g Factors and the tetragonal distortion of ligand octahedron for Co2+ in Rb2MgF4 crystal

Spectrochimica Acta Part A 56 (2000) 2061 – 2066 www.elsevier.nl/locate/saa

g Factors and the tetragonal distortion of ligand octahedron for Co2 + in Rb2MgF4 crystal Zheng Wen-Chen a,b,*, Wu Shao-Yi a,b b

a Department of Material Science, Sichuan Uni6ersity, Chengdu 610064, People’s Republic of China International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110015, People’s Republic of China

Received 28 January 2000; accepted 26 February 2000

Abstract The calculation formulas of g-factors g// and gÞ for 3d7 ion in tetragonal octahedral crystals are established from a cluster approach. In these formulas, the parameters related to covalency effect, configuration interaction and low-symmetry crystal field can be determined from the optical spectra and the structural data of the studied system. Based on these formulas, the structural parameters of ligand octahedra of Co2 + in Rb2MgF4 crystal are obtained by fitting the calculated g// and gÞ to the observed values. The result suggests that the CoF6 (and hence MgF6) octahedra in Rb2MgF4:Co2 + are tetragonal compressed. The relationship between the sign of Dg( = gÞ − g//) and the sign of distortion (elongated or compressed) of ligand octahedron and the causes of the mistakes of octahedron distortion for Rb2MgF4:Co2 + in the previous papers are discussed. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Cluster structure in molecular; Electron paramagnetic resonance (EPR); Optical spectra; Crystal- and ligand-field; Rb2MgF4; Co2 +

1. Introduction The compounds A2MF4 (A denotes alkali metal ion and M divalent cation) are isomorphous with K2NiF4. This structure is of considerable current interest because it is believed to be the ideal examples of two-dimensional antiferromagnets (when M= Mn, Co) and the relevant structure for the presently most frequently studied superconducting phase in the high TC oxide superconduc* Corresponding author. Fax: +86-28-5405541. E-mail address: [email protected] (Z. WenChen).

tors. In the compounds, M2 + ion is at a site with tetragonal point symmetry (D4h) being surrounded by an F− octahedron. The sign of tetragonal distortion of MF6 octahedra (elongation where R// − RÞ \ 0 or compression where R// − RÞ B 0, R// and RÞ stand for the M2 + –F− bonding lengths parallel with and perpendicular to C4 axis, respectively) for some A2MF4 crystals were reported [1,2]. However, for Rb2MgF4 crystal, the tetragonal distortion has not been known. Since the electron paramagnetic resonance (EPR) parameters (zero-field splitting and g factors) of paramagnetic ion are sensitive to their immediate environments, one can study the tetragonal distor-

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tion of MF6 octahedra by using the paramagnetic ions as probes. In the previous papers [3,4], we pointed out that for 3d3 and 3d5 ions in A2MF4 crystals, when the zero-field splitting D( = b 02)B 0, the octahedral environments of 3dn ion are compressed and vice versa. From the splitting D= − 526×10 − 4 cm − 1 for Rb2MgF4:Cr3 + [5] and D= − 53.2× 10 − 4 cm − 1 for Rb2MgF4:Mn2 + [2], one can conclude that the ligand octahedra of 3dn ions are compressed. This result suggests that the MF6 octahedra in pure Rb2MgF4 may also be compressed. On the other hand, for Co2 + (3d7) ion in octahedral site, some authors [6–8] thought that when Dg( =gÞ −g//) \0, the ligand octahedron is elongated. Thus, the observed Dg(=6.63− 3.34 \0, [9]) for Rb2MgF4:Co2 + suggests that the ligand octahedra of Co2 + are elongated. The suggestion is consistent with that obtained by analyzing the absorption spectra of Rb2MgF4:Co2 + in [10]. However, since the suggestion is opposite to that obtained in Rb2MgF4:Cr3 + and Rb2MgF4:Mn2 + , it should be made clear whether the opposition is because the distortion sign of ligand octahedron is different from impurity to impurity, or because there are some mistakes in the analysis of EPR and optical spectra in [6 – 8,10]. In order to make it clear, the quantitative and satisfactory explanation of g// and gÞ for Rb2MgF4:Co2 + is required. Abragam and Pryce [6] worked out the perturbation formulas of g// and gÞ for 3d7 (Co2 + ) ion in an axially distorted octahedral site. These formulas are often quoted and used by many researchers [7 – 9,11]. However, in these formulas, the covalency (CO) effect is neglected and some parameters related to configuration interaction (CI) and low-symmetry crystal field are adjustable. Tinkham [12], Rei [13] and Robbroeck et al. [14] considered the contributions of CO effect. However, in their formulas, the treatments of the contributions from CI, CO effects and the lowsymmetry field are oversimplified. Even so, at least four parameters related to the above three contributions are adjustable. So, none of these formulas are able to get the quantitative explanation of g factors, but they are often used to estimate some adjustable parameters from the observed g factors. In this Report, we first of all

