4 f and 5d levels of Ce3+ in D2 eightfold oxygen coordination
Luis Seijo1, 2, ∗ and Zoila Barandiarán1, 2
1
Departamento de Química, Universidad Autónoma de Madrid, 28049 Madrid, Spain
2
Instituto Universitario de Ciencia de Materiales Nicolás Cabrera,
Universidad Autónoma de Madrid, 28049 Madrid, Spain
(Dated: January 15, 2014)
The effects of the first coordination shell geometry of the trivalent Cerium ion (Ce3+ ) on its 4 f and
5d levels in Ce-doped oxides with a D2 8-fold site, like garnets, are studied with embedded cluster, wave
function based ab initio methods. The only deformations of a D2 CeO8 moiety that are found to shift the
lowest 4 f → 5d transition to the red (longer wavelengths) are the symmetric Ce-O bond compression and
the tetragonal symmetric bond bending. In a first approximation, the lowest 5d level of Ce3+ in garnets
can be understood as resulting from the cubic E g level with a strong E g × e g Jahn-Teller coupling. These
results are analyzed in terms of 5d − 4 f centroid energy differences and ligand field stabilizations. The
splittings of the upper 5d levels and of the 4 f levels are also discussed.
Keywords: Cerium, Ce3+ , 4f, 5d, D2 , oxides, garnets, 8-fold coordination
I. INTRODUCTION
Yttrium aluminum garnet Y3 Al5 O12 (YAG) doped with
Ce3+ is a well known phosphor with a blue Ce3+ 4 f → 5d
absorption and a corresponding yellow 5d → 4 f emisssion,1 which is used in InGaN blue LED based, white light
solid-state lighting devices.2,3 In the search for alternative phosphors with the best efficiencies and tailored colorrendering indexes, doping Ce3+ in other artificial garnets
has been a line of work. It led, for instance, to the discoveries of the Lu2 CaMg2 Si3 O12 :Ce3+ orange phosphor4 and
the Ca3 Sc2 Si3 O12 :Ce3+ green phosphor.5
Most garnets can be described in terms of a 160 atom
body-centered cubic unit cell (80 atom primitive cell) of
the I a3̄d (230) space group, which contains eight formula units of A3 B′2 B′′3 O12 . A, B′ and B′′ are cations of
different nominal charges in different symmetry sites. In
Y3 Al5 O12 , for instance, A≡Y, B′ ≡Al, and B′′ ≡Al,6 all with
oxidation states +3. In Ca3 Sc2 Si3 O12 , A≡Ca, B′ ≡Sc, and
B′′ ≡Si,7 with respective oxidation sates +2, +3, and +4.
In Lu2 CaMg2 Si3 O12 , 2/3 of the A sites are occupied by Lu
and 1/3 by Ca atoms,4 B′ ≡Mg, and B′′ ≡Si, with respective
oxidation states +3, +2, +2 and +4. In all cases, A occupy
24(c) sites of 8-fold coordination, B′ 16(a) sites of 6-fold
coordination, and B′′ 24(d) sites of 4-fold coordination, all
of them at fixed positions, with the remaining 96 O atoms
in (h) sites, which depend on three x, y and z internal parameters.6 Optically active Ce3+ impurities substitute for A
atoms at (c) sites, which have the local symmetry of the D2
point group and a first coordination shell made of 8 oxygen
atoms.
The energies of the 4 f and 5d levels of Ce3+ in particular garnets and in other oxides depend on the bonding
and electrostatic interactions between Ce and the hosts,
and they can be calculated in a one-by-one basis, with reasonable accuracy, by means of ab initio methods, both in
D2 local symmetry like in Ce-doped YAG8 and in lower local symmetries like in Ce,La-doped YAG9 and Ce,Ga-doped
YAG.10 These energies are dominated, up to first-order, by
the bonding interactions between Ce and its first oxygen
coordination shell, subject to the basic confinement embedding effects of the host. Other specific host embedding effects are important at higher orders of approximation and
they refine the results.
This paper is aimed at providing a basic study of the energies of the levels of the 4 f 1 and 5d 1 manifolds of Ce3+
in D2 8-fold oxygen coordination. We report the results
of an ab initio calculation of them after consideration of
the bonding interactions with the first oxygen coordination
shell only, under the effects of a cubic, non host specific,
confinement embedding potential. The present results are
a common reference for the 4 f and 5d levels of Ce3+ in specific garnets (and other oxides with 8-fold coordination),
which can be considered as resulting from them after including the host specific embedding effects.
