ISSN 10637745, Crystallography Reports, 2010, Vol. 55, No. 6, pp. 1060–1066. © Pleiades Publishing, Inc., 2010. Original Russian Text © A.P. Dudka, R. Chitra, R.R. Choudhury, Yu.V. Pisarevsky, V.I. Simonov, 2010, published in Kristallografiya, 2010, Vol. 55, No. 6, pp. 1119–1125.

STRUCTURAL STUDIES Dedicated to the memory of B.N. Grechushnikov

Accurate Crystal Structure Refinement of La3Ta0.25Zr0.50Ga5.25O14 A. P. Dudkaa, R. Chitrab, R. R. Choudhuryb, Yu. V. Pisarevskya, and V. I. Simonova a

Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiі pr. 59, Moscow, 119333 Russia email: [email protected] b Bhabha Atomic Research Center, Trombay, India Received June 3, 2010

Abstract—An accurate Xray diffraction study of a La3Ta0.25Zr0.50Ga5.25O14 single crystal (a = 8.2574(4) Å, c = 5.1465(4) Å, sp. gr. P321, Z = 1, R/Rw = 0.62/0.57% for 4144 unique reflections and 91 parameters) has been performed with a simultaneous neutron diffraction analysis. Tantalum, zirconium, and gallium atoms are found to occupy the mixed octahedral position (symmetry 32). Gallium atoms and a few zirconium atoms are in the position on axis 2 in the tetrahedron. The tetrahedral position on axis 3 is completely occupied by gallium atoms, while the large polyhedron on axis 2 is occupied by lanthanum atoms. The high resolution and averaging of the results obtained in two independent Xray experiments with the same sample provided accu rate structural data, in particular, on the anharmonicity of thermal atomic vibrations (atomic displacements). The Xray and neutron diffraction data on the atomic displacements are compared. DOI: 10.1134/S1063774510060246

INTRODUCTION Optical studies of langasite family crystals were ini tiated by B.N. Grechushnikov. He and his colleagues were the first to report the data on the absorption spec tra, linear dichroism, refractive indices, birefringence, and optical activity of Ca3Ga2Ge4O14, Sr3Ga2Ge4O14, La3Ga2Ge4O14, La3Nb0.5Ga5.5O14, and La3Ta0.5Ga5.5O14 (LTG) crystals [1]. In this paper we report the results of a structural study of a La3Ta0.25Zr0.50Ga5.25O14 (LTZG) crystal, which is another representative of a wide class of piezoelectric materials belonging to the family of lan gasite La3Ga5SiO14 (structure type Ca3Ga2Ge4O14, sp. gr. Р321, Z = 1 [2, 3]). The physical properties of LTZG have not been studied in detail. A systematic accurate structural study of langasite crystals based on highresolution data is important for determining the relationship between the physical properties of crystals and their structure and establish ing the possibility of controlling these properties by isomorphic substitutions [4]. For LTZG it was impor tant to establish the distribution of cations over crys tallographic positions and determine the fine struc tural features related to the corresponding isomorphic substitutions. To increase the reliability and accuracy of our study, two sets of Xray diffraction data and one set of neutron diffraction data were collected. EXPERIMENTAL We analyzed a transparent faceted LTZG single crystal grown by the Czochralski method by Mill’ et al. [5]. The sample for Xray diffraction analysis, pre

pared by spinning, had a shape close to ellipsoidal and was 0.23–0.25 mm in size. Two sets of intensities of the diffraction reflections at different initial orientations of the sample were collected on an Xcalibur S diffrac tometer (Oxford Diffraction) equipped with a CCD area detector [6] in the full sphere of reciprocal space. The details of data collection and structure refinement 1

are listed in Table 1. A sample for neutron diffraction study in the form of a 3mm cube was cut from the same crystal. In each Xray experiment, the available reciprocal space was almost completely covered. The data obtained were processed in the same way as in [7], where the ASTRA software package [8, 9] was used. A correction for the thermal diffuse scattering was intro duced according to [10], with the elastic constants (in 1010 N m–2) c11 = 18.85, c12 = 10.58, c13 = 10.01, c14 = 1.38, c33 = 26.17, and c44 = 5.24 obtained by extrapo lating the corresponding values for LTG [11]; the absorption was taken into account according to [12, 13]; a correction for the nonuniform CCD detector sensitivity over area [14] was introduced; the extinc tion effect [15, 16] was refined; and the contribution of the halfwavelength radiation to the scattering inten sity [17] was taken into consideration. The model of the LTZG atomic structure was refined using the F2 values from the crossset in Table 2, which was obtained by averaging two data sets [18]. The atomic scattering curves were taken from [19]. The Friedel 1 The

