ISSN 10637834, Physics of the Solid State, 2011, Vol. 53, No. 3, pp. 554–560. © Pleiades Publishing, Ltd., 2011. Original Russian Text © E.N. Tumaev, K.S. Avadov, 2011, published in Fizika Tverdogo Tela, 2011, Vol. 53, No. 3, pp. 518–523.

OPTICAL PROPERTIES

Optical Properties of Trivalent Chromium Ions in the LiNbO3 Crystal E. N. Tumaev and K. S. Avadov* Kuban State University, ul. Stavropol’skaya 149, Krasnodar, 350040 Russia * email: [email protected] Received May 24, 2010

Abstract—The formulas of the crystalfield theory have been adapted to a system with the symmetry group C3v. A simple method has been proposed for including the polarization of the local environment of the Cr3+ impurity ion in LiNbO3. A model dependent on one parameter has been proposed for a distortion of the nio bium octahedron due to the incorporation of the trivalent chromium ion. This parameter has been deter mined from experimental data. The parameters of the intraionic and interionic interactions have been obtained for the Cr3+ ion in the lithium and niobium positions of the crystal lattice of lithium niobate. DOI: 10.1134/S1063783411030334

1. INTRODUCTION

6

At present, considerable attention has been focused on solidstate laser active media based on het erodesmic compounds doped with transition metal ions. A weak quenching of impurity ions makes these active media very promising for the design of solid state lasers in the nearIR range, which have found wide application in medicine, ecology, communica tion, spectroscopy, condensed matter physics, opto electronics, military purposes, etc. The development of highefficiency active media based on doped het erodesmic compounds is associated with the calcula tion of spectroscopic properties of impurity ions. However, crystals with mixed bonding are frequently characterized by a low symmetry of positions occupied by impurity ions of transition metals. At the same time, the methods used for calculating the spectro scopic properties, as a rule, are intended for investigat ing highsymmetry positions, which have an octahe dral or tetrahedral environment of ligands. Therefore, the study of the spectroscopic properties of ions in lowsymmetry matrices is an important problem. Crystals of lithium niobate LiNbO3 with Cr3+ impurity ions have been extensively investigated by optical and electron paramagnetic resonance (EPR) spectroscopy. A deviation of the lithium niobate crys tal structure from the stoichiometry and the necessity to compensate the excess charge in the case of doping lead to the formation of various lattice defects. As a consequence, there arise several types of Cr3+ impurity centers, which are very difficult to distinguish from optical spectra. At temperatures below the ferroelectric Curie point (TC ~ 1480 K), LiNbO3 crystals have a trigonal sym

metry with space group R3c ( C 3v ). The oxygen frame work built up according to the closest hexagonal pack ing motif forms a local environment with the symmetry group C3 (insignificantly different from the symmetry C3v) at the Li+ and Nb5+ positions. We can assume with a high accuracy that the local environment of the impurity ion has the symmetry group C3v [1]. In LiNbO3 : Cr3+ crystals, chromium ions predom inantly replace lithium ions. However, when LiNbO3 crystals contain not only Cr3+ impurity ions but also Mg2+ ions, there appears a new type of impurity cen ters. Beginning with some magnesium concentration, the optical spectrum of the crystal indicates that, apart from chromium centers in the lithium positions, there arise Cr3+ ions in the Nb5+ positions. This threshold concentration depends on the crystal composition and is approximately equal to 2 mol % MgO for the com position close to the stoichiometry of the lithium nio bate [2]. Leushin and Irinyakov [3] proposed a sufficiently reliable and complete interpretation of the optical and EPR spectra of impurity γ centers (Cr3+ ion in an undistorted lithium octahedron) in a lithium niobate crystal. However, the problem associated with the interpretation of the spectra of LiNbO3 : Cr3+ crystals has not been completely solved. In [3], the inference was made that attempts to explain the experimental data in the cubicfield approximation cannot be suc cessful. Therefore, the adaptation of the formulas of the crystalfield theory to symmetry C3v is an impor tant problem. The main purpose of the present work is to solve this problem.

