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Enhanced approach to the Eu3+ ion 5D0 → 7F0 transition intensity
Optical Materials 35 (2013) 1633–1635
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Enhanced approach to the Eu3+ ion 5D0 ? 7F0 transition intensity A.S. Souza ⇑, Y.A.R. Oliveira, M.A. Couto dos Santos Federal University of Sergipe, Physics Department, 49100-000 São Cristóvão, SE, Brazil
a r t i c l e
i n f o
Article history: Received 21 June 2012 Received in revised form 1 April 2013 Accepted 8 April 2013 Available online 25 May 2013
a b s t r a c t The dependence of the 5D0 ? 7F0 transition intensity with the linear terms of the crystal field parameter and polarizability of the Eu3+ ion is predicted. It is shown that the breakdown of the closure approximation mechanism is the most important one if the Eu3+ ion has a finite dipole moment in the ground state. The Eu:BFCl crystal is used for comparison with a satisfactory agreement. Ó 2013 Elsevier B.V. All rights reserved.
Keywords: 5 D0 ? 7F0 transition Crystal field theory Judd–Ofelt theory
1. Introduction The standard interpretation of the f-spectra of the lanthanide ions in various materials is based on Judd–Ofelt theory and the 5 D0 ? 7F0 (0 ? J) transition of Eu3+ ion in several sites is a very beautiful example [1,2]. The 0 ? 0 electric dipole transition is used in technological applications, because it reveals non-equivalent sites in a given host matrix [3–5]. However, this transition is forbidden by a selection rule of the Judd–Ofelt theory [6,7]. Various mechanisms were proposed in order to describe the dependence on 0 ? 0 transition with the environment. The J-mixing mechanism leads to changes in intensity of the 0 ? 0 transition of nonequivalent sites by mixing of the J = 0 state to states with J – 0 [7,8]. In some cases the strongest transitions cannot simply be ascribed to J-mixing effect, and are explained by contribution of the so-called Wybourne–Downer mechanism (WDM) [9] and the breakdown of the closure approximation in the Judd–Ofelt theory (BCJO) mechanisms [8]. The J-mixing effect contribution to the 0 ? 0 transition describes the participation of the 7F2, 7F4 and 7F6 states in the final composition of the 7F0 states. This contribution is easily obtained only for symmetry where the crystal field parameters B21 ¼ 0 and selection rules are available only from group theory. Besides, in the case of the Eu3+:BaFCl and of the Sm2+ doped fluoride glass ab initio model is practically impossible. The usual way of considering these two mechanisms has been semi-empirical methods, because it is difficult to obtain theoretical values of the radial integrals h4f|rk|nli and charge-transfer bands. The present paper is devoted to the theoretical description of the 0 ? 0 transition ⇑ Corresponding author. Tel.: +55 79 98087125. E-mail address: [email protected] (A.S. Souza). 0925-3467/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optmat.2013.04.010
intensity using the relationship with crystal field invariants and polarizability of the Eu3+ ion. 2. Theory The experimental observation and selection rules from point group theory show that the 0 ? 0 transition can occur only in point symmetries containing the linear terms in the crystal field Hamiltonian ðA1p Þ [10,11]. This term is present in the Cs, Cn, Cnv, (n = 1, 2, 3, 4, 6) point symmetries. The theory to calculate the intensities of 0 ? 0 transition is based on electrostatic interaction of the 4f electrons with the crystal field. In terms of the polar coordinates of the 4f electrons the crystal field Hamiltonian can be expressed as [11]:
HCF ¼
X
Akq Dkq þ
kðev enÞ q¼k...k
X
Atp Dtp
ð1Þ
tðoddÞ t¼p...p
Dkq and Dtp and are operators that acts on states of the configurations of the Eu3+ ions. Akq and Atp depends on the environment and the site symmetry determines its nonzero terms and represent only the structural parts of the CF. The even part of the CF removes the degeneracy in J and mix states with J different. The odd part of the CF induces f–f transitions by mixing configurations of opposite parities to 4f6. The electric dipole moment for a n state is given by l = ehn|D1|ni, where e is the elementary charge and D1 is the electric dipole operator. This may be written in terms of cartesian coordinates as D10 ¼ z, D11 ¼ x iy. The matrix elements of the electric dipole moment operator between 5D0 and 7F0 states can be expressed by:
T q ¼ hAiq þ hBiq
ð2Þ
1634
A.S. Souza et al. / Optical Materials 35 (2013) 1633–1635
hAiq is the matrix elements due to the J-mixing effect and hBiq is due to the WDM and BCJO mechanisms. q = 0 gives the z-component and q = ±1 gives the x ± iy components [1,12] of the transition. The 0 ? 0 transition is also forbidden by spin-selection rule (DS = 0). Such selection rule is relaxed by the mixing between the 7F0 and 5D0 states via the spin–orbit interaction. The LS-coupling is a good approximation for describing the 7FJ states [13]. On the other hand, for the 5D0 level, the LS-coupling is not appropriate, and it must be written in the intermediate coupling [12,13].
