Structural relaxation around substitutional Cr3+ in MgAl2O4
Structural relaxation around substitutional Cr3+ in MgAl2O4
Amélie Juhin,∗ Georges Calas, Delphine Cabaret, and Laurence Galoisy
Institut de Minéralogie et de Physique des Milieux Condensés,
UMR CNRS 7590
Université Pierre et Marie Curie, Paris 6
140 rue de Lourmel, F-75015 Paris, France
Jean-Louis Hazemann†
Laboratoire de Cristallographie, CNRS, 25 avenue des Martyrs, BP 166, 38042 Grenoble cedex 9, France
(Dated: October 26, 2018)
The structural environment of substitutional Cr3+ ion in MgAl2O4 spinel has been investigated
by Cr K-edge Extended X-ray Absorption Fine Structure (EXAFS) and X-ray Absorption Near
Edge Structure (XANES) spectroscopies. First-principles computations of the structural relaxation
and of the XANES spectrum have been performed, with a good agreement to the experiment. The
Cr-O distance is close to that in MgCr2O4, indicating a full relaxation of the first neighbors, and the
second shell of Al atoms relaxes partially. These observations demonstrate that Vegard’s law is not
obeyed in the MgAl2O4-MgCr2O4 solid solution. Despite some angular site distortion, the local D3d
symmetry of the B-site of the spinel structure is retained during the substitution of Cr for Al. Here,
we show that the relaxation is accomodated by strain-induced bond buckling, with angular tilts of
the Mg-centred tetrahedra around the Cr-centred octahedron. By contrast, there is no significant
alteration of the angles between the edge-sharing octahedra, which build chains aligned along the
three four-fold axes of the cubic structure.
PACS numbers: 61.72.Bb, 82.33.Pt, 78.70.Dm, 71.15.Mb
I. INTRODUCTION
Most multicomponent materials belong to complete or
partial solid solutions. The presence of chemical sub-
stitutions gives rise to important modifications of the
physical and chemical properties of the pure phases. For
instance, the addition of a minor component can im-
prove significantly the electric, magnetic or mechanical
behaviour of a material.1,2,3 Another evidence for the
presence of impurities in crystals comes from the modifi-
cation of optical properties such as coloration. Transition
metal ions like Cr3+ cause the coloration of wide band
gap solids, because of the splitting of 3d-levels under the
action of crystal field.4 Despite the ubiquitous presence
of substitutional elements in solids, their accommoda-
tion processes and their structural environment are still
discussed,5 since they have important implications. For
example, the interpretation of the color differences be-
tween Cr-containing minerals (e.g. ruby, emerald, red
spinel) requires to know the structural environment of
the coloring impurity.4,6,7,8 The ionic radius of a sub-
stitutional impurity being usually different from that of
the substituted ion, the accommodation of the mismatch
imposes a structural relaxation of the crystal structure.
Vegard’s law states that there is a linear relationship
between the concentration of a substitutional impurity
and the lattice parameters, provided that the substi-
tuted cation and impurity have similar bonding proper-
ties. Chemically selective spectroscopies, like Extended
X-ray Absorption Fine Structure (EXAFS), have pro-
vided evidence that diffraction studies of solid solutions
give only an average vision of the microscopic states and
that Vegard’s law is limited.9,10,11 Indeed, a major result
concerns the existence of a structural relaxation of the
host lattice around the substitutional cation. This im-
plies the absence of modification of the site occupied by
a doping cation, when decreasing its amount in a solid
solution. This important result has been observed in var-
ious materials, including III-V semi-conductors or mixed
salts:12,13 e. g., in mixed alkali halides, some important
angular buckling deviations have been observed.13 Re-
cently, the use of computational tools, as a complement
of EXAFS experiments, has been revealed successful for
the study of oxide/metal epilayers.14 In oxides contain-
ing dilute impurities, this combined approach is manda-
tory. It has been recently applied to the investigation of
the relaxation process around Cr dopant in corundum:
in the α-Al2O3 - α-Cr2O3 system, the radial relaxation
was found to be limited to the first neighbors around Cr,
while the angular relaxation is weak.8,15
In this work, we investigate the relaxation caused by
the substitution of Al3+ by Cr3+ in spinel MgAl2O4,
which gives rise to a solid solution, as observed for corun-
dum α-Al2O3. The spinel MgAl2O4 belongs to an impor-
tant range of ceramic compounds, which has attracted
considerable interest among researchers for a variety of
applications, great electrical, mechanical, magnetic and
optical properties.16 The spinel structure is based on a
cfc close-packing, with a Fd3̄m space group symmetry.
