Научная статья на тему 'PR of the V4+ ion in single crystals of pyrovanadates β-Mg2V2O7, α-Zn2V2O7: Spin-Hamiltonian parameters'

PR of the V4+ ion in single crystals of pyrovanadates β-Mg2V2O7, α-Zn2V2O7: Spin-Hamiltonian parameters Текст научной статьи по специальности «Физика»

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V4+ ion / Spin-Hamiltonian parameters / g-matrix / A-matrix / EPR / pyrovanadates

Аннотация научной статьи по физике, автор научной работы — S. K. Misra, S. I. Andronenko

The angular variation of V4+ electron paramagnetic resonance (EPR) line positions were recorded in single crystals of β-Mg2V2O7 and α-Zn2V2O7 at 120 K and 295 K in three mutually perpendicular planes in the temperature range from 120 to 295 K and at some intermediate temperatures. Least-squares fitting was used by diagonalization of the Spin-Hamiltonian (SH) matrix to determine the SH parameters and the orientations of the principal axes of the gand A-matrices from the angular variations of the EPR line positions. Although the V4+ SH parameters were found to be similar in the two crystals, the orientations of the principal axes of the gand A-matrices were not found to be coincident in the two crystals.

Текст научной работы на тему «PR of the V4+ ion in single crystals of pyrovanadates β-Mg2V2O7, α-Zn2V2O7: Spin-Hamiltonian parameters»

ISSN 2072-5981

aänetic Resonance in Solids

Electronic Journal

Volume 20, Issue 1 Paper No 18101,

1-8 pages 2018

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Yoshio Kitaoka (Osaka University,

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Boris Malkin (KFU, Kazan) Alexander Shengelaya (Tbilisi State University, Tbilisi) Jörg Sichelschmidt (Max Planck Institute for Chemical Physics of Solids, Dresden) Haruhiko Suzuki (Kanazawa University, Kanazava) Murat Tagirov (KFU, Kazan) Dmitrii Tayurskii (KFU, Kazan) Valentine Zhikharev (KNRTU,

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* In Kazan University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.

Short cite this: Magn. Reson. Solids 20, 18101 (2018)

EPR of the V4+ ion in single crystals of pyrovanadates P-Mg2V2O7, a-Zn2V2O7: Spin-Hamiltonian parameters

S.K. Misra 1, S.I. Andronenko 2 *

1 Concordia University, 1455 de Maisonneuve Blvd West, Montreal, Quebec, H3G 1M8, Canada

2 Kazan Federal University, Kremlevskaya 18, Kazan 420008, Russia

*E-mail: Sergey.Andronenko@gmail.com

(Received November 28, 2017; revised March 19, 2018; accepted March 19, 2018; published March 30, 2018)

The angular variation of V4+ electron paramagnetic resonance (EPR) line positions were recorded in single crystals of p-Mg2V2O7 and a-Zn2V2O7 at 120 K and 295 K in three mutually perpendicular planes in the temperature range from 120 to 295 K and at some intermediate temperatures. Least-squares fitting was used by diagonalization of the Spin-Hamiltonian (SH) matrix to determine the SH parameters and the orientations of the principal axes of the g- and A-matrices from the angular variations of the EPR line positions. Although the V4+ SH parameters were found to be similar in the two crystals, the orientations of the principal axes of the g- and A-matrices were not found to be coincident in the two crystals.

PACS: 75.10.Dg, 76.30.-v, 75.20

Keywords: V4+ ion, Spin-Hamiltonian parameters, g-matrix, A-matrix, EPR, pyrovanadates

