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Extracting structural information from low symmetry crystal field parameters: Pr4+ in BaPrO3
Journal of Alloys and Compounds 451 (2008) 694–696
Extracting structural information from low symmetry crystal field parameters: Pr4+ in BaPrO3 Czesław Rudowicz ∗ , Paweł Gnutek Institute of Physics, Szczecin University of Technology, Al. Piast´ow 17, 70–310 Szczecin, Poland Available online 13 April 2007
Abstract Recent literature survey has revealed several crystal field parameter (CFP) datasets for rare-earth (RE) ions at orthorhombic, monoclinic, and triclinic symmetry sites, which cannot be directly compared. Triclinic CFPs include for a given rank (k = 2, 4, 6) all components (−k ≤ q ≤ +k). Depending on the axis system chosen, the low symmetry effects involved in the symbolic, fitted, or model CFPs may be only apparent and not actual. In order to extract useful structural information involved in the low symmetry CFPs, an approach comprising three methods is proposed: (1) the principal values of the 2nd-rank CFPs and the orientation of their principal axis system (AS) w.r.t. the original or crystallographic AS; (2) the pseudosymmetry axes method; (3) the closeness factors. This approach is applied to reanalyze the CFPs for Pr4+ in BaPrO3 . © 2007 Elsevier B.V. All rights reserved. Keywords: Rare-earth alloys and compounds; Crystal and ligand fields; Optical properties; Pr4+ ions; BaPrO3
1. Introduction This study benefits from cross-fertilization between the crystal field (CF) theory and the spin Hamiltonian theory used in the electron magnetic resonance (EMR) area. We present briefly a comprehensive approach to analysis of the crystal field parameters (CFPs), which comprises three methods employed in the EMR studies: (i) the procedure 3DD for diagonalization of the 2nd-rank CFPs [1], (ii) an extension of the pseudosymmetry axes method (PAM) [2] to lower symmetry cases [3], and (iii) quantitative comparison of CFP datasets and other quantities using the closeness factors [4]. This approach is applied to the theoretical triclinic-like CFPs for Pr4+ ion in BaPrO3 [5]. Due to space limitations, we provide here only the major results, whereas full presentation is given in Ref. [6]. For details of the methods and CFP notations, we refer to [1,3,4]. Section 2 provides brief discussion of the reliability of the CFPs [5]. Section 3 presents analysis of these CFPs and the low symmetry effects. 2. Crystal field parameters for Pr4+ ion in BaPrO3 The BaPrO3 unit cell contains four magnetically inequivalent Pr4+ ions (Fig. 1(a)) [5]. The local site symmetry of each ∗
Corresponding author. E-mail address: [email protected] (C. Rudowicz).
0925-8388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2007.04.084
octahedral Pr site is Ci [5]. The cubic approximation was used in optical studies of Pr4+ in BaPrO3 [5,7,8,9]. Based on the exchange charge model (ECM) [10] and the crystallographic data at T = 300 K [11], Popova et al. [5] calculated the CFPs defined by the CF Hamiltonian: HCF = Bpq ηp Oqp . (1) p,q
q
No definitions of “the Stevens operators (Op ) and the reduced matrix elements (ηp )” were provided [5]. The HCF notation [5,10] differs [6] from the well-established notations for the usual and extended Stevens operators and the associated CFPs [12,13]: q q q q q q HCF = Ak r k θk Ok = Ck θk Ok = B k Ok , (2) k,q
k,q
k,q
with k denoting the rank and q denoting the component. We follow the convention [5] of omitting the Stevens factors ηp . However, since the original CFPs Bpq in Eq. (1) and Bpk (with p as the rank and k as the component) in Table 2 of Ref. [5] resemble the Wybourne notation [12,13], to avoid confusion we q use the notation Ck defined in Eq. (2). The spin–orbit coupling (λ) and the octahedral CF split the f1 multiplets 2 F5/2 and 2 F7/2 into three Kramers doublets and two quartets yielding four possible transitions [5,7,8]. Lower symmetry CF splits the quartets, yielding total six transitions
C. Rudowicz, P. Gnutek / Journal of Alloys and Compounds 451 (2008) 694–696
695
Table 1 q The CFP Ck for Pr4+ in BaPrO3 (in cm−1 ) #1
#2
#3
#4
#5
α β γ
