Investigation of the Cr3+ centers in Rb2ZnF4, Rb2CdF4, and Rb2MgF4 fluorine compounds: A semi-empirical analysis

Journal of Fluorine Chemistry 175 (2015) 152–159

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Investigation of the Cr3+ centers in Rb2ZnF4, Rb2CdF4, and Rb2MgF4 fluorine compounds: A semi-empirical analysis Muhammed Ac¸ıkgo¨z * Faculty of Engineering and Natural Sciences, Bahcesehir University, Bes¸iktas¸, 34353 I˙stanbul, Turkey

A R T I C L E I N F O

A B S T R A C T

Article history: Received 18 February 2015 Received in revised form 27 April 2015 Accepted 28 April 2015 Available online 8 May 2015

The trivalent chromium centers, namely tetragonal (TE) center I and orthorhombic (OR) centers III and IV, have been explored by means of semi-empirical analyses in Rb2MF4 crystals. The local structures for all Cr3+ centers are comprehensively investigated through the zero-field splitting (ZFS) parameters (ZFSPs), which are calculated theoretically and modeled following several modeling approaches. Based on the correlation between the experimental and theoretical ZFSPs, the local structure distortions around the Cr3+ centers are obtained. This study shows that the semi empirical calculations provide a powerful way to determine the local structure in the vicinity of all Cr3+ centers in Rb2MF4 crystals. ß 2015 Elsevier B.V. All rights reserved.

Keywords: Magnetic materials Computational techniques Defects Magnetic properties

1. Introduction The Rb2MF4 crystals, where M is a diamagnetic (Zn, Mg, Cd) divalent ion, are isomorphic with layered perovskite K2NiF4-type crystals, with local structures belonging to tetragonal (D17 4h , I4/ mmm) group symmetry. These crystals are named as rubidium tetrafluorometallates. In general, the structure of fluoride compounds is less complex than those of oxide or sulfide due to the ionic character of the fluorine bond and the relatively small polarizability possessed by the fluoride ion [1]. There is a close relation between the K2NiF4-type structure and perovskite structure. It is known that paramagnetic impurity ions, transition metal (TM) and rare earth (RE) ions, form various magnetic impurity centers when those replace to host cation sites. TM and RE ions, such as Co2+, Fe3+, Cr3+, Mn2+, and Gd3+ doped Rb2MF4 crystals have been investigated in several experimental and theoretical studies, e.g. [2–7]. EPR and optical absorption techniques have been generally used to study the effect of the paramagnetic impurities at the host lattices of the diamagnetic ions. Basically, it is known that three types of Cr3+ centers are formed with different symmetries when Cr3+ ions replace to the M2+ cation sites in A2MF4 crystals [8,9]. One of them is a tetragonal (TE) center (center I), which is assigned to be the charge uncompensated Cr3+

* Tel.: +90 212 3810564; fax: +90 212 3810300. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jfluchem.2015.04.017 0022-1139/ß 2015 Elsevier B.V. All rights reserved.

center. Another one is the monoclinic center (center II) associated with a nearest A+ vacancy along [1 1 1]-axis [8]. Two of these centers are orthorhombic (OR) centers associated with a cation vacancy (Cr3+–VM) or a Li+ ion placing at the nearest M cation site along [1 0 0] or [0 1 0] axis [9]. The latter one provides the charge compensation. Not only the negativity in the effective charge of VM or Li+ in this M2+ site but also the difference in ionic radii of M2+ cation ion and dopant Cr3+ ion affect the local structure around the Cr3+ centers by means of altering the distances between the F ligands and the central Cr3+ ion. In particular, as a result of EPR investigations, it was observed that there exist two types of Cr3+ centers in Rb2MF4 crystals; TE center I and OR centers III and IV. EPR measurements at room temperatures were carried out in Cr3+ doped Rb2ZnF4 and Rb2CdF4 [2] and in Rb2MgF4 [3] for crystals co-doped with Li+. Spin Hamiltonian parameters (g-factor and zero-field splitting (ZFS) parameters (ZFSPs)) were experimentally determined in these studies. Later, a couple of theoretical investigations were devoted to these crystals. OR Cr3+ centers, with the structural cases of Cr3+– VM and Cr3+–Li+, in these three crystals were investigated by Wu et al. by establishing the high-order perturbation formulas of spin Hamiltonian parameters [6]. Furthermore, the local structure of immediate environment around the TE Cr3+, Mn2+ and Fe3+ ions in Rb2ZnF4 crystals was studied by analyzing their EPR data [7]. However, in order to have a comprehensive understanding of the role of Cr3+ ions in such crystals we believe that a more accurate modeling of ZFSPs for all reported Cr3+ centers in Rb2MF4 crystals is still needed. It is worth to say that modeling of ZFSPs may provide a

