AEC 原子环境计算：230空间群列表

Atomic Environment Calculation list by the 230 space groups

          已经发布的AEC，说明和下载链接置于对应的空间群号下面，是红色字体，否则就是缺省的待发布标记。

已经发布了4个空间群       

No. 1；No.167；No. 194；No. 225

        No.1空间群到No. 34空间群，列出了空间群表的原子坐标，按照国际空间群表的格式：Multiplicity, Wyckoff letter, Coordinates：1 a, (x,y,z)

       所谓的Multiplicity，实际上就是晶胞原子数；Wyckoff letter表示点位；原子坐标是晶体学坐标，就是所谓的分数坐标，取值在0和1之间，包括0和1.   需要强调的是，在特殊取值下，空间群的点位实际上已经发生了变化，有时和空间群表对不上了，有时和晶体学CIF数据对不上了。因此，原子环境数据才是唯一的和绝对可靠的。只有在包含x,y,z的情况下，原子坐标才是晶体学变量，否则被称为Wyckoff不变量。如果不含x,y,z参数，原子占位确定了，原子坐标也同时被确定了。对原子坐标全部是Wyckoff不变量的时候，晶体的原子环境数据只取决于点阵参数。例如，对于简单的面心立方晶体，如Al, Ni, Cu, 等，原子环境数据只取决于晶格常数。

         国际空间群表的信息量太大，大到吓人的程度。首先，每个空间群号下面的那个对称元素投影图，就把90%的人给整蒙了，不信你(材料科学，材料物理，材料化学)去看看。接下来的Origin选择，还有Asymmetric unit，Symmetry operations，Generators selected，Reflection conditions，230个空间群，100多年的积累，一大堆的信息，把普通读者(材料科学，材料物理，材料化学)压得喘不过气来。

         我通过我的魔方方程，采用民间“是骡子是马拉出来遛遛”的策略，从积累沉淀了100多年的经过多次修订的繁茂芜杂的有700多页的国际空间群表，蒸馏出尽可能少的充分必要的信息，支撑的原子环境计算(AEC)。

🟫 三斜晶系（Triclinic, a ≠ b ≠ c, α ≠ β ≠ γ）

三斜晶系有2个空间群：1-2

C1（1）点群(order =1)

空间群号： 1  

Multiplicity, Wyckoff letter, Coordinates：1 a, (x,y,z)

 科学网—No.1 空间群的原子环境计算：三斜晶体，对称之初 - 李世春的博文

Ci（‑1）点群(order =2)

空间群号： 2

1a, (0,0,0);1b, (0,0,1/2);1c, (0,1/2,0);1d, (1/2,0,0);1e, (1/2,1/2,0);1f, (1/2,0,1/2);1g, (0,1/2,1/2);1h, (1/2,1/2,1/2);2i, (x,y,z).

🟧 单斜晶系(Monoclinic)

单斜晶系有13个空间群：3-15

          (Uniq-b: a ≠ b ≠ c, α =90°, β ≠ 90°, γ = 90°)

          (Uniq-c: a ≠ b ≠ c, α = 90°, β = 90°, γ ≠ 90°)

C2（2）点群(order =2)

空间群号： 3–5

No. 3

UNIQUE AXIS b (P121)

1a, (0,y,0);1b, (0,y,1/2);1c, (1/2,y,0);1d, (1/2,y,1/2);2e, (x,y,z).

UNIQUE AXIS c (P112)

1a, (0,0,z);1b, (1/2,0,z);1c, (0,1/2,z);1d, (1/2,1/2,z);2e, (x,y,z).

No. 4

 UNIQUE AXIS b (P1211)

2a, (x,y,z).

 UNIQUE AXIS c (P1121)

2a, (x,y,z).

No. 5

 UNIQUE AXIS b, CELL CHOICE 1 (C121)

  2a, (0,y,0);2b, (0,y,1/2);4c, (x,y,z).

