Skip to main content
SpringerLink
Log in

Thermodynamics and phonon dispersion of pyrope and grossular silicate garnets from ab initio simulations

  • Original Paper
  • Published:
Physics and Chemistry of Minerals Aims and scope Submit manuscript

Abstract

The phonon dispersion and thermodynamic properties of pyrope (\(\hbox {Mg}_3\hbox {Al}_2\hbox {Si}_3\hbox {O}_{12}\)) and grossular (\(\hbox {Ca}_3\hbox {Al}_2\hbox {Si}_3\hbox {O}_{12}\) ) have been computed by using an ab initio quantum mechanical approach, an all-electron variational Gaussian-type basis set and the B3LYP hybrid functional, as implemented in the Crystal program. Dispersion effects in the phonon bands have been simulated by using supercells of increasing size, containing 80, 160, 320, 640, 1280 and 2160 atoms, corresponding to 1, 2, 4, 8, 16 and 27 \(\mathbf {k}\) points in the first Brillouin zone. Phonon band structures, density of states and corresponding inelastic neutron scattering spectra are reported. Full convergence of the various thermodynamic properties, in particular entropy (S) and specific heat at constant volume (\(C_\mathrm{{V}}\)), with the number of \(\mathbf {k}\) points is achieved with 27 \(\mathbf {k}\) points. The very regular behavior of the S(T) and \(C_\mathrm{{V}}(T)\) curves as a function of the number of \(\mathbf {k}\) points, determined by high numerical stability of the code, permits extrapolation to an infinite number of \(\mathbf {k}\) points. The limiting value differs from the 27-\(\mathbf {k}\) case by only 0.40 % at 100 K for S (the difference decreasing to 0.11 % at 1000 K) and by 0.29 % (0.05 % at 1000 K) for \(C_\mathrm{{V}}\). The agreement with the experimental data is rather satisfactory. We also address the problem of the relative entropy of pyrope and grossular, a still debated question. Our lattice dynamical calculations correctly describe the larger entropy of pyrope than grossular by taking into account merely vibrational contributions and without invoking “static disorder” of the Mg ions in dodecahedral sites. However, as the computed entropy difference is found to be larger than the experimental one by a factor of 2–3, present calculations cannot exclude possible thermally induced structural changes, which could lead to further conformational contributions to the entropy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Alfè D (2009) PHON: a program to calculate phonons using the small displacement method. Comput Phys Comm 180(12):2622–2633

    Article  Google Scholar 

  • Anderson DL (1986) Transition region of the earth’s upper mantle. Nature 320:321–328

    Article  Google Scholar 

  • Anderson DL (1989) Theory of the earth. Blackwell Scientific Publications, Boston

    Google Scholar 

  • Artioli G, Pavese A, Moze O (1996) Dispersion relations of acoustic phonons in pyrope garnet; relationship between vibrational properties and elastic constants. Am Mineral 81(1–2):19–25

    Article  Google Scholar 

  • Baroni S, de Gironcoli S, Corso AD, Giannozzi P (2001) Phonons and related crystal properties from density-functional perturbation theory. Rev Mod Phys 73:515

    Article  Google Scholar 

  • Becke AD (1993) Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys 98(7):5648–5652

    Article  Google Scholar 

  • Broyden CG (1970) The convergence of a class of double-rank minimization algorithms 1. General considerations. J Inst Math Appl 6:76–90

    Article  Google Scholar 

  • Bush I, Tomić S, Searle B, Mallia G, Bailey C, Montanari B, Bernasconi L, Carr J, Harrison N (2011) Parallel implementation of the ab initio CRYSTAL program: electronic structure calculations for periodic systems. Proc R Soc A Math Phys Eng Sci 467:2112–2126

    Article  Google Scholar 

  • Chaplin T, Price DG, Ross NL (1998) Computer simulation of the infrared and Raman activity of pyrope garnet, and assignment of calculated modes to specific atomic motions. Am Mineral 83:841

    Article  Google Scholar 

  • Chopelas A (2006) Modeling the thermodynamic parameters of six endmember garnets at ambient and high pressures from vibrational data. Phys Chem Miner 33(6):363–376

    Article  Google Scholar 

  • Civalleri B, D’Arco P, Orlando R, Saunders VR, Dovesi R (2001) Hartree–Fock geometry optimization of periodic system with the CRYSTAL code. Chem Phys Lett 348:131

