# A complete 4fn energy level diagram for all trivalent lanthanide ions

**ABSTRACT** We describe the calculations of the 4fn energy levels, reduced matrix elements for 4fn-4fn transitions and the simulation of absorption and emission spectra. A complete 4fn energy level diagram is calculated for all trivalent lanthanide ions in LaF3: The calculated energy levels are compared with experimentally obtained energies. For Ce, Pr, Nd, Eu, Gd, Ho, Er, Tm and Yb many, and in some cases all, energy levels have been observed. This work provides a starting point for future investigation of as yet unobserved VUV energy levels.

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Journal of Solid State Chemistry 178 (2005) 448–453

A complete 4fnenergy level diagram for all trivalent lanthanide ions

P.S. Peijzela,?, A. Meijerinka, R.T. Wegha, M.F. Reidb, G.W. Burdickc

aDepartment of Condensed Matter and Interfaces, Debye Institute, Utrecht University, P.O. Box 80 000, Utrecht 3508 TA, The Netherlands

bDepartment of Physics and Astronomy, University of Canterbury, Christchurch, New Zealand

cDepartment of Physics, Andrews University, Berrien Springs, MI 49104, USA

Received 5 May 2004; received in revised form 21 July 2004; accepted 28 July 2004

Available online 18 September 2004

Abstract

We describe the calculations of the 4fnenergy levels, reduced matrix elements for 4fn-4fntransitions and the simulation of

absorption and emission spectra. A complete 4fnenergy level diagram is calculated for all trivalent lanthanide ions in LaF3: The

calculated energy levels are compared with experimentally obtained energies. For Ce, Pr, Nd, Eu, Gd, Ho, Er, Tm and Yb many,

and in some cases all, energy levels have been observed. This work provides a starting point for future investigation of as yet

unobserved VUV energy levels.

r 2004 Elsevier Inc. All rights reserved.

Keywords: Lanthanide; Spectroscopy; VUV; Energy levels; Carnall

1. Introduction

Extensive measurements of energy levels of the 4fn

configurations of lanthanide ions in various host lattices

were carried out in the 1950s and 1960s. Much of this

work was carried out by Dieke and co-workers and the

data summarized in his 1968 book [1] (published

posthumously). The energy-level diagram for trivalent

lanthanide ions presented in that book is commonly

referred to as a ‘‘Dieke diagram’’. These diagrams are

useful because the energies of the J multiplets vary by

only a small amount in different host crystals. The

diagram allows rapid identification of the energy levels

in new hosts, and has been a crucial tool in the design of

materials suitable for phosphors or lasers.

As this diagram developed in the 1960s there was a

fruitful interplay between measurements, theoretical

models, and computational modeling of energy levels.

Two- and three-body operators representing configura-

tion interaction corrections to the Coulomb interaction

were found to be necessary to accurately reproduce the

observed spectra [2–4]. Anomalies in the crystal-field

splitting were also noted, but the modeling of correla-

tion effects on the crystal-field levels were not performed

until the late 1970s [5,6].

Dieke’s experimental data were largely gathered using

the LaCl3 host. The high-symmetry ðC3hÞ sites in this

crystal mean that only a small number of crystal-field

parameters (four) are required to fit the spectra, which

made it attractive for early studies. However, other

hosts with better optical properties, especially in the UV

region, were sought. The LaF3 host-lattice has the

advantage of being optically transparent up into the

VUV, and the chemical stability in air makes it easy to

handle.

An important legacy of the work by Bill Carnall and

his co-workers was a detailed study of the spectra of

trivalent lanthanide ions in LaF3: They compared the

absorption spectra of all lanthanides in LaF3 with

calculated energies for C2v site symmetry, which is a

good approximation of the actual C2 site [7]. For

configurations with an odd number of 4f-electrons each

multiplet with quantum number J splits into J þ 1=2

ARTICLE IN PRESS

www.elsevier.com/locate/jssc

0022-4596/$-see front matter r 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.jssc.2004.07.046

?Corresponding author. Faculteit Scheikunde, Sectie Geconden-

seerde materie en Grensvlakken, Utrecht Universiteit, 3584 CC-

Utrecht, Netherlands. Fax: +31-30-253-24-03.

E-mail address: p.s.peijzel@phys.uu.nl (P.S. Peijzel).

Page 2

crystal field levels in any symmetry lower than cubic. In

the absence of a magnetic field the energy levels are

doubly degenerate due to Kramers’ degeneracy. For

configurations with an even number of 4f-electrons, the

symmetry must be lower than D3 in order to split up

into the maximum number of 2J þ 1 crystal field levels.

