Elementary Symmetry Operation Derivation of Plane Lattices Crystal Structure and Crystallography of Materials Chapter 6: Elementary Symmetry Operation and Derivation of Plane Lattices

How should we understand crystal structure? Crystal Structure: periodic arrangement of atoms (types of atoms and the amount of atoms) the only thing we have to remember (about the definition of crystal structure) is long-range order in 3-dimension space, namely repetitiveness Thus, instead of investigating all the arrangement of atoms, we approach it geometrically, and when doing that symmetry is one of the most important thing to understand it systematically. We will first consider the arrangement of geometrical points, which we call lattice points, in 3-D space in accordance with the definition of crystal structure – repetitiveness Then, we will assign the actual arrangement of atoms in each lattice point. – basis Here, all the lattice points is chemically and physically identical each other in the sense of atomic arrangements.

Symmetry Elements: ► Point symmetry : Consistent with lattice Rotational Symmetry Reflection Symmetry Inversion Center (Center of symmetry) Rotation-Inversion Symmetry

(A) Rotational Symmetry: - n-fold rotational symmetry if it is identical when rotated by 360o/n 2-fold symmetry (diad) ex) water H O 180° (2) 3-fold symmetry (triad) ex) BCl3 B Cl 120° 120°

(A) Rotational Symmetry: - n-fold rotational symmetry if it is identical when rotated by 360o/n (3) 4-fold symmetry (tetrad) 90° (4) 6-fold symmetry (hexad) 60°

(A) Rotational Symmetry: Note that molecule can have 5,7,12 rotational symmetry but they can’t be arranged in crystal with that symmetry → When the rotational symmetry does occur in crystals, then severe restrictions on the value of α are imposed by the simultaneous occurrence of repetition by rotation and translation For instance, t A A' B B' AA'= t translation BB'= t also rotation

(A) Rotational Symmetry: Ex) 3-fold axis Ex) 5-fold axis t BB′= 2AA′=2t A A′ 2t B B′ 72° BB′= t′ new translation AA′= t t′

(A) Rotational Symmetry: In general, BB′= mt (m=integer) t A A′ B B′ α° b t cosα BB′= b = t – 2 t cos α = mt m=1 – 2 cos α 2 cos α = 1- m = M cos α = M/2 M cos α α n b -3 -1.5 - -2 -1 π 2 3t -0.5 2π/3 3 2t π/2 4 t 1 0.5 π/3 6 ∞=1 -t

(A) Rotational Symmetry: Magnitudes of b for various crystallographic values of n Specialized plane nets for various crystallographic values of n

(B) Simultaneous Rotational Symmetry: Two rotations about intersecting axes → inevitably create a third rotation equivalent to the combination Rotation around A axis to α By the vector notation ; 1 3 2 A C B α β γ Crystal could, conceivably, be symmetrical with respect to many different intersection n-fold axis

(B) Simultaneous Rotational Symmetry: Euler’s Construction The combined motions of Aα and Bβ have the following effect : 1 : Aα brings C to C′ 2 : Bβ restores C′ to C Thus, the combination of rotations Aα and Bβ leaves C unmoved. Therefore, if there is a motion of points on the sphere due to Aα and Bβ, it must be a rotation about an axis OC To calculate ; 1 : Aα leaves A unmoved 2 : Bβ moves A to A’ Now consider A M M′ N′ N B C C′ A′ α β γ u v w

(B) Simultaneous Rotational Symmetry: Euler’s Construction v w U=α/2 V=β/2 W=γ/2

(B) Simultaneous Rotational Symmetry: Euler’s Construction Axis at A, B, or C Throw of axis, α, β, or γ U (=α/2) V(= β/2) or W(= γ/2) cos U, V, W sin U, V, W 1-fold 360º 180º -1 2-fold 90º 1 3-fold 120º 60º 4-fold 45º 6-fold 30º

(B) Simultaneous Rotational Symmetry: Euler’s Construction

(B) Simultaneous Rotational Symmetry: Euler’s Construction

(B) Simultaneous Rotational Symmetry: Euler’s Construction

Matrix Operation of the Rotational Symmetry: P(x,y,z) P(x’,y’,z’) a f Determinant of R(nz)=1

