Fe–Mg substitution in aluminate spinels: effects on elastic properties investigated by Brillouin scattering

  • Enrico Bruschini
  • Sergio Speziale
  • Ferdinando Bosi
  • Giovanni B. Andreozzi
Original Paper
First Online:
Received:
Accepted:
  • 73 Downloads

Abstract

We investigated by a multi-analytical approach (Brillouin scattering, X-ray diffraction and electron microprobe) the dependence of the elastic properties on the chemical composition of six spinels in the series (Mg1−x,Fe x )Al2O4 (0 ≤ x ≤ 0.5). With the exception of C12, all the elastic moduli (C11, C44, KS0 and G) are insensitive to chemical composition for low iron concentration, while they decrease linearly for higher Fe2+ content. Only C12 shows a continuous linear increase with increasing Fe2+ across the whole compositional range under investigation. The high cation disorder showed by the sample with x = 0.202 has little or no influence on the elastic parameters. The range 0.202 < x < 0.388 bounds the percolation threshold (pc) for nearest neighbor interaction of Fe in the cation sublattices of the spinel structure. Below x = 0.202, the iron atoms are diluted in the system and far from each other, and the elastic moduli are nearly constant. Above x = 0.388, Fe atoms form extended interconnected clusters and show a cooperative behavior thus affecting the single-crystal elastic moduli. The elastic anisotropy largely increases with the introduction of Fe2+ in substitution of magnesium in spinel. This behavior is different with respect to other spinels containing transition metals such as Mn2+ and Co2+.

Keywords

Mg–Fe2+ substitution Spinels Elasticity Brillouin scattering Crystal chemistry Percolation threshold Elastic anisotropy 
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Notes

Acknowledgements

We are very grateful to the two anonymous reviewers for their constructive comments which greatly improved our work and to C. McCammon for handling the manuscript. E.B. acknowledges support from Sapienza University of Rome (“Avvio alla ricerca 000047_13_GR_BRUSC”). G.B.A. acknowledges funding from “Progetto di ricerca Università 2015”.

