1、第十一講第十一講 空間群空間群(3)(3)：非點式空間群非點式空間群1 1、非點式空間群非點式空間群舉例分析舉例分析2 2、空間群國際表空間群國際表舉例分析舉例分析3 3、二維空間群二維空間群( (全部全部) )第十二講第十二講 空間群空間群(4)(4)：空間群中的特殊位置空間群中的特殊位置1 1、空間群國際表中的特殊位置空間群國際表中的特殊位置舉例分析舉例分析2 2、二維空間群二維空間群( (全部全部) )P4nc4mm Tetragonal+, + + + +,8 c 1 x,y,z; x,y,z; +x,-y,+z; -x,+y,+z; y,x,z; y,x,z; +y,+x,+z; -

2、y,-x,-z.4 b 2 0,z; ,0,z; 0,+z; ,0,+z.2 a 4 0,0,z; , , +z.Origin on 4Number of positions, Wyckoff notation, and point symmetryCo-ordinates of equivalent positions6C4vNo. 104P4ncConditions limiting possible reflectionsGeneral:hkl: No conditions 0kl: k + l = 2n hkl: l = 2nSpecial:hkl: h + k = 2n; l = 2

3、n hkl: h + k + l = 2n+1 a 1 x,y,z2 i 1 x,y,z; x,y,z.1 h 1 ,0,.1 g 1 0,.1 f 1 ,0,.1 e 1 ,0.1 d 1 ,0,0.1 c 1 0,0.1 b 1 0,0,.1 a 1 0,0,0.+,_+,_+,_+,_Wyckoff 符號符號P1(C1, No. 1)1P1(Ci, No. 2)1Origin at 1等效位置數等效位置數特殊位置特殊位置位置對稱性位置對稱性Fm3m (Oh, No. 225)5Im3m (Oh, No. 229)9192 l 1 x,y,z; .8 c 43m ,; ,.4 b m3m

4、, .4 a m3m 0,0,0. (0,0,0; 0,; ,0,; ,0) +96 l 1 x,y,z; .8 c 3m ,; ,; ,; ,. 6 b 4/mmm 0,; ,0,; ,0.2 a m3m 0,0,0. (0,0,0; ,) +點式空間群：點式空間群：由全部作用于同一個公共點上的對稱操作完全確由全部作用于同一個公共點上的對稱操作完全確定，或者說僅由點對稱操作和平移對稱操作組合而產生。定，或者說僅由點對稱操作和平移對稱操作組合而產生。? 螺旋軸或滑移面不是其螺旋軸或滑移面不是其基本操作基本操作。至少有一個至少有一個基本操作基本操作為為非點式操作即是非點式空間群非點式操作即是非點

5、式空間群? 點式空間群在單胞中一定至少有一個位置具有與空間群點點式空間群在單胞中一定至少有一個位置具有與空間群點群相同的群相同的位置對稱性位置對稱性空間群操作：空間群操作：r = R|tr = Rr + t (賽茲算符賽茲算符)對非點式操作對非點式操作 t = ，對于點式操作對于點式操作t = = 0R|t、 1|tn、 R|0 、R| 一般來說，對于給定的一組等效位置，一般來說，對于給定的一組等效位置，等效位置數等效位置數乘以位置對稱性位置對稱性點群的階點群的階等于空間群點群的階空間群點群的階P62m (D3h, No. 189)3+,_,+,_,+,_,+,_,+,_,+,_,+,_,+,

6、_,+,_,+,_,+,_,+,_,Origin at 62m12 l 1 x,y,z; y,x-y,z; y-x,x,z; x,y,z; y,x-y,z; y-x,x,z; y,x,z; x,y-x,z; x-y,y,z; y,x,z; x,y-x,z; x-y,y,z.1 b 62m 0,0,.1 a 62m 0,0,0.Wyckoff 符號符號P2+2 e 1 x,y,z; x,y,z.1 d 2 , , z.1 c 2 , 0, z.1 b 2 0, , z.1 a 2 0, 0, z.P21+ + + + +2 a 1 x,y,z; x,y,+z.Pm+-,+-,+-,+-,-+,+

