A theoretical calculation of vibronic coupling strength: the trend in the lanthanide ion series and the host-lattice dependence

Journal of Physics and Chemistry of Solids 61 (2000) 1489–1498 www.elsevier.nl/locate/jpcs

A theoretical calculation of vibronic coupling strength: the trend in the lanthanide ion series and the host-lattice dependence A.F. Campos a, A. Meijerink b, C. de Mello Donega´ c,*, O.L. Malta c a Departamento de Quı´mica, UFRPE-52171-900, Recife PE, Brazil Debye Institute, Utrecht University, P.O. Box 80000, 3508TA Utrecht, Netherlands c Universidade Federal de Pernambuco, Departamento de Quı´mica Fundamental, UFPE 50670-901, Recife PE, Brazil b

Received 11 November 1999; accepted 19 December 1999

Abstract This paper reports a comparison between theoretical and experimental aspects of the vibronic coupling strength in lanthanide ion compounds. The theoretical calculations also have the objective of applying and testing a model developed by one of us. Due to the amount of information and results no detailed comparison with calculations based on previous existing models is made. The emphasis is given to the 3 P0 ! 3 H4 ; 5 D0 ! 7 F2 ; 6 P7=2 ! 8 S7=2 and 1 D2 ! 3 H6 transitions of the Pr 3⫹, Eu 3⫹, Gd 3⫹ and Tm 3⫹ ions, respectively. The divalent lanthanide ions Eu 2⫹ and Sm 2⫹ in LiBaF3 are also included in the analysis as model systems. In these systems the agreement between the theoretical and the experimental vibronic rates is quite satisfactory. The calculations reproduce the observation that the vibronic coupling is stronger in the beginning (Pr 3⫹) and at the end (Tm 3⫹) of the lanthanide ion series and weaker at the center (Eu 3⫹, Gd 3⫹), and also the experimentally observed host-lattice dependence. Furthermore, the model also predicts a stronger vibronic coupling for the Eu 2⫹ and Sm 2⫹ ions than for the isoelectronic Gd 3⫹ and Eu 3⫹ ions, as experimentally observed in fluoride host lattices. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: Vibronic transitions; Lanthanide ions; Vibronic theory

1. Introduction The 4f electrons of lanthanide ions are shielded from the surroundings by the outer 5s 2 and 5p 6 shells and, therefore, experience a weak electron–phonon (or vibronic) coupling. As a result, the intra-4f n absorption and luminescence spectra of lanthanide ions are characterized by sharp lines, whose positions are only slightly dependent on the host lattice. Although weak, the interaction of the 4f electrons with the surroundings has important consequences, making the 4f–4f transitions partially electric-dipole allowed and giving rise to many phenomena, such as vibronic transitions, single- and multiphonon non-radiative relaxation, line broadening, phonon assisted energy transfer and crossrelaxation processes. The understanding of the electron– phonon coupling in lanthanide ions is thus a worthwhile pursuit. Still, until recently, only a few systematic investi* Corresponding author. Tel.: ⫹ 55-81-271-8440; fax: ⫹ 55-81271-8442. E-mail address: [email protected] (C.M. Donega´).

gations have been carried out with this purpose [1–4]. Three different approaches have been used to probe the electron– phonon coupling in lanthanide compounds: vibronic transition probabilities [5–12], multiphonon relaxation rates [12–14] and, more recently, temperature-dependent line broadening [12,15–16]. The first systematic investigation of vibronic transitions in lanthanide (Ln) ions was carried out by Hellwege [1] in 1941, by measuring the low temperature (80 K) spectra of all trivalent Ln ions (except Pm 3⫹) in Ln2(SO4)3·8H2O. He concluded that the vibronic coupling was the strongest in the beginning of the series (Pr 3⫹ and Nd 3⫹), reached a minimum at the center (Eu 3⫹, Gd 3⫹) and then increased again from Gd 3⫹ to Tm 3⫹ [1]. A similar trend was reported decades later by Krupke [2], based on the relative intensities of the vibronic lines in the room temperature absorption spectra of single crystals of Y2O3:Ln (Ln 3⫹ ˆ Pr, Nd, Eu, Er, Tm) and LaF3:Ln (Ln 3⫹ ˆ Pr and Nd). Both studies were based on relative intensities, rather than on quantitative measurements of the vibronic coupling strength. A good way of quantitatively expressing vibronic intensities

0022-3697/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(00)00007-X

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A.F. Campos et al. / Journal of Physics and Chemistry of Solids 61 (2000) 1489–1498

