Theoretical investigation of zero field splitting parameters for Mn2+ centers in ammonium tartrate

Physica B 406 (2011) 3917–3921

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Theoretical investigation of zero field splitting parameters for Mn2 þ centers in ammonium tartrate Ram Kripal n, Shri Devi Pandey EPR Laboratory, Department of Physics, University of Allahabad, Allahabad 211002, India

a r t i c l e i n f o

abstract

Article history: Received 22 October 2010 Received in revised form 13 July 2011 Accepted 14 July 2011 Available online 23 July 2011

Zero-field splitting (ZFS) parameters D and E for Mn2 þ centers in ammonium tartrate single crystal are calculated with perturbation formulae using the superposition model. The theoretically calculated ZFS parameters for Mn2 þ at site I and site II of ammonium ion are compared with the experimental values obtained by electron paramagnetic resonance (EPR) at room temperature. The superposition model gives the ZFS parameters similar to those from experiment. The energy band positions of optical absorption spectrum of Mn2 þ in ammonium tartrate are calculated using the CFA package and crystal field parameters from superposition model. These are in good agreement with experimental energy band positions. & 2011 Elsevier B.V. All rights reserved.

Keywords: Organic crystals Crystal and ligand fields Spin–orbit effects Electron paramagnetic resonance

1. Introduction Theoretical studies in the past few decades on the spin Hamiltonian of d5 (6S) ions suggested that various mechanisms contribute to the ground state splitting of the magnetic ions interacting with the lattices. The Hamiltonian of a d5 ion can be written as the sum of the free-ion Hamiltonian, the crystal field, the spin–orbit and spin–spin couplings. As usual, the crystal field can be written as the sum of cubic part and a low symmetry component. Because of the weakness of spin–spin coupling, important effect is expected due to the spin–orbit interaction. This effect is called a spin–orbit (SO) mechanism. Blume and Orbach [1] treated the calculation within the ground d5 configuration taking spin–orbit coupling and low symmetry component of the crystal field as a part of perturbation. For calculation of spin–orbit coupling effect in d5 configuration, one may treat the total crystal field as one of the perturbation terms together with spin–orbit interaction [2] or make the calculation in the strong-field as done by Macfarlane for F-state ions [3]. Perturbation involving the spin–spin interaction is the spin–spin mechanism [4] (SS), which contribute values to the spin Hamiltonian parameters much smaller in magnitude than those due to the SO mechanism [5–7].

n

Corresponding author. Tel.: þ91 532 2470532; fax: þ91 532 2460993. E-mail addresses: [email protected] (R. Kripal), [email protected] (S.D. Pandey). 0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.07.026

Three alternative perturbation procedures for the SO mechanism are shown to be equivalently correct [8]. The calculation of the crystal field parameters plays a crucial role in the evaluation of contributions of SO mechanism. Two models, namely the pointcharge model and the superposition model, are generally used to calculate the crystal field parameters. These two models can establish relations between the crystal field parameters and crystal structure properties. The superposition model [9] has been shown to be quite successful in explaining the crystal field splitting of 4fn ions. More recently, this model has been employed to deal with some 3dn ions (e.g. Cr3 þ in Al2O3 [10], Fe2 þ in garnets [11] and Mn2 þ in BiVO4) [12] and the results are satisfactory. The EPR and optical absorption are two powerful tools to study the dynamic aspects of crystalline state, site symmetry of the impurity and nature of bonding in crystals. EPR of Mn2 þ impurity has been widely studied in a variety of single crystals [13–15], as the zero field splitting of this ion is sensitive to even small distortion in the lattice [16]. The EPR study of Mn2 þ impurity in ammonium tartrate (AT) single crystal at room temperature has been reported [17]. There are two possibilities for the site of Mn2 þ center in the single crystal of ammonium tartrate— substitution at ammonium ion and/or structural vacancy. It is interesting and worthwhile to determine the site of this paramagnetic impurity. It was proposed that Mn2 þ ion substitutes ammonium ion with charge compensation. In this paper, we present the calculated ZFS parameters for Mn2 þ ion under the assumption that this ion is present at two sites of ammonium ion, using the point-charge model and the superposition model. The result derived from the superposition model is consistent with the experimental observation.