Its chemical composition is expressed as AB2X4, where
A and B are tetrahedral and octahedral cations, respec-
tively, and X is an anion. These two types of cations
define two different cationic sublattices, which may in-
duce a very different relaxation process than in corun-
dum. In the normal spinel structure, the octahedra host
http://arxiv.org/abs/0704.0878v1
trivalent cations and exhibit D3d site symmetry. This
corresponds to a small distortion along the [111] direc-
tion, arising from a departure of the position of oxy-
gen ligands from a cubic arrangement. Small amounts
of chromium oxide improve the thermal and mechanical
properties of spinel.1 A color change from red to green
is also observed with increasing Cr-content. In this arti-
cle, we report new results on the local geometry around
Cr3+ in spinel MgAl2O4, using a combination of EXAFS
and X-ray Absorption Near Edge Structure (XANES).
The experimental data are compared to those obtained
by theoretical calculations, based on the Density Func-
tional Theory in the Local Spin Density Approximation
(DFT-LSDA): this has enabled us to confirm the local
structure around substitutional Cr3+ and investigate in
detail the radial and angular aspects of the relaxation.
The paper is organized as follows. Section II is dedi-
cated to the methods, including the sample description
(Sec. II A), the X-ray absorption measurements and
analysis (Sec. II B), and the computational details (Sec.
II C). Section III is devoted to the results and discussion.
Conclusions are given in Sec. IV.
II. MATERIALS AND METHODS
A. Sample description
Two natural gem-quality red spinel single crystals from
Mogok, Burma (Cr-1, Cr-2) were investigated. They
contain respectively 70.0, 71.4 wt %-Al2O3, 0.70, 1.03
wt%-Cr2O3 and 26.4, 25.3 wt%-MgO. These composi-
tions were analyzed using the Cameca SX50 electron mi-
croprobe at the CAMPARIS analytical facility of the Uni-
versities of Paris 6/7, France. A 15 kV voltage with a 40
nA beam current was used. X-ray intensities were cor-
rected for dead-time, background, and matrix effects us-
ing the Cameca ZAF routine. The standards used were
α-Al2O3, α-Cr2O3 and MgO.
B. X-ray Absorption Spectroscopy measurements
and analysis
Cr K-edge (5989 eV) X-ray Absorption Spectroscopy
(XAS) spectra were collected at room temperature at
beamline BM30b (FAME), at the European Synchrotron
Radiation Facility (Grenoble, France) operated at 6 GeV.
The data were recorded using the fluorescence mode with
a Si (111) double crystal and a Canberra 30-element Ge
detector.17 We used a spacing of 0.1 eV and of 0.05 Å−1,
respectively in the XANES and EXAFS regions. Data
treatment was performed using ATHENA following the
usual procedure and the EXAFS data were analyzed us-
ing IFEFFIT, with the support of ARTEMIS.18 The de-
tails of the fitting procedure can be found elsewhere.19
An uvarovite garnet, Ca3Cr2Si3O12, was used as model
compound to derive the value of the amplitude reduction
factor S20 (0.81) needed for fitting. For each sample, a
multiple-shell fit was performed in the q-space, including
the first four single scattering paths: the photoelectron is
backscattered either by the first (O), second (Al or Cr),
third (O) or fourth (Mg) neighbors. Treating identically
the third and fourth paths, we used a unique energy shift
∆e0 for all paths, three different path lengths R and three
independent values of the Debye-Waller factor σ2. In a
first step, the number of neighbors N was fixed to the
path degeneracy. In a second time, a single amplitude
parameter was fitted for the last three shells, assuming
a proportional variation of the number of atoms on each
shell.
