7大晶系的力学稳定性判据及其应用：以SiO<sub>2</sub>为例

高娟; 刘其军; 蒋城露; 樊代和; 张淼; 刘福生; 唐斌

doi:10.11858/gywlxb.20220575

7大晶系的力学稳定性判据及其应用：以SiO₂为例

doi: 10.11858/gywlxb.20220575

高娟^1,,
刘其军^1,*, ,,
蒋城露²,
樊代和¹,
张淼¹,
刘福生¹,
唐斌³

1.
西南交通大学物理科学与技术学院, 四川成都　610031
2.
四川农业大学水利水电学院, 四川雅安　625014
3.
西北工业大学材料学院, 陕西西安　710072

基金项目: 国家自然科学基金（12147208）；中央高校基本科研业务费专项资金（2682020ZT102）

详细信息

作者简介:
高　娟（1995－），女，博士研究生，主要从事材料高压结构与物性研究.E-mail：gao.juan.xnjd@my.swjtu.edu.cn

通讯作者:
刘其军（1985－），男，博士，副教授，主要从事计算材料科学、亚稳材料、相变动力学、冲击波与爆轰物理研究. E-mail：qijunliu@home.swjtu.edu.cn

中图分类号: O521.2; O469
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出版历程
- 收稿日期: 2022-05-05
- 修回日期: 2022-05-19
- 录用日期: 2022-05-19
- 网络出版日期: 2022-10-11
- 刊出日期: 2022-10-11

Criteria of Mechanical Stability of Seven Crystal Systems and Its Application: Taking Silica as an Example

1.
School of Physical Science and Technology, Southwest Jiaotong University, Chengdu 610031, Sichuan, China
2.
College of Water Conservancy and Hydropower Engineering, Sichuan Agricultural University, Ya’an 625014, Sichuan, China
3.
School of Materials, Northwestern Polytechnical University, Xi’an 710072, Shaanxi, China

摘要

摘要: 根据二次型的正定性，在Mouhat和Coudert推导的力学稳定性判据的基础上补充了单斜和三斜晶系的力学稳定性判据，给出了7大晶系的力学稳定性判据。基于密度泛函理论的第一性原理方法，计算了属于不同晶系的9种SiO₂在0 GPa下的弹性常数。结合力学稳定性判据，对立方P2₁3、六角P6₃/mmc、三角P3₁21和$R{\bar {3}}$、四方P4₁2₁2和$I{\bar 4}$、正交Pbcn、单斜P2₁/c以及三斜$P\bar 1$晶系的SiO₂进行了稳定性判定，结果显示这9种SiO₂在0 GPa下都是力学稳定的。
- 弹性常数 /
- 力学稳定性判据 /
- SiO₂ /
- 7大晶系
Abstract: According to the positive definiteness of the quadratic type, and based on the mechanical stability criteria derived by Mouhat and Coudert, the mechanical stability criteria of monoclinic and triclinic crystal systems are supplemented. The mechanical stability criteria of the seven crystal systems are given. Based on the density functional theory of the first principle method, the elastic constants of nine kinds of SiO₂ belonging to different crystal systems are calculated. Combined with the mechanical stability criteria, the stabilities of SiO₂ in different space groups, including cubic P2₁3, hexagonal P6₃/mmc, trigonal P3₁21 and $R\overline 3 $, tetragonal P4₁2₁2 and $I\overline 4 $, orthorhombic Pbcn, monoclinic P2₁/c, and triclinic $P\bar 1$, are determined. The calculated results show that the nine types of SiO₂ are all mechanically stable at 0 GPa.
- elastic constants /
- mechanical stability criterion /
- SiO₂ /
- seven kinds of crystal systems

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图 1 9种SiO₂的晶体结构（红色为氧原子，蓝色为硅原子）

Figure 1. Nine kinds of SiO₂ crystal structures (The oxygen atom is red and the silicon atom is blue.)

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表 1 7大晶系的点群个数及符号、空间群个数及编号以及独立弹性常数的个数

Table 1. Quantites and symbols of point groups and space groups, and the quantities of independent elastic constants for the seven crystal systems