0 0 0
0.43 67.08 0.55
5.04 168.4 0.33
50.3 81.8 −8.2
139.1 98.1 −8.3
k, q 2, 2 2, 1 2, 0 2, −1 2, −2 4, 4 4, 3 4, 2 4, 1 4, 0 4, −1 4, −2 4, −3 4, −4
130.3 123.7 3.4 −1.1 2.4 −4708 −4184 307.4 −3839 830.3 −341.5 60.9 −1075 −1601.9
57.2 0.0 76.5 0.0 0.0 3347 −9567 −4321 1886 −329.1 −1341 −314.3 −1775 −46.1
138.7 66.6 −6.3 3.1 19.6 −5193 −101.0 −75.7 14.5 1033 5.6 −3.1 39.3 0.0
−19.3 −44.9 −6.7 −277.3 24.6 5173 −43.7 14.5 −6.2 1036 −37.8 −28.4 265.1 0.0
0.3 49.3 13.0 277.3 22.4 5168 99.3 11.4 14.1 1037 37.7 −12.5 −265.2 0.0
#1: ECM triclinic set [5]; #2: standardized set after diagonalization of the 2ndrank CFPs; #3–5: PAM TEI approximation for the 4th-rank CFPs.
Fig. 1. (a) BaPrO3 unit cell for the crystallographic data at T = 300 K [11] and (b) buckling of the four adjacent Pr–O6 octahedra.
used for fitting the CFPs. Hence, it is impossible to fit all triclinic 27 CFPs for Pr4+ . The ECM [5] employed instead three model parameters Gk (k = 2, 4, 6) and λ. The values of Gk and λ were obtained by matching theoretical energy levels with optical data. The cubic CFPs determined by various authors for Pr4+ in BaPrO3 are reviewed in Ref. [6]. Doubts arise concerning the numerical conversion in Ref. [5] of the cubic CFPs for Pr4+ in BaCeO3 [8]. Hence, the cubic, and by implication triclinic-like CFPs [5], should be treated with caution. The prevailing relations for the cubic CFPs are [12]: C44 = 5C40 and C64 = −21C60 . The authors [5] seemingly use: C44 = −5C40 and C64 = 21C60 . Note that a rotation Oz/45◦ changes signs in these relations [6] and has structural implications (Section 3). From the energy differences [8], we obtain [6] C4 = 1025 and C6 = 51 yielding C44 = 5125 and C64 = −1071, instead of the values listed in [5]: C44 = −5075 (this value seems misprinted in [5] for −5125) and C64 = 1071 (in cm−1 ). The data [9] yield [6] [C60 , C60 , C44 , C64 ] (in cm−1 ): [959.8, 48.0, 4799.0, −1007.8]; the data [7] yield: [1103, 71, 5516, −1491]. In view of the signs disagreement, the cubic CFPs, and thus by implication the triclinic-like CFPs [5], should be treated with caution [6]. 3. Analysis of the CFPs for Pr4+ ions in BaPrO3 The program 3DD diagonalizes the 2nd-rank CFPs analogously to the method applied to the conventional 2nd-rank ZFS terms in the EMR area [1]. It provides the principal values
(PVs) of the 2nd-rank CFPs and the orientation of their principal axis systems (PASs) w.r.t. the original axis system (OAS). The program PAM based on an extension of the cubic/axial pseudosymmetry axes method [2] to monoclinic, orthorhombic, tetragonal I and trigonal I approximations provides the pseudosymmetry axes (PAs) obtained by minimizing an appropriate CFP combination w.r.t. the Euler angles (α, β, γ). The program PAM calculates the 3-D surface of the function εsym w.r.t. the angles α and β and the contours of εsym 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. The closeness factors [4] 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 (n-D) ‘vectors’. The original triclinic-like CFPs for Pr1 [5] are listed in Table 1 (set #1). We provide here only the k = 2 and 4 terms, whereas k = 6 terms are listed in Ref. [6]. The set #1 yields the rotational invariants [4]: S2 = 37.28, S4 = 452.00 and S6 = 44.48 (in cm−1 ), which are the same for all transformed CFP sets in Table 1. S4 and S6 are close to the cubic approximation ones in [7–9]. For the sets #3–5 we also calculate S4∗ = 451.96 and S6∗ = 44.18 (in cm−1 ) based on the cubic CFPs only, which indicate the goodness of the higher symmetry approximation. Applying the program 3DD [1] to set #1 yields several sets of (i) the PVs of the 2nd-rank CFP ‘tensor’, (ii) the respective (α, β, γ), and (iii) the accordingly transformed 4th- and 6th-rank CFPs. In Table 1, we provide only one set—#2 with the rhombicity ratio [13,14] λ = C22 /C20 = 0.747. Other related sets are discussed in Ref. [6]. Correlation of the PAS of the 2nd-rank CFPs and the PAS of the g-factor [5] indicates their closeness. The relative orientations of the g-factor PAS [5], 3DD/PAS of the 2nd-rank CFPs, and PAM orthorhombic axes for the 4th-rank CFPs are considered in Ref. [6].
