Low symmetry aspects inherent in electron magnetic resonance (EMR) data for transition ions at triclinic and monoclinic symmetry sites: EMR of Fe3+ and Gd3+ in monoclinic zirconia revisited

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Physica B 403 (2008) 2349–2366 www.elsevier.com/locate/physb

Low symmetry aspects inherent in electron magnetic resonance (EMR) data for transition ions at triclinic and monoclinic symmetry sites: EMR of Fe3+ and Gd3+ in monoclinic zirconia revisited Czes"aw Rudowicz, Pawe" Gnutek Institute of Physics, Szczecin University of Technology, Al. Piasto´w 17, 70–310 Szczecin, Poland Received 14 December 2007; accepted 17 December 2007

Abstract Our recent survey of the EMR literature has revealed a number of experimental zero-field splitting (ZFS) parameter datasets for transition ions at triclinic symmetry sites in various compounds, which include for a given rank (k ¼ 2, 4, 6) all components kpqp+k. Model calculations of ZFS parameters, which usually employ the crystallographic axis system centered at the transition ion, may also yield triclinic-like ZFS parameter datasets. Closer analysis is required in order to distinguish the actual low symmetry aspects from the apparent ones. A comprehensive approach is proposed to analyze the low symmetry aspects involved in the monoclinic- or triclinic-like spin Hamiltonian (SH) terms, especially the ZFS ones. The approach comprises three methods: (i) finding the principal values of the various 2nd-rank SH terms and the orientation of the respective principal axis systems w.r.t. the laboratory or crystallographic axis system, (ii) extending the cubic/axial pseudosymmetry axes method (PAM) to lower symmetry cases and finding the pseudosymmetry axis system for the 4th-rank ZFS parameters, and (iii) employing the closeness factors C and the norms ratios R ¼ NA/NB for quantitative comparison of several ZFS parameter datasets. Each method is facilitated by recently developed computer programs. Application of this approach, with focus on the PAM, for the ZFS parameter datasets for Fe3+ and Gd3+ in monoclinic zirconia (m-ZrO2) is presented. Our considerations enable better understanding of the low symmetry aspects as well as correlation of the principal axis systems and/or pseudosymmetry axis systems with the symmetry-adapted axis systems, which may be approximately related to the respective metal ion–ligands bonds. The equivalence between various physically equivalent ZFS parameter datasets generated by the first and second method, especially those transformed to the standard range by proper rotations of the axis systems, is also studied. This equivalence may be utilized in the multiple correlated fitting technique to improve the reliability of the fitted results. The final standardized ZFS parameters are best to be used as the starting parameters for simulations and fittings of the EMR spectra for transition ions in structurally similar hosts. Importantly, the three methods proposed here may be also applied to the crystal field parameters studied by optical spectroscopy. r 2007 Elsevier B.V. All rights reserved. Keywords: Electron magnetic resonance (EMR); Spin Hamiltonian; Zero-field splitting (ZFS) parameters; 3d-ions; 4f-ions; Low symmetry effects; Monoclinic zirconia (m-ZrO2); Inorganic compounds (doped with 3d- and 4f-ions)

1. Introduction An adequate analysis and interpretation of the EMR spectral data for transition-metal and rare-earth ions at triclinic symmetry sites in crystals (see, e.g. Refs. [1–4]) require consideration of the low symmetry aspects involved in the respective spin Hamiltonian (SH) terms (for a Corresponding author.

E-mail address: [email protected] (C. Rudowicz). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.12.017

review, see Refs. [5,6]). Two major SH terms are the Zeeman electronic (Ze) interaction and the zero-field splitting (ZFS) terms. Various notations employed in the literature for the operators appearing in the ZFS Hamiltonians and the associated ZFS parameters have been reviewed in Refs. [7,8]. Our recent comparative survey of the low symmetry cases in the up-to-now EMR-related literature have revealed a number of experimental ZFS parameter datasets for transition ions in various compounds, which include for a given rank k=2, 4, and 6 all

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components kpqp+k; see, e.g., Refs. [9–14]. Such full ZFS parameter datasets are, in principle, admissible by group theory for triclinic symmetry [2–8]. Apart from the experimentally determined triclinic-like ZFS parameter datasets, also various model calculations of ZFS parameters, e.g. the superposition model (for references, see, Refs. [7,8]), often yield monoclinic- or triclinic-like ZFS parameter datasets. The model calculations usually employ the crystallographic axis system centered at the transition ion located at a specific symmetry site. If the crystallographic axis system does not coincide with the symmetryadapted axis system suitable for the given metal ion– ligands complex, the theoretically determined ZFS parameters would indicate lower symmetry that the actual one. Such apparent lower symmetry cases should be treated with caution. Our experience shows that in order to consider the low symmetry aspects [5,6] inherent in the SH parameter datasets exhibiting monoclinic or triclinic symmetry, proper transformations and standardization [15–19] of the parameters must be carried out. Most important is the knowledge of the principal values of the various SH terms and the orientation of the respective principal axis systems w.r.t. the crystallographic axis system or the laboratory axis system. The principal quantities of the Zeeman electronic Hamiltonian HZe can be obtained by diagonalization of the 3  3 matrix. In the case of the ZFS Hamiltonian HZFS, which may be expressed in any of several existing conventional or tensor operator notations [7,8] and may include the rank k=2, 4, and 6 terms, such diagonalization may be carried out only for the rank k=2 terms. Additional complication arises, namely, how to transform accordingly the rank k=4 and 6 terms to the principal axis system of the 2nd-rank ZFS parameters. The authors, who employ the computer program EPR-FOR or its more recent version EPR-NMR [20], provide often the 2nd-rank ZFS terms in their principal axis system, while leave the 4th- and 6th-rank ZFS terms in the original laboratory axis system. The ZFS parameter datasets presented in such dual way (see, e.g. Refs. [21–23]) cannot be directly compared with the data taken from other sources, which are often expressed in a different, albeit consistent, axis systems. In this paper, we present a comprehensive approach to analysis of the monoclinic- and/or triclinic-like ZFS parameter datasets appearing in the literature. The approach comprises three methods. The first method consists in finding the principal values of the various 2nd-rank SH terms and the orientation of the respective principal axis systems w.r.t. the laboratory or crystallographic axis system based on the idea of their diagonalization, in a similar way as in the case of the conventional ZFS terms [7,8]. The implications of such diagonalization, yielding orthorhombic-like crystal field and spin Hamiltonians, for the fitting procedures used in the optical and EMR spectroscopy, respectively, have been discussed in Ref. [19]. The second method is based on an extension of

the pseudosymmetry axis method (PAM) [9,10]. It consists in finding the pseudosymmetry axis system for the 4th-rank ZFS parameters, which best reflect the symmetry of the nearest ligands in the paramagnetic complex. The third method consists in a comparative analysis of several ZFS parameter datasets using the closeness factors Cp and the norms ratios Rp=NA/NB for the respective ZFS terms: p=k=2, 4, 6, and the global (p=gl) ones [19,24,25]. The quantities C and R quantify the closeness of any two datasets of the same type, e.g. the SH parameters or the energy levels, as well as enable meaningful comparison of datasets taken from different sources and identification of compatible yet numerically distinct datasets. The computations are facilitated by the recently developed computer package DPC [26], which comprises three modules: (i) module 3DD for diagonalization of any 3  3 matrices representing, e.g. the 2nd-rank ZFS or other SH tensors [27], (ii) module PAM for extension of the pseudosymmetry axes method to lower symmetry cases that enables finding the pseudosymmetry axis system for the 4th-rank ZFS parameters, and (iii) module CFNR for calculation of the closeness factors and the norms ratios [28]. Our considerations, with focus on application of the pseudosymmetry axis method, enable better understanding of the low symmetry aspects as well as correlation of the principal axis systems with the symmetry-adapted axis system, which may be approximately related to the respective bonds between the transition ion and the coordinating ligands. The equivalence between various physically equivalent ZFS parameter datasets transformed to the standard range by proper rotations of the axis systems [15,16] and other non-standard yet physically equivalent ZFS parameter datasets is also studied. This equivalence may be utilized in the multiple correlated fitting technique [16] to improve the reliability of the fitted results [6,18,19]. The final standardized ZFS parameter datasets are best to be used as the initial parameters for simulations and fittings of the EMR spectra for transition ions in structurally similar hosts. As an illustrative example of application of the proposed approach we consider the EMR studies of Fe3+ and Gd3+ ions doped in single crystals of monoclinic zirconia (m-ZrO2) [9,10]. Interest in the zirconia compounds is still active due to important technological applications of, e.g. yttria-stabilized zirconia as inert matrix for actinide immobilization or transmutation [29–31] as well as compacted monoclinic ZrO2 as an oxygen-ion conductor used in oxygen sensors and pumps or fuel cells for electricity generation [32]. The paper is organized as follows. The basic SH forms and nomenclatures are briefly outlined in Section 2. The theoretical and computational backgrounds underlying the three methods within our approach are discussed in Section 3. Follow-up analysis of the EMR data for Fe3+ and Gd3+ ions in m-ZrO2 [9,10] based on our approach is presented in Section 4. Summary and conclusions are given in Section 5. Importantly, the three methods proposed here may be also applied to the crystal field parameters studied by

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optical spectroscopy. Comprehensive analysis of the other low symmetry ZFS parameter datasets and/or crystal field ones for transition ions in various crystals will be dealt with elsewhere. 2. Spin Hamiltonian forms and nomenclatures For clarity, the definitions of the basic spin Hamiltonian forms and nomenclatures are succinctly outlined in this section. This enables easier understanding of the nature of each method as well as specific intricacies involved in the low symmetry studies dealt with in Sections 3 and 4, respectively. For paramagnetic species with the spin SX1 at sites with triclinic symmetry various forms of HZFS have been used in the EMR literature [1–4]. The general conventional form of HZFS, which is suitable and most widely used for the spin 1pSp3/2 systems, is given by [7,8] H ZFS ¼ S:D:S.