II. METHOD
In order to know the 4 f and 5d energy levels of Ce3+
under the effects of its interactions with a coordination
shell of eight oxygens in a D2 symmetry site of a solid oxide, we calculated the corresponding energy levels of the
(CeO8 )13− cluster embedded in a Oh host potential. We
will describe later the details of the ab initio wave function
based quantum mechanical calculation. With the choice
of a host embedding potential of cubic symmetry, all the
energy splittings and changes due to D2 distortions are ascribable to the interactions between Ce and the O atoms.
It is convenient to look at the D2 energy levels as derived
from the more familiar Oh levels after geometry distortions
and this is what we did in this work. In consequence with
this, we decided to take a cubic CeO8 Oh moiety as a reference and to chose six atomic displacement coordinates
S1 − S6 that transform according to irreducible representations of the Oh point symmetry group for the six degrees
of freedom of the CeO8 D2 moiety. They are defined in Table I and represented in Fig. 1; more details are given in the
Appendix. S1 represents the breathing (or symmetric bond
stretching) of the reference cube and it transforms accord-
Published in Opt. Mater., 35, 1932 (2013)
ing to the totally symmetric irreducible representation a1g ;
S2 represents an asymmetric bond stretching of the crosses
a1 − a2 − a3 − a4 and e1 − e3 − e2 − e4 and it transforms
according to the e g ε sub-species of the doubly degenerate
e g irreducible representation;11 S3 and S4 are the symmetric (e g θ ) and asymmetric (e g ε) bond bendings of the a and
e crosses; finally, S5 and S6 are the symmetric (eu θ ) and
asymmetric (eu ε) twistings of the crosses.
The details of the quantum mechanical calculation are
the following. We performed ab initio calculations on
a (CeO8 )13− cluster which include Ce-O bonding effects,
static and dynamic electron correlation effects, scalar and
spin-orbit coupling relativistic effects, and cubic host embedding effects. For each D2 nuclear configuration of
the (CeO8 )13− embedded cluster, we performed, in a
first step, scalar relativistic calculations with the manyelectron second-order Douglas-Kroll-Hess (DKH) Hamiltonian.12,13 These were state-average complete-active-space
self-consistent-field14–16 (SA-CASSCF) calculations with the
active space that results from distributing the open-shell
electron in 13 active molecular orbitals with main character Ce 4 f , 5d, 6s, which provided occupied and empty
molecular orbitals to feed subsequent multi-state secondorder perturbation theory calculations (MS-CASPT2),17–20
where the dynamic correlation of 73 electrons (the
5s, 5p, 4 f , 5d, 6s electrons of Ce and 2s, 2p electrons of the
O atoms) were taken into account. In a second step, spinorbit coupling effects were included by adding the AMFI
approximation of the DKH spin-orbit coupling operator21
to the Hamiltonian and performing restricted-active-space
state-interaction spin-orbit calculations (RASSI-SO)22 with
the previously computed SA-CASSCF wave functions and
MS-CASPT2 energies. All are all-electron calculations with
atomic natural orbital (ANO) relativistic basis sets for
Cerium23 and Oxygen.24
In all these calculations, the Hamiltonian of the
(CeO8 )13− cluster is supplemented with a cubic embedding
Hamiltonian. For this we chose the ab initio model potential embedding Hamiltonian (AIMP)25 of a SrO host in its
high pressuere CsCl lattice structure26 (P m3m, no. 221),
which provides a perfect 8-fold cubic oxide coordination of
the cationic site. We used a lattice constant a = 2.87 Å
(the one that gives a Ce-O equilibrium distance in the cubic
(CeO8 )13− embedded cluster of 2.34 Å at the MS-CASPT2
level of calculation, which is the Ce-O distance of the reference CeO8 cube in an hypothetical undistorted Ce-doped
YAG6 ). The embedding potential of this 8-fold coordinated
cubic oxide was computed in Hartree-Fock self-consistent
embedded-ions calculations (SCEI),27 in which the input
embedding AIMPs used for the Sr2+ and O2− ions of SrO
are consistent with the output AIMPs obtained out of the
HF orbitals of the embedded Sr2+ and O2− ions.
