- Email: [email protected]
Measurements and simulation of the Zeeman splitting in NdOCl and Nd3+:LiYF4
Journal of Alloys and Compounds 250 (1997) 336–341
L
Measurements and simulation of the Zeeman splitting in NdOCl and Nd 31 :LiYF 4 Marcos A. Couto dos Santos 1 , Laure Beaury, Jacqueline Derouet, Pierre Porcher Laboratoire de Chimie Metallurgique et Spectroscopie des Terres Rares, 1 Place Aristide Briand, 92195, Meudon, France
Abstract Measurements and simulation of the Zeeman effect in NdOCl and 1%Nd 31 :LiYF 4 single crystals were carried out. The rare earth local symmetries are C 4v and D 2d respectively. The isomorphism between these two point groups allows discussion on symmetry assignments. The attention is focused on the 4 G 7 / 2 and 4 G 9 / 2 levels for NdOCl and on the 4 F 9 / 2 and 4 G 7 / 2 levels for Nd 31 :LiYF 4 , because they are representative of the general splitting behaviour. We have first revisited the simulation of the energy level scheme in order to deal with confident wave functions. In the experimental magnetic field (B) range we have obtained linear, non-linear, undefined crossing and anti-crossing levels. These features depend on the orientation of B, on the irreducible representation and on the transition closeness. The line widths have been the main obstacle in confirming crossing regions, because their average values are 8 cm 21 at liquid helium temperature. The simulations were performed by introducing the magnetic Hamiltonian B B(L1ge S) in the secular determinant before diagonalisation. A magnetic accidental degeneracy is considered when discussing the crossing and anti-crossing cases. Keywords: Neodymium; Crystal field; Zeeman effect
1. Introduction The observed electronic transitions inside a 4f N configuration mean that there is an effective interaction between the Stark 4f N states and more excited configurations of opposite parity (e.g. 4f N 21 5d and 4f N 21 5g), which permits the violation of Laporte’s rule. In calculations as well as in theoretical approaches, the wave functions associated with the energy levels are described on the basis of the ground configuration states, the interaction with excited configurations being taken into account by means of effective operators [1,2]. In fact, experiment, theory and fitting procedures are put together with the aim of testing phenomenological wave functions by applying them to the interpretation of spectroscopic and / or magnetic properties of RE ions [1–4]. The Zeeman effect can be measured by magneto-optical techniques and is a way of confirming line intensities for different polarisations. In this sense, this work is dedicated to measurements and simulation of the Zeeman effect of the Nd 31 ion in the NdOCl and 1%Nd 31 :LiYF 4 systems. For NdOCl, the calculations were developed 1
Postdoctoral period, CNPq-Brazil.
within the C 4v site symmetry, whereas the approximate D 2d local symmetry, very close to the S 4 real one, has been used for the Nd 31 :LiYF 4 [5,6]. This is justified by the small value of the imaginary crystal field parameter S 46 [6–9].
2. Experiment The NdOCl sample was a platelet perpendicular to the c crystallographic axis. In this way, only magnetic measurements with B parallel to c were developed for this crystal. The Nd 31 :LiYF 4 sample was a small cube with dimensions 53535 mm 3 . The Zeeman spectra were recorded in two steps. In the first step, the NdOCl was analysed at 10 K in an experimental apparatus for MCD measurements under magnetic fields up to 1.8 T [5]. In the second step, the NdOCl and Nd 31 :LiYF 4 crystals were analysed at 4.2 K and the magnitudes of B up to 6.2 T were produced by superconductor coils placed in the Helmholtz configuration. In both cases only the visible wavelength range was studied and the light was analysed through a Czerny–Turner type HR 1000 Jobin Yvon mono-
0925-8388 / 97 / $17.00 1997 Elsevier Science S.A. All rights reserved PII S0925-8388( 96 )02546-7
M. A. Couto dos Santos et al. / Journal of Alloys and Compounds 250 (1997) 336 – 341
chromator equipped with an R374 Hamamatsu photomultiplier. The resolution was smaller than 0.07 nm.
