Announcements 1)Revised Lab timings: 1-3 PM (all groups) 2) Quiz 1, 28 th Jan 2014, Tuesday 7:30 PM, WS 209, WS 213

Chapter 3 Crystal Geometry and Structure Determination

Recap Lattice, Motif/Basis Crystal = Lattice + Motif e.g. Brass, diamond, ZnS Miller indices of direction: components of vector w.r.t to basis vector a, b and c

Miller Indices of directions and planes William Hallowes Miller (1801 – 1880) University of Cambridge

5. Enclose in parenthesis Miller Indices for planes 3. Take reciprocal 2. Find intercepts along axes in terms of respective lattice parameters 1. Select a crystallographic coordinate system with origin not on the plane 4. Convert to smallest integers in the same ratio (111) x y z O

Miller Indices for planes (contd.) origin intercepts reciprocals Miller Indices A B C D O ABCD O 1 ∞ ∞ (1 0 0) OCBE O* 1 -1 ∞ (1 1 0) _ Plane x z y O* x z E Zero represents that the plane is parallel to the corresponding axis Bar represents a negative intercept

Courtesy: H Bhadhesia

Crystallographically equivalent planes

Miller indices of a family of symmetry related planes = (hkl ) and all other planes related to (hkl ) by the symmetry of the crystal {hkl } All the faces of the cube are equivalent to each other by symmetry Front & back faces: (100) Left and right faces: (010) Top and bottom faces: (001) {100} = (100), (010), (001)

{100} cubic = (100), (010), (001) {100} tetragonal = (100), (010) (001) Cubic Tetragonal Miller indices of a family of symmetry related planes x z y z x y

CUBIC CRYSTALS [hkl]  (hkl) Angle between two directions [h 1 k 1 l 1 ] and [h 2 k 2 l 2 ]: C [111] (111)

Some IMPORTANT Results Weiss zone law True for ALL crystal systems Not in the textbook If a direction [uvw] lies in a plane (hkl) then uh+vk+wl = 0 [uvw] (hkl)

d hkl Interplanar spacing between ‘successive’ (hkl) planes passing through the corners of the unit cell O x (100) B O x z E

[uvw]Miller indices of a direction (i.e. a set of parallel directions) (hkl)Miller Indices of a plane (i.e. a set of parallel planes) Miller indices of a family of symmetry related directions {hkl}Miller indices of a family of symmetry related planes Summary of Notation convention for Indices

How do we determine the structure of a piece of crystalline solid? You can probe the atomic arrangements by X-ray diffraction (XRD)

Incident Beam X-Ray Diffraction Transmitted Beam Diffracted Beam Sample Braggs Law (Part 1): For every diffracted beam there exists a set of crystal lattice planes such that the diffracted beam appears to be specularly reflected from this set of planes. ≡ Bragg Reflection

Braggs Law (Part 1): the diffracted beam appears to be specularly reflected from a set of crystal lattice planes. Specular reflection: Angle of incidence =Angle of reflection (both measured from the plane and not from the normal) The incident beam, the reflected beam and the plane normal lie in one plane X-Ray Diffraction i   plane r

X-Ray Diffraction i   r d hkl Bragg’s law (Part 2):

i  r Path Difference =PQ+QR P Q R   d hkl

Path Difference =PQ+QR i r P Q R   Constructive inteference Bragg’s law

Extinction Rules: Table 3.3 Bravais LatticeAllowed Reflections SCAll BCC(h + k + l) even FCCh, k and l unmixed DC h, k and l are all odd Or if all are even then (h + k + l) divisible by 4

Diffraction analysis of cubic crystals   2 sin  222 )lkh( constant Bragg’s Law: Cubic crystals (1) (2) (2) in (1) =>

X Ray Diffractometer

You do not get indices of plane!!

Cu target, Wavelength = Angstrom 2θ2θ Unknown sample, cubic Determine: 1)The crystal structure 2)Lattice parameter

5 step program for the determination of crystal structure 1)Start with 2θ values and generate a set of sin 2 θ values 2)Normalise the sin 2 θ values by dividing it with first entry 3)Clear fractions from normalised column: Multiply by common number 4) Speculate on the hkl values that, if expressed as h 2 +k 2 +l 2, would generate the sequence of the “clear fractions” column 5) Compute for each sin 2 θ /(h 2 +k 2 +l 2 ) on the basis of the assumed hkl values. If each entry in this column is identical, then the entire process is validated.

2θ2θSin 2 θSin 2 θ/Sin 2 θ 1 Clear fractions (hkl)?sin 2 θ /(h 2 +k 2 +l 2 )

William Henry Bragg (1862–1942), William Lawrence Bragg (1890–1971) Nobel Prize (1915) A father-son team that shared a Nobel Prize

h 2 + k 2 + l 2 SCFCCBCCDC , , ,

Two equivalent ways of stating Bragg’s Law 1 st Form 2 nd Form

X-rays Characteristic Radiation, K  Target Mo Cu Co Fe Cr Wavelength, Å