Introduction

The color of minerals is due to the interaction of light with matter. When light goes through ruby (α-Al2O3:Cr3+), the whole yellow-green and violet radiations are absorbed, while red and few blue radiations are transmitted. Therefore, ruby is red with a slight purple overtone. On the contrary, red light is absorbed in eskolaite (α-Cr2O3), so that this mineral looks green. Although it is now well known that color in ruby and eskolaite is due to the chromium ions (Nassau 1983; Burns 1993), a question remains unexplained. Why does the same chromium chromophore ion, with the same valence state and the same kind of distorted octahedral site, yield a red color in ruby and a green one in eskolaite?

In such oxide minerals, color is generally interpreted within the ligand field theory. This model is based on an electrostatic interaction between the central cation and the ligands of its coordination sphere. It is based mainly on the geometry and the symmetry around the central cation. This theory has been applied successfully to predict the number of 3d–3d transitions in optical absorption spectra and to identify the corresponding absorption bands (Lever 1984). Moreover, this theory enables an empirical calculation of the transition energies (Liehr 1963). The UV–VIS absorption spectra of Cr3+-containing materials present two main absorption bands, which are already well explained by the ligand field theory applied to the system Al2O3:Cr3+ (Poole 1964; Poole and Itzel 1963; McClure 1962, 1963; Reinen 1969; Sugano and Peter 1961; Liehr 1963; MacFarlane 1963; Graham 1960). The spectroscopic term of the fundamental state for the free Cr3+ coloring ion in spherical symmetry is 4F. In the octahedral symmetry, 4F is split into three states of increasing energies that are 4A2g , 4T2g and 4T1g . The symmetry of the ground state is 4A2g . The absorption band at lower energy, labeled ε1, arises from 4A2g toward 4T2g transitions. The broad band at higher energy, labeled ε2, arises from 4A2g toward 4T1g transitions. The crystal field parameter Δ and the Racah parameter B are extracted from the values of ε1 and ε2 within the framework of the crystal field model. The Δ parameter is given by the energy ε1. The B parameter is given by \(B=[(\epsilon_2-\epsilon_1)/3][(2\epsilon_1-\epsilon_2)/(9\epsilon_1 - 5\epsilon_2)]\) (Marfunin 1979; Reinen 1969). The optical data given by Reinen (1969) lead to Δ=2.24 eV (18,070 cm−1) for ruby and Δ=2.07 eV (16,700 cm−1) for α-Cr2O3, while the Racah parameter B is B=0.080 eV (645 cm−1) for ruby and B=0.058 eV (468 cm−1) for α-Cr2O3. The ratio \(\Delta/B\) increases when the concentration x of chromium in Cr x Al2−x O3 increases.

Despite the success of the ligand field theory, some points remained misunderstood. Liehr (1963) tried to sort the minerals by increasing values of the crystal field Δ and the Racah parameter B given by optical spectroscopy. He wanted to make an analogy with the spectroscopic and the nephelauxetic series in solutions but found irregular changes for the crystal field Δ and for the Racah parameter B in his crystalline series. The variations of these parameters were also studied as a function of the temperature T, the pressure P and the amount x of chromium in Cr x Al2−x O3 (Poole 1964; Poole and Itzel 1963). Starting from the idea that the crystal field Δ and the Racah parameter B depend only on the Cr–O distance, three different laws for their variations were found, depending on which parameter, T, P or x, was at the origin of the variation of the Cr–O distance. These results tend to prove that the value of the crystal field as measured by the 4A2g →4T2g transition energy is not simply a function of the Cr–O distance. McClure (1962) studied the optical absorption spectra of ruby (Al2O3:Cr3+) in detail. To interpret some features, he needed to consider complex models. He calculated the potential for a chromium impurity displaced by 0.1 Å toward or away from the nearest cation in the structure and proposed a tetragonal distortion of the excited state. These high-quality and detailed studies of Al2O3–Cr2O3 solid solution series were made within the virtual crystal approximation, in which the chromium atom is supposed to substitute exactly for the aluminum atom in the structure, so that the Cr–O distances in ruby are the same as the Al–O distances in α-Al2O3. This leads to mistakes and to misunderstandings. Actually, when chromium substitutes for aluminum, the chromium site is expected to be different from the aluminum site in α-Al2O3, due to the larger ionic radius of chromium (r 3+Cr =0.615 Å) compared to that of aluminum (r 3+Al =0.535 Å) in an octahedral site (Shannon 1976). As a consequence, the understanding of the local environment around the chromium atom is absolutely needed to explain the evolution of physical properties with respect to the chromium content x in Cr x Al2−x O3.

Subsequently, many studies have been undertaken to determine the chromium site in ruby (α-Al2O3:Cr3+). Single crystal X-ray diffraction (McCauley and Gibbs 1972; Moss and Newnham 1964), UV–VIS spectroscopy (McClure 1962; Langer 2001; Andrut et al. 2004) and EPR spectroscopy (Büscher et al. 1987) have shown the existence of relaxations around the chromium atom in α-Al2O3. Extended X-ray absorption fine structure (EXAFS) has been performed at the Cr K-edge on powders (Kizler et al. 1996) or single crystals (Emura et al. 1993) of ruby. In a previous piece of work, we determined Cr–O bond lengths in the coordination shell by linear dichroic EXAFS at the chromium K-edge in α-Al2O3:Cr3+ (Gaudry et al. 2003). However, to our knowledge, no EXAFS experiment has been undertaken on the whole Al2O3–Cr2O3 solid solution.