is by determining vibronic transition probabilities (AVIB). In recent years, this has been systematically done for a number of Ln ions in several host lattices: Gd3⫹ [5], Pr3⫹ [6–8], Tm3⫹ [9], Eu2⫹ [10] and Sm 2⫹ [11]. The results show that the vibronic transition probabilities can vary more than two orders of magnitude depending on the host lattice. For the same lanthanide ion, the AVIB values increase as the degree of covalence of the host lattice increases [5,6,8,9] and the position of opposite-parity states shifts to lower energies [6,10,11]. Also, the vibronic transition probabilities are orders of magnitude larger for Pr 3⫹ and Tm 3⫹ than for Gd3⫹ in the same host lattices [5,6,9,12], in agreement with the trend proposed by Hellwege [1] and Krupke [2]. Recent measurements of the temperature-dependent line broadening of several 4f transitions for nine Ln 3⫹ ions (Ce, Pr, Nd, Eu, Gd, Tb, Er, Tm and Yb) [12,15,16] lead to a similar trend: the electron–phonon coupling is stronger in the beginning and at the end of the lanthanide series, and weaker in the middle, and, for the same ion, it increases with the degree of covalence of the host lattice. Also, the electron–phonon coupling strength differs for different transitions of the same ion [12,15,16]. These conclusions are corroborated by recently reported multiphonon relaxation rates for Gd 3⫹ and Nd 3⫹ in LiYF4 and La2O3 [12], for Pr 3⫹ in several host lattices [13] and for Tm 3⫹, Ho 3⫹, Nd 3⫹ and Pr 3⫹ in many hosts [14]. The intensity of vibronic transitions of lanthanide ions can be determined by two mechanisms, which have been labeled M and D processes [17]. The former describes vibrationally induced forced electric dipole transitions [18]. Coupling with odd-parity vibrations leads to admixture of opposite-parity configurations into the 4f n configuration, thus making the 4f–4f transitions partially electric-dipole allowed [18]. The D process for vibronic transitions, also known as Franck–Condon replicas, results from a difference between the equilibrium Ln–ligand distances in the ground state and in the excited state, which leads to non-zero Franck–Condon vibrational overlap factors between the ground and excited states [18]. This process does not lead to additional intensity but simply redistributes the total transition probability between purely electronic (zerophonon) and vibronic (n-phonon replica) lines. For a transition within a 4f n configuration the D process contribution is, in many situations neglected, since the Huang–Rhys coupling factor is assumed to be very small even though the observation of two-photon vibronics for Pr 3⫹in SrMoO4 [7] and two-phonon replicas for Pr 3⫹ in Na5La(MO4)4 (M ˆ W, Mo) [8] gives a direct proof that the D process contributes to the vibronic coupling strength of this ion in these hosts. However, in the present analysis this contribution has not been considered for reasons that will be later justified. Currently, there are a few theoretical approaches to describe the intensities of vibronic transitions in lanthanide ions. Faulkner and Richardson [19] developed a general theory for vibrationally induced electric-dipole intensities

of f–f transitions in octahedrally (Oh) coordinated Ln 3⫹ ions. Judd [20] developed a similar approach to the same system using tensorial techniques. These authors considered both the static and dynamic coupling (DC) mechanisms. Stavola et al. [21] developed a theoretical treatment of cooperative vibronic transitions, which was later applied by Dexpert-Ghys and Auzel [22]. Recently, one of us [23] developed a theoretical approach that also takes into account the contributions of the forced electric dipole and dynamic coupling mechanisms, which can be expressed in terms of usual ligand-field parameters for any given point symmetry. This model has been applied for the Eu 3⫹ ion in LiYF4 and Cs2NaEuCl6 [23] and for the 3 P0 ! 3 H4 transition of Pr 3⫹ in LiYF4, Na5La(WO4)4 and Na5La(MoO4)4 [24]. In this paper, we use this theoretical model to calculate the vibronic transition rates for the 6 P7=2 ! 8 S7=2 transition of Eu 2⫹ in LiBaF3 and the 5 D0 ! 7 F2 transition of Sm 2⫹ in LiBaF3. These two systems were chosen as model systems [10,11] for reasons which will be presented below. The host-lattice dependence of AVIB and its trend through the lanthanide series are also addressed by calculating the vibronic transition rates for the 3 P0 ! 3 H4 transition of Pr 3⫹, the 5 D0 ! 7 F2 transition of Eu 3⫹, the 6 P7=2 ! 8 S7=2 transition of Gd 3⫹ and the 1 D2 ! 3 H6 transition of Tm 3⫹ in different host lattices, ranging from ionic to covalent compounds (LiYF4, YOCl and Na5La(WO4)4). The 3 P0 ! 3 H4 transition of Pr 3⫹ and the 1 D2 ! 3 H6 transition of Tm 3⫹ are particularly interesting to investigate since the previous existing theoretical models [19–22] have been applied under the selection rules imposed by the unit tensor operator U (2), which for these transitions has vanishing matrix elements. In all cases the calculated rates are compared to experimental values already published by some of us.

2. Theory The 4f–4f vibronic transitions in lanthanide compounds have been treated theoretically in more detail, during the last two decades, by several authors [19–22,25]. In most of these treatments both the forced electric dipole and dynamic coupling mechanisms are considered. Due to the inherent complexity of the vibronic interactions, these treatments in general require considerable computational effort and they have been mainly applied to centrosymmetric compounds. Many interpretations of experimental data have been made on the basis of the U (2) unit tensor operator only, which leads to rather strict selection rules on the total angular momentum J of the 4f states [19,20], in contrast to what is indeed experimentally observed for the lanthanide ions series [4– 9,11,21,22]. The theoretical approach used in the present work is an alternative treatment which minimizes the computational demand, particularly concerning normal coordinate analysis, while keeping a general character in the sense that it can

A.F. Campos et al. / Journal of Physics and Chemistry of Solids 61 (2000) 1489–1498

be used for any symmetry, without the restrictions on the J quantum number selection rules imposed by the U (2) unit tensor operator [23]. Further, it is based on a more reliable representation of the ligand field Hamiltonian than the crude electrostatic approximation. As emphasized in Ref. [23], the model is expected to be more appropriate to the case of integrated vibronic intensities.

 b4 ˆ ⫺3



3 4p

ប  2v0 

1=2 

1=2

…l ⫹ 1†…2l ⫹ 3† …2l ⫹ 1†

X

1=2

1491

a…1 ⫺ s l †具r l 典具 f 储C …l† 储 f 典

…2K ⫹ 1† × …⫺1† 2 ge r2K⫹1 K;p 0 ;m; j p0

!