(1a)

In the principal axis system (x, y, z) for monoclinic and triclinic symmetry as well as for orthorhombic symmetry HZFS is expressed as [1–8]: h i   H ZFS ¼ D S 2z  13SðS þ 1Þ þ E S2x  S2y . (1b) The axes (x, y, z) in Eq. (1a) may be chosen for orthorhombic symmetry in different ways w.r.t. the symmetry-adapted axis system [15], whereas for monoclinic [16] and triclinic [19] symmetry, the orientation of the principal axes w.r.t. the laboratory axis system (X, Y, Z) must be provided for a meaningful data comparison. For the spin SX2 higher-rank ZFS terms are required. For such systems, apart from the conventional notation in Eqs. (1a) and (1b), researchers have employed a number of various tensor operator notations belonging to one of the two distinct classes: the spherical tensor operators or the tesseral tensor operators. Various types of the spherical tensor operators and the tesseral tensor operators used in the EMR and optical spectroscopy area have been categorized in the reviews [7,8]. The operators that comprise in a unified way also the components Oqk with negative q are the extended Stevens (ES) operators Oqk defined in Ref. [33] and generalized in Ref. [34]). These operators belong to the tesseral tensor operator class and have gained a dominant position in the EMR literature in the last several decades [7,8]. They were introduced in Ref. [33] as an extension of the usual (or conventional) Stevens operators [35], which were originally defined only for qX0 [1–4]. As the general reference form, below we adopt the ZFS Hamiltonian in terms of the extended Stevens operators expressed in the compact form (to be distinguished from the alternative expanded form applicable to the spherical tensor operators [7,8]): X q q X H ZFS ¼ Bk Ok ðSx ; S y ; Sz Þ ¼ f k bqk Oqk ðS x ; Sy ; S z Þ. kq

kq

(2)

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The second form of HZFS in Eq. (2) has been more often used in EMR studies of transition ions due to its numerical convenience. The ‘scaling’ factors fk [7,8] of ZFS parameters bqk are most commonly defined in a uniform way as [1–4,36]: 1 1 1 f4 ¼ ; f6 ¼ . (3) f2 ¼ ; 3 60 1260 Note that as discussed in Ref. [36] in some papers the factors fk are inadvertently omitted from HZFS thus leading to an ambiguity concerning the values of the ZFS parameters. Although the relations between the orthorhombic ZFS parameters in Eqs. (1b) and (2) are well known [1,3,4,7]: D ¼ b02 ðESÞ ¼ 3B02 ðESÞ;

3E ¼ b22 ðESÞ ¼ 3B22 ðESÞ,

(4)

we provide them here to warn readers about the incorrect relations appearing in Refs. [2,37] as pointed out in Ref. [36]. Note that the form of the crystal field Hamiltonian is mathematically equivalent to the first HZFS form in Eq. (2), the only difference being the implicit replacement of the spin operators S ¼ (Sx, Sy, Sz) by the orbital L or total J angular momentum operators [7,8,38]. As reviewed in Ref. [38], this fact has lead to a wide spread confusion between the physically different crystal fieldrelated quantities and ZFS ones [7,8]. The explicit definitions of the extended Stevens operators (see, Ref. [33] and references therein) yield for triclinic symmetry the following relations [39,40] between the Dij components in Eq. (1a) and the ZFS parameters in Eq. (2): B02 ðESÞ ¼

Dzz ; 2

B1 2 ðESÞ ¼ 2Dyz ;

B12 ðESÞ ¼ 2Dxz , B22 ðESÞ ¼

ðDxx  Dyy Þ , 2

B2 2 ðESÞ ¼ Dxy .

ð5Þ Bqk

and Dij exist in the Note that incorrect relations between literature [41,42] as discussed in Ref. [36]. The relations in Eq. (5) and P the traceless property of the tensor D in Eq. (1a): Dij ¼ 0, are utilized in the module 3DD [27] as i¼j

described in Section 3. 3. Theoretical and computational backgrounds The three methods used in our approach utilize several computational procedures, namely, diagonalization, transformations, standardization, pseudosymmetry axis method, and comparative analysis of low symmetry ZFS parameters. The underlying theoretical backgrounds for each method are provided in this section. The low symmetry aspects inherent in the monoclinic- or tricliniclike ZFS parameter datasets may be either actual or apparent. Consideration of such aspects requires a method that yields the ZFS parameters expressed in a respective principal axis system. This can be achieved by an effective ‘diagonalization’ of the 2nd-rank ZFS term expressed in any tensor operator notation, e.g. the extended Stevens

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operators in Eq. (2). By converting the 2nd-rank ZFS terms (expressed either in the spherical or tesseral tensor operators) to the conventional notation, one can employ the method [19] commonly applied in the EMR area [7,8] to the conventional ZFS term in Eq. (1a). Such diagonalization yields the principal values of the 2nd-rank ZFS parameters and the orientation of the principal axes w.r.t. the original axis system [19]. Such method is built into the program EPR-NMR [20] and yields the principal values, denoted Yk, ordered consecutively from the largest value (k ¼ 1) to the lowest one (k ¼ 3). The original nomenclature [20] defines the ‘directions of matrices’ Y ¼ g, D, A, and P in the ‘crystal Cartesian coordinate system’ (see, e.g., Refs. [21,22]), which are given by a set of six angles (yk, fk). Hence, the relationships between the original axes (x, y, z) and the resulting principal axes are rather not directly evident. Moreover, due to the internal constraints the program EPR-NMR [20] seems to return only one subset of all possible solutions (see Section 4). Note that for the spin values higher than S47/2 the program EPR-NMR [20] returns incorrect matrix elements of the Stevens operators [34], however, no practical application of this program for such spin systems has appeared as yet. In general, for any type of the 2nd-rank SH terms, their diagonalization corresponds to diagonalization of a 3  3 matrix. Any such matrix can be diagonalized by three consecutive rotations in the three-dimensional (3D) space. Based on this idea appropriate transformation expressions [33,43] have been derived by us and incorporated into the module 3DD [27] together with the necessary conversion relations [7]. Computational details are provided elsewhere [27]. Briefly, using the relations in Eq. (5), the module 3DD internally converts the ZFS parameters in the extended Stevens operator notation (as well as the crystal field parameters [19] in the Wybourne notation [7]) to the equivalent Dij components (i, j ¼ x, y, z), subsequently numerically carries out the 3D diagonalization (3DD), which yields all possible solutions for the principal axes, and then reconverts the ZFS parameters as appropriate. Numerical solutions for the two principal values and the orientation of the respective principal axis systems given in terms of the three Euler angles (a, b, g) are obtained using MathCAD [27]. Applications of the module 3DD [27] for the 2nd-rank ZFS terms reveal the existence of several physically equivalent ZFS parameter datasets expressed in the principal axis systems of the 2nd-rank terms. The higherrank k ¼ 4 and 6 ZFS parameters must be transformed accordingly. The transformation computations as well as standardization discussed below are facilitated by employing the extended package CST [44,45] for conversions, standardization, and transformations of the spin Hamiltonians and the crystal field Hamiltonians. Note that the package CST may be obtained from one of us (CZR) on collaborative basis. It turns out that the various physically equivalent sets of the orthorhombic-like parameters B02 and B22 returned by the module 3DD [27] may include both the

standard and non-standard sets [15]. In general, to enable meaningful comparison with other ZFS parameter datasets available in the literature, the non-standard datasets require transformations to the standard range by proper rotations of the axis systems [15,16]. The standardization idea enables to limit the ratio l0 ¼ B22 =B02 of the ZFS parameters in the extended Stevens operator notation [33] to the range (0, 1). Different ways of choosing appropriate transformations, depending on the initial ratio l0 , were classified in Ref. [15]. Orthorhombic [15] and monoclinic [16] standardization enable consistent presentation of the ZFS (crystal field) parameters and has been widely applied for the non-standard sets appearing in the literature for the transition ions in various ion-host systems (see, Refs. [15–19] and references therein). This method enables meaningful analysis and comparison of the different ZFS parameter datasets, since the standardized sets are referred to the same nominal axis system [19]. In the present case, the standard and non-standard sets of the orthorhombic-like ZFS parameters B02 and B22 are obtained by a different route than via the standardization transformations within the package CST [44,45], namely, by diagonalization of the 2nd-rank ZFS parameter terms using the module 3DD [27]. The Euler angles (a, b, g) returned by the module 3DD are subsequently utilized as input to compute the corresponding 4th- and 6th-rank ZFS parameters in the principal axis system of the 2nd-rank ZFS parameters. In general, the module 3DD yields for monoclinic- and triclinic-like ZFS parameter datasets 12 and 24 possible physically distinct yet equivalent sets, each comprising 2+10 and 4+20 standard and non-standard sets, respectively. We have verified that starting from any 2nd-rank ZFS parameter dataset generated by the module 3DD, all other sets, to within small rounding-off differences, can also be obtained using the standardization transformation module of the package CST [44,45]. This reinforces the reliability of the two programs, each based on an independent computational procedure. Importantly, all equivalent ZFS parameter datasets yield the same rotational invariants Sk [19] or equivalently the norms Nk [33] of the ZFS parameters. Here we provide only the definition of Sk for the extended Stevens operators [33]: 1 Nk, 2k þ 1 k  2 X N k ¼ B0k þ

ðS k Þ2 ¼

q¼1

1 ckq

!2



Bqk

2

 2  þ Bq , k

ð6Þ

where the coefficients ckq are listed in Refs. [7,33]. The quantities Sk and Nk in Eq. (6) are invariant with respect to any arbitrary rotation of the axis system. Hence, they measure the ZFS ‘strength’ and also provide an additional check of the reliability of experimentally fitted ZFS parameters as well as the consistency of the transformed ZFS parameters expressed in different axis systems by various authors [19]. Automatic calculations of the

