DONG Lian-Ke, Lü Guo-Hao, WANG Ke-Gang, WANG Xiao-Wei. Scaling Theory and Fractal Geometry[J]. Chinese Journal of High Pressure Physics, 1990, 4(3): 187-193 . doi: 10.11858/gywlxb.1990.03.004
Citation: LI Peiyun, HUANG Haijun, LI Yanli. First-Principles Calculations of the Equation of State and Sound Velocity of Fe-3.24%Si: Implications for the Composition of Earth’s Inner Core[J]. Chinese Journal of High Pressure Physics, 2019, 33(6): 060101. doi: 10.11858/gywlxb.20190781

First-Principles Calculations of the Equation of State and Sound Velocity of Fe-3.24%Si: Implications for the Composition of Earth’s Inner Core

doi: 10.11858/gywlxb.20190781
  • Received Date: 22 May 2019
  • Rev Recd Date: 26 Jun 2019
  • Issue Publish Date: 25 Sep 2019
  • Silicon (Si) is considered as one major light element in Earth’s inner core, but its content is still controversy. In order to constrain its content in the inner core, using first-principles calculation method, we constructed four different supercells of Fe-3.24%Si and investigated the effects of cell size and spin on geometry optimization. It is found that the spin doesn’t affect the equation of state of Fe-3.24%Si above 100 GPa, and below 100 GPa, the calculated results with the spin are closer to the experimental data. Based on the equation of state, the sound velocity at 0 K and the corresponding thermodynamic parameters, the density and sound velocity of Fe-3.24%Si are obtained under the conditions of the inner core. The density of Fe-3.24%Si is lower than that of pure iron and slightly higher than that of the inner core. The sound velocities of longitudinal wave and shear wave for Fe-3.24%Si are very close to that of pure iron, but both are significantly higher than that of the inner core. Therefore, we could exclude the possibility that Earth’s inner core contains a large amount of Si.

     

  • 随着科学技术的发展,人们越来越关注材料在高温、高应变率等极端环境下的动态力学行为[1-3]。分离式Hopkinson压杆(Split Hopkinson Pressure Bar,SHPB)作为一种高应变率(102~104 s-1)加载装置[4-5],广泛应用于材料的动态力学行为研究。20世纪60年代,有学者尝试利用SHPB技术研究高温下材料的动态力学性能[6-7]。到目前为止,一般采取两种实验方案。一种是对试样与一部分压杆同时加热,然而由于入射杆和透射杆均为热的良导体,在入射杆和透射杆上将不可避免地形成温度梯度。例如:Latella等[8]基于SHPB装置,采用同时加热方式测量了2.25Gr-1Mo高强钢的杨氏模量,发现它由室温时的212.4 GPa降为600 ℃时的169.0 GPa;Lankford[9]在SHPB装置中引入快速加温技术,以减少温度在杆中的变化梯度;Gilat等[10]采用对热不敏感的铝材料制作波导杆,利用局部快速升温设备测试材料的动态高温扭转性能,试样的温度范围为650~1 060 ℃。另一种方案是将试样与入射杆及透射杆分离,将试样单独加热到预定温度,实验杆与试样接触后立即进行高应变率加载。20世纪90年代,Nemat-Nasser等[11-13]在SHPB上引入一种实现高温的方法,其特点是无需特殊的高温材料压杆,通过建立一个与撞击杆同步启动的连杆机构,实现对试样的高温-高应变率耦合作用。他们首先将试样与弹性杆分离,然后加热试样,达到预定温度后,开启空气炮,在应力波到达试样与入射杆界面时,由驱动系统推动试样与入射杆及透射杆紧密接触,并由此进行了Ta和Ta-W合金材料在1 000 ℃下的高应变率实验。Apostol等[14]提出了冷接触时间(Cold Contact Time,即从杆与试样接触到入射波到达试样端面的时间)的概念,并认为冷接触时间应控制在50~100 ms。李玉龙等[15-16]通过实验测量了不同冷接触时间下材料的动态力学性能,结果表明,冷接触时间大于50 ms时,其对实验结果有较大的影响,实验数据不能真实地反映材料在高温下的动态力学性能。最近,Kajberg等[17]利用电磁推进方式推动入射杆和透射杆同步组装,测得不锈钢和含碳钢在700~4 400 s-1的高应变率下、温度在900~1 200 ℃之间的应力-应变曲线。

