Crystal Structures Crystal is constructed by the continuous repetition in space of an identical structural unit. Lattice: a periodic array of mathematical points Basis: periodically repeated arrangement of a set of atoms, ions, or molecules latticebasis lattice + basis = unit cell repeated by translations to cover the whole crystal

cubictetragonalorthorhombichexagonal rhombohedralmonoclinictriclinic The Bravais Lattices 14 different types (translational symmetry) 7 crystal systems (type of conventional unit cell) Bravais lattice: arrangement of structureless points so that each point bears the same geometrical relation to all other points.

cubic (simple) rhombohedral tetragonal (simple) hexagonal orthorhombic monoclinic 7 crystals systems and 14 Bravais lattices triclinic body centeredface centered base centered b-c f-c

a a a a a a a a a simple cubicbody-centered Cubicface-centered cubic Number of Bravais latticesConditions simple body-centered cubic face-centered cubic a 1 = a 2 = a 3  Cubic

Number of Bravais latticesConditions simple body-centered a 1 = a 2 ≠ a 3 c a a c a a simple tetragonal body-centered tetragonal  Tetragonal

Number of Bravais latticesConditions simple body-centered face-centered base-centered a 1 ≠ a 2 ≠ a 3 c b a c b a simplebody-centeredface-centered c b a c b a base-centered  Orthorhombic:

Number of Bravais latticesConditions simple simple hexagonal  Hexagonal a 1 = a 2 ≠ a 3

simple rhombohedral Number of Bravais latticesConditions simple a 1 = a 2 = a 3  Rhombohedral

Number of Bravais latticesConditions simple base-centered a 1 ≠ a 2 ≠ a 3 simple monoclinicbase-centered monoclinic  Monoclinic 4 rectangular faces and 2 parallelogram faces

Number of Bravais latticesConditions simple a 1 ≠ a 2 ≠ a 3 simple triclinic  Triclinic

 Primitive Vectors: vector between any two lattice points m, n, l : integers ex) a simple cubic lattice a a a simple cubic Assignment of primitive vectors is not unique.

a a a face-centered cubic rotated by 90º primitive Cell a a a a body-centered cubic unit cell Primitive Unit Cell Parallelepipes formed by the three primitive vectors Volume of the unit cell remains the same no matter how the primitive vectors are chosen.

 Wigner-Seitz Unit Cell bccfccdiamond

Indexing procedure 1 z x y 2 3 ( 632 )  Miller Index 1) Divide each intercept value by the unit cell length along the respective coordinate axis 1, 2, 3 2) Invert the intercept values 1/1, 1/2, 1/3 3) Using an appropriate multiplier, convert the ( 1 /intercept) set to the smallest possible set of whole numbers 4) Enclose the whole-number set in curvilinear brackets (632) h, k, l integers orientation of crystal plane direction perpendicular to crystal plane

z x y (110) a a a Crystal planes of body-centered cubic z x y (100) a a a

Basis Made of Two or More Atoms basis Arranging Na + and Cl - ions alternatively at the lattice points of a simple cubic lattice Each ion surrounded by six nearest neighbors of the opposite charge FCC with basis Na + at (1/2,1/2,1/2), Cl - at (0, 0, 0) Simple Cubic with basis Cs + at (0,0,0), Cl - at (1/2, 1/2, 1/2)  Cesium Chloride ( CsCl )  Sodium Chloride ( NaCl )

 Diamond structure (Zincblende) FCC with basis one at (0,0,0) the other at (1/4, 1/4, 1/4)

 YBa 2 Cu 3 O 7 (Yttrium-barium-copper oxide) high temperature superconductor: superconducting at temperature below 91 K simple orthorhombic lattice with basis containing 13 atoms

top view  Graphite ( C ) band structure effective mass of an electron, very high conductivity (theory).

jpg Fullerene ( C x, usually C 60 ), CNT (Carbon Nano Tube), and graphite SWNT (Single Wall Nano Tube)

covalent bond between atoms loose bond between planes graphite chiral vector  CNT axis: perpendicular to chiral vector magnitude of : perimeter of tube tube diameter Cross-sectional area of SWCNT graphene layer