Supplementary material

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References

  1. Andreozzi GB, Lucchesi S (2002) Intersite distribution of Fe2+ and Mg in the spinel (sensu stricto)—hercynite series by single crystal X-ray diffraction. Am Miner 87(8–9):1113–1120CrossRefGoogle Scholar
  2. Andreozzi GB, Princivalle F (2002) Kinetics of cation ordering in synthetic MgAl2O4 spinel. Am Miner 87(7):838–844CrossRefGoogle Scholar
  3. Andreozzi GB, Princivalle F, Skogby H, Della Giusta A (2000) Cation ordering and structural variations with temperature in MgAl2O4 spinel: an X-ray single-crystal study. Am Miner 85(9):1164–1171CrossRefGoogle Scholar
  4. Angel RJ (2000) Equations of state. Rev Miner Geochem 41(1):35–59CrossRefGoogle Scholar
  5. Angel RJ (2004) Equations of state of plagioclase feldspars. Contrib Miner Petrol 146(4):506–512CrossRefGoogle Scholar
  6. Angel RJ, Hazen RM, McCormick TC, Prewitt CT, Smyth JR (1988) Comparative compressibility of end-member feldspars. Phys Chem Miner 15(4):313–318CrossRefGoogle Scholar
  7. Askarpour V, Manghnani MH, Fassbender S, Yoneda A (1993) Elasticity of single-crystal MgAl2O4 spinel up to 1273 K by Brillouin spectroscopy. Phys Chem Miner 19(8):511–519CrossRefGoogle Scholar
  8. Bass JD, Sinogeikin SV, Li B (2008) Elastic properties of minerals: a key for understanding the composition and temperature of Earth’s interior. Elements 4(3):165–170CrossRefGoogle Scholar
  9. Berry RL, Raynor GV (1953) The crystal chemistry of the Laves phases. Acta Crystallogr 6(2):178–186CrossRefGoogle Scholar
  10. Birch F (1938) The effect of pressure upon the elastic parameters of isotropic solids according to Murnaghan’s theory of finite strain. J Appl Phys 9(4):279–288CrossRefGoogle Scholar
  11. Birch F (1947) Finite elastic strain of cubic crystals. Phys Rev 71(11):809CrossRefGoogle Scholar
  12. Birch F (1952) Elasticity and constitution of the Earth’s interior. J Geophys Res 57(2):227–286CrossRefGoogle Scholar
  13. Birch F (1961) Composition of the Earth’s mantle. Geophys J R Astron Soc 4:295CrossRefGoogle Scholar
  14. Bosi F, Hålenius U, Skogby H (2010) Crystal chemistry of the MgAl2O4–MgMn2O4–MnMn2O4 system: analysis of structural distortion in spinel-and hausmannite-type structures. Am Miner 95(4):602–607CrossRefGoogle Scholar
  15. Broadbent SR, Hammersley J (1957) Percolation processes: I. Crystals and mazes. In Math. Proc. Cambridge University Press, Cambridge. pp 629–641Google Scholar
  16. Brown JM, McQueen RG (1986) Phase transitions, Grüneisen parameter, and elasticity for shocked iron between 77 GPa and 400 GPa. J Geophys Res Sol Earth 91:7485–7494CrossRefGoogle Scholar
  17. Bruschini E, Speziale S, Andreozzi GB, Bosi F, Hålenius U (2015) The elasticity of MgAl2O4–MnAl2O4 spinels by Brillouin scattering and an empirical approach for bulk modulus prediction. Am Miner 100(2–3):644–651CrossRefGoogle Scholar
  18. Buchen J, Marquardt H, Ballaran TB, Kawazoe T, McCammon C (2017) The equation of state of wadsleyite solid solutions: constraining the effects of anisotropy and crystal chemistry. Am Miner 102(12):2494–2504CrossRefGoogle Scholar
  19. Cummins HZ, Schoen PE (1972) Linear scattering from thermal fluctuations. Laser Handb 2(S1031)Google Scholar
  20. Davies GF, Dziewonski AM (1975) Homogeneity and constitution of the Earth’s lower mantle and outer core. Phys Earth Planet Interior 10:336–343CrossRefGoogle Scholar
  21. Duan Y, Li X, Sun N, Ni H, Tkachev SN, Mao Z (2018) Single-crystal elasticity of MgAl2O4-spinel up to 10.9 GPa and 1000 K: implication for the velocity structure of the top upper mantle. Earth Planet Sci Lett 481:41–47CrossRefGoogle Scholar
  22. Duffy TS, Vaughan MT (1988) Elasticity of enstatite and its relationship to crystal structure. J Geophys Res Sol Earth 93:383–391CrossRefGoogle Scholar