M. Ac¸ıkgo¨z / Journal of Fluorine Chemistry 175 (2015) 152–159

better insight into properties of these crystals by enabling correlation of crystallographic, spectroscopic, and magnetic data for transition ions in crystals. For this, in this study, we intended to study the Cr3+ centers in Rb2MF4 crystals by applying semi-empirical calculations in a more quantitative manner based on the EPR results. The modeling of the ZFSPs has been carried out by means of the correlation of the crystallographic and spectroscopic data. The effect of the dopant Cr3+ ions on the local structure of the substituted M2+ sites has been investigated through various modeling approaches using superposition model (SPM) technique [10,11]. We have also discussed the obtained results thoroughly in view of previous experimental and theoretical inferences. 2. Crystal structure It is known that the environment of M2+ ions in A2MF4 crystals is closely related to those of AMF3-type cubic perovskite structure [3]. The structure of these A2MF4 crystals can be regarded as consisting of MF2 layers separated by two AF layers [12]. The unitcell of this type of crystals consists of two molecules as shown in Fig. 1. The center of the unit cell is surrounded by slightly distorted octahedra of F ions whereas the divalent ions are located at the corners of the unit cell. In these central MF64 octahedra there are two equivalent F sites along the z-axis as the axial ligands (F1) and four equivalent F sites in the x-y plane as the equatorial ligands (F2). The unit-cell parameters and the ligand distances for MF64 octahedra are given in Table 1 for all three Rb2MF4 crystals at two different temperatures. Most of these crystal structure parameters were provided by Schrama [12]. However, only the ligand distances M–F2 (R2) of F2 from the central M2+ ion were provided in [12]. Actually, knowing lattice parameter a is enough to determine R2 since the site of the binary cations M2+ is exactly at the center of the unit-cell of Rb2MF4 crystals, i.e. a/2 equals to R2.

153

In order to interpret the results of SPM calculations and to determine the local structure distortion around the Cr3+ centers it is pertinent to estimate M–F1 (R1) distances for all these crystals. Thus, we estimated M–F1 distances for MF64 octahedra following several methods: (i) using VESTA software: we read R1 values after drawing the typical TE K2NiF4 crystal structure (see Fig. 1) using the relevant unit-cell parameters and ionic radii of the Rb+ and M2+ ions. (ii) Estimating R1from a factor between c and R1: we used R1 = 0.151c from the X-ray data provided for Mn2+ doped K2ZnF4 [4], for which both R1 and R2 are known experimentally. (iii) We estimated R1 using the formula R1 = R2/tana, where a is the angle characterizes the TE distortion of the octahedral environment with a0 = 450. If a  a0 > 0, then R2 > R1 which corresponds to the compression of the MF6 octahedron and if a  a0 < 0, the MF6 octahedron is elongated so R1 > R2. The a value can be determined from the ZFSP D using D  3(a  a0)G11, where G11 is the spin– lattice coupling coefficient. Here, the R1 values are obtained using 0.6 cm1 for G11 from MgO:Cr3+ system [13] and the D values of the TE Cr3+ centers in each crystal. (iv) We consider the TE distortion ratio (DR/R1 = (R2  R1)/R1), which is usually used to estimate the local TE distortion. We obtained R1 values using DR/R1 = 0.004 observed experimentally for Cr3+ doped RbCdF3 [14]. Note that in (iii) and (iv) the reported parameter values of very similar crystals have been used. The obtained R1 values from these four methods are given in Table 1.

3. Method for calculations Electron magnetic resonance spectra for Rb2MF4 crystals can be analyzed using the spin Hamiltonian (H) describing the energy levels of the ground spin state of transition metal ions doped into crystals. The energy levels splittings of Cr3+ ion can be interpreted by following a spin-Hamiltonian H of the form given in Eq. (1) [15–17]: H ¼ HZe þ HZFS ¼ mB B  g  S þ

X

bqk Oqk

(1)

where the first term consists of Bohr magneton mB, the applied magnetic field B, the spectroscopic splitting factor g, and the effective spin operator S, and the second term consists of ZFSPs bqk and the extended Stevens (ES) operators Oqk defined in [18,19]. The presence of three different Cr3+ centers with two different site symmetries, TE and OR, was shown experimentally in Rb2MF4 crystals. Explicit form of the ZFS term in Eq. (1) can be written for each symmetry case [20–22]. ZFS of a d3 configuration Cr3+ center with TE symmetry can be analyzed by the following explicit expression of the spin-Hamiltonian:   1 HZFS ¼ b02 O02 ¼ D S2z  SðS þ 1Þ 3

(2)

Table 1 The unit-cell parameters of Rb2MF4 crystals and the ligand distances for MF64 octahedra. (i)–(iv) denotes the structure analyses as discussed in text. All values are in [nm] units. Rb2ZnF4

Fig. 1. The unit cell of Rb2MF4 crystals.

