Research Article  Open Access
Urszula D. Wdowik, Agnieszka Twardowska, Bogusław Rajchel, "Vibrational Spectroscopy of Binary Titanium Borides: FirstPrinciples and Experimental Studies", Advances in Condensed Matter Physics, vol. 2017, Article ID 4207301, 9 pages, 2017. https://doi.org/10.1155/2017/4207301
Vibrational Spectroscopy of Binary Titanium Borides: FirstPrinciples and Experimental Studies
Abstract
Vibrational dynamics of binary titanium borides is studied from firstprinciples. Polarized and unpolarized Raman spectra of TiB, TiB_{2}, and Ti_{3}B_{4} are reported along with the experimental spectra of commercial powder and bulk TiB_{2} containing less than 1 wt.% of impurity phases. The Xray diffraction spectroscopy, applied for phase composition examination of both bulk and powder materials, identifies only the TiB_{2} phase. The simulated Raman spectra together with literature data support interpretation and refinement of experimental spectra which reveal components arising from titanium dioxide (TiO_{2}) and amorphous boron carbide (B_{4}C) impurity phases as well as graphitic carbon. These contaminations are the byproducts of synthesis, consolidation, and sintering aids employed to fabricate powder and bulk titanium diboride.
1. Introduction
Generally, the Ti–B system comprises three compounds, namely, TiB, TiB_{2}, and Ti_{3}B_{4} [1]. These borides have attracted much experimental and theoretical research because of their unique properties such as high melting point, high hardness, high elastic modulus, good thermal and electrical conductivity, excellent oxidation resistance, and considerable chemical stability [2]. The combination of these properties makes titanium borides promising materials for multifunctional applications, for example, electrode materials, cutting tools, wearresistant parts, protecting coatings, and all kinds of hightemperature structural components [3].
The most extensively studied TiB_{2} compound has recently gained renewed experimental interest due to its application for deposition of thin Ti–B films [4–6]. We note that chemical composition of Ti–B film varies with applied conditions of deposition, and hence the phase composition of deposited material differs from that desired and expected. An analysis of the phase composition of thin and frequently amorphous (or partially amorphous) Ti–B films by the Xray diffraction (XRD) method remains uncertain due to the content of light boron element. Therefore, for characterization of deposited thin Ti–B films, some complementary methods, such as the Raman spectroscopy, have to be applied. On the other hand, reference Raman spectra are usually based on measurements carried out on samples prepared in different conditions, and thus the resulting Raman spectra show differences in position, intensities, and even the number of peaks among the spectra recorded for the same phase and analyzed in very similar experimental conditions. In order to resolve these ambiguities one may calculate positions and intensities of the Ramanactive modes for a given system using the currently available theoretical tools such as those based on stateoftheart density functional theory (DFT). Results of numerical simulations can then be used for interpretation and refinement of respective experimental spectra.
So far, a number of theoretical and experimental works have been done to investigate the structural, electronic, and elastic properties of titanium borides [7–11], leaving their dynamical properties highly unexplored [12, 13]. This research extends and supplements the present knowledge on titanium borides by providing information on their vibrational properties. In particular, the positions and intensities of the Ramanactive phonons of TiB, TiB_{2}, and Ti_{3}B_{4} are determined form firstprinciples calculations by using the DFT theory and the direct method. We also provide interpretation of the Raman spectra measured for commercially available powder and bulk samples of titanium diboride. Results of these studies are hoped to stimulate further experimental and theoretical progress in the field of Ti–B system.
