《固体物理学》课程教学课件（PPT讲稿）Chapter 1 Crystal Structure 1.9 Elements of Symmetry 1.10 Space groups 1.11 7 crystal system and 14 Bravis Lattice

1.9 Elements of Symmetry SYMMETRY INVERSION REFLECTION ROTATION Operation Element Inversion Point Reflection Plane Rotation Axis Rotoinversion Axes Crystal Structure

Crystal Structure 1 SYMMETRY INVERSION REFLECTION ROTATION 1.9 Elements of Symmetry Operation Element Inversion Point Reflection Plane Rotation Axis Rotoinversion Axes

Symmetry elements Symmetry operation:the distance between two points is unchanged。 Point Symmetry operation:at least one point is fixed Symmetrical elements:point,inversion center,line, rotating axis,plane,reflection plane et al

Symmetry elements Symmetry operation：the distance between two points is unchanged。 Point Symmetry operation：at least one point is fixed Symmetrical elements：point，inversion center, line, rotating axis, plane, reflection plane et al

Inversion:a point at the center of the molecule(x,y,z) >（X,y,-z) Center of Inversion:M Reflection:when a mirror reflection in this plane is performed,the cell remains invariant

Inversion:a point at the center of the molecule(x,y,z) -> (-x,-y,-z) Reflection: when a mirror reflection in this plane is performed, the cell remains invariant

Rotational Symmetries 90 120 180 This is an axis such that,if the cell is rotated around it through some angles,the cell remains invariant. The axis is called n-fold if the angle of rotation is 2/n. Crystal Structure

Crystal Structure 4 This is an axis such that, if the cell is rotated around it through some angles, the cell remains invariant. The axis is called n-fold if the angle of rotation is 2π/n. 90 ° 120 ° 180 ° Rotational Symmetries

Orthogonal transformation the distance between two points is unchanged Rotation about an axis a11 12 3 a11 12 d21 022 d23 y A,= a21 a22 023 z' 、a31 32 33 2 a31 a32 033 cos0 -sin 0 A= sin0 cos0 0 0

Orthogonal transformation • the distance between two points is unchanged • Rotation about an axis 11 12 13 21 22 23 31 32 33 ' ' ' x a a a x y a a a y z a a a z             =                   11 12 13 21 22 23 31 32 33 i j a a a A a a a a a a     =       cos sin 0 sin cos 0 0 0 1 Ai j       −   =      

International symbols Inversion -1 ,(①) A x,y,z→-X,-y,-2 0 Reflection 1 m,o (Z=0 plane) 0 71 A 0 1(E) 0 01

1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 ij ij ij A A A   −   = −   −     =   −     =     Inversion 1,(i) Reflection m, （Z=0 plane） 1(E) x y z x y z , , , , → − − − International symbols

Cube:48 symmetrical operations)(including 1 and 1,(i)) π 2，2 2π4π T 3’3 B 3*3=9 6*1=6 4*2=8 High symmetry demands more symmetrical operation

Cube: 48 symmetrical operations)(including 1 and ) 2 4 , 3 3   3 , , 2 2    4*2=8 π 3*3=9 6*1=6 1,(i) High symmetry demands more symmetrical operation

24 symmetrical operations

24 symmetrical operations

Possible ten symmetry elements for crystals B'A=nAB B'A=nAB 1,2,3,4,6 B'A'=AB(1-2cos0) 1,2,34,6 1-2c0s0=n c0s0:-1~+1 n=-1,0,1,2,3 0=0°，60°，90°，120°，180° XCH001031 1,2,3,4,6

1, 2, 3, 4, 6 1, 2, 3, 4, 6 Possible ten symmetry elements for crystals

Axes of Rotation 6 6 6 6 6 0 △ 9 9 9 1-fold 2-fold 3-fold 4-fold 6-fold Objects with symmetry: a Z + identity Crystal Structure

Crystal Structure 10 Axes of Rotation