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Abstract

In Part I (Chapter 3 in this volume - Wilder et al., 2004) we described the basic principles of crystal field theory (CFT) based on group theory and symmetry. The usefulness of CFT resides in the fact that it can predict the type and number of electronic transitions and their relative energies for transition metal ions in crystals. Hence CFT enables interpretation of the optical absorption spectra. The crystal (or ligand) field induced on the central ion depends on the type and positions of the ligands (i.e., bond angles and distances R) and on the point symmetry of the resulting coordination polyhedron. The number of exited crystal field (CF) states and the type of the ground state arising from a given free-ion dN configuration depends solely upon molecular symmetry, i.e. the site symmetry in case of crystals, and is independent of any model used to describe the metal-ligand bonds. Although the exact energies cannot be calculated ab initio, it is possible to extract empirical parameters from experimental electronic absorption spectra which describe the interaction between metal and ligand. For a given dNX6 complex with an ideal octahedral coordination (symmetry Oh), the cubic CF splitting parameter 10Dq, together with Racah parameters B and C, provide basis for a reasonably complete description of the electronic spectra (Lever, 1984). In most crystals the site symmetry is, however, lower than Oh. This requires introduction of additional, so-called distortion parameters to describe the lower symmetry CF components

Introduction

In Part I (Chapter 3 in this volume - Wilder et al., 2004) we described the basic principles of crystal field theory (CFT) based on group theory and symmetry. The usefulness of CFT resides in the fact that it can predict the type and number of electronic transitions and their relative energies for transition metal ions in crystals. Hence CFT enables interpretation of the optical absorption spectra. The crystal (or ligand) field induced on the central ion depends on the type and positions of the ligands (i.e., bond angles and distances R) and on the point symmetry of the resulting coordination polyhedron. The number of exited crystal field (CF) states and the type of the ground state arising from a given free-ion dN configuration depends solely upon molecular symmetry, i.e. the site symmetry in case of crystals, and is independent of any model used to describe the metal-ligand bonds. Although the exact energies cannot be calculated ab initio, it is possible to extract empirical parameters from experimental electronic absorption spectra which describe the interaction between metal and ligand. For a given dNX6 complex with an ideal octahedral coordination (symmetry Oh), the cubic CF splitting parameter 10Dq, together with Racah parameters B and C, provide basis for a reasonably complete description of the electronic spectra (Lever, 1984). In most crystals the site symmetry is, however, lower than Oh. This requires introduction of additional, so-called distortion parameters to describe the lower symmetry CF components (Ballhausen, 1962; König & Kremer, 1977). The number of these parameters increases with the lowering of symmetry up to a total of 14 CF parameters for dN ions at triclinic site symmetry. A higher number of parameters requires for their fitting also a higher number of observable quantities, which can in practice be extracted from the optical absorption spectra. It is by no means an easy task since the observables have to be properly and undoubtedly assigned to their respective transitions. Structure determinations show that regular high-symmetry coordination polyhedra of transition metal ions are rather an exception. Nevertheless, an acceptable description of the spectra of transition metal ions is obtained approximating the actual site symmetry by a higher pseudo-symmetry (see Part I). In many cases the calculations of the CF splittings were performed on the basis of the approximated Oh symmetry. The procedure for extraction of the pertinent quantitative relationships from optical spectra is given in sections 2 and 5.1 Compared to the actual symmetry and distortion of coordination polyhedra in minerals, the assumption of a regular coordination polyhedron is a crude oversimplification. In fact, only a few transition metal ions tend to occupy nearly regular coordination polyhedra, i.e. d3 and d8 ions (Cr3+, Ni2+). In those cases the Oh approximation can yield reasonable results (e.g. Abs-Wurmbach et al., 1985; Langer & Andrut, 1996).

On the other hand, with increasing distortion of the coordination polyhedra, calculated approximated (i.e., cubic) energies are subject to increasingly significant errors. Therefore, any subsequent quantitative conclusions concerning crystal chemistry based on the such derived crystal field stabilisation energy (CFSE) become doubtful. Thus, by applying the Oh symmetry approximation, often only qualitative statements can be made, whereas important crystal chemical information is lost. Consequently, an approach that can take the actual local site symmetry and polyhedral geometry into account is needed. This leads to the so-called semi-empirical methods, one of which – the superposition model of crystal fields (SPM) – will be discussed in section 5. A comparison of the qualitative results derived from the simplified Oh approximation and those from the semi-empirical SPM evaluations will also be given.

As a starting point, the discussion in section 2 is still entirely based on the point charge electrostatic model assuming a 1/R5 dependence of the field strength, while more realistic, adjustable power-law exponent parameters are introduced with the SMP and discussed in sections 5 and 6.

The relationship between 10Dq and the interatomic distance R for a regular octahedron

In applying CFT to problems in geosciences several approximations have been made, especially concerning the calculation of the CFSE of the transition metal ions at various structural sites. The CFSE is calculated from the cubic CF splitting parameter 10Dq, which is theoretically derived from the point charge model of CF as (e.g.Schläfer & Gliemann, 1967; Lever, 1984):  

formula
where q is the effective charge of the ligands, forumla is the mean central ion-ligand distance and forumla is the mean value of the forth power of the radial distance of the d electrons from the nucleus. Equation 1, widely applied in geosciences, is only valid under very restrictive conditions, namely:
  1. a) one type of ligand is assumed;

  2. b) the ligands behave as point electric charges;

  3. c) the ligands surrounding the central metal ion form a regular octahedron (Oh symmetry), i.e., exhibit equal bond lengths and angles.

Equation 1 states that 10Dq is inversely proportional to the fifth power of the mean central ion-ligand distance. Thus, this equation relates a spectroscopic quantity (10Dq), which can be estimated from optical absorption spectra, with the interatomic distance of a specific type of coordination polyhedron occupied by a transition metal in a crystal structure. This relationship has often been used in the literature to investigate or predict the behaviour of a given 3dN ion at a certain coordination site as a function of pressure, temperature, or occupation (i.e., composition) (e.g. Burns, 1993; Langer, 2001). Then, as a first approximation, it is assumed that q and 〈r4〉, which are related to the bonding character, retain their free-ion values in crystals (e.g. Burns, 1993). Equation 1 is then simplified to:  

formula

If the local symmetry of the coordination polyhedron is reduced from Oh, additional terms have to be included in Equation 2 for a proper description. Examples for trigonal local symmetries of the octahedron in the structures of garnet and ruby will be given in sections 6.2 and 6.3.

Applications and consequences

According to the rule of proportion using Equation 2, we can express the dependence of and 10Dq on pressure P, temperature T, or composition X, as described below.

Pressure dependence

Determining the mean octahedral distances at high pressure forumla relative to that at ambient pressure forumla, Equation 2 yields:  

formula

Based upon Equation 3, Burns (1987) extracted polyhedral bulk moduli kpoly from pressure-dependent electronic absorption spectra. The bulk modulus k is an important parameter that relates the change of volume V with pressure. By definition, k is the reciprocal of the volume compressibility βv defined as:  

formula
with  
formula
where the subscripts T and X refer to partial derivatives at constant temperature and composition. similarly, the linear compressibility β1 is defined as:  
formula
where d denotes the interatomic distance metal (M) to ligand (L). Substituting d by the mean interatomic distance forumla we obtain the linear polyhedral compressibility βl,poly expressed as:  
formula
We can simplify Equation 7 by using the differential quotient and write  
formula
with forumla and the pressure difference forumla. A more detailed description is provided by Burns (1985, 1993). The assumptions mentioned above, which bear on the accuracy of Equation 3, also affect Equations 4 and 8. Besides, one has to keep in mind that the bulk modulus of a crystal depends upon the bulk moduli of its component coordination polyhedra and on the manner in which these polyhedra are linked in the structure (e.g.Hazen, 1985). For a detailed description of the so-called “polyhedral approach” the reader is referred to Hazen & Finger (1982).

Spectroscopically derived data on polyhedral bulk moduli are scarce. Langer et al. (1997) investigated Cr3+-centred octahedra in spinel, garnet, ruby and kyanite. They found discrepancies between the spectroscopically determined polyhedral bulk moduli and those obtained by high-pressure single crystal diffraction, especially for ruby and pyrope. In both minerals Cr3+ and Al3+ occupy the octahedral positions. Thus, the diffraction methods yield polyhedral bulk moduli averaged over all polyhedra of a specific site. On the other hand, absorption spectroscopy methods distinguish between the individual polyhedra occupied by a transition metal. Hence, the results can be interpreted by lattice strain induced by the Cr3+/Al3+ substitution (Langer et al., 1997). An interesting approach to explain the pressure dependence of the CF and SPM parameters has recently been proposed by Mulak (2003).

Temperature dependence

The temperature variation of the mean interatomic distance can be described in a similar way as the pressure dependence:  

formula
Since R is proportional to V1/3, Equation 9 can be used to relate the volume coefficient α of linear thermal expansion with 10Dq (Burns, 1985, 1993):  
formula

Similarly as for compressibility, the volume coefficient of linear thermal expansion αV describing the dimensional changes of a crystal can be expressed as (e.g. Hazen, 1985)  

formula
where the subscripts P and X refer to partial derivatives at constant pressure and composition.

Taran et al. (1994) determined the local thermal expansion coefficients of CrO6 octahedra in various oxygen-based minerals from absorption spectroscopy data. The authors compared their results with the corresponding ones obtained by the high temperature crystal structure refinements. Again, the differences observed were due to the averaging nature of the diffraction experiments over sites that were occupied by at least two different species.

Compositional dependence

The influence of the composition on the mean interatomic distance in a specific coordination polyhedron can be systematically studied in binary systems. The compositional dependence of the mean interatomic distance in a binary solid solution can be calculated according to the following equation in analogy to the aforementioned pressure and temperature dependence as:  

formula
Prerequisite for applying this relationship is the knowledge of the parameter 10DqX=1 and forumla of the transition metal bearing end-member. The value of forumla for the solid solution may be experimentally extracted from optical absorption spectra. The respective mean interatomic distance forumla of the coordination polyhedron hosting the transition metal can be then calculated applying Equation 12. An example is given below (section 6.2.7).

Vegard's rule (and its relationship to local interatomic distances)

In solid solutions the lattice parameters generally vary with composition. Vegard (1928) discovered that the unit cell parameter aX in a binary solid solution A1–XBXC can be expressed as a function of its composition in terms of an additivity rule:  

formula
where a1 and a2 represent the lattice parameter for the end-members AC and BC, respectively, and X1, X2 are their respective molar fractions. In the past decades, the number of characterised binary phase systems increased together with experimental accuracy. Only some experiments have indeed substantiated Vegard's rule, whereas several other results were favouring a non-linear behaviour. To explain these deviations, different models were proposed, which proved to be more or less successful (see e.g. references in Urusov, 1992).

In case of a coupled substitution, an intra-crystalline fractionation may take place, since two or more non-equivalent structural sites are involved (see Part I). If at least one of these sites is preferentially occupied, a deviation from Vegard's rule is likely to occur (Newton & Wood, 1980). For further details the paper by Urusov (1992) may be consulted.

Local interatomic distances in coordination polyhedra

In this section we will outline the change of bond lengths at a specific structural site in a binary solid solution A1–XBXC as a function of its composition. We assume here that the interatomic distance forumla of the end-member AC is smaller than the respective interatomic distance forumla of BC. When a given cation is substituting for another one in a solid, at first instance two extreme cases can be distinguished as discussed below.

Concept of the virtual crystal approximation

This model accounts for the variation of distances as a function of composition and was developed and applied first for alloys (Martins & Zunger, 1984). It assumes that all bond lengths of the MLn polyhedra are equal regardless of their individual occupancy by A or B. Thus, these interatomic distances RX correspond to an averaged value according to Vegard's rule (see the dotted line in Fig. 1):  

formula
where XAC and XBC are the mole fractions of the end-members AC and BC, respectively. In short, this model does not take into account the individual polyhedra occupied by A or B, but describes the structure as composed of “averaged” polyhedra at the substituted sites. Accordingly, structural relaxation is completely neglected in this model.

Fig.1.

Dependence of the individual bond lengths R(AC) and R(BC) in a binary solid solution A1–XBXC according to different approaches. No relaxation (ε = 0) exists for the virtual crystal approximation (VCA), which corresponds to a behaviour according to Vegard's rule. In case of full relaxation (ε = 1), the individual bond lengths Rrelax are equal to those of the end-member. Real changes of the individual bond length are expressed in terms of Rexp, which is a function of composition (after Urusov, 1992).

Fig.1.

Dependence of the individual bond lengths R(AC) and R(BC) in a binary solid solution A1–XBXC according to different approaches. No relaxation (ε = 0) exists for the virtual crystal approximation (VCA), which corresponds to a behaviour according to Vegard's rule. In case of full relaxation (ε = 1), the individual bond lengths Rrelax are equal to those of the end-member. Real changes of the individual bond length are expressed in terms of Rexp, which is a function of composition (after Urusov, 1992).

Hard sphere model

In this model the cations A and B retain their respective sizes when incorporated into a system, i.e., they behave like hard spheres. The substitution induces strain that shall be fully absorbed by other parts of the structure. Hence, this is the case of maximum structural relaxation. Accordingly, the individual bond lengths RAC and RBC for A and for B in the solid solution are equal to their respective end-member values forumla and forumla The individual bond lengths will not change with composition (represented by the horizontal dashed lines in Fig. 1). Consequently, two geometries for the substitution site exist simultaneously in the structure of a solid solution, while the frequency of their occurrence corresponds to the mole fraction of the respective cation.

Diffraction methods applied to such solid solution series will not discern between the two geometries of the polyhedra, which are symmetry equivalent. These methods will yield only averaged values. Thus the resultant mean bond lengths obtained by diffraction techniques also correspond to Vegard's rule (Equation 14) (see the dotted line in Fig. 1).

The degree of relaxation ε for substituting B for A in a solid solution is expressed after Urusov (1992) by  

formula
forumla and forumla are the interatomic distances in the end-member in question, and RBC is the interatomic distance for the impurity B in the host lattice AC (i.e., XA ≈ 1). The denominator of this quotient is a constant, and we have to investigate the behaviour of RBC to determine the data range for ε. In case of full relaxation we obtain ε = 1, because the interatomic distance of RBC will not change with composition: RBC = forumla In the absence of relaxation ε = 0, one obtains RBC = forumla

In practice, the behaviour of cations in a solid solution will be between the two extreme cases described above (solid lines in Fig. 1), i.e., 0 < ε < 1.