establish the calculation formulas of g// and gÞ for 3d7 ion in tetragonal octahedral site from a cluster approach. In these formulas, the contributions from CI, CO effects and low-symmetry field are considered more completely and the parameters related to the three contributions can be obtained from the optical spectra and the structural parameters of the studied system. From these formulas, the structural parameter R// for Rb2MgF4:Co2 + is estimated by fitting the calculated g// and gÞ to the observed values. The result suggests that the ligand octahedra of Co2 + , as those of Cr3 + and Mn2 + in Rb2MgF4, are compressed. The relationship between the sign of Dg and the sign of tetragonal distortion for Co2 + octahedra and the causes of mistakes in [6–8,10] are discussed.

2. Calculation formulas According to LCAO molecular-orbital model, for a 3dn octahedral cluster, the one-electron basis functions can be expressed as [15,16] g= N 1/2 g ( dg − lg pg ),

(1)

as one-electron basis function, where g=t2g or eg denotes the irreducible representation of Oh group. dg  and pg  are, respectively, the d orbit of 3dn ion and p orbit of ligands. Ng and lg are the normalization factors and the orbital mixing coefficients of molecular orbitals. They can be determined from a semiempirical LCAO method [15,16], that yields the approximate relation fg = N 2g (1+l 2g S 2dp(g)− 2lgSdp(g)),

(2)

and the normalization correlation Ng (1−2lg Sdp(g)+ l 2g )= 1,

(3)

where Sdp(g) are the group overlap integrals, which can be calculated theoretically. fg [:(B/ B0 + C/C0)/2] is the ratio of Racah parameters for a 3dn ion in crystal to those of free ion. The spin–orbit (SO) coupling coefficients z, z% and the orbital reduction factors k, k% due to the CO effect can be written as [15,16]

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z =Nt(z 0d +l 2t z 0p/2), z%= (NtNe)1/2(z 0d −ltlez 0p/2), k = Nt(1+l 2t /2), k%= (NtNe)1/2(1 − ltle/2) (4) where z 0d and z 0p are, respectively, the SO coupling coefficient of 3dn ion and ligand in free ion. Thus, from the optical spectral parameters B and C of the studied system, the parameters z, z’, k and k% can be obtained. For a Co2 + (3d7) ion in a cubic octahedral field, the ground orbital state is triplet 4T1, it splits into six Kramers doublets by SO interaction and tetragonal crystal field and the lowest Kramers doublet corresponds to 9 12. For this system, the Hamiltonian can be expressed as H=H0 +H% H0 = HCoul(B, C) +V ac (Dq, Ds, Dt) H% =Hz(k, k%, a, a%) + HSO(z, z%) +V bc (Dq, Ds, Dt) + Hhf(P, P%)

(5)

2063

where x is determined from the energy splitting D(= E{4A2[4T1(F)]}− E{4E[4T1(F)]}) of 4T1 ground state in tetragonal field by the following expression D=





za%2 3 4 za + − (x+ 3) 3a x x+2 6

yi are defined as

  

 

(7)



y1 =

k%z 15f 21 2q 21 + , 3 2E1x E2x

y3 =

k%z 15f1 f2 2q1q2 − 3 2E1X E2X

y4 =

k%z 15f 22 4q 22 + , 3 E1X E2X

y5 =

k%z%q 23 3E2Z

y6 =

k%z 15f 23 2q 23 8r 2 + + , 3 2E1Z E2Z E3



y7 = y3/2

(8) where E1X, E1Z, E2X, E2Z and E3 are, respectively, energy differences between the ground state 4 E[4T1(F)] and the excited states 4E[4T1(P)], 4 4 4 A2[4T1(P)], E[4T2(F)], B2[4T2(F)] and 4 B1[4A2(F)]. They and also the splitting D can be calculated from the optical spectral parameters by using the d–d transition energy matrices of 3d7 ion in tetragonal symmetry. From the perturbation method, the parameters fi, qi, a and a% in the above formulas can be written as

where HCoul is the Coulomb repulsion interaction. V ac (Dq, Ds, Dt) and V bc (Dq, Ds, Dt) are, respec3 1 a= o 2 − 15or + r 2 − t 2, tively, diagonal and off-diagonal parts of the crys2 2 tal-field. Dq is the cubic crystal-field parameter, 3