1. Introduction

Vanadium-mixed oxides (V-Mg-O, V-Zn-O) are important in catalytic processes such as oxidative dehydrogenation of hydrocarbons [1] and selective catalytic reduction of NO by ammonia [2]. The class of vanadia known as vanadates is of great interest now, because these compounds are used in the synthesis of the supported V2O7 catalyst [3], insulin-mimetic agents [4] and rechargeable Li batteries [5]. A further point of interest is the thermochromic nature of a-Zn2V2O7, which is light yellow in the a phase and changes to red in the p phase [6]. Ioffe et al. [7] found that the electrical conductivity of Mg2V2O7 and Zn2V2O7 pyrovanadates strongly depends on the impurity ions and thermal treatment, which governs the formation of V4+ defects. They also obtained qualitative V4+ electron paramagnetic resonance (EPR) spectra in Ca, Cd, Mg and Zn pyrovanadates. Crystallochemistry of these compounds was studied experimentally by solid state nuclear magnetic resonance (NMR) [8, 9, 10], and theoretically by using point-monopole approximation and ab initio calculations [10, 11]. The Mn2+ EPR spectra in single crystals of Cd2V2O7 were investigated by Stager [12], whereas the Mn2+ EPR spectra in single crystals of Ca2V2O7 and Mg2V2O7 were investigated by Andronenko et al. [13]. Later, the Mn2+ EPR spectra in a-Zn2V2O7 single crystals were investigated by multifrequency EPR [14]. Recently, the use of nanoparticles of titanium and vanadium oxides as catalysts in Ti-O [15] or V-O [16], has attracted great interest because of its effectiveness in nanostate as compared to that in bulk materials. The doping of catalysts, such as Mg2V2O7, with transition metals (Mn, Co, Ni, Fe) also increases its effectiveness [17]. Therefore, investigation of different defects in these compounds, which play a key role in catalysis, is very important to understand the effectiveness of catalytic properties of such oxides.

In this paper we present a precise determination of Spin-Hamiltonian (SH) parameters, specifically the g- and A-matrices and the orientation of their principal axes of V4+ ions in p-Mg2V2O7 and a-Zn2V2O7 single crystals, and a determination of the position of V4+ ions in the crystal structure.

2. Sample preparation and crystal structure

Synthesis

The phase diagrams of ZnO-V2O5 and MgO-V2O5 systems were investigated to determine the conditions of crystallization of low and high-temperature phases of Zn2V2O7 and Mg2V2O7 [18, 19]. Single crystals of Zn2V2O7 and Mg2V2O7 were grown by the spontaneous-crystallization method during slow cooling of the melt with stoichiometric composition using the chemicals V2O5 (extreme pure), ZnO (chemically

pure) and MgCO3 (chemically pure). Crystals of Zn2V2O7 grew as large rectangular slabs with well-defined (110) cleavage planes. All crystals were twinned, as determined by X-ray diffraction. The growth habits of Zn2V2O7 and Mg2V2O7 single crystals are shown in Fig. 1 with respect to the orientations of respective laboratory axes XYZ. Note, that the authors of [10] investigated NMR spectra of both a and p-Mg2V2O7 crystals at room temperature, where p-Mg2V2O7 was obtained from a-Mg2V2O7 simply with annealing at 850°C during 48 hours, followed by rapid cooling to room temperature. Therefore, the phase transition becomes irreversible and the high-temperature phase is stable at room temperature.

Figure 1. The crystal growth habits of Mg2V2O7 and Zn2V2O7 crystals in relation to the laboratory axes XYZ.

Crystal structure of ZmVO7

The Zn2V2O7 crystal undergoes a fast reversible structural phase transformation at 615°C from the high-temperature thortveitite P-phase (HT phase) with the space group C2/m to the low-temperature a-phase (LT phase) of Zn2V2O7, possessing monoclinic symmetry characterized by the space group C2/c with the unit-cell parameters: a = 7.429 À, b = 8.340 À, c = 10.098 À and P = 114.4° and Z = 4 [20]. The main difference between the high (HT) and low (LT) temperature phase structures is that in the former the coordination of Zn ions is six-fold, while in the latter the cations reduce their coordination to five oxygen atoms. In the LT-phase, the ZnO5 group is a distorted trigonal bipyramid with the longer Zn-O bonds oriented in the axial direction. Vanadium and oxygen ions form V2O7 pyrogroups in the structure of a-Zn2V2O7. The low-temperature structure contains layers of oxygen atoms stacked perpendicular to the [001] axis, and Zn ions and V-O-V groups lie in octahedrally coordinated sites in alternate layers of oxygen atoms.

Crystal structure of Mg2VO7

The high-temperature P-phase of Mg2V2O7 was synthesized at higher temperatures, T > 800°C, above the phase transition between a- and P-phases, at T = 760°C [21]. This phase is stable at room temperature

and possesses triclinic space symmetry P1, with the unit-cell parameters being: a = 13.767 À, b = 5.414 À, c = 4.912 À, a = 81.42°, p = 106.82°, y = 130.33°, Z = 2 [22]. Only the high-temperature P-phase was investigated here, which remains stable at room temperature. although it is below the phasetransition temperature. The structure of P-Mg2V2O7 consists of chains of V2O7 groups formed from two VO4 tetrahedra, which share one common oxygen ion. The adjacent V2O7 chains form sheets lying in the (001) plane. They are separated by Mg cations which share oxygen atoms with these sheets.