The LaF3host-lattice is very suitable for comparison of

calculations and measurements of energy levels as the C2

site symmetry in LaF3[8] causes all multiplets to split up

completely for lanthanide ions with an even number of

4f-electrons.

In 1977 Carnall, Crosswhite, and Crosswhite pub-

lished the ‘‘Blue Report’’ [9], which, curiously, contains

no date and no report number. This was an important

guide to workers in the late 1970s and 1980s for the

analysis of spectra in other hosts, and the matrix

elements were used by many workers in their analysis

of transition intensities.

A decade later the report by Carnall et al. [10] and the

subsequent paper [11] provided what is arguably still the

most thorough study of the energy levels of the entire

series of lanthanide ions in a host crystal. An important

feature of this work was the detailed comparison

between the computational modeling and the experi-

mental data, with the analysis directed by the require-

ment for the Hamiltonian parameters to vary smoothly

across the series. The parameters derived in that work

have been widely used as starting points for analysis of

other systems, and the data have been used to test

extensions to the models, such as the inclusion of

correlation crystal field effects, see, for example, Li and

Reid [12].

The experimental data used by Carnall and co-

workers rarely extended above 40000cm?1: Parameter

values were optimized by least squares fitting of energy

level calculations to experimentally obtained energies.

Since the absorption spectra recorded concern the

energy region up to about 40000cm?1; calculations

are expected to be less accurate in the VUV region,

especially for gadolinium which has 1716 (doubly

degenerate) energy levels, where the parameters were

obtained by fitting to only the lowest 70 levels measured

by Carnall [11]. While there was some work in the 1970

and 1980s using synchrotron radiation [13,14], and

multi-photon spectra have been used to probe a few

levels, it is only relatively recently that extensive,

detailed, high-resolution spectra have become available.

We are now in the process of extending Carnall’s

energy-level diagram for trivalent lanthanide ions in

LaF3by assembling and analyzing data from synchro-

tron-radiation measurements [15–17], and laser techni-

ques, including two-photon absorption and excited-state

absorption [18]. This paper reports preliminary results

of comparisons between calculations based on Carnall’s

parameters and the new experimental data. We show

that the model is capable of providing a good

description of the new experimental measurements,

and comment on some of the technical issues involved

in the calculations.

2. Description of the calculations

This section describes the energy level calculations

using programs written by Reid and coworkers [19,20],

based on the model of Carnall [10] and Crosswhite and

Crosswhite [21]. Besides energy level calculations the

program also offers the possibility to calculate transition

intensities and reduced matrix elements for transitions

between multiplets and it is possible to simulate

absorption and emission spectra. Recently, Edvardsson

and A˚berg [22] reported the complete energy level

diagram of all actinides, using a similar calculation

program.

The free-ion energy level calculation uses a matrix

containing all allowed electronic states for a certain 4fn

configuration. The matrix elements are

hCln½tSL?JjHjCln0½t0S0L0?J0i;

ð1Þ

where Cln½tSL?Jand Cln0½t0S0L0?J0 are basis functions for the

4fnconfiguration and H is the parameterized Hamilto-

nian. This expression is valid only for free-ion calcula-

tions, where J is a good quantum number. ½tSL? are

‘‘nominal’’ identifiers for states, as t; S and L are not

good quantum numbers, where t (sometimes designated

as a) is seniority, and S and L are spin and angular

momentum quantum numbers, respectively. In a crystal-

line lattice, the non-spherical elements of the crystal field

intermix states of different J and MJ; and the only good

quantum number is the group theoretical irreducible

representation (‘‘irrep’’) of the site symmetry, G and the

corresponding basis functions are Cln½tSLJMJ?G:

Calculation of the angular part of the matrix elements

can be done exactly following the methods of Cross-

white and Crosswhite [21]. The radial parts of the matrix

elements are included in the calculation parameters.

The full Hamiltonian Hfull has separable contribu-

tions from the ‘‘free-ion’’ terms and from the crystal

field.

Hfull¼ Hfree-ionþ HCF:

ð2Þ

The expression for Hfree-ionis

Hfree-ion¼ EAVGþ

X

k¼2;4;6

Fkfkþ zð4fÞASO

þ aLðL þ 1Þ þ bGðG2Þ þ gGðR7Þ

þ

i¼2;3;4;6;7;8

þ

j¼0;2;4

X

Titiþ

X

k¼2;4;6

Pkpk

X

Mjmj:

ð3Þ

ARTICLE IN PRESS

P.S. Peijzel et al. / Journal of Solid State Chemistry 178 (2005) 448–453

449

Page 3

and is given using the notation of standard practice

[11,23].