1

l⫹1

K

m

⫺p

p0

1 K 0

l⫹1

0

0

    2Rj 1 dK;l ⫹ dK;l⫹2 × g…K†ⴱ p 0 … j† 2Q m R2

2.1. Vibronic transition probabilities

…6†

The vibronic transition probability (AVIB) stands for a quantitative measure of the vibronic coupling and is given by the following expression [26]: " # X 64pe2 AVIB ˆ x s 3 V lVIB 具J储U …l† 储J 0 典2 …2J ⫹ 1†⫺1 …1† 3h l where x is the local field correction term, 具J储U …l† 储J 0 典 the reduced matrix element of the unit tensor operator U (l ) connecting the electronic states J and J 0 and s the transition energy in cm ⫺1. The intensity parameters V lVIB …l ˆ 2; 4 and 6) are directly related to the BVIB l;t;p quantities through the equation [27,28]:

V lVIB

2 X 兩BVIB l;t;p 兩 : ˆ …2l ⫹ 1† t;p …2t ⫹ 1†

…2†

The Bl ,t,ps, according to the present model, contain the following contributions [23]: l⫹2

具r 典 Q …l ⫹ 1; l† DE

b1 ˆ ⫺ ‰…l ⫹ 2†…2l ⫹ 3†…2l ⫹ 5†Š1=2 

ប 2v 0

! 1=2

X

1

p 0 ;m; j

m

×

l⫹1 l⫹2 p0

p

!

g p…l0 ⫹2†ⴱ … j†

2Rj 2Q

! m

…3† "

…l ⫹ 1†…l ⫹ 2†…2l ⫹ 5† b2 ˆ …2l ⫹ 3† …2l ⫹ 1†

#1=2

a具r l 典 ប 具 f 储C…l† 储 f 典 × 2v0 ge2 r2l⫹3 ! ! 1 l⫹1 l⫹2 2Rj … l⫹2†ⴱ gp0 … j† 2Q m m p p0

p 0 ;m; j

3 b3 ˆ ⫺ n p  

!1=2

0 2Rj 2Q

0 !

0 2 m

…4†

X 0 具rl⫹2 典 Q…l ⫹ 1; l† …⫺1† p …2K ⫹ 1† DE K;p 0 ;m; j

l⫹1

1 K

R2

!

ប 2v 0

dK;l ⫹

!1=2

1 d 2 K;l⫹2

ប 2v 0



! 1=2

X p 0 ; m; j

…7†   0 n 具r l 典 ប 1=2 X b6 ˆ ⫺ p Q …l ⫺ 1; l† …⫺1†p …2K ⫹ 1† p DE 2v 0 0 K;p ;m; j

1 K



0

 

0

2

l⫺1 0

dK;l⫺2 ⫹

R2

! ×

1 d 2 K;l

1

l⫺1

K

m 

⫺p

p0

!

g …K†ⴱ p 0 … j†



2R j 2Q

 m

(8)

where DE is the energy difference between the barycenters of the excited 4f n⫺15d and ground 4f n configurations, 具rl 典 is a radial expectation value, Q…t; l† is a numerical factor, s l is a shielding factor, v0 is the angular frequency of the normal mode Q, C (l ) is a Racah tensor operator of rank l and R 2 the average of the squared distances Rj. The quantities g tp 0 s are the so-called ligand field parameters, which, from the simple overlap model (SOM) [29], are given by X g pt 0 s ˆ r j …2b j † t ⫹1 gpt … P:C:; j † …9† where rj is the magnitude of the total overlap between the 4f and the ligand wave functions bj ˆ 1=…1 ⫹ rj † and   4p 1=2 g j e2 t g pt … P:C:; j † ˆ Y p …V j † …10† 2t ⫹ 1 Rt⫹1 j

 …1 ⫺ s l † X

具rl 典 Q …l ⫺ 1; l† DE ! ! 1 l⫺1 K 2Rj …l†ⴱ × g … j† 0 p 2Q m m p p0

b5 ˆ ⫺‰l…2l ⫺ 1†…2l ⫹ 1†Š1=2

j

!1=2



!

1

l⫹1

K

m !

⫺p

p0

!

g…K† p 0 … j† …5†

is the ligand field parameter due to the jth ligand with charge ⫺gj e as given by the point charge (P.C.) electrostatic model. Eq. (9) should be interpreted as a ligand field parameter produced by effective charges ⫺rj gj e located around the midpoints of the lanthanide–ligand chemical bonds. In this way the charge factors gj are more appropriately treated as parameters that no longer have to be given by the valences of the ligand atoms. It is assumed that the overlap rj varies as !n R rj ˆ r0 0 …11† Rj

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3. Results and discussion 3.1. Theoretical vibronic rates for the LiBaF3:Eu 2⫹ and LiBaF3:Sm 2⫹ systems

Fig. 1. Energy level schemes for the isoelectronic (4f 7) Gd 3⫹ and Eu 2⫹ ions (after Ref. [10]). The J values are 3/2, 5/2 and 7/2, respectively. Note the breaks in the energy scale.

where R0 is the smallest among the Rj s, with r0 ˆ 0:05 and n ˆ 3:5 [30]. The quantities b1, b3, b5 and b6 stand for the contributions from the forced electric dipole mechanism, and the quantities b2 and b4 are the contributions from the dynamic coupling mechanism. A relevant point is that these contributions may have opposite signs [23]. Special attention is given to the b3 and b4 contributions. The b3 contribution takes into account the dependence of the effective charges that produce the ligand field interaction on the distances between central ion and ligands. The b4 contribution takes into account the distance dependence of the ligand polarizabilities. These quantities have not been considered by the previous theoretical models [19–22,25]. In our calculations we assume the approximation introduced p in Ref. [23], i.e. …2Rj =2Q†m ˆ 1= 3Mj ; where Mj is an effective reduced mass. This approximation makes the sums over j in Eqs. (3)–(8) equal to the so-called even-rank ligand field parameters defined in Eq. (9).