2. Methodology
2.1. Experimental
Experiments were performed for commercial TiB_{2} powder (H. C. Starck, Germany) and bulk (target, Goodfellow, UK) samples. The grain size of TiB_{2} powder with purity of about 99 wt.% was in the range 2.5–3.5 m. The target sample of 30 mm in diameter and 4 mm in thickness was mechanically polished on one side using diamond grinding (9, 6, and 3 m) and finally polished in 1m suspension. At each polishing step, the surface was degreased by 2Propanol and then ultrasonically cleaned in acetone bath for 5 minutes. After drying (in air), the target was mounted in a vacuum chamber to perform ion cleaning at room temperature and pressure of Pa. The iron cleaning was done by a beam of Ar^{+} ions of energy of 10 keV directed at sample at an angle of 65° (measured to the normal of target surface). Such preparation procedure is required for the Raman measurements as the spectrometer used in our studies is equipped with the confocal (light) microscope. Moreover, the target is further used for deposition of TiB_{2} thin films by the PVD method (results not discussed in the present paper).
Phase identification was performed by the Xray diffraction (XRD) method using PANalytical Empyrean diffractometer. Theradiation (intended Å, intensity ratio, kV, I = 30 mA) in the BraggBrentano configuration was used for this purpose. The XRD patterns were collected in 2geometry over the scattering angles ranging from 20° to 82° with a step size of 0.02°. Analysis was performed according to the ICSD database and the Rietveld method which took into account the ratio.
The Raman spectroscopy was applied to refine the phase composition of both powder and bulk (target) samples. To excite Raman spectra, the Nd:YAG laser beam with wavelength of 532 nm and a power of 6.25 mW was used. Unpolarized Raman spectra in backscattering geometry were collected at room temperature using the ThermoNicolet Raman ALMEGA XR dispersive confocal spectrometer operating in the microRaman mode. Raman spectra were recorded with normal (4 cm^{−1}) and highspectral (2 cm^{−1}) resolutions.
2.2. Theoretical
Calculations were carried out within the DFT method implemented in the VASP code [14, 15]. Electronion interaction was represented by the projector augmented wave (PAW) method. The generalized gradient approximation with parametrization of Perdew, Burke, and Ernzerhof (GGAPBE) [16, 17] was applied for the exchange and correlation potential. The wavefunctions were expanded in a planewave basis set with a cutoff energy of 420 eV. Reference configurations for valence electrons were (3d^{3}4s^{1}) for Ti and (2s^{2}2p^{1}) for B. Lattice constants and internal atomic positions of TiB, TiB_{2}, and Ti_{3}B_{4} unit cells were fully optimized with convergence criteria for the residual HellmanFeynmann (HF) forces and the system’s total energy of 10^{−5} eV Å^{−1} and 10^{−7} eV, respectively. The Brillouin zones of TiB, TiB_{2}, and Ti_{3}B_{4} were sampled using, respectively, 54, 96, and 50 irreducible kpoints generated according to the MonkhorstPack scheme. Phonon calculations were performed within the direct method approach [18, 19] and harmonic approximation. The HF forces were obtained by displacing the symmetry nonequivalent Ti and B atoms from their equilibrium positions by Å in the supercells containing 64 atoms (TiB), 92 atoms (TiB_{2}), and 112 atoms (Ti_{3}B_{4}). The HF forces were calculated with reduced number of kpoints. The total number of calculated displacements amounted to 12 for TiB, 6 for TiB_{2}, and 24 for Ti_{3}B_{4}. Peak intensities of the nonresonant Raman spectrum (in Stokes process) were calculated from the wellknown expression [20]:, whereis the population factor for Stokes scattering withdenoting the BoseEinstein thermal factor,() is the polarization of the incident (scattered) radiation, andis the Raman susceptibility tensor. The components oftensor () were determined from derivatives of the electric polarizability tensor over the atomic displacements [19, 21, 22]. The electric polarizabilities were calculated within the linearresponse method [23] and for each symmetry nonequivalent atom was displaced from its equilibrium position by Å. Details of calculations can also be found elsewhere [24, 25]. We also note that anharmonic effects leading to changes in phonon frequencies and reflected by shifts of the Raman peaks’ positions have been neglected. This is mainly because our measurements are performed at room temperature, where the effects related to the thermal expansion of compounds from the Ti–B system are negligible. Also, the effect of anharmonicity on the widths of Raman peaks is not considered in the present work. Thus, the Raman peaks are simulated by Lorentzian functions with artificial FWHMs corresponding to energy resolution of the Raman spectrometer used in our studies.
