Sayyadul Arafin* and Ram N. Singh

1. A Correlation betweenSpecific Heat Capacity,Grüneisen Parameter andPhase changeswithin400 –700 km of the Earth

2. Sayyadul Arafin*andRam N. Singh Physics Department, College of Science, Sultan Qaboos University, Box: 36, Al-Khoudh 123, Sultanate of Oman *Corresponding author: sayfin@squ.edu.om

3. INTRODUCTION Thermophysical properties of rocks and minerals are useful to • study the deep interior of the Earth, • derive mineralogical and compositional models of the mantle. • determine the tectonic and petrogenic processes involved in the generation of oceanic lithosphere. • enhance our knowledge about the formation and transformation of rocks and minerals.

4. INTRODUCTION • Due to complexity of X-ray diffraction and other modern spectroscopic techniques, these have not been widely used in characterizing upper mantle rocks [1]. • Most of the Earth is solid, and much of it is at temperatures and pressures that are difficult to achieve in the laboratory. • Usual practice is to formulate a reasonable model to extrapolate the laboratory measurements.

5. INTRODUCTION • Mineral physics, partial melting and solid-solid phase changes play a vital role in understanding the composition of the mantle. • Variations in densities and seismic velocities of rocks with temperature, pressure and composition are usually very small. • However an extreme change in mineralogy can change density and seismic velocity appreciably. • Proportions and compositions of various phases or minerals that determine the physical properties of rocks depend on temperature, pressure and composition as well.

6. INTRODUCTION In the present work • We have used p- and s-wave velocities of Jeffreys and Bullen [2] and the density data of [3] to obtain thermo-physical properties like Debye temperature, adiabatic compressibility, Grüneisen parameter and specific heat capacity of the Earth’s mantle. • Purpose of using smooth data is to investigate if the thermo-physical properties determined from these data can shed some light in identifying the established mantle discontinuities.

7. PHASE CHANGES IN THE UPPER MANTLE • Mantle is composed of silicate minerals rich in magnesium. . • Low velocity layer (50-220km) beneath the lithosphere is termed as aesthenosphere. • Olivine constitutes about 80% of the upper mantle materials. Large-scale mineralogical zoning in the upper mantle has an important effect on seismic velocities and density.

8. PHASE CHANGES IN THE UPPER MANTLE • Below the depth range of 390-450km, main minerals are garnet and spinel. • Compression of olivine's atomic structure to its spinel phases under extreme pressure causes a seismic discontinuity at approximately 390-450 km. This discontinuity corresponds to a transition from the α- to β-structures of multi-component olivine. • At depth around 670km, the transformation is fundamentally different from the α- to β- transition, because it involves a change in chemical composition[4].

9. PHASE CHANGES IN THE UPPER MANTLE • Olivine and pyroxene-garnet components transform into magnesiowüstite, (Mg,Fe)O, and perovskite with ~10% density increase across the 1-2 km thick phase change region[4,5].

10. FORMALISM Debye Temperature, Adiabatic Compressibility, Grüneisen Parameter and Specific Heat Capacity are used here to investigate their contributions to the understanding of the phase changes within the upper mantle. • Debye Temperature Debye temperature is the temperature corresponding to the highest normal mode of vibration. It’s given by h is Planck’s constant, kis Boltzmann constant, and νm is the Debye frequency. .

11. Debye Temperature • It facilitates the characterization of rocks, minerals and even geological processes such as metamorphism[6]. • Any change in the external conditions of rocks and other material formation and transformation can lead to a change in the Debye characteristic temperature. • Difference between Debye temperatures of separated facies of metamorphic rocks is • maximum in case of regional metamorphism, • lower at intermediate level of metamorphism and • minimum in case of extremely altered rocks.

12. Debye Temperature Working formula for Debye temperature for isotropic solids is given as [7](Arafin et al. 2008) μ is the mean atomic weight given by M/p, M is the molecular weight, and p is the atomic number. μ for minerals important to the earth's mantle doesn't vary much from 21amu [8], ρ is the density in g/cm3..

13. Debye Temperature vm is the generalized mean velocity in km/s of the seismic p- and s-wave velocities given by: vpand, vs are the primary and secondary seismic velocities respectively.

14. Adiabatic Compressibility • The adiabatic or isentropic compressibility, κS can be calculated by using the measured values of vp and vs, • For rocks, minerals and other solids values of adiabatic and isothermal compressibilities differ by about 1 percent [8].

