The effects of crystal-fields on the optical properties of Pr:ZBLAN glass

Journal of Non-Crystalline Solids 325 (2003) 282–294

www.elsevier.com/locate/jnoncrysol

The effects of crystal-fields on the optical propertiesof Pr:ZBLAN glass

Hiroyuki Inoue a,*, Kohei Soga b, Akio Makishima c

a Department of Materials Engineering, School of Engineering, The University of Tokyo, 7-3-1, Hongo,

Bunkyo-ku, Tokyo 113-8656, Japanb Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, 7-3-1 Hongo,

Bunkyo-ku, Tokyo 113-8656, Japanc Center for New Materials, Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Tatsunokuchi,

Nomi, Ishikawa 923-1292, Japan

Received 15 April 2002

Abstract

Absorption and emission spectra of Pr3þ-doped ZBLAN glass at room temperature were estimated from even- and

odd-parity components of crystal-field parameters based on crystal-field theory and Judd–Ofelt theory. On the basis of

point-charge approximation, the crystal-field parameters were obtained from structural models prepared by using

molecular dynamics (MD) simulation. The calculated oscillator strengths for the 1D2 and3PJ states were insufficient for

the reproduction of the observed oscillator strengths for those states. These spectra were also estimated on the basis of

the modified Judd–Ofelt theory, in which the variation of the energy difference between the 4f2 and the excited 4f15d1 or

4f15g1 configurations were taken into account. It was found that the oscillator strengths increased and the values were

comparable to the observed ones. The effect of the even-parity components of the crystal-field on the splitting of the

energy levels and the rate of the forced electric-dipole transition were investigated. The contribution of the crystal-field

and the energy variation of the 4f2 configurations on the Judd–Ofelt intensity parameters was evaluated.

� 2003 Elsevier B.V. All rights reserved.

1. Introduction

A understanding of the optical properties of the

Pr3þ ion in glasses is of great importance due to its

potential technological applications in optical

* Corresponding author. Present address: Institute of Indus-

trial Science, The University of Tokyo, 4-6-1, Komaba,

Meguro-ku, Tokyo 153-8505, Japan. Tel.: +81-3 5452 6315;

fax: +81-3 5452 6316.

E-mail address: [email protected] (H. Inoue).

0022-3093/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/S0022-3093(03)00316-8

amplifiers [1–5] and fiber lasers [6,7]. ZBLAN glass

seems to be a particularly useful host material for

rare earth ions due to its optical qualities, chemical

stability and low phonon energy. Since lower

phonon energy of the host material provides more

efficient emission, there are more alternatives for

the emission levels in ZBLAN glass than in con-

ventional oxide glasses.The Judd–Ofelt theory [8,9] is a widely used and

remarkably successful theory for quantitatively

characterizing optical properties of rare earth ions

in glasses. Negative values of the phenomenological

ed.

mail to: [email protected]

H. Inoue et al. / Journal of Non-Crystalline Solids 325 (2003) 282–294 283

X2 intensity parameter of the Judd–Ofelt theoryfor the Pr3þ ion have been reported [10–12]. The

results are in contradiction to the definition of the

Xk parameters. It has also been indicated that the

prediction of the oscillator strength for the Pr3þ

ions on the basis of the Judd–Ofelt theory is in

poor agreement with the experimental results. For

example, the oscillator strengths from the ground

state (3H4) to the 1D2 and3P2 states were only 49%

and 36% of the experimental values [11]. Quimby

and Miniscalco [11] introduced a modified Judd–

Ofelt treatment which incorporated the measured

fluorescence branching ratios into the evaluation.

This procedure shows that the X2 changed from

negative values to positive ones. Kornienko et al.

[13] proposed a different modification for the

Judd–Ofelt treatment in which the dependence onthe energies of the 4f states was taken into ac-

count. Medeiros et al. [14] reported the improve-

ment between the estimated and measured

oscillator strengths and the radiative lifetimes ac-

cording to the modified Judd–Ofelt treatment of

Kornienko.

The calculation of the energy levels of rare earth

ions in glasses has been proposed [15] and devel-oped [16,17]. We have investigated the structure

analysis and performed a computer simulation of

the atomic structure of the glass based on ZrF4

[18,19] and have described the optical properties

of Eu3þ, Er3þ and Tm3þ ions in ZBLAN glass [20–

23]. The evaluation of the crystal-field was useful

for the description and prediction of the optical

properties of the rare earth ions in the glass. Wehave reported that the splitting of the energy level

and the transition rate between the energy levels of

4f electrons in the rare earth ions could be calcu-

lated from the crystal-field parameters derived

from the structural models, which were prepared

from molecular dynamic simulations. The shapes

of absorption and emission spectra of Eu3þ, Er3þ

and Tm3þ ions in the ZBLAN glass were repro-ducible. It was found that the relation between the

splitting of the energy level and the crystal-field

parameter is different at each energy level. There-

fore, the understanding of the relation is important

for not only the analysis of the structure around

the rare earth ions but also the design of the op-

tical properties.

In the present paper, we estimate the absorptionand emission spectra of the Pr3þ ion in ZBLAN

glass by considering the crystal-field derived from

the structural models and describe the effect of the

crystal-field parameters on the splitting of the 4f

energy levels and transition rates between them.

We also describe the values of the oscillator

strength calculated from the modified Judd–Ofelt

treatment in which the crystal-field and the energyvariation of the 4f2 states were taken into account.