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Moreover, the problem regarding the interpretation of the optical spectrum of chromium ions in the nio bium positions in LiNbO3 crystals is of considerable interest. When incorporated into the niobium octahe dron, the Cr3+ ion (together with the neighboring Mg2+ ion) distorts this octahedron. Yang et al. [2] pro posed two models in order to investigate this distor tion. Model I suggests a change in the positions of oxy gen ions near the fixed Cr3+ ion, whereas model II sug gests a displacement of the Cr3+ ion from the Nb5+ position toward the center of the octahedron. The cal culations included the procedure of complete diago nalization of the Hamiltonian of the 3d3 configuration with due regard for the crystal field, as well as the Cou lomb and spin–orbit interactions. A detailed analysis demonstrates that model I more adequately describes the properties of the impurity center as compared to model II. It is worth noting that model I, which is eval uated as the most accurate model, describes the distor tion of the niobium octahedron with the use of two fit ting parameters. However, the effects associated with the lattice polarization and the spatial charge distribu tion were disregarded in [2]. In this respect, the devel opment of the model that takes into account the above effects and uses only one fitting parameter for describ ing the distortion is an important problem.

6

Ve ( r ) =

∑

i=1

2

Ze  , Ri – r

where r is the radius vector of the electron and R is the radius vector of the ith ligand, is expanded in spherical harmonics ∞

Ve ( r ) =

k

∑∑rq k

kp Y kp ( θ,

2. MODELS AND CALCULATION METHOD Initially, let us consider the oneelectron wave functions. We assume that the ligands surrounding the central ion are immobile point charges and the wave functions of the only electron have the form ϕ(r) =

∑R

3d ( r )Y 2m ( θ,

ϕ ).

(1)

m

The crystal potential PHYSICS OF THE SOLID STATE

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(2)

It is assumed that the wave functions can be written in the form ϕ nlm ( r, θ, ϕ ) = R nl ( r )Y lm ( θ, ϕ ).

(3)

In order to determine the correction ΔE to the electron energy due to the interaction with the ligands, the crystal field Ve(r) is treated as a perturbation. Then, the corrections ΔE are determined as eigenvalues of the matrix in the form V mm' = 〈ϕ nlm|V e |ϕ nl'm'〉,

(4)

where l' = l = 2. Substituting expressions (2) and (3) into relationship (4) gives ∞

k

m'

k ( –1 ) k0 kp 5 B p  V mm' =   C 20, 20 C 2m; 2, –m' , 4π k = 0 p = –k 2k + 1

∑∑

k k 4πZe k where B p = q kp 〈 r 〉 = –   〈r 〉 2k + 1

∑

(5)

Y kp ( θ i, ϕ i ) 6   i=1 k+1 Ri *

.

The matrix elements Vmm' are nonzero only at k = 0, 2, and 4. For these values of k in formula (5), the quan tity 〈rk〉 is calculated in the approximation of hydrogen like functions, which adequately describe the electron density distribution for outer electrons; that is, 2

It is obvious that the larger the number of different quantum interactions included in the model and the smaller the number of semiempirical parameters used in the expression for the final result, the higher the value of the model. One of the advantages of the results obtained in our work lies in the decrease in the number of these parameters.

ϕ ).

k = 0 p = –k

2

As a rule, the determination of the polarization of the crystal lattice involves substantial difficulties. The values obtained can differ by one order of magnitude from real values. This requires the introduction of semiempirical parameters. The inclusion of covalent bonds and effects of overlap of the transition ion with the nearest environment also frequently leads to an increase in the number of parameters of the problem.

555

a 2 〈 r 〉 = 126 B, 2 Zi

4

a 4 〈 r 〉 = 25515 B, 4 Zi

where aB = 0.529 Å is the Bohr magneton and Zi is the charge of the atomic core of the impurity ion (Zi ≈ 6 for the Cr3+ ion) (the parameter Zi determines the oblate ness of the Rydberg orbital for multiply charged nuclei, is equal to the charge of the nucleus, and does not equal to the ion charge equal to three). The differ ence of Zi from 6 is associated with the absence of a clearcut boundary in the spatial distribution of the charge of the atomic core. The coordinate system related to the impurity cen ter is introduced so that the elements of the symmetry group of the crystal environment have the following 2π form: the rotation C3ψ(r, ϕ, θ) = ψ ⎛ r, ϕ – , θ⎞ and ⎝ 3 ⎠ the reflection in the vertical (passing through the threefold axis) plane σψ(r, ϕ, θ) = ψ ( r, – ϕ, θ ) .