3. Results and Discussion
3.1. Structural Properties
Titanium monoboride (TiB) crystallizes in the orthorhombic FeB structure with the space group(no. 62) [26], where both Ti and B atoms occupylattice sites. Its primitive unit cell consists of 8 atoms (4 formula units). The main building block of TiB is the trigonal prism with the B atom at the center and the Ti atoms in corners. The transverse stacking of the trigonal prisms in columnar arrays leads to a zigzag chain of B atoms along thedirection, as schematically shown in Figure 1.
Titanium diboride (TiB_{2}) has hexagonal, layered structure of AlB_{2}type (space group, no. 191) with Ti and B atoms located, respectively, at and Wyckoff positions [27]. The primitive unit cell of TiB_{2} consists of 3 atoms (1 formula unit). The TiB_{2} crystal structure is presented in Figure 2. Each Ti atom is surrounded by 12 equidistant B atoms, whereas each B atom has 3 B atoms at a short distance and 6 Ti atoms at a much longer distance. The Bsublattice resembles that of graphitic carbon. The Tisublattice is nested in the interstices provided by the Bsublattice.
Typical Xray diffraction spectra of commercial powder and bulk samples of TiB_{2} are shown in Figure 3. The XRD patterns of powder and target TiB_{2} are very similar to each other and the Rietveld analysis confirms that the samples contain titanium diboride as a majority phase with the lattice constants Å, Å, Å, and Å which correspond to other experimental studies [27] ( Å and Å).
(a)
(b)
The crystal structure of Ti_{3}B_{4} is orthorhombic (space group, no. 71) and isomorphous with that of Ta_{3}B_{4} [28]. There are 2 nonequivalent B atoms at () and () lattice sites. Also Ti atoms reside in 2 different Wyckoff positions, namely, () and (). Thus, the primitive unit cell of Ti_{3}B_{4} contains 14 atoms. The crystal structure of Ti_{3}B_{4} is displayed in Figure 4.
Parameters of the TiB, TiB_{2}, and Ti_{3}B_{4} structures determined at the ground state are summarized in Table 1, along with the available experimental data for comparison. In general, the calculated structural parameters of the Ti–B compounds remain in very good agreement with results of the previous experiments [26–28]. Therefore, our theoretical bond lengths between boron atoms (B–B), titanium and boron atoms (Ti–B), and between titanium atoms (Ti–Ti), which are collected in Table 2, closely correspond to those observed in experimental studies. In all considered titanium borides, the shortest bond lengths (~1.8 Å) are found between B atoms. The Ti–B bonds are much longer (~2.4 Å) as compared to B–B bonds, but shorter than the Ti–Ti bonds (~2.9 Å). The values of interatomic distances reflect the nature of bonding in titanium borides. This has already been discussed in numerous theoretical studies considering the electronic structure of these compounds [7–11]. Results of the present research confirm that the chemical bonding in TiB, TiB_{2}, and Ti_{3}B_{4} is a mixture between covalent, ionic, and metallic bonding. Strong covalent bonds exist between B atoms, while mixed metalliccovalent bonds are between Ti atoms. There is also a mixed ioniccovalent interaction between Ti and B atoms.