In a first step, we evaluate the influence of the crystal field Δ and the Racah parameter B on the colors of the green eskolaite and the red ruby. These two parameters are usually deduced from the positions of large features in the UV–VIS spectra, using a semi-quantitative theory. Our approach is to analyze the isotropic X-ray absorption spectra and the X-ray magnetic circular dichroic (XMCD) spectra at the Cr L2,3-edges with the quantitative ligand field multiplet (LFM) theory following Theo Thole’s developments (Cowan 1981; Butler 1981; Thole et al. 1985), in order to address the values of Δ and B from a different point of view. In addition, this analysis gives access to the nature of the Cr–O bonding in α-Cr2O3 and Cr0.07Al1.93O3. To our knowledge, no X-ray absorption spectroscopy (XAS) at the Cr L2,3-edges in Cr x Al2−x O4 has yet been made. The structural environment around chromium in the whole solid solution Al2O3–Cr2O3 is also specified by XAS at the chromium K-edge. We especially focused on the determination of the mean Cr–O distances for several values of x in Cr x Al2−x O3, in order to make a link between the color change and the local environment around chromium in the Al2O3–Cr2O3 series. Given that the same technique is used for the whole solid solution series, the results for different chromium concentrations are directly comparable.

Materials and methods

Synthesis of the samples

Corundum, α-Al2O3, and eskolaite, α-Cr2O3, belong to the \(R\bar{3}2/c\) space group (Wyckoff 1964). They are a stacking of distorted octahedra made of six oxygen atoms, at the center of which lies an aluminum atom (in α-Al2O3) or a chromium atom (in α-Cr2O3). The six oxygen atoms are gathered into two groups of three atoms to comply with the C 3 local symmetry. These two groups lie at two different distances from the central atom. The nearest three oxygen atoms, labeled O1, are between edge-shared octahedra and lie at 1.96 Å in α-Cr2O3 and 1.86 Å in α-Al2O3. The farther three oxygen atoms, labeled O2, are between face-shared octahedra and lie at 2.01 Å in α-Cr2O3 and 1.97 Å in α-Al2O3. The closer Cr–Cr pairs in α-Cr2O3 lie at 2.65 Å (one pair) and 2.88 Å (three pairs), while the closer Al–Al pairs in α-Al2O3 lie at 2.65 Å (one pair) and 2.79 Å (three pairs) (Pearson 1962; Finger and Hazen 1980).

The Al2−x Cr x O3 powders were prepared by co-precipitation and subsequent calcination (Bondioli et al. 2000). The amorphous gels of Al3+ and Cr3+ hydroxides were co-precipitated with ammonia (NH4OH, RPE, Carlo Erba, Milan, Italy) from aluminum nitrate [Al(NO3)3, RPE, Carlo Erba] and chromium nitrate [Cr(NO3)3, RPE, Carlo Erba] solutions at pH 9. The amorphous co-precipitated hydroxides, carefully washed with distilled water, were dried in a conventional furnace at 120°C and then ground in an agate mortar. The powders obtained were calcined in air, in an electric furnace, for 3 h at 1,300°C. All the heat-treated samples form a highly pure and crystalline solid solution, whatever the level of chromium content. The samples have been characterized by many techniques presented elsewhere (Bondioli et al. 2000).

X-ray absorption spectroscopy

Cr K-edge EXAFS

Fine powders of the Al2−x Cr x O3 samples (x=2.00, 1.00, 0.45, 0.29, 0.07) were mixed with cellulose and mounted on a sample holder for EXAFS data collection at beamline D44 of DCI storage ring at LURE (France). Spectra were collected at the Cr K-edge (5,989 eV) in transmission mode using a pair of Si(111) monochromator crystals. The beamline is equipped with a pair of borosilicate parallel mirrors working in specular reflection to reject harmonic reflections. The monochromator was calibrated by assigning an energy value of 5,989 eV to the first inflection point in the absorption edge of a Cr metal reference foil. Samples were mounted in a liquid nitrogen cryostat (≃77 K) to decrease thermal disorder. Three spectra were collected for all samples except Al1.93Cr0.07O3, for which ten spectra were registered. A standard procedure is used to extract the normalized EXAFS signal from the absorption spectra (Winterer 1997). The EXAFS spectra at the Cr K-edge of the five samples are shown in Fig. 1.

Fig. 1
figure 1

EXAFS k 3χ(k) functions (left) and Fourier transforms (right) for the Cr x Al2−x O3 powders (x =0.07, 0.29, 0.45, 1.00, 2.00)