2. Experimental conditions

2.1. Glass preparation

Glass with a composition of 52ZrF4 Æ 20BaF2 Æ3.5LaF3 Æ3AlF3 Æ20NaF Æ 0.5InF3 Æ 1PrF3 (molar ra-

tio) was prepared. The powders were mixed and

melted in a gold crucible at 900 �C for 15 min. The

melt was cast into a preheated aluminum mold. All

of these processes were conducted in a glove box

under an atmosphere of dry nitrogen gas. The

glass obtained was cut into a 10 · 25 · 5 mm shape

and polished.Absorption spectrum was measured with a self-

recording spectrophotometer (U3410, Hitachi) in

the wavelength range of 200–2600 nm. Fluores-

cence spectrum was measured by 479 nm excita-

tion using an OPO laser. The emission from the

sample was focused on the entrance slit of the

spectrometer and detected with a photomultiplier

tube. All of the measurements were carried outat room temperature.

2.2. Molecular dynamics simulation of Pr3þ-doped

ZBLAN glass

The structure of Pr3þ-doped ZBLAN glass was

simulated using a molecular dynamics (MD)

method. Three hundred ninety three ions (Zr4þ:53,Ba2þ:20, La3þ:3, Al3þ:3, Naþ:20, Pr3þ:1, F�:293)

were placed randomly in a cubic cell with periodic

boundary conditions. A cell parameter of 1.759 nm

was determined from the experimental density of

the glass. Simulations were carried out at a con-

stant volume. Born–Mayer type potentials were

used with formal ionic charges, and the parameters

Table 1

Potential and parameters Aij (10�16 J) used in MD simulations Born–Mayer potential Uij ¼ ðe2=4pe0ÞðZiZj=ðrijÞ þ Aij expð�rij=qÞÞq ¼ 0:03 nm

Na Ba Pr La Al Zr F

Na 1.00 4.84 4.60 5.83 1.29 2.90 1.04

Ba 8.34 18.67 23.76 5.71 13.64 5.07

Pr 4.82 5.59 5.65 10.68 4.81

La 6.51 7.01 13.25 6.11

Al 1.93 3.91 1.30

Zr 2.49 3.02

F 0.84

284 H. Inoue et al. / Journal of Non-Crystalline Solids 325 (2003) 282–294

used are listed in Table 1. Our previous values wereused as the parameters except for the ionic pair

with the Pr3þ ion [22,23]. The parameters for the

ionic pair with the Pr3þ ion were estimated from the

ionic radius of the Pr3þ ion reported by Shannon

and Prewitt [24]. The Coulomb force was evaluated

by the Ewald summation. To obtain the variation

of the Pr3þ sites in the glass structure, MD simu-

lation was performed for 300 different sets of ran-dom initial coordinates. The temperature of the

simulation was lowered from 3000 to 300 K with a

time step of 1fs for 10 000 time steps (10 ps). After

5000 time steps (5 ps) at 300 K, the co-ordinates of

the last step were used for further calculation.

Computations were made with a Hitac SR2201

computer in the Information Technology Center,

the University of Tokyo.

2.3. Calculation of the splitting of 4f energy levels

and the transition rate between them

A detailed method of the calculations can be

found in the literature [25–27], and we have re-

cently described them as well [22,23]. The Hamil-

tonian describing the electrostatic field (crystal-field) at the Pr3þ ion can be written as follows:

HCF ¼Xkq

A�kq

Xi

rki CkqðriÞ; ð1Þ

where CkqðriÞ is an irreducible spherical tensor

operator of rank k, operating on the ith electron

whose radius is ri, and Akq is a crystal-field pa-

rameter. On the basis of the point charge ap-

proximation, one can write

Akq ¼ � e2

4pe0

Xj

qjRkþ1j

CkqðRjÞ; ð2Þ

where qj is the charge on the jth ion in the glass, and

Rj is its distance from the Pr3þ ion. Only operators

of rank 2, 4 and 6 are needed to determine relative

energies within the f2 manifold. Once the ion posi-

tions and charges are known, the only undeter-

mined variables are the values of the powers of the

electron radii rki . The values were obtained by use of

a DV-Xa calculation [28] for a trivalent Pr ion towhich the 5d and 5g orbits were added. It is known

that the modification of the values is necessary for

the reproduction of the observation [29]

hrki ¼ akhrkiDV-Xa; ð3Þwhere ak is a phenomenological parameter for the

modification. The sum in Eq. (2) was evaluated

with the co-ordinates taken from the MD simula-tion in a sphere with a radius of 20 nm from each

Pr3þ ion. The values of the energy-level parameters

for SLJ terms determined by Carnall et al. [30] for

the Pr3þ ion in an LaF3 crystal were used. The

eigenstates and eigenvalues were obtained from

the diagonalization of the crystal-field Hamilto-

nian. The resulting 91 eigenstates were a linear

combination of basis states SLJM of the form

jwi ¼XSLJM

aðSLJMÞj4f2aSLJMi: ð4Þ

These states will henceforth be referred to as

SLJM.

The electric-dipole transition rate between an

initial level SLJM and a final level S0L0J 0M 0 was

calculated by two kinds of calculation methods.