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crystallographic data for the structure studied were depos ited at the Inorganic Crystal Structure Database ICSD (CSD no. 421887).

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Table 1. Crystallographic characteristics of the La3Ta0.25Zr0.50Ga5.25O14 crystal and details of two Xray diffraction exper iments Experiment Calculated sample sizes, mm Sp. gr., Z Т, K a, Å c, Å μ, mm–1 Diffractometer Radiation; λ, Å θmax, deg Ranges of indices h, k, l Number of reflections measured Number of unique reflections, F 2 > 2σ(F 2) Number of rejected unique reflec tions, F 2 < 2σ(F 2) Redundancy 〈σ(F 2)/F 2〉 R1av(F 2), % wR2av(F 2), % Number of parameters refined R1(|F|), % wR2(|F|), % S Δρmax, Δρmin, e/Å3 Program packages

I

II

0.232(1), 0.248(1), 0.251(1)

0.230(1), 0.248(1), 0.253(1) P321, 1 295

8.2571(4) 5.1462(4) 24.49 Xcalibur S MoKα; 0.71073

71.9 74.0 –18 ≤ h ≤ 21, –22 ≤ k ≤ 21, –13 ≤ l ≤ 13 –16 ≤ h ≤ 17, –22 ≤ k ≤ 22, –13 ≤ l ≤ 13 46813 47488 3971 4131 78

80

11.55 0.017 2.12 3.40

11.28 0.019 1.98 3.37

91 0.647 0.694 0.625 0.645 1.010 1.007 +0.30/–0.30 +0.43/–0.29 CrysAlis, ASTRA, JANA2006

pairs in reflection sets were not averaged, which made it possible to establish the enantiomorphic modifica tion of the crystal under study. It turned out to be righthanded, according to the terminology [20]. The Flack parameter [21] was close to zero (0.02(2)); i.e., the crystal was singledomain. The meaning and importance of the anharmonic components of atomic thermal vibrations were checked by constructing the probability density function for atomic displacements from the equilibrium position to a given point in space and Fourier difference syntheses; the latter were con structed using the JANA2006 program [22]. Neutron diffraction data were collected on an automatic fourcircle diffractometer installed in the horizontal beam (λ = 0.995 Å, reflection (220) from a Cu monochromator single crystal) of the Dhruva reac tor (Trombay, India). Pointtopoint measurements of Bragg peak intensities were performed in the θ/2θ scan mode with a 2θ step of 0.1°. The background was measured at distances of ±1° from the peak maximum. CRYSTALLOGRAPHY REPORTS

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RESULTS AND DISCUSSION The structures of langasite crystals of different compositions have been repeatedly investigated [23, 24]. We took the model from [4] as the initial one for Table 2. Characteristics of the LLa3Ta0.25Zr0.50Ga5.25O14 structure refinement based on the reflections from the cross set a, Å

8.2574(4)

c, Å

5.1465(4)

Number of unique reflections sinθ/λ|max,

Å–1

R1av(|F|), %, R factor for averaging unique sets Number of parameters refined

4144 1.35 0.541 91

R1(|F|), %

0.620

wR2(|F|), %

0.569

Δρmax,

e/Å3

0.44

Δρmin,

e/Å3

–0.31

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O2

[(Ga, Zr)O4]

O2 O3

O3

O3 2.911

O3

O3

2.038 2.038 2.038 2.038 2.038 2.038

O2

2.454

O3

O3

2.421

O1 2.631

2.421

O3

2.454

O3

2.911

O3 O3 O2 [(Ta, Zr Ga)O6] c [LaO8] b a

1.843

O2 O2

1.870

O3

[(Ta, Zr Ga)O6]