together with the properties of the Eq. (7), the Eq. (4) can be written as:
3. The J-mixing effect
are rotational invariants and it is not necessary to know which irreducible representation labels correspond to each 7FJC and the site symmetry [15]. The Eq. (8) is applicable to any symmetry. Therefore all information of symmetry of the site is in the sum over q and p of the Eq. (9). Using any CF Hamiltonian [11], if the A1p is zero then Aq = 0. This confirms that 0 ? 0 transition is allowed only in point symmetries containing the A1p . The equation also confirms the selection rule for q polarization according to the group theory selection rules. Besides, the selection rule does not depend on how many J states are involved in the mixing. For example, using the CF potential for a C4v symmetry, the Eq. (9) shows that only for q = 0 the 0 ? 0 transition is allowed. Thus, knowing the positions of the ligands now on it is no longer necessary to consult the tables of the point group theory to obtain the selection rules of the 0–0 transition. Moreover, it is also possible to determinate the Bkq parameters that do not contribute to the intensities.
As the 5D0–5DJ energy separation is greater than the 7F0–7FJ and the h[5D]0||U(k)||[5D]ki matrix element is smaller than h7F0||U(k)||7 Fki [14], the mixing of the 5D0 with 5DJ states is negligible [7]. The square brackets mean that the quantities between parentheses are not good quantum numbers. Thus, as the wave function of the ground state contains considerable J-mixing effect, the matrix elements are given by hAiq ffi hf 67 F ½0 jD1q j½5 D0 i. The J-mixing effect depend on even part of the CF due the two states involved in mix have the same parity. Thus, using the firstorder perturbation the ground state is given by:
jf 6 7 F ½0 i ¼ jf 6 7 F 0 i þ
X hf 6 7 F JC jAkq Dkq jf 6 7 F 0 i Eð7 F 0 Þ Eð7 F J Þ
k; q JC
jf 6 7 F JC i
ð3Þ
where E(7FJ) is the barycentre energy of the 7FJ multiplet, P jf 6 7 F JC i ¼ M jf 67 F JM ihf 6 7 F JM jf 6 7 F JC i denotes the symmetrised crystal field states, C identify the irreducible representation and M is the projection of the total angular momentum J. The sum over the quantum number C is converted into a sum over M through of P the projection operator, M|JMihJM|, that acts onto the subspace of the free-ion states. Thus, using the Eq. (3), the hAiq matrix elements are given by:
hAiq ¼
X
Bkq
J–0 jqJM
7 h7 F JM jC ðkÞ q j F0i 67 hf 6 ½5 D0 jDð1Þ q jf F JM i 7 Eð F 0 Þ Eð7 F J Þ
ð4Þ
The Bkq are the crystal field parameters. Using the Wigner–Eckart theorem, the first matrix element of the Eq. (4) can be written as
h7 F JM jC qðkÞ j7 F 0 i ¼ ð1ÞJM
J
k 0
M
q 0
h7 F J jjC ðkÞ jj7 F 0 i
ð5Þ
The second matrix element of the Eq. (4) is the electric dipole matrix elements of the 5D0 ? 7FJ–0 transition in approach of the Judd–Ofelt theory. Thus, using the Eq. (13) of the Ref. [6], and the Wigner–Eckart theorem, the second matrix element of the Eq. (4) can be written as: 67 hf 6 ½5 D0 jDð1Þ q jf F JM i ¼
X 1 ð2k þ 1Þð1Þqþp Atp k;p;t
0
k
J
0 qþp M
k
t
q q p p h½5 D0 jjU ðkÞ jj7 F J iNðk; tÞ ð6Þ
Nðk; tÞ includes the monoelectronic radial integrals, the energy dif6
5
ference between the 4f and 4f nl configurations that characterize the interconfigurational 4f6–4f5nl interaction [6]. U ðkÞ is the unit tensor operator [12,14]. By the properties of the 3-j symbol, 0
j j m m0
0 0
!