One is the standard Judd–Ofelt theory [8,9,27], inwhich all energy differences between the 4fn state

and the excited 4f15d1 or 4f1n0g1 state, Dðn0l0Þ, areassumed to be equal. The electric-dipole transition

rate is given by


AðSLJM ;S0L0J 0M 0Þ

¼ e2

4pe0

64p4�mm3

3hc3nðn2þ2Þ2

9jhSLJM jPEDjS0L0J 0M 0ij2;

hSLJM jPEDjS0L0J 0M 0i¼

XaSLJMa0S0L0J 0M 0

baðaSLJMÞa0ða0S0L0J 0M 0Þ

�ð4f2aSLJM jPEDj4f2a0S0L0J 0M 0Þc;ð4f2aSLJM jPEDj4f2a0S0L0J 0M 0Þ

¼Xkqk

ð(

�1ÞqþqþJ�MþS0þL0þJþk½J ;J 0�1=2½k�

�1 k k

q �ðqþqÞ q

� �J k J 0

�M qþq M 0

!

�J J 0 k

L0 L S

( )AkqNðk;kÞð4f2aSLjU kj4f2a0S0L0Þ

);

Nðk;kÞ¼ 2Xn0l0

½l;l0�ð�1Þlþl0

�1 k k

l l0 l

� �l 1 l0

0 0 0

!l0 k l

0 0 0

!

�h4f jrjn0l0ihn0l0jrkj4f iDðn0l0Þ ; ð5Þ

where �mm is the frequency of the transition, c is the

velocity of light, q is the polarization of the tran-

sition. The doubly reduced matrix elements of thespherical tensor operator U k have been tabulated

by Nielson and Koster [31]. The value of refractive

index n at wavelength k0 was calculated using the

relation n ¼ Aþ B=k02 taking A ¼ 1:50 and

B ¼ 3500 (nm2) for ZBLAN glass [32]. The n0l0

configuration were assumed to be 5d and 5g con-

figurations. The radial integrals h4f jrjn0l0i and

h4f jrkjn0l0i of the free-ion state of the Pr3þ ion wereestimated from the DV-Xa calculation. The values

of Dð5dÞ and Dð5gÞ configurations were assumed

to be 54 000 and 1 62 000 cm�1, respectively [33]. A

phenomenological parameter, bðk; kÞ, was intro-

duced because the correction was also necessary

for the value of Nðk; kÞ.Nðk; kÞ ¼ bðk; kÞNDV-Xaðk; kÞ: ð6Þ

In the Pr3þ ion, the first excited 4f15d1 state lies

close to the 4f2 state. Therefore, we calculated the

electric-dipole transition rate by the modified

Judd–Ofelt method as a second method, in whichthe variation of the energy difference between the

4f2 state and the excited 4f15d1 or 4f15g1 state was

taken into account. In this method, the electric-

dipole transition rate can be written as

ð4f2aSLJM jPEDj4f2a0S0L0J 0M 0Þ

¼Xkqk

ð(

� 1ÞqþqþJ�MþS0þL0þJþk½J ; J 0�1=2½k�

�1 k k

q �ðqþ qÞ q

!J k J 0

�M qþ q M 0

!

�J J 0 k

L0 L S

( )Akqð4f2aSLjU kj4f2a0S0L0Þ

�Xn0l0

Nn0l0 ðk; kÞ1

Eð4f1n0l01Þ � Eð4f2aSLJMÞ

"

þ ð�1Þ1þkþk

Eð4f1n0l01Þ � Eð4f2aS0L0J 0M 0Þ

#);

Nn0l0 ðk; kÞ ¼ ½l; l0�ð�1Þlþl01 k k

l l0 l

( )l 1 l0

0 0 0

!

�l0 k l

0 0 0

!h4f jrjn0l0ihn0l0jrkj4f i:

ð7ÞThe term Eð4f1n0l01Þ � Eð4f2aSLJMÞ was assumed

to be the energy difference between the energy

position of the SLJM state of the 4f2 configuration

and the 5d or 5g states whose energies lie 54 000

and 162 000 cm�1 above the ground state of the 4f2

state, respectively. The same values of the bðk; kÞdetermined by the former method were used in this

method. The value of bðk; kþ 1Þ or bðk; k� 1Þ wasused as a value of bðk; kÞ in which k was odd.

The magnetic-dipole transition rate was ob-

tained by use of the method in the literature [34].

The calculation method of the spectrum from the

transition rate was described elsewhere [22,23].

The averages of the spontaneous emission ratesand the cross-section of absorption and emission

of 300 spectra obtained from 300 structural models

were used in the following section.

ctio

n (1

024

m2)

1.5

0.0

0.5

1.0

1.5(a)

(b)

ctio

n (1

0-m

)

1.5

0.0

0.5

1.0

1.5(a)

(b)

ctio

n (1

0m

)

1.5

0.0

0.5

1.0

1.5

1.5

0.0

0.5

1.0

1.5(a)

(b)


3. Results

3.1. Molecular dynamics simulation

The pair distribution function for the Pr–F pair

in the simulated structural models for the ZBLAN

glass at 300 K is shown in Fig. 1 together with the

accumulated coordination number. The peak ofthe Pr–F pair was at 0.248 nm with 0.024 nm of

FWHM. The valley of the peak was found around

0.31 nm. Thus, we determined the F� ions within

this distance as the first coordination polyhedron.

The Pr3þ ions, whose fluorine coordination num-

bers were 7, 8, 9 and 10, were 7%, 43%, 38% and

10% in the 300 structural models, respectively. The

average fluorine coordination number of the Pr3þ

ions in the models was 8.55.