3.2. ZoneCenter Phonon Modes
The optically active zonecenter phonon modes in TiB, TiB_{2}, and Ti_{3}B_{4} are either Ramanactive (gerade) or infrared (IR) active (ungerade) due to the presence of inversion symmetry in these systems. Thepoint phonon modes in TiB can be decomposed into the irreducible representations of the point groupas follows:. Among them 3 modes () are lattice translational modes and ones are silent (optically inactive). The modes with symmetries , , , and are Ramanactive, whereas modes , , and are IRactive. Both Ti and B atoms occupying the () lattice positions contribute to the Raman and IRactive modes. The optical IR modes of and symmetries correspond to the oscillations of the dipole moment within the crystal acplane, whereas those of symmetry to the oscillations parallel the crystal baxis. The and phonons involve vibrations of the Ti and Bsublattices within the acplane, while the and phonons arise from atomic vibrations along the baxis. The frequencies of the Raman and IRactive phonon modes predicted by our calculations for TiB are listed in Table 3. The silent modes are found at 280.2 and 453.1 cm^{−1}. The IR modes gather into 2 bands with lowerfrequency band located at~250 cm^{−1}and the higherfrequency band extending from about 470 to 560 cm^{−1}. Similarly, the Raman modes are also concentrated within 2 bands. The lowerfrequency band ranges from about 260 to 350 cm^{−1} and the higherfrequency one from 570 to 780 cm^{−1}.

Phonons at the Brillouin zone center of the TiB_{2} structure can be classified according to the irreducible representations of the point group as follows: . The modes with and symmetries are IRactive, the modes of symmetry are Ramanactive, and the mode is silent. Modes and remain doubly degenerate. The phonons constitute lattice translational modes. The IRactive and modes are related to the dipole moment oscillations perpendicular and parallel to the crystal hexagonal plane, respectively. In the Ramanactive modes the Ti atoms are at rest, and hence these modes are only associated with the B atoms vibrating within the hexagonal plane. The Raman phonon appears at 883.1 cm^{−1} and the infrared and phonons have frequencies of 515.1 cm^{−1} and 521.5 cm^{−1}, respectively. The calculated frequency of the silent amounts to 557.9 cm^{−1}. The frequencies of the Raman and infrared modes in TiB_{2} crystal determined in the present DFT studies closely correlated with those obtained previously [12, 13].
Thepoint phonon modes in Ti_{3}B_{4} can be decomposed into the irreducible representations of the point groupin the following way:, where the Raman modes have symmetries of ,, and. The,, andmodes are infraredactive. There are 3 acoustic modes constituted by the IR phonons (). The IRactive,, andare associated with the oscillations of the dipole moment along the crystallographic c, b, and a axes, respectively. The Ti_{1} atoms residing in () sites do not contribute to the Raman modes. Therefore, the , , and phonons results from the displacements of Ti_{2}, B_{1}, and B_{2} atoms along the c, a, and b axes of the Ti_{3}B_{4} crystal. Respective frequencies of the Raman and IR modes are collected in Table 4.

3.3. Raman Spectra
The Raman tensors of the , , , and phonon modes in TiB have the following nonzero components:and the polarization selection rules [29] for the point groupallow the polarized Raman scattering experiments to determine phonons having particular symmetries. In the backscattering geometry, where the wave vector of incident () and scattered () radiations remain antiparallel, the modes of symmetry can be measured, for example, at scattering configuration (in Porto’s notation). In order to observe the,, andmodes one needs to apply the , , andscattering geometries, respectively. The polarized backscattering Raman spectra at scattering configurations outlined above are shown in Figure 5. One notes that not all Ramanactive modes of TiB are intense enough to be experimentally observed.
(a)
(b)
(c)
(d)
The Raman spectrum of TiB_{2} single crystal is characterized by a single peak due to the mode of symmetry, which can be detected at scattering geometry. The corresponding Raman tensor of the doubly degenerate phonon mode has the following form:
The Raman tensors of the , , and modes in Ti_{3}B_{4} crystal are defined in the same manner as for the TiB crystal (see (1)). Therefore, one determines the , , and phonons in Ti_{3}B_{4} by using the same scattering geometries as those given for orthorhombic TiB crystal. The resulting Raman spectra are presented in Figure 6.
