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A simple model for the f–d transition of actinide and heavy lanthanide ions in crystals
Current Applied Physics 6 (2006) 359–362 www.elsevier.com/locate/cap www.kps.or.kr
A simple model for the f–d transition of actinide and heavy lanthanide ions in crystals Chang-Kui Duan a
a,b,*
, Michael F. Reid
b,c
, Gang Ruan
a
Institute of Modern Physics, Chongqing University of Post and Telecommunications, Chongqing 400065, China b Department of Physics and Astronomy, University of Canterbury, Christchurch, New Zealand c MacDiarmid Institute for Advanced Materials and Nanotechnology, New Zealand Available online 28 December 2005
Abstract The original simple model for 4f–5d transitions [C.K. Duan, M.F. Reid, G.W. Burdick, Phys. Rev. B 66 (2002) 155108; C.K. Duan, M.F. Reid, J. Solid State Chem. 171 (2003) 299], which assumes a stronger exchange Coulomb interaction between 4f orbitals and 5d orbitals than the 4f spin–orbit interaction, is extended to the case where the effect of the f (4f for heavy lanthanide and 5f for actinide ions) spin–orbit interaction is stronger than that of the exchange Coulomb interaction. The extended model is used to discuss the 4f–5d spectra of heavy lanthanide and actinide ions in crystals. Ó 2005 Elsevier B.V. All rights reserved. PACS: 78.55.m; 78.40.Pg; 42.70.a; 42.62.Fi; 71.20.Eh Keywords: f–d transition; The simple model; Lanthanide and actinide ions; VUV spectra; Crystal field
1. Introduction New luminescent phosphors for vacuum ultraviolet (VUV) excitation are required for plasma display panels and mercury-free fluorescent tubes, where the VUV emission from a noble gas xenon discharge is used to generate visible luminescence. Other applications where the VUV spectroscopy of lanthanides is relevant are scintilator materials and VUV lasers. Due to these potential applications, and the availability of VUV excitations by synchrotron radiation, the VUV spectroscopy of lanthanide ions and actinide ions in crystals have recently become an important field of research. The VUV spectra of lanthanide and actinide ions are dominated by parity-allowed nf N–nf N1(n + 1)d transitions (n = 4 for lanthanides and 5 for actinides). In contrast to * Corresponding author. Address: Institute of Modern Physics, Chongqing University of Post and Telecommunications, Chongqing 400065, China. Tel.: +86 23 62471434. E-mail address: [email protected] (C.-K. Duan).
1567-1739/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2005.11.019
4f N–4f N transitions, the spectra of transition between 4f N and 4f N15d configurations have broad-band vibronic structures, with zero-phonon lines only resolvable at low temperatures. The vibronic structure is due to the difference in vibrational equilibrium between 4f N15d and 4f N configurations. The phenomenological crystal-field Hamiltonian for 4f N configuration has been extended to 4f N15d configuration by adding the crystal-field and spin–orbit interactions for the 5d orbitals, and the Coulomb interactions between the 4f N1 core and the 5d orbitals [1]. 4f N–4f N15d transitions are electric dipole allowed, and the relative rates can be calculated straightforwardly. Extensive calculations have been carried out for trivalent lanthanide ions in crystals [1,2] and these give satisfactory agreement with experimental spectra. However, the calculations give hundreds or thousands of transition lines which convolve into a small number of broad bands after the vibronic structure is taken into account. This makes straightforward interpretations of measured spectra by the simulations difficult. Recently Duan and co-workers [3–5] simplified the calculations by considering only the largest interactions in the
360
C.-K. Duan et al. / Current Applied Physics 6 (2006) 359–362
4f N15d configuration. The model gave quantum numbers to characterize various groups of states, transition selection rules, and an expression of the relative intensities with these quantum numbers. Application of the model to Eu2+ and Sm3+ in various crystals successfully explained the measured spectra. However, since there is an implicit assumption that the exchange Coulomb interaction between the 4f and 5d orbitals is stronger than the spin–orbit interaction in the 4f N1 core, which no long holds for heavy lanthanide ions or actinide ions in crystals, the model cannot be applied to heavy lanthanide ions or actinides ions. The present work extends the original model to the case where the effect of the f (4f or 5f) spin–orbit interaction is stronger than that of the f–d exchange Coulomb interaction. Section 2 presents the energy eigenstates of the f N and f N1d configurations in terms of several relevant angular-momentum quantum numbers. Section 3 the electric dipole moment is written as a coupling of the creation and annihilation spherical tensor operators to derive selection rules and line strengths for transitions between energy eigenstates of f N1d and f N configurations. This is followed by a brief discussion of the spectra of the f–d transitions using the framework of the simple model. 2. Energy eigenstates of the f N and f N1d configurations 2.1. The f N configuration In the spectra of the transitions between f N and f N1d configurations, the crystal-field energy levels of the f N configuration are usually either buried in the vibronic bands or hard to distinguish from vibronic lines. We aim to give a model to interpret the positions and relative intensities of the broad bands, but not the details of f crystal-field splitting. Therefore, we neglect the f N crystal-field interaction. In this case, the energy eigenstates are highly degenerate and can be written as X j½gSLJ i ¼ C 0 jgSLJ i þ ci jgi S i Li J i. ð1Þ iP1
In most cases the states may be approximated by the main jgSLJi component. 2.2. The f N1d configuration The interactions that split the f N1d configuration can be written as H ðf N 1 dÞ ¼ H Coul ðff Þ þ H cf ðdÞ þ H Coul ðfdÞ þ H so ðf Þ þ H so ðdÞ þ H cf ðf Þ.
