第五講—點(diǎn)群(1)

1、第五講第五講 點(diǎn)群點(diǎn)群(I)360o/n (n = 1,2,3,4,6)1 (E, L1)2 (C2, L2)3 (C3, L3)4 (C4, L4)6 (C6, L6)1 (i, C)2 (, P), m3 (S65, Li3)4 (S43, Li4)6 (S35, Li6)+,+_,旋轉(zhuǎn)軸，旋轉(zhuǎn)軸， n 旋轉(zhuǎn)反演軸，旋轉(zhuǎn)反演軸， n 點(diǎn)對(duì)稱操作點(diǎn)對(duì)稱操作對(duì)稱條件對(duì)稱條件晶系晶系特點(diǎn)特點(diǎn)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1(E)或或1(i)2(C2)或或2(m)兩個(gè)兩個(gè)2(C2)或或2(m)4(C4)或或4(S43)3(C3)或或3(S65)6(C6)或或

2、6(S35)a b c, abc, = = 90o abc, = = = 90oa = bc, = = = 90oa = bc, = = 90o, = 120oa = b = c, = = = 90oa = b = c, = = 菱形菱形a = bc, = = 90o, = 120o全對(duì)稱點(diǎn)群全對(duì)稱點(diǎn)群12/mmmm4/mmm6/mmmm3m3m1(L1)m(P)1(C)42m (Li42L22P)2(L2)2/m (L2PC)222(3L2)mm2 (L22P)mmm (3L23PC)4 (Li4)422 (L44L2)4/mmm(L44L25PC)4mm (L44P)4/m (L4PC)4

3、(L4)62m (Li63L23P)6 (Li6)622 (L66L2)6/mmm (L66L27PC)6mm (L66P)6/m (L6PC)6(L6)23(3L24L3)m3 (3L24L33PC)432 (3L44L36L2)m3m (3L44L36L29PC)3m (Li33L23P)3(L3)3m (L33P)32(L33L2)43m (3Li44L36P)3(Li3)1(C1)m (C1h)1(Ci)42m (D2d)2(C2)2/m (C2h)222(D2)mm2 (C2v)mmm (D2h)4 (S4)422 (D4)4/mmm(D4h)4mm (C4v)4/m (C4h)4(

4、C4)62 (D3h)6 (C3h)622 (D6)6/mmm (D6h)6mm (C6v)6/m (C6h)6(C6)23(T)m3 (Th)432 (O)m3m (Oh)3m(D3d)3(C3)3m (C3v)32(D3)43m (Td)3(S6)點(diǎn)群各符號(hào)的順序點(diǎn)群各符號(hào)的順序123三斜三斜單斜單斜正交正交四方四方三方三方六方六方立方立方只用一個(gè)符號(hào)只用一個(gè)符號(hào)第一種定向：第一種定向：c是唯一軸；第二種定向：是唯一軸；第二種定向：b是唯一軸是唯一軸2或或2沿沿a2或或2沿沿b2或或2沿沿c4或或4沿沿c2或或2沿沿a和和b2或或2沿沿ab3或或3沿沿c2或或2沿沿a、b和和a+b6或或6

5、沿沿c2或或2 a、b和和a+b3或或3沿沿 2或或2沿沿 2或或2 a、b和和a+b2或或2沿沿a、b和和a+b4、4、2或或2沿沿 極射赤面（平）投影極射赤面（平）投影一般形一般形xy四方晶系：四方四面體晶類四方晶系：四方四面體晶類4 (Li4)復(fù)習(xí)：復(fù)習(xí)：第四講第四講 點(diǎn)對(duì)稱操作、點(diǎn)對(duì)稱操作、晶系晶系第四講第四講(I)(I) 點(diǎn)對(duì)稱操作點(diǎn)對(duì)稱操作在操作過(guò)程中保持空間有一不動(dòng)點(diǎn)在操作過(guò)程中保持空間有一不動(dòng)點(diǎn)的對(duì)稱操作。的對(duì)稱操作。比較：平移操作比較：平移操作(t)對(duì)稱操作、點(diǎn)對(duì)稱操作、參考軸、對(duì)稱算符對(duì)稱操作、點(diǎn)對(duì)稱操作、參考軸、對(duì)稱算符對(duì)分子或晶體，對(duì)分子或晶體，使其各個(gè)原子的位置發(fā)生變

6、換使其各個(gè)原子的位置發(fā)生變換的操作，但其結(jié)果是使分子或的操作，但其結(jié)果是使分子或晶體的狀態(tài)與操作前的狀態(tài)正晶體的狀態(tài)與操作前的狀態(tài)正好相同。好相同。在操作過(guò)程中保在操作過(guò)程中保持空間有一不動(dòng)點(diǎn)的對(duì)稱操作。持空間有一不動(dòng)點(diǎn)的對(duì)稱操作。bac1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, ABABttt = mt鏡面鏡面m001，反映，反映恒等恒等 1x y zx y z-1 -0 -0-0 -1 -0-0 -0 -1=x y zx y z=-1 -0 -0-0 -1 -0-0 -0 -1參考軸：參考軸：對(duì)稱算符對(duì)稱算符r = Rra, b, c (無(wú)需正交無(wú)需正交)x y