696
C. Rudowicz, P. Gnutek / Journal of Alloys and Compounds 451 (2008) 694–696
Fig. 2. The z-axis for each of the three solutions #3–5 obtained using the program PAM in the tetragonal I (TEI) approximation.
Applying the program PAM [1] to set #1 yields a number of sets for various approximations. In Table 1, we present only the tetragonal I (TEI) approximation results (sets #3–5). These 4thand 6th-rank CFPs are very close to the cubic CFPs for Pr4+ in BaCeO3 [8]. The dominant cubic 4th-rank CFPs and relatively small non-cubic ones in Table 1 suggest that that 4th-rank CFPs [5] were reliably determined. The relative values of the 4th- and 6th-rank CFPs in the extended and normalized Stevens notation [12] are discussed in Ref. [6]. The closeness factors were calculated [6] for: (i) the cubic 4th- and 6th-rank CFPs [7–9], (ii) the CFPs (k = 2, 4, 6; global) obtained for various PAM approximations, (iii) the Pr4+ energy levels, and (iv) the structural parameters for Pr–O6 . The closeness factors show quantitatively the correlations between various sets of comparable quantities. The CFPs and respective Euler angles in Table 1 enable us to extract useful structural information. In the CAS [5], the Pr–O6 complex (Fig. 1) appears rotated by 45◦ /Oz (Section 2). The axes x and y directed towards ligands represent orthorhombic symmetry of the 1st-kind, whereas the 45◦ /Oz rotation that of the 2nd-kind [15]; these axis systems apply also to cubic symmetry. The PAM/TEI sets #4 and #5 correspond to the former case, whereas the set #3 corresponds to the latter case. The zaxes for the sets #3–5 are mutually perpendicular to each other (Fig. 2), whereas the orientations of the x- and y-axes vary. The ratios C44 /C40 ∼ = ±5 and C64 /C60 ∼ = ∓21 for sets #3–5 are very close to pure cubic ones [12]. This reveals very high degree of cubic octahedral symmetry with a slight triclinic distortion for Pr4+ in BaPrO3 and agrees well with the hypothesis [7]. The low symmetry effects (LSE) are most evident in the highly triclinic 2nd-rank CFPs—sets #1, #3–5. This may represent the actual or apparent LSE inherent in the 2nd-rank CFPs. The q suspected improper estimations of C2 [5] may lead to appar2 ent LSE. The orthorhombic CFPs C2 and C20 in set #4 and #5 are very small as for the cases very close to cubic symmetry. The large non-zero C21 , C2−1 , C2−2 raise doubts concerning their accuracy. Aspects bearing on the actual and/or apparent LSE inherent in the triclinic-like ECM-calculated CFP sets and their reliability have been discussed in Ref. [6]. In summary, application of the three-method approach to the triclinic-like CFPs for Pr4+ ion in BaPrO3 [5] confirms its use-
