Review of Last Class 1
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2 Review of Last Class 1 X-Ray diffraction of crystals: the Bragg formulation Condition of diffraction peak: 2dd sin θθ = nnλλ
3 Review of Last Class 2 X-Ray diffraction of crystals: the Von Laue formulation Condition of diffraction peak: kk kk = KK kk KK = 1 2 KK 2 Bragg formulation Laue formulation
4 Experimental geometries: Review of Last Class 3 Condition needed: kk kk = KK Ewald construction
5 Experimental geometries: Review of Last Class 4 The Laue method: Non-monochromatic X-rays
6 Experimental geometries: Review of Last Class 5 The rotating crystal method
7 Experimental geometries: Review of Last Class 5 The rotating crystal method
8 Experimental geometries: Review of Last Class 5 The powder or Debye-Scherrer Method
9 Diffraction by Real Crystals 1 Monoatomic lattice with a basis: Consider the scattering from one primitive unit cell
10 Diffraction by Real Crystals 2 Monoatomic lattice with a basis: Geometrical structure factor SS KK = ee iikk dd jj jj=1 Example: BCC lattice considered as a simple cubic lattice with a basis nn dd jj labels atoms in the basis
11 Diffraction by Real Crystals 3 Polyatomic lattice: Atomic form factor nn dd jj labels atoms in the basis SS KK = ff jj (KK)ee iikk dd jj jj=1 ff jj (KK) : atomic form factor: internal structure of the ion defined as ff jj KK = 1 ee ddrr eeiikk rr ρρ jj (rr ) Fourier transform of charge distribution
12 Interesting Example 1 From Philip Nelson s book (to appear)
13 Interesting Example 2 How did we know that the DNA molecules have a double helix structure? From Philip Nelson s book (to appear)
14 Classification of Bravais Lattices A Bravais lattice is characterized by the specification of all rigid operations that take the lattice into itself (symmetry operations) Rigid operations: operations that keep the distances between all lattice points Operations taking the lattice into itself: symmetry group / space group of the Bravais lattice These operations include: All translations through lattice vectors Rotations Reflections (mirror image) Inversions (rr rr)
15 Symmetry Operations Any symmetry operation of a Bravais lattice can be compounded out of: a translation TT RR through a lattice vector RR a rigid operation leaving at least one lattice point fixed (point group) Example:
16 Symmetry Operations
17 Crystal Systems and Bravais Lattices Bravais lattices can be classified based on their point group (rigid operation leaving at least one lattice point fixed) 7 distinct point groups 7 crystal systems (3D) Adding translational symmetry 14 Bravais lattices (3D) Symmetry Group Point group operations 7 crystal systems (3D) Translational operations 14 Bravais lattices (3D)
18 The 14 Bravais Lattices
19 Crystal Lattices with Basis
20 Why So Few? Q: Which kind of rotational symmetry can exist in 2D Bravais lattices? A: Two-, Three-, Four-, Six-fold rotational symmetry. Why there can t be other rotational symmetries? Hidden assumption: Bravais lattice is a discrete lattice, and lattice vectors must be of finite length. What if there s a Five-fold symmetry in a Bravais lattice?
21 Enumerate the Bravais Lattices in 2D 5 Bravais Lattices in 2D: oblique rectangular centered rectangular hexagonal square Similar construction follows in 3D
22 Symmetry of Different Phases Symmetry operations of empty space: Continuous translations Continuous rotations Inversion Mirror image crystal Symmetry operations of crystals: Translations with Bravais lattice vectors Finite number of operations (instead of continuous) including rotations, inversion, mirror image, depending on the given crystal. Symmetry operations of simple liquids: Same as that of empty space (statistically) liquid
23 Liquid Crystal Transition disordered phase liquid First order transition as temperature lowered ordered phase solid Breaking of continuous translational symmetry Breaking of continuous rotational symmetry Can we only partially break these symmetries?
24 Liquid Crystals Microscopic components of liquid crystals: Molecule: We can approximate them as rods:
25 Liquid Crystalline Order disordered phase ordered phase Lower temperature Isotropic Phase Nematic Phase Breaking of continuous rotational symmetry Continuous translational symmetry remain
26 More Liquid Crystalline Order At even lower temperature: Smectic-A Phase: In addition to the breaking of continuous rotational symmetry, Breaking of translational symmetry in the N direction. Translational symmetry in other two directions remain. Smectic-C Phase: Even lower symmetry than that of the smectic-a phase.
