Optical absorption spectroscopy in geosciences: Part II: Quantitative aspects of crystal fields

Published:January 01, 2004
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 freeion d^{N} 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 metalligand 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 d^{N}X_{6} complex with an ideal octahedral coordination (symmetry O_{h}), 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 O_{h}. This requires introduction of additional, socalled 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 freeion d^{N} 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 metalligand 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 d^{N}X_{6} complex with an ideal octahedral coordination (symmetry O_{h}), 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 O_{h}. This requires introduction of additional, socalled 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 d^{N} 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 highsymmetry 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 pseudosymmetry (see Part I). In many cases the calculations of the CF splittings were performed on the basis of the approximated O_{h} 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. d^{3} and d^{8} ions (Cr^{3+}, Ni^{2+}). In those cases the O_{h} approximation can yield reasonable results (e.g. AbsWurmbach 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 O_{h} 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 socalled semiempirical 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 O_{h} approximation and those from the semiempirical 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/R^{5} dependence of the field strength, while more realistic, adjustable powerlaw 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):
where q is the effective charge of the ligands, is the mean central ionligand distance and 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:a) one type of ligand is assumed;
b) the ligands behave as point electric charges;
c) the ligands surrounding the central metal ion form a regular octahedron (O_{h} 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 ionligand 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 3d^{N} 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 〈r^{4}〉, which are related to the bonding character, retain their freeion values in crystals (e.g. Burns, 1993). Equation 1 is then simplified to:
If the local symmetry of the coordination polyhedron is reduced from O_{h}, 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 relative to that at ambient pressure , Equation 2 yields:
Based upon Equation 3, Burns (1987) extracted polyhedral bulk moduli k_{poly} from pressuredependent 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 β_{v} defined as:
with where the subscripts T and X refer to partial derivatives at constant temperature and composition. similarly, the linear compressibility β_{1} is defined as: where d denotes the interatomic distance metal (M) to ligand (L). Substituting d by the mean interatomic distance we obtain the linear polyhedral compressibility β_{l,poly} expressed as: We can simplify Equation 7 by using the differential quotient and write with and the pressure difference . 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 socalled “polyhedral approach” the reader is referred to Hazen & Finger (1982).Spectroscopically derived data on polyhedral bulk moduli are scarce. Langer et al. (1997) investigated Cr^{3+}centred octahedra in spinel, garnet, ruby and kyanite. They found discrepancies between the spectroscopically determined polyhedral bulk moduli and those obtained by highpressure single crystal diffraction, especially for ruby and pyrope. In both minerals Cr^{3+} and Al^{3+} 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 Cr^{3+}/Al^{3+} 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:
Since R is proportional to V^{1/3}, Equation 9 can be used to relate the volume coefficient α of linear thermal expansion with 10Dq (Burns, 1985, 1993):Similarly as for compressibility, the volume coefficient of linear thermal expansion α_{V} describing the dimensional changes of a crystal can be expressed as (e.g. Hazen, 1985)
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 CrO_{6} octahedra in various oxygenbased 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:
Prerequisite for applying this relationship is the knowledge of the parameter 10Dq_{X=1} and of the transition metal bearing endmember. The value of for the solid solution may be experimentally extracted from optical absorption spectra. The respective mean interatomic distance 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 a_{X} in a binary solid solution A_{1–}_{X}B_{X}C can be expressed as a function of its composition in terms of an additivity rule:
where a_{1} and a_{2} represent the lattice parameter for the endmembers AC and BC, respectively, and X_{1}, X_{2} 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 nonlinear 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 intracrystalline fractionation may take place, since two or more nonequivalent 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 A_{1–X}B_{X}C as a function of its composition. We assume here that the interatomic distance of the endmember AC is smaller than the respective interatomic distance 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 ML_{n} polyhedra are equal regardless of their individual occupancy by A or B. Thus, these interatomic distances R_{X} correspond to an averaged value according to Vegard's rule (see the dotted line in Fig. 1):
where X_{AC} and X_{BC} are the mole fractions of the endmembers 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.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 R_{AC} and R_{BC} for A and for B in the solid solution are equal to their respective endmember values and 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
and are the interatomic distances in the endmember in question, and R_{BC} is the interatomic distance for the impurity B in the host lattice AC (i.e., X_{A} ≈ 1). The denominator of this quotient is a constant, and we have to investigate the behaviour of R_{BC} to determine the data range for ε. In case of full relaxation we obtain ε = 1, because the interatomic distance of R_{BC} will not change with composition: R_{BC} = In the absence of relaxation ε = 0, one obtains R_{BC} =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 4f^{N} and 5f^{N}^{} ions to describe the interaction between the openshell 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 3d^{N} 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 3d^{N} ion in a crystal (cf. Part I; for details see e.g.Ballhausen, 1962; Schläfer & Gliemann, 1967; Gerloch & Slade, 1973):
As outlined in Part I, the free ion Hamiltonian H_{FI} consists of the spherical free ion Hamiltonian H_{spher} (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 nonspherical part) amongst the d electrons H_{ee}, the spinorbit and spinspin interactions H_{SO} and H_{SS}, and – if applicable – also the Trees correction H_{Trees}, describing the twobody orbitorbit polarisation interaction (Gerloch & Slade, 1973; in Part I H_{Trees} has been omitted for clarity). These parts of the Hamiltonian are generally expressed in terms of the Racah parameters A (H_{spher}), B and C (H_{ee}), the spinorbit coupling constant ζ (H_{SO}), the spinspin coupling constant ρ (H_{SS}), and Trees correction parameter α (H_{Trees}). Two notations for general forms of CF Hamiltonians based on the tensor operators are now prevailing in the literature, namely: (i) the Wybourne operators ^{} (Wybourne, 1965; Rudowicz, 1987) and (ii) the extended Stevens (ES) operators (Rudowicz, 1985a, 1987). The CF Hamiltonian H_{CF} is given in the Wybourne notation by: where B_{kq} are the CF parameters representing the electron radial integrals and the CF interaction strength, and are the renormalised 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) H_{CF} can also be expressed in terms of the ES operators (Rudowicz, 1985a, 1987) as (Rudowicz & Misra, 2001): where the nature of the Stevens' operators (J_{x}, J_{y}, J_{z}) in Equation 18 is explicitly indicated as being the functions of the total angular J (or total orbital L) momentum operators. Thus the parameters (CF) in Equation 18 should not be confused with the zerofield splitting parameters (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 nonvanishing crystal field parameters to be taken into account for various local symmetries of the crystal field?
In general, the parameters B_{kq} with q ≠ 0 may be complex, i.e., B_{kq} = Re B_{kq} + i Im B_{kq}, whereas are all real (Rudowicz, 1985b). It must be emphasised that not all parameters of B_{kq} () are independent (see e.g.5.1 and refer to Table 1). Since H_{CF} as an operator of energy must be a Hermitian operator (see e.g.Newman & Ng, 1989; Rudowicz, 1987), the Wybourne parameters B_{kq} satisfy the relations (Rudowicz, 1985b): and The former B_{kq} are often referred to as “real” B_{kq}, the latter as “imaginary” B_{kq}.
Group theory, especially the irreducible tensor methods and the WignerEckart theorem, enable predicting the nonzero matrix elements of the CF operators for each rank k in Equations 17 and 18 on the basis of the oneelectron as well as multielectron 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 noncentrosymmetric 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 codomain 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 nonvanishing 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 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 lowsymmetry 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 lowsymmetry cases are described in the recent papers of Rudowicz & Qin (2003, 2004a, 2004b, 2004c).
crystallographic point groups .  nonvanishing B_{kq} .  

.  .  .  B_{20} .  B_{21} .  B_{22} .  B_{40} .  B_{41} .  B_{42} .  B_{43} .  B_{44} . 
.  
cubic  O_{h}  m3m  +  –  
O  432  +    
T_{d}  3m  +    
T_{h}  m  +    
T  23  +    
tetragonal  D_{4}_{h}  4/mmm  +  +  +  
D_{4}  422  +  +  +  
D_{2}_{d}  2m  +  +  +  
D_{4}_{v}  4mm  +  +  +  
D_{4}_{h}  4/m  +  +  ±  
S_{4}  +  +  ±  
C_{4}  4  +  +  ±  
hexagonal  D_{6}_{h}  6/mmm  +  +  
D_{6}  622  +  +  
D_{3}_{h}  m2  +  +  
C_{6}_{v}  6mm  +  +  
C_{6}_{h}  6/m  +  +  
C_{3}_{h}  +  +  
C_{6}  6  +  +  
trigonal  D_{3}_{d}  m  +  +  +  
D_{3}  32  +  +  +  
C_{3}_{v}  3m  +  +  +  
C_{3}_{i}  +  +  ±  
C_{3}  3  +  +  ±  
orthorhombic  D_{2}_{h}  mmm  +  +  +  +  +  
D_{2}  222  +  +  +  +  +  
C_{2}_{v}  mm2  +  +  +  +  +  
monoclinic  C_{2}_{h}  2/m  +  ±  +  ±  ±  
C_{s}  m  +  ±  +  ±  ±  
C_{2}  2  +  ±  +  ±  ±  
triclinic  C_{i}  +  ±  ±  +  ±  ±  ±  ±  
C_{1}  1  +  ±  ±  +  ±  ±  ±  ± 
crystallographic point groups .  nonvanishing B_{kq} .  