1.870 1.843

[(Ga, Zr)O4]

O3

Fig. 1. Fragment of the La3Ta0.25Zr0.50Ga5.25O14 crystal structure. (Ta, Zr, Ga) octahedron, La polyhedron, cavity, and translationally identical (Ta, Zr, Ga) octahedron are located along the cell axis a from left to right.

refinement and refined the La3Ta0.25Zr0.50Ga5.25O14 structure within the harmonic approximation for atomic displacements. The distribution of atoms over crystallographic positions was found to be as follows (Fig. 1). The largest La atom is located on axis 2. Atoms of three types (Ta0.25Zr0.41Ga0.34) are located at the origin of coordinates (position of symmetry 32) in an octahedral environment. The tetrahedron on axis 2 is occupied by gallium and zirconium atoms (Ga0.97Zr0.03) and the tetrahedron on axis 3 is occupied by gallium. The refinement of this model gave R1(|F|)/wR2(|F|) = 1.11/1.37% and Δρmax, Δρmin = +1.14, –0.91 e/Å3. An alternative model, where all zirconium is placed in the octahedral position, yielded worse results: R1(|F|)/wR2(|F|) = 1.16/1.41% and Δρmax, Δρmin = +1.29, 0.83 e/Å3. Then, to construct a model taking into account the anharmonicity of atomic vibrations, we used the Expert programs for anharmonic displacements (automatic Hamilton–Fisher test [25]) and the repro

ducibility test for the model parameters obtained in independent refinements (normal probability plot [26]), which was included in the ASTRA package [8, 9], and analyzed the extrema in the Fourier differ ence synthesis. As a result, the displacements of lan thanum atoms were described by anharmonic tensors up to the fourth rank. For (Ta, Zr, Ga) atoms, only harmonic components of vibrations are statistically significant. A statistically significant anharmonicity up to the fourth order was established for (Ga, Zr) atoms in the position on axis 2 from a set of criteria. The Ga atoms on axis 3 are characterized by third order anharmonicity. The oxygen atom on axis 3 exhibits harmonic vibrations, while the two oxygen atoms in the shared positions are characterized by thirdorder anharmonicity. Having assumed the sequence of atoms corresponding to the table of atomic coordinates (Table 3), we can label their vibra tions as follows: 4243233. The consideration of the anharmonicity of atomic vibrations is fairly important; in particular, it resulted in a decrease in the reliability factor R1(|F|) from 1.111 to 0.620%. Finally, we attempted to refine the occupancies of atomic positions in the crystal based on the Xray dif fraction data. A free refinement of cation occupancies without imposing constraints (two additional parame ters) reduces R1(|F|) to 0.590%, and the occupancies take the form (Ta0.235Zr0.409Ga0.356) and (Ga0.976Zr0.024) instead of (Ta0.25Zr0.41Ga0.34) and (Ga0.97Zr0.03) for the octahedral and tetrahedral positions, respectively. The drawbacks of this version manifest themselves in the difference (zero) Fourier synthesis, which retains a significant peak (0.95 e/Å3) at the origin of coordi nates. Thus, when refining occupancies, we failed to simultaneously obtain a low R factor and an accept able difference synthesis. Then it was established that the peak at the origin of coordinates in the difference synthesis has a minimum amplitude when this posi tion is occupied by tantalum by 0.25 and the valence balance is retained after locating a small amount of zirconium in the tetrahedron on axis 2 and restricting

Table 3. Atomic coordinates and equivalent thermal parameters Ueq, Å2 Atom

Symmetry

Multiplicity

Occupancy

x/a

y/b

z/c

Ueq

La Ta1 Zr1 Ga1 Ga2 Zr2 Ga3 O1 O2 O3

2 32 32 32 2 2 3 3 1 1

3 1 1 1 3 3 2 2 6 6

1.0 0.250 0.381(5) 0.369(5) 0.960(5) 0.040(5) 1.0 1.0 1.0 1.0

0.425967(8) 0 0 0 0.75862(2) 0.75862(2) 1/3 1/3 0.45783(10) 0.2231(1)

0 0 0 0 0 0 2/3 2/3 0.3113(1) 0.0791(1)