27 More Liquid Crystalline Order Chiral Molecules Cholesteric Phases
28 How to Observe Liquid Crystals? Liquid crystal molecules are bifriengent: different n along and perpendicular to the director direction. As a result: Incident linear-polarized light Outgoing eliptical-polarized light
29 How to Observe Liquid Crystals? Use cross polarizers:
30 Are There Other Types of Order? Shechtman et al. Metallic phase with long-range orientational order and no translational symmetry, PRL, 53, 1951 (1984). Material: Al-Mn alloy What can we know about the rotational symmetry of this material from the X-ray diffraction pattern?
31 Five-Fold Rotational Symmetry But how can we have five-fold rotational symmetry in a crystal? aa 0, aa 1, aa 2, aa 3, aa 4 are Bravais lattice vectors It can not be a periodic structure!
32 Structure of Quasicrystals Q: If a finite set of tiles can tile the whole plane, can we always tile the plane with them periodically? (Hao Wang, 1961) A: No. Counter-example: Penrose tiling (1976) 2 tiles, can only tile the space aperiodically Fourier transform of this tile sharp peaks Quasicrystals have this type of structure
33 Other Types of Condensed Matter? Some solids are trivial from the symmetry point of view E.g. Glasses
34 Other Types of Condensed Matter? Some solids are trivial from the symmetry point of view E.g. Colloidal gel Plastic particles of size 10 6 mm suspended in liquid, with short-ranged attraction
35 How do We Characterize These Disordered Structures? Density operator: nn rr = δδ(rr rr jj ) jj Thermal average of density nn(rr) Crystal: periodic (long-range order) Simple Fluid: homogeneous nn(rr) = nn = NN/VV Disordered solids: no long-range order, homogeneous at large-scale Density-density correlation function CC nnnn rr 1, rr 2 = nn rr 1 nn(rr 2 ) = δδ(rr rr ii ) δδ(rr rr jj ) ii,jj Position of j-th particle
36 Density-Density Correlation Function Density-density correlation function CC nnnn rr 1, rr 2 = nn rr 1 nn(rr 2 ) = δδ(rr rr ii ) δδ(rr rr jj ) Written in Fourier space II qq = nn qq nn( qq) where ii,jj nn qq = dddd ee iiiiii nn rr = ee iiiirr jj jj This is actually what is measured in scattering experiment
37 Relation to Scattering Experiment Another way to write the scattering amplitude: kk UU kk = dddd ee iiiiii UU xx ee iikk rr From last slide: II qq = nn qq nn( qq) nn qq = dddd ee iiiiii nn rr = ee iiiirr jj jj = ee iiiiii UU jj (rr rr jj )ee iikk rr jj = UU jj (kk kkk) The scattering intensity is then jj ee ii kk kk rr jj kk UU kk 2 = UU jj qq UU jj qq ee iiii(rr ii rr jj ) ii,jj where qq = kk kkk So we have (suppose potential of each particle is the same) kk UU kk 2 = UU qq 2 II(qq) Scattering experiments directly measure density-density correlation functions
38 Pair Distribution Function A variation of the density-density correlation function gg rr 1, rr 2 = nn rr 1 nn(rr 2 ) nn rr 1 δδ(rr 1 rr 2 ) nn rr 1 nn(rr 2 ) In systems with (statistical) translational and rotational invariance: scalar form 1 gg rr = nn 4ππrr 2 δδ(rr rr jj rr 1 ) jj 1 Nearest neighbor Next-nearest neighbor g(r) for glasses look very similar to this. What does it look like in crystals? Homogeneous at large scales Hard-sphere liquid
39 Other Disordered Solids? Colloidal gels Fractal aggregate of gold particles
40 Pair Distribution Function in Fractal Structures Pair distribution function: 1 gg rr = nn 4ππrr 2 δδ(rr rr jj rr 1 ) jj 1 Koch curve: Density of particles in a shell (*area of shell) at radius r from particle 1 What is this quantity like in fractals? MM LL DD DD = log MM log LL :fractal dimension Density in a shell: dddd dddd LLDD 1 Pair distribution function for a fractal embedded in 3-dimensions: gg(rr) rr DD 3 Flat if DD = 3 (compact instead of fractal)
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