 UNIQUE AXIS b, CELL CHOICE 2 (A121)

  2a, (0,y,0);2b, (1/2,y,1/2);4c, (x,y,z).

 UNIQUE AXIS b, CELL CHOICE 3 (I121)

  2a, (0,y,0);2b, (1/2,y,0);4c, (x,y,z).

 UNIQUE AXIS c, CELL CHOICE 1 (A112)

  2a, (0,0,z);2b, (1/2,0,z);4c, (x,y,z).

 UNIQUE AXIS c, CELL CHOICE 2 (B112)

  2a, (0,0,z);2b, (1/2,1/2,z);4c, (x,y,z).

 UNlQUE AXIS c, CELL CHOICE 3 (I112)

  2a, (0,0,z);2b, (0,1/2,z);4c, (x,y,z).

Cs（m）点群(order =2)

空间群号： 6–9

No. 6

 UNIQUE AXIS b (P1m1)

  1a, (x,0,z);1b, (x,1/2,z);2c, (x,y,z).

 UNIQUE AXIS c (P11m)

  1a, (x,y,0);1b, (x,y,1/2);2c, (x,y,z).

No. 7

 UNIQUE AXIS b, CELL CHOICE 1 (P1c1)

   2a, (x,y,z).

UNIQUE AXIS b, CELL CHOICE 2 (P1n1)

   2a, (x,y,z).

UNIQUE AXIS b, CELL CHOICE 3 (P1a1)

  2a, (x,y,z).

 UNIQUE AXIS c, CELL CHOICE 1 (P11a)

  2a, (x,y,z).

 UNIQUE AXIS c, CELL CHOICE 2 (P11n)

  2a, (x,y,z).

 UNIQUE AXIS c, CELL CHOICE 3 (P11b)

  2a, (x,y,z).

No. 8

UNIQUE AXIS b, CELL CHOICE 1 (C1m1)

   2a, (x,0,z);4b, (x,y,z).

UNIQUE AXIS b, CELL CHOICE 2 (A1m1)

   2a, (x,0,z);4b, (x,y,z).

UNIQUE AXIS b, CELL CHOICE 3 (I1m1)

  2a, (x,0,z);4b, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 1 (A11m)

  2a, (x,y,0);4b, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 2 (B11m)

  2a, (x,y,0);4b, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 3 (I11m)

  2a, (x,y,0);4b, (x,y,z).

No. 9

UNIQUE AXIS b, CELL CHOICE 1 (C1c1)

  4a, (x,y,z).

UNIQUE AXIS b, CELL CHOICE 2 (A1n1)

  4a, (x,y,z).

UNIQUE AXIS b, CELL CHOICE 3 (l1a1)

  4a, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 1 (A11a)

  4a, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 2 (B11n)

  4a, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 3 (I 11b)

C2h（2/m）点群(order =4)

空间群号： 10–15

No. 10

 UNIQUE AXIS b (P12/m1)

  1a, (0,0,0);1b, (0,1/2,0);1c, (0,0,1/2);1d, (1/2,0,0);1e, (1/2,1/2,0);1f, (0,1/2,1/2);1g, (1/2,0,1/2);1h, (1/2,1/2,1/2);2i, (0,y,0) ;2j, (1/2,y,0) ;2k, (0,y,1/2) ;2l, (1/2,y,1/2) ;2m, (x,0,z) ;2n, (x,1/2,z) ;4o, (x,y,z).

UNIQUE AXIS c (P112/m)

  1a, (0,0,0);1b, (0,0,1/2);1c, (1/2,0,0);1d, (0,1/2,0);1e, (0,1/2,1/2);1f, (1/2,0,1/2);1g, (1/2,1/2,0);1h, (1/2,1/2,1/2);2i, (0,0,z) ;2j, (0,1/2,z) ;2k, (1/2,0,z) ;2l, (1/2,1/2,z) ;2m, (x,y,0) ;2n, (x,y,1/2) ;4o, (x,y,z).

