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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