Optical spectrum and crystal-field analysis of uranium trifluoride

Article abstract of DOI:10.1016/j.chemphys.2007.08.017

The first low temperature absorption spectrum and crystal-field analysis of uranium trifluoride are presented. The measurements have been performed on a polycrystalline sample in the 4000-30,000 cm^-1 range at 4.2 K. One may notice the largest crystal-field splitting of SLJ multiplets among all so far reported data for uranium(III) as well as in polycrystalline samples and U³⁺ doped single crystals. In the presented analysis 94 experimental crystal-field levels, observed in 0-24,116 cm^-1 energy range, have been identified and fitted to a semiempirical Hamiltonian representing the combined atomic and crystal-field interactions at the C₂ site symmetry. In order to exclude a possible influence of inaccuracy in the description of the free-ion multiplets on crystal-field parameters a number of additional freely varied parameters associated with the centroids of the free-ion multiplets were introduced. In the final step of the fitting procedure eight free-ion and 14 CF parameters were freely varied giving B₀² = - 833 (45), Re B₂² = 272 (36), B₀⁴ = 1817 (89), Re B₂⁴ = 186 (106), Im B₂⁴ = 776 (82), Re B₄⁴ = 523 (104), Im B₄⁴ = + 1460 (58), B₀⁶ = 1081 (105), Re B₂⁶ = - 841 (89), Im B₂⁶ = - 2459 (74), Re B₄⁶ = - 665 (88), Im B₄⁶ = - 1024 (74), Re B₆⁶ = 1416 (56), Im B₆⁶ = - 1682 (93) and r.m.s. = 24 cm^-1. The performed calculation procedure enabled the determination of good starting values, reliable crystal-field parameters and a small r.m.s. deviation.

Full text of DOI:10.1016/j.chemphys.2007.08.017

Available online at www.sciencedirect.com
Chemical Physics 340 (2007) 187–196
www.elsevier.com/locate/chemphys
Optical spectrum and crystal-field analysis of uranium trifluoride
*
_ _ ´
M. Karbowiak , J. Drozdzynski
Faculty of Chemistry, University of Wrocław, ul. F. Joliot-Curie 14, 50-383 Wrocław, Poland
Received 6 April 2007; accepted 16 August 2007
Available online 23 August 2007
Abstract
The first low temperature absorption spectrum and crystal-field analysis of uranium trifluoride are presented. The measurements have
been performed on a polycrystalline sample in the 4000–30,000 cmꢀ1 range at 4.2 K. One may notice the largest crystal-field splitting of
SLJ multiplets among all so far reported data for uranium(III) as well as in polycrystalline samples and U3+ doped single crystals. In the
presented analysis 94 experimental crystal-field levels, observed in 0–24,116 cmꢀ1 energy range, have been identified and fitted to a semi-
empirical Hamiltonian representing the combined atomic and crystal-field interactions at the C2 site symmetry. In order to exclude a
possible influence of inaccuracy in the description of the free-ion multiplets on crystal-field parameters a number of additional freely
varied parameters associated with the centroids of the free-ion multiplets were introduced. In the final step of the fitting procedure eight
free-ion and 14 CF parameters were freely varied giving B20 ¼ ꢀ833ð45Þ, ReB22 ¼ 272ð36Þ, B04 ¼ 1817ð89Þ, ReB42 ¼ 186ð106Þ,
ImB42 ¼ 776ð82Þ, ReB44 ¼ 523ð104Þ, ImB44 ¼ þ1460ð58Þ, B06 ¼ 1081ð105Þ, ReB62 ¼ ꢀ841ð89Þ, ImB26 ¼ ꢀ2459ð74Þ, ReB46 ¼ ꢀ665ð88Þ,
ImB46 ¼ ꢀ1024ð74Þ, ReB66 ¼ 1416ð56Þ, ImB66 ¼ ꢀ1682ð93Þ and r.m.s. = 24 cmꢀ1. The performed calculation procedure enabled the
determination of good starting values, reliable crystal-field parameters and a small r.m.s. deviation.
Ó 2007 Elsevier B.V. All rights reserved.
Keywords: Uranium trifluoride; Low temperature absorption spectrum; Energy levels; Crystal-field analysis; Crystal-field parameters
1. Introduction
with a bimolecular hexagonal cell. The polyhedra are fully
capped trigonal prisms (CN = 11) in which the fluorine
atoms are located on all corners (bond lengths: 2.48–
The first optical studies on U3+ ions in a fluoride envi-
ronment were reported for U3+ doped CaF2, BaF2 and
SrF2 single crystals [1,2]. The crystals have been extensively
investigated for its potential use in infrared lasers, but an
interpretation of the optical spectra has not been per-
formed. More recently 700–1500 ppm U3+ doped LiYF4
single crystals were obtained by irradiation of U4+:LiYF4
crystals with a 60Co source for 48 h [3,4]. Besides, a room
temperature spectrum of UF3 in a teflon disc is also avail-
able [5].
˚
3.01 A) and outside of the two triangular and three square
˚
boundary planes (bond lengths: 2.42–2.48 A). The main
difference between the two structures is a slight displace-
ment of the atoms forming the prism and the atoms outside
the triangular surfaces normal to the c axis [6]. Due to the
distortion of the polyhedra, the site symmetry of the U3+
ion is C2. In this paper we are presenting the first low tem-
perature absorption spectrum as well as the first crystal-
field analysis of uranium trifluoride.
UF3 exhibits the LaF3 type structure, but the symmetry
4
ꢀ
is reported to be either trigonal (space group P3c1-D3d , No.
185) or hexagonal (space group P63cm-C36v, No. 165). Both
structures may be considered as distorted ideal polyhedra
2. Experimental
For the preparation of uranium trifluoride stoichio-
metric quantities of cleaned uranium turnings and UF4
were placed in a nickel tube and heated to about 250 °C
in a stream of pure hydrogen. The finely divided UH3
*
Corresponding author. Tel.: +48 71 3757 304; fax: +48 71 3282 348.
E-mail address: karb@wchuwr.pl (M. Karbowiak).
0301-0104/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.chemphys.2007.08.017

188
M. Karbowiak, J. Droz_dz_yn´ski / Chemical Physics 340 (2007) 187–196
was decomposed at 400 °C, after which the tube has been
shaken in order to obtain an intimate mixture that was
then heated to 700–900 °C to give pure uranium trifluoride.