ð2Þ
The first two terms are the strongest terms and have the following form: X X H 0 ðf N 1 dÞ ¼ F k ðff Þ C k ðiÞ C k ðjÞ k¼2;4;6
þ
X X
16i
k¼2;4 k6q6k
Bkq C ðkÞ q ðdÞ.
ð3Þ
Here Fk(ff) are Slater integrals, which are usually treated as adjustable parameters. Bkq are parameters for the d crystalfield Hamiltonian. Note that only those Bkq with (k, q) allowed by the site symmetry are nonzero. H0(f N1d) contributes to the splitting of the f N1 core into energy levels characterized with spin and orbit angular momenta S and L, and d orbital into strong crystal-field energy levels characterized with site-symmetry labels. The contribution to the splitting from the third term of (2), i.e., the Coulomb interaction between the f N1 core and the d orbitals, can be approximated with an isotropic exchange term H exc f N 1 d ¼ J exc S f S d ; ð4Þ where J exc ¼
6 1 8 3 20 5 G ðfdÞ þ G ðfdÞ þ G ðfdÞ. 35 105 231
ð5Þ
Here G1,3,5(fd) are Slater integrals for the f–d Coulomb exchange interaction, which can be obtained using the atomic calculations [6]. Alternatively, we can simply treat Jexc as a fitting parameter. It is straightforward to show that Hexc commutes with the total spin of f N1d. This interaction splits the degenerate eigenstates of H0(f N1d) into states of high spin and low spin. The following approximation is often used for the f-electron spin–orbit interaction within a given zero-order f N1 core energy level characterized by gSL: H so ðf Þ ¼
N 1 X
nnl si l i kgSL S L;
ð6Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lðl þ 1Þð2l þ 1Þ ¼ SðS þ 1Þð2S þ 1ÞLðL þ 1Þð2L þ 1Þ f N 1 aSLV 11 f N 1 aSL nnl .
ð7Þ
i¼1
where kgSL
The interaction Hso(f) splits the degenerate eigenstates of H0(f N1d) into states of different total f N1 angular momentum Jf. In the case, where Sf takes the largest possible value for the f N1 configuration, kgf S f Lf is simply sign(8 N)nf/ 2Sf [4,7]. The negative sign of kgSL for heavy lanthanide ions means that the lower-energy multiplets have larger Jf and separate more from each other. In general, Hso(f) commutes with the total angular momentum of f N1 core, no matter the above approximation in Eq. (6) is used or not. The other terms in Eq. (2) are not important in the interpretation of the broad bands in f–d spectra and may be neglected. Therefore, we write the effective Hamiltonian as the sum of the above important terms, i.e., H eff ¼ H 0 f N 1 d þ H exc f N 1 d þ H so ðf Þ. ð8Þ Heff does not commute with the total angular-momentum operator since it contains d crystal-field interactions. It can be shown that Heff commutes with the following effective angular-momentum operator J eff ¼ S f þ S d þ Lf .