fulness in optical studies of transition ions at low symmetry sites. The axes determined by (i) diagonalization of the 2ndrank CFPs method and (ii) pseudosymmetry axes method are related to the structural data as well as the principal axes of the g- and A-tensors [5]. The apparent nature of low symmetry effects involved in the CFPs [5] becomes evident due to application of the PAM extended to orthorhombic and axial cases. Our considerations confirm the hypothesis concerning the cooperative buckling of the Pr–O6 octahedra reducing the local site symmetry, while preserving the nearly octahedral geometry for the Pr site [7]. The closeness factors facilitate quantitative comparison of CFPs and other quantities. The several physically equivalent CFP sets obtained due to the 3DD method and PAM enable application of the multiple correlated fitting technique (MCFT) [4]. Using these CFPs as starting values for additional fittings would yield several correlated datasets. Thus MCFT offers ways to increase accuracy and reliability of final fitted CFPs, identify global minima, and eliminate spurious sets [4]. MCFT requires transformations of CFP sets, which are facilitated by the package CST [16]. Full results for Pr4+ ion in BaPrO3 are given in [6]. Our approach may help experimentalists to better interpret and analyze optical data as well as to extract useful structural information from CFP datasets. Other applications for RE ions in various hosts will be considered elsewhere. Importantly, this approach is applicable both to the CF parameter datasets and the zero-field splitting ones used in the EMR studies. Acknowledgment This work was partially supported by the research grant from the Polish Ministry of Science and Tertiary Education in the years 2006–2009. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12] [13] [14] [15] [16]
C. Rudowicz, P. Gnutek, J. Phys. Condens. Matter, in press. G. Bacquet, et al., J. Phys. C 7 (1974) 1551. C. Rudowicz, P. Gnutek, J. Magn. Res., in press. C. Rudowicz, J. Qin, J. Lumin. 110 (2004) 39. M.N. Popova, S.A. Klimin, L.A. Kasatkina, G. Cao, J. Crow, Phys. Lett. A 223 (1996) 308. C. Rudowicz, P. Gnutek, J. Alloys Compd., doi:10.1016/j.jallcom.2007. 02.088, in press. M. Bickel, et al., J. Solid State Chem. 76 (1988) 178. Y. Hinatsu, N. Edelstein, J. Solid State Chem. 112 (1994) 53. S. Kern, C.-K. Loong, G.H. Lander, Phys. Rev. B32 (1985) 3051. B.Z. Malkin, in: A.A. Kaplyanskii, R.M. Macfarlane (Eds.), Spectroscopy of Solids Containing Rare-earth Ions, North-Holland, Amsterdam, 1987, p. 13. N. Rosov, et al., Phys. Rev. B 45 (1992) 982. C. Rudowicz, Magn. Res. Rev. 13 (1987) 1; C. Rudowicz, Magn. Res. Rev. 13 (1988) 335. C. Rudowicz, R. Bramley, J. Chem. Phys. 83 (1985) 5192. C. Rudowicz, J. Chem. Phys. 84 (1986) 5045. C. Rudowicz, J. Phys. Chem. Solids 38 (1977) 1243. C. Rudowicz, Q. Jian, Comp. Chem. 26 (2002) 149.
