镁铝尖晶石声子色散关系及热力学性质的第一性原理研究
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摘要
镁铝尖晶石是地幔和类地行星中常见的氧化物类矿物.高温高压实验表明尖晶石的高压相可以存在于从地壳到下地幔广泛的深度范围内.作为含铝矿物的端元组分,尖晶石可能是地幔中铝元素的主要富集矿物.尖晶石的高压相还可能是俯冲大洋板块的主要组成物质.获得尖晶石的热力学性质为计算尖晶石的相变奠定了基础,为通过地震波解释地幔结构提供了理论依据.本文利用冻结声子法计算了镁铝尖晶石的声子色散关系和常压下的各种热力学性质.首先,用基于密度泛函理论的从头计算方法计算绝对零度下平衡体积的晶体结构.然后,通过构建超晶胞并对超晶胞内的原子进行微小的位移获得原子力,利用原子力来构造力常数矩阵,进一步构造动力学矩阵,将温度扩展为有限温度,计算出尖晶石的声子色散关系.最后,根据准谐近似理论得到常压下的各种热力学性质.第一布里渊区中心的声子振动频率与红外、拉曼光谱的实验数据相吻合.各种热力学性质,标准熵、等压热容和热膨胀系数与实验值相吻合.
The phonon dispersion relations of magnesium aluminium spinel MgAl2O4are calculated by using frozen phonon method.MgAl2O4 is the dominant component of the spinel phase minerals on the Earth.It is also an important rock-forming mineral,which constitutes peridotites originating from the uppermost part of the Earth's mantle.As MgAl2O4 is an end composition of the alumina-rich mineral,its high-pressure polymorphs may be the most dominant host of aluminium element in the mantle.It is very important to understand the stability and phase transitions of MgAl2O4.The calculation of the vibrational frequencies of the crystalline lattice is fundamental to the understanding of phase stability and phase transitions and of the thermodynamics of crystalline materials.The methods for first principle calculations of phonon frequencies fall into two broad classes: the linear response approach and the frozen phonon approach.Since the frozen phonon approach is easy to be implemented into any state-of-the-art electron structure calculation software,we have chosen this method to investigate the thermodynamic properties of MgAl2O4.The force constants are calculated via a self-consistent supercell approach in terms of the Hellmann-Feynman forces induced by the small displacement of a single atom in the supercell.From the forces calculated via the Hellmann-Feynman theorem,certain elements of the force constant matrix can be evaluated.By displacing entire planes of atoms,interplanar force constants may be calculated.This force constants approach is based on periodically repeated supercell.The amplitude of atomic displacements is limited to 0.02.One limitation of this method is that the LO-TO splitting of polar materials can not be calculated directly.But it is proved that this limitation has little effect on thermodynamic properties calculation.The calculated phonon dispersion spectrum is almost the same as the previous theoretical study.The phonon frequencies at the center of Brillion zone are in very good agreement with Raman and Infrared data.Within the quasi-harmonic approximation,some thermodynamic properties of magnesium aluminium spinel at ambient pressure,such as the specific heat capacity at constant pressure,thermal expansion coefficient and standard entropy,are calculated which all agree well with experimental results.