.  .  .  B_{20} .  B_{21} .  B_{22} .  B_{40} .  B_{41} .  B_{42} .  B_{43} .  B_{44} . 
.  
cubic  O_{h}  m3m  +  –  
O  432  +    
T_{d}  3m  +    
T_{h}  m  +    
T  23  +    
tetragonal  D_{4}_{h}  4/mmm  +  +  +  
D_{4}  422  +  +  +  
D_{2}_{d}  2m  +  +  +  
D_{4}_{v}  4mm  +  +  +  
D_{4}_{h}  4/m  +  +  ±  
S_{4}  +  +  ±  
C_{4}  4  +  +  ±  
hexagonal  D_{6}_{h}  6/mmm  +  +  
D_{6}  622  +  +  
D_{3}_{h}  m2  +  +  
C_{6}_{v}  6mm  +  +  
C_{6}_{h}  6/m  +  +  
C_{3}_{h}  +  +  
C_{6}  6  +  +  
trigonal  D_{3}_{d}  m  +  +  +  
D_{3}  32  +  +  +  
C_{3}_{v}  3m  +  +  +  
C_{3}_{i}  +  +  ±  
C_{3}  3  +  +  ±  
orthorhombic  D_{2}_{h}  mmm  +  +  +  +  +  
D_{2}  222  +  +  +  +  +  
C_{2}_{v}  mm2  +  +  +  +  +  
monoclinic  C_{2}_{h}  2/m  +  ±  +  ±  ±  
C_{s}  m  +  ±  +  ±  ±  
C_{2}  2  +  ±  +  ±  ±  
triclinic  C_{i}  +  ±  ±  +  ±  ±  ±  ±  
C_{1}  1  +  ±  ±  +  ±  ±  ±  ± 
In practice, the actual number of nonvanishing 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 B_{kq} 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 (C_{6}, C_{3}_{h}, C_{6h}), tetragonal II (C_{4}, S_{4}, C_{4}_{h}), trigonal II (C_{3}, S_{6}), monoclinic (C_{2,} C_{s}, C_{2}_{h}), and triclinic (C_{i}, C_{1}) 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 semiempirical 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 B_{kq} 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:
where (R_{i}, θ_{i}, φ_{i}) are the polar coordinates of the i^{th} ligand, The intrinsic parameters represent the strength of the k^{th} rank CF contributions from a given ligand type. The geometrical information is described by the coordination factors K_{kq}. The expressions for i.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 K_{kq} 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 powerlaw dependence (Newman & Ng, 1989):
where R_{0} 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 R_{0} 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 t_{k} > 0 for the powerlaw exponents (Newman & Ng, 1989). Equation 19 then transforms into a general expression, which enables to derive the SPM relations for a particular symmetry: 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 twoterm powerlaw (Hikita, 1992; Donnerberg et al., 1993) have met with limited success due to overparameterisation.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 R_{i} (with i = 1 to 6), and exhibit the following polar angles (θ_{i}, φ_{i}): (0°, 0°), (90°, 0°); (90°, 90°); (90°, 180°); (90°, 270°); (180°, 0°).
The parameter B_{40} is the only independent CF parameter for the point group O_{h}, since B_{44} is related to B_{40} (see e.g. Schläfer & Gliemann, 1967) as:
Wybourne (1965) provides the conversion relations for the conventional CF parameters of Ballhausen (1962); extracting data from these sources yields:
and thus: 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 R_{i} = R_{1} and sum over six ligands in Equations 25 and 26. Inserting the angular polar coordinates for each ligand, i.e., the angles (θ_{i}, φ_{i}) given above, into the expression for K_{40}, the Equation 25 simplifies to: 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 R_{1} equals R_{0}, Equation 27 simplifies to 6Dq_{cub} = . At a closer look, Equation 27 reveals a similarity with Equation 2 derived by conventional CFT. Actually, CFT predicts a powerlaw exponent t_{4} = 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 t_{4} are not necessarily in agreement with this prediction. Fitting t_{4} to experimentally observed crystal field energy levels may lead to more realistic values for the particular system under investigation. For example, t_{4} ≈ 11 was reported for tetravalent actinide ions with chlorine and oxygen ligands (cf. Newman & Ng, 2000), and in section 6 we derive empirical powerlaw parameters for Co^{2+} and Cr^{3+} in oxygen crystal fields within the range 1 ≤ t_{4} ≤ 7.After rearrangement of some factors in Equation 27, one obtains an equivalent form of Equation 2 in case of a regular octahedron:
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 O_{h} is a rare exception among the numberless structure types offering sixfold coordinated sites, and thus can only serve as a first approximation. Taking into account the actual symmetry of a coordination polyhedron distorted from cubic O_{h} 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 leastsquares fittings. This requirement is not always met in case of the 3d^{N} 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 f^{N} 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 spinorbit 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 semiempirical 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 (D_{3}_{d}). For D_{3}_{d} 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 Dq_{trig} in the following. An octahedron with symmetry D_{3}_{d} 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 B_{kq} simplify, since all the imaginary terms Im B_{kq} as well as most real terms Re B_{kq} are zero (compare with Table 1). Also the coordination factors K_{kq}(θ_{i}, φ_{i}) of each nonzero B_{kq} become equal, thus leading to the Equations 29–31. The corresponding relationships to Ballhausen's (1962) conventional CF and distortion parameters for trigonal symmetry, i.e., Dq_{trig}, Dτ, Dσ, apply:
For simplification, the conversion factors from Stevens to Wybourne notation are formally included into the coordination factors K_{kq}(θ_{i}, φ_{i}) Similarity as for a regular octahedron, we can use the CF parameter B_{40} (Eqn. 30) for trigonal symmetry to determine the dependence of the CF splitting. This can now be expressed in terms of 10Dq_{trig}, as a function of the mean interatomic distance: To show the similarity between Equation 32 and Equations 2 and 28, certain factors were grouped together. In comparison to the O_{h} 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 powerlaw exponent t_{4} = 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:
Here, the factors that are specific for the system under investigation, like the intrinsic parameter and the reference distance R_{0}, vanish. The subscripts X = 1 and Xi refer to the endmember and an intermediate composition of the solid solution, respectively. Since we are dealing with only one interatomic distance R_{i} in point group D_{3}_{d}, we can set <R_{i}> = R_{1}. Equation 33 shows a dependence of R_{Xi} 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 D_{3}_{d} 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 O_{h} 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 H_{CF} as well as the free ion contribution H_{FI}. The free ion parameters comprise the Racah parameters A, B and C, the spinorbit 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 Xray crystal structure investigations. Hence, only the intrinsic parameters and powerlaw exponents t_{k} 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 powerlaw SPM parameters for several 3d^{N} 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 3d^{N} ion fully occupies its respective crystallographic site, i.e., the structural information is obtained from endmember 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 doublesided 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 B_{kq} 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 spinHamiltonian (MSH) module has recently been developed (Rudowicz et al., 2003; Yang et al., 2003). Hence with the HCFLDN2module of the CF program package (Chang et al., 1994), we are able to calculate energy levels for any d^{N} cation at arbitrary low site symmetries. Some preliminary calculations for spinallowed energy levels in three or fourfold symmetry
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 coordinates; (ii) the systematic variation of the free ion parameters and of the intrinsic and powerlaw SPM parameters, as well as the corresponding communication with the HCFLDN2 program; (iii) the SPM calculation itself yielding the values for the B_{kq}'s and (iv) the interpretation of the HCFLDN2 output results in terms of a reliability index for the agreement of calculated and observed spinallowed transition energies.
In order to avoid any bias in the parameter determination from scratch for a newly investigated cation, we only regard > 0 and t_{k} ≥ 0 as prerequisites. Hence, a systematic variation of the intrinsic and powerlaw parameters covering a very wide range of parameter values is necessary to avoid any wrong pseudominima 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 metalligand distance R_{0} has to be chosen for each 3d^{N} 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” t_{k} crucially depends on the bond length distortion of the polyhedron under consideration. For weak or zero bond length distortion, the t_{k} will be doubtful or even meaningless and have to be fixed at appropriate values (e.g. at the “electrostatic values” t_{2} = 3 and t_{4} = 5). However, if the individual bond lengths R_{i} are moreover (nearly) equal to R_{0}, the t_{k} have no influence on the SPM calculations at all.
In this way, a set of freeion parameters, intrinsic parameters and powerlaw exponents t_{k} 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 spinforbidden transitions. Within 3d^{N} systems, spinorbit 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 spectralstructural correlations,
(b) used for systematic investigations aiming to establish a basic “global” SPM parameter set for a newly investigated d^{N} cation, e.g. Co^{2+} (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 optoelectronic 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 Fe^{2+} in the dodecahedral site of pyropealmandine garnets (Newman et al., 1978). Up to now, data for d block elements focus on Cr^{3+} and Co^{2+} ions only, and there exist just a handful of such papers correlating spectroscopic and structural properties of minerals. Hence, the necessary intrinsic and powerlaw exponent parameters of 3d^{N}^{} 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. Cr^{3+} in alumosilicates; Qin et al., 1994; Yeung et al., 1994a, 1994b, and references within these papers). Present efforts to provide reliable SPM parameters for Cr^{3+} and Co^{2+} from synthetic endmember 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 Co^{2+} in oxygenbased 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 Co^{2+} doped in CdCl_{2} (Edgar, 1976). However, that paper provides no powerlaw parameters t_{k} as well as no reference distance R_{0}.
For our recent research on crystal fields of Co^{2+}, several mineraltype or related oxygenbased Co^{2+} compounds with endmember compositions were synthesised at lowhydrothermal conditions. Their crystal structures were thoroughly characterised by single crystal Xray 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 metalligand distance R_{0} was fixed at 2.1115 Å, the overall mean Co–O bond length in sixfold coordination (Wildner, 1992).
Phase  Sym.  t_{4}  t_{2}  references  

