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材料科学与工程学院多媒体课件. 课程名称：. Crystallography . 学时： 40. 梁叔全 教授. 主讲教师：. Crystallography Dr.S.Q.Liang Part I: Introduction 1. Ethics of Advanced Study Why?

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  1. 材料科学与工程学院多媒体课件 课程名称： Crystallography 学时：40 梁叔全 教授 主讲教师：

  2. Crystallography Dr.S.Q.Liang Part I: Introduction 1.Ethics of Advanced Study Why? 2.Stratagy And Philosophy for Advanced Study How? a:be hard,but should be creative b:be active,but not impetuous c: be patient, step by step Simple Complex Specific General Macro Micro Theoretical Applied

  3. 3.About Science (Personal Opinions) Natural Phenomena: Language Description concepts,definitions,laws… Physical Description properties, pictures, models,….. Mathematic Description symbology,formulas, equations…. Should be clear as much as possible

  4. E.G. • Apples dropped to the ground ,……. F=Gm1m2/r2 m1 m2 • Mass-Energy-light Interaction,….. E, m ,v E= mc2 /[1-(v/c)1/2] • Gas power , Zhuyi ??? Metaphysics

  5. 4.About crystals Before 1900’s Steno(1669) SiO2, Hauy(1801) CaCO3….. …………………………. Physical Descriptions a:density>uniform(isotropy) b:optical,mechanical,…>un-uniform(anisotropy) c:geometry>symmetry F+v=E+2 d:heating>has a fixed melting point

  6. Chemical Descriptions Reactions to solvents>has a limited solubility • Definition for the first time about crystal Crystals were the matters which were of ………..

  7. From 1805 to 1912 The two most important things Lattice Theory (WEISS Germany , …1805~1809) It was a very famous assumption.The geometric crystallography was built, which included 7 crystal systems; 14 Space Lattice(Bravis lattice); a lot of famous laws, like the crystal band law,…………

  8. Group Theory(Point, Space ) (Hessel, Schonflies,1830-1890) It was constructed based on the symmetry of the crystal, which discovered a lot of detail characteristics in both structure and properties of the crystals. • 32 point groups • 230 space group

  9. In 1912 A very famous experiment was performed—XRD by Laue , see Fig The result was the first direct evidence for the lattice theory

  10. Definition II for the crystal: Crystal is matter, the structure of which could be described by a lattice plus a basis.

  11. In order to describe the lattice , a coordinate system is needed. c a  b The parameters of the coordinate system are a, b, c, , , . How many chooices we have?. Some limits should be given. What are they?

  12. What results will be produced? • Triclinic:abc,     ; • Monoclinic:abc,  ==90   ; • Orthorhombic : abc,  = ==90; • Tetragonal:a=b  c,  = ==90; • Rhombohedral:a=b=c,  = =90; • Hexagonal: a=b  c,  ==90,  =120; • Cubic: a=b = c,  = ==90

  13. Questions: (1)Why only have 7 crystal systems, not 8 or more? (2)What the relationship betweencrystal systemandBravis lattice ? In fact, if we add some additional lattice points in to a specific crystal system. Some new lattice will be produced. By this way,we obtain 14 Space Lattices. They are:AP; mP,mC; oP,oC,oI,oF; tP,tI; hP; rP; cP,cI,cF, See Fig.

  14. We knew them already, but maybe don’t know why ? (3)In practice, we have a=b=c=same number,===90, but the crystal system is not cubic.

  15. Knowledge we knew: • 7 crystal systems; 14 Space Lattice(Bravis lattice); • symbology for lattice points(x,y,z), lattice direction[uvw], lattice plane(hkl), their relationships; • stereogram , Wulf net etal; • Packing theories, interaction to the various rays and so on.

  16. Knowledge we did not know: • Why 7C.S.? Why 14B.L.? • What is the key basis for the classification of C.S.,S.L.? • What are 32 point groups? • What are 230 space group? • What are the relationships between them and the crystal structures, properties of crystals….?

  17. In the study of above knowledge there is a key link(纲）----symmetry of the crystals. Remember: the symmetry even is the key to open the door for everything from universe to people , to molecules, to crystals.