Rb2MgF4

T (K) 295 4.2 295 a=b 0.41364 [12] 0.41125 0.40584 [12] c 1.37060 [12] 1.36390 1.3799 [12] M–F1 (i) 0.22259 0.22410 (ii) 0.20696 0.20837 (iii) 0.20667 0.202658 (iv) 0.20600 0.20211 M–F2 0.20682 [12] 0.20563 0.20292 [12]

Rb2CdF4 4.2 295 0.40447 0.44017 [12] 1.37430 1.33980 [12] 0.21758 0.20231 0.21987 0.21921 0.20224 0.22009 [12]

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On the other hand, for Cr3+ center with OR symmetry we can use HZFS as:   1 HZFS ¼ b02 O02 þ b22 O22 ¼ D S2z  SðS þ 1Þ þ EðS2x  S2y Þ 3

(3)

The ZFSPs in Eqs. (2) and (3) can be expressed following the general definitions for the SPM quantities outlined recently in [23,24] as: bqk ¼

X i

 tk R0 b¯ k ðR0 Þ  Kkq ðui ; fi Þ Ri

(4)

where Kkq ðu i ; fi Þ are the coordination factors [25] as functions of the position angles ui and wi of ligands, R0 is the reference distance, Ri are the ligand distances in the ML6 complex; b¯ k ðR0 Þ is the intrinsic parameter, whereas tk is the power law exponent which are treated as adjustable parameters. In SPM applications, the b¯ k ðR0 Þ, tk, and the reference distance R0 are generally combined into a set, known as SPM parameter set [26]. The values of the SPM parameters for the ligand system of the Cr3+–F bond configuration were provided in [27] as: b¯2 ðR0 Þ ¼ ð46; 770  800Þ  104 cm1 , t2 = 0.24  0.03, and R0 = 0.2113 nm. In our calculations the set (b¯2 ðR0 Þ, t2, R0): (46,770, 0.24, 0.2113 nm) was adopted. 4. Results and discussion Corresponding to the TE environment around the M2+ sites, the distances of the axial ligands are slightly different than those of equatorial ligands. It is normally expected that the substitution of Cr3+ ions for M2+ sites induces some local structural distortions, which appear to be either TE or rhombic distortion according to the TE or OR centers, respectively. It is seen that, contrary to the some other A2BF4 crystals such as K2ZnF4, Cr3+ ions do not induce any monoclinic Cr3+ center in Rb2MF4 crystals. The experimental values of ZFSPs are given in Table 2 for three different Cr3+ centers in these crystals. The rhombicity ratio l, which measures the deviation from axial symmetry, for the OR centers is also given in Table 2. In terms of the conventional ZFSPs in EPR and ESR area the ratio l = E/D yields 0  l = E/D  1/3 [28]. As can be seen in the table, the rhombicity of the center III in Rb2CdF4 is quite larger than that of the other OR centers in other crystals. 4.1. Tetragonal (TE) center I No vacancy (Rb+ or M2+) was considered for A2MF4 type crystals associating with the TE Cr3+ centers. These TE centers are also known as the charge uncompensated Cr3+ centers due to inequality in the charges of dopant Cr3+ ions and host M2+ cation ions. Thus, the distortion regarding of these TE Cr3+ centers may be expected to be larger than those of other charge compensated OR centers III and IV. In the frame of SPM, we can calculate the ZFSP D ¼ b02 and

determine the induced local TE distortion in the structure of M2+ sites due to Cr3+. SPM provides the following ZFSPs expression for the TE Cr3+ centers in 6-fold coordination in terms of SPM parameters: b02 ¼ D ¼ b¯ 2 ðR0 Þ

 n  X R0 t2 i¼1

Ri

ð3cos2 u i  1Þ

(5)

