Study of lattice dynamics of Fe2SiO4- and Mg2SiO4-spinels

Fe2SiO4 is an end member of Mg-rich (Mg, Fe)2SiO4, which is believed to be a major mineral of the Earth's transition zone [1, 2]. The presence of iron has a pronounced effect on elastic and thermodynamic properties of rock-forming minerals. These properties play crucial role in the interpretation of the geophysical data and thus have a large influence on our knowledge of the earth's interior. An understanding of the elastic properties of silicate will be further helpful to the interpretation of seismological data, in particular the variation in the depth range of transition zone of earth's interior. The spinel form of magnesium-iron orthosilicate, (Mg, Fe)2SiO4, is believed to be one of the most abundant minerals in the mantle's transition zone and is found to be stable in ambient conditions and therefore, a detailed study of zone centre phonons of this stable phase of orthosilicates (Mg, Fe)2SiO4 is of high interest. Hence, in the present study, the zone centre phonons of antiferromagnetic Fe2SiO4-spinel and Mg2SiO4-spinel have been studied by using short range force constant model involving interatomic interactions upto first three neighbours. The zone centre phonons of Fe2SiO4 is compared with that of Mg2SiO4 in order to study the effect of the cation exchange on the dynamic and thermodynamic properties of (Mg, Fe)2SiO4-spinel. The calculated results are compared and analyzed with exiting experimental results.


Introduction
The spinel form of magnesium-iron orthosilicate, (Mg, Fe) 2 SiO 4 is the most abundant mineral in the Earth upper mantle and is believed to have dominant influence on crucial geophysical processes in this part of the Earth [1,2]. The olivine-spinel phase transition and crystal structure of silicate spinel have been extensively investigated for understanding possible phase transformations and crystal chemistry of the orthosilicate. Detailed characterization of this mineral and its end members, Mg 2 SiO 4 and Fe 2 SiO 4 , is, therefore, crucial for future understanding of the main Earth processes. (Mg,Fe) 2 SiO 4 is known to exists in three phases:(i) low-pressure  orthorhombic (olivine), (ii) moderate-pressure  orthorhombic (wadslayite), and (iii) high-pressure  cubic (ringwoodite) [3] So far, several experimental [4] and theoretical [5,6] studies have been reported concerning the structural, dynamical, and thermodynamical properties of Mg 2 SiO 4 , crystallizing in all three {,,} phases. Whereas, Fe 2 SiO 4 is known to exist only in two stable phases ( fayalite and  spinel) represents much more challenging material as it belongs to the group of transition metal orthosilicates, additionally possessing nonvanishing magnetic moment [7].
The knowledge of the properties of all phases of (Mg,Fe) 2 SiO 4 is of great importance for earth sciences, since it determines crucial geophysical properties of the earth's interior such as possible phase transitions and thermodynamics of the rock forming minerals. These properties play crucial role in the interpretation of the geophysical data, and thus have a large influence on our knowledge of the earth's interior. However, it is very difficult and expensive to reproduce the pressure conditions of the deep regions of the earth in the laboratory and it is more difficult to perform precise measurements in such conditions. The fundamental link between microscopic (atomistic) behaviour of minerals and macroscopic properties can be made through the lattice dynamics of the minerals which is determined by interatomic interactions, and this in turn determine the basic thermodynamic properties of the minerals. In comparison to the Fe 2 SiO 4 fayalite, its high-pressure-spinel phase is a much less studied system. Therefore, in the present investigation the interatomic interaction and zone center phonons of high-pressure-spinel phase of Fe 2 SiO 4 has been studied by using an angular force constant model [8].
We compare them with those obtained for Mg 2 SiO 4 -spinel in order to understand the influence of cation exchange on lattice dynamics and thermodynamics in the ringwoodite (Fe,Mg) 2 SiO 4 .

Crystal structure
Spinel phase of (Mg,Fe) 2 SiO 4 is cubic and exists in a -spinel structure that belongs to space group Fd3m (O h 7 ) [9]. This cubic structure consists of isolated SiO 4 tetrahedra, with Mg/Fe atoms occupying the interstitial sites between SiO 4 groups. The primitive cell has 2 formula units including 14 atoms (figure 1), so there are 42 normal modes for each point in the Brillouin zone, among which 3 are acoustic and 39 are optical modes. The optical modes at the BZ centre may be divided by symmetry as  = A 1g + 2A 2u + E g + 2E u + T 1g + 4T 1u + 2T 2u + 3T 2g . Here subscripts g and u denote symmetric and antisymmetric modes with respect to the centre of inversion, while R and IR represent Raman and infrared active modes. All E modes are doubly degenerate, and all T modes are ternary degenerate. A 1g , E g , and all T 2g modes are Raman active, while all T 1u modes are infrared active.

Interatomic interaction
The determination of the strengths of the interatomic interaction in theses orthosilicates is important from the point of view of their thermo dynamical properties. One of the few procedures suitable are the interatomic force constant investigations. Therefore, in the present investigation a de Launey angular force constant (DAF) model [8] has been applied for the calculation of the interatomic interaction of (-spinel). In this model, the relative displacement of the reference atom and one of its neighbours is considered. The restoring force on the reference atom is taken to be proportional to the component of the relative displacement perpendicular to a line joining the two atoms. In the present analysis , the short range force constants between the first three neighbours  1 ,  2 ,  3 (central) for the interatomic interactions for Si-O, Mg(Fe)-O , and Mg(Fe)-Mg(Fe) are evaluated by fitting the measured zone-center phonon frequencies [5,7,[10][11] to the corresponding analytical expressions for A 1g , E g , and T 1u obtained in Ref. 12 by solving a dynamical matrix of order (42x42) at k = 0 for a compound with an ideal spinel structure. The force constants, thus calculated are listed in Table 1. It is obvious from the table 1 that the short range Si-O forces of the SiO 4 tetrahedra are larger than other interatomic forces in both orthosilicates. The interatomic interactions between Si-O is covalent in nature may be the reason of dominance over the other interactions. The rattling of cations is the reason for this kind of interactions in minerals [13]. There is almost negligible change (0.9%) in force

Zone centre phonons
The short range force constants thus calculated are taken as input parameters to compute zone-centre (ZC) phonon frequencies for (Mg/Fe) 2 SiO 4 . The calculated ZC phonons along with other results [5,7,[10][11] are given in table 2. In the case of the spinel with a cubic Fd3m symmetry, there are three independent displacements, one for each non-equivalent atom (Mg/Fe, Si, and O).