15 and Ds and Dt, tetragonal ones. Hz is the Zeeman a% = oo% − o%r−tt%, 2 2 term with the effective lande factors a and a% in the axial and perpendicular directions. HSO is the 2 r tt%% r2 − f1 = − oo%%% 1+ − 2 , SO interaction term. Using the perturbation o

15 o oo%% method similar to that in [6], we obtain the per3 r 2 t%t%% turbation formulas of gi based on the cluster f2 = − o%o%%% 1− + , 7 approach for 3d ion in tetragonal octahedra as

15 o 15 o%o%% follows: 3 9 a 3 4 4 4 − +2 2 − y −2 y 4(ka +2) 2 − x (x +2)2 x (x+ 2)2 1 a% x x+2 3 g// =2+ a 2 6 8 + 2+ a% x (x+ 2)2 a 2 2ka 12 a 2 8 12 a 4 4 + + + y4 + y + y6 − y 2 5 a% x +2 x(x + 2) a% (x+ 2) x(x+ 2) a% (x+ 2) 7 gÞ = a 2 6 8 + 2+ a% x (x+ 2)2

 



 

       



    



  





(6)

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f3 =o 1+

r

 

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3. Calculations for Rb2MgF4:Co2 +

.







15o 5 t 15 t%% tr rt%% q1 = − oo%% + − 2 − , 2o 2 o%% o 2oo%%



For Rb2MgF4:Co2 + , from the optical spectra [10], we obtain Dq : − 765 cm − 1,

t% 3t 15 t%% , q2 = − o%o%% + − o% 2o 2 o%% q3 = oo%



t 3t% 15 t%r + + o 2o% 2 o%o



B: 990 cm − 1, (9)

where o%, t% and r are the mixing (or CI) coefficients which characterize the admixture among the wave functions of ground state and excited states. According to the normalization relationships, we have o 2 +t 2 + r 2 =1, o%2 + t%2 =1

C: 3980 cm − 1,

(12)

For free Co2 + ion [17], B0 : 1115 cm − 1, C0 : 4366 cm − 1, z 0d : 533 cm − 1 and for free F− ion [18], z 0p : 220 cm − 1. Thus, we have fg : 0.8997. The integrals Sdp(g) can be obtained from the Slater-type SCF functions [19,20] and the average metal–ligand distance R( (= (2RÞ + R//)/3) and so the parameters z, z%, k and k% can be obtained. From the superposition model [21], we have Ds = (4/7)A( 2(R0)((R0/RÞ)t2 − (R0/R//)t2),

1 2 2

Dt = (16/21)A( 4(R0)((R0/RÞ)t4 − (R0/R//)t4)

o%%(1+ (t/o)2 +(t%%/o%%) ) =1 1

o%%%(1 +(r/o)2 +(t%%/o%%)2)2 =1

(10)

From the perturbation procedure [6] and the above d–d transition energy matrices, we have





25 D t 12 t # , o 25 −30Dq +75B + 9 Ds − Dt 12 −20Dq +6 Ds −









5 −2 15 Ds + Dt 4 r , : o − 80Dq −75Dt +4Ds

t%% − 2 15(4Ds +5Dt) , : o%% −40Dq +300B +(75Dt +28Ds)

where t2 and t4 are the power-law exponents. We take t2 : 3 and t4 : 5 here because of the ionic nature of bonds [21,22]. A( 2(R0) and A( 4(R0) are the intrinsic parameters with the reference distance R0(= R( ). For 3dn octahedral clusters, A( 4(R0): (3/4)Dq [21,22]. The ratio A( 2(R0)/A( 4(R0) is in the range of 9–12 for 3dn ions in many crystals [22–24] and we take A( 2(R0): 9A( 4(R0) here. The bond length RÞ = a/2: 0.20292 nm [2]. The bond length R// is not reported. Using all the above parameters to the calculation formulas of g// and gÞ and fitting the calculated g// and gÞ to the observed values, we obtain for Rb2MgF4:Co2 + R// : 0.1970 nm

t% −20Dq −12Ds : , o% − 30Dq +75B − 18Ds

(13)

(14)

The comparison of g// and gÞ between calculation and experiment is shown in Table 1. In the (11)

Thus, all of the parameters in the calculation formulas of g// and gÞ can be determined from the optical spectra parameters B, C, Dq, Ds and Dt (note: the tetragonal field parameters Ds and Dt are often calculated from the structural parameters because it is difficult to measure them exactly) of the studied system.