Superposition model of crystal fields (SPM)

The superposition model (SPM) was originally developed to separate the geometrical and physical information contained in CF parameters for lanthanides (Newman, 1971). Subsequently, its area of application has been extended to the CF parameters for transition metal ions as well as to the phenomenological parameterisation of the zero field splitting (ZFS) Hamiltonian for all transition ions. A comprehensive review of the studies dealing with SPM is given by Newman & Urban (1975) and Newman & Ng (1989, 2000). Due to the strong interest in laser host materials, intensive research has focused on 4fN and 5fN ions to describe the interaction between the open-shell electrons of these transition ions and their surrounding crystalline environment (Morrison, 1992). So far, there have been only a few applications of the SPM for 3dN ions (e.g. Rudowicz & Yu, 1991; Rudowicz et al., 1992; Chang et al., 1993; Yeom et al., 1994; Yeung et al., 1994a, 1994b; Qin et al., 1994; Chang et al., 1995; Wildner & Andrut, 1999; Andrut & Wildner, 2001a, 2001b; Rudowicz et al., 2002; Yang et al., 2002; Andrut & Wildner, 2002).

The parameterisation used in SPM concerns only the crystal field part of the Hamiltonian for a 3dN ion in a crystal (cf. Part I; for details see e.g.Ballhausen, 1962; Schläfer & Gliemann, 1967; Gerloch & Slade, 1973):  

formula
As outlined in Part I, the free ion Hamiltonian HFI consists of the spherical free ion Hamiltonian Hspher (comprising the kinetic energy of electrons, their Coulomb attraction with the nucleus, and the spherical part of the electrostatic repulsion), the residual electrostatic repulsion (i.e. its non-spherical part) amongst the d electrons Hee, the spin-orbit and spin-spin interactions HSO and HSS, and – if applicable – also the Trees correction HTrees, describing the two-body orbit-orbit polarisation interaction (Gerloch & Slade, 1973; in Part I HTrees has been omitted for clarity). These parts of the Hamiltonian are generally expressed in terms of the Racah parameters A (Hspher), B and C (Hee), the spin-orbit coupling constant ζ (HSO), the spin-spin coupling constant ρ (HSS), and Trees correction parameter α (HTrees). Two notations for general forms of CF Hamiltonians based on the tensor operators are now prevailing in the literature, namely: (i) the Wybourne operators forumla (Wybourne, 1965; Rudowicz, 1987) and (ii) the extended Stevens (ES) operators (Rudowicz, 1985a, 1987). The CF Hamiltonian HCF is given in the Wybourne notation by:  
formula
where Bkq are the CF parameters representing the electron radial integrals and the CF interaction strength, and forumla are the re-normalised spherical tensor operators of the electron angular momenta (Morrison & Leavitt, 1982). The indices k and q denote the rank and component, respectively, of the crystal field contributions. Within a given J or L multiplet (see Part I) HCF can also be expressed in terms of the ES operators (Rudowicz, 1985a, 1987) as (Rudowicz & Misra, 2001):  
formula
where the nature of the Stevens' operators forumla (Jx, Jy, Jz) in Equation 18 is explicitly indicated as being the functions of the total angular J (or total orbital L) momentum operators. Thus the parameters forumla (CF) in Equation 18 should not be confused with the zero-field splitting parameters forumla (ZFS), which have different physical nature (Rudowicz, 1987; Rudowicz & Misra, 2001; Rudowicz & Sung, 2001).

For the background theory and derivation of the CF Hamiltonian the interested reader is referred to respective textbooks (e.g. Schläfer & Gliemann, 1967; Newman & Ng, 2000; Mulak & Gajek, 2000). Here we only summarise relevant facts that are necessary for the subsequent understanding, especially concerning the question: what are the non-vanishing crystal field parameters to be taken into account for various local symmetries of the crystal field?

In general, the parameters Bkq with q ≠ 0 may be complex, i.e., Bkq = Re Bkq + i Im Bkq, whereas forumla are all real (Rudowicz, 1985b). It must be emphasised that not all parameters of Bkq (forumla) are independent (see e.g.5.1 and refer to Table 1). Since HCF as an operator of energy must be a Hermitian operator (see e.g.Newman & Ng, 1989; Rudowicz, 1987), the Wybourne parameters Bkq satisfy the relations (Rudowicz, 1985b): forumla and forumla The former Bkq are often referred to as “real” Bkq, the latter as “imaginary” Bkq.

Group theory, especially the irreducible tensor methods and the Wigner-Eckart theorem, enable predicting the non-zero matrix elements of the CF operators for each rank k in Equations 17 and 18 on the basis of the one-electron as well as multi-electron wavefunctions for transition ions in crystals. The triangular rule limits k to the values between 0 ≤ k ≤ 2l, where l is the orbital quantum number. Thus, in case of the d and f states, with l = 2 and 3, respectively, we obtain 0 ≤ k ≤ 4 and 0 ≤ k ≤ 6. The allowed values of q run from –k to +k. Note that for non-centrosymmetric point groups the CF terms of odd k are allowed, but were hardly used in the literature, whereas for centrosymmetric point groups only the CF terms of even k are admissible, thus further reducing the co-domain of rank k. The CF operator with rank k = 0 corresponds to a spherical (i.e. isotropic) potential, which does not produce any CF splitting, and can be therefore omitted.

Apart from the general rules for the rank k, group theory determines which of the tensor operator components q for a given k in Equations 17 and 18 have to be included in the CF Hamiltonian for a given site symmetry of the transition metal ion under investigation. For the lowest, i.e. triclinic symmetry (or for higher symmetries in case of an arbitrary axis setting), the number of the CF parameters admissible by group theory amounts up to a total of 14 for the d electron systems, and up to 27 for the f electron systems, whereas the non-vanishing CF parameters for other symmetry cases are listed in Table 1.

In the course of evaluation, certain conventions should be considered that relate the symmetry operators at the site of a transition metal ion (and its coordination polyhedron) to the reference Cartesian coordinate system of the tensor operators forumla with the axes labelled x, y and z. For example, the principal axis with the highest rotational symmetry of the coordination polyhedron is generally chosen as the z axis, leaving only one degree of freedom in the xy plane. In case of perpendicular axes in a coordination polyhedron (i.e., in cubic, tetragonal and orthorhombic systems), these axes will be aligned parallel to x and y, respectively. However, in low-symmetry cases there are alternative choices of the coordinate system. Such different sets equally well describe the physical properties of the transition metal complex; however, they may lead to different forms of CF Hamiltonians as well as different values for the respective CF parameters (see e.g.Rudowicz, 1987). Each physically equivalent but distinct CF parameter set can be transformed from one coordinate system into another one. The peculiarities of such choices have been discussed for orthorhombic (Rudowicz & Bramley, 1985) and monoclinic (Rudowicz, 1986) symmetry cases, whereas general intricacies of the CF approach for low-symmetry cases are described in the recent papers of Rudowicz & Qin (2003, 2004a, 2004b, 2004c).

Table 1.

Non-vanishing crystal field parameters Bkq with rank k = 2 and 4 for the 32 crystallographic point group symmetries. “+” indicates a non-vanishing real part, while “±” denotes non-zero real and imaginary parts. “–” indicates that this parameter is not independent and is generated due to the symmetry (see text).

crystallographic point groupsnon-vanishing Bkq
B20B21B22B40B41B42B43B44

cubic Oh m3m       – 
 O 432       
 Td forumla3m       
 Th forumlam       
 T 23       

 
tetragonal D4h 4/mmm      
 D4 422      
 D2d forumla2m      
 D4v 4mm      
 D4h 4/m      ± 
 S4 forumla      ± 
 C4      ± 

 
hexagonal D6h 6/mmm       
 D6 622       
 D3h forumlam      
 C6v 6mm       
 C6h 6/m       
 C3h forumla       
 C6       

 
trigonal D3d forumlam      
 D3 32      
 C3v 3m      
 C3i forumla     ±  
 C3     ±  

 
orthorhombic D2h mmm    
 D2 222    
 C2v mm   

 
monoclinic C2h 2/m  ±  ±  ± 
 Cs m  ±  ±  ± 
 C2  ±  ±  ± 

 
triclinic Ci forumla ± ± ± ± ± ± 
 C1 ± ± ± ± ± ± 
crystallographic point groupsnon-vanishing Bkq
B20B21B22B40B41B42B43B44

cubic Oh m3m       – 
 O 432       
 Td forumla3m       
 Th forumlam       
 T 23       

 
tetragonal D4h 4/mmm      
 D4 422      
 D2d forumla2m      
 D4v 4mm      
 D4h 4/m      ± 
 S4 forumla      ± 
 C4      ± 

 
hexagonal D6h 6/mmm       
 D6 622       
 D3h forumlam      
 C6v 6mm       
 C6h 6/m       
 C3h forumla       
 C6       

 
trigonal D3d forumlam      
 D3 32      
 C3v 3m      
 C3i forumla     ±  
 C3     ±  

 
orthorhombic D2h mmm    
 D2 222    
 C2v mm   

 
monoclinic C2h 2/m  ±  ±  ± 
 Cs m  ±  ±  ± 
 C2  ±  ±  ± 

 
triclinic Ci forumla ± ± ± ± ± ± 
 C1 ± ± ± ± ± ± 

In practice, the actual number of non-vanishing crystal field parameters in Equations 17 and 18 is determined by the site symmetry (as listed in Table 1) as well as the appropriate choice of the coordinate system. The phenomenological approach aims to find those Bkq that best fit the experimental data for the transition metal ion under investigation. Recent findings providing a deeper insight into the feasibility of experimental determination of CF parameters for the continuous symmetry cases are worth noting. It turns out that for hexagonal II (C6, C3h, C6h), tetragonal II (C4, S4, C4h), trigonal II (C3, S6), monoclinic (C2, Cs, C2h), and triclinic (Ci, C1) symmetry cases the number of CF parameters that can be fitted from optical spectra is less by one than the number of CF parameters admissible by group theory (Rudowicz & Qin, 2003, 2004b).

On the other hand, the number of CF parameters for most site symmetry cases occurring in real crystals exceeds the number of observed, discernible, and properly assigned transitions in d–d spectra, leading to an ambiguous solution. This dilemma can be circumvented to a certain extent – especially for lower symmetry cases – by applying semi-empirical methods like the SPM or the AOM (angular overlap model), where geometrical information about the crystal field is supplied as input (but see chapter 6.3). Hence, although the number of the admissible Bkq is fixed for a given symmetry (Table 1), the number of the input parameters to be used for fitting experimental spectra can be effectively reduced.

The SPM of crystal fields is based on the assumption that the CF can be expressed as the sum of separate axially symmetric contributions from all nearest neighbour ligands of the transition metal ion (Newman & Urban, 1975; Newman & Ng, 1989, 2000). This leads to the following parameterisation of the crystal field:  

formula
where (Ri, θi, φi) are the polar coordinates of the ith ligand, The intrinsic parameters forumla represent the strength of the kth rank CF contributions from a given ligand type. The geometrical information is described by the coordination factors Kkq. The expressions for forumlai.e., the coordination factors in the extended Stevens notation (Rudowicz, 1985a) are listed explicitly by Rudowicz (1987), whereas the respective conversion factors to the Wybourne notation applicable also for Kkq are found in Newman & Ng (1989) and Rudowicz (1987).

Strictly speaking, the SPM assumption is valid only for the electrostatic point charge contributions to the crystal field. In reality, other contributions also exist due to, e.g., overlap, exchange and covalency processes. Newman (1971) undertook an analysis of these individual contributions. For details we refer to the review (Newman, 1971) and references therein. He concluded that “the superposition principle remains valid for a far more realistic model of the crystal field, in which overlap and covalency dominate the contributions from neighbouring ions” as well as that in practice only the nearest neighbour ligands have to be taken into account. The effectiveness of this concept has, to a great extent, been confirmed by experiments (for a review, see Newman & Ng, 2000). A convenient way of expressing the distance dependence of the intrinsic parameters is to assume an adjustable power-law dependence (Newman & Ng, 1989):  

formula
where R0 is the reference distance, i.e., a more or less arbitrarily fixed standard M–L distance for a particular cation. As a matter of course, the value of R0 should be meaningful and consistently used for all investigations on a particular cation, thus enabling an easy comparison of the results. The expected reduction in the values of the intrinsic parameters with increasing ligand distance is then reflected by the inequality tk > 0 for the power-law exponents (Newman & Ng, 1989). Equation 19 then transforms into a general expression, which enables to derive the SPM relations for a particular symmetry:  
formula
Equation 21 will be used as a basis for the SPM calculations for specific symmetry cases discussed in the following subsections. Note that attempts to extend SPM by introducing two-term power-law (Hikita, 1992; Donnerberg et al., 1993) have met with limited success due to over-parameterisation.

The regular octahedron

In the following we shall examine the case of a regular octahedron, and present the respective equations of conventional CFT. In Figure 1 the orientation of the octahedron with respect to the polar coordinate system is shown. In this case, all six ligands have the same M–L distances Ri (with i = 1 to 6), and exhibit the following polar angles (θi, φi): (0°, 0°), (90°, 0°); (90°, 90°); (90°, 180°); (90°, 270°); (180°, 0°).

The parameter B40 is the only independent CF parameter for the point group Oh, since B44 is related to B40 (see e.g. Schläfer & Gliemann, 1967) as:  

formula

Fig. 2.

Polar coordinates of the ligands of a regular octahedron.

Fig. 2.

Polar coordinates of the ligands of a regular octahedron.

Wybourne (1965) provides the conversion relations for the conventional CF parameters of Ballhausen (1962); extracting data from these sources yields:  

formula
 
formula
and thus:  
formula
 
formula
The same relation as in Equation 22 is obeyed by Equations 25 and 26. Therefore, we will only use Equation 25 for calculations. For a regular octahedron with equal interatomic M–L distances, we can set Ri = R1 and sum over six ligands in Equations 25 and 26. Inserting the angular polar coordinates for each ligand, i.e., the angles (θi, φi) given above, into the expression for K40, the Equation 25 simplifies to:  
formula
Equation 27 can be used to the estimate the starting values of the SPM parameters for a related system (Newman & Ng, 2000). In case that R1 equals R0, Equation 27 simplifies to 6Dqcub = forumla. At a closer look, Equation 27 reveals a similarity with Equation 2 derived by conventional CFT. Actually, CFT predicts a power-law exponent t4 = 5 in the case of a point charge electrostatic model, and, in principle, such a result could be obtained even when overlap, covalency etc. were properly taken into account. However, empirically determined values of t4 are not necessarily in agreement with this prediction. Fitting t4 to experimentally observed crystal field energy levels may lead to more realistic values for the particular system under investigation. For example, t4 ≈ 11 was reported for tetravalent actinide ions with chlorine and oxygen ligands (cf. Newman & Ng, 2000), and in section 6 we derive empirical power-law parameters for Co2+ and Cr3+ in oxygen crystal fields within the range 1 ≤ t4 ≤ 7.

