4(c) inversion triad ( 3 )3= Octahedral site inan octahedron
5(d) inversion tetrad ( 4 ) x= tetrahedral site in atetrahedron(e) inversion hexad ( 6 )xHexagonal close-packed (hcp) lattice
63-2. Fourteen Bravais lattice structures D lattice3 types of symmetry can be arranged in a 1-D lattice(1) mirror symmetry (m)(2) 2-fold rotation (2)(3) center of symmetry ( 1 )
7Proof: There are only 1, 2, 3, 4, and 6 fold rotation symmetries for crystal withtranslational symmetry.graphicallyThe space cannotbe filled!
8Start with the translation Add a rotationlatticepointlattice pointlattice pointA translation vector connecting two lattice points! It must be some integerof or we contradicted the basicAssumption of our construction.T: scalarp: integerTherefore, is not arbitrary! The basic constrain has to be met!
9To be consistent with the original translation t: tcostcosAA’TTTo be consistent with theoriginal translation t:p must be integerM must be integerp M cos n (= 2/) bM >2 orM<-2:no solution4321-1 T/ T/ T/ (1) TAllowable rotationalsymmetries are 1, 2,3, 4 and 6.
10Look at the case of p = 2 n = 3; 3-fold 3-fold lattice. angle = 120o3-fold lattice.Look at the case of p = 1n = 4; 4-fold = 90o4-fold lattice.
11Exactly the same as 3-fold lattice. Look at the case of p = 3 n = 6; 6-fold = 60oExactly the same as 3-fold lattice.Look at the case of p = 3n = 2; 2-foldLook at the case of p = -1n = 1; 1-fold12
12These are the lattices obtained by combining ParallelogramCan accommodate1- and 2-foldrotational symmetries1-fold2-fold3-fold4-fold6-foldHexagonal NetCan accommodate3- and 6-foldrotational symmetriesSquare NetCan accommodate4-fold rotationalSymmetry!These are the lattices obtained by combiningrotation and translation symmetries?How about combining mirror and translationSymmetries?
13Combine mirror line with translation: constrainmmUnlessOr0.5TPrimitive cellcentered rectangularRectangular
26The second layer coincident to the middle of the first layer and maintain 2-fold symmetryNote: other ways to maintain 2-fold symmetryacbA-face centeredmonoclinic latticeIf relabeling lattice coordinationA-face centered monoclinic = B-face centered
27(2) Body centered monoclinic lattice Body centered monoclinic = Base centered monoclinic
28So monoclinic has two types 1. Primitive monoclinic2. Base centered monoclinic
29(3) Orthorhombic system a(𝑎≠𝑏≠𝑐; 𝛼=𝛽=𝛾 = 90o)3 -diad axes
30(1) Derived from rectangular lattice (𝑎≠𝑏; 𝛾= 90o) to maintain 2 fold symmetryThe second layer superposes directly on the first layer(a) Primitive orthorhombic latticeabc
31(b) B- face centered orthorhombic = A -face centered orthorhombic (c) Body-centered orthorhombic (I- cell)
32body-centered orthorhombic based centered orthorhombic rectangularbody-centered orthorhombic based centered orthorhombic
33(2) Derived from centered rectangular lattice (𝑎≠𝑏; 𝛾= 90o) (a) C-face centered OrthorhombicabcC- face centered orthorhombic= B- face centered orthorhombic
34(b) Face-centered Orthorhombic (F-cell) Up & DownLeft & RightFront & Back
35Orthorhombic has 4 types 1. Primitive orthorhombic2. Base centered orthorhombic3. Body centered orthorhombic4. Face centered orthorhombic
41Hexagonal has 1 types1. Primitive hexagonalRhombohedral (trigonal)2. Primitive rhombohedral (trigonal)
42(6) Cubic systemca = b = c; = = = 90oba4 triad axes ( triad axis = cube diagonal )Cubic is a special form of Rhombohedral latticeCubic system has 4 triad axes mutually inclined along cube diagonal
43(a) Primitive cubic = 90oca = b = c; = = = 90oba(b) Face centered cubic = 60oa = b = c; = = = 60o
44(c) body centered cubic a = b = c; = = = 109o
451. Primitive cubic (simple cubic) 2. Body centered cubic (BCC) cubic (isometric)Special case of orthorhombic with a = b = cPrimitive (P)Body centered (I)Face centered (F)Base center (C)Tetragonal (I)?Tetragonal (P)a = b cCubic has 3 types1. Primitive cubic (simple cubic)2. Body centered cubic (BCC)3. Face centered cubic (FCC)
46http://www.theory.nipne.ro/~dragos/Solid/Bravais_table.jpg = P = I = T P