Li_{2}Co_{3}(SeO_{3})_{4}  1  4740  3.1  7000  5.5  [1]^{1,2,3} 
Co(OH)_{2}, 290K  5260  *  4920  *  [2]^{1}, [3]^{1,2,3}  
Co(OH)_{2}, 90K  5320  3900  
CoSO_{4}·H_{2}O  4840  1.9  5300  4.0  [4]^{1}, [5]^{2}, [6]^{3}  
CoSeO_{4}·H_{2}O  5000  1.5  6890  2.7  [7]^{1}, [5]^{2}, [6]^{3}  
NaCo_{2}(SeO_{3})_{2}(OH)  m  4760  1.0  5040  2.4  [8]^{1,2}, [6]^{3} 
CoSe_{2}O_{5}  2  4960  *  4270  *  [9]^{1}, [10]^{2}, [6]^{3} 
CoSeO_{3}·2H_{2}O  1  5090  5.4  8000  0  [11]^{1}, [10]^{2}, [12]^{3} 
Phase  Sym.  t_{4}  t_{2}  references  

Li_{2}Co_{3}(SeO_{3})_{4}  1  4740  3.1  7000  5.5  [1]^{1,2,3} 
Co(OH)_{2}, 290K  5260  *  4920  *  [2]^{1}, [3]^{1,2,3}  
Co(OH)_{2}, 90K  5320  3900  
CoSO_{4}·H_{2}O  4840  1.9  5300  4.0  [4]^{1}, [5]^{2}, [6]^{3}  
CoSeO_{4}·H_{2}O  5000  1.5  6890  2.7  [7]^{1}, [5]^{2}, [6]^{3}  
NaCo_{2}(SeO_{3})_{2}(OH)  m  4760  1.0  5040  2.4  [8]^{1,2}, [6]^{3} 
CoSe_{2}O_{5}  2  4960  *  4270  *  [9]^{1}, [10]^{2}, [6]^{3} 
CoSeO_{3}·2H_{2}O  1  5090  5.4  8000  0  [11]^{1}, [10]^{2}, [12]^{3} 
^{1} crystal structure, ^{2} polarised absorption spectra, ^{3} SPM analysis
* fixed at t_{4} = 5 and t_{2} = 3 (see text)
[1] Wildner & Andrut (1999); [2] Pertlik (1999); [3] Andrut & Wildner (2001a); [4] Wildner & Giester (1991); [5] Wildner (1996a); [6] this paper; [7] Giester & Wildner (1992); [8] Wildner (1995); [9] Hawthorne et al. (1987); [10] Andrut & Wildner, unpublished; [11] Wildner (1990); [12] Wildner & Andrut (2001a)
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 Co^{2+} cations generally pose some problems, especially concerning the characteristic structures and splitting of the intense ^{4}T_{1g}(^{4}F) → ^{4}T_{1g}(^{4}P) 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 intensityenhanced spinforbidden transitions to the ^{4}T_{1g}(P) band, splitting of this band due to low symmetry components of the crystal field or due to spinorbit 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 Co^{2+} was extracted from polarised electronic absorption spectra of Li_{2}Co_{3}(SeO_{3})_{4}, which is characterised by strong distortions of two crystallographically different CoO_{6} polyhedra with low symmetry C_{1} and C_{i} (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, cisO–Co–O = 76–99°) makes it an ideal candidate for the extraction of reliable as well as t_{k}. 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 t_{2} < t_{4} (Yeung & Newman, 1986). The complete energy level schemes for both CoO_{6} polyhedra, calculated with these SPM parameters, are presented in Figure 1.
In brucitetype Co(OH)_{2} the Co^{2+} cations occupy a highsymmetry site (D_{3d}) within a compressed hexagonal close packing of oxygen atoms. Structural data and polarised absorption spectra were obtained at 90 and 290 K. For D_{3}_{d} symmetry all Co–O distances are equal and hence the t_{k} 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 (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. O_{3}Se^{2–} vs. OH^{−}) affects the intrinsic However, an attempt to account for such influences, especially in mixedligand 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 Co^{2+} kieserites, CoSO_{4}·H_{2}O and CoSeO_{4}·H_{2}O, resemble the results for Li_{2}Co_{3}(SeO_{3})_{4} and corroborate that t_{4} < t_{2} for this 3d^{N} transition ion. The CoO_{4}(H_{2}O)_{2} octahedron with point symmetry C_{i} exhibits a distinct pseudotetragonal shape with an elongation along the H_{2}O–H_{2}O axis.
The structure of NaCo_{2}(SeO_{3})_{2}(OH) is built up from olivinelike octahedral chains. The optical absorption spectra are characterised by d–d transitions at the acentric Co(2) site with symmetry C_{s}, showing a pseudotetragonal compression of its CoO_{5}(OH) coordination polyhedron. The derived powerlaw parameters are rather small, but again t_{4} < t_{2}.
Spectroscopic and SPM analyses of two further synthetic Co^{2+} compounds yielded more ambiguous results concerning both the assignment of the energy level components split by lowsymmetry CF and the subsequent extraction of SPM parameters. In CoSe_{2}O_{5} the CoO_{6} octahedron (symmetry C_{2}) exhibits distinct bond angle distortions but
negligible bond length distortion, and hence the t_{k} had to be fixed. In CoSeO_{3}·2H_{2}O the SPM calculations for the CoO_{4}(H_{2}O)_{2} polyhedron (symmetry C_{1}) yielded reasonable intrinsic parameters but an implausible powerlaw parameter t_{2} = 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 Co^{2+} cations: ≈ 4900 cm1, t_{4} ≈ 2.5, 6000 cm^{1}, t_{2} ≈ 4.0.
Extraction of crystal field parameters for Cr^{3+} from the binary solid solution uvarovitegrossular
Introduction
Due to the relevance of garnettype 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 Albearing 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 rockforming garnets are cubic, and the overwhelming majority of crystal structure investigations have been performed on synthetic or natural crystals in space group
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]}X_{3}^{[6]}Y_{2}^{[4]}Z_{3}^{[4]}O_{12}, among the latter are common rockforming minerals (Z = Si) subsumed in the pyralspite group (Y = Al: pyrope X = Mg, almandine X = Fe^{2+}, spessartine X = Mn^{2+}) and the ugrandite group (X = Ca: uvarovite Y = Cr^{3+}, grossular Y = Al, andradite Y = Fe^{3+}). For a survey of the group of Ca/Crbearing garnets see e.g. Wildner & Andrut (2001b).
In space group 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 fourfold 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 ZO_{4} tetrahedron is usually occupied by Si but may be replaced by O_{4}H_{4} groups (“hydrogarnets”).
The ZO_{4} tetrahedra and YO_{6} 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.
(tricl)  Fddd (orth)  (cub)  

atom  m  s  atom  m  s  atom  m  s 
Ca1  Ca1_{1256}  32  1  
…  4  1  Ca2_{3}  8  222  Ca_{1–6}  24  222 
Ca6  Ca3_{4}  8  222  
Y1  Y1_{1367}  
…  2  16  Y_{1–8}1643  16  
Y8  Y2_{2458}  
Si1  Si1_{1256}  32  1  
…  4  1  Si_{1–6}  24  
Si6  Si2_{34}  16  2  
O1  O1  
…  4  1  …  32  1  o  96  1 
O24  O6 
(tricl)  Fddd (orth)  (cub)  

atom  m  s  atom  m  s  atom  m  s 
Ca1  Ca1_{1256}  32  1  
…  4  1  Ca2_{3}  8  222  Ca_{1–6}  24  222 
Ca6  Ca3_{4}  8  222  
Y1  Y1_{1367}  
…  2  16  Y_{1–8}1643  16  
Y8  Y2_{2458}  
Si1  Si1_{1256}  32  1  
…  4  1  Si_{1–6}  24  
Si6  Si2_{34}  16  2  
O1  O1  
…  4  1  …  32  1  o  96  1 
O24  O6 
Birefringent garnets
Natural garnets in general, but especially those belonging to the grossularandradite and grossularuvarovite 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 nonintrinsically structural origins (e.g. external strain) are assumed, while maintaining the cubic symmetry and space group . However, some structure investigations on such birefringent garnets have been performed in order to verify a symmetry deviation from due to cation ordering on the octahedral and/or dodecahedral sites. Violation of the symmetry beyond doubt has been reported by Takéuchi et al. (1982) and Wildner & Andrut (2001b).
Sample material
Synthetic, optically isotropic flux grown uvarovite, Ca_{3}Cr_{2}[SiO_{4}]_{3} (Uwsyn22), and six natural birefringent uvarovitegrossular garnets from three localities (Saranov [Sardesy, Sarkl2, Sar899, Sarw2], Veselovsk [Ves2] and Saranka [Ska1], Ural Mountains, Russia) were characterised by optical methods, electron microprobe analysis, and UVVISIR microspectrometry 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 Cr^{3+} content in mol% is given in Table 4. The crystal structures were investigated using single crystal Xray CCD diffraction data (Wildner & Andrut, 2001b).
Crystal structures
Crystal structures of natural birefringent uvarovitegrossular solid solutions
The Xray intensity data as well as the obtained lattice parameters attest to the violation of the cubic garnet space group and the symmetry reduction to subgroups with
.  Cr^{3+} .  ^{4}A_{2g} → .  ^{4}A_{2g} → .  ^{4}A_{2g} → .  ^{4}A_{2g} → .  .  .  R_{mean}^{$} . 

