空间群的确定

    §1.4.2   空间群的确定  注意事项：  (1)、在 分析 定性数据统计分析pdf销售业绩分析模板建筑结构震害分析销售进度分析表京东商城竞争战略分析 消光类型时，应从格子类型、滑移面、螺旋轴的顺序分析。因为一种消光规 律可能会掩盖另一种消光规律。如体心格子，则有 h+k+l=2n，这就意味着 hk0中，必有 h+k=2n； 0kl中，k+l=2n；h0l中，h+l=2n。同时在 h00中，必有 h=2n；0k0中,  k=2n；00l中，l=2n。 这时并不能确定在三个轴向上是否真正存在 n 滑移面和 21螺旋轴，所以像空间群 I222 与 I212121就无法用系统消光来区别它们。  (2)、仅从消光规律，一般无法确定晶体是否含有旋转轴、对称面或对称中心。在 230个 空间群中，只有 50 个空间群(如 P21/c、P212121) 与消光规律具有一一对应关系，可以根据 消光规律毫不含糊地确定它们。而剩余的 180 个空间群与消光规律之间不具有一一对应关 系，一种消光规律可对应于两个或两个以上的空间群，如 I222、Immm、Imm2 和 I212121， 这四个空间群具有完全相同的系统消光。这 180个空间群分属于 72种消光规律，所以根据 消光规律只能把 230个空间群区分成 122种衍射群(diffraction symbols)。  一些比较特殊的情况有，如平行(100)、(010)和(001)取向的 d滑移面只出现在正交晶系和 立方晶系的面心格子(oF,  cF)中, 由于面心格子要求 hkl 满足全是奇数或全是偶数，因此，d 滑移面所要求的 4n，则意味着 h=2n、k=2n和(或)l=2n，在表 2中用圆括号(h, k=2n)、(h, l=2n) 或(k,  l=2n)表示之。对于四方原始格子(tP)，如 hhl和 h‐hl中存在 l=2n，则意味着在 c轴方向 交替存在 c和n滑移面。习惯上不管原点取在何处，在空间群国际 标准 excel标准偏差excel标准偏差函数exl标准差函数国标检验抽样标准表免费下载红头文件格式标准下载 符号(Hermann‐Mauguin) 中只用 c滑移面来表示，如 P4cc (No.103), P‐42c(No.112)和 P4/nnc (No.126)。而在立方晶系， 如 hhl和 h‐hl中，存在 l=2n衍射条件，则意味着 c轴方向交替存在 c和 n滑移面(其它等效 方向类同)，在空间群国际标准符号中依据原点所选取的位置是否含有 c 或 n 滑移面，分别 定为对应的滑移面，如原点选在 n 滑移面上的有空间群 P‐43n(No.218)、Pn‐3n(No.  222)和 Pm‐3n (No.223)，相反原点选 c滑移面上的空间群有F‐43c(No. 219)、Fm‐3c (No.226)和Fd‐3c(No.  228)。  表 1   不同格子类型与 hkl衍射条件的关系  格子类型  点阵点分布特征  hkl衍射条件  P(原始格子)  (0,0,0)+ ; 下同(省略)  无限制  I(体心格子)  (1/2,1/2,1/2)+;  h + k + l = 2n  F(面心格子)  (0,1/2,1/2)+; (1/2,0,1/2)+; (1/2,1/2,0)+;  k + l = 2n, h + l = 2n, h + k = 2n  A(底心格子)  (0,1/2,1/2)+;  k + l = 2n  B(底心格子)  (1/2,0,1/2)+;  h + l = 2n  C(底心格子)  (1/2,1/2,0)+;  h + k = 2n  R(菱面体格子)  (2/3,1/3,1/3)+; (1/3,2/3,2/3)+;  ‐h + k + l = 3n (正定向)  R(菱面体格子)  (1/3,2/3,1/3)+; (2/3,1/3,2/3)+;  h ‐ k + l = 3n (负定向)    表 2   滑移面与衍射条件的关系  滑移面  滑移距离  符号  衍射类型  衍射条件  晶系  (100)  b/2  b  0kl  k = 2n  斜方/立方(b‐‐)、四方(‐b‐)      (100)  c/2  c  0kl  l = 2n  斜方/立方(c‐‐)、四方(‐c‐)  (100)  (b + c)/2  n  0kl  k + l = 2n  斜方/立方(n‐‐)、四方(‐n‐)  (100)  (b ± c)/4  d  0kl  k + l = 4n, (h,l=2n) 斜方/立方(d‐‐) (F格子)  (010)  a/2  a  h0l  h = 2n  单斜/斜方/四方/(‐a‐)、立方(a‐‐) (010)  c/2  c  h0l  l = 2n  单斜/斜方/四方/(‐c‐)、立方(c‐‐) (010)  (a + c)/2  n  h0l  h + l = 2n  单斜/斜方/四方/(‐n‐)、立方(n‐‐) (010)  (a ± c)/4  d  h0l  h + l = 4n, (k,l=2n) 斜方(‐d‐)、立方(d‐‐)(F格子)  (001)  a/2  a  hk0  h = 2n  斜方(‐‐a)、四方/立方(a‐‐)  (001)  b/2  b  hk0  k= 2n  斜方(‐‐b)、四方/立方(b‐‐)  (001)  (a + b)/2  n  hk0  h + k = 2n  斜方(‐‐n)、四方/立方(n‐‐)  (001)  (a ± b)/4  d  hk0  h + k = 4n, (k,l=2n) 斜方(‐‐d)、立方(d‐‐)(F格子)  (11‐20)  c/2  c  h‐h0l  l = 2n  三方/六方(‐c‐), ⊥[110]或 d轴    (‐2110)  c/2  c  0k‐kl  l = 2n  三方/六方(‐c‐), ⊥[100]或 a轴  (1‐210)  c/2  c  ‐h0hl  l = 2n  三方/六方(‐c‐),⊥[010]或 b轴  (1‐100)  c/2  c  hh.‐2h.l  l = 2n  三方/六方(‐‐c), ⊥[1‐10]或//dc面 (01‐10)  c/2  c  ‐2h.hhl  l = 2n  三方/六方(‐‐c), ⊥[120]或//ac面 (‐1010)  c/2  c  h.‐2h.hl  l = 2n  三方/六方(‐‐c), ⊥[210]或//bc面 (110),(1‐10)  c/2  c, n  hhl,h‐hl  l = 2n  四方(‐‐c)*、立方(‐‐n)  (110),(1‐10)  (a ± b ± c)/4  d  hhl,h‐hl  2h + l =4n  四方/立方(‐‐d)  (011),(01‐1)  a/2  a, n  hkk,hk‐k  h = 2n  立方(‐‐n) , 立方(‐‐a)(F格子)  (011),(01‐1)  (±a + b± c)/4  d  hkk,hk‐k  2k + h =4n  立方(‐‐d)  (101),(‐101)  b/2  b, n  hkh,‐hkh  k= 2n  立方(‐‐n), 立方(‐‐b)(F格子)  (101),(‐101)  (±a ± b+ c)/4  d  hkh,‐hkh  2h + k=4n  立方(‐‐d)    表 3   螺旋轴与衍射条件的关系  方向  平移距离 符号  衍射类型 衍射条件 晶系  [100]  a/2  21    h00  h = 2n  斜方/立方(21‐‐)、四方(‐21‐)  [100]  a/2  42  h00  h = 2n  立方(42‐‐)  [100]  a/4  41, 43  h00  h = 4n  立方(41‐‐)/(43‐‐)  [010]  b/2  21    0k0  k = 2n  斜方/四方(‐21‐)、立方(21‐‐)  [010]  b/2  42  0k0  k = 2n  立方(42‐‐)  [010]  b/4  41, 43  0k0  k = 4n  立方(41‐‐)/(43‐‐)  [001]  c/2  21    00l  l = 2n  斜方(‐‐21)、四方(‐21‐)、立方(21‐‐)  [001]  c/2  42  00l  l = 2n  立方(42‐‐)  [001]  c/4  41,43  00l  l = 4n  立方(41‐‐/43‐‐)  [001]  c/2  63  000l  l = 2n  六方(63‐‐)  [001]  c/3  31, 32, 62, 64  000l  l = 3n  三方(31‐‐/32‐‐)、六方(62‐‐/64‐‐)  [001]  c/6  61, 65  000l  l = 6n  六方(61‐‐/65‐‐)      2.2. Contents and arrangement of the tables BY TH. HAHN AND A. LOOIJENGA-VOS 2.2.1. General layout The presentation of the plane-group and space-group data in Parts 6 and 7 follows the style of the previous editions of International Tables. The entries for a space group are printed on two facing pages as shown below; an example (Cmm2, No. 35) is provided inside the front and back covers. Deviations from this standard sequence (mainly for cubic space groups) are indicated on the relevant pages. Left-hand page: (1) Headline (2) Diagrams for the symmetry elements and the general position (for graphical symbols of symmetry elements see Chapter 1.4) (3) Origin (4) Asymmetric unit (5) Symmetry operations Right-hand page: (6) Headline in abbreviated form (7) Generators selected; this information is the basis for the order of the entries under Symmetry operations and Positions (8) General and special Positions, with the following columns: Multiplicity Wyckoff letter Site symmetry, given by the oriented site-symmetry symbol Coordinates Reflection conditions Note: In a few space groups, two special positions with the same reflection conditions are printed on the same line (9) Symmetry of special projections (not given for plane groups) (10) Maximal non-isomorphic subgroups (11) Maximal isomorphic subgroups of lowest index (12) Minimal non-isomorphic supergroups Note: Symbols for Lattice complexes of the plane groups and space groups are given in Tables 14.2.3.1 and 14.2.3.2. Normalizers of space groups are listed in Part 15. 2.2.2. Space groups with more than one description For several space groups, more than one description is available. Three cases occur: (i) Two choices of origin (cf. Section 2.2.7) For all centrosymmetric space groups, the tables contain a description with a centre of symmetry as origin. Some centrosym- metric space groups, however, contain points of high site symmetry that do not coincide with a centre of symmetry. For these 24 cases, a further description (including diagrams) with a high-symmetry point as origin is provided. Neither of the two origin choices is considered standard. Noncentrosymmetric space groups and all plane groups are described with only one choice of origin. Examples (1) Pnnn (48) Origin choice 1 at a point with site symmetry 222 Origin choice 2 at a centre with site symmetry �1. (2) Fd�3m �227� Origin choice 1 at a point with site symmetry �43m Origin choice 2 at a centre with site symmetry �3m. (ii) Monoclinic space groups Two complete descriptions are given for each of the 13 monoclinic space groups, one for the setting with ‘unique axis b’, followed by one for the setting with ‘unique axis c’. Additional descriptions in synoptic form are provided for the following eight monoclinic space groups with centred lattices or glide planes: C2 �5�, Pc �7�, Cm �8�, Cc �9�, C2�m �12�, P2�c �13�, P21�c �14�, C2�c �15�. These synoptic descriptions consist of abbreviated treatments for three ‘cell choices’, here called ‘cell choices 1, 2 and 3’. Cell choice 1 corresponds to the complete treatment, mentioned above; for comparative purposes, it is repeated among the synoptic descrip- tions which, for each setting, are printed on two facing pages. The cell choices and their relations are explained in Section 2.2.16. (iii) Rhombohedral space groups The seven rhombohedral space groups R3 (146), R�3 �148�, R32 (155), R3m (160), R3c (161), R�3m (166), and R�3c (167) are described with two coordinate systems, first with hexagonal axes (triple hexagonal cell) and second with rhombohedral axes (primitive rhombohedral cell). For both descriptions, the same space-group symbol is used. The relations between the cell parameters of the two cells are listed in Chapter 2.1. The hexagonal triple cell is given in the obverse setting (centring points 23 , 1 3 , 1 3 � 1 3 , 2 3 , 2 3). In IT (1935), the reverse setting (centring points 13 , 2 3 , 1 3 � 2 3 , 1 3 , 2 3) was employed; cf. Chapter 1.2. 2.2.3. Headline The description of each plane group or space group starts with a headline on a left-hand page, consisting of two (sometimes three) lines which contain the following information, when read from left to right. First line (1) The short international (Hermann–Mauguin) symbol for the plane or space group. These symbols will be further referred to as Hermann–Mauguin symbols. A detailed discussion of space- group symbols is given in Chapter 12.2, a brief summary in Section 2.2.4. Note on standard monoclinic space-group symbols: In order to facilitate recognition of a monoclinic space-group type, the familiar short symbol for the b-axis setting (e.g. P21�c for No. 14 or C2�c for No. 15) has been adopted as the standard symbol for a space-group type. It appears in the headline of every description of this space group and thus does not carry any information about the setting or the cell choice of this particular description. No other short symbols for monoclinic space groups are used in this volume (cf. Section 2.2.16). (2) The Schoenflies symbol for the space group. Note: No Schoenflies symbols exist for the plane groups. 17 International Tables for Crystallography (2006). Vol. A, Chapter 2.2, pp. 17–41. Copyright © 2006 International Union of Crystallography (3) The short international (Hermann–Mauguin) symbol for the point group to which the plane or space group belongs (cf. Chapter 12.1). (4) The name of the crystal system (cf. Table 2.1.2.1). Second line (5) The sequential number of the plane or space group, as introduced in IT (1952). (6) The full international (Hermann–Mauguin) symbol for the plane or space group. For monoclinic space groups, the headline of every description contains the full symbol appropriate to that description. (7) The Patterson symmetry (see Section 2.2.5). Third line This line is used, where appropriate, to indicate origin choices, settings, cell choices and coordinate axes (see Section 2.2.2). For five orthorhombic space groups, an entry ‘Former space-group symbol’ is given; cf. Chapter 1.3, Note (x). 