Crystal Structure_basic introduction.pptx

It is a power point presentation on the crystal structure.

1. What is a Crystal? Generally, a crystal structure is defined as a regular arrangement of atoms or molecules. But thermal vibration will always be there at a finite temperature, so it is better to consider the average position as equivalent to the real position. Def. A crystal structure is a time-invariant, three-dimensional regular arrangement of atoms or molecules on a lattice.

2. Mathematics of Crystal Structure O         Assumptions: 1. spans the whole 3-dimensional space. 2. is the smallest unit of expansion.   , , are basis vectors. Each corner point of in this space is a lattice point. is the set of all vectors such that any point within the space spanned by is related as . is the lattice translation vector. are the lattice parameters, which makes the skeleton of the crystal. Therefore, Crystal structure = Lattice + Basis  

3. Unit Cell O         Assumptions: 1. spans the whole 3-dimensional space. 2. is the smallest unit of expansion.   If this has the smallest possible volume and its repetition makes the whole crystal, then is the unit cell of the crystal. If a unit cell contains only one lattice site, then it is primitive, otherwise non-primitive. The choice of the unit cell is not unique.  

4. How to Choose a Unit Cell In Fig. (a) a lattice point B is added at which changes the surroundings of A, so within this basis, a unit cell can not be defined. In Fig. (b) a lattice point B is added at and after that also the environment of A, B and C is invariant. So we can define a unit cell within this basis. The grey-shaded area has a smaller area than the conventional one, but the conventional rectangular unit cell has higher degree of symmetry .   The unit cell will be non-primitive rectangular Ref. Struc . of Mate., McHenry, Cambridge Univ. Press

5. The Five 2-D Crystal System Ref. Struc . of mate., McHenry, Cambridge Univ. Press The 2-D lattices have 4 unique unit cells: Oblique Rectangular Hexagonal Square The Rectangular unit cell with an atom at constitutes a non-primitive unit cell called a centered rectangular unit cell. These 4 primitive and one non-primitive unit cells are called 2-D crystal systems or 2-D Bravais lattices.  

6. The 3-D Crystal System There are 7 primitive unit cells in a 3-dimensional crystal system. We can add extra lattice points in 3 ways to make new unit cells out of these 7 unit cells: Body centering: for any site , there is an additional site . is called the body centering vector. Face centering: for every site , there are 3 additional sites , and . , , are face centering vectors. Base centering: we can add a lattice point at , and . C- centered if plane is centered. Simiar argument for A- centered and B- centered . Therefore, A, B, C, I and F- centered lattices along with the 7 primitive unit cells constitutes total 14 Bravais lattices in the 3-D space.  

7. The 3-D Bravais lattices 7 primitive unit cells with the new 5 different centered unit cells can generate different possible unit cells. Many of them are reducible to a primitive unit cell. Only 7 of them: are irreducible. These all constitute the 14 Bravais Lattices.  

8. We can Always choose a Primitive Unit Cell We can choose the basis of cF (Fig. a) in such a way that edge of the unit cell becomes (Fig. b). The only advantage is that cF transforms into a primitive unit cell such that there will be only one lattice point per unit cell.   Wigner-Seitz unit cell is a way to construct a unit cell in such a way that each unit cell corresponds to only one lattice point.

9. Miller Indices Consider a 2x2x2 cF Bravais lattice. Planes indicated by grey colour incorporates equivalent lattice points. The planes containing the set of equivalent lattice points are called crystal planes. These planes are characterised by the Miller indices. Each plane is denoted by ( hkl ), e.g., (110) Family of planes are denoted by { hkl }, e.g., {110} Family of directions is denoted by < uvw >