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These normalized EXAFS signals at the Cr K-edge were further analyzed by two complementary methods. The contributions from oxygen shells in the coordination sphere were isolated and analyzed by a Fourier series of plane wavelets. The typical k range used to obtain the radial structure function in real space was 3.90–15.60 Å−1. The EXAFS function was k 2-weighted. An appropriate r range was selected to separate the contributions of the nearer shells from the total experimental signal using back Fourier transform. Typically, the r range was 0.90–1.96 Å. EXAFS fitting was performed using the program EXAFS (Bonnin et al. 1985) with theoretical phases, amplitudes and mean free paths calculated with the FEFF8 code (Ankudinov et al. 1998). Starting models for these calculations included α-Cr2O3 and a structural model resulting from an ab initio energy minimization (see below). Four adjustable parameters (r j , σ j , N j , ΔE j0 ) were fitted for one shell. The isotropic signal was analyzed with only one mean metal–oxygen distance for the fit, although the difference between the shorter Cr–O1 and the longer Cr–O2 distances should lead to two separate Cr–O distances. Indeed, this difference was expected to be less than 0.09 Å for both α-Cr2O3 and Cr0.07Al1.93O3 (Pearson 1962; Gaudry et al. 2003), which would induce a beating effect on the spectra, with a first node for 2kΔr=π, where k=18 Å−1 for Δr=0.09 Å. Since our spectra were recorded on a shorter k range, this beating effect cannot be seen and the terms corresponding to the Cr–O1 and Cr–O2 shells can then be mixed together to give a single term corresponding to a mean Cr–O distance, with an additional contribution to σstatic (Teo 1985). This point has been extensively discussed in Fig. 5 of Gaudry et al. (2003). The σstatic values can be found in Table 1. As a second method, we applied a global simulation of χ(k) using structural models and calculation of the Cr K-edge (see below).

Table 1 Mean Cr–O distances in the coordination shell of chromium in the whole solid solution series Al2O3–Cr2O3
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Cr L2,3-edges

The experiments were carried out both at beamline ID8 of the European Synchrotron Radiation Facility (ESRF) in Grenoble (France) and at beamline SU23 of the Super-ACO storage ring at LURE in Orsay (France). On ID8 the photon source was an Apple II undulator that delivers a high flux of circularly polarized light with a polarization rate of about 100%. The monochromator was a Dragon type one with spherical grating. Isotropic and XMCD signals of sample Cr0.07Al1.93O3 have been measured at ≈6 K in a magnetic induction of 7 T. On SU23 the photon source was an asymmetric wiggler that delivers a moderate flux of elliptically polarized light above and below the orbit plane of the storage ring. The polarization rate was estimated to be around 45%. The photons were monochromatized by a plane grating with fixed exit slit (Arrio et al. 1999). The isotropic cross-section has been measured at 4.2 K for Cr2O3 and for Cr0.14Al1.86O3. XMCD on Cr0.14Al1.86O3 has also been measured. The results concerning Cr0.14Al1.86O3 are in line with those measured at ESRF on Cr0.07Al1.93O3 and are not presented here. Since α-Cr2O3 is antiferromagnetic (Brown et al. 2002), its XMCD signal is zero. The spectra at the Cr L2,3-edges are reported in Fig. 2.

Fig. 2
figure 2

Isotropic (top) and XMCD (bottom) spectra at the Cr L2,3-edge in α-Cr2O3 (dashed line) and Cr0.07Al1.93O3 (solid lines). The spectra are normalized to 1.0 at the maximum of the isotropic L3-edge

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Theoretical calculations

Cr K-edge

X-ray absorption spectra are calculated within a real space multiple scattering approach (FEFF8) for a muffin-tin potential with a screened core hole (Ankudinov et al. 1998). Structural models are validated by a direct comparison between the experimental and the calculated k 3χ(k). The structural model for α-Cr2O3 is a rhombohedral unit cell defined by a=5.35 Å and α=55.12° (Finger and Hazen 1980). The EXAFS signal for this structural model is computed for a cluster size of 6.3 Å around the absorbing atom. The total number of paths whose intensities are larger than 4% of the EXAFS contribution from the first three oxygen neighbors was found to be equal to 150. For all these paths, the number of legs was strictly less than six. The FEFF8 EXAFS curve was then multiplied by a broadening Gaussian function to take into account the Debye Waller broadening (σ=0.06 Å). The structural model for Cr0.07Al1.93O3 is given by an ab initio energy minimization using the density functional theory and the spin polarized local density approximation (Car and Parrinello 1985). The cell used in the density functional calculations is a super-cell built on the vectors \(2\vec{a}_{\rm R}, 2\vec{b}_{\rm R}, 2\vec{c}_{\rm R}\) ( \(\vec{a}_{\rm R}, \vec{b}_{\rm R}, \vec{c}_{\rm R}\) are the base vectors of the rhombohedral unit cell) and contained 80 atoms as follows: 1 chromium atom, 31 aluminum atoms and 48 oxygen atoms. The lattice constants are those resulting from the calculation of reference (Duan et al. 1998): a R=5.11 Å and θ=55.41°. The super-cell is large enough to minimize interaction between two paramagnetic ions: the minimum distance between two of these is 10.22 Å. The spin multiplet degeneracy imposed on the trivalent paramagnetic ions is four for Cr3+. The calculations are performed with the cpmd program (Hutter et al. 1996). From this structural model, a theoretical EXAFS was calculated by the FEFF8 code. For this calculation, we used the same set of parameters as those detailed for Cr2O3 except for the Debye Waller factor that we choose equal to σ=0.04 Å.