Based on the axis system (zjjc, yjjb, xjja) for the Ri, ui and wi of F ligands in the [Cr–F6]3 cluster for this TE Cr3+ center I, Eq. (5) turns out to the following simplest form: "   t2 # R0 t 2 R0  b02 ¼ D ¼ 4b¯ 2 ðR0 Þ (6) R1 R2 The results of the SPM calculations for the ZFSP b02 ¼ D and the distortion parameters are listed in Table 3 for the TE Cr3+ centers in Rb2MF4 crystals. Since R1 is not known the calculations have been performed by changing R2 between 0.004 nm and 0.004 nm, which corresponds to almost 2% of R2. From the results in Table 3, it is clearly seen that R2 > R1 for all three crystals. This indicates that the compression of the MF6 octahedron in these crystals occurs when Cr3+ substitute for M2+ ion. When we change R2 by 0.004 nm, the corresponding change in R1 becomes 0.00395 nm for Rb2ZnF4, 0.00393 nm for Rb2MgF4, and 0.00391 nm for Rb2CdF4. Furthermore, the TE distortion ratio (DR/R1 = (R2  R1)/R1) indicates that the TE distortion on Rb2CdF4 is quite large than that on the other crystals, even it is almost two times that on Rb2ZnF4. The findings in Table 3 for R1 can be discussed in the view of the results of the structural analyses ((ii)–(iv)) in Section 2. They provide a value around 0.2065 nm for Rb2ZnF4, 0.2024 nm for Rb2MgF4, and 0.2195 nm for Rb2CdF4. As can be seen in Table 3, these values are out of ranges obtained for Rb2MgF4 and Rb2CdF4, unlike Rb2ZnF4. This is another indication for higher TE distortion in Rb2MgF4 and Rb2CdF4 with respect to that in Rb2ZnF4. As can also be seen in Table 3, for all three crystals, both R1 and R2 react in the same manner for any distortion on the other one, i.e. when R2 decreases (increases) R1 also decreases (increases). To decide about the type of distortion for the TE Cr3+ center we may consider two points: one of them is the difference in the size of the host M2+ ions (0.074 nm for Zn2+, 0.066 nm for Mg2+, and 0.097 nm for Cd2+) and dopant Cr3+ ion (0.062 nm [29]), which is smaller than all M2+ ions (the profound difference for Rb2CdF4). So the ligands are expected to move toward the central Cr3+ ion. The other point is the role of the effective charges in electrostatic interaction. For Cr3+ substitution for M2+ ions, due to the increase in the positivity of the effective charge for the central ion, the electrostatic attraction increases and thus all the ligands are expected to move toward the central Cr3+. Based on these arguments, it results in the compression of the MF6 octahedron for TE Cr3+ center in Rb2MF4 crystals. The configurations of the

Table 2 The values of the previously determined experimental ZFSPs [2,3] for various Cr3+ centers in Rb2MF4 crystals. The ZFSPs D ¼ b02 and 3E ¼ b22 are given in units of [104] cm1. Compounds

Center

D ¼ b02

3E ¼ b22

l = E/D

Refs.

Rb2ZnF4

I (TE) III (OR) IV (OR)

369.0 +401.5 +433.1

– 20.4 66.9

– 0.017 0.051

[2] [2] [2]

Rb2MgF4

I (TE) III (OR) IV (OR)

526.1 +580.7 +587.1

– 9.3 69.3

– 0.005 0.039

[3] [3] [3]

Rb2CdF4

I (TE) III (OR) IV (OR)

666.4 +554.0 +611.2

– +334.2 +79.5

– 0.201 0.043

[2] [2] [2]

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Table 3 The SPM calculated ZFSPs b02 ¼ D (in 104 cm1) and the obtained distortion parameters for the TE Cr3+ centers in Rb2MF4 crystals. All R values are in [nm]. I (TE) Rb2ZnF4

Rb2MgF4

Rb2CdF4

D ¼ b02

R1

R2

DR2

D ¼ b02

R1

R2

DR2

D ¼ b02

R1

R2

DR2

369.5 369.2 368.9 369.8 369.5 369.2 368.9 368.7 369.5

0.19947 0.20046 0.20145 0.20243 0.20342 0.20441 0.20540 0.20639 0.20737

0.20282 0.20382 0.20482 0.20582 0.20682 0.20782 0.20882 0.20982 0.21082

0.004 0.003 0.002 0.001 0 +0.001 +0.002 +0.003 +0.004

526.4 526.7 526.9 526.0 526.2 526.5 526.7 526.9 526.1

0.19423 0.19521 0.19619 0.19718 0.19816 0.19914 0.20012 0.20110 0.20209

0.19892 0.19992 0.20092 0.20192 0.20292 0.20392 0.20492 0.20592 0.20692

0.004 0.003 0.002 0.001 0 +0.001 +0.002 +0.003 +0.004

666.5 666.3 666.0 666.9 666.6 666.4 666.2 666.0 666.8

0.20978 0.21076 0.21174 0.21271 0.21369 0.21467 0.21565 0.21663 0.21760

0.21609 0.21709 0.21809 0.21909 0.22009 0.22109 0.22209 0.22309 0.22409

0.004 0.003 0.002 0.001 0 +0.001 +0.002 +0.003 +0.004

atoms around TE Cr3+ center and the movements of the ligands are shown in Fig. 2. Both the compression and elongation of the MF6 octahedron are illustrated in this figure. Furthermore, the dependencies of ZFSP D on the distortion parameters DR are presented in Fig. 3 for these Cr3+ centers in Rb2MF4 crystals. For this, the distortion on R1 and R2 has been considered separately. It is clearly seen that the change in DR affects D linearly.