The absorption spectrum of a polycrystalline sample of
UF3 has been recorded in the 4000–30,000 cmꢀ1 spectral
range on a Cary-5 UV–Vis–NIR spectrophotometer at
4.2 K and 100 K. In order to obtain the spectrum a well
ground mixture of UF3 with some chlorinated naphthalene
oil (index of refraction 1.635) was placed between two
quartz plates, approximately 1 cm in diameter, pressed to
get a transparent layer, and inserted in the cell compart-
ment of an Oxford Instrument Model CF1204 cryostat.
An X-ray powder diffraction analysis, performed for the
investigated compound reveals that the obtained lattice
In Fig. 3 the UF3 spectrum is presented in the range of
the I9/2 ! H29/2 and I9/2 ! (4I15/2 + 4G5/2) transitions,
measured at 4.2 and 100 K. A comparison of the spectra
at the two different temperatures enabled the identification
of a hot band centered at ca. 99 cmꢀ1 from the main band
which is indicating the energy of the second component of
4
2
4
4
the I9/2 ground multiplet.
The presented spectrum shows similarity to that
reported for UF3 in a teflon disc at 300 K [5] but exhibits
distinct differences in energy of the 5f3 ! 5f3 absorption
bands as compared to earlier reported spectra of the U3+
ion [7]. It is particularly noticeable in the visible range
where the SLJ levels are shifted of ca.1000 cmꢀ1 towards
4
higher wave numbers. For example the G7/2 multiplet in
the spectrum of U3+:LaCl3 is located between 13,230 and
13,350 cmꢀ1 whereas in that of UF3 it appears between
14,108 and 14,379 cmꢀ1. Since the shift is not uniform
˚
constants for the trigonal cell a = 7.164 A and c =
˚
7.338 A, are close to those reported in the literature,
˚
˚
a = 7.173 A and c = 7.341 A [6].
4
and amounts e.g. to 255 cmꢀ1 for I11/2 and 934 cmꢀ1 for
3. Results and discussion
4G7/2, one may notice a change in the sequence of some
of the SLJ levels of UF3 relative to those of U3+ in LaCl3,
3.1. Characterization of the absorption spectrum
especially above 16,000 cmꢀ1
.
The energy level structure of UF3 in comparison to that
of the U3+ ion in a non-fluorine environment e.g.
U3+:LaCl3 (Fig. 4), also exhibits a much larger splitting of
the SLJ multiplets, caused by a larger crystal-field effect of
the fluorine ligands and shorter uranium-ligand distances.
The absorption spectrum of UF3 at 4.2 K is presented in
Figs. 1 and 2. The edge of strong 5f3 ! 5f26d1 absorption
bands appears at about 24,000 cmꢀ1 (Fig. 2). Since these
bands obscure the 5f3 ! 5f3 transitions an assignment of
the 5f3 levels above this energy was not possible. Hence,
the analysis have to be limited to the 4000–24,116 cmꢀ1
range, for which the theory predicts 115 intraconfiguration-
al 5f3 ! 5f3 transitions. As can be seen from Table 1, all of
the 92 recorded bands have been assigned.
˚
The shortest U–F distance in UF3 is equal to 2.420 A
˚
whereas that of La(U)–Cl amounts to 2.950 A. The splitting
4
4
of the I11/2 and F3/2 multiplets are, for instance, equal to
164 and 18 cmꢀ1 for U3+ doped in LaCl3 single crystals
and to 716 and 73 cmꢀ1 for UF3, respectively.
4I13/2
4I11/2
4I15/2 + 4S3/2
4G5/2 + 4F7/2
+
4F5/2
2H29/2
4F3/2
4000
5000
6000
7000
8000
9000
10000
11000
12000
13000
14000
Energy (cm-1)
4
Fig. 1. Electronic absorption spectrum recorded at 4.2 K for polycrystalline UF3. All transitions start from the I9/2 ground state of U3+. The terminal
multiplets are labelled.

M. Karbowiak, J. Droz_dz_yn´ski / Chemical Physics 340 (2007) 187–196
189
4G7/2
4F9/2
2G 17/2+4D5/2
2H211/2+2K13/2
4D3/2+4D1/2
+
4D3/2+2H1
+
11/2
4G9/2
2K15/2
2D1
2L15/2
5/2
4D7/2
2D23/2
2G 19/2 +
2I11/2 +2P1/2
14000
16000
18000
20000
Energy (cm-1)
22000
24000
4
Fig. 2. Electronic absorption spectrum recorded at 4.2 K for polycrystalline UF3. All transitions start from the I9/2 ground state of U3+. The terminal
multiplets are labelled.
3.2. Energy level calculations
where CðqkÞðiÞ are spherical tensors of rank k and Bkq are
crystal-field parameters (CFPs).
The crystal-field (CF) analysis has been performed by
applying the effective operator model [8–10]. For the crys-
tal-field calculations the f-shell empirical program of Reid
(University of Canterbury, New Zealand) [11] has been
used.
In order to facilitate a comparison between the obtained
sets of crystal-field parameters in the different steps of the
calculations, the sk (k = 2,4,6) crystal-field strength param-
eters as well as the total crystal-field strength parameter scf,
defined as [12]
"
"
!#
The eigenvectors and eigenvalues of the crystal-field
levels were obtained by diagonalization of the combined
free-ion and CF energy matrices. The applied Hamiltonian
included the following terms:
1=2
X
2
2
2
sk ¼ ð2k þ 1Þꢀ1 ðB0kÞ þ 2
ððReBkqÞ þ ðImBkqÞ Þ
q>0
#
1=2
X
and scf ¼ ð1=3Þ
sk2
ð4Þ
b
b
b
H ¼ H A þ H CF
where H A contains the isotropic (atomic) parts of H and is
ð1Þ
k
b
b
have been applied.
defined as
The applied fitting procedure encompasses several
stages, which are described in detail in the following chap-
ters. At first we have focused on evaluation of the starting
Bkq parameters. For this purpose we have used rotational
invariants combined with the superposition (SPM) model.