(a)
(b)
(c)
In majority of cases, experimental characterization of the Ti–B material by using the Raman spectroscopy is based on measurements performed on powder samples, and hence the resulting spectra of polycrystalline materials may differ from those for single crystals. Indeed, the simulated unpolarized Raman spectra in backscattering geometry of TiB, TiB_{2}, and Ti_{3}B_{4} polycrystals, which are shown in Figure 7, remain quite distinct from the polarized spectra of the respective single crystals given in Figures 5 and 6. First of all, not all Ramanactive modes are observed due to their weak intensities. The peaks of TiB and Ti_{3}B_{4} polycrystals originate from phonons of the symmetry. Therefore, the unpolarized Raman spectrum of multiphase Ti–B system may consist of three bands. The lowfrequency (240–360 cm^{−1}) and middlefrequency (520–680 cm^{−1}) bands are expected to be dominated by the modes of TiB and Ti_{3}B_{4} phases, whereas the highfrequency band (800–900 cm^{−1}) is expected to be dominated by the modes of TiB_{2} and Ti_{3}B_{4} phases.
According to the group symmetry analysis, the TiB_{2} compound exhibits a single doubly degenerate Ramanactive mode of symmetry, which should be revealed by the Raman spectra of either a single crystal or polycrystalline samples. However, a typical Raman spectrum of TiB_{2} commercial powder sample shows more rich experimental pattern than that predicted theoretically. Such spectrum is presented in Figure 8. It shows a small intensity peak at 143 cm^{−1} and broad highintensity peaks at 260, 420, and 610 cm^{−1}. These peaks are characteristic for rutile titanium dioxide (TiO_{2}) phase, whose vibrational spectrum has 4 vibrational bands centered around 145 cm^{−1} , 445 cm^{−1} , 610 cm^{−1} , and 240 cm^{−1} for secondorder scattering effect (SOE) [30, 31]. We note that similar spectrum was also obtained for commercial powder TiB_{2}, although with slightly different positions of the Raman peaks (260, 409, and 598 cm^{−1}), and it was assigned to the anatase phase of TiO_{2} [32]. Our spectrum is unlikely to represent powder anatase TiO_{2}, as such spectrum usually shows 5 Raman modes centered around 144 cm^{−1} , 196 cm^{−1}, 394 cm^{−1} , 516 cm^{−1} , and 638 cm^{−1} [30, 33, 34]. Moreover, the characteristic feature of anatase spectrum is the highintensity peak at 144 cm^{−1} which dominates over the remaining peaks having comparably smaller intensities. In addition, the highfrequency range of our spectrum reveals two quite intense peaks at about 1360 and 1570 cm^{−1} indicating the presence of graphitic carbon withandbonds [35]. Nevertheless, we confirm that even a small amount of contaminating phase, such as TiO_{2} and unreacted carbon, being byproducts of the carbothermal reduction process employed to fabricate TiB_{2} powder according to the following reaction [36]:can prevail in the Raman spectrum of commercial TiB_{2} powder.
TiB_{2} can also be prepared by reduction of TiO_{2} by boron carbide (B_{4}C) and carbon as follows [37]:The above procedure is frequently applied to produce commercial targets of TiB_{2}. Thus, the Raman spectrum of such target usually shows a dominant contribution from the byproducts of synthesis and consolidation reactions, as shown in Figure 9. Here, the main peaks appearing at 271, 318, 480, 532, 728, 827, 1000, and 1089 cm^{−1} are associated with the amorphous B_{4}C phase which displays characteristic Raman bands at 270, 320, 481, 531, 728, 830, 1000, and 1088 cm^{−1} [38–40]. Additionally one detects a weak feature at about 970 cm^{−1} which is also visible in the previously measured spectra of crystalline and amorphous boron carbide [39, 40]. Besides, our Raman spectrum of target sample reveals a peak at 881 cm^{−1} being an evidence of the presence of TiB_{2} phase, for which a Raman peak was predicted by our DFT calculations at 883 cm^{−1}.
