1. Solid State Physics Yuanxu Wang School of Physics and Electronics Henan University wyxhenu@gmail.com 双语教学示范课程

2. Page  2 Review of previous sections  Bravais lattice  Primitive cell and conversional cell  Simple lattice and complex lattice  Close-packing

3. Page  3 §4 Crystal array and plane indices 1. Crystal array Drawing a straight line through any two grids, we call that crystal column, and call the orientation of the crystal columns, crystal orientation. A set of numbers which describing the orientation of the crystal are called crystal array index(crystal array index).

4. Page  4 Why do we introduce "crystal column," "crystal plane" ? C33 = 1107 GPa C11 = 687 GPa

5. Page  5 Thermoelectric materials Ca 5 Ga 2 As 6

6. Page  6 (1) The crystal array family are composed of crystal arrays,and they contain all the grids. (2) The distribution of the grids in the crystal array is periodical. (3)The distribution of the grids in each crystal array of Crystal array family is the same. (4) In the same plane, the distance between the adjacent crystal columns is equal. Note:

7. Page  7 2.Crystal array index (1) Represented by the basis vectors of the primitive cell If the position vector which is from a grid in the crystal array to an arbitrary grid in the direction of the crystal orientation described by: is the basis vectors of the primitive cell is an integer,and will be turned into an coprime integer,and it is recorded as [ ] ， [ ] is the crystal column index of the crystal column. If it is a negative number, we should add a horizontal line above the number 。 Such as [121] ;.

8. Page  8 (2)Represented by the basis vectors of the bravais primitive cell If the position vector which is from a grid in the crystal array to an arbitrary grid in the direction of the crystal orientation. is rational, and will be turned into an co-prime integer m, n, p,and it is recorded as [ mnp ] ， [ mnp ] is the crystal column index of the crystal column. Case 1: As it is shown in figure of the cube :, D is the midpoint of the BC.To solve the crystal column index of the BE and the AD. The solution: The crystal column index of the crystal column ： [011]

9. Page  9 To solve the crystal column index of AD. O A B C D E Case 1: As it is shown in figure of the cube :, D is the midpoint of the BC.To solve the crystal column index of the BE and the AD. The solution: The crystal column index of the crystal column ：

10. Page  10 (1) Crystal column index must be a set of coprime integers; (2) Crystal column index is expressed in brackets [ ]; (3) Encountering a negative number,we should add a horizontal line above the number. (4) The quivalent orientation of crystal. Attention:

11. Page  11 Equivalent to the crystal [ 100 ] [ 001 ] [ 010 ] [ 100 ] [ 010 ][ 001 ] 3.Crystal plane and miller index （ 1 ） Crystal plane In the lattice, a plane is made through any three gards which are not in the same line,and it is called crystal face, a set of numbers describe the orientation of crystal called crystal orientation index. （ 2 ） Crystal plane index Crystal orientation The normal direction of the crystal plane (The angle of the normal direction and three axes ) The intercept of the crystal plane and the three axes

12. Page  12 A2A2 A3A3 O A1A1 N dd Take as the natural length unit ： The ratio of the cosine of the angle of the normal direction of the crystal face and three axes (vector), is equal to the ratio of the reciprocal value of the intercept of the crystal face and the three axes.

13. Page  13 We can prove that: r, s, t is a set of rational numbers. Set :The grids at the end of on the crystal faces of being in the distance of h 1 d 、 h 2 d 、 h 3 d from the origin,respectively. Here,h1, h2, h3 is integer. ① All grids are included in a set of the crystal face; So there will be a crystal of the families of the crystal faces passing through the origin of the coordinate system ； The grids at the end of the basis vector also must fall on the crystal faces of the families. ② The interplanar spacings of the parallel and adjacent crystal face of the same family of crystal faces are equal, and so, there is an integer number of crystal faces between the origin and at the end of basis vector.

14. Page  14 The ratio of the cosine of the angle of the normal direction of the crystal face and three basic vectors ： Because the h1, h2, h3 is an integer, so r, s, t will be rational. h 1 ， h 2 ， h 3, are called the crystal face index, and are recorded as( h 1 h 2 h 3 ) The meaning of the crystal plane index : ② T he Co-prime ratio of the reciprocal value of the intercept of the crystal plane on the coordinate axis,which we get With taking as the unit length of each axis. ① Basis vector are divided into the same parts of h1, h2, h3 by the parallel planes of the crystal ; ③ The ratio of the cosine of the angle of the normal direction of the crystal face and the basic vector.

15. Page  15 Example 2: drawing the crystal plane of (210) 、 in the cubic crystal system. The intercepts of the crystal plane and the three axes are below, respectively: A B C D E F G (210) 1 1 1 The crystal face whose the miller index is (210) is the plane of ABCD. The crystal face whose the miller index is is the plane of EFG. (121) Crystal plane index represented by the bravais primitive cells basic vectors whose the axis is is called miller index, (hkl).