Crystal system	Point groups quantity and symbol	Space groups quantity and symbol	$n{_{C{_{ij} } }}$
Triclinic	2 (1,$\bar 1$)	2 (1–2)	21
Monoclinic	3 (2, m, 2/m)	13 (3–15)	13
Orthorhombic	3 (222, mm2, mmm)	59 (16–74)	9
Tetragonal (Ⅱ)	3 (4, $\bar 4$, 4/m)	14 (75–88)	7
Tetragonal (Ⅰ)	4 (422, 4mm, $\bar 42m$, 4/mmm)	54 (89–142)	6
Trigonal (Ⅱ)	2 (3, $\bar 3$)	6 (143–148)	7
Trigonal (Ⅰ)	3 (32, 3m, $\bar 3m$)	19 (149–167)	6
Hexagonal	7 (6, $\bar 6$, 6/m, 622, 6mm, $\bar 6m2$, 6/mmm)	27 (168–194)	5
Cubic	5 (23, $m\bar 3$, 432, $\bar 43m$, $m\bar 3m$)	36 (195–230)	3

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表 2 计算得到的9种SiO₂的弹性常数

Table 2. Calculated elastic constants of nine kinds of SiO₂ GPa　

Space group	${C{_{11} } }$	${C{_{12} } }$	${C{_{13} } }$	${C{_{14} } }$	${C{_{15} } }$	${C{_{16} } }$	${C{_{22} } }$	${C{_{23} } }$	${C{_{24} } }$	${C{_{25} } }$	${C{_{26} } }$
P2₁3	119.5	64.5	64.5	0	0	0	119.5	64.5	0	0	0
P6₃/mmc	50.3	15.7	52.3	0	0	0	50.3	52.3	0	0	0
P3₁21	99.8	15.3	24.7	−9.5	0	0	99.8	24.66	9.5	0	0
$R\overline 3 $	404.1	198.4	104.7	−27.9	−13.2	0	404.1	104.7	27.9	13.2	0
P4₁2₁2	73.9	9.4	−3.4	0	0	0	73.9	−3.4	0	0	0
$I\overline 4 $	59.8	51.2	8.1	0	0	8.2	59.8	8.1	0	0	−8.2
Pbcn	525.9	156.6	151.3	0	0	0	559.7	159.5	0	0	0
P2₁/c	117.9	30.4	31.3	0	4.5	0	88.0	−0.85	0	1.8	0
$P\bar 1$	79.7	47.7	15.0	−5.6	10.4	0.2	78.3	16.6	−10.8	6.7	−1.9

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Space group	${C{_{33} } }$	${C{_{34} } }$	${C{_{35} } }$	${C{_{36} } }$	${C{_{44} } }$	${C{_{45} } }$	${C{_{46} } }$	${C{_{55} } }$	${C{_{56} } }$	${C{_{66} } }$
P2₁3	119.5	0	0	0	63.8	0	0	63.8	0	63.8
P6₃/mmc	148.4	0	0	0	33.4	0	0	33.4	0	17.3
P3₁21	122.8	0	0	0	67.4	0	0	67.4	−9.5	42.3
$R\overline 3 $	348.3	0	0	0	29.8	0	13.2	29.8	−27.9	102.9
P4₁2₁2	55.3	0	0	0	75.4	0	0	75.4	0	22.2
$I\overline 4 $	59.0	0	0	0	32.7	0	0	32.7	0	32.0
Pbcn	623.7	0	0	0	194.7	0	0	219.6	0	237.8
P2₁/c	75.9	0	−8.7	0	23.0	0	6.1	37.3	0	71.8
$P\bar 1 $	14.7	−3.9	4.4	2.8	15.1	−0.8	4.2	12.0	−4.7	45.0

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参考文献(37)