7、-+,+Pb2 c 1 x,y,z; x,y,z.1 b m x, y, .1 a m x, y, 0.2 a 1 x,y,z; x,+y,z.特征特征1：特殊位置：特殊位置與與非點式操作非點式操作Origin on 2+_+_+_+_+_+_+_+_+_+_+_+_8 p 1 x,y,z; x,y,z; x,y,z; x,y,z; y,x,z; y,x,z; y,x,z; y,x,z.1 a 42 0,0,0.P422 (D4, No. 89)1P4212 (D4, No. 90)28 g 1 x,y,z; x,y,z; -x,+y,z; +x,-y,z; y,x,z; y,x,z; +y,

8、-x,z; -y,+x,z.2 a 222 0,0,0; ,0.特征特征2：點式點式SG與與非點式非點式SG+_+_+_+_+_+_+,+,+,+,+,+Pmm2 (C2v, No. 25)1Origin on mm24 i 1 x,y,z; x,y,z; x,y,z; x,y,z.2 h m ,y,z; ,y,z.2 g m 0,y,z; 0,y,z2 f m x,z; x,z.2 e m x,0,z; x,0,z1 d mm ,z.1 c mm ,0,z.1 b mm 0,z.1 a mm 0,0,z.Pcc2 (C2v, No. 27)3,+,+,+,+,+,+,+,+4 e 1 x,y

9、,z; x,y,z; x,y,+z; x,y,+z.2 d 2 ,z; ,+z.2 c 2 ,0,z; ,0,+z2 b 2 0,z; 0,+z.2 a 2 0,0,z; 0,0,+zOrigin on 2Pban (D2h, No. 50)4P 2/b 2/a 2/n+,+,-,-,-+-+-+-+Origin at 222, at , , 0 from 1+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,P4mm (C4v, No. 99)1Origin on 4mm8 c 1 x,y,z; x,y,z; x,y,z; x,y,z; y,x,z; y,x,z; y,x,z;

10、y,x,z.4 f m x,z; x,z; ,x,z; ,x,z.4 e m x,0,z; x,0,z; 0,x,z; 0,x,z. 4 d m x,x,z; x,x,z; x,x,z; x,x,z.2 c mm ,0,z; 0,z.1 b 4mm ,z.1 a 4mm 0,0,z.P4nc4mm Tetragonal+, + + + +,8 c 1 x,y,z; x,y,z; +x,-y,+z; -x,+y,+z; y,x,z; y,x,z; +y,+x,+z; -y,-x,-z.4 b 2 0,z; ,0,z; 0,+z; ,0,+z.2 a 4 0,0,z; ,+z.Origin on

11、46C4vNo. 104P4nc4 d 1 x,y,z; x,y,z; y,x,z; y,x,z. 2 c 2 0,z; ,0,z.1 b 4 ,z.1 a 4 0,0,z.P4Pmmm (D2h, No. 47)1+,-+,-,+,-+,-+,-,+,-+,-+,-,+,-+,-+,-,+,-+P 2/m 2/m 2/m8 1 x,y,z; x,y,z; x,y,z; x,y,z; x,y,z; x,y,z; x,y,z; x,y,z.1 a mmm 0,0,0.16 r 1 x,y,z; x,y,z; x,y,z; x,y,z; x,y,z; x,y,z; x,y,z; x,y,z.2 a

12、 mmm 0,0,0.Cmmm (D2h, No. 65)19Origin at center (mmm)特征特征3 3：初基：初基與與有心有心 等效位置數等效位置數(0,0,0; ,0) +Fm3m (Oh, No. 225)5Im3m (Oh, No. 229)9192 l 1 x,y,z; .8 c 43m ,; ,.4 b m3m , .4 a m3m 0,0,0. (0,0,0; 0,; ,0,; ,0) +96 l 1 x,y,z; .8 c 3m ,; ,; ,; ,. 6 b 4/mmm 0,; ,0,; ,0.2 a m3m 0,0,0. (0,0,0; ,) +? 螺旋軸或