2.2.4. International (Hermann–Mauguin) symbols for plane groups and space groups (cf. Chapter 12.2) 2.2.4.1. Present symbols Both the short and the full Hermann–Mauguin symbols consist of two parts: (i) a letter indicating the centring type of the conventional cell, and (ii) a set of characters indicating symmetry elements of the space group (modified point-group symbol). (i) The letters for the centring types of cells are listed in Chapter 1.2. Lower-case letters are used for two dimensions (nets), capital letters for three dimensions (lattices). (ii) The one, two or three entries after the centring letter refer to the one, two or three kinds of symmetry directions of the lattice belonging to the space group. These symmetry directions were called blickrichtungen by Heesch (1929). Symmetry directions occur either as singular directions (as in the monoclinic and orthorhombic crystal systems) or as sets of symmetrically equivalent symmetry directions (as in the higher-symmetrical crystal systems). Only one representative of each set is required. The (sets of) symmetry directions and their sequence for the different lattices are summarized in Table 2.2.4.1. According to their position in this sequence, the symmetry directions are referred to as ‘primary’, ‘secondary’ and ‘tertiary’ directions. This sequence of lattice symmetry directions is transferred to the sequence of positions in the corresponding Hermann–Mauguin space-group symbols. Each position contains one or two characters designating symmetry elements (axes and planes) of the space group (cf. Chapter 1.3) that occur for the corresponding lattice symmetry direction. Symmetry planes are represented by their normals; if a symmetry axis and a normal to a symmetry plane are parallel, the two characters (symmetry symbols) are separated by a slash, as in P63�m or P2�m (‘two over m’). For the different crystal lattices, the Hermann–Mauguin space- group symbols have the following form: (i) Triclinic lattices have no symmetry direction because they have, in addition to translations, only centres of symmetry, �1. Thus, only two triclinic space groups, P1 (1) and P�1 �2�, exist. (ii) Monoclinic lattices have one symmetry direction. Thus, for monoclinic space groups, only one position after the centring letter is needed. This is used in the short Hermann–Mauguin symbols, as in P21. Conventionally, the symmetry direction is labelled either b (‘unique axis b’) or c (‘unique axis c’). In order to distinguish between the different settings, the full Hermann–Mauguin symbol contains two extra entries ‘1’. They indicate those two axial directions that are not symmetry directions of the lattice. Thus, the symbols P121, P112 and P211 show that the b axis, c axis and a axis, respectively, is the unique axis. Similar considerations apply to the three rectangular plane groups pm, pg and cm (e.g. plane group No. 5: short symbol cm, full symbol c1m1 or c11m). (iii) Rhombohedral lattices have two kinds of symmetry directions. Thus, the symbols of the seven rhombohedral space groups contain only two entries after the letter R, as in R3m or R3c. (iv) Orthorhombic, tetragonal, hexagonal and cubic lattices have three kinds of symmetry directions. Hence, the corresponding space-group symbols have three entries after the centring letter, as in Pmna, P3m1, P6cc or Ia�3d. Lattice symmetry directions that carry no symmetry elements for the space group under consideration