The spherically symmetric part of the perturbations of

the free ion and the crystal field are represented together

by EAVG: The value of EAVG shifts the energy of the

entire 4fnconfiguration, and represents the energy

difference between the ground state energy and the

configuration center of gravity (barycenter).

Diagonalization of the energy matrix yields eigenvec-

tors describing the free ion levels, and eigenvalues of the

matrix are the multiplet energies. At this point, reduced

matrix elements for transition intensities between J-

multiplets can be calculated. These ðUðlÞÞ2reduced

matrix elements can be used in Judd–Ofelt calculations

[24] for transition intensities and branching ratios.

The crystal field Hamiltonian is parameterized by

HCFand is expressed in Wybourne notation as [25]

HCF¼

X

qparameters define the radially dependent part

of the one-electron crystal field interaction, and CðkÞ

the many-electron spherical tensor operators for the 4fn

configuration. For 4f-electron configurations the values

of k are restricted to 2, 4 and 6. The applicable values of

q depend on the site-symmetry of the lanthanide ion in

the host-lattice. For the C2veffective site symmetry used

for LaF3the restrictions are: q ¼ even and 0pqpk:

The crystal field splits each multiplet into individual

Stark components. Eigen-vectors obtained from diag-

onalization of the full Hamiltonian are used to obtain

dipole strengths for transitions between individual

crystal field levels.

Diagonalization of the matrix containing the com-

plete Hamiltonian yields all states of the 4fnconfigura-

tion. The gadolinium wavefunction matrix has a size of

3432 ? 3432 (which can be block-diagonalized into

smaller submatrices), all other energy matrices are

smaller. In Table 1 the number of electrons, SL states,

SLJ multiplets and SLJM energy levels are listed for the

trivalent lanthanide ions. The number of energy levels in

the table does not take Kramers’ degeneracy into

account, as in the absence of a magnetic field all energy

levels of configurations with an odd number of 4f-

electrons are doubly degenerate.

With the current computer systems it is no longer

necessary to perform ‘‘truncated’’ calculations, where

only energy levels up to a certain energy are included in

the calculation and the complete set of wavefunctions

are not used. In order to keep the calculations manage-

able, Carnall et al. [11] used truncation for configura-

tions where the Hamiltonian matrices were greater than

200 ? 200: This truncation may not only have a

significant impact upon reported wavefunctions, but

also may produce shifts in the calculated energies when

compared to a full energy level calculation. We

k;q

Bk

qCðkÞ

q;

ð4Þ

where Bk

q are

performed energy level calculations for the 4f7energy

levels of Gd3þin LaF3up to 51000 cm?1; which is the

energy range studied by Carnall et al. [10] using

truncation at different energies. Fig. 1 shows the energy

shift for each level caused by truncation of the

calculation as a function of the difference in energy

between the calculated level and the truncation energy.

From this figure it can be concluded that in order to

have no significant effect on the calculated energies the

truncation should be at least 30000cm?1above the

highest energy level of interest. For modern computers

with a high clock speed and a large amount of memory

there is no longer a need for truncation when calculating

the 4fnlevels of lanthanides. Another speed improving

factor is the use of a ‘‘relatively robust’’ tridiagonaliza-

tion routine [26], which for large matrices can result in a

factor of 20 increase in speed. Especially in calculations

where multiple diagonalizations are used this 20-fold

increase in speed is important.

If a sufficient number of experimentally obtained

values for energy levels is known, a least squares fitting

routine can be used to adjust some or all of the

parameters to give a better agreement between calcula-

tion and experiment. Care has to be taken using this

ARTICLE IN PRESS

Table 1

Number of electrons, SL states, SLJ multiplets and SLJM energy

levels for all trivalent lanthanide ions

Ce PrNd

(Er)

Pm

(Ho)

Sm

(Dy)

EuGd

(Yb)(Tm)(Tb)

n

SL

SLJ

SLJM

1

1

2

2

7

34567

17

41

364

47

107

1001

73

198

2002

119

295

3003

119

327

3432

13

9114

Note. Configurations with 14-n electrons (listed in parentheses) have

the same number of states as configurations with n electrons.

Fig. 1. Energy shifts for energy levels of Gd3þup to 51000cm?1due to

truncation of the wavefunctions at different energies.