3. The local structure of V5+ ions in V2O74- pyrogroups

The V4+ and O2- ions compose V2O74- pyrogroups, which consist of two VO4 tetrahedra, connected through common O2- ion. This pyrogroup is shown for P-Mg2V2O7 in Fig. 2. The point symmetry of the ion in Mg2V2O7 is Ci and there are two structurally inequivalent sites for V5+ ions in its lattice. In this pyrogroup one V5+ ion is 5-fold coordinated and the second V5+ ion is 4-fold tetrahedrally coordinated. The corresponding quadrupole coupling parameters {Co and //o) are different for these two vanadium nuclei, Cq = 10.1 MHz for the 5-fold coordinated vanadium ions, much larger, than that for the other 4-fold coordinated vanadium ion (Cq = 4.8 MHz) [9, 10]. There is only one structurally inequivalent site for V5+ ions in the Zn2V2O7 structure with the point symmetry Ci. The vanadium ion is 4-fold coordinated and the V-O bond lengths as well as the value for Cq = 3.9 MHz [10] are similar to those in

Mg2V2O7 for the 4-fold coordinated ion. Therefore, similar the V

■4+

Figure 2. The pyrogroup V2O7 in P-Mg2V2Û7 and a-Zn2V2Û7 crystals (without O'(2)).

hyperfine (HF) EPR spectra are expected for the V4+ ion in these sites in both Mg2V2O7 and Zn2V2O7 single crystals, and, thus, similar values for V4+ hyperfine parameters and the orientations of their principal axes are also expected. The second structurally inequivalent V5+ ion, which is situated in distorted tetrahedral configuration associated with the fifth oxygen ion was not observed by EPR in Mg2V2O7. Ioffe et al. [7] observed another V4+ EPR spectrum in Mg2V2O7 after annealing it in reduced atmosphere (CO or NH3 gas at 450°C). They deduced that each hyperfine line of the V4+ EPR spectrum split into 8 components by the superhyperfine (SHF) interaction (Ashf = 6 G) with the nearby vanadium nuclei. However, such EPR spectrum was not observed in the presently investigated Mg2V2O7 single crystal. The formation of V4+ ions (3d1 state) in V-O polyhedra, in which the vanadium ion is in 5-valent state, can be due to the presence of uncontrolled nonmagnetic impurities, or proton (H+) as an impurity [7]. The EPR spectra for the V4+ ion have been observed in many vanadium compounds (CaV2O6,V2O5) [23, 24]. On the other hand, the V4+ EPR spectra were not observed in orthovanadates (YVO4, PrVO4) [25], implying that the V5+ state is stable in 4-fold configuration of VO4 polyhedra.

4. V4+ EPR spectra: determination of SH parameters

Experiment

A Bruker ER-200D SRC EPR X-band spectrometer equipped with nitrogen-flow Bruker variable temperature assembly was used to investigate the EPR spectra in single crystals of P-Mg2V2O7 and a-Zn2V2O7. Usual setting of EPR spectrometer: modulation field is 1-5 G / 100 kHz and microwave power is 20 Db (max output power is 200 mW). The EPR spectra of the V4+ ions were recorded at X-band (9.6 GHz) in the temperature range 120-300 K. Only one magnetically inequivalent V4+ ion was observed in the two crystals in the temperature range accessible in the present experiment. Detailed angular variations of V4+ X-band EPR line positions were recorded at 120 K and 290 K in three mutually perpendicular planes in the two single crystals. They are shown in Figs. 3a,b,c and 3d,e,f respectively. The angular variations of V4+ X-band EPR spectra recorded at 290 K in tree mutually perpendicular planes in P-Mg2V2O7 single crystal are similar to those shown in Figs. 3 for a-Zn2V2O7 single crystal, and not shown here. In each plane, the magnetic field orientation was varied at 5° intervals. The orientation of the principal axes corresponding to the largest principal g-value, i.e. the direction of the Zeeman field for which the positions of the lines are at their minimum, in this plane was chosen to be the magnetic Z'-axis, which was found to be approximately perpendicular to largest flat surface of the crystal. The Z-axis and Z'-axis are not coincident. For EPR measurements in the laboratory ZY and XY planes the specimen was oriented in such a way that it could be rotated about the X and Z-axes, keeping the external static magnetic field fixed.