III. RESULTS
The dependence of the 4 f → 4 f and 4 f → 5d transitions on the D2 distortions of the first coordination shell
are shown in Fig. 2 without spin-orbit coupling and in
Fig. 3 with spin-orbit coupling interactions included. The
chosen ranges of the S1 − S6 coordinates span their usual
values in natural and artificial garnets, which are shown
in Table II and in the Figures. Besides the transitions,
the Figures include two additional lines: One is the difference between the 5d and 4 f energy centroids (average energies of the five 5d 1 levels and the seven 4 f 1
levels), ∆Ecent r oid = Ecent r oid (5d 1 ) − Ecent r oid (4 f 1 ); the
other is the difference between the ligand field stabilization energies of the 5d and 4 f lowest levels ∆El i g and f iel d =
[Ecent r oid (5d 1 )−E1 (5d 1 )]−[Ecent r oid (4 f 1 )−E1 (4 f 1 )]. (See
Fig. 2 in Ref. 9.) The fact that the lowest 4 f → 5d
transition equals their subtraction, E1 (5d 1 ) − E1 (4 f 1 ) =
∆Ecent r oid −∆El i g and f iel d , has been used to analyze the reasons behind red and blue shifts of this transition.9,10
Let us first focus on the lowest 4 f → 5d transition,
whose reverse is the luminescence of Ce3+ in garnets and
other oxides. It is clear that only two coordinates have an
important impact on it: the breathing mode S1 , which does
not distort the cubic symmetry, and the symmetric bond
bending mode S3 , which gives a e g θ tetragonal distortion
of the CeO8 cube.
Symmetric compression of the Ce-O bonds lowers the
4 f → 5d transition. This is almost entirely due to the
increase of the 5d ligand field it produces, which lowers
the first 5d 1 level with respect to its centroid. This effect
is slightly compensated with a small increase due to the
centroids: bond compression slightly shifts the 5d centroid
upwards with respect to the 4 f centroid. The latter observation has been made before10,28 and it contradicts the
predictions of the usual model for the centroid energy difference,29–31 according to which it should be smaller for
smaller bond lengths; nevertheless, the model has been
useful to rationalize 5d → 4 f luminescences.32 The contribution of the breathing mode S1 to the shift of the lowest 4 f → 5d transition energy represented in Fig. 3 fits
8255 S1 − 4803 S12 (S1 in Å, energy shift in cm−1 ); for cup
bic CeO8 moieties S1 = 2 2∆d, ∆d being the Ce-O bond
length change with respect to the reference cube; in our
case the reference cube has Ce-O bond length of 2.34 Å
and a lowest 4 f → 5d transition of 26000 cm−1 .
Symmetric bond bending along the S3 coordinate lowers
de 4 f → 5d transition. Among the D2 distortions of the
cube, it is the only tetragonal one. Also, it is the only one
with a significant impact on the transition (for the ranges
of S2 − S6 shown by garnets). The enhancement of tetragonal distortions of the cube as responsible for the red-shifts
of the Ce3+ luminescence has been suggested by Cheetham
and coworkers33 and it is supported by the present investigation. The image emerging from an analysis in terms of
configurational energy centroids and ligand field stabilization is one in which the lowering of the first 5d 1 level with
respect to the ground state is due in almost equal amounts
to an increase of the ligand field (of its tetragonal component) and to a reduction of the centroid energy difference. Again, the latter effect contradicts the prediction of
Published in Opt. Mater., 35, 1932 (2013)
the Judd-Morrison model: a very small increase of the centroid energy difference out of a very small increase of the
Ce-O distances. The contribution of the tetragonal mode
S3 (in Å) to the shift of the lowest 4 f → 5d transition
(in cm−1 ) represented in Fig. 3 fits 1844 S3 − 2885 S32 for
S3 < 0 and −1876 S3 − 141 S32 for S3 > 0. This contribution is due in almost equal amounts to the lowering of the
centroid energy difference and the increasing of the ligand
field. The centroid effect fits 10 S3 − 2695 S32 for S3 < 0
and 12 S3 − 2581 S32 for S3 > 0; the ligand field effect fits
1834 S3 − 190 S32 for S3 < 0 and −1887 S3 + 2440 S32 for
S3 > 0.