  18. Some Ref.Books for Reading [1]. 《固体科学中的空间群》 G.本斯著，俞文海译，高等教育出版社,1981 [2] Space Groups for Solid State Scientists G.Burns, A.M.Glazer,Academic Press,1978 [3] 《晶体学》，俞文海，中国科大出版社 [4]《近代晶体学基础》，张克从，科学出 版社，1987

  19. 5.About symmetry Symmetry:(1)exact match in size and shape. (2)pleasingly regular way in which parts are arranged. In Language : poem, antithetical couplets In Geometry : geometric patterns In Mathematics: Metric symmetry, simple formula to describe complex things In Physics: Physical symmetry ( movement , principle, universe….) In Chemistry: molecule In crystallography:atoms in 3-D space

  20. 常德德山山有德 三绝诗书画 才饮长沙水 长沙沙水水无沙 一官归来去 又食武昌鱼 y=sin( 2n + ) V=k.exp[-E/RT]

  21. Symmetry of the movement

  22. Geometry definition : After giving a specific physical operation and interchanging the various parts of the object, results in the object appearing exactly same as before operation. We call this characteristics of the objects is Symmetry. There are several kinds of symmetry: Point symmetry, in which geometric patterns repeated regularly in 3 Dimension space except for one point at least, see Fig.

  23. Translational symmetry, in which geometric patterns repeated regularly in 3 Dimension space without any exception, see Fig. Some important concepts in the symmetry: Symmetry operations(physical actions): Rotation, Inversion, Reflection and their composites, Translation. symmetry related positions:geometric patterns, symmetry elements : geometric point, line(axe), Plane(mirror plane).

  24. Mathematic description of the point symmetry The point which does’t move in the symmetry operation could be selected as the origin of the reference coordinate system r(x,y,z) r’(x’,y’,z’)   

  25. Point(x,y,z) r point (x’,y’,z’) r’ symmetrical operation r ,=R r x, a11 a12 a13 x y, a21 a22 a23 y z, a31 a32 a33 z R symmetry operator Matrix

  26. So up to now we have 4 ways to describe the point symmetry . Language way, Definition Geometry method, clear picture Mathematic way, formula symbology method , notation

  27. Part II The point symmetry in crystal The most important aspect which makes the crystals are different from other matter is that the atoms in crystals are arranged regularly in 3-D spaces. This means the crystals have symmetry in both macro level (external appearance )and micro (atom )level. The first group of symmetry we will discuss includes Identity, Rotations,Inversion, Rotation-inversion axes.

  28. Part II The point symmetry in crystal a)Identity : Language Description :Rotate the crystal 360, or do noting to the crystal. Trivial? But the most important! Why say so? Symmetry operator r’ (x,y,z)=R r(x,y,z) 1 0 0 R={1[000]}={E[000]}= 0 1 0 0 0 1

  29. {1[000]}, {E [000]} is the symbology forms of the Identity symmetry 1- international notation, E-schoenflies notation. [000] indicates the direction of the rotation axe. 2.Rotations n (Cn) Language Description: Rotate the crystal 2π/n about an axe, results in the crystal appearing exactly same as before rotation. Since the symmetry axe could be described by S=ua+vb+wc The symmetry operators are n[uvw] ,or Cn [uvw]

  30. For normal objects, n can take any number. What about the crystal? What number can take? The answer is the n only can take 1,2, 3, 4, 6. Why? This is determined by the inside structure of the crystal----Lattice structure t’  t t t

  31. Therefore, we can write t’=m.t where m is some integer From the diagram t’= -2t cos  +t Combining these two equations ,we get cos =(1-m)/2 Now if m is an integer, then 1-m=M where M is an integer. Furthermore,  must lie in 0-180 for the diagram, I.e. cos  lies between +1 and –1 cos 1 Therefore M 2 And then M= -2, -1, 0, 1,or 2

  32. This means that the only acceptable values of  are 2/2 , 2/3, 2/4, 2/6, or 2 /1 Hence,the only allowed n are 1, 2, 3, 4, 6 Mathematic description of the rotation: symmetry operator n[uvw] . In Cartesian system, In Hexagonal system n[001] n [001] cos -sin  0cos  +3/3sin -2 3/3sin 0 sin cos 0 2 3/3sin cos  -3/3sin 0 0 0 1 0 0 1

  33. 2 C2Point Symmetries Symbology Geometry Mathematics (language) Intern.Notations Schoenflies Notations Symmetric Operators Geometric symbols Specific patterns General patterns -1 0 0 {2[001]} = 0 -1 0 0 0 1 2 C2

  34. 3(C3),4(C4) Point Symmetries Symbology Geometry Mathematics (language) 0 –1 0 H {3[001]}= 1 - 1 0 0 0 1 • C3 -1 1 0 H {32[001]}= 1 0 0 0 0 1 0 -1 0 {4[001]} = 1 0 0 0 0 1 4 C4 {42[001]}={2[001]} 0 1 0 {43[001]}= -1 0 0 0 0 1

  35. 6(C6)Point Symmetries Symbology Geometry Mathematics (language) 1 -1 0 H {6[001]}= 1 0 0 0 0 1 H{62[001]}={3[001]} 6 C6 H{63[001]}={2[001]} H{64[001]}={32[001]} 0 1 0 H{65[001]}=-1 0 1 0 0 1