0 0 0 0 1/2 1/2 0.53109(3) 0.18072(8) 0.3055(1) 0.7591(1)

0.01144(2) 0.009844(6) 0.009844(6) 0.009844(6) 0.0121(1) 0.0121(1) 0.010232(7) 0.01316(4) 0.01679(7) 0.01602(7)

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Table 4. Parameters of anisotropic atomic displacements, Å2 Atom

U11

U22

U33

U12

U13

U23

La Ta1 Zr1 Ga1 Ga2 Zr2 Ga3 O1 O2 O3

0.01218(1) 0.01051(1) 0.01051(1) 0.01051(1) 0.01229(7) 0.01229(7) 0.01098(1) 0.01497(7) 0.01177(7) 0.01600(8)

0.01092(3) 0.01051(1) 0.01051(1) 0.01051(1) 0.0122(2) 0.0122(2) 0.01098(1) 0.01497(7) 0.0236(1) 0.0187(1)

0.01079(1) 0.00850(1) 0.00850(1) 0.00850(1) 0.01166(4) 0.01166(4) 0.00874(1) 0.00955(8) 0.01440(7) 0.01575(7)

0.00546(1) 0.00526(1) 0.00526(1) 0.00526(1) 0.00610(2) 0.00610(2) 0.00549(1) 0.00749(4) 0.00842(7) 0.01049(8)

0.000194(9) 0.0 0.0 0.0 –0.00177(2) –0.00177(2) 0.0 0.0 0.00137(6) 0.00324(7)

0.00039(1) 0.0 0.0 0.0 –0.00355(4) –0.00355(4) 0.0 0.0 0.00662(8) 0.00730(7)

the total zirconium amount. As a result, the crystal composition regained the initial form: La3Ta0.25Zr0.50Ga5.25O14. In this case, the occupancies of the octahedral and tetrahedral positions are (Ta0.250Zr0.381Ga0.369) and (Ga0.960Zr0.040), respectively. This version gives an electrically neutral chemical for mula with acceptable residual peaks in the Fourier dif ference synthesis and acceptable R factor. The transi tion of a small fraction of Zr atoms to the 3f position on the axis 2 is rather unexpected, because Zr4+ ions are characterized by coordination numbers of 6, 7, and 8. The observed small increase in the volume of the 3f tetrahedron at the transition from La3Ta0.5Ga5.5O14 to LTZG does not clearly indicate a significant occupancy of the 3f position by Zr4+ ions; at the same time, it does not exclude this situation. A model was obtained independently in the neutron dif fraction experiment, where zirconium (Ga0.974Zr0.026) also enters the 3f position on axis 2. This is a weighty argument in favor of the model proposed. The use of 91 parameters in the structural model and refinement over 4144 unique reflections of the crossset yielded R1(F)/wR2(F) = 0.620/0.569% and Δρmax/Δρmin = +0.44/–0.31 e/Å3. The number of reflections in the cross set exceeds that in each individual set, because the cross set contains not only general reflections but also reflections that are unique for individual sets. The final coordinates of basis atoms and the parameters of their atomic displacements are given in Tables 3 and 4. The basic interatomic distances are listed in Table 5. The results of a neutron diffraction study of LTZG will be reported in detail elsewhere. Here we will only compare the Ueq values (which characterize the aver aged atomic displacements) determined from Xray and neutron diffraction data. X rays are known to be scattered by electrons, and the scattered intensities depend on both the redistribution of valence elec trons, which form chemical bonds in crystal, and on the displacement of all electrons due to thermal vibra tions. As a result, it is rather difficult to select the effect CRYSTALLOGRAPHY REPORTS

