2013晶体学第七章

ByheartSomeplaysaresosuccessfulthattheyrunforyearsonend,Inmanyways,thisisunfortunateforthepooractorswhoarerequiredtogoonrepeatingthesamelinesnightafternight.Onewouldexpectthemtoknowtheirpartsbyheartandneverhavecausetofalter.Yetthisisnotalwaysthecase.AfamousactorinahighlysuccessfulplaywasoncecastintheroleofanaristocratwhohadbeenimprisonedintheBastillefortwentyyears.Inthelastact,agaolerwouldalwayscomeontothestagewithaletterwhichhewouldhandtotheprisoner.Eventhoughthenoblewasexpectedtoreadtheletterateachperformance,healwaysinsistedthatitshouldbewrittenoutinfull.Onenight,thegaolerdecidedtoplayajokeonhiscolleaguetofindoutif,aftersomanyperformances,hehadmanagedtolearnthecontentsoftheletterbyheart.Thecurtainwentuponthefinalactoftheplayandrevealedthearistocratsittingalonebehindbarsinhisdarkcell.Justthen,thegaolerappearedwiththepreciousletterinhishands.Heenteredthecellandpresentedthelettertothearistocrat.Butthecopyhegavehimhadnotbeenwrittenoutinfullasusual.Itwassimplyablanksheetofpaper.Thegaolerlookedoneagerly,anxioustoseeifhisfellowactorhadatlastlearnthislines.Thenoblestaredattheblanksheetofpaperforafewseconds.

Then,squintinghiseyes,hesaid:'Thelightisdim.Readthelettertome'.Andhepromptlyhandedthesheetofpapertothegaoler.Findingthathecouldnotrememberawordofthelettereither,thegaolerreplied:'Thelightisindeeddim,sire,Imustgetmyglasses.'Withthis,hehurriedoffthestage.Muchtothearistocrat'samusement,thegaolerreturnedafewmomentslaterwithapairofglassesandtheusualcopyoftheletterwithheproceededtoreadtotheprisoner.第七章

晶

体

学

空

间

群

7.1点式空间群

若平移群

T

在点群

P

的任意操作的相似变换下不变,则

T

与

P

可构成半直积群,即点式空间群(symmorphicspacegroup)。

G=T

∧

P

Syngony中心非中心非中心triclinic1

(C1)monoclinic2/m(C2h)2

(C2)m(Cs)orthorhombicmmm222(D2)mm2(C2v)trigonal

3

(C3)32(D3)3m(C3v)tetragonal4/m

(C4h)4

(C4)

4/mmm(D4h)422(D4)4mm(C4v)hexagonal6/m

(C6h)6(C6)

6/mmm(D6h)622(D6)6mm(C6v)cubic23(T)

432(O)SyngonyBravaislatticeLatticepointgroupPointgrouptriclinicaP1monoclinicmP,mS2/m

2/m,

2,morthorhombicoP,oI,oS,oFmmmmmm,222,

mm2tetragonaltP,tI4/mmm4/m,4,

4/mmm,422,4mm,

cubiccP,cI,cF

23

432,hexagonalhP6/mmm6/m,6,

6/mmm,622,6mm,

trigonalhP6/mmm332,3m,hR晶系点阵点群P空间群三斜aP()1

P1(1)

(2)单斜mP(2/m)mC(2/m)2,m2/m

P2(3)

Pm

(6)

P2/m

(10)

C2(5)

Cm(8)

C2/m

(12)正交

oP(mmm)oC(mmm)

oF(mmm)oI(mmm)222mm2mmm

P222(16)

Pmm2(25)

Pmmm

(47)

C222(21)

Cmm2(35)

Amm2(38)

Cmmm

(65)F222(22)

Fmm2(42)

Fmmm

(69)

I222(23)

Imm2(44)

Immm(71)

P

7.1.1三斜晶系空间群例举

反演与平移的组合定理以

o点为反演心先反演,再平移t,两步操作等效于对一个新反演心o’的反演,o’位于

t/2

(t/2

的始点为

o)。证:设平移矢

a位于纸面,o为对称心。

对

o点{0,0,0}的反演

等效于绕与纸面垂直并过

o点的

2重轴

C2{0,0,z}的旋转与以纸面为反映面的反映之组合,a{0,0,0}=aC2{0,0,z}mh

=C2'{a/2,0,z}mh

={1/2,0,0}

_cell_length_a6.4227(4)_cell_length_b9.4246(5)_cell_length_c16.9326(9)_cell_angle_alpha83.890(3)_cell_angle_beta81.735(3)_cell_angle_gamma