Fig. 2. Energy level schemes for the isoelectronic (4f 6) Eu 3⫹ and Sm 2⫹ ions (after Ref. [11]). Note the breaks in the energy scale.

For lanthanide ions in centrosymmetric sites the electric dipole transitions are forbidden in the first-order. However, vibronic coupling occurs with odd-parity vibrations that break the parity selection rule for the f–f transition. These vibronics are called vibrationally induced forced electric dipole transitions, or M-process vibronics [17]. Thus, hosts in which the Ln 3⫹ ion occupy a centrosymmetric site are ideal for quantitative studies of vibronic transitions, since the vibronic lines can be more easily identified and there is no interference from purely electronic lines (apart from some weak magnetic dipole allowed transitions). The compound LiBaF3 has the perovskite structure, in which divalent lanthanide ions substitute for the Ba 2⫹ ion and, therefore, occupy a cubic twelve-coordinated site [10]. The vibronic intensities for Eu 2⫹ and Sm 2⫹ in LiBaF3 single crystals have been previously investigated by one of us at temperatures down to 4.2 K [10,11]. The Eu 2⫹ ion is isoelectronic with Gd 3⫹ (4f 7), and the Sm 2⫹ ion is isoelectronic with Eu 3⫹ (4f 6). Thus, the energy level schemes of these two sets of isoelectronic ions are similar (Figs. 1 and 2), apart from the shift of the energy levels to lower energies in the divalent ions. The most striking difference is the much lower position of the 4f n⫺15d excited states in the divalent ions. Depending on the host lattice these states can be at such low energies that emission from the 6P7/2 state of Eu 2⫹ is no longer observed. In LiBaF3:Eu 2⫹ this is not the case, allowing the investigation of the 6 P7=2 ! 8 S7=2 transition [10]. Furthermore, the vibronic lines observed for Eu 2⫹ and Sm 2⫹ in LiBaF3 are strong and very narrow …Dn1=2 ⬃ 3:5 cm⫺1 †; allowing for an accurate determination of the vibronic transition probabilities in emission and excitation, at temperatures from 4.2–150 K [10,11]. As mentioned above, the states belonging to the 4f 7 and 4f 65d configurations of Eu 2⫹ are shifted to lower energies relative to those of Gd 3⫹ (Fig. 1). A similar behavior is observed for the Sm 2⫹ ion (Fig. 2). These facts have an important consequence. Currently, there is experimental evidence that the vibronic transition probabilities (AVIB) depend strongly on the degree of covalence and on the energy difference between the opposite parity configurations and the 4f configuration [5,6,8–12]. Thus, considering that the opposite parity 4f n⫺15d states are closer to the 4f n ones in the divalent ions than in the trivalent isoelectronic ions, one expects the AVIB values for Eu 2⫹ and Sm 2⫹ in LiBaF3 to be larger than those for Gd 3⫹ and Eu 3⫹ in similar fluoride lattices. This is confirmed by the experimental observation that the AVIB values are two orders of magnitude higher for the two divalent ions than for the isoelectronic trivalent ions [10,11]. The characteristics mentioned above make LiBaF3:Eu 2⫹ and LiBaF3:Sm 2⫹ model systems for testing theories on

A.F. Campos et al. / Journal of Physics and Chemistry of Solids 61 (2000) 1489–1498 Table 1 Numerical values of the quantities used in the calculation of the vibronic transition probabilities AVIB. The ligand field parameters k g…K† p are in units of erg=具r 典. The term DE is the energy difference between 4f n ground configuration and 4f n⫺1 5d excited configuration. s is the frequency of the zero-phonon transition of the Ln 2⫹ ion ( 6 P 7=2 ! 8 S7=2 for Eu 2⫹ and 5 D0 ! 7 F2 for Sm 2⫹). The term v 0 gives the frequency differences between the zero-phonon line and the observed vibronic lines. The values of DE, s and v 0 are from Refs. [10,11].The values for the radial integrals 具r l 典 are also given [41]. The refraction index n0 is taken as 1.5 for both host lattices

g00 g20 g40 g44 g60 g64 v 0 (cm ⫺1) DE (cm ⫺1) s (cm ⫺1) 具r 2 典 具r4 典 具r6 典

LiBaF3:Eu 2⫹

LiBaF3:Sm 2⫹

3:344 × 10⫺12 0.0 1:088 × 1020 6:25 × 1019 – – 60, 102, 144, 270, 465 18 000 27 854 3:07 × 10⫺17 2:64 × 10⫺33 5:17 × 10⫺49

3:344 × 10⫺12 0.0 1:088 × 1020 6:25 × 1019 – – 61,141 16 000 13 915 3:29 × 10⫺17 3:02 × 10⫺33 6:26 × 10⫺49

vibronic transitions of Ln ions. Therefore it seemed worthwhile to apply the theoretical model described above to calculate the vibronic transition rates for the 6 P7=2 ! 8 S7=2 transition of Eu 2⫹ in LiBaF3 and the 5 D0 ! 7 F2 transition of Sm 2⫹ in LiBaF3. The experimental AVIB values are determined from decay time measurements and the ratio between the total vibronic intensity and the total zero-phonon line intensity, as described in detail in Refs. [10,11]. The estimated error is about 5%. The theoretical AVIB values have been calculated as described above. Values of radial integrals, DE; ligand field parameters g Kq ; the frequency differences between the zero-phonon line and the vibronic lines observed for the Eu 2⫹ and Sm 2⫹cases and the refraction index used in the calculations are presented in Table 1. The g Kq values were taken from Refs. [31,32]. In these references g Kq correspond to the Cs2NaBkCl6 cubic system.