3.2. Optical spectra

The observed absorption cross-section is shown

in Fig. 2(a). The absorption bands can be ascribed

to the transitions from the ground state, 3H4, to

the upper levels of the Pr3þ ion. The parameters akand bðk; kÞ were determined by the comparison of

the observed absorption spectra and the spectra

obtained from the standard Judd–Ofelt theory.

0.1 0.2 0.3 0.40

100

200

0

5

10

15

Pr-F

pai

r di

stri

butio

n cu

rve

(a.u

.)

Acc

umul

ated

coo

rdin

atio

n nu

mbe

r

Pr-F distance (nm)

0.1 0.2 0.3 0.40

100

200

0

5

10

15

Pr-F

pai

r di

stri

butio

n cu

rve

(a.u

.)

Acc

umul

ated

coo

rdin

atio

n nu

mbe

r

Pr-F distance (nm)

Fig. 1. Pr–F pair distribution curve and accumulated coordi-

nation number in the structural models.

The determined values are listed in Table 2 to-gether with the parameters obtained from the

DV-Xa method. The calculated absorption cross-

section on the basis of the standard Judd–Ofelt

theory is shown in Fig. 2(b). The positions of the3FJ and 3PJ states were 150 and 100 cm�1 higher

than the observed ones, respectively. The positions

of the 1G4 and 1D2 states were 50 and 200 cm�1

lower, respectively. As the position of the ab-sorption to the 1I6 state was shifted to the lower

energy side, the first peak of the absorption to the3PJ state was higher and the second one was lower

than the observed ones.

Wave number (cm-1)

Abs

orpt

ion

Cro

ss S

e

0 10000 20000

2.0

0.0

0.5

1.0

0.0

0.5

1.0

1.5

(c)

Wave number (cm-1)

Abs

orpt

ion

Cro

ss S

e

0 10000 20000

2.0

0.0

0.5

1.0

0.0

0.5

1.0

1.5

(c)

Wave number (cm-1)

Abs

orpt

ion

Cro

ss S

e

0 10000 200000 10000 20000

2.0

0.0

0.5

1.0

0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

0.0

0.5

1.0

1.5

(c)

Fig. 2. The observed (a), the calculated (b) and (c) absorption

cross-section of Pr3þ-dope ZBLAN glass. The calculated spec-

tra by the standard Judd–Ofelt theory (b) and by the modified

Judd–Ofelt one (c).

Table 2

The parameters of hrki, Nðk; kÞ and Nn0l0 (k; k) for the Pr3þ ion calculated from a DV-Xa method and the modification parameters

of ak and bðk; kÞ usedk hrki ak

2 3.215· 10�21 m2 0.40

4 2.342· 10�41 m4 2.46

6 3.099· 10�61 m6 9.33

k, k Nðk; kÞ bðk; kÞ1, 2 )1.313· 10�3 m2 J�1 0.80

3, 2 0.850· 10�23 m4 J�1 1.36

3, 4 0.937· 10�23 m4 J�1 1.36

5, 4 )0.649· 10�43 m6 J�1 4.61

5, 6 )1.582· 10�43 m6 J�1 4.61

7, 6 0.941· 10�63 m6 J�1 7.67

k, k N5d (k; k) N5g(k; k) bðk; kÞ1, 1 0.856· 10�21 m2 )1.718· 10�21 m2 0.80

1, 2 )0.514· 10�21 m2 )0.573· 10�21 m2 0.80

3, 2 0.233· 10�41 m4 0.669· 10�41 m4 1.36

3, 3 )0.412· 10�41 m4 0.473· 10�41 m4 1.36

3, 4 0.455· 10�41 m4 0.143· 10�41 m4 1.36

5, 4 )0.124· 10�61 m6 )0.674· 10�61 m6 4.61

5, 5 0.371· 10�61 m6 )0.337· 10�61 m6 4.61

5, 6 )0.829· 10�61 m6 )0.058· 10�61 m6 4.61

7, 6 0.0 1.514· 10�81 m8 7.67


Morrison and Leavitt [29] have advocated theequation for the correction of the value of hrkigiven by

hrki ¼ 1� rk

skhrkiHF; ð8Þ

where rk is a shielding factor of the crystal-field by

the outer shells and s is a scaling parameter for the

expansion of the wave function calculated from the

Hartree–Fock expectation when the ion is intro-

duced into a solid. The typical values were r2 � 0:8,r4 ¼ r6 � 0:1 and s � 0:75 [29]. According to the

values of rk and s of Morrision et al., the value of

the parameters a2, a4 and a6 were 0.36, 2.84 and5.06, respectively. The value of the a6 in Table 2 was

larger than that of Morrison�s. The relation of the

line width and oscillator strength of these values is

discussed in the following discussion. The calcu-

lated values of the oscillator strength are listed in

Table 3. As can be seen from the figures and Table

3, the values of the oscillator strength of the

absorption to the 1D2 and 3PJ states were about70% and 60% of the observation, respectively. It

was found that the agreement with the oscillator

strength was greatly improved, because the typical

estimated values of the 1D2 and3PJ states were only

49% and 26% of the experimental values in the

conventional Judd–Ofelt treatment [11].

