a b b a, b, c 1 norm of b a b a c b a b c c a c b 0 (1) 1 (2) a a b b c c Reciprocal lattice in crystallography Where i and k may each equal 1,2 and 3

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2 Reciprocal lattice in crystallography 0, ai bk ik ik 1, ik 0 bk ai 1 norm of b ik Reciprocal lattice b a, b, c k i i i k k Where i and k may each equal 1,2 and 3 Each b k is perpendicular to two a i conventional crystallographic notation Direct lattice a b c * * * a,, k 0 (1) 1 (2) * * * * * * a b a c b a b c c a c b * * * a a b b c c

3 a p b c p : constant (3) * Using (1) 1, * p p V V * a a 1 p b c a pv Equation (3) may be written now: * 1 a b c V * 1 a bc sin V V: Volume of the unit cell in direct space V*: Volume of the reciprocal unit cell (4) V V V * * * a b c b c a c a b

4 1 1 1 V V V * * * a b c b c a c a b (5) a b c b c a c a b * * * V V V * * * * * * (6)

5 Using (5) and (6): Summary * * V a a * * * * * * * * a b c sin sin V b c Vb c * V sin abcsin sin

6 Some relations - The reciprocals of direct lattices (7 crystal systems) preserve the symmetry of the lattice. 1. Orthogonal lattices: orthorhombic, tetragonal and cubic lattices: * * * a a, b b, c c * * * a 1, b 1, c 1 a b c * * * 2 2. Monoclinic lattices: * * * b b, while a and c are in the plane: a, c * 1 * 1 * a, b, c 1 a sin b c sin 2 * * *,

7 Some relations 3. Trigonal and hexagonal lattices: Solution: * * * c c, while a and b are in the plane: a, b * * * a 1 2 b, c 1 a sin a 3 c * * *, 2 3

8 Some relations - The reciprocal of an I lattice is an F lattice and viceversa. Primitive lattice of a bcc lattice a 2 a 2 a 2 a i j k ˆ ˆ ˆ ˆ ˆ b i j k c i ˆ ˆ j k ˆ ˆ Primit ive lattice of a fcc lattice a ˆ 2 ˆ a ˆ 2 ˆ a ˆ 2 ˆ a j k b i k c i j a lattice constant Exercise, prove that The reciprocal of : a a i ˆ ˆj k ˆ 2 * 1 is: a ˆj kˆ a

10 Some properties of the reciprocal lattice 1. Scalar product of two vectors defined in different bases: r r xa yb zc x a y b z c 2. The reciprocal vector : * * * * * * * x x y y z z X X * * * T * r ha kb lc * * * * H is normal to the family of lattice planes (hkl) in direct space b a c a c b k h l h l k * * * * r b a H B A ha kb lc 0 k h r C A r C B 0 B A, C A, C B H * * H

11 Some properties of the reciprocal lattice 3. r * 1 H d H d H : spacing of the planes (hkl) a ON d ˆ ˆ H n n : unit vector h * rh nˆ * rh * a rh 1 d H * * h r r H H

$Diffraction of Materials, by V.$

12 The reciprocal lattice Useful properties derived from the definition of the reciprocal lattice: Taken from: Fundamentals of Powder Diffraction of Materials, by V.K Pecharsky, et al.

13 Some properties of the reciprocal lattice 4. The metric matrix G* for the reciprocal lattice: r x a y b z c, r x a y b z c * * * * * * * * r r X G X * * * T * * * * * * * * a a a b a c *2 * * * * * * a a b cos a c cos * * * * * * * * * * *2 * * * G b a b b b c a b cos b b c cos * * * * * * * * * * * * *2 c a c b c c a c cos b c cos c G G * * T * *2 * 1 1. det G V and V V * 1 1 * 2. G G G G G G I Identity matrix

14 Calculate the reciprocal metric matrix (metric tensor) for: Hexagonal reciprocal basis Tetragonal reciprocal basis

15 Some properties of the reciprocal lattice r X G X where r ha kb lc *2 * T * * * * * * H From property (3) The quadratic forms in reciprocal space H 2 cos 2 cos 2 cos 1 d h a k b l c hka b hla c klb c 2 *2 2 *2 2 *2 * * * * * * * * * 2 d h a k a l a hka hla kla H Cubic: Triclinic: H d h k l a Tetragonal: dh h k a11 l a33 Hexagonal: d h hk k a l a H d h a k a l a hla hka lka H a a a a a a a b c *2 *2 *2 * * * 2a b cos * * * 2a c cos * * * 2b c cos

16 The quadratic forms in direct space

1 2 * * * d * H1d H 2rH 1 rh 2 rh 1 rh 2 r r X G X * * * * * H1 H 2 H1 H 2 r nˆ *

17 5. Some geometrical calculation concerning directions and planes a) The angle between two planes: H 1 = (h 1 k 1 l 1 ) and H 2 = (h 2 k 2 l 2 ) rh rh cos * * * 1 2 * * * d * H1d H 2rH 1 rh 2 rh 1 rh 2 r r X G X * * * * * H1 H 2 H1 H 2 r nˆ * H1 H1 Exercise : Calculate the angle * between planes (100) and (010) for a Hexagonal base

18 b) Crystallographic planes parallel to a given direction, defined by: are said to belong to the same zone. A family of planes H = (hkl) belongs to the zone [uvw] if: r ua vb wc U r r hu kv lw * H U 0 c) If planes H 1, H 2 and H 3 belong to the same zone then: r, r, r * * * H1 H 2 H 3 are coplanar

19 Stereographic projection Zone axis: [001] Cubic P lattice

20 Zone axis: [001] Cubic P lattice - Reciprocal lattice section normal to [001]

21 Zone axis: [001] Cubic P lattice - Reciprocal lattice section normal to [001]

22 Zone axis: [001] Cubic P lattice - Reciprocal lattice section normal to [001] Recirpocal space: EDP

23 Zone axis: [111] Cubic P lattice - Reciprocal lattice section normal to [111]

24 Zone axis: [111] Cubic P lattice - Reciprocal lattice section normal to [111]

25 Zone axis: [111] Cubic P lattice - Reciprocal lattice section normal to [111] Recirpocal space: EDP

26 Transformation of the coordinate system Examples of coordinate system transformations: Rombohedral- hexagonal transformations: BCC - P transformation FCC - P transformation

27 Transformation of the coordinate system Linear part of the transformation + Shift of the origin = General transformation (affine transformation) Matrix P P P P P P P P P P Shift vector + p p1a p2b p3c = General transformation Matrix P, p

28 Direct space General transformation: P, p a) a, b, c a, b, c P, with p 0 b) if, than occur only: c) a, b, c a, b, c P p P I p O O p Inverse general transformation 1 1 Q, q, Q P and q P p a) a, b, c a, b, c Q, with q 0 b) if, than occur only: c) a, b, c a, b, c Q q Q I q O O q Reciprocal space General transformation: Q, q * * a a * * a) b Q b, with q 0 * * c c b) if, than occur only: * * a a c) * * Q I q O O q * * b Q b q * * c c Inverse general transformation * * a a * * P, p, a) b P b p * * c c

2-D D Symmetry. Symmetry (Part II) Point Group m 3 4. The set of symmetry operations that leave the appearance of the crystal structure

2-D D Symmetry. Symmetry (Part II) Point Group m 3 4. The set of symmetry operations that leave the appearance of the crystal structure Symmetry (Part II) Lecture 6 Point Group The set of symmetry operations that leave the appearance of the crystal structure unchanged. There are 32 possible point groups (i.e., unique combinations of symmetry