Spin-Hamiltonian _ parameters

The spin-Hamiltonian of the V4+ ion, describing the interaction of its magnetic moment with the external magnetic field B, and the hyperfine (HF) interaction with its own 51V nucleus, is written in following form [26]:

H = ^BgS + SAI, (1)

where /j.b is the Bohr magneton, S = '/2 is electronic spin of the V4+ ion, and g is the g-matrix, [26]. The 51V nucleus (99.76% natural abundance) has the nuclear spin I = 7/2 (gn = 1.468); thus, each line splits into eight HF lines at X-band. In Eq. (1) A is HF interaction matrix; the principal axes of the g and A matrices are, in general, not coincident with each other for low (monoclinic and triclinic) symmetries. A rigorous least-squares fitting of EPR line positions in three mutually perpendicular planes to the SH parameters enabled determination of the orientations of the principal axes of the g and A matrices [27, 28]. Two fitting programs were used here in the evaluation of the SH parameters, one for fitting the principal values and their orientations of the g-matrix and the second one for fitting the principal values of the A matrix. The results are listed in Tables 1-6.

The orientations of the principal axes of the g-matrix are denoted as Z'X'Y', whereas the principal axes of the A-matrix are denoted as Z"X"Y". The principal values of g are dimensionless, while those of A are expressed in GHz. The indicated errors are estimated by the use of a statistical method as outlined

by Misra and Subramanian [29]. The direction cosines of the principal axes of the g-matrix (X\ Y\ Zr) are given with respect to the laboratory, XYZ-axes defined in section 2, whereas those of the A-matrix (X"YZ~) are expressed relative to (XYZ").

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Orientation of magnetic field, degrees Orientation of magnetic field, degrees

Figure 3. Angular variations of the V4+ EPR spectra at 120 K in single crystals of Mg2V2Û7 and Zn2V2Û7, respectively, in three mutually perpendicular planes in the laboratory coordinate system; panels (a) and (d) correspond to ZX; (b) and (e) correspond to Z7; (c) and (f) correspond to XY.

Temperature dependence and unresolved SHF splitting

The EPR spectra in the temperature range from 120 to 295 K for the specific orientations of the magnetic fields in P-Mg2V2Û7 and a-Zn2V2Û7 are shown in Figs. 4a and 4b, respectively. The EPR spectra were recorded for Mg2V2Û7 for the orientations of the external magnetic field in the ZY plane, whereas those

for Zn2V2O7 for the orientations of the external magnetic field in the ZX plane. There was observed no significant temperature dependence of the EPR linewidth for V4+ ions in the temperature range 120-290 K. The average V4+ EPR linewidth is rather large, about 30-35 G. It is due to the superhyperfine (SHF) interaction of spin of the V4+ ion with nearest V nucleus (I = 7/2), which splits each HF line into eight unresolved SHF lines. If the individual EPR linewidth is larger than 5 G, then unresolved SHF structure will appear. The SHF interaction constant can be estimated to be 5-6 G, which is reasonable, similar to that observed by Ioffe et al. [7] in pyrovanadates for the "second" EPR V4+ center (Ashf = 6 G).

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Figure 4. The EPR spectra in Mg2V2O7 and Zn2V2O7 single crystals at various temperatures for the particular orientation of the external magnetic field. Fig. 4a: p = 150°, ZY plane (as shown in Fig. 3b), minimum HF splitting and Fig. 4b: a = 170°, ZX plane (as shown in Fig. 3d), maximum HF splitting, respectively. The EPR lines for Cu2+, also present as impurity, are seen at lower magnetic fields.

Table 1. The principal values of the g-matrices of the V4+ ions in single crystals of P-Mg2V2O7 and a-Zn2V2O7 at 120 K and 290 K. The numbers of V4+ EPR lines fitted to EPR line positions at 120 and 290 K under consideration in P-Mg2V2O7 are 808 and 744, and those in a-Zn2V2O7 are 768 and 768. SMD (GHz2) = Z, (AE, /h - v)2, where (AE, is the calculated energy difference in GHz between the levels participating in resonance for the ith line position; v is the corresponding klystron frequency in GHz, h is Planck's constant; RMSL(GHz) = (SMD/n)1/2 is average mean-square deviation of energy level difference from klystron frequency.

