POINT SYMMETRY AND TYPES OF CRYSTAL LATTICE Abdul Rashid Mirza Associate Professor of Physics. Govt. College of Science, wahdatroad, Lahore. 1
WHAT ARE CRYSTALS? The word crystal means icy or frozen water. Up to 17 th century, crystal---- a body with icy transparency and having regular faces(facets) at characteristic angles. 19 TH CENTURY Solids with order at microscopic level. Ordermeans --- periodicity (filling all space when unit cells are put together side by side). 2
1913-- DISCOVERY OF X RAY DIFFRACTION Max Von Laue and Bragg Crystals achieve there order through periodicity. Since then, Crystals have been defined as, solids composed of periodic arrangement of identical unit cells. s PROPERTIES 3
Two Basic Terms; LATTICE & BASIS A three-dimensional arrangement of points, where atoms could be placed to develop a crystal structure, is called LATTICE.. An atom ( or a group of atoms ) located at each points is called BASIS. 4
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TYPES OF CRYSTAL STRUCTURE Various types are due to SYMMETRY PROPERTIES (Looking identical after an operation). An operation after which a crystal remains identical, is called SYMMETRY OPERATION. SYMMETRY OPERATIONS ARE OF TWO TYPES. POINT SYMMETRY TRANSLATIONAL SYMMETRY 7
POINT SYMMETRY OPERATIONS THE OPERATIONS THAT LEAVE A PARTICULAR POINT OF THE LATTICE FIXED. THESE ARE; 1-ROTATION. 2-INVERTION. 3-ROTARY INVERTION. (ROTATION+INVERTION) 4-MIRROR REFLECTION. A SET OF SUCH OPERATIONS IS CALLED POINT GROUP TRANSLATIONAL SYMMETRY IDENTICAL APPEARANCE OF SURROUNDING AFTER TRANSLATION FROM ONE LATTICE POINT TO ANOTHER. 8
POINT SYMMETRY + TRANSLATIONAL SYMMETRY ==== SPACE GROUP 9
N-FOLD ROTATIONAL SYMMETRY A crystal will have N-Fold Symmetry if it stays same after a rotation through 360/N degrees. N =1,2,3,4,and 6.(N=5 is forbidden) 10
1 + 1 = 1 2 + 1 = 2 3 + 1 = 3 4 + 1 = 4 6 + 1 = 6 0NLY 10 SYMMETRY OPERATIONS Rotational Symmetry Operations--- ---1,2,3,4,6. Inversion.---------------- 1 is same as Mirror Reflection------ ------m The proper Rotary Invertions. ----.3,4,5 Total= 10 11
POINT GROUPS Collections of symmetry elements passing through each lattice point. There are 32 Point Groups,, and are divided into 7 Sets. In each set,all point groups are compatible. 1 CRYSTAL SYSTEM COMMON SYMMETRY ELEMENT POINT GROUPS A Triclinic 1(or1) 1, 1 B Monoclinic One 2(or2) C Orthorhombic Threemutually perpendicular 2-fold axes 2,m,2/m 222, 2mm, 2/m 2/m 2/m D Triogonal One 3-fold axis 3,3, 3m, 32,3 2/m E Tetragonal One 4-foldaxis 4,4, 4/m, 422, 4mm, 4/m 2/m, 42/m F Hexagonal One six fold axis 6,6, 6/m, 6mm, 622, 62m, 6/m 2/m 2/m G Cubic 4, 3-fold axis at 72ᵒ,32 to each other 23, 2/m3, 4 3m, 432, 4/m 3 2/m 12
IN EACH SET, THE MEMBER GROUPS HAVE A COMMON SYMMETRY COMPONENT. (OR ELEMENT) Set A 1-fold rotation Set B one 2-fold2 Set C three-two two fold perpendicular to each other Set D one 3-fold3 Set E 4-fold axis Set F 6-fold axis Set G four, 3-fold 3 axis at 72⁰,32 to one another (parallel to the body diagonal ) 13
IT IS,THEREFORE, LOGICAL TO DIVIDE THE UNIT CELLS INTO SEVEN GROUPS ON THE BASIS OF THE COMMON SYMMETRY ELEMENT. SO, WE HAVE 7 CRYSTAL SYSTEMS. BRAVAIS LATTICES 7 CRYSTAL SYSTEMS But,how many LATTICE TYPES ---? CRITERION : Each lattice type has to obey POINT SYMMETRY. GEOMETRICAL SIMPLICITY. 14
GEOMETRICAL SIMPLICITY I a,b,c have similar values. II α,β,as close as possible to 90ᵒ. HOW TO DETERMINE NO.OF LATTICE TYPES?? Firstly,weshall shall look at the PRIMITIVE CELL of the system,whetherit obeys POINT SYMMETRY of that system. If, it does, it is lattice type of that system. If,itdoes not,we,weshall chase a closest possible unit cell having GEOMETRICAL SIMPLICITY. 15