Table 1 EPR parameters g// and gÞ for Co2+ in Rb2MgF4

g// gÞ a b

Cal.a

Cal.b

Expt. [9]

6.78 3.14

6.60 3.30

6.63 3.34

Calculated from the parameters Ds and Dt in [10]. Calculated from the parameters Ds and Dt in this work.

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calculation, we obtain Ds :268 cm − 1 and Dt : 67 cm − 1 from the above parameters. The results are close to the observed Ds :230 cm − 1 and Dt : 20 cm − 1 from the optical spectral measurements [10]. The small difference of Ds and Dt between calculation and experiment [10] may be due to the large errors in the observed values obtained from the optical spectra because of the electron– phonon and spin – orbit interactions. The calculated values of g// and gÞ by using the Ds and Dt obtained from the optical spectra in [10] are also shown in Table 1. The results are not as good as those from the structural parameters.

4. Discussion From the above studies, it can be seen that the ligand octahedron of Co2 + in Rb2MgF4 is compressed (i.e. R// BRÞ, even the Ds and Dt obtained in [10] are used, from Eq. (13), we also have R// BRÞ). The result is consistent with those obtained for Cr3 + and Mn2 + in Rb2MgF4 crystals and so the disagreement mentioned in Section 1 is removed. Since the ligand octahedra of various paramagnetic ion probes are compressed, we suggest that the MgF6 octahedron in pure Rb2MgF4 is also compressed. In the calculations, we find that when R// BRÞ, Dg B0; R// =RÞ, Dg = 0 and R// \RÞ, Dg \0. The relation between the sign of Dg and DR( = R// −RÞ) are therefore obtained. Although our above opinions are opposite to those obtained in [6 – 8,10], they are supported by the following points: (1) EPR measurements [25] showed that for K2ZnF4:Co2 + , Dg( : 3.13–6.0: −3.17)B0 and susceptibility analysis [25] showed that for K2CoF4, Dg( : 3.13 − 6.5 : −3.20)B 0. According to the X-ray diffraction studies [1,2], the (MF6)4 − octahedra in both crystals are compressed. (2) For 3dn ion in octahedra with tetragonal symmetry, we have [3] Dg : 2(u − u0)(F11 −F12)



:2



RÞ − R// (F11 −F12) RÞ + R//

(15)

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where F11 and F12 are the spin–lattice coupling coefficients. u is the angle defined as tan u=RÞ/ R// and u0(= p/4) is the same angle in cubic symmetry. From the observed values of Dg and the above R//, RÞ for Rb2MgF4:Co2 + , we obtain F11 − F12 : − 110. The value is comparable with the observed values of Co2 + in other crystals, e.g. for MgO:Co2 + , F11 − F12 : − 101 [26] and for KMgF3:Co2 + , F11 − F12 : − 97 [27]. If for Rb2MgF4:Co2 + , R// \ RÞ, from the observed Dg, we obtain F11 − F12 \ 0. The result is not reasonable. The causes of mistakes in [6–8,10] are as follows: The relation between Dg and DR obtained from [6–8] are based on the studies of g factors of the hydrate (Tutton) salts, where the tetragonal parameters Ds, Dt and hence the separation D are adjustable and the tetragonal distortion of M2 + (H2O)6 is mistaken as elongated. In fact, from the X-ray diffraction experiments [28,29], the hydrate Co2 + Tutton salts, such as the studied Mg(NH4SO4)2·6H2O:Co2 + and Co(NH4SO4)2·6H2O:Co2 + in [6–8], are approximately compressed. Similar mistakes also occur in [10], where by analyzing the signs of parameters Ds and Dt obtained from optical spectra, Ferguson et al. thought that the MF6 octahedra in K2CoF4 is elongated and that in K2MgF4 is compressed. However, the results obtained from X-ray diffraction [1,2] are just opposite to their opinions. In fact, from the crystalfield theory calculation, one can find that for Co2 + in elongated octahedra, the ground state due to the splitting of 4T1 by tetragonal field is 4 A2 rather than 4E given in [6–8,10] (note: for 3d3,7 ions in octahedra, the ground 4F term splits into 4T1, 4T2 and 4A2 states. For 3d7 ion [30], the energy level of 4A2 is higher than that of 4T2 and so Dq B 0 in the 3d3,7 energy matrices). So, in our opinion, the misassignment of ground state of Co2 + in elongated (or compressed) octahedra (which may be due to the misuse of the sign of Dq) in [6–8,10] is the important cause of the mistakes. It appears that the formulas based on the cluster approach can be used to study quantitatively the g factors and to obtain the useful information about the tetragonal distortion of ligand octahedra for Co2 + in tetragonal crystals.

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