  36. c) Inversionthrough a center, also called Inversion or center.反演 Language description: A symmetry operation by which every position in space given by the coordinates (x,y,z) goes through the origin to the position(-x,-y,-z).( In crystallography,we put the minus signs above the coordinates. That is (-x,-y,-z)=(x, y, z ) Symbology Geometry Mathematics right hand {1(i)}(x,y,z) 1 i = (x,y,z) left hand -1 0 0 Symmetry element? Where? Operation? 0 –1 0 0 0 -1 ,

  37. d)Mirror plane: Reflection across a plane Language description: A symmetry operation by which every position in space given by the coordinates (x,y,z) goes to the position(x,y,z)by reflection across a plane. Symbology Geometry Mathematics in ac plane in ab plane {m[010]}(x,y,z) 2 ,m ,() = (x,y,z) Symmetry element? Where? Operation? 1 0 0 0 –1 0 0 0 1 , ，

  38. From theinternationalSymbols of the Inversion(1) and Mirror Plane(2), we can make the following two questions. (1)1,2,3,4,6 1, 2, 3? 4? 6? (2)What is the relationship between 1,2 and 1,2 ? The answer for (1) is YES The answer for (2) is that 1,2 are produced by 1.1 and 1.2 . This means 1,2 are composite symmetry operations. So, this symmetry group is called Rotatory Inversion Axes. 1.3， 1.4， 1.6 1 1.1 2 1.2 , ,

  39. e) Rotation Inversion axes ( improper rotation) Symbology Geometry Mathematics (language) Intern.Notations m.n Schoenflies Notations Geometric symbols General patterns Symmetric Operators 0 1 0 H{3[001]}= -1 1 0 0 0 1 1 –1 0 3 S 65 H{35[001]}= 1 0 0 0 0 –1 H {32[001]}=H{32[001]}, H{33[001]}={1[001]}, H{34[001]}=H{3[001]} , , , , ,

  40. , Symbology Geometry Mathematics (language) 0 1 0 {4[001]}=-1 0 0 0 0 1 • S43 {42[001]}={2[001]} 0 -1 0 {43[001]}= 1 0 0 0 0 -1 -1 1 0 H {6[001]}= -1 0 0 0 0 -1 • S35 {62[001]}={3[001]} {63[001]}={m[001]} {64[001]}={32[001]} 0 –1 0 {65[001]}= 1 -1 0 0 0 -1 , , , , , , ,

  41. In above table,we can find that in Schoenflies system we use S65, S43, S35 to symbolize 3,4,6. This means they don’t seem to be in tune with the international notations. In fact, in Schoenflies system for 3,4,6, they are produced by 2. 3, 2.4, 2.6. See Fig.s In International system In Schoenflies system Compared with the international approach, the positions were obtained successively in a counterclockwise manner.

  42. Summary of the notations International Schoenflies 1 E 2 C2 31 C31 3 32 C3 C32 33 =1 C33 =E 41 C41 4 42=2 C4 C42=C2 43 C43 44 C44=E

  43. Summary of the notations International Schoenflies 61 C61 62=31 C62=C31 63 C63 =C2 6 64 =32 C6 C64=C32 65 C65 66=1 C66=E 1 i 2=m 

  44. Summary of the notations International Schoenflies 31 S65 32=32 , S64=C32 33 =1 , S63 =i 3 34 =31 . S65 S62 =C31 35 S6 36=1 S66=E 41 S43 4 42=2 S43 S42=C2 43 S41 44 S44=E , , , , ,

  45. summary of the notations International Schoenflies 61 S35 62=31 S34=C31 63 =m S33 =  6 64 =32 - S35 S32 =C32 65 S3 66=1 S36=E The relationship between international notations and Schoenflies nations in3 (S65 ),4 (S43), 6(S35 ) should be treated carefully. , , ,

  46. Some other important things about symmetry • The concept of the Inverse symmetry For symmetry operator R, there is an another inversion symmetry operator R-1, meet the condition of R.R-1=1 ,or vice versa. (2)some calculation rules of the symmetry notations Snm=(hCn)(hCn)…….( hCn)=(hCn)m m symmetry operator Inverse symmetry operator Cnm Cnn-m Snm all m and even n Snn-m

  47. Snm odd m and odd n Sn2n-m (3)If A,B are symmetry operation , and C =AB, C is another symmetry operation. But if C is a symmetry operation, and C=A.B, no need for A,B to be a symmetry. For Example, for the case of 3 =1.3, 3, 1 and 3 are symmetry, but for the case of 4=1.4, and 6=1.6, 1, 4, 6 are not symmetry

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