10. Miller Indices to Reciprocal Lattice Reciprocal basis vectors are defined by: Within a 3-D reciprocal space, where, volume V is defined as The volume of the unit cell in the real space is reciprocal to the volume of the reciprocal unit cell. The number of basis vectors in both the spaces are the same, but there is a difference in the quantity dimension, i.e., when real space has the dimension length the reciprocal space will have .   A reciprocal lattice vector can be defined as: It can be shown that, . This implies, with components ( h,k,l ) is perpendicular to the plane with Miller indices ( hkl ) . serves the same purpose as as in the direct lattice. Therefore,  

11. Symmetries in Crystal Lattice Fundamental symmetry operations for any physical object are: Translation 2. Rotation 3. Reflection Any crystal system possesses two fundamental types of symmetries: Translational symmetry 2. Point group symmetry Point group symmetry and translational symmetry result in space group symmetry. In the 7-crystal system, there are 32 point group symmetries which along with the translational symmetry result in 230 space group symmetries for the 14 Bravais lattices.

12. Point Group Symmetry Point symmetries are symmetries that all pass through a given point and this point does not change with the application of a symmetry operation. The symmetry elements which constitute the crystallographic point groups are: 1. Proper rotation axes (n) 2. Mirror planes (m) 3. Inversion center (1, or no explicit symbol) 4. Rotary inversion axes ( ) Only n-fold axes where n = 1, 2, 3, 4, 6 are allowed for space-filling 3-dimensional objects. 32 unique crystallographic point groups are obtained from combining the various allowed rotation axes, mirror planes, and inversions. 11 of the 32 crystallographic point groups are centrosymmetric .  

13. Point Group Symmetry Proper Rotation Axes Rotation about an axis is defined by where n=1,2,3,4 and 6. It is the symmetry operation of the First kind. This operation does not change the handedness of the body.  

14. Point Group Symmetry Mirror Plane It creates the reflected object concerning the mirror. This is the symmetry operation of the second kind. In this operation the handedness of the body changes.

15. Point Group Symmetry Inversion Centre Transforms . This is a symmetry operation of the second kind. Changes the handedness of the object.  

16. Rotation of followed by an inversion. It is a symmetry element of the second kind. This changes handedness of the object.   Point Group Symmetry Roto-Inversion Axis

17. 32 Crystallographic Point groups There are 11 Laue groups associated with the 32-point groups among 14 Bravais lattices. The correct choice of the Laue group identifies the crystal system of the sample and narrows the selection of the relevant space group.

18. Space groups vs Point groups Point groups describe symmetry of isolated objects Space groups describe symmetry of infinitely repeating space filling objects Space groups include point symmetry elements Space groups include additional translational symmetry elements The presence of translational symmetry elements causes systematic absences in the diffraction pattern Space Groups

19. Lattice Translations Trivial unit cell translations Translations due to centering vectors from non-primitive Bravais lattices Screw Axes – combine a rotation with translation Glide Planes – combine a reflection with translation Translational Symmetry Elements

20. Choosing a non-primitive vs. primitive lattice is a matter of convention and observable symmetry It is always possible to choose a primitive triclinic lattice Choose the lattice and crystal system that conforms to the observable symmetry Designation Extra lattice point(s) Mnemonic Device Centering Vector A bc face Abc (0, ½, ½) B ac face aBc ( ½, 0, ½) C ab face abC ( ½, ½, 0) F each face center (0, ½, ½); (½, 0, ½); (½, ½, 0) I body centre ( ½, ½, ½) Non-primitive Lattice Translation Vectors

21. Combines rotation and translation. Designated as n m ( e.g. 2 1 , 4 1 , 3 2 , 4 3 ). Rotation as . Symmetry element of the first kind. Translation as m/n of a unit cell (n > m). Certain pairs of screw axes correspond to right and left handed screws ( e.g. 3 1 and 3 2 ) and are enantiomorphs.   Screw Axes

22. Combine reflection with translation. Symmetry element of the second kind. Designated as a , b , c , d , n and letter gives direction of translational component. Glide Planes