4.2. Orthorhombic (OR) centers III and IV The following ZFSPs expressions are provided by SPM for the OR Cr3+ centers in 6-fold coordination:  n  X R0 t2 ð3cos2 ui  1Þ b02 ¼ D ¼ b¯ 2 ðR0 Þ Ri i¼1 (7)  n  X R0 t2 sin2 ui cos2’i b22 ¼ 3E ¼ 3b¯ 2 ðR0 Þ Ri i¼1

Fig. 2. Configurations of the atoms around the TE Cr3+ center in Rb2MF4 crystal. (a) elongation and (b) compression of the MF6 octahedron. Arrows indicate the movements of the ligands after Cr3+ substitution for M2+ site.

Fig. 3. Variation of the ZFSP D with the distortions on R1 and R2 for TE Cr3+ centers in the Cr3+ Rb2MF4 crystals.

M. Ac¸ıkgo¨z / Journal of Fluorine Chemistry 175 (2015) 152–159

156

It should be noted that, for three OR symmetry point groups D2, C2v, and D2h, there are three mutually perpendicular and equivalent symmetry axes. However, ZFS of these groups can be expressed with the same ZFSPs [23]. In order to analyze these OR centers we have considered two different structural models: one of them is based on only the ligand-length distortions. The other one consists of both ligand-length and angular distortions. 4.2.1. Model 1 This model is based on three ligand-lengths, R 1 , R 2, and R 3 . Here, R 3 is for the intervening F between the central Cr 3+ and M 2+ vacancy site along [1 0 0] direction. Thus, the angular positions of F  ligands in the [Cr–F 6 ] 3 cluster are taken with respect to the axis system (zjjc, yjjb, xjja) as those of TE center I in Section 4.1. The corresponding spherical coordinates for the ligands are: (R 1 , 0, 0); (R 1 , 180, 0); (R 3 , 90, 0); (R 2 , 90, 90); (R 2, 90, 180); (R 2 , 90, 270). Using these coordinates the SPM expressions given in Eq. (7) turn out to be the below ones in terms of R 1, R 2, and R 3 : "    t2  t2 # R0 t2 R0 R0 b02 ¼ D ¼ b¯ 2 ðR0 Þ 4 3  R1 R2 R3 "   t2 # R0 t 2 R0 b22 ¼ 3E ¼ 3b¯ 2 ðR0 Þ  R3 R2

(8)

We obtained the SPM calculated ZFSPs b02 ¼ D and 3E ¼ b22 and the distortion parameters for both OR Cr3+ centers in Rb2MF4 crystals. The results of the calculations are tabulated in Table 4 for various modeling approaches. It is clearly seen that the ligand distance R3 gets larger in Rb2ZnF4 and Rb2MgF4, whereas it gets

shorter in Rb2CdF4 for the center III. On the other hand, for the center IV, R 3 gets shorter in Rb 2CdF 4 and Rb 2MgF 4 but it gets larger in Rb2 ZnF 4 as for the center III. Furthermore, for the center III, the profound distortion on R 3 was found for Rb 2CdF 4 in Calc. b, which corresponds almost 2% change with respect to host structure. However, a quite slight distortion was obtained for Rb 2MgF 4 relative to that for Rb2 CdF 4 (i.e. only 0.05% change). For the center IV, the distortion on R 3 occurs considerably small and similar in all three crystals with about 0.4% change. Moreover, the results for D R2 reveal an inward relaxation of the equatorial ligands in Rb2 ZnF 4, whereas an outward relaxation in Rb 2CdF 4 for both OR centers. As another important point, it is seen that an outward relaxation occurs on R 1 for both OR centers in all three crystals. It is also possible to compare our results with the results obtained by Zheng et al. [7] for the OR centers in these crystals. In [7], the structural data were calculated for both OR centers based on model 1. The distortion DR 3 was obtained 0.0045 (0.0051) nm for Rb2ZnF4 , 0.0057 (0.0062) nm for Rb 2MnF 4, and 0.0037 (0.0060) nm for Rb 2CdF 4, which correspond to 2.18% (2.47%), 2.81% (3.06%), and 1.68% (2.73%) changes for OR center III (IV) with respect to host structures, respectively. It is clear that these are quite larger values than those obtained here. As a further analysis, the dependence of the ZFSPs on the distortion parameters has been explored by changing R 1, R 2, and R 3 in the range of 0.003 nm and +0.003 nm. The results are shown in Fig. 4 for all three crystals. In general, it is clearly seen that both b02 ¼ D and 3E ¼ b22 are linearly dependent on D R. Also, we see that 3E ¼ b22 is not affected by the axial distortion (D R) and D ¼ b02 changes with DR 2 and DR 3 in the same manner whereas 3E ¼ b22 are affected in an opposite manner.