Apart from S1 and S3 , only the asymmetric stretching S2
has an effect on the first 4 f → 5d transition, although it is a
very small one (it fits −2033 S22 in the range of the figures).
This result allows one to say that, in a first approximation
and in spite of the large values of the Si coordinates applicable to garnets, the lowest 5d level can be understood
as resulting from the cubic E g level with a strong E g × e g
Jahn-Teller coupling.34
Regarding higher 4 f → 5d transitions, several remarks
can be made from Figs. 2 and 3. The splitting of the first
and second 5d levels related with the cubic 2 E g is due to
the symmetric bending S3 and to the symmetric twisting
S5 , with a significantly higher contribution from the latter
in garnets; however, different splittings in different garnets
would be due to S3 . The splittings of the three upper levels,
mostly related with the cubic 2 T2g , come basically from the
symmetric bending S3 and the asymmetric stretching S2 ,
with a small contribution from the symmetric twisting S5
and negligible contributions from the asymmetric bending
and twisting S4 and S6 . The larger S1 and the absolute
values of S2 and S3 , the higher probability for the third 5d
level to appear below the conduction band of the garnet.
Finally, the 4 f levels are shown in more detail in Figs. 4
and 5, without and with spin-orbit coupling respectively.
These are interesting because the 5d → 4 f emission is
made of the superposition of the seven individual emissions
and the full width and shape of the emission band depends
on the relative positions of the 4 f levels, although not only
on them. Since the emission to the highest 4 f level (7Γ5
or 4 f7 ) has a minor contribution to the full emission band
shape,8 we can have a predictive hint by looking at the six
lowest 4 f levels. (Note that the experimental 5d → 4 f
emission exhibits two peaks or shoulders, which are often
taken as the difference between the 2 F7/2 and 2 F5/2 in the
garnets; this difference is around 2000 cm−1 in Fig.5.) According to Fig. 5, the emission band width seems to be controlled by S2 and S3 , so that the garnets with higher absolute values of these two coordinates would tend to have
wider emission bands because of their higher 4 f level separation. Of course, the width of each individual 5d → 4 f
emission will depend on the 5d and 4 f equilibrium offsets.
Since only the first coordination shell effects in a hypothetical undistorted CeA substitutional defect are properly
taken into account in this work, whereas the host specific
electronic effects and the structural relaxation effects af-
ter the substitution are disregarded, direct comparisons of
the present results with experiments must be taken with
care. Nevertheless, it is interesting to see how these results compare with experiments in the most studied case
of Ce3+ -doped YAG: It gives us a hint on the size of the
disregarded contributions. The results are summarized in
Table III. The host specific embedding and structural relaxation effects are not large but are significant. Different
signs and sizes are observed in different states, which are
a consequence of effects on the effective field of the second and more distant neighbors, as well as of local (mostly
first neighbor) relaxations. These additional effects on the
effective ligand field are shown, for instance, in the different impacts on the total splitting of the 5d shell (30150
vs. 33200 cm−1 , -10%), on the splitting of the four first 5d
levels (28500 vs. 27880 cm−1 , +2%; with 26900 cm−1 experimental), and on the splitting of the two first 5d levels
(7620 vs. 4000 cm−1 , +48%; with 7400 cm−1 experimental). The only consideration of the first coordination shell
of Cr3+ gives a good qualitative and even semi-quantitive
description of the four 5d levels of Ce3+ in YAG that have
been experimentally observed, which are quite reasonably
reproduced by the ab initio calculations in Ref. 8.
IV.
CONCLUSIONS
The levels of the 4 f 1 and 5d 1 manifolds of Ce3+ in D2
8-fold oxygen coordination have been calculated ab initio
as a function of D2 deformation coordinates of a reference
cube, with the goal of pinpointing the effects of the geometry of the first coordination shell. A (CeO8 )13− cluster under the effects of a confinement AIMP embedding potential
of cubic symmetry has been used at a level of calculation
all-electron DKH for the Hamiltonian and SA-CASSCF/MSCASTP2/RASSI-SO for the wave functions. The results
include bonding, correlation, scalar relativistic, spin-orbit
coupling, and embedding interactions.
