清华材料科学基础课件(英文)skja_05.ppt

1、2020/8/12,材料科学基础,材料科学基础Fundamental of Materials,Lesson five,2020/8/12,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.,2020/8/12,材料科学基础,1. St

2、eps 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,2020/8/12,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. Enclose

3、 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.,2020/8/12,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.,2020/8/12,材料科学基础,2. The important planes in cubic crystals,(110),(112),(111),(001),2020/8/12,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: 4,2020/8/12,材料科学基础,Total: 12,Total: 43！=24,2020/8/12,材料科学基础,. 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 i

6、n questions. Reduce to three smallest integers: u, v, w. Enclose in square brackets u v w.,2020/8/12,材料科学基础,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 equivale

7、nt directions, denoted e.g.,2020/8/12,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.,2020/8/12,材料科学基础,2020/8/12,材料科学基础,. Plane indices (hkil) It can be proved: i (hk)

8、,2020/8/12,材料科学基础,Important planes :,2020/8/12,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,2020/8/12,材料科学基础,Transformation of indices Transformation of 3 to 4 indices, or vice versa. Suppose we have a vector, who

9、se 3 indices u v w, and 4 indices u v t w.,We have,Since,2020/8/12,材料科学基础,or:,2020/8/12,材料科学基础,For example:,2020/8/12,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,2020/8/12,材料科

10、学基础,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.,2020/8/12,材料科学基础,4. The atomic a

11、rrangement and planar density of the important direction in cubic crystal.,2020/8/12,材料科学基础,5. The atomic arrangement and linear density of the important direction in cubic crystal.,2020/8/12,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,2