Table 4 The SPM calculated ZFSPs b02 ¼ D and 3E ¼ b22 (in 104 cm1) and the obtained distortion parameters for the OR Cr3+ centers at different temperatures regarding of model 1. All R values are in [nm] and u values are in [degrees]. III (OR) D ¼ b02

3E ¼ b22

R1

R2

R3

Rb2ZnF4

a

Calc. Calc.b Calc.c Calc.d

34.6 401.5 401.6 401.4

0 20.4 20.3 20.4

0.2065 0.21050 0.210755 0.210625

0.20682 0.20682 0.207071 0.206693

0.20682 0.206568 0.20682 0.206945

0 0 0.000251 0.000125

0 0.000252 0 0.000127

Rb2MgF4

Calc.a Calc.b Calc.c Calc.d

57.0 580.6 580.8 580.6

0 9.3 9.3 9.3

0.2024 0.208243 0.208360 0.208300

0.20292 0.20292 0.203033 0.202976

0.20292 0.202807 0.20292 0.202863

0 0 0.000113 0.000056

0 0.000113 0 0.000057

Rb2CdF4

Calc.a Calc.b Calc.c Calc.d

60.8 554.0 554.1 554.0

0 334.1 334.1 334.1

0.2195 0.226622 0.222234 0.224428

0.22009 0.22009 0.215798 0.217944

0.22009 0.224447 0.22009 0.222269

0 0 0.004292 0.002146

0 0.004357 0 0.002179

Calc.a Calc.b Calc.c Calc.d

34.6 433.0 433.2 433.1

0 66.9 66.9 66.8

0.2065 0.210650 0.211490 0.211070

0.20682 0.20682 0.207647 0.207233

0.20682 0.205995 0.20682 0.206408

0 0 0.000827 0.000413

0 0.000825 0 0.000412

Rb2MgF4

Calc.a Calc.b Calc.c Calc.d

57.0 587.1 587.0 587.1

0 69.3 69.2 69.2

0.2024 0.208548 0.207688 0.208119

0.20292 0.20292 0.202079 0.202500

0.20292 0.203765 0.20292 0.203342

0 0 0.000841 0.000420

0 0.000845 0 0.000422

Rb2CdF4

Calc.a Calc.b Calc.c Calc.d

60.8 611.2 611.1 611.2

0 79.4 79.7 79.6

0.2195 0.226347 0.225295 0.225822

0.22009 0.22009 0.219060 0.219575

0.22009 0.221120 0.22009 0.220605

0 0 0.001030 0.000515

0 0.001030 0 0.000515

Compounds

IV (OR) Rb2ZnF4

a b c d

DR2

DR3

Calculations based on the direct matching of R1 to the experimental ZFSP D ¼ b02 with the host crystal structure data for the equatorial ligands (DR2 = DR3 = 0). Calculations based on the direct matching of R1 and the distortion parameter DR3 to the experimental ZFSPs D ¼ b02 and 3E ¼ b22 . Calculations based on the direct matching of R1 and the distortion parameter DR2 to the experimental ZFSPs D ¼ b02 and 3E ¼ b22 . Calculations based on the direct matching of R1 and the distortion parameters DR2 and DR3 to the experimental ZFSPs D ¼ b02 and 3E ¼ b22

M. Ac¸ıkgo¨z / Journal of Fluorine Chemistry 175 (2015) 152–159

(a)

157

(b)

(c)

Fig. 4. Dependence of the ZFSPs D ¼ b02 and 3E ¼ b22 on the distortion parameters (a) DR1, (b) DR2, and (c) DR3 for the OR Cr3+ centers in model 1.

