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Course A: Atomic Structure of Materials A. Crystals and Atomic Arrangements A1. Historical Introduction A2. The packing of atoms in 2D A3. The packing of atoms in 3D > Hexagonal closed-packed (hcp) > Cubic closed-packed(ccp) > Stacking fault > Packing Efficiency A4. Unit cells of the hcp and ccp structures

A5. Square layers of atoms > Body-centred cubic (bbc) A6. Interstitial structures > Anion > Cation > Goldschmidt’s Packing Principle > Co-ordinate > Interstitial site

B. 2D Patterns, Lattice and Symmetry B1. 2D patterns and lattices > Lattice > Motif

Explanation - Although Kepler (1611) was the first to discuss the 6-fold symmetry of the snowflake, it was Hooke(1665) who was the first to consider the structure of crystalline materials in his seminal book ‘Microgrphia’. - ABAB - ABC - mistakes in the stacking sequence - the percentage of volume occupied by atoms - > Simple Hexagonal > hcp

> ccp

- bbc

-Relatively big atoms. - Relatively small atoms. - The number of anions surround a cation tends to be as large as possible, subject to the condition that all anions touch the cation. - Surrounding atoms that form a polyhedral interstice. - Interstitial atom > Interstice type depends on the radius ratio (rx/rA) and the associated coordination number. *In bcc crystals, similar interstitial sites exist but they are “distorted”.

- An infinite array of points repeated periodically throughout space. The view from each lattice point is the same as from any other. - The element of a structure associated with any lattice point. It is a unit of pattern. Structure = Lattice + Motif > By convention, we choose a lattice to give a unit cell with angles closest to a right angle.


B2. Symmetry elements

- --------------------------------------------- Rotational Symmetry------------------------------------------- ----Mirror Symmetry----> DIAD (2-fold) > TRIAD (3-fold) > TETRAD (4-fold) > HEXAD (6-fold) > Mirror

> Crystallographic/ Plane point groups

- A combination of rotation axes and mirrors. There are 10 types of 2D crystallographic. - There is no 5- or 7- fold rotation axes because their patterns are non-repeating, i.e. non-periodic. e.g.

B3. 2D lattices

- The need for periodic tiling of the 2D surface limits the number of possible 2D lattices to just 5. > P = “Primitive” > C = “Centred” - “Glide line”

> A further symmetry element

B4. The 2D plane groups C. Describing Crystals C1. Indexing lattice directions (zone axes) C2. The angles between lattice vectors ( interzonal angles) C3. Indexing lattice planes ( Miller indices) > A lattice plane > Miller Indices C4. Miller indices and lattice directions in cubic crystals

- 5 2D lattices+ 10 crystallographic 2D point groups = 17 “2D Plane Groups”.

- The basis vectors a, b, c are not necessary to be orthogonal or equal. Direction [UVW] in the term of basis vectors: r = Ua + Vb + Wc 1 -   cos (

pq ) | p || q |

- A plane which passes through any 3 lattice points which are not in a straight line. -Miller Indices = (hkl). Remember there are a set of parallel planes and the origin can be moved. - In cubic, the axes are crystallographically equivalent and interchange able. Hence > For <UVW>, there are 48 (24 pairs) variant > For Miller indices we write, {hkl}. > Only in cubic, the direction are perpendicular to planes with the same numerical indices.


C5. Interplaner spacings

h cos     d hkl a k - cos     d hkl b l cos     d hkl c -For cubic crystals d hkl 

C6. Weiss zone law > Zone > Zone Axis > Weiss zone law

2

a h  k2  l2 2

- A set of faces or planes in a crystal whose intersections are all parallel. - The common direction of the intersection - If rUVW is contained in a plane of the set (hkl) then

hU  kV  lW  0

This can be used to find the common direction of the intersections.

hU  k1V  l1W  0 1 h2U  k2V  l2W  0 D. Lattices and Crystal Systems in 3D - Bravais Lattice/Lattice Type - Crystal systems/Crystal classes

2

1 h k   l  For orthogonal axes        d hkl a b c

e1

e2

e3

=> the solutions are h1

k1

l1  0

h2

k2

l2

- 14 crystallographically distinct 3D space lattices. - 7 distinctly-grouped Bravais lattices. > some crystals with a hexagonal lattice have triad instead of hexad, then it is trigonal system E.g. α-quartz.