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quantities Sk for all major tensor operator notations are provided in the package CST [44,45]. The rotational invariants are very useful for comparison of the experimentally or theoretically determined ZFS parameters for the related ion-host systems taken from various sources. Additionally, the module 3DD computes also the values ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h   2 2 i  0 2 S 2 ¼ ð1=5Þ B2 þ ð1=3Þ B2 for the original ZFS parameters, i.e. omitting the low symmetry ZFS parameters except of the orthorhombic ones. The difference between total invariant quantity S2 and S 2 provides a measure of the deviation of the original axis system, in which the ZFS parameters are expressed, and the principal axis system of the 2nd-rank ZFS terms. Thus indirectly it accounts for the strength of low symmetry aspects inherent in the 2nd-rank ZFS parameters. Additionally, for the monoclinic-like ZFS parameters one can also apply directly the monoclinic standardization module within the package CST [44,45]. The ZFS parameters standardized in this way should agree nearly exactly with the corresponding parameters obtained using the module 3DD for the 2nd-rank ZFS parameters and then the package CST for the 4th- and 6th-rank ones. We have carried such tests on several monoclinic ZFS parameter datasets taken from recent literature [6,46,47], thus confirming again the validity of both programs. Note that all physically equivalent ZFS parameter datasets obtained in our method must yield the same energy levels. Conversely, fittings of experimental spectra may, in general, return several distinct ZFS parameter datasets. Much in this regard depends on the choice of the initial parameters used for least-squares fittings [19]. The equivalence of various ZFS (crystal field) parameter datasets for the same ion-host systems has occasionally been recognized in the EMR (optical spectroscopy) related literature, see, e.g. Refs. [15,16,48,49]. However, often quite disparate ZFS (crystal field) parameter datasets are reported without realization of their inherent equivalence. Examples of such sets are provided, e.g., for ZFS parameters for Mn2+: BiVO4 in Ref. [50] and crystal field parameters for RE3+:LiYF4 in Ref. [51]. Various ZFS (crystal field) parameter datasets may only be meaningfully compared and thus analyzed for any possible common trends among them, if the correlations between them as well as the actual choice of the respective axis systems are unambiguously established. Utilization of the equivalence between the correlated ZFS parameter datasets in the multiple correlated fitting technique [16,18,19] is discussed in Section 4. A word of caution should be said here about the relationship between the non-zero ZFS (crystal field) parameters and the local site symmetry. Generally, the lower local site symmetry, the greater the number of the non-zero ZFS (crystal field) parameters; with the maximum 14 and 27 ZFS (crystal field) parameters in the case of 3dN and 4fN ions at triclinic sites, respectively. On the other hand, the maximum number of the non-zero ZFS

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(crystal field) parameters determined either experimentally from fittings or theoretically from model calculations does not automatically mean that the local site symmetry around an impurity ion is triclinic. Our crystal field considerations [52,53] illustrate that the apparent low symmetry ZFS parameter datasets or crystal field ones may indicate the choice of specific axis system that does not coincide with the symmetry-adapted axis system. In order to verify which case actually applies, one needs to carry out either detailed considerations of the crystal structure and the respective axis systems or analyze the observed degeneracy of the energy levels—complete or partial removal of degeneracy may indicate that the symmetry is actually as low as monoclinic or triclinic. Concerning the pseudosymmetry axis method originally proposed by Bacquet et al. [9], the basic idea is that the 4th-rank ZFS parameters reflect approximately the symmetry of the nearest-neighbor ligands around the paramagnetic ion. Determination of the ZFS parameters corresponding approximately to a selected higher symmetry case may be helpful in identification of the actual site symmetry as well as the nature of the local distortion responsible for the lowering of symmetry. The PAM consists in finding an axis system in which the ZFS parameters, admissible by group theory for the selected higher symmetry case and denoted by the components qsym, gain maximal values at the expense of the remaining lower symmetry ones, denoted by the components q6¼qsym, that become minimal. This can be achieved by numerical minimalization w.r.t. the Euler angles (a, b, g) of the quantity esym [9] expressed in terms of Bq4 ða; b; gÞ being functions of (a, b, g). The definition of the Euler angles adopted in the package CST [44,45] and the present package DPC follows the convention: (a/Oz, b/Oy0 , g/Oz00 ); see, e.g. Ref. [54]. Here, we provide the extended definition of the quantity esym generalized for arbitrary symmetry as: P sym ¼

qaqsym

 2 ðBq4 ða; b; gÞÞ=c4q Nk

,

(7)

Table 1 Components qsym of the 4th-rank ZFS parameters (or crystal field ones) relevant for the PAM for various point symmetry cases classified into 10 Laue groups—for cubic and hexagonal symmetry both type I and II cases are included Symmetry type [abbreviation]

q components

Monoclinic* C2Jz [MO] Orthorhombic* [OR] Tetragonal I* or cubic* [TEI] or [CU] Tetragonal II [TEII] Trigonal I* or cubic* [TGI] or [CU] Trigonal II [TGII] Hexagonal* [HE]

0, 0, 0, 0, 0, 0, 0

2, 2, 4 4, 3 3,

2, 4, 4 4 4 3

An asterisk (*) indicates the cases considered by us for the first time.

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where the summation runs over all q6¼qsym listed in Table 1 and Nk is defined in Eq. (6). Note that only the parameters associated with the normalized operators, e.g. the normalized Stevens (tesseral tensor) operators or any type of the spherical tensor operators, may be used in the expressions equivalent to Eq. (7). Since the extended Stevens operators are not normalized, the factors c4q arising from conversion of Bq4 in the extended Stevens notation to the normalized Stevens (NS) one [7,33] must be included in esym. In the original PAM formulation [9] expressed in terms of the ZFS parameters in the normalized Stevens notation Eq. (7) was defined only for tetragonal and trigonal type II symmetry. The authors [9] aimed at determination of the two types of pseudosymmetry axes only, i.e. four-fold (e4) and three-fold (e3) axes. Note that the pure cubic case is represented by a definitive ratio: B44 =B04 ¼ 5 for the p four-fold axes (tetragonal I case), whereas B34 =B04 ¼ ffiffiffi 20 2 for the three-fold axes (trigonal I case); the sign depends on the trigonal axes chosen [7]. For monoclinic symmetry, three equivalent forms of ZFS Hamiltonian exist [6,16]. We have included only the case with the monoclinic axis C2Jz, since the other cases C2Jy and C2Jx are related to the basic case C2Jz by appropriate transformations [16]. For practical applications of Eq. (7), the general transformation relations for the extended Stevens operators or the normalized Stevens operators are indispensable. We utilize the pertinent expressions for the extended Stevens operators derived by computer in Ref. [33] and the conversion factors between the extended Stevens and normalized Stevens parameters [7,33]. To carry out the numerical calculations we have developed the module PAM, which minimizes esym w.r.t. the Euler angles for a given symmetry case. The minimalization is carried out in steps p/8 in the range of the angle a and b from 0 to 2p. The angle g is searched for a given solution for a and b as the value closest to zero radians to ensure the final solution falls in the range p to +p. The major advantages of the module PAM, as compared with the original approach of Bacquet et al. [9], are (i) extension to other symmetry cases and (ii) automatic transformations of the ZFS parameters for each rank k ¼ 2, 4, and 6, expressed in the axis system defined by the Euler angles returned by this module as solutions of Eq. (7). For the latter purpose, the Euler angles (a, b, g) obtained from the module PAM are utilized as an input to obtain the corresponding 2nd- and 6th-rank ZFS parameters in the pseudosymmetry axis system of the 4th-rank ZFS parameters. Initially, for these computations we have used the transformation module (TRANS) within the package CST [44,45]. However, to avoid problems with the accuracy of (a, b, g) being inputted into the package CST and to facilitate computations, the necessary transformation expressions have been incorporated into the latest version of the package DPC [26]. Our tests show that the results of the two programs CST/TRANS and DPC/PAM turn out to be identical. Tests have also been carried out by transforming a selected ZFS parameter set

by arbitrary Euler angles using the package CST and then applying the module PAM to the transformed set. The sets returned by the module PAM turn out to be identical with the original test sets to within small rounding-off differences in the last 8th or 10th digits, thus confirming the correctness of both programs. The working of the module PAM and its output is illustrated below. For both Fe3+ and Gd3+ ions in m-ZrO2 the following approximations are considered MO, OR, TEI, TEII, TGI, and TGII (see Table 1). Note that the 6thrank ZFS parameters admissible in Eq. (2) for Gd3+ have not been determined in Ref. [9]. Concerning the 6th-rank ZFS parameters, the present version of our module PAM includes their transformations, whereas the function esym (6th-rank) can be easily built into an extended version, if needed. The transformation properties of Bq4 ða; b; gÞ require that for the case MO, TEII, and TGII the angle g must be equal 0, since the ZFS parameters (both in the extended Stevens and normalized Stevens notation) appear in pairs with +q and q. Hence, there is no need to consider these cases in detail, since the respective solutions for a and b can be read out from the corresponding ones for the cases OR, TEI, and TGI, respectively. For the latter cases, any transformation by the Euler angles (a, b, g) yields additionally the parameters with qo0, which must be minimized by an appropriate g-rotation. Thus we obtain g6¼0, which bears on the orientation of the x- and y-axes for the given orientation of the z-axis. This yields several equivalent (a, b, g) solutions. The number of minima obtained in the case TEI is smaller than that for the case OR. It turns out that each TEI minimum has its counterpart among solutions for the case OR, with each TEI minimum being close to an OR minimum. Our extensive computations for various ZFS parameter datasets and crystal field ones indicate that if symmetry is purely orthorhombic, than both approximations OR and TEI should yield exactly the same minima for the four-fold axes. Since this is our first detailed paper dealing with applications of the extended pseudosymmetry axes method, it is illustrative to provide pertinent computational details. For this purpose, using a selected ZFS parameter set, we present in Fig. 1 the three-dimensional graphs of the function esym (Eq. (7)) in the approximation TEI and TEII, whereas in Fig. 2 the two-dimensional graphs of the contours w.r.t. the angles a (x-axis) and b (y-axis) of esym in various approximations. Comparison of Figs. 1a and b indicates variation of the location of the absolute minima due to the non-zero g for the case TEI w.r.t. the case TEII with g ¼ 0. For simplicity, in Fig. 2 for the approximation OR, TEI, and TGI we have used the graphs corresponding to g ¼ 0, as is the case by default for the approximation MO, TEII, and TGII. The minima and maxima are represented by darker and lighter color in the black and white version, whereas by blue and red in the color version (online only). Comparison of Figs. 2a (MO), b (TEII), d (OR) and e (TEI) shows the compatibility of the counterpart minima in the respective cases. For the TE (TG) cases I and II, the

ARTICLE IN PRESS C. Rudowicz, P. Gnutek / Physica B 403 (2008) 2349–2366

π ¾π ½π α ¼π 0

0

½π

¼π

π

¾π

π ¾π ½π α ¼π

π

0

β

2355

¼π

½π

0

¾π β

π

π

¾π

¾π

¾π

½π

½π

½π

¼π

¼π

0 0

¼π

½π α

¾π

¼π

0

π

0

¼π

½π α

¾π

0

π

π

π

¾π

¾π

¾π

½π

½π

½π

¼π

β

π

β

β

β

π

β

β

Fig. 1. Sample three-dimensional graphs of the function esym w.r.t. the angles a and b obtained by the pseudosymmetry axis method in the approximation TEI (a) and TEII (b) using the ZFS parameters for Gd3+ in m-ZrO2 (set #Gd1 in Table 2); the values of esym on the z-axis change from 0 to 1 (color online).