Temperature (K) gz gx gY n RMSL (GHz)

Mg2V2O7 295 1.930 ± 0.001 1.977 ± 0.001 1.996 ± 0.001 744 0.006

120 1.932 ± 0.001 1.969 ± 0.001 2.002 ± 0.001 808 0.011

Zn2V2O7 295 1.928 ± 0.001 1.969 ± 0.001 2.013 ± 0.001 768 0.016

120 1.932 ± 0.001 1.976 ± 0.001 2.011 ± 0.001 768 0.014

Table 2. The principal values of the A-matrices of the V4+ ions in single crystals of P-Mg2V2O7 and a-Zn2V2O7 at 120 K and 295 K.

Temperature (K) Az (GHz) Ax (GHz) Ay (GHz) n RMSL (GHz)

Mg2V2O7 295 0.480 ± 0.005 0.185 ± 0.005 0.147 ± 0.005 744 0.058

120 0.490 ± 0.005 0.170 ± 0.005 0.156 ± 0.005 808 0.055

Zn2V2O7 295 0.504 ± 0.005 0.189 ± 0.005 0.170 ± 0.005 768 0.083

120 0.500 ± 0.005 0.194 ± 0.005 0.179 ± 0.005 768 0.060

Table 3. The principal values and direction cosines of the principal axes of the g-matrices for the V4+ ions in single crystals of P-Mg2V2O7 at 120 K and 295 K.

Temp. (K) g^ gX, gY Direction cosines

Z/Z' X/X' Y/Y'

295 gZ = 1.930 0.924 0.210 0.318

gx = 1.977 -0.105 -0.660 0.744

gY = 1.996 -0.366 -0.721 0.588

120 gZ = 1.932 0.927 -0.083 -0.365

gx = 1.969 0.215 -0.680 0.701

gY = 2.002 0.306 0.729 0.612

Table 5. The principal values and direction cosines of the principal axes of the g-matrices for the V4+ ion in a single crystal of a-Zn2V2O7 at 120 K and 295 K.

Table 4. The principal values and direction cosines of the principal axes of the A-matrices for the V4+ ion in a single crystal of P-Mg2V2O7 at 120 K and 295 K.

Temp. (K) Az, Ax, Ay Direction cosines

Z/Z' X/X' Y/Y'

295 Az = 0.480 0.863 0.433 -0.260

Ax = 0.185 -0.424 0.901 0.094

Ay = 0.147 0.275 0.029 0.961

120 Az = 0.490 0.891 -0.359 0.278

Ax = 0.170 0.241 0.893 0.381

Ay = 0.156 -0.385 -0.272 0.882

Table 6. The principal values and direction cosines of the principal axes of the A-matrices for the V4+ ion in a single crystal of a-Zn2V2O7 at 120 K and 295 K.

Temp. g^ gx, gY Direction cosines

(K) Z'/Z" X'/X" Y'/Y"

gz = 1.928 0.780 0.373 -0.502

295 gx = 1.969 0.623 -0.537 0.569

gY = 2.013 0.057 0.757 0.651

gz = 1.932 0.846 0.294 -0.444

120 gx = 1.976 0.528 -0.570 0.629

gY = 2.011 0.068 0.767 0.638

Temp. Az, Ax, Ay Direction cosines

(K) Z'/Z' X'/X" Y'/Y"

Az = 0.504 0.944 -0.057 0.324

295 Ax = 0.189 -0.269 -0.434 0.860

Ay = 0.170 -0.189 -0.899 0.394

Az = 0.500 0.966 -0.031 0.258

120 Ax = 0.194 -0.086 0.899 0.430

Ay = 0.180 -0.246 0.437 0.865

5. Coordination of the V4+ ion (3d1) in VO4 polyhedra

The point symmetry of the vanadium ion is Ci in Mg2V2O7 for the two magnetically inequivalent sites for V5+ ions. The V5+ ion is situated in the first VO4 tetrahedron, V(1) is 4-fold tetrahedrally coordinated, with the V(1) - O(n) bonding lengths varying from 1.682 to 1.784 A. The V5+ ion is situated in the second VO4 tetrahedron, V(2), with the bonding lengths from 1.629 to 1.817 A [22]. It is distorted with the additional bonding to the fifth oxygen ion (V(2) - O(5), with the bonding length being 2.44 A [22]). In a-Zn2V2O7, the V5+ ion possesses Ci point symmetry. There is only one magnetically inequivalent site for the V4+ ion, which occupies a V5+ site. This V4+ ion, situated at a regular V5+ site, is 4-fold tetrahedrally coordinated with the bonding lengths varying from 1.658-1.775 A [20]. The 3d1 configuration of the V4+ ion is split in cubic crystal field into a r3 doublet and a r5 triplet [26]. In tetrahedral coordination, the r3 doublet lies lower, representing the ground state [26]. The spin-orbit