2


E. Crystal Symmetry in 3D > Crystallographic point groups E1. Centres of symmetry and rotoinversion axes > Centre of symmetry

> Stereographic Projection

- There are 32 distinct crystallographic point groups.

- Any line passing through the centre of the crystal connects equivalent faces. In other words, if an atom is located at (x,y,z) then the same atom must be located at (-x,-y,-z). The origin O is called a centre of symmetry/inversion centre. That crystal is centrosymmetric. - Project object in northern hemisphere as  . Project object in southern hemisphere as  . _

_

> It is given the symbol 1 (we don’t have 2 because it is a miller). > Roto-inversion axes E2. Crystallographic point groups E3. Crystal symmetry and properties E4. Translational symmetry elements > Glide Plane > Screw Axes Rn E5. Space groups

- Roto-inversion = centre of symmetry + rotation axes - There are 32 possible crystallographic point groups.

- The 3D version of a glide line. - R rotations + n translation => there are 11 in totals - 14 Bravais + 32 point groups + translational elements = 230 pattern!

F. Introduction to Diffraction F1. Interference and diffraction F2. Bragg’s law > Bragg’s law F3. The intensities of Bragg reflections > Atomic form factor

-   2d sin  - f 

Amplitude _ of _ Scattering _ by _ Atom Amplitude _ of _ Scattering _ by _ an _ electron

> for an atom of atomic number Z: at the angle=0, f=z > The shape of f depends upon Z and λ as well. > F , atomic form factor for many atoms

Fhkl 2   f n cos 2 (hxn  kyn  lzn )    f n sin 2 (hxn  kyn  lzn )  2

-

2


G. The Reciprocal Lattice >Reciprocal Lattice Vectors

- a  *

1 d 100

,c  *

1 d 001

,  *  180  

In general, * dhkl  ha*  kb*  lc*

>Weiss Zone Law Proof

- if rUVW is contained in a plane (hkl), then

(ha*  kb*  lc* )  (Ua  Vb  Wc)  0 (hU  kV  lW )  0 > The absent of reciprocal lattice

H. The Geometry of X-ray Diffraction H1. The Ewald sphere

- Depends on the choice of a non-primitive real space lattice Real Space Lattice | Reciprocal Lattice I | F F | I

-Bragg’s law in reciprocal space

1 * 1 d  sin  2 

> Fixed λ varying θ method => H2. Single crystal X-ray diffractometry

-> 2-Circle Diffractometer

> Laue Technique H3. X-ray powder diffractometry > Texture > Intensity & diffraction angle

- Fixed θ varying λ method => use white radiation.

H4. Neutron diffraction

> 4-Circle Diffractometer

- Some orientations are preferred. * - I  mhkl Fhkl  Fhkl

and sin 2  

2 4a

2

N ; N  h2  k 2  l 2

- The scattering factors are independent of scattering angle - Light elements (e.g. H2, O2) have relatively large scattering amplitude. It allow their locations to be determined easily than with x-ray diffraction.


I. Electron Microscopy and Diffraction I1. Electron diffraction

I2. Microscopy and image formation > Abbe’s Theorem I3. Electron microscopy

- > Have much larger radius of the Ewald sphere (lower λ). > A specimen is a very thin film (~10 nm) because electrons interact very strongly and the reciprocal lattice are broadened into rods.

-d

 2n sin 

, where d is a limit of resolution.

-> The limit of resolution for electron microscope is ~0.1 nm > By placing an aperture around the direct beam only, we can form a bright-field image. If an aperture is placed around a diffracted beam only, a dark-field (DF) image is formed.

> If the aperture is large and placed around many beams you form a high resolution “lattice image”.


Atomic Structure of Materials