It is found that, within the range of D2 distortions covering natural and artificial garnets, the lowest 4 f → 5d transition shifts significantly to the red only as a consequence of
symmetric Ce-O bond compression and tetragonal symmetric bond bending. Then, in a first approximation, the lowest
5d level of Ce3+ in garnets can be understood as resulting
from the cubic E g level with a strong E g × e g Jahn-Teller
coupling.
The increase of the 5d cubic ligand field dominates the
effects of the bond compression, whereas the effect of the
tetragonal bond bending distortion is divided in almost
equal amounts between an increase of the 5d tetragonal
ligand field and a reduction of the energy difference between the 5d and 4 f energy centroids.
The splittings of the upper 5d levels and of the 4 f levels
have also been studied.
Published in Opt. Mater., 35, 1932 (2013)
Acknowledgments
This work was partly supported by a grant from Ministerio de Economía y Competitivad, Spain (Dirección General de Investigación y Gestión del Plan Nacional de I+D+I,
MAT2011-24586).
APPENDIX
We have chosen Oxygen atoms a1 and e1 (see Table I
and Fig. 1) as the symmetry independent atoms. The
other atoms are obtained upon application of the D2 point
group symmetry operations. So, any particular D2 atomic
configuration of the CeO8 moiety is defined with ~ra1 =
(x a1 , ya1 , za1 ), ~ra2 = Ĉ2x ~ra1 , ~ra3 = Ĉ2 y ~ra1 , ~ra4 = Ĉ2z ~ra1 ,
~re1 = (x e1 , ye1 , ze1 ), ~re2 = Ĉ2x ~re1 , ~re3 = Ĉ2 y ~re1 , ~re4 = Ĉ2z ~re1 .
Taking an arbitrary reference cube with Ce-O distance d,
in which the chosen positions for a1 and e1 are ~ra1,r e f =
p
p
p
p
( 2, 0, 1)d/ 3 and ~re1,r e f = (0, 2, 1)d/ 3, the D2
atomic configurations of the CeO8 moiety can also be defined in terms of displacements of the symmetry
indepenp
dent atoms δ~ra1 = ~ra1 − ~ra1,r e f = (x a1 − d 2/3, ya1 , za1 −
p
p
d/ 3) and δ~re1 = ~re1 − ~re1,r e f = (x e1 , ye1 − d 2/3, ze1 −
p
d/ 3), together with the displacements of the symmetry
dependent atoms δ~ra2 = Ĉ2x δ~ra1 , δ~ra3 = Ĉ2 y δ~ra1 , δ~ra4 =
Ĉ2z δ~ra1 , δ~re2 = Ĉ2x δ~re1 , δ~re3 = Ĉ2 y δ~re1 , δ~re4 = Ĉ2z δ~re1 .
Substitution of the values of these atomic displacements in
the equations in Table I gives the values of the D2 totally
symmetric displacement coordinates of the CeO8 moiety
S1 , S2 , . . . S6 .
∗
1
2
3
4
5
6
7
8
9
10
11
12
Corresponding author; Electronic address:
uam.es
luis.seijo@
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Published in Opt. Mater., 35, 1932 (2013)
TABLE I: Definitions of the D2 totally symmetric displacements of the CeO8 moiety chosen in this work: Stretching, bending, and twisting
Oh symmetry coordinates. The labels of the oxygen atoms and the
axespare defined
pchosen cartesian
p
p in Fig. 1: In the reference cube, the
symmetry independent oxygen atoms a1 and e1 are located at ( 2, 0, 1)d/ 3 and (0, 2, 1)d/ 3 respectively, with d being the Ce-O
distance. R̂ is the D2 symmetrization operator: the normalized addition of the group symmetry operations (the identity and the three
180◦ rotations around the cartesian axes). δx a1 is the x cartesian displacement of the Oxygen atom a1 from its position in the reference
cube: δx a1 = x a1 − x a1,r e f ; identical definitions stand for the other cartesian displacements.
