First-principles investigation of the Raman spectroscopy of perovskite-like crystal K3B6O10Cl

Computational Materials Science 83 (2014) 86–91

Contents lists available at ScienceDirect

Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

First-principles investigation of the Raman spectroscopy of perovskite-like crystal K3B6O10Cl Xiaoyang Gong, Xiaoyan Zhao, Zhenlong Lv, Tongwei Li, Jinghan You, Hui Wang ⇑ School of Physics and Engineering, Henan University of Science and Technology, Luoyang 471003, China

a r t i c l e

i n f o

Article history: Received 28 August 2013 Received in revised form 12 October 2013 Accepted 18 October 2013 Available online 23 November 2013 Keywords: Raman spectroscopy Density-functional theory Non-linear optical material

a b s t r a c t K3B6O10Cl is a newly synthesized perovskite-like non-linear optical crystal, which exhibits a large second harmonic generation response and deep UV absorption edge. We find 31 infrared active modes (12A1 + 19E) in it, although only ten peaks were observed in experiment. Based on the density-functional theory, we calculate the vibrational frequencies of K3B6O10Cl and classify them according to the internal and external modes of B6O10 and KCl6 groups. We further investigate the direction dispersion and find large LO–TO splitting in it. The calculated Raman spectra for different geometry configuration are depicted in detail. Our calculation results provide extraordinary insights into the Raman spectroscopy of this non-linear optical material. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction

2. Calculation methods

Borate system and distorted perovskite are two most widely used non-linear optical (NLO) materials [1–4]. By incorporating alkali cations, halide anions and borate into a perovskite structure, Wu et al. successfully synthesized a perovskite-like NLO crystal K3B6O10Cl [5,6]. This newly synthesized crystal exhibits a large second harmonic generation (SHG) response, which evolves from a BaB2O4-like NLO material to a LiNbO3-like one [7]. Similar to many NLO borate crystals, such as b-BaB2O4 [8,9] and CsB3O5 [10], K3B6O10Cl shows transparency from deep UV to middle-IR region [5]. These unique properties make K3B6O10Cl an ideal NLO material. Although the powder sample was measured to reveal its infrared spectrum, little is known about the lattice vibration of K3B6O10Cl crystal. Infrared and Raman spectra are widely used as fast, cheap and nondestructive methods to investigate NLO materials. They can resolve the local structure of borate glasses [11,12], reveal the formations of ring-type metaborate units [13,14] and probe the existence of non-bridging oxygen [15,16]. In order to get much insight into the lattice vibration of K3B6O10Cl, we calculate its vibrational frequencies of the Brillouin zone center, obtain the infrared intensity and Raman tensor, depict the vibrational modes and give the direction dispersion.

Our calculation results were obtained under the framework of density-functional theory (DFT) as implemented in the Quantum Espresso package [17]. The local density approximation was adopted to describe the exchange–correlation functional and the norm-conserving pseudopotential was used to represent the electron–ion interaction [18]. The Brillouin zone integral was sampled on a 7  7  7 mesh and the 75 Ry plane-wave cutoff was used to expand the wave function. Born effective charges, optical-frequency dielectric tensor and C point phonon frequencies were calculated within the framework of density-functional perturbation theory [19].

⇑ Corresponding author. Tel.: +86 15137903591. E-mail address: [email protected] (H. Wang). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.10.027

3. Results and discussion 3.1. Structure of K3B6O10Cl K3B6O10Cl crystal has a polar rhombohedral symmetry [5]. As shown in Fig. 1, it is composed of three building blocks: BO3 triangle, BO4 tetrahedron and ClK6 octahedron. The BO4 tetrahedrons are linked together by the BO3 triangles, while the KCl6 octahedrons are connected by sharing potassium atoms. The borate-oxygen groups fill the vacant space between the KCl6 octahedrons. The structure of K3B6O10Cl is similar to that of ABO3: the B site atoms are located in the oxygen octahedrons and the A site atoms occupy the vacant space between oxygen octahedrons [4]. The calculated lattice parameters of the conventional unit cell are a = b = 10.084

87

X. Gong et al. / Computational Materials Science 83 (2014) 86–91

Table 1 Calculated frequencies (cm1), IR intensity ((D/Å)2/amu) and Raman tensor (Å2/ amu0.5) of K3B6O10Cl. The experimental frequencies are obtained from the IR spectrum of powder sample [5]. Cal. freq.