¼π

0 0

¼π

½π α

¾π

π

0

¼π

½π α

¾π

π

0

¼π

½π α

¾π

π

¼π

0 0

¼π

½π α

¾π

π

0

Fig. 2. Two-dimensional contours of esym w.r.t. the angles a and b obtained by the pseudosymmetry axis method in various approximations defined in Table 1: (a) MO, (b) TEII, (c) TGII, (d) OR, (e) TEI, and (f) TGI, using the ZFS parameters (set #Gd1 in Table 2) for Gd3+ in m-ZrO2 (color online).

minima correspond to the pseudo-four-fold (three-fold) axes. At the initial stages of development of the module PAM, the graphs as in Figs. 1 and 2 have served mainly to test its correctness and to find the best form of the numerical output to be produced. The final version of the module for the selected approximation returns automatically an Excel table containing the Euler angles (a, b, g) for the minima of esym and the accordingly transformed Bqk ðESÞ. The graphs as in Figs. 1 and 2 can be produced on demand and may be helpful for comparison of the final

solutions for various approximations and analysis of their symmetry properties. Additionally, for the rank k=4 and 6, the module PAM computes also the values S k as in Eq. (6) but involving only the components qsym listed in Table 1: 31=2 !2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 k    X     1 1 2 2 2 q q 4 B0 þ 5 . Sk ¼ Bk þ Bk k 2k þ 1 ckq q sym

ARTICLE IN PRESS 2356

C. Rudowicz, P. Gnutek / Physica B 403 (2008) 2349–2366

Comparison of the quantities S k with the total rotational invariants defined in Eq. (6) provides a measure of the goodness of a given higher symmetry approximation. It turns out that for the ZFS parameter datasets under investigations S 4 are very close to S4 (see Section 4.3). Concerning the comparative analysis of low symmetry ZFS parameters, in order to assess the closeness or divergence of any two same-nature datasets, represented as ‘vectors’ A ¼ {Ai} and B ¼ {Bi} in the multi-parameter space, the normalized scalar product has originally been utilized [24,25]. For this purpose, the closeness factors C [19,24,25] and more recently [28] the ratios of the norms NA and NB: R ¼ NA/NB have been defined as: Pn Pn Ai Bi i¼1 Ai Bi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . (8) ¼ i¼1 Pn P N AN B n 2 2 A B i¼1 i i¼1 i Hence, CCk for the ZFS parameters of a given rank k ¼ 2, 4, and 6, whereas CCgl for the corresponding global factor for all k terms. The closeness factors C are in the range (0, 1) and pertain to the angle between A and B thus describing their closeness-if close to 1, or divergence-if different from 1 [19]. The ratios of the vectors’ norms R ¼ NA/NB pertain to the relative length of A and B thus describing their closeness or divergence—either in percents (100–0%) or rescaled to the range (1, 0) for NApNB and (1, N) for NAXNB. Note that the definition of ‘norms’ in Eq. (8) is mathematically more appropriate than in Ref. [28]. The quantities NA, NB, and R in Ref. [28] are equivalent to (NA)2, (NB)2, and (R)2 used here, respectively, so both R and (R)2 describe adequately the closeness. These factors enable quantitative comparison of datasets from different sources and also identification of compatible yet numerically distinct datasets of a given type. Their intricate properties for non-standard ZFS parameters and crystal field ones have been discussed in details in Ref. [28]. 4. Analysis of EMR data for Fe3+ and Gd3+ in monoclinic zirconia 4.1. Crystal structure and experimental EMR data The EMR experiments [9] were carried out on single crystals of monoclinic zirconia (m-ZrO2) doped with Fe3+ and Gd3+ ions. The crystal has the space group P21/c with a unit cell containing four ZrO2. The crystal structure parameters were determined [9] as: a ¼ 0.51454, b ¼ 0.52075, c ¼ 0.53107 (in nm), and b ¼ 991140 . In Fig. 3 we have plotted a 3D diagram of the structure of ZrO2 using the general position of P21/c: ð4eÞ  ðx; y; z; x; y þ 12; 12  zÞ with the Zr at x ¼ 0.2758, y ¼ 0.0411 and z ¼ 0.2082; O1 at x ¼ 0.070, y ¼ 0.342 and z ¼ 0.341; and O2 at x ¼ 0.442, y ¼ 0.755 and z ¼ 0.179 [9]. Our diagram matches well with the projections of the ZrO2 unit cell in Fig. 2 of Ref. [9]. The impurity M3+ ions, which substitute for Zr4+ are located at sevenfold coordinated sites (see Fig. 3b). The surrounding

Fig. 3. Crystal structure of m-ZrO2: (a) the unit cell and the crystallographic axis system defined in the space group P21/c and the laboratory axis system: xJa0 , yJb (out of the page), zJc and (b) the seven-fold coordinated complex around one of the four Zr4+ positions.

complex has no symmetry and hence, formally the site symmetry cannot be assigned as triclinic. However, the general triclinic form of SH (see below) should also apply in this case. The unit cell of ZrO2 contains four crystallographically equivalent Zr4+ ions, which substituted by M3+ ions yield four magnetically inequivalent impurity sites. Since the crystallographic axes (a, b, c) do not form a Cartesian system, an axis a0 was defined as perpendicular to the plane (bc) [9] and thus the a-axis is in the (a0 c) plane at about 991 from the c-axis. Bacquet et al. [9] used the general spin Hamiltonian form: HS ¼

3 X 3 X

mB H i gij Sj þ

n X X n

i¼1 j¼1

0

0

Bnm Onm ;

(9)

m¼n

0

where Onm are the normalized Stevens operators, n ¼ 2 and 4 for Fe3+(S ¼ 5/2), whereas n ¼ 2, 4, and 6 for Gd3+ (S ¼ 7/2). The SH parameters for Fe and Gd ions were determined for the two types of spectra 1 and 2, each consisting of 2S lines, which were recorded for any orientation of the crystal in the static magnetic field [9]. The authors [9] stated the following symmetry properties of the spectra and thus of the respective ZFS parameters, quote: 0

0

‘‘Bnm ðspectrum 1Þ ¼ ð1Þm Bnm ðspectrum 2Þ

ðfor any mÞ (10)

if the z-axis is chosen parallel to the binary b-axis or 0

0

Bnm ðspectrum 1Þ ¼ Bnm ðspectrum 2Þ

ðfor mX0Þ;

(11a)

and 0

0

Bnm ðspectrum 1Þ ¼ ð1Þm Bnm ðspectrum 2Þ ðfor mo0Þ; (11b) if the z-axis lies in a plane perpendicular to the b-axis.’’ Later in Table 3 of Ref. [9] the definition of the reference

ARTICLE IN PRESS C. Rudowicz, P. Gnutek / Physica B 403 (2008) 2349–2366

frame, i.e. the laboratory axis system, for the ZFS parameters is provided as: (OxJOa0 , OyJOb, OzJOc). Note that in Eqs. (10), (11a) and (11b), the ZFS parameters adopt implicitly different values, so the same symbol is used. The relations in Eqs. (10), (11a) and (11b) may be extended to account for other transformations specific for orthorhombic and lower symmetry. Such relations are derived below and then utilized for a meaningful interpretation of the transformed sets in later sections. In general, any axis system (x, y, z) may be chosen w.r.t. the set of fixed nominal reference axes (X, Y, Z) in four possible ways, i.e. (x, y, z), (x, y, z), (x, y, z), and (x, y, z). Each choice satisfies the condition (xJX, yJY, zJZ) and constitutes a right-handed axis system. For orthorhombic or higher symmetry the choice of one of the four orientations of the axes w.r.t. (X, Y, Z) is irrelevant, since the ZFS parameters (or crystal field ones) remain invariant under such transformations. However, for lower symmetry cases the orientation of the axes (x, y, z) is important. Three equivalent monoclinic symmetry cases exist [6,16], i.e. C2JZ, C2JY, or C2JX; here C2 denotes either the monoclinic axis or direction. Note that two equivalent choices of the axis systems apply for each monoclinic case, e.g., for C2JZ we have (x, y, z) and (x, y, z) for C2 along the +Z-axis, whereas (x, y, z) and (x, y, z) for C2 along the Z-axis. For triclinic case symmetry, since no symmetry exists, neither of the four axis systems (x, y, z), (x, y, z), (x, y, z), and (x, y, z) are mutually equivalent. Choice of another non-equivalent axis system for lower than orthorhombic symmetry results in a sign change of some non-axial parameters as follows. For a complete set of ZFS parameters (or crystal field ones) with kpqp+k, the transformation of an original axis system (x, y, z) by Euler angles (p, 0, 0), i.e. a rotation by p about the z-axis, yields (x, y, z) and the following relationships between the original (L.H.S) and transformed (R.H.S.) parameters are obtained: Bqk ¼ Bqk ;

for q even ð6; 4; 2; 0; 2; 4; 6Þ

(12a)

for q odd ð5; 3; 1; 1; 3; 5Þ .

(12b)

and Bqk ¼ Bqk ;

For the transformation by Euler angles (0, p, 0), i.e. p/Oy, in the axis system (x, y, z) we obtain: Bqk ¼ Bqk ;

for q positive ð0; 1; 2; 3; 4; 5; 6Þ

(13a)

and Bqk ¼ Bqk ;

for q negative ð6; 5; 4; 3; 2; 1Þ . (13b)

For the transformation by Euler angles (p, p, 0), i.e. p/Ox, in the axis system (x, y, z) we obtain: Bqk ¼ Bqk ;

for q ¼ 5; 3; 1; 0; 2; 4; 6

(14a)

2357

and Bqk ¼ Bqk ;

for q ¼ 6; 4; 2; 1; 3; 5 .

