现代结构分析方法2009-5

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,现代结构分析方法,(09,10,年度第一学期,),第五讲,对称操作,点对称操作：至少空间一点保持不动，,1,，,2,，,3,，,4,，,6,，,m,，,-1,，,-2,，,-3,，,-4,，,-6,平移对称操作：,14,种点阵（,7,晶系,+P, (A, B, C), I, F,）,非点对称操作：含平移的对称操作分,螺旋旋转（来自纯旋转轴）和滑移反映（来自,m,）,2,1,3,1, 3,2,4,1, 4,2, 4,3,6,1, 6,2, 6,3, 6,4, 6,5,3,1,周期,(d),空间群的表示方法及例子,符号：,Hermann-,Mauguin,点阵类型对称元素符号,(,与点群相同,),例：三斜晶系：,1,P 1,，,1,P 1,，,单斜晶系：,2,P 2,，,C 2,，,P 2,1,；,m,P m,，,P c,，,C m,，,C c,；,2/m,P 2/m,，,P 2,1,/m,，,P 2/c,，,P2,1,/c,，,C 2/m,，,C 2/c,。,一般每个点群对应着几个空间群。,用一般等效位置投影图表示：,点群平移群（,Bravais,点阵）的组合空间群,例C mm2,C mm2mm2,No 35Cmm2Patterson,对称性,Cmmm,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,a,b,c,例C mm2,多重性,Wyckoff,字母 点对称性 坐标反射条件,（,0,，,0,，,0,）（,1/2,，,1/2,，,0,）一般,:,8 f 1(1)x, y, z (2) x, y, z (3) x, y, z (4) x, y, z,4 e m0, y, z0, y, z,4 d mx, 0, zx, 0, z,4 c 2 1/4, 1/4, z1/4, 3/4, z,2 b mm2 0, 1/2, z,2 a mm20, 0, z,a,b,0, y, z,1/4, 1/4, z,0, 0, z,x, y, z,0, 1/2, z,晶体结构结构单元周期平移,原子位置由对称性联系,点对称性种类：,1,，,2,，,3,，,4,，,6,，,m,-1,，,-2, -3, -4, -6,平移规律点阵,非点对称性,7,种晶系、,4,种类型,点对称性构成,32,种组合（点群）,14,种,Bravais,点阵,230,种空间群,一些基本概念：,crystals,AlFe,AuCu3,Cu3N,CaO3Ti,NbO,Pearson sym.,cP2,cP4,cP4,cP5,cP6,Struct,. type,CsCl,AuCu3,O3Re,CaO3Ti,NbO,Space Group,Pm m,Pm m,Pm m,Pm m,Pm m,lattice parameters,0.29,0.37484,0.3819,0.38,0.42101,Atoms,Al,Fe,Au,Cu,Cu,N,Ca,O,Ti,Nb,O,Wyckoff notation,1(a),1(b),1(a),3(c),3(d),1(a),1(a),3(c),1(b),3(c),3(d),Symmetry,m m,m m,m m,4/mmm,4/mmm,m m,m m,4/mmm,m m,4/mmm,4/mmm,X,0.0,0.5,0.0,0.0,0.5,0.0,0.0,0.0,0.5,0.0,0.5,Y,0.0,0.5,0.0,0.5,0,0.0,0.0,0.5,0.5,0.5,0,Z,0.0,0.5,0.0,0.5,0,0.0,0.0,0.5,0.5,0.5,0,occupancy,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,相结构命名：希腊字母成分,b,-AlFe,Pearson,符号：,hP6, cF24,以典型结构作为结构类型,空间群（对称性分布）,P 4/m 3 2/m,，,号码,点阵常数,原子位置（几组等效位置）,一些基本概念：,crystals,AlFe,AuCu3,Cu3N,CaO3Ti,NbO,Pearson sym.,cP2,cP4,cP4,cP5,cP6,Struct,. type,CsCl,AuCu3,O3Re,CaO3Ti,NbO,Space Group,Pm m,Pm m,Pm m,Pm m,Pm m,lattice parameters,0.29,0.37484,0.3819,0.38,0.42101,Atoms,Al,Fe,Au,Cu,Cu,N,Ca,O,Ti,Nb,O,Wyckoff notation,1(a),1(b),1(a),3(c),3(d),1(a),1(a),3(c),1(b),3(c),3(d),Symmetry,m