    Article  Google Scholar 

  • Cressey G (1981) Entropies and enthalpies of aluminosilicate garnets. Contrib Mineral Petrol 76(4):413–419

    Article  Google Scholar 

  • De La Pierre M, Orlando R, Maschio L, Doll K, Ugliengo P, Dovesi R (2011) Performance of six functionals (LDA, PBE, PBESOL, B3LYP, PBE0, and WC1LYP) in the simulation of vibrational and dielectric properties of crystalline compounds. The case of forsterite \(\text{ Mg }_2\text{ SiO }_4\). J Comput Chem 32(9):1775–1784

    Article  Google Scholar 

  • De La Pierre M, Orlando R, Ferrabone M, Zicovich-Wilson CM, Dovesi R (2014) Exploitation of symmetry in periodic self-consistent-field ab initio calculations: application to large three-dimensional compounds. Sci China Chem 57(10):1418–1426

    Article  Google Scholar 

  • Demichelis R, Civalleri B, D’Arco P, Dovesi R (2010a) Performance of 12 DFT functionals in the study of crystal systems: Al2SiO5 orthosilicates and Al hydroxides as a case study. Int J Quantum Chem 110(12):2260–2273

    Article  Google Scholar 

  • Demichelis R, Civalleri B, Ferrabone M, Dovesi R (2010b) On the performance of eleven DFT functionals in the description of the vibrational properties of aluminosilicates. Int J Quantum Chem 110(2):406–415

    Article  Google Scholar 

  • Doll K (2001) Implementation of analytical Hartree–Fock gradients for periodic systems. Comput Phys Commun 137:74

    Article  Google Scholar 

  • Doll K, Harrison NM, Saunders VR (2001) Analytical Hartree–Fock gradients for periodic systems. Int J Quantum Chem 82:1

    Article  Google Scholar 

  • Dove MT (1993) Introduction to lattice dynamics. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Dovesi R, De La Pierre M, Ferrari AM, Pascale F, Maschio L, Zicovich-Wilson CM (2011) The IR vibrational properties of six members of the garnet family: a quantum mechanical ab initio study. Am Mineral 96(11–12):1787–1798

    Article  Google Scholar 

  • Dovesi R, Orlando R, Erba A, Zicovich-Wilson CM, Civalleri B, Casassa S, Maschio L, Ferrabone M, De La Pierre M, D’Arco P, Nol Y, Caus M, Rrat M, Kirtman B (2014a) CRYSTAL14: a program for the ab initio investigation of crystalline solids. Int J Quantum Chem 114(19):1287–1317

    Article  Google Scholar 

  • Dovesi R, Saunders VR, Roetti C, Orlando R, Zicovich-Wilson CM, Pascale F, Doll K, Harrison NM, Civalleri B, Bush IJ, D’Arco P, Llunell M, Causà M, Noël Y (2014b) CRYSTAL14 User’s Manual. Università di Torino, Torino. http://www.crystal.unito.it

  • Erba A (2014) On combining temperature and pressure effects on structural properties of crystals with standard ab initio techniques. J Chem Phys 141(124):115

    Google Scholar 

  • Erba A, Ferrabone M, Orlando R, Dovesi R (2013) Accurate dynamical structure factors from ab initio lattice dynamics: the case of crystalline silicon. J Comput Chem 34:346

    Article  Google Scholar 

  • Erba A, Mahmoud A, Belmonte D, Dovesi R (2014a) High pressure elastic properties of minerals from ab initio simulations: the case of pyrope, grossular and andradite silicate garnets. J Chem Phys 140(124):703

    Google Scholar 

  • Erba A, Mahmoud A, Orlando R, Dovesi R (2014b) Elastic properties of six silicate garnet end members from accurate ab initio simulations. Phys Chem Miner 41(2):151–160

    Article  Google Scholar 

  • Erba A, Mahmoud A, Orlando R, Dovesi R (2014c) Erratum to: elastic properties of six silicate garnet end-members from accurate ab initio simulations. Phys Chem Miner 41:161–162

    Article  Google Scholar 

  • Erba A, Maul J, De La Pierre M, Dovesi R (2015a) Structural and elastic anisotropy of crystals at high pressure and temperature from quantum-mechanical methods: the case of Mg\(_2\)SiO\(_4\) forsterite. J Chem Phys 142(204):502