Extracting structural information from low symmetry crystal field parameters: Pr4+ in BaPrO3 Czesław Rudowicz ∗ , Paweł Gnutek Institute of Physics, Szczecin University of Technology, Al. Piast´ow 17, 70–310 Szczecin, Poland Available online 13 April 2007
Abstract Recent literature survey has revealed several crystal field parameter (CFP) datasets for rare-earth (RE) ions at orthorhombic, monoclinic, and triclinic symmetry sites, which cannot be directly compared. Triclinic CFPs include for a given rank (k = 2, 4, 6) all components (−k ≤ q ≤ +k). Depending on the axis system chosen, the low symmetry effects involved in the symbolic, fitted, or model CFPs may be only apparent and not actual. In order to extract useful structural information involved in the low symmetry CFPs, an approach comprising three methods is proposed: (1) the principal values of the 2nd-rank CFPs and the orientation of their principal axis system (AS) w.r.t. the original or crystallographic AS; (2) the pseudosymmetry axes method; (3) the closeness factors. This approach is applied to reanalyze the CFPs for Pr4+ in BaPrO3 . © 2007 Elsevier B.V. All rights reserved. Keywords: Rare-earth alloys and compounds; Crystal and ligand fields; Optical properties; Pr4+ ions; BaPrO3
1. Introduction This study benefits from cross-fertilization between the crystal field (CF) theory and the spin Hamiltonian theory used in the electron magnetic resonance (EMR) area. We present briefly a comprehensive approach to analysis of the crystal field parameters (CFPs), which comprises three methods employed in the EMR studies: (i) the procedure 3DD for diagonalization of the 2nd-rank CFPs [1], (ii) an extension of the pseudosymmetry axes method (PAM) [2] to lower symmetry cases [3], and (iii) quantitative comparison of CFP datasets and other quantities using the closeness factors [4]. This approach is applied to the theoretical triclinic-like CFPs for Pr4+ ion in BaPrO3 [5]. Due to space limitations, we provide here only the major results, whereas full presentation is given in Ref. [6]. For details of the methods and CFP notations, we refer to [1,3,4]. Section 2 provides brief discussion of the reliability of the CFPs [5]. Section 3 presents analysis of these CFPs and the low symmetry effects. 2. Crystal field parameters for Pr4+ ion in BaPrO3 The BaPrO3 unit cell contains four magnetically inequivalent Pr4+ ions (Fig. 1(a)) [5]. The local site symmetry of each ∗
Corresponding author. E-mail address: [email protected] (C. Rudowicz).
0925-8388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2007.04.084
octahedral Pr site is Ci [5]. The cubic approximation was used in optical studies of Pr4+ in BaPrO3 [5,7,8,9]. Based on the exchange charge model (ECM) [10] and the crystallographic data at T = 300 K [11], Popova et al. [5] calculated the CFPs defined by the CF Hamiltonian: HCF = Bpq ηp Oqp . (1) p,q
q
No definitions of “the Stevens operators (Op ) and the reduced matrix elements (ηp )” were provided [5]. The HCF notation [5,10] differs [6] from the well-established notations for the usual and extended Stevens operators and the associated CFPs [12,13]: q q q q q q HCF = Ak r k θk Ok = Ck θk Ok = B k Ok , (2) k,q
k,q
k,q
with k denoting the rank and q denoting the component. We follow the convention [5] of omitting the Stevens factors ηp . However, since the original CFPs Bpq in Eq. (1) and Bpk (with p as the rank and k as the component) in Table 2 of Ref. [5] resemble the Wybourne notation [12,13], to avoid confusion we q use the notation Ck defined in Eq. (2). The spin–orbit coupling (λ) and the octahedral CF split the f1 multiplets 2 F5/2 and 2 F7/2 into three Kramers doublets and two quartets yielding four possible transitions [5,7,8]. Lower symmetry CF splits the quartets, yielding total six transitions
C. Rudowicz, P. Gnutek / Journal of Alloys and Compounds 451 (2008) 694–696
695
Table 1 q The CFP Ck for Pr4+ in BaPrO3 (in cm−1 ) #1
#2
#3
#4
#5
α β γ
0 0 0
0.43 67.08 0.55
5.04 168.4 0.33
50.3 81.8 −8.2
139.1 98.1 −8.3
k, q 2, 2 2, 1 2, 0 2, −1 2, −2 4, 4 4, 3 4, 2 4, 1 4, 0 4, −1 4, −2 4, −3 4, −4
130.3 123.7 3.4 −1.1 2.4 −4708 −4184 307.4 −3839 830.3 −341.5 60.9 −1075 −1601.9
57.2 0.0 76.5 0.0 0.0 3347 −9567 −4321 1886 −329.1 −1341 −314.3 −1775 −46.1
138.7 66.6 −6.3 3.1 19.6 −5193 −101.0 −75.7 14.5 1033 5.6 −3.1 39.3 0.0
−19.3 −44.9 −6.7 −277.3 24.6 5173 −43.7 14.5 −6.2 1036 −37.8 −28.4 265.1 0.0
0.3 49.3 13.0 277.3 22.4 5168 99.3 11.4 14.1 1037 37.7 −12.5 −265.2 0.0
#1: ECM triclinic set [5]; #2: standardized set after diagonalization of the 2ndrank CFPs; #3–5: PAM TEI approximation for the 4th-rank CFPs.