sample .  content .  ^{4}T_{2g}(^{4}F) .  ^{4}T_{1g}(^{4}F) .  ^{2}E_{g}(^{2}G) R_{1} .  ^{2}E_{g}(^{2}G) R_{2} .  .  R_{i}_{,individual}^{§} .  (Vegard's rule) . 
Ska1  48.3  16310  22650  14264  14368  1.9848  1.9852  1.9576 
Ves2  65.5  16235  22640  14260  14367  1.9867  1.9877  1.9693 
Sarw2  67.6  16235  22710  14262  14368  1.9867  1.9880  1.9707 
Sarkl2  68.5  16220  22640  14264  14365  1.9870  1.9882  1.9713 
Sar899  68.8  16240  22630  14262  14366  1.9865  1.9882  1.9716 
Sardesy  70.5  16200  22630  14262  14368  1.9875  1.9885  1.9727 
Uwsyn  100.0  15930  22775  14272  14374  1.9942  1.9928  1.9928 
.  Cr^{3+} .  ^{4}A_{2g} → .  ^{4}A_{2g} → .  ^{4}A_{2g} → .  ^{4}A_{2g} → .  .  .  R_{mean}^{$} . 

sample .  content .  ^{4}T_{2g}(^{4}F) .  ^{4}T_{1g}(^{4}F) .  ^{2}E_{g}(^{2}G) R_{1} .  ^{2}E_{g}(^{2}G) R_{2} .  .  R_{i}_{,individual}^{§} .  (Vegard's rule) . 
Ska1  48.3  16310  22650  14264  14368  1.9848  1.9852  1.9576 
Ves2  65.5  16235  22640  14260  14367  1.9867  1.9877  1.9693 
Sarw2  67.6  16235  22710  14262  14368  1.9867  1.9880  1.9707 
Sarkl2  68.5  16220  22640  14264  14365  1.9870  1.9882  1.9713 
Sar899  68.8  16240  22630  14262  14366  1.9865  1.9882  1.9716 
Sardesy  70.5  16200  22630  14262  14368  1.9875  1.9885  1.9727 
Uwsyn  100.0  15930  22775  14272  14374  1.9942  1.9928  1.9928 
Upon symmetry reduction, the unique X, Y, Z and O positions of garnets split into as many as six crystallographically independent X', eight Y', six Z' and 24 O' positions in space group . Although the Si and Ca atoms occupy general positions in , they hardly deviate from their respective special positions in The Y positions retain a centre of symmetry in Fddd and , and the deviation of the YO_{6} octahedra from the symmetry in cubic garnets is extremely small: the average bond length distortion is only 0.92·10^{−6}, ranging from (0.03–3.43)·10^{−6}; Brown & Shannon, 1973). Table 3 lists the atomic site symmetries and multiplicities together with the respective labelling for the garnet symmetries , Fddd, and (Wildner & Andrut, 2001b).
Crystal structure of synthetic uvarovite
In agreement with the isotropic behaviour of synthetic endmember uvarovite under crossed polarisers, all criteria for space group determination from Xray data – as applied and discussed by Wildner & Andrut (2001b) – confirm “usual” cubic garnet symmetry , in contrast to the results for natural birefringent uvarovitegrossular solid solutions. Hence, the crystal structure of synthetic uvarovite was refined at room temperature in space group (a = 11.9973 Å, Cr–O = 1.9942(6) Å, Si–O = 1.6447(6) Å, Ca–O_{a} = 2.3504(6) Å, Ca–O_{b} = 2.4971(6) Å; Andrut & Wildner, 2002). The structure of Ca_{3}Cr_{2}[SiO_{4}]_{3} complies with crystal chemical expectations for ugrandite group garnets in general as well as with predictions drawn from “cubically averaged” data of noncubic uvarovitegrossular solid solutions (Wildner & Andrut 2001b). According to the Cr^{3+} site symmetry (C_{3}_{i}), the octahedral point symmetry is (D_{3}_{d}). As a common feature of all ugrandite garnets, the edges of the YO_{6} octahedron shared with the CaO_{8} polyhedron are longer than the unshared ones (e.g.Novak & Gibbs, 1971), corresponding to an octahedral compression along the C_{3} axis.
Individual octahedral size and Cr^{3+} occupation
For each natural crystal, a nearly perfect linear correlation of the individual octahedral size with its Cr occupancy was observed in Xray diffraction experiments. Considering the dependence on the bulk Cr mole fraction, the <Cr/Al–O> distance at an individual octahedral site in noncubic uvarovitegrossular solid solutions is represented by (Wildner & Andrut, 2001b):
Equation 34 refers to noncubic garnets exhibiting different Cr^{3+}centred octahedral positions. This equation can be used to describe the actual Cr^{3+}–O interatomic distances in cubic binary garnet solid solutions as well as the behaviour according to Vegard's law.In the former case X_{Cr,individual} is set to 1, which leads to:
Thus, Equation 35 can be used to predict the “real” size of a CrO_{6} polyhedron within any uvarovitegrossular solid solution with high reliability (see Wildner & Andrut, 2001b). Hence, the relaxation ε for a CrO_{6} octahedron in grossular (i.e., X_{Cr,bulk} = 0) is calculated as ε = 0.77. Furthermore, if X_{Cr,bulk} = 1, the Cr–O bond length in endmember 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 X_{Cr,bulk} = X_{Cr,individual} in Equation 34, one obtains the behaviour according to Vegard's rule (section 3):
Equation 36 describes the behaviour of the averaged <Cr/Al–O> bond lengths as a function of the bulk Cr^{3+} content. Thus, this relationship describes the interatomic cationoxygen distance in binary solid solutions that represent the averaged respective individual values of the endmembers weighted by the mole fraction.The different cases are displayed in Figure 1. For reliable single crystal structure investigations of uvarovitegrossular solid solutions, Figure 1 summarises the relation between the Cr content and the cubic cell lengths compared with a Vegard's law plot
joining the synthetic endmember 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 α_{i} – seems to be responsible for deviations from a continuous function (Wildner & Andrut, 2001b).
Singlecrystal absorption spectra
Polarised absorption spectra were measured at room temperature in the UVVIS range between 28000 cm^{−1} and 10000 cm^{−1} on a Bruker IFS 66v/S FTIR spectrometer using the attached mirror optics microscope IRScopeII. The spectral bandwidth was 20 cm^{−1}, the local resolution was 60 μm.
As a representative example, the polarised UVVIS spectra of sample Sarw2 are displayed in Figure 1 as linear absorption coefficient α vs. wavenumber . They are characterised by two broad absorption bands at around 16250 and 22600 cm^{−1}, which are typical for Cr^{3+} in octahedral coordination by oxygen atoms (e.g.Lever, 1984). In agreement with the structural results, the comparatively low intensity of the spinallowed d–d bands of Cr^{3+} in the UVVIS region is indicative of an inversion centre at the Cr^{3+} sites, permitting only dynamic violation of the Laporte selection rule due to uneven octahedral vibrations. There is no significant band polarisation in the UVVIS energy range, even though the Cr/Al cation distribution and the resulting orientation dependence of “highCr” and “lowCr” octahedra is clearly noncubic, 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 O_{h} symmetry. The absorption bands located around 16235 cm^{−1} (v_{1}) and 22710 cm^{−1} (v_{2}) in sample Sarw2, with typical FWHM values of 2200 cm^{−1} and 3200 cm^{−1} are assigned to the spinallowed d–d transitions ^{4}A_{2g}(^{4}F) →^{4}T_{2g}(^{4}F) and ^{4}A_{2g}(^{4}F) → ^{4}T_{1g}(^{4}F), respectively. The third spinallowed transition ^{4}A_{2g}(^{4}F) → ^{4}T_{1g}(^{4}P) is calculated to occur around 35700 cm^{−1} and is hence hidden under the absorption edge which represents the lowenergy wing of an intense absorption caused by metaloxygen charge transfer. For a cation with d^{3} electron configuration in octahedral coordination, the first spinallowed transition v_{1} is equivalent to the crystal field splitting parameter 10Dq. Racah B_{35} (cf. Part I) is a measure of the degree of interelectronic d–d repulsion and is derived from the following relationship (Lever, 1968):
The crystal field stabilisation energy for Cr^{3+} in a crystal field with O_{h} symmetry is calculated from: At their lowenergy wings, the spinallowed transitions exhibit shoulders at 14560 cm^{−1}, 15340 cm^{−1}, 15750 cm^{−1}, 21050 cm^{−1} and 22000 cm^{−1}. Furthermore, a spinforbidden 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 ^{2}E_{g}(^{2}G) level, split up due to the spinorbit 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 3d^{N} ion to ligand distance. The derivation of Equations 12 and 33 for symmetries O_{h} and D_{3}_{d}, 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 Cr^{3+} cations. Consequently, the crystal field strength is reduced due to the larger mean interatomic distances, thus shifting the first spinallowed d–d absorption band ^{4}A_{2g}(^{4}F) → ^{4}T_{2g}(^{4}F) to lower wavenumbers. Figure 1 shows the relation between the energy of this band (= 10Dq) and the Cr^{3+} content for several grossularuvarovite solid solutions. Excluding outlying points (grey symbols), the correlation is 10Dq [cm^{−1}] = 16668 – 6.75X_{Cr}3+ [mol%] with r^{2} = 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 spinallowed transition (Fig. 8). Similarly, the second spinallowed transition is also shifted to lower wave numbers with increasing Cr^{3+} content, but with a different slope due to configurational interaction with the ^{4}T_{1g}(^{4}P) state of alike symmetry (see Figs. 6 and 7 in Part I). On the other hand, the spinforbidden transitions are crystal field independent to a first approximation (Table 4), in accordance with the corresponding TanabeSugano diagram (Tanabe & Sugano, 1954; cf. Part I, Fig. 7 and section 3.3.4).
Crystal field SPM calculations
Due to the comparable high point symmetry 33m of the CrO_{6} octahedron with six equal CrO bonds in synthetic uvarovite, the equations for the crystal field parameters B_{kq} simplify (see discussion in section 5.2), leading to Equations 29–31.
The cubic crystal field parameter Dq_{cub} can be calculated from to the trigonal Dq_{trig} and the distortion parameter Dτ via the following equation (König & Kremer, 1977):
For a given angle θ between the threefold axis and the metalligand vector of an octahedron of D_{3}_{d} symmetry, the distortion parameter Dτ can be predicted according to König & Kremer (1977) with: (compare with the corresponding relation, expressed in terms of Dq_{trig}, in Part I, section 3.3.5).The determination of the SPM parameter set for uvarovitegrossular 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) R_{i} ≈ R_{0} in the case of synthetic uvarovite, where the particular values of t_{k} have only a marginal influence on the calculations.
(i) In the beginning of the fitting process, the powerlaw exponents were fixed at t_{4} = 5 and t_{2} = 3, but the values were varied over a wide range to avoid any wrong minimum during the fitting process. Nevertheless, the range of values for is restricted, since both spinallowed absorption bands show no significant band splitting. From Equation 31 it is evident that the energy of the first spinallowed transition is directly related to , whereas (Eqn. 29) predominantly governs the splitting of the energy levels.
(ii) After having determined the approximate magnitude of the for synthetic uvarovite, the value of the powerlaw exponent t_{4} was constrained using additional data sets of the six natural uvarovitegrossular solid solutions with R_{i} < R_{0}, (Table 4). In particular, the actual mean octahedral Cr–O bond length was calculated by Equation 35 using the respective value of X_{Cr,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 and obtained on synthetic uvarovite were applied in the individual fitting of the exponential parameter t_{4} for each natural garnet to describe the respective first spinallowed transition (the choice of t_{2} has no influence on the calculation of the first spinallowed band, compare Equations 29 to 31. The values of t_{4} determined in this way for the six natural samples scatter only slightly between 6.5 and 7.1.
(iii) The mean value t_{4} = 6.7 was used in the final SPM calculations, including a recalculation for synthetic uvarovite. The spinorbit 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 spinallowed bands were attained with = 9532 cm^{−1}, = 4650 cm^{−1}, t_{4} = 6.7, t_{2} (fixed) = 3. Racah B_{35} was fitted individually for all investigated samples. The Racah parameters B_{55} and C (see Part I, section 3.6) were calculated from the energies of the ^{4}A_{2g}(^{4}F) → ^{2}T_{2g}(^{2}G) and the mean value of the ^{4}A_{2g}(^{4}F) → ^{2}E_{g}(^{2}G) transition, v_{2T2} and v_{2E}, respectively, via
Both parameters were varied in additional calculations, but only a limited agreement with the experimental data for all spinforbidden transitions was achieved. Table 5 summarises all relevant parameters for synthetic uvarovite, i.e., SPM parameters, nonzero B_{kq} (Wybourne notation), corresponding conventional CF parameters (Ballhausen, 1962), rotational invariants s_{k} (Leavitt, 1982; Yeung & Newman, 1985), and resulting cubic crystal field parameters Dq_{cub}. Furthermore, s_{4} is related to the commonly used cubic crystal field parameter Dq_{cub} by and can thus be compared with crystal field analyses based on other formalismsUvasyn22  Sardesy  Sarkl2  Sar899  Sarw2  Ves2  Ska1  