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Table 1. Coordinates of the oxygen environment of the lithium position of the Cr3+ ion with the point symmetry C3v Position 3+

(Cr )Li O1 O2 O3 O4 O5 O6

R, Å

θ, deg

ϕ, deg

0 2.259 2.259 2.259 2.055 2.055 2.055

– 44.28 44.28 44.28 109.66 109.66 109.66

– –120 120 0 60 –60 180

Table 2. Coordinates of the oxygen environment of the nio bium position of the Cr3+ ion with the point symmetry C3v Position

R, Å

θ, deg

ϕ, deg

Cr O1 O2 O3 O4 O5 O6

0 2.125 2.125 2.125 1.879 1.879 1.879

– 132.09 132.09 132.09 61.93 61.93 61.93

– –60 180 60 120 0 –120

In particular, for the chromium ion in the lithium and niobium oxygen octahedra in LiNbO3, these spherical and related right Cartesian rectangular coor dinates can be introduced as follows. In the lithium oxygen octahedron, the origin of the coordinates is related to the chromium ion (occupying the Li+ posi

tion) and the Oz axis is directed along the hexagonal axis of the crystal so that the niobium ion is located at the point with the spherical coordinates R = 2.999 Å, ϕ = 0, and θ = 0. The Ox and Oy axes are directed so that the oxygen ions forming the lithium octahedron have the coordinates given in Table 1. The angle ϕ is reckoned from the Ox axis toward the Oy axis. The coordinates of the oxygen atoms in Table 1 dif fer insignificantly from the real coordinates. The angles ϕ of the O4, O5, and O6 atoms are 7.4° smaller than their real values. This approximation insignifi cantly distorts the energy levels of the system but allows us to operate with the position having the sym metry group C3v rather than the symmetry group C3 (hereafter, the designations of point groups corre spond to the Schönflies notation [4]). The coordinate system introduced according to Table 1 is shown in Fig. 1. In the niobium oxygen octahedron, the origin of the coordinates is also related to the chromium ion occupying the niobium position. The Oz axis is directed along the hexagonal axis of the crystal so that the lithium ion is located at the point with the spheri cal coordinates R = 2.999 Å, ϕ = 0, and θ = 2π. The Ox and Oy axes are directed so that the oxygen ions adjacent to the chromium ion have the coordinates given in Table 2. The angle ϕ is reckoned from the Ox axis toward the Oy axis. As for the lithium position, the atomic coor dinates in Table 2 also differ insignificantly from the real coordinates corresponding to the symmetry C3: the angles ϕ of the O1, O2, and O3 atoms are 1.24°

z

z

O1 Mg2+

O6 O3 O2

O5

Cr3+

x

Cr3+

O5 y

O6

O4 O1

x O4 O2

y

O3 Fig. 1. Lithium oxygen octahedron in LiNbO3.

Fig. 2. Niobium oxygen octahedron in LiNbO3. PHYSICS OF THE SOLID STATE

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OPTICAL PROPERTIES OF TRIVALENT CHROMIUM IONS IN THE LiNbO3 CRYSTAL

smaller than their real values. The coordinate system introduced according to Table 2 is shown in Fig. 2. The group C3v acts on functions (1). This action generates the representation Γ, which is expanded in the direct sum of irreducible representations in the form Γ = A1 + 2E. The characteristic equation of the operator V act ing on functions (1) has one root with a multiplicity of 1 and two roots with a multiplicity of 2. The matrix V k

is expressed through the coefficients B p . The quintu ply degenerate oneelectron level 3d under the action of the operator V is split into one nondegenerate level a1 and the doubly degenerate levels e and e'. The numerical experiment shows that, for the oxygen octa hedra close to the lithium octahedron in the lithium niobate, the 3d level is split into the closely spaced lev els a1 and e (transforming according to the representa tions A1 and E, respectively) and the e' level located considerably higher than the a1 and e levels. The sequence of the location of these levels and the accu rate distances between them depend on the coordi nates of oxygen atoms.