Fig. 1. Building blocks and crystal structure of K3B6O10Cl. (a) Three building blocks of K3B6O10Cl: BO3 triangle, BO4 tetrahedron and ClK6 octahedron. The borate atoms located in BO3 triangle and BO4 tetrahedron are represented as B1 and B2, respectively. (b) B6O10 framework is composed of the BO3 triangles (yellow) and BO4 tetrahedra (green). Three inequivalent oxygen atoms are labeled with O1, O2 and O3. (c) The crystal structure of K3B6O10Cl, in which the vacant space between the KCl6 octahedrons is filled with the B6O10 groups. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

and c = 8.876 Å, which agree well with the experimental results [5–7]. 3.2. Lattice vibration frequencies of K3B6O10Cl We use factor group analysis method to classify the vibrational modes of K3B6O10Cl, which is isomorphic with the point group C3v. The calculated results show that there are 38 vibrational modes (13A1 + 7A2 + 20E), including acoustic modes A1 + E. Among the 12A1 + 7A2 + 19E optical modes, the polar A1 and E modes are both infrared and Raman active and the non-polar A2 modes are silent. Therefore, there are 31 infrared and Raman active modes, although only 10 peaks were observed in the infrared spectroscopy experiment [5]. The calculated vibrational frequencies, infrared intensities and Raman tensors are listed in Table 1. Based on the calculated frequency and infrared intensity, the assignment of the experimental frequencies is made. It is below 10 cm1 for most of the deviation between the calculated and experimental frequencies. Raman spectroscopy is a complementary method to infrared spectroscopy [20–22]. The Raman tensor of the A1(z) mode is

0

1 a 0 0 B C @ 0 a 0 A: 0 0

ð1Þ

b

There are two independent elements. The Raman tensor of the E(y) mode is

0

c

0

B @ 0 c 0 d

0

1

C d A:

ð2Þ

0

As long as the independent elements of the A1 and E modes are known, the Raman spectrum can be obtained directly [23]. We calculate the Raman tensors and list them in Table 1. As only 10 peaks are observed in the infrared spectrum, Raman spectrum will give more valuable lattice vibration information of K3B6O10Cl. 3.3. Lattice vibrational modes of B6O10 The vibrational modes of many borate-based NLO crystals can be divided into external and internal modes [24–26]. As the

60 76 112 121 126 131 156 186 194 214 270 311 329 356 401 429 481 493 556 568 599 632 682 707 724 796 828 866 972 987 997 1006 1161 1191 1324 1328 1341 1437

Expt. freq.

491

597 634 686 734 825 877

1006 1176 1315

Sym.

A1 E A1 E A2 E E A2 A1 E E A1 A2 E E A2 A1 E A2 A1 E A1 E A1 E E E E A1 A2 A1 E A1 E E A1 A2 E

IR intensity

1.5  102 3.5  102 1.2 7.5  101 0 5.4  101 1.8  101 0 1.6 1.7 9.5  103 2.2  101 0 4.7  101 8.3  102 0 2.5  102 7.5  101 0 3.9  101 1.1 3.6 2.7 2.1  101 6.6 1.2  101 4.2 22 67 0 2.5 30 1.8 9.4 51 17 0 3.9  101

Raman tensor a(c)

b(d)

6.4  104 3.5  104 8.0  105 1.2  103 0 2.2  103 4 7.7  10 0 2.3  103 1.2  103 2.8  103 1.3  102 0 1.9  103 4 6.2  10 0 7.7  103 5.4  103 0 4.0  103 3 7.6  10 1.4  102 5.2  103 3.6  103 4.8  103 2.9  103 2.0  103 1.0  103 1.7  103 0 2.3  102 0.0076 2.2  102 3.3  103 2.0  103 1.8  102 0 4.5  103

2.4  104 2.9  104 5.1  103 7.0  104 0 2.8  104 3 1.7  10 0 5.0  104 2.1  103 1.7  103 9.1  103 0 5.0  104 4 7.0  10 0 6.0  103 4.5  103 0 1.8  103 3 5.0  10 5.2  103 7.5  103 2.5  102 5.4  103 4.1  103 4.0  103 4.7  103 9.0  103 0 2.9  102 6.0  104 3.2  103 9.1  103 7.4  103 2.2  103 0 7.0  104