    Google Scholar 

  • Erba A, Maul J, Demichelis R, Dovesi R (2015b) Assessing thermochemical properties of materials through ab initio quantum-mechanical methods: the case of \(\alpha -\text{ Al }_2\text{ O }_3\). Phys Chem Chem Phys 17:11670–11677

    Article  Google Scholar 

  • Erba A, Maul J, Itou M, Dovesi R, Sakurai Y (2015c) Anharmonic thermal oscillations of the electron momentum distribution in lithium fluoride. Phys Rev Lett 115:117402

    Article  Google Scholar 

  • Erba A, Navarrete-López AM, Lacivita V, D’Arco P, Zicovich-Wilson CM (2015d) Katoite under pressure: an ab initio investigation of its structural, elastic and vibrational properties sheds light on the phase transition. Phys Chem Chem Phys 17:2660–2669

    Article  Google Scholar 

  • Erba A, Shahrokhi M, Moradian R, Dovesi R (2015e) On how differently the quasi-harmonic approximation works for two isostructural crystals: thermal properties of periclase and lime. J Chem Phys 142(044):114

    Google Scholar 

  • Fletcher R (1970) A new approach to variable metric algorithms. Comput J 13(3):317–322

    Article  Google Scholar 

  • Geiger CA (2013) Static disorders of atoms and experimental determination of Debye temperature in pyrope: low-and high-temperature single-crystal X-ray diffraction study—discussion. Am Mineral 98:780–782

    Article  Google Scholar 

  • Gibbs GV, Smith JV (1965) Refinement of the crystal structure of synthetic pyrope. Am Mineral 50:2032–2039

    Google Scholar 

  • Goldfarb D (1970) A family of variable-metric methods derived by variational means. Math Comput 24:23–26

    Article  Google Scholar 

  • Gonze X, Lee C (1997) Dynamical matrices, born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory. Phys Rev B 55(10):355

    Google Scholar 

  • Gonze X, Charlier JC, Allan D, Teter M (1994) Interatomic force constants from first principles: the case of \(\alpha\)-quartz. Phys Rev B 50:13035–13038

    Article  Google Scholar 

  • Gramaccioli CM, Pilati T (2003) Interpretation of single-crystal vibrational spectra and entropy of pyrope and almandine using a rigid-ion lattice-dynamical model. J Phys Chem A 107(22):4360–4366

    Article  Google Scholar 

  • Haselton H Jr, Westrum E Jr (1980) Low-temperature heat capacities of synthetic pyrope, grossular, and \(\text{ pyrope }_{60} \text{ grossular }_{40}\). Geochim Cosmochim Acta 44(5):701–709

    Article  Google Scholar 

  • Hofmeister AM, Chopelas A (1991) Thermodynamic properties of pyrope and grossular from vibrational spectroscopy. Am Mineral 76(5–6):880–891

    Google Scholar 

  • Hudson BS (2001) Inelastic neutron scattering: a tool in molecular vibrational spectroscopy and a test of ab initio methods. J Phys Chem A 105(16):3949–3960

    Article  Google Scholar 

  • Kieffer SW (1979) Thermodynamics and lattice vibrations of minerals: 3. Lattice dynamics and an approximation for minerals with application to simple substances and framework silicates. Rev Geophys 17(1):35–59

    Article  Google Scholar 

  • Kolesov BA, Geiger CA (1998) Raman spectra of silicate garnets. Phys Chem Min 25(2):142–151

    Article  Google Scholar 

  • Kolesov BA, Geiger CA (2000) Low-temperature single-crystal Raman spectrum of pyrope. Phys Chem Min 27(9):645–649

    Article  Google Scholar 

  • Krupka KM, Robie RA, Hemingway BS (1979) High-temperature heat capacities of corundum, periclase, anorthite, \(\text{ CaAl }_2\text{ Si }_2\text{ O }_8\) glass, muscovite, pyrophyllite, KAlSi\(_3\)O\(_8\) glass, grossular, and \(\text{ NaAlSi }_3\text{ O }_8\) glass. Am Mineral 64(1–2):86–101

