˙
˙ ´
2802
J. Chem. Phys., Vol. 117, No. 6, 8 August 2002
Karbowiak, Drozdzynski, and Sobczyk
2
Si→f
ϭ
͉
͉
ˆ eff
͉
͉
2ϩ
͉
͉
mˆ
͉ ͉
͘
f
,
͑8͒
where K runs through the even integers from 0 to 12, i
͗
͘
͗
i
f
i
(K)
distinguishes different gˆ operators with identical K, and Q
iQ
where ˆ eff is an effective electric-dipole operator, mˆ eff is the
magnetic-dipole operator, ⌿i and ⌿f are eigenvectors of the
initial and final levels for the i→f transition. ˆ eff is defined
as an even-parity operator that operates only within the 5f3
electronic configuration. The ˆ eff operator includes the com-
bined perturbation of odd-parity crystal-field interactions and
odd-parity electric-dipolar radiation field interactions on the
5f electrons of the system. The radial dependence of
electric-dipole matrix elements is entirely absorbed in the
parametric form of the ˆ eff operator. The qth component of
the ˆ eff operator in a spherical basis representation is given
by
is restricted by the crystal-field symmetry.
Since there are 41 independent correlation crystal-field
͑CCF͒ parameters it is not possible to include all of them in
a fit with a set of 67 experimental data only. However, Li and
Reid17 in an analysis of a number of Nd3ϩ doped crystals
have shown that the inclusion of only a few of them have
markedly improved the fits and could resolve problems with
poorly fitted levels by the one-electron crystal-field operator
HCF . On the other hand, there arises the question which of
the 41 parameters should be chosen. One possibility is to find
those with the largest influence on the problematic levels.
Although such a selection may be statistically justified, most
probably it would have little physical meaning. Judd18 had
invented the simple ␦-function model in which the correla-
tion effects are considered from paired electrons within the
same angular f orbital only ͑with opposite spin͒. The validity
of this model has been tested and proved by Quagliano
et al.19 for neodymium nonahydrate tris͑trifluoromethane-
sulfonate͒. Consequently, in the case of the 4f3 configuration
only multiplets of a predominantly doublet character should
be significantly influenced by two-electron correlation-
crystal-field interactions. In the present analysis we have
q
ˆ
ˆ ͒ ϭϪe Ϫ1͒
At,p l,1Ϫq
͉
tp U ,
͑9͒
͑
͑
͗
͘
͚
eff q
l
,t,p,l
where ϭ2,4,6; tϭ, Ϯ1; pϭ0,Ϯ1,Ϯ2,...,Ϯt, and 1
ˆ ()
ϭqϩp. Ul are interconfigurational many-electron unit
tensor operators that act within the 5f3 electronic configura-
tion, and Atp are parameters that contain structural and
mechanistic details about the interaction of the odd-parity
crystal-field and the electric-dipolar radiation field with the
5f electrons of the U3ϩ ion.
2
For the C3h site symmetry the p values are restricted to
found the levels of the H29/2 multiplet to be the most prob-
Ϯ3. The Atp parameters are related by the expression
lematic ones ͑see Sec. IV͒ and therefore we reduced the set
of CCF parameters by applying the ␦-function model restric-
tions.
␥
tϩpϩl
*
(At,p) ϭ(Ϫ1)
A
. There are seven independent
t,Ϫp
complex parameters: A23,3, A34,3, A44,3, A54,3, A65,3, A66,3, and
A67,3. In the line-strength calculations the
The ␦-function model are only contributing the g1(k)
,
ˆ ()
͉
Ul
͉
matrix
͘
f
͗
i
g(2k) , g(3k) , and g4(k) operators from among which the g(1k)
contribution is already incorporated in HCF . We have
checked the effect of all these parameters and have proved
that only the three G410A,0, G140B,0, and G24,0 fourth-rank pa-
rameters and one G610B,0 sixth-rank parameter are statistically
significant.
elements were directly evaluated by using the ⌿i and ⌿f
eigenfunctions obtained from the energy-level calculations.
In the fitting procedure of the calculated and experimental
line strengths the Atp parameters are treated as variables. The
magnetic-dipole contributions were obtained from direct
evaluation of the ⌿
͉
mˆ q
͉
⌿
matrix elements. This contri-
͘
͗
i
f
butions are considerable only for transitions to levels of the
4I11/2 multiplet.
C. Transition line strengths
The experimental line strengths of transitions from the
lowest Stark component of the I9/2 ground multiplet to ex-
IV. RESULTS AND DISCUSSION
A. Energy level calculations
4
cited levels have been determined from the 7 K and
-polarized absorption spectra. The observed transition line
profiles were integrated and the line strengths Si→f in
squared Debye units (D2) (Dϵ3.3356ϫ10Ϫ30 C m) deter-
mined from the equation
In his latest analysis2 of the U3ϩ:LaCL3 absorption
spectrum, Carnall had included in the fitting procedure 82
experimental energy levels which could be recorded up to
25 723 cmϪ1. However above 22 000 cmϪ1 the f – f bands
are obscured by strong and broad f –d bands and also one
cannot be fully sure as well about the origin and assignment
of these sharp lines superimposed on the envelope of the
f –d bands. Since the main purpose of our analysis was to
obtain reliable eigenfunctions with a quality high enough for
reproducing transition intensities, we have confined our
analysis to the 0–22 000 cmϪ1 region, i.e., below the ap-
pearance of the f –d bands. Hence, from 7 K unpolarized as
well as - and -polarized absorption spectra we could iso-
late and assign 67 energy levels which were fitted to the
parameters of the phenomenological Hamiltonian. The initial
values of the free-ion and crystal-field parameters were taken
from Carnall’s analysis.2 The calculations were done for the
full 5f3 electron configuration leading to a 364ϫ364 energy
9.186ϫ10Ϫ3
Si→f D2͒ϭ
˜͒d
͑
˜,
͑6͒
͑
͵
i→f
˜
ed
where assuming a predominance of electric-dipole contribu-
tion to the transition intensities, the correction factor for bulk
refractivity is defined as
ed
2
n2ϩ2͒
͑
edϭ
,
͑7͒
9n
and is for nϭ1.849 equal to 1.7645.
Assuming, that the observed line strengths arise exclu-
sively from electric- and magnetic-dipole transition mecha-
nisms, the line strengths may be calculated by the evaluation
of