B6O10 group in K3B6O10Cl can be approximated to a planar B6O10 group, we focus on an isolate planar B6O10 group. The point group of the planar B6O10 molecular is D3h and its vibrational representation is 5A01 þ A001 þ 5A02 þ 5A002 þ 11E0 þ 5E00 . The non-degenerate and doubly-degenerate optical modes are depicted in Figs. 2 and 3, respectively. As shown in Fig. 2, the non-degenerate vibrational modes can be divided into three groups [25]. Firstly, Fig. 2(a–c, h and i) belong to the first group, in which all the atoms move perpendicular to the B6O10 plane. In Fig. 2(a), the three vertex oxygen atoms move opposite to their nearest neighbors, but the other atoms move in phase with each other. Consequently, its frequency is as low as 85 cm1. Different from Fig. 2(a), all atoms move opposite to their nearest neighbors in Fig. 2(i), which is a characteristic out-of-plane bending mode. Secondly, the vibrational modes of Fig. 2(d, f, k and m) belong to the second group, which can be viewed as rotation modes. In Fig. 2(d), the three vertex oxygen atoms clockwise rotate around the three fold axis, while the B1 and other oxygen atoms anti-clockwise rotate. Thirdly, Fig. 2(e, g, j, l and n) belong to the third group. They have a common feature: most atoms move along the radial direction. Fig. 2(e) is a characteristic breathing mode, in which all atoms move in phase along the radial direction [27–30]. On the contrary, all borate atoms of Fig. 2(n) move in the opposite direction to counter atoms in Fig. 2(e). Therefore, the frequency of Fig. 2(n) is over two times higher than that of Fig. 2(e).

88

X. Gong et al. / Computational Materials Science 83 (2014) 86–91

Fig. 2. The non-degenerate vibrational modes of isolate planar B6O10 molecule. Arrows represent displacement directions. Frequencies and symmetries are labeled on each graph. Three inequivalent oxygen atoms are labeled with O1, O2 and O3.

Fig. 3. The doubly-degenerate vibrational modes of isolate planar B6O10 molecule. Arrows represent displacement directions. Frequencies and symmetries are labeled on each graph. Three inequivalent oxygen atoms are labeled with O1, O2 and O3.

The doubly-degenerate modes shown in Fig. 3 can be divided into two groups. Fig. 2(a, c, g and h) belong to the first group, in which all the atoms move perpendicular to the B6O10 plane. Their frequencies are located in two intervals: below 200 cm1 and around 650 cm1. Similar frequency regions also occur in the non-degenerate modes. The other ten vibrational modes belong to the second group, in which all atoms move in the B6O10 plane. 3.4. Lattice vibrational modes of K3B6O10Cl As K3B6O10Cl are composed of borate-oxygen and KCl6 frames, the vibrational modes of K3B6O10Cl can be divided into two parts: external and internal modes [25]. The external modes are relative vibrations between neighboring B6O10 and KCl6 groups. On the contrary, the internal modes mainly originate from the relative movement among B6O10 or KCl6 group. The non-degenerate and

doubly-degenerate optical modes are depicted in Fig. 4 and 5, respectively. As shown in Fig. 4, the non-degenerate modes can be divided into external and internal modes. The external modes comprise translational and librational motions. Fig. 4(a and e) are translational motions, in which the B6O10 group moves in the opposite direction to Cl or K atom. Fig. 4(c and d) are librational motions because B6O10 or KCl6 group rotate around the three fold axis. Except for Fig. 4(a and e), the others are internal modes. A relative vibration between K and Cl atoms occur in Fig. 4(b), which is an internal mode of KCl6 group. The rest modes, from Fig. 4(f–s), are the B6O10 internal modes. Although the B6O10 group in K3B6O10Cl is not a plane, some B6O10 internal modes of K3B6O10Cl can still be approximately attributed to that of isolated B6O10 plane shown in Fig. 2 [25]. Let us focus on Fig. 4(f, g, i, m and n), in which the vibrational direction are almost perpendicular to the B6O10

X. Gong et al. / Computational Materials Science 83 (2014) 86–91

89

Fig. 4. The non-degenerate vibrational modes of K3B6O10Cl. Arrows represent displacement directions. Frequencies and symmetries are labeled on each graph.