Here, the index α indicates different representa tions equivalent to Γ. Then, the threeelectron wave functions are constructed from the twoelectron func tions through antisymmetrization of the expression

∑

1, 〈EνEu|A 1 e 1〉 = –  2

i , 〈EuEν|A 2 e 2〉 = –  2

i , 〈EνEu|A 2 e 2〉 =  2

〈EuEu|Eν〉 = 1, 〈A 2 e 2 Eu|Eu〉 = 1,

1 × S 0 m 0  m 3 SM 〈Γ 0 γ 0 Γ 3 γ 3|Γγ〉. 2 Below, the main wave functions necessary for inter preting the spectrum of LiNbO3 : Cr3+ are written through the Slater determinants. They belong to the states whose mixing due to the Coulomb and spin– orbit interaction gives rise to the levels between which there occur transitions observed in the optical spectra: 2 1

2 2

Ψ ( SΓ = a 1 e A2 ) 1 =  ( 2 e 1 u 1 ν 1 – e 1 u 1 ν 1 + e 1 ν 1 u 1 ), 6 2 2

Ψ ( SΓ = a 1 e E ) = e 1 u 1 u 1 ,

1 × 1 m 1  m 2 SM 〈Γ 1 γ 1 Γ 2 γ 2|Γγ〉. 2 2 PHYSICS OF THE SOLID STATE

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3

4

Ψ ( SΓ = a 1 e E e' E ) = e 1 u 1 u 2 , 2 3

〈A 2 e 2 Eν|Eν〉 = – 1,

ϕ ( Γ 1 m 1 γ 1 )ϕ ( Γ 2 m 2 γ 2 )

3 4 1 Ψ ( SΓ = a 1 e E e' A1 ) =  ( e 1 u 1 ν 2 + e 1 ν 1 u 2 ), 2

4

Ψ ( SΓ = e A2 e' E ) = u 1 ν 1 u 2 .

By using them and the Wigner coefficients (coeffi cients of vector addition for spin functions) 〈s1m1s2m2|SM〉, we find the twoelectron wave func tions (basis functions γ) transforming according to the representation Γ of the group C3v:

∑

2 4

Ψ ( SΓ = a 1 e A2 ) = e 1 u 1 ν 1 ,

3 2

〈EνEν|Eu〉 = 1,

m 1, m 2, γ 1, γ 2

2

Ψ ( SΓ = a 1 A1 e E ) = e 1 e 1 u 1 ,

Ψ ( SΓ = e E ) = u 1 ν 1 u 1 ,

where the basis vectors of the representations are as follows: u and ν for the representation E, e1 for the rep resentation A1, and e2 for the representation A2. The other coefficients 〈Eγ1Eγ2|Γγ〉 and 〈A2γ1Eγ2|Γγ〉, except for the above coefficients, are equal to zero.

Ψ ( αSΓMγ ) =

Ψ 2 ( αS 0 Γ 0 M 0 γ 0 )ϕ ( Γ 3 m 3 γ 3 )

M 0, m 3, γ 0, γ 3

The coefficients of vector addition (the Clebsch– Gordan coefficients) 〈Γ1γ1Γ2γ2|Γγ〉 for the wave func tions transforming according to the representations of the group C3v can be chosen in a different manner. In our calculations, we used the coefficients 1, 〈EuEν|A 1 e 1〉 = –  2

557

Here, e1 is the basis function of the representation a1, u1 and ν1 are the basis functions of the representa tion e, and u2 and ν2 are the basis functions of the rep resentation e'. The quantities S and Γ characterize spin and orbital symmetries of the energy level 2S + 1Γ also designated as SΓ. The bar means that the correspond ing spin coordinate is equal to –1/2; otherwise, this coordinate is equal to 1/2. The matrix elements of the Coulomb interaction operator between the threeelectron functions can be conventionally expressed [5] through the Racah parameters B and C. The matrix of the spin–orbit interaction in the explicit form can be easily found in the basis set of oneelectron functions. This matrix allows one to determine the matrix elements of the spin–orbit inter action between the threeelectron functions [5]. These elements are expressed through the spin–orbit inter action constant ζ. Its specific value in the case of the EPR spectrum of chromium in lithium niobate can be easily determined with a high accuracy from experi

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mental data, because the spin–orbit interaction is considerably weaker than the other interactions.