Afterwards the second rank B2q parameters as well as the
B4 and B6 intrinsic parameters were additionally adjusted
by fitting to the experimental energy levels. In the next step
of calculation 15 (a full approach) or 14 (a reduced
approach) CFPs have been fitted to the experimental
energy levels. To exclude a possible influence of the free-
ion parameters on the CFPs a number of additional para-
meters connected with the multiplet centroids have been
applied. This afford possibilities for a discussion on the
influence of free-ion parameters on CFPs, which is pre-
sented in the last section.
X
k
^
b
b
b b
H A ¼ Eave
þ
F ðnf ; nf Þf k þ f5f ASO þ aLðL þ 1Þ
k¼2;4;6
X
i
b
b
^
þ bGðG2Þ þ cGðR7Þ þ
T ti
i¼2;3;4;6;7;8
Pk^pk
X
X
þ
Mjm^j þ
ð2Þ
j¼0;2;4
k¼2;4;6
where the operators and interaction parameters are defined
b
according to the conventional practice [8–10]. The H CF
term of the Hamiltonian represents the one electron crys-
tal-field interactions and is defined in the expended form as
X
X
q
Bk0Ck0 þ
fBqk½Cqk þ ðꢀ1Þ Cꢀk q
ꢁ
b
H CF
¼
k
k;q
q
þ iBkꢀq½Ckq ꢀ ðꢀ1Þ Ckꢀqꢁg
ð3Þ

190
M. Karbowiak, J. Droz_dz_yn´ski / Chemical Physics 340 (2007) 187–196
Table 1 (continued)
Table 1
Calculated (using CFPs set Fit IIIA of Table 3) and experimental energy
levels of UF3
LSJ multipleta
Calculated Experimental Experimental ꢀ
Calculated
LSJ multipleta
Calculated
Experimental
Experimental ꢀ
15,836
15,965
16,027
16,151
16,254
16,392
16,437
16,538
16,613
16,683
16,742
16,939
17,194
17,258
15,861
15,922
16,024
16,147
16,289
16,403
16,445
16,542
16,615
16,680
16,751
16,916
–
25
ꢀ43
ꢀ3
ꢀ4
35
11
8
Calculated
4I9/2
ꢀ10
110
310
633
1032
0
99
10
ꢀ11
–
–
–
–
–
–
4
2
4I11/2
4472
4644
4698
4763
5031
5162
4441
4666
4704
4728
5074
5157
ꢀ31
22
ꢀ3
6
9
ꢀ35
ꢀ23
–
–
43
ꢀ5
9
ꢀ6
ꢀ10
ꢀ8
12
ꢀ11
ꢀ6
20
–
4F3/2
4I13/2
7513
7601
7522
7595
4G9/2
17,382
17,425
17,550
17,601
17,814
–
–
17,413
17,545
17,617
17,828
ꢀ12
ꢀ5
16
8203
8329
8480
8571
8706
8967
9118
8193
8321
8492
8560
8700
8987
9122
14
2G17/2 + 4D5/2
17,930
18,001
18,107
18,212
18,353
18,458
18,567
–
–
17,970
18,101
–
18,364
–
–
ꢀ31
ꢀ6
–
11
–
4
2H29/2
9788
9907
10,047
10,211
10,307
9817
9870
10,032
10,225
10,314
29
ꢀ37
ꢀ15
14
–
2K15/2
18,972
19,054
19,096
19,179
19,379
19,519
19,652
19,709
18,943
19,069
19,100
–
19,369
19,525
19,646
19,729
ꢀ29
15
4
7
4F5/2
10,407
10,508
10,629
10,440
10,501
10,604
33
ꢀ7
–
ꢀ10
ꢀ25
6
4S3/2 + 4G5/2
+ 4F7/2 + 4I15/2
11,234
11,325
11,469
11,613
11,701
11,818
11,908
12,051
12,084
12,171
12,237
12,341
12,407
12,491
12,688
13,033
13,191
11,251
11,317
11,482
11,622
–
11,814
11,906
–
12,098
12,178
–
12,317
12,410
12,502
12,655
13,020
13,198
17
ꢀ8
13
9
ꢀ6
20
4D3/2 + 2H111/2 + 2D15/2 20,103
–
–
20,260
20,283
20,418
20,511
20,630
20,675
20,774
20,879
20,980
21,103
–
–
ꢀ10
–
20,273
20,434
20,487
–
20,686
20,784
20,858
20,964
21,134
ꢀ4
ꢀ2
–
14
7
16
ꢀ24
–
12
10
–
ꢀ21
ꢀ16
30
ꢀ24
3
11
ꢀ33
ꢀ13
7
2I11/2 + 2P1/2 + 2G19/2
21,439
21,625
21,718
21,839
21,988
22,072
22,194
22,309
22,443
22,558
22,723
22,874
21,464
21,590
–
–
–
22,086
22,182
22,309
22,424
22,580
–
25
ꢀ35
–
–
–
14
ꢀ12
0
ꢀ20
22
–
4G7/2
14,112
14,215
14,263
14,403
14,108
14,205
14,301
14,379
ꢀ4
ꢀ10
38
ꢀ24
2
–
4F9/2
15,158
15,269
15,325
15,391
15,557
15,160
–
15,302
15,409
15,562
ꢀ23
18
–
–
5
2D23/2
22,982
23,193
22,979
23,199
ꢀ3
2H211/2 + 2K13/2
+ 4D3/2 + 4D1/2
15,677
15,792
–
–
ꢀ22
6
15,770

M. Karbowiak, J. Droz_dz_yn´ski / Chemical Physics 340 (2007) 187–196
191
X
Table 1 (continued)
1
2
2
ðr2ðaSLJÞÞ ¼
s2khaSLJkCðkÞkaSLJi
ð6Þ
LSJ multipleta
Calculated
Experimental
Experimental ꢀ
2J þ 1
k¼2;4;6
Calculated
4D7/2 + 2L15/2
23,627
23,726
23,876
24,021
24,110
. . .
23,621
–
–
24,041
24,116
ꢀ6
–
–
20
6
From these equations, by minimizing the differences be-
tween the experimental and calculated values of the r2 qua-
dratic moments in the least squares fits, the values of the
rotational invariants sk have been determined. The applied
reduced matrix elements haSLJkC(k)kaSLJi were computed
using free-ion parameters, obtained in a fitting procedure
of the above mentioned 13 multiplets. For the calculation
the free-ion part of the Hamiltonian formulated by Eq.