Fig. 1. (a) BaPrO3 unit cell for the crystallographic data at T = 300 K [11] and (b) buckling of the four adjacent Pr–O6 octahedra.
used for fitting the CFPs. Hence, it is impossible to fit all triclinic 27 CFPs for Pr4+ . The ECM [5] employed instead three model parameters Gk (k = 2, 4, 6) and λ. The values of Gk and λ were obtained by matching theoretical energy levels with optical data. The cubic CFPs determined by various authors for Pr4+ in BaPrO3 are reviewed in Ref. [6]. Doubts arise concerning the numerical conversion in Ref. [5] of the cubic CFPs for Pr4+ in BaCeO3 [8]. Hence, the cubic, and by implication triclinic-like CFPs [5], should be treated with caution. The prevailing relations for the cubic CFPs are [12]: C44 = 5C40 and C64 = −21C60 . The authors [5] seemingly use: C44 = −5C40 and C64 = 21C60 . Note that a rotation Oz/45◦ changes signs in these relations [6] and has structural implications (Section 3). From the energy differences [8], we obtain [6] C4 = 1025 and C6 = 51 yielding C44 = 5125 and C64 = −1071, instead of the values listed in [5]: C44 = −5075 (this value seems misprinted in [5] for −5125) and C64 = 1071 (in cm−1 ). The data [9] yield [6] [C60 , C60 , C44 , C64 ] (in cm−1 ): [959.8, 48.0, 4799.0, −1007.8]; the data [7] yield: [1103, 71, 5516, −1491]. In view of the signs disagreement, the cubic CFPs, and thus by implication the triclinic-like CFPs [5], should be treated with caution [6]. 3. Analysis of the CFPs for Pr4+ ions in BaPrO3 The program 3DD diagonalizes the 2nd-rank CFPs analogously to the method applied to the conventional 2nd-rank ZFS terms in the EMR area [1]. It provides the principal values
(PVs) of the 2nd-rank CFPs and the orientation of their principal axis systems (PASs) w.r.t. the original axis system (OAS). The program PAM based on an extension of the cubic/axial pseudosymmetry axes method [2] to monoclinic, orthorhombic, tetragonal I and trigonal I approximations provides the pseudosymmetry axes (PAs) obtained by minimizing an appropriate CFP combination w.r.t. the Euler angles (α, β, γ). The program PAM calculates the 3-D surface of the function εsym w.r.t. the angles α and β and the contours of εsym 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. The closeness factors [4] 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 (n-D) ‘vectors’. The original triclinic-like CFPs for Pr1 [5] are listed in Table 1 (set #1). We provide here only the k = 2 and 4 terms, whereas k = 6 terms are listed in Ref. [6]. The set #1 yields the rotational invariants [4]: S2 = 37.28, S4 = 452.00 and S6 = 44.48 (in cm−1 ), which are the same for all transformed CFP sets in Table 1. S4 and S6 are close to the cubic approximation ones in [7–9]. For the sets #3–5 we also calculate S4∗ = 451.96 and S6∗ = 44.18 (in cm−1 ) based on the cubic CFPs only, which indicate the goodness of the higher symmetry approximation. Applying the program 3DD [1] to set #1 yields several sets of (i) the PVs of the 2nd-rank CFP ‘tensor’, (ii) the respective (α, β, γ), and (iii) the accordingly transformed 4th- and 6th-rank CFPs. In Table 1, we provide only one set—#2 with the rhombicity ratio [13,14] λ = C22 /C20 = 0.747. Other related sets are discussed in Ref. [6]. Correlation of the PAS of the 2nd-rank CFPs and the PAS of the g-factor [5] indicates their closeness. The relative orientations of the g-factor PAS [5], 3DD/PAS of the 2nd-rank CFPs, and PAM orthorhombic axes for the 4th-rank CFPs are considered in Ref. [6].
696
C. Rudowicz, P. Gnutek / Journal of Alloys and Compounds 451 (2008) 694–696
Fig. 2. The z-axis for each of the three solutions #3–5 obtained using the program PAM in the tetragonal I (TEI) approximation.