V D ( r ) = 10e

Moreover, we ignore the distortion of the lithium oxygen octahedron due to the incorporation of the chromium ion and consider that the polarization is caused only by the change in the Coulomb interaction potential. In other words, since the charge of the Li+ ion is equal to e and the charge of the Cr3+ ion is equal to +3e, the incorporation of chromium into the lith ium position is treated as the appearance of an addi tional charge of +2e in the Li+ position. The Cr3+ ion enters into the niobium position provided that the Mg2+ ion enters into the neighboring empty octahe dron. in the above approximation, this is similar to the appearance of a charge –2e in the Nb5+ position and a charge of +2e in the empty oxygen octahedron. O2–

The polarizability of the ion in the lithium nio bate is taken to be α0 = 2.267 × 10–24 cm3 [6, 7]. In order to take into account the error in the approximation of the proposed model, polarizability α0 is multiplied by the dimensionless semiempirical coefficient ωD. Each of the six oxygen ions surrounding the chro mium ion has a dipole moment pi. The center of the electron cloud with a charge of –10e is displaced with respect to the center of the nucleus with a charge of +8e. A charge of –2e at the center of the nucleus was considered in the pointcharge model. In this respect, it is necessary to examine the additional dipole formed by a charge of 10e at the center of the nucleus and a charge of –10e at the center of the electron cloud. Let Δr be the vector directed from the center of the immobile nucleus to the center of the electron cloud. Then, the equality Δr = –p/(10e) can be written for the dipole moment p. The energy of the 3d electron of the Cr3+ ion in the field of the dipoles under consideration can be repre sented in the form

∑∑

i = 1k = 0

In the evaluation of the lattice polarization, the O2– ions can be approximately considered as dipoles formed as a result of the displacement of the center of the electron cloud with respect to the center of the nucleus due to the electrostatic interaction with neighboring atoms. The location of oxygen ions in the lithium niobate differs insignificantly from the closest hexagonal packing. A high symmetry of this configu ration makes it possible to consider that the polariza tion of oxygen ions in the stoichiometric compound LiNbO3 is negligible as compared to the polarization caused by the incorporation of the Cr3+ ion and to take into account only the latter polarization.

∞

6

2

k r 1  ⎛ ⎞ R i + Δr ⎝ R i + Δr ⎠ ∞

6

× P k ( cos ( ω i + δ i ) ) – 10e

2

(6)

∑ ∑ R ⎛⎝ R⎞⎠

i = 1k = 0

1 r i

k

P k ( cos ( ω i ) ),

i

where ωi and ωi + δi are the angles between the vector r and the vectors Ri and Ri + pΔx, respectively. From expression (6), we obtain ∞

Vd =

k

6

∑ ∑ ∑r Y k

k, m ( θ,

k = 0 m = –k i = 1

2 4π m ϕ )10e  ( – 1 ) 2k + 1

∂ 1 k+1 ×  ⎛ ⎛ ⎞ Y k, –m ( θ i, ϕ i )⎞ Δr + o ( Δr ) . ⎠ ∂n Δr ⎝ ⎝ R i⎠ By rejecting the term o(Δr) and performing the differ entiation, we find that expression (6) with a high accu racy is approximated by the relationship ∞

Vd =

k

∑ ∑rY k

k, m ( θ,

ϕ )q kmD ,

k = 0 m = –k

where 6

q kmD =

 ⎛ ⎞ ∑e  2k + 1 ⎝ R ⎠ 4π

i=1

1

k+2

i

( k + 1 ) ( Ri ⋅ p ) Y k*, m(θ i, ϕ i)   Ri

im –  ( sin ϕ i p x – cos ϕ i p y ) sin θ i – iϕ i 1 + ⎛  k ( k + 1 ) + m ( –m + 1 ) Y k*, m – 1 ( θ i, ϕ i )e ⎝2 iϕ i 1 –  k ( k + 1 ) + m ( –m – 1 ) Y k*, m + 1 ( θ i, ϕ i )e ⎞ ⎠ 2

 × ( cos ϕ i cos θ i p x + sin ϕ i cos θ i p y – sin θ i p z ) . This expansion is similar to expansion (2), and the lat tice polarization can be taken into account in the k

k

numerical calculations by using the sum B p + B pD k

k

k

(where B pD = q kpD 〈 r 〉 ) instead of B p in the point charge model. The effects associated with the overlap of the wave functions of the Cr3+ and O2– ions are taken into account using the simple overlap model [8]: we intro duce the field of effective positive charges ωO|gje|ρj, where ωO is the coefficient determined from experi mental data, gje is the charge of the jth ligand, ρj is the value of the complete overlap integral between the wave functions of the 3d ion and the jth ligand, and e > 0 is the elementary charge. The introduced charges have the radius vector rj = Rj/2. These charges form