C. Computations
1. Structural relaxation
In order to complement the structural information
from EXAFS, a simulation of the structural relaxation
was performed to quantify the geometric surrounding
around an isolated Cr3+. The calculations were done in a
neutral supercell of MgAl2O4, using a first-principles to-
tal energy code based on DFT-LSDA.20 We used Plane
Wave basis set and norm conserving pseudopotentials21
in the Kleiman Bylander form.22 For Mg, we considered
3s, 3p, 3d as valence states (core radii of 1.05 a.u, ℓ=2
taken as local part) and those of Ref.15 for Al, Cr, O.
We first determined the structure of bulk MgAl2O4. We
used a unit cell, which was relaxed with 2×2×2 k-point
grid for electronic integration in the Brillouin Zone and
cut-off energy of 90 Ry. We obtained a lattice constant of
7.953 Å and an internal parameter of 0.263 (respectively
-1.6 % and +0.3 % relative to experiment),23 which
are consistent with previous calculations.16. In order to
simulate the Cr defect, we used a 2×2×2 supercell, built
using the relaxed positions of the pure phase. It con-
tains 1 neutral Cr, 31 Al, 16 Mg and 64 O atoms. It was
chosen large enough to minimize the interaction between
two paramagnetic ions, with a minimal Cr-Cr distance
of 11.43 Å. While the size of the supercell is kept fixed,
all atomic positions are relaxed in order to investigate
long-range relaxation. We used the same cut-off energy
and a single k-point sampling. The convergence of the
calculation was verified by comparing it to a computa-
tion with a 2×2×2 k-point grid, and discrepancies in the
atomic forces are lower than 0.3 mRy/a.u. In order to
compare directly the theoretical bond distances to those
obtained by EXAFS spectroscopy, the inital slight un-
derestimation of the lattice constant (systematic within
the LDA)24 was removed by rescaling the lattice param-
eter by -1.6 %. This rescaling is homothetic and does not
affect the relative atomic positions.
2. XANES simulations
As the analysis of the experimental XANES data is
not straightforward, ab initio XANES simulations are re-
quired to relate the experimental spectral features to the
local structure around the absorbing atom. The method
used for XANES calculations are described in Ref. 25,26.
The all-electron wave-functions are reconstructed within
the projector augmented wave framework.27 In order to
allow the treatment of large systems, the scheme uses a
recursion method to construct a Lanzcos basis and then
compute the cross section as a continued fraction.28,29
The XANES spectrum is calculated in the electric dipole
approximation, using the same first-principles total en-
ergy code as the one used for the structural relaxation. It
was carried out in the relaxed 2×2×2 supercell (i.e 112
atoms), which contains one Cr atom and results from
ab initio energy minimization mentioned in the previ-
ous subsection. The pseudopotentials used are the same
as those used for structural relaxation, except for Cr.
Indeed, in order to take into account the core-hole ef-
fects, the Cr pseudopotential is generated with only one
1s electron. Convergence of the XANES calculation is
reached for the following parameters: a 70 Ry energy
cut-off for the plane-wave expansion, one k-point for the
self-consistent spin-polarized charge density calculation,
and a Monkhorst-Pack grid of 3×3×3 k-points in the
Brillouin Zone for the absorption cross-section calcula-
tion. The continued fraction is computed with a con-
stant broadening γ=1.1 eV, which takes into account the
core-hole lifetime.30
III. RESULTS AND DISCUSSION
Figure 1 shows the k3-weighted experimental EXAFS
signals for Cr-1 and Cr-2 samples and the Fourier Trans-
forms (FT) for the k-range 3.7-11.9 Å−1. The similari-
ties observed suggest a close environment for Cr in the
two samples (0.70 and 1.03 wt%-Cr2O3), which is con-
firmed by fitting the FT in the R-range 1.0-3.1 Å (see
Table I). The averaged Cr-O distance derived from EX-
TABLE I: Structural parameters obtained from the EXAFS
analysis in the R range [1.0-3.1 Å] for Cr-1 and Cr-2 samples.
The energy shifts ∆e0 were found equal to 1.3 ± 1.5 eV. The
obtained RF factors were 0.0049 and 0.0045.