m,m m,m m,4/mmm,4/mmm,m m,m m,4/mmm,m m,4/mmm,4/mmm,X,0.0,0.5,0.0,0.0,0.5,0.0,0.0,0.0,0.5,0.0,0.5,Y,0.0,0.5,0.0,0.5,0,0.0,0.0,0.5,0.5,0.5,0,Z,0.0,0.5,0.0,0.5,0,0.0,0.0,0.5,0.5,0.5,0,occupancy,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,a,b,d,c,d,c,c,d,a,b,一些基本概念：,crystals,AlFe,AuCu3,Cu3N,CaO3Ti,NbO,Pearson sym.,cP2,cP4,cP4,cP5,cP6,Struct,. type,CsCl,AuCu3,O3Re,CaO3Ti,NbO,Space Group,Pm m,Pm m,Pm m,Pm m,Pm m,lattice parameters,0.29,0.37484,0.3819,0.38,0.42101,Atoms,Al,Fe,Au,Cu,Cu,N,Ca,O,Ti,Nb,O,Wyckoff notation,1(a),1(b),1(a),3(c),3(d),1(a),1(a),3(c),1(b),3(c),3(d),Symmetry,m m,m m,m m,4/mmm,4/mmm,m m,m m,4/mmm,m m,4/mmm,4/mmm,X,0.0,0.5,0.0,0.0,0.5,0.0,0.0,0.0,0.5,0.0,0.5,Y,0.0,0.5,0.0,0.5,0,0.0,0.0,0.5,0.5,0.5,0,Z,0.0,0.5,0.0,0.5,0,0.0,0.0,0.5,0.5,0.5,0,occupancy,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,c,c,c,a,d,c,d,c,c,d,a,b,一些基本概念：,crystals,AlFe,AuCu3,Cu3N,CaO3Ti,NbO,Pearson sym.,cP2,cP4,cP4,cP5,cP6,Struct,. type,CsCl,AuCu3,O3Re,CaO3Ti,NbO,Space Group,Pm m,Pm m,Pm m,Pm m,Pm m,lattice parameters,0.29,0.37484,0.3819,0.38,0.42101,Atoms,Al,Fe,Au,Cu,Cu,N,Ca,O,Ti,Nb,O,Wyckoff notation,1(a),1(b),1(a),3(c),3(d),1(a),1(a),3(c),1(b),3(c),3(d),Symmetry,m m,m m,m m,4/mmm,4/mmm,m m,m m,4/mmm,m m,4/mmm,4/mmm,X,0.0,0.5,0.0,0.0,0.5,0.0,0.0,0.0,0.5,0.0,0.5,Y,0.0,0.5,0.0,0.5,0,0.0,0.0,0.5,0.5,0.5,0,Z,0.0,0.5,0.0,0.5,0,0.0,0.0,0.5,0.5,0.5,0,occupancy,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,d,d,d,a,d,c,d,c,c,d,a,b,一些基本概念：,crystals,AlFe,AuCu3,Cu3N,CaO3Ti,NbO,Pearson sym.,cP2,cP4,cP4,cP5,cP6,Struct,. type,CsCl,AuCu3,O3Re,CaO3Ti,NbO,Space Group,Pm m,Pm m,Pm m,Pm m,Pm m,lattice parameters,0.29,0.37484,0.3819,0.38,0.42101,Atoms,Al,Fe,Au,Cu,Cu,N,Ca,O,Ti,Nb,O,Wyckoff notation,1(a),1(b),1(a),3(c),3(d),1(a),1(a),3(c),1(b),3(c),3(d),Symmetry,m m,m m,m m,4/mmm,4/mmm,m m,m m,4/mmm,m m,4/mmm,4/mmm,X,0.0,0.5,0.0,0.0,0.5,0.0,0.0,0.0,0.5,0.0,0.5,Y,0.0,0.5,0.0,0.5,0,0.0,0.0,0.5,0.5,0.5,0,Z,0.0,0.5,0.0,0.5,0,0.0,0.0,0.5,0.5,0.5,0,occupancy,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,c,c,c,a,b,d,c,d,c,c,d,a,b,一些基本概念：,crystals,AlFe,AuCu3,Cu3N,CaO3Ti,NbO,Pearson