are represented by the symbol ‘1’, as in P3m1 and P31m. If no misinterpretation is possible, entries ‘1’ at the end of a space-group symbol are omitted, as in P6 (instead of P611), R�3 (instead of R�31), I41 (instead of I4111), F23 (instead of F231); similarly for the plane groups. Table 2.2.4.1. Lattice symmetry directions for two and three dimensions Directions that belong to the same set of equivalent symmetry directions are collected between braces. The first entry in each set is taken as the representative of that set. Lattice Symmetry direction (position in Hermann– Mauguin symbol) Primary Secondary Tertiary Two dimensions Oblique Rotation point in planeRectangular [10] [01] Square �10� �01� � � �1�1� �11� � � Hexagonal �10� �01� ��1�1� �� � �� � �1�1� �12� ��2�1� �� � �� � Three dimensions Triclinic None Monoclinic* [010] (‘unique axis b’) [001] (‘unique axis c’) Orthorhombic [100] [010] [001] Tetragonal [001] �100� �010� � � �1�10� �110� � � Hexagonal [001] �100� �010� ��1�10� �� � �� � �1�10� �120� ��2�10� �� � �� � Rhombohedral (hexagonal axes) [001] �100� �010� ��1�10� �� � �� � Rhombohedral �rhombohedral axes� [111] �1�10� �01�1� ��101� �� � �� � Cubic �100� �010� �001� �� � �� � �111� �1�1�1� ��11�1� ��1�11� � � � � � � �1�10� �110� �01�1� �011� ��101� �101� �� � �� � * For the full Hermann–Mauguin symbols see Section 2.2.4.1. 18 2. GUIDE TO THE USE OF THE SPACE-GROUP TABLES Short and full Hermann–Mauguin symbols differ only for the plane groups of class m, for the monoclinic space groups, and for the space groups of crystal classes mmm, 4�mmm, �3m, 6�mmm, m�3 and m�3m. In the full symbols, symmetry axes and symmetry planes for each symmetry direction are listed; in the short symbols, symmetry axes are suppressed as much as possible. Thus, for space group No. 62, the full symbol is P21�n 21�m 21�a and the short symbol is Pnma. For No. 194, the full symbol is P63�m 2�m 2�c and the short symbol is P63�mmc. For No. 230, the two symbols are I41�a �3 2�d and Ia�3d. Many space groups contain more kinds of symmetry elements than are indicated in the full symbol (‘additional symmetry elements’, cf. Chapter 4.1). A complete listing of the symmetry elements is given in Tables 4.2.1.1 and 4.3.2.1 under the heading Extended full symbols. Note that a centre of symmetry is never explicitly indicated (except for space group P�1); its presence or absence, however, can be readily inferred from the space-group symbol. 2.2.4.2. Changes in Hermann–Mauguin space-group symbols as compared with the 1952 and 1935 editions of International Tables Extensive changes in the space-group symbols were applied in IT (1952) as compared with the original Hermann–Mauguin symbols of IT (1935), especially in the tetragonal, trigonal and hexagonal crystal systems. Moreover, new symbols for the c-axis setting of monoclinic space groups were introduced. All these changes are recorded on pp. 51 and 543–544 of IT (1952). In the present edition, the symbols of the 1952 edition are retained, except for the following four cases (cf. Chapter 12.4). (i) Two-dimensional groups Short Hermann–Mauguin symbols differing from the corre- sponding full symbols in IT (1952) are replaced by the full symbols for the listed plane groups in Table 2.2.4.2. For the two-dimensional point group with two mutually