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of the redistribution of valence electrons in crystal [27]. Unlike X rays, neutrons are scattered by nuclei and thus yield information specifically about the ther mal vibrations of atomic nuclei. For (Ta, Zr, Ga) atoms, which have an octahedral environment and exhibit harmonic thermal vibrations, the Ueq values determined from the neutron and Xray diffraction data are approximately the same: 0.0099(9) and 0.00984(1) Å2. For atoms that have more asymmetric chemical bonds and, accordingly, exhibit anharmonic thermal vibrations, the Ueq values obtained from the neutron data are smaller than the corresponding Xray diffraction values: 0.0096(4) and 0.01144(2) Å2 for lanthanum, 0.0103(4) and 0.0121(1) Å2 for (Ga, Zr), and 0.0091(4) and 0.01023(1) Å2 for Ga, respectively. The atomic displacement parameters obtained in the Xray diffraction experiment are known to be overesti mated in comparison with the neutron diffraction data [28]. In the case under consideration, this difference is additionally increased due to thermal diffuse scatter ing, which is taken into account only in the Xray dif fraction analysis. To establish the nature of the anharmonic compo nents of atomic displacements, which were established based on the Xray diffraction data, they were recalcu lated into the probability density distribution, which determines the probability that an atom will be at a given point in space during vibrations. When recalcu lating only anharmonic components, the probability density distribution contains both positive and nega tive regions. The latter disappear when the harmonic components of atomic vibrations are added. Among the structureforming atoms, the most pronounced anharmonicity was established for (Ga, Zr) atoms, which have a tetrahedral environment. Stronger chemical bonds with the cation were observed for two O3 atoms spaced 1.842 Å from it. The (Ga, Zr)–O2 bonds are characterized by distances of 1.870 Å and, therefore, are weaker than the (Ga, Zr)–O3 bonds. This circumstance manifests itself in the probability density distributions of atomic positions during ther

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Table 5. Interatomic distances in the La3Ta0.25Zr0.50Ga5.25O14 structure (Å) La polyhedron

[(Ta,Zr,Ga)O4] octahedron

La–O1 × 2

2.6308(3)

(Ta,Zr,Ga)–O3 × 6

2.0380(9)

O2 × 2

2.4534(8)

O3–O3 × 3**

2.725(1)

O2' × 2*

2.9105(9)

O3' × 6

2.802(2)

O3 × 2

2.422(1)

O3" × 3

3.223(1)

(La–O)av

2.604

O1–O2(O2') × 4

2.9924(8)

O2 × 2

3.6193(10)

O3 × 2

3.199(1)

((Ga,Zr)–O)av

1.856

2.842(2)

O2–O2

2.899(1)

O2–O2' × 2**

[(Ga,Zr)O4] tetrahedron (3f) (Ga,Zr)–O2 × 2

1.870(1)

O3 × 2

1.8425(8)

O2

3.779(1)

O3 × 2

2.854(1)

O2–O3 × 2

3.929(1)

O3' × 2

3.028(1)

O2'–O3 × 2

2.977(1)

O3–O3

3.370(1)

O3' × 2

3.410(1)

O3–O3**

[GaO4] tetrahedron (2d)

2.725(1)

Ga–O1 O2 × 3

1.8032(3) 1.8438(5)

(Ga–O)av

1.834

O1–O2 × 3

3.1119(8)

O2–O2' × 3**

2.842(2)

* Symmetry transformations: O2' (x – y, –y, –z); O3' – (–x, –x + y, 1 – z); O3" (–x + y, –x, –1 + z). ** Shared edges of polyhedra.

mal vibrations, which are shown in Figs. 2a and 2b. Figure 2a demonstrates that the more strongly bound (Ga, Zr)–O3 atoms vibrate consistently. Another character of vibrations of (Ga, Zr)–O2 atoms is shown in Fig. 2b. The lanthanum atom located on axis 2 in the eight vortex polyhedron also exhibits anharmonic vibra tions. Figure 1 shows a structural fragment with the environment of all cations and an indication of the corresponding interatomic distances. The distances between the lanthanum atom and the two nearest O3 atoms are 2.421 Å. The line these atoms lie on is an edge of the [(Ta, Zr, Ga)O6] octahedron. There are also two O2 atoms located at a distance of 2.454 Å (the next in value). These four oxygen atoms, which are nearest to lanthanum, form a tetrahedron. The next pair of oxygen atoms, O1, is spaced by 2.631 Å from lanthanum, and the last two oxygen atoms (O2) form ing the eightvertex polyhedron are at the largest dis tances (2.911 Å). The probability density functions for the four oxygen atoms that are nearest to lanthanum are shown in Figs. 2c and 2d. In the direction from the origin of coordinates to the lanthanum atom along axis 2, there is a cavity in the structure after the point where this axis intersects with the O2–O2 edge of the La polyhedron (where the