4.2.2. Model 2 This model is based on the axis system (xjjc; zjj½1¯ 1¯ 0; yjjb), thus it consists of compression or elongation of the octahedra around ¯ zjj½1¯ 10-axis, i.e. with both ligand-length and angular distortions. With respect to this axis system the spherical coordinates of F ligands in the [Cr–F6]3 cluster for the Cr3+ centers III and IV are: F1(R1, 908, 08); F2(R1, 908, 1808); F3(R2, 458, 2708); F4(R2, 1358, 2708); F5(R2, 1358, 908); F6(R2, 458, 908). For this model the SPM expressions of the ZFSPs in Eq. (7) turns out to the following explicit expressions: "   #  t2 R0 t2 R0 þ2 ½3cos2 ðu Du Þ  1 b02 ¼ D ¼ 2b¯ 2 ðR0 Þ  R1 R2 "  # (9)  t 2 t2 R R 0 0 b22 ¼ 3E ¼ 6b¯ 2 ðR0 Þ 2 sin2 ðu DuÞ R1 R2 where Du represents the angular distortion for the equatorial ¯ ligands with respect to zjj½1¯ 10-axis. Following several modeling approaches we have carried out the SPM calculations for the OR centers in this model. The results for the ZFSPs and the distortion parameters are tabulated in Table 5. It is clearly seen for both OR centers in all crystals that Du yields a compression of the CrF6 ¯ octahedron along zjj½1¯ 10-axis. Also, Du is rather small and changing between 0.06 and 0.10 degree. The largest distortion is obtained for Rb2CdF4, which can be attributed to the big size of Cd2+ ion relative to that of Cr3+. Furthermore, it is found that F1

ligands relax outward unlike F2 ligands as another common point for both OR centers and all crystals. Comparing OR center III and IV by means of distortions we see that Du s for the center III are larger than those for the center IV in Rb2ZnF4 and Rb2MgF4, unlike the situation in Rb2CdF4. Furthermore, the dependence of the ZFSPs on the distortion parameters are given in Fig. 5 by changing R1 and R2 in the range of 0.003 nm and +0.003 nm whereas changing u between 0.15 and +0.15 degree. As can be clearly seen in Fig. 5, both b02 ¼ D and 3E ¼ b22 are linearly dependent on not only DR but also Du and the change resulted in DR1 and DR2 are equal in magnitude but in opposite manner for all three crystals. Also, b02 ¼ D and 3E ¼ b22 are inversely affected by DR1 and DR2, while they likewise change with Du. 4.2.3. Discussions on models Considering the role of the M2+ vacancy (Cr3+–VM) and Li+ ion at this vacancy site (Cr3+–Li+) the following points can be mentioned: Once the Cr3+ ion substitutes for M2+ ion site an inward relaxation of all the ligands may be expected due to increase in electrostatic attraction between the central Cr3+ and F-ligands as well as the reduction resulted from the size difference between the dopant Cr3+ and host M2+ ions. However, the formation of Cr3+–VM and Cr3+–Li+ may lead to different structural changes in Rb2MF4 crystals. Here, we think that the key role for the structural changes around OR center III (Cr3+–VM) belongs to the electrostatic

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158

Table 5 The SPM calculated ZFSPs b02 ¼ D and 3E ¼ b22 (in 104 cm1) and the obtained distortion parameters for the OR Cr3+ centers in Rb2MF4 crystals according to model 2. All R values are in [nm] and u values are in [degrees]. III (OR) D ¼ b02

3E ¼ b22

R1

R2

DR2

u*

Du

Rb2ZnF4

a

Calc. Calc.b Calc.c

401.5 401.2 401.1

1204.5 20.6 20.7

0.199485 0.204874 0.204477

0.20682 0.20682 0.20642

0 0 0.0004

45 44.9393 44.9393

0 0.0607 0.0607

Rb2MgF4

Calc.a Calc.b Calc.c

580.6 580.6 584.0

1741.8 9.3 9.3

0.192530 0.200241 0.199846

0.20292 0.20292 0.20252

0 0 0.0004

45 44.9107 44.9107

0 0.0893 0.0893

Rb2CdF4

Calc.a Calc.b Calc.c

554.2 554.0 553.8

1662.5 334.0 334.0

0.209528 0.219024 0.218627

0.22009 0.22009 0.21969

0 0 0.0004

45 44.8991 44.8991

0 0.1009 0.1009

Calc.a Calc.b Calc.c

432.9 432.9 432.8

1298.7 66.9 67.0

0.198920 0.204516 0.204119

0.20682 0.20682 0.20642

0 0 0.0004

45 44.9368 44.9368

0 0.0632 0.0632

Rb2MgF4

Calc.a Calc.b Calc.c

586.9 586.9 586.6

1760.6 69.0 69.0

0.192420 0.200567 0.200172

0.20292 0.20292 0.20252

0 0 0.0004

45 44.9057 44.9057

0 0.0943 0.0943

Rb2CdF4

Calc.a Calc.b Calc.c

611.0 611.2 611.0

1833.0 79.4 79.4

0.208467 0.217520 0.217125

0.22009 0.22009 0.21969

0 0 0.0004

45 44.9033 44.9014

0 0.0967 0.0986

Compounds

IV (OR) Rb2ZnF4

a b c

Calculations based on the direct matching of R1 to the experimental ZFSP D ¼ b02 with when there is no distortion for the equatorial ligands (DR2 = Du = 0). Calculations based on the direct matching of R1 and the distortion parameter Du to the experimental ZFSPs D ¼ b02 and 3E ¼ b22 . Calculations based on the direct matching of R1 and the distortion parameters DR2 and Du to the experimental ZFSPs D ¼ b02 and 3E ¼ b22 .