perpendicular mirror lines, the symbol mm is changed to 2mm. For plane group No. 2, the entries ‘1’ at the end of the full symbol are omitted: No. 2: Change from p211 to p2. With these changes, the symbols of the two-dimensional groups follow the rules that were introduced in IT (1952) for the space groups. (ii) Monoclinic space groups Additional full Hermann–Mauguin symbols are introduced for the eight monoclinic space groups with centred lattices or glide planes (Nos. 5, 7–9, 12–15) to indicate the various settings and cell choices. A complete list of symbols, including also the a-axis setting, is contained in Table 4.3.2.1; further details are given in Section 2.2.16. For standard short monoclinic space-group symbols see Sections 2.2.3 and 2.2.16. (iii) Cubic groups The short symbols for all space groups belonging to the two cubic crystal classes m�3 and m�3m now contain the symbol �3 instead of 3. This applies to space groups Nos. 200–206 and 221–230, as well as to the two point groups m�3 and m�3m. Examples No. 205: Change from Pa3 to Pa�3 No. 230: Change from Ia3d to Ia�3d. With this change, the centrosymmetric nature of these groups is apparent also in the short symbols. (iv) Glide-plane symbol e For the recent introduction of the ‘double glide plane’ e into five space-group symbols, see Chapter 1.3, Note (x). 2.2.5. Patterson symmetry The entry Patterson symmetry in the headline gives the space group of the Patterson function P(x, y, z). With neglect of anomalous dispersion, this function is defined by the formula P�x, y, z� � 1 V h k l �F�hkl��2 cos 2��hx� ky� lz�� The Patterson function represents the convolution of a structure with its inverse or the pair-correlation function of a structure. A detailed discussion of its use for structure determination is given by Buerger (1959). The space group of the Patterson function is identical to that of the ‘vector set’ of the structure, and is thus always centrosymmetric and symmorphic.* The symbol for the Patterson space group of a crystal structure can be deduced from that of its space group in two steps: (i) Glide planes and screw axes have to be replaced by the corresponding mirror planes and rotation axes, resulting in a symmorphic space group. (ii) If this symmorphic space group is not centrosymmetric, inversions have to be added. There are 7 different Patterson symmetries in two dimensions and 24 in three dimensions. They are listed in Table 2.2.5.1. Account is taken of the fact that the Laue class �3m combines in two ways with the hexagonal translation lattice, namely as �3m1 and as �31m. Note: For the four orthorhombic space groups with A cells (Nos. 38–41), the standard symbol for their Patterson symmetry, Cmmm, is added (between parentheses) after the actual symbol Ammm in the space-group tables. The ‘point group part’ of the symbol of the Patterson symmetry represents the Laue class to which the plane group or space group belongs (cf. Table 2.1.2.1). In the absence of anomalous dispersion, the Laue class of a crystal expresses the point symmetry of its diffraction record, i.e. the symmetry of the reciprocal lattice weighted with I(hkl). Table 2.2.4.2. Changes in Hermann–Mauguin symbols for two- dimensional groups No. IT (1952) Present edition 6 pmm p2mm 7 pmg p2mg 8 pgg p2gg 9 cmm c2mm 11 p4m p4mm 1