No. 11

 UNIQUE AXIS b (P121/m1)

  2a, (0,0,0);2b, (1/2,0,0);2c, (0,0,1/2);2d, (1/2,0,1/2);2e, (x,1/4, z);4f, (x,y,z).

 UNIQUE AXIS c (P1121/m)

2a, (0,0,0);2b, (0,1/2,0);2c, (1/2,0,0);2d, (1/2,1/2,0);2e, (x,y,1/4);4f, (x,y,z).

No. 12

UNIQUE AXIS b, CELL CHOICE 1 (C12/m1)

2a, (0,0,0);2b, (0,1/2,0);2c, (0,0,1/2);2d, (0,1/2,1/2);4e, (1/4,1/4,0);4f, (1/4,1/4,1/2);4g, (0,y,0);4h, (0,y,1/2);4i, (x,0,z) ;8j, (x,y,z).

UNIQUE AXIS b, CELL CHOICE 2 (A12/m1)

2a, (0,0,0);2b, (0,1/2,0);2c, (1/2,0,1/2);2d, (1/2,1/2,1/2);4e, (0,1/4,1/4);4f, (1/2,1/4,3/4);4g, (0,y,0);4h, (1/2,y,1/2);4i, (x,0,z) ;8j, (x,y,z).

UNIQUE AXIS b, CELL CHOICE 3 (I12/m1)

2a, (0,0,0);2b, (0,1/2,0);2c, (1/2,0,0);2d, (1/2,1/2,0);4e, (3/4,1/4,3/4);4f, (1/4,1/4,3/4);4g, (0,y,0);4h, (1/2,y,0);4i, (x,0,z) ;8j, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 1 (A112/m)

2a, (0,0,0);2b, (0,0,1/2);2c, (1/2,0,0);2d, (1/2,0,1/2);4e, (0,1/4,1/4);4f, (1/2,1/4,1/4);4g, (0,0,z);4h, (1/2,0,z);4i, (x,y,0) ;8j, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 2 (B112/m)

2a, (0,0,0);2b, (0,0,1/2);2c, (1/2,1/2,0);2d, (1/2,1/2,1/2);4e, (1/4,0,1/4);4f, (3/4,1/2,1/4);4g, (0,0,z);4h, (1/2,1/2,z);4i, (x,y,0) ;8j, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 3 (I112/m)

2a, (0,0,0);2b, (0,0,1/2);2c, (0,1/2,0);2d, (0,1/2,1/2);4e, (3/4,3/4,1/4);4f, (3/4,1/4,1/4);4g, (0,0,z);4h, (0,1/2,z);4i, (x,y,0) ;8j, (x,y,z).

No. 13

 UNIQUE AXIS b, CELL CHOICE 1 (P12/c1)

2a, (0,0,0);2b, (1/2,1/2,0);2c, (0,1/2,0);2d, (1/2,0,0);2e, (0,y,1/4);2f, (1/2,y,1/4);4g, (x,y,z).

 UNIQUE AXIS b, CELL CHOICE 2 (P12/n1)

2a, (0,0,0);2b, (0,1/2,1/2);2c, (0,1/2,0);2d, (0,0,1/2);2e, (3/4,y,3/4);2f, (3/4,y,1/4);4g, (x,y,z).

 UNIQUE AXIS b, CELL CHOICE 3 (P12/a1)

2a, (0,0,0);2b, (1/2,1/2,1/2);2c, (0,1/2,0);2d, (1/2,0,1/2);2e, (1/4,y,0);2f, (3/4,y,1/2);4g, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 1 (P112/a)

2a, (0,0,0);2b, (0,1/2,1/2);2c, (0,0,1/2);2d, (0,1/2,0);2e, (1/4,0,z);2f, (1/4,1/2,z);4g, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 2 (P112/n)

2a, (0,0,0);2b, (1/2,0,1/2);2c, (0,0,1/2);2d, (1/2,0,0);2e, (3/4,3/4,z);2f, (1/4,3/4,z);4g, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 3 (P112/b)

2a, (0,0,0);2b, (1/2,1/2,1/2);2c, (0,0,1/2);2d, (1/2,1/2,0);2e, (0,1/4,z);2f, (1/2,3/4,z);4g, (x,y,z).