Š
€
R̂ = 12 Î + Ĉ2x + Ĉ2 y + Ĉ2z
Displacement
Oh irrep
Definition
Š—
Š €p
”€p
symmetric stretching
a1g
S1 = p16 R̂
2 δx a1 + δza1 +
2 δ ye1 + δze1
Š €p
Š—
”€p
asymmetric stretching
eg ε
S2 = p16 R̂
2 δx a1 + δza1 −
2 δ ye1 + δze1
Š
€
Š—
”€
p
p
symmetric bending
eg θ
S3 = p16 R̂ −δx a1 + 2 δza1 + −δ ye1 + 2 δze1
Š
€
Š—
”€
p
p
asymmetric bending
eg ε
S4 = p16 R̂ −δx a1 + 2 δza1 − −δ ye1 + 2 δze1
symmetric twisting
eu θ
S5 = p12 R̂ −δ ya1 + δx e1
asymmetric twisting
eu ε
S6 = p12 R̂ δ ya1 + δx e1
TABLE II: Crystallographic data of some garnets: lattice constant a and special position (h) of the I a3d (230) space group x O , yO , zO . Local D2 8-fold oxygen coordination of the
fixed position (c) occupied by Ce as a dopant: cation-oxygen distance in the reference cube d r e f and D2 displacements S1 - S6 as defined in Table I; S1 is defined with respect
to a cube with cation-oxygen distance d =2.34 Å (note that the use of d r e f , S1 = 0, and S2 − S6 or d r e f =2.34 Å, and S1 − S6 are two valid alternatives to define the oxygen
coordinations). a, d r e f and S1 - S6 are given in Å and x O , yO , zO in fractional cell units. The lowest 4 f → 5d transitions calculated for the (CeO8 )13− clusters, embedded in a
common cubic confinement potential and having the local structures of the undistorted garnets, are given in the last column in cm−1 ; their stabilizations with respect to a reference
cube of d =2.34 Å are given in parenthesis.
Lu3 Al5 O12
Yb3 Al5 O12
Er3 Al5 O12
Y3 Al5 O12
Gd3 Al5 O12
Ref.
6
6
36
6
6
a
11.9060
11.9310
11.9928
12.0000
12.1130
xO
-0.02940
-0.02960
-0.03039
-0.03060
-0.03110
yO
0.05370
0.05290
0.05124
0.05120
0.05090
zO
0.15090
0.15040
0.14915
0.15000
0.14900
dr e f
2.303
2.313
2.339
2.339
2.368
S1
-0.1042
-0.0759
-0.0039
-0.0019
0.0788
S2
-0.1683
-0.1819
-0.2047
-0.2094
-0.2060
S3
0.0555
0.0497
0.0297
0.0156
0.0101
S4
-0.0866
-0.0906
-0.0944
-0.1018
-0.0932
S5
0.9766
0.9858
1.0046
1.0099
1.0180
S6
-0.1044
-0.1286
-0.1854
-0.1613
-0.1957
4 f → 5d
24910 (-1090)
25140 ( -860)
25710 ( -290)
25680 ( -320)
26340 ( +340)
LuGG
YbGG
YGG
HoGG
DyGG
TbGG
GdGG
SmGG
NdGG
Lu3 Ga5 O12
Yb3 Ga5 O12
Y3 Ga5 O12
Ho3 Ga5 O12
Dy3 Ga5 O12
Tb3 Ga5 O12
Gd3 Ga5 O12
Sm3 Ga5 O12
Nd3 Ga5 O12
6
6
37
38
38
39
39
39
39
12.1880
12.2040
12.2730
12.2900
12.3060
12.3474
12.3829
12.4361
12.5051
-0.02520
-0.02590
-0.02740
-0.02740
-0.02780
-0.02820
-0.02890
-0.02920
-0.03000
0.05700
0.05630
0.05460
0.05520
0.05490
0.05470
0.05420
0.05300
0.05220
0.15060
0.15190
0.14930
0.15020
0.14990
0.14950
0.14940
0.14900
0.14850
2.322
2.328
2.363
2.362
2.369
2.381
2.394
2.412
2.434
-0.0520
-0.0336
0.0647
0.0616
0.0823
0.1166
0.1538
0.2031
0.2667
-0.1378
-0.1561
-0.1629
-0.1550
-0.1558
-0.1538
-0.1578
-0.1816
-0.1896
0.1961
0.1596
0.1338
0.1293
0.1194
0.1115
0.0901
0.0777
0.0554
-0.0727
-0.0899
-0.0757
-0.0771
-0.0752
-0.0712
-0.0719
-0.0818
-0.0819
0.9580
0.9744
0.9876
0.9861
0.9896
0.9933
1.0019
1.0190
1.0321
-0.0755
-0.0459
-0.1440
-0.1101
-0.1229