The observed emission spectrum for an excita-

tion wavelength at 479 nm is shown in Fig. 3 to-

gether with the emission spectra from the 3P0 and3P1 states calculated by the standard Judd–Ofelttheory. As can be seen from the figure, the observed

emission spectrum was comprised of those from the3P0 and 3P1 states. The calculated spectrum from

the 3P1 state was described by assuming that the

population of the 3P1 state was 25% of that of the3P0 state. The electric-dipole transitions to the 3F3

and 3H5 states from the 3P0 state were observed.

These electric-dipole transitions are forbiddenunder the intermediate coupling eigenstates and

allowed by the crystal-field. Though the calculated

intensities of these transitions were a little smaller

than the observed ones, it could be estimated that

the transitions were allowed by the crystal-field.

It was found that the shape of the absorption

and emission spectra of the Pr3þ ion in ZBLAN

glass could be reproduced to some extent using the

Table 3

The observed and calculated oscillator strengths (10�6) of the Pr3þ ion in ZBLAN glass

Level Observed MD Calculated

Standard Modified

ED Total ED Total

3H5 – 0.20 1.90 2.10 1.98 2.183H6 3.08 0.01 1.19 3.55 1.30 3.883F2 0.01 2.35 2.563F3 7.89 0.01 4.58 8.13 5.23 9.323F4 0.01 3.52 4.071G4 0.20 0.01 0.33 0.33 0.40 0.411D2 2.05 0.00 1.45 1.45 2.07 2.073P0 0.00 1.76 2.661I6 17.10 0.00 1.54 10.35 2.60 17.523P1 0.00 1.26 2.263P2 0.00 5.79 10.00

MD: Magnetic-dipole transition, ED: Electric-dipole transition.


standard Judd–Ofelt theory, in which the crystal-

field was taken into consideration.

The calculated absorption cross-section on the

basis of the modified Judd–Ofelt theory is shown

in Fig. 2(c), and the calculated oscillator strength

is listed in Table 3. The values of the oscillator

strength of the absorption to the 1D2 and 3PJ

states increased 43% and 69% from those based on

the standard Judd–Ofelt theory. As can be seen

from Eq. (7), the increasing ratio was larger at the

upper level. Both oscillator strength values of the

absorption to the 1D2 and 3PJ states were com-

parable to the experimental oscillator strengths.

Concurrently, the estimated oscillator strengths

for the 3H6,3F2 and

3F2,3F4 levels increased, and

the values were about 20% in excess of the exper-

imental ones. The emission intensities from the 3P0

and 3P1 states also increased by using the modified

Judd–Ofelt theory, and the shapes of the emission

spectra were similar to those of the standard Judd–

Ofelt theory.

4. Discussion

4.1. The effect of the crystal-field on the splitting of

the energy level

The effect of the crystal-field on the energy

levels of the 4f electrons can be evaluated on the

basis of the structural models. The effect is classi-

fied into three parts: (1) The SLJ state, which is

described by the intermediate coupling eigenstate,

is mixed with other S0L0J 0 states of the 4f electron

by even-parity components of the crystal-field. The

effect was observed as a splitting of the SLJ state

into SLJM states. (2) Odd-parity components of

the crystal-field mixes the SLJ state with the ex-cited 4f15d1 (or 4f1n0g1) states. The effect was ob-

served as a forced electric-dipole transition. (3)

The rate of the forced electric-dipole transition

changes with the change from the SLJ state to the

SLJM state as an additional effect.

At first we investigate the relation between the

even-parity components of the crystal-field and the

splitting of the energy level. It is too complicatedto describe the relation between the structure

around the Pr3þ ion and the crystal-field. Here, we

describe the relation between the splitting of each

energy level and the crystal-field parameters. The

crystal-field parameters were classified into three

groups by their orders; A2q, A4q and A6q terms. The

energy splitting of each level calculated only from

the parameters of each group was compared withthe original splitting, which was calculated from

full sets of the crystal-field parameters. If a certain

group of the crystal-field parameter dominates the

splitting of an energy level, the calculated energy

only from that particular group must be correlated

with the original value of the energy splitting. Fig.

4 shows the relation between the energy levels of

the 3F3 state calculated from the full sets of the

600 800 1000 1200600

800

1000

1200

600 800 1000 1200600

800

1000

1200

600 800 1000 1200600

800

1000

1200

Energy (full set) (cm-1)

Ene

rgy

(A6q

) (c

m-1

)E

nerg

y (A

4q)

(cm

-1)

Ene

rgy

(A2q

) (c

m-1

)

(a)

(b)

(c)

600 800 1000 1200600

800

1000

1200

600 800 1000 1200600

800

1000

1200

600 800 1000 1200600

800

1000

1200

600 800 1000 1200600

800

1000

1200

600 800 1000 1200600 800 1000 1200600

800

1000

1200

600 800 1000 1200600

800

1000

1200

600 800 1000 1200600 800 1000 1200600

800

1000

1200

600 800 1000 1200600

800

1000

1200

600 800 1000 1200600 800 1000 1200600

800

1000

1200

Energy (full set) (cm-1)

Ene

rgy

(A6q

) (c

m-1

)E

nerg

y (A

4q)

(cm

-1)

Ene

rgy

(A2q

) (c

m-1

)

(a)

(b)

(c)

Fig. 4. The relation between the energy position of Stark

components calculated from individual A2q (a), A4q (b) and A6q

(c) terms and the energy positions calculated from the full set

of the crystal-field parameters.

14000 16000 18000 200000.0

0.2

0.4

0.6

0.8

1.00.0

0.2

0.4

0.6

0.8

1.0

Wave number (cm -1)

Inte

nsity

(a.

u.)