3. Calculations In order to get confident phenomenological wave functions in the absence of a magnetic field, the energy level scheme was first fitted. The Hamiltonian describing the electrostatic, spin–orbit, interconfigurational (HFI ) and crystal field (HCF ) contributions is given in Eqs. (1,2): HF1 5 H0 1
v50,1,2,3
E v (nf,nf )e v 1 A SO 1 L(L 1 1)
1 G(G2 ) 1 G(R 7 ) 1
52,3,4,6,7,8
T t
(1)
k
fB (C 1 (21) C k q
HCF 5
k q
q
k 2q
)
k 52,4,6q50 k
k
q
k
1 iS q (C q 2 (21) C 2q ) g
(2)
H0 is the spherically symmetric part, E v , , , , and T are respectively Racah, spin–orbit coupling constant, two electrons and Judd’s three-body free ion parameters, multiplied by their respective angular part. B kq and S kq are real and imaginary parts of the crystal field parameters and C kq are the spherical harmonic tensors [10]. The two point groups considered here are isomorphous, in the sense that their character tables are identical as well as the even part of their crystal field potential expansion. The difference lies in the odd part, responsible for the transition intensities as well as for the different polarisation rules. In spite of the fact that the correlation crystal field has not been considered here [11,12], good rms deviation ( ) values are found for the 2 H(2) 11 / 2 level of the Nd 31 ion. The contribution of the 2 H(2) 11 / 2 states to the value of is 11% [6], whereas the best result found in the literature is 19% [1]. Thus, no further parameters have been included, but a more complete energy level scheme has been simulated (137 experimental levels against a maximum of 129 in the literature [13]). Table 1 shows the sets of free ion and crystal field parameters used in the simulation for both symmetries, including the rms deviations . In the presence of a magnetic field, the appropriate Hamiltonian (Eq. (3)) is introduced in the secular determinant before diagonalising it. This makes possible a discussion of the magnetic strength effects all together. At this point, the free ion and crystal field parameters previously obtained are kept constant and B varies from 0 to 8 or 10 T: HM 5 B B(L 1 ge S)
(3)
337
Table 1 Free ion, crystal field parameters and values for NdOCl [5] and Nd:LiYF 4 [6] (units cm 21 ) Parameter
NdOCl C 4v
Nd:LiYF 4 D 2d
E0 E1 E2 E3 T2 T3 T4 T6 T7 T8 B 20 B 40 B 44 B 60 B 64
11616 4697 22.92 471.2 870.3 19.6 2649 1791 398 32 75 2240 296 330 2920 2333 2819 934 2209
23726 4822 23.72 485.6 874.5 21.8 2604 1513 365 41 86 2251 321 373 421 2985 21146 7 21074
No. of levels
105 20
137 17.7
where B is the Bohr magneton, ge is the gyromagnetic factor, L and S are the total orbital and total spin angular momentum operators respectively. The wave functions are written on the basis of all the configuration states, Kramers’ degeneracy thus being lifted (Eq. (4)):
5
a
SLJM
SLJM
SLJM
(4)
where a SLJM are the probability amplitudes of the wave function expansion.
4. Results and discussion To compare the simulation with experiment, we have considered the 4 I 9 / 2 \ 4 G 7 / 2 and 4 G 9 / 2 transitions of the NdOCl crystal [5] and the 4 I 9 / 2 \ 4 F 9 / 2 and 4 G 7 / 2 transitions with both orientations of B in the case of Nd 31 :LiYF 4 [6]. The magnetic splitting of these transitions makes explicit linear, non-linear, crossing and anti-crossing degeneracy lifting behaviours. We have characterised each state by the M value of the most important ket in their wave function composition, connected with the crystal quantum number (M 5 (mod q)), convenient for describing the irreducible representations of isomorphic point groups [14]. Their initial and final M values are written on the diagrams of the magnetic splitting to underline the evolution of the main ket. In some cases, although the wave functions remain orthogonal, the main coefficients are the same for different sublevels. Calcula-
338
M. A. Couto dos Santos et al. / Journal of Alloys and Compounds 250 (1997) 336 – 341
tions have shown that, even for the highest experimental value of B, the magnetic separation of the ground state is smaller than the average value of the experimental line width at 4.2 K. In practice, to determine the energy positions, the ground level has been considered as unsplit.
4.1. Observed and simulated level splitting versus B 4.1.1. NdOCl 4 G 7 / 2 : this level is a good example of the simulation reproducing satisfactorily the behaviour of all states. Three components have a rather linear splitting, but the third one (M 5 ; 1 / 2) is practically unsplit (Fig. 1). 4 G 9 / 2 : the third state (M 5 ; 9 / 2) has not been observed. The other states are split and in part superimposed, which makes the assignment difficult. The simulation shows a well-defined sublevel crossing pattern with a magnetic energy separation comparable with the crystal field splitting (Fig. 2). Many of the split transitions not shown on the figures are also well simulated along with their g-factor values (e.g. for the 4 F 9 / 2,3 / 2 and 2 D(1) 5 / 2,25 / 2 components these values are 1.31 and 6.42 respectively as Fig. 2. 4 G 9 / 2 splitting with B / /c for NdOCl. Experimental (1) and calculated (———).
experimental results, calculation gives 1.19 and 5.95 [5]). The badly simulated cases are discussed below.