(14b)

Eqs. (12a)–(14b) indicate that, apart from the six basic orthorhombic standardization transformations and the respective axis systems Si defined in Refs. [15,16], there exist other choices of the equivalent axis systems. Hence, for triclinic (monoclinic) symmetry we obtain total of 24 (12) options for selection of an equivalent original axis system. Any two equivalent axis systems are related by a specific transformation as listed above. 4.2. Diagonalization of the 2nd-rank ZFS parameters The original ZFS parameters [9] are listed in Table 2 for Fe3+ spectrum 1 (2)-set 1 (2) and Gd3+ spectrum 1 (2)-set 3 (4). The values of the rotational invariants are very close for the two types of spectra both for Fe3+ and Gd3+ (see Table 2). In order to compare these ZFS parameter datasets, we calculate the respective closeness factors C and the norms ratios R defined in Eq. (8). The values of C and R are listed in Table 3. The results seem to indicate a divergence between the ZFS parameters for the spectrum 1 and 2 both for Fe3+ and Gd3+ (see column 1 in Table 3), since they are close in absolute magnitudes but differ in signs. However, as follows from Eqs. (14a) and (14b), excluding the 6th-rank ZFS parameters not determined in Ref. [9], the ZFS parameters with q ¼ 4, 2, 1, and 3 for the spectrum 2 change sign after the transformation (p, p, 0). After this transformation, the ZFS parameters for a given q for spectra 1 and 2, i.e. sets Fe1 & Fe2t and Gd1 & Gd2t in Table 2, acquire same signs. The one-to-one correspondence in the signs of the relevant sets after the transformation in Eqs. (14a) and (14b) illustrates the usefulness of these relationships for a meaningful comparison of the sets in Table 2. The original ZFS parameter sets 1 and 3 turn out to be very close to the respective transformed (t) ones as evidenced by the values of Ck and Rk (see, column 2 in Table 3). Since most of the experimental ZFS parameters available in the literature are expressed in the usual Stevens notation, see, e.g. the books [1–5], or for low symmetry cases in the extended Stevens notation [7,33], for presentation of our results in Sections 4.2 and 4.3 we have converted the original ZFS parameters [9] expressed in the normalized Stevens notation to the extended Stevens notation using the package CST [44,45]. This enables direct comparison with experimental data of others. However, it has turned out that the extended Stevens notation has some inherent drawbacks as discussed in Section 4.3. The module 3DD [27] enables diagonalization of the 2nd-rank ZFS terms and yields several sets of the Euler angles (a, b, g) and the respective principal values of the 2nd-rank ZFS terms. Using these angles (a, b, g), the 4thand 6th-rank ZFS parameters must be then transformed accordingly. Initially, for this purpose we have used the

ARTICLE IN PRESS C. Rudowicz, P. Gnutek / Physica B 403 (2008) 2349–2366

2358

Table 2 The original [9] and transformed ZFS parameters for Fe3+ ions (spectrum 1 and 2) and Gd3+ ions (spectrum 1 and 2) in m-ZrO2 in the normalized Stevens (NS) and extended Stevens (ES) notations (in 104 cm1) Fe3+(spectrum 1)

Fe3+(spectrum 2)

Gd3+(spectrum 1)

Gd3+(spectrum 2)

a b g

– – –

– – –

58.3 92.7 63.0

– – –

180 180 0

58.4 92.4 62.4

– – –

– – –

96.3 113.8 93.4

– – –

180 180 0

96.1 113.6 93.2

Set k, q 2, 2 2, 1 2, 0 2, 1 2, 2

1 NS 411.3 100.0 690.6 182.3 842.9

Fe1 ES 712.4 346.4 690.6 631.5 1459.9

Fe1(ST) ES 387.8 0 1161.8 0 0

2 NS 414.4 111.3 686.8 172.4 838.0

Fe2t ES 717.8 385.6 686.8 597.2 1451.5

Fe2t(ST) ES 391.3 0 1156.2 0 0

3 NS 93.8 8.7 80.8 129.1 23.4

Gd1 ES 162.5 30.1 80.8 447.2 40.5

Gd1(ST) ES 87.4 0 173.4 0 0

4 NS 93.9 8.6 81.6 128.7 22.7

Gd2t ES 162.6 29.8 81.6 445.8 39.3

Gd2t(ST) ES 88.2 0 173.3 0 0

l0 4, 4 4, 3 4, 2 4, 1 4, 0 4, 1 4, 2 4, 3 4, 4

– 0.0043 0.0443 0.0243 0.0235 0.0113 0.0132 0.0288 0.0028 0.0082

– 0.0256 0.7418 0.1088 0.1486 0.0113 0.0833 0.1289 0.0474 0.0483

0.334 0.1957 0.5880 0.0655 0.1176 0.0085 0.1770 0.0315 0.1390 0.1286

– 0.0027 0.0422 0.0207 0.0240 0.0137 0.0073 0.0187 0.0353 0.0080

– 0.0158 0.7056 0.0924 0.1518 0.0137 0.0464 0.0835 0.5912 0.0473

0.338 0.1129 0.5392 0.1566 0.1020 0.0012 0.2014 0.0667 0.1908 0.1231

– 0.2853 0.2280 0.0635 0.2685 0.3308 0.2773 0.0920 0.0530 0.0052

– 1.6881 3.8152 0.2840 1.6981 0.3308 1.7540 0.4114 0.8869 0.0306

0.504 0.7492 0.5678 2.0153 0.1412 0.0075 1.3406 0.7965 4.1047 1.3170

– 0.2848 0.2402 0.0813 0.2503 0.3268 0.2698 0.0888 0.0413 0.0195

– 1.6851 4.0188 0.3637 1.5832 0.3268 1.7066 0.3973 0.6916 0.1154

0.509 0.6886 0.5120 1.9718 0.2979 0.0115 1.4521 0.8138 3.8537 1.2908

S 2 S2 S4

359.5 529.1 0.0220

359.5 529.1 0.0220

529.1 529.1 0.0220

358.7 526.8 0.0228

358.7 526.8 0.0228

526.8 526.8 0.0228

55.4 80.8 0.2127

55.4 80.8 0.2127

80.8 80.8 0.2127

55.6 80.8 0.2102

55.6 80.8 0.2102

80.8 80.8 0.2102

The Euler angles for the principal axis system of the 2nd-rank ZFS parameters obtained from the module 3DD are in degrees; for explanation of the sets see text.

Table 3 Closeness factors Cp and norms ratios Rp for the pairs (X, Y) of selected experimental [9] and transformed ZFS parameter sets for Fe3+ and Gd3+ ions in m-ZrO2 Pair

C2 C4 Cgl R2 R4 Rgl

C2 C4 Cgl R2 R4 Rgl

Column 1 (Fe1, Fe2)

2 (Fe1, Fe2t)

3 (Fe1(ST), Fe2t(ST))

0.029627 0.481475 0.029627 0.9831 0.9831 0.8540

0.999911 0.865477 0.999911 0.9831 0.9831 0.8540

0.999997 0.869241 0.999997 0.9829 0.9829 0.8549

(Gd1, Gd2)

(Gd1, Gd2t)

(Gd1(ST), Gd2t(ST))

0.999980 0.998554 0.999980 0.9998 0.9998 0.9539

0.999997 0.998316 0.999997 0.9992 0.9992 0.9539

0.962868 0.351214 0.962861 0.9998 0.9998 0.9539

package CST [44,45]. However, as in the case of PAM calculations, there was a problem with the accuracy of the angles (a, b, g) used as input into the package CST. Hence, the latest version of the package DPC [26] incorporates the necessary transformation expressions [33] as in the CST

package and thus the calculations employ internal the high precision values of (a, b, g) returned by the module 3DD [27]. For the sake of space, for presentation of the results in the tables, we use appropriate rounding. The 3DD solutions include, in addition to the standard (ST) sets listed in Table 2, also several non-standard sets [15,16]. The latter sets could independently also be obtained from the standard sets using the orthorhombic standardization [15,16] module within the package CST. Since the nonstandard sets are not important for these considerations we refrain from listing them here; the respective tables may be obtained from the authors upon request. A word of caution is pertinent concerning the relative values of the ZFS parameters. Comparison of the set 1(NS) and the converted set Fe1(ES) in Table 2 reveals that apparently some low symmetry ZFS parameters appear quite large in the extended Stevens notation, whereas become relatively less important in the normalized Stevens notation. Similar observations apply also for other pertinent calculations. Implications of the relative ratios of the ZFS parameters expressed in the extended Stevens and normalized Stevens notation are more pronounced for considerations of the descent/ascent in symmetry approach within the pseudosymmetry axes method dealt with in Section 4.3. The standardized ratios l0 ¼ B22 =B02 limited to the range (0, 1), which describe the ‘rhombicity’ of the 2nd-rank (ES)

ARTICLE IN PRESS C. Rudowicz, P. Gnutek / Physica B 403 (2008) 2349–2366

ZFS parameters [15,16], are also listed in Table 2. The values of l0 in Table 2 indicate varying degree of the ‘rhombicity’ for the Fe3+ spectra as compared with the Gd3+ spectra. Notably, the standardized ZFS parameter sets reveal even greater closeness (see column 3 in Table 3). Thus the spectrum 1 and 2 for each ion, which yield the respective ZFS parameters, appear to be very closely related. It confirms that the two spectra for Fe3+ and Gd3+ ions originate from crystallographically the same, so magnetically inequivalent sites. This analysis also proves the advantage of comparing ZFS parameter datasets that are first properly transformed into the same nominal axis system [19] with regard both to the orientation of axes and their direction, i.e. sets expressed in the region of the multiparameter space corresponding to the standard range (0, 1) of the rhombicity ratio l0 . For any two properly transformed ZFS parameter datasets, one may judge directly from their values of l0 how close the two sets are one to the other. In our case we obtain for the spectrum 1 and 2: l0 ¼ 0.334 and 0.338 for Fe3+, whereas l0 ¼ 0.504 and 0.509 for Gd3+, respectively. The comparable values of l0 reveal the closeness of the two types of spectra for each ion. A large discrepancy between S2 for the original triclinic-like ZFS parameters and the total S2, observed for all cases considered, indicates significant deviations of the laboratory axis system, in which the original ZFS parameters are expressed, and the principal axis system of the 2nd-rank ZFS. Thus the strength of low symmetry aspects inherent in the 2nd-rank ZFS parameters appears quite large. The high closeness of the transformed sets, i.e. Fe1(ST) & Fe2t(ST) and Gd1(ST) & Gd2t(ST) in Table 3, derived independently from the two EMR spectra [9] indicate that indeed the spectra correspond to two magnetically inequivalent sites for each ion. It would be useful to discuss the non-coincidence of the principal axes of the 2nd-rank ZFS tensor and the g-tensor. Such non-coincidence is predicted from group theory and may serve as a signature of the triclinic site symmetry [5]. However, since for Fe3+ and Gd3+ ions in m-ZrO2 only the isotropic values of the g-tensor were determined [9], consideration of this non-coincidence must await till more accurate EMR data become available. The principal axes of the 2nd-rank ZFS tensor for Fe3+ and Gd3+ sites turn out to be much different, thus indicating that either the two ions enter distinct sites or much larger distortions occur due to the larger size of Gd3+ ions. The Euler angles obtained from the module 3DD for the sets (ST) listed in Table 2 represent the orientation of the principal axis system of the 2nd-rank standardized ZFS parameters w.r.t. the laboratory axis system, which is related to the crystallographic axis system [9]. It would be worthwhile to compare this orientation and the orientation of the bonds in the crystal structure of m-ZrO2 to see if any correlations exist. For a better visualization of the orientation of the axis systems involved, in Fig. 4 we present the axis systems arising from the 3DD considerations in