  23. Duffy TS, Zha CS, Downs RT, Mao HK, Hemley RJ (1995) Elasticity of forsterite to 16 GPa and the composition of the upper mantle. Nature 378:170–173CrossRefGoogle Scholar
  24. Every AG (1980) General closed-form expressions for acoustic waves in elastically anisotropic solids. Phys Rev B 22:1746CrossRefGoogle Scholar
  25. Fan D, Mao Z, Yang J, Lin JF (2015) Determination of the full elastic tensor of single crystals using shear wave velocities by Brillouin spectroscopy. Am Miner 100(11–12):2590–2601CrossRefGoogle Scholar
  26. Fei Y (1995) Thermal expansion. In: Ahrens TJ (ed) Mineral physics and crystallography: a handbook of physical constants. American Geophysical Union, Washington DC, pp 29–44CrossRefGoogle Scholar
  27. Finger LW, Hazen RM, Hofmeister AM (1986) High-pressure crystal chemistry of spinel (MgAl2O4) and magnetite (Fe3O4): comparisons with silicate spinels. Phys Chem Miner 13:215–220CrossRefGoogle Scholar
  28. Fiorani D, Viticoli S (1979a) Experimental evidence of a critical concentration for the long-range magnetic order in the A-sublattice of spinels. Solid State Commun 29:239–241CrossRefGoogle Scholar
  29. Fiorani D, Gastaldi L, Lapiccirella A, Viticoli S (1979b) Monte Carlo simulation of percolative phenomena in the cationic B-sublattice of spinels. Solid State Commun 32:831–832CrossRefGoogle Scholar
  30. Galasso FS (2016). Structure and properties of inorganic solids: international series of monographs in solid state physics. vol 7. Elsevier, OxfordGoogle Scholar
  31. Giri AK, Mitra GB (1986) Elastic constants of solid solutions of ionic compounds. J Phys D Appl Phys 19:L5CrossRefGoogle Scholar
  32. Goodenough JB (1964) Jahn–Teller distortions induced by tetrahedral-site Fe2+ ions. J Phys Chem Solids 25:151–160CrossRefGoogle Scholar
  33. Grimsditch M (2001) Brillouin scattering. In Levy, Stern, (eds), Handbook of elastic properties of solids, liquids and gases, vol 1. Academic Press, LondonGoogle Scholar
  34. Hålenius U, Skogby H, Andreozzi GB (2002) Influence of cation distribution on the optical absorption spectra of Fe3+-bearing spinel ss-hercynite crystals: evidence for electron transitions in VIFe2+–VIFe3+ clusters. Phys Chem Miner 29:319–330CrossRefGoogle Scholar
  35. Hazen R, Navrotsky A (1996) Effects of pressure on order–disorder reactions. Am Miner 81:1021–1035CrossRefGoogle Scholar
  36. Hazen RM, Yang H (1999) Effects of cation substitution and order-disorder on P-V-T equations of state of cubic spinels. Am Mineral 84(11–12):1956–1960CrossRefGoogle Scholar
  37. Hubsch J, Gavoile G, Bolfa J (1978) Percolation and magnetic order in diluted spinels. J Appl Phys 49:1363–1365CrossRefGoogle Scholar
  38. Isaak DG, Graham EK (1976) The elastic properties of an almandine-spessartine garnet and elasticity in the garnet solid solution series. J Geophys Res 81:2483–2489CrossRefGoogle Scholar
  39. Ishikawa Y, Syono Y (1971) Giant magnetostriction due to Jahn–Teller distortion in Fe2TiO4. Phys Rev Lett 26:1335CrossRefGoogle Scholar
  40. Jackson I (1983) Some geophysical constraints on the chemical composition of the Earth’s lower mantle. Earth Planet Sci Lett 62:91–103CrossRefGoogle Scholar
  41. Jackson I (2000) The Earth’s mantle: composition, structure, and evolution. Cambridge University Press, CambridgeGoogle Scholar
  42. Jackson I, Rigden SM (1996) Analysis of PVT data: constraints on the thermoelastic properties of high-pressure minerals. Phys Earth Planet Inter 96:85–112CrossRefGoogle Scholar
  43. Jackson I, Liebermann RC, Ringwood AE (1978) The elastic properties of (MgxFe1–x)O solid solutions. Phys Chem Miner 3:11–31CrossRefGoogle Scholar
  44. Jacobsen SD, Reichmann HJ, Spetzler HA, Mackwell SJ, Smyth JR, Angel RJ, McCammon CA (2002) Structure and elasticity of single-crystal (Mg,Fe)O and a new method of generating shear waves for gigahertz ultrasonic interferometry. Sol Earth J Geophys Res:107Google Scholar
