対称性と群論 (2) 空間群のヒエラルキーと分類 Massimo NESPOLO: Hierarchy and Classification of Space Groups ヒエラルキーと分類図 1 International Tables for Cr
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1 対称性と群論 (2) 空間群のヒエラルキーと分類 Massimo NESPOLO: Hierarchy and Classification of Space Groups ヒエラルキーと分類図 1 International Tables for Crystallography, Volume A ITA wallpaper group plane group crystal system space-group type crystal class crystal familycrystal class crystal class crystal family 1 図 1 2 object object 図 1 object object object object 2. 群のヒエラルキー W w m m W W w W 図 群対群タイプ point-group types Bravais
2 ソンケ空間 ( イ ) アフィン 無限 ソンケ ビーベル( ア ) バッハ第 2 種対称操作含有 第 2 種対称操作除外 無限 図 1 ITA mm 6mm 6/mmm m3m m3m /mmm 3 4 c a b c
3 1723 crystallographic space-group type 514 lattice type lattice system 2.2 席対称群 x y z Wyckoff position ITA 3 multiplicity W w w w loc location w intr intrinsic site-symmetry group 4 w intrw w loc P 図 2 表 12a x 1 x x 1 x x ( I) 1 = ; ( II) 1 = 1 1 I II a 2 II 2x 2 isomorphic 2 P isomorphic group x ½ 1 x x 1 x x ( I) 1 = ;( II) 1 = 1 ½ ½ 1 ½ ½ IIx ½ x ½ 4 stabilizer x ½ x ½ 1 x x 1 ( I) 1 = ;( II) 1 ½ ½ 1 x x = 1 1 ½ ½ I II 2 z ½ 2 x½ 2x ( ア ) ( イ ) 図 2 P222 1 C222 1 c 表 1 3 P e 1 x y z x y z ½ x y z ½ x y z 2 d.2. ½ y ¼ ½ y ¾ 2 c.2. y ¼ y ¾ 2 b 2.. x ½ x ½ ½ 2 a 2.. x x ½ 53
4 2.3 焦電群 焦電群 pyroelectric groups 2 1 m m mm2 3m 4mm 6mm468 1 polar direction 1 correlation polar groups 2.4 ビーベルバッハ空間群 ビーベルバッハ群 Bieberbach groups 不動点をもたない群 fixed-point free groups ねじれのない群 torsion-free groups 2 p1 pg 13 P1 P2 1 Pc Cc P Pca2 1 Pna2 1 P4 1 P4 3 P3 1 P3 2 P6 1 P キラル空間群, アキラル空間群, ソンケ空間群 chirality 演習問題 Sohncke groups 22 P アフィン空間群 P3 1 P3 2 3 object P3 1 P affine space-group types 2.7 共型と非共型空間群 a p q 1 r p q r 表 2 P n np 2 22 p C ½ ½ C 表 3C 2 2n 2 2n 1 P222 1p2 2 p p p 2p ½ C222 1 a b r 図 2c J-Stage 6 54
5 表 2 P p q r p q r p q r 2 x 1 2 x½ 1 2 x ½ x x½ x ½ x x½ x ½ x ½ ½ 2 2 x x 1 ½ x ½ ½ x x 1 ½ x ½ ½ x x 1 ½ 表 3 C p q r p q r p q r 2 x ½ ½ 2 1 x ¼ 1 2 x½ x ³/2 ½ 2 3 x ¼ x½ x 5 /2 ½ 2 5 x ¼ x½ ½ ½ x ¼ ½ 1 2 x ½ ½ ³/2 2 1 x ¾ ³/2 ½ x ¼ ½ x ½ ³/2 ³/2 2 3 x ¾ 5 /2 ½ x ¼ ½ x ½ 5 /2 ³/2 2 5 x ¾ x ½ ½ ½ ³/2 1 2 x ¾ ½ 2 2 x x ½ ½ ³/2 ³/ x ¾ ½ x x ½ ½ 5 /2 ³/ x ¾ ½ x symmorphic 2 1 w intr P222 1 C a bw intr 2 non-symmorphic 図 2 a2 z 2n ½ n z ½ 1 ³/2 b 2 z 2n ¼z¼ ¾ 5 / a2 b2 P C a b c P222 C P222 4 C P222 P222 1 P P a 1b 1c 1d 222 2a 2b 2c 2d a 2b..2 4a1 7 w intr p1 P1 P1 w intr 55
6 P4mm z½½z ½z ½z p q r g m m 11 m x x z m x x z gx ½ x z P4mm p q r 非共型空間群の区分 SS n S n n n n n /2 S 2S 2 半共型空間群タイプ hemisymmorphic space-group types 異共型空間群タイプ asymmorphic space-group types Pmmm S mmm 2 n 8 15 n 4S Pnnn Pccm Pban 12 3 crystal pattern 図 3 mmm Pmmm Pnnn 2 Pmna 2Pcca 4 7 reproduced with permission. VESTA 9 56
7 1 2 International Tables for Crystallography 図 3 mmm Pmmm mmm 8 Pnnn 榍 2 2 Pmna.2/m. 2 Pcca 群の分類 object 図 4 42m 3.1 代数的結晶類 arithmetic crystal class 7 P2/m P2 1/m P2/c P2 1/c 4 2/m 7 1 図 4 4 2m mp 2/mP P2/m C2/m C2/c 2 2/m mc2/mc 1373 演習問題 高調的結晶類 harmonic crystal classes International Tables for Crystallography 3 Souvignier 1 P42m P4m2 4 2m 42m mP 1266 演習問題 ブラベー類 5 14 J-Stage 57
8 表 /m 2 m 2/m 2 m 2mm m mmm 222 mmm mmm 222 mmm 4/m 422 4mm 4mm 4 4/mmm 4/m m 4 4 4/mmm 422 4mm 4 2m 3m m m 32 3m 6mm 6 3m 3 6/m 622 6mm 6/mmm 6/m m 6 6 6/mmm 622 6mm 6 2m m3 23 m3m m3 4 3m m3m 4 3m 432 Bravais classes 2p 2mmp 2mmc 4mmp 6mmp1P 2/mP 2/mC mmmp mmmc mmmi mmmf 4/mmmP 4/mmmI 3mR 6/mmmP m3mp m3mi m3mf 3.4 幾何的結晶類 geometric crystal class object centrosymmetric ラウエ結晶類 Laue class 8 完面像 holohedry 欠面像 merohedry 47 表 晶系 crystal system ITA 3 8 図 1 cp cf ci 323 m m m3mm3m 5 5 P P'P P' P' P 3 m 6/m2/m2/m 3 m3m trigonal mm 62m 6/m 6/m2/m2/m hexagonal crystal family Krystallisationssysteme