No. 14

 UNIQUE AXIS b, CELL CHOICE 1 (P121c1)

2a, (0,0,0);2b, (1/2,0,0);2c, (0,0,1/2);2d, (1/2,0,1/2);4e, (x,y,z).

 UNIQUE AXIS b, CELL CHOICE 2 (P121n1)

2a, (0,0,0);2b, (0,0,1/2);2c, (1/2,0,1/2);2d, (1/2,0,0);4e, (x,y,z).

 UNIQUE AXIS b, CELL CHOICE 3 (P121a1)

2a, (0,0,0);2b, (1/2,0,1/2);2c, (1/2,0,0);2d, (0,0,1/2);4e, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 1 (P1121/a)

2a, (0,0,0);2b, (0,1/2,0);2c, (1/2,0,0);2d, (1/2,1/2,0);4e, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 2 (P1121/n)

2a, (0,0,0);2b, (1/2,0,0);2c, (1/2,1/2,0);2d, (0,1/2,0);4e, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 3 (P1121/b)

2a, (0,0,0);2b, (1/2,1/2,0);2c, (0,1/2,0);2d, (1/2,0,0);4e, (x,y,z).

No. 15

UNIQUE AXIS b, CELL CHOICE 1 (C12/c1)

4a, (0,0,0);4b, (0,1/2,0);4c, (1/4,1/4,0);4d, (1/4,1/4,1/2);4e, (0,y,1/4) ;8f, (x,y,z).

UNIQUE AXIS b, CELL CHOICE 2 (A12/n1)

4a, (0,0,0);4b, (0,1/2,0);4c, (0,1/4,1/4);4d, (1/2,1/4,3/4);4e, (3/4,y,3/4) ;8f, (x,y,z).

UNIQUE AXIS b, CELL CHOICE 3 (I12/a1)

4a, (0,0,0);4b, (0,1/2,0);4c, (3/4,1/4,3/4);4d, (1/4,1/4,3/4);4e, (1/4,y,0) ;8f, (x,y,z).

 UNIQUE AXIS c, CELL CHOICE 1 (A112/a)

4a, (0,0,0);4b, (0,0,1/2);4c, (0,1/4,1/4);4d, (1/2,1/4,1/4);4e, (1/4,0,z) ;8f, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 2 (B112/n)

4a, (0,0,0);4b, (0,0,1/2);4c, (1/4,0,1/4);4d, (3/4,1/2,1/4);4e, (3/4,3/4,z) ;8f, (x,y,z).

UNIQUE AXIS c, CELL CHOICE 3 (I112/b)

4a, (0,0,0);4b, (0,0,1/2);4c, (3/4,3/4,1/4);4d, (3/4,1/4,1/4);4e, (0,1/4,z) ;8f, (x,y,z).

🟨 正交晶系(Orthorhombic, a ≠ b ≠ c, α = β = γ= 90°)

正交晶系有59个空间群：16-74

D2（222）点群(order =4)

空间群号： 16–24

No. 16

1a, (0,0,0);1b, (1/2,0,0);1c, (0,1/2,0);1d, (0,0,1/2);1e, (1/2,1/2,0);1f, (1/2,0,1/2);1g, (0,1/2,1/2);1h, (1/2,1/2,1/2);2i, (x,0,0) ;2j, (x,0,1/2) ;2k, (x,1/2,0) ;2l, (x,1/2,1/2) ;2m, (0,y,0) ;2n, (0,y,1/2) ;2o, (1/2,y,0) ;2p, (1/2,y,1/2) ;2q, (0,0,z) ;2r, (1/2,0,z) ;2s, (0,1/2,z) ;2t, (1/2,1/2,z) ;4u, (x,y,z).