After rearrangement of some factors in Equation 27, one obtains an equivalent form of Equation 2 in case of a regular octahedron:  

formula
Despite of different kinds of grouped factors that are assumed to be constant, both, Equations 28 and 2, are equivalent. In analogy to the procedure outlined in section 2.1.3 and considering the restrictions mentioned therein, Equation 28 can be used via the rule of proportion to determine local interatomic distances as well.

The distorted octahedron – one selected example

As mentioned above, a regular octahedron with actual point symmetry Oh is a rare exception among the numberless structure types offering six-fold coordinated sites, and thus can only serve as a first approximation. Taking into account the actual symmetry of a coordination polyhedron distorted from cubic Oh symmetry, some additional CF terms are needed to describe the distortions. The resulting increased number of CF parameters also requires a higher number of firmly assigned observables for meaningful least-squares fittings. This requirement is not always met in case of the 3dN elements. In general, the FWHM of optical absorption bands of the d block elements is rather large and in the order of 2000 cm−1, compared with the “sharp line” spectra of fN transition metal compounds. Hence, the deconvolution of a band envelope consisting of split energy levels that are separated by only a few 100 cm−1 becomes rather ambiguous. Minor splitting due to spin-orbit coupling can further broaden and blur spectral features. In addition, the expression of the symmetry reduction in terms of distortion parameters can be ambiguous and confusing. In the past, different groups of scientists developed their “own” set of parameters, which was often valid for a particular symmetry only. A comparison or even transformation of such different parameter sets is confusing or even impossible, since only limited conversion relations are established (compare Part I and see e.g. König & Kremer, 1977; Lever, 1984). These problems can be overcome, to a certain extent, by the application of semi-empirical models like the SPM or AOM, which use the individual polar coordinates to treat the distortion of a local polyhedron (refer to Eqn. 21). Hence, these models can account for all types of coordination polyhedra and their distortions, avoiding the introduction of further parameters.

Below, the advantages of the SPM over the conventional CFT are exemplified for a trigonally distorted octahedron with point symmetry forumla (D3d). For D3d and most other trigonal point groups, Ballhausen (1962) introduced the hitherto well established distortion parameters Dτ and Dσ in addition to Dq. Note, however, that this Dq is no longer a cubic quantity but represents the equatorial field strength only, and hence we label it Dqtrig in the following. An octahedron with symmetry D3d may be either compressed or elongated along its threefold axis and has six equal M–L bond lengths. Consequently, the equations for the crystal field parameters Bkq simplify, since all the imaginary terms Im Bkq as well as most real terms Re Bkq are zero (compare with Table 1). Also the coordination factors Kkq(θi, φi) of each non-zero Bkq become equal, thus leading to the Equations 2931. The corresponding relationships to Ballhausen's (1962) conventional CF and distortion parameters for trigonal symmetry, i.e., Dqtrig, Dτ, Dσ, apply:  

formula
 
formula
 
formula
For simplification, the conversion factors from Stevens to Wybourne notation are formally included into the coordination factors Kkq(θi, φi) Similarity as for a regular octahedron, we can use the CF parameter B40 (Eqn. 30) for trigonal symmetry to determine the dependence of the CF splitting. This can now be expressed in terms of 10Dqtrig, as a function of the mean interatomic distance:  
formula
To show the similarity between Equation 32 and Equations 2 and 28, certain factors were grouped together. In comparison to the Oh approximation (Eqn. 2), a distortion term appears in addition to Dq, while the coordination factor involves the angular distortion. Note that the value of the power-law exponent t4 = 5 in Equations 2 and 28 is predicted by the point charge electrostatic model.

Applying the rule of proportion we can use Equation 32 to extract local mean interatomic polyhedral distances in solid solutions. The respective relation then reads:  

formula
Here, the factors that are specific for the system under investigation, like the intrinsic parameter forumla and the reference distance R0, vanish. The subscripts X = 1 and Xi refer to the end-member and an intermediate composition of the solid solution, respectively. Since we are dealing with only one interatomic distance Ri in point group D3d, we can set <Ri> = R1. Equation 33 shows a dependence of RXi not only on the interatomic bond length R, but also on the polar angle θ. Consequently, it takes the respective elongation or compression of the polyhedron into account, while maintaining the point symmetry. Contrary, a cubic symmetry approximation expressed in the basic Equations 2 and 12 exhibits further simplifications than those mentioned in section 5.1. Basically, Equation 2 takes into account only one mean interatomic distance that may change as a function of pressure, temperature or composition, while no attention is paid to the actual individual bond lengths, or to the change of bond angles. Whether such an approximation is justified depends upon the particular degree of the polyhedral distortion. The example given just above for D3d point symmetry still represents an easy and straightforward example from the theoretical point of view. Nevertheless, due to the introduction of further variables, like distortion parameters and coordination factors, the practical application for deriving mean interatomic distances becomes ambiguous via a simple rule of proportion, because more than one geometrical parameter is unknown (i.e., length and/or angle). Instead, the SPM calculations can yield a better description of the polyhedral changes. Comparison of the results obtained by SPM and those of the Oh approximation will be given in section 6.2.7.

Strategy for the determination of crystal field parameters using SPM

As outlined in Part I, a thorough description of the energy levels of a transition ion complex has to take into account both the crystal field contribution HCF as well as the free ion contribution HFI. The free ion parameters comprise the Racah parameters A, B and C, the spin-orbit coupling parameter ζ, and the (often negligible) Trees correction α. The spherical parameter Racah A has no influence on relative energy splittings and hence is usually omitted. Since the free ion parameters are less sensitive to the crystalline environment, they may be taken in a first approximation directly from the literature (e.g. Morrison, 1992). The crystal field parameters strongly depend on the ligand type as well as on the point symmetry and geometry of a transition metal complex. In the SPM approach the geometry and symmetry of the ligand field is completely described by the polar coordinates of the ligands, which can be extracted, e.g., from X-ray crystal structure investigations. Hence, only the intrinsic parameters forumla and power-law exponents tk have to be fitted to a set of polarised optical spectra. In case a sufficient number of properly assigned bands and band components have been observed, all necessary parameters including the free ion parameters may be fitted.

The principal steps of an SPM investigation are outlined in a flow chart diagram in Figure 1. As the initial step towards the application of the SPM in geosciences, our approach aims at the determination of reliable intrinsic and power-law SPM parameters for several 3dN ions on the basis of the exact polyhedral geometry extracted with high accuracy from single crystal structure investigations. It has to be emphasised that the given 3dN ion fully occupies its respective crystallographic site, i.e., the structural information is obtained from end-member phases, either synthetic or natural. In case of natural samples, the chemical composition has to be determined by microprobe analysis. The samples have to be oriented parallel to the axes of the optical indicatrix and prepared as double-sided polished crystal plates. In the case of optically biaxial crystals at least two oriented crystal slabs have to be prepared. A set of polarised optical absorption spectra has to be measured in the appropriate spectral range, if necessary using a spectrometer with attached microscope. The correct assignment of the spectral features to the respective electronic transitions is most crucial for the reliability of the SPM parameters. This interpretation is performed on the basis of the appropriate symmetry selection rules as outlined in Part I, corresponding to the local symmetry of the transition metal polyhedron as obtained by prior crystal structure analysis.

For the actual energy level calculations from the Bkq and the free ion parameters, we use the crystal field computer package developed by Yeung & Rudowicz (1992) for orthorhombic and higher symmetries; the latest version of this program (Chang et al., 1994) includes the imaginary CF terms and is applicable also for low symmetry. A Visual Basic version (VBA) including a microscopic spin-Hamiltonian (MSH) module has recently been developed (Rudowicz et al., 2003; Yang et al., 2003). Hence with the HCFLDN2-module of the CF program package (Chang et al., 1994), we are able to calculate energy levels for any dN cation at arbitrary low site symmetries. Some preliminary calculations for spin-allowed energy levels in three- or four-fold symmetry

Fig. 3.

Flow chart diagram depicting the course of spectroscopic and SPM investigations.

Fig. 3.

Flow chart diagram depicting the course of spectroscopic and SPM investigations.

were also done with the program TETRIG (Wildner, 1996b), which employs the energy matrices given by Perumareddi (1967). For the setup of the quantitative evaluation we developed supplementary programs (Wildner & Andrut, unpublished) which are used for (i) the transformation of atomic to polyhedral polar co-ordinates; (ii) the systematic variation of the free ion parameters and of the intrinsic and power-law SPM parameters, as well as the corresponding communication with the HCFLDN2 program; (iii) the SPM calculation itself yielding the values for the Bkq's and (iv) the interpretation of the HCFLDN2 output results in terms of a reliability index for the agreement of calculated and observed spin-allowed transition energies.

In order to avoid any bias in the parameter determination from scratch for a newly investigated cation, we only regard forumla > 0 and tk ≥ 0 as prerequisites. Hence, a systematic variation of the intrinsic and power-law parameters covering a very wide range of parameter values is necessary to avoid any wrong pseudo-minima in course of the “fitting process”. CPU power and time may be a limiting factor for the number of the computationally often very demanding calculations. Furthermore, a meaningful reference metal-ligand distance R0 has to be chosen for each 3dN cation, which then should be used for all subsequent SPM calculations to ensure the transferability of SPM parameter sets from one compound/host material to another. The reliability of the “refined” tk crucially depends on the bond length distortion of the polyhedron under consideration. For weak or zero bond length distortion, the tk will be doubtful or even meaningless and have to be fixed at appropriate values (e.g. at the “electrostatic values” t2 = 3 and t4 = 5). However, if the individual bond lengths Ri are moreover (nearly) equal to R0, the tk have no influence on the SPM calculations at all.

In this way, a set of free-ion parameters, intrinsic parameters forumla and power-law exponents tk can be extracted for the compound under investigation. In practice, the number of variables should be kept as small as possible to reduce calculation times and avoid ambiguities. Usually, Racah C can be subsequently estimated from the positions of some more or less safely identified spin-forbidden transitions. Within 3dN systems, spin-orbit coupling often plays a minor role and can be taken into account in a final step for the calculation of multiplet mixing coefficients.

In the context of our investigations (examples are given below), the obtained crystal field parameters may be

(a) applied for the interpretation of spectral-structural correlations,

(b) used for systematic investigations aiming to establish a basic “global” SPM parameter set for a newly investigated dN cation, e.g. Co2+ (see below), and

(c) used as starting values for structurally less well defined systems, e.g. solid solutions, thus being a basis for the investigation of real geoscientific systems.

Applications of the superposition model in geosciences

The continuing interest and the relevance of the superposition model approach to crystal fields since its introduction by Newman (1971) is demonstrated by the number of publications and books in this research field and by its application in the development of opto-electronic systems and magnetic materials. The recently published “Crystal Field Handbook”, edited by Newman & Ng (2000), gives an overview about the current status of research. Most investigations concentrated on the lanthanide and the actinide series (with partly filled 4f and 5f shells, respectively), whereas scarce information is still available for the transition metal cations with partly filled 3d or 4d shells. The first application of the SPM in geosciences was an attempt to explain the order of energy levels for Fe2+ in the dodecahedral site of pyrope-almandine garnets (Newman et al., 1978). Up to now, data for d block elements focus on Cr3+ and Co2+ ions only, and there exist just a handful of such papers correlating spectroscopic and structural properties of minerals. Hence, the necessary intrinsic and power-law exponent parameters of 3dN systems relevant for geosciences are practically missing or were obtained from natural, dilute phases, where the local structure around the particular transition ion is not exactly known (e.g. Cr3+ in alumosilicates; Qin et al., 1994; Yeung et al., 1994a, 1994b, and references within these papers). Present efforts to provide reliable SPM parameters for Cr3+ and Co2+ from synthetic end-member compounds or thoroughly characterised natural solid solutions, as well as the subsequent application to a particular geoscientific problem, are discussed in sections 6.1 to 6.3 below.

Superposition model parameters for Co2+ in oxygen-based crystal fields

For cobalt cations in particular, there seems to exist only one superposition model analysis yielding the intrinsic SPM parameters prior to our investigations, namely for EPR spectra of Co2+ doped in CdCl2 (Edgar, 1976). However, that paper provides no power-law parameters tk as well as no reference distance R0.

For our recent research on crystal fields of Co2+, several mineral-type or related oxygen-based Co2+ compounds with end-member compositions were synthesised at low-hydrothermal conditions. Their crystal structures were thoroughly characterised by single crystal X-ray diffraction methods. Polarised optical absorption spectra were measured using microscope spectrometric techniques and devices as mentioned above and in Part I. Experimental details can be extracted from relevant publications of the authors (e.g. given in Table 2). For the subsequent SPM analyses the reference metal-ligand distance R0 was fixed at 2.1115 Å, the overall mean Co–O bond length in six-fold coordination (Wildner, 1992).

Table 2.

SPM parameters for Co2+ extracted from synthetic mineral-type or related compounds.

PhaseSym.forumlat4forumlat2references

Li2Co3(SeO3)4147403.170005.5[1]1,2,3
Co(OH)2, 290Kforumla5260*4920*[2]1, [3]1,2,3
Co(OH)2, 90K53203900
CoSO4·H2Oforumla48401.953004.0[4]1, [5]2, [6]3
CoSeO4·H2Oforumla50001.568902.7[7]1, [5]2, [6]3
NaCo2(SeO3)2(OH)m47601.050402.4[8]1,2, [6]3
CoSe2O524960*4270*[9]1, [10]2, [6]3
CoSeO3·2H2O150905.480000[11]1, [10]2, [12]3
PhaseSym.forumlat4forumlat2references

Li2Co3(SeO3)4147403.170005.5[1]1,2,3
Co(OH)2, 290Kforumla5260*4920*[2]1, [3]1,2,3
Co(OH)2, 90K53203900
CoSO4·H2Oforumla48401.953004.0[4]1, [5]2, [6]3
CoSeO4·H2Oforumla50001.568902.7[7]1, [5]2, [6]3
NaCo2(SeO3)2(OH)m47601.050402.4[8]1,2, [6]3
CoSe2O524960*4270*[9]1, [10]2, [6]3
CoSeO3·2H2O150905.480000[11]1, [10]2, [12]3

1 crystal structure, 2 polarised absorption spectra, 3 SPM analysis

* fixed at t4 = 5 and t2 = 3 (see text)

As a matter of course, the extraction of reliable SPM parameters crucially depends on the correct interpretation and assignment of the spectral features. In this regard, electronic absorption spectra of octahedrally coordinated Co2+ cations generally pose some problems, especially concerning the characteristic structures and splitting of the intense 4T1g(4F) → 4T1g(4P) band system (see e.g. Fig. 11 in Part I).