interactions between the neighboring ions. It can be expected that the intervening F ion move toward the central Cr3+ ion due to the decrease in electrostatic attraction between the F ion and M2+ ion

(a)

that was at the vacancy site formerly. However, there is another electrostatic interaction (i.e. repulsion) between pffiffiffi the intervening F and the two axial F ligands over the distance 2R. The magnitude

(b)

(c)

Fig. 5. Dependence of the ZFSPs D ¼ b02 and 3E ¼ b22 on the distortion parameters (a) DR1, (b) DR2, and (c) Du for the OR Cr3+ centers in model 2.

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159

Fig. 6. Configurations of the atoms around the OR Cr3+ centers in Rb2MF4 crystals with a nearest M2+(Li+) vacancy (substitution) along ajj[1 0 0]-axis; (a) model 1 and (b) model 2.

of this repulsion is rather less than decrease in the so-called attraction for OR center III, while it is almost equal to that of attraction between the intervening F and Li+ ions for the OR center IV. Thus, the distortion for the OR center IV (Cr3+–Li+) strongly depends on the difference in the radii of the Cr3+ and host M2+ ions. Also, the small distortions for the center IV relative to the center III are normally expected when we consider the role of Li+ ion at vacancy site. The effect of this difference should be clearly seen for the example of Rb2CdF4 crystal (with 0.097 nm radius for Cd2+), which is expected to yield a profound compression of the surrounding F ligands along both axial and equatorial axes. To predict this effect in question we may utilize the following expression: Ri  Rhi þ 1=2ðr s  r h Þ [30], where Ri is the distance of the ith ligand and Rhi is the cation–anion distance in the host lattice, rs and rh are the radius of the substitution atom and the host atom, respectively. Accordingly, the size difference results in a reduction of 0.006 nm for Rb2ZnF4 and 0.002 nm for Rb2MnF4 but 0.018 nm for Rb2CdF4. When we compare the results of the models 1 and 2 for the OR centers, we see that (i) for all three crystals the distortion on R1 occurs to be an outward (inward) relaxation for both centers in model 1 (model 2) whereas its magnitude for the center IV is generally greater than that for the center III. Note that the comparison of the results for R1 have been done for the R1 values 0.2065 nm for Rb2ZnF4, 0.2024 nm for Rb2MnF4, and 0.2195 nm for Rb2CdF4 taking them as the host structure data (see Section 2 for explanations about R1 values); (ii) it is clear that in model 1 the distortions for the center IV are quite larger than those for the center III in Rb2ZnF4 and Rb2MnF4, while it arises opposite for Rb2CdF4. However, in model 2 the distortions on both u and R are larger for the center III than those in the center IV for Rb2CdF4, unlike Rb2ZnF4 and Rb2MnF4. Thus, whether regarding the structural changes in model 1 or model 2 we see that it is possible some unexpected structural changes around both OR centers for Rb2ZnF4 and Rb2MnF4, however, for Rb2CdF4 both model 1 and 2 seem rather feasible by means of above points. The configurations of the atoms around the OR Cr3+ centers with the chosen principal axes and a nearest M2+ vacancy along ajj[1 0 0]-axis are depicted in Fig. 6 for both models.

5. Conclusions Within the frames of semi-empirical analyses all the Cr3+ centers have been explored in Rb2MF4 crystals. The zero-field splitting (ZFS) parameters (ZFSPs) are used to investigate comprehensively the local structures for all Cr3+ centers. The compression of the MF6 octahedron taking place for TE Cr3+ center is shown in Rb2MF4 crystals. Also, it is seen that the axial F-ligands relax away from the central impurity ion for both OR centers III and

IV in all three Rb2MF4 crystals, while the magnitude of the relaxation is generally more for the center IV. By means of the correlation of the crystallographic and spectroscopic data, it was theoretically shown that the substitution of Cr3+ ions for the M2+ sites can be reasonably explained by the distortion models predicted based on SPM analysis and with low distortion values compared to those obtained from other methods. A clear linear relation between the zero-field splitting (ZFS) and the local crystal structure parameters was also demonstrated for all centers. The results obtained from theoretical analyses in this study shows that the semi empirical calculations provide a powerful way to determine the local structure in the vicinity of all Cr3+ centers in Rb2MF4 crystals. Acknowledgment The present work was supported by the Research Fund of Bahcesehir University, Turkey. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

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