(2) has been employed.
a
Nominal spectroscopic symbols for the energy states associated with
the group of CF levels (main component of eigenvectors).
3.2.1. Evaluation of starting values of the Bkq parameters
In the reported absorption spectra of the U3+ ion nearly
all well separated SLJ multiplets are observed up to the
2H211/2 level only. Above this multiplet, i.e. over ca.
16,000 cmꢀ1, the CF components of the multiplets are to
a large extent over mixed or superimposed by strong
5f3 ! 5f26d1 bands and may not be straightforwardly
assigned. For that reason they may be taken into account
for the determination of the free-ion parameters to a lim-
ited degree only. However, we have shown [13] that even
on the basis of a truncated, but properly assigned set of
experimental CF levels one can quite well predict the posi-
tions of the problematic levels. Hence, in order to receive
reliable values of the starting parameters we have first
taken into consideration only 13 well assigned SLJ multi-
plets, observed in the 0–16,000 cmꢀ1 spectral range and
specified in Table 2.
The s2k values, estimated from Eq. (6) using reduced
matrix elements obtained in the intermediate coupling
scheme, were subsequently applied for the evaluation of
the J-mixing coefficients, a(wJ,w0J0). The calculations were
performed using the formulas derived in Ref. [16]
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
2
0
ðkÞ
0
X
hwJjjU jjw J i
2
jaðwJ; w0J0Þj ¼
sk
ð7aÞ
ð7bÞ
2
0
0
EðwJÞ ꢀ Eðw J Þ
k¼2;4;6
2
3
5
2
0
0
X
jaðwJ; w J Þj
2J þ 1
2
4
jaðwJ; wJÞj ¼ ð2J þ 1Þ 1 ꢀ
w0J0
The new values of the matrix elements, obtained by taking
into account the J-mixing effect were used for the determi-
nation of the following final values of the sk rotational
invariants: s2 = 372 cmꢀ1
,
s4 = 1094 cmꢀ1 and s6 =
1540 cmꢀ1. The r2 quadratic moments calculated for these
values are listed in Table 2.
For the calculations, a coordination system in which the
z and y axes are parallel to the a and c axes of the crystal,
respectively, has been chosen. The coordination system was
subsequently rotated by 30° around the y axis in order to
The quadratic rotational invariants, s2k, are related to the
superposition model (SPM) intrinsic parameters, Bk, by the
Eq. (8) taken from Ref. [14]
˚
place the fluorine atom, observed at a distance of 2.42 A
X
1
s2k ¼
BkðRiÞBkðRjÞCðkÞðxijÞ
ð8Þ
from the uranium atom, on the z axis. In this way the C2
axis coincides with the z axis and the C2 symmetry of the
uranium site has been clearly displayed. Such an orienta-
tion of the coordination system corresponds also to that
reported in Ref. [14] for Er3+:LaF3 single crystals.
The starting values of the CF parameters have been
obtained on the basis of the procedure given in Ref. [14].
For the calculation the derived [14,15] relationships
between the Bkq crystal-field parameters and the sk crystal-
field invariants of Eq. (4) as well as the so-called ‘‘quadratic
crystal-field splitting moments’’, r2, have been applied. The
quadratic moments, r2, were evaluated from the equation
(Ref. [15])
0
2k þ 1
i;j
~
where the angle, xij, between the direction Ri ¼ ðRi; hi; uiÞ
~
of the ligand i and that of Rj ¼ ðRj; hj; ujÞ of the ligand j,
is given by
cos xij ¼ sin hi sin hj cosðui ꢀ ujÞ þ cos hi cos hj
ð9Þ
This equation together with the distance dependence
assumed in the form of
tk
BkðRÞ ¼ BkðR0ÞðR0=RÞ
ð10Þ
enabled to plot jBk=skj as a function of the power low expo-
nents, tk, provided that the polar coordinates of the ligands
are known.
X
1
2
ðr2ðaSLJÞÞ ¼
½EiðaSLJÞ ꢀ E0ðaSLJÞꢁ
ð5Þ
Similar lattice constants for the isostructural lanthanum
and uranium trifluorides point out that also the positions
of the fluoride ligands in UF3 should be very close to those
observed in LaF3. Consequently one may presume for both
compounds a very similar jBk=skj versus tk dependence.
Those for Er3+:LaF3 are presented in Figs. 1–3 of Ref.
[14]. On the basis of these curves, assuming that the tk
power low exponents are sited in the 5–15 range, the values
of the intrinsic parameters B2, B4 and B6 for the U–F
2J þ 1
i
where E0 is the mean energy of the appropriate aSLJ mul-
tiplet and Ei(aSLJ) is the energy of the corresponding crys-
tal-field component (i) of this multiplet. The experimental
r2 values are listed in column 3 of Table 2.
The quadratic r2 moments of the multiplets may be
expressed in terms of the sk rotational invariants by Eq.
(6), where k = 2, 4 or 6 [15]:

192
M. Karbowiak, J. Droz_dz_yn´ski / Chemical Physics 340 (2007) 187–196
99 cm-1
100 K
4.2 K
11100
11150
11200
11250
Energy (cm-1)
11300
11350
11400
99 cm-1
100 K
4.2 K
9600
9700
9800
9900
10000
10100
Energy (cm-1)
Fig. 3. A comparison of the UF3 absorption spectra measured at 4.2 and 100 K. The hot band displaced by 99 cmꢀ1 from the low temperature lines is
indicated.
˚
distance of R = 2.42 A could have been evaluated. The B4
and B6 parameters may assume values enclosed in relatively
In order to narrower the ranges of values of the intrinsic
parameters obtained by means of the rotational invariants
sk, the SPM parameters have been adjusted by carrying out
direct fits to the experimental data. At this SPM stage we
have included 52 experimental energy levels observed in
the 0–16,150 cmꢀ1 spectral range. The energy levels were
calculated by simultaneous diagonalization of the com-
bined free-ion and CF interaction matrices. The relative
values of the fourth- and six-rank Bkq parameters do not
narrow ranges of 1620–1860 cmꢀ1 and 1220–1720 cmꢀ1
,
respectively, presuming that the s4 and s6 values were deter-
mined with an accuracy of 50 cmꢀ1. Since the jB2=s2j
ratio exhibits a considerably stronger dependence from tk
values and taking into account also a presumed accuracy
of 100 cmꢀ1 in the determination of s2, the expected value
for B2 should be limited to the 1700–4330 cmꢀ1 range.

