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空間群、格、點群、晶體的分類 来自维基百科,自由的百科全书
晶体通常可分为七种晶系,即立方晶系、六方晶系、四方晶系、三方晶系、正交晶系、单斜晶系、三斜晶系。其中的立方晶系具有各向同性,属于高级晶族。
晶系的特征与细分关系如下表:
晶族 | 晶系 | 点群的对称性 | 点群 | 空间群 | 布拉菲晶格 | 特征 | 晶格系统 |
---|---|---|---|---|---|---|---|
三斜 | 无 | 2 | 2 | 1 | α≠β≠γ≠90°,a≠b≠c | 三斜 | |
单斜 | 1个两次对称轴 或 1个对称面 | 3 | 13 | 2 | α=γ=90°,β≠90°,a≠b≠c | 单斜 | |
正交/斜方 | 3个两次对称轴 或 1个两次对称轴+2个对称面 | 3 | 59 | 4 | α=β=γ=90°,a≠b≠c | 正交/斜方 | |
四方/正方 | 1个四次对称轴 | 7 | 68 | 2 | α=β=γ=90°,a=b≠c | 四方/正方 | |
六方 | 三方 | 1个三次对称轴 | 5 | 7 | 1 | α=β=γ≠90°,a=b=c | 三方 |
18 | 1 | α=β=90°,γ=120°,a=b≠c | 六方 | ||||
六方 | 1个六次对称轴 | 7 | 27 | ||||
立方/等轴 | 4个三次对称轴 | 5 | 36 | 3 | α=β=γ=90°,a=b=c | 立方/等轴 | |
6 | 7 | 共计 | 32 | 230 | 14 | 7 |
这14种布拉菲晶格可分成7种晶系,每种晶系又可依中心原子在晶胞中的位置不同再分成6种晶格:
7种不同晶系与每种晶系的6种不同晶格共有7 × 6 = 42种组合,但是有些组合其实是相同的,都能组成14种布拉菲晶格。例如,单斜晶系的体心晶格可以通过单斜晶系的底心(C)晶格选择不同的晶轴得到,所以这两种其实是同一种;同样,所有的底心(A)、底心(B)晶格都相当于底心(C)或简单(P)晶格。因此,去除相同的组合,可以得到14种不同的布拉菲晶格,列于下表(晶格图下方是代表该布拉菲晶格的皮尔逊符号,表中空白的格表示于已有的晶格重复):
每一个单位晶格的体积可以由计算得知。其中,和是晶格向量。各种布拉菲晶格的体积如下:
晶系 | 体积 | |||
三斜晶系 | ||||
单斜晶系 | ||||
斜方晶系 | ||||
四方晶系 | ||||
三方晶系 | ||||
六方晶系 | ||||
等轴晶系 | 解析失败 (SVG(MathML可通过浏览器插件启用):从服务器“http://localhost:6011/zh.wikipedia.org/v1/”返回无效的响应(“Math extension cannot connect to Restbase.”):): {\displaystyle a^3} |
在熊夫利中,点群是用字母符号加上数字下标表示的。下面简述晶体学中使用的这种符号的意义[1]:
根据晶体局限定理,在二维或三维空间中n的取值只有1、2、3、4和6。
n | 1 | 2 | 3 | 4 | 6 |
---|---|---|---|---|---|
Cn | C1 | C2 | C3 | C4 | C6 |
Cnv | C1v=C1h | C2v | C3v | C4v | C6v |
Cnh | C1h | C2h | C3h | C4h | C6h |
Dn | D1=C2 | D2 | D3 | D4 | D6 |
Dnh | D1h=C2v | D2h | D3h | D4h | D6h |
Dnd | D1d=C2h | D2d | D3d | D4d | D6d |
S2n | S2 | S4 | S6 | S8 | S12 |
D4d和D6d实际上是不存在的,因为它们分别包含了n=8和12的旋转反映轴。表格中剩下的27种点群与T、Td、Th、O和Oh共同组成32种晶体学点群。
赫尔曼–莫甘记号的一种简略形式广泛用于表示空间群,也用于描述晶体学点群。群的名称列在下表中;点群间相互之关系可见右图。
1 | 1 | |||||
2 | 2⁄m | 222 | m | mm2 | mmm | |
3 | 3 | 32 | 3m | 3m | ||
4 | 4 | 4⁄m | 422 | 4mm | 42m | 4⁄mmm |
6 | 6 | 6⁄m | 622 | 6mm | 62m | 6⁄mmm |
23 | m3 | 432 | 43m | m3m |
晶族 | 晶系 | 赫尔曼–莫甘 (完整记号) |
赫尔曼–莫甘 (简写记号) |
舒勃尼科夫[2] | 熊夫利 | 轨形记号 | 考克斯特记号 | 阶 |
---|---|---|---|---|---|---|---|---|
1 | 1 | C1 | 11 | [ ]+ | 1 | |||