R(Å) N σ2 (Å2)
Cr-O 1.98 6.0 0.0031 Cr-1
1.98 6.0 0.0026 Cr-2
Cr-Al 2.91 5.3 0.0032 Cr-1
2.91 5.4 0.0033 Cr-2
Cr-O 3.39 1.8 0.0079 Cr-1
3.37 1.8 0.0077 Cr-2
Cr-Mg 3.39 5.3 0.0079 Cr-1
3.39 5.4 0.0077 Cr-2
FIG. 1: Fourier-transform of k3-weighted EXAFS function for
Cr-1 and Cr-2 samples (dashed and solid lines respectively).
Inset: background-subtracted data
FIG. 2: (a) Inverse-FT of EXAFS data (dots) and fitted signal
(solide line) for R=1.0-3.1 Å. (b) Inverse-FT of EXAFS data
(dots) for R=2.0-3.1 Å, multi-shell fit with Cr-Al pairs (solid
line) and theoretical function with Cr-Cr pairs (dashed line)
in the same structural model.
TABLE II: First, second and third neighbor mean distances (in Å) from central M3+ in the different structures considered in
this work.
MgAl2O4: Cr
3+ exp MgAl2O4: Cr
3+ calc MgAl2O4 exp
a MgCr2O4 exp
Cr-O 1.98 1.99 — 1.99
Al-O — — 1.93 —
Cr-Al 2.91 2.88 — —
Cr-Cr — — — 2.95
Al-Al — — 2.86 —
Cr-O 3.37 3.34 — 3.45
Al-O — — 3.34 —
Cr-Mg 3.39 3.36 — 3.45
Al-Mg — — 3.35 —
afrom Ref.23
bfrom Ref.34
AFS data is equal to 1.98 Å (± 0.01 Å), with six oxygen
first neighbors. The second shell is composed of six Al
atoms, located at 2.91 Å (± 0.01 Å). Two oxygen and
six magnesium atoms compose the further shells, at dis-
tances of 3.38 Å and 3.39 Å (± 0.03 Å). We investigated
in detail the chemical nature of these second neighbors,
by fitting the second peak on the FT (2.0-3.1 Å) with ei-
ther a Cr or an Al contribution, this latter corresponding
to a statistical Cr-distribution (Cr/Al ∼ 0.01). The only
satisfactory fits were obtained in the latter case (Fig. 2).
Calculated and experimental interatomic distances are
in good agreement (Table II), a confirmation of the
EXAFS-derived radial relaxation around Cr3+ after sub-
stitution. The symmetry of the relaxed Cr-site is re-
tained from the Al-site in MgAl2O4 and is similar to
the Cr-site in MgCr2O4. It belongs to the D3d point
group, with an inversion center, three binary axes and
a C3 axis (Fig. 3a). This result is consistent with opti-
cal absorption31 and Electron-Nuclear Double Resonance
experiments32 performed on MgAl2O4: Cr
3+. Our first-
principles calculations also agree with a previous inves-
tigation of the first shell relaxation, using Hartree-Fock
formalism on an isolated cluster.33 As it has been men-
tioned previously, the simulation can provide comple-
mentary distances (Fig. 3b): the Al1-O distances, equal
to 1.91 Å, are slightly smaller than Al-O distances in
MgAl2O4. The Al1-Al2 distances are equal to 2.85 Å,
which is close to the Al-Al distances in MgAl2O4.
Apart from the radial structural modifications around
Cr, significant angular deviations are observed in the
doped structure. Indeed, the Cr-centred octahedron is
slightly more distorted in MgAl2O4: Cr
3+, with six O-
Cr-O angles of 82.1◦ (and six supplementary angles of
97.9◦): O-Cr-O is more acute than O-Cr-O in MgCr2O4
(84.5◦, derived from refined structure)34 and than O-Al-
O in MgAl2O4 (either calculated in the present work,
83.5◦, or derived from refined structure, 83.9◦) (Fig. 3a).
