A Lattice Dynamical Investigation of Raman and Infrared Wavenumbers of Mn2SiO4

A Lattice Dynamical Investigation of Raman and Infrared Wavenumbers of Mn2SiO4

Harleen Kaur

Dept. of Applied Sciences

Baba Banda Singh Bahadur Engineering College Fatehgarh Sahib, India

M. M. Sinha

Dept. of Physics

Sant Longowal Institute of Engineering & Technology Longowal, Sangrur, India

Abstract Wilson's GF matrix method has been applied for the investigation of optical phonons of Mn2SiO4 in orthorhombic phase having space group Pbnm using normal coordinate analysis. The calculation of zone center phonons have been made with fifteen stretching and ten bending force constants. The calculated values of Raman and infrared wavenumbers are in good agreement with the experimental values. The contribution of each force constant towards the zone centre phonons has been determined in terms of potential energy distribution.

Keywords Phonon spectra; lattice dynamics; force constant; potential energy distribution.

1. INTRODUCTION

   Olivine, (Mg,Fe)2SiO4, is the major rock-forming mineral in the Earths upper mantle [1]. The family of olivine compounds, including forsterite (Mg2SiO4), fayalite (Fe2SiO4), tephroite (Mn2SiO4) and solid solutions between them, play a significant role in geosciences since olivines are the most common silicate phases in the Earth's mantle. Manganese orthosilicate, Mn2SiO4, the mineral tephroite, has been the subject of structural, crystal, chemical and thermodynamical study because of its petrological and geophysical importance [2]. Apart from its importance in Earth sciences, this family of olivines is potentially significant from technological point of view [3-4]. Silicate olivines show predominant occurence in igneous rock and have been used as an important composition in some refractory materials, grit blasting materials, ceramic pigments, additives in cement concrete, flux and slag conditioner in the steel industry [5-6], and so on.

   The study of phonon properties of these minerals is crucial for understanding the phenomenon of phase transition in such compounds. The complete information of the macroscopic behavior of the minerals can be best obtained from a detailed knowledge of microscopic nature and this relation is best made via their vibrational spectra. However, the experimental task required for studying the lattice dynamical behavior and properties of minerals is exigent because of the technical difficulties involved in reproducing the temperature and pressure conditions that are relevant to the Earths interior and in carrying out controlled experiments at such conditions. Therefore, the theoretical prediction of vibrational properties through accurate modeling is the only feasible solution. It is well known that many interatomic force dependent properties of solids can be described very

   successfully through harmonic models. Hence, in the present work, the lattice dynamical investigation of Mn2SiO4 olivine has been undertaken within the harmonic approximation.

   Previous studies reveal that Mg2SiO4 and Fe2SiO4 olivines have been a subject of thorough study to investigate the lattice dynamical properties both experimentally and theoretically [7-9], but few studies have been conducted to investigate the spectral activity in Mn2SiO4. However, none of the observations [10-11] could assign all the observed Raman and infrared modes in Mn2SiO4. Also to our knowledge, no theoretical calculation of optical phonons has been made in the orthorhombic phase of Mn2SiO4. Hence, in this paper a short range force constant model has been applied to investigate the optical phonons using normal coordinate analysis involving fifteen stretching and ten bending force constants. The theoretically obtained wavenumbers are in very good agreement with the experimental ones. Also, an effort is also made to assign experimental wavenumbers to their respective optical phonon modes. The potential energy distribution (PED) has also been investigated for determining the significance of contribution from each force constant towards the Raman and infrared wavenumbers.

2. THEORY

   Mn2SiO4, the mineral tephroite, crystallizes in the orthorhombic olivine structure with space group Pbnm (no

   62) and D2h symmetry with four formula units. The structure is composed of an almost hexagonally close packed array of oxygen ions. One eighth of the tetrahedral sites are occupied by silicon ions and one half of the octahedral positions are filled with manganese ions. The Mn octahedra have common edges and form chains along the c-axis. Mn ions occupy two crystallographically non-equivalent octahedral positions with different site symmetries, M1 and M2 (as given in Fig 1) with M1 having 4a site and M2 occupying 4c site. The oxygen ions occupy three distinct positions i.e. O1, O2 and O3. Si, O1and O2 reside at 4c and O3 ions occupy 8d site. The crystal structure consists of SiO4 tetrahedra linked by the divalent manganese cations in six fold oxygen anion coordination. Thus there are twenty eight atoms in primitive cell resulting in eighty four vibrational modes. The detailed analysis of total number of modes at zone centre (k=0) is:

   total= 11Ag+11B1g+7B2g+7B3g+10Au+10B1u+14B2u+14B3u

   where B1u+B2u+B3u are acoustical modes, 11Ag, 11B1g, 7B2g and 7B3g are Raman active, 9B1u, 13B2u and 13B3u are infrared active and 10Au are inactive modes.