ð9Þ
C.-K. Duan et al. / Current Applied Physics 6 (2006) 359–362
This operator is not the total angular-momentum operator for f N1d configuration, since it omits the orbit angular momentum of the d orbitals. However, this angular momentum is usually quenched in low symmetry sites. Since Jeff is the sum of three commutative angular-momentum operators, there are two coupling schemes to construct the eigenstates of Jeff. In this paper, we consider the following two cases: Case I, where the effect of Hexc(f N1d) is much stronger than Hso(f) and the coupling scheme with Hexc(f N1d) diagonalized is used, and Case II, where the effect of the Hso(f) is much stronger than Hexc(f N1d) and the coupling scheme with Hso(f) diagonalized is used. Our former work [3,4] corresponds to Case I, where the S L energy levels were written using the ðSO3 f SOS3 d Þ SO3 f coupling scheme as E f N1 gS f Lf ; 2 d i ; SJ ¼ E0 ðgS f Lf Þ þ d i
J exc 3 SðS þ 1 S f ðS f þ 1Þ 4 2 2S þ 1 kgf S f Lf þ 2 2S f þ 1 J ðJ þ 1Þ SðS þ 1Þ Lf ðLf þ 1Þ . 2 ð10Þ However, the strength of Hexc(f N1d) decreases as the nucleus charge increases, while at the same time Hso(f) increases. In addition to this, compared to the light lanthanide ions with N 6 7, the spin–orbit splitting for f N1 with N P 9 is reversed, so that the most-concerned low-energy f N1d states with N P 9 feel a stronger effect from Hso(f). Therefore, for the heavy lanthanides we must S consider Case II. In this case, the coupling ðSO3 f Lf Sd SO3 Þ SO3 is preferred. Then the approximate eigenstates can be written as jf N1gSfLfJf,2di;Ji, where J is the quantum number for the Jeff in Eq. (9). The eigenvalues can be written as E f N 1 gS f Lf J f ;2 d i ; J ¼ E0 f N 1 gS f Lf ;2 d i þ kgS f Lf
ðJ f ðJ f þ 1Þ S f ðS f þ 1Þ Lf ðLf þ 1Þ 2 J f ðJ f þ 1Þ þ S f ðS f þ 1Þ Lf ðLf þ 1Þ J exc 2J f ðJ f þ 1Þ
J ðJ þ 1Þ J f ðJ f þ 1Þ 34 . ð11Þ 2 3. Line strengths of one-photon transition between f N and f N1d The f N to f N1d transitions are electric dipole allowed. Here we consider only this mechanism. The electric dipole moment is a spin-independent rank 1 tensor in the spaces of both total orbital angular momentum and the total angular momentum. The three components of this tensor Dq (q = 1, 0, 1) are:
N X
Dq ¼
361
rq ðiÞ.
ð12Þ
i¼1
The second quantization technique is much clearer for calculations of the line strengths [4] than the conventional method [5,8]. Using the second quantization techniques, the electric dipole moment can be written in the conventional S–L coupling form as Dq ¼
h ið01Þ1q ið01Þ1q h pffiffiffi 1 1 2hf jrjdi ðaþ Þð23Þ ~að22Þ ðaþ Þð1=22Þ ~að1=23Þ ;
ð13Þ where sms lml
ð~aÞ
¼ ð1Þ
sms þlml sms lml
a
;
ð14Þ
þ ðsms lml Þ
ða Þ are components of tensors that transform under rotations the same way as jsmslmli, and hfjrjdi is a radial integral. The coupling of two creation-annihilation operators is just a coupling of two tensors to give a new tensor. In Eq. (13), a+ couples with ~a and results in a rank (0 Æ 1) tensor of spin and orbital angular momenta and a rank 1 tensor of the total angular momentum. Using coupling and recoupling techniques, we can rewrite the electric dipole moment tensor in the two coupling schemes, which are convenient for the derivation of selection rules and the calculation of line strengths, as follows: h ið032Þð3q1 ;2q2 Þ X Dq pffiffiffi h3q1 2q2 j1qi ðaþ Þð1=23Þ ~ þ að1=22Þ ¼ 2hf jrjdi q1 ;q2
ð15Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2jf þ 1h þ ð1=23Þjf ð1=22Þ ið32Þð3q1 ;2q2 Þ ~ ¼ h3q1 2q2 j1qi þ a ða Þ 14 q ;q j ¼5=2 7=2 X
1
2
f
ð16Þ
Here the neglected terms ( ) do not contribute since we put f N states on the left and f N1d states on the right in the calculation of the electric dipole matrix elements. In Eqs. (15) and (16), we have used the coupling schemes s L s L ðSO3f SOs3d Þ SO3 f and ðSO3f SO3 f Þ SOs3d , respectively. The f N energy eigenstates in Eq. (1) are also eigenstates of the Jeff operator for the two schemes, since there is no d electron and so the Sd quantum number is zero. Hence, the matrix elements of Dq between states in f N and f N1d configurations can be calculated via the Wigner-Eckart theorem [9] by using one of the forms given in Eqs. (15) and (16) appropriate to the coupling schemes used in the energy eigenstates. Supposing that the transition concerned is between two energy levels I and F, and i and f are partners of I and F to distinguish degenerate states, we adopt for the isotropic line strengths Siso(I M F) the following definition: SðI $ F Þ ¼