ð1Þjm ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi dmm0 djj0 2j þ 1
ð7Þ
the Eq. (5) is nonzero only if J = k and M = q and the Eq. (6) is nonzero only if k ¼ k and (q + p) = M. Using the Eqs. (6) and (5)
hAiq ¼
X Nðk; tÞh½5 D jjU ðkÞ jj7 F k ih7 F k jjC ðkÞ jj7 F 0 i 1 0 Skt 7 F Þ Eð7 F Þ Eð 0 k t;k
ð8Þ
where
S1kt ¼
X
Bkq Atp
q;p
1
k
t
ð9Þ
q q p
4. The breakdown of the closure approximation Using perturbation theory the hBqi transition matrix elements can be expressed as:
hBiq ¼ A1q
X hf 67 F 0 jD1q jjihjjD1q jf 6 ½5 D0 i EðjÞ
j
þ A1q
X hf j
67
F 0 jD1q jjihjjD1q jf 6 ½5 D0 i Eð5 D0 Þ EðjÞ
ð10Þ
the sum in j includes all states of the Eu3+. As the initial and final J are zero the sum over all j states is independent of both q and the coupling scheme of the |j wave functions. Besides, the level and its wave function includes the 7F0 level component due to spin–orbit interaction within the 4f6 configuration [13]. The appropriate wave function is given by:
jf 6 ½5 D0 i ¼ c1 jf 6 7 F 0 i þ c2 jf 6 5 D0 i;
ð11Þ
where c1 and c2 are normalized coefficients. A good approximation for the second denominator in Eq. (10) is 1 5 ðEð5 D0 Þ Ej Þ1 ffi E1 j ½Eð D0 Þ=Ej Ej Ej is the barycentre energy of the first excited configuration. In the case of the Eu3+ ion, Eð5 D0 Þ=Ej is negligible. This is not true for the Sm2+ ion case. From perturbation theory the static dipole polarizability a of an n0 atomic state of an atom or ion is defined as.
X hnjD1q jn0 ihn0 jD1q jni
app0 ¼ 2e
n
EðnÞ Eðn0 Þ
;
where n are all states of the atom. Thus, one does not need to know the nature of such j states to perform sum in the Eq. (9) and it can be developed by using the static dipole polarizability (aEu) for the ground state of the Eu3+ ion. The first sum can be written as
A1q
X hf 6 ½5 D0 jD1q jjihjjD1q jf 6 7 F 0 i Eð7 F 0 Þ EðjÞ
j
¼ c1
aEu 2e2
A1q þ c1 A1q
X hf 6 5 D0 jD1q jjihjjD1q jf 6 7 F 0 i j
Eð7 F 0 Þ EðjÞ
ð12Þ
A.S. Souza et al. / Optical Materials 35 (2013) 1633–1635
aEu also considers the sum in states of the continuous spectrum and the spin–orbit interaction in the excited-state configuration. In other words, it includes the WDM. Now, if one uses only the average energy denominator the second term in the right side of the Eq. (12) is zero [6]. Thus, combining Eqs. (12) and (10), the hBiq matrix element is written in the form: hBiq ffi c1
aEu e2
A1q
ð13Þ
hBiq depends on A1q and the Eq. (13) confirms the selection rule for q polarization. The Eq. (10) is independent of the J-mixing effect in 4f6 configurations and the intensities can be expressed in term of jA1q j2 . A recent estimation of the dipole polarizability of the Eu3+ ion is 0.8 Å3 [16] and c1 = 0.23 [13]. Using the Simple Overlap Model P (SOM) [17], A1q ¼ L e2 g L d½2=ð1 dÞ2 C 1q ðXL ÞR2 L Þ. d is the overlap between the 4f orbitals and the s and p orbitals of a RL and XL define the position of the Lth ligand and gL its net valence. The oscillator strength is given by 4.702 1029 m |T|2e2,m is the P 2 wavenumber at the absorption maximum (in cm1), |T|2 = q|Tq| 2 (cm ) and e (esu) [18]. The contribution to the oscillator strength due to the BCJO is compared with the observed one (107). From Eq. (10), hBiq can also be written in terms of the electric dipole moment for the ground state as follows:
ehBiq ffi c1 lq
ð14Þ
Thus, one expects that strong oscillator strengths of the 0 ? 0 transition are associated with a large dipole expectation value of the ground state of the Eu3+ in a crystal field [19]. 5. Application to the Eu:BaFCl crystal Strong 0 ? 0 transition intensities are often found in systems such as Eu3+:BaFCl [3]. When comparing the non-distorted site (site I, as it stands in the stoichiometric formulae), experimental and theoretical considerations indicate that this enhancement is associated with the distortion caused by the charge compensator O2 substituting one F ion in the first vicinity of the Eu3+ ion (site 2) [20]. However, a quantitative analysis cannot be developed due to theoretical difficulties in defining the quantum states and covalence of the charge transfer states. The BCJO explains the intensities due to the shortening of the Eu3+O2 distance (0.265 nm), which induces polarization in the Eu3+ ion. This observation indicates the