Cr L2,3-edges: ligand field multiplet calculations

The L2,3-edges are analyzed with the LFM calculations. These calculations describe the transition for a single chromium ion in a given symmetry, from a 2p63d3 ground state to a 2p53d4 excited state (Cowan 1981; Butler 1981; Thole et al. 1985). The calculation is parameterized. The parameters are adjusted to minimize the differences between the calculated and the experimental spectra. Coulomb interactions, exchange interactions and 2p and 3d spin–orbit interactions in the initial (F 2dd , F 4dd , ζ3d) and final states (F 2dd , F 4dd , F 2pd , G 1pd , G 3pd , ζ2p, ζ3d) are calculated in the spherical symmetry. The electron–electron interactions are reduced by a factor labeled κ X that takes into account effects such as covalency and charge transfer. All electron–electron integrals are reduced by the same factor. Although the local Cr3+ site symmetry in α-Cr2O3 is a slightly distorted octahedron (C 3 symmetry), with three Cr–O distances of 1.96 Å and three Cr–O distances of 2.01 Å (Finger and Hazen 1980), we described the local symmetry of the chromium site in Cr x Al2−x O3 by O h . In fact, the isotropic absorption spectrum does not depend on such small distortions (Theil et al. 1999). In O h symmetry, the monoelectronic 3d level is split into e g and t 2g . The energy separation between these two states is the crystal field parameter, labeled \(\Delta_X=\epsilon_{e_g}-\epsilon_{t_{2g}},\) for the X-ray energy range. For LFM calculation applied to L2,3-edges, one is dealing with transitions between 2p63d3 toward 2p53d4. One then needs two sets of parameters for the initial and the final states. The initial states parameters are set to the values obtained by UV–VIS spectroscopy, i.e., Δ=2.24 eV (18,070 cm−1) and \(B=(9F^2-5F^4)/441=0.080\,\hbox{eV}\) (645 cm−1) for Cr0.07Al1.93O3 and Δ=2.07 eV (16,700 cm−1) and B=0.058 eV (468 cm−1) for α-Cr2O3. In fact we checked that the LFM calculations do not depend much on the parameters for the initial state (2p63d3). On the contrary the set of parameters Δ X and B X for the final state (2p53d4) strongly modifies the intensities and energies of the transitions. One then proceeds to a fit between experiment and calculations to determine the set of parameters for the final state. They are not expected to be exactly identical to those found from UV–VIS spectroscopy, due to the 2p core hole. The broadening of the line spectrum is determined by the lifetime of the final states and is mainly determined by Auger decay from this final state (2p53d4). The relaxation rate of each individual final state is unknown. Individual spectral features and shoulders can have different broadening factors. The calculated spectra are broadened using a Lorentzian with a 2Γ full width at half maximum (FWHM), with 2Γ=0.2 eV (L3) and 2Γ=0.6 eV (L2) and convoluted with a Gaussian with 0.5 eV FWHM, to describe the total lifetime and instrumental broadening processes.

In the UV–VIS range, the LFM calculations are also used to evaluate the dependence of various band energies of chromium in the 2p63d3 initial state (4 T 1g , 2 T 2g , 4 T 2g , 2 T 1g , 2 E 1g ) as a function of \(\Delta/B.\)

Results and discussion

Our ultimate purpose is to connect the green color of eskolaite (α-Cr2O3) and the red color of ruby (α-Al2O3:Cr3+) to the local organization around chromium. To do so, it is worth knowing the effects of the structural and electronic parameters around chromium on the positions of the transmission and absorption bands.

Color and band energies in Cr x Al2−x O3

In this part, LFM calculations are performed to (1) evaluate the contributions of the electronic parameters Δ and B on the positions of the UV–VIS transmission bands of α-Cr2O3 and Cr0.07Al1.93O3 and (2) interpret the X-ray absorption spectra at the L2,3-edges in α-Cr2O3 and Cr0.07Al1.93O3.

Band energies

The colors of eskolaite and ruby are mainly due to the positions of the transmission bands. One of them lies approximately at the middle of the two absorption bands of energies ε1 and ε2. When one affirms that the position of the transmission band is given by \(\epsilon_{\rm m} = (\epsilon_1 + \epsilon_2)/2,\) one makes the assumption that the transitions 4 A 2g (O h ) towards 4 T 2g (O h ) and 4 A 2g (O h ) towards 4 T 1g (O h ) have similar intensities and similar widths for powders. The difference of the exact position of the transmission maximum with our estimation εm of the transmission maximum is evaluated to be around 0.025 eV (200 cm−1).

We evaluate the dependence of \(\epsilon_{\rm m} = (\epsilon_1 + \epsilon_2)/2\) and εd=ε1−ε2 by multiplet calculations, taking into account the spin–orbit coupling and the trigonal distortion of the chromium site. We performed the calculation with spin–orbit coupling and trigonal distortion to see how the splittings due to these effects would spread the various transitions. The trigonal distortion arises from the fact that in M2O3 compounds, the metallic atom (M) lies in a C 3 symmetry site (Hahn 1995). However, it is allowed to treat it as if it were C 3v in the absence of external magnetic field (Sugano and Tanabe 1958). In C 3v symmetry, the 4 A 2g (O h ) spectroscopic term becomes 4 A 2(C 3v ), while the 4 T 2g (O h ) spectroscopic term is split into 4 E(C 3v ) and 4 A 1(C 3v ) and the 4 T 1g (O h ) spectroscopic term is split into 4 E(C 3v ) and 4 A 2(C 3v ). So, for the trigonal splitting, we introduced a C 3v distortion yielding the experimentally observed splittings on the 4 T 2g (O h ) and 4 T 1g (O h ) levels. Spin–orbit coupling on the 3d level was extracted from atomic calculations (ζ3d=35 meV). The various levels above the ground state are reported in Fig. 3. We do not consider the negligible 4 A 2g (O h ) ground-state splitting (<0.08 meV). From Fig. 3, one notices that the energy spread for 4 T 2g (O h ) levels is quite large. For the particular set of parameters of α-Cr2O3, the 4 T 2g (O h ) and 2 T 2g (O h ) states can hybridize efficiently. This means that the ε1 transition cannot be assigned to the single transition 4 A 2g (O h ) toward 4 T 2g (O h ) any more, but rather to a set of nine different transitions spread around 125 meV (≈1,000 cm−1, 6% of Δ).