1 X 1 X 2 jhIijDq jFf ij . 3 q¼1 i;f
ð17Þ
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C.-K. Duan et al. / Current Applied Physics 6 (2006) 359–362
Using orthonormal relations of coupling and recoupling coefficients [9], after a lengthy but straightforward analytic calculation, we finally obtain the isotropic line strengths of the f–d transition for the two cases. For Case I, the line strength is S iso jf N gSLJ $ f N 1 gf S f Lf ;2 d i ; S 0 J 0 2
¼
2 N hf jrjdi dSS 0 ½d i ½L; J ; J 0 hf N gSL f N 1 gf S f Lf 35
2 L Lf 3 ; J0 J S
ð18Þ
where the [di] is the degeneracy of the di crystal field levels, and the [S], etc. is short for (2S + 1), etc. For Case II, the line strength is E N 1 S iso f N gSLJ i $ gf S f Lf J f ;2 d i ; J 0 2 ¼ N hf jrjdi2 ½d i ½S; L; J f ; J ; J 0 hf N gSL jf N 1 gf S f Lf 9 81 0 12 ) S f S >( 7 > 2 1 = < 2 jf 3 2 C 7 BX @ ð1Þjf 2 ½jf 3 Lf L A. > J0 Jf J > ; : jf ¼52 jf J f J ð19Þ There are substantial differences between the two cases, even qualitatively. For Case I, as shown in earlier work [3,4], the low-energy states of f N1d fall into two sets, one with high spin and the other with low spin. When N 6 7, the ground-state absorption to the high-spin states are allowed and shows up as several peaks, whose line strengths can be simulated with Eq. (18), while the low-spin states cannot be observed in the excitation spectra in general, but may be important for excited-state dynamics [3] or as intermediate states for f–f one or two-photon transitions [3]. For Case II, the low-energy states of f N1d are first grouped by the spin–orbit splitting of the f N1 core into sets characterized by the Jf quantum number. The set of states with a given Jf is further split into two groups of states characterized by the quantum number Jf ± 1/2 of the effective angular momentum operator Jeff. The states are in general a mixture of different spins, except those with maximum or minimum Jeff quantum numbers, which, under the frame of the simple model, happen to be pure high spin or pure low spin states. For the heavy lanthanides Case II is more realistic. The group of 4f N15d states of lowest energy have a maximum Jeff and hence are states of higher spin than that of the ground 4f N states. There-
fore, absorptions to them are spin-forbidden and appear as very weak peaks in the excitation spectra. They also have very long lifetimes. Other groups of low-energy 4f N15d states do not have a definite spin, and so the transition between them and the ground states is generally spin-allowed and appears in the spectra as strong peaks. If Case I were applied to the heavy lanthanide ions instead, there would be several groups of high-spin states in the low energy region of 4f N15d configuration, corresponding to several very weak peaks in the low energy region of the 4f N ! 4f N15d excitation spectra, disagreeing with experiment [10]. 4. Conclusion Using Racah–Wigner algebra and the second quantization techniques, the model for 4f N–4f N15d transitions proposed earlier [4] has been extended to the case where the effect of the f spin–orbit interaction is stronger than the isotropic exchange interaction between f N1 and d. The result is expected to be useful for actinide ions, but there are few experimental measurements for actinides. In this paper, we have shown that this extended model also serves as a good qualitative approximation for heavy lanthanides. Acknowledgments C.K.D. acknowledges support of this work by the National Natural Science Foundation of China, Grant Nos. 10474092 and 10274079. References [1] M.F. Reid, L. van Pieterson, R.T. Wegh, A. Meijerink, Phys. Rev. B 62 (2000) 14744. [2] P.A. Tanner, C.S.K. Mak, M.D. Faucher, W.M. Kwok, D.L. Phillips, V. Mikhailik, Phys. Rev. B 67 (2003) 115102. [3] C.K. Duan, M.F. Reid, G.W. Burdick, Phys. Rev. B 66 (2002) 155108. [4] C.K. Duan, M.F. Reid, J. Solid State Chem. 171 (2003) 299. [5] L.X. Ning, C.K. Duan, S. Xia, M.F. Reid, P.A. Tanner, J. Alloy. Compd. 366 (2004) 34. [6] R.D. Cowan, The theory of atomic structure and spectra, University of California Press, Berkeley, 1981. [7] B.G. Wybourne, Spectroscopic Properties of Rare Earths, Interscience Publishers, John Wiley & Sons Inc., New York, 1965. [8] C.K. Duan, S.D. Xia, M.F. Reid, G. Ruan, Phys. Stat. Sol. B 242 (2005) 2503. [9] P.H. Butler, Point Group Symmetry Applications, Plenum Press, New York, 1981. [10] L. van Pieterson, M.F. Reid, G.W. Burdick, A. Meijerink, Phys. Rev. B 65 (2002) 045114.