The phonon dispersion relations of magnesium aluminium spinel MgAl2O4are calculated by using frozen phonon method.MgAl2O4 is the dominant component of the spinel phase minerals on the Earth.It is also an important rock-forming mineral,which constitutes peridotites originating from the uppermost part of the Earth's mantle.As MgAl2O4 is an end composition of the alumina-rich mineral,its high-pressure polymorphs may be the most dominant host of aluminium element in the mantle.It is very important to understand the stability and phase transitions of MgAl2O4.The calculation of the vibrational frequencies of the crystalline lattice is fundamental to the understanding of phase stability and phase transitions and of the thermodynamics of crystalline materials.The methods for first principle calculations of phonon frequencies fall into two broad classes: the linear response approach and the frozen phonon approach.Since the frozen phonon approach is easy to be implemented into any state-of-the-art electron structure calculation software,we have chosen this method to investigate the thermodynamic properties of MgAl2O4.The force constants are calculated via a self-consistent supercell approach in terms of the Hellmann-Feynman forces induced by the small displacement of a single atom in the supercell.From the forces calculated via the Hellmann-Feynman theorem,certain elements of the force constant matrix can be evaluated.By displacing entire planes of atoms,interplanar force constants may be calculated.This force constants approach is based on periodically repeated supercell.The amplitude of atomic displacements is limited to 0.02.One limitation of this method is that the LO-TO splitting of polar materials can not be calculated directly.But it is proved that this limitation has little effect on thermodynamic properties calculation.The calculated phonon dispersion spectrum is almost the same as the previous theoretical study.The phonon frequencies at the center of Brillion zone are in very good agreement with Raman and Infrared data.Within the quasi-harmonic approximation,some thermodynamic properties of magnesium aluminium spinel at ambient pressure,such as the specific heat capacity at constant pressure,thermal expansion coefficient and standard entropy,are calculated which all agree well with experimental results.
引文
[1]Irifune T,Naka H,Sanehira T,et al.In situ X(ray observations of phase transitions inMgAl2O4spinel to 40 GPa using multianvil ap-paratus with sintered diamond anvils.Physicsand Chemistry of Minerals,2002,29:645~654.
[2]Ono S,Kikegawa T,Ohishi Y.The stabilityand compressibility of MgAl2O4high-pressurepolymorphs.Physics and Chemistry of Miner-als,2006,33:200~206.
[3]Irifune T,Ringwood A E.Phase transforma-tions in subducted oceanic crust and buoyancyrelationships at depths of 600-800 kmin themantle.Earth and Planetary Science Letters,1993,117:101~110.
[4]Kesson S E,FitzGerald J D,Shelley J M.Min-eral chemistry and density of subducted basalticcrust at lower-mantle pressures.Nature,1994,372:767~769.
[5]Wang D,Zhou H Q,Zhang W X,et al.Pa-rameterization of Fovce Field of Chromite fromDensity Functional Theory Data.Journal ofNanjing University(Natural Sciences),2007,43(5):520~525.(王迪,周会群,张文轩等.铬铁矿的分子动力学势能参数反演.南京大学学报(自然科学),2007,43(5):520~525).
[6]Gillan M J,AlfèD,Brodholt J,et al.First-principles modeling of Earth and planetary ma-terials at high pressures and temperatures.Re-ports on Progress in Physics,2006,69:2365~2441.
[7]Baroni S,Gironcoli S D,Dal Corso A,et al.Plane wave self consistent field codes.http://www.pwscf.org,2008-6-3.
[8]Perdew J P.Unified theory of exchange andcorrelation beyond the local density approxi ma-tion.Ziesche P,Esching H.Electronic Struc-ture of Solids’91.Berlin:Akademic Verlag,1991,11~20.
[9]Vanderbilt D.Soft self-consistent pseudopoten-tials in a generalized eigenvalue formalism.Physical Review B,1990,41:7892~7895.
[10]Monkhorst H J,Pack J D.Special points forBrillouin-zone integrations.Physical Review B,1976,13:5188~5192.
[11]Togo A.fropho:Frozen Phonon analyzer forperiodic boundary condition materials.http://fropho.sourceforge.net/doc/index.ht ml,2008-6-3.
[12]Murnaghan F D.The compressibility of mediaunder extreme pressures.Proceedings of theNational Academy of Sciences of the UnitedStates of America,1944,30:244~247.
[13]Finger L W,Hazen R M,Hof meister A M.High-pressure crystal chemistry of spinel(MgAl2O4)and magnetite(Fe3O4):Compari-sons with silicate spinels.Physics and Chemis-try of Minerals,1986,13:215~220.
[14]Levy D,Pavese A,Hanfland M.SyntheticMgAl2O4(spinel)at high-pressure conditions(0.0001-30 GPa):A synchrotron X-raypowder diffraction study.American Mineralo-gist,2003,88:93~98.