These have been attributed by various authors to one or more of several reasons, e.g., admixture of intensity-enhanced spin-forbidden transitions to the 4T1g(P) band, splitting of this band due to low symmetry components of the crystal field or due to spin-orbit coupling, and contribution of vibrational components. For the detailed band assignments of the investigated compounds the reader is referred to the original literature cited above.

The first complete SPM parameter set for Co2+ was extracted from polarised electronic absorption spectra of Li2Co3(SeO3)4, which is characterised by strong distortions of two crystallographically different CoO6 polyhedra with low symmetry C1 and Ci (Wildner & Andrut, 1999). According to the Laporte selection rule, the acentric polyhedron is expected to dominate the absorption spectra. Its severe bond length and angle distortions (Co–O = 2.01–2.39 Å, cis-O–Co–O = 76–99°) makes it an ideal candidate for the extraction of reliable forumla as well as tk. Hence, the extracted parameter set given in Table 2 may be assumed as the most reliable up to now, despite the fact that it does not comply with the general expectation that t2 < t4 (Yeung & Newman, 1986). The complete energy level schemes for both CoO6 polyhedra, calculated with these SPM parameters, are presented in Figure 1.

In brucite-type Co(OH)2 the Co2+ cations occupy a high-symmetry site (D3d) within a compressed hexagonal close packing of oxygen atoms. Structural data and polarised absorption spectra were obtained at 90 and 290 K. For D3d symmetry all Co–O distances are equal and hence the tk had to be fixed. The angle θ between the trigonal axis and the Co–O bonds is around 60.5°. In general, the magnitude of the intrinsic forumla (Table 2) grossly complies with those of the other compounds. On closer inspection, however, it appeared that the specific position of closely related ligands within the spectrochemical series (e.g. O3Se2– vs. OH) affects the intrinsic forumla However, an attempt to account for such influences, especially in mixed-ligand coordinations, by introducing an empirical ligand type correction factor into the SPM calculations (Wildner & Andrut, 2001a) – resembling the f factor formerly introduced by Jørgensen (1962) – did not improve the results but rather aggravated ambiguities in the parameter refinements.

SPM analyses of the absorption spectra of synthetic Co2+ kieserites, CoSO4·H2O and CoSeO4·H2O, resemble the results for Li2Co3(SeO3)4 and corroborate that t4 < t2 for this 3dN transition ion. The CoO4(H2O)2 octahedron with point symmetry Ci exhibits a distinct pseudo-tetragonal shape with an elongation along the H2O–H2O axis.

The structure of NaCo2(SeO3)2(OH) is built up from olivine-like octahedral chains. The optical absorption spectra are characterised by d–d transitions at the acentric Co(2) site with symmetry Cs, showing a pseudo-tetragonal compression of its CoO5(OH) coordination polyhedron. The derived power-law parameters are rather small, but again t4 < t2.

Spectroscopic and SPM analyses of two further synthetic Co2+ compounds yielded more ambiguous results concerning both the assignment of the energy level components split by low-symmetry CF and the subsequent extraction of SPM parameters. In CoSe2O5 the CoO6 octahedron (symmetry C2) exhibits distinct bond angle distortions but

Fig. 4.

Graphical representation of observed and calculated energy levels up to 30000 cm−1 for the Co2+ sites in Li2Co3(SeO3)4. Parental free-ion term labels and quartet split labels for Oh symmetry are included. All bold lines, numbers and labels refer to spin-allowed, normal to spin-forbidden levels. Numerical results are given for the spin-allowed and some selected spin-forbidden levels. Percentage values in parentheses denote approximate quartet admixtures to spin-forbidden doublet levels.

Fig. 4.

Graphical representation of observed and calculated energy levels up to 30000 cm−1 for the Co2+ sites in Li2Co3(SeO3)4. Parental free-ion term labels and quartet split labels for Oh symmetry are included. All bold lines, numbers and labels refer to spin-allowed, normal to spin-forbidden levels. Numerical results are given for the spin-allowed and some selected spin-forbidden levels. Percentage values in parentheses denote approximate quartet admixtures to spin-forbidden doublet levels.

negligible bond length distortion, and hence the tk had to be fixed. In CoSeO3·2H2O the SPM calculations for the CoO4(H2O)2 polyhedron (symmetry C1) yielded reasonable intrinsic parameters but an implausible power-law parameter t2 = 0. Hence the results of the analysis in the latter two cases must be treated with caution.

Taking into account the different reliability of the parameters presented in Table 2, we propose the following SPM parameter set as a starting point for future applications of the SPM to Co2+ cations: forumla ≈ 4900 cm-1, t4 ≈ 2.5, forumla 6000 cm-1, t2 ≈ 4.0.

Extraction of crystal field parameters for Cr3+ from the binary solid solution uvarovite-grossular

Introduction

Due to the relevance of garnet-type compounds in geosciences as essential constituents of the Earth's crust, mantle and transition zone as well as to their importance for technical applications, this structure type has attracted the interest of geoscientists, crystallographers and physicists since the first structure determinations of natural garnets by Menzer (1926, 1928). Comprehensive surveys on the crystal chemistry of garnets are provided, e.g., by Geller (1967), Meagher (1980) and Griffen (1992), and structural systematics of oxide garnets have been summarised by Hawthorne (1981). In the present book, a review of spectroscopic investigations on Al-bearing garnets is given by Geiger (2004).

Despite some garnets are known to show anomalous birefringence frequently (e.g. Akizuki, 1984; Griffen, 1992; Hofmeister et al., 1998), it is generally accepted that the common rock-forming garnets are cubic, and the overwhelming majority of crystal structure investigations have been performed on synthetic or natural crystals in space group forumla

As a result of its chemical flexibility, the garnet structure type comprises a large number of synthetic and natural compounds with the general formula unit [8]X3[6]Y2[4]Z3[4]O12, among the latter are common rock-forming minerals (Z = Si) subsumed in the pyralspite group (Y = Al: pyrope X = Mg, almandine X = Fe2+, spessartine X = Mn2+) and the ugrandite group (X = Ca: uvarovite Y = Cr3+, grossular Y = Al, andradite Y = Fe3+). For a survey of the group of Ca/Cr-bearing garnets see e.g. Wildner & Andrut (2001b).

In space group forumla the number of formula units is Z = 8. The cations X, Y and Z occupy special positions, while the oxygen atoms occupy a general site. Table 3 summarises the respective site symmetries of the cations. The X cation is coordinated by eight oxygens with two different X–O distances (each 4 times) forming a triangular dodecahedron. Six oxygen ligands form a slightly distorted octahedron around the Y cation with six equal Y–O bond lengths. The Z cation is four-fold coordinated by the oxygen ligands, forming a tetrahedron distorted to a tetragonal disphenoid with one Z–O bond length. Usually, the X site hosts larger divalent cations, the Y site smaller trivalent cations, and the ZO4 tetrahedron is usually occupied by Si but may be replaced by O4H4 groups (“hydrogarnets”).

The ZO4 tetrahedra and YO6 octahedra share edges constituting a three dimensional framework structure. The resulting voids of this framework are occupied by the large X cations. Each oxygen is coordinated to one Z, one Y and two X cations. For further details and figures of cubic garnets refer to the respective chapter by Geiger (2004) in this book.

Table 3.

Labelling, multiplicity m and site symmetry s of the positions occupied in the garnet structure for the space groups under consideration.

forumla (tricl)Fddd (orth)forumla (cub)
atommsatommsatomms

Ca1Ca11256321
41Ca238222Ca1–624222
Ca6Ca348222

Y1Y11367
2forumla16forumlaY1–8164316forumla
Y8Y22458

Si1Si11256321
41Si1–624forumla
Si6Si234162

O1O1
41321o961
O24O6
forumla (tricl)Fddd (orth)forumla (cub)
atommsatommsatomms

Ca1Ca11256321
41Ca238222Ca1–624222
Ca6Ca348222

Y1Y11367
2forumla16forumlaY1–8164316forumla
Y8Y22458

Si1Si11256321
41Si1–624forumla
Si6Si234162

O1O1
41321o961
O24O6

Birefringent garnets

Natural garnets in general, but especially those belonging to the grossular-andradite and grossular-uvarovite series, often exhibit weak birefringence. Various reasons for this anomalous optical behaviour of normally cubic garnets are discussed in the literature (e.g.Akizuki, 1984; Allen & Buseck, 1988; Kingma & Downs, 1989; Hofmeister et al., 1998; see a summary in Andrut & Wildner, 2001b). Often non-intrinsically structural origins (e.g. external strain) are assumed, while maintaining the cubic symmetry and space group forumla. However, some structure investigations on such birefringent garnets have been performed in order to verify a symmetry deviation from forumla due to cation ordering on the octahedral and/or dodecahedral sites. Violation of the forumla symmetry beyond doubt has been reported by Takéuchi et al. (1982) and Wildner & Andrut (2001b).

Sample material

Synthetic, optically isotropic flux grown uvarovite, Ca3Cr2[SiO4]3 (Uwsyn-22), and six natural birefringent uvarovite-grossular garnets from three localities (Saranov [Sar-desy, Sar-kl2, Sar-899, Sar-w2], Veselovsk [Ves-2] and Saranka [Ska-1], Ural Mountains, Russia) were characterised by optical methods, electron microprobe analysis, and UV-VIS-IR micro-spectrometry by Andrut & Wildner (2001b, 2002). Microprobe analyses reveal that all investigated garnets are chemically homogenous. For the detailed analyses we refer to the original literature. The Cr3+ content in mol% is given in Table 4. The crystal structures were investigated using single crystal X-ray CCD diffraction data (Wildner & Andrut, 2001b).

Crystal structures

Crystal structures of natural birefringent uvarovite-grossular solid solutions

The X-ray intensity data as well as the obtained lattice parameters attest to the violation of the cubic garnet space group forumla and the symmetry reduction to subgroups with

Table 4.

Cr3+ content [mol%] and observed energies [cm1] of the bands due to [6]Cr3+d–d transitions (in Oh symmetry), as well as local Cr–O bond distances [Å] as determined by X-ray analysis (Wildner & Andrut, 2001b) and calculated according to different models for the investigated uvarovite samples.

Cr3+4A2g4A2g4A2g4A2gRmean$
samplecontent4T2g(4F)4T1g(4F)2Eg(2G) R12Eg(2G) R2forumlaRi,individual§(Vegard's rule)

 
Ska-1 48.3 16310 22650 14264 14368 1.9848 1.9852 1.9576 
Ves-2 65.5 16235 22640 14260 14367 1.9867 1.9877 1.9693 
Sar-w2 67.6 16235 22710 14262 14368 1.9867 1.9880 1.9707 
Sar-kl2 68.5 16220 22640 14264 14365 1.9870 1.9882 1.9713 
Sar-899 68.8 16240 22630 14262 14366 1.9865 1.9882 1.9716 
Sar-desy 70.5 16200 22630 14262 14368 1.9875 1.9885 1.9727 
Uw-syn 100.0 15930 22775 14272 14374 1.9942 1.9928 1.9928 
Cr3+4A2g4A2g4A2g4A2gRmean$
samplecontent4T2g(4F)4T1g(4F)2Eg(2G) R12Eg(2G) R2forumlaRi,individual§(Vegard's rule)

 
Ska-1 48.3 16310 22650 14264 14368 1.9848 1.9852 1.9576 
Ves-2 65.5 16235 22640 14260 14367 1.9867 1.9877 1.9693 
Sar-w2 67.6 16235 22710 14262 14368 1.9867 1.9880 1.9707 
Sar-kl2 68.5 16220 22640 14264 14365 1.9870 1.9882 1.9713 
Sar-899 68.8 16240 22630 14262 14366 1.9865 1.9882 1.9716 
Sar-desy 70.5 16200 22630 14262 14368 1.9875 1.9885 1.9727 
Uw-syn 100.0 15930 22775 14272 14374 1.9942 1.9928 1.9928 

#According to Equation 12

§ According to Equation 35

$ According to Equation 36

triclinic forumla symmetry (samples Ska-1, Ves-2, Sar-desy, Sar-kl2 and Sar-w2) and orthorhombic Fddd symmetry (sample Sar-899). For details about the refinement strategies as well as the criteria for space group selection the reader is referred to the paper by Wildner & Andrut (2001b). The structure refinements reveal that partial long-range Cr3+/Al ordering on the octahedral sites is the most prominent non-cubic feature. Besides, these uvarovites structurally incorporate traces of hydrous component (< 1 wt% H2O) predominantly as O4H4 “hydrogarnet” substitution in a non-cubic way, thus leading to further subtle deviations from cubic symmetry (Andrut & Wildner, 2001b; Andrut et al., 2002).

Upon symmetry reduction, the unique X, Y, Z and O positions of forumla garnets split into as many as six crystallographically independent X', eight Y', six Z' and 24 O' positions in space group forumla. Although the Si and Ca atoms occupy general positions in forumla, they hardly deviate from their respective special positions in forumla The Y positions retain a centre of symmetry in Fddd and forumla, and the deviation of the YO6 octahedra from the forumla symmetry in cubic garnets is extremely small: the average bond length distortion forumla is only 0.92·10−6, ranging from (0.03–3.43)·10−6forumla; Brown & Shannon, 1973). Table 3 lists the atomic site symmetries and multiplicities together with the respective labelling for the garnet symmetries forumla, Fddd, and forumla (Wildner & Andrut, 2001b).

Crystal structure of synthetic uvarovite

In agreement with the isotropic behaviour of synthetic end-member uvarovite under crossed polarisers, all criteria for space group determination from X-ray data – as applied and discussed by Wildner & Andrut (2001b) – confirm “usual” cubic garnet symmetry forumla, in contrast to the results for natural birefringent uvarovite-grossular solid solutions. Hence, the crystal structure of synthetic uvarovite was refined at room temperature in space group forumla (a = 11.9973 Å, Cr–O = 1.9942(6) Å, Si–O = 1.6447(6) Å, Ca–Oa = 2.3504(6) Å, Ca–Ob = 2.4971(6) Å; Andrut & Wildner, 2002). The structure of Ca3Cr2[SiO4]3 complies with crystal chemical expectations for ugrandite group garnets in general as well as with predictions drawn from “cubically averaged” data of non-cubic uvarovite-grossular solid solutions (Wildner & Andrut 2001b). According to the Cr3+ site symmetry forumla (C3i), the octahedral point symmetry is forumla (D3d). As a common feature of all ugrandite garnets, the edges of the YO6 octahedron shared with the CaO8 polyhedron are longer than the unshared ones (e.g.Novak & Gibbs, 1971), corresponding to an octahedral compression along the C3 axis.