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Fig. 4. Projections onto the (a0 c) plane of: (a) the ions and bonds in the Zr4+–O7 complex in m-ZrO2 and (b) the axes (x0 , y0 , z0 ) obtained from the module 3DD based on the standard extended Stevens (ES) sets in Table 2 for Fe3+ (red) and Gd3+ (blue) ions in m-ZrO2; circles denote points for Fe1 & Gd1 and crosses those for Fe2t & Gd2t; the values of the ycoordinate of each axis (x0 , y0 , z0 ) indicating their location above (positive) or below (negative) the (a0 c) plane are: (0.450, 0.275, 0.850), (0.448, 0.274, 0.851), (0.085, 0.406, 0.910), (0.084, 0.404, 0.911) for Fe1, Fe2t, Gd1, and Gd2t, respectively (color online).

this subsection, together with those arising from an analysis of the nearest ligands coordinates. Note that Fig. 4b depicts all sets considered in Table 2. Having these data indicated in the same figure may better visualize the local symmetry of the sites under investigation. Comparison of Figs. 4a and b indicates that no obvious correlation exists between and the respective principal axis systems of the 2nd-rank ZFS terms for the Fe and Gd ions and the orientation of the bonds in the crystal structure of m-ZrO2. This may evidence the significant low symmetry distortions experienced by the Fe and Gd ions doped m-ZrO2. In fact, as discussed in Section 4.1 this complex has no symmetry [9]. The bonds between the central ion Zr4+ and the oxygen ligands in ZrO2 (in nm) were calculated by us using the data [9] as: O11—0.20764, O12—0.21538, O13—0.20399, O21—0.18525, O22—0.29005, O23—0.29464, and O24— 0.17370, whereas the respective bond angles (in degrees) as: O11–Zr–O24—164.3, O12–Zr–O23—162.0, O13–Zr–O21— 165.9; the subscripts denote the ions as indicated in Fig. 3b. The M–O7 complex in Fig. 4a may possibly be idealized as a distorted eight-fold coordinated system with one extra ion in the vicinity playing the role of the eighth ion or alternatively a missing ion playing the role of a virtual vacancy. Such idealization is considered in Section 4.3 taking into account also the PAM results. Note that the major difficulty in applying this model arises from the fact that the studied samples [9] were obtained from mixtures of ZrO2 and M2O3, were M3+QFe3+ or Gd3+. The actual crystallographic parameters for this system may be somewhat different from those for pure m-ZrO2. Defect–defect interactions in nitrogen-doped zirconia have been recently studied by Lee et al. [55]. Further studies along the lines proposed in Ref. [55] may be carried out in future.

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4.3. Pseudosymmetry axis method for the 4th-rank ZFS parameters Applications of the module PAM within a given approximation (defined in Section 3) for the original 4thrank ZFS parameters yield several solutions for the Euler angles (a, b, g) and the corresponding 4th-rank ZFS parameters expressed in the given pseudosymmetry axis system. The original 2nd- and 6th-rank ZFS parameters must be transformed accordingly. As discussed in Section 3 the latest version of the package DPC, including the module PAM, incorporates the necessary transformation expressions as in the CST package [44,45]. For the sake of space, only the basic PAM results, i.e. the quantities (appropriately rounded): (a, b, g), esym, B44 =B04 , and Sk obtained for the sets Fe1, Fe2t, Gd1, and Gd2t from Table 2 are presented in Tables 4–6 for the orthorhombic, tetragonal (TEI), and trigonal (TGI) approximation defined in Table 1, respectively. Some selected ZFS parameter sets, including k ¼ 2 and 4 and all q values (kpqp+k), obtained from the module PAM are listed

in Table 7, whereas full tables including all ZFS parameter sets considered may be obtained from the authors upon request. Note that all PAM solutions obtained for a given original ZFS parameter set are physically equivalent due to the transformation properties of the ZFS parameters. The PAM results in Tables 4–6 enable the following general observations. Comparison of the ZFS parameters for the spectrum 1 and 2 (described in Section 4.1) indicates much larger differences between the two spectra in the case of Fe ions than Gd ions. The PAM results visualize these differences in a more pronounced way, especially concerning the values of (a, b, g), which turn out to be significantly different for Fe ions, whereas very close for Gd ions (see Tables 4–6). This pattern is revealed for all solutions within each approximation considered. The possible reasons for the differences between the spectrum 1 and 2 in the case of Fe ions may lie in the assignment of resonance lines in Ref. [9]. The experimental accuracy of the ZFS parameters for Fe ions is twice less than that for Gd ions. The 2nd-rank ZFS parameters for Fe ions are much larger than for Gd ions, whereas the opposite

Table 4 The PAM solutions for the Euler angles (a, b, g) (in degrees) and the cubic ratio of the 4th-rank ZFS parameters in orthorhombic (OR) approximation for Fe3+ (upper sets 1–8) and Gd3+ (lower sets 1–9) ions in m-ZrO2 Set 1

2

3

4

5

6

7

8

48.2 67.8 11.7 0.058 11.56 0.0213

76.9 155.1 26.4 0.058 2.47 0.0213

142.6 79.2 22.6 0.058 2.97 0.0213

123.9 120.4 26.6 0.060 3.38 0.0213

4.6 52.2 10.1 0.105 33.18 0.0208

168.3 39.0 12.8 0.105 2.97 0.0208

22.9 108.9 33.4 0.145 42.53 0.0203

86.6 37.8 58.1 0.145 27.40 0.0203

39.2 62.8 12.2 0.057 6.76 0.0222 Set

64.6 150.3 22.4 0.057 7.49 0.0222

134.9 79.1 27.7 0.057 7.52 0.0222

113.0 119.0 31.3 0.047 5.18 0.0223

168.0 131.2 2.2 0.012 24.34 0.0227

164.6 41.3 2.5 0.012 3.52 0.0227

16.2 102.5 30.3 0.070 15.61 0.0220

85.9 32.6 23.7 0.070 12.14 0.0220

Fe1 a b g esym B44 =B04 S4 Fe2t a b g esym B44 =B04 S4

1

2

3

4

5

6

7

8

9

Gd1 a b g esym B44 =B04 S4

9.8 148.1 18.4 0.034 8.40 0.2094

168.4 120.1 11.1 0.034 7.51 0.2094

72.1 38.0 21.5 0.071 11.73 0.2053

98.7 124.9 16.0 0.071 25.60 0.2053

85.6 80.4 13.6 0.083 6.60 0.2040

177.9 76.6 9.9 0.083 5.03 0.2040

42.2 92.7 16.2 0.088 38.68 0.2034

131.5 73.8 2.8 0.088 50.50 0.2034

141.4 163.6 9.6 0.088 3.72 0.2034

Gd2 a b g esym B44 =B04 S4

10.2 148.3 18.0 0.035 11.54 0.2065

169.2 120.0 10.8 0.035 8.25 0.2065

72.9 38.0 21.5 0.074 11.12 0.2022

99.5 124.9 16.0 0.074 22.78 0.2022

86.3 80.6 13.6 0.097 6.17 0.1998

178.6 76.6 9.7 0.097 5.17 0.1998

42.9 92.8 16.1 0.099 33.44 0.1995

132.1 73.9 2.9 0.099 37.05 0.1995

142.5 163.6 10.0 0.099 3.89 0.1995

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relation is obtained for the 4th-rank ZFS parameters. Hence, the smaller 4th-rank ZFS parameters for Fe ions are determined with relatively much larger experimental uncertainty than those for Gd ions. Our extensive computations for various crystal field parameter datasets and ZFS ones indicate that within the orthorhombic approximation, in general, one may expect for a given set a maximum of twelve solutions for (a, b, g), so in some cases lesser number of solutions may be obtained due to computational conditions. Some solutions

Table 5 The PAM solutions for the Euler angles (a, b, g) (in degrees) and the cubic ratio of the 4th-rank ZFS parameters in tetragonal (TEI) approximation for Fe3+ and Gd3+ ions in m-ZrO2 Set 1

2

3

4

5

Fe1 a b g esym B44 =B04 S4

48.1 67.2 11.7 0.060 11.72 0.0213

125.5 121.2 20.7 0.171 3.47 0.0200

Fe2 167.5 38.9 10.8 0.152 3.04 0.0202

39.2 62.9 12.2 0.057 6.75 0.0222

113.5 119.2 14.2 0.077 5.21 0.0219

Gd1 a b g esym B44 =B04 S4

85.5 80.5 13.9 0.089 6.60 0.2033

141.0 163.6 9.2 0.089 3.71 0.2034

6

164.4 41.6 2.3 0.052 3.52 0.0222

Gd2 178.0 76.4 9.4 0.106 5.01 0.2014

86.3 80.7 13.9 0.102 6.16 0.1992

142.2 163.6 9.8 0.099 3.88 0.1995

178.7 76.3 9.3 0.112 5.15 0.1981

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may fall at the borders of the angles range, i.e. 01 or equivalently 1801, thus representing apparent double solutions. Certain solutions correspond to the similar or identical values of esym(OR). In the present case we obtain eight solutions for Fe ions and nine for Gd ions. Some of the OR solutions in Table 4 shall coincide with some of the TEI ones in Table 5. This is the case for solutions OR 1–3 and TEI 1–3, respectively. For the tetragonal and trigonal approximation, in general, three and four solutions may be expected. The orthorhombic approximation visualizes both the four-fold and two-fold pseudosymmetry axes determined by PAM w.r.t. the laboratory axis system, whereas the tetragonal (trigonal) approximation visualizes the three (four) equivalent four-fold (three-fold) pseudosymmetry axes. In Fig. 5 we present illustrative cases of the axis systems arising from the PAM considerations in this subsection. Diagrams as in Fig. 5 were worked out for all sets considered in Tables 4–6, however, they are not presented here. Analysis of the values of (a, b, g) and other quantities listed in Tables 4–6, and the intercorrelations between the values for various sets for a given ion as well as analysis of the corresponding diagrams as in Fig. 5 enable us to find out which solution represents the axes of a given type, namely, either the four-, three-, or two-fold pseudosymmetry axes. This analysis shows that the Fe sets in Table 4: 1, 4, and 6 represent the four-fold pseudosymmetry axes, whereas 3, 5, 7, and 8 represent the two-fold pseudosymmetry axes. For the Fe and Gd all sets in Table 5 (6) represent the four-fold (three-fold) pseudosymmetry axes. The analysis for the Gd sets yields similar allocation of the pseudosymmetry axes. Qualitative comparison of our PAM results with the corresponding ones of Ref. [9] has been carried out using the stereographic projections of only

