清华大学：《材料科学基础》课程教学资源（PPT课件讲稿）Chapter 2.5 Indices of crystal planes and directions

32.5 Indices of crystal planes and directions I What's crystal planes and direct i ons? The atomic planes and directions passing through the crystal are called(crystal) planes and directions espectively

§2.5 Indices of crystal planes and directions ◆Ⅰ.What’s crystal planes and directions? The atomic planes and directions passing through the crystal are called (crystal) planes and directions respectively

◆‖.P| ane indices 1. Steps to determinate the plane indices O Establish a set of coordinate axes 2 Find the intercepts of the planes to be indexed on a, b and c axes (r,y,=) b

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). a c b x y z ◆Ⅱ. Plane indices

3 Take the reciprocals of the intercepts 1/x, 1/y, 1/z 4 Clear fractions but do not reduce to lowest integers 5 Enclose them in parentheses,(h k1) 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

 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(020)=(020) e Planes and their multiples are not identical 3 In cubic systems, a direction that has the same indices as a plane is perpendicular to that plane

Several important aspects of the Miller indices for planes should be noted:  Planes and their negatives are identical. Therefore .  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. (020) = (020)

2. The important planes in cubic crystals (00 (110) (111 112)

2. The important planes in cubic crystals (110) (112) (111) (001)

3. A family of planes consists of equivalent planes so far as the atom arrangement is concerned 10}=(110)+(110)+(101)+ (101)+(011)+(01 Total: 6 111=(111)+(111)+(111)+ Total: 4

3. A family of planes consists of equivalent planes so far as the atom arrangement is concerned. (101) (011) (011) {110} (110) (110) (101) + + = + + + Total: 6 (111) {111} = (111) + (111) + (111) + Total: 4

112}=(112)+(112)+(112)+(112)+ (121)+(121)+(121)+(121)+ (211)+(211)+(211)+(211) otal 123}=(123)+(123)+(123)+(123)+(132)+ 132)+(132)+(132)+(231)+(231)+ (231)+(231)+(213)+(213)+(213)+ 213)+(312)+(312)+(312)+(312)+ (321)+(321)+(321)+(321) Tota:4×3!=24

(211) (211) (211) (211) (121) (121) (121) (121) {112} (112) (112) (112) (112) + + + + + + + = + + + + (321) (321) (321) (321) (213) (312) (312) (312) (312) (231) (231) (213) (213) (213) (132) (132) (132) (231) (231) {123} (123) (123) (123) (123) (132) + + + + + + + + + + + + + + + + + + = + + + + + Total: 12 Total: 4×3！=24

l. Direction Indices 1. Derivation for the crystallographic direction o As the first above, set the origin on the direction to be indexed (2 Find the coordinates of another point on the direction in questions 3 Reduce to three smallest integers: u, v, w (4 Enclose in square brackets Ju vwI

◆ Ⅲ. Direction Indices 1. Derivation for the crystallographic direction ① As the first above, set the origin on the direction to be indexed. ② Find the coordinates of another point on the direction in questions. ③ Reduce to three smallest integers: u, v, w. ④ 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 e9.[1001+[010]+[001+ [100]+[010]+[001]

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 e.g. [100] [010] [001] 100 [100] [010] [001] + +  = + + +

32.6 Hexagonal axes for hexagonal crystals Why choose four -ax is system? Four indices has been devised for hexagonal unit cells because of the unique symmetry of the system

§2.6 Hexagonal axes for hexagonal crystals ◆ Ⅰ. Why choose four-axis system? Four indices has been devised for hexagonal unit cells because of the unique symmetry of the system