A simple model for the f–d transition of actinide and heavy lanthanide ions in crystals Chang-Kui Duan a
a,b,*
, Michael F. Reid
b,c
, Gang Ruan
a
Institute of Modern Physics, Chongqing University of Post and Telecommunications, Chongqing 400065, China b Department of Physics and Astronomy, University of Canterbury, Christchurch, New Zealand c MacDiarmid Institute for Advanced Materials and Nanotechnology, New Zealand Available online 28 December 2005
Abstract The original simple model for 4f–5d transitions [C.K. Duan, M.F. Reid, G.W. Burdick, Phys. Rev. B 66 (2002) 155108; C.K. Duan, M.F. Reid, J. Solid State Chem. 171 (2003) 299], which assumes a stronger exchange Coulomb interaction between 4f orbitals and 5d orbitals than the 4f spin–orbit interaction, is extended to the case where the effect of the f (4f for heavy lanthanide and 5f for actinide ions) spin–orbit interaction is stronger than that of the exchange Coulomb interaction. The extended model is used to discuss the 4f–5d spectra of heavy lanthanide and actinide ions in crystals. Ó 2005 Elsevier B.V. All rights reserved. PACS: 78.55.m; 78.40.Pg; 42.70.a; 42.62.Fi; 71.20.Eh Keywords: f–d transition; The simple model; Lanthanide and actinide ions; VUV spectra; Crystal field
1. Introduction New luminescent phosphors for vacuum ultraviolet (VUV) excitation are required for plasma display panels and mercury-free fluorescent tubes, where the VUV emission from a noble gas xenon discharge is used to generate visible luminescence. Other applications where the VUV spectroscopy of lanthanides is relevant are scintilator materials and VUV lasers. Due to these potential applications, and the availability of VUV excitations by synchrotron radiation, the VUV spectroscopy of lanthanide ions and actinide ions in crystals have recently become an important field of research. The VUV spectra of lanthanide and actinide ions are dominated by parity-allowed nf N–nf N1(n + 1)d transitions (n = 4 for lanthanides and 5 for actinides). In contrast to * Corresponding author. Address: Institute of Modern Physics, Chongqing University of Post and Telecommunications, Chongqing 400065, China. Tel.: +86 23 62471434. E-mail address: [email protected] (C.-K. Duan).
1567-1739/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2005.11.019
4f N–4f N transitions, the spectra of transition between 4f N and 4f N15d configurations have broad-band vibronic structures, with zero-phonon lines only resolvable at low temperatures. The vibronic structure is due to the difference in vibrational equilibrium between 4f N15d and 4f N configurations. The phenomenological crystal-field Hamiltonian for 4f N configuration has been extended to 4f N15d configuration by adding the crystal-field and spin–orbit interactions for the 5d orbitals, and the Coulomb interactions between the 4f N1 core and the 5d orbitals [1]. 4f N–4f N15d transitions are electric dipole allowed, and the relative rates can be calculated straightforwardly. Extensive calculations have been carried out for trivalent lanthanide ions in crystals [1,2] and these give satisfactory agreement with experimental spectra. However, the calculations give hundreds or thousands of transition lines which convolve into a small number of broad bands after the vibronic structure is taken into account. This makes straightforward interpretations of measured spectra by the simulations difficult. Recently Duan and co-workers [3–5] simplified the calculations by considering only the largest interactions in the
360
C.-K. Duan et al. / Current Applied Physics 6 (2006) 359–362
4f N15d configuration. The model gave quantum numbers to characterize various groups of states, transition selection rules, and an expression of the relative intensities with these quantum numbers. Application of the model to Eu2+ and Sm3+ in various crystals successfully explained the measured spectra. However, since there is an implicit assumption that the exchange Coulomb interaction between the 4f and 5d orbitals is stronger than the spin–orbit interaction in the 4f N1 core, which no long holds for heavy lanthanide ions or actinide ions in crystals, the model cannot be applied to heavy lanthanide ions or actinides ions. The present work extends the original model to the case where the effect of the f (4f or 5f) spin–orbit interaction is stronger than that of the f–d exchange Coulomb interaction. Section 2 presents the energy eigenstates of the f N and f N1d configurations in terms of several relevant angular-momentum quantum numbers. Section 3 the electric dipole moment is written as a coupling of the creation and annihilation spherical tensor operators to derive selection rules and line strengths for transitions between energy eigenstates of f N1d and f N configurations. This is followed by a brief discussion of the spectra of the f–d transitions using the framework of the simple model. 2. Energy eigenstates of the f N and f N1d configurations 2.1. The f N configuration In the spectra of the transitions between f N and f N1d configurations, the crystal-field energy levels of the f N configuration are usually either buried in the vibronic bands or hard to distinguish from vibronic lines. We aim to give a model to interpret the positions and relative intensities of the broad bands, but not the details of f crystal-field splitting. Therefore, we neglect the f N crystal-field interaction. In this case, the energy eigenstates are highly degenerate and can be written as X j½gSLJ i ¼ C 0 jgSLJ i þ ci jgi S i Li J i. ð1Þ iP1
In most cases the states may be approximated by the main jgSLJi component. 2.2. The f N1d configuration The interactions that split the f N1d configuration can be written as H ðf N 1 dÞ ¼ H Coul ðff Þ þ H cf ðdÞ þ H Coul ðfdÞ þ H so ðf Þ þ H so ðdÞ þ H cf ðf Þ.