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2. Theoretical investigation In this section, the ZFS parameters of Mn2 þ ions located at site I and site II in AT (Fig. 1) are calculated using the microscopic spin Hamiltonian (MSH) theory. These calculated values are then compared with the experimental results. The spin Hamiltonian (SH) in the conventional form [18,19] for the spin S¼5/2 ground state of Mn2 þ ion is taken as [20]   1 H ¼ g mB BS þD Sz2  SðS þ 1Þ þEðSx2 Sy2 Þ 3   a  1 4 4 Sx þ Sy þ Sz4  SðS þ 1Þð3S2 þ 3S1Þ þ 6 5 F f35Sz4 30SðS þ1ÞSz2 þ 25Sz2 6SðS þ 1Þ þ 3S2 ðS þ1Þ2 g 180 K þ ½f7Sz2 SðS þ1Þ5gðS þ 2 þ S2 Þ 4 þ

þ ðS þ 2 þS2 Þf7Sz2 SðS þ 1Þ5g þ ASzIz þBðSxIx þSyIyÞ

ð1Þ

where g is the isotropic spectroscopic splitting factor, mB—the Bohr magneton, B—the external field. The parameters D and E are the second-rank axial and rhombic ZFS parameters, whereas a, F and K are the fourth-rank cubic, axial and rhombic ones, respectively. For relationships between the conventional ZFS parameters and those in the (extended) Stevens notation refer to Refs. [18,19]. It may also be useful to consult the review [18] on the spin Hamiltonian formalisms, the review [19a] on the often confused interrelations between the CF and ZFS quantities and the note

[19b] on the incorrect orthorhombic ZFSPs relations. The last two terms in Eq. (1) represent the hyperfine (I¼ 5/2) interaction. The F and K ZFS terms are omitted here as their effect are small [19a,21,22]. The isotropic approximation used for the electronic Zeeman interaction is generally valid for Mn2 þ ions [21,23]. The two approximations in question may slightly affect the fitted value of a [24]. The direction of the maximum overall splitting of EPR spectrum is taken as the z-axis and that of the minimum as the x-axis [25]. The laboratory axes (x, y, z) determined from EPR spectra are found to coincide with the crystallographic axes (CA). The z-axis of the local site symmetry axes, i.e. the symmetry adapted axes (SAA) is along the metal oxygen bond and the other two axes (x, y) are perpendicular to the z-axis. The crystal field theory has been extensively applied to study the spin Hamiltonian parameters [26–30] of d5 ions. These works discuss contributions from different mechanisms to ZFS of the d5 (6S) ion. With these formulae ZFS parameters are determined by the electrostatic, spin–orbit coupling and the crystal field parameters of the d5 ion in a crystal. When the transition metal ion substitutes a host ion, the electrostatic and spin–orbit coupling parameters for the transition metal–ligand bonds with equal ligand ions are similar. Therefore, the electrostatic and spin– orbit coupling parameters of Mn2 þ ion for site I are similar to those of Mn2 þ ion for site II, because Mn2 þ at both sites have same kind of ligand (oxygen ions). However, the local ionic arrangement around Mn2 þ at both sites is different. This difference provides different crystal fields; consequently ZFS parameters of Mn2 þ ion at both sites are different.

Fig. 1. Crystal structure of ammonium tartrate projected down the c-axis.

R. Kripal, S. Devi Pandey / Physica B 406 (2011) 3917–3921

In ammonium tartrate, ammonium ion is located within a distorted octahedron of oxygen ions with average bond length of NH4 þ –O2  as 2.87 and 2.98 A˚ for sites I and II, respectively, and the local symmetry is approximately orthorhombic of first kind (OR-I) [31]. In an OR-I symmetry, the ZFS parameters D and E are derived as [8] 2