4. Summary and Conclusions
The present work reports on experimental Raman spectra of commercially available powder and bulk samples of titanium diboride. It is shown that microRaman spectroscopy enables identification of impurity phases contained in the samples even though their concentration remains below 1 wt.%. Detailed analysis uncover contamination of commercial TiB_{2} powder and bulk samples by TiO_{2} and B_{4}C phases, respectively, which are the byproducts of chemical reactions applied to produce samples. Additionally, the graphitic carbon is identified as a fingerprint of sintering aids used in production process of bulk TiB_{2}. Vibrational properties of titanium borides (TiB, TiB_{2}, and Ti_{3}B_{4}), in particular, positions and intensities of the Ramanactive phonons, were gained from theoretical approach based on the DFT method. Theoretical Raman spectra were simulated at conditions close to those encountered in experiments and for ideal crystals, namely, free of defects and residual stresses which are always present in real samples due to their preparation procedure. Thus, our ab initio results can serve not only as a guide for interpretation of experimental Raman spectra or symmetry mode assignment in particular titanium borides, but also for estimation of effects connected with macrostresses and their influence on positions and intensities of the measured Raman peaks.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
Interdisciplinary Center for Mathematical and Computational Modeling (ICM), Warsaw University, Poland, is acknowledged for providing the computer facilities under Grant no. G2812.
References
 J. L. Murray, P. K. Liao, and K. E. Spear, “The BTi (BoronTitanium) system,” Bulletin of Alloy Phase Diagrams, vol. 7, no. 6, pp. 550–555, 1986. View at: Publisher Site  Google Scholar
 V. I. Matkovich, Boron and Refractory Borides, Springer, Berlin, Germany, 1977. View at: Publisher Site
 R. G. Munro, “Material properties of titanium diboride,” Journal of Research of the National Institute of Standards and Technology, vol. 105, no. 5, pp. 709–720, 2000. View at: Publisher Site  Google Scholar
 M. Gu, C. Huang, B. Zou, and B. Liu, “Effect of (Ni, Mo) and TiN on the microstructure and mechanical properties of TiB_{2} ceramic tool materials,” Materials Science and Engineering A, vol. 433, no. 12, pp. 39–44, 2006. View at: Publisher Site  Google Scholar
 P. H. Mayrhofer, C. Mitterer, L. Hultman, and H. Clemens, “Microstructural design of hard coatings,” Progress in Materials Science, vol. 51, no. 8, pp. 1032–1114, 2006. View at: Publisher Site  Google Scholar
 A. Twardowska, “On mechanical and frictionwear properties of TiBx coated alumina sinters,” Mechanik, no. 56, pp. 492–493, 2016. View at: Publisher Site  Google Scholar
 D.C. Tian and X.B. Wang, “Electronic structure and equation of state of TiB2,” Journal of Physics: Condensed Matter, vol. 4, no. 45, pp. 8765–8772, 1992. View at: Publisher Site  Google Scholar
 P. Vajeeston, P. Ravindran, C. Ravi, and R. Asokamani, “Electronic structure, bonding, and groundstate properties of AlB_{2}type transitionmetal diborides,” Physical Review B, vol. 63, no. 4, Article ID 045115, 12 pages, 2001. View at: Publisher Site  Google Scholar
 K. B. Panda and K. S. R. Chandran, “First principles determination of elastic constants and chemical bonding of titanium boride (TiB) on the basis of density functional theory,” Acta Materialia, vol. 54, no. 6, pp. 1641–1657, 2006. View at: Publisher Site  Google Scholar