M. Karbowiak, J. Droz_dz_yn´ski / Chemical Physics 340 (2007) 187–196
193
2H2
16000
14000
12000
10000
8000
6000
4000
2000
0
11/2
4F9/2
4G7/2
4F7/2
4I15/2
4S3/2
4G5/2
4F5/2
2H2
9/2
4I13/2
4F3/2
4I11/2
4I9/2
SLJ
SLJ
CF
CF
U3+
:LaCl3
UF3
Fig. 4. Schematic representation showing energy levels of the U3+ ion in UF3 and U3+:LaCl3 single crystals.
Table 2
varied. In order to test the possible existence of different
minima in the space of the fitting results a number of cal-
culations with various starting values of the B2q parameters
within ranges limited by B2 ¼ 1700–4330 cmꢀ1 and t2 = 5–
15, have been performed. The fits have shown that indepen-
dently from the starting values of the B2q parameters the
same results, within the parameters uncertainties, have
been obtained. A number of fits have been carried out also
with t4 and t6 values in the 5–15 range, which correspond to
various distance dependences of the B4(RL) and B6(RL)
parameters in accordance with Eq. (10). From these fits
to a set of 52 experimental data the following parameter
values have been obtained (in cmꢀ1): B02 ¼ ꢀ544 ꢂ 50,
ReB22 ¼ ꢀ147 ꢂ 20, ImB22 ¼ ꢀ420 ꢂ 165, B4ðR0 ¼ 2:42 Þ ¼
1695 ꢂ 40 and B6ðR0 ¼ 2:42 Þ ¼ 1440 ꢂ 180. The above
specified parameter ranges arise from the applied assump-
tion concerning the location of the t4 and t6 power low
exponents in the 5–15 range as well as from the uncertain-
ties of the parameter values. The fitting enabled a further
narrowing of the possible B4 and B6 parameter value ranges
as compared to those of 1620–1860 cmꢀ1 and 1220–
1720 cmꢀ1, respectively, predicted by the values of the sk
rotational invariants.
Barrycenters, experimental and fitted r2 quadratic moments for the first 13
low-lying multiplets of UF3 (values in cmꢀ1
)
SLJ multiplet
Barrycentera
r2 (experimental)
r2 (calculated)
4I9/2
415
4795
7559
389
246
36
313
193
68
417
180
35
271
261
51
4I11/2
4F3/2
4I13/2
2H29/2
4F5/2
4S3/2
4G5/2
4I15/2
4F7/2
4G7/2
4F9/2
2H211/2
a
8625
10,052
10,515
11,746
11,757
12,225
12,297
14,248
15,340
15,881
70
52
316
674
165
102
107
186
314
621
182
155
150
237
Calculated from experimental energy levels; in cases when experi-
mental level was not observed the calculated value (Table 1) was taken
into account.
depend strongly on the values of the B4 and B6 SPM
parameters and the power low exponents tk. By contrast,
the second-order B2q parameters may even change their sign
if one varies the B2 value in the 1700–4330 cmꢀ1 range and
shift t2 from 5 to 15. Therefore, since the t2 value is
unknown one may not be able to estimate the reliable ini-
tial values of the B2q parameters, even with the relatively
well determined B2 value. Hence, in the fit to the experi-
mental energy levels the B4 and B6 intrinsic parameters as
well as the B20, ReB22 and ImB22 CF parameters were freely
A fitting procedure of the B2q, B4 and B6 parameters to all
94 observed energy levels shows no essential differences
between the obtained parameter values and those per-
formed with the truncated set of experimental levels. For
the B4 and B6 parameters the differences remain in the lim-
its of the fitting errors and those of B2q were also insignifi-
cantly small. It is worth to notice that this ‘‘5 parameter’’

194
M. Karbowiak, J. Droz_dz_yn´ski / Chemical Physics 340 (2007) 187–196
Table 3
model describes 94 experimental CF energy levels of UF3
with a relatively low r.m.s. deviation of 30–33 cmꢀ1
.
Received values of the crystal-field parameters in the performed fitting
procedures
CF
Fit Ib Fit
IIc,d,e,f
Fit
Fit
Fit
Fit
3.2.2. Fitting of Bkq parameters
The starting values of the CF parameters were estimated
by means of the above determined superposition model
parameters and the following equation (Ref. [17]):
parametera
IIIc,d,f,g IVe,f,h
IIAd,e,f,i IIIAd,f,i,j
B02
ꢀ544
ꢀ147
ꢀ420
445
ꢀ813
144
ꢀ812
301
ꢀ902
153
ꢀ833
136
ꢀ833(45)
272(36)
0
ReB22
ImB22
jB22j
B04
ꢀ265
0
ꢀ199
ꢀ236
X
tk
302
301
251
272
272
Bkq ¼
BkðR0ÞðR0=RLÞ gk;qðhL; uLÞ
ð11Þ
1610
1172
ꢀ226
1194
1235
1924
636
1923
ꢀ24
737
737
531
1860
730
1817
765
1817(89)
186(106)
776(82)
798
523(104)
1460(58)
1551
1081(105)
ꢀ841(89)
ꢀ2459(74)
2599
ꢀ665(88)
ꢀ1024(74)
1221
L
ReB42
ImB42
jB42j
ReB44
ImB44
jB44j
B06
372
737
1055
321
797
838
226
798
1001
where the gk,q coordination factors were calculated by tak-
ing into account the positions of the ligands L ꢃ Fꢀ.
On the basis of the obtained values for B4 ¼ 1695 cmꢀ1
and B6 ¼ 1440 cmꢀ1 the values for Bq4 and B6q have been
generated by means of Eq. (11) (see Table 3, Fit I). As a
matter of fact one may also obtain somewhat different Bq4
and B6q parameters, depending on the tk value and by taking
into account the possible range of changes of the intrinsic
B4 and B6 parameter values. However, due to a small dis-
tance dependence the differences are not large and, as we
have checked, do not cause the obtaining of an other min-
imum in the fitting procedure.