coupling constant X is positive for tetrahedral coordination. Further, the r3 doublet is split into a T/ singlet (wavefunction |3z2 — r2)) and T singlet (wavefunction |x2 — y2^) [26]. Unfortunately, the V4+ ion possesses a very low symmetry in p-Mg2V2O7 and a-Zn2V2O7 crystals, thus, it is not possible to

determine its actual ground state wavefunction from the available experimental data. The principal values of the g- and A-matrices, obtained here for Mg2V2O7 and Zn2V2O7 are close to those obtained by Ioffe et al. [23] for the V4+ ion in Ca2V2O7 single crystal, which is isostructural to the triclinic Mg2V2O7 (with the parameters gZ = 1.948; gx = 1.966; gj = 1.975; and Az = 0.475 GHz; Ax = 0.150 GHz; Ay = 0.138 GHz), from which they determined the ground state function of the V4+ ion to be |x2 — y2). They did not determine the orientations of the principal axes of g- and A-matrices.

The principal values of the g- and A-matrices are very similar to each other for the V4+ ions in Mg2V2O7 and Zn2V2O7 single crystals. This is because the V(1) ion in Mg2V2O7 is 4-fold coordinated and the average bonding length V(1) - O(n) is 1.730 A, very close to the average bonding length V - O(n) in Zn2V2O7, which is 1.716 A. The values of g- and A-matrices depend strongly on overlap of the wave functions of the V4+ ions and neighboring oxygen ligands. Therefore, one can deduce that the V4+ ion occupies the V(1) crystallographic position in Mg2V2O7 single crystals. This conclusion was supported by NMR of 51V nucleus in p-Mg2V2O7 and a-Zn2V2O7 [9, 10]. The values for quadrupole coupling parameters are: Cq = 4.8 MHz for V(1) and Cq = 10.1 for V(2) in p-Mg2V2O7 and Cq = 3.68 MHz for a-Zn2V2O7 [9, 10]. Both the V(1) ion in p-Mg2V2O7 and the V ion in a-Zn2V2O7 are 4-fold coordinated and possess similar values of Cq. This result was proved also by theoretical calculations of Cq [11]. Therefore, the environments of V(1) in p-Mg2V2O7 and V ions in a-Zn2V2O7 are similar and the V4+ ions, which occupy these sites, expect to have similar SH parameters. The orientations of the principal axes of the g-matrices Z', X', Y' of the V4+ ions relative to the crystal faces are also similar in the two crystals, with the Z'-axis being perpendicular to the (110) cleavage plane (XY plane, see Fig. 1) for a-Zn2V2O7, which implies that the cleavage plane of Mg2V2O7 is also the (110) plane.

The V4+ EPR spectra in several vanadium compounds have been observed over a very large temperature range up to room temperature. For its temperature stability the O3 - V4+- O - V5+ - O3 pyrogroup should be charge-compensated with the positive charge being in the vicinity. Ioffe et al. [7] indeed showed that the charge compensation is due to proton (H+) being in an interstitial position to render the V4+ ion stable.

6. Conclusions

The main conclusions of the V4+ EPR investigations in p-Mg2V2O7 and a-Zn2V2O7 single crystals are as follows:

(i) The principal values of the g and A matrices of the V4+ ion and their orientations relative to the crystal faces system have been determined in these crystals.

(ii) The SH parameters of the V4+ ion in p-Mg2V2O7 and a-Zn2V2O7 single crystals are found to be similar because the V4+ ions occupy similar tetrahedrally coordinated crystallographic sites in them.

(iii) The principal axes of the g-matrix are not coincident with those of the A-matrix because of the low point symmetry Ci.

Acknowledgments

This work is supported by Natural Sciences and Engineering Research Council of Canada (NSERC) (SKM). SIA is grateful for partial support in the frame of research project allocated to Kazan Federal University, Russia for the state assignment in the sphere of scientific activities (#3.2166.2017/4.6).

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