-0.1377
-0.1473
-0.1748
-0.2009
25870 ( -130)
25860 ( -140)
26640 ( +640)
26610 ( +610)
26740 ( +740)
26980 ( +980)
27180 (+1180)
27410 (+1410)
27720 (+1720)
Pyrope
Almandine
Spessartine
Grossular
Andradite
Mg3 Al2 Si3 O12
Fe3 Al2 Si3 O12
Mn3 Al2 Si3 O12
Ca3 Al2 Si3 O12
Ca3 Fe2 Si3 O12
Ca3 Sc2 Si3 O12
40
41
42
40
7
7
11.4566
11.5250
11.6190
11.8515
12.0578
12.2500
-0.03290
-0.03401
-0.03491
-0.03760
-0.03940
-0.04004
0.05010
0.04901
0.04791
0.04540
0.04870
0.05010
0.15280
0.15278
0.15250
0.15130
0.15540
0.15887
2.243
2.267
2.295
2.368
2.398
2.427
-0.2731
-0.2070
-0.1278
0.0790
0.1648
0.2454
-0.2191
-0.2334
-0.2490
-0.2752
-0.2102
-0.1972
-0.0768
-0.1126
-0.1414
-0.2203
-0.2895
-0.3362
-0.1215
-0.1279
-0.1337
-0.1378
-0.1260
-0.1401
0.9900
1.0084
1.0281
1.0725
1.0718
1.0894
-0.0892
-0.1028
-0.1243
-0.1909
-0.0351
0.0842
22750 (-3250)
23380 (-2620)
24100 (-1900)
25670 ( -330)
26210 ( +210)
26580 ( +580)
Lu2 CaMg2 Si3 O12
4
11.9750
-0.03510
0.05380
0.15780
2.333
-0.0200
-0.1522
-0.1619
-0.1139
1.0153
0.0956
25240 ( -760)
Published in Opt. Mater., 35, 1932 (2013)
Garnet
LuAG
YbAG
ErAG
YAG
GdAG
Published in Opt. Mater., 35, 1932 (2013)
TABLE III: 4 f and 5d energy levels of Ce3+ -doped YAG (in cm−1 ).
4f
Level
1Γ5 4 f1
2Γ5 4 f2
3Γ5 4 f3
4Γ5 4 f4
5Γ5 4 f5
6Γ5 4 f6
7Γ5 4 f7
This work a
0
145
900
2280
2300
2540
5650
Ref. 8 b
0
370
820
2360
2570
2810
4320
Difference c
0
+225
–80
+80
+270
+270
–1330
Level
8Γ5 5d1
9Γ5 5d2
10Γ5 5d3
11Γ5 5d4
12Γ5 5d5
a
Spin-orbit coupling calculation: Undistorted relaxation, first coordination shell effect only.
b
Ab initio calculations also including host specific embedding and structural relaxation.
c
Estimated effects of host specific embedding plus structural relaxation.
d
After the assignment analysis in Reference 35.
e
Reference 1.
f
Reference 43.
This work a
25680
29680
52350
53560
58900
5d
Ref. 8 b
24040
31660
48070
52540
54190
Difference c
–1640
+1980
–4280
–1020
–4710
Experiment d
22000 e
29400 e
44000 e
48900 f
Published in Opt. Mater., 35, 1932 (2013)
Figure captions
FIG. 1: D2 totally symmetric displacements of the CeO8 moiety: Stretching, bending, and twisting Oh symmetry coordinates S1 (a1g
symmetric stretching -breathing-), S2 (e g ε asymmetric stretching), S3 (e g θ symmetric bending), S4 (e g ε asymmetric bending), S5 (eu θ
symmetric twisting), and S6 (eu ε asymmetric twisting). See Table I for the detailed definitions.
FIG. 2: 4 f (dashed lines) and 5d (full lines) energy levels of the (CeO8 )13− embedded cluster (relative to the ground state) as a function
of the S1 - S6 D2 oxygen displacement coordinates, calculated without spin-orbit coupling. The differences between the 5d and 4 f
energy centroids (dot-dashed lines) and between the ligand field stabilization energies of the 5d and 4 f lowest levels (doted lines) are
also shown; the lowest 4 f → 5d transition equals the subtraction of these two. Experimental values of S1 - S6 of 21 pure garnets are
shown as small vertical lines.