Individual octahedral size and Cr3+ occupation

For each natural crystal, a nearly perfect linear correlation of the individual octahedral size with its Cr occupancy was observed in X-ray diffraction experiments. Considering the dependence on the bulk Cr mole fraction, the <Cr/Al–O> distance at an individual octahedral site in non-cubic uvarovite-grossular solid solutions is represented by (Wildner & Andrut, 2001b):  

formula
Equation 34 refers to non-cubic garnets exhibiting different Cr3+-centred octahedral positions. This equation can be used to describe the actual Cr3+–O interatomic distances in cubic binary garnet solid solutions as well as the behaviour according to Vegard's law.

In the former case XCr,individual is set to 1, which leads to:  

formula
Thus, Equation 35 can be used to predict the “real” size of a CrO6 polyhedron within any uvarovite-grossular solid solution with high reliability (see Wildner & Andrut, 2001b). Hence, the relaxation ε for a CrO6 octahedron in grossular (i.e., XCr,bulk = 0) is calculated as ε = 0.77. Furthermore, if XCr,bulk = 1, the Cr–O bond length in end-member uvarovite is predicted to be 1.9928 Å, in excellent agreement with the experimentally determined value of 1.9942(6) Å (Andrut & Wildner, 2002).

In the latter case, by setting XCr,bulk = XCr,individual in Equation 34, one obtains the behaviour according to Vegard's rule (section 3):  

formula
Equation 36 describes the behaviour of the averaged <Cr/Al–O> bond lengths as a function of the bulk Cr3+ content. Thus, this relationship describes the interatomic cationoxygen distance in binary solid solutions that represent the averaged respective individual values of the end-members weighted by the mole fraction.

The different cases are displayed in Figure 1. For reliable single crystal structure investigations of uvarovite-grossular solid solutions, Figure 1 summarises the relation between the Cr content and the cubic cell lengths compared with a Vegard's law plot

Fig. 5.

Calculated local interatomi c Cr3+–O distances in the solid solution based upon Vegard's rule (Eqn. 36, no relaxation; dotted line), and Equation 35 (based on experimental values by Wildner & Andrut, 2001b, thus representing partial relaxation; solid line). In addition, the respective curves for the hard sphere model (full relaxation; dashed lines) are given (compare to section 4.2). Data points (white circles) are calculated from the optical spectra of the natural garnet solid solutions (Wildner & Andrut, 2001b), assuming a 10Dq ∝ 1/R5 relation according to Equation 12. The experimental data points for synthetic uvarovite (Andrut & Wildner, 2002) and grossular (Geiger & Armbruster, 1997) are shown as grey and black circles, respectively.

Fig. 5.

Calculated local interatomi c Cr3+–O distances in the solid solution based upon Vegard's rule (Eqn. 36, no relaxation; dotted line), and Equation 35 (based on experimental values by Wildner & Andrut, 2001b, thus representing partial relaxation; solid line). In addition, the respective curves for the hard sphere model (full relaxation; dashed lines) are given (compare to section 4.2). Data points (white circles) are calculated from the optical spectra of the natural garnet solid solutions (Wildner & Andrut, 2001b), assuming a 10Dq ∝ 1/R5 relation according to Equation 12. The experimental data points for synthetic uvarovite (Andrut & Wildner, 2002) and grossular (Geiger & Armbruster, 1997) are shown as grey and black circles, respectively.

Fig. 6.

Cell edge lengths of synthetic uvarovite (Andrut & Wildner, 2002) and natural uvarovite-grossular solid solutions (Wildner & Andrut, 2001b) as a function of the Y site Cr3+ occupancy, compared with a linear grossular-uvarovite join. The water content is given as integral absorption coefficient αi (cm−2) in the OH stretching region, a (thin) regression line links the three “high-water” uvarovites (0.31–0.34 wt% H2O, cf. Andrut & Wildner, 2001b, 2002). Reliable literature data are also included.

Fig. 6.

Cell edge lengths of synthetic uvarovite (Andrut & Wildner, 2002) and natural uvarovite-grossular solid solutions (Wildner & Andrut, 2001b) as a function of the Y site Cr3+ occupancy, compared with a linear grossular-uvarovite join. The water content is given as integral absorption coefficient αi (cm−2) in the OH stretching region, a (thin) regression line links the three “high-water” uvarovites (0.31–0.34 wt% H2O, cf. Andrut & Wildner, 2001b, 2002). Reliable literature data are also included.

joining the synthetic end-member grossular (a = 11.847 Å; Geiger & Armbruster, 1997) and uvarovite (a = 11.997 Å; Andrut & Wildner, 2002). For intermediate binary compositions, the content of hydrous component – where available given as integral absorption coefficient αi seems to be responsible for deviations from a continuous function (Wildner & Andrut, 2001b).

Single-crystal absorption spectra

Polarised absorption spectra were measured at room temperature in the UV-VIS range between 28000 cm−1 and 10000 cm−1 on a Bruker IFS 66v/S FTIR spectrometer using the attached mirror optics microscope IR-ScopeII. The spectral bandwidth was 20 cm−1, the local resolution was 60 μm.

Fig. 7.

Polarised UV-VIS absorption spectra of sample Sar-w2 and assignment of d–d transitions of Cr3+. The absorptions show an isotropic behaviour. Spectra are offset for clarity. The insert enlarges the spectral range from 14100 to 14500 cm−1, displaying the spin-forbidden 4A2g(4F) → 2Eg(2G) transition split by spin-orbit coupling.

Fig. 7.

Polarised UV-VIS absorption spectra of sample Sar-w2 and assignment of d–d transitions of Cr3+. The absorptions show an isotropic behaviour. Spectra are offset for clarity. The insert enlarges the spectral range from 14100 to 14500 cm−1, displaying the spin-forbidden 4A2g(4F) → 2Eg(2G) transition split by spin-orbit coupling.

As a representative example, the polarised UV-VIS spectra of sample Sar-w2 are displayed in Figure 1 as linear absorption coefficient α vs. wavenumber forumla. They are characterised by two broad absorption bands at around 16250 and 22600 cm−1, which are typical for Cr3+ in octahedral coordination by oxygen atoms (e.g.Lever, 1984). In agreement with the structural results, the comparatively low intensity of the spin-allowed d–d bands of Cr3+ in the UV-VIS region is indicative of an inversion centre at the Cr3+ sites, permitting only dynamic violation of the Laporte selection rule due to uneven octahedral vibrations. There is no significant band polarisation in the UV-VIS energy range, even though the Cr/Al cation distribution and the resulting orientation dependence of “high-Cr” and “low-Cr” octahedra is clearly non-cubic, governing the orientation of the optical indicatrix axes parallel to the lattice axes of the orthorhombic cell, even in the triclinic crystals (Wildner & Andrut, 2001b). The bands show a slight asymmetric shape, but no energy splittings are observed. Therefore, the spectra were at first interpreted on the basis of an effective local crystal field with Ohforumla symmetry. The absorption bands located around 16235 cm−1 (v1) and 22710 cm−1 (v2) in sample Sar-w2, with typical FWHM values of 2200 cm−1 and 3200 cm−1 are assigned to the spin-allowed d–d transitions 4A2g(4F) 4T2g(4F) and 4A2g(4F) → 4T1g(4F), respectively. The third spin-allowed transition 4A2g(4F) → 4T1g(4P) is calculated to occur around 35700 cm−1 and is hence hidden under the absorption edge which represents the low-energy wing of an intense absorption caused by metal-oxygen charge transfer. For a cation with d3 electron configuration in octahedral coordination, the first spin-allowed transition v1 is equivalent to the crystal field splitting parameter 10Dq. Racah B35 (cf. Part I) is a measure of the degree of inter-electronic d–d repulsion and is derived from the following relationship (Lever, 1968):  

formula
The crystal field stabilisation energy for Cr3+ in a crystal field with Oh symmetry is calculated from:  
formula
At their low-energy wings, the spin-allowed transitions exhibit shoulders at 14560 cm−1, 15340 cm−1, 15750 cm−1, 21050 cm−1 and 22000 cm−1. Furthermore, a spin-forbidden quartet → doublet transition is clearly observed in the optical spectra. Sharp peaks at 14260 cm−1 and 14370 cm−1 are attributed to components of the 2Eg(2G) level, split up due to the spin-orbit coupling. Table 4 summarises the data for the investigated garnets.

In addition to the ligand type and the symmetry and geometry of the coordination polyhedron, the crystal field splitting parameter 10Dq is determined by the mean 3dN ion to ligand distance. The derivation of Equations 12 and 33 for symmetries Oh and D3d, respectively, was given in sections 5 and 5.2.

An increase of the bulk chromium content generally expands the octahedral sites in the garnet structure by replacing Al with the larger Cr3+ cations. Consequently, the crystal field strength is reduced due to the larger mean interatomic distances, thus shifting the first spin-allowed d–d absorption band 4A2g(4F) → 4T2g(4F) to lower wavenumbers. Figure 1 shows the relation between the energy of this band (= 10Dq) and the Cr3+ content for several grossular-uvarovite solid solutions. Excluding outlying points (grey symbols), the correlation is 10Dq [cm−1] = 16668 – 6.75XCr3+ [mol%] with r2 = 0.88. The chromium content in our samples ranges from 48 to 100 mol% uvarovite component, and this variation results in a band shift of 380 cm−1 for the first spin-allowed transition (Fig. 8). Similarly, the second spin-allowed transition is also shifted to lower wave numbers with increasing Cr3+ content, but with a different slope due to configurational interaction with the 4T1g(4P) state of alike symmetry (see Figs. 6 and 7 in Part I). On the other hand, the spin-forbidden transitions are crystal field independent to a first approximation (Table 4), in accordance with the corresponding Tanabe-Sugano diagram (Tanabe & Sugano, 1954; cf. Part I, Fig. 7 and section 3.3.4).

Fig. 8.

Crystal field splitting parameter 10Dq for [6]Cr3+ in grossular-uvarovite solid solutions. Circles represent samples of this study (Andrut & Wildner, 2001b, 2002; Sar-w2t was heated to 1000 °C, cf. Andrut & Wildner, 2001b). Inverted triangles refer to literature data by Amthauer (1976; Gross Magog, Outokumpu), Taran et al. (1994; Gross Ural, Uv Ural), Abu Eid (1976, Uv syn Abu), and Langer & Abu Eid (1977, Uv syn L&A). The dotted triangle (Uv syn JDB) represents a reinvestigated synthetic crystal from Bass (1986). Grey symbols were not used for calculating the regression line, i.e., 10Dq [cm−1] = 16668 – 6.75XCr3+[mol%].

Fig. 8.

Crystal field splitting parameter 10Dq for [6]Cr3+ in grossular-uvarovite solid solutions. Circles represent samples of this study (Andrut & Wildner, 2001b, 2002; Sar-w2t was heated to 1000 °C, cf. Andrut & Wildner, 2001b). Inverted triangles refer to literature data by Amthauer (1976; Gross Magog, Outokumpu), Taran et al. (1994; Gross Ural, Uv Ural), Abu Eid (1976, Uv syn Abu), and Langer & Abu Eid (1977, Uv syn L&A). The dotted triangle (Uv syn JDB) represents a reinvestigated synthetic crystal from Bass (1986). Grey symbols were not used for calculating the regression line, i.e., 10Dq [cm−1] = 16668 – 6.75XCr3+[mol%].

Crystal field SPM calculations

Due to the comparable high point symmetry 33m of the CrO6 octahedron with six equal Cr-O bonds in synthetic uvarovite, the equations for the crystal field parameters Bkq simplify (see discussion in section 5.2), leading to Equations 2931.

The cubic crystal field parameter Dqcub can be calculated from to the trigonal Dqtrig and the distortion parameter Dτ via the following equation (König & Kremer, 1977):  

formula
For a given angle θ between the threefold axis and the metal-ligand vector of an octahedron of D3d symmetry, the distortion parameter Dτ can be predicted according to König & Kremer (1977) with:  
formula
(compare with the corresponding relation, expressed in terms of Dqtrig, in Part I, section 3.3.5).

The determination of the SPM parameter set for uvarovite-grossular solid solutions had to be performed in a few successive steps, because of (a) the small number of observables derived from the optical spectra, and (b) RiR0 in the case of synthetic uvarovite, where the particular values of tk have only a marginal influence on the calculations.

(i) In the beginning of the fitting process, the power-law exponents were fixed at t4 = 5 and t2 = 3, but the forumla values were varied over a wide range to avoid any wrong minimum during the fitting process. Nevertheless, the range of values for forumla is restricted, since both spin-allowed absorption bands show no significant band splitting. From Equation 31 it is evident that the energy of the first spin-allowed transition is directly related to forumla, whereas forumla (Eqn. 29) predominantly governs the splitting of the energy levels.

(ii) After having determined the approximate magnitude of the forumla for synthetic uvarovite, the value of the power-law exponent t4 was constrained using additional data sets of the six natural uvarovite-grossular solid solutions with Ri < R0, (Table 4). In particular, the actual mean octahedral Cr–O bond length was calculated by Equation 35 using the respective value of XCr,bulk determined by microprobe analysis (Table 4). The corresponding polyhedral shape for the SPM calculations was modelled by applying respective hard constraints in the crystal structure refinements (cf.Andrut & Wildner, 2002). Hence, the values of forumla and forumla obtained on synthetic uvarovite were applied in the individual fitting of the exponential parameter t4 for each natural garnet to describe the respective first spin-allowed transition (the choice of t2 has no influence on the calculation of the first spin-allowed band, compare Equations 29 to 31. The values of t4 determined in this way for the six natural samples scatter only slightly between 6.5 and 7.1.