[15]Tropf WJ,Thomas M E,Haris T J.Proper-ties of crystals and glasses:Handbook of Op-tics,Devises,Measurements and PropertiesSecond Edition.New-York:McGraw-Hill,1995,Chapter 33.
[16]Cynn H,Sharma S K,Cooney T F,et al.High-temperature Raman investigation of or-der-disorder behavior in the MgAl2O4spinel.Physical Review B,1992,45:500~502.
[17]Cynn H,Anderson O L,Nicol M.Effects ofcation disordering in a natural MgAl2O4spinelobserved by rectangular parallelepiped ultrason-ic resonance and Raman measurements.Pureand Applied Geophysics,1993,141:415~444.
[18]Chopelas A,Hof meister A M.Vibrationalspectroscopy aluminate spinels and of MgAl2O4to over 200 kbar.Physics and Chemistry ofMinerals,1991,18:279~293.
[19]Ishii M,Hiraishi J,Yamanaka T.Structureand lattice vibrations of Mg-Al spinel solid so-lution.Physics and Chemistry of Minerals,1982,8:64~68.
[20]Striefler ME,Boldish S I.Transverse and lon-gitudinal optic frequencies of spinel MgAl2O4.Journal of Physics C:Solid State Physics,1978,11:237~241.
[21]Fraas L M,Moore J E,Salzberg J B.Ramancharacterization studies of synthetic and naturalMgAl2O4crystals.Journal of Chemical Phys-ics,1973,58:3585.
[22]O’Horo MP,Frisillo A L,White WB.Latticevibrations of MgAl2O4spinel.Journal of Phys-ics and Chemistry of Solids,1973,34:23~28.
[23]Preudhomme J,Tarte P.Infrared studies ofspinels—Ⅲ:The normalⅡ—Ⅲspinels.Spec-trochi mica Acta Part A.1971,27:1817~1835.
[24]de Wijs G A,Fang C M,Kresse G,et al.First-principle calculation of the phonon spec-trum of spinel.Physical Review B,2002,65:094305.
[25]Robie R A,Hemingway B S,Fisher J R.Ther-modynamic properties of minerals and relatedsubstances at 298.15 Kand 1 bar and at highertemperatures.U.S.Geological Survey Bulle-tin,1978,1452:456.
[26]Klemme S,Ahrens M.Low-temperature heatcapacities of MgAl2O4and spinels of theMgCr2O4-MgAl2O4solid solution.Physicsand Chemistry of Minerals,2007,34:59~72.
[27]Touloukian Y S,Kirby R K,Taylor R E,etal.Thermal expansion,Nonmetallic Solids:Thermophysical Properties of Matter.Plenum:New York-Washington,1977,1658.
[2]Ono S,Kikegawa T,Ohishi Y.The stabilityand compressibility of MgAl2O4high-pressurepolymorphs.Physics and Chemistry of Miner-als,2006,33:200~206.
[3]Irifune T,Ringwood A E.Phase transforma-tions in subducted oceanic crust and buoyancyrelationships at depths of 600-800 kmin themantle.Earth and Planetary Science Letters,1993,117:101~110.
[4]Kesson S E,FitzGerald J D,Shelley J M.Min-eral chemistry and density of subducted basalticcrust at lower-mantle pressures.Nature,1994,372:767~769.
[5]Wang D,Zhou H Q,Zhang W X,et al.Pa-rameterization of Fovce Field of Chromite fromDensity Functional Theory Data.Journal ofNanjing University(Natural Sciences),2007,43(5):520~525.(王迪,周会群,张文轩等.铬铁矿的分子动力学势能参数反演.南京大学学报(自然科学),2007,43(5):520~525).
[6]Gillan M J,AlfèD,Brodholt J,et al.First-principles modeling of Earth and planetary ma-terials at high pressures and temperatures.Re-ports on Progress in Physics,2006,69:2365~2441.
[7]Baroni S,Gironcoli S D,Dal Corso A,et al.Plane wave self consistent field codes.http://www.pwscf.org,2008-6-3.
