l7 - CIVE 2700 Civil Engineering Materials Winter 2007...

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CIVE 2700 – Civil Engineering Materials, Winter 2007 1 CRYSTALLOGRAPHIC DIRECTIONS AND PLANES Professor George Hadjisophocleous, Ph.D., P.Eng. Carleton University Department of Civil and Environmental Engineering Ottawa, Ontario CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.2 Outline Crystallographic directions Crystallographic planes
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CIVE 2700 – Civil Engineering Materials, Winter 2007 2 CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.3 Objectives Learn to compute crystallographic directions and planes CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.4 Crystallographic Directions A crystallographic direction is defined as a line between two points, or a vector. Three integers or indexes are used to designate directions and planes Basis for determining index values is the unit cell, with a coordinate system (x,y,z) axes located at one corner of the cell and coinciding with cell edges.
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CIVE 2700 – Civil Engineering Materials, Winter 2007 3 CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.5 Unit Cell Coordinates The numbers refer to the distance from the origin in terms of the numbers of lattice parameters. CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.6 Crystallographic Directions The following steps are used to determine the three directional indices: 1. A vector of convenient length is positioned such that it passes through the origin of the coordinate system. Any vector may be translated throughout the crystal lattice without alteration, if parallelism is maintained. 2. 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. 3. These three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values.
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CIVE 2700 – Civil Engineering Materials, Winter 2007 4 CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.7 Crystallographic Directions 4. The three indices, not separated by commas, are enclosed in square brackets, thus: [uv w]. The u, v, and w integers correspond to the reduced projections along the x, y, and z axes, respectively . For each of the three axes, there will exist both positive and negative coordinates. Thus negative indices are also possible, which are represented by a bar over the appropriate index . CIVE 2700 – Civil Engineering Materials, Winter 2007, 6.8 Crystallographic Directions For example, the [1 ! 1] direction would have a component in the -y direction. Changing the signs of all indices produces an antiparallel direction; that is, [ ! 1 ! ] is directly opposite to [1 ! 1].
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