3H5

3H6

3F2

3F3

3F4

3H53H6

3F23F33F4

(b)

(c)

(d)

14000 16000 18000 2000014000 16000 18000 200000.0

0.2

0.4

0.6

0.8

1.00.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Wave number (cm -1)

Inte

nsity

(a.

u.)

3H5

3H6

3F2

3F3

3F4

3H53H6

3F23F33F4

(b)

(c)

(d)

14000 16000 18000 200000.0

0.2

0.4

0.6

0.8

1.0

Wave number (cm-1)

Inte

nsit

y (a

.u.)

(a)

14000 16000 18000 200000.0

0.2

0.4

0.6

0.8

1.0

Wave number (cm-1)

Inte

nsit

y (a

.u.)

14000 16000 18000 200000.0

0.2

0.4

0.6

0.8

1.0

14000 16000 18000 2000014000 16000 18000 200000.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Wave number (cm-1)

Inte

nsit

y (a

.u.)

(a)

Fig. 3. The observed (a) and calculated (b) emission spectra of

Pr3þ-doped ZBLAN glass excited at 479 nm. The calculated

spectra from the 3P0 (c) and the 3P1 (d) states. The intensity of

the emission spectrum from the 3P1 state was described at 25%.


crystal-field parameters and from each group of

the crystal-field, for example. The energy distri-

bution calculated from the A2q or A4q terms was

smaller than that calculated from the full sets of

the crystal-field parameters. The distribution cal-

culated from the A6q terms was larger than those

from the A2q or A4q terms and the value of the slope

was closer to unity. The slope represented the cor-

relation between the splitting calculated from thefull sets and that from each group of the crystal-

field parameters. As can be seen from Fig. 4, the

positions of the components of the 3F3 state were

mainly determined by the A6q terms. Similarly, the

Table 4

The correlation coefficients CA2q, CA4q and CA6q between the

energy positions of Stark components of each level calculated

from individual A2q, A4q and A6q terms and the energy positions

calculated from the full set of the crystal-field parameters

CA2q CA4q CA6q

3H4 0.457 0.694 0.5343H5 0.561 0.677 0.4153H6 0.440 0.637 0.6463F2 0.454 0.295 0.2203F3 0.342 0.151 0.6513F4 0.114 0.914 0.3741G4 0.013 0.675 0.6631D2 0.439 0.870 0.1013P0 – – –1I6 0.602 <0.0 0.1883P1 0.825 0.019 0.0213P2 0.824 0.222 0.054

Table 5

Square of matrix elements of U ðkÞ for the Pr3þ ion [36]

U2 U4 U6

3H5 0.1096 0.2035 0.61063H6 0.0002 0.0322 0.14073F2 0.5079 0.4048 0.11963F3 0.0658 0.3487 0.70023F4 0.0162 0.0528 0.49011G4 0.0019 0.0044 0.01191D2 0.0020 0.0165 0.04933P0 0.0 0.1713 0.01I6 0.0081 0.0447 0.02033P1 0.0 0.1721 0.03P2 0.0001 0.0362 0.1373


splitting of the other states was evaluated and the

correlation coefficients are listed in Table 4.

The value of the correlation coefficients of the3HJ state was within 0.4–0.7 for each group of the

crystal-field term. The splitting of the 3HJ states

broadly represents the crystal-field. On the con-

trary, the value of the 3FJ state for one group waslarger than those for others. There was little con-

tribution of the A2q terms for the splitting of the1G4 state. The splitting of 1I6,

3P1 and 3P2 states

was mainly determined by the A2q terms.

The values of the correlation coefficient chan-

ged by the value of the ak used. However, the

variation of the ak did not change the fact that

the contribution of the A6q terms is larger for thesplitting of 3H6,

3F3 and 1G4 states than those of

other states, and the contribution of the A4q terms

is larger for the splitting of the 3F4 and1D2 states.

From Eq. (2), the value of the Akq term with a

higher k is dominated by the closer structure

around the Pr3þ ion. Therefore, it must be possible

to obtain the local structural information around

the Pr3þ ion in the ZBLAN glass from the sys-tematic examination of the splitting of the 3H6,

3F3

and 1G4 states.

4.2. The oscillator strength and the electric-dipole

transition rate


is another important part of the effect of the

crystal-field. According to the Judd–Ofelt theory,the oscillator strength of the forced electric-dipole

transitions, fED, was given by [35]

fED ¼ 8p2mcx3hð2J þ 1Þ

ðn2 þ 2Þ2

9n

�ðX2U2 þ X4U4 þ X6U6Þ; ð9Þ

where J is the total angular momentum of an

initial state, x is the transition energy (in wave

numbers), n is the refractive index of the medium,

Xk is the Judd–Ofelt intensity parameters, and Uk

is the square of the doubly reduced matrix elementU k. The Xk is given by [35]

Xk ¼Xkq

ð2kþ 1ÞA2kq

2k þ 1N2ðk; kÞ; ð10Þ

where Akq is the odd-parity components of the

crystal-field parameters. Therefore, the relation

between the oscillator strength and the odd-paritycomponents of the crystal-field can be seen from

the values of the Uk, which are listed in Table 5

[36].