Dð4Þ ðSOÞ ¼

Eð4Þ ðSOÞ ¼

2

3x x ðB220 21B20 þ B222 Þ þ ð5B240 4B242 þ 14B244 Þ 70P 2 D 63P 2 G ð2Þ

pffiffiffi 2  pffiffiffi 6x x2  pffiffiffiffiffiffi ð2B20 21xÞB20 þ 3 10B40 þ 2 7B44 B42 2 2 70P D 63P G ð3Þ

where Wybourne notation is used for the crystal-field parameters, Bkq [32–34]; P¼ 7Bþ7C, G¼10Bþ5C, D¼17Bþ5C, B and C are Racah parameters describing the electron–electron repulsion, x is spin–orbit coupling parameter. Meanwhile, the first-, second-, third-, fifth-order perturbations of D and E are zero, and sixth-order term is small enough to be negligible [30]. Thus only fourth-order term is considered here. Eqs. (2) and (3) are appropriate for weakfield cases. These are still correct even when the low-symmetry components are comparable with the cubic part [8]. For transition metal ion in a crystal, the crystal field can be written as X Hc ¼ Bkq CqðkÞ ð4Þ k,q

where CqðkÞ are orbital angular momentum operators and Bkq are crystal field parameters. For the rhombic crystal field, Bkq a0 only for k¼2, 4; q¼0, 2, 4. The crystal field parameters Bkq , closely related to the structure of AT, are calculated using the superposition model. 2.1. Superposition model The SM expresses the crystal field parameters [8] as X Bkq ¼ Ak ðRj ÞKkq ðyj , fj Þ

ð5Þ

j

where the co-ordination factor Kkq ðyj , jj Þ is an explicit function of the angular position of the ligand [32–34]. The intrinsic parameter Ak ðRj Þis given by [16,27] tk R0 Ak ðRj Þ ¼ Ak ðR0 Þ ð6Þ Rj where Rj is the distance between the dn ion and the ligand, Ak ðR0 Þ is the intrinsic parameter of the reference crystal and tk is the power law exponent. The parameter tk and Ak ðR0 Þ can be obtained from the crystal field splitting. The crystal field splitting of an ion in the same bond is usually similar for different crystals. Likewise, the superposition model parameters of the bond in different crystals are similar to each other. The expressions for nonzero crystal field parameters derived for Mn2 þ ion are given in Appendix-A.

3. Analysis and discussion Within the superposition model B2q and B4q are proportional to R  3 and R  7, respectively. Furthermore the formulae of D and E contain (Bkq)2 term. Therefore, although we consider only the nearest ligand oxygen ions, the calculated parameters using the superposition model may be used to identify the site of Mn2 þ center. The bond length and values of cos y are presented in Table 1.

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Table 1 The spherical co-ordinates of O2  for sites I and II at room temperature. NH4 þ (I) site

NH4 þ (II) site

˚ R (A)

Cos y

˚ R (A)

Cos y

9.54 2.83 2.74 9.10 11.6 –

 0.5937  0.1573  0.5579 0.3994  0.5858 –

2.87 15.2 7.5 3.01 4.45 2.87

0.2204  0.1701 0.0844  0.0388 0.3471 0.08045

For free Mn2 þ ion [35,36]: B0 ¼960, C0 ¼ 3325 cm  1, x0 ¼ 336.6 cm  1 and /r 2 S0 ¼ 2:7755 au /r 4 S0 ¼ 23:2594 au, where B0 and C0 are Racah electrostatic parameters, x0 is spin–orbit coupling parameter and /r k S0 is the expectation value of rk for the free ion. The values of B, C, x and /r k S for the transition-metal ion in a crystal are different from those of free ion due to covalency. In fact, the experimental values in a crystal are less than those of the free ion. The average covalency parameter N takes into account the covalency. The Racah parameters, spin– orbit coupling parameter and the expectation values of rk are given by the relation [29,37,38]: B¼N4B0, C ¼N4C0, x ¼N2x0 and /r 2 S ¼ N2 /r 2 S0 , /r 4 S ¼ N2 /r 4 S0 , where 0oN o1(N ¼1 for pure ionic bond). The parameter N depends on the property of the transition-metal–ligand bond. From the experimental data [17] the values B ¼752 cm  1 and C ¼2438 cm  1 are used to calculate average covalency parameter N from expression for Racah parameters. Thus, we have N ¼0.925, 0.941 and average of these values N ¼0.933 is also taken. The ZFS parameters D and E of Mn2 þ ion in ammonium tartrate are calculated from the expressions (2) and (3) using parameters B, C, x and Bkq from the superposition model. In this calculation, the effect of spin–orbit interaction is considered as a part of the perturbation to the free ion Hamiltonian. However, the spin–spin interaction is neglected because its contribution to the spin-Hamiltonian parameters is much smaller than that of the spin–orbit interaction. In the superposition model, the non-zero crystal field parameters Bkq (Appendix-A) of Mn2 þ ion at sites I and site II are calculated by considering the parameters A2 and A4 as well as the arrangement of O2  ions around both sites of ammonium ion. For Mn2 þ –O2  bonds [39], t2 ¼3, t4 ¼7 and for tetrahedral coordination A4 correlates Dq as follows [8]: A4 ðR0 Þ ¼ 