At a local scale around the dopant, the sequence of edge-
sharing octahedra is hardly modified by the substitution
(Fig. 3b): the Cr-O-Al1 angles (95.1◦) are similar to
Cr-O-Cr in MgCr2O4 (95.2
◦) and Al-O-Al in MgAl2O4
FIG. 3: (color online) (a) Cr-centred octahedron before re-
laxation (green) and after (red). (b) Model of structural dis-
tortions around Cr (red) in MgAl2O4: Cr
3+. The O first
neighbors (black) and the Al1 (green) second neighbors are
displaced outward the Cr dopant in the direction of arrows.
FIG. 4: Cr K-edge XANES spectra in MgAl2O4: Cr
3+. The
experimental signal (thick line) is compared with the theo-
retical spectra calculated in the relaxed structure (solid line)
and in the non-relaxed structure (dotted line)
(95.8◦). However, the six Al-centred octahedra connected
to the Cr-octahedron are slightly distorted (with six O-
Al1-O angles of 86.7◦), compared to O-Cr-O angles in
MgCr2O4 (84.5
◦) and O-Al-O angles in MgAl2O4 (83.9
This modification affects in a similar way the three types
of chains composed of edge-sharing octahedra, in agree-
ment with the conservation of the C3 axis. On the
contrary, the relative tilt angle between the Mg-centred
tetrahedra and the Cr-centred octahedron is very differ-
ent in MgAl2O4: Cr
3+ (with Cr-O-Mg angle of 117.4◦)
than in MgCr2O4 and MgAl2O4 (with respectively, Cr-
O-Mg and Al-O-Mg angles of 124.5◦ and 121.0◦)
The experimental XANES spectrum of natural
MgAl2O4: Cr
3+ is shown in Fig. 4. It is similar to that of
a synthetic Cr-bearing spinel.35 A good agreement with
the one calculated from the ab initio relaxed structure is
obtained, particularly in the edge region : the position,
intensity and shape of the strong absorption peak (peak
c) is well reproduced by the calculation. The small fea-
tures (peaks a and b) exhibited at lower energy are also
in good agreement with the experimental ones. In our
calculation, the pre-edge features (visible at 5985 eV on
the experimental data) cannot be reproduced, since we
only considered the electric dipole contribution to the X-
ray absorption cross-section: indeed, as it has been said
previously, the Cr-site is centrosymmetric in the relaxed
structure, which implies that the pre-edge features are
due to pure electric quadrupole transitions. The sen-
sitivity of the XANES calculation to the relaxation is
evaluated by computing the XANES spectrum for the
non-relaxed supercell, in which one Cr atom substitutes
an Al atom in its exact position. The result is plotted in
Fig. 4: the edge region (peaks a, b and c) is clearly not
as well reproduced as in the relaxed model, and peak e
is not visible at all. Therefore, we can conclude that the
structural model obtained from our ab initio relaxation
is reliable.
The Cr-O distance is larger than the Al-O distance
in MgAl2O4, but is similar to the Cr-O distance in
MgCr2O4 (Table II). This demonstrates the existence of
an important structural relaxation around the substitu-
tional Cr3+ ion, which is expected since Cr3+ has a larger
ionic radius than Al3+ (0.615 Å vs 0.535 Å).36 The size
mismatch generates indeed a local strain, which locally
expands the host structure. As a result, the O atoms
relax outward the Cr defect. This radial relaxation is ac-
companied with a slight angular deviation of the O first
neighbors, as compared to the host structure. The mag-
nitude of the radial relaxation may be quantified by a
relaxation parameter ζ, defined by the relation:10
RCr−O(MgAl2O4 : Cr
3+)− RAl−O(MgAl2O4)
RCr−O(MgCr2O4)− RAl−O(MgAl2O4)
We find ζ = 0.83 (taking the Cr-O experimental dis-
tance), close to the full relaxation limit (ζ = 1), which
is more than in ruby α-Al2O3: Cr
3+ (ζ = 0.76).8 Veg-
ard’s law, which corresponds to ζ = 0, is thus not obeyed
at the atomic scale. The Cr-Al distance is intermedi-
ate between the Al-Al and Cr-Cr distances in MgAl2O4
and MgCr2O4, which accounts for a partial relaxation
of the second neighbors, but the third and fourth shells
(O, Mg) do not relax, within the experimental and com-
putational uncertainties. The chains of Al-centred octa-
hedra are radially affected only at a local scale around
Cr: the Al second neighbors relax partially outward Cr,
with a Al1-O bond slightly shortened. The angular devi-
ations are also moderate (below 1◦), since the sequence
of octahedra is not modified, but these Al-centred oc-
tahedra are slighlty distorted. Indeed, these octahedra
being edge-shared, the number of degrees of freedom is
reduced, and the polyhedra can either distort or tilt a
little, one around another. It is interesting to point out
that the three chains of octahedra are orientated along
the three four-fold axes of the cubic structure, which are
highly symmetric directions. On the contrary, an angular
relaxation (3.5◦) is observed for the Mg atoms, but with
the absence of radial modifications. This must be con-
nected to the fact that the tetrahedra share a vertex with
the Cr-centred octahedron, a configuration which allows
more flexibility for relative rotation of the polyhedra.