 L. Sun, Y. Gao, B. Xiao, Y. Li, and G. Wang, “Anisotropic elastic and thermal properties of titanium borides by firstprinciples calculations,” Journal of Alloys and Compounds, vol. 579, pp. 457–467, 2013. View at: Publisher Site  Google Scholar
 G. Wang, Y. Li, Y. Gao, Y. Cheng, and S. Ma, “Theoretical study of structural, mechanical, thermal and electronic properties of Ti_{3}B_{4} with Ta_{3}B_{4} structure under high pressure,” Computational Materials Science, vol. 104, pp. 29–34, 2015. View at: Publisher Site  Google Scholar
 R. Heid, B. Renker, H. Schober, P. Adelmann, D. Ernst, and K.P. Bohnen, “Lattice dynamics and electronphonon coupling in transitionmetal diborides,” Physical Review B, vol. 67, no. 18, Article ID 180510, pp. 1–4, 2003. View at: Google Scholar
 E. Deligoz, K. Colakoglu, and Y. Ciftci, “Lattice dynamical properties of ${\text{SCB}}_{2}$, ${\text{TiB}}_{2}$, and ${\text{VB}}_{2}$ compounds,” Solid State Communications, vol. 149, no. 4142, pp. 1843–1848, 2009. View at: Publisher Site  Google Scholar
 G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set,” Physical Review B, vol. 54, no. 16, pp. 11169–11186, 1996. View at: Publisher Site  Google Scholar
 G. Kresse and J. Furthmüller, “Efficiency of abinitio total energy calculations for metals and semiconductors using a planewave basis set,” Computational Materials Science, vol. 6, no. 1, pp. 15–50, 1996. View at: Publisher Site  Google Scholar
 J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Physical Review Letters, vol. 77, no. 18, pp. 3865–3868, 1996. View at: Publisher Site  Google Scholar
 J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple [Phys. Rev. Lett. 77, 3865 (1996)],” Physical Review Letters, vol. 78, no. 7, p. 1396, 1997. View at: Publisher Site  Google Scholar
 K. Parlinski, Z. Q. Li, and Y. Kawazoe, “Firstprinciples determination of the soft mode in cubic ZrO_{2},” Physical Review Letters, vol. 78, no. 21, article 4063, 1997. View at: Publisher Site  Google Scholar
 K. Parlinski, Software PHONON ver. 6.15, Cracow, Poland, 2015.
 M. Cardona, Ed., Light Scattering in Solids I, vol. 8 of Topics in Applied Physics, Springer, Berlin, Germany, 1983.
 P. Umari, A. Pasquarello, and A. Dal Corso, “Raman scattering intensities in αquartz: a firstprinciples investigation,” Physical Review B, vol. 63, no. 9, Article ID 094305, 2001. View at: Publisher Site  Google Scholar
 P. Umari, X. Gonze, and A. Pasquarello, “Concentration of small ring structures in vitreous silica from a firstprinciples analysis of the Raman spectrum,” Physical Review Letters, vol. 90, no. 2, Article ID 027401, 2003. View at: Google Scholar
 M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, “Linear optical properties in the projectoraugmented wave methodology,” Physical Review B, vol. 73, no. 4, Article ID 045112, 2006. View at: Publisher Site  Google Scholar
 U. D. Wdowik, A. Twardowska, and M. MedalaWasik, “Lattice dynamics of binary and ternary phases in TiSiC system: a combined Raman spectroscopy and density functional theory study,” Materials Chemistry and Physics, vol. 168, pp. 58–65, 2015. View at: Publisher Site  Google Scholar
 U. D. Wdowik, M. Wasik, and A. Twardowska, “Influence of carbon dopants on the structure, elasticity and lattice dynamics of Ti_{5}Si_{3}C_{x} Nowotny phases,” Modelling and Simulation in Materials Science and Engineering, vol. 24, Article ID 025001, 2016. View at: Publisher Site  Google Scholar