    本研究设计了一种更便捷的双同步实验装置,重新设计高温炉,从而获得更高的工作温度。考虑到某些材料在高温下极易与空气中的某些气体(如氧气)发生反应,所设计的超高温加热炉可通入保护气体,在炉内形成惰性气体环境,并且配有观察窗口,可利用高速摄像机记录实验过程。采用此实验装置,测量TC4钛合金和SiC陶瓷的超高温动态力学性能,并对TC4钛合金在无氧环境下进行高温动态测试,验证实验装置在材料超高温动态力学性能测试方面的可行性。

    在SHPB的基础上重新设计了实验装置,如图 1所示。利用两个活塞组成双同步系统;利用超高温炉对试样加热;加热炉设有观察窗口和保护气通气孔,可通过通气孔将保护气通入高温炉中形成惰性气体环境,可通过观察窗口记录试样的超高温动态测试过程;发射阀同时控制气室和同步气室,用于进行同步组装和发射撞击杆。

    图  1  带有超高速摄像机的超高温双同步SHPB装置示意图
    Figure  1.  Schematic illustration of the ultra-high temperature SHPB with a double-synchronically assembled heating system and high speed camera

    为实现超高温加热环境,使用加热源为MoSi2、带智能控制装置的超高温加热炉加热试样,利用位于试样上方的热电偶测量炉内温度,超高温加热系统的工作温度最高达1 650 ℃。在加热炉前方设有观察窗,在观察口前安装石英玻璃。石英玻璃的软化温度在1 600 ℃左右,当加热温度超过石英玻璃的稳定工作温度(1 400 ℃)时,利用隔热材料将观察口堵住以保护石英玻璃。在加热炉下方及上方各设置一个通气孔,用于将保护气通入加热炉内,形成惰性气体氛围,使材料不与加热炉中的气体发生反应。试样的固定方式如图 1中插图所示,利用半圆形氧化铝陶瓷管和高温莫来石纤维泡沫块对试样进行固定,其工作温度均超过高温炉的加热温度。高温泡沫的厚度小于试样厚度,高温泡沫的强度很低,对试样的加载方向没有影响,高温泡沫使试样的其他3个面也没有约束,从而将试样的轴向约束降到最低。为了更好地固定试样,将高温胶涂于高温石棉与陶瓷管及试样的接触面上。

    原有的单同步系统只有透射杆处一个活塞,活塞推动透射杆与试样一起朝入射杆运动,使入射杆及透射杆与试样紧密接触,同时发射子弹撞击入射杆。在实验过程中,由于试样由透射杆推动并与入射杆接触,因此试样与透射杆的冷接触时间长于试样与入射杆的冷接触时间,造成试样温度分布不均匀。尤其当使用较大的加热炉时,入射杆与透射杆之间的距离较远,如果透射杆的运动速度过低,则会使透射杆与试样间的冷接触时间过长;如果透射杆的运动速度过高,则会对试样产生较大的预冲击,影响实验结果。因此单同步组装系统难以应对较大体积的加热炉,从而限制了实验环境。

    针对单同步系统的不足,重新设计了双同步系统(如图 1所示),即在入射杆和透射杆处各设置一个同步装置。实验前,利用控制器将推进气管与同步气室连接,通过发射阀同时发射撞击杆和双同步组装系统,入射杆和透射杆同步朝试样运动,撞击杆同时朝入射杆运动。在入射波到达试样端面前,通过双同步系统使入射杆和透射杆夹紧试样。实验后,由控制器将回拉气管与同步气室连接,通过活塞带动实验杆离开加热炉,避免实验杆长时间处于加热炉内而影响性能,特别是当加热炉温度达到1 600 ℃时。同时也可以看到,此系统避免了加热炉体积较大时透射杆与试样间冷接触时间过长的问题,因此可应用不同的加热炉。采用此方法,在标定同步气压并调整活塞的条件下,冷接触时间可控制在10 ms以内,实现了入射杆和透射杆同时与试样接触。利用Abaqus有限元软件建立热分析模型,发现当冷接触时间保持在10 ms以下时,双同步组装对实验中试样温度的影响不大[18]

    为观察试样在实验中的变化情况,利用高速摄像机通过高温加热炉的光学观察口拍摄实验过程,如图 2所示。可见,在实验过程中高温炉辐射出很高的热量。

    图  2  高速摄像机布局(插图为高温炉内部图像)
    Figure  2.  High speed camera (The inset shows the internal of the furnace.)