The extension of the relaxation process up to the sec-
ond shell is not observed in the corundum solid solution,
in which it is limited to the first coordination shell.15
Such a difference between these two solid solutions can
be related to the lattice rigidity: the bulk modulus B
is smaller in MgAl2O4 than in α-Al2O3, 200 GPa and
251 GPa, respectively.37 This difference directly arises
from the peculiarity of the structure of these two crystals:
in the spinel structure, one octahedron is edge-shared
to 6 Al octahedra and corner-shared to 6 Mg-centred
tetrahedra (Fig. 3b). In corundum, each octahedron is
face-shared with another, in addition to corner and edge-
sharing bonds: this is at the origin of the rigidity of the
corundum structure, which is less able to relax around a
substitutional impurity such as Cr3+, and relaxation is
thus limited to the first neighbors.
IV. CONCLUSIONS
This study provides a direct evidence of the struc-
tural relaxation during the substitution of Cr for Al in
MgAl2O4 spinel. The local structure determined by X-
ray Absorption Spectroscopy and first-principles calcu-
lations show similar Cr-O distances and local symmetry
in dilute and concentrated spinels. This demonstrates
that, at the atomic scale, Vegard’s law is not obeyed in
the MgAl2O4-MgCr2O4 solid solution. Though this re-
sult has been obtained in other types of materials (semi-
conductors, mixed salts), it is particularly relevant for
oxides like spinel and corundum: indeed, the application
of Vegard’s law has long been a structural tool to in-
terpret, within the so-called ”point charge model”,4 the
color of minerals containing transition metal ions. In
spinel, the full relaxation of the first shell is partially ac-
comodated by strain-induced bond buckling, which was
found to be weak in corundum: important angular tilts
of the Mg-centred tetrahedra around the Cr-centred oc-
tahedron have been calculated, while the angles between
Cr- and Al-bearing edge-sharing octahedra are hardly af-
fected. The improved thermal and mechanical properties
of Cr-doped spinel may be explained by remanent local
strain fields induced by the full relaxation of the structure
around chromium, as it has been observed in other solid
solutions.2 Another important consequence of relaxation
concerns the origin of the partition of elements between
minerals and liquids in geochemical systems.5 Finally, the
data obtained in this study will provide a structural basis
for discussing the origin of color in red spinel and its vari-
ation at high Cr-contents. Indeed, the origin of the color
differences between Cr-containing minerals (ruby, emer-
ald, red spinel, alexandrite) is still actively debated.6,8,38
Acknowledgments
The authors are very grateful to O. Proux (FAME
beamline) for help during experiment. The theoret-
ical part of this work was supported by the French
CNRS computational Institut of Orsay (Institut du
Développement et de Recherche en Informatique Scien-
tifique) under project 62015. This work has been greatly
improved through fruitful discussions with E. Balan, F.
Mauri, M. Lazzeri and Ph. Sainctavit. This is IPGP
Contribution n◦XXXX.
∗ Electronic address: amelie.juhin@impmc.jussieu.fr
† Electronic address: hazemann@grenoble.cnrs.fr
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