[1]	王皖燕. 晶体学和晶体材料研究的进展 [J]. 科技导报, 2002, 20(23): 26–28. doi: 10.3321/j.issn:1000-7857.2002.03.007 WANG W Y. Progress in the research of crystal [J]. Science and Technology Review, 2002, 20(23): 26–28. doi: 10.3321/j.issn:1000-7857.2002.03.007
[2]	施倪承, 李国武. 对称与晶体学 [J]. 自然杂志, 2008, 30(1): 44–49. doi: 10.3969/j.issn.0253-9608.2008.01.009 SHI N C, LI G W. Symmetry and crystallography [J]. Chinese Journal of Nature, 2008, 30(1): 44–49. doi: 10.3969/j.issn.0253-9608.2008.01.009
[3]	刘其军. 固体物理学教材在中国的发展历程 [J]. 大学物理, 2015, 34(2): 51–55. doi: 10.16854/j.cnki.1000-0712.2015.02.009 LIU Q J. Development process of solid state physics textbooks in China [J]. College Physics, 2015, 34(2): 51–55. doi: 10.16854/j.cnki.1000-0712.2015.02.009
[4]	麦振洪. X射线晶体学的创立与发展 [J]. 物理, 2014, 43(12): 787–800. doi: 10.7693/wl20141202 MAI Z H. The founding and development of X-ray crystallography [J]. Physics, 2014, 43(12): 787–800. doi: 10.7693/wl20141202
[5]	马礼敦. X射线晶体学的百年辉煌 [J]. 物理学进展, 2014, 34(2): 47–117. doi: 10.13725/j.cnki.pip.2014.02.001 MA L D. Splendid century of X-ray crystallography [J]. Progress in Physics, 2014, 34(2): 47–117. doi: 10.13725/j.cnki.pip.2014.02.001
[6]	龙光芝, 陈瀛, 陈敬中. 晶体学与准晶体学点群的母子群关系 [J]. 物理学报, 2006, 55(6): 2838–2845. doi: 10.3321/j.issn:1000-3290.2006.06.031 LONG G Z, CHEN Y, CHEN J Z. Subgroups and supergroups of crystallographic and quasi-crystallographic point groups [J]. Acta Physica Sinica, 2006, 55(6): 2838–2845. doi: 10.3321/j.issn:1000-3290.2006.06.031
[7]	BORN M. On the stability of crystal lattices. Ⅰ [J]. Mathematical Proceedings of the Cambridge Philosophical Society, 1940, 36(2): 160–172. doi: 10.1017/S0305004100017138
[8]	玻恩, 黄昆. 晶格动力学理论 [M]. 葛惟锟, 贾惟义, 译. 北京: 北京大学出版社, 1989: 135−176. BORN M, HUANG K. Dynamical theory of crystal lattices [M]. Translated by GE W K, JIA W Y. Beijing: Peking University Press, 1989: 135−176.
[9]	BECKSTEIN O, KLEPEIS J E, HART G L W, et al. First-principles elastic constants and electronic structure of α-Pt₂Si and PtSi [J]. Physical Review B, 2001, 63(13): 134112. doi: 10.1103/PhysRevB.63.134112
[10]	赵昆, 张坤, 王家佳, 等. Heusler合金Pd₂CrAl四方变形、磁性及弹性常数的第一性原理计算 [J]. 物理学报, 2011, 60(12): 127101. doi: 10.7498/aps.60.127101 ZHAO K, ZHANG K, WANG J J, et al. A first principles study on tetragonal distortion, magnetic property and elastic constants of Pd₂CrAl Heusler alloy [J]. Acta Physica Sinica, 2011, 60(12): 127101. doi: 10.7498/aps.60.127101
[11]	GRIMVALL G, MAGYARI-KÖPE B, OZOLIŅŠ V, et al. Lattice instabilities in metallic elements [J]. Reviews of Modern Physics, 2012, 84(2): 945–986. doi: 10.1103/RevModPhys.84.945
[12]	LIU Q J, ZHANG N C, LIU F S, et al. Structural, electronic, optical, elastic properties and Born effective charges of monoclinic HfO₂ from first-principles calculations [J]. Chinese Physics B, 2014, 23(4): 047101. doi: 10.1088/1674-1056/23/4/047101
[13]	LIU Q J, RAN Z, LIU F S, et al. Phase transitions and mechanical stability of TiO₂ polymorphs under high pressure [J]. Journal of Alloys and Compounds, 2015, 631: 192–201. doi: 10.1016/j.jallcom.2015.01.085
[14]	MOUHAT F, COUDERT F X. Necessary and sufficient elastic stability conditions in various crystal systems [J]. Physical Review B, 2014, 90(22): 224104. doi: 10.1103/PhysRevB.90.224104
[15]	HAINES J, LÉGER J M, GORELLI F, et al. Crystalline post-quartz phase in silica at high pressure [J]. Physical Review Letters, 2001, 87(15): 155503. doi: 10.1103/PhysRevLett.87.155503
[16]	TETER D M, HEMLEY R J, KRESSE G, et al. High pressure polymorphism in silica [J]. Physical Review Letters, 1998, 80(10): 2145–2148. doi: 10.1103/PhysRevLett.80.2145
[17]	LIU W, WU X B, LIANG Y F, et al. Multiple pathways in pressure-induced phase transition of coesite [J]. Proceedings of the National Academy of Sciences of the United States of America, 2017, 114(49): 12894–12899. doi: 10.1073/pnas.1710651114
