材料化学全英文课件ppt课件

材料科学基础 Fundamental of Materials Science,Prof: Tian Min Bo Tel: 62795426 ，62772851 E-mail: Department of Material Science and Engineering Tsinghua University. Beijing 100084,Lesson five,2.5 Indices of Crystal Planes and Directions,2.5.1 Whats crystal planes and directions?,The atomic planes and directions passing through the crystal are called Crystal Planes and Crystal Directions respectively.,1. Steps to determinate the plane indices: Establish a set of coordinate axes. Find the intercepts of the planes to be indexed on a, b and c axes (x, y, z).,2.5.2 Plane indices, Take the reciprocals of the intercepts 1/x, 1/y, 1/z. Clear fractions but do not reduce to lowest integers. Enclose them in parentheses, (h k l) Example: 1/2,1,2/3 2,1,3/2 (423) Plane indices referred to three axes a, b and c are also called Miller Indices.,Several important aspects of the Miller indices for planes should be noted: Planes and their negatives are identical. Therefore, . In some situation, planes and their multiples are not identical. In cubic systems, a direction that has the same indices as a plane is perpendicular to that plane.,Total: 12,Total: 43！=24,3.5.3 Direction indices,1. Derivation for the crystallographic direction As the first above, set the origin on the direction to be indexed. Find the coordinates of another point on the direction in questions. Reduce to three smallest integers: u, v, w. Enclose in square brackets u v w.,2. The important direction in cubic crystals: : crystal axes : face diagonal : body diagonal : apices to opposite face-centers 3. Family of directions consists of crystallographically equivalent directions, denoted ,2.6 Hexagonal Axes for Hexagonal Crystals,2.6.1 Why choose four-axis system? Four indices has been devised for hexagonal unit cells because of the unique symmetry of the system.,2.6.2 Plane indices (hkil) It can be proved: i (hk),So four axis of plane indices can be deduced directly.,Important planes :,2.6.3 Direction indices u v t w To make the indices unique, an additional condition is imposed. - Let t(uv),Important directions,For example : three-axis: four-axis: , , are all allowed, but not unique.,Transformation of indices Transformation of 3 to 4 indices, or vice versa. Suppose we have a vector, whose 3 indices U V W, and 4 indices u v t w.,We have,Since,let,or:,For example:,2.6.4 Discussion about Direction indices u v t w ,1. Why let,according to three indices:,So the direction indices of OP could be:,、 、 、,To make the direction indices be unique, we let:,2. Another method to get direction indices u v t w ,For any point that on the direction, we get out its vertical projection on a1,a2,a3,c. and multiply the first three indices by 2/3. Clear fractions and reduce them to lowest integers.,1. Quick way for indexing the directions in cubic crystals: The value of a direction depends on its feature while the sign on direction.,Discussion,For example: ,2. The coordinate origin can be set arbitrarily (for example on apices, body-center, face-centers etc.), but never on plane in questions, otherwise the intercepts would be 0,0,0 . 3. The coordinate system can be transferred arbitrarily, but rotation is forbidden.,4. The atomic arrangement and planar density of the important direction in cubic crystal.,5. The atomic arrangement and linear density of the important direction in cubic crystal.,6. Calculate the planar density and planar packing fraction for the (010) and (020) planes cubic Polonium, which has a lattice parameter of 0.334nm.,Solution,However, no atoms are centered on the (020) planes. There fore, the planar density and the planar packing fraction are both zero.,Exercise,1. Write out the indices of all the tantamount planes and directions of family of 123 plane and family of direction in cubic crystals.,2. Draw up the planes and directions in unit cell of cubic crystals as follow:,3. Label the indices of planes and directions in cubic crystals below.,4. Write out the indices of all the planes and directions in family of plane and fa