13、滑移面不是螺旋軸或滑移面不是點式空間群點式空間群的的基本操作基本操作。至少有一個至少有一個基本操作基本操作為為非點式操作非點式操作即為即為非非點式空間群點式空間群? 點式空間群在單胞中一定至少有一個位置具有與空間群點點式空間群在單胞中一定至少有一個位置具有與空間群點群相同的群相同的位置對稱性位置對稱性空間群操作：空間群操作：r = R|tr = Rr + t (賽茲算符賽茲算符)對非點式操作對非點式操作 t = ，對于點式操作對于點式操作t = = 0R|t、 1|tn、 R|0 、R| 一般來說，對于給定的一組等效位置，一般來說，對于給定的一組等效位置，等效位置數等效位置數乘以位置對稱性位置

14、對稱性點群的階點群的階等于空間群點群的階。空間群點群的階。 對有心單胞，為對有心單胞，為2h, 3h, 2h, 3h, 或或4h4hOrthorhombic 222P 21 21 21No. 19Origin halfway between three pairs of non-intersecting screw axes+_+_P 212121 D24Number of positions, Wyckoff notation, and point symmetryCo-ordinates of equivalent positions 4 a 1 x,y,z; -x, y,+z; +x,-

15、y, z; x,+y,-z; Orthorhombic mmmP 21/n 21/m 21/aNo. 62Origin at 1_,_,+,_,_,+,_,+,+,+_P nma D2h16Number of positions, Wyckoff notation, and point symmetryCo-ordinates of equivalent positions 8 d 1 x,y,z; +x,-y,-z; x,+y,z; -x,y,+z; x,y,z; -x,+y,+z; x,-y,z; +x,y,-z.4 c m x,z; x,z; -x,+z; +x,-z.4 b 1 0,0

16、,; 0,; ,0,0; ,0.4 a 1 0,0,0; 0,0; ,0,; ,. P 21 21 21P 2/m 2/m 2/mOrigin at 1 P 21/n 21/m 21/aabc_,_,+,_,_,+,_,+,+,+_ 21/a 21/m 21/n ,Number of positions, Wyckoff notation, and point symmetryCo-ordinates of equivalent positions_,_,+,_,_,+,_,+,+,+_P nmaGdFeO3 GdFeO baccbabcaabc+GdFeO3 GdFeO bca (4c) (

17、8d) Pm3m (Oh, No. 221)1abc48 n 1 x,y,z; .3 c 4/mmm 0,; ,0,; ,0.1 b m3m , .1 a m3m 0,0,0. ()() Fm3m (Oh, No. 225)5Im3m (Oh, No. 229)9192 l 1 x,y,z; .8 c 43m ,; ,.4 b m3m , .4 a m3m 0,0,0. (0,0,0; 0,; ,0,; ,0) +96 l 1 x,y,z; .8 c 3m ,; ,; ,; ,. 6 b 4/mmm 0,; ,0,; ,0.2 a m3m 0,0,0. (0,0,0; ,) +Ferromag

18、netism :Superconductivity:Ferroelectricity:Multiferroics(La,Sr)MnO3(La,Ca)MnO3SrRuO3YBa2Cu3O7(La,Sr)CuO4SrTiO3BaTiO3Pb(Zr,Ti)O3Pb(Mg1/3Nb2/3)O3BiFeO3DyMnO3Perovskite Structure1 1、非點式空間群非點式空間群舉例分析舉例分析2 2、空間群國際表空間群國際表舉例分析舉例分析3 3、二維空間群二維空間群( (全部全部) )一維情況：一維情況：aa點陣，點陣，Latticep p1p pmp p1p pmOblique, a b

19、 90oRectangular, a b = 90oSquare, a = b = 90o60o angle rhombus,Hexagonal, a = b = 120o斜方斜方長方長方有心長方有心長方正方正方六角六角晶系晶系點群點群布拉菲布拉菲點陣點陣三三 斜斜單單 斜斜正正 交交四四 方方三三 方方六六 方方立立 方方PPPPPPP1,1m,2,2/m222, mm2, mmm42m,4,422,4/mmm4mm,4/m,4,3m3, 3m, 32,3,622,6/mmm6mm,6/m,6,62m,6,23, m3,432, m3m43m,P1,P1Pm,P2,P2/mP222,Pmm2