Fig. 1. 4 G 7 / 2 splitting with B / /c for NdOCl. Experimental (1) and calculated (———).
4.1.2. Nd 31 : LiYF4 4 F 9 / 2 : when B is perpendicular to c, the first two states (M 5 ; 5 / 2, ;7 / 2) present a non-linear magnetic repulsion. The third (M 5 ; 7 / 2) and fourth states (M 5 ; 3 / 2) also present such behaviour, but the fourth one shows only a blue shift (Fig. 3(a)). The last state (M 5 ; 9 / 2) is isolated and only slightly blue-shifted. The simulation reproduces satisfactorily the non-linear splitting of the first three lines, and the last two lines are partially simulated (Fig. 3(b)). 4 G 7 / 2 : for B parallel to c the first two transitions (M 5 ; 1 / 2, ;3 / 2) split non-linearly. For the upper two lines (M 5 ; 5 / 2, ;7 / 2) a blue shift has been measured. The simulation gives a negligible splitting for the first transition (M 5 ; 1 / 2) and reproduces well the second line (M 5 ; 3 / 2) splitting. For the last two transitions the fitting is linear and exhibits a crossing (Fig. 4). Among the transitions not presented here in the figures, the 4 G 5 / 2 level with B parallel to c is well
M. A. Couto dos Santos et al. / Journal of Alloys and Compounds 250 (1997) 336 – 341
339
Fig. 3. (a) 4 F 9 / 2 absorption spectra with Bqc for Nd 31 :LiYF 4 . (b) 4 F 9 / 2 splitting with Bqc for Nd 31 :LiYF 4 . Experimental (1) and calculated (———).
simulated with a small experimental and calculated splitting. On the contrary, for the 4 G 7 / 2 level with B perpendicular to c some lines present a very pronounced experimental magnetic splitting, whereas an almost B independent effect should be expected from their simulation [6].
4.2. Discussion When the energy splitting induced by the magnetic field is small in comparison with the crystal field splitting, Kramers’ degeneracy lifting is proportional to the magnetic strength (normal Zeeman effect) and
340
M. A. Couto dos Santos et al. / Journal of Alloys and Compounds 250 (1997) 336 – 341
possess the same irreducible representation and the magnetic field leads them close to each other, a repulsion occurs because of a classical group theoretical rule [19]. In this situation, the wave functions exchange their kets after the repulsion point. When B is perpendicular to c the secular determinant has non-diagonal elements, even for weak magnetic fields, because of the triangularity rules between the ;1 components of the magnetic field tensor and the ket components. Non-linear behaviour as a function of B with or without crossing features is the general case. A linear splitting occurs only on isolated levels, when the mixing of kets is negligible. When B is parallel to c (component 0 of the magnetic field tensor), the splittings are linear except when two close sublevels have the same irreducible representation [20]. This is the case for the crystal field states associated with 5 ; 1 / 2 and 5 ; 3 / 2 irreducible representations of the 2 H(2) 9 / 2 level in NdOCl (situation (ii), Fig. 5). For NdOCl, the complete calculation allows us to explain most of the experimental splitting features, including some cases poorly reproduced in Ref. [5]. However, some situations remain badly simulated (e.g. the experimental and calculated g-factors for the 4 G 5 / 2,25 / 2 and 4 G 5 / 2,1 / 2 components are 1.49, 1.23 and 0.7, 0.19 respectively) as well as in the case of Fig. 4. 4 G 7 / 2 splitting with B / /c for Nd 31 :LiYF 4 . Experimental (1) and calculated (———).
perturbation theory can be applied normally, especially to calculate the g values. If not, that is if the magnetic energy separation is higher by around 10% with respect to the crystal field energy separation, it is necessary to include the magnetic Hamiltonian in the secular determinant. This procedure is used here because sublevel linear and non-linear as well as sublevel crossing and anti-crossing patterns are found experimentally. From all transitions presented above, two cases can occur. (i) Crossing case. Accurate measurements are difficult to perform, because the average line width at liquid helium temperature is about 8 cm 21 . However, we can simulate it. A wide analysis was made in a series of articles with the Ho 3 ion doped into several hosts [2,15–18], where measurements have been made at temperatures under 2 K. The line widths were smaller than 5 cm 21 . It was then possible to observe experimental crossing patterns, reproduced by the calculation. The states with different irreducible representations can cross, which means a magnetic accidental degeneracy, corresponding to a precise value of the magnetic field. (ii) Anti-crossing case. When neighbouring states
Fig. 5. 2 H(2) 9 / 2 splitting with B / /c for NdOCl.