(iii) The mean value t4 = 6.7 was used in the final SPM calculations, including a recalculation for synthetic uvarovite. The spin-orbit coupling parameter ζ was estimated from literature data to be ζ = 135 cm−1, and kept constant for all subsequent calculations. Thus, best fit results for the spin-allowed bands were attained with forumla = 9532 cm−1, forumla = 4650 cm−1, t4 = 6.7, t2 (fixed) = 3. Racah B35 was fitted individually for all investigated samples. The Racah parameters B55 and C (see Part I, section 3.6) were calculated from the energies of the 4A2g(4F) → 2T2g(2G) and the mean value of the 4A2g(4F) → 2Eg(2G) transition, v2T2 and v2E, respectively, via  

formula
 
formula
Both parameters were varied in additional calculations, but only a limited agreement with the experimental data for all spin-forbidden transitions was achieved. Table 5 summarises all relevant parameters for synthetic uvarovite, i.e., SPM parameters, non-zero Bkq (Wybourne notation), corresponding conventional CF parameters (Ballhausen, 1962), rotational invariants sk (Leavitt, 1982; Yeung & Newman, 1985), and resulting cubic crystal field parameters Dqcub. Furthermore, s4 is related to the commonly used cubic crystal field parameter Dqcub by  
formula
and can thus be compared with crystal field analyses based on other formalisms

Table 5.

Summary of SPM and crystal field parameters (non-zero Bkq in Wybourne notation, intrinsic forumla and tk, rotational invariants sk, cubic field strengths Dqcub), inter-electronic repulsion parameters and conventional CF parameters for the Cr3+-bearing garnets investigated at 290 K. The respective values of the crystal field strength Dqcub and Racah B35, derived from spectrum analysis using an Oh symmetry approach are given for comparison. Last rows: application of the SPM with calculated band positions (mean), experimental values are given for comparison. Values are in cm1 (except tk, β, C/B).

Uvasyn-22Sar-desySar-kl2Sar-899Sar-w2Ves-2Ska-1

forumla9532
forumla4650
t46.7
t2 (fixed)3
ζ (fixed)135
B20–559–637–657–665–653–659–662
B40–21621–21892–21888–21893–21927–21937–22105
B43–26908–27397–27431–27454–27460–27495–27707
S414588.914834.114847.414857.714863.814881.414996.0
S2250.0285.0293.6297.3292.2294.6295.9
Dqcub (from s4)1591.81618.51620.01621.11621.81623.71636.2
Racah B35703642641638648636620
Racah B55714715718717716717715
Racah C3165315231443148315031483149
β35 (B0 = 995[1])0.710.640.640.640.650.640.62
C/B354.514.914.904.934.864.955.08
Dqcub (from Dqtrig)1591.51618.21619.61620.81621.41623.41635.9
Dqtrig1608.01637.31639.31640.71641.01643.11655.8
–42.4–49.0–50.6–51.2–50.4–50.8–51.3
79.991.093.895.093.494.194.5
Dqcub (from spectra [2])1593.01620.01622.01624.01623.51623.51631.0
Racah B35 [2]703641640636647638628
application of SPM:
4A2g(F) → 4T2g(F)
calc. (mean)15930161931620816219162261624416369
observed15930162001622016240162351623516310
4A2g(F) → 4T1g(F)
calc. (mean)22776226302263822627227102264122653
observed22775226302264022630227102264022650
Uvasyn-22Sar-desySar-kl2Sar-899Sar-w2Ves-2Ska-1

forumla9532
forumla4650
t46.7
t2 (fixed)3
ζ (fixed)135
B20–559–637–657–665–653–659–662
B40–21621–21892–21888–21893–21927–21937–22105
B43–26908–27397–27431–27454–27460–27495–27707
S414588.914834.114847.414857.714863.814881.414996.0
S2250.0285.0293.6297.3292.2294.6295.9
Dqcub (from s4)1591.81618.51620.01621.11621.81623.71636.2
Racah B35703642641638648636620
Racah B55714715718717716717715
Racah C3165315231443148315031483149
β35 (B0 = 995[1])0.710.640.640.640.650.640.62
C/B354.514.914.904.934.864.955.08
Dqcub (from Dqtrig)1591.51618.21619.61620.81621.41623.41635.9
Dqtrig1608.01637.31639.31640.71641.01643.11655.8
–42.4–49.0–50.6–51.2–50.4–50.8–51.3
79.991.093.895.093.494.194.5
Dqcub (from spectra [2])1593.01620.01622.01624.01623.51623.51631.0
Racah B35 [2]703641640636647638628
application of SPM:
4A2g(F) → 4T2g(F)
calc. (mean)15930161931620816219162261624416369
observed15930162001622016240162351623516310
4A2g(F) → 4T1g(F)
calc. (mean)22776226302263822627227102264122653
observed22775226302264022630227102264022650

The refined set of SPM parameters for Cr3+ was then applied to describe all observed spin-allowed transitions of the natural garnets, i.e., only the Racah parameter B35 had to be aligned. Racah B55 and C of the natural samples were again estimated using Equations 41 and 42. The respective results, i.e. non-zero Bkq, rotational invariants, corresponding Dqcub, and Racah parameters are also listed in

Table 5. This Table also shows the energies calculated from SPM evaluation compared with the observed energies of the first two spin-allowed transitions. Slight deviations, if any, only occur for the first spin-allowed transition, the maximum error is less than 0.5% for sample Ska-1. Calculated energy splittings within the first and second spin-allowed band are in the order of 150 cm−1 and 350 cm−1, respectively. These values are too small to cause a resolvable splitting of the experimental bands with FWHM's in the order of 2000–3000 cm−1.

The results of the SPM analyses for synthetic and natural uvarovites do not meet the general expectation forumla (Yeung & Newman, 1986). However, the magnitude of the forumla tolerably complies with data by Stedman (1969) and Yeung & Newman (1986) obtained on ruby and kyanite, respectively, taking the different reference distances R0 used into account. The possible range of forumla was restricted in the present case to be in the order of a few thousand cm−1, due to the fact that the observed spin-allowed transitions exhibit no energy splitting. Using for example forumla = 35000 cm−1 proposed for Cr3+ in kyanite and ruby (Yeung & Newman, 1986) or forumla up to ∼ 70000 cm−1 we extracted for eskolaite (see below), leads to a stronger energy splitting of the spin-allowed bands, which is not in accordance with the experimental results.

It is noteworthy that spin-forbidden and spin-allowed transitions cannot be described with a single set of SPM parameters, although different Racah B values are taken into account. The observed splitting of the 2Eg level of ∼ 100 cm−1 cannot be explained by the spin-orbit coupling parameter ζ alone, but only by a simultaneous increase of forumla up to about 40000 cm−1. However, as pointed out above, the concomitant splitting of the spin-allowed bands would be in contradiction to the experimental results.

In contrast to the previous SPM investigations on Cr3+-bearing phases (Yeung & Newman, 1986; Yeung et al., 1994a, 1994b; Qin et al., 1994), our data were derived using a consistent set of thoroughly characterised compounds with respect to both chemical composition and interatomic distances of the chromium-hosting octahedron. In addition, the SPM parameters determined on uvarovite comply very well with the conventional CF parameters.

Using Equation 35 with XCr,bulk = 0, the individual Cr–O bond distance in grossular is predicted to be 1.9781 Å, which corresponds to a rather high degree of relaxation with ε = 0.77. Via Equation 12 the corresponding extrapolated 10Dq value is estimated to be 16590 cm−1. Hence, over the whole solid solution series, a total band shift of 660 cm−1 is predicted for the first spin-allowed band. Due to the comparable small band shift, the quotient 10DqXCr3+ = 1/10DqXCr3+ deviates only marginally from 1 (see Eqns. 12 and 33). A slight slope of the curve that may indicate an exponential behaviour will not be recognised. Only in case of a larger change of 10Dq with composition, the exponential behaviour could be detected. Thus, in first approximation, one observes for 10Dq a linear relationship as a function of the Cr3+ mole fraction (Fig. 8).

Conclusions

The structural relationships drawn from natural birefringent uvarovite-grossular solid solutions are fully obeyed by synthetic uvarovite, showing that these relationships are suited to establish precise structural predictions. In particular, a set of SPM parameters, extracted from the electronic absorption spectrum of synthetic uvarovite, could be refined using the predicted actual Cr–O bond lengths in uvarovite-grossular solid solutions (forumla = 9532 cm−1, forumla = 4650 cm−1, t4 = 6.7, t2 (fixed) = 3). This SPM parameter set is valid for the complete uvarovite-grossular solid solution series and enables to calculate the Cr–O bond length Ri solely from a single UV-VIS spectroscopic observation (or vice versa, cf. Figs. 5 and 8). In turn, a perfect agreement between the calculated and observed d–d transition energies corroborates the applicability of the SPM concept to d block element bearing compounds in geosciences.

Local interatomic bond lengths and angles derived from optical absorption spectra: The CrO6 polyhedron in ruby, Al2O3:Cr3+

Introduction

Rubies, Al2O3:Cr3+, are of particular interest due to their continuing importance in laser technology (e.g. Morrison, 1992) and for determining the pressure in diamond anvil cell experiments using the pressure dependency of the ruby R1 line (e.g. Eggert et al., 1989, and references therein). Therefore, the vast majority of respective spectroscopic studies concentrated on the narrow energy range of the spin-forbidden ruby lines between 13000 and 15000 cm−1.

Superposition model analyses for Cr3+ in ruby have been performed so far by Stedman (1969), Clare & Devine (1983), Müller & Berlinger (1983), and Yeung & Newman (1986) by fitting the SPM parameter to the EPR data of strained ruby, but only Stedman (1969) additionally analysed the optical absorption spectrum.

Most of the optical spectroscopic studies performed at ambient conditions in the binary system Cr2O3–Al2O3 have been devoted to either reflection measurements of powdered samples for the whole solid solution series (Poole & Itzel, 1963; Schmitz-DuMont & Reinen, 1959; Neuhaus, 1960; Reinen, 1969) or polarised single crystal studies of the end-member eskolaite Cr2O3 (McClure, 1963) and ruby, i.e., Cr3+-doped corundum (McClure, 1962; MacFarlane, 1963; Langer & Andrut, 1996). Additionally, optical spectra of rubies have been measured as a function of temperature (e.g. Taran et al., 1994) and pressure (Stephens & Drickamer, 1961; Langer et al., 1997).

Brief description of the crystal structures of corundum and eskolaite

The end-member compounds corundum and eskolaite crystallise isomorphously in space group forumla (e.g.Newnham & de Haan, 1962; Finger & Hazen, 1978, 1980; Sawada, 1994a, 1994b). The structures are usually described as a slightly distorted hexagonal close packing of oxygen, where the cations occupy 2/3 of the octahedral voids. The site symmetry of the M3+ cation is C3, thus enabling (a) a displacement of the cation along the threefold axis relative to the centre of the octahedron, resulting in two different X–O distances, (b) different sizes of the two oxygen triangles lying perpendicular to the threefold axis, and (c) a mutual rotation of these two triangles ≠ 60°. A comparison of the end-member structures reveals that the AlO6 and CrO6 polyhedra differ in size as well as in distortion, as shown in Figure 1. Consequently, crystal structure refinements of (Al,Cr)2O3 solid solutions based on standard diffraction experiments will average over all polyhedra of a particular crystallographic site and are, thus, not specific for, e.g., the Cr3+-centred sites themselves. Hence, single crystal structure analyses of rubies yield the mean (Cr,Al)–O bond distances as <Cr/Al–O> ≈ 1.92 Å, which is close to that for pure Al2O3 (McCauley & Gibbs, 1972; Moss & Newnhan, 1964; Tsirel'son et al., 1983). In some of these investigations the authors proposed (contradictory) results indicating a shift of the Cr3+ cation position along the threefold axis as compared with Al (Tsirel'son et al., 1985).

Fig. 9.

Comparison of the AlO6 (left hand side) and Cr3+O6 (right hand side) polyhedra in corundum and eskolaite, respectively. Polar coordinates (trigonal setting) are labelled at the AlO6 polyhedron. Numerical polyhedral data are included for the respective polyhedra in the end-members (in bold Arial font, X-ray data from Finger & Hazen, 1978, 1980), as well as for an actual CrO6 polyhedron in a ruby crystal from Longido containing 1.5 mol% Cr2O3 (in italic Times New Roman font, data extracted using SPM analysis from polarised optical absorption spectra of eskolaite and ruby, present paper; see Table 6 for further data).

Fig. 9.

Comparison of the AlO6 (left hand side) and Cr3+O6 (right hand side) polyhedra in corundum and eskolaite, respectively. Polar coordinates (trigonal setting) are labelled at the AlO6 polyhedron. Numerical polyhedral data are included for the respective polyhedra in the end-members (in bold Arial font, X-ray data from Finger & Hazen, 1978, 1980), as well as for an actual CrO6 polyhedron in a ruby crystal from Longido containing 1.5 mol% Cr2O3 (in italic Times New Roman font, data extracted using SPM analysis from polarised optical absorption spectra of eskolaite and ruby, present paper; see Table 6 for further data).

In contrast, absorption spectroscopic methods indicate more realistic mean Cr–O distances. Up to now, a <Cr–O> bond length of 1.957 Å was estimated from polarised optical absorption spectroscopy (Langer, 2001), whereas EXAFS studies even revealed individual Cr–O distances (1.93 and 2.02 Å, Sainctavit et al., 2001; 1.92 and 2.01 Å, Gaudry et al., 2003). Our present investigations aim at a full geometrical characterisation of a local CrO6 polyhedron in ruby, extracted solely from the SPM analyses of polarised optical absorption spectra of ruby and eskolaite. In the following, the experimental and computational procedures are outlined and the most important results are summarised. The full details of these investigations will be presented in a forthcoming paper by Andrut & Wildner.

Absorption spectroscopic investigations

Polarised absorption spectra of synthetic rubies were measured at room temperature in the UV-VIS range between 32000 cm−1 and 10000 cm−1 on a Bruker IFS 66v/S FT- spectrometer. Figure 1 in Part I shows a representative example (sample syn2) of the spectra polarised parallel and perpendicular to the threefold c axis. For a detailed interpretation and band assignment we refer to Part I.

In addition, the polarised absorption spectra of eskolaite by McClure (1963) and those of natural rubies by Langer & Andrut (1996) were re-evaluated concerning peak centres and barycentres of the spin-allowed bands, in order to assure a consistent treatment with respect to our present measurements. The results are summarised in Table 6. In Figure 1 the energy of the first spin-allowed transition v1 of Cr3+ is displayed as a function of the composition of the corundum-eskolaite solid solution series. Data obtained by powder reflection measurements are also included for comparison (Schmitz-Dumont & Reinen, 1959; Neuhaus, 1960; Reinen, 1969).

Table 6.