strong charge transfer induced by lattice distortion and the strength of the observed intensity can be associated with the enhancement of the ground state dipole moment of the Eu3+ ion. Thus, the contribution of the BCJO mechanism to the oscillator strength is 6 106. This prediction has be obtained using g = 2 as the oxygen charge factor and 0.8 cm3 as the Eu3+ ion polarizability, the positions in Ref. [20] and the SOM for calculating the A1q parameters. Using site 1 as the non-distorted Eu site in the BaFCl crystal, one has defined R = I0-0(site 2)/I0-0(site 1) ratio, for quantifying the BCJO contribution. Thus, RBCJO 15. From table 1 of the Ref. [20], Rexp 32. Indeed, the prediction developed here is underestimated, because the O2 coordinates has been kept the same of the F in the original crystal. However, the average Eu3+ ? O2 distance in any system (crystal, glass, complex) is in the 0.21 to 0.24 nm range
1635
[21,22]. By taking the superior limit of the range, one obtains RBCJO 28, which is in good agreement with Rexp. 6. Conclusions The 0 ? 0 transition intensity was studied from theoretical point of view through J-mixing and BCJO mechanisms. The equations reveal the dependence with symmetry and with some measurable quantities, e.g., the polarizability of the Eu3+ ion. The predictions confirmed the experimental evidence of the dependence with the linear term of the crystal-field expansion [10]. The selection rules were confirmed for both mechanisms and reproduce the same rules from the point group theory, as expected. These two mechanisms have the same order of magnitude. For the sake of comparison, the oscillator strength of the 0 ? 0 transition of the Eu:BaFCl crystal, pure and distorted by the O ? F substitution, were calculated using the BCJO. Using g = 2 and 0.8 cm3 as the Eu3+ ion charge and polarizability, respectively, one has obtained the intensity ratios 15 and 28 when comparing the structures with and without oxygen and entering an overestimated and a more realistic Eu–O distance. The last one is in good agreement with the experiment (Rexp 32). Acknowledgements The authors deeply acknowledge CAPES and CNPq (Brazilian agencies) for financial support. References [1] G.K Liu, Electronic energy level structure, in: G.K. Liu, B. Jacquier (Eds.), Spectroscopic Properties of Rare Earths in Optical Materials, Tsinghua University Press and Springer-Verlag, Berlin, Heidelberg, 2005 (Chapter 1). [2] S.V. Eliseevaa, J.C.G. Bunzli, Chem. Soc. Rev. 39 (2010) 189. [3] X.Y. Chen, G.K. Liu, J. Solid State Chem. 178 (2005) 419. [4] H. Wen, G. Jia, C. Duanw, P.A. Tanner, Phys. Chem. Chem. Phys. 12 (2010) 9933. [5] L. Smentek, A. Ke˛dziorski, J. Alloys Compd. 488 (2009) 586. [6] B.R. Judd, Phys. Rev. 127 (1962) 750. [7] T. Kushida, J. Lumin. 100 (2002) 73. [8] M. Tanaka, G. Nishimura, T. Kushida, Phys. Rev. B 49 (1994) 16917. [9] M.C. Downer, G.W. Burdick, D.K. Sardar, J. Chem. Phys. 89 (1988) 1787. [10] K.F. Francis, A.V. Lawrence, R.E. de Wames, Phys. Rev. 170 (1968) 412. [11] C. Görller-Walrand, K. Binnemans, Rationalization of crystal-field parametrization, in: K.A. Gschneidner Jr., L. Eyring (Eds.), Handbook on the Physics and Chemistry of Rare Earths, vol. 25, Elsevier, Amsterdam, 1996. [12] B.G. Wybourne, Spectroscopic Properties of Rare Earths, John Wiley, New York, 1965. [13] G.S. Ofelt, J. Chem. Phys. 38 (1963) 2171. [14] W.T. Carnall, H. Crosswhite, H.M. Crosswhite, Energy Level Structure and Transition Probabilities of the Trivalent Lanthanides in LaF3, The Johns Hopkins University, 1977. [15] Y.Y. Yeung, D.J. Newman, J. Chem. Phys. 84 (1986) 4470. [16] C. Clavaguéra, J.P. Dognon, Chem. Phys. 311 (2005) 169. [17] O.L. Malta, Chem. Phys. Lett. 88 (1982) 353. [18] C. Gorller-Walrand, K. Binnemans, Spectral intensities of the transitions, Handbook on the Physics and Chemistry of Rare Earths, vol. 25, Elsevier, Amsterdam, 1998. [19] R.M. MacFarlane, J. Lumin. 125 (2007) 156. [20] X.Y. Chen, W. Zhao, R.E. Cook, G.K. Liu, Phys. Rev. B 70 (2004) 205122. [21] M.F.O. Bezerra, M.A. Couto dos Santos, A. Monteil, S. Chaussedent, Opt. Mater. 30 (2008) 1013–1016. [22] M.A. Beltrão, M.L. Santos, M.E. Mesquita, L.S. Barreto, N.B. da Costa Jr., R.O. Freire, M.A. Couto dos Santos, J. Lumin. 116 (2006) 132.