ð2Þ
The first two terms are the strongest terms and have the following form: X X H 0 ðf N 1 dÞ ¼ F k ðff Þ C k ðiÞ C k ðjÞ k¼2;4;6
þ
X X
16i
k¼2;4 k6q6k
Bkq C ðkÞ q ðdÞ.
ð3Þ
Here Fk(ff) are Slater integrals, which are usually treated as adjustable parameters. Bkq are parameters for the d crystalfield Hamiltonian. Note that only those Bkq with (k, q) allowed by the site symmetry are nonzero. H0(f N1d) contributes to the splitting of the f N1 core into energy levels characterized with spin and orbit angular momenta S and L, and d orbital into strong crystal-field energy levels characterized with site-symmetry labels. The contribution to the splitting from the third term of (2), i.e., the Coulomb interaction between the f N1 core and the d orbitals, can be approximated with an isotropic exchange term H exc f N 1 d ¼ J exc S f S d ; ð4Þ where J exc ¼
6 1 8 3 20 5 G ðfdÞ þ G ðfdÞ þ G ðfdÞ. 35 105 231
ð5Þ
Here G1,3,5(fd) are Slater integrals for the f–d Coulomb exchange interaction, which can be obtained using the atomic calculations [6]. Alternatively, we can simply treat Jexc as a fitting parameter. It is straightforward to show that Hexc commutes with the total spin of f N1d. This interaction splits the degenerate eigenstates of H0(f N1d) into states of high spin and low spin. The following approximation is often used for the f-electron spin–orbit interaction within a given zero-order f N1 core energy level characterized by gSL: H so ðf Þ ¼
N 1 X
nnl si l i kgSL S L;
ð6Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lðl þ 1Þð2l þ 1Þ ¼ SðS þ 1Þð2S þ 1ÞLðL þ 1Þð2L þ 1Þ f N 1 aSLV 11 f N 1 aSL nnl .
ð7Þ
i¼1
where kgSL
The interaction Hso(f) splits the degenerate eigenstates of H0(f N1d) into states of different total f N1 angular momentum Jf. In the case, where Sf takes the largest possible value for the f N1 configuration, kgf S f Lf is simply sign(8 N)nf/ 2Sf [4,7]. The negative sign of kgSL for heavy lanthanide ions means that the lower-energy multiplets have larger Jf and separate more from each other. In general, Hso(f) commutes with the total angular momentum of f N1 core, no matter the above approximation in Eq. (6) is used or not. The other terms in Eq. (2) are not important in the interpretation of the broad bands in f–d spectra and may be neglected. Therefore, we write the effective Hamiltonian as the sum of the above important terms, i.e., H eff ¼ H 0 f N 1 d þ H exc f N 1 d þ H so ðf Þ. ð8Þ Heff does not commute with the total angular-momentum operator since it contains d crystal-field interactions. It can be shown that Heff commutes with the following effective angular-momentum operator J eff ¼ S f þ S d þ Lf .