   Wilsons GF matrix method [12] has been employed to calculate the Raman and infrared wavenumbers by using normal coordinate analysis. In this method the concept of internal coordinates is being introduced which makes the problem more logical. These internal coordinates includes the parameters like bond distances and angles or interatomic distances. The zone centre modes are determined in terms of kinetic and potential energies of the system. The Kinetic energy T is dependent on geometrical arrangement of the atoms and their masses mij where as the potential energy V which originate due to interactions within the molecule is defined in terms of force constants Fij .

   36 36

   The present study includes fifteen stretching Ki and and ten bending force constants Hi to formulate F matrix. The short range forces are significant upto certain neighbors and their magnitude generally decreases after the second neighbor interaction. The reason for inclusion of bending forces in our calculations is that the stretching forces only are not sufficient to explain transverse vibrations. Short range stretching forces between nearest neighbors Si-O, Mn-O, O- O and bending forces between O-Mn-O and O-Si-O are used in the present analysis. The input parameters used for the study are masses of the atoms, unit cell dimensions [13], symmetry coordinates and available Raman and infrared frequencies [11].

3. RESULTS AND DISCUSSION

   The force constants as given in Table I have been optimized and empirically scaled so as to make the

   = 1

   2

   (1)

   vibrational wavenumbers in good agreement with the observed Raman and infrared numbers.

   =1 =1

   = 1

   Mn1-O2

   Force constant	Between atoms	Coordination Number	Distance (Ã…)/ Angle (degree)	Force constant value
   K1	Si-O1	4	1.619	3.812
   K2	Si-O3	8	1.639	3.466
   K3	Si-O2	4	1.657	3.388
   K4	8	2.167	0.474
   K5	Mn1-O1	8	2.200	0.414
   K6	Mn1-O3	8	2.249	0.328
   K7	Mn2-O2	4	2.139	0.522
   K8	Mn2-O3	8	2.154	0.354
   K9	Mn2-O1	4	2.278	0.311
   K10	O2-O3	8	2.580	0.638
   K11	O1-O3	8	2.753	0.305
   K12	O1-O3	8	3.154	0.203
   K13	O2-O3	8	3.031	0.172
   K14	O2-O3	8	3.358	0.153
   K15	O2-O3	8	3.585	0.146
   H1	O2-Si-O3	8	103.03	0.423
   H2	O3-Si-O3	4	105.53	0.524
   H3	O1-Si-O2	4	113.11	0.272
   H4	O1-Si-O3	8	115.32	0.179
   H5	O3-Mn2-O3	8	87.64	0.156
   H6	O3-Mn2-O3	4	115.59	0.182
   H7	O1-Mn2-O3	8	90.70	0.156
   H8	O3-Mn2-O3	4	68.47	0.484
   H9	O2-Mn1-O3	4	108.51	0.212
   H10	O1-Mn1-O3	8	84.71	0.159

   2

   3N6 3N6

   TABLE I. INTERATOMIC FORCE CONSTANT VALUES (in N cm-1)

   i=1 j=1

   where G-1 stands for the inverse of the G matrix which is describing the kinetic energies in terms of mass-weighted

   cartesian displacements and =

   . The secular equation

   for calculating the frequencies is given by

   FG E 0

   Here G is a matrix connected with the vibrational kinetic energy and F is a matrix of force constants to represent potential energy required for each vibration and thus gives an idea of the bond strength. Molecules are constructed in cartesian coordinates and then transformed to internal coordinates with changes in bond distances and bond angles. Both F and G are symmetrical in nature and E is a unit matrix and is related to the frequency given by = 42c22.

   Fig. 1. Unit cell of Mn2SiO4 in orthorhombic phase

   With the force constants taken as input, the eigen value equation involving 84X84 matrix was solved. It is obvious from Table I that the stretching force constants for Si-O i.e. K1, K2 and K3 show a systematic variation with interatomic distance. The value of K1 is the highest among all corresponding to the smallest interatomic distance Si-O1 in comparison to K2 and K3. For comparison, the results of previous experimental [11] studies are listed in Table II along

   with the present calculated results. There is a good agreement between the theory and experimental results thus establishing the validity of present calculations. The potential energy distribution for each normal mode in Mn2SiO4 is also investigated and the two dominant contributions from force constant are given in the results. The assignment of specific modes have been made observing the atomic displacements in the eigen vectors.