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Optical Materials journal homepage: www.elsevier.com/locate/optmat
Enhanced approach to the Eu3+ ion 5D0 ? 7F0 transition intensity A.S. Souza ⇑, Y.A.R. Oliveira, M.A. Couto dos Santos Federal University of Sergipe, Physics Department, 49100-000 São Cristóvão, SE, Brazil
a r t i c l e
i n f o
Article history: Received 21 June 2012 Received in revised form 1 April 2013 Accepted 8 April 2013 Available online 25 May 2013
a b s t r a c t The dependence of the 5D0 ? 7F0 transition intensity with the linear terms of the crystal field parameter and polarizability of the Eu3+ ion is predicted. It is shown that the breakdown of the closure approximation mechanism is the most important one if the Eu3+ ion has a finite dipole moment in the ground state. The Eu:BFCl crystal is used for comparison with a satisfactory agreement. Ó 2013 Elsevier B.V. All rights reserved.
Keywords: 5 D0 ? 7F0 transition Crystal field theory Judd–Ofelt theory
1. Introduction The standard interpretation of the f-spectra of the lanthanide ions in various materials is based on Judd–Ofelt theory and the 5 D0 ? 7F0 (0 ? J) transition of Eu3+ ion in several sites is a very beautiful example [1,2]. The 0 ? 0 electric dipole transition is used in technological applications, because it reveals non-equivalent sites in a given host matrix [3–5]. However, this transition is forbidden by a selection rule of the Judd–Ofelt theory [6,7]. Various mechanisms were proposed in order to describe the dependence on 0 ? 0 transition with the environment. The J-mixing mechanism leads to changes in intensity of the 0 ? 0 transition of nonequivalent sites by mixing of the J = 0 state to states with J – 0 [7,8]. In some cases the strongest transitions cannot simply be ascribed to J-mixing effect, and are explained by contribution of the so-called Wybourne–Downer mechanism (WDM) [9] and the breakdown of the closure approximation in the Judd–Ofelt theory (BCJO) mechanisms [8]. The J-mixing effect contribution to the 0 ? 0 transition describes the participation of the 7F2, 7F4 and 7F6 states in the final composition of the 7F0 states. This contribution is easily obtained only for symmetry where the crystal field parameters B21 ¼ 0 and selection rules are available only from group theory. Besides, in the case of the Eu3+:BaFCl and of the Sm2+ doped fluoride glass ab initio model is practically impossible. The usual way of considering these two mechanisms has been semi-empirical methods, because it is difficult to obtain theoretical values of the radial integrals h4f|rk|nli and charge-transfer bands. The present paper is devoted to the theoretical description of the 0 ? 0 transition ⇑ Corresponding author. Tel.: +55 79 98087125. E-mail address: [email protected] (A.S. Souza). 0925-3467/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optmat.2013.04.010
intensity using the relationship with crystal field invariants and polarizability of the Eu3+ ion. 2. Theory The experimental observation and selection rules from point group theory show that the 0 ? 0 transition can occur only in point symmetries containing the linear terms in the crystal field Hamiltonian ðA1p Þ [10,11]. This term is present in the Cs, Cn, Cnv, (n = 1, 2, 3, 4, 6) point symmetries. The theory to calculate the intensities of 0 ? 0 transition is based on electrostatic interaction of the 4f electrons with the crystal field. In terms of the polar coordinates of the 4f electrons the crystal field Hamiltonian can be expressed as [11]:
HCF ¼
X
Akq Dkq þ
kðev enÞ q¼k...k
X
Atp Dtp
ð1Þ
tðoddÞ t¼p...p
Dkq and Dtp and are operators that acts on states of the configurations of the Eu3+ ions. Akq and Atp depends on the environment and the site symmetry determines its nonzero terms and represent only the structural parts of the CF. The even part of the CF removes the degeneracy in J and mix states with J different. The odd part of the CF induces f–f transitions by mixing configurations of opposite parities to 4f6. The electric dipole moment for a n state is given by l = ehn|D1|ni, where e is the elementary charge and D1 is the electric dipole operator. This may be written in terms of cartesian coordinates as D10 ¼ z, D11 ¼ x iy. The matrix elements of the electric dipole moment operator between 5D0 and 7F0 states can be expressed by:
T q ¼ hAiq þ hBiq
ð2Þ
1634
A.S. Souza et al. / Optical Materials 35 (2013) 1633–1635
hAiq is the matrix elements due to the J-mixing effect and hBiq is due to the WDM and BCJO mechanisms. q = 0 gives the z-component and q = ±1 gives the x ± iy components [1,12] of the transition. The 0 ? 0 transition is also forbidden by spin-selection rule (DS = 0). Such selection rule is relaxed by the mixing between the 7F0 and 5D0 states via the spin–orbit interaction. The LS-coupling is a good approximation for describing the 7FJ states [13]. On the other hand, for the 5D0 level, the LS-coupling is not appropriate, and it must be written in the intermediate coupling [12,13].