[18]	TSUCHIYA T, TSUCHIYA J. Prediction of a hexagonal SiO₂ phase affecting stabilities of MgSiO₃ and CaSiO₃ at multimegabar pressures [J]. Proceedings of the National Academy of Sciences of the United States of America, 2011, 108(4): 1252–1255. doi: 10.1073/pnas.1013594108
[19]	桂质亮, 赵东方. 奥古斯丁·路易斯·柯西: 杰出的数学家纪念柯西诞辰 [J]. 华中师范大学学报(自然科学版), 1989, 23(4): 597–602. GUI Z L, ZHAO D F. Augustin-Louis Cauchy: a famous mathematician [J]. Journal of Central China Normal University (Natural Sciences), 1989, 23(4): 597–602.
[20]	LIU Z L, LI H J, FAN C Z, et al. Necessary and sufficient elastic stability conditions in 21 quasicrystal Laue classes [J]. European Journal of Mechanics A: Solids, 2017, 65: 30–39. doi: 10.1016/j.euromechsol.2017.02.007
[21]	SINGH S, LANG L, DOVALE-FARELO V, et al. MechElastic: a python library for analysis of mechanical and elastic properties of bulk and 2D materials [J]. Computer Physics Communications, 2021, 267: 108068. doi: 10.1016/j.cpc.2021.108068
[22]	WALLACE D C. Thermoelastic theory of stressed crystals and higher-order elastic constants [J]. Solid State Physics, 1970, 25: 301–404.
[23]	SIN’KO G V, SMIRNOV N A. Ab initio calculations of elastic constants and thermodynamic properties of bcc, fcc, and hcp Al crystals under pressure [J]. Journal of Physics: Condensed Matter, 2002, 14(29): 6989–7005. doi: 10.1088/0953-8984/14/29/301
[24]	WANG J H, LI J, YIP S, et al. Mechanical instabilities of homogeneous crystals [J]. Physical Review B, 1995, 52(17): 12627–12635. doi: 10.1103/PhysRevB.52.12627
[25]	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): 2717–2744. doi: 10.1088/0953-8984/14/11/301
[26]	HOHENBERG P, KOHN W. Inhomogeneous electron gas [J]. Physical Review, 1964, 136(3B): B864. doi: 10.1103/PhysRev.136.B864
[27]	KOHN W, SHAM L J. Self-consistent equations including exchange and correlation effects [J]. Physical Review, 1965, 140(4A): A1133–A1138. doi: 10.1103/PhysRev.140.A1133
[28]	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): 6671–6687. doi: 10.1103/PhysRevB.46.6671
[29]	BORISOV S V, PERVUKHINA N V, MAGARILL S A. Crystallographic analysis of SiO₂ polymorphs [J]. Journal of Structural Chemistry, 2019, 60(12): 1946–1958. doi: 10.1134/S0022476619120102
[30]	VAKULA N I, KURAMSHINA G M, GORB L G, et al. Adsorption and diffusion of a silver atom and its cation on α-SiO₂ (0 0 1): comparison of a pure surface with a surface containing an Al defect [J]. Chemical Physics Letters, 2013, 567: 27–33. doi: 10.1016/j.cplett.2013.02.067
[31]	郭杰荣, 马磊. 石英α-SiO₂晶体基本特性的原子模拟 [J]. 材料科学, 2019, 9(4): 355–360. doi: 10.12677/MS.2019.94047 GUO J R, MA L. Atomic simulation of basic properties for α-SiO₂ crystal [J]. Material Sciences, 2019, 9(4): 355–360. doi: 10.12677/MS.2019.94047
[32]	LIU C, SHI J Y, GAO H, et al. Mixed coordination silica at megabar pressure [J]. Physical Review Letters, 2021, 126(3): 035701. doi: 10.1103/PhysRevLett.126.035701
[33]	COH S, VANDERBILT D. Structural stability and lattice dynamics of SiO₂ cristobalite [J]. Physical Review B, 2008, 78(5): 054117. doi: 10.1103/PhysRevB.78.054117
[34]	DOWNS R T, PALMER D C. The pressure behavior of α cristobalite [J]. American Mineralogist, 1994, 79(1/2): 9–14.
[35]	BOISEN M B, GIBBS G V, BUKOWINSKI M S T. Framework silica structures generated using simulated annealing with a potential energy function based on an H₆Si₂O₇ molecule [J]. Physics and Chemistry of Minerals, 1994, 21(5): 269–284. doi: 10.1007/BF00202091
[36]	HOCHGRÄFE M, MARLER B, GIES H, et al. The structure of the low temperature (20 K) form of zeolite ZSM-11 derived from ²⁹Si MAS NMR spectroscopy, lattice energy minimization and Rietveld refinement [J]. Zeitschrift für Kristallographie-Crystalline Materials, 1996, 211(4): 221–227.
[37]	MIYAHARA M, KANEKO S, OHTANI E, et al. Discovery of seifertite in a shocked lunar meteorite [J]. Nature Communications, 2013, 4(1): 1737. doi: 10.1038/ncomms2733