oxygen atoms are spaced 2.454 Å from lanthanum) (Fig. 1). It was previously noted [3] that the presence of this cavity is an important structural feature of lan gasite crystals. The cavity geometry is determined by the edgelength ratio for the octahedron, eightvertex polyhedron, and tetrahedron located on axis 2, which in turn depends on the type of cations in these polyhe dra. The role of the tetrahedron on axis 3 is least signif icant in this context. The largest spread of cation– anion distances is characteristic of the La polyhedron. Figure 1 clearly presents the polyhedron edges O3–O3 (2.725 Å) and O2–O2 (3.779 Å), which are perpendic ular to axis 2 and limit the La polyhedron along this axis. These edges also determine the cavity shape. The anharmonicity of thermal vibrations of the lanthanum atom indicates the asymmetry of the crystal field around this atom. Calculations showed that the largest contribution (60–80%) to the piezoelectric properties of langasite crystals is from the atoms occupying eight vertex polyhedra [29]; i.e., lanthanum atoms in our case. The geometry of atomic arrangement (Fig. 1) and the probability density maps (Figs. 2c, 2d) clearly demonstrate how the lanthanum cation will shift after applying pressure or an electric field along symmetry axis 2. Here, the correlation between the size ratio for polyhedra in the langasite structure and the piezoelec

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(b) (Ga, Zr)

A (a) 1.5

1.5 O3

O2

0.5

0.5 O3 (Ga, Zr) –0.5 A

–0.5

–1.5 O2

0

O2 A –0.5 A 3

1.5

(c)

1.5

0.5 (d)

2.5

A

La

2.0 2 1.5 O3 0.5

–0.5 –1

1 La

O3 O2 0

1

2

0 0

3 A

1

2

3 A

Fig. 2. Anharmonic components of probability density of finding atoms at a given point of space during thermal vibrations. (a, b) Atoms forming the (Ga, Zr) tetrahedron, with the following interatomic distances: (a) (O3–(Ga, Zr)) 1.842 Å and (O3–O3) 3.370 Å and (b) (O2–(Ga, Zr) 1.870 Å and (O2–O2) 2.899 Å. (c, d) Four oxygen atoms closest to the La atom spaced by the following distances: (c) (O2–La) 2.454 Å and (O2–O2) 3.779 Å and (d) (O3–La) 2.421 Å and (O3–O3) 2.725 Å.

tric modulus d11 (which was noted in [30, 31]) is implemented due to the peculiar cavity shape and the possibilities of preferred atomic displacements under various effects on crystals. CONCLUSIONS An accurate Xray diffraction study of the La3Ta0.25Zr0.50Ga5.25O14 crystal made it possible to establish and refine the occupancies of two positions in which cations of different type are statistically located. Atoms of three types (Ta, Zr, and Ga) are located in the octahedron, while the tetrahedron on axis 2 con tains Ga and Zr atoms. The anharmonicity of thermal vibrations was established and analyzed for a number of atoms in the structure. The average atomic displace ments, which were determined based on the Xray and neutron diffraction data, were compared. CRYSTALLOGRAPHY REPORTS

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ACKNOWLEDGMENTS We are grateful to B.V. Mill for supplying the sam ples, actively participating in the discussions of the results, and helping us edit the manuscript. This study was supported by the Russian Foundation for Basic Research, project nos. 080291302IND_a and 09 0200444a, and grant NSh4034.2010.5 for the Sup port of Leading Scientific Schools.

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REFERENCES 1. O. A. Baturina, B. N. Grechushnikov, A. A. Kaminskii, et al., Kristallografiya 32 (2), 406 (1987) [Sov. Phys. Crystallogr. 32 (2), 236 (1987)]. 2. E. L. Belokoneva and N. V. Belov, Dokl. Akad. Nauk SSSR, 260 (6), 1363 (1981). 3. B. V. Mill and Yu. V. Pisarevsky, Proc. 2000 IEEE/EIA Intern. Frequency Control Symp., Kansas City, Missouru, USA, p. 133.

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DUDKA et al.