surface. Fig. 4(f) is related to Fig. 2(a), in which the O1 atom moves opposite to the O3 atom. The O1 and O3 atoms move out of phase with the O2 atom in Fig. 2(b), which is similar to Fig 4(i). Fig. 4(g) originates from Fig. 2(c) because neighboring O2 atoms move out of phase with each other. The B1 and B2 atoms move out of phase and in phase with each other in Fig. 2(h and i), which are similar to Fig. 4(m and n). The vibrational modes of Fig. 4(h, j–l, and o–s) are closely related to those of Fig. 2(d, f, e, g, m, j, l, n and k), in which the vibrational modes are mainly due to the stretching and in-plane bending of B–O bonds. The breathing mode frequency is 439 cm1 in the isolated plane B6O10 (Fig. 2(e)), while it increases to 568 cm1 in K3B6O10Cl (Fig. 4(k)). The frequencies of Fig. 4(o, q and r) decrease relative to their counterparts in Fig. 2. These frequency shifts are due to the bending of B6O10 plane and the crystal environment. Similar to the non-degenerate modes, doubly-degenerate modes can also be divided into external and internal modes [25]. Fig. 5(a) is a translational motion because the B6O10 group moves in the opposite direction to Cl and K atoms. Fig. 5(e and i) are librational motions. Fig. 5(b) is an internal K–Cl stretching mode. Except the above modes, the others are the B6O10 internal modes. The

vibrational modes of Fig. 5(c, d and l) originate from those of Fig. 3(c, a and h), in which atomic movement directions are perpendicular to the B6O10 plane. The vibrational modes of Fig. 5(g, h, j, n and q) are closely related with those of Fig. 3(d, b, e, i and m), in which all atoms move parallel to the B6O10 plane. The KCl6 group only has major contribution below 300 cm1, while the vibration frequencies above 1000 cm1 are mainly due to the stretching and in-plane bending of B–O bond. It is similar to CsB3O5 [25]. 3.5. LO–TO splitting, direction dispersion and Raman spectroscopy Born effective charge (BEC) is widely used as a ‘‘key concept’’ to characterize ferroelectric perovskite [31–33]. It is defined as the derivative of polarization with respect to the atomic position at zero macroscopic electric field. Therefore, it is not a static but a dynamical quantity. It can also be understood as the ratio of the electric force acting on the ion over the electric field strength. Even a small electric field can exert large force on an ion with large BEC. Using the rectangular coordinate system shown in Fig. 1(b), the calculated Born effective charges are listed below

90

X. Gong et al. / Computational Materials Science 83 (2014) 86–91

Fig. 5. Doubly-degenerate vibrational modes of K3B6O10Cl. Arrows represent displacement directions. Frequencies are labeled on each graph.

0

þ1:05

B Z K ¼ @ 0:0

0:01 0

þ2:37

B Z B2 ¼ @ 0:0

þ0:20 0

0:0 þ1:00 0:0 0:0 þ2:46 0:0

0:03

1

0

þ2:62

þ0:69

1

0

2:00

C B 0:0 A; Z O1 ¼ @ 0:0 þ2:46 0:0

1:23 0:20 0:09

1

0

0:03 0:67 1:11 1:03

B Z Cl ¼ @ 0:0 0:0

0:0

þ0:96

1

C B C 0:0 A; Z B1 ¼ @ 0:0 þ2:35 0:0 A; þ1:02 þ1:12 0:0 þ1:21

2:12

B C B Z O2 ¼ @ 0:14 2:13 0:63 A; Z O3 ¼ @ 0:0

0

0:0

0:0

1:00

0:0

0:0

1

0:0

C 0:0 A; 0:84

0:0

1:02

2:00

0:99 0:0

1

C 0:0 A; 1:87

1

C 1:03 0:0 A: 0:0 0:97

As the Cl and O1 atoms are located in the three fold axis, their BECs only have diagonal elements. After diagonalization, the three diagonal elements of O2, O3, B1 are B2 are (2.47, 1.21, 0.79), (3.02, 0.97, 0.99), (+3.20, +0.63, +2.35) and (+2.04, +2.79, +2.46), respectively. Therefore, the BECs of K, B, O and Cl are similar