sym.,cP2,cP4,cP4,cP5,cP6,Struct,. type,CsCl,AuCu3,O3Re,CaO3Ti,NbO,Space Group,Pm m,Pm m,Pm m,Pm m,Pm m,lattice parameters,0.29,0.37484,0.38,0.42101,Atoms,Al,Fe,Au,Cu,Cu,N,Ca,O,Ti,Nb,O,Wyckoff notation,1(a),1(b),1(a),3(c),3(d),1(a),1(a),3(c),1(b),3(c),3(d),Symmetry,m m,m m,m m,4/mmm,4/mmm,m m,m m,4/mmm,m m,4/mmm,4/mmm,X,0.0,0.5,0.0,0.0,0.5,0.0,0.0,0.0,0.5,0.0,0.5,Y,0.0,0.5,0.0,0.5,0,0.0,0.0,0.5,0.5,0.5,0,Z,0.0,0.5,0.0,0.5,0,0.0,0.0,0.5,0.5,0.5,0,occupancy,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0,d,c,d,c,c,d,a,b,d,c,d,c,c,d,四 、倒易空间,（一）、,倒易点阵,1,、定义：对于一个由 定义的正点阵，都有一个对应的倒易点阵，其基轴满足,构成倒易点阵。,又称波矢空间。,a,a*,=,b,b,*,=,c,c,*,=,1,a,b*,=,a,c,*,=,b,c,* =,0,晶面,是经过阵点的平面，由描述，,平面方程,hx,+,ky,+,lz,= m,h,k,l.x,y,z,=m,，即俩,矢量点积,当,a,.,a*,=,b,.,b,*,=,c,.,c,*,=,1,，,且,a,.,b*,=,a. c,*,=,b. c,* =,0,，,为单位矩阵,因此有：,N,为晶面法线,点积,2,、,性质：,表示晶面的一种方式：考虑晶面（,hkl,）,晶面方程,hx,+,ky,+,lz,= m,其法线和从原点指向面上任意一点,(x, y, z),的矢量分别为,则 ，,R,为,N,与晶面的交点，,O,为原点。,及,当,m=1,时,I,OR,I,=,面间距,d,hkl,c,b,i,a,k,j,m/l,m/h,N,R,n,O,r,m/k,2,、,性质：,c,b,i,a,k,j,m/l,m/h,N=r,*,hkl,R,n,O,r,m/k,2,、性质,c,b,i,a,k,j,m/l,m/h,N,R,m/k,r1,r2,3,、晶体几何计算公式,其形式取决于晶系,晶面间距计算公式,立方倒易点阵：,a*=b*=c*,且,正交系倒易点阵：基轴,3,、晶体几何计算公式,单斜系：,a,=,b,=90,a,b,c,（二）、倒易点阵与正点阵的关系,1,、,简单点阵,b,a,000,d,100,r*,100,100,r*,100,=1/d,100,正交点阵沿,c,轴投影图,a,b,c,（二）、倒易点阵与正点阵的关系,1,、,简单点阵,b,a,000,r*,100,100,r*,010,=1/d,010,d,010,r*,010,a,b,c,a,b,c,（二）、倒易点阵与正点阵的关系,1,、,简单点阵,r*,110,b,a,b*,a*,000,100,010,110,r*,110,d,110,a*,=,r*,100,= 1/,d,100,= 1/,a,b*,=,r*,010,= 1/,d,010,= 1/,b,c*,=,r*,001,= 1/,d,001,= 1/,c,a,b,c,a,b,c,（二）、倒易点阵与正点阵的关系,1,、,简单点阵,b,a,b*,a*,000,100,010,110,220,注意：对于简单型正点阵，有公因子的指数，如（,220,）等，不对应于真正的晶面。,(020)?,020,2,、,简单单斜点阵,a,b,r*,100,r*,100,g,简单单斜,点阵沿,c,轴投影图,b,a,a,b,c,c,g,g,d,100,2,、,简单单斜点阵,a,b,r*,100,r*,010,r*,100,r*,010,g,g,*,a*,=,r*,100,= 1/,d,100,= 1/(,a,cos,g,-90)=,1/(,a,sin,g),b*,=,r*,010,= 1/,d,010,= 1/(,b,cos,g,-90)=,1/(,b,sin,g),c*,=,r*,001,= 1/,d,001,= 1/,c,g,*,= 180-g,简单单斜,点阵沿,c,轴投影图,b,a,a,b,c,c,g,g,d,010,2,、,简单单斜点阵,a*,=,r*,100,= 1/,d,100,= 1/(,a,cos,g,-90)=,1/(,a,sin,g),b*,=,r*,010,= 1/,d,010,= 1/(,b,cos,g,-90)=,1/(,b,sin,g),c*,=,r*,001,= 1/,d,001,= 1/,c,a,b,r*,100,r*,010,r*,100,r*,010,110,g,g,*,g,*,= 180-g,单斜,点阵沿,c,轴投影图,b,a,a,b,c,c,g,g,3,、,底心点阵,b,a,(020)!,消光！