Fig. 3
figure 3

Variations of the ratio \(E/B,\) where E is the energy of various bands (4 T 1g , 2 T 2g , 4 T 2g , 2 T 1g , 2E g in O h symmetry), as a function of \(\Delta/10B.\) From UV–VIS spectroscopy, \(\Delta/10B=3.475\) for Cr2O3, and \(\Delta/10B=2.784\) for ruby

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From LFM calculation, the variations dεm of εm and dεd of εd as functions of the variations dΔ of Δ and dB of B are

$${\rm d}\epsilon_{\rm m}\simeq4.2\,{\rm d}B + 1.02{\rm d}\Delta$$
(1)

and

$${\rm d}\epsilon_{\rm d}\simeq8.5\,{\rm d}B + 0.057 \frac{B}{0.068\,(\hbox{eV})}{\rm d}\Delta.$$
(2)

Since the parameter Δ ranges from 2.24 eV (18,070 cm−1) in ruby to 2.07 eV (16,700 cm−1) in eskolaite and the Racah parameter B from 0.080 eV (645 cm−1) in ruby to 0.058 eV (468 cm−1) in eskolaite (Reinen 1969), 35% of the variation of εm is due to the variation of B and the remaining 65% to the variation of Δ, while the variation of εd is mainly due to the variation of B (95%). The spin–orbit coupling and the trigonal distortion lead to a broadening of the states. For example, due to spin–orbit coupling, the 4 T 2g (O h ) is mixed with the 2 T 2g (O h ). The broadness of these bands can be greater than the range of variation of one state as a function of x in the whole Cr x Al2−x O3 composition.

The covalence of the Cr–O bonding (Racah parameter B) efficiently conditions the color (35%). The analysis of the X-ray absorption spectra at the L2,3-edges is then meaningful because this technique is a probe for the 3d orbitals that are effective in the appearance of coloration.

XAS at the Cr L2,3-edges

The isotropic signal of the α-Cr2O3 and the Cr0.07Al1.93O3 samples are quite identical because they are the signature of a Cr3+ ion with the 3d3 ionic configuration in an octahedral environment (Fig. 2). Eight similar features appear on the spectra of both samples. However, the positions and intensities of some of them are different. The difference E g−E c between the c (L3-edge) and g (L2-edge) features is 8.2 eV for α-Cr2O3 while it is 7.9 eV for Cr0.07Al1.93O3. The difference E b−E c between the b and c features of the L3-edge is larger (1.20 eV) for Cr0.07Al1.93O3 than it is for α-Cr2O3 (1.05 eV). The position and the shape of peak a is different for the two samples. These isotropic L2,3-edges can be analyzed with LFM calculations.

We are interested in the determination of two parameters, namely Δ X and B X . The last parameter is closely linked to the κ X parameter by \(\kappa_X=B_X/B_X^0\) where B 0 X is the calculated Racah parameter for the free ion (B 0 X =0.154 eV). From the optical spectra of free ions and extrapolation, Sugano and Tanabe (1958) give B 0=0.114 eV. The theoretical B 0 X and experimental B 0 values differ, due to the presence of the 2p core hole in the X-ray range. For the isotropic signals, it is worth noticing that when the parameter Δ X increases, the peak a appears progressively. When the parameter κ X increases, the peak a disappears steadily, the peak b is displaced toward lower energies and increases in intensity. To have a precise idea of the influence of the Δ X and κ X parameters on the position of the b feature, we have plotted the difference E c−E b between the calculated b and c features for different values of Δ X and κ X (see Fig. 4). The difference E c−E b is almost independent of the Δ X parameter for a given value of κ X in the energy range 1.6–2.2 eV, but it increases steadily with the κ X parameter, for a given value of Δ X . It is then possible to determine the κ X value from the experimental value of E c−E b, which is about 1.05 eV for α-Cr2O3 and 1.20 eV for Cr0.07Al1.93O3. It implies a κ X value of 0.55 for α-Cr2O3 and 0.65 for Cr0.07Al1.93O3. The Δ X value is determined by the shape of the spectra, in particular by the relative intensity of the a feature. Therefore, we have compared the calculated and experimental Cr L2,3-edges. The calculations were made for 14 different values of Δ X , in the range 1.5–2.8 eV, with either κ X =0.65 or κ X =0.55. From all these calculations, we deduce Δ X (α-Cr2O3)=1.8 eV and Δ X (Cr0.07Al1.93O3)=2.1 eV. The dichroic signal provides useful information on these parameters too. It is worth noticing that when the Δ X parameter increases, the \(I_{\rm a}/I_{\rm b}\) ratio increases, where I a and I b are the intensities of peak a and b, respectively (the labeling of the spectral features is defined in Fig. 2), while when the κ X parameter increases, the \(I_{\rm a}/I_{\rm b}\) ratio decreases. In addition, the difference E c−E b is almost independent of the variation of the Δ X parameter, while it increases with the κ X parameter. Similar to the isotropic signals analysis, we can deduce values of κ X and Δ X that agree with the previous values. This corroborates the previous Δ X and κ X parameters for Cr0.07Al1.93O3.