1493

g Kq

for the LiBaF3 system, we use the factor To obtain the ˚ being the distance Bk– g Kq ˆ g Kq …Cl†…2:58=2:83†K⫹1 ; 2.58 A ˚ the distance Ba–F in LiBaF3. Cl in Cs2NaBkCl6 and 2.83 A It is important to mention that for the AVIB calculations, the b2 and b4 contributions contain the shielding factors s l . The s l values for the trivalent lanthanide ions can be found in Refs. [33–39]. However, there are no suitable values for the divalent ions Eu 2⫹ and Sm 2⫹ (in these cases, s 2). To obtain the s 2 values for these ions, we used the following approximation: …1 ⫺ s l † ˆ rj …2bj † l⫹1

…12†

This approximation has been applied to the systems Eu(TTA)3·2DBSO (TTA ˆ thenoyltrifluoroacetonate and DBSO ˆ dibenzyl sulfoxide) [30], for l ˆ 0; and LiYF4:Nd 3⫹ [40], for l ˆ 2; 4 and 6, in the analysis of intensity parameters and intramolecular energy transfer processes. The AVIB values for the LiBaF3:Eu 2⫹ and LiBaF3:Sm 2⫹ systems were obtained with polarizability (a ) and charge factor (g) values, in both cases, equal to 1:0 × 10⫺24 cm3 and 1.0, respectively. A comparison between the experimental and theoretical vibronic rates for the isoelectronic lanthanide ions in fluoride host lattices is presented in Table 2. Further details on the evaluation of the theoretical vibronic intensity parameters will be given in the next section. It is important to comment that the existing theoretical models predict high vibronic rates when the square of the reduced matrix element of U (2) has high values [19,20]. For the 6 P7=2 ! 8 S7=2 transition of the Eu 2⫹ and Gd 3⫹ ions, (U …2† †2 ˆ 0:0011 (Table 3), and for the 5 D0 ! 7 F2 transition of the Sm 2⫹ and Eu 3⫹ ions, (U …2† †2 ˆ 0:0032 (Table 3). Thus, on this basis, one can expect low vibronic rates for these two sets of ions. The experimental results reported in Refs. [10,11] and the theoretical values presented in Table 2 show that this does not occur. The AVIB value for the 6 P7=2 ! 8 S7=2 transition of Eu 2⫹ in LiBaF3 is about two orders of magnitude larger than the AVIB value for the same transition of Gd 3⫹ in LiYF4 [10], despite that the squared reduced matrix element of U (2) is about the same. This is possibly due to the fact that the opposite-parity excited states of the Sm 2⫹ and Eu 2⫹ ions are shifted to lower energies relative to

Table 2 Experimental [10,11] and theoretical vibronic rates (AVIB) for the isoelectronic lanthanide ions Sm 2⫹ and Eu 3⫹ (5 D0 ! 7 F2 transition), and Eu 2⫹ and Gd 3⫹ (6 P7=2 ! 8 S7=2 transition) in fluoride host lattices. The experimental values were obtained at 4.2 K [10,11]. The AVIB values concern the integrated intensities over all the observed vibronic lines. The values for the polarizability and charge factor used in the calculations are also given Host lattice

LiBaF3:Sm 2⫹ LiYF4:Eu 3⫹ LiBaF3:Eu 2⫹ LiYF4:Gd 3⫹

AVIB (s ⫺1) Experimental

Theoretical

15 ⱕ5 245 6

10 2.8 193 2.9

Polarizability (a × 10 ⫺24 cm 3)

Charge factor

1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0

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A.F. Campos et al. / Journal of Physics and Chemistry of Solids 61 (2000) 1489–1498

Table 3 Square of the reduced matrix elements for some selected transitions of the Pr 3⫹, Gd 3⫹, Eu 3⫹ and Tm 3⫹ ions [31] Transition

(U (2)) 2

(U (4)) 2

(U (6)) 2

P0 ! 3 H4 (Pr 3⫹) P7=2 ! 8 S7=2 (Gd 3⫹) 5 D0 ! 7 F2 (Eu 3⫹) 1 D2 ! 3 H6 (Tm 3⫹)

0.0 0.0011 0.0032 0.0

0.173 0.0 0.0 0.3074

0.0 0.0 0.0 0.093

3 6

those of the isoelectronic Eu 3⫹ and Gd 3⫹ ions, as shown in Figs. 1 and 2. This favors the admixture of opposite-parity states resulting in an increase of the vibronic rates. It is likely that the energy differences between excited states of the 4f n⫺15d and 4f n configurations are not the only reason since the vibronic rates depend on several other factors. However, in these systems the degree of covalence and the polarizability of the ligands are similar and no large variation of AVIB due to host-lattice effects is expected. Finally, the present model predicts vibronic rates for the LiBaF3:Eu 2⫹ and LiBaF3:Sm 2⫹ systems that are in good agreement with the experimental values. It is important to mention that these systems are considered as model systems to test the theoretical models of vibronic intensities. Furthermore, it reproduces the fact that the vibronic coupling is stronger for the Eu 2⫹ and Sm 2⫹ ions when compared with the Gd 3⫹ and Eu 3⫹ ions in fluoride host lattices. 3.2. The trend of the vibronic transition probabilities through the lanthanide ion series (Pr 3⫹ ! Tm 3⫹) and its host-lattice dependence As it was mentioned above, the trend in the vibronic coupling strength has been investigated experimentally.[1,2,12,15,16] The results pointed out that the vibronic coupling is stronger in the beginning (Pr 3⫹, Nd 3⫹) and at the end (Er 3⫹, Tm 3⫹) of the lanthanide ion series and weaker at Table 4 Theoretical values for ratio V lDC =V lED for the Pr 3⫹ ion in LiYF4, YOCl and Na5La(WO4)4, assuming different values for the oxygen polarizability Polarizability a oxygen (10 ⫺24 cm 3)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