9532  
4650  
t_{4}  6.7  
t_{2} (fixed)  3  
ζ (fixed)  135  
B_{20}  –559  –637  –657  –665  –653  –659  –662 
B_{40}  –21621  –21892  –21888  –21893  –21927  –21937  –22105 
B_{43}  –26908  –27397  –27431  –27454  –27460  –27495  –27707 
S_{4}  14588.9  14834.1  14847.4  14857.7  14863.8  14881.4  14996.0 
S_{2}  250.0  285.0  293.6  297.3  292.2  294.6  295.9 
Dq_{cub} (from s_{4})  1591.8  1618.5  1620.0  1621.1  1621.8  1623.7  1636.2 
Racah B_{35}  703  642  641  638  648  636  620 
Racah B_{55}  714  715  718  717  716  717  715 
Racah C  3165  3152  3144  3148  3150  3148  3149 
β_{35} (B_{0} = 995[1])  0.71  0.64  0.64  0.64  0.65  0.64  0.62 
C/B_{35}  4.51  4.91  4.90  4.93  4.86  4.95  5.08 
Dq_{cub} (from Dq_{trig})  1591.5  1618.2  1619.6  1620.8  1621.4  1623.4  1635.9 
Dq_{trig}  1608.0  1637.3  1639.3  1640.7  1641.0  1643.1  1655.8 
Dτ  –42.4  –49.0  –50.6  –51.2  –50.4  –50.8  –51.3 
Dσ  79.9  91.0  93.8  95.0  93.4  94.1  94.5 
Dq_{cub} (from spectra [2])  1593.0  1620.0  1622.0  1624.0  1623.5  1623.5  1631.0 
Racah B_{35} [2]  703  641  640  636  647  638  628 
application of SPM:  
^{4}A_{2}_{g}(F) → ^{4}T_{2}_{g}(F)  
calc. (mean)  15930  16193  16208  162191  622616  24416369  
observed  15930  16200  16220  16240  16235  16235  16310 
^{4}A_{2}_{g}(F) → ^{4}T_{1}_{g}(F)  
calc. (mean)  22776  22630  22638  22627  22710  22641  22653 
observed  22775  22630  22640  22630  22710  22640  22650 
Uvasyn22  Sardesy  Sarkl2  Sar899  Sarw2  Ves2  Ska1  