9 表 5 表 5 ITA 3.6 格子系 lattice system trigonal 3m 2 rhombohedral hexagonal système cristallin kristallsystem trigonal rhombohedral rhombohédrique τρίγωνον trígōnon rhombus 結晶族 crystal family 9 ITA ῥóμβος rhómbos 2 4/mmm 3 m 6/mmm 4/mmm 3 m 3 m 6/mmm ブラベー群れ Bravais flock object 514 図 5 図 5 1m3 mp object ITA flockflock 59
10 ( ア ) ( イ ) ( ウ ) ( エ ) 図 5 m3mp 6
11 表 6 * a anortic triclinic asymmetric tetartoprismatic m monoclinic clinorhombic monosymmetric binary hemiprismatic monoclinoedric o orthorhombic rhombic trimetric terbinary prismatic anisometric t tetragonal quadratic dimetric monodimetric quaternary h hexagonal senary monotrimetric c cubic isometric monometric triquaternary regular tesseral tessural... 6 a g 9 4 a b g 9 3 a b a b g 9 a b a b 9 g 12 a b c a b g Triclinic Monoclinic Orthorhombic Tetragonal Trigonal ternary*** Hexagonal Cubic * ** S *** Triclinic Monoclinic Orthorhombic Tetragonal Rhombohedral Hexagonal Cubic ** ap mp mbms mc ma mi mf op os oi of tp tcti tf hr hp cp ci cf ITA 4. 対称性による格子定数の制限 b b 1 2 ac β 912 ac1 b 9 α γ 2 1 a c b g a b a g 表 6 a b c β 9 β 9 1 β a b c β 9 mp b cos 1 a/2c B 61
12 mi b cos 1 a/c C a 2 b 2 c 2 a 2 accosb b 2 b 2 accosb a 2 c 2 3b 2 9a 2 c 3acosb a 2 3b 2 9c 2 a 3ccosb ITA 議論 metric specialization 2 1 6/mmm1 1 m3mc a 1 R3c a m3m 2 P2 b 9 mmm 3 Pccm a b4/mmm 16 P4mm a a c 4/mmm a c 1 c a 1 c a 図 1 今後 謝辞 Bernd Souvignier 図 2 Wiley 図 3 文献 1 58, M. I. Aroyo, Ed.: International Tables for Crystallography Volume A Space-group Symmetry, Sixth edition, Wiley 216 4, 13, M. Nespolo and B. Souvignier: Point groups in crystallography, Z. Kristallogr. 224, M. Nespolo: A note on the notion of chirality, Cryst. Res. Techn. 5, M. Nespolo: A practical approach to symmorphism, Cryst. Res. 62
13 Techn. 52, B. K. Vainsthein: Fundamentals of Crystals. Symmetry, and Methods of Structural Crystallography. Second, Enlarged Edition. Springer, Modern Crystallography, K. Momma and F. Izumi: VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data, J. Appl. Crystallogr. 44, B. Souvignier: Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6, Acta Crystallogr. A59, J. F. C. Hessel: Kristallometrie oder Kristallonomie und Kristallographie Ostwald s Klassiker der exacten Wissenschaften Engelmann, Leipzig 1897, nos 88, J. J. Bernhardi: Darstellung einer neuen Methode, Kristalle zu besischreiben, Gehlen s J. Chem. Phys. Mineral. Berlin, 5, 157; 492; C. S. Weiss: Übersichtliche Darstellung der verschiedenen natürlichen Abtkeilungen der Krystallisations-Systeme, Abhandlungen der Königlichen Akademie der Wissenschaften, Berlin, p H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus: Crystallographic groups of four-dimensional space, John Wiley & Sons, New York, Chichester, Brisbane, Toronto, xiv M. Nespolo: The ash heap of crystallography. Restoring forgotten basic knowledge, J. Appl. Crystallogr. 48, M. Nespolo and G. Ferraris: Twinning by syngonic and metric merohedry. Analysis, classification and effects on the diffraction pattern, Z. Kristallogr. 215, 77 2 プロフィール Massimo NESPOLO CRM2 UMR CNRS 736, Institut Jean Barriol, FR 2843, Faculté des Sciences et Technologies, Université de Lorraine BP 7239, Boulevard des Aiguillettes, F5456 Vandoeuvre-lès-Nancy cedex France 63
対称性 群論トレーニングコトレーニングコース 総研大 学融合レクチャーレクチャー 記号のの表記 uvw 実格子点の座標の座標 [uvw] 方向指数 ( 必ず互いに素)ず互いに素)互いに素)いに素)素 ) uvw 等価な方向のうち な方向のうち 方向のうち 1 本の方向の指数の方向の指数 [uvw]*
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