LiYF4

YOCl

Na5La(WO4)4

0.086 – – – – – – – – –

0.021 0.060 0.125 0.216 0.334 0.480 0.656 0.863 1.101

0.014 0.056 0.123 0.215 0.331 0.470 0.629 0.809 1.010

the center (Eu 3⫹, Gd 3⫹).[1,2,12,15,16] Therefore, it would be interesting to verify if the present model reproduces this behavior. The compounds LiYF4, YOCl and Na5La(WO4)4 were chosen because there are experimental results for Pr 3⫹, Tm 3⫹, Eu 3⫹ and Gd 3⫹ in these hosts, providing a good range from typically ionic to covalent host lattices. LiYF4 and Na5La(WO4)4 have the scheelite structure, space group I41/a, where the Ln 3⫹ ion occupies a site with S4 point symmetry [42,43]. YOCl has the layered PbFCl structure, space group P4/nmm, with site symmetry C4v for the Ln 3⫹ ion [44]. In order to obtain information on which contribution (forced electric dipole or dynamic coupling) is dominant in the AVIB values for the LiYF4, YOCl and Na5La(WO4)4 systems with the Pr 3⫹, Eu 3⫹, Gd 3⫹ and Tm 3⫹ ions, we estimate the V lDC =V lED ratios in all these cases where V lDC stands for the vibronic intensity parameter that contain the b2 and b4 contributions, and V lED takes into account the b1, b3, b5 and b6 contributions. To illustrate the calculations we consider the case of the 3 P0 ! 3 H4 transition of the Pr 3⫹ ion in the YOCl host (C4v point symmetry). For this transition in this host, according to the point symmetry and DJ selection rules, the BVIB l;t;p s of interest are B4,5,0, B4,5,1, B4,5,3, B4,3,0, B4,3,1 and B4,3,3. The terms B4,5,0, B4,5,1 and B4,5,3 are due to t ˆ l ⫹ 1; and B4,3,0, B4,3,1 and B4,3,3 are due to t ˆ l ⫺ 1: The expressions for the corresponding intensity parameters are

V 4ED …v0 † ˆ

9 11

⫹

V 4DC …v0 † ˆ

9 11

……B4;5;0 †2 ⫹ 2…B4;5;1 †2 ⫹ 2…B4;5;3 †2 † 9 7

……B4;3;0 †2 ⫹ 2…B4;3;1 †2 ⫹ 2…B4;3;3 †2 †

……B4;5;0 †2 ⫹ 2…B4;5;1 †2 ⫹ 2…B4;5;3 †2 †

…13† …14†

In Eq. (13), B4,5,0, B4,5,1 and B4,5,3 correspond to the sum of b1 and b3, while B4,3,0, B4,3,1 and B4,3,3 correspond to the sum of the b5 and b6 electric dipole contributions for a given frequency, v 0, of normal mode. In Eq. (14), B4,5,0, B4,5,1 and B4,5,3 correspond to the sum of the b2 and b4 dynamic coupling contributions, also for the same given frequency of normal mode. The factor (two) that multiplies some of the VIBⴱ p VIB BVIB l;t;p s takes into account the fact that Bl;t;p ˆ …⫺1† Bl;t;⫺p DC;ED [23]. The total V l is the sum of the V l …v0 †s over all the normal modes considered. The results are presented in Table 4. We should call attention to the fact that in the evaluation of the total V lVIB parameter, Eqs. (1) and (2), the Bl ,t,ps corresponding to the dynamic coupling and electric dipole contributions, for a given normal mode, should be added up before taking the square in Eq. (2). After that, the sum over the normal modes (v 0) is performed. Tables 4–7 show the V lDC =V lED ratios for the Pr 3⫹, Eu 3⫹, Gd 3⫹ and Tm 3⫹ ions in the LiYF4, YOCl and Na5La(WO4)4 hosts. From both quantum mechanical calculations and experimental determinations, it has been noticed that the

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Table 5 Theoretical values for the ratio V lDC =V lED for the Eu 3⫹ ion in LiYF4, YOCl and Na5La(WO4)4, assuming different values for the oxygen polarizability

Table 7 Theoretical values for the ratio V lDC =V lED for the Tm 3⫹ ion in LiYF4, YOCl and Na5La(WO4)4, assuming different values for the oxygen polarizability

Polarizability a oxygen (10 ⫺24 cm 3)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

LiYF4

YOCl

Na5La(WO4)4

Polarizability a oxygen (10 ⫺24 cm 3)