In the investigated case the Bkq parameters are complex.
In Table 3 we are separately presenting the real ðReBkqÞ
and imaginary ðImBkqÞ values of Bkq. However, since the
relative values of ReBqk and ImBkq depend on the orienta-
tion of the x and y axes we have also included in Table 3
ꢀ586 ꢀ1318
1603
1689
1061
ꢀ822
ꢀ1085 ꢀ1184
1367
1826
1688
1061
1371
1111
1550
1081
ReB62
ImB62
jB62j
ReB64
ImB64
jB64j
ReB66
ImB66
jB66j
s2
ꢀ2438 ꢀ2589
ꢀ2541 ꢀ2551
ꢀ139
2442
ꢀ308
1166
1206
ꢀ471 ꢀ2500
ꢀ426
2576
ꢀ538
1232
1344
ꢀ499
2599
ꢀ552
1089
1221
2631
2632
ꢀ498
ꢀ667
1026 ꢀ1118
1224
1224
1498
ꢀ1361 ꢀ1373
ꢀ1444 ꢀ1412
1416(56)
ꢀ1682(93)
2199
1612
2110
372
1010
1443
52
1647 ꢀ1534
1622
2171
433
971
1456
94
1685
2198
410
1021
1450
94
2144
411
1079
1445
94
2144
410
1080
1446
94
410
1021
1450
94
s4
s6
n
r.m.s.
a
–
24
24
33
24
24
2
2 1=2
the jBkqj ¼ ½ðReBqkÞ þ ðImBkqÞ ꢁ
values, which are con-
2
2 1=2
jBkqj ¼ ½ðReBqkÞ þ ðImBkqÞ ꢁ
.
b
stant magnitudes, independent from the rotation of
the coordination system around the z axis, and so facilitate
a comparison of parameters, obtained in the various
fittings.
The B4q and B6q parameters have been calculated on the basis of Eq. (11)
from the fitted SPM intrinsic parameters: B4ðR0 ¼ 2:42 Þ ¼ 1695 cmꢀ1 and
B6ðR0 ¼ 2:42 Þ ¼ 1440 cmꢀ1 for t4 = 10 and t6 = 10. The values of B2q CF
parameters were obtained in the direct fitting to 94 experimental energy
levels (see text).
c
Values of the free-ion parameters were taken from Ref. [13] (in cmꢀ1):
Thus, for the investigated system of C2 point group sym-
metry, in which the twofold axis coincides with the z axis,
there exists a continuous minimum in the space of solutions
and the ReBkq and ImBkq parameters may assume any values
F2 = 38,269, F4 = 30,530, F6 = 19,770,
f5f = 1612, a = 31, b = ꢀ886,
c = 2059, T2 = 394, T3 = 48, T4 = 169, T6 = ꢀ192, T7 = 358, T8 = 300,
M0 = 0.672, M2 = 0.376, M4 = 0.255, P2 = 1579, P4 = 1184, P6 = 790.
d
Calculated with additional parameters which independently shifted the
2
2
provided that the sum of ðReBkqÞ þ ðImBqkÞ keeps a con-
stant value. The imaginary part of one of the crystal-field
parameters (usually ImB22Þ can be set to zero by an appro-
priate coordination rotation about the z axis. This method
corresponds to the ‘‘reduced’’ approach of Rudowicz [20].
However, the values of the second-order parameters
strongly depend on those of B4q and Bq6, which so far have
been calculated from the fitted B4 and B6 SPM parameters
and may markedly differ from the final ones. For that rea-
son, at this stage, the B2q parameters obtained in the hith-
erto fitting procedures could not be used for the
orientation of the axis system, e.g. by establishing the zero
value for ImB22. Incorrectness in the evaluation of the
parameters at the superposition model (SPM) stage could
cause a manifold increase of errors in the estimation of
the off-diagonal fourth- and sixth-order parameters.
Hence, in the next steps all 15Bkq parameters were allowed
to vary freely along with the earlier mentioned additional
‘‘centroid’’ parameters. All CF levels observed in the 0–
24,116 cmꢀ1 spectral range have been gradually included
in the fitting procedure.
centroid of each free-ion multiplet to the initially determined place.
e
In the fitting procedure 15 CF parameters have been freely varied.
f
The fitting procedure has been performed to 94 crystal-field levels
located in the 0–24,116 cmꢀ1 spectral range.
g
The fitting procedure has been performed with starting parameters
listed in column Fit II, after the rotation of the coordination system about
the z axis by an angle of 30.7°; 14 CF parameters were freely varied.
h
As starting parameters have been applied those listed in column Fit II;
eight free-ion and 15 crystal-field parameters were allowed to freely vary
(the Ti, Mj and Pk parameters were kept at constant values taken from
Ref. [13] – see footnote c). The following free-ion parameters have been
obtained: Eavg = 20,194(6), F2 = 39,074(67), F4 = 31,670(193), F6 =
19,092(181), f5f = 1639(3), a = 26.8(0.6), b = ꢀ815(32) and c = 1973(44)
and (all values in cmꢀ1).
i
The free-ion parameters listed in the footnote h have been applied.
As starting parameters have been applied those listed in column Fit
j
IIA, after the rotation of the system about the z axis by an angle of 30.0°;
14 crystal-field parameters were freely varied.
The parameters given in the column denoted as Fit I in
Table 3 were applied as starting values in the fittings. The
results are listed in column Fit II. The axial parameters
exhibit significant changes; the absolute values of B20 and

M. Karbowiak, J. Droz_dz_yn´ski / Chemical Physics 340 (2007) 187–196
195
B40 have distinctly increased whereas that of B60 is twice
smaller. The ReB22 and ImB24 parameters have changed
their signs, but their absolute values are small. In order
to carry out a comparison between the obtained set of
CF parameters (Fit II) with those obtained from the
SPM phase (Fit I) one may treat the Bkq parameters as com-
ponents of the Bk vector: Bk ¼ fBkꢀq; . . . ; B0k; . . . ; Bkqg
[21,22]. By this means a comparison between the lengths
of the Bk(i) and Bk(j) vectors as well as of the angles
between these two vectors, obtained for the (i) and (j) sets
of parameters, is possible. The lengths of the vectors are
directly connected with the rotational invariants sk of Eq.