4. A. P. Dudka, B. V. Mill, and Yu. V. Pisarevsky, Kristal lografiya 54 (4), 599 (2009) [Crystallogr. Rep. 54 (4), 558 (2009)]. 5. B. V. Mill, A. V. Butashin, G. G. Khodzhabagyan, et al., Dokl. Akad. Nauk SSSR, 264 (6), 1385 (1982). 6. Oxford Diffraction. Xcalibur CCD System. CrysAlisPro Software System. 2007. Version 1.171.32 (Oxford Dif fraction Ltd., Oxford, 2007). 7. A. P. Dudka and B. V. Mill, Kristallografiya 55 (5), 798 (2010) [Crystallogr. Rep. 55 (5), 748 (2010)]. 8. A. P. Dudka, Kristallografiya 47 (1), 163 (2002) [Crys tallogr. Rep. 47 (1), 152 (2002)]. 9. A. Dudka, J. Appl. Crystallogr. 40, 602 (2007). 10. A. P. Dudka, M. Kh. Rabadanov, and A. A. Loshmanov, Kristallografiya 34 (4), 818 (1989) [Sov. Phys. Crystal logr. 34 (4), 490 (1989)]. 11. P. A. Senushenkov, Yu. V. Pisarevsky, P. A. Popov, et al., Proc IEEE International Frequency Control Symposium, 1996, p. 653. 12. A. P. Dudka, Kristallografiya 50 (6), 1148 (2005) [Crys tallogr. Rep. 50 (6), 1068 (2005)]. 13. A. P. Dudka, Kristallografiya 51 (1), 165 (2006) [Crys tallogr. Rep. 51 (1), 157 (2006)]. 14. A. P. Dudka, Proc. Nat. Conf. on Application of XRays, Synchrotron Radiation, Neutrons, and Electrons, XNSE NBIC2009, Moscow, November 17–23, 2009, p. 486. 15. P. J. Becker and P. Coppens, Acta Crystallogr. A 30, 129 (1974). 16. Y. Le Page and E. J. Gabe, J. Appl. Crystallogr. 11, 254 (1978).

17. A. Dudka, J. Appl. Crystallogr. 43, 27 (2010). 18. A. P. Dudka, Kristallografiya 47 (1), 156 (2002) [Crys tallogr. Rep. 47 (1), 152 (2002)]. 19. Z. Su and P. Coppens, Acta Crystallogr. A 54, 646 (1998). 20. V. N. Molchanov, B. A. Maksimov, D. F. Kondakov, et al., Pis’ma Zh. Eksp. Teor. Fiz. 74 (4), 244 (2001). 21. H. D. Flack, Acta Crystallogr. A 39, 876 (1983). 22. V. Petricek, M. Dusek, and L. Palatinus, JANA2006. The Crystallographic Computing System (Institute of Physics, Prague, 2006). 23. E. L. Belokoneva, S. Yu. Stefanovich, Yu. V. Pisarevsky, et al., Zh. Neorg. Khim. 45 (11), 1786 (2000). 24. B. V. Mill, A. A. Klimenkova, B. A. Maksimov, et al., Kristallografiya 52 (5), 841 (2007) [Crystallogr. Rep. 52 (5), 785 (2007)]. 25. W. C. Hamilton, Acta Crystallogr. 18, 502 (1965). 26. S. C. Abrahams and E. T. Keve, Acta Crystallogr. A 27, 157 (1971). 27. P. R. Mallinson, T. Koritsanszky, E. Elkam, et al., Acta Crystallogr. A 44, 336 (1988). 28. R. H. Blessing, Acta Crystallogr. B 51, 816 (1995). 29. J. Chen, Y. Zheng, H. Kong, and E. Shi, Appl. Phys. Lett. 89, 012 901 (2006). 30. T. Iwataki, H. Oshato, K. Tanaka, et al., J. Eur. Ceramic Soc. 21, 1409 (2001). 31. N. Araki, H. Oshato, K. Kakimoto, et al., J. Eur. Ceramic Soc. 27, 4099 (2007).

CRYSTALLOGRAPHY REPORTS

Translated by Yu. Sin’kov

Vol. 55

No. 6

2010