together with the properties of the Eq. (7), the Eq. (4) can be written as:
3. The J-mixing effect
are rotational invariants and it is not necessary to know which irreducible representation labels correspond to each 7FJC and the site symmetry [15]. The Eq. (8) is applicable to any symmetry. Therefore all information of symmetry of the site is in the sum over q and p of the Eq. (9). Using any CF Hamiltonian [11], if the A1p is zero then Aq = 0. This confirms that 0 ? 0 transition is allowed only in point symmetries containing the A1p . The equation also confirms the selection rule for q polarization according to the group theory selection rules. Besides, the selection rule does not depend on how many J states are involved in the mixing. For example, using the CF potential for a C4v symmetry, the Eq. (9) shows that only for q = 0 the 0 ? 0 transition is allowed. Thus, knowing the positions of the ligands now on it is no longer necessary to consult the tables of the point group theory to obtain the selection rules of the 0–0 transition. Moreover, it is also possible to determinate the Bkq parameters that do not contribute to the intensities.
As the 5D0–5DJ energy separation is greater than the 7F0–7FJ and the h[5D]0||U(k)||[5D]ki matrix element is smaller than h7F0||U(k)||7 Fki [14], the mixing of the 5D0 with 5DJ states is negligible [7]. The square brackets mean that the quantities between parentheses are not good quantum numbers. Thus, as the wave function of the ground state contains considerable J-mixing effect, the matrix elements are given by hAiq ffi hf 67 F ½0 jD1q j½5 D0 i. The J-mixing effect depend on even part of the CF due the two states involved in mix have the same parity. Thus, using the firstorder perturbation the ground state is given by:
jf 6 7 F ½0 i ¼ jf 6 7 F 0 i þ
X hf 6 7 F JC jAkq Dkq jf 6 7 F 0 i Eð7 F 0 Þ Eð7 F J Þ
k; q JC
jf 6 7 F JC i
ð3Þ
where E(7FJ) is the barycentre energy of the 7FJ multiplet, P jf 6 7 F JC i ¼ M jf 67 F JM ihf 6 7 F JM jf 6 7 F JC i denotes the symmetrised crystal field states, C identify the irreducible representation and M is the projection of the total angular momentum J. The sum over the quantum number C is converted into a sum over M through of P the projection operator, M|JMihJM|, that acts onto the subspace of the free-ion states. Thus, using the Eq. (3), the hAiq matrix elements are given by:
hAiq ¼
X
Bkq
J–0 jqJM
7 h7 F JM jC ðkÞ q j F0i 67 hf 6 ½5 D0 jDð1Þ q jf F JM i 7 Eð F 0 Þ Eð7 F J Þ
ð4Þ
The Bkq are the crystal field parameters. Using the Wigner–Eckart theorem, the first matrix element of the Eq. (4) can be written as
h7 F JM jC qðkÞ j7 F 0 i ¼ ð1ÞJM
J
k 0
M
q 0
h7 F J jjC ðkÞ jj7 F 0 i
ð5Þ
The second matrix element of the Eq. (4) is the electric dipole matrix elements of the 5D0 ? 7FJ–0 transition in approach of the Judd–Ofelt theory. Thus, using the Eq. (13) of the Ref. [6], and the Wigner–Eckart theorem, the second matrix element of the Eq. (4) can be written as: 67 hf 6 ½5 D0 jDð1Þ q jf F JM i ¼
X 1 ð2k þ 1Þð1Þqþp Atp k;p;t
0
k
J
0 qþp M
k
t
q q p p h½5 D0 jjU ðkÞ jj7 F J iNðk; tÞ ð6Þ
Nðk; tÞ includes the monoelectronic radial integrals, the energy dif6
5
ference between the 4f and 4f nl configurations that characterize the interconfigurational 4f6–4f5nl interaction [6]. U ðkÞ is the unit tensor operator [12,14]. By the properties of the 3-j symbol, 0
j j m m0
0 0
!
