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 − =