    Google Scholar 

  • Lacivita V, Erba A, Dovesi R, D’Arco P (2014) Elasticity of grossular–andradite solid solution: an ab initio investigation. Phys Chem Chem Phys 16:15331–15338

    Article  Google Scholar 

  • Lacivita V, Mahmoud A, Erba A, D’Arco P, Mustapha S (2015) Hydrogrossular, \(\text{ Ca }_3\text{ Al }_2(\text{ SiO }_4)_{3-x}(\text{ H }_4\text{ O }_4)_{x}\): an ab initio investigation of its structural and energetic properties. Am Mineral. doi:10.2138/am-2015-5334

    Google Scholar 

  • Lucas MS, Kresch M, Stevens R, Fultz B (2008) Phonon partial densities of states and entropies of Fe and Cr in bcc Fe–Cr from inelastic neutron scattering. Phys Rev B 77(184):303

    Google Scholar 

  • Mahmoud A, Erba A, Doll K, Dovesi R (2014) Pressure effect on elastic anisotropy of crystals from ab initio simulations: the case of silicate garnets. J Chem Phys 140(234):703

    Google Scholar 

  • Maschio L, Ferrabone M, Meyer A, Garza J, Dovesi R (2011) The infrared spectrum of spessartine \(\text{ Mn }_3\text{ Al }_2\text{ Si }_3\text{ O }_{12}\): An ab initio all electron simulation with five different functionals (LDA, PBE, PBESOL, B3LYP and PBE0). Chem Phys Lett 501(4):612–618

    Article  Google Scholar 

  • Maul J, Erba A, Santos IMG, Sambrano JR, Dovesi R (2015) In silico infrared and Raman spectroscopy under pressure: The case of \(\text{ CaSnO }_3\) perovskite. J Chem Phys 142(1):014505

    Article  Google Scholar 

  • Mittal R, Chaplot SL, Choudhury N (2001) Lattice dynamics calculations of the phonon spectra and thermodynamic properties of the aluminosilicate garnets pyrope, grossular, and spessartine M3Al2Si3O12 \((M=\text{ Mg },\) Ca, and Mn). Phys Rev B 64(094):302

    Google Scholar 

  • Orlando R, Delle Piane M, Bush IJ, Ugliengo P, Ferrabone M, Dovesi R (2012) A new massively parallel version of CRYSTAL for large systems on high performance computing architectures. J Comput Chem 33(28):2276–2284

    Article  Google Scholar 

  • Orlando R, De La Pierre M, Zicovich-Wilson CM, Erba A, Dovesi R (2014) On the full exploitation of symmetry in periodic (as well as molecular) self-consistent-field ab initio calculations. J Chem Phys 141(10):104108

    Article  Google Scholar 

  • Osborn R, Goremychkin EA, Kolesnikov AI, Hinks DG (2001) Phonon density of states in \(\text{ MgB }_{2}\). Phys Rev Lett 87(017):005

    Google Scholar 

  • Ottonello G, Bokreta M, Sciuto PF (1996) Parametrization of energy and interactions in garnets: end-member properties. Am Mineral 81:429–447

    Article  Google Scholar 

  • Parlinski K, Li ZQ, Kawazoe Y (1997) First-principles determination of the soft mode in cubic \(\text{ ZrO }_{2}\). Phys Rev Lett 78:4063–4066

    Article  Google Scholar 

  • Pascale F, Zicovich-Wilson CM, Gejo FL, Civalleri B, Orlando R, Dovesi R (2004) The calculation of the vibrational frequencies of the crystalline compounds and its implementation in the CRYSTAL code. J Comput Chem 25:888–897

    Article  Google Scholar 

  • Pascale F, Zicovich-Wilson CM, Orlando R, Roetti C, Ugliengo P, Dovesi R (2005) Vibration frequencies of \(\text{ Mg }_3\text{ Al }_2\text{ Si }_3\text{ O }_{12}\) pyrope. An ab initio study with the CRYSTAL code. J Phys Chem B 109(13):6146–6152

    Article  Google Scholar 

  • Pavese A, Artioli G, Prencipe M (1995) X-ray single-crystal diffraction study of pyrope in the temperature range 30–973 K. Am Mineral 80(5–6):457–464

    Article  Google Scholar 

  • Pavese A, Artioli G, Moze O (1998) Inelastic neutron scattering from pyrope powder: experimental data and theoretical calculations. Eur J Mineral 10(1):59–70