Fig. 4
figure 4

Influence of the Δ X and κ X parameters on the difference E c−E b between the features b and c of the calculated isotropic absorption spectra at the Cr L 2,3-edges

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We have plotted in Fig. 5 the final fit of our data. The parameters used for the Cr3+ (2p53d4) final configuration are Δ X =2.1 eV (16,940 cm−1), κ X =0.65, ζ2p=5.902 eV and ζ3d=0.047 eV for Cr0.07Al1.93O3, and Δ X =1.8 eV (14,520 cm−1), κ X =0.55, ζ2p=6.034 eV and ζ3d=0.047 eV for α-Cr2O3. The Racah parameter B X deduced from κ X is greater in Cr0.07Al1.93O3 (0.100 eV) than in eskolaite (0.085 eV). The crystal field Δ X is greater in Cr0.07Al1.93O3 (2.1 eV) than in eskolaite (1.8 eV). These results are in line with the ones given by UV–VIS spectroscopy, which are Δ=2.24 eV (18,070 cm−1) for ruby and Δ=2.07 eV (16,700 cm−1) for eskolaite and B=0.080 eV (645 cm−1) for ruby and B=0.058 eV (468 cm−1) for eskolaite. From the fitting procedure, one sees that Δ X varies by +15% between α-Cr2O3 and Cr0.07Al1.93O3 and B X by +17%. This is the same general trend as the one observed in UV–VIS spectroscopy. Indeed, the crystal field Δ increases by about +8% when the concentration x decreases from 2 to 0, while variation of the Racah parameter B is +32% (Reinen 1969). However, the values of the variations of the amplitudes given by UV–VIS spectroscopy or XAS are different. This difference is related to the fact that the parameters Δ X and B X are by definition different from Δ and B, due to the 2p core hole.

Fig. 5
figure 5

Fitted isotropic and dichroic spectra of Cr2O3 and Cr0.07Al1.93O3. The straight line is the fit and the dashed line is the experimental spectra (see text for calculation parameters)

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Color and structural environment around Cr in Cr x Al2−x O3

Since the variation of εm, causing the green color of α-Cr2O3 and the red color of ruby, is mainly due to the variation of the crystal field Δ (65%), we examine in the following whether the only variation of the Cr–O distance in the solid solution series can explain the variation of Δ.

XAS at the Cr K-edge

The EXAFS functions of the five samples are shown in Fig. 1 (left panel). The spectra of α-Cr2O3 and Cr0.07Al1.93O3, the two poles of the solid Al2O3–Cr2O3 solid solution, are very different from each other, suggesting a different local environment for chromium atoms in both structures. The spectra of α-Cr2O3 contains more features. In the k range 5-13 Å−1, characteristic features disappear steadily from α-Cr2O3 to Cr0.07Al1.93O3. For instance, the three sharp features at about 8–9 Å−1, which are present in the spectra of α-Cr2O3, become two features in the spectra of Cr0.07Al1.93O3. It is the same for the three features at about 6–7 Å−1 on the spectra of α-Cr2O3. The FEFF8 calculations show that these EXAFS features are mainly due to single Cr–Cr scattering paths (up to 6 Å) in α-Cr2O3. The substitution of chromium atoms by Al atoms from α-Cr2O3 to Cr0.07Al1.93O3 could explain the disappearance of these features, because aluminum atoms have lower back-scattering amplitude than chromium atoms. However, structural changes between α-Cr2O3 and Cr0.07Al1.93O3 could also explain the changes observed in EXAFS spectra. On Fig. 1 (left panel), one notices that there is a continuous change of the Cr K-edge X-ray absorption signals from α-Cr2O3 to Cr0.07Al1.93O3. An attempt to fit EXAFS spectra for intermediate composition with a linear combination of the EXAFS spectra of the two poles (α-Cr2O3 and Cr0.07Al1.93O3) by a least square fitting procedure failed. This result indicates that the local environment around chromium in Cr x Al2−x O3 is not a ponderated average of chromium environment in Cr0.07Al1.93O3 and Cr2O3. This result can be interpreted as continuous modification of chromium environment within the series. It also shows that there is no chromium clustering in the samples, in agreement with previous work (Bondioli et al. 2000). The first peak of the Fourier transform that contains the contributions of two Cr–O pairs, with quite similar Cr–O distances, is comparable from α-Cr2O3 to Cr0.07Al1.93O3, suggesting that there is no significant structural change in the chromium coordination sphere in the whole range Cr2O3–Al2O3. From α-Cr2O3 to CrAlO3, the other peaks are very similar (Fig. 1, right panel), but their intensity decreases considerably. This decrease would be due to the increase of aluminum concentration from α-Cr2O3 to CrAlO3. This hypothesis is supported by the canceling of the Cr–Cr and Cr–Al pairs contributions, shown in Fig. 6. In Cr0.45Al1.55O3 and Cr0.29Al1.71O3, the intensities of the second and third peaks in the Fourier transform are almost null. The canceling of the Cr–Cr and Cr–Al pairs contributions is added to a probably large spreading of bonds lengths in the structure. In Cr0.07Al1.93O3, the second peak in the Fourier transform increases. The structure around chromium becomes again enough organized, but in a different way as that occurred in the α-Cr2O3 structure.