1 | 1 | Ci = S2 | x | [2+,2+] | 2 | |||
2 | 2 | C2 | 22 | [2]+ | 2 | |||
m | m | Cs = C1h | * | [ ] | 2 | |||
2/m | C2h | 2* | [2,2+] | 4 | ||||
222 | 222 | D2 = V | 222 | [2,2]+ | 4 | |||
mm2 | mm2 | C2v | *22 | [2] | 4 | |||
mmm | D2h | *222 | [2,2] | 8 | ||||
4 | 4 | C4 | 44 | [4]+ | 4 | |||
4 | 4 | S4 | 2x | [2+,4+] | 4 | |||
4/m | C4h | 4* | [2,4+] | 8 | ||||
422 | 422 | D4 | 422 | [4,2]+ | 8 | |||
4mm | 4mm | C4v | *44 | [4] | 8 | |||
42m | 42m | D2d | 2*2 | [2+,4] | 8 | |||
4/mmm | D4h | *422 | [4,2] | 16 | ||||
3 | 3 | C3 | 33 | [3]+ | 3 | |||
3 | 3 | S6 = C3i | 3x | [2+,6+] | 6 | |||
32 | 32 | D3 | 322 | [3,2]+ | 6 | |||
3m | 3m | C3v | *33 | [3] | 6 | |||
3 | 3m | D3d | 2*3 | [2+,6] | 12 | |||
6 | 6 | C6 | 66 | [6]+ | 6 | |||
6 | 6 | C3h | 3* | [2,3+] | 6 | |||
6/m | C6h | 6* | [2,6+] | 12 | ||||
622 | 622 | D6 | 622 | [6,2]+ | 12 | |||
6mm | 6mm | C6v | *66 | [6] | 12 | |||
6m2 | 6m2 | D3h | *322 | [3,2] | 12 | |||
6/mmm | D6h | *622 | [6,2] | 24 | ||||
23 | 23 | T | 332 | [3,3]+ | 12 | |||
3 | m3 | Th | 3*2 | [3+,4] | 24 | |||
432 | 432 | O | 432 | [4,3]+ | 24 | |||
43m | 43m | Td | *332 | [3,3] | 24 | |||
3 | m3m | Oh | *432 | [4,3] | 48 |
二维空间具有相同数量的晶系、晶族和晶格。在二维空间有四种晶系:斜晶系、矩晶系、方晶系、六方晶系。
四维晶胞由四个边长(a、b、c、d)和六个轴间角(α、β、γ、δ、ε、ζ)定义。以下晶格参数条件定义了23种晶系。
No. | 晶系(1985年Whittaker命名[3]) | 边长 | 轴间角 |
---|---|---|---|
1 | Hexaclinic | a ≠ b ≠ c ≠ d | α ≠ β ≠ γ ≠ δ ≠ ε ≠ ζ ≠ 90° |
2 | Triclinic | a ≠ b ≠ c ≠ d | α ≠ β ≠ γ ≠ 90° δ = ε = ζ = 90° |
3 | Diclinic | a ≠ b ≠ c ≠ d | α ≠ 90° β = γ = δ = ε = 90° ζ ≠ 90° |
4 | Monoclinic | a ≠ b ≠ c ≠ d | α ≠ 90° β = γ = δ = ε = ζ = 90° |
5 | Orthogonal | a ≠ b ≠ c ≠ d | α = β = γ = δ = ε = ζ = 90° |
6 | Tetragonal monoclinic | a ≠ b = c ≠ d | α ≠ 90° β = γ = δ = ε = ζ = 90° |
7 | Hexagonal monoclinic | a ≠ b = c ≠ d | α ≠ 90° β = γ = δ = ε = 90° ζ = 120° |
8 | Ditetragonal diclinic | a = d ≠ b = c | α = ζ = 90° β = ε ≠ 90° γ ≠ 90° δ = 180° − γ |
9 | Ditrigonal (dihexagonal) diclinic | a = d ≠ b = c | α = ζ = 120° β = ε ≠ 90° γ ≠ δ ≠ 90° cos δ = cos β − cos γ |
10 | Tetragonal orthogonal | a ≠ b = c ≠ d | α = β = γ = δ = ε = ζ = 90° |