The calculated oscillator strengths to the 1D2

and 3P2 states by the standard Judd–Ofelt theory

were from 30% to 40% insufficient for the obser-

vation. The contribution of the X6U6 term is

dominant in the oscillator strengths. Therefore, theoscillator strengths to both states increased with

an increase of the values of parameters bð5; 6Þ andbð7; 6Þ. However, if we used the larger values of the

bðk; 6Þ parameters, the oscillator strengths to other

states, such as the 3F4,3H6,

3H5 and 1G4 states,

increased. Consequently, the priority of the


agreements of lower energy levels caused the in-sufficiency to the 1D2 and

3P2 states. The setting of

the parameters such as, ak and bðk; kÞ will be an

important future problem, which is not limited to

the Pr3þ ion.


between the SLJ states changes also by mixing the

states with other states of the 4f2 configurations

due to the even-parity components of the crystal-field. In other words, the oscillator strengths and

transition rates are affected by the factor ak for theeven-parity components of the crystal-field. The

oscillator strengths obtained from the several

conditions of ak are listed in Table 6. The oscillator

strengths, except to the 3H5 and 3H6 states, de-

creased with an increase of the value of a2. Theoscillator strengths to the 1I6,

3P0 and 3P1 statesincreased with an increase of the value of a4 or a6.It was found that the calculated oscillator strength

with the even-parity components of the crystal-

field changed from )37% to +44% for those ob-

tained from the intermediate coupling states

(ak ¼ 0). These changes are not negligibly small.

The rates of the electric transition from the 3P0

state obtained from the several conditions of the akare listed in Table 7. The rates of the electric-

dipole transitions from the 3P0 state to the 3F3

and 3H5 states, which are forbidden under the in-

termediate coupling eigenstates, increased with

increasing the values of the ak parameter. Ti-

Table 6

The calculated oscillator strengths (10�6) of the Pr3þ ion under vario

k a

2 0.0 0.40 1.00 0

4 0.0 0.0 0.0 2

6 0.0 0.0 0.0 0

Level3H5 2.04 2.06 2.20 13H6 0.91 0.93 1.07 03F2 2.63 2.54 2.51 23F3 7.24 6.71 6.34 63F4 4.89 4.71 4.58 31G4 0.37 0.36 0.36 01D2 1.33 1.14 0.99 13P0 1.23 1.13 1.02 11I6 1.16 1.13 1.06 13P1 1.26 1.18 1.13 13P2 4.72 4.08 3.57 6

khomirov et al. [37], and Tikhomirov and Ti-khomirova [38] reported that the transition from

the 3P0 state to 3F2 state was hypersensitive to the

host materials. The 3P0–3F2 hypersensitive transi-

tion has been explained by the dynamic coupling

of the electric quadrupole and dipoles. The rate of

the transition decreased with an increase of the

values of ak parameter. On the contrary, the rate of

the transition from the 3P0 state to the 3H6 statedecreased with an increase of the a2 parameter and

increased with an increase of a4 and a6. This

transition was used as a reference of a non-hy-

persensitive transition. Both rates of the 3P0–3F2

and 3P0–3H6 transitions changed by the crystal-

field, and the manner of the changes were different.

Therefore, it is better to consider not only the ef-

fect of the dynamic coupling but also the effect ofthe static crystal-field.

4.3. The Judd–Ofelt intensity parameters

It was found that there are three kinds of con-

tributions to the oscillator strength of the electric-

dipole transition: (a) the odd-parity components of

the crystal-field, (b) the even-parity components ofthe crystal-field and (c) the energy difference be-

tween the 4f2 and 4f15d1 or 4f15g1 configurations.

Among the above effects, (b) and (c) are not con-

sidered in the original Judd–Ofelt theory. By sub-

tracting these factors, the Judd–Ofelt parameters

us condition of the ak

k

.0 0.0 0.0 0.0

.46 3.60 0.0 0.0

.0 0.0 9.33 13.33

.75 1.72 2.26 2.34

.92 0.98 1.19 1.39

.75 2.80 2.32 2.21

.30 5.92 5.62 5.09

.87 3.71 4.55 4.41

.34 0.34 0.40 0.45

.62 1.71 1.31 1.29

.58 1.69 1.58 1.67

.25 1.32 1.49 1.58

.45 1.52 1.10 1.11

.01 6.37 5.28 5.46

Table 7

The transition rates (s�1) from 3P0 state of the Pr3 ion under various condition of the ak

k ak

2 0.0 0.40 1.00 0.0 0.0 0.0 0.0

4 0.0 0.0 0.0 2.46 3.60 0.0 0.0

6 0.0 0.0 0.0 0.0 0.0 9.33 13.33

Level3H6 7215 7200 7045 7223 7211 7052 69443H5 0 21 121 98 209 139 2733H6 6352 6178 5860 6406 6468 6498 67393F2 7021 6942 6698 6806 6674 6805 64783F3 0 6 31 20 36 10 253F4 1377 1403 1429 1381 1321 1383 13741G4 324 331 368 362 386 302 2981D2 6 6 5 5 5 6 7

Table 8

Judd–Ofelt intensity parameters X2, X4 and X6 (10�24 m2) ob-

tained from the calculated oscillator strengths

Vcrys Dð4f2Þ 3P2 X2 X4 X6

) ) ) 2.33 2.18 8.42

) ) + 2.30 2.16 8.44

+ ) ) 2.89 2.46 4.94

+ ) + 1.74 1.75 5.73

) + ) 1.10 3.82 9.18

) + + )0.43 2.87 9.76

+ + ) 1.45 4.56 5.18

+ + + )1.38 2.80 6.26

Vcrys: The even-parity of the crystal-field.