to their formal charges. The diagonal elements of calculated optical 1 1 dielectric tensor e1 of K3B6O10Cl are e1 xx ¼ exx ¼ 2:3 and ezz ¼ 2:8, while the non-diagonal elements are zero. Based on BECs and optical dielectric tensor e1, we calculate the LO–TO splitting and direction dispersion [34–36] of K3B6O10Cl and give them in Fig. 6. As the LO–TO splitting is small in the low frequency region, we only give the results above 850 cm1. As shown by the inset of Fig. 6(a), borate atoms move opposite to oxygen atoms along the z axis. When the wave vector is parallel to the z(x) axis, it is a longitude (transverse) optical mode. The LO–TO splitting of this A1 mode is as high as 114 cm1. The vibrational mode shown in the inset of Fig. 6(b) is a doubly-degenerate mode, in which the B1 atoms move opposite to the oxygen and B2 atoms in the xy plane. Therefore, this doubly-degenerate mode is a transverse (longitude) optical mode when the wave vector is parallel to the z(x) axis. We calculate the non-resonant Raman intensity in zðxxÞz and zðxyÞz configurations. Fig. 7 depicts the Raman spectrum of K3B6O10Cl at 300 K with 532 nm excitation laser [36]. The largest peak is located at 313 cm1, which is a A1(LO) mode and cannot be detected in zðxyÞz configuration. As shown by the inset of Fig. 6(a), the oxygen atom located in the three fold axis move along the z axis. Consequently, it is a longitude optical mode in zðxxÞz configurations. The LO–TO splitting of this A1 mode is as small as 2 cm1.

X. Gong et al. / Computational Materials Science 83 (2014) 86–91

91

Acknowledgements We are grateful for insightful discussions with Professor Yufang Wang. This work is supported by the National Natural Science Foundation of China (11304080, 51302065, U1304111, U1304605) and the Young Scientist Foundation of Henan University of Science and Technology (2013QN026). This work is supported by the Young Scientist Foundation of Henan University of Science and Technology (2013QN026). The crystal structure was drawn using the VESTA software [37]. References

Fig. 6. Direction dispersion of K3B6O10Cl. The horizontal axis (h) is the angle between wave vector and z axis. The ‘LO’ and ‘TO’ in brackets represent ‘longitude optical mode’ and ‘transverse optical mode’, respectively. The biggest LO–TO splitting in (a) and (b) are 114 cm1 (A1 mode) and 89 cm1 (E mode), respectively. The insets are vibrational modes of corresponding frequencies.

Fig. 7. Raman spectrum of K3B6O10Cl with zðxxÞz and zðxyÞz configurations. Inset depicts the vibrational mode of the largest peak.

The second largest peak is located at 131 cm1, which is a E(TO) mode and have the same amplitude in both zðxxÞz and zðxyÞz configurations.

4. Conclusions Using factor group analysis method, we find 31 infrared and Raman active modes in K3B6O10Cl, although only 10 peaks were observed in the infrared spectroscopy experiment. Base on density-functional theory, we calculate the vibrational frequencies, infrared intensity and Raman tensor and propose an assignment of the observed peaks in experiment. We analysis these vibrational modes and find some of them can be approximately assigned to the plane B6O10 molecule. Furthermore, we calculate the Born effective charges and optical dielectric tensor and in turn obtain the LO–TO splitting and direction dispersion of polar vibrational modes. In the end, we present the Raman spectrum of K3B6O10Cl. Our calculated vibrational frequencies, infrared intensity, Raman spectroscopy, direction dispersion and classified vibrational modes give much more valuable insight of this newly synthesized material.