    图 3给出了根据普朗克公式计算得到的在400~1 200 ℃温度范围内热辐射能量与辐射波长之间的关系曲线。从图 3中可以看出:随着温度的升高,各波段的热辐射能量均大幅增加,热辐射总能量迅速增大;并且波长越长,辐射热量越多。为了减少辐射对设备及成像的影响,采用带滤红外波段的石英玻璃将波长较长的红外辐射光滤掉,既控制了热辐射,又保证了可见光通过观察口以满足高速摄像机的拍摄需求。

    图  3  不同温度下波长与热辐射能量的关系曲线
    Figure  3.  Wavelength vs.thermal radiation energy at different temperatures

    由于拍摄动态变形所需的相机工作帧频较高,曝光时间很短,为了获得清晰的图像,使用高强度闪光灯。考虑到一般闪光灯的照明时间非常短,且亮度随时间变化,在触发一段时间后才能达到最大亮度并稳定保持一段时间,为此闪光灯的触发时间须与相机的拍照时间配合,使动态加载过程中闪光灯的亮度保持较高水平。本研究将数据采集器的信号输出端分别与闪光灯触发器及高速相机触发端相连,数据采集器通过入射波信号触发,触发同时给闪光灯触发器和高速摄像机输出触发信号。

    以TC4钛合金和SiC陶瓷作为研究对象,进行超高温动态力学性能测试。在TC4钛合金实验中,应变率ε˙为2 000 s-1,温度范围为20~1 400 ℃;在SiC实验中,应变率为250 s-1,温度范围为20~1 200 ℃。此外,将氩气通入高温炉中,测量TC4钛合金试样在无氧化情况下的动态力学性能,并利用高速摄像机观察试样的变形过程。

    图 4为TC4钛合金在应变率为2 000 s-1、温度为20~1 400 ℃时的应力-应变曲线。从图 4中可以看出:温度软化现象非常明显;20 ℃时试样的峰值应力达到1.6 GPa,当温度达到1 400 ℃时,流动应力降为150 MPa。此外从图 4中还可以看出,在不同温度下TC4钛合金基本没有出现塑性强化现象。需要注意的是,在最初应变很小的阶段,应力测量不够准确,这归因于在开始阶段试样两端的应力状态未达到平衡,试样两端受力不均匀。

    图  4  TC4在应变率为2 000 s-1、温度范围为20~1 400 ℃条件下的应力-应变曲线
    Figure  4.  Stress-strain curves of TC4 at strain rate of 2 000 s-1 and temperature range of 20 ℃ to 1 400 ℃

    考虑到某些材料在高温下易与空气中的氧气等气体发生反应,对实验装置进行了改进,以实现惰性气体环境下的高温动态力学实验。实验时,先加热超高温炉,同时以较小流量通入氩气,达到排尽炉内氧气的目的。达到预定温度后,加大氩气流量并放入试样,保温一段时间后,进行动态压缩实验。在1 100 ℃下测得TC4钛合金在氩气环境下的动态压缩实验结果,如图 5所示,其中高速摄像机的采样速率为0.1 MHz。

    图  5  TC4在应变率为2 000 s-1、温度为1 100 ℃、不同气体环境下的应力-应变曲线及形貌
    Figure  5.  Stress-strain curves and morphologies of TC4 at 2 000 s-1 and 1 100 ℃ in different gas environments

    图 5中可以看到:在空气环境中,40 μs时TC4钛合金试样表面有氧化层裂开现象,100 μs时部分氧化层与试样表面分离;而在氩气环境下,试样表面未发生氧化层裂开或分离现象,说明氩气的通入防止了TC4钛合金试样的氧化。实验表明,本实验装置能进行保护气环境下的超高温动态力学性能测试,并能同步采集实验过程图像。

    图 5还可以看出,在空气环境下TC4钛合金的流动应力为332 MPa,而在氩气环境下则上升至361 MPa,增幅约8.5%,说明TC4钛合金氧化后其动态压缩强度明显降低。图 5中的高速摄影图像显示,TC4钛合金氧化后其表面会形成一层氧化层。氧化层的形成一方面使材料的力学性能下降,另一方面使动态压缩实验中的真实试样变为被氧化层包裹的试样,其截面积等几何形态发生变化,从而对测量结果产生影响。反映在实验结果上则是氩气环境中的流动应力明显高于空气环境中的流动应力,因此研究惰性气体环境下材料的高温动态力学性能具有实际意义。