D(4f2): The energy position of the 4f2 configuration.

The mark of + and ) indicate the oscillator strength calculated

with and without the item, respectively.


can be obtained in the original meaning, and thevalues of the odd-parity components of the crystal-

field, Dð5dÞ and Nðk; kÞ can be discussed. The

values of the Judd–Ofelt intensity parameters X2,

X4 and X6 were obtained from the calculated os-

cillator strengths. In order to avoid the negative

value of the X2 parameter, the 3P2 state is often

omitted in a least-squares fitting of the conven-

tional Judd–Ofelt treatment for the Pr3þ ion[10,11,14,32]. We present results for eight cases for

the absorption oscillator strengths calculated from

standard and modified theory with and without

even-parity components of the crystal-field as well

as the 3H4–3P2 transition both included and ex-

cluded. We classified the states into eight groups,3H5, (

3H63F2), (

3F33F4),

1G4,1D2,

3P0, (1I6,

3P1)

and 3P2 for a least-squares fitting. Table 8 showsthe resulting sets of the Judd–Ofelt parameters

ðX2;X4;X6Þ. The sets of (2.3, 2.2 and 8.4) were

obtained from the oscillator strengths without ei-

ther of the contributions of (b) and (c). It is shown

that the Judd–Ofelt theory has been established

well for this condition, because these values of the

parameter sets were not influenced by the consid-

eration of the 3P2 state. On the other hand, thevalues of the parameter set which was obtained

from the oscillator strengths calculated with the

contributions (b) and/or (c) was influenced in the

consideration of the 3P2 state. The value of X2 was

always obtained as a decrease by the consideration

of the 3P2 state. It is worth noting that the negative

X2 parameter was obtained from the oscillator

strengths calculated with the contribution of (c).

Quimby and Miniscalco [11] reported that the

observed parameter set of (1.89, 5.05 and 5.28) was

changed to ()0.14, 4.92 and 7.10) by the consid-

eration of the 3P2 state. Our calculated set of (1.45,

4.56 and 5.18) was changed to ()1.38, 2.80 and6.26). The values of X2 and X4 decreased and the

value of X6 increased. Though the amount of the

change was slightly different, its tendency was re-

produced. Fig. 5 shows the relation between the

increasing ratio by the factor (c) and the energy

position of the 4f2 state together with those of

the transition rate from the 3P0 and 3P1 states.

As can be seen from the figure, the linear rela-tion between the increasing ratio and ð1=ð54000�Eð4f2SLJÞÞÞ þ ð1=ð54 000 � Eð4f2S0L0J 0ÞÞÞ was

4.0 5.0 6.01.0

1.5

2.0

Incr

easi

ng r

atio

)'''(54000

1

)(54000

1

JLSESLJE −+

− (10-5 cm)

4.0 5.0 6.01.0

1.5

2.0

4.0 5.0 6.01.0

1.5

2.0

Incr

easi

ng r

atio

)'''(54000

1

)(54000

1

JLSESLJE −+

− (10-5 cm))'''(54000

1

)(54000

1

JLSESLJE −+

− (10-5 cm)

Fig. 5. The relation between the increasing ratio by the modi-

fied Judd–Ofelt theory and the energy difference between the 4f2

and 5d levels: (�) Oscillator strength and (M) transition rate

from 3P0 and 3P1 states.


roughly valid. Therefore, it is possible to subtract

the contribution of the energy position of the 4f2

state from the observed oscillator strengths if these

increasing ratios are considered. For example,

these ratios comprise 30% and 42% of the contri-

bution of (c) in the calculated oscillator strengthsto the 1D2 and

3P2 states, respectively. If they can

be subtracted, they still comprise 8% and 19% of

the contribution of (b), the even-parity compo-

nents of the crystal-field in the residual oscillator

strengths, respectively. The contribution of the

even-parity components of the crystal-field cannot

be estimated only from the observed oscillator

strengths. The parameter sets of (2.89, 2.46 and4.94) or (1.74, 1.75 and 5.73) are clearly different

from the original set of (2.3, 2.2 and 8.4). There-

fore, it is necessary to pay careful attention to the

discussion of the obtained Judd–Ofelt parameter.

It should be emphasized that this situation is not

specific for the Pr3þ ion.

5. Conclusion

Absorption and emission spectra of Pr3þ-doped

ZBLAN glass at room temperature were calcu-

lated based on crystal-field theory. The calculated

oscillator strengths to the 1D2 and3PJ states were

insufficient for the observation. It was found thatconsidering the variation of the energy difference

between the 4f2 and the excited 4f15d1 or 4f15g1

configurations, these oscillator strengths increased

and the values were comparable to the observed

ones. It was possible to estimate the effect of the

energy variation of the 4f2 configurations in a

linear equation. Therefore, the effect was sub-

tracted from the observed oscillator strengths andthe Judd–Ofelt intensity parameters of the Pr3þ ion

was obtained by a least-squares fitting of the

conventional Judd–Ofelt treatment as well as other

rare earth ions.

Acknowledgements

This study was supported financially by a

Grant-in-Aid from the Ministry of Education with

the contract number #09450239. The authors

would like to thank Morita Chemical Industries

Co. Ltd. and Central Glass Co. Ltd. for the supply

of fluorides.

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Documents

The effects of crystal-fields on the optical properties of Pr:ZBLAN glass