Chemical compositions, energies of the spin-allowed transitions of Cr3+, conventional CF parameters (C3v approximation; program TETRIG, Wildner, 1996b) and polyhedral geometries determined for Cr3+ in the investigated rubies (SPM analyses) and eskolaite (X-ray) compared with the AlO6 polyhedron in corundum (X-ray).

sampleeskolaiteruby Longidoruby Mysoresynthetic rubysynthetic rubycorundum
[1, 2]syn2syn1[3]

ChemistryCr2O3Al1.97Cr0.03O3Al1.99Cr0.01O3Al2.00Cr0.00O3Al1.98Cr0.02O3Al2O3
EMPAl2O3: 97.2Al2O3: 99.3Al2O3: 99.8Al2O3: 98.5
[wt%]Cr2O3: 1.82Cr2O3: 0.81Cr2O3: 0.14Cr2O3: 1.48
Fe2O3: 0.28Fe2O3: 0.07
Spectra
Eobs 4E1677517800179251789017860
[cm−1] 4A1639018200182001836018300
4E2142024180242602434024320
4A2185025000250602512025060
CF
Dq [cm−1]1703.31847.81853.31857.81850.7
Dτ [cm−1]–117.3–138.1–134.0–132.5–124.9
Dσ [cm−1]549.161.2129.25.35.7
B35 [cm−1]457.2629.6629.6631.5633.2
GeometryX-raySPMSPMSPMSPMX-ray
R1 [Å]2.0091.9721.9641.9631.9611.971
R2 [Å]1.9621.9251.9301.9281.9341.856
forumla1.9861.94851.9471.94551.94751.914
θ1 [°]48.7547.6847.8747.7447.9247.67
θ2 [°]118.50118.01117.98118.01118.12116.85
φ2φ1 [°]55.9756.1055.8455.7855.9056.11
sampleeskolaiteruby Longidoruby Mysoresynthetic rubysynthetic rubycorundum
[1, 2]syn2syn1[3]

ChemistryCr2O3Al1.97Cr0.03O3Al1.99Cr0.01O3Al2.00Cr0.00O3Al1.98Cr0.02O3Al2O3
EMPAl2O3: 97.2Al2O3: 99.3Al2O3: 99.8Al2O3: 98.5
[wt%]Cr2O3: 1.82Cr2O3: 0.81Cr2O3: 0.14Cr2O3: 1.48
Fe2O3: 0.28Fe2O3: 0.07
Spectra
Eobs 4E1677517800179251789017860
[cm−1] 4A1639018200182001836018300
4E2142024180242602434024320
4A2185025000250602512025060
CF
Dq [cm−1]1703.31847.81853.31857.81850.7
Dτ [cm−1]–117.3–138.1–134.0–132.5–124.9
Dσ [cm−1]549.161.2129.25.35.7
B35 [cm−1]457.2629.6629.6631.5633.2
GeometryX-raySPMSPMSPMSPMX-ray
R1 [Å]2.0091.9721.9641.9631.9611.971
R2 [Å]1.9621.9251.9301.9281.9341.856
forumla1.9861.94851.9471.94551.94751.914
θ1 [°]48.7547.6847.8747.7447.9247.67
θ2 [°]118.50118.01117.98118.01118.12116.85
φ2φ1 [°]55.9756.1055.8455.7855.9056.11
Fig. 10.

Energy of the first spin-allowed transition v1, 4A2g(4F) → 4T2g(4F) (Oh), of Cr3+ as a function of the composition of the solid solution corundum-eskolaite. Comparison of data obtained from single crystal investigations on the basis of a local symmetry C3 (grey triangle symbols: this work; eskolaite: McClure, 1963) with mean values (white circles), and data obtained by powder reflection measurements (dark grey circles; Schmitz-DuMont & Reinen, 1959; Neuhaus, 1960; Reinen, 1969). The dotted lines indicate the inversion of the trigonal split components (4A, 4E) along the solid solution series.

Fig. 10.

Energy of the first spin-allowed transition v1, 4A2g(4F) → 4T2g(4F) (Oh), of Cr3+ as a function of the composition of the solid solution corundum-eskolaite. Comparison of data obtained from single crystal investigations on the basis of a local symmetry C3 (grey triangle symbols: this work; eskolaite: McClure, 1963) with mean values (white circles), and data obtained by powder reflection measurements (dark grey circles; Schmitz-DuMont & Reinen, 1959; Neuhaus, 1960; Reinen, 1969). The dotted lines indicate the inversion of the trigonal split components (4A, 4E) along the solid solution series.

As a result of the different ionic radii in six-fold coordination (rCr3+ = 0.615 Å and rAl = 0.535 Å; Shannon, 1976) an increase of the bulk chromium content leads to an expansion of the octahedral sites in the structure. Due to the concomitant reduction of the crystal field strength, the trigonal split levels of the first spin-allowed d–d band 4A2g 4T2g (Oh) shift to lower wavenumbers. Over the whole corundum-eskolaite solid solution series a band shift of 1900 and 1300 cm−1 is observed for the split components 4A → 4A and 4A → 4E, respectively. It is important to note that due to their different slopes these two split levels change their order as a function of the Cr3+ content: in ruby the energy sequence is 4A → 4E < 4A → 4A, while in eskolaite 4A → 4E > 4A → 4A is observed. The split levels of the second spin-allowed transition are also shifted to lower wave numbers with increasing Cr3+ content, but they keep their relative order 4A → 4E < 4A → 4A. In addition, the slope of the second spin-allowed band (and its respective split levels) is influenced by the configurational interaction with the 4T1g(4P) split states of alike symmetry.

Crystal field SPM calculations

In accordance with the C3 point symmetry of the CrO6 octahedron in eskolaite, the electronic z axis of the crystal field was chosen parallel to the threefold axis. The

reference metal-ligand distance R0 was set to 1.995 Å, the sum of the ionic radii of [6]Cr3+ and [3–4]O2– (Shannon, 1976). Preliminary calculations showed that spin-orbit coupling effects can be neglected when dealing with the spin-allowed bands only. In this case, the only necessary free-ion parameter is Racah B35.

Due to the C3 point symmetry of the CrO6 octahedron with only two different Cr–O bonds, the equations for the crystal field parameters Bkq simplify, in that only two coordination factors Kkq(θi, φi) for each Bkq are needed and – proper orientation provided – only the real B20, B40, B43 and the imaginary B43 are non-zero (compare Table 1). Hence, the relationships for B20, B40, and B43 to the conventional CF and distortion parameters for higher trigonal symmetries can be taken as a fairly good approximation (compare to section 5.2, Eqns. 2931). Nevertheless, it is emphasised that all SPM calculations described below have been performed with the full set of crystal field parameters Bkq to allow a free variation of the geometrical data in the fitting process.

We started the SPM analyses with synthetic eskolaite, extracting the band positions from the polarised optical absorption spectra published by McClure (1963), and employing the structural data by Finger & Hazen (1980; see Table 6). In the course of the fitting process it became obvious that the influence of t2 was marginal with respect to forumla and the given ratio of R0/Ri. Therefore, this power-law exponent was fixed at t2 = 3 in the calculations. Thus, best fit results for the spin-allowed bands were attained with the SPM parameters forumla = 10380 cm−1, forumla = 71900 cm−1, t4 = 5.0, t2 (fixed) = 3, and with Racah B35 = 457 cm−1. Table 6 summarises relevant spectroscopic, CF and structural parameters for eskolaite.

As the next step, the supplementary programs for the SPM parameter variation and the actual SPM calculations (Wildner & Andrut, unpublished) had to be modified in order to allow a variation of the polyhedral polar coordinates at fixed SPM input parameters. According to the C3 site symmetry of the CrO6 polyhedron, two different Cr–O bond lengths, two different polar angles θ and one polar angle φ (for the relative rotation of one oxygen triangle) had to be fitted.

Hence, the refined set of SPM parameters for [6]Cr3+ in synthetic eskolaite was then applied to the investigated natural and synthetic rubies. The local CrO6 polyhedra in these ruby samples were modelled by minimising the difference between the observed and calculated spin-allowed energy levels.

The final fully optimised geometrical data for the local CrO6 polyhedron in some natural and synthetic rubies are listed in Table 6 and compared with the respective data for eskolaite and corundum. The results indicate that on insular incorporation of Cr3+ in the Al2O3 matrix, the longer R1 distance (∼ 1.97 Å) does not change significantly, whereas the short R2 bond length with Al–O = 1.856 Å strongly increases to Cr–O ≈ 1.93 Å, resulting in a mean <Cr–O> distance of about 1.947 Å. Similarly, the θ1 angle between the R1 bonds and the threefold axis remains rather constant compared to the AlO6 polyhedron in corundum, while the θ2 angle related to the shorter R2 bonds increases to ∼ 118°. The mutual rotation φ of the two oxygen triangles seems to be rather constant over the whole solid solution series. Generally, the CrO6 polyhedra in ruby are less distorted than the AlO6 ones, a trend which is also confirmed by the structural data for the end-member eskolaite. The results of our SPM analyses reveal a moderate degree of relaxation of ε ≈ 0.46 for Cr3+ within the Al2O3 matrix (Eqn. 15). This value is comparable to the results by Langer (2001) who found, from conventional CF considerations, an average <Cr–O> bond length of 1.957 Å, i.e., ε ≈ 0.60. On the other hand, EXAFS was used to determine the two individual Cr–O distances to be 1.93 and 2.02 Å, giving <Cr–O> = 1.975 Å (Sainctavit et al., 2001), whereas 1.92 and 2.01 Å with <Cr–O> = 1.965 Å (Gaudry et al., 2003). The values of Sainctavit et al. (2001) correspond to a very high degree of relaxation with ε ≈ 0.85, and would imply an extremely strong power-law dependence of the CF strength from the bond lengths in the order of t4 ≈ 15. Furthermore, the longer Cr–O distance even exceeds that found in end-member eskolaite using diffraction methods (Cr–O = 2.01 Å). The values by Gaudry et al. (2003) yield ε ≈ 0.71. An ab initio DFT calculation by the same authors yield distances of 1.95 and 2.00 Å, which again correspond to a high relaxation of ε ≈ 0.85. These authors consequently concluded that the modification of the colour in ruby and eskolaite does not originate from the difference in the Cr–O distances. However, their findings are in contrast to our present results revealing a moderate relaxation with ε < 0.5. Figure 1 summarises the present results and compares them with the cited EXAFS data as well as with relevant single crystal structure investigations (see 6.3.2).

Fig. 11.

Comparison of the individual interatomic distances for Cr3+O6 and AlO6 polyhedra as a function of the composition of the solid solution series corundum-eskolaite, derived from X-ray single crystal data (white [Cr] and black symbols [Al]) and from spectroscopic investigations (grey symbols [Cr]). The lines describe the behaviour of the Cr3+O6 polyhedron under the assumption of full relaxation (dashed), partial relaxation (solid) and no relaxation (dotted).

Fig. 11.

Comparison of the individual interatomic distances for Cr3+O6 and AlO6 polyhedra as a function of the composition of the solid solution series corundum-eskolaite, derived from X-ray single crystal data (white [Cr] and black symbols [Al]) and from spectroscopic investigations (grey symbols [Cr]). The lines describe the behaviour of the Cr3+O6 polyhedron under the assumption of full relaxation (dashed), partial relaxation (solid) and no relaxation (dotted).

Conclusions

The results of the SPM analyses for eskolaite obey the general expectation forumla (Yeung & Newman, 1986). Considering the different reference distances R0 used, the magnitudes of forumla and forumla tolerably comply with data by Stedman (1969) and Yeung & Newman (1986) obtained for ruby and kyanite, respectively. On the contrary, for Cr3+ in uvarovite a much smaller value for forumla in the order of 5000 cm−1 has been determined, corresponding to an indiscernible trigonal energy splitting of the spin-allowed transitions (see 6.2.7 and Andrut & Wildner, 2002).

As we pointed out earlier, until the present investigations, the sets of SPM parameters for Cr3+ in ruby have already been presented by several authors (Stedman, 1969; Clare & Devine, 1983; Müller & Berlinger, 1983; Yeung & Newman, 1986), which were subsequently transferred by Yeung and co-workers (Yeung et al., 1994a, 1994b; Qin et al., 1994) to Cr3+ in kyanite. However, in all these cases chromium represented a trace element replacing aluminium. A critical survey reveals that in those investigations the geometries and sizes of the CrO6 octahedra were only estimated and the respective SPM parameters were evaluated lacking precisely determined interatomic distances. Thus, the actual distortion of the Cr3+ polyhedron was not appropriately taken into account in these investigations. Besides, the SPM parameters were related to a rather arbitrary reference distance R0, making a direct comparison with our data difficult. It is furthermore noteworthy that these earlier investigations aimed at describing the ground multiplet level splitting, disregarding the respective results for the higher multiplet levels. A closer inspection of the calculated energies for these excited levels shows discrepancies with the experimentally determined band positions. In addition, rather large energy splittings in the order of at least 3500 cm−1 are calculated for the spin-allowed transitions, which are not in agreement with the experimental data reported so far.

However, our set of SPM parameters for Cr3+ was derived from absorption spectra and structural data of a well-characterised synthetic end-member eskolaite, which was then applied to chemically and spectroscopically characterised natural and synthetic rubies. In this way, a complete description of the local structure of a CrO6 polyhedron in an Al2O3 matrix solely from optical absorption spectra could be realised for the first time.

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Acknowledgements

The authors thank D.J. Newman, Southampton, for useful comments which helped us to improve the manuscript. MW and MA gratefully acknowledge financial support to MA by a research fellowship from the Austrian Science Fund (FWF) for the project “Superposition model analysis for application in mineralogy”, no. P13976-CHE.

Figures & Tables

Fig.1.

Dependence of the individual bond lengths R(AC) and R(BC) in a binary solid solution A1–XBXC according to different approaches. No relaxation (ε = 0) exists for the virtual crystal approximation (VCA), which corresponds to a behaviour according to Vegard's rule. In case of full relaxation (ε = 1), the individual bond lengths Rrelax are equal to those of the end-member. Real changes of the individual bond length are expressed in terms of Rexp, which is a function of composition (after Urusov, 1992).

Fig.1.

Dependence of the individual bond lengths R(AC) and R(BC) in a binary solid solution A1–XBXC according to different approaches. No relaxation (ε = 0) exists for the virtual crystal approximation (VCA), which corresponds to a behaviour according to Vegard's rule. In case of full relaxation (ε = 1), the individual bond lengths Rrelax are equal to those of the end-member. Real changes of the individual bond length are expressed in terms of Rexp, which is a function of composition (after Urusov, 1992).

Fig. 2.

Polar coordinates of the ligands of a regular octahedron.

Fig. 2.

Polar coordinates of the ligands of a regular octahedron.

Fig. 3.