M. A. Couto dos Santos et al. / Journal of Alloys and Compounds 250 (1997) 336 – 341
Nd 31 :LiYF 4 , when the wave functions contain important components of kets with different multiplicity corresponding to cases for which the magnetic splitting is almost null and / or when a non-linear anti-crossing degeneracy lifting versus B is calculated. One possible explanation lies in the fact that the spin correlated crystal field effect has not been included in the simulation [13]. For the Nd 31 :LiYF 4 system there are cases where only one transition is observed (e.g. 4 F 9 / 2 (M 5 ; 9 / 2, ;3 / 2) (Fig. 3(a) Fig. 3(b)) and 4 G 7 / 2 (M 5 ; 7 / 2, ;5 / 2) (Fig. 4)). For each doublet, we have observed that the intensity of one Zeeman component is relatively weak. When only transition to one component is measured, the transition probability to the other component could be close to zero. This is clearly shown in Fig. 3(a), corroborated by the fact that the observed peak positions are well simulated.
5. Conclusion Measurements and simulation of the Zeeman effect of NdOCl and 1%Nd 31 :LiYF 4 crystal hosts have been analysed in the visible spectral range for testing phenomenological wave functions. The conventional set of free ion and crystal field parameters has been used. Most of the observed behaviour under magnetic field (linear, non-linear, crossing and anti-crossing) is reproduced by introducing the magnetic Hamiltonian in the secular determinant. The crossing patterns are discussed and interpreted by considering a magnetic accidental degeneracy. However, at this moment we cannot explain by our simulations the poor agreement between the experimental and simulated splitting of the 4 G 5 / 2 level in NdOCl, and the lower component of the 4 G 7 / 2 level in Nd 31 :LiYF 4 , in spite of the fact that the energy positions of its crystal field components are very well simulated in the absence of the magnetic field.
341
Acknowledgments One of us (MACS) thanks the CNPq (Brazilian agency). The authors are deeply grateful to Professor Paul Caro for clarifying discussions.
References ¨ [1] C. Gorller-Walrand, M. Behets, P. Porcher and I. Laursen, J. Chem. Phys., 83 (9) (1985) 4329. [2] M. Enderle, B. Pilawa, W. Schlaphof and H.G. Kahle, J. Phys.: Condens. Matter, 2 (1990) 4685. [3] O.L. Malta, S.J.L. Ribeiro, M. Faucher and P. Porcher, J. Chem. Phys. Solids, 52 (4) (1991) 587. [4] F. Ramaz, J.C. Vial and R.M. MacFarlane, J. Lumin., 53 (1992) 244. [5] L. Beaury, J. Derouet, P. Porcher, P. Caro and P.G. Feldman, J. Less-Common Met., 126 (1986) 263. [6] M.A. Couto dos Santos, P. Porcher, J.-C. Krupa and J.-Y. Gesland, J. Phys.: Condens. Matter, 8 (1996) 4653. [7] L.Ya. Shekun, Opt. Spectrosc., 23 (1967) 422. [8] D.E. Wortman, J. Phys. Chem. Solids, 33 (1972) 311. [9] H.P. Christensen, Phys. Rev. B, 19 (12) (1979) 6564. [10] W.T. Carnall, G.L. Goodman, K. Rajnak and R.S. Rana, A systematic analysis of the spectra of the lanthanides doped into the single crystal LaF3 , Argonne National Laboratory Report, Argonne, IL, 1988. [11] M. Faucher, D. Garcia, P. Caro, J. Derouet and P. Porcher, J. Phys. Fr., 50 (1989) 219. [12] C.L. Li and M.F. Reid, Phys. Rev. B, 42 (4) (1990) 1903. [13] C.K. Jayasankar, F.S. Richardson, M.F. Reid, P. Porcher and P. Caro, Inorg. Chim. Acta, 139 (1987) 287. [14] B.G. Wybourne, Spectroscopic Properties of Rare Earths, Wiley, New York, 1965. [15] M. Enderle, B. Pilawa, M. Schwab and H.G. Kahle, J. Phys.: Condens. Matter, 2 (1990) 4701. [16] M. Enderle, B. Pilawa and H.G. Kahle, J. Phys.: Condens. Matter, 2 (1990) 4711. [17] B. Pilawa, J. Phys.: Condens. Matter, 2 (1990) 5555. [18] B. Pourat, B. Pilawa and H.G. Kahle, J. Phys.: Condens. Matter, 3 (1991) 6069. [19] B.R. Judd, Cours de Spectroscopie Cristalline, Orsay, 1962 (unpublished). ´ ´ [20] A. Herpin, Theorie du Magnetisme, Presses Universitaires de France, Paris, 1968.