,020,b*,000,C,底心正交点阵沿,c,轴投影图,a,b,c,a,b,c,b*,=,r*,020,= 1/,d,020,= 2/,b,(010)?,3,、,底心点阵,020,b,a,b*,000,c*,=,r*,001,= 1/,d,001,= 1/,c,a*,200,消光！,a*,=,r*,200,= 1/,d,200,= 2/,a,d,100,?,d,200,经,周期平移获得这个倒易点阵,3,、,底心点阵,r*,110,020,d,110,b,a,b*,a*,000,200,110,r*,110,a*,=,r*,200,= 1/,d,200,= 2/,a,b*,=,r*,020,= 1/,d,020,= 2/,b,c*,=,r*,001,= 1/,d,001,= 1/,c,消光！,对于,C,底心型，指数,h, k,和为偶数的晶面才出现；,3,、,底心单斜点阵：,b*,=,r*,010,= 1/,d,010,= 1/(b,.cos,g,-90)=,1/(,b.sin,g),a,b,r*,200,r*,010,r*,200,r*,010,220,g,g,*,*,*,*,*,*,*,*,*,*,*,*,*,*,*,*,*,*,*,*,*,*,*,*,*,1/2,1/2,B,底心单斜,点阵沿,c,轴投影图,b,a,a,b,c,c,g,g,a*,=,r*,200,= 1/,d,200,= 2/(,a.cos,g,-90)=,2/(,a.sin,g),c*,=,r*,002,= 1/,d,002,= 2/,c,g,*,= 180-g,010,010?,100?,200,（二）、倒易点阵与正点阵的关系,4,、,体,心点阵,对于体心型，指数和为偶数的晶面才出现；,(110),（二）、倒易点阵与正点阵的关系,5,、对于面心型，指数同为偶数或奇数的晶面才出现；,(200),(111),(220),（三）、倒易点阵小结,1,、,均为无限的周期点阵，,2,、,正点阵的晶面对应于倒易点阵的阵点（除有公因子指数外）；,3,、,晶系不变，为,11,种中心对称的劳厄点群；,4,、,P-P*, C-C*, I-F*, F-I*,，,即对复合单胞出现倒易点阵系统消光，立方系指数表见下表,h,2,+k,2,+l,2,（,hkl,）,简单立方,体心立方,面心立方,1,100,100,2,110,110,110,3,111,111,111,4,200,200,200,200,5,210,210,6,211,211,211,8,220,220,220,220,立方系指数表 （续）,9,221, 300,221, 300,10,310,310,310,11,311,311,311,12,222,222,222,222,13,320,320,14,321,321,321,16,400,400,400,400,17,322, 410,322, 410,18,330, 411,330, 411,330, 411,19,331,331,331,20,420,420,420,420,21,421,421,22,332,332,332,24,422,422,422,422,立方系指数表 （续）,25,430, 500,430, 500,26,431, 501,431, 501,431, 501,27,333, 511,333, 511,333, 511,29,432, 520,432, 520,30,521,521,521,32,440,440,440,440,33,441, 522,441, 522,34,433, 530,433, 530,433, 530,35,531,531,531,36,442, 600,442, 600,442, 600,442, 600,37,610,610,38,532, 611,532, 611,532, 611,40,620,620,620,620,41,443, 540, 541, 621,443, 540, 621,42,541,541,43,533,533,533,44,622,622,622,622,45,542, 630,542, 630,46,631,631,631,48,444,444,444,444,Indexing of cubic reciprocal lattices,000,100,010,001,200,020,002,111,cP,cP,*,cF,cI,*,cI,cF,*,200,020,002,110,*,*,*,