P.S. Peijzel et al. / Journal of Solid State Chemistry 178 (2005) 448–453

450

Page 4

procedure to ensure that the set of parameters giving the

best agreement yields physically reasonable parameter

values. The standard deviation s is used as a measure for

the quality of this fit.

s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

N ? P

In this formula Eexpand Ecalcare the experimental and

calculated energies for the energy levels. N is the number

of experimental levels and P is the number of

parameters being varied. A parameter set giving a value

of s smaller than 20cm?1is considered to be a good fit

[27].

After calculation of the crystal field levels a simulated

absorption or emission spectrum may be plotted that is

based on the calculated transition energies and inten-

sities that takes into consideration the temperature and

spectral linewidths. This simulated spectrum may be

directly compared with experimental traces, and may

help to facilitate interpretation of the experimental

results.

PðEexp? EcalcÞ2

s

:

ð5Þ

3. Results and discussion

Using the parameter values published by Carnall et al.

[11] we calculate the complete 4fnenergy level diagram

for all lanthanide ions in LaF3 and compare the

calculations with the recent extension of the Dieke

diagram with experimentally observed energy levels up

to 70000cm?1: The highest level calculated was the

1Sð1Þ0level of terbium at approximately 193000cm?1:

The complete 4fnenergy level diagram is depicted in

Fig. 2.

Fig. 3 shows the energy levels calculated in the region

39000 up to 75000cm?1: This is the region that is

experimentally accessible using VUV spectroscopy. In

this figure the calculated absorption energies of the

lowest 4fn-4fn?15d transitions in LaF3(based upon Ref.

[29]) are indicated.

It is now well understood that for configurations with

more than 7 4f-electrons there exist so-called low-spin

and high-spin 4fn?15d states [28]. For example, the

trivalent terbium ion has 8 4f-electrons. When one of

the 4f-electrons is promoted to a 5d-orbital, the

remaining 4f7core has all spins parallel and the spin

multiplicity is 8. There are two possibilities for the

orientation of the spin of the d-electron with this 4f7

core. In the case of a parallel spin of the d-electron, a

high-spin configuration is formed, with a spin-multi-

plicity of 9 which is lower in energy than the low-spin

configuration with the spin of the d-electron opposite to

the parallel 4f-electrons giving a spin multiplicity of 7.

Transitions from the ground state to the low-spin state

do not change the spin of the electrons, and are thus

‘‘spin-allowed’’ and therefore more intense than the

‘‘spin-forbidden’’ transitions to the high-spin states.

Dorenbos [29] reviewed the position of the 4fn?15d

states in many host-lattices and showed that the position

of the first 4fn?15d absorption band in a host-lattice

could be estimated if the position is known for one of

the lanthanide ions. In Fig. 3 the calculated positions of

spin-allowed and spin-forbidden 4fn?15d absorptions

for Ln3þin LaF3are indicated with circles and squares,

respectively.

Not all 4fn-levels in Fig. 3 are expected to be

observed. The first reason for this is that the relatively

weak 4fn-4fntransitions are obscured by intense 4fn-

4fn?15d absorptions that occur in this energy region.

Another reason for the fact that not all energy levels can

be observed is the occurrence of transitions to energy

levels with a change in J of more than six with respect to

the ground state. Transitions to these states are

‘‘forbidden’’ and tend not to be seen. A large number

of energy levels in this region have been observed using

excitation with synchrotron radiation [15,16], an over-

view of these energy levels is given in Fig. 4. If emission

has been observed from this level this is indicated with a

semicircle. Emission originating from a certain energy

level can occur when the energy gap to the next lower

level is more than four or five times the maximum

ARTICLE IN PRESS

Fig. 2. Complete 4fnenergy level diagram for the trivalent lanthanides

in LaF3calculated using parameters reported by Carnall [10].

P.S. Peijzel et al. / Journal of Solid State Chemistry 178 (2005) 448–453

451

Page 5

phonon energy of the host-lattice. If the energy gap is

smaller, then multi-phonon relaxation dominates and

emission is no longer seen. In LaF3 the maximum

phonon energy is about 350cm?1and emission can be

expected from energy levels with an energy gap of

1800cm?1or more to the next lowest level.