L
Measurements and simulation of the Zeeman splitting in NdOCl and Nd 31 :LiYF 4 Marcos A. Couto dos Santos 1 , Laure Beaury, Jacqueline Derouet, Pierre Porcher Laboratoire de Chimie Metallurgique et Spectroscopie des Terres Rares, 1 Place Aristide Briand, 92195, Meudon, France
Abstract Measurements and simulation of the Zeeman effect in NdOCl and 1%Nd 31 :LiYF 4 single crystals were carried out. The rare earth local symmetries are C 4v and D 2d respectively. The isomorphism between these two point groups allows discussion on symmetry assignments. The attention is focused on the 4 G 7 / 2 and 4 G 9 / 2 levels for NdOCl and on the 4 F 9 / 2 and 4 G 7 / 2 levels for Nd 31 :LiYF 4 , because they are representative of the general splitting behaviour. We have first revisited the simulation of the energy level scheme in order to deal with confident wave functions. In the experimental magnetic field (B) range we have obtained linear, non-linear, undefined crossing and anti-crossing levels. These features depend on the orientation of B, on the irreducible representation and on the transition closeness. The line widths have been the main obstacle in confirming crossing regions, because their average values are 8 cm 21 at liquid helium temperature. The simulations were performed by introducing the magnetic Hamiltonian B B(L1ge S) in the secular determinant before diagonalisation. A magnetic accidental degeneracy is considered when discussing the crossing and anti-crossing cases. Keywords: Neodymium; Crystal field; Zeeman effect
1. Introduction The observed electronic transitions inside a 4f N configuration mean that there is an effective interaction between the Stark 4f N states and more excited configurations of opposite parity (e.g. 4f N 21 5d and 4f N 21 5g), which permits the violation of Laporte’s rule. In calculations as well as in theoretical approaches, the wave functions associated with the energy levels are described on the basis of the ground configuration states, the interaction with excited configurations being taken into account by means of effective operators [1,2]. In fact, experiment, theory and fitting procedures are put together with the aim of testing phenomenological wave functions by applying them to the interpretation of spectroscopic and / or magnetic properties of RE ions [1–4]. The Zeeman effect can be measured by magneto-optical techniques and is a way of confirming line intensities for different polarisations. In this sense, this work is dedicated to measurements and simulation of the Zeeman effect of the Nd 31 ion in the NdOCl and 1%Nd 31 :LiYF 4 systems. For NdOCl, the calculations were developed 1
Postdoctoral period, CNPq-Brazil.
within the C 4v site symmetry, whereas the approximate D 2d local symmetry, very close to the S 4 real one, has been used for the Nd 31 :LiYF 4 [5,6]. This is justified by the small value of the imaginary crystal field parameter S 46 [6–9].
2. Experiment The NdOCl sample was a platelet perpendicular to the c crystallographic axis. In this way, only magnetic measurements with B parallel to c were developed for this crystal. The Nd 31 :LiYF 4 sample was a small cube with dimensions 53535 mm 3 . The Zeeman spectra were recorded in two steps. In the first step, the NdOCl was analysed at 10 K in an experimental apparatus for MCD measurements under magnetic fields up to 1.8 T [5]. In the second step, the NdOCl and Nd 31 :LiYF 4 crystals were analysed at 4.2 K and the magnitudes of B up to 6.2 T were produced by superconductor coils placed in the Helmholtz configuration. In both cases only the visible wavelength range was studied and the light was analysed through a Czerny–Turner type HR 1000 Jobin Yvon mono-
0925-8388 / 97 / $17.00 1997 Elsevier Science S.A. All rights reserved PII S0925-8388( 96 )02546-7
M. A. Couto dos Santos et al. / Journal of Alloys and Compounds 250 (1997) 336 – 341
chromator equipped with an R374 Hamamatsu photomultiplier. The resolution was smaller than 0.07 nm.