ð1Þjm ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi dmm0 djj0 2j þ 1
ð7Þ
the Eq. (5) is nonzero only if J = k and M = q and the Eq. (6) is nonzero only if k ¼ k and (q + p) = M. Using the Eqs. (6) and (5)
hAiq ¼
X Nðk; tÞh½5 D jjU ðkÞ jj7 F k ih7 F k jjC ðkÞ jj7 F 0 i 1 0 Skt 7 F Þ Eð7 F Þ Eð 0 k t;k
ð8Þ
where
S1kt ¼
X
Bkq Atp
q;p
1
k
t
ð9Þ
q q p
4. The breakdown of the closure approximation Using perturbation theory the hBqi transition matrix elements can be expressed as:
hBiq ¼ A1q
X hf 67 F 0 jD1q jjihjjD1q jf 6 ½5 D0 i EðjÞ
j
þ A1q
X hf j
67
F 0 jD1q jjihjjD1q jf 6 ½5 D0 i Eð5 D0 Þ EðjÞ
ð10Þ
the sum in j includes all states of the Eu3+. As the initial and final J are zero the sum over all j states is independent of both q and the coupling scheme of the |j wave functions. Besides, the level and its wave function includes the 7F0 level component due to spin–orbit interaction within the 4f6 configuration [13]. The appropriate wave function is given by:
jf 6 ½5 D0 i ¼ c1 jf 6 7 F 0 i þ c2 jf 6 5 D0 i;
ð11Þ
where c1 and c2 are normalized coefficients. A good approximation for the second denominator in Eq. (10) is 1 5 ðEð5 D0 Þ Ej Þ1 ffi E1 j ½Eð D0 Þ=Ej Ej Ej is the barycentre energy of the first excited configuration. In the case of the Eu3+ ion, Eð5 D0 Þ=Ej is negligible. This is not true for the Sm2+ ion case. From perturbation theory the static dipole polarizability a of an n0 atomic state of an atom or ion is defined as.
X hnjD1q jn0 ihn0 jD1q jni
app0 ¼ 2e
n
EðnÞ Eðn0 Þ
;
where n are all states of the atom. Thus, one does not need to know the nature of such j states to perform sum in the Eq. (9) and it can be developed by using the static dipole polarizability (aEu) for the ground state of the Eu3+ ion. The first sum can be written as
A1q
X hf 6 ½5 D0 jD1q jjihjjD1q jf 6 7 F 0 i Eð7 F 0 Þ EðjÞ
j
¼ c1
aEu 2e2
A1q þ c1 A1q
X hf 6 5 D0 jD1q jjihjjD1q jf 6 7 F 0 i j
Eð7 F 0 Þ EðjÞ
ð12Þ
A.S. Souza et al. / Optical Materials 35 (2013) 1633–1635
aEu also considers the sum in states of the continuous spectrum and the spin–orbit interaction in the excited-state configuration. In other words, it includes the WDM. Now, if one uses only the average energy denominator the second term in the right side of the Eq. (12) is zero [6]. Thus, combining Eqs. (12) and (10), the hBiq matrix element is written in the form: hBiq ffi c1
aEu e2
A1q
ð13Þ
hBiq depends on A1q and the Eq. (13) confirms the selection rule for q polarization. The Eq. (10) is independent of the J-mixing effect in 4f6 configurations and the intensities can be expressed in term of jA1q j2 . A recent estimation of the dipole polarizability of the Eu3+ ion is 0.8 Å3 [16] and c1 = 0.23 [13]. Using the Simple Overlap Model P (SOM) [17], A1q ¼ L e2 g L d½2=ð1 dÞ2 C 1q ðXL ÞR2 L Þ. d is the overlap between the 4f orbitals and the s and p orbitals of a RL and XL define the position of the Lth ligand and gL its net valence. The oscillator strength is given by 4.702 1029 m |T|2e2,m is the P 2 wavenumber at the absorption maximum (in cm1), |T|2 = q|Tq| 2 (cm ) and e (esu) [18]. The contribution to the oscillator strength due to the BCJO is compared with the observed one (107). From Eq. (10), hBiq can also be written in terms of the electric dipole moment for the ground state as follows:
ehBiq ffi c1 lq
ð14Þ
Thus, one expects that strong oscillator strengths of the 0 ? 0 transition are associated with a large dipole expectation value of the ground state of the Eu3+ in a crystal field [19]. 5. Application to the Eu:BaFCl crystal Strong 0 ? 0 transition intensities are often found in systems such as Eu3+:BaFCl [3]. When comparing the non-distorted site (site I, as it stands in the stoichiometric formulae), experimental and theoretical considerations indicate that this enhancement is associated with the distortion caused by the charge compensator O2 substituting one F ion in the first vicinity of the Eu3+ ion (site 2) [20]. However, a quantitative analysis cannot be developed due to theoretical difficulties in defining the quantum states and covalence of the charge transfer states. The BCJO explains the intensities due to the shortening of the Eu3+O2 distance (0.265 nm), which induces polarization in the Eu3+ ion. This observation indicates the