For Ce3þ[30,31] and Pr3þ[32,33] all 4fnlevels have

been observed. As can be seen when comparing Fig. 3

with Fig. 4 there are a few levels of Nd3þaround

68000cm?1that have not been measured as they are

situated within the 4f25d absorption band. Downer et

al. [34] successfully applied two-photon excitation to

investigate the 4f7energy levels of Eu2þin CaF2that are

situated within the 4f65d absorption band. Since Nd3þ

shows resonant one-photon absorption at approxi-

mately 34000cm?1; two-photon excitation from the

ground state is not probable. Two-photon absorption

from an excited state of Nd3þmay be used to measure

the position of the VUV levels. Pm3þis a radioactive

ion, with a lifetime of about 2.5 years. As far as we know

there are no reports on the VUV energy levels of Pm3þ:

Below the energy level calculated at 61000cm?1there is

an energy gap of more than 2000cm?1and this indicates

the possibility for emission from this level in LaF3:

However, as Pm3þdecays to Sm3þ(b decay), the Sm3þ

eventually becomes a dopant in the crystal, and we

expect efficient Pm3þ! Sm3þenergy transfer to occur,

quenching the Pm3þemission. For Sm3þsome energy

levelshave been measured

58000cm?1; and there are many levels that still have

to be measured [15]. Many of the Eu3þlevels in the VUV

have been identified, but above 60000cm?1a charge

transfer transition hampers the observation of Eu3þ

excitation lines.

The VUV levels of Gd3þhave been measured up to

68000cm?1and most of the calculated multiplets have

been observed. Recently, high-resolution excited state

absorption measurements showed differences between

calculatedand experimental

150cm?1for the levels probed [18]. This is not un-

expected as the parameter values calculated by Carnall

et al. [10] are based on 70 levels in the UV region of the

spectrum with only a few VUV levels included in the fit.

Moreover, the crystal field parameter values used for the

calculation of the Gd3þenergy levels were set to the

values for Tb3þby Carnall. Upon inclusion of the VUV

levels of Gd3þthe parameter values changed to some

extent, but not more than a few percents.

between52000 and

energiesof upto

ARTICLE IN PRESS

Fig. 3. Energy level diagram for the lanthanides in LaF3in the region

39000–75000cm?1calculated using parameters reported by Carnall

[10]. The calculated lowest positions of the low-spin and high-spin

4fn?15d states in LaF3(based upon Ref. [29]) are indicated with filled

circles and squares, respectively.

Fig. 4. Energy level diagram showing all experimentally observed 4fn

energylevelsfor thetrivalent

39000–70000cm?1: Levels from which emission is observed are

marked with a semicircle.

lanthanidesin therange

P.S. Peijzel et al. / Journal of Solid State Chemistry 178 (2005) 448–453

452

Page 6

The observation of the 4f8levels of Tb3þin the VUV

region of the spectrum is not possible using one photon

techniques as the 4f75d bands start absorbing in the

UV. We expect that two-photon excitation can be used

to probe these levels, in analogy with the Eu2þ

experiments described by Downer et al. [34]. Not all

4f9levels of Dy3þhave been observed yet and for this

ion the 4f85d bands start to absorb at 58000cm?1:

Many of the 4f10levels of Ho3þhave been measured

and recently emission from the

situated at 63000 cm?1has been observed in YF3[17].

Nearly all energy levels of Er3þhave been measured.

The2Fð2Þ5=2level situated at 63000cm?1is the highest

emitting level observed for Er3þin LaF3[16]. In Fig. 3

the2Gð2Þ7=2level is situated around 66500cm?1; just

below the calculated position of the high-spin 4f-5d

absorption. This position is, however, the maximum of

the f-d absorption band so the onset will be at lower

energy and therefore the2Gð2Þ7=2cannot be observed.

Three multiplets situated at even higher energies, the

2Gð2Þ9=2;2Fð1Þ5=2and2Fð1Þ7=2are calculated at 70000,

93000 and 97800cm?1; respectively. For Tm3þthe1S0

level is the only energy level that has not been observed

yet. It is calculated at approximately 73000cm?1and is

situated above the onset of the 4f115d absorption bands.

Finally, Yb3þhas no 4f levels in the VUV.

3Pð1Þ2level of Ho3þ

4. Conclusions

The Hamiltonian parameters for trivalent lanthanide

ions derived by Carnall and co-workers for the LaF3

host are a valuable starting point for the analysis of the

VUV levels of the 4fnconfigurations. Advances in

computer technology now allow routine calculations for

complete 4fnconfigurations. We have generated a

complete 4fnenergy level diagram and begun the

process of analyzing the available experimental data in

the VUV region. Comparison of the calculated energy

level scheme with experimentally obtained energies

allows identification of many 4fnlevels that have been

measured using synchrotron radiation, and predicts

regions where high energy levels of various lanthanides

that have not been observed yet may be probed by one-

or two-photon spectroscopy. Using the extensive VUV

data it should now be possible to further refine the

parameter values.

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