3. Calculations In order to get confident phenomenological wave functions in the absence of a magnetic field, the energy level scheme was first fitted. The Hamiltonian describing the electrostatic, spin–orbit, interconfigurational (HFI ) and crystal field (HCF ) contributions is given in Eqs. (1,2): HF1 5 H0 1
v50,1,2,3
E v (nf,nf )e v 1 A SO 1 L(L 1 1)
1 G(G2 ) 1 G(R 7 ) 1
52,3,4,6,7,8
T t
(1)
k
fB (C 1 (21) C k q
HCF 5
k q
q
k 2q
)
k 52,4,6q50 k
k
q
k
1 iS q (C q 2 (21) C 2q ) g
(2)
H0 is the spherically symmetric part, E v , , , , and T are respectively Racah, spin–orbit coupling constant, two electrons and Judd’s three-body free ion parameters, multiplied by their respective angular part. B kq and S kq are real and imaginary parts of the crystal field parameters and C kq are the spherical harmonic tensors [10]. The two point groups considered here are isomorphous, in the sense that their character tables are identical as well as the even part of their crystal field potential expansion. The difference lies in the odd part, responsible for the transition intensities as well as for the different polarisation rules. In spite of the fact that the correlation crystal field has not been considered here [11,12], good rms deviation ( ) values are found for the 2 H(2) 11 / 2 level of the Nd 31 ion. The contribution of the 2 H(2) 11 / 2 states to the value of is 11% [6], whereas the best result found in the literature is 19% [1]. Thus, no further parameters have been included, but a more complete energy level scheme has been simulated (137 experimental levels against a maximum of 129 in the literature [13]). Table 1 shows the sets of free ion and crystal field parameters used in the simulation for both symmetries, including the rms deviations . In the presence of a magnetic field, the appropriate Hamiltonian (Eq. (3)) is introduced in the secular determinant before diagonalising it. This makes possible a discussion of the magnetic strength effects all together. At this point, the free ion and crystal field parameters previously obtained are kept constant and B varies from 0 to 8 or 10 T: HM 5 B B(L 1 ge S)
(3)
337
Table 1 Free ion, crystal field parameters and values for NdOCl [5] and Nd:LiYF 4 [6] (units cm 21 ) Parameter
NdOCl C 4v
Nd:LiYF 4 D 2d
E0 E1 E2 E3 T2 T3 T4 T6 T7 T8 B 20 B 40 B 44 B 60 B 64
11616 4697 22.92 471.2 870.3 19.6 2649 1791 398 32 75 2240 296 330 2920 2333 2819 934 2209
23726 4822 23.72 485.6 874.5 21.8 2604 1513 365 41 86 2251 321 373 421 2985 21146 7 21074
No. of levels
105 20
137 17.7
where B is the Bohr magneton, ge is the gyromagnetic factor, L and S are the total orbital and total spin angular momentum operators respectively. The wave functions are written on the basis of all the configuration states, Kramers’ degeneracy thus being lifted (Eq. (4)):
5
a
SLJM
SLJM
SLJM
(4)
where a SLJM are the probability amplitudes of the wave function expansion.
4. Results and discussion To compare the simulation with experiment, we have considered the 4 I 9 / 2 \ 4 G 7 / 2 and 4 G 9 / 2 transitions of the NdOCl crystal [5] and the 4 I 9 / 2 \ 4 F 9 / 2 and 4 G 7 / 2 transitions with both orientations of B in the case of Nd 31 :LiYF 4 [6]. The magnetic splitting of these transitions makes explicit linear, non-linear, crossing and anti-crossing degeneracy lifting behaviours. We have characterised each state by the M value of the most important ket in their wave function composition, connected with the crystal quantum number (M 5 (mod q)), convenient for describing the irreducible representations of isomorphic point groups [14]. Their initial and final M values are written on the diagrams of the magnetic splitting to underline the evolution of the main ket. In some cases, although the wave functions remain orthogonal, the main coefficients are the same for different sublevels. Calcula-
338
M. A. Couto dos Santos et al. / Journal of Alloys and Compounds 250 (1997) 336 – 341
tions have shown that, even for the highest experimental value of B, the magnetic separation of the ground state is smaller than the average value of the experimental line width at 4.2 K. In practice, to determine the energy positions, the ground level has been considered as unsplit.
4.1. Observed and simulated level splitting versus B 4.1.1. NdOCl 4 G 7 / 2 : this level is a good example of the simulation reproducing satisfactorily the behaviour of all states. Three components have a rather linear splitting, but the third one (M 5 ; 1 / 2) is practically unsplit (Fig. 1). 4 G 9 / 2 : the third state (M 5 ; 9 / 2) has not been observed. The other states are split and in part superimposed, which makes the assignment difficult. The simulation shows a well-defined sublevel crossing pattern with a magnetic energy separation comparable with the crystal field splitting (Fig. 2). Many of the split transitions not shown on the figures are also well simulated along with their g-factor values (e.g. for the 4 F 9 / 2,3 / 2 and 2 D(1) 5 / 2,25 / 2 components these values are 1.31 and 6.42 respectively as Fig. 2. 4 G 9 / 2 splitting with B / /c for NdOCl. Experimental (1) and calculated (———).
experimental results, calculation gives 1.19 and 5.95 [5]). The badly simulated cases are discussed below.
