1. 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

2. 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.

3. 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

4. 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

5. 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

6. 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:

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

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

9. 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

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

11.  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.

12. 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.

13.  Wigner-Seitz Unit Cell bccfccdiamond

14. 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

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

16. 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 )

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

18.  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

19. top view http://www.als.lbl.gov/pics/154graphene01.png  Graphite ( C ) band structure effective mass of an electron, very high conductivity (theory).

20. http://www.rsc.org/images/FEATURE-graphene-390_tcm18- 116226.jpg Fullerene ( C x, usually C 60 ), CNT (Carbon Nano Tube), and graphite SWNT (Single Wall Nano Tube)

21. 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