ð9Þ
C.-K. Duan et al. / Current Applied Physics 6 (2006) 359–362
This operator is not the total angular-momentum operator for f N1d configuration, since it omits the orbit angular momentum of the d orbitals. However, this angular momentum is usually quenched in low symmetry sites. Since Jeff is the sum of three commutative angular-momentum operators, there are two coupling schemes to construct the eigenstates of Jeff. In this paper, we consider the following two cases: Case I, where the effect of Hexc(f N1d) is much stronger than Hso(f) and the coupling scheme with Hexc(f N1d) diagonalized is used, and Case II, where the effect of the Hso(f) is much stronger than Hexc(f N1d) and the coupling scheme with Hso(f) diagonalized is used. Our former work [3,4] corresponds to Case I, where the S L energy levels were written using the ðSO3 f SOS3 d Þ SO3 f coupling scheme as E f N1 gS f Lf ; 2 d i ; SJ ¼ E0 ðgS f Lf Þ þ d i
J exc 3 SðS þ 1 S f ðS f þ 1Þ 4 2 2S þ 1 kgf S f Lf þ 2 2S f þ 1 J ðJ þ 1Þ SðS þ 1Þ Lf ðLf þ 1Þ . 2 ð10Þ However, the strength of Hexc(f N1d) decreases as the nucleus charge increases, while at the same time Hso(f) increases. In addition to this, compared to the light lanthanide ions with N 6 7, the spin–orbit splitting for f N1 with N P 9 is reversed, so that the most-concerned low-energy f N1d states with N P 9 feel a stronger effect from Hso(f). Therefore, for the heavy lanthanides we must S consider Case II. In this case, the coupling ðSO3 f Lf Sd SO3 Þ SO3 is preferred. Then the approximate eigenstates can be written as jf N1gSfLfJf,2di;Ji, where J is the quantum number for the Jeff in Eq. (9). The eigenvalues can be written as E f N 1 gS f Lf J f ;2 d i ; J ¼ E0 f N 1 gS f Lf ;2 d i þ kgS f Lf
ðJ f ðJ f þ 1Þ S f ðS f þ 1Þ Lf ðLf þ 1Þ 2 J f ðJ f þ 1Þ þ S f ðS f þ 1Þ Lf ðLf þ 1Þ J exc 2J f ðJ f þ 1Þ
J ðJ þ 1Þ J f ðJ f þ 1Þ 34 . ð11Þ 2 3. Line strengths of one-photon transition between f N and f N1d The f N to f N1d transitions are electric dipole allowed. Here we consider only this mechanism. The electric dipole moment is a spin-independent rank 1 tensor in the spaces of both total orbital angular momentum and the total angular momentum. The three components of this tensor Dq (q = 1, 0, 1) are:
N X
Dq ¼
361
rq ðiÞ.
ð12Þ
i¼1
The second quantization technique is much clearer for calculations of the line strengths [4] than the conventional method [5,8]. Using the second quantization techniques, the electric dipole moment can be written in the conventional S–L coupling form as Dq ¼
h ið01Þ1q ið01Þ1q h pffiffiffi 1 1 2hf jrjdi ðaþ Þð23Þ ~að22Þ ðaþ Þð1=22Þ ~að1=23Þ ;
ð13Þ where sms lml
ð~aÞ
¼ ð1Þ
sms þlml sms lml
a
;
ð14Þ
þ ðsms lml Þ
ða Þ are components of tensors that transform under rotations the same way as jsmslmli, and hfjrjdi is a radial integral. The coupling of two creation-annihilation operators is just a coupling of two tensors to give a new tensor. In Eq. (13), a+ couples with ~a and results in a rank (0 Æ 1) tensor of spin and orbital angular momenta and a rank 1 tensor of the total angular momentum. Using coupling and recoupling techniques, we can rewrite the electric dipole moment tensor in the two coupling schemes, which are convenient for the derivation of selection rules and the calculation of line strengths, as follows: h ið032Þð3q1 ;2q2 Þ X Dq pffiffiffi h3q1 2q2 j1qi ðaþ Þð1=23Þ ~ þ að1=22Þ ¼ 2hf jrjdi q1 ;q2
ð15Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2jf þ 1h þ ð1=23Þjf ð1=22Þ ið32Þð3q1 ;2q2 Þ ~ ¼ h3q1 2q2 j1qi þ a ða Þ 14 q ;q j ¼5=2 7=2 X
1
2
f
ð16Þ
Here the neglected terms ( ) do not contribute since we put f N states on the left and f N1d states on the right in the calculation of the electric dipole matrix elements. In Eqs. (15) and (16), we have used the coupling schemes s L s L ðSO3f SOs3d Þ SO3 f and ðSO3f SO3 f Þ SOs3d , respectively. The f N energy eigenstates in Eq. (1) are also eigenstates of the Jeff operator for the two schemes, since there is no d electron and so the Sd quantum number is zero. Hence, the matrix elements of Dq between states in f N and f N1d configurations can be calculated via the Wigner-Eckart theorem [9] by using one of the forms given in Eqs. (15) and (16) appropriate to the coupling schemes used in the energy eigenstates. Supposing that the transition concerned is between two energy levels I and F, and i and f are partners of I and F to distinguish degenerate states, we adopt for the isotropic line strengths Siso(I M F) the following definition: SðI $ F Þ ¼