70.108(3)_cell_volume951.96(9)_cell_formula_units_Z27.1.2单斜晶系空间群例举

_symmetry_cell_settingMonoclinic_symmetry_space_group_name_H-MP2

_cell_length_a7.6924(15)_cell_length_b6.6543(13)_cell_length_c34.269(7)_cell_angle_alpha90.00_cell_angle_beta91.742(2)_cell_angle_gamma90.00_cell_volume1753.3(6)_cell_formula_units_Z2正交晶系空间群例举

四方晶系例举

Wyckoffletter

区别晶胞中对称地位不同的位置,称为

Wyckoffposition。acFeHg(SCN)4ac定理7-2:晶体结构中的分子(或原子团)只能处于位置点群为该分子的点群之子群的

Wyckoff位置,即分子点群至少具有所在

Wyckoff位置的对称性。

三方晶系例举

_chemical_formula_moiety'C36H27N3‘_chemical_formula_weight501.61_symmetry_cell_settingHexagonal_symmetry_space_group_name_H-MR-3_cell_length_a22.6693(6)_cell_length_b22.6693(6)_cell_length_c8.9507(3)_cell_angle_alpha90.00_cell_angle_beta90.00_cell_angle_gamma120.00_cell_volume3983.5(2)_cell_formula_units_Z6_cell_measurement_temperature150(2)_refine_ls_number_reflns2008_refine_ls_number_parameters154_refine_ls_number_restraints0_refine_ls_R_factor_all0.0611_refine_ls_R_factor_gt0.0438_refine_ls_wR_factor_ref0.1145_refine_ls_wR_factor_gt0.1076_refine_ls_restrained_S_all0.969_refine_ls_shift/su_max0.001_diffrn_measured_fraction_theta_max987_diffrn_reflns_theta_full27.46_refine_diff_density_max0.198_refine_diff_density_min-0.1507.2非点式空间群G=T∧P'=T∧{(W1=I,w1=0);(W2,w2);(W3,w3);...,(Ws,ws)}=T+T(W2,w2)+T(W3,w3)+...+T(Ws,ws)晶系点阵与点阵点群点群P空间群单斜mP(2/m)mC(2/m)2,m,2/m

P2,Pm,P2/m,C2,Cm,C2/m

P21,Pc,P21/m,P2/c,P21/c

Cc,C2/c

定理7-3:

源于中心对称点群的点式和非点式空间群都是中心对称的。空间群的中心对称与否取决于其点群的中心对称与否。

晶系点阵点群P空间群单斜mP(2/m)mC(2/m)2,m,2/m

P2,Pm,P2/m,C2,Cm,C2/m

P21,Pc,P21/m,P2/c,P21/c

Cc,C2/c

定理7-3:空间群的中心对称与否取决于其点群的中心对称与否。P2/m

W2W1=W3P21/c

(W2,w2)(W1,w1)=(W2W1,W2w1+w2)P21群

由

P2=T∧{1;2{0,y,0}},可引伸出

P21=T∧{1;(2{0,y,0},b/2)}C2=T∧{1,2}={

(ua+vb+wc)∪(ua+vb+wc+a/2+b/2)}∧{1;2{0,y,0}}C21=T∧{1,21}={

(ua+vb+wc)∪(ua+vb+wc+a/2+b/2)}∧{1;(2{0,y,0},b/2)}

C21

与

C2为同一种空间群

ETHgCl3Pc(Pc,Pa,Pn)晶系点阵点群P空间群单斜mP(2/m)mC(2/m)2,m,2/m

P2,Pm,P2/m,C2,Cm,C2/m

P21,Pc,P21/m,P2/c,P21/c

Cc,C2/c

Whatistheanglebetweenthetwomoleculardipolemomentintheunitcell?Whatisthedirectionofthepolarizationvector?_symmetry_cell_settingMonoclinic_symmetry_space_group_name_H-MPn_symmetry_equiv_pos_as_xyz'x,y,z''x+1/2,-y,z+1/2'_cell_length_a3.9630(3)_cell_length_b11.0210(9)_cell_length_c8.7098(7)_cell_angle_alpha90.00_cell_angle_beta101.456(5)_cell_angle_gamma90.00_cell_volume372.83(5)_cell_formula_units_Z2S1S0.4980(5)0.13296(18)0.7880(2)S2S0.6837(3)0.08389(15)0.48270(17)S3S0.8720(4)0.17942(15)0.1787(2)S4S0.6976(3)0.33170(13)0.60625(16)S5S0.8828(4)0.43139(16)0.3049(2)O1O1.0496(11)0.3738(3)0.0207(5)C1C0.6216(13)0.1800(5)0.6355(8)C2C0.7833(17)0.1996(6)0.3650(8)C3C0.9477(17)0.3377(7)0.1491(8)C4C0.7882(18)0.3143(6)0.4211(8)r(O1,S1)=7.893Cm={

(ua+vb+wc)∪(ua+vb+wc+a/2+b/2)}∧{1;