    对SiC进行超高温动态压缩实验,其中SiC材料是由SiC粉末烧结而成的微孔陶瓷材料,其密度较小,孔隙率较大。SiC试样尺寸为8 mm×8 mm×10 mm,高速摄像机的采样速率为1 MHz。由于SiC为脆性材料,因此需对入射波进行整形以达到应力平衡状态[19-20]。在本实验中,采用圆形树脂整形片进行整形,试样的应变率为250 s-1。为了观察试样表面的变形情况,常用喷漆制作散斑。然而,喷漆只能耐200 ℃以下的温度,随着温度的增加会出现以下3类问题:(1)脱色——在冲击载荷下脱色;(2)脱层——喷漆与试样表面的黏结变松,散斑变形与试样表面变形不一致;(3)脱落——随着温度的继续升高,喷漆从试样表面剥落,所采集的图像完全失真。为此,本研究利用高温无机胶和氧化铝粉末在试样表面制作随机分布的高温散斑,以增强图像的对比度。其制作方式如图 6所示,将高温无机胶和氧化铝粉末一起放入喷枪中,利用高压在试样表面形成随机分布的散斑。在高温下散斑依然清晰可见,并紧密地附着在试样表面。

    图  6  高温散斑
    Figure  6.  High temperature speckle

    图 7中可以看到:SiC的强度随温度的升高而降低;但压缩强度下降得较少,从室温时的250 MPa下降到1 200 ℃时的220 MPa,说明压缩强度对温度不敏感。图 7中红色圆点代表裂纹产生后采集的图像所对应的应力值,由此可以看出:在室温下,当SiC试样应力为199 MPa时,有肉眼可见的初始裂纹产生,随着应变的增大,真实应力持续增加,达到250 MPa时试样完全失去承载能力,裂纹产生时的应力为压缩强度的80%(199 MPa/250 MPa);在高温下,随着温度的升高,出现初始裂纹时的应力随之增加,在800 ℃时,初始裂纹产生时的应力为208 MPa,是压缩强度的90%(208 MPa/231 MPa),而在1 200 ℃时,初始裂纹产生时的应力为218 MPa,是压缩强度的99%(218 MPa/220 MPa)。高温下初始裂纹的出现时刻随着温度的升高而推迟,在本实验中表现为初始裂纹产生时的应力越来越接近压缩强度。对图像进行数字图像相关(DIC)计算,可以看到试样表面的应力状态基本均匀。实验结果显示,所制作的高温散斑能很好地反映高温下试样的变形情况。本研究的高温高速原位图像采集方法可以用来获取高温动态实验中试样的变形信息,为材料高温动态性能研究提供很大的帮助。

    图  7  SiC在应变率为250 s-1,温度为20、800和1 200 ℃时的应力-应变曲线及形貌
    Figure  7.  Stress-strain curves and morphologies of SiC at strain rate of 250 s-1 and temperature of 20, 800 and 1 100 ℃

    在现有SHPB的基础上,设计了一种能用于超高温(1 600 ℃)环境的动态力学性能测试装置。该装置采用两个活塞组成双同步系统,选用加热源为MoSi2的超高温加热炉,由半圆形陶瓷管和高温泡沫固定试样,通过高速摄像机结合滤光片实时记录高温试样的变形过程。

    采用该装置,测量了TC4钛合金在应变率为2 000 s-1、温度为20~1 400 ℃时的应力-应变曲线,以及SiC在应变率为250 s-1、温度为20~1 200 ℃时的压缩强度,获得了TC4钛合金和SiC的高温动态变形图像,得到以下结论:

    (1) 在空气环境下,当温度达到1 400 ℃时,TC4钛合金的流动应力从20 ℃时的1.6 GPa降为150 MPa;而在1 100 ℃的氩气环境下,相比于空气环境,TC4钛合金的流动应力增幅约8.5%;

    (2) 在空气环境下,TC4钛合金试样表面有氧化层裂开现象,部分氧化层与试样表面分离,而在氩气环境下,试样表面没有氧化层裂开或分离情况;

    (3) SiC的压缩强度从室温时的250 MPa降到1 200 ℃时的220 MPa,初始裂纹出现时刻随着温度的升高而推迟,并且初始裂纹出现时的应力由常温下压缩强度的80%增加到1 200 ℃时的99%。

