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 closedpacked (hcp) > Cubic closedpacked(ccp) > Stacking fault > Packing Efficiency A4. Unit cells of the hcp and ccp structures
A5. Square layers of atoms > Bodycentred cubic (bbc) A6. Interstitial structures > Anion > Cation > Goldschmidt’s Packing Principle > Coordinate > 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 6fold 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 (2fold) > TRIAD (3fold) > TETRAD (4fold) > HEXAD (6fold) > 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 nonrepeating, i.e. nonperiodic. 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 (
pq )  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 distinctlygrouped Bravais lattices. > some crystals with a hexagonal lattice have triad instead of hexad, then it is trigonal system E.g. αquartz.
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). > Rotoinversion axes E2. Crystallographic point groups E3. Crystal symmetry and properties E4. Translational symmetry elements > Glide Plane > Screw Axes Rn E5. Space groups
 Rotoinversion = 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 Xray Diffraction H1. The Ewald sphere
 Depends on the choice of a nonprimitive 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 Xray diffractometry
> 2Circle Diffractometer
> Laue Technique H3. Xray powder diffractometry > Texture > Intensity & diffraction angle
 Fixed θ varying λ method => use white radiation.
H4. Neutron diffraction
> 4Circle 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 xray 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 brightfield image. If an aperture is placed around a diffracted beam only, a darkfield (DF) image is formed.
> If the aperture is large and placed around many beams you form a high resolution “lattice image”.