Table 6 The PAM solutions for the Euler angles (a, b, g) (in degrees) and the cubic ratio of the 4th-rank ZFS parameters in trigonal (TGI) approximation for Fe3+ and Gd3+ ions in m-ZrO2 Set 1

2

3

4

5

6

Fe1 a b g esym B34 =B04 S4

18.4 17.9 23.8 0.169 44.58 0.0200

60.6 125.6 12.9 0.130 15.48 0.0205

32.8 60.4 18.3 0.041 62.47 0.2087

53.7 126.3 20.3 0.058 17.45 0.2068

8

101.1 65.4 13.4 0.072 23.63 0.0220

166.3 95.9 1.7 0.074 32.79 0.0220

128.3 38.6 5.6 0.109 24.91 0.1984

133.7 108.9 3.3 0.111 30.99 0.1982

Fe2 107.8 69.6 19.2 0.194 29.26 0.0197

176.8 92.7 7.4 0.189 35.43 0.0198

172.6 166.7 6.8 0.048 44.96 0.0223

50.5 118.0 11.9 0.052 19.64 0.0222

Gd1 a b g esym B34 =B04 S4

7

Gd2 127.9 38.5 5.3 0.104 25.34 0.2016

133.2 108.9 3.3 0.106 29.44 0.2014

33.5 60.7 18.2 0.040 66.21 0.2060

54.5 126.2 20.3 0.052 16.60 0.2046

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Table 7 The full listing of the ZFS parameters (k, q), including k ¼ 2, 4 and –kpq p+k, for selected sets from Tables 4 and 5 used for presentation of the axes in Fig. 5

a b g

OR1(ES)

OR2(ES)

TE1(ES)

TE2(ES)

48.189 67.825 11.670

39.195 62.788 12.246

48.108 67.241 11.705

39.192 62.900 12.228

TE2(NS) 39.192 62.900 12.228

OR3(ES)

OR4(ES)

85.559 80.363 13.600

86.289 80.563 13.616

TE3(ES) 85.545 80.466 13.863

TE4(ES) 86.282 80.668 13.910

TE4(NS) 86.282 80.668 13.910

Set

Fe1

Fe2t

Fe1

Fe2t

Fe2t

Gd1

Gd2t

Gd1

Gd2t

Gd2t

2, 2 2, 1 2, 0 2, 1 2, 2

231.5 2786.3 811.9 194.9 467.0

294.7 3004.5 617.1 828.6 711.0

246.1 2831.6 797.5 181.0 466.8

291.2 2997.6 619.8 834.3 711.1

168.1 865.3 619.8 240.8 410.6

17.9 561.8 75.4 14.9 38.9

15.0 559.0 77.7 23.5 35.9

17.74 561.02 75.85 17.04 38.97

14.9 558.1 78.2 26.0 35.9

8.6 161.1 78.2 7.5 20.7

4, 4 4, 3 4, 2 4, 1 4, 0 4, 1 4, 2 4, 3 4, 4

0.3361 0.2091 0.0146 0.0594 0.0291 0.0100 0.0085 0.0260 0.0003

0.2960 0.2024 0.0057 0.0412 0.0438 0.0155 0.0310 0.0844 0.0005

0.3371 0.1962 0.0126 0.0657 0.0288 0.0093 0.0085 0.0278 0.0000

0.2958 0.2049 0.0053 0.0395 0.0438 0.0161 0.0310 0.0833 0.0000

0.0500 0.0122 0.0012 0.0062 0.0438 0.0025 0.0069 0.0050 0.0000

Fig. 5. Projections onto the (a0 c) plane of the axes (x0 , y0 , z0 ) obtained from the module PAM based on the ZFS parameters for Fe3+ (red) and Gd3+ (blue) ions in m-ZrO2: (a) OR sets 1(Fe) & 5(Gd) in Table 4 and (b) TEI sets 1 & 4 in Table 5; circles denote points for Fe1 & Gd1 and crosses those for Fe2t & Gd2t; the values of the y-coordinate of each of the axes (x0 , y0 , z0 ) indicating their location above (positive) or below (negative) the (a0 c) plane: (a) (0.410, 0.596, 0.690), (0.447, 0.696, 0.562), (0.180, 0.036, 0.983), (0.174, 0.024, 0.948), (b) (0.417, 0.595, 0.686), (0.445, 0.696, 0.563), (0.179, 0.036, 0.983), (0.177, 0.041, 0.983) for Fe1, Fe2t, Gd1, and Gd2t, respectively (color online).

the z-axes for Fe3+ and Gd3+ ions in m-ZrO2 presented in Figs. 5 and 6 of Ref. [9], respectively. It turns out that these projections match well with the results of our TEII approximation for the Euler angles a and b, which can be obtained directly from Table 5 for TEI approximation substituting g ¼ 0. The seven-fold coordinated metal–ligands complex in mZrO2 (Fig. 3) has actually no symmetry, however, a question arises which is the best closest structural approximation. Since depending on the actual geometry of the complex around the central ion various cases may

2.6876 1.6637 0.2267 0.3325 0.4069 0.0310 0.6513 0.1766 0.0481

2.5508 1.7191 0.2114 0.3222 0.4135 0.0140 0.7112 0.0859 0.0516

2.6867 1.6788 0.2202 0.3165 0.4071 0.0379 0.6540 0.2008 0.0000

2.5498 1.7352 0.2034 0.3055 0.4138 0.0204 0.7139 0.1160 0.0000

0.4310 0.1037 0.0455 0.0483 0.4138 0.0032 0.1596 0.0069 0.0000

occur, below we consider if such complex may possibly be idealized as either a distorted eight-fold coordinated system with one extra ion in the vicinity playing the role of the eighth ion, or a distorted six-fold coordinated system with one oxygen ion virtually missing, i.e. playing no significant role. In general, for cubic site symmetry some actual symmetry axes are parallel to the central ion—ligands bonds as follows: for an octahedron with the six-fold coordination these are the four-fold symmetry axes, whereas for a regular cube with the eight-fold coordination and a tetrahedron with the four-fold coordination these are the three-fold symmetry axes. Considerable differences may be noted between the effect of Gd3+ and Fe3+ ions in this regard as discussed below. In the present case, the tetragonal approximation (see Table 5) yields one solution corresponding to one of the three possible four-fold pseudosymmetry axes. The results in Table 5 indicate that the existing low symmetry of the significantly distorted metal sites in the pure host appears to be modified by Gd ions towards a higher symmetry. The values of the cubic ratio B44 =B04 ¼ 6:599 [6.162], 3.708 [3.881], and 5.014 [5.151] for Gd1 [Gd2] are close to the cubic ones, i.e. 5 or 5. Similar findings are obtained using the trigonal approximation (see Table 6), which yields one solution corresponding to one of the four possible three-fold pseudosymmetry axes. The results in Table 6 show that for most of the sets the ratio B34 =B04 is effectively close to the cubic value, i.e. 28 or 28. Only the sets 1 (Fe1 and Gd1) and the corresponding sets 5 (Fe2 and Gd2) in Table 6 indicate a larger distortion, i.e. a markedly higher ratio B34 =B04 . Comparison of the TEI pseudo-fourfold symmetry axes for Gd ions in Fig. 5 and the Zr–O7 complex in Fig. 4a shows that these three axes are

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approximately parallel to the axes bisecting the central ion–ligands bonds. In general, the negative values of B04 and B44 indicate [56] either the eight-fold or four-fold coordination. Hence, the Gd complex in m-ZrO2 may be considered approximately as a cubic eight-fold coordinated complex with large distortion. From the point of view of the descent of symmetry this distortion may be represented by lowering of symmetry along one of the trigonal axes with smaller triclinic components. This idealized model agrees with the structural data. The eighth oxygen ion at a distance 0.358 nm does exist in m-ZrO2, however, it was not considered in the coordination polyhedron [9]. It appears that due to the larger size of Gd3+ than both Fe3+ and Zr4+ and thus more pronounced and more symmetric distortion of the local surrounding upon substitution for Zr ions, the effect of the eighth more distant oxygen becomes significant. Hence, the overall conclusion is that the Gd ions are located at sites exhibiting substantial degree of cubicity corresponding to a cubic eight-fold coordinated complex. Similar analysis for Fe ions shows no clear consistency neither in the values of the cubic ratios B44 =B04 (TEI) and B34 =B04 (TGI) nor the orientation of the pseudosymmetry axes. In this case disparate values of the ratios are obtained, whereas the pseudosymmetry axes appear not to be related to any specific higher symmetry complex. It appears that due to the smaller size of Fe3+ ions, when entering the Zr sites they fit more easily into the existing very low symmetry surroundings. From the point of view of the ascent of symmetry, this indicates that an approximation to a higher symmetry does not work well for Fe ions. Having now considered the structural approximations for the metal–ligands complex in m-ZrO2, a remaining question is which approximation represents best the ascent in symmetry. Since the total values of S4 in Table 2, which are identical for all transformed sets, are very close to the values S k in Tables 4–6, the goodness of the all approximations is very high. Due to the small differences between these values for the three approximations, one cannot firmly differentiate which approximation is the best. Comparison of Figs. 4a, 5a and b reveals that overall the orthorhombic symmetry approximation may be considered as best reflecting the local surrounding of Fe3+ and Gd3+ ions in m-ZrO2. Comparison of Figs. 4 and 5 indicates that no obvious correlation exists for the Fe and Gd ions between and their respective principal axis systems of the 2nd-rank ZFS terms (3DD) and those representing the pseudosymmetry axes (PAM). As mentioned in Section 4.2, the differing relative ratios of the ZFS parameters, expressed in the extended Stevens notation and the normalized Stevens notation, bear on meaningful interpretation of the results. The descent/ascent in symmetry approach illustrated in Table 7 indicates several dominant ZFS parameters in the extended Stevens notation. However, it turns out that such dominance is only apparent, since some low symmetry ZFS parameters,