Flow chart diagram depicting the course of spectroscopic and SPM investigations.

Fig. 3.

Flow chart diagram depicting the course of spectroscopic and SPM investigations.

Fig. 4.

Graphical representation of observed and calculated energy levels up to 30000 cm−1 for the Co2+ sites in Li2Co3(SeO3)4. Parental free-ion term labels and quartet split labels for Oh symmetry are included. All bold lines, numbers and labels refer to spin-allowed, normal to spin-forbidden levels. Numerical results are given for the spin-allowed and some selected spin-forbidden levels. Percentage values in parentheses denote approximate quartet admixtures to spin-forbidden doublet levels.

Fig. 4.

Graphical representation of observed and calculated energy levels up to 30000 cm−1 for the Co2+ sites in Li2Co3(SeO3)4. Parental free-ion term labels and quartet split labels for Oh symmetry are included. All bold lines, numbers and labels refer to spin-allowed, normal to spin-forbidden levels. Numerical results are given for the spin-allowed and some selected spin-forbidden levels. Percentage values in parentheses denote approximate quartet admixtures to spin-forbidden doublet levels.

Fig. 5.

Calculated local interatomi c Cr3+–O distances in the solid solution based upon Vegard's rule (Eqn. 36, no relaxation; dotted line), and Equation 35 (based on experimental values by Wildner & Andrut, 2001b, thus representing partial relaxation; solid line). In addition, the respective curves for the hard sphere model (full relaxation; dashed lines) are given (compare to section 4.2). Data points (white circles) are calculated from the optical spectra of the natural garnet solid solutions (Wildner & Andrut, 2001b), assuming a 10Dq ∝ 1/R5 relation according to Equation 12. The experimental data points for synthetic uvarovite (Andrut & Wildner, 2002) and grossular (Geiger & Armbruster, 1997) are shown as grey and black circles, respectively.

Fig. 5.

Calculated local interatomi c Cr3+–O distances in the solid solution based upon Vegard's rule (Eqn. 36, no relaxation; dotted line), and Equation 35 (based on experimental values by Wildner & Andrut, 2001b, thus representing partial relaxation; solid line). In addition, the respective curves for the hard sphere model (full relaxation; dashed lines) are given (compare to section 4.2). Data points (white circles) are calculated from the optical spectra of the natural garnet solid solutions (Wildner & Andrut, 2001b), assuming a 10Dq ∝ 1/R5 relation according to Equation 12. The experimental data points for synthetic uvarovite (Andrut & Wildner, 2002) and grossular (Geiger & Armbruster, 1997) are shown as grey and black circles, respectively.

Fig. 6.

Cell edge lengths of synthetic uvarovite (Andrut & Wildner, 2002) and natural uvarovite-grossular solid solutions (Wildner & Andrut, 2001b) as a function of the Y site Cr3+ occupancy, compared with a linear grossular-uvarovite join. The water content is given as integral absorption coefficient αi (cm−2) in the OH stretching region, a (thin) regression line links the three “high-water” uvarovites (0.31–0.34 wt% H2O, cf. Andrut & Wildner, 2001b, 2002). Reliable literature data are also included.

Fig. 6.

Cell edge lengths of synthetic uvarovite (Andrut & Wildner, 2002) and natural uvarovite-grossular solid solutions (Wildner & Andrut, 2001b) as a function of the Y site Cr3+ occupancy, compared with a linear grossular-uvarovite join. The water content is given as integral absorption coefficient αi (cm−2) in the OH stretching region, a (thin) regression line links the three “high-water” uvarovites (0.31–0.34 wt% H2O, cf. Andrut & Wildner, 2001b, 2002). Reliable literature data are also included.

Fig. 7.

Polarised UV-VIS absorption spectra of sample Sar-w2 and assignment of d–d transitions of Cr3+. The absorptions show an isotropic behaviour. Spectra are offset for clarity. The insert enlarges the spectral range from 14100 to 14500 cm−1, displaying the spin-forbidden 4A2g(4F) → 2Eg(2G) transition split by spin-orbit coupling.

Fig. 7.

Polarised UV-VIS absorption spectra of sample Sar-w2 and assignment of d–d transitions of Cr3+. The absorptions show an isotropic behaviour. Spectra are offset for clarity. The insert enlarges the spectral range from 14100 to 14500 cm−1, displaying the spin-forbidden 4A2g(4F) → 2Eg(2G) transition split by spin-orbit coupling.

Fig. 10.

Energy of the first spin-allowed transition v1, 4A2g(4F) → 4T2g(4F) (Oh), of Cr3+ as a function of the composition of the solid solution corundum-eskolaite. Comparison of data obtained from single crystal investigations on the basis of a local symmetry C3 (grey triangle symbols: this work; eskolaite: McClure, 1963) with mean values (white circles), and data obtained by powder reflection measurements (dark grey circles; Schmitz-DuMont & Reinen, 1959; Neuhaus, 1960; Reinen, 1969). The dotted lines indicate the inversion of the trigonal split components (4A, 4E) along the solid solution series.

Fig. 10.

Energy of the first spin-allowed transition v1, 4A2g(4F) → 4T2g(4F) (Oh), of Cr3+ as a function of the composition of the solid solution corundum-eskolaite. Comparison of data obtained from single crystal investigations on the basis of a local symmetry C3 (grey triangle symbols: this work; eskolaite: McClure, 1963) with mean values (white circles), and data obtained by powder reflection measurements (dark grey circles; Schmitz-DuMont & Reinen, 1959; Neuhaus, 1960; Reinen, 1969). The dotted lines indicate the inversion of the trigonal split components (4A, 4E) along the solid solution series.

Table 2.

SPM parameters for Co2+ extracted from synthetic mineral-type or related compounds.

PhaseSym.forumlat4forumlat2references

Li2Co3(SeO3)4147403.170005.5[1]1,2,3
Co(OH)2, 290Kforumla5260*4920*[2]1, [3]1,2,3
Co(OH)2, 90K53203900
CoSO4·H2Oforumla48401.953004.0[4]1, [5]2, [6]3
CoSeO4·H2Oforumla50001.568902.7[7]1, [5]2, [6]3
NaCo2(SeO3)2(OH)m47601.050402.4[8]1,2, [6]3
CoSe2O524960*4270*[9]1, [10]2, [6]3
CoSeO3·2H2O150905.480000[11]1, [10]2, [12]3
PhaseSym.forumlat4forumlat2references

Li2Co3(SeO3)4147403.170005.5[1]1,2,3
Co(OH)2, 290Kforumla5260*4920*[2]1, [3]1,2,3
Co(OH)2, 90K53203900
CoSO4·H2Oforumla48401.953004.0[4]1, [5]2, [6]3
CoSeO4·H2Oforumla50001.568902.7[7]1, [5]2, [6]3
NaCo2(SeO3)2(OH)m47601.050402.4[8]1,2, [6]3
CoSe2O524960*4270*[9]1, [10]2, [6]3
CoSeO3·2H2O150905.480000[11]1, [10]2, [12]3

1 crystal structure, 2 polarised absorption spectra, 3 SPM analysis

* fixed at t4 = 5 and t2 = 3 (see text)

Table 3.

Labelling, multiplicity m and site symmetry s of the positions occupied in the garnet structure for the space groups under consideration.

forumla (tricl)Fddd (orth)forumla (cub)
atommsatommsatomms

Ca1Ca11256321
41Ca238222Ca1–624222
Ca6Ca348222

Y1Y11367
2forumla16forumlaY1–8164316forumla
Y8Y22458

Si1Si11256321
41Si1–624forumla
Si6Si234162

O1O1
41321o961
O24O6
forumla (tricl)Fddd (orth)forumla (cub)
atommsatommsatomms

Ca1Ca11256321
41Ca238222Ca1–624222
Ca6Ca348222

Y1Y11367
2forumla16forumlaY1–8164316forumla
Y8Y22458

Si1Si11256321
41Si1–624forumla
Si6Si234162

O1O1
41321o961
O24O6
Table 5.

Summary of SPM and crystal field parameters (non-zero Bkq in Wybourne notation, intrinsic forumla and tk, rotational invariants sk, cubic field strengths Dqcub), inter-electronic repulsion parameters and conventional CF parameters for the Cr3+-bearing garnets investigated at 290 K. The respective values of the crystal field strength Dqcub and Racah B35, derived from spectrum analysis using an Oh symmetry approach are given for comparison. Last rows: application of the SPM with calculated band positions (mean), experimental values are given for comparison. Values are in cm1 (except tk, β, C/B).

Uvasyn-22Sar-desySar-kl2Sar-899Sar-w2Ves-2Ska-1

forumla9532
forumla4650
t46.7
t2 (fixed)3
ζ (fixed)135
B20–559–637–657–665–653–659–662
B40–21621–21892–21888–21893–21927–21937–22105
B43–26908–27397–27431–27454–27460–27495–27707
S414588.914834.114847.414857.714863.814881.414996.0
S2250.0285.0293.6297.3292.2294.6295.9
Dqcub (from s4)1591.81618.51620.01621.11621.81623.71636.2
Racah B35703642641638648636620
Racah B55714715718717716717715
Racah C3165315231443148315031483149
β35 (B0 = 995[1])0.710.640.640.640.650.640.62
C/B354.514.914.904.934.864.955.08
Dqcub (from Dqtrig)1591.51618.21619.61620.81621.41623.41635.9
Dqtrig1608.01637.31639.31640.71641.01643.11655.8
–42.4–49.0–50.6–51.2–50.4–50.8–51.3
79.991.093.895.093.494.194.5
Dqcub (from spectra [2])1593.01620.01622.01624.01623.51623.51631.0
Racah B35 [2]703641640636647638628
application of SPM:
4A2g(F) → 4T2g(F)
calc. (mean)15930161931620816219162261624416369
observed15930162001622016240162351623516310
4A2g(F) → 4T1g(F)
calc. (mean)22776226302263822627227102264122653
observed22775226302264022630227102264022650
Uvasyn-22Sar-desySar-kl2Sar-899Sar-w2Ves-2Ska-1

forumla9532
forumla4650
t46.7
t2 (fixed)3
ζ (fixed)135
B20–559–637–657–665–653–659–662
B40–21621–21892–21888–21893–21927–21937–22105
B43–26908–27397–27431–27454–27460–27495–27707
S414588.914834.114847.414857.714863.814881.414996.0
S2250.0285.0293.6297.3292.2294.6295.9
Dqcub (from s4)1591.81618.51620.01621.11621.81623.71636.2
Racah B35703642641638648636620
Racah B55714715718717716717715
Racah C3165315231443148315031483149
β35 (B0 = 995[1])0.710.640.640.640.650.640.62
C/B354.514.914.904.934.864.955.08
Dqcub (from Dqtrig)1591.51618.21619.61620.81621.41623.41635.9
Dqtrig1608.01637.31639.31640.71641.01643.11655.8
–42.4–49.0–50.6–51.2–50.4–50.8–51.3
79.991.093.895.093.494.194.5
Dqcub (from spectra [2])1593.01620.01622.01624.01623.51623.51631.0
Racah B35 [2]703641640636647638628
application of SPM:
4A2g(F) → 4T2g(F)
calc. (mean)15930161931620816219162261624416369
observed15930162001622016240162351623516310
4A2g(F) → 4T1g(F)
calc. (mean)22776226302263822627227102264122653
observed22775226302264022630227102264022650
Table 6.

Chemical compositions, energies of the spin-allowed transitions of Cr3+, conventional CF parameters (C3v approximation; program TETRIG, Wildner, 1996b) and polyhedral geometries determined for Cr3+ in the investigated rubies (SPM analyses) and eskolaite (X-ray) compared with the AlO6 polyhedron in corundum (X-ray).

sampleeskolaiteruby Longidoruby Mysoresynthetic rubysynthetic rubycorundum
[1, 2]syn2syn1[3]

ChemistryCr2O3Al1.97Cr0.03O3Al1.99Cr0.01O3Al2.00Cr0.00O3Al1.98Cr0.02O3Al2O3
EMPAl2O3: 97.2Al2O3: 99.3Al2O3: 99.8Al2O3: 98.5
[wt%]Cr2O3: 1.82Cr2O3: 0.81Cr2O3: 0.14Cr2O3: 1.48
Fe2O3: 0.28Fe2O3: 0.07
Spectra
Eobs 4E1677517800179251789017860
[cm−1] 4A1639018200182001836018300
4E2142024180242602434024320
4A2185025000250602512025060
CF
Dq [cm−1]1703.31847.81853.31857.81850.7
Dτ [cm−1]–117.3–138.1–134.0–132.5–124.9
Dσ [cm−1]549.161.2129.25.35.7
B35 [cm−1]457.2629.6629.6631.5633.2
GeometryX-raySPMSPMSPMSPMX-ray
R1 [Å]2.0091.9721.9641.9631.9611.971
R2 [Å]1.9621.9251.9301.9281.9341.856
forumla1.9861.94851.9471.94551.94751.914
θ1 [°]48.7547.6847.8747.7447.9247.67
θ2 [°]118.50118.01117.98118.01118.12116.85
φ2φ1 [°]55.9756.1055.8455.7855.9056.11
sampleeskolaiteruby Longidoruby Mysoresynthetic rubysynthetic rubycorundum
[1, 2]syn2syn1[3]

ChemistryCr2O3Al1.97Cr0.03O3Al1.99Cr0.01O3Al2.00Cr0.00O3Al1.98Cr0.02O3Al2O3
EMPAl2O3: 97.2Al2O3: 99.3Al2O3: 99.8Al2O3: 98.5
[wt%]Cr2O3: 1.82Cr2O3: 0.81Cr2O3: 0.14Cr2O3: 1.48
Fe2O3: 0.28Fe2O3: 0.07
Spectra
Eobs 4E1677517800179251789017860
[cm−1] 4A1639018200182001836018300
4E2142024180242602434024320
4A2185025000250602512025060
CF
Dq [cm−1]1703.31847.81853.31857.81850.7
Dτ [cm−1]–117.3–138.1–134.0–132.5–124.9
Dσ [cm−1]549.161.2129.25.35.7
B35 [cm−1]457.2629.6629.6631.5633.2
GeometryX-raySPMSPMSPMSPMX-ray
R1 [Å]2.0091.9721.9641.9631.9611.971
R2 [Å]1.9621.9251.9301.9281.9341.856
forumla1.9861.94851.9471.94551.94751.914
θ1 [°]48.7547.6847.8747.7447.9247.67
θ2 [°]118.50118.01117.98118.01118.12116.85
φ2φ1 [°]55.9756.1055.8455.7855.9056.11

Contents

GeoRef

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