   TABLE II. CALCULATED AND EXPERIMENTAL RAMAN ACTIVE WAVENUMBERS (in cm-1) FOR Mn2SiO4

   frequencies lying between 400 cm-1 to 565 cm-1 i.e. 565cm-1, 511cm-1, 408cm-1 in Ag mode, 560 cm-1, 530cm-1, 403cm-1 in B1g mode, 550cm-1, 427cm-1 in B2g mode and 548cm-1 and 395cm-1 in B3g are mainly contributed by O-Si-O bending force constants given by H1, H2, H3 and H4. Thus we find that all the frequencies > 400 cm-1 are associated with the internal vibrations within the SiO4 tetrahedron. The frequencies < 400 cm-1are low frequency modes which have the contribution of multiple force constants (K4 to K15 and H5 to H10). These frequencies are due to Mn-O stretching force and O-O repulsive force among SiO4 tetrahedra. These

   Mode	Expt [11]	Present result	Two dominant contribution as per PED
   Ag	935	944	K1-40%, K2-29%
   	840	853	K3-57%, K2-24%
   	808	812	K1-40%, K2-11%
   	575	565	H1-33%, K10-13%
   	515	511	H2-48%, K13-9%
   	389	408	H3-23%, K13-13%
   	291	318	H6-19%, K8-12%
   	256.	283	K13-21%, K9-15%
   	244	247	K8-30%,K10-15%
   	167	182	H7-27%, K7-24%
   	124	129	K7-21%, K8-15%
   B1g	–	942	K1-41%, K2-30%
   	–	858	K3-52%, K2-27%
   	820	819	K1-39%, K3-16%
   	588	560	H1-31%, K10-14%
   	546	530	H2-50%, K13-8%
   	393	403	H3-31%, H4-15%
   	307	312	K7-18%, H6-15%
   	288	273	K9-18%, H8-10%
   	271	253	H8-21%, K13-12%
   	203	192	H7-28%, H6-14%
   	155	168	K7-31%, H7-12%
   B2g	–	882	K2-82%, K11-3%
   	553	550	H1-43%, K10-24%
   	401	427	H4-37%, K12-17%
   	319	338	H5-29%, K8-19%
   	274	288	K13-21%, K12-13%
   	188	186	K8-37%, K9-30%
   	119	123	H11-20%, K14-12%
   B3g	892	866	K2-84%, K10-4%
   	555	548	H1-45%, K10-24%
   	378	395	H4-36%, K11-21%
   	304	318	H7-26%, K8-28%
   	276	279	K9- 29%, K13-14%
   	223	226	K8-20%, K15-13%
   	137	165	H5-25% , K14-20%

   The inferences drawn from the PED are described below: It is quite evident from Table II that the highest frequencies

   modes are external vibrations which involve the rotation and

   translation of SiO4 tetrahedra and Mn cations. As per our knowledge no other theoretical data is available to compare the calculated infrared Raman and infrared wavenumbers. A precise determination of infrared wavenumbers is required to further establish the present results.

   TABLE III. CALCULATED AND EXPERIMENTAL INFRARED ACTIVE WAVENUMBERS (in cm-1) FOR Mn2SiO4

   Mode	Expt [11]	Present result	Two dominant contribution as per PED
   B1u	875	883	K2-81%, K10-4%
   	480	500	H1-46%, K10-26%
   	430	437	H4-32%, K11-17%
   	350	365	H5-22%, K6-20%
   	–	306	K5-31%, H10-26%
   	300	283	K5-39%, H10-18%
   	240	228	H10-34%, K6-17%
   	187	172	H9-31%, K13-19%
   	–	129	K6-24%, K5-16%
   B2u	945	931	K1-45%, K2-33%
   	860	884	K3-51%, K2-20%
   	816	801	K1-29%, K3-22%
   	–	564	H1-30%, K10-11%
   	512	514	H2-40%, H10-13%
   	454	440	H3-22%, K13-11%
   	–	373	K12-18%, K6-14%
   	340	322	H9-31%, H10-19%
   	276	283	K4-31%, H6-15%
   	242	236	H8-26%, K7-23%
   	–	198	K7-23%, H7-29%
   	157	164	K5-25%, H9-14%
   	–	100	H8-24%, H5-16%
   B3u	950	934	K1-47%, K2-34%
   	912	889	K3-53%, K2-20%
   	815	792	K1-30%, K2-19%
   	562	550	H1-32%, K10-14%
   	490	502	H2-42%, K10-11%
   	–	444	H3-22%, K13-14%
   	–	378	K5-24%, H10-17%,
   	365	341	H10-32%, K7-12%
   	297	273	H7-20%, H8-15%
   	–	204	K4-33%, H9-24%
   	–	188	K6-33%, K5-20%
   	177	174	K12-20%, K5-17%
   	–	129	K5-22%, K14-15%