No. 17

2a, (x,0,0);2b, (x,1/2,0);2c, (0,y,1/4);2d, (1/2,y,1/4);4e, (x,y,z).

No. 18

2a, (0,0,z);2b, (0,1/2,z);4c, (x,y,z).

No. 19

4a,(x,y,z).

No. 20

4a,(x,0,0); 4b,(0,y,1/4); 8c,(x,y,z).

No. 21

2a, (0,0,0);2b, (0,1/2,0);2c, (1/2,0,1/2);2d, (0,0,1/2);4e, (x,0,0);4f, (x,0,1/2);4g, (0,y,0);4h, (0,y,1/2);4i, (0,0,z) ;4j, (0,1/2,z) ;4k, (1/4,1/4,z) ;8l, (x,y,z).

No. 22

4a, (0,0,0);4b, (0,0,1/2);4c, (1/4,1/4,1/4);4d, (1/4,1/4,3/4);8e, (x,0,0);8f, (0,y,0);8g, (0,0,z);8h, (1/4,1/4,z);8i, (1/4,y,1/4) ;8j, (x,1/4,1/4) ;16k, (x,y,z).

No. 23

2a, (0,0,0);2b, (1/2,0,0);2c, (0,0,1/2);2d, (0,1/2,0);4e, (x,0,0);4f, (x,0,1/2);4g, (0,y,0);4h, (1/2,y,0);4i, (0,0,z) ;4j, (0,1/2,z) ;8k, (x,y,z).

No. 24

4a, (x,0,1/4);4b, (1/4,y,0);4c, (0,1/4,z);8d, (x,y,z).

C2v（mm2）点群(order =4)

空间群号： 25–46

No. 25

1a, (0,0,z);1b, (0,1/2,z);1c, (1/2,0,z);1d, (1/2,1/2,z);2e, (x,0,z);2f, (x,1/2,z);2g, (0,y,z);2h, (1/2,y,z);4i, (x,y,z).

No. 26

2a, (0,y,z);2b, (1/2,y,z);4c, (x,y,z).

No. 27

2a, (0,0,z);2b, (0,1/2,z);2c, (1/2,0,z);2d, (1/2,1/2,z);4e, (x,y,z).

No. 28

2a, (0,0,z);2b, (0,1/2,z);2c, (1/4,y,z);4d, (x,y,z).

No. 29

4a, (x,y,z).

No. 30

2a, (0,0,z);2b, (1/2,0,z);4c, (x,y,z).

No. 31

2a, (0,y,z);4b, (x,y,z).

No. 32

2a, (0,0,z);2b, (0,1/2,z);4c, (x,y,z).

No. 33

4a, (x,y,z).

No. 34

2a, (0,0,z);2b, (0,1/2,z);4c, (x,y,z).

No. 35

No. 36

No. 37

No. 38

No. 39

No. 40

No. 41

No. 42

No. 43

No. 44

No. 45

No. 46

D2h（mmm）点群(order =8)

空间群号： 47–74

No. 48:

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 50

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 59

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 70

ORIGIN CHOICE 1

ORIGIN CHOICE 2

🟩 四方晶系(Tetragonal, a = b ≠ c, α = β = γ= 90°)

四方晶系有68个空间群：75-142

C4（4）点群(order =4)

空间群号： 75–80

S4（‑4）点群(order =4)

空间群号： 81–82

C4h（4/m）点群(order =8)

空间群号： 83–88

No. 85

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 86

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 88

ORIGIN CHOICE 1

ORIGIN CHOICE 2

D4（422）点群(order =8)

空间群号： 89–98

C4v（4mm）点群(order =8)