4.642 – – – – – – – – –

0.021 0.028 0.083 0.168 0.271 0.385 0.500 0.616 0.732

0.390 1.026 1.804 2.598 3.168 4.033 4.647 5.195 5.679

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

polarizabilities of the F ⫺ and Cl ⫺ ions are practically independent of the chemical environment. On the other hand, the polarizability of the O 2⫺ ion may change considerably with the nature and structure of the chemical environment [41– 49]. For the fluorine and chlorine ions we have taken polarizability values equal to 1:0 × 10⫺24 cm3 and 3:0 × 10⫺24 cm3 ; respectively [45]. In the case of the O 2⫺ ion we have performed several calculations by assuming polarizability values in the physically acceptable range from 0:5 × 10⫺24 to 4:5 × 10⫺24 cm3 : It can be concluded from Table 4 that the electric dipole mechanism is clearly dominant in LiYF4:Pr 3⫹. In the other two systems (YOCl:Pr 3⫹ and Na5La(WO4)4:Pr 3⫹), the V lDC =V lED ratios increase with the oxygen polarizability. However, only for aoxygen ˆ 4:5 × 10⫺24 cm3 ; the dynamic coupling and the electric dipole contributions have practically the same magnitude. Similar conclusions can be drawn from Table 7. However, in this case the dynamic coupling mechanism is dominant for smaller values of the oxygen polarizability. For the systems YOCl:Gd 3⫹, Table 6 Theoretical values for the ratio V lDC =V lED for the Gd 3⫹ ion in LiYF4, YOCl and Na5La(WO4)4, assuming different values for the oxygen polarizability Polarizability a oxygen (10 ⫺24 cm 3)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

LiYF4

YOCl

Na5La(WO4)4

1.845 – – – – – – – – –

0.026 0.090 0.243 0.432 0.631 0.826 1.012 1.191 1.352

0.638 1.795 3.181 4.587 5.962 7.148 8.256 9.247 10.13

LiYF4

YOCl

Na5La(WO4)4

0.531 – – – – – – – – –

0.196 0.540 1.027 1.703 2.478 3.233 4.315 5.333 6.437

0.094 0.366 0.796 1.369 2.065 2.869 3.766 4.732 5.779

Na5La(WO4)4:Gd 3⫹, YOCl:Eu 3⫹ and Na5La(WO4)4:Eu 3⫹ (Tables 5 and 6) one observes the same behavior. For the LiYF4:Gd 3⫹ and LiYF4:Eu 3⫹ systems, the dynamic coupling mechanism is clearly dominant when compared to the systems LiYF4:Pr 3⫹ and LiYF4:Tm 3⫹ where the electric dipole mechanism is more relevant. The model, therefore, predicts that the vibronic rates increase with the ligand polarizabilities, and thus also, although from a rather crude chemical point of view, with covalence, in agreement with the experimentally observed behavior [5,6,8,9,12]. Table 8 shows the experimental and theoretical vibronic transition probabilities (AVIB(exp) and AVIB(theor)) for the 3 P0 ! 3 H4 ; 5 D0 ! 7 F2 ; 6 P7=2 ! 8 S7=2 and 1 D2 ! 3 H6 transitions of the Pr 3⫹, Eu 3⫹, Gd 3⫹ and Tm 3⫹ ions, respectively, in LiYF4, YOCl and Na5La(WO4)4. The AVIB value for the Pr 3⫹ ion in YOCl was obtained with the oxygen polarizability (a ) and charge factor (g) equal to 2:5 × 10⫺24 cm3 and 2.0, respectively. These values were 2:0 × 10⫺24 cm3 and 0.8, respectively, for Pr 3⫹ in Na5La(WO4)4. The charge factors for the oxygen are different in the two compounds because in YOCl all the chemical bonds are ionic, whereas in the tungstate the oxygen anions are covalently bound to the W(VI) ion. For the Eu 3⫹ ion in these same hosts the polarizability and charge factor values were the same as those for the Pr 3⫹ ion. In the case of the Gd 3⫹ ion, the oxygen polarizability values were 4:0 × 10⫺24 and 2:0 × 10⫺24 cm3 , respectively, in YOCl and Na5La(WO4)4, with charge factors equal to 2.0 and 0.8, respectively. Finally, for the Tm 3⫹ ion the AVIB values were obtained with the oxygen polarizabilities in YOCl and Na5La(WO4)4 equal to 2:5 × 10⫺24 and 3:5 × 10⫺24 cm3 ; respectively. The charge factor values were also 2.0 and 0.8, respectively. For the LiYF4 host, independently of the lanthanide ion, the values of polarizability and charge factor were taken as 1:0 × 10⫺24 cm 3 and 1.0. The experimental vibronic transition probabilities (AVIB(exp)) were all obtained at 4.2 K, by measuring the intensity of the vibronic lines relative to the zero-phonon

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Table 8 Experimental (exp) and theoretical (theor) vibronic transition probabilities (AVIB) for the 3 P0 ! 3 H4 transition of Pr 3⫹, the 5 D0 ! 7 F2 transition of Eu 3⫹, the 6 P7=2 ! 8 S7=2 transition of Gd 3⫹ and the 1 D2 ! 3 H6 transition of Tm 3⫹ in LiYF4, YOCl and Na5La(WO4)4. The AVIB values concern the integrated intensities over all the observed vibronic lines. The experimental AVIB values were obtained from measurements performed at 4.2 K (LiYF4:Pr 3⫹, [6] Na5La(WO4)4:Pr 3⫹,[8] YOCl:Pr 3⫹,[50] Gd 3⫹,[5] LiYF4:Eu 3⫹,[11] Na5La(WO4)4:Eu 3⫹[9] and Tm 3⫹ [9]) Host lattices

LiYF4 YOCl Na5La(WO4)4

Pr 3⫹ AVIB (s ⫺1)

Gd 3⫹ AVIB (s ⫺1)

Eu 3⫹ AVIB (s ⫺1)