Fig. 1. 4 G 7 / 2 splitting with B / /c for NdOCl. Experimental (1) and calculated (———).
4.1.2. Nd 31 : LiYF4 4 F 9 / 2 : when B is perpendicular to c, the first two states (M 5 ; 5 / 2, ;7 / 2) present a non-linear magnetic repulsion. The third (M 5 ; 7 / 2) and fourth states (M 5 ; 3 / 2) also present such behaviour, but the fourth one shows only a blue shift (Fig. 3(a)). The last state (M 5 ; 9 / 2) is isolated and only slightly blue-shifted. The simulation reproduces satisfactorily the non-linear splitting of the first three lines, and the last two lines are partially simulated (Fig. 3(b)). 4 G 7 / 2 : for B parallel to c the first two transitions (M 5 ; 1 / 2, ;3 / 2) split non-linearly. For the upper two lines (M 5 ; 5 / 2, ;7 / 2) a blue shift has been measured. The simulation gives a negligible splitting for the first transition (M 5 ; 1 / 2) and reproduces well the second line (M 5 ; 3 / 2) splitting. For the last two transitions the fitting is linear and exhibits a crossing (Fig. 4). Among the transitions not presented here in the figures, the 4 G 5 / 2 level with B parallel to c is well
M. A. Couto dos Santos et al. / Journal of Alloys and Compounds 250 (1997) 336 – 341
339
Fig. 3. (a) 4 F 9 / 2 absorption spectra with Bqc for Nd 31 :LiYF 4 . (b) 4 F 9 / 2 splitting with Bqc for Nd 31 :LiYF 4 . Experimental (1) and calculated (———).
simulated with a small experimental and calculated splitting. On the contrary, for the 4 G 7 / 2 level with B perpendicular to c some lines present a very pronounced experimental magnetic splitting, whereas an almost B independent effect should be expected from their simulation [6].
4.2. Discussion When the energy splitting induced by the magnetic field is small in comparison with the crystal field splitting, Kramers’ degeneracy lifting is proportional to the magnetic strength (normal Zeeman effect) and
340
M. A. Couto dos Santos et al. / Journal of Alloys and Compounds 250 (1997) 336 – 341
possess the same irreducible representation and the magnetic field leads them close to each other, a repulsion occurs because of a classical group theoretical rule [19]. In this situation, the wave functions exchange their kets after the repulsion point. When B is perpendicular to c the secular determinant has non-diagonal elements, even for weak magnetic fields, because of the triangularity rules between the ;1 components of the magnetic field tensor and the ket components. Non-linear behaviour as a function of B with or without crossing features is the general case. A linear splitting occurs only on isolated levels, when the mixing of kets is negligible. When B is parallel to c (component 0 of the magnetic field tensor), the splittings are linear except when two close sublevels have the same irreducible representation [20]. This is the case for the crystal field states associated with 5 ; 1 / 2 and 5 ; 3 / 2 irreducible representations of the 2 H(2) 9 / 2 level in NdOCl (situation (ii), Fig. 5). For NdOCl, the complete calculation allows us to explain most of the experimental splitting features, including some cases poorly reproduced in Ref. [5]. However, some situations remain badly simulated (e.g. the experimental and calculated g-factors for the 4 G 5 / 2,25 / 2 and 4 G 5 / 2,1 / 2 components are 1.49, 1.23 and 0.7, 0.19 respectively) as well as in the case of Fig. 4. 4 G 7 / 2 splitting with B / /c for Nd 31 :LiYF 4 . Experimental (1) and calculated (———).
perturbation theory can be applied normally, especially to calculate the g values. If not, that is if the magnetic energy separation is higher by around 10% with respect to the crystal field energy separation, it is necessary to include the magnetic Hamiltonian in the secular determinant. This procedure is used here because sublevel linear and non-linear as well as sublevel crossing and anti-crossing patterns are found experimentally. From all transitions presented above, two cases can occur. (i) Crossing case. Accurate measurements are difficult to perform, because the average line width at liquid helium temperature is about 8 cm 21 . However, we can simulate it. A wide analysis was made in a series of articles with the Ho 3 ion doped into several hosts [2,15–18], where measurements have been made at temperatures under 2 K. The line widths were smaller than 5 cm 21 . It was then possible to observe experimental crossing patterns, reproduced by the calculation. The states with different irreducible representations can cross, which means a magnetic accidental degeneracy, corresponding to a precise value of the magnetic field. (ii) Anti-crossing case. When neighbouring states
Fig. 5. 2 H(2) 9 / 2 splitting with B / /c for NdOCl.