11 | Hexagonal orthogonal | a ≠ b = c ≠ d | α = β = γ = δ = ε = 90°, ζ = 120° |
12 | Ditetragonal monoclinic | a = d ≠ b = c | α = γ = δ = ζ = 90° β = ε ≠ 90° |
13 | Ditrigonal (dihexagonal) monoclinic | a = d ≠ b = c | α = ζ = 120° β = ε ≠ 90° γ = δ ≠ 90° cos γ = −1/2cos β |
14 | Ditetragonal orthogonal | a = d ≠ b = c | α = β = γ = δ = ε = ζ = 90° |
15 | Hexagonal tetragonal | a = d ≠ b = c | α = β = γ = δ = ε = 90° ζ = 120° |
16 | Dihexagonal orthogonal | a = d ≠ b = c | α = ζ = 120° β = γ = δ = ε = 90° |
17 | Cubic orthogonal | a = b = c ≠ d | α = β = γ = δ = ε = ζ = 90° |
18 | Octagonal | a = b = c = d | α = γ = ζ ≠ 90° β = ε = 90° δ = 180° − α |
19 | Decagonal | a = b = c = d | α = γ = ζ ≠ β = δ = ε cos β = −1/2 − cos α |
20 | Dodecagonal | a = b = c = d | α = ζ = 90° β = ε = 120° γ = δ ≠ 90° |
21 | Diisohexagonal orthogonal | a = b = c = d | α = ζ = 120° β = γ = δ = ε = 90° |
22 | Icosagonal (icosahedral) | a = b = c = d | α = β = γ = δ = ε = ζ cos α = −1/4 |
23 | Hypercubic | a = b = c = d | α = β = γ = δ = ε = ζ = 90° |
由1985年Whittaker命名[3]。
名字几乎与Brown等人[4]的命名相同,只有9、13、22名称不同。括号是他们命的名。
晶族序 | 晶族(英文) | 晶系(英文) | 晶系序 | 点群 | 空间群 | 布拉菲晶格 | 晶格 |
---|---|---|---|---|---|---|---|
I | Hexaclinic | 1 | 2 | 2 | 1 | Hexaclinic P | |
II | Triclinic | 2 | 3 | 13 | 2 | Triclinic P, S | |
III | Diclinic | 3 | 2 | 12 | 3 | Diclinic P, S, D | |
IV | Monoclinic | 4 | 4 | 207 | 6 | Monoclinic P, S, S, I, D, F | |
V | Orthogonal | Non-axial orthogonal | 5 | 2 | 2 | 1 | Orthogonal KU |
112 | 8 | Orthogonal P, S, I, Z, D, F, G, U | |||||
Axial orthogonal | 6 | 3 | 887 | ||||
VI | Tetragonal monoclinic | 7 | 7 | 88 | 2 | Tetragonal monoclinic P, I | |
VII | Hexagonal monoclinic | Trigonal monoclinic | 8 | 5 | 9 | 1 | Hexagonal monoclinic R |
15 | 1 | Hexagonal monoclinic P | |||||
Hexagonal monoclinic | 9 | 7 | 25 | ||||
VIII | Ditetragonal diclinic* | 10 | 1 (+1) | 1 (+1) | 1 (+1) | Ditetragonal diclinic P* | |
IX | Ditrigonal diclinic* | 11 | 2 (+2) | 2 (+2) | 1 (+1) | Ditrigonal diclinic P* | |
X | Tetragonal orthogonal | Inverse tetragonal orthogonal | 12 | 5 | 7 | 1 | Tetragonal orthogonal KG |