Applying the program PAM [1] to set #1 yields a number of sets for various approximations. In Table 1, we present only the tetragonal I (TEI) approximation results (sets #3–5). These 4thand 6th-rank CFPs are very close to the cubic CFPs for Pr4+ in BaCeO3 [8]. The dominant cubic 4th-rank CFPs and relatively small non-cubic ones in Table 1 suggest that that 4th-rank CFPs [5] were reliably determined. The relative values of the 4th- and 6th-rank CFPs in the extended and normalized Stevens notation [12] are discussed in Ref. [6]. The closeness factors were calculated [6] for: (i) the cubic 4th- and 6th-rank CFPs [7–9], (ii) the CFPs (k = 2, 4, 6; global) obtained for various PAM approximations, (iii) the Pr4+ energy levels, and (iv) the structural parameters for Pr–O6 . The closeness factors show quantitatively the correlations between various sets of comparable quantities. The CFPs and respective Euler angles in Table 1 enable us to extract useful structural information. In the CAS [5], the Pr–O6 complex (Fig. 1) appears rotated by 45◦ /Oz (Section 2). The axes x and y directed towards ligands represent orthorhombic symmetry of the 1st-kind, whereas the 45◦ /Oz rotation that of the 2nd-kind [15]; these axis systems apply also to cubic symmetry. The PAM/TEI sets #4 and #5 correspond to the former case, whereas the set #3 corresponds to the latter case. The zaxes for the sets #3–5 are mutually perpendicular to each other (Fig. 2), whereas the orientations of the x- and y-axes vary. The ratios C44 /C40 ∼ = ±5 and C64 /C60 ∼ = ∓21 for sets #3–5 are very close to pure cubic ones [12]. This reveals very high degree of cubic octahedral symmetry with a slight triclinic distortion for Pr4+ in BaPrO3 and agrees well with the hypothesis [7]. The low symmetry effects (LSE) are most evident in the highly triclinic 2nd-rank CFPs—sets #1, #3–5. This may represent the actual or apparent LSE inherent in the 2nd-rank CFPs. The q suspected improper estimations of C2 [5] may lead to appar2 ent LSE. The orthorhombic CFPs C2 and C20 in set #4 and #5 are very small as for the cases very close to cubic symmetry. The large non-zero C21 , C2−1 , C2−2 raise doubts concerning their accuracy. Aspects bearing on the actual and/or apparent LSE inherent in the triclinic-like ECM-calculated CFP sets and their reliability have been discussed in Ref. [6]. In summary, application of the three-method approach to the triclinic-like CFPs for Pr4+ ion in BaPrO3 [5] confirms its use-
fulness in optical studies of transition ions at low symmetry sites. The axes determined by (i) diagonalization of the 2ndrank CFPs method and (ii) pseudosymmetry axes method are related to the structural data as well as the principal axes of the g- and A-tensors [5]. The apparent nature of low symmetry effects involved in the CFPs [5] becomes evident due to application of the PAM extended to orthorhombic and axial cases. Our considerations confirm the hypothesis concerning the cooperative buckling of the Pr–O6 octahedra reducing the local site symmetry, while preserving the nearly octahedral geometry for the Pr site [7]. The closeness factors facilitate quantitative comparison of CFPs and other quantities. The several physically equivalent CFP sets obtained due to the 3DD method and PAM enable application of the multiple correlated fitting technique (MCFT) [4]. Using these CFPs as starting values for additional fittings would yield several correlated datasets. Thus MCFT offers ways to increase accuracy and reliability of final fitted CFPs, identify global minima, and eliminate spurious sets [4]. MCFT requires transformations of CFP sets, which are facilitated by the package CST [16]. Full results for Pr4+ ion in BaPrO3 are given in [6]. Our approach may help experimentalists to better interpret and analyze optical data as well as to extract useful structural information from CFP datasets. Other applications for RE ions in various hosts will be considered elsewhere. Importantly, this approach is applicable both to the CF parameter datasets and the zero-field splitting ones used in the EMR studies. Acknowledgment This work was partially supported by the research grant from the Polish Ministry of Science and Tertiary Education in the years 2006–2009. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12] [13] [14] [15] [16]
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