9532  
4650  
t_{4}  6.7  
t_{2} (fixed)  3  
ζ (fixed)  135  
B_{20}  –559  –637  –657  –665  –653  –659  –662 
B_{40}  –21621  –21892  –21888  –21893  –21927  –21937  –22105 
B_{43}  –26908  –27397  –27431  –27454  –27460  –27495  –27707 
S_{4}  14588.9  14834.1  14847.4  14857.7  14863.8  14881.4  14996.0 
S_{2}  250.0  285.0  293.6  297.3  292.2  294.6  295.9 
Dq_{cub} (from s_{4})  1591.8  1618.5  1620.0  1621.1  1621.8  1623.7  1636.2 
Racah B_{35}  703  642  641  638  648  636  620 
Racah B_{55}  714  715  718  717  716  717  715 
Racah C  3165  3152  3144  3148  3150  3148  3149 
β_{35} (B_{0} = 995[1])  0.71  0.64  0.64  0.64  0.65  0.64  0.62 
C/B_{35}  4.51  4.91  4.90  4.93  4.86  4.95  5.08 
Dq_{cub} (from Dq_{trig})  1591.5  1618.2  1619.6  1620.8  1621.4  1623.4  1635.9 
Dq_{trig}  1608.0  1637.3  1639.3  1640.7  1641.0  1643.1  1655.8 
Dτ  –42.4  –49.0  –50.6  –51.2  –50.4  –50.8  –51.3 
Dσ  79.9  91.0  93.8  95.0  93.4  94.1  94.5 
Dq_{cub} (from spectra [2])  1593.0  1620.0  1622.0  1624.0  1623.5  1623.5  1631.0 
Racah B_{35} [2]  703  641  640  636  647  638  628 
application of SPM:  
^{4}A_{2}_{g}(F) → ^{4}T_{2}_{g}(F)  
calc. (mean)  15930  16193  16208  162191  622616  24416369  
observed  15930  16200  16220  16240  16235  16235  16310 
^{4}A_{2}_{g}(F) → ^{4}T_{1}_{g}(F)  
calc. (mean)  22776  22630  22638  22627  22710  22641  22653 
observed  22775  22630  22640  22630  22710  22640  22650 
The refined set of SPM parameters for Cr^{3+} was then applied to describe all observed spinallowed transitions of the natural garnets, i.e., only the Racah parameter B_{35} had to be aligned. Racah B_{55} and C of the natural samples were again estimated using Equations 41 and 42. The respective results, i.e. nonzero B_{kq}, rotational invariants, corresponding Dq_{cub}, 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 spinallowed transitions. Slight deviations, if any, only occur for the first spinallowed transition, the maximum error is less than 0.5% for sample Ska1. Calculated energy splittings within the first and second spinallowed 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 (Yeung & Newman, 1986). However, the magnitude of the tolerably complies with data by Stedman (1969) and Yeung & Newman (1986) obtained on ruby and kyanite, respectively, taking the different reference distances R_{0} used into account. The possible range of was restricted in the present case to be in the order of a few thousand cm^{−1}, due to the fact that the observed spinallowed transitions exhibit no energy splitting. Using for example = 35000 cm^{−1} proposed for Cr^{3+} in kyanite and ruby (Yeung & Newman, 1986) or up to ∼ 70000 cm^{−1} we extracted for eskolaite (see below), leads to a stronger energy splitting of the spinallowed bands, which is not in accordance with the experimental results.
It is noteworthy that spinforbidden and spinallowed 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 ^{2}E_{g} level of ∼ 100 cm^{−1} cannot be explained by the spinorbit coupling parameter ζ alone, but only by a simultaneous increase of up to about 40000 cm^{−1}. However, as pointed out above, the concomitant splitting of the spinallowed bands would be in contradiction to the experimental results.
In contrast to the previous SPM investigations on Cr^{3+}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 chromiumhosting octahedron. In addition, the SPM parameters determined on uvarovite comply very well with the conventional CF parameters.
Using Equation 35 with X_{Cr,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 spinallowed band. Due to the comparable small band shift, the quotient 10Dq_{XCr}3+ = 1/10Dq_{X}_{Cr}3+ 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 Cr^{3+} mole fraction (Fig. 8).
Conclusions
The structural relationships drawn from natural birefringent uvarovitegrossular 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 uvarovitegrossular solid solutions ( = 9532 cm^{−1}, = 4650 cm^{−1}, t_{4} = 6.7, t_{2} (fixed) = 3). This SPM parameter set is valid for the complete uvarovitegrossular solid solution series and enables to calculate the Cr–O bond length R_{i} solely from a single UVVIS 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 CrO_{6} polyhedron in ruby, Al_{2}O_{3}:Cr^{3+}
Introduction
Rubies, Al_{2}O_{3}:Cr^{3+}, 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 R_{1} 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 spinforbidden ruby lines between 13000 and 15000 cm^{−1}.
Superposition model analyses for Cr^{3+} 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 Cr_{2}O_{3}–Al_{2}O_{3} have been devoted to either reflection measurements of powdered samples for the whole solid solution series (Poole & Itzel, 1963; SchmitzDuMont & Reinen, 1959; Neuhaus, 1960; Reinen, 1969) or polarised single crystal studies of the endmember eskolaite Cr_{2}O_{3} (McClure, 1963) and ruby, i.e., Cr^{3+}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 endmember compounds corundum and eskolaite crystallise isomorphously in space group (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 M^{3+} cation is C_{3}, 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 endmember structures reveals that the AlO_{6} and CrO_{6} polyhedra differ in size as well as in distortion, as shown in Figure 1. Consequently, crystal structure refinements of (Al,Cr)_{2}O_{3} 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 Cr^{3+}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 Cr^{3+} cation position along the threefold axis as compared with Al (Tsirel'son et al., 1985).
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 CrO_{6} 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 UVVIS 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 reevaluated concerning peak centres and barycentres of the spinallowed 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 spinallowed transition v_{1} of Cr^{3+} is displayed as a function of the composition of the corundumeskolaite solid solution series. Data obtained by powder reflection measurements are also included for comparison (SchmitzDumont & Reinen, 1959; Neuhaus, 1960; Reinen, 1969).
sample  eskolaite  ruby Longido  ruby Mysore  synthetic ruby  synthetic ruby  corundum 

[1, 2]  syn2  syn1  [3]  
Chemistry  Cr_{2}O_{3}  Al_{1.97}Cr_{0.03}O_{3}  Al_{1.99}Cr_{0.01}O_{3}  Al_{2.00}Cr_{0.00}O_{3}  Al_{1.98}Cr_{0.02}O_{3}  Al_{2}O_{3} 
EMP  Al_{2}O_{3}: 97.2  Al_{2}O_{3}: 99.3  Al_{2}O_{3}: 99.8  Al_{2}O_{3}: 98.5  
[wt%]  Cr_{2}O_{3}: 1.82  Cr_{2}O_{3}: 0.81  Cr_{2}O_{3}: 0.14  Cr_{2}O_{3}: 1.48  
Fe_{2}O_{3}: 0.28  Fe_{2}O_{3}: 0.07  
Spectra  
E_{obs} ^{4}E  16775  17800  17925  17890  17860  – 
[cm^{−1}] ^{4}A  16390  18200  18200  18360  18300  – 
^{4}E  21420  24180  24260  24340  24320  – 
^{4}A  21850  25000  25060  25120  25060  – 
CF  
Dq [cm^{−1}]  1703.3  1847.8  1853.3  1857.8  1850.7  – 
Dτ [cm^{−1}]  –117.3  –138.1  –134.0  –132.5  –124.9  – 
Dσ [cm^{−1}]  549.1  61.2  129.2  5.3  5.7  – 
B_{35} [cm^{−1}]  457.2  629.6  629.6  631.5  633.2  – 
Geometry  Xray  SPM  SPM  SPM  SPM  Xray 
R_{1} [Å]  2.009  1.972  1.964  1.963  1.961  1.971 
R_{2} [Å]  1.962  1.925  1.930  1.928  1.934  1.856 
1.986  1.9485  1.947  1.9455  1.9475  1.914  
θ_{1} [°]  48.75  47.68  47.87  47.74  47.92  47.67 
θ_{2} [°]  118.50  118.01  117.98  118.01  118.12  116.85 
φ_{2} – φ_{1} [°]  55.97  56.10  55.84  55.78  55.90  56.11 
sample  eskolaite  ruby Longido  ruby Mysore  synthetic ruby  synthetic ruby  corundum 