27 Dq 16

ð7Þ

The value of Dq are obtained from optical data. In our case we have usedA4 ðR0 Þ ¼ 1290:93 cm1 where R0 ¼2.15 A˚ is the Mn2 þ –O2  bond length in AT, which is slightly larger than the value taken by Yeom et al. [12]. For 3dn ions the ratio A2 =A4 is constant and for 3d5 its value lies between 8 and 10 [10,11,40]. The values A2 =A4 ¼11.4, 11.7, 12 and A2 =A4 ¼8.7, 8.8, 8.9 are taken to evaluate A2 ðR0 Þ for site I and site II, respectively. Then the values of crystal field parameters are used for calculation of ZFS parameters D and E. The calculated ZFS parameters of Mn2 þ ion at sites I and II of AT are summarized in Table 2. The ZFS parameters at sites I and II are calculated with N ¼0.925, 0.941 and 0.933. At site I, for N ¼0.933 and A2 =A4 ¼ 11:7 the theoretically calculated ZFS parameters are much closer to the experimental values of ZFS parameters than the values obtained for other N and A2 =A4 taken for calculation. For N¼0.933 and A2 =A4 ¼ 11:7 at site I the crystal field parameters (CFPs) are B20 ¼ 6689.19 cm  1, B22 ¼ 2986.46 cm  1, B40 ¼ 258.37 cm  1, B42 ¼ 1073.11 cm  1 and B44 ¼ 1204.96 cm  1, which are non standard.

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Table 2 Superposition model calculation of zero field splitting parameters (in unit of 10  4 cm  1) for Mn2 þ ion in AT. Parameters

N ¼ 0.933

N ¼ 0.925

NH4þ (I) site A2 =A4 D E

11.7

NH4þ (II) site 8.8

180 58

NH4þ (I) site

N ¼0.941 NH4þ (II) site

12

186 63

8.7

179 60

Transition from 6A1g(s)

T1g(G) Eg(G) 4 A1g(G) 4 Eg(D) 4 T2g(D) 4 T1g(P) 4 T1g(F) 4

NH4þ (II) site 8.9

180 55

190 64

NH4þ (I) site

NH4þ (II) site

–

–

180 57

191 61

good agreement with the experimental data. This indicates that Mn2 þ ions substitute in place of NH4þ in AT.

Wave number (cm  1) Experimental

4

NH4þ (I) site 11.4

183 62

Table 3 Experimental and theoretical energy values of different transitions of Mn2 þ ion (in units of cm  1) in AT.