  45. Jiang F, Speziale S, Duffy TS (2004) Single-crystal elasticity of grossular-and almandine-rich garnets to 11 GPa by Brillouin scattering. J Geophys Res Sol Earth 109:B10Google Scholar
  46. Karki BB, Stixrude L, Clark SJ, Warren MC, Ackland GJ, Crain J (1997) Structure and elasticity of MgO at high pressure. Am Miner 82:51–60CrossRefGoogle Scholar
  47. Kurnosov A, Marquardt H, Frost DJ, Ballaran TB, Ziberna L (2017) Evidence for a Fe3+-rich pyrolitic lower mantle from (Al,Fe)-bearing bridgmanite elasticity data. Nature 543(7646):543CrossRefGoogle Scholar
  48. Kyono A, Gramsch SA, Nakamoto Y, Sakata M, Kato M, Tamura T, Yamanaka T (2015) High-pressure behavior of cuprospinel CuFe2O4: Influence of the Jahn–Teller effect on the spinel structure. Am Miner 100:1752–1761CrossRefGoogle Scholar
  49. Lavina B, Salviulo G, Della Giusta A (2002) Cation distribution and structure modelling of spinel solid solutions. Phys Chem Miner 29:10–18CrossRefGoogle Scholar
  50. Li B, Liebermann RC (2007) Indoor seismology by probing the Earth’s interior by using sound velocity measurements at high pressures and temperatures. Proc Natl Acad Sci 104:9145–9150CrossRefGoogle Scholar
  51. Li B, Zhang J (2005) Pressure and temperature dependence of elastic wave velocity of MgSiO3 perovskite and the composition of the lower mantle. Phys Earth Planet Inter 151:143–154CrossRefGoogle Scholar
  52. Li Z, Fisher ES, Liu JZ, Nevitt MV (1991) Single-crystal elastic constants of Co–Al and Co–Fe spinels. J Mater Sci 26(10):2621–2624CrossRefGoogle Scholar
  53. Li L, Carrez P, Weidner D (2007) Effect of cation ordering and pressure on spinel elasticity by ab initio simulation. Am Miner 92:174–178CrossRefGoogle Scholar
  54. Lin JF, Speziale S, Mao Z, Marquardt H (2013) Effects of the electronic spin transitions of iron in lower mantle minerals: implications for deep mantle geophysics and geochemistry. Rev Geophys 51:244–275CrossRefGoogle Scholar
  55. Lu C, Mao Z, Lin JF, Zhuravlev KK, Tkachev SN, Prakapenka VB (2013) Elasticity of single-crystal iron-bearing pyrope up to 20 GPa and 750 K. Earth Planet Sci Lett 361:134–142CrossRefGoogle Scholar
  56. Mao Z, Fan D, Lin JF, Yang J, Tkachev SN, Zhuravlev K, Prakapenka VB (2015) Elasticity of single-crystal olivine at high pressures and temperatures. Earth Planet Sci Lett 426:204–215CrossRefGoogle Scholar
  57. Marquardt H, Speziale S, Jahn S, Ganschow S, Schilling FR (2009) Single-crystal elastic properties of (Y,Yb)3Al5O12. J Appl Phys 106:3519Google Scholar
  58. Martignago F, Andreozzi GB, Dal Negro A (2006) Thermodynamics and kinetics of cation ordering in natural and synthetic Mg(Al,Fe3+)2O4 spinels from in situ high-temperature X-ray diffraction. Am Miner 91:306–312CrossRefGoogle Scholar
  59. Mori-Sánchez P, Marqués M, Beltrán A, Jiang JZ, Gerward L, Recio JM (2003) Origin of the low compressibility in hard nitride spinels. Phys Rev B 68(6):064115CrossRefGoogle Scholar
  60. Murakami M, Ohishi Y, Hirao N, Hirose K (2012) A perovskitic lower mantle inferred from high-pressure, high-temperature sound velocity data. Nature 485(7396):90CrossRefGoogle Scholar
  61. Nakamura S, Fuwa A (2014) Local and dynamic Jahn–Teller distortion in ulvöspinel Fe2TiO4. Hyperfine Interact 226:267–274CrossRefGoogle Scholar
  62. Nestola F, Ballaran TB, Liebske C, Bruno M, Tribaudino M (2006) High-pressure behaviour along the jadeite NaAlSi2O6–aegirine NaFeSi2O6 solid solution up to 10 GPa. Phys Chem Miner 33:417–425CrossRefGoogle Scholar
  63. Nestola F, Boffa Ballaran T, Balic-Zunic T, Princivalle F, Secco L, Dal Negro A (2007) Comparative compressibility and structural behavior of spinel MgAl2O4 at high pressures: the independency on the degree of cation order. Am Miner 92:1838–1843CrossRefGoogle Scholar