Fig. 6
figure 6

F j (k) sin (2kr+Φ j (k)) functions for Cr–Cr pairs in α-Cr2O3 and for Cr–Al pairs in the structural model of ruby for the distance r=2.85 Å

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The local environment of chromium in the Al2O3–Cr2O3 solid solution series is specified by the EXAFS analysis. The results for the first shell are presented in Table 1. The parameter ΔE 0 can be different for each sample and for each shell, because it depends on electronic effects of the chemical bond taken into account (Teo 1985). The influence of this parameter on the final Cr–O distance was checked by fitting the Cr2O3 first shell with a mean Cr–O distance set at 1.980 Å and with various fixed values for ΔE 0. Results indicate that the variation of the goodness of fit for each fit, defined as

$$\xi^2 = \sum \left(\frac{k^3\chi_{\rm exp}}{\sqrt{\sum k^6 \chi_{\rm exp}^2}} - \frac{k^3\chi_{\rm fit}}{\sqrt{\sum k^6 \chi_{\rm exp}^2}}\right)^2,$$

is lower than 0.01 for −5 ≤ ΔE 0 ≤ 0 (Fig. 7, right). The influence of ΔE 0 on the final Cr–O distance is then negligible. In the same way, the error on the mean Cr–O distance was evaluated by fitting the Cr2O3 first shell with ΔE 0 set at −1 eV and with various fixed values for the mean Cr–O distance. Results indicate that a majoration of 0.005 Å for the mean Cr–O distance yields a degradation of ≈100% for the goodness of fit (Fig. 7, left). Considering these results, the relative precision on the mean Cr–O distances can be estimated to be 0.005 Å for the whole series.

Fig. 7
figure 7

Left panel variation of the goodness of fit (GOF) as a function of the Cr–O distance (ΔE0 is fixed with ΔE0=−1.0 eV) in α-Cr2O3 and Cr0.07Al1.93O3. Right panel variation of the goodness of fit as a function of the ΔE0 parameter in α-Cr2O3 (the Cr–O bond length is 1.98 Å)

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Concerning the α-Cr2O3 pole, the mean Cr–O distance (1.980 Å) is in good agreement with the one (1.985 Å) given by Finger and Hazen (1980). Concerning the Cr0.07Al1.93O3 pole, our mean Cr–O distance (1.965 Å) is similar to EXAFS results on powders (1.96 Å) given by Kizler et al. (1996) and from optical spectra (1.96 Å) given by Langer (2001). It is larger than the one (1.95 Å) given by Andrut (2004) from the application of the superposition model to optical absorption data. Our mean Cr–O distance is also similar to the one (1.97 Å) determined from a previous work on single crystals (Gaudry et al. 2003). The EXAFS analysis shows that the average Cr–O bond length in Cr x Al2−x O3 (0.07≤x< 2) increases by ≈1% in the whole Al2O3–Cr2O3 series (Fig. 8). This average Cr–O distance in Cr x Al2−x O3 is close to the mean Cr–O distance in α-Cr2O3: 1.980 Å from this EXAFS study or 1.985 Å from Finger and Hazen (1980). It is longer than the averaged Al–O bond (1.915 Å) in α-Al2O3 in the whole range Al2O3–Cr2O3. Such a behavior matches the Pauling (1967) model. The relaxation parameter

$$\zeta = \frac{R_{{\rm Cr}-{\rm O}}(\hbox{Cr}_{0.07}\hbox{Al}_{1.93}\hbox{O}_3) - R_{{\rm Al}-{\rm O}}(\hbox{Al}_2\hbox{O}_3)}{R_{{\rm Cr}-{\rm O}}(\hbox{Cr}_2\hbox{O}_3) - R_{{\rm Al}-{\rm O}}(\hbox{Al}_2\hbox{O}_3)}$$

defined by Martins and Zunger (1984) is equal to ≈0.76. This is less than the value found for Co(II), Zn(II), Pb(II) or Mn(II) in calcite (0.8–0.9) (Reeder et al. 1999; Lee et al. 2002). It is also less than the ones found for S or Se in CdS x Se1−x (0.87–0.88) (Levelut et al. 1991). The ζ value for Cr0.07Al1.93O3 is, however, larger than ζ=0.5 in FeO–MgO solid solutions (Waychunas et al. 1994) and values found by studies on alkali halide solid solutions lying between 0.4 and 0.6 (Martins and Zunger 1984; Boyce and Mikkelsen 1985; Frenkel et al. 1996). Actually, the corundum structure is made of edge and face-sharing octahedra that allow less flexibility than the corner-sharing octahedra structure of calcite, but more flexibility than the compact rock salt structure of MgO, FeO and alkali halide solid solutions.

Fig. 8
figure 8

Mean Cr–O distances (Å) from EXAFS analysis on Cr x Al2−x O3 powders. Black dots are experimental points deduced from this study. Triangles are XRD data from Finger and Hazen (1980) and from Pearson (1962)

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For α-Cr2O3, the structure from Finger and Hazen (1980) allowed the calculation of the χ(k) signal for a chromium atom in such a crystal. This is compared in the upper part of Fig. 9 with the raw (non-Fourier filtered) experimental χ(k) signal. For the Cr0.07Al1.93O3 sample, the structural model is built as follows: a 6.3 Å diameter cluster around chromium is determined from the ab initio energy minimization calculation, except for the average Cr–O distance of the chromium coordination shell, which is reduced by 0.5% to rescale it to the one found from isotropic EXAFS analysis (1.965 Å). The lower part of Fig. 9 shows that this model for Cr0.07Al1.93O3 is consistent with the experimental χ(k) signal. It can be noticed that the FEFF8 EXAFS calculation without this Cr–O rescaling yields the same features as those in Fig. 9 (lower panel).