空间群号： 99–110

D2d（‑42m）点群(order =8)

空间群号： 111–122

D4h（4/mmm）点群(order =16)

空间群号： 123–142

No. 125

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 126

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 129

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 130

ORIGIN CHOICE I

ORIGIN CHOICE 2

No. 133

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 134

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 137

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 138

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 141

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 142

ORIGIN CHOICE 1

ORIGIN CHOICE 2

🟦 三方晶系（Rhombohedral, a = b = c, α = β = γ ≠ 90°）

三方晶系有25个空间群：143-167

C3（3）点群(order =3)

空间群号： 143–146

No. 146

HEXAGONAL AXES

RHOMBOHEDRAL AXES

C3i（‑3）点群(order =6)

空间群号： 147–148

No. 148

HEXAGONAL AXES

RHOMBOHEDRAL AXES

D3（32）点群(order =6)

空间群号： 149–155

No. 155

HEXAGONAL AXES

RHOMBOHEDRAL AXES

C3v（3m）点群(order =6)

空间群号： 156–161

No. 160

HEXAGONAL AXES

RHOMBOHEDRAL AXES

No. 161

HEXAGONAL AXES

RHOMBOHEDRAL AXES                                      

D3d（‑3m）点群(order =12)

空间群号： 162–167

No. 166

HEXAGONAL AXES

RHOMBOHEDRAL AXES                                      

No. 167

HEXAGONAL AXES

RHOMBOHEDRAL AXES

科学网—167号空间群的AEC：Hexagonal axes & Rhombohedral axes - 李世春的博文

科学网—Al解读Hexa axes 和 Rhom axes 的实验数据：Al2O3局域团簇结构 - 李世春的博文

科学网—Al2O3晶体的壳层结构：Hexa vs Rhom - 李世春的博文                                                                       

🟦 六方晶系（Hexagonal, a = b ≠ c, α = β = 90°, γ = 120°）

六方晶系有27个空间群：168-194

C6（6）点群(order =6)

空间群号： 168–173

C3h（‑6）点群(order =6)

空间群号： 174

C6h（6/m）点群(order =12)

空间群号： 175–176

D6（622）点群(order =12)

空间群号： 177–182

C6v（6mm）点群(order =12)

空间群号： 183–186

D3h（‑6m2）点群(order =12)

空间群号： 187–190

D6h（6/mmm）点群

空间群号： 191–194

科学网—194号空间群：原子环境计算程序(AEC) - 李世春的博文

科学网—Al 解读 Laves 相MgZn2的局域团簇结构 - 李世春的博文

🟦 立方晶系(Cubic, a=b=c, α = β = γ = 90°)

立方晶系有36个空间群：195-230

T（23）点群(order =12)

空间群号： 195–199

Th（m‑3）点群(order =24)

空间群号： 200–206

No. 201

ORIGIN CHOICE 1

ORIGIN CHOICE 2

No. 203

ORIGIN CHOICE 1

ORIGIN CHOICE 2

O（432）点群(order =24)

空间群号： 207–214

Td（‑43m）点群(order =24)

空间群号： 215–220

Oh（m‑3m）点群(order =48)

No.221

No.222

 ORIGIN CHOICE 1

 ORIGIN CHOICE 2

No.223

No.224

 ORIGIN CHOICE 1

 ORIGIN CHOICE 2

No.225

科学网—原子环境计算程序(AEC)：No.225 空间群 - 李世春的博文

科学网—Al 对我的 AEC 原子环境计算 的 评价 - 李世春的博文

科学网—余瑞璜 EET 经验电子理论完整的键络参数模板 - 李世春的博文

No.226

No.227

 ORIGIN CHOICE 1

 ORIGIN CHOICE 2

No.228

 ORIGIN CHOICE 1

 ORIGIN CHOICE 2

No.229

No.230

部分空间群的原子坐标列表待完成。

支持单位：