MSE230 Structure and Bonding Chapter 3 Continued Aisling Coughlan Crystallographic Directions Defined as a line between points or a vector How to determine the three directional indices A vector of convenient length is positioned such that it passes through the origin of the coordinate system Any vector my be translated throughout the crystal lattice without alteration if parallelism is maintained The length of the vector projection on each of the three axes is determined these are measured in terms of the unit cell dimensions a b and c These three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values The three indices not separated by commas are enclosed in square brackets thus uvw The u v and w integers correspond to the reduced projections along the x y and z axes respectively What if Specific Crystallographic Direction Crystallographic Planes Orientation representation hkl Miller Indices Parallel planes have same miller indices Determine hkl A plane can not pass the chosen origin A plane must intersect or be parallel any axis If the above is not met translation of the plane or origin is needed To get intercepts a b c infinite if the plane is parallel to an axis Take the reciprocal Smallest integer rule hkl hkl in opposite side of the origin Cubic plane orientations and directions with same indices are perpendicular to one another Crystallographic Planes Miller Indices are the reciprocals of the fractional intercepts with fractions cleared which the plane makes with the crystallographic x y z axes of the three nonparallel edges of the cubic unit cell 1 1 1 Miller Indices with fractions cleared a b c Spacing between planes in a cubic crystal Miller Indices Example Miller Indices Example Miller Indices Example Why are planes in a lattice important A Determining crystal structure Diffraction methods directly measure the distance between parallel planes of lattice points This information is used to determine the lattice parameters in a crystal and measure the angles between lattice planes B Plastic deformation Plastic permanent deformation in metals occurs by the slip of atoms past each other in the crystal This slip tends to occur preferentially along specific lattice planes in the crystal Which planes slip depends on the crystal structure of the material C Transport Properties In certain materials the atomic structure in certain planes causes the transport of electrons and or heat to be particularly rapid in that plane and relatively slow away from the plane Example Graphite Conduction of heat is more rapid in the sp2 covalently bonded lattice planes than in the direction perpendicular to those planes Example YBa2Cu3O7 superconductors Some lattice planes contain only Cu and O These planes conduct pairs of electrons called Cooper pairs that are responsible for superconductivity These superconductors are electrically insulating in directions perpendicular to the Cu O lattice planes
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