[1, 2]  syn2  syn1  [3]  
Chemistry  Cr_{2}O_{3}  Al_{1.97}Cr_{0.03}O_{3}  Al_{1.99}Cr_{0.01}O_{3}  Al_{2.00}Cr_{0.00}O_{3}  Al_{1.98}Cr_{0.02}O_{3}  Al_{2}O_{3} 
EMP  Al_{2}O_{3}: 97.2  Al_{2}O_{3}: 99.3  Al_{2}O_{3}: 99.8  Al_{2}O_{3}: 98.5  
[wt%]  Cr_{2}O_{3}: 1.82  Cr_{2}O_{3}: 0.81  Cr_{2}O_{3}: 0.14  Cr_{2}O_{3}: 1.48  
Fe_{2}O_{3}: 0.28  Fe_{2}O_{3}: 0.07  
Spectra  
E_{obs} ^{4}E  16775  17800  17925  17890  17860  – 
[cm^{−1}] ^{4}A  16390  18200  18200  18360  18300  – 
^{4}E  21420  24180  24260  24340  24320  – 
^{4}A  21850  25000  25060  25120  25060  – 
CF  
Dq [cm^{−1}]  1703.3  1847.8  1853.3  1857.8  1850.7  – 
Dτ [cm^{−1}]  –117.3  –138.1  –134.0  –132.5  –124.9  – 
Dσ [cm^{−1}]  549.1  61.2  129.2  5.3  5.7  – 
B_{35} [cm^{−1}]  457.2  629.6  629.6  631.5  633.2  – 
Geometry  Xray  SPM  SPM  SPM  SPM  Xray 
R_{1} [Å]  2.009  1.972  1.964  1.963  1.961  1.971 
R_{2} [Å]  1.962  1.925  1.930  1.928  1.934  1.856 
1.986  1.9485  1.947  1.9455  1.9475  1.914  
θ_{1} [°]  48.75  47.68  47.87  47.74  47.92  47.67 
θ_{2} [°]  118.50  118.01  117.98  118.01  118.12  116.85 
φ_{2} – φ_{1} [°]  55.97  56.10  55.84  55.78  55.90  56.11 
As a result of the different ionic radii in sixfold coordination (r_{Cr}3+ = 0.615 Å and r_{Al} = 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 spinallowed d–d band ^{4}A_{2g} → ^{4}T_{2g} (O_{h}) shift to lower wavenumbers. Over the whole corundumeskolaite solid solution series a band shift of 1900 and 1300 cm^{−1} is observed for the split components ^{4}A → ^{4}A and ^{4}A → ^{4}E, respectively. It is important to note that due to their different slopes these two split levels change their order as a function of the Cr^{3+} content: in ruby the energy sequence is ^{4}A → ^{4}E < ^{4}A → ^{4}A, while in eskolaite ^{4}A → ^{4}E > ^{4}A → ^{4}A is observed. The split levels of the second spinallowed transition are also shifted to lower wave numbers with increasing Cr^{3+} content, but they keep their relative order ^{4}A → ^{4}E < ^{4}A → ^{4}A. In addition, the slope of the second spinallowed band (and its respective split levels) is influenced by the configurational interaction with the ^{4}T_{1g}(^{4}P) split states of alike symmetry.
Crystal field SPM calculations
In accordance with the C_{3} point symmetry of the CrO_{6} octahedron in eskolaite, the electronic z axis of the crystal field was chosen parallel to the threefold axis. The
reference metalligand distance R_{0} was set to 1.995 Å, the sum of the ionic radii of ^{[6]}Cr^{3+} and ^{[3–4]}O^{2–} (Shannon, 1976). Preliminary calculations showed that spinorbit coupling effects can be neglected when dealing with the spinallowed bands only. In this case, the only necessary freeion parameter is Racah B_{35}.
Due to the C_{3} point symmetry of the CrO_{6} octahedron with only two different Cr–O bonds, the equations for the crystal field parameters B_{kq} simplify, in that only two coordination factors K_{kq}(θ_{i}, φ_{i}) for each B_{kq} are needed and – proper orientation provided – only the real B_{20}, B_{40}, B_{43} and the imaginary B_{43} are nonzero (compare Table 1). Hence, the relationships for B_{20}, B_{40}, and B_{43} 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. 29–31). Nevertheless, it is emphasised that all SPM calculations described below have been performed with the full set of crystal field parameters B_{kq} 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 t_{2} was marginal with respect to and the given ratio of R_{0}/R_{i}. Therefore, this powerlaw exponent was fixed at t_{2} = 3 in the calculations. Thus, best fit results for the spinallowed bands were attained with the SPM parameters = 10380 cm^{−1}, = 71900 cm^{−1}, t_{4} = 5.0, t_{2} (fixed) = 3, and with Racah B_{35} = 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 C_{3} site symmetry of the CrO_{6} 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]}Cr^{3+} in synthetic eskolaite was then applied to the investigated natural and synthetic rubies. The local CrO_{6} polyhedra in these ruby samples were modelled by minimising the difference between the observed and calculated spinallowed energy levels.
The final fully optimised geometrical data for the local CrO_{6} 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 Cr^{3+} in the Al_{2}O_{3} matrix, the longer R_{1} distance (∼ 1.97 Å) does not change significantly, whereas the short R_{2} 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 θ_{1} angle between the R_{1} bonds and the threefold axis remains rather constant compared to the AlO_{6} polyhedron in corundum, while the θ_{2} angle related to the shorter R_{2} 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 CrO_{6} polyhedra in ruby are less distorted than the AlO_{6} ones, a trend which is also confirmed by the structural data for the endmember eskolaite. The results of our SPM analyses reveal a moderate degree of relaxation of ε ≈ 0.46 for Cr^{3+} within the Al_{2}O_{3} 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 powerlaw dependence of the CF strength from the bond lengths in the order of t_{4} ≈ 15. Furthermore, the longer Cr–O distance even exceeds that found in endmember 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).
Conclusions
The results of the SPM analyses for eskolaite obey the general expectation (Yeung & Newman, 1986). Considering the different reference distances R_{0} used, the magnitudes of and tolerably comply with data by Stedman (1969) and Yeung & Newman (1986) obtained for ruby and kyanite, respectively. On the contrary, for Cr^{3+} in uvarovite a much smaller value for in the order of 5000 cm^{−1} has been determined, corresponding to an indiscernible trigonal energy splitting of the spinallowed transitions (see 6.2.7 and Andrut & Wildner, 2002).
As we pointed out earlier, until the present investigations, the sets of SPM parameters for Cr^{3+} 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 coworkers (Yeung et al., 1994a, 1994b; Qin et al., 1994) to Cr^{3+} 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 CrO_{6} octahedra were only estimated and the respective SPM parameters were evaluated lacking precisely determined interatomic distances. Thus, the actual distortion of the Cr^{3+} polyhedron was not appropriately taken into account in these investigations. Besides, the SPM parameters were related to a rather arbitrary reference distance R_{0}, 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 spinallowed transitions, which are not in agreement with the experimental data reported so far.
However, our set of SPM parameters for Cr^{3+} was derived from absorption spectra and structural data of a wellcharacterised synthetic endmember 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 CrO_{6} polyhedron in an Al_{2}O_{3} matrix solely from optical absorption spectra could be realised for the first time.
References
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. P13976CHE.
Figures & Tables
Phase  Sym.  t_{4}  t_{2}  references  

Li_{2}Co_{3}(SeO_{3})_{4}  1  4740  3.1  7000  5.5  [1]^{1,2,3} 
Co(OH)_{2}, 290K  5260  *  4920  *  [2]^{1}, [3]^{1,2,3}  
Co(OH)_{2}, 90K  5320  3900  
CoSO_{4}·H_{2}O  4840  1.9  5300  4.0  [4]^{1}, [5]^{2}, [6]^{3}  
CoSeO_{4}·H_{2}O  5000  1.5  6890  2.7  [7]^{1}, [5]^{2}, [6]^{3}  
NaCo_{2}(SeO_{3})_{2}(OH)  m  4760  1.0  5040  2.4  [8]^{1,2}, [6]^{3} 
CoSe_{2}O_{5}  2  4960  *  4270  *  [9]^{1}, [10]^{2}, [6]^{3} 
CoSeO_{3}·2H_{2}O  1  5090  5.4  8000  0  [11]^{1}, [10]^{2}, [12]^{3} 
Phase  Sym.  t_{4}  t_{2}  references  

Li_{2}Co_{3}(SeO_{3})_{4}  1  4740  3.1  7000  5.5  [1]^{1,2,3} 
Co(OH)_{2}, 290K  5260  *  4920  *  [2]^{1}, [3]^{1,2,3}  
Co(OH)_{2}, 90K  5320  3900  
CoSO_{4}·H_{2}O  4840  1.9  5300  4.0  [4]^{1}, [5]^{2}, [6]^{3}  
CoSeO_{4}·H_{2}O  5000  1.5  6890  2.7  [7]^{1}, [5]^{2}, [6]^{3}  
NaCo_{2}(SeO_{3})_{2}(OH)  m  4760  1.0  5040  2.4  [8]^{1,2}, [6]^{3} 
CoSe_{2}O_{5}  2  4960  *  4270  *  [9]^{1}, [10]^{2}, [6]^{3} 
CoSeO_{3}·2H_{2}O  1  5090  5.4  8000  0  [11]^{1}, [10]^{2}, [12]^{3} 
^{1} crystal structure, ^{2} polarised absorption spectra, ^{3} SPM analysis
* fixed at t_{4} = 5 and t_{2} = 3 (see text)
[1] Wildner & Andrut (1999); [2] Pertlik (1999); [3] Andrut & Wildner (2001a); [4] Wildner & Giester (1991); [5] Wildner (1996a); [6] this paper; [7] Giester & Wildner (1992); [8] Wildner (1995); [9] Hawthorne et al. (1987); [10] Andrut & Wildner, unpublished; [11] Wildner (1990); [12] Wildner & Andrut (2001a)
(tricl)  Fddd (orth)  (cub)  

atom  m  s  atom  m  s  atom  m  s 
Ca1  Ca1_{1256}  32  1  
…  4  1  Ca2_{3}  8  222  Ca_{1–6}  24  222 
Ca6  Ca3_{4}  8  222  
Y1  Y1_{1367}  
…  2  16  Y_{1–8}1643  16  
Y8  Y2_{2458}  
Si1  Si1_{1256}  32  1  
…  4  1  Si_{1–6}  24  
Si6  Si2_{34}  16  2  
O1  O1  
…  4  1  …  32  1  o  96  1 
O24  O6 
(tricl)  Fddd (orth)  (cub)  

atom  m  s  atom  m  s  atom  m  s 
Ca1  Ca1_{1256}  32  1  
…  4  1  Ca2_{3}  8  222  Ca_{1–6}  24  222 
Ca6  Ca3_{4}  8  222  
Y1  Y1_{1367}  
…  2  16  Y_{1–8}1643  16  
Y8  Y2_{2458}  
Si1  Si1_{1256}  32  1  
…  4  1  Si_{1–6}  24  
Si6  Si2_{34}  16  2  
O1  O1  
…  4  1  …  32  1  o  96  1 
O24  O6 
Uvasyn22  Sardesy  Sarkl2  Sar899  Sarw2  Ves2  Ska1  

9532  
4650  
t_{4}  6.7  
t_{2} (fixed)  3  
ζ (fixed)  135  
B_{20}  –559  –637  –657  –665  –653  –659  –662 
B_{40}  –21621  –21892  –21888  –21893  –21927  –21937  –22105 
B_{43}  –26908  –27397  –27431  –27454  –27460  –27495  –27707 
S_{4}  14588.9  14834.1  14847.4  14857.7  14863.8  14881.4  14996.0 
S_{2}  250.0  285.0  293.6  297.3  292.2  294.6  295.9 
Dq_{cub} (from s_{4})  1591.8  1618.5  1620.0  1621.1  1621.8  1623.7  1636.2 
Racah B_{35}  703  642  641  638  648  636  620 
Racah B_{55}  714  715  718  717  716  717  715 
Racah C  3165  3152  3144  3148  3150  3148  3149 
β_{35} (B_{0} = 995[1])  0.71  0.64  0.64  0.64  0.65  0.64  0.62 
C/B_{35}  4.51  4.91  4.90  4.93  4.86  4.95  5.08 
Dq_{cub} (from Dq_{trig})  1591.5  1618.2  1619.6  1620.8  1621.4  1623.4  1635.9 
Dq_{trig}  1608.0  1637.3  1639.3  1640.7  1641.0  1643.1  1655.8 
Dτ  –42.4  –49.0  –50.6  –51.2  –50.4  –50.8  –51.3 
Dσ  79.9  91.0  93.8  95.0  93.4  94.1  94.5 
Dq_{cub} (from spectra [2])  1593.0  1620.0  1622.0  1624.0  1623.5  1623.5  1631.0 
Racah B_{35} [2]  703  641  640  636  647  638  628 
application of SPM:  
^{4}A_{2}_{g}(F) → ^{4}T_{2}_{g}(F)  
calc. (mean)  15930  16193  16208  162191  622616  24416369  
observed  15930  16200  16220  16240  16235  16235  16310 
^{4}A_{2}_{g}(F) → ^{4}T_{1}_{g}(F)  
calc. (mean)  22776  22630  22638  22627  22710  22641  22653 
observed  22775  22630  22640  22630  22710  22640  22650 
Uvasyn22  Sardesy  Sarkl2  Sar899  Sarw2  Ves2  Ska1  

