1、2019/5/28,材料科学基础,材料科学基础 Fundamental of Materials,Lesson five,2019/5/28,材料科学基础,2.5 Indices of crystal planes and directions,.Whats crystal planes and directions?,The atomic planes and directions passing through the crystal are called (crystal) planes and directions respectively.,2019/5/28,材料科学基础,1. S
2、teps 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).,. Plane indices,2019/5/28,材料科学基础, Take the reciprocals of the intercepts 1/x, 1/y, 1/z. Clear fractions but do not reduce to lowest integers. Enclos
3、e 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.,2019/5/28,材料科学基础,Several important aspects of the Miller indices for planes should be noted: Planes and their negatives are identical. Therefore . Planes and
4、 their multiples are not identical. In cubic systems, a direction that has the same indices as a plane is perpendicular to that plane.,2019/5/28,材料科学基础,2. The important planes in cubic crystals,(110),(112),(111),(001),2019/5/28,材料科学基础,3. A family of planes consists of equivalent planes so far as the
5、 atom arrangement is concerned.,Total: 6,Total: 4,2019/5/28,材料科学基础,Total: 12,Total: 43！=24,2019/5/28,材料科学基础,. Direction Indices,1. Derivation for the crystallographic directionAs the first above, set the origin on the direction to be indexed. Find the coordinates of another point on the direction in
6、 questions. Reduce to three smallest integers: u, v, w. Enclose in square brackets u v w.,2019/5/28,材料科学基础,2. The important direction in cubic crystals: crystal axes: face diagonal: body diagonal: apices to opposite face-centers3. Family of directions consists of crystallographically equivalent dire
7、ctions, denoted e.g.,2019/5/28,材料科学基础,2.6 Hexagonal axes for hexagonal crystals,. Why choose four-axis system?Four indices has been devised for hexagonal unit cells because of the unique symmetry of the system.,2019/5/28,材料科学基础,2019/5/28,材料科学基础,. Plane indices (hkil)It can be proved: i (hk),2019/5/2
8、8,材料科学基础,Important planes :,2019/5/28,材料科学基础,. Direction indices u v t w To make the indices unique, an additional condition is imposed. - Let t(uv),Important directions,2019/5/28,材料科学基础,Transformation of indicesTransformation of 3 to 4 indices, or vice versa. Suppose we have a vector, whose 3 indic
9、es u v w, and 4 indices u v t w.,We have,Since,2019/5/28,材料科学基础,or:,2019/5/28,材料科学基础,For example:,2019/5/28,材料科学基础,1. Quick way for indexing the directions in cubic crystals:The value of a direction depends on its feature while the sign on direction.,Examples and Discussions,2019/5/28,材料科学基础,2. The
10、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.,2019/5/28,材料科学基础,4. The atomic arrangement a
11、nd planar density of the important direction in cubic crystal.,2019/5/28,材料科学基础,5. The atomic arrangement and linear density of the important direction in cubic crystal.,2019/5/28,材料科学基础,Exercise,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,2019/5/28,材料科学基础,However, no atoms are centered on the (020) planes. There fore, the planar density and the planar packing fraction are both zero.,2019/5/28,材料科学基础,Thank you !,5,