20、,PmmmP42m,P4,P422,P4/mmm,P4mm,P4/m,P4,P31m,P3, P3m1, P312, P3,P23, Pm3,P432, Pm3mP43m,Bm,B2,B2/mC222, Cmm2,Cmmm,I222,Imm2,ImmmF222,Fmm2,FmmmAmm2BCIFIP4m2I42m,I4,I422,I4/mmm,I4mm,I4/m,I4,I4m2RR3mR3, R3m, R32, R3,P321, P3m1P31m,P6m2,P6,P622,P6/mmm,P6mm,P6/m,P6,P62mIFI23, Im3,I432,Im3mI43m,F23, Fm3,F43

21、2, Fm3mF43m,二維空間群二維空間群點陣點陣點群點群空間群空間群序號序號斜形斜形1p11 1對應圖像對應圖像p矩形矩形p, c正方形正方形p六方形六方形p2p211p4433m66mmp3p3m1p31mp6p6mm2 2p2mmp2mgP2gg (n)c2mm2mm6 67 78 89 91010p4mmC4gm (g)4mm1111121213131414151516161717p1m1p1g1c1m1m3 34 45 5T The he 1717 Two-dimensional Space Groups Two-dimensional Space GroupsEquivalent

22、 positions, Symmetry, and Possible reflectionsp1p2 p2111 a 1 x,y.Origin on 1Number of positions, Wyckoff notation, and point symmetryOrigin at 22 e 1 x, y, x, y.1 d 2 1/2, 1/2.1 c 2 1/2, 0.1 b 2 0,1/2.1 a 2 0,0.pm p1m1pg p1g1cm c1m1,Origin on m2 c 1 x, y; x,y.1 b m 1/2, y.1 a m 0, y.Origin on m4 b 1

23、 x, y; x, y.2 a m 0, y.,Co-ordinates of equivalent positions(0,0; 1/2,1/2)+,Origin on g2 a 1 x, y; x,1/2+y.pmm p2mmpmg p2mgOrigin at 2mmOrigin at 24 i 1 x,y; x,y; x,y; x,y.2 h m ,y; ,y.2 g m 0,y; 0,y.2 f m x,1/2; x,1/2.2 e m x,0; x,0.1 d mm 1/2, 1/2.1 c mm 1/2, 0.1 b mm 0, 1/2.1 a mm 0, 0.,4 d 1 x,y

24、; x,y; +x,y; -x,y.2 c m ;y; ,y. 2 b 2 0,; ,. 2 a 2 0,0; ,0.pgg p2ggOrigin at 2,4 c 1 x,y; x,y; +x,-y; -x,+y.2 b 2 ,0; 0,. 2 a 2 0,0; , .cmm c2mmOrigin at 2mm,Co-ordinates of equivalent positions(0,0; 1/2,1/2)+8 f 1 x,y; x,y; x,y; x,y.4 e m 0,y; 0,y.4 d m x,0; x,0.4 c 2 ,; ,.2 b mm 0,.2 a mm 0,0.p4 p

25、4p4m p4mmOrigin at 44 d 1 x,y; x,y; y,x; y,x.2 c 2 ,0; 0,.1 b 4 ,.1 a 4 0,0.Origin at 4mm,8 g 1 x,y; x,y; y,x; y,x; x,y; x,y; y,x; y,x.4 f m x,x; x,x; x,x; x,x.4 e m x,; x,; ,x; ,x.4 d m x,0; x,0; 0,x; 0,x.2 c mm ,0; 0,.1 b 4mm ,.1 a 4mm 0,0.abcP4mm+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,P4bmp4g p4gmOrigin at 4,8 d 1 x,y; y,x; -x,+y; -y,-x; x,y; y,x; +x,-y; +y,+x.4 c m x,+x; x,-x; +x,x; -x,x.2 b mm2 ,0; 0,.2 a 4 0,0; ,.P4bm, Plus “+, and