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which are quite large in the extended Stevens notation, become relatively less important in the normalized Stevens notation. For illustration in Table 7 we provide also the ZFS parameters in the normalized Stevens notation. The extended Stevens parameters (k ¼ 4, q ¼ 3) and (k ¼ 4, q ¼ 3) in the column TE2 significantly reduce their value when converted to the normalized Stevens notation (column TE2(NS)). Similar finding applies to the parameters (k ¼ 4, q ¼ 3) and (k ¼ 4, q ¼ 2) in the column TE4 and TE4(NS). Overall these relations may incorrectly suggest more pronounced low symmetry aspects for the ZFS parameters expressed in the extended Stevens notation. The considerably different relative ZFS parameter values are due to the lack of consistent normalization inherent in the extended Stevens notation, unlike for the normalized Stevens notation [7,33]. Hence, it turns out that the normalized Stevens notation provides more accurate representation of the relative values of the ZFS parameters (or crystal field ones [26]), whereas the extended Stevens notation yields apparently misleading ratios of the parameters due to inconsistent normalization factors used for various q-components [7,33]. The above considerations of the relative ratios of the ZFS parameters expressed in the two notations reveal potentially important problem with the extended Stevens operators. It turns out that these are the normalized Stevens operators and not the extended Stevens operators that enable directly more meaningful interpretation of the ZFS parameters. As an aside it would seem that this would justify a re-evaluation of the operators that should be adopted as a standard with the EMR community, contrary to the earlier proposal favoring the extended Stevens operators [57,58]. It seems it would be sensible to move to the normalized Stevens operators or the two-vector tesseral operators [59]. If a change of notation is required a freely available conversion utility, as e.g. the package CST [44,45] (or a future improved version combined with the package DPC) as well as definite reviews, as e.g. Refs. [7,8], would seem to be indispensable. Finally, we discuss potential applications of the 3DD and PAM results for the multiple correlated fitting technique [16]. There are additional advantages of the comprehensive approach proposed in Section 3 and implemented above for the EMR data for Fe3+ and Gd3+ ions in zirconia. The first method (3DD) and the second one (PAM) generate a number of numerically distinct yet physically equivalent ZFS parameter datasets. The equivalence between such correlated ZFS parameter datasets may be utilized in the multiple correlated fitting technique [16] to improve the reliability of the final results [6,18,19]. Following the procedure of the multiple correlated fitting technique [16,19], the equivalent ZFS parameter sets may be used as starting values for additional fittings. Thus, this procedure would provide several independently fitted and inherently correlated ZFS parameter sets. Hence, this technique offers ways to increase accuracy and reliability of the final fitted ZFS parameter datasets, identify global minima, and

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eliminate spurious sets (computational artifacts). However, finding the intercorrelations between the sets fitted in different regions of the multiparameter space requires knowledge of the transformation properties of ZFS parameters [33,43]. Here, the package CST [44,45], which incorporates the general transformations, comes as a handy computational tool. Practical applications of the multiple correlated fitting technique to ZFS parameter datasets (crystal field ones) require access to raw experimental EMR (optical spectroscopy) data. Several applications will be deal with elsewhere. 5. Summary and conclusions A comprehensive approach to a more detailed analysis of the monoclinic- and/or triclinic-like experimental spin Hamiltonian (SH) parameter datasets, most importantly the zero-field splitting (ZFS) parameters, obtained in electron magnetic resonance (EMR) studies is proposed. This approach comprises three methods aimed at extracting useful structural information from ZFS parameter datasets for transition ions at low symmetry sites. First method consists in diagonalization of any 2nd-rank SH term in an analogous way as in the case of the conventional ZFS term S.D.S. The calculations are carried out using the computer module 3DD developed by us. This method enables finding the principal values of the various 2nd-rank SH terms, including the ZFS terms, and the orientation of the principal axis systems w.r.t. the laboratory or crystallographic axis system. Expressing the 2nd-rank ZFS parameters in their principal axis system enables consideration of the inherent low symmetry aspects and correlation of the principal axis system with the symmetry-adapted axis system, which, for low symmetry cases, may be approximately related to the central ion–ligands bonds. Subsequently, the Euler angles obtained from the module 3DD are utilized as input to obtain the 4th- and 6th-rank ZFS terms in the principal axis system of the 2ndrank ZFS terms, using either to the transformation module of the package CST [44,45] or internally within the package DPC [26]. This enables consistent presentation of the full ZFS parameter dataset in one axis system. Second method consists in extending the cubic and axial pseudosymmetry axes method (PAM) to lower symmetry cases and finding the pseudosymmetry axis system for the 4th-rank ZFS parameters. The calculations are carried out using the computer module PAM developed by us. Since the 4thrank ZFS parameters reflect the symmetry of the nearestneighbor ligands around the paramagnetic ion in a given higher symmetry approximation, the method PAM enables determination of the ZFS parameters corresponding approximately to a selected higher symmetry case. Hence, it may be helpful in identification of the actual site symmetry as well as the nature of the local distortion responsible for lowering of symmetry. Third method consists in employing the closeness factors C and the norms ratios R ¼ NA/NB for quantitative comparison of

several ZFS parameter datasets. The calculations are carried out using the computer module CFNR developed by us. The quantities C and R quantify the closeness of any two datasets of the same type, e.g. the SH parameters or the energy levels, as well as enable meaningful comparison of datasets taken from different sources and identification of compatible yet numerically distinct datasets. The three modules based on each method have been incorporated into a combined computer package DPC to facilitate practical applications. Work is now in progress to incorporate the package CST (FORTRAN) and DPC (MathCAD) into a comprehensive user-friendly package based on an object-oriented programming. The present versions of both packages are available from the authors on collaborative basis. As an illustrative example of application of our approach, the EMR results [9] for Fe3+ and Gd3+ ions at low symmetry sites in monoclinic zirconia (m-ZrO2) are reanalyzed. Using the module 3DD, the principal values of the 2nd-rank ZFS parameters and the orientation of the principal axes of the 2nd-rank ZFS parameters as well as the appropriately transformed 4th-rank (and 6th-rank, if available) ZFS parameters are obtained. Similarly, using the module PAM within a given higher symmetry, the principal values of the 4th-rank ZFS parameters and the orientation of the principal axes of the 4th-rank ZFS parameters as well as the appropriately transformed 2nd-rank (and 6th-rank, if available) ZFS parameters are obtained. The module CFNR enables quantitative comparison of the original ZFS parameter datasets with those obtained using the module 3DD and PAM. Various principal axis systems generated by our calculations are visualized graphically and compared with the local environment of the Fe3+ and Gd3+ ions in mZrO2. This enables us to extract structural information not hitherto available. It appears that the seven-fold coordinated metal–ligands complex in m-ZrO2 may be approximated as a cubic eight-fold coordinated complex with large distortion for Gd3+ ions. The results for the Fe3+ ions indicate, however, that from the point of view of the ascent of symmetry, an approximation to a higher symmetry does not work well for Fe ions. The two ZFS parameter datasets associated with the two spectra observed for Fe3+ and Gd3+ ions at magnetically inequivalent sites are also reanalyzed regarding: (i) their symmetry properties-using the relations derived for specific transformations and (ii) their closeness-employing the closeness factors and the norms ratios. Our considerations enable better understanding of the low symmetry aspects as well as correlation of the principal axis systems with the symmetry-adapted axis systems. The relationships between the principal axis system of the 2nd-rank ZFS terms and the respective bonds between the transition ion and the coordinating ligands are also studied. To the best of our knowledge, this study is first of this kind in the EMR literature. The present study provides a deeper and more comprehensive analysis of the low symmetry aspects inherent in the EMR data for Fe3+ and Gd3+ ions in m-ZrO2.

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Additionally, calculations using the module 3DD and/or PAM within the package DPC reveal the existence of a number of ZFS parameter datasets belonging to the standard as well as non-standard ranges. All such correlated yet numerically distinct datasets correspond to different regions in the multivariable ZFS parameter space. It turns out that such datasets are equivalent, since by proper rotations of the axis systems they can be transformed into each other, e.g. using the package CST. Consideration of the equivalence between the ZFS parameter datasets transformed to the standard range and other physically equivalent ZFS parameter datasets provides a better insight into the intricate properties of the ZFS parameters for low symmetry cases. Such aspects bear on the feasibility of obtaining various ZFS parameter datasets from fittings of the experimental EMR spectra. It is suggested that the equivalence between various correlated ZFS parameter datasets is utilized in the multiple correlated fitting technique to improve the reliability of the final results. The final standardized ZFS parameters obtained in this way may be used as the initial parameters for fitting the experimental EMR spectra for transition ions in structurally similar hosts. In summary, the comprehensive approach proposed here comprising three methods (3DD, PAM, and closeness factors) together with the multiple correlated fitting technique proves to be useful in EMR studies of low symmetry systems. Importantly, the three methods and thus the combined package DPC may be also applied to the crystal field parameters studied by optical spectroscopy. This approach enables to extract useful structural information from SH parameter datasets for transition ions at low symmetry sites. Several other applications of our approach to triclinic ZFS parameter sets available in literature may be envisaged. Thus, the methods proposed here may help experimentalists to better interpret and analyze EMR and related data appearing in the literature. Another advantage of the package DPC is that it enables consistent presentation of the full ZFS parameter dataset for any rank (k ¼ 2, 4, 6) in the same axis system, e.g. the principal axis system of the 2nd-rank ZFS terms (obtained using the module 3DD) or the approximated higher symmetry 4th-rank ZFS parameters (obtained using the module PAM). Acknowledgments This work was partially supported by the research grant from the Polish Ministry of Science and Tertiary Education in the years 2006–2009. P. Gnutek acknowledges gratefully the opportunity of Ph.D. program (under the CZR supervision) at the Institute of Physics, SUT. References [1] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Dover, New York, 1986.

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