[8]Perdew J P.Unified theory of exchange andcorrelation beyond the local density approxi ma-tion.Ziesche P,Esching H.Electronic Struc-ture of Solids’91.Berlin:Akademic Verlag,1991,11~20.
[9]Vanderbilt D.Soft self-consistent pseudopoten-tials in a generalized eigenvalue formalism.Physical Review B,1990,41:7892~7895.
[10]Monkhorst H J,Pack J D.Special points forBrillouin-zone integrations.Physical Review B,1976,13:5188~5192.
[11]Togo A.fropho:Frozen Phonon analyzer forperiodic boundary condition materials.http://fropho.sourceforge.net/doc/index.ht ml,2008-6-3.
[12]Murnaghan F D.The compressibility of mediaunder extreme pressures.Proceedings of theNational Academy of Sciences of the UnitedStates of America,1944,30:244~247.
[13]Finger L W,Hazen R M,Hof meister A M.High-pressure crystal chemistry of spinel(MgAl2O4)and magnetite(Fe3O4):Compari-sons with silicate spinels.Physics and Chemis-try of Minerals,1986,13:215~220.
[14]Levy D,Pavese A,Hanfland M.SyntheticMgAl2O4(spinel)at high-pressure conditions(0.0001-30 GPa):A synchrotron X-raypowder diffraction study.American Mineralo-gist,2003,88:93~98.
[15]Tropf WJ,Thomas M E,Haris T J.Proper-ties of crystals and glasses:Handbook of Op-tics,Devises,Measurements and PropertiesSecond Edition.New-York:McGraw-Hill,1995,Chapter 33.
[16]Cynn H,Sharma S K,Cooney T F,et al.High-temperature Raman investigation of or-der-disorder behavior in the MgAl2O4spinel.Physical Review B,1992,45:500~502.
[17]Cynn H,Anderson O L,Nicol M.Effects ofcation disordering in a natural MgAl2O4spinelobserved by rectangular parallelepiped ultrason-ic resonance and Raman measurements.Pureand Applied Geophysics,1993,141:415~444.
[18]Chopelas A,Hof meister A M.Vibrationalspectroscopy aluminate spinels and of MgAl2O4to over 200 kbar.Physics and Chemistry ofMinerals,1991,18:279~293.
[19]Ishii M,Hiraishi J,Yamanaka T.Structureand lattice vibrations of Mg-Al spinel solid so-lution.Physics and Chemistry of Minerals,1982,8:64~68.
[20]Striefler ME,Boldish S I.Transverse and lon-gitudinal optic frequencies of spinel MgAl2O4.Journal of Physics C:Solid State Physics,1978,11:237~241.
[21]Fraas L M,Moore J E,Salzberg J B.Ramancharacterization studies of synthetic and naturalMgAl2O4crystals.Journal of Chemical Phys-ics,1973,58:3585.
[22]O’Horo MP,Frisillo A L,White WB.Latticevibrations of MgAl2O4spinel.Journal of Phys-ics and Chemistry of Solids,1973,34:23~28.
[23]Preudhomme J,Tarte P.Infrared studies ofspinels—Ⅲ:The normalⅡ—Ⅲspinels.Spec-trochi mica Acta Part A.1971,27:1817~1835.
[24]de Wijs G A,Fang C M,Kresse G,et al.First-principle calculation of the phonon spec-trum of spinel.Physical Review B,2002,65:094305.
[25]Robie R A,Hemingway B S,Fisher J R.Ther-modynamic properties of minerals and relatedsubstances at 298.15 Kand 1 bar and at highertemperatures.U.S.Geological Survey Bulle-tin,1978,1452:456.
[26]Klemme S,Ahrens M.Low-temperature heatcapacities of MgAl2O4and spinels of theMgCr2O4-MgAl2O4solid solution.Physicsand Chemistry of Minerals,2007,34:59~72.
[27]Touloukian Y S,Kirby R K,Taylor R E,etal.Thermal expansion,Nonmetallic Solids:Thermophysical Properties of Matter.Plenum:New York-Washington,1977,1658.
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