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the “additional” crystal potential, which can be easily included from the formulas of the pointcharge approximation if the sum instead of the parameters

k Bp

k Bp ,

+

k B pD

+

k B pO

Table 3. Energy levels of chromium in the lithium position Level

is used

4A

k

= q kpO 〈 r 〉 ,

q kpO

4πg j ρ j e = –   2k + 1

a1ee 2A2 a1ee 2E(2T1) a1a1e 2E(2T2)

20244

4E

* ( θ i, ϕ i ) Y kp  . k+1 r i i=1

∑

2E

3. RESULTS AND DISCUSSION We developed the program that makes it possible to calculate all 120 threeelectron wave functions of the 3d3 configuration with the symmetry C3v of the local environment, to find the matrix of the total Hamilto nian for all these states (with due regard for the influ ence of the crystal field, the Coulomb and spin–orbit interactions, the lattice polarization, and the overlap effects), to diagonalize the matrix obtained, and to bring the levels in correspondence with specific elec tronic states. In this way, the calculated positions of energy levels are determined for each set of variable parameters. The final optimum values of fitting parameters are found by minimizing the residual func tion, which is the sum of the squares of the deviations of calculated values from experimental results. The most considerable contribution to the spec trum of LiNbO3 : Cr3+ crystals is always made by the chromium centers in the undistorted lithium octahe dra (γ centers). The best agreement between the exper imental data and the results of the model of γ centers is observed for the following parameters of the problem: the spin–orbit constant is ζ = 152.35576 cm–1, the effective charge of the atomic core of the Cr3+ ion is Zi = 5.7684, the Racah parameters characterizing the strength of the Coulomb interaction between the elec trons of the impurity ion are B = 383.10792 cm–1 and C = 3759.07891 cm–1, the factor determining the mag nitude of the overlap effects of wave functions is ωO = 0.022882, and the constant determining the strength of the crystal lattice polarization is ωD = 0.200563. The experimental energy levels of chromium and those calculated with the use of the developed program with the Maple and MATLAB systems are presented in Table 3. In [3], the band at 19238 cm–1 in the excitation spectrum was assigned to the levels of the split cubic orbital triplet 2T2; however, it was emphasized that this assignment has a hypothetical character. Our attempts to attribute the level at 19238 cm–1 to the states corre sponding to the orbital triplet 2T2 with octahedral sym metry resulted in considerably worse agreement between the theory and experiment. The minimum PHYSICS OF THE SOLID STATE

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Experiment, cm–1 0 0.773 13544 13570 13772 13810 14053 14635

2

where 2 6

k B pO

559

Theory, cm–1 0 1.544 13509.1 13535.5 13991.1 14028.8 13808.9 14803.8 14807.3 20131.7 20177.3

value of the residual function increased by one order of magnitude. This means that, most probably, the assignment of the band at 19238 cm–1 to the level 2T2 is erroneous. Our level energies obtained indicate that the band at 19238 cm–1 is most likely associated with some foreign impurity centers. The semiempirical parameters for the niobium position are determined in the same way as for the lith ium position. However, the Cr3+ ion enters into the niobium position only in the case where the neighbor ing vacant oxygen octahedron is occupied by the Mg2+ ion. This leads to the distortion of the niobium octahe dron, which is taken into account as follows: let us assume that three oxygen ions of the niobium octahe dron (closest to the Mg2+ ion) are displaced parallel to the line connecting the chromium and magnesium ions in the direction of the Mg2+ ion. The overlap inte gral for this displacement is approximated by the approximate formula [8]: ρ = ρ0(R0/R)3.5, where ρ0 is the initial value of the overlap integral and R0 and R are the initial and final interatomic distances, respectively. The parameter ωD determines the compensation of the arising dipole moment of the ligand due to the fac tors primarily associated with the properties of the closest hexagonal packing of oxygen atoms. There fore, with a high degree of accuracy, the value of ωD for chromium in the niobium position can be set equal to the corresponding value of ωD for chromium in the lithium position. The parameter ωO is the proportionality coefficient between the charge and the overlap integral in the sim ple overlap model used in our work. This coefficient is predominantly determined by the properties of the chromium–oxygen interaction, and, therefore, the values of ωO for chromium in the niobium and lithium positions are taken to be equal to each other. Further