  64. Nestola F, Ballaran TB, Koch-Müller M, Balic-Zunic T, Taran M, Olsen L, Princivalle F, Secco L, Lundegaard L (2010) New accurate compression data for γ-Fe2SiO4. Phys Earth Planet Inter 183:421–425CrossRefGoogle Scholar
  65. Nestola F, Periotto B, Andreozzi GB, Bruschini E, Bosi F (2014) Pressure–volume equation of state for chromite and magnesiochromite: a single-crystal X-ray diffraction investigation. Am Miner 99:1248–1253CrossRefGoogle Scholar
  66. Nestola F, Periotto B, Anzolini C, Andreozzi GB, Woodland AB, Lenaz D, Alvaro M, Princivalle F (2015) Equation of state of hercynite, FeAl2O4, and high-pressure systematics of Mg–Fe–Cr–Al spinels. Miner Mag 79:285–294CrossRefGoogle Scholar
  67. Newnham RE (2005) Properties of materials: anisotropy, symmetry, structure. Oxford University Press, OxfordGoogle Scholar
  68. Pamato MG, Kurnosov A, Ballaran TB, Frost DJ, Ziberna L, Giannini M, Speziale S, Tkachev SN, Zhuravlev KK, Prakapenka VB (2016) Single crystal elasticity of majoritic garnets: stagnant slabs and thermal anomalies at the base of the transition zone. Earth Planet Sci Lett 451:114–124CrossRefGoogle Scholar
  69. Poirier JP (2000) Introduction to the physics of the Earth’s interior, 2nd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  70. Pouchou JL, Pichoir F (1984) A new model for quantitative X-ray microanalysis. I. Application to the analysis of homogeneous samples. Rech Aerosp 3:167–192Google Scholar
  71. Recio JM, Franco R, Pendás AM, Blanco MA, Pueyo L, Pandey R (2001) Theoretical explanation of the uniform compressibility behavior observed in oxide spinels. Phys Rev B 63:184101CrossRefGoogle Scholar
  72. Reichmann HJ, Jacobsen SD (2006) Sound velocities and elastic constants of ZnAl2O4 spinel and implications for spinel-elasticity systematics. Am Miner 91:1049–1054CrossRefGoogle Scholar
  73. Reichmann HJ, Jacobsen SD, Mackwell SJ, McCammon CA (2000) Sound wave velocities and elastic constants for magnesiowüstite using gigahertz interferometry. Geophys Res Lett 27:799–802CrossRefGoogle Scholar
  74. Reichmann HJ, Jacobsen SD, Boffa Ballaran T (2013) Elasticity of franklinite and trends for transition-metal oxide spinels. Am Miner 98:601–608CrossRefGoogle Scholar
  75. Scholl F, Binder K (1980) Selective sublattice dilution in ordered magnetic compounds: a new kind of percolation problem. Z Phys B Condens Matter 39:239–247CrossRefGoogle Scholar
  76. Schreiber E (1967) Elastic moduli of single-crystal spinel at 25 °C and to 2 kbar. J Appl Phys 38(6):2508–2511CrossRefGoogle Scholar
  77. Shannon RD (1976) Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Crystallogr A 32(5):751–767CrossRefGoogle Scholar
  78. Sheldrick GM (2013) SHELXS 2013, program for crystal structure solution. University of Göttingen, GöttingenGoogle Scholar
  79. Shimizu H (1995) High-pressure Brillouin scattering of molecular single-crystals grown in a diamond-anvil cell. In: Senoo M, Suito K, Kobayashi T, Kubota H (eds) High pressure research on solids. Elsevier, Amsterdam, pp 1–17Google Scholar
  80. Sickafus KE, Wills JM, Grimes NW (1999) Structure of spinel. J Am Ceram Soc 82:3279–3292CrossRefGoogle Scholar
  81. Skogby H, Hålenius U (2003) An FTIR study of tetrahedrally coordinated ferrous iron in the spinel–hercynite solid solution. Am Miner 88:489–492CrossRefGoogle Scholar
  82. Slack GA, Ham FS, Chrenko RM (1966) Optical absorption of tetrahedral Fe2+(3d6) in cubic ZnS, CdTe, and MgAl2O4. Phys Rev 152:376CrossRefGoogle Scholar
  83. Slagle OD, McKinstry HA (1967) Temperature dependence of the elastic constants of the alkali halides. I. NaCl, KCl, and KBr. J Appl Phys 38:437–446CrossRefGoogle Scholar
  84. Speziale S, Duffy TS, Angel RJ (2004) Single-crystal elasticity of fayalite to 12 GPa. J Geophys Res Sol Earth 109:B12CrossRefGoogle Scholar
