7.1.3 CrystalsystemsandBravais Latticesa.CrystalsystemsThereareatotal of seventypes ofBLacrystal systemsdiffering insymmetryVUnitcellChoice of axisLatticCharacteristicCrystal systemsPointsymmetryparametersGroupelementsNilCiTriclinica+b+cαβC2hb // C2MonoclinicC2, Oha+b+cα=y=90°±βD2hOrthorhombica, b, c // 3C23C2, 2 Oha+b+cα=β==90°Unit cell is chosen in such a waythat it contains as many symmetryelements ofthe latticeas possible andhas thesmallest volume
a. Crystal systems Crystal systems Characteristic symmetry elements Unit cell parameters Choice of axis Lattic Point Group Triclinic Nil abc Ci Monoclinic C2, h abc ==90 b // C2 C2h Orthorhombic 3C2, 2 h abc ===90 a, b, c // 3C2 D2h 7.1.3 Crystal systems and Bravais Lattices There are a total of seven types of crystal systems differing in symmetry. Unit cell is chosen in such a way that it contains as many symmetry elements of the lattice as possible and has the smallest volume
C3D3dRhombohedralTrigonala=b=c,α=β=<120°→90°DhHexagonala=b≠c,α=β=90°=120°C4c// C4D4hTetragonala=b#c,α=β==90°C6c // C6D6hHexagonala=b+cα=β=90°=120°4C3C3CubicFourOna=b=caxesparallel toα=β==90°arethebodyofdiagonalsthe cube
Trigonal C3 Rhombohedral a=b=c, ==<12090 D3d Hexagonal a=bc, ==90 =120 D6h Tetragonal C4 a=bc, ===90 c // C4 D4h Hexagonal C6 a=bc ==90, =120 c // C6 D6h Cubic 4C3 a=b=c ===90 Four C3 axes are parallel to the body diagonals of the cube Oh
b.BravaisLattices:(14)[developed byBravaisin1850!]Primitive cell: minimal size, one lattice point only!Unit Cell of Bravais Lattice: having the highest symmetry &minimal size, may contain more than one lattice pointSymmetry of LatticeCrystal Systems (7)Unit Cell of LatticeBravais Lattices (14)(Primitive orCentered)TriclinicOrthor-MonoclinicTetragonalCubicTrigonalHexagonalhombicaPhPmP, mCoP,oCtP, tIhR, hPcP, cl,cFo, o1Lowercaseletter(crystal system)+Capital letter(Typeofcell)e.g., hR--R-centred hexagonal, mC-- C-centred monoclinic
b. Bravais Lattices: (14) [developed by Bravais in 1850 !] Symmetry of Lattice Crystal Systems (7) Bravais Lattices (14) Unit Cell of Lattice (Primitive or Centered) Triclinic Monoclinic Orthorhombic Tetragonal Cubic Trigonal Hexagonal aP mP, mC oP, oC oF, oI tP, tI cP, cI, cF hR, hP hP Lowercase letter (crystal system) + Capital letter (Type of cell) e.g., hR- R-centred hexagonal, mC- C-centred monoclinic Primitive cell: minimal size，one lattice point only! Unit Cell of Bravais Lattice: having the highest symmetry & minimal size, may contain more than one lattice point
*Triclinicab+cαβBabYaP(Primitive)Triclinic crystal system hasthelowest symmetryIt canbesimplyrepresented byaprimitive cell.Symmetry element:
* Triclinic abc aP (Primitive) • Triclinic crystal system has the lowest symmetry. • It can be simply represented by a primitive cell. • Symmetry element: i
A primitive cell contains onelattice pointand aMonoclinicC-centered unit cell contains two lattice pointsa#bcα'+90°Sym.Elements:+90°α==90°i, C2(//b), Ona.CB90.β090°90°bbmP (Primitive)mC(C-centeredorA-centered)The primitive cell of a C-or A-centered monoclinic is triclinic!
b 90° 90° a c mC (C-centered or A-centered) b 90° c 90° * Monoclinic abc = =90º mP (Primitive) A primitive cell contains one lattice point and a C-centered unit cell contains two lattice points. The primitive cell of a C-or A-centered monoclinic is triclinic! Sym. Elements: i, C2 (//b), h , b' c' a' 90 90