   i.e., 944cm-1, 853cm-1, 812cm-1 in A mode, 942cm-1, 858cm-

   1 -1

   g

   -1 -1

   , 819cm

   in B1g mode, 882cm in B2g mode and 866cm in

   B3g mode has the dominant contribution of Si-O stretching interaction i.e K1, K2 and K3. This result is in confirmation with the inferences drawn by Mouri et al. [10] and Stidham et al. [11]. It has been observed that the M-site cations do not participate in the higher frequency modes. A close look at the reveals that identical results are obtained for the highest infrared wavenumbers in each mode and is attributable mainly due to the Si-O stretching. All the middle order

4. CONCLUSION

Normal coordinate analysis has been performed on Mn2SiO4 olivine to calculate the Raman and the infrared wavenumbers with twenty five short range force constants. It was found that the higher order frequencies are mainly contributed SiO4 tetrahedra while the lower frequencies are dominated by Mn atoms. Eigen vectors associated with Raman and infrared frequencies for tephroite has also been determined. The theoretical results are found to be in good agreement with experimental observed values. The contribution of each force constant has been determined from the potential energy distribution towards the different vibrational modes.

REFERENCES

1. W. Deer, R. Howie and J. Zussman, 1992. An Introduction to the Rock Forming Minerals. John Wiley, New York.

2. A.M. Dziewonski, A.L. Hales and E.R. Lapwood, "Parametrically Simple Earth Models Consistent With Geophysical Data", Physics of earth and planetary interiors,vol 10 (1), pp. 1248, 1975.

3. S. Ni, L. Chou and J. Chang, "Preparation and characterization of Forsterite (Mg2SiO4) bioceramics", Ceram Int., vol 33, pp. 83-88, 2007.

4. M. Kharaziha and M.H. Fathi, "Improvement of mechanical properties and biocompatibility of forsterite bioceramic addressed to bone tissue engineering materials", J. Mech. Behav. Biomed. Mater., vol 3(7), pp. 530-537, 2010.

5. R. Pawley, "The reaction talc + forsterite = enstatite +H2O; new experimental results and petrological implications.", Am. Mineralogist, vol 83 (1), pp. 5157,1998.

6. N. Maliavski, O. Dushkin and J. Markina, "Forsterite powder prepared from water soluble hybrid precursor", AIChE J., vol 43 (11A), pp. 2832 2836, 1997.

7. Y. Kudoh and H.Takeda, "Single crystal X-ray diffraction study on the bond compressibility of fayalite, Fe2SiO4 and rutile, TiO2 under high pressure", Physica B + C, vol 139, pp. 333-336, 1986.

8. A. M. Hofmeister and K. M. Pitman, "Evidence for kinks in structural and thermodynamic properties across the forsterite fayalite binary from thin film IR absorption spectra" , Phys Chem Miner, vol 34, pp. 319- 333, 2007.

9. E. Dachs, C. Geiger, V.von Seckendorff and M. Grodzicki, " A low temperature calorimetric study of synthetic (forsterite-fayalite)

{(Mg2SiO4-Fe2SiO4)} solid solutions: An analysis of vibrational, magnetic and electronic contributions to the molar heat capacity and entropy of mixing", J. Chem Thermodyn, vol 39, pp. 906-933, 2007.

[10] T. Mouri and M. Enami, "Raman spectroscopic study of olivine-group material", Journal of Minerological and Petrological Sciences, vol 103, pp.100104, 2008.

[11] H.D. Stidham, , J.B. Bates and C.B. Finch,"Vibrational spectra of synthetic single crystal tephroite, Mn2SiO4", Journal of Physical Chemistry, vol 80, pp.1226-1234,1976.

[12] T. Shimanouchi, , M. Tsuiboi and T. J. Miyazawa, "Optically active lattice vibrations as treated by the GFmatrix method", Chem. Phys, vol 35, pp.1597-1609, 1961.

[13] K. Fujino, S. Sasaki, Y. Takeuchi and R. Sadanaga, "X-ray determination of electron distributions in forsterite, fayalite, and tephroite", Acta Crystallographica, vol B37, pp. 513-518, 1981.