[1] D. Xue, K. Betzler, H. Hesse, D. Lammers, Solid State Communications 114 (2000) 21. [2] Z.S. Lin, J. Lin, Z.Z. Wang, Y.C. Wu, N. Ye, C.T. Chen, R.K. Li, Journal of Physics: Condensed Matter 13 (2001) R369. [3] I. Shoji, T. Kondo, R. Ito, Optical and Quantum Electronics 34 (2002) 797. [4] L.E. Myers, R.C. Eckardt, M.M. Fejer, R.L. Byer, W.R. Bosenberg, J.W. Pierce, Journal of the Optical Society of America B 12 (1995) 2102. [5] H. Wu et al., Journal of the American Chemical Society 133 (2011) 7786. [6] H. Wu, S. Pan, H. Yu, D. Jia, A. Chang, H. Li, F. Zhang, X. Huang, CrystEngComm 14 (2012) 799. [7] H. Wang, L. Kong, X. Zhao, Z. Lv, T. Li, W.W. Ju, J. You, Y. Bai, Applied Physics Letters 103 (2013) 101902. [8] P.P. Fedorov, A.E. Kokh, N.G. Kononova, Russian Chemical Reviews 71 (2002) 651. [9] R. Degl’Innocenti, A. Guarino, G. Poberaj, P. Günter, Applied Physics Letters 89 (2006) 041103. [10] Y. Wu, T. Sasaki, S. Nakai, A. Yokotani, H. Tang, C. Chen, Applied Physics Letters 62 (1993) 2614. [11] B.N. Meera, J. Ramakrishna, Journal of Non-Crystalline Solids 159 (1993) 1. [12] W.L. Konijnendijk, J.M. Stevels, Journal of Non-Crystalline Solids 18 (1975) 307. [13] E.I. Kamitsos, G.D. Chryssikos, Journal of Molecular Structure 247 (1991) 1. [14] B.N. Meera, A.K. Sood, N. Chandrabhas, J. Ramakrishna, Journal of NonCrystalline Solids 126 (1990) 224. [15] C. Julien, M. Massot, W. Balkanski, A. Krol, W. Nazarewicz, Materials Science and Engineering: B 3 (1989) 307. [16] A.H. Verhoef, H.W. den Hartog, Journal of Non-Crystalline Solids 182 (1995) 221. [17] P. Giannozzi et al., Condensed matter: an Institute of Physics journal 21 (2009) 395502. [18] N. Troullier, J.L. Martins, Physical Review B 43 (1991) 1993. [19] S. Baroni, S. de Gironcoli, A. Dal Corso, Reviews of Modern Physics 73 (2001) 515. [20] H. Wang, J. You, L. Wang, M. Feng, Y. Wang, Journal of Raman Spectroscopy 41 (2010) 125. [21] H. Wang, Y. Wang, X. Cao, M. Feng, G. Lan, Journal of Raman Spectroscopy 40 (2009) 1791. [22] P.H. Tan et al., Nature Materials 11 (2012) 294. [23] D. Porezag, M. Pederson, Physical Review B 54 (1996) 7830. [24] Y. Wang, M. Feng, H. Wang, P. Fu, J. Wang, X. Cao, G. Lan, Journal of Physics: Condensed Matter 19 (2007) 436207. [25] H. Wang, J. Zhu, Y. Wang, X. Cao, M. Feng, G. Lan, Physica B: Condensed Matter 403 (2008) 4189. [26] H. Wang, Y. Wang, X. Cao, L. Zhang, M. Feng, G. Lan, Physica Status Solidi (b) 246 (2009) 437. [27] A. Jorio, R. Saito, J. Hafner, C. Lieber, M. Hunter, T. McClure, G. Dresselhaus, M. Dresselhaus, Physical Review Letters 86 (2001) 1118. [28] E. Dobardzˇic´, I. Miloševic´, B. Nikolic´, T. Vukovic´, M. Damnjanovic´, Physical Review B 68 (2003) 045408. [29] H. Wang, X. Cao, M. Feng, Y. Wang, Q. Jin, D. Ding, G. Lan, Spectrochimica Acta Part A, Molecular and Biomolecular Spectroscopy 71 (2009) 1932. [30] H. Wang, Chinese Physics B 22 (2013) 086301. [31] W. Zhong, R.D. King-Smith, D. Vanderbilt, Physical Review Letters 72 (1994) 3618. [32] P. Ghosez, J.P. Michenaud, X. Gonze, Physical Review B 58 (1998) 6224. [33] H. Wang, J.G. Che, EPL (Europhysics Letters) 96 (2011) 67012. [34] G.Y. Zhang, G.X. Lan, Y.F. Wang, Lattice Vibration Spectroscopy, High Education Press, China, 2001. [35] D. Feldman, J. Parker, W. Choyke, L. Patrick, Physical Review 170 (1968) 698. [36] P. Alonso-Gutiérrez, M. Sanjuán, Physical Review B 78 (2008) 045212. [37] K. Momma, F. Izumi, Journal of Applied Crystallography 44 (2011) 1272.