BRAVAIS LATTICE Infinite array of discrete points arranged (and oriented) in such a way that it looks exactly the same from whichever point the array is looked at. BL describes the periodic nature of the atomic arrangements (units) in a X’l. X’l structure is obtained when we attach a unit to every lattice point and repeat in space Unit – Single atoms (metals) / group of atoms (NaCl) [BASIS] Lattice + Basis = X’l structure

2-D honey comb net P Q R Not a BL Lattice arrangement looks the same from P and R, bur rotated through 180º when viewed from Q

Primitive Translation Vectors If the lattice is a BL, then it is possible to find a set of 3 vectors a, b, c such that any point on the BL can be reached by a translation vector R = n 1 a + n 2 b + n 3 c where a, b, c are PTV and n i ’s are integers eg: 2D lattice a1a1 a2a2 a1a1 a2a2 a1a1 a2a2 (A) (B) (C) (D) a1a1 a2a2 (A), (B), (C) define PTV, but (D) is not PTV

j i k 3-D Bravais Lattices (a) Simple Cube a2a2 a1a1 a PTV : a3a3

F4 F6 F1 F2 F3 F5 A B C2 C1 All atoms are either corner points or face centers and are EQUIVALENT Face centered cubic a1a1 a2a2 a3a3 For Cube B, C1&C2 are Face centers; also F2&F3 PTV

(0,0,0) (1,0,0) (1,1,0) (0,1,0) (1,2,0) (0,0,1) (0,-1,2) (0,0,2) (-1,0,2)(-1,-1,2) (0,1,1) (-1,-1,3) a1a1 a2a2 a3a3 PTV

Alternate choice of PTV a1a1 a2a2 a3a3

Oblique Lattice : a ≠ b, α ≠ 90 Only 2-fold symmetry 2

Rectangular Lattice : a ≠ b, α = 90

a b 2 mirror Rectangular Lattice : a ≠ b, α = 90 : Symmetry operations

a b Hexagonal Lattice : a = b, α = 120

PUC and Unit cell for BCC Unit Cell Primitive Unit Cell

A is the body center A B is the body center All points have identical surrounding Body-centered cubic: 2 sc lattices displaced by (a/2,a/2,a/2) B PUC

PUC and Unit cell for FCC Unit Cell PUC

PUC and Unit cell for FCC : alternate PTV

P PCP P (Trigonal) PICF P PI I F 7 X’l Systems 14 BL

b a c

2-D Lattice a1a1 a2a2 60º A lattice which is not a BL can be made into a BL by a proper choice of 2D lattice and a suitable BASIS A B The original lattice which is not a BL can be made into a BL by selecting the 2D oblique lattice (blue color) and a 2-point BASIS A-B

BCC Structure

FCC Structure

NaCl Structure

Diamond Structure

(0,0,0) (¼, ¼,¼) x y z (¾, ¾,¼) (¼, ¾, ¾) (¾, ¼, ¾) No. of atoms/unit cell = 8 Corners – 1 Face centers – 3 Inside the cube – 4

Hexagonal Close Packed (HCP) Structure

HCP = HL (BL) + 2 point BASIS at (000) and (2/3,1/3,1/2)

The Simple Hexagonal Lattice

The HCP Crystal Structure

4-circle Diffractometer

Reciprocal Lattice (000)

2π/λ2π/λ (001) (002) (00 -1) (102) (202)(302) (101)(201)(301) (100)(200) (300) (30 -1)  k = k ´- k = G 201 θ 201 (201) plane k´ Incident beam k

a* b* (000)(200)(-200) Rotaion = 0º 2π/λ2π/λ Incident beam

a* b* (000)(200)(-200) 2π/λ2π/λ Rotaion = 5º

a* b* (000)(200)(-200) Rotaion = 10º

a* b* (000)(200)(-200) Rotaion = 20º 2π/λ2π/λ

a* b* (000)(200)(-200) Rotaion = 5º 2π/λ2π/λ Incident beam 2π/λ2π/λ Rotaion = 20º

Schematic diagram of a four-circle diffractometer.

2θ2θ I Scattering Intensities and Systematic Absence

Diffraction Intensities Scattering by electrons Scattering by atoms Scattering by a unit cell Structure factors Powder diffraction intensity calculations – Multiplicity – Lorentz factor – Absorption, Debye-Scherrer and Bragg Brentano – Temperature factor

Scattering by atoms We can consider an atom to be a collection of electrons. This electron density scatters radiation according to the Thomson approach (classical Scattering). However, the radiation is coherent so we have to consider interference between x-rays scattered from different points within the atom – This leads to a strong angle dependence of the scattering – FORM FACTOR.

Form factor (Atomic Scattering Factor) We express the scattering power of an atom using a form factor (f) – Form factor is the ratio of scattering from the atom to what would be observed from a single electron f Cu sinθ/λ Form factor is expressed as a function of (sinθ)/λ as the interference depends on both λ and the scattering angle Form factor is equivalent to the atomic number at low angles, but it drops rapidly at high (sinθ)/λ

X-ray and neutron form factor The form factor is related to the scattering density distribution in an atoms - It is the Fourier transform of the scattering density - Neutrons are scattered by the nucleus not electrons and as the nucleus is very small, the neutron form factor shows no angular dependence F-F- C Li+ f sinθ/λ 1H1H 7 Li 3 He b X-RAY NEUTRON

Scattering by a Unit Cell – Structure Factor The positions of the atoms in a unit cell determine the intensities of the reflections Consider diffraction from (001) planes in (a) and (b) If the path length between rays 1 and 2 differs by λ, the path length between rays 1 and 3 will differ by λ/2 and destructive interfe- rence in (b) will lead to no diffracted intensity (a) a b c (b) (a)