(4) and enable thereby a comparison of the crystal-field
strengths for different sets of CF parameters, whereas the
angle may be expressed as [21]
3.2.3. Influence of free-ion parameters
In the hitherto stages of the calculation the energy levels
were calculated by simultaneous diagonalization of the
combined free-ion and CF interaction parameters. How-
ever, in such a procedure the obtained free-ion parameter
values may not be necessarily the same as those resulting
from a fitting of the SLJ multiplets merely. It has been
proved that the Fk Slater parameters strongly depend on
the type of ligand and they may also in a part absorb the
crystal-field effect. One may, of course, as well expect a
reverse interaction of the free-ion parameters on the values
of crystal-field parameters. Such effects have been theoret-
ically predicted [18,19] and may be easily proved by appro-
priate simulation calculation. Hence, not fully correct
values of the free-ion parameters may to some extent per-
turb the CF values.
In order to exclude a possible influence of inaccuracy in
the setting of the free-ion multiplets on CFPs, we have first
fixed the free-ion parameters on the well determined values
for U3+:LaCl3 reported in Ref. [13] (footnote c in Table 3).
Afterwards, with each of SLJ free-ion multiplet included in
the combined energy matrixes, an additional freely varied
parameter has been associated in order to shift the centroid
of every multiplet, independently on each other, in such a
way that their positions could be precisely adjusted. Then
a reliable fitting of the experimental CF energy levels could
be achieved.
In the next step we have performed an attempt to
exclude the additional ‘‘centroid’’ parameters in a fit in
which as well the free-ion and CF parameters were allowed
to freely vary. A fitting procedure with 15 Bkq CF parame-
ters (with starting values as in Fit II) and eight free-ion
parameters Eave, Fk (k = 2,4,6), a, b, c and f5f has shown,
however, that for the 2G17/2, 2D15/2, 2I11/2, 2D23/2 and 2L15/2
multiplets the application of the additional freely varied
‘‘centroid’’ parameters was still indispensable. The
obtained r.m.s. deviation of 33 cmꢀ1 was distinctly larger
in comparison with the former results (24 cmꢀ1), but still
reasonably good as for a U3+ ion. The obtained values of
the free-ion parameters are presented in footnote h of
Table 3. As one should expect for the less covalent fluoride
environment, the values of the F2, F4 and 15f parameters are
larger, and those of the two-body interaction parameters
are smaller, in comparison to the corresponding values
(footnote c of Table 3) for U3+:LaCl3 single crystals [13].
However, that of F6 is somewhat smaller.
The obtained Bkq crystal-field parameters are presented in
the column Fit IV of Table 3. With the exception of jB44j
and jB64j the difference between these values and those
obtained in Fit II is less than 100 cmꢀ1. As was to be
expected we have now also obtained a somewhat different
ImB22=ReB22 ratio. The / angle, required for the setting of
the ImB22 parameter to zero, is now equal to 26.2°. Never-
theless, one may state that the CF parameter values,
obtained with the application of the additional ‘‘centroid’’
parameters for all multiplets (Fit II), should be treated as
more reliable.
BkðiÞ ꢄ BkðjÞ
1=2
cos hij ¼
where jBkj ¼ ðBk ꢄ BkÞ
:
ð12Þ
jBkðiÞj ꢄ jBkðjÞj
From this relation it follows that with the increase of the
agreement between both sets, coshij approaches to unity.
A comparison of the results obtained in the Fit I and Fit
II shows that the s2 and s4 values have increased by 10%
and 7%, respectively, whereas that of s6 remains almost
unchanged. The cosines of the hij angle between the B4
and B6 vectors, obtained from Fit I and Fit II, are equal
to 0.906 and 0.970, respectively. Thus, one may state that
the correlation between the two sets of parameters is satis-
factory as well that the observed differences result rather
from the fitting process of the Bkq parameters, in which they
have been allowed to vary independently in comparison to
that one in which Bkq were expressed by the intrinsic param-
eters, and not by a change of the orientation of the coordi-
nation system. In the consequence one may assume that the
coordination system in which fitted CF parameters (Fit II)
are obtained holds approximately the same orientation as
the crystallographic axis system [14] used for calculation
of Fit I parameters in the SPM stage.
In order to fix the arbitrary orientation angle of the z
axis implicitly we have applied the commonly used proce-
dure in which one of the parameters, usually ImB22, is set
to zero. For this purpose the coordination system of the
fitted parameters has been rotated around the z axis by
an angle / of 30.7°, calculated by using the expression
tanð2/Þ ¼ ImB22=ReB22 [23]. The obtained parameters have
been used as starting values in the next step of calculations,
similar to that of Fit II but in which 14 CF parameters were
freely varied and ImB22 was set to zero. The obtained values
(Fit III) differ by less than 2 cmꢀ1 from those of the starting
values. The calculated centers of gravity of all SLJ multi-
plets are shifted towards higher energies from 30 cmꢀ1
4
2
for I9/2 to 1188 cmꢀ1 for G17/2 in comparison to those
obtained on the basis of free-ion parameters reported for
U3+:LaCl3 [13]. The magnitude of the shift, induced by
the ‘‘centroid’’ parameters, corresponds to that observed
between the experimental SLJ multiplets of UF3 and
U3+:LaCl3.

196
M. Karbowiak, J. Droz_dz_yn´ski / Chemical Physics 340 (2007) 187–196
In order to check the influence of the free-ion para-
parameters) and IV (with varied free-ion parameters) are
equal to approximately 320 and 360 cmꢀ1, respectively.
The rotational invariants, sk, reveal practically constant
values for s6 and very similar for s4, with the largest differ-
ences of 0.9% and 10.6%, respectively. Those of s2 differ by
not more than 61 cmꢀ1 (15.2%).