Fig. 9
figure 9

The k 3-weighted experimental EXAFS (solid lines) and theoretical EXAFS (dashed line) for both the α-Cr2O3 and the α-Al2O3:Cr3+ structural models. Isotropic signals at Cr K-edge

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Influence of the Cr–O bonding on the crystal field

Since we determined how the variation of Δ induces the color change, we now turn to the question of the relationship between the variation of the Cr–O distance with the variation of Δ. In the point charge model, the dependence of the crystal field Δ on the mean metal–ligand distance is given by

$$\Delta = \frac{5}{3} Z_{\rm L} e^2 {\langle r^4\rangle} R_{{\rm Cr}-{\rm O}}^{-5},$$
(3)

where R Cr–O is the mean Cr–O distance, Z L e 2 the effective charge of the ligands and \({\langle r^4\rangle}=\int r^2\,{\rm d}r\,R_2(r)^2r^4,\) where R 2(r) is the radial part of the d orbitals (Dunn et al. 1965). According to Langer (2001), the quantities Z L e 2 and 〈r 4〉 would be constant along the Al2O3–Cr2O3 solid solution series, because chromium lies in the same kind of site, with the same oxygen ligands in the whole range. Equation 3 is then rewritten as

$$\Delta = a R_{{\rm Cr}-{\rm O}}^{-5},$$
(4)

where a is a constant. As shown from the EXAFS analysis at the Cr K-edge, the variation of the Cr–O distance in the whole composition range is ≈1%, while it is 8% for the Δ crystal field. According to Eq. 4, a variation of the Cr–O distance by 1.6% (more than 0.03 Å) would be necessary to explain the variation of Δ (8% in the Al2O3–Cr2O3 series). This variation of the Cr–O distance is much larger than the one found (0.015 Å), which indicates that the variation of the mean Cr–O distance is not the only explanation for the variation of Δ. A different metal–ligand charge transfer between α-Cr2O3 and ruby probably occurs, so that a is no longer a constant in the whole solid solution series. The different metal–ligand charge transfer between α-Cr2O3 and ruby might be responsible for some percent in the variation of the crystal field. This statement is in agreement with the smaller value for the Racah parameter B in eskolaite than in ruby, and with the theory of Sugano and Peter (1961), for which the energy of the 4 A 2g → 4 T 2g transition is Δ+10ξ B, where ξ is a covalency parameter that stands for the different charge transfer between σ(e g ) and π(t 2g ) Cr–O bonds. The greater Cr–O bonding covalency in α-Cr2O3 than in α-Al2O3:Cr3+ comes from a modification of the electrostatic repulsions, and to configuration interaction. This means that the ground state of the chromium atom is described by the configuration 2p63d3 interacting with \(2{\rm p}^63{\rm d}^4\underline{L}\) where \(\underline{L}\) represents a hole on a ligand.

Recent publications (Andrut et al. 2004; Brik et al. 2004) suggested that the variations of Δ was not strictly given by Eq. 4. Through a fitting procedure, Andrut et al. found that the exponent in Eq. 4 was in between 5 and 7.1, depending on the crystal host. By adding to the point charge model a contribution coming from both the covalent bond formation and the exchange interaction, Brik et al. found that the exponent was strictly larger than 5. On the other hand, Duclos et al. found that the pressure dependence of the Δ parameter was well described by \(\Delta \propto V^{-5/3}\) (Duclos et al. 1990) in the short range of pressure between 0 and 35 GPa. If for such moderate pressure, the unit cell volume is scaling as the cube of the averaged Cr–O distance, then Eq. 4 would be strictly obeyed. These pressure-dependent results are in line with those already published by Poole et al. (Poole 1964; Poole and Itzel 1963). These various findings can be harmonized in the following unifying picture. When one is comparing small variations of Δ parameter without changing the crystal host or the impurity concentration, the point charge model (i.e., Eq. 4) might be valid. When one is comparing variations of Δ due to changes of lattice host or impurity concentration, then the point charge model is clearly not sufficient and the theory needs to be modified to take into account the drastic chemical modifications around the impurity.

Conclusion

This study indicates that the color of the red ruby and the green eskolaite is partially due to both the crystal field and the covalence of the Cr–O bonding. In fact, the position of the transmission band is due 65% to the Δ value and 35% to the Racah B value. The variation in Cr–O distances from α-Cr2O3 to Cr0.07Al1.93O3 estimated in this work is 0.015 Å (≈1%). It is too low to explain the variation of Δ in the point charge model. Following Sugano and Peter (1961), the variation of Δ could also be ascribed to the increase of the covalence of the Cr–O bond. Indeed, the Cr–Cr coupling in α-Cr2O3 might play a role in the covalence of the Cr–O bond by broadening the 3d band.

The study of the origin of the difference of color between the green emerald and the red ruby would be interesting. The Cr–O distances in ruby and emerald are probably close, the concentration of chromium is quite identical and very low in both minerals, so that no Cr–Cr coupling is expected to interfere with the color appearance. Such a study would be able to separate covalency effects from crystal field ones.