Tm 3⫹ AVIB (s ⫺1)

Theor

Exp

Theor

Exp

Theor

Exp

Theor

Exp

750 5190 2903

100 4600 1400

2.9 14 88

6 114 –

2.8 1.0 54

ⱕ4 – 40

513 813 1065

650 850 1100

line and the radiative transition probability for the zero-phonon line, which can be obtained from decay time measurements. This procedure is described in detail in Refs. [6,8,9]. It is important to notice that each AVIB value has an experimental error, which can vary considerably depending on the lanthanide ion and the host lattice. For Gd3⫹ and Pr3⫹ the values are relatively accurate since the vibronic lines are observed next to one zero-phonon line of a transition to a non-degenerate level and at low temperatures (6 P7=2 ! 8 S7=2 for Gd3⫹ and 3 H…1† 4 ! 3 3 1 P0 for Pr3⫹). However, the 3 H…1† 4 ! P1 ; I6 excitation lines of Pr 3⫹ start at about 400 cm ⫺1 higher energy, thus imposing a limit on the observable vibronics [6]. For Tm 3⫹ the error is larger due to the ligand field splitting of the final level and uncertainties in the contribution of the 1 D2 ! 3 H6 transition upon 3 H6 ! 1 D2 excitation [9]. Therefore, the estimated error for the AVIB values is about 10–20% for Gd 3⫹ in all three hosts [5] and for Pr 3⫹ in LiYF4 and YOCl [6,50], ⬃30% for Pr 3⫹ in Na5La(WO4)4 [8], and up to 50% for Tm 3⫹ in Na5La(WO4)4 [9].

A comparison between the experimental and theoretical values for all the investigated cases is presented in Fig. 3. The correlation between theory and experiment for the trend of the vibronic coupling strength through the lanthanide ion series is very good. The experimentally observed hostlattice dependence of the vibronic rates is also well reproduced by the theoretical calculations. The agreement between theoretical and experimental vibronic rates is fairly good in the cases of the Eu 3⫹, Gd 3⫹ and Tm 3⫹ ions. For the Pr 3⫹ ion there is a rather large discrepancy between the experimental and theoretical values, but the host-lattice dependence is still well reproduced. As it was discussed by some of us in a previous work [24], the inclusion in the calculations of higher order corrections from electronic correlation and the so-called D process [17] may lead to theoretical vibronic rate values that are in better agreement with the experimental ones. However, the inclusion of these contributions would require an enormous increase in the computational demand, and since the focus of the present

Fig. 3. Experimental and theoretical vibronic transition probabilities (AVIB) for the 3 P0 ! 3 H4 transition of Pr 3⫹, the 5 D0 ! 7 F2 transition of Eu 3⫹, the 6 P7=2 ! 8 S7=2 transition of Gd 3⫹ and the 1 D2 ! 3 H6 transition of Tm 3⫹ in LiYF4, YOCl and Na5La(WO4)4. The experimental values were obtained at 4.2 K (LiYF4:Pr 3⫹,[6] Na5La(WO4)4:Pr 3⫹,[8] YOCl:Pr 3⫹,[50] Gd 3⫹,[5] LiYF4:Eu 3⫹,[11] Na5La(WO4)4:Eu 3⫹[9] and Tm 3⫹[9]).

A.F. Campos et al. / Journal of Physics and Chemistry of Solids 61 (2000) 1489–1498

work is on the trend of the coupling strength along the lanthanide ion series we have decided not to include them. The predicted vibronic rates for the Pr 3⫹ and Tm 3⫹ ions are larger than the vibronic rates for the Gd 3⫹ and Eu 3⫹ ions, in agreement with experiment. One factor that contributes strongly to this behavior is the small values of the reduced matrix elements (U (2)) for these latter ions. However, this cannot be the only reason, otherwise one would expect larger vibronic rates for the Tm 3⫹ ion than for the Pr 3⫹ ion, since the matrix element (U (4)) for the Tm 3⫹ ion is almost twice that for the Pr 3⫹ ion. Thus, we have noticed in our calculations that factors such as the energy difference between the barycenters of the excited 4f n⫺15d and ground 4f n configurations (DE), polarizabilities and charge factors may give a dominant contribution to this behavior. We have also noticed in these calculations that for the Tm 3⫹ ion the contribution from the rank 6 effective operator is negligible.

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

4. Concluding remarks

[15]

The theoretical calculations carried out in the present work reproduce successfully the experimental trend of the vibronic coupling strength through the lanthanide ion series: stronger in the beginning (Pr 3⫹) and the end (Tm 3⫹) of the series and weaker at the center (Eu 3⫹, Gd 3⫹). The experimentally observed host-lattice dependence is also well reproduced. In addition, the model, which conveniently expresses the vibronic intensity parameters in terms of the usual even-rank ligand field parameters, has proven to be useful in the prediction of vibronic intensities for 兩DJ兩 ⱕ 6; in any point symmetry, as in the case of the 3 P0 ! 3 H4 and 1 D2 ! 3 H6 transitions of the Pr 3⫹ and Tm 3⫹ ions, respectively. In the case of the divalent lanthanide ions in LiBaF3, the model leads to vibronic rates in good agreement with the experimental values. Furthermore, it predicts that the vibronic coupling is stronger for the Eu 2⫹ and Sm 2⫹ ions when compared with the Gd 3⫹ and Eu 3⫹ ions in fluoride host lattices, as experimentally observed. The AVIB values concern the integrated intensities over all the observed vibronic lines.

[16] [17]

[18]

[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

Acknowledgements [31]

The authors acknowledge the Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq, Brazilian agency) for financial support.

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