 B. F. Decker and J. S. Kasper, “The crystal structure of TiB,” Acta Crystallographica, vol. 7, no. 1, pp. 77–80, 1954. View at: Publisher Site  Google Scholar
 S. Möhr, H. MüllerBuschbaum, Y. Grim, and H. G. von Schnering, “H–TiO oder TiB_{2}??eine Korrektur,” Zeitschrift für Anorganische und Allgemeine Chemie, vol. 622, no. 2, pp. 1035–1037, 1996. View at: Publisher Site  Google Scholar
 K. E. Spear, P. Mcdowell, and F. Mcmahon, “Experimental evidence for the existence of the Ti_{3}B_{4} phase,” Journal of the American Ceramic Society, vol. 69, no. 1, pp. C4–C5, 1986. View at: Publisher Site  Google Scholar
 R. Loudon, “The Raman effect in crystals,” Advances in Physics, vol. 50, no. 7, pp. 813–864, 2001. View at: Publisher Site  Google Scholar
 O. Frank, M. Zukalova, B. Laskova, J. Kürti, J. Koltai, and L. Kavan, “Raman spectra of titanium dioxide (anatase, rutile) with identified oxygen isotopes (16, 17, 18),” Physical Chemistry Chemical Physics, vol. 14, no. 42, pp. 14567–14572, 2012. View at: Publisher Site  Google Scholar
 Y. Zhang, C. X. Harris, P. Wallenmeyer, J. Murowchick, and X. Chen, “Asymmetric lattice vibrational characteristics of rutile TiO_{2} as revealed by laser power dependent raman spectroscopy,” Journal of Physical Chemistry C, vol. 117, no. 45, pp. 24015–24022, 2013. View at: Publisher Site  Google Scholar
 Ľ. Bača and N. Stelzer, “Adapting of sol–gel process for preparation of TiB2 powder from lowcost precursors,” Journal of the European Ceramic Society, vol. 28, no. 5, pp. 907–911, 2008. View at: Publisher Site  Google Scholar
 V. Swamy, A. Kuznetsov, L. S. Dubrovinsky, R. A. Caruso, D. G. Shchukin, and B. C. Muddle, “Finitesize and pressure effects on the Raman spectrum of nanocrystalline anatase TiO_{2},” Physical Review B  Condensed Matter and Materials Physics, vol. 71, no. 18, Article ID 184302, 2005. View at: Publisher Site  Google Scholar
 X. Chen and S. S. Mao, “Titanium dioxide nanomaterials: synthesis, properties, modifications and applications,” Chemical Reviews, vol. 107, no. 7, pp. 2891–2959, 2007. View at: Publisher Site  Google Scholar
 F. Tuinstra and J. L. Koenig, “Raman spectrum of graphite,” The Journal of Chemical Physics, vol. 53, no. 3, pp. 1126–1130, 1970. View at: Publisher Site  Google Scholar
 R. V. Krishnarao and J. Subrahmanyam, “Studies on the formation of TiB_{2} through carbothermal reduction of TiO_{2} and B_{2}O_{3},” Materials Science and Engineering A, vol. 362, no. 12, pp. 145–151, 2003. View at: Publisher Site  Google Scholar
 C. Subramanian, T. S. R. C. Murthy, and A. K. Suri, “Synthesis and consolidation of titanium diboride,” International Journal of Refractory Metals and Hard Materials, vol. 25, no. 4, pp. 345–350, 2007. View at: Publisher Site  Google Scholar
 X. Q. Yan, W. J. Li, T. Goto, and M. W. Chen, “Raman spectroscopy of pressureinduced amorphous boron carbide,” Applied Physics Letters, vol. 88, no. 13, Article ID 131905, 2006. View at: Publisher Site  Google Scholar
 J. Guo, L. Zhang, T. Fujita, T. Goto, and M. Chen, “Pressureinduced depolarization and resonance in Raman scattering of singlecrystalline boron carbide,” Physical Review B, vol. 81, no. 6, Article ID 060102, 2010. View at: Publisher Site  Google Scholar
 K. M. Reddy, P. Liu, A. Hirata, T. Fujita, and M. W. Chen, “Atomic structure of amorphous shear bands in boron carbide,” Nature Communications, vol. 4, article 2483, 2013. View at: Publisher Site  Google Scholar
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