Experimental

15,513 21,201 23,529 25,454 28,292 32,150 36,650

4. Conclusion

Theoretical NH4þ (I) site

NH4þ (II) site

– 21,211 24,322 25,461 27,016 32,131 36,645

– 21,223 23,954 25,455 28,948 32,216 36,546

Using transformation S3 [25] the standardized CFPs are obtained as B20 ¼6689.19 cm  1, B22 ¼2986.46 cm  1, B40 ¼258.37 cm  1, B42 ¼ 1073.11 cm  1 and B44 ¼  1204.96 cm  1. At site II, for N¼0.941 and A2 =A4 ¼ 8:9 the theoretically calculated ZFS parameters are much closer to the experimental values of ZFS parameters as compared to the values obtained for other N and A2 =A4 taken for calculation. For N¼0.941 and A2 =A4 ¼ 8:9 at site II the CFPs are B20 ¼4826.74 cm  1, B22 ¼  7361.85 cm  1, B40 ¼  332.34 cm  1, B42 ¼ 712.23 cm  1 and B44 ¼ 1257.98 cm  1, which are also non standard. Using transformation S4 [25] the standardized CFPs are obtained as B20 ¼ 6094.29 cm  1, B22 ¼ 3559.18 cm  1, B40 ¼ 192.85 cm  1, B42 ¼1815.96 cm  1 and B44 ¼  2234.44 cm  1. We could not compare these CFPs with the literature data as no data are available for similar ion host system. From Table 2 one can find that the calculated ZFS parameters using the superposition model are consistent with the experimental results for both sites (I and II) of Mn2 þ in AT. However, the values of D and E using the point-charge model [15] are found as, site I: 354  10  4, 51  10  4 cm  1 for N ¼0.933; 331  10  4, 48  10  4 cm  1 for N ¼0.925; 379  10  4, 55  10  4 cm  1 for N ¼0.941; site II: 444  10  4, 91  10  4 cm  1 for N ¼0.933; 433  10  4, 88  10  4 cm  1 for N ¼0.925; 461  10  4, 4 1 94  10 cm for N ¼0.941. These are inconsistent with the experimental values. Thus the superposition model seems to be an appropriate model for theoretical investigation of D and E in the present system. Using the crystal field parameters of the superposition model and CFA package [41,42] considering OR-I symmetry of the crystal field the energy band positions of optical absorption spectrum of Mn2 þ in AT are calculated. The CFA program allows obtaining the complete energy level scheme for any 3dN ion in a crystal field of arbitrarily low symmetry within the whole basis of 3dN states. The energy levels of the impurity ion are obtained by diagonalization of the complete Hamiltonian within the 3dN basis of states in the intermediate crystal field coupling scheme. The Hamiltonian includes the Coulomb interaction (in terms of Racah parameters B and C), Trees correction, the spin–orbit interaction, the crystal field Hamiltonian, the spin–spin interaction and the spin–other orbit interaction. The calculated energy band positions are given in Table 3, which show

The EPR ZFS parameters have been investigated using the superposition model. The superposition model gives the ZFS parameters for Mn2 þ ion in AT similar to the experimental result, but the point-charge model gives unsatisfactory results. From the results of the superposition model and the analysis of optical spectra we conclude that Mn2 þ substitutes for the two sites of ammonium ion in AT. Our results support the inference derived from the experimental data.

Acknowledgement The authors are thankful to the Head of Physics department for providing departmental facilities. One of the authors, Shri Devi Pandey, is thankful to the University Grants Commission for providing research scholarship.

Appendix A Crystal field parameters within the superposition model for the Mn2 þ positioned at NH4þ sites of AT are given as 0  t 2 B B20 ¼ A2 ðR0 ÞB @

ð3Cos2 y1 1Þþ  t2 ð3Cos2 y02 1Þ þ RR00 R0 R1

 t2 R0 R01

ð3Cos2 y01 1Þ þ

 t2 R0 R2

1 ð3Cos2 y2 1Þ C C A

2

ðA1Þ B22 ¼

! t2 t2 t2 t2 pffiffiffi R0 R0 R0 R0 2 2 2 2 6A2 ðR0 Þ Sin y1 þ 0 Sin y01  Sin y2  0 Sin y02 R1 R1 R2 R2

ðA2Þ 1 ð35Cos4 y01 30Cos2 y01 þ3Þ C C  t4  t4 A þ RR02 ð35Cos4 y2 30Cos2 y2 þ 3Þ þ RR00 ð35Cos4 y02 30Cos2 y02 þ 3Þ

0  t 4 B B40 ¼ A4 ðR0 ÞB @

R0 R1

ð35Cos4 y1 30Cos2 y1 þ 3Þþ

  t4 R0 R01

1

ðA3Þ 1  t4 2 2 0 2 2 0 R0 R0 B R1 Sin y1 ð7Cos y1 1Þ þ R01 Sin y1 ð7Cos y1 1Þ C pffiffiffiffiffiffi C B !t4 C B42 ¼ 10A4 ðR0 ÞB C B R0 t4 2 2 0 2 2 0 R0 @ Sin y2 ð7Cos y2 1Þ Sin y2 ð7Cos y2 1Þ A R2 R’ 0  t 4

2

ðA4Þ B44 ¼

1 pffiffiffiffiffiffi 70A4 ðR0 Þ 2

! t4 t4 t4 t4 R0 R0 R0 R0 4 4 4 4 Sin y1 þ 0 Sin y01 þ Sin y2 þ 0 Sin y02 R1 R1 R2 R2

ðA5Þ

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