1 X 1 X 2 jhIijDq jFf ij . 3 q¼1 i;f
ð17Þ
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C.-K. Duan et al. / Current Applied Physics 6 (2006) 359–362
Using orthonormal relations of coupling and recoupling coefficients [9], after a lengthy but straightforward analytic calculation, we finally obtain the isotropic line strengths of the f–d transition for the two cases. For Case I, the line strength is S iso jf N gSLJ $ f N 1 gf S f Lf ;2 d i ; S 0 J 0 2
¼
2 N hf jrjdi dSS 0 ½d i ½L; J ; J 0 hf N gSL f N 1 gf S f Lf 35
2 L Lf 3 ; J0 J S
ð18Þ
where the [di] is the degeneracy of the di crystal field levels, and the [S], etc. is short for (2S + 1), etc. For Case II, the line strength is E N 1 S iso f N gSLJ i $ gf S f Lf J f ;2 d i ; J 0 2 ¼ N hf jrjdi2 ½d i ½S; L; J f ; J ; J 0 hf N gSL jf N 1 gf S f Lf 9 81 0 12 ) S f S >( 7 > 2 1 = < 2 jf 3 2 C 7 BX @ ð1Þjf 2 ½jf 3 Lf L A. > J0 Jf J > ; : jf ¼52 jf J f J ð19Þ There are substantial differences between the two cases, even qualitatively. For Case I, as shown in earlier work [3,4], the low-energy states of f N1d fall into two sets, one with high spin and the other with low spin. When N 6 7, the ground-state absorption to the high-spin states are allowed and shows up as several peaks, whose line strengths can be simulated with Eq. (18), while the low-spin states cannot be observed in the excitation spectra in general, but may be important for excited-state dynamics [3] or as intermediate states for f–f one or two-photon transitions [3]. For Case II, the low-energy states of f N1d are first grouped by the spin–orbit splitting of the f N1 core into sets characterized by the Jf quantum number. The set of states with a given Jf is further split into two groups of states characterized by the quantum number Jf ± 1/2 of the effective angular momentum operator Jeff. The states are in general a mixture of different spins, except those with maximum or minimum Jeff quantum numbers, which, under the frame of the simple model, happen to be pure high spin or pure low spin states. For the heavy lanthanides Case II is more realistic. The group of 4f N15d states of lowest energy have a maximum Jeff and hence are states of higher spin than that of the ground 4f N states. There-
fore, absorptions to them are spin-forbidden and appear as very weak peaks in the excitation spectra. They also have very long lifetimes. Other groups of low-energy 4f N15d states do not have a definite spin, and so the transition between them and the ground states is generally spin-allowed and appears in the spectra as strong peaks. If Case I were applied to the heavy lanthanide ions instead, there would be several groups of high-spin states in the low energy region of 4f N15d configuration, corresponding to several very weak peaks in the low energy region of the 4f N ! 4f N15d excitation spectra, disagreeing with experiment [10]. 4. Conclusion Using Racah–Wigner algebra and the second quantization techniques, the model for 4f N–4f N15d transitions proposed earlier [4] has been extended to the case where the effect of the f spin–orbit interaction is stronger than the isotropic exchange interaction between f N1 and d. The result is expected to be useful for actinide ions, but there are few experimental measurements for actinides. In this paper, we have shown that this extended model also serves as a good qualitative approximation for heavy lanthanides. Acknowledgments C.K.D. acknowledges support of this work by the National Natural Science Foundation of China, Grant Nos. 10474092 and 10274079. References [1] M.F. Reid, L. van Pieterson, R.T. Wegh, A. Meijerink, Phys. Rev. B 62 (2000) 14744. [2] P.A. Tanner, C.S.K. Mak, M.D. Faucher, W.M. Kwok, D.L. Phillips, V. Mikhailik, Phys. Rev. B 67 (2003) 115102. [3] C.K. Duan, M.F. Reid, G.W. Burdick, Phys. Rev. B 66 (2002) 155108. [4] C.K. Duan, M.F. Reid, J. Solid State Chem. 171 (2003) 299. [5] L.X. Ning, C.K. Duan, S. Xia, M.F. Reid, P.A. Tanner, J. Alloy. Compd. 366 (2004) 34. [6] R.D. Cowan, The theory of atomic structure and spectra, University of California Press, Berkeley, 1981. [7] B.G. Wybourne, Spectroscopic Properties of Rare Earths, Interscience Publishers, John Wiley & Sons Inc., New York, 1965. [8] C.K. Duan, S.D. Xia, M.F. Reid, G. Ruan, Phys. Stat. Sol. B 242 (2005) 2503. [9] P.H. Butler, Point Group Symmetry Applications, Plenum Press, New York, 1981. [10] L. van Pieterson, M.F. Reid, G.W. Burdick, A. Meijerink, Phys. Rev. B 65 (2002) 045114.