23. n and d Glide Planes n glides translate along face diagonals, ( a+b )/2, ( a+c )/2, or ( b+c )/2. d glides only occur in F and I-centered lattices. d glides translate along face diagonals at ¼ along each direction, i.e. ( a+b )/4, ( a+c )/4, or ( b+c )/4. Glide Planes

24. Space group symbols consist of several parts Bravais lattice type List of symbols denoting the type and orientation of symmetry elements Must know the Crystal System to interpret the space group symbol correctly. Interpretation of Space Group Symbols

25. Perform the following steps: Identify the point group of the crystal Remove Bravais lattice type symbol Iba2 →”ba2” Convert all translational symmetry elements to their point counterparts (glides →mirror; screw axes → rotation axes) “ba2” → mm2 Look up the crystal system that corresponds to that point group (mm2 → orthorhombic) Body centered b glide reflecting across (100) a glide reflecting across (010) 2-fold proper rotation parallel to [001] mm2 is an achiral point group. Therefore, Iba2 is an achiral space group. Interpretation of Space Group Symbols

26. Some simple examples of Space groups

27. The End

28. Notations Proper Rotations: 2         6 4 3 Screw axis:                      

29. Glide planes:

30. Some examples of the point group

31. Space groups in 3 dimensions are a combination of 7 crystal system, 14 Bravais lattices, 32 point groups, along with glide symmetries. A total of 230 different ways in which a certain object can be arranged in 3-dimensional space. Crystal system Lattice type Number of Space groups Triclinic P 2 Monoclinic P, C or I 13 Orthorhombic P, F, I, A 59 Tetragonal P, I 68 Trigonal P 25 Hexagonal P 27 Cubic P, I, F 36

32. Space groups Centrosymmetric (has centre of inversion) Noncentrosymmetric (does not have any centre of inversion) Achiral (contains m and glide symmetry) Chiral (does not contain m and glide symmetry)

33. Meaning of the Space group notations

34. Understanding Hermann-Mauguin Notation for Point Groups Crystal System 1 st Position 2 nd Position 3 rd Position Point Groups Triclinic Only one position, denoting all directions in crystal 1 , 1 Monoclinic Only 1 symbol: 2 or 2 ║ to Y ( b is principal axis) 2/m , 2, m Orthorhombic 2 and/or 2 ║ to X 2 and/or 2 ║ to Y 2 and/or 2 ║ to Z mmm , mm2, 222 Tetragonal 4 and/or 4 ║ to Z 2 and/or 2 ║ to X and Y 2 and/or 2 ║ to [110] 4/mmm , 42m, 4mm, 422, 4/m, 4, 4 Trigonal 3 and/or 3 ║ to Z 2 and/or 2 ║ to X, Y, U 3m , 3m, 32, 3, 3 Hexagonal 6 and/or 6 ║ to Z 2 and/or 2 ║ to X, Y, U 2 and/or 2 along [110] 6/mmm , 6m2, 6mm, 622, 6/m, 6, 6 Cubic 2 and/or 2 ║ to X, Y, Z 3 and/or 3 ║ to [111] m3, 23 4 and/or 4 ║ to X, Y, Z 2 and/or 2 along face diagonals m3m , 43m, 432

35. Notable Features of Space Groups Combining point symmetry and translational symmetry elements with the 14 Bravais lattices yields 230 unique space groups 73 of these are symmorphic space groups. These have no translational symmetry elements ( e.g. P222, F23, Immm) 11 enantiomorphous pairs. If a (+) chiral molecule crystallizes in one of these space groups, the (-) enantiomer will crystallize in the other of the pair. E.g. P6 1 22 and P6 5 22 Enantiopure compounds will crystallize in space groups which only contain symmetry elements of the first kind. There are 65 of these space groups

36. Graphical Representation of Symmetry Elements Proper rotations depicted as symbols with the number of vertices which corresponds to n Screw axes have same symbol, but have “tails” Enantiomeric pairs of screw axes ( e.g. 6 1 and 6 5 ) are mirror images of each other

37. Example of International Tables Entry (P2 1 /c)