strong charge transfer induced by lattice distortion and the strength of the observed intensity can be associated with the enhancement of the ground state dipole moment of the Eu3+ ion. Thus, the contribution of the BCJO mechanism to the oscillator strength is 6 106. This prediction has be obtained using g = 2 as the oxygen charge factor and 0.8 cm3 as the Eu3+ ion polarizability, the positions in Ref. [20] and the SOM for calculating the A1q parameters. Using site 1 as the non-distorted Eu site in the BaFCl crystal, one has defined R = I0-0(site 2)/I0-0(site 1) ratio, for quantifying the BCJO contribution. Thus, RBCJO 15. From table 1 of the Ref. [20], Rexp 32. Indeed, the prediction developed here is underestimated, because the O2 coordinates has been kept the same of the F in the original crystal. However, the average Eu3+ ? O2 distance in any system (crystal, glass, complex) is in the 0.21 to 0.24 nm range
1635
[21,22]. By taking the superior limit of the range, one obtains RBCJO 28, which is in good agreement with Rexp. 6. Conclusions The 0 ? 0 transition intensity was studied from theoretical point of view through J-mixing and BCJO mechanisms. The equations reveal the dependence with symmetry and with some measurable quantities, e.g., the polarizability of the Eu3+ ion. The predictions confirmed the experimental evidence of the dependence with the linear term of the crystal-field expansion [10]. The selection rules were confirmed for both mechanisms and reproduce the same rules from the point group theory, as expected. These two mechanisms have the same order of magnitude. For the sake of comparison, the oscillator strength of the 0 ? 0 transition of the Eu:BaFCl crystal, pure and distorted by the O ? F substitution, were calculated using the BCJO. Using g = 2 and 0.8 cm3 as the Eu3+ ion charge and polarizability, respectively, one has obtained the intensity ratios 15 and 28 when comparing the structures with and without oxygen and entering an overestimated and a more realistic Eu–O distance. The last one is in good agreement with the experiment (Rexp 32). Acknowledgements The authors deeply acknowledge CAPES and CNPq (Brazilian agencies) for financial support. References [1] G.K Liu, Electronic energy level structure, in: G.K. Liu, B. Jacquier (Eds.), Spectroscopic Properties of Rare Earths in Optical Materials, Tsinghua University Press and Springer-Verlag, Berlin, Heidelberg, 2005 (Chapter 1). [2] S.V. Eliseevaa, J.C.G. Bunzli, Chem. Soc. Rev. 39 (2010) 189. [3] X.Y. Chen, G.K. Liu, J. Solid State Chem. 178 (2005) 419. [4] H. Wen, G. Jia, C. Duanw, P.A. Tanner, Phys. Chem. Chem. Phys. 12 (2010) 9933. [5] L. Smentek, A. Ke˛dziorski, J. Alloys Compd. 488 (2009) 586. [6] B.R. Judd, Phys. Rev. 127 (1962) 750. [7] T. Kushida, J. Lumin. 100 (2002) 73. [8] M. Tanaka, G. Nishimura, T. Kushida, Phys. Rev. B 49 (1994) 16917. [9] M.C. Downer, G.W. Burdick, D.K. Sardar, J. Chem. Phys. 89 (1988) 1787. [10] K.F. Francis, A.V. Lawrence, R.E. de Wames, Phys. Rev. 170 (1968) 412. [11] C. Görller-Walrand, K. Binnemans, Rationalization of crystal-field parametrization, in: K.A. Gschneidner Jr., L. Eyring (Eds.), Handbook on the Physics and Chemistry of Rare Earths, vol. 25, Elsevier, Amsterdam, 1996. [12] B.G. Wybourne, Spectroscopic Properties of Rare Earths, John Wiley, New York, 1965. [13] G.S. Ofelt, J. Chem. Phys. 38 (1963) 2171. [14] W.T. Carnall, H. Crosswhite, H.M. Crosswhite, Energy Level Structure and Transition Probabilities of the Trivalent Lanthanides in LaF3, The Johns Hopkins University, 1977. [15] Y.Y. Yeung, D.J. Newman, J. Chem. Phys. 84 (1986) 4470. [16] C. Clavaguéra, J.P. Dognon, Chem. Phys. 311 (2005) 169. [17] O.L. Malta, Chem. Phys. Lett. 88 (1982) 353. [18] C. Gorller-Walrand, K. Binnemans, Spectral intensities of the transitions, Handbook on the Physics and Chemistry of Rare Earths, vol. 25, Elsevier, Amsterdam, 1998. [19] R.M. MacFarlane, J. Lumin. 125 (2007) 156. [20] X.Y. Chen, W. Zhao, R.E. Cook, G.K. Liu, Phys. Rev. B 70 (2004) 205122. [21] M.F.O. Bezerra, M.A. Couto dos Santos, A. Monteil, S. Chaussedent, Opt. Mater. 30 (2008) 1013–1016. [22] M.A. Beltrão, M.L. Santos, M.E. Mesquita, L.S. Barreto, N.B. da Costa Jr., R.O. Freire, M.A. Couto dos Santos, J. Lumin. 116 (2006) 132.