FIG. 3: 4 f and 5d energy levels of the (CeO8 )13− embedded cluster as a function of S1 - S6 , calculated with spin-orbit coupling. All
levels are D2′ Γ5 Kramer doublets. See Fig.2 caption.
FIG. 4: 4 f energy levels of the (CeO8 )13− embedded cluster as a function of S1 - S6 , calculated without spin-orbit coupling. Dotted lines:
2
A levels; full lines: 2 B1 levels; dashed lines: 2 B2 and 2 B3 levels. Experimental values of S1 - S6 of 21 pure garnets are shown as small
vertical lines.
FIG. 5: 4 f energy levels of the (CeO8 )13− embedded cluster as a function of S1 - S6 , calculated with spin-orbit coupling. All levels are
D2′ Γ5 Kramer doublets. Experimental values of S1 - S6 of 21 pure garnets are shown as small vertical lines.
Figure 1. Seijo and Barandiarán
65000
2
60000
2
55000
2
Vertical excitation / cm
-1
50000
T2g
A
2
2
B2, B3
2
B2, B3
2
45000
2
2
2
B2, B3
2
A
2
B2, B3
2
A
A
40000
35000
2
30000
Eg
2
2
2
B1
A
25000
2
20000
2
A
2
2
B1
B1
A
0.4
-1
B1
2
A
15000
10000
5000
0
2.2
2.3
2.4
-0.4 -0.2
d(Ce-O) / Å
-0.4
-0.2
0
0
0.2
0.2
0
-0.5
S3 / Å
0.5
1
S5 / Å
0.4
S1 / Å
65000
60000
2
2
-1
Vertical excitation / cm
B3
A
2
B3
2
55000
50000
2
B2
2
A
2
A 2B
2
3
2
2
B2
B2
2
B3 A
2
B2
2
2
A B2 B3
45000
40000
35000
30000
25000
2
2
20000
2
B1
B1
2
A
2
A
2
2
B1 A
2
2
B1 A
2
A B1
15000
10000
5000
0
-0.4
-0.2
0
S2 / Å
0.2
0.4 -0.2
-0.1
0
0.1
0.2 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
S4 / Å
Figure 2. Seijo and Barandiarán
S6 / Å
65000
60000
55000
Vertical excitation / cm
-1
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
2.2
2.3
2.4
-0.4 -0.2
d(Ce-O) / Å
-0.4
-0.2
0
0.2
0
0.2
0.4
-1
S3 / Å
-0.5
0
0.5
1
S5 / Å
0.4
S1 / Å
65000
60000
55000
Vertical excitation / cm
-1
50000
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
-0.4
-0.2
0
S2 / Å
0.2
0.4 -0.2
-0.1
0
0.1
0.2 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
S4 / Å
Figure 3. Seijo and Barandiarán
S6 / Å
7000
6000
Vertical excitation / cm
-1
2
A2u
5000
4000
3000
2000
1000
2
T1u
0
2.2
2
T2u
2.3
2.4
-0.4 -0.2
d(Ce-O) / Å
-0.4
-0.2
0
0.2
0
0.2
0.4
-1
S3 / Å
-0.5
0
0.5
1
S5 / Å
0.4
S1 / Å
7000
Vertical excitation / cm
-1
6000
5000
4000
3000
2000
1000
0
-0.4
-0.2
0
S2 / Å
0.2
0.4 -0.2
-0.1
0
0.1
0.2 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
S4 / Å
Figure 4. Seijo and Barandiarán
S6 / Å
7000
Vertical excitation / cm
-1
6000
5000
4000
3000
2000
1000
0
2.2
2.3
2.4
-0.4 -0.2
d(Ce-O) / Å
-0.4
-0.2
0
0.2
0
0.2
0.4
-1
S3 / Å
-0.5
0
0.5
1
S5 / Å
0.4
S1 / Å
7000
Vertical excitation / cm
-1
6000
5000
4000
3000
2000
1000
0
-0.4
-0.2
0
S2 / Å
0.2
0.4 -0.2
-0.1
0
0.1
0.2 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
S4 / Å
Figure 5. Seijo and Barandiarán
S6 / Å