351 | 5 | Tetragonal orthogonal P, S, I, Z, G | |||||
Proper tetragonal orthogonal | 13 | 10 | 1312 | ||||
XI | Hexagonal orthogonal | Trigonal orthogonal | 14 | 10 | 81 | 2 | Hexagonal orthogonal R, RS |
150 | 2 | Hexagonal orthogonal P, S | |||||
Hexagonal orthogonal | 15 | 12 | 240 | ||||
XII | Ditetragonal monoclinic* | 16 | 1 (+1) | 6 (+6) | 3 (+3) | Ditetragonal monoclinic P*, S*, D* | |
XIII | Ditrigonal monoclinic* | 17 | 2 (+2) | 5 (+5) | 2 (+2) | Ditrigonal monoclinic P*, RR* | |
XIV | Ditetragonal orthogonal | Crypto-ditetragonal orthogonal | 18 | 5 | 10 | 1 | Ditetragonal orthogonal D |
165 (+2) | 2 | Ditetragonal orthogonal P, Z | |||||
Ditetragonal orthogonal | 19 | 6 | 127 | ||||
XV | Hexagonal tetragonal | 20 | 22 | 108 | 1 | Hexagonal tetragonal P | |
XVI | Dihexagonal orthogonal | Crypto-ditrigonal orthogonal* | 21 | 4 (+4) | 5 (+5) | 1 (+1) | Dihexagonal orthogonal G* |
5 (+5) | 1 | Dihexagonal orthogonal P | |||||
Dihexagonal orthogonal | 23 | 11 | 20 | ||||
Ditrigonal orthogonal | 22 | 11 | 41 | ||||
16 | 1 | Dihexagonal orthogonal RR | |||||
XVII | Cubic orthogonal | Simple cubic orthogonal | 24 | 5 | 9 | 1 | Cubic orthogonal KU |
96 | 5 | Cubic orthogonal P, I, Z, F, U | |||||
Complex cubic orthogonal | 25 | 11 | 366 | ||||
XVIII | Octagonal* | 26 | 2 (+2) | 3 (+3) | 1 (+1) | Octagonal P* | |
XIX | Decagonal | 27 | 4 | 5 | 1 | Decagonal P | |
XX | Dodecagonal* | 28 | 2 (+2) | 2 (+2) | 1 (+1) | Dodecagonal P* | |
XXI | Diisohexagonal orthogonal | Simple diisohexagonal orthogonal | 29 | 9 (+2) | 19 (+5) | 1 | Diisohexagonal orthogonal RR |
19 (+3) | 1 | Diisohexagonal orthogonal P | |||||
Complex diisohexagonal orthogonal | 30 | 13 (+8) | 15 (+9) | ||||
XXII | Icosagonal | 31 | 7 | 20 | 2 | Icosagonal P, SN | |
XXIII | Hypercubic | Octagonal hypercubic | 32 | 21 (+8) | 73 (+15) | 1 | Hypercubic P |
107 (+28) | 1 | Hypercubic Z | |||||
Dodecagonal hypercubic | 33 | 16 (+12) | 25 (+20) | ||||
共计 | 23 (+6) | 33 (+7) | 227 (+44) | 4783 (+111) | 64 (+10) | 33 (+7) |
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