M. A. Couto dos Santos et al. / Journal of Alloys and Compounds 250 (1997) 336 – 341
Nd 31 :LiYF 4 , when the wave functions contain important components of kets with different multiplicity corresponding to cases for which the magnetic splitting is almost null and / or when a non-linear anti-crossing degeneracy lifting versus B is calculated. One possible explanation lies in the fact that the spin correlated crystal field effect has not been included in the simulation [13]. For the Nd 31 :LiYF 4 system there are cases where only one transition is observed (e.g. 4 F 9 / 2 (M 5 ; 9 / 2, ;3 / 2) (Fig. 3(a) Fig. 3(b)) and 4 G 7 / 2 (M 5 ; 7 / 2, ;5 / 2) (Fig. 4)). For each doublet, we have observed that the intensity of one Zeeman component is relatively weak. When only transition to one component is measured, the transition probability to the other component could be close to zero. This is clearly shown in Fig. 3(a), corroborated by the fact that the observed peak positions are well simulated.
5. Conclusion Measurements and simulation of the Zeeman effect of NdOCl and 1%Nd 31 :LiYF 4 crystal hosts have been analysed in the visible spectral range for testing phenomenological wave functions. The conventional set of free ion and crystal field parameters has been used. Most of the observed behaviour under magnetic field (linear, non-linear, crossing and anti-crossing) is reproduced by introducing the magnetic Hamiltonian in the secular determinant. The crossing patterns are discussed and interpreted by considering a magnetic accidental degeneracy. However, at this moment we cannot explain by our simulations the poor agreement between the experimental and simulated splitting of the 4 G 5 / 2 level in NdOCl, and the lower component of the 4 G 7 / 2 level in Nd 31 :LiYF 4 , in spite of the fact that the energy positions of its crystal field components are very well simulated in the absence of the magnetic field.
341
Acknowledgments One of us (MACS) thanks the CNPq (Brazilian agency). The authors are deeply grateful to Professor Paul Caro for clarifying discussions.
References ¨ [1] C. Gorller-Walrand, M. Behets, P. Porcher and I. Laursen, J. Chem. Phys., 83 (9) (1985) 4329. [2] M. Enderle, B. Pilawa, W. Schlaphof and H.G. Kahle, J. Phys.: Condens. Matter, 2 (1990) 4685. [3] O.L. Malta, S.J.L. Ribeiro, M. Faucher and P. Porcher, J. Chem. Phys. Solids, 52 (4) (1991) 587. [4] F. Ramaz, J.C. Vial and R.M. MacFarlane, J. Lumin., 53 (1992) 244. [5] L. Beaury, J. Derouet, P. Porcher, P. Caro and P.G. Feldman, J. Less-Common Met., 126 (1986) 263. [6] M.A. Couto dos Santos, P. Porcher, J.-C. Krupa and J.-Y. Gesland, J. Phys.: Condens. Matter, 8 (1996) 4653. [7] L.Ya. Shekun, Opt. Spectrosc., 23 (1967) 422. [8] D.E. Wortman, J. Phys. Chem. Solids, 33 (1972) 311. [9] H.P. Christensen, Phys. Rev. B, 19 (12) (1979) 6564. [10] W.T. Carnall, G.L. Goodman, K. Rajnak and R.S. Rana, A systematic analysis of the spectra of the lanthanides doped into the single crystal LaF3 , Argonne National Laboratory Report, Argonne, IL, 1988. [11] M. Faucher, D. Garcia, P. Caro, J. Derouet and P. Porcher, J. Phys. Fr., 50 (1989) 219. [12] C.L. Li and M.F. Reid, Phys. Rev. B, 42 (4) (1990) 1903. [13] C.K. Jayasankar, F.S. Richardson, M.F. Reid, P. Porcher and P. Caro, Inorg. Chim. Acta, 139 (1987) 287. [14] B.G. Wybourne, Spectroscopic Properties of Rare Earths, Wiley, New York, 1965. [15] M. Enderle, B. Pilawa, M. Schwab and H.G. Kahle, J. Phys.: Condens. Matter, 2 (1990) 4701. [16] M. Enderle, B. Pilawa and H.G. Kahle, J. Phys.: Condens. Matter, 2 (1990) 4711. [17] B. Pilawa, J. Phys.: Condens. Matter, 2 (1990) 5555. [18] B. Pourat, B. Pilawa and H.G. Kahle, J. Phys.: Condens. Matter, 3 (1991) 6069. [19] B.R. Judd, Cours de Spectroscopie Cristalline, Orsay, 1962 (unpublished). ´ ´ [20] A. Herpin, Theorie du Magnetisme, Presses Universitaires de France, Paris, 1968.