    实验结果表明,所设计的实验装置能够进行保护气环境下的超高温动态力学性能测试,所使用的图像采集方法能在高温高速加载下进行原位图像采集,从而获取试样在高温下的动态变形信息。所提出的实验方法为研究材料的高温动态性能及破坏形式提供很大的帮助。

  • [1]
    RINGWOOD A E. On the chemical evolution and densities of the planets [J]. Geochimica et Cosmochimica Acta, 1959, 15(4): 257–283. doi: 10.1016/0016-7037(59)90062-6
    [2]
    BIRCH F. Density and composition of mantle and core [J]. Journal of Geophysical Research, 1964, 69(20): 4377–4388. doi: 10.1029/JZ069i020p04377
    [3]
    TAKAFUJI N, HIROSE K, MITOME M, et al. Solubilities of O and Si in liquid iron in equilibrium with (Mg, Fe)SiO3 perovskite and the light elements in the core [J]. Geophysical Research Letters, 2005, 32(6).
    [4]
    FISCHER R A, CAMPBELL A J, REAMAN D M, et al. Phase relations in the Fe-FeSi system at high pressures and temperatures [J]. Earth and Planetary Science Letters, 2013, 373: 54–64. doi: 10.1016/j.jpgl.2013.04.035
    [5]
    FISCHER R A, CAMPBELL A J, CARACAS R, et al. Equations of state in the Fe-FeSi system at high pressures and temperatures [J]. Journal of Geophysical Research: Solid Earth, 2014, 119(4): 2810–2827. doi: 10.1002/2013JB010898
    [6]
    TATENO S, KUWAYAMA Y, HIROSE K, et al. The structure of Fe-Si alloy in Earth’s inner core [J]. Earth and Planetary Science Letters, 2015, 418: 11–19. doi: 10.1016/j.jpgl.2015.02.008
    [7]
    OZAWA H, HIROSE K, YONEMITSU K, et al. High-pressure melting experiments on Fe-Si alloys and implications for silicon as a light element in the core [J]. Earth and Planetary Science Letters, 2016, 456: 47–54. doi: 10.1016/j.jpgl.2016.08.042
    [8]
    KNITTLE E, JEANLOZ R. Earth’s core-mantle boundary: results of experiments at high pressures and temperatures [J]. Science, 1991, 251(5000): 1438–1443. doi: 10.1126/science.251.5000.1438
    [9]
    DUBROVINSKY L, DUBROVINSKAIA N, LANGENHORST F, et al. Iron-silica interaction at extreme conditions and the electrically conducting layer at the base of Earth’s mantle [J]. Nature, 2003, 422(6927): 58. doi: 10.1038/nature01422
    [10]
    LIN J F, CAMPBELL A J, HEINZ D L, et al. Static compression of iron-silicon alloys: implications for silicon in the Earth’s core [J]. Journal of Geophysical Research: Solid Earth, 2003, 108(B1).
    [11]
    ASANUMA H, OHTANI E, SAKAI T, et al. Static compression of Fe0.83Ni0.09Si0.08 alloy to 374 GPa and Fe0.93Si0.07 alloy to 252 GPa: implications for the Earth’s inner core [J]. Earth and Planetary Science Letters, 2011, 310(1/2): 113–118. doi: 10.1016/j.jpgl.2011.06.034
    [12]
    BADRO J, FIQUET G, GUYOT F, et al. Effect of light elements on the sound velocities in solid iron: implications for the composition of Earth’s core [J]. Earth and Planetary Science Letters, 2007, 254(1/2): 233–238.
    [13]
    ANTONANGELI D, SIEBERT J, BADRO J, et al. Composition of the Earth’s inner core from high-pressure sound velocity measurements in Fe-Ni-Si alloys [J]. Earth and Planetary Science Letters, 2010, 295(1/2): 292–296.
    [14]
    MAO Z, LIN J F, LIU J, et al. Sound velocities of Fe and Fe-Si alloy in the Earth’s core [J]. Proceedings of the National Academy of Sciences, 2012, 109(26): 10239–10244. doi: 10.1073/pnas.1207086109
    [15]
    LIU J, LIN J F, ALATAS A, et al. Seismic parameters of hcp-Fe alloyed with Ni and Si in the Earth’s inner core [J]. Journal of Geophysical Research: Solid Earth, 2016, 121(2): 610–623. doi: 10.1002/2015JB012625
    [16]
    SAKAIRI T, SAKAMAKI T, OHTANI E, et al. Sound velocity measurements of hcp Fe-Si alloy at high pressure and high temperature by inelastic X-ray scattering [J]. American Mineralogist, 2018, 103(1): 85–90. doi: 10.2138/am-2018-6072
    [17]