Reflection (Mirror) Symmetry: Mirror plane in x-axis Inversion Center: L R Inversion center

Roto-Inversion Symmetry: - Rotate by 360/n and invert LH RH (1) 1- fold rotation-inversion axis ( )

Roto-Inversion Symmetry: (2) 2-fold rotation inversion (2) (3) Inversion triad (3) RH LH

Roto-Inversion Symmetry: (4) Inversion tetrad (4) (5) Inversion hexad (6)

Derivation of Plane Lattices In the crystallography, you have to know how to derive, 7 Crystal Systems – triclinic, monoclinic, trigonal, orthorhombic, tetragonal, hexagonal, cubic 14 Bravais Lattices 32 Point Groups 230 Space Groups

Symmetrical Plane Lattices: Combination of a rotation and a perpendicular translation α P1 P2 P3 A B A′ 2 1 3 t Aα causes P1 → P2 then t causes P2 → P3 ∴ Aα · t causes P1 → P3 ⇒ The net motion of the line is therefore equivalent to a rotation about B. Furthermore, since the line is embedded in space, and since the operations Aα and t act to all spaces, All space must also be rotated about B by this combination of operations. Point B : lines on the bisector of AA′ distance (AA′/2)cot(α /2) from AA′ This can be expressed analytically as Aα · t = Bα

Symmetrical Plane Lattices: Combination of the rotation axes with a plane lattice : General principles 1) A rotation axis implies, in general, several related rotations When the symmetry axis is n-fold, the smallest rotation is α = 2π/n The rotation axis then implies the rotations α, 2α, … , nα, where nα = 2π Each of these rotations is to be combined with the translation 2) Each rotation must be combined with the various translations of the plane lattice such as t1, t2, t1+t2

(A) 2-fold axis: rotation A ∴ Aπ · t = Bπ ∴ B lies in the bisector (AA'/2) cot (α/2) =0 A A′ B C D Has 4 non-equivalent 2-fold axis π

(B) 3-fold axis: A 2π/3 A′ B1 B2 C1 C2 t1 t1+t2 t2 Rotation at A Translation t1 t1+t2 1 2π/3 2π/3 at B1 2π/3 at C1 -2π/3 -2π/3 at B2(C1) -2π/3 at C2(B1) -2π/3 A B C It has three non-equivalent 3-fold axis located in the cell at A, B, C.

(C) 4-fold axis: The operations of the 4-fold axis are rotations of 1, π/2, π, 3π/2( = -π/2) A π/2 -π/2 π A′ B1 B2 C1 C2 t1 t1+t2 t2 A″ B3 at B1 : 4-fold B2 : 2-fold Rotation at A translations t1 t1+t2 1 π/2 π/2 at B1 π/2 at C1 π π at B2 π at C2 (B1) -π/2 - π/2 at B3(B1) -π/2 at C3 (A´) A1 B1 B2

(D) 6-fold axis: When a lattice plane has 6-fold axis, then t1 = t2 = t1+t2 →Thus, only consider t1 A 2π/3 -2π/3 -π π π/3 -π/3 Rotation : 1, π /3, 2π/3, π, -2π/3, - π /3 A′ B1 B2 t1 t1+t2 t2 B3 B4 B5 Rotation at A Translations ( t1) 1 π/3 π/3 at B1 (A) 2π/3 2π/3 at B2 π π at B3 -2π/3 -2π/3 at B4 (B2) -π/3 -π/3 at B5

Lattice types consistent with plane symmetries of the 2nd sort: Lemma : For n>2, the shape of a plane lattice consistent with pure axial symmetry n is also consistent with corresponding symmetry nmm containing reflection planes m → Therefore, new lattice-plane mesh types can only be found by causing a general lattice type to be consistent either with symmetry m (or 2mm) m t Fig.A Fig.B

Lattice types consistent with plane symmetries of the 2nd sort: Fig.C Fig.D m t Fig.E

Lattice types consistent with plane symmetries of the 2nd sort: Fig.F Fig.G ( Diamond Rhombous)

The distribution of rotation axes and mirrors in the five plane lattice types