66

11.2

S66

B

B44

B

367.7

B

B

B

B

B

B

0

B

B

S12

0

B

B04

–426.5

–148.4

B24

279.0

S14

–515.2

B44

235.7

S34

B06

–368.2

76.7

B26

–163.9

S16

–8.9

B26

155.7

B46

830.7

S36

–46.0

B46

–93.2

B66

126.0

S56

47.2

B66

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B12

0

238.8

B14

–429.5

–549.0

B34

321.0

–122.4

B16

–134.2

760.1

B36

–20.3

130.5

B56

142.8

B

B

B

B

B

B

622

orhombic approximation (to be presented in our further study[9]) enables us to extract additional structural information from the SPM determined 4th and 6th-rank CFPs. Finally, the closeness factors and norms ratios[7,8] were calculated. These quantities provide means for quantitative comparison of various sets of n quantities of the same nature, e.g. the CFPs or the energy levels, considered as n-dimensional ‘vectors’. The complete results of PAM calculations as well as the closeness factors and norms ratios will be presented in our further study[9]. Here, for illustration Fig. 2 presents the ErO12 complex in YAlO3 with the respective axis systems indicated: (1) the pseudo-symmetry 4-fold axis system (x’, y’, z’) obtained from PAM based on the SPM determined CFPs from Table 2 set (a) and (2) the crystallographic axis system (a, b, c), which coincides in this case with the symmetry adapted axis system (x, y, z). Comparison with an ideal YO12 ‘cuboctahedron’ with the cubic 4-fold axis system (x, y, z) shows that the ErO12 complex exhibits cubic symmetry with a strong orthorhombic distortion.

Fig. 2 ErO12 complex in YAlO3 with the respective axis systems indicated (see text)

3 Conclusion The results of this study proved the usefulness of the proposed comprehensive approach to analysis and modeling of the CF parameter (CFP) sets in optical spectroscopy studies of transition ions at low symmetry sites. As a case study, applications of this approach to CFPs for Er3+ and Nd3+ ions in YAlO3 at monoclinic symmetry sites were considered. Superposition model (SPM) was used to calculate CFPs in the crystallographic axis system. The axes determined for Er3+:YAlO3 by (1) diagonalization of the 2nd-rank CFPs method and (2) pseudosymmetry axes method were related to the structural data. The importance of standardization of CFP sets and clear definitions of the various axis systems

JOURNAL OF RARE EARTHS, Vol. 27, No. 4, Aug. 2009

used for meaningful comparisons of the pertinent CFP sets was elucidated. Detailed results for both Er3+ and Nd3+ ions in YAlO3 would be presented in our further work. Our approach may help experimentalists to better interpret and analyze optical spectroscopy data as well as to extract useful structural information from CFP sets. Other applications for RE ions in various hosts will be considered elsewhere.

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