    ANTONANGELI D, MORARD G, PAOLASINI L, et al. Sound velocities and density measurements of solid hcp-Fe and hcp-Fe-Si (9 wt.%) alloy at high pressure: constraints on the Si abundance in the Earth’s inner core [J]. Earth and Planetary Science Letters, 2018, 482: 446–453. doi: 10.1016/j.jpgl.2017.11.043
    [18]
    TSUCHIYA T, FUJIBUCHI M. Effects of Si on the elastic property of Fe at Earth’s inner core pressures: first principles study [J]. Physics of the Earth and Planetary Interiors, 2009, 174(1): 212–219.
    [19]
    CÔTÉ A S, VOČADLO L, DOBSON D P, et al. Ab initio lattice dynamics calculations on the combined effect of temperature and silicon on the stability of different iron phases in the Earth’s inner core [J]. Physics of the Earth and Planetary Interiors, 2010, 178(1/2): 2–7.
    [20]
    MARTORELL B, WOOD I G, BRODHOLT J, et al. The elastic properties of hcp-Fe1− xSi x at Earth’s inner-core conditions [J]. Earth and Planetary Science Letters, 2016, 451: 89–96. doi: 10.1016/j.jpgl.2016.07.018
    [21]
    HOHENBERG P, KOHN W. Inhomogeneous electron gas [J]. Physical Review, 1964, 136(3B): B864. doi: 10.1103/PhysRev.136.B864
    [22]
    PERDEW J P. Exchange and correlation in atoms, molecules, and solids: the density functional picture [M]//Electron Correlations and Materials Properties. Boston: Springer, 1999: 287–298.
    [23]
    GROSS E K U, DREIZLER R M. Density functional theory: an approach to the quantum many-body problem [M]. Berlin: Springer, 1990.
    [24]
    KOHN W, SHAM L J. Quantum density oscillations in an inhomogeneous electron gas [J]. Physical Review, 1965, 137(6A): A1697. doi: 10.1103/PhysRev.137.A1697
    [25]
    LANGRETH D C, PERDEW J P. Theory of nonuniform electronic systems. I. analysis of the gradient approximation and a generalization that works [J]. Physical Review B, 1980, 21(12): 5469. doi: 10.1103/PhysRevB.21.5469
    [26]
    PERDEW J P, CHEVARY J A, VOSKO S H, et al. Atoms, molecules, solids, and surfaces: applications of the generalized gradient approximation for exchange and correlation [J]. Physical Review B, 1992, 46(11): 6671. doi: 10.1103/PhysRevB.46.6671
    [27]
    SEGALL M D, LINDAN P J D, PROBERT M J, et al. First-principles simulation: ideas, illustrations and the CASTEP code [J]. Journal of Physics: Condensed Matter, 2002, 14(11): 2717. doi: 10.1088/0953-8984/14/11/301
    [28]
    VOIGT W. The relation between the two elastic moduli of isotropic materials [J]. Annals of Physics (Leipzig), 1889, 33: 573.
    [29]
    REUSS A. Calculation of the flow limits of mixed crystals on the basis of the plasticity of monocrystals [J]. Zeitschrift für Angewandte Mathematik und Mechanik, 1929, 9: 49–58. doi: 10.1002/zamm.19290090104
    [30]
    HILL R. The elastic behaviour of a crystalline aggregate [J]. Proceedings of the Physical Society Section A, 1952, 65(5): 349. doi: 10.1088/0370-1298/65/5/307
    [31]
    WANG C S, KLEIN B M, KRAKAUER H. Theory of magnetic and structural ordering in iron [J]. Physical Review Letters, 1985, 54(16): 1852. doi: 10.1103/PhysRevLett.54.1852
    [32]
    ASADA T, TERAKURA K. Cohesive properties of iron obtained by use of the generalized gradient approximation [J]. Physical Review B, 1992, 46(20): 13599. doi: 10.1103/PhysRevB.46.13599
    [33]
    COHEN R E, MUKHERJEE S. Non-collinear magnetism in iron at high pressures [J]. Physics of the Earth and Planetary Interiors, 2004, 143: 445–453.
    [34]
    BROWN J M, FRITZ J N, HIXSON R S. Hugoniot data for iron [J]. Journal of Applied Physics, 2000, 88(9): 5496–5498. doi: 10.1063/1.1319320
    [35]
    冯磊. 高压下温度对Fe-8.6Si声速的影响 [D]. 武汉: 武汉理工大学, 2017: 72–83.