7、 zx y za11 a21 a31a11 a21 a31a11 a21 a31=r = xa + yb + zcr = xa + yb + zc(x,y,z)(x,y,z)rrabc = 90o 直角坐標(biāo)直角坐標(biāo) = 120o 六角坐標(biāo)六角坐標(biāo)對(duì)稱心對(duì)稱心(i)，反演，反演x y zx y z-1 -0 -0-0 -1 -0-0 -0 -1=m001x y zx y z=-1 -0 -0-0 -1 -0-0 -0 -1m100 x y zx y z=-1 -0 -0-0 -1 -0-0 -0 -1m010 x y zx y z=-1 -0 -0-0 -1 -0-0 -0 -1m110 x y

8、 zx y z=-0 -1 -0-1 -0 -0-0 -0 -1m110 x y zx y z=-0 -1 -0-1 -0 -0-0 -0 -1baba直角坐標(biāo)直角坐標(biāo) 六角坐標(biāo)六角坐標(biāo) m001 ( = 90o)x y zx y z=-1 -0 -0-0 -1 -0-0 -0 -1m001 ( = 120o)x y zx y z=-1 -0 -0-0 -1 -0-0 -0 -1對(duì)稱心對(duì)稱心(i)，反演，反演x y zx y z-1 -0 -0-0 -1 -0-0 -0 -1=m100 ( = 90o)x y zx y z=-1 -0 -0-0 -1 -0-0 -0 -1m100 ( = 12

9、0o)x y zx y z=-1 -0 -0-1 -1 -0-0 -0 -11 (E, L1)x y zx y z=-1 -0 -0-0 -1 -0-0 -0 -12 (C2, L2)+_+_3 (C3, L3)3 (C3, L3); 33; 333x y zx y z=-0 -1 -0-1 -1 -0-0 -0 -14 (C4, L4)4 (C4, L4); 42 = 2; 43; 44 = 1x y zx y z=-0 -1 -0-1 -0 -0-0 -0 -16 (C6, L6)6 (C6, L6); 62 = 3; 63 = 2; 64; 65; 66 = 1+_,1 (i, C)+_

10、,+_,x y zx y z=-1 -0 -0-0 -1 -0-0 -0 -12 (, P), m+,+_,+_,3 (S65, Li3)3 (S65, Li3) = 3+Cx y zx y z=-0 -1 -0-1 -1 -0-0 -0 -14 (S43, Li4)x y zx y z=-0 -1 -0-1 -0 -0-0 -0 -16 (S35, Li6)6 (S35, Li6) = 3+P360o/n (n = 1,2,3,4,6)1 (E, L1)2 (C2, L2)3 (C3, L3)4 (C4, L4)6 (C6, L6)1 (i, C)2 (, P), m3 (S65, Li3)

11、4 (S43, Li4)6 (S35, Li6)+_,+_,旋轉(zhuǎn)軸，旋轉(zhuǎn)軸， n 旋轉(zhuǎn)反演軸，旋轉(zhuǎn)反演軸， n 3 (S65, Li3)6 (S35, Li6) S6, S62(C3), S63(i), S64(C32), S65, S66(E) 35, 34, 33, 32, 31, 36 S3, S32(C32), S33(h), S34(C3), S35, S36(E) 65, 64, 63, 62, 6, 66對(duì)稱操作元素對(duì)稱操作元素4 (S43, Li4)4 (C4, L4)4 (C4, L4); 42 = 2; 43; 44 = 14 (S43, Li4); 42 (S42) =

12、2; 43 (S4); 44 = 1xy四方雙錐晶類四方雙錐晶類4/m (L4PC)4 (L4); 42 = 2; 43 ; 44 = 1; m (P); 1(C); 4(Li4); 43C41; C42 = C2; C43; C44 = E; ; i; S43; S41 (E, L1)2 (C2, L2)3 (C3, L3)4 (C4, L4)6 (C6, L6)1 (i, C)2 (, P), m3 (S65, Li3)4 (S43, Li4)6 (S35, Li6)(C41, C42, C43, C44 )(C61, C62, C63, C64 , C65, C66 )(C31, C32

13、, C33)n = 1n (iCn), Sn = Cn(h, v, d) S6, S62(C3), S63(i), S64(C32), S65, S66(E) 35, 34, 33, 32, 31, 36 S3, S32(C32), S33(h), S34(C3), S35, S36(E) 65, 64, 63, 62, 6, 66 S4(43), S42(42), S43(4), S44(E)點(diǎn)對(duì)稱操作符號(hào)點(diǎn)對(duì)稱操作符號(hào)!第四講第四講(II)(II) 晶系晶系七種晶系的提出，正是由于將各種不同的真旋轉(zhuǎn)七種晶系的提出，正是由于將各種不同的真旋轉(zhuǎn)和非真旋轉(zhuǎn)操作應(yīng)用于單胞的各個(gè)軸或點(diǎn)陣平移和非真旋