  85. Speziale S, Nestola F, Jiang F, Duffy T (2016) Single-crystal elastic constants of spinel (MgAl2O4) to 11.1 GPa by Brillouin scattering. In: Abstract MR23A-2658 2016 Fall Meeting. AGU, San FranciscoGoogle Scholar
  86. Steinle-Neumann G, Stixrude L, Cohen RE, Gülseren O (2001) Elasticity of iron at the temperature of the Earth’s inner core. Nature 413:57–60CrossRefGoogle Scholar
  87. Stixrude L, Lithgow-Bertelloni C (2005) Thermodynamics of mantle minerals—I. Physical properties. Geophys J Int 162:610–632CrossRefGoogle Scholar
  88. Sumino Y, Kumazawa M, Nishizawa O, Pluschkell W (1980) The elastic constants of single crystal Fe1–xO, MnO and CoO, and the elasticity of stoichiometric magnesiowüstite. J Phys Earth 28(5):475–495CrossRefGoogle Scholar
  89. Suzuki I, Ohno I, Anderson O (2000) Harmonic and anharmonic properties of spinel MgAl2O4. Am Miner 85:304–311CrossRefGoogle Scholar
  90. Sykes MF, Essam JW (1964) Critical percolation probabilities by series methods. Phys Rev 133:A310CrossRefGoogle Scholar
  91. Syono Y, Fukai Y, Ishikawa Y (1971) Anomalous elastic properties of Fe2TiO4. J Phys Soc Jpn 31:471–476CrossRefGoogle Scholar
  92. Taran MN, Langer K (2001) Electronic absorption spectra of Fe2+ ions in oxygen-based rock-forming minerals at temperatures between 297 and 600 K. Phys Chem Miner 28:199–210CrossRefGoogle Scholar
  93. Tatli A, Özkan H (1987) Variation of the elastic constants of tourmaline with chemical composition. Phys Chem Miner 14:172–176CrossRefGoogle Scholar
  94. Uchida N, Saito S (1972) Elastic constants and acoustic absorption coefficients in MnO, CoO, and NiO single crystals at room temperature. J Acoust Soc Am 51(5B):1602–1605CrossRefGoogle Scholar
  95. van der Marck SC (1997) Percolation thresholds and universal formulas. Phys Rev Earth 55:1514CrossRefGoogle Scholar
  96. Verma RK (1960) Elasticity of some high-density crystals. J Geophys Res 65(2):757–766CrossRefGoogle Scholar
  97. Wang H, Simmons G (1972) Elasticity of some mantle crystal structures: 1. Pleonaste and hercynite spinel. J Geophys Res 77:4379–4392CrossRefGoogle Scholar
  98. Waskowska A, Gerward L, Olsen JS, Steenstrup S, Talik E (2001) CuMn2O4: properties and the high-pressure induced Jahn–Teller phase transition. J Phys Condens Matter 13:2549CrossRefGoogle Scholar
  99. Watt JP, Davies GF, O’Connell RJ (1976) The elastic properties of composite materials. Rev Geophys 14:541–563CrossRefGoogle Scholar
  100. Webb SL, Jackson I, Gerald JF (1988) High-pressure elasticity, shear-mode softening and polymorphism in MnO. Phys Earth Planet Interior 52(1–2):117–131CrossRefGoogle Scholar
  101. Whitfield CH, Brody EM, Bassett WA (1976) Elastic moduli of NaCl by Brillouin scattering at high pressure in a diamond anvil cell. Rev Sci Instrum 47:942–947CrossRefGoogle Scholar
  102. Yang J, Mao Z, Lin JF, Prakapenka VB (2014) Single-crystal elasticity of the deep-mantle magnesite at high pressure and temperature. Earth Planet Sci Lett 392:292–299CrossRefGoogle Scholar
  103. Yoneda A (1990) Pressure derivatives of elastic constants of single crystal MgO and MgAl2O4. J Phys Earth 38(1):19–55CrossRefGoogle Scholar
  104. Zhang JS, Bass JD (2016) Sound velocities of olivine at high pressures and temperatures and the composition of Earth’s upper mantle. Geophys Res Lett 43(18):9611–9618CrossRefGoogle Scholar
  105. Zhang J, Reeder RJ (1999) Comparative compressibilities of calcite-structure carbonates: deviations from empirical relations. Am Miner 84:861–870CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Earth SciencesSapienza University of RomeRomeItaly
  2. 2.Helmholtz Centre Potsdam-GFZ German Research Centre for GeosciencesPotsdamGermany

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