    FENG L. Effect of temperature on Fe-8.6Si sound velocity at high pressure [D]. Wuhan: Wuhan University of Technology, 2017: 72–83.
    [36]
    经福谦. 实验物态方程导引 [M]. 2版. 北京: 科学出版社, 1999: 188–197.

    JING F Q. Introduction to experimental equation of state [M]. 2nd ed. Beijing: Science Press, 1999: 188–197.
    [37]
    BROWN J M, MCQUEEN R G. Phase transitions, Grüneisen parameter, and elasticity for shocked iron between 77 GPa and 400 GPa [J]. Journal of Geophysical Research: Solid Earth, 1986, 91(B7): 7485–7494. doi: 10.1029/JB091iB07p07485
    [38]
    BONESS D A, BROWN J M, MCMAHAN A K. The electronic thermodynamics of iron under Earth core conditions [J]. Physics of the Earth and Planetary Interiors, 1986, 42(4): 227–240. doi: 10.1016/0031-9201(86)90025-7
    [39]
    FEI Y, MURPHY C, SHIBAZAKI Y, et al. Thermal equation of state of hcp-iron: constraint on the density deficit of Earth’s solid inner core [J]. Geophysical Research Letters, 2016, 43(13): 6837–6843. doi: 10.1002/2016GL069456
    [40]
    ANDERSON O L. The power balance at the core-mantle boundary [J]. Physics of the Earth and Planetary Interiors, 2002, 131(1): 1–17. doi: 10.1016/S0031-9201(02)00009-2
    [41]
    BIRCH F. Elasticity and constitution of the Earth’s interior [J]. Journal of Geophysical Research, 1952, 57(2): 227–286. doi: 10.1029/JZ057i002p00227
    [42]
    HIROSE K, LABROSSE S, HERNLUND J. Composition and state of the core [J]. Annual Review of Earth and Planetary Sciences, 2013, 41: 657–691. doi: 10.1146/annurev-earth-050212-124007
    [43]
    ZHANG Y, SEKINE T, LIN J F, et al. Shock compression and melting of an Fe-Ni-Si alloy: implications for the temperature profile of the Earth’s core and the heat flux across the core-mantle boundary [J]. Journal of Geophysical Research: Solid Earth, 2018, 123(2): 1314–1327. doi: 10.1002/2017JB014723
    [44]
    ANTONANGELI D, KOMABAYASHI T, OCCELLI F, et al. Simultaneous sound velocity and density measurements of hcp iron up to 93 GPa and 1100 K: an experimental test of the Birch’s law at high temperature [J]. Earth and Planetary Science Letters, 2012, 331: 210–214.
    [45]
    ANTONANGELI D, OHTANI E. Sound velocity of hcp-Fe at high pressure: experimental constraints, extrapolations and comparison with seismic models [J]. Progress in Earth and Planetary Science, 2015, 2(1): 3. doi: 10.1186/s40645-015-0034-9
    [46]
    LIN J F, STURHAHN W, ZHAO J, et al. Sound velocities of hot dense iron: Birch’s law revisited [J]. Science, 2005, 308(5730): 1892–1894. doi: 10.1126/science.1111724
    [47]
    SAKAMAKI T, OHTANI E, FUKUI H, et al. Constraints on Earth’s inner core composition inferred from measurements of the sound velocity of hcp-iron in extreme conditions [J]. Science Advances, 2016, 2(2): e1500802. doi: 10.1126/sciadv.1500802
    [48]
    CHEN B, LAI X, LI J, et al. Experimental constraints on the sound velocities of cementite Fe3C to core pressures [J]. Earth and Planetary Science Letters, 2018, 494: 164–171. doi: 10.1016/j.jpgl.2018.05.002
    [49]
    GAO L, CHEN B, WANG J, et al. Pressure-induced magnetic transition and sound velocities of Fe3C: implications for carbon in the Earth’s inner core [J]. Geophysical Research Letters, 2008, 35(17).
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