    Article  Google Scholar 

  • Pilati T, Demartin F, Gramaccioli CM (1996) Atomic displacement parameters for garnets: a lattice-dynamical evaluation. Acta Crystallogr Sect B 52(2):239–250

    Article  Google Scholar 

  • Sears V (1992) Neutron scattering lengths and cross sections. Neutron News 3(3):29–37

    Article  Google Scholar 

  • Shanno DF (1970) Conditioning of quasi-Newton methods for function minimization. Math Comput 24:647–656

    Article  Google Scholar 

  • Téqui C, Robie RA, Hemingway BS, Neuville DR, Richet P (1991) Melting and thermodynamic properties of pyrope (\(\text{ Mg }_3\text{ Al }_2\text{ Si }_3\text{ O }_{12}\)). Geochim Cosmochim Acta 55(4):1005–1010

    Article  Google Scholar 

  • Togo A, Oba F, Tanaka I (2008) First-principles calculations of the ferroelastic transition between rutile-type and CaCl2-type SiO2 at high pressures. Phys Rev B 78(134):106

    Google Scholar 

  • Ungaretti L, Leona M, Merli M, Oberti R (1995) Non-ideal solid-solution in garnet; crystal-structure evidence and modelling. Eur J Mineral 7:1299–1312

    Article  Google Scholar 

  • Wallace D (1972) Thermodynamics of crystals. Wiley, New York

    Google Scholar 

  • Wang Y, Wang J, Wang Y, Mei ZG, Shang SL, Chen LQ, Liu ZK (2010) A mixed-space approach to first-principles calculations of phonon frequencies for polar materials. J Phys Condens Matter 22(202):201

    Google Scholar 

  • Wentzcovitch RM, Yu YG, Wu Z (2010) Thermodynamic properties and phase relations in mantle minerals investigated by first principles quasiharmonic theory. Rev Mineral Geochem 71:59–98

    Article  Google Scholar 

  • Winkler B, Milman V, Akhmatskaya EV, Nobes RH (2000) Bonding and dynamics of Mg in pyrope: a theoretical investigation. Am Mineral 85(3–4):608–612

    Article  Google Scholar 

  • Yu YG, Wentzcovitch RM, Vinograd VL, Angel RJ (2011) Thermodynamic properties of \(\text{ MgSiO }_3\) majorite and phase transitions near 660 km depth in \(\text{ MgSiO }_3\) and \(\text{ Mg }_2\text{ SiO }_4\): a first principles study. J Geophys Res 116(B02):208

    Google Scholar 

  • Zhao J, Gaskell P, Cormier L, Bennington S (1997) Vibrational density of states and structural origin of the heat capacity anomalies in \(\text{ Ca }_3\text{ Al }_2\text{ Si }_3\text{ O }_{12}\) glasses. Phys B 241:906–908

    Article  Google Scholar 

  • Zicovich-Wilson CM, Pascale F, Roetti C, Orlando VRSR, Dovesi R (2004) The calculation of the vibration frequencies of \(\alpha\)-quartz: the effect of Hamiltonian and basis set. J Comput Chem 25:1873–1881

    Article  Google Scholar 

  • Zicovich-Wilson CM, Torres FJ, Pascale F, Valenzano L, Orlando R, Dovesi R (2008) Ab initio simulation of the IR spectra of pyrope, grossular, and andradite. J Comput Chem 29(13):2268–2278

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Chimica and Centre of Excellence NIS (Nanostructured Interfaces and Surfaces), Università di Torino, Via P. Giuria 5, 10125, Turin, Italy

    Jacopo Baima, Matteo Ferrabone, Roberto Orlando, Alessandro Erba & Roberto Dovesi

Authors
  1. Jacopo Baima
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matteo Ferrabone
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Roberto Orlando
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Alessandro Erba
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Roberto Dovesi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jacopo Baima.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 2173 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baima, J., Ferrabone, M., Orlando, R. et al. Thermodynamics and phonon dispersion of pyrope and grossular silicate garnets from ab initio simulations. Phys Chem Minerals 43, 137–149 (2016). https://doi.org/10.1007/s00269-015-0781-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00269-015-0781-6

Keywords

Search

Navigation