15. Grüneisen Parameter • Grüneisen parameter is a property of materials that establishes a link between thermal behaviorand the elastic response to thermally induced stress of materials. • It measures anharmonic interactions

16. Grüneisen Parameter Thermodynamic approach to determine Grüneisen parameter follows • mathematical expressions for entropy of a pure substance • Maxwell's equations • coefficient of volume expansion • isothermal compressibility.

17. Grüneisen Parameter The working formula for Grüneisen parameter, ξ is given by • β(K) coefficient of volume expansion • ρ density • κT,κS isothermal and adiabatic compressibilities • cv, cP specific heat at const. vol. and const. pressure • Functions on the RHS of Eq. 5 can be determined from experiments and hence ξ is directly determinable.ξ generally lies between 1 & 2 for materials of geophysical interest[8].

18. Specific Heat Capacity Specific heat at constant pressure (cP) and constant volume (cV) can be derived as (6) • V molar volume • β coefficient of volume expansion • κSadiabatic compressibility • Specific heat ratio, For rocks and minerals κSis larger than κT by about 1 percent [8]. Adiabatic compressibility can be determined from Eq. 4 and hence γ can be estimated. The values of β as a function of temperature are taken from [9]. An average value of β for the upper mantle material may be taken as 2.66 x 10-5 K-1.

19. RESULTS AND DISCUSSIONSDebye Temperature

20. RESULTS AND DISCUSSIONSAdiabatic compressibility

21. RESULTS AND DISCUSSIONSCoefficient of volume expansion

22. RESULTS AND DISCUSSIONSGruneisen parameter and Sp. heat

23. Summary and Conclusions

24. Summary and Conclusions • Comprehensive diagram (Fig.7) shows all the plotting of the thermo-elastic and thermodynamic properties alongside the different discontinuities of the upper mantle. • Minimum and maximum values of each thermo-physical quantity are shown. • Our analysis is focused on mantle materials below depth 200km. • Specific heat capacity and Grüneisen parameter are more deterministic in the sense that they show prominent minima and maxima which are easily identifiable than the inflexion points.

25. Summary and Conclusions • The inflexion points are easily identifiable from the curves of Debye temperature and adiabatic compressibility unlike the curves of Grüneisen parameter and specific heat capacity, which show the maximum and minimum very clearly. • The CP values suddenly drop at depths 400 km and 612 km which correspond to humps of the Grüneisen parameter as two maxima at 400km and 615km respectively.

26. Summary and Conclusions • The average depths determined from the depth dependence of these thermo-physical properties are 414km and 645km which are in good agreement with the accepted depths of 410km and 670km for the corresponding discontinuities in the upper mantle. • The study has shown that it is possible to determine the two upper mantle discontinuities from the thermo-physical properties by utilizing geophysical data.

27. References • [1]. D.C. Rubie, T.S. Duffy, E. Ohtani, New developments in high pressure mineral physics and applications to the Earth’s interior, Phys. Earth Planet. Inter. 143–144 (2004) 1–3. • [2]. Fowler C.M.R. 1990 The Solid Earth: An Introduction to Global Geophysics, Cambridge University press. • [3]. Ganguly J., Freed A.M. & Saxena S.K. 2009. Density profiles of oceanic slas and surrounding mantle: Integrated thermodynamic and thermal modeling, and implications for the fate of slabs at the 660km discontinuity, Physics of the Earth and Planetary Interiors 172, 257 – 267. • [4]. Green D.H. & Ringwood A.E. 1963. Mineral assemblages in a model mantle composition, Journal of Geophysical Research 68, 937-945. • [5]. Katsura T. & Ito E. 1989. The system Mg2SiO4 – Fe2SiO4 at high pressures and temperatures – precise determination of stabilities of olivine, modified spinel and spinel, Journal of Geophysical Research 94(B11), 15663-15670

28. References • [6]. Dergachov A.L. & Starostin V.I. 1981. The Debye characteristic temperature as indicator for formation and transformation conditions of rocks and ores, Ore Deposit Geology 6, 67–75. • [7]. Arafin S., Singh R.N., George A.K. & Al-Lazki A. 2008. Thermoelastic and thermodynamic properties of harzburgite - An upper mantle rock, Journal of Physics and Chemistry of Solids 69, 1766–1774.  • [8].O.L. Anderson, Equations of State of Solids for Geophysics and Ceramic Science, Oxford University Press, New York, 1995. • [9]. O.L. Anderson, D.G. Isaak, Elastic constants of mantle minerals at high temperature, in: Mineral Physics and Crystallography, A Handbook of Physical Constants, AGU Reference Shelf 2, 1995. • George A.K., Singh R.N., Arafin S., Carboni C. & Al-Harthi S.H. 2010. Physica B 405, 4586–4593

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