14、轉(zhuǎn)操作應(yīng)用于單胞的各個(gè)軸或點(diǎn)陣平移矢量所得到的結(jié)果。矢量所得到的結(jié)果。晶系是由晶體的對(duì)稱性來(lái)劃分的。單胞幾何形狀晶系是由晶體的對(duì)稱性來(lái)劃分的。單胞幾何形狀(晶胞參數(shù)晶胞參數(shù))的限制是對(duì)稱性要求的結(jié)果。的限制是對(duì)稱性要求的結(jié)果。(x,y,z)(x,y,z)rrabc = 90o 直角坐標(biāo)直角坐標(biāo) = 120o 六角坐標(biāo)六角坐標(biāo)參考軸：參考軸：對(duì)稱算符對(duì)稱算符r = Rra, b, c (無(wú)需正交無(wú)需正交)x y zx y za11 a21 a31a11 a21 a31a11 a21 a31=r = xa + yb + zcr = xa + yb + zc這里這里(x, y, z) 和和 (x,

15、y, z) 可以是單胞軸長(zhǎng)的分可以是單胞軸長(zhǎng)的分?jǐn)?shù)，表示原子位置。數(shù)，表示原子位置。三斜晶系：三斜晶系：1(E)或或1(i)恒等恒等 Ex y zx y z-1 -0 -0-0 -1 -0-0 -0 -1=對(duì)稱心對(duì)稱心(i)，反演，反演x y zx y z-1 -0 -0-0 -1 -0-0 -0 -1=兩種操作中，單胞軸彼此之間沒(méi)有任何約束關(guān)系，兩種操作中，單胞軸彼此之間沒(méi)有任何約束關(guān)系，所以對(duì)單胞的幾何形狀沒(méi)有特別限制，即：所以對(duì)單胞的幾何形狀沒(méi)有特別限制，即：a b c, 單斜晶系：?jiǎn)涡本担?(C2)或或2(m)m001x y zx y z=-1 -0 -0-0 -1 -0-0 -0

16、-12001x y zx y z=-1 -0 -0-0 -1 -0-0 -0 -1r = 2001r = xa yb + zcr = m001r = xa + yb zc2 2次旋轉(zhuǎn)不變要求：次旋轉(zhuǎn)不變要求： xa zc = xa zc a b c, = = 90o a c, = 90o對(duì)稱條件對(duì)稱條件晶系晶系特點(diǎn)特點(diǎn)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1(E)或或1(i)2(C2)或或2(m)兩個(gè)兩個(gè)2(C2)或或2(m)4(C4)或或4(S43)3(C3)或或3(S65)6(C6)或或6(S35)a b c, a b c, = = 90o a b c, =

17、 = = 90oa = b c, = = = 90oa = b c, = = 90o, = 120oa = b = c, = = = 90oa = b = c, = = 菱形菱形a = b c, = = 90o, = 120o第四講第四講 3232種種點(diǎn)群點(diǎn)群(I)結(jié)晶學(xué)點(diǎn)群：結(jié)晶學(xué)點(diǎn)群：3232種種對(duì)稱操作的一個(gè)集合對(duì)稱操作的一個(gè)集合，滿足以下四條件，就構(gòu)成一個(gè)，滿足以下四條件，就構(gòu)成一個(gè)群：群：1 1、封閉性、封閉性2 2、有恒等操作、有恒等操作3 3、每一個(gè)操作都有逆操作、每一個(gè)操作都有逆操作4 4、操作的乘法滿足結(jié)合律、操作的乘法滿足結(jié)合律結(jié)晶學(xué)點(diǎn)群是指一些點(diǎn)對(duì)稱操作的集合，對(duì)稱元素結(jié)

18、晶學(xué)點(diǎn)群是指一些點(diǎn)對(duì)稱操作的集合，對(duì)稱元素的組合可用來(lái)描述三維晶體的宏觀對(duì)稱性。的組合可用來(lái)描述三維晶體的宏觀對(duì)稱性。研究點(diǎn)群的意義？研究點(diǎn)群的意義？1 (L1)極射赤面（平）投影極射赤面（平）投影一般形一般形1 (C)2 (L2)m (P)2/m (L2PC)2 (L2)m (P)2/m (L2PC)222 (3L2)xyxymm2 (mm;L22P)xymmm (3L23PC)xy4 (L4)xy4/m (L4PC)4mm (L44P)xy4/mmm (L44L25PC)xy422 (42, L44L2)xyxy4 (Li4)42m (Li42L22P)xy1 (L1)m (P)1 (C)42m (Li42L22P)2 (L2)2/m (L2PC)222 (3L2)mm2 (L22P)mmm (3L23PC)4 (Li4)422 (42, L44L2)4/mmm (L44L25PC)4mm (L44P)4/m (L4PC)4 (L4)1(C1)m (C1h)1(Ci)42m (D2d)2(C2)2/m (C2h)222(D2)mm2 (C2v)mmm (D2h)4 (S4)422 (D4)4/mmm(D4h)4mm (C4v)4/m (C4h)4(C4)62 (D3h)6 (C3h)622 (D6)6/mmm (