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TUMAEV, AVADOV

more, the spin–orbit interaction constant ζ is deter mined by the selfconsistent field, which acts on the electrons of the outer shell of the Cr3+ ion. Under the assumption that the Cr3+ is rigid, the value of ζ in the niobium position is assumed to be equal to the value of ζ in the lithium position. The minimization of the deviation of the calculated values from the experimental data leads to the follow ing parameters of the problem: ζ = 152.35576 cm–1, Zi = 6.08519, B = 313.23172 cm–1, C = 3844.84037 cm–1, ωO = 0.022882, ωD = 0.200563, and the elongation of the niobium octahedron along of the Oz axis is d = 0.119485 Å. For the above parameters, we obtain the following calculated energy levels: the splitting of the ground level 4A2 is equal to 1.236 cm–1, the components of the split level 2E amount to 13599.9 and 13636.4 cm–1, the position of the level 4E (to which there corresponds the level 4T2 in the cubic symmetry approximation) corre sponds to 12442.6 cm–1. These results are in good agreement with the exper imental data. In particular, it was experimentally established that the splitting of the ground level 4A2 is equal to zero [2]. The energy of the level 2E is approx imately equal to 13595 cm–1 [9]. Moreover, the exper imental splitting of the level 2E is 36.5 cm–1 [2], which exactly coincides with the calculated spacing between these sublevels. The position of the level 4T2 was esti mated to be 12440 cm–1 from the experimental data on the influence of pressure [9], which is very close to the calculated value. 4. CONCLUSIONS Thus, in this work, it has been demonstrated that the changeover in the calculations to the representa tion C3v with allowance made for the main interactions between the impurity center with ligands provides good agreement between the theory and the experi ment. In [3], seven fitting parameters were used for inter preting the spectrum of chromium in the lithium posi

tion. In our work, we used only six fitting parameters, which is a progress in the solution of the problem. The model proposed in [2] for the distortion of the nio bium octahedron, which most accurately describes experimental data, uses two semiempirical parame ters. For the corresponding parametrization, we used one quantity. In [2, 3], the energy levels of the system were deter mined with due regard only for the crystal field and the Coulomb and spin–orbit interactions. The advantage of the proposed method is that, in addition to the aforementioned interactions, we also took into account the polarization of ligands and effects associ ated with the overlap of wave functions. REFERENCES 1. Yu. S. Kuz’minov, ElectroOptical and Nonlinear Opti cal Lithium Niobate Crystals (Nauka, Moscow, 1987) [in Russian]. 2. Z.Y. Yang, C. Rudowicz, and J. Qin, Physica B (Amsterdam) 318, 188 (2002). 3. A. M. Leushin and E. N. Irinyakov, Fiz. Tverd. Tela (St. Petersburg) 47 (10), 1788 (2005) [Phys. Solid State 47 (10), 1859 (2005)]. 4. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: NonRelativistic Theory (Nauka, Moscow, 1989; Butterworth–Heine mann, Oxford, 1991). 5. S. Sugano, Y. Tanabe, and H. Kamimura, Multiples of TransitionMetal Ions in Crystals (Academic, New York, 1970). 6. R. I. Shostak and A. V. Yatsenko, Uch. Zap. Tavricheskogo Nats. Univ. im. V. I. Vernadskogo, Ser. Fiz. 20, 78 (2007). 7. A. V. Yatsenko, Fiz. Tverd. Tela (St. Petersburg) 42 (9), 1673 (2000) [Phys. Solid State 42 (9), 1722 (2000)]. 8. O. L. Malta, A. E. Mauro, M. P. D. Mattioli, V. Sargen telli, and H. F. Brito, J. Braz. Chem. Soc. 9, 243 (1998). 9. S. A. Basun, A. A. Kaplyanskii, A. B. Kutsenko, V. Dierolf, T. Troster, S. E. Kapphan, and K. Polgar, Fiz. Tverd. Tela (St. Petersburg) 43 (6), 1010 (2001) [Phys. Solid State 43 (6), 1043 (2001)].

Translated by O. BorovikRomanova

PHYSICS OF THE SOLID STATE

Vol. 53

No. 3

2011