9532  
4650  
t_{4}  6.7  
t_{2} (fixed)  3  
ζ (fixed)  135  
B_{20}  –559  –637  –657  –665  –653  –659  –662 
B_{40}  –21621  –21892  –21888  –21893  –21927  –21937  –22105 
B_{43}  –26908  –27397  –27431  –27454  –27460  –27495  –27707 
S_{4}  14588.9  14834.1  14847.4  14857.7  14863.8  14881.4  14996.0 
S_{2}  250.0  285.0  293.6  297.3  292.2  294.6  295.9 
Dq_{cub} (from s_{4})  1591.8  1618.5  1620.0  1621.1  1621.8  1623.7  1636.2 
Racah B_{35}  703  642  641  638  648  636  620 
Racah B_{55}  714  715  718  717  716  717  715 
Racah C  3165  3152  3144  3148  3150  3148  3149 
β_{35} (B_{0} = 995[1])  0.71  0.64  0.64  0.64  0.65  0.64  0.62 
C/B_{35}  4.51  4.91  4.90  4.93  4.86  4.95  5.08 
Dq_{cub} (from Dq_{trig})  1591.5  1618.2  1619.6  1620.8  1621.4  1623.4  1635.9 
Dq_{trig}  1608.0  1637.3  1639.3  1640.7  1641.0  1643.1  1655.8 
Dτ  –42.4  –49.0  –50.6  –51.2  –50.4  –50.8  –51.3 
Dσ  79.9  91.0  93.8  95.0  93.4  94.1  94.5 
Dq_{cub} (from spectra [2])  1593.0  1620.0  1622.0  1624.0  1623.5  1623.5  1631.0 
Racah B_{35} [2]  703  641  640  636  647  638  628 
application of SPM:  
^{4}A_{2}_{g}(F) → ^{4}T_{2}_{g}(F)  
calc. (mean)  15930  16193  16208  162191  622616  24416369  
observed  15930  16200  16220  16240  16235  16235  16310 
^{4}A_{2}_{g}(F) → ^{4}T_{1}_{g}(F)  
calc. (mean)  22776  22630  22638  22627  22710  22641  22653 
observed  22775  22630  22640  22630  22710  22640  22650 
sample  eskolaite  ruby Longido  ruby Mysore  synthetic ruby  synthetic ruby  corundum 

[1, 2]  syn2  syn1  [3]  
Chemistry  Cr_{2}O_{3}  Al_{1.97}Cr_{0.03}O_{3}  Al_{1.99}Cr_{0.01}O_{3}  Al_{2.00}Cr_{0.00}O_{3}  Al_{1.98}Cr_{0.02}O_{3}  Al_{2}O_{3} 
EMP  Al_{2}O_{3}: 97.2  Al_{2}O_{3}: 99.3  Al_{2}O_{3}: 99.8  Al_{2}O_{3}: 98.5  
[wt%]  Cr_{2}O_{3}: 1.82  Cr_{2}O_{3}: 0.81  Cr_{2}O_{3}: 0.14  Cr_{2}O_{3}: 1.48  
Fe_{2}O_{3}: 0.28  Fe_{2}O_{3}: 0.07  
Spectra  
E_{obs} ^{4}E  16775  17800  17925  17890  17860  – 
[cm^{−1}] ^{4}A  16390  18200  18200  18360  18300  – 
^{4}E  21420  24180  24260  24340  24320  – 
^{4}A  21850  25000  25060  25120  25060  – 
CF  
Dq [cm^{−1}]  1703.3  1847.8  1853.3  1857.8  1850.7  – 
Dτ [cm^{−1}]  –117.3  –138.1  –134.0  –132.5  –124.9  – 
Dσ [cm^{−1}]  549.1  61.2  129.2  5.3  5.7  – 
B_{35} [cm^{−1}]  457.2  629.6  629.6  631.5  633.2  – 
Geometry  Xray  SPM  SPM  SPM  SPM  Xray 
R_{1} [Å]  2.009  1.972  1.964  1.963  1.961  1.971 
R_{2} [Å]  1.962  1.925  1.930  1.928  1.934  1.856 
1.986  1.9485  1.947  1.9455  1.9475  1.914  
θ_{1} [°]  48.75  47.68  47.87  47.74  47.92  47.67 
θ_{2} [°]  118.50  118.01  117.98  118.01  118.12  116.85 
φ_{2} – φ_{1} [°]  55.97  56.10  55.84  55.78  55.90  56.11 
sample  eskolaite  ruby Longido  ruby Mysore  synthetic ruby  synthetic ruby  corundum 

[1, 2]  syn2  syn1  [3]  
Chemistry  Cr_{2}O_{3}  Al_{1.97}Cr_{0.03}O_{3}  Al_{1.99}Cr_{0.01}O_{3}  Al_{2.00}Cr_{0.00}O_{3}  Al_{1.98}Cr_{0.02}O_{3}  Al_{2}O_{3} 
EMP  Al_{2}O_{3}: 97.2  Al_{2}O_{3}: 99.3  Al_{2}O_{3}: 99.8  Al_{2}O_{3}: 98.5  
[wt%]  Cr_{2}O_{3}: 1.82  Cr_{2}O_{3}: 0.81  Cr_{2}O_{3}: 0.14  Cr_{2}O_{3}: 1.48  
Fe_{2}O_{3}: 0.28  Fe_{2}O_{3}: 0.07  
Spectra  
E_{obs} ^{4}E  16775  17800  17925  17890  17860  – 
[cm^{−1}] ^{4}A  16390  18200  18200  18360  18300  – 
^{4}E  21420  24180  24260  24340  24320  – 
^{4}A  21850  25000  25060  25120  25060  – 
CF  
Dq [cm^{−1}]  1703.3  1847.8  1853.3  1857.8  1850.7  – 
Dτ [cm^{−1}]  –117.3  –138.1  –134.0  –132.5  –124.9  – 
Dσ [cm^{−1}]  549.1  61.2  129.2  5.3  5.7  – 
B_{35} [cm^{−1}]  457.2  629.6  629.6  631.5  633.2  – 
Geometry  Xray  SPM  SPM  SPM  SPM  Xray 
R_{1} [Å]  2.009  1.972  1.964  1.963  1.961  1.971 
R_{2} [Å]  1.962  1.925  1.930  1.928  1.934  1.856 
1.986  1.9485  1.947  1.9455  1.9475  1.914  
θ_{1} [°]  48.75  47.68  47.87  47.74  47.92  47.67 
θ_{2} [°]  118.50  118.01  117.98  118.01  118.12  116.85 
φ_{2} – φ_{1} [°]  55.97  56.10  55.84  55.78  55.90  56.11 
Contents
Spectroscopic methods in mineralogy
Spectroscopic methods provide information about the local structure of minerals. The methods do not depend on longrange periodicity or crystallinity. The geometric arrangement of atoms in a mineral phase is only one aspect of its constitution. Its vibrational characteristic, electronic structure and magnetic properties are of greatest importance when we consider the behaviour of minerals in dynamic processes. The characterisation of the structural and physicochemical properties of a mineral requires the application of several complementary spectroscopic techniques. However, it is one of the main aims of this School to demonstrate that different spectroscopic methods work on the same basic principles. Spectroscopic techniques represent an extremely rapidly evolving area of mineralogy and many recent research efforts are similar to those in materials science, solid state physics and chemistry. Applications to different materials of geoscientific relevance have expanded by the development of microspectroscopic techniques and by in situ measurements at low to hightemperature and highpressure conditions.
 Abstract
 Introduction
 The relationship between 10Dq and the interatomic distance R for a regular octahedron
 Applications and consequences
 Vegard's rule (and its relationship to local interatomic distances)
 Local interatomic distances in coordination polyhedra
 Superposition model of crystal fields (SPM)
 The regular octahedron
 The distorted octahedron – one selected example
 Strategy for the determination of crystal field parameters using SPM
 Applications of the superposition model in geosciences
 Superposition model parameters for Co^{2+} in oxygenbased crystal fields
 Extraction of crystal field parameters for Cr^{3+} from the binary solid solution uvarovitegrossular
 Local interatomic bond lengths and angles derived from optical absorption spectra: The CrO_{6} polyhedron in ruby, Al_{2}O_{3}:Cr^{3+}
 References
 Acknowledgements