From a comparison of the final parameter values (Fit
IIIA) with those calculated on the basis of the SPM param-
eters (Fit I) one may notice an increase of the absolute val-
ues of B20 and a decrease of that of B60 as well as a good
agreement between the sk values. The largest difference
has been observed for s2, equal to 38 cmꢀ1 (9.3%). A good
agreement between the sk values obtained in the final fitting
procedure and by means of the quadratic r2 moments of
the multiplets has also been observed.
meters on the CF parameters, we have repeated the above
steps of calculations by applying the freely varied ‘‘cen-
troid’’ parameters and the free-ion parameters fixed at
the obtained, new values. The results are shown in column
Fit II A of Table 3. One may state no essential differences
in the computed CF and sk values. The new s2 and s6 values
are almost the same as before (Fit II) whereas s4 is some-
what smaller. The orientation of the coordination system
remains almost unchanged too. The cosinuses of the hij
angles between the Bk(i) ꢃ Bk (Fit II) and Bk(j) ꢃ Bk (Fit
IIA) vectors are equal to 0.999, 0.996 and 0.999 for
k = 2, 4 and 6, respectively. In order to fix the value of
ImB22 at zero the system has to be rotated around the z axis
by / = 30.0°, which is close to that of 30.7° used in Fit II.
Hence, one may state that the applied procedure enabled
the calculation of reliable CF parameters, not affected by
possible inaccuracies of the free-ion parameter values.
The so obtained parameters have been used, after the
rotation of the system by an angle of 30.0°, as starting val-
ues in the final fitting procedure to 94 experimental crystal-
field levels observed in 0–24,116 cmꢀ1 energy range. The
results are presented in Table 3 (Fit IIIA) and the experi-
mental and calculated energy levels are gathered in Table
1. The CFP set Fit IIIA may be considered as the final
result of our CF analysis for the U3+ ion in UF3.
One may conclude that the proposed calculation proce-
dure enabled the determination of good starting values,
reliable crystal-field parameters and
deviation.
a small r.m.s.
References
[1] P.P. Sorokin, M.J. Stevenson, Phys. Rev. Lett. 5 (1960) 557.
[2] G.D. Boyd, R.J. Collins, S.P.S. Porto, A. Yariv, W.A. Hargreaves,
Phys. Rev. Lett. 8 (1962) 269.
[3] S. Hubert, E. Simoni, M. Louis, W.P. Zhang, J.Y. Gesland, J. Lumin.
65–61 (1994) 245.
[4] E. Simoni, M. Louis, J.Y. Gesland, S. Hubert, J. Lumin. 65 (1995)
153.
[5] H. Schmieder, E. Dornberger, B. Kanelakopulos, Appl. Spectrsc. 24
(1970) 499.
4. Conclusions
For the first time a low temperature absorption spec-
trum and crystal-field analysis of uranium trifluoride are
presented. In comparison to the spectra of the U3+ ion in
other environments a much larger splitting of the SLJ mul-
tiplets, including the so far largest value of the 4I9/2 ground
level splitting, has been noticed. The calculated total crys-
tal-field strength parameter scf = 1051 cmꢀ1 is as expected
more than two times larger than that for NdF3
(scf = 404 cmꢀ1) [24]. The calculated value of 1032 cmꢀ1
for the 4I9/2 ground level splitting is also in agreement with
the expected one for U3+ ions in a fluoride environment.
In the performed analysis 94 experimental crystal-field
levels, observed in 0–24,116 cmꢀ1 energy range, have been
identified and fitted to a semiempirical Hamiltonian repre-
senting the combined atomic and crystal-field interactions
at the C2 site symmetry. The inclusion of a number of
parameters associated with the centroids of the free-ion
multiplets has excluded a possible influence of inaccuracy
in the description of the free-ion levels on the crystal-field
parameters.
[6] J.C. Taylor, Coord. Chem. Rev. 20 (1976) 197.
_
_ ´
[7] J. Drozdzynski, Coord Chem. Rev. 249 (2005) 2351.
[8] B.G. Wybourne, Spectroscopic Properties of Rare Earths, Inter-
science, New York, 1965.
[9] W.T. Carnall, H. Crosswhite, H.M. Crosswhite, J.P. Hessler, N.M.
Edelstein, J.G. Conway, G.V. Shalimoff, R. Sarup, J. Chem. Phys. 72
(1980) 5089.
[10] H.M. Crosswhite, H. Crosswhite, J. Opt. Soc. Am. B 1 (1984) 246.
[11] Y.Y. Yeung, M.F. Reid, D.J. Newman, Appendix 3: accessible
program packages, in: D.J. Newman, B. Ng (Eds.), Crystal Field
Handbook, Cambridge Univ. Press, 2000, p. 254.
[12] N.C. Chang, J.B. Gruber, R.P. Leavitt, C.A. Morrison, J. Chem.
Phys. 76 (1982) 3877.
_
_ ´
[13] M. Karbowiak, J. Drozdzynski, M. Sobczyk, J. Chem. Phys. 117
(2002) 2800.
[14] Y.Y. Yeung, D.J. Newman, J. Chem. Phys. 82 (1985) 3747.
[15] Y.Y. Yeung, Invariants and Moments, in: D.J. Newman, B. Ng
(Eds.), Crystal Field Handbook, Cambridge Univ. Press, 2000, p. 160.
[16] X. Shangda, C. Yimin, J. Lumin. 31–32 (1984) 204.
[17] D.J. Newman, B. Ng, Superposition model, in: D.J. Newman, B. Ng
(Eds.), Crystal Field Handbook, Cambridge Univ. Press, 2000, p. 83.
[18] K. Rajnak, B.G. Wybourne, J. Chem. Phys. 41 (1964) 565.
[19] S.S. Bishton, D.J. Newman, J. Phys. C 3 (1970) 1753.
[20] C. Rudowicz, J. Chem. Phys. 84 (1986) 5045.
[21] C. Rudowicz, J. Qin, J. Lumin. 110 (2004) 39.
[22] M. Faucher, D. Garcia, Phys. Rev. B 26 (1982) 5451.
[23] J. Mulak, Physica B 348 (2004) 362.
[24] E. Rukmini, C.K. Jayasankar, M.F. Reid, J. Phys.: Condens. Matter
6 (1994) 5919.
A comparison of the B0k and jBkqj parameters obtained in
the different fitting procedures (Fit I–Fit IV) reveals a good
agreement with differences not larger than 120 cmꢀ1. An
exception are the jB44j and jB62j parameter values for which
the differences between the Fits III (with varied ‘‘centroid’’