SYSTEM 1--TRICLINIC Essential Symmetry; Nil or 1. a b Of infinite no. of unit cells, we shall select one having geometrical simplicity. Only One Lattice------ ------PRIMITIVE SYSTEM-2 MONOCLINIC Symmetry---- one 2-fold rot. Axis parallel to b. a ==90. Primitive Cell---- ----one lattice type 16
Another Primitive Cell a=b,= Having One 2-fold rot.axis,it it may fall in, Monoclinic. But, it does not! WHY-- --? No GEOMETRICAL SIMPLICITY However, if we select a different set of axes,we get a cell with specifications, a, ==90, Which is bace-centred centred mono-clinic. 17
TWO TYPES OF LATTICESIN MONO-CLNIC 1. PRIMITIVE 2 - BACE-CENTRED CENTRED SYSTEM 3 RHOMBOHEDERAL a=b=c,, = = SYMMETRY; One 3-fold 3 rot. axis parallel to the un-equal body- diagonal. Long body diagonal for <90 and Short body diagonal for >90. Only One Lattice Type----- -----Primitive 18
SYSTEM 4 CUBIC a=b=c = ==90 Symmetry;Four3-fold axes! at 70 32 " #!$% & $%&' $ 54 4.to the cell edges. The unit cell is PRIMITIVE. ANOTHER CELL OF THE SAME SYMMETRY---? Yes! a=b=c = = = 109,28 It has four 3- fold rot. Axes, one parallel to the short body and others parallel to the three cell edges. Is it a 2 nd lattice type in CUBIC SYSTEM.? NO! It does not have GEOMETRICAL SIMPLICITY. 19
BY SELECTING A DIFFERENT SET OF AXES, WE CAN HAVE A UNIT CELL OF THE REQUIRED GEOMETRICAL SIMPLICITY. THE NEW CELL IS BODY- CENTRED AND IS, 2 ND LATTICE TYPE OF CUBIC SYSTEM. 20
YET ANOTHER---? YES! a=b=c ===60 Four 3-fold rot. axes; Parallel to body diagonal, at 70,32 $ &% $%&' NO GEOMETRICAL SIMPLICITY Again we look for another cell by selecting other set of axes. Clearly, it has GEO. SIMPLICITY. It is FACE-CENTRED. CENTRED. 21
THREE LATTICE TYPES IN CUBIC GROUP 1.Simple Cubic (Primitive) Primitive)-------- --------.(P) 2. Body-centered centered------------------------- -------------------------(I) 3. Face-centered centered--------------------------- ---------------------------(F) 22
TETRAGONAL a=b c, α=β=ỵ=90ᵒ Symmetry: One 4-fold axis parallel to C axis. The unit cell is primitive Another primitive cell a=b=c, α=β Ỵ Symmetry: ONE 4-fold AXIS PARALLEL TO C FACE DIAGONAL AXIS. THE UNIT CELL IS PRIMITIVE(HEAVY LINES). NO GEOMETRICAL SIMPLICITY. IF WE EXTEND WE HAVE A BODY CENTERED CELL. 23
Thus TETRAGONAL system has two lattice types 1. Primitive -----------------------P 2. Body centered --------------I ORTHORHOMBIC a b c, α=β=ỵ=90 Symmetry Three 2-folds axes parallel to a,b, c Primitive cell. 24
When primitive cell are placed side by side we get a base-centered Heavy lines show a base-centered( C ) in the same lattice. Yet another primitive cell(heavy lines) a=b=c α β Ỵ has a 2-fold axis parallel to one of the diagonals After extension we get body-centered (I) 25
Yet another primitive cell(heavy lines) a=b c α=β Ỵ 2-fold rotation axis parallel to each of the C face diagonal After extension we get face-centered(f) Thus there are four lattice types in Orthorhombic 1. Primitive (P) 2. Base-centered ( C ) 3. Body-centered (I) 4. Face-centered (F) 26
HEXAGONAL a=b c, α=β=90ᵒ, Ỵ=120ᵒ Symmetry One 6-folds axis parallel to c Only one primitive cell We have derived 14 different unit cells through point symmetry. 27
TRICLINIC : P MONOCLINIC : P and C ORTHORHOMBIC: P,C,I and F RHOMBOHEDRAL: P HEXAGONAL : P TETRAGONAL : P and I CUBIC : P,I and F Why 14 types???? 7 crystal systems Each could have 4 types similar cells 1. P 2. I 3. C 4. F 28
There should be (7 4=) 28 lattice types However only 14 are different from one another- remaining 14 are redundant. Cubic System. Only three types (P, F, I) No base-centered 29
If there were a base-centered, there would be only one 4-fold axis parallel to z edge (as required by tetragonal) instead of three 4- fold axes along x, y, z (required for cubic) Thus, a base-centered cubic has already been identified as tetragonal Similar considerations can show that there are only 14 lattice types 30
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