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1 lectures accompanying the book: Solid State Physics: An Introduction,by Philip Hofmann (1st edition, October 2008, ISBN-10: 3-527-40861-4, ISBN-13: 978-3-527-40861-0, Wiley-VCH Berlin.Philip Hofmann Wiley-VCH Berlin www.philiphofmann.net

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2 This is only the outline of a lecture, not a final product. Many “fun parts” in the form of pictures, movies and examples have been removed for copyright reasons. In some cases, www addresses are given for particularly good resources (but not always). I have left some ‘presenter notes’ in the lectures. They are probably of very limited use only. README

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3 Crystals and crystalline solids solid made from small, identical building blocks (unit cells) which also show in the macroscopic shape

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4 Bravais lattice, unit cell and basis Miller indices Real crystal structures X-ray diffraction: Laue and Bragg conditions, Ewald construction The reciprocal lattice Electron microscopy and diffraction at the end of this lecture you should understand.... Crystals and crystalline solids: contents

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5 Crystals and crystalline solids The total energy for the crystalline state must be very low because the ordered states exists also at elevated temperatures, despite its low entropy. The high symmetry greatly facilitates the solution of the Schrödinger equation for the solid because the solutions also have to be highly symmetric.

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6 Crystals and crystalline solids most real solids are polycrystalline still, even if the grains are small, only a very small fraction of the atoms is close to the grain boundary the solids could also be amorphous (with no crystalline order)

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7 Some formal definitions

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8 The crystal lattice: Bravais lattice (2D) A Bravias lattice is a lattice of points, defined by The lattice looks exactly the same from every point

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9 The crystal lattice: Bravais lattice (2D) A Bravias lattice is a lattice of points, defined by The lattice looks exactly the same from every point

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10 The crystal lattice: Bravais lattice (2D) Not every lattice of points is a Bravais lattice

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11 The crystal lattice: Bravais lattice (3D) A Bravias lattice is a lattice of points, defined by This reflects the translational symmetry of the lattice

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12 Bravais lattice (2D) The number of possible Bravais lattices (of fundamentally different symmetry) is limited to 5 (2D) and 14 (3D). Θ

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13 The crystal lattice: primitive unit cell Primitive unit cell: any volume of space which, when translated through all the vectors of the Bravais lattice, fills space without overlap and without leaving voids a1a1 a2a2

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14 The crystal lattice: primitive unit cell Primitive unit cell: any volume of space which, when translated through all the vectors of the Bravais lattice, fills space without overlap and without leaving voids a1a1 a2a2 14

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15 The crystal lattice: Wigner-Seitz cell Wigner-Seitz cell: special choice of primitive unit cell: region of points closer to a given lattice point than to any other.

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16 The crystal lattice: basis We could think: all that remains to do is to put atoms on the lattice points of the Bravais lattice. But: not all crystals can be described by a Bravais lattice (ionic, molecular, not even some crystals containing only one species of atoms.) BUT: all crystals can be described by the combination of a Bravais lattice and a basis. This basis is what one “puts on the lattice points”.

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17 The crystal lattice: one atomic basis The basis can also just consist of one atom.

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18 The crystal lattice: basis Or it can be several atoms.

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19 The crystal lattice: basis Or it can be molecules, proteins and pretty much anything else.

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20 The crystal lattice: one more word about symmetry The other symmetry to consider is point symmetry. The Bravais lattice for these two crystals is identical: four mirror lines 4-fold rotational axis inversion no additional point symmetry

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21 Nasty stuff... adding the basis keeps translational symmetry but can reduce point symmetry but it can also add new symmetries (like glide planes here)

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22 Labelling crystal planes (Miller indices) 1. determine the intercepts with the axes in units of the lattice vectors 2. take the reciprocal of each number 3. reduce the numbers to the smallest set of integers having the same ratio. These are then called the Miller indices. step 1: (2,1,2) step 2: ((1/2),1,(1/2)) step 3: (1,2,1)

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23 Example

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24 Real crystal structures

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25 What structure do the solids have? Can we predict it? Consider inert elements (spheres). This could be anything with no directional bonding (noble gases, simple and noble metals). Just put the spheres together in order to fill all space. This should have the lowest energy. A simple cubic structure?

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26 Simple cubic The simple cubic structure is a Bravais lattice. The Wigner-Seitz cell is a cube The basis is one atom. So there is one atom per unit cell. a1a1 a2a2 a3a3

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27 Simple cubic We can also simply count the atoms we see in one unit cell. But we have to keep track of how many unit cells share these atoms. 1/8 atom

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28 Simple cubic A simple cubic structure is not a good idea for packing spheres (they occupy only 52% of the total volume). Only two elements crystallise in the simple cubic structure (F and O).

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29 Better packing In the body-centred cubic (bcc) structure 68% of the total volume is occupied. The bcc structure is also a Bravais lattice but the edges of the cube are not the correct lattice vectors.

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30 Better packing In the body-centred cubic (bcc) structure 68% of the total volume is occupied. The bcc structure is also a Bravais lattice but the edges of the cube are not the correct lattice vectors.

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31 Close-packed structures

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32 Close-packed structures: fcc and hcp hcp ABABAB... fcc ABCABCABC...

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33 Close-packed structures: fcc and hcp hcp ABABAB... fcc ABCABCABC...

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34 Close-packed structures: fcc and hcp hcp ABABAB... fcc ABCABCABC...

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35 Close-packed structures: fcc and hcp hcp ABABAB... fcc ABCABCABC...

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36 Close-packed structures: fcc and hcp hcp ABABAB... fcc ABCABCABC...

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37 Close-packed structures: fcc and hcp hcp ABABAB... fcc ABCABCABC... The hexagonal close-packed (hcp) and face-centred cubic (fcc) and structure have the same packing fraction

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38 The fcc structure In the face-centred cubic (fcc) structure 74% of the total volume is occupied (slightly better than bcc with 68%) This is probably the optimum (Kepler, 1611) and grocers.

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39 The crystal lattice: Bravais lattice (3D) The fcc lattices is also a Bravais lattice but the edges of the cube are not the correct lattice vectors. The cubic unit cell contains more than one atom.

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40 The crystal lattice: Bravais lattice (3D) The fcc lattices is also a Bravais lattice but the edges of the cube are not the correct lattice vectors. The cubic unit cell contains more than one atom. 1/8 atom 1/2 atom

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41 The crystal lattice: Bravais lattice (3D) The fcc and bcc lattices are also Bravais lattices but the edges of the cube are not the correct lattice vectors. When choosing the correct lattice vectors, one has only one atom per unit cell.

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42 The crystal lattice: Bravais lattice (3D) The fcc and bcc lattices are also Bravais lattices but the edges of the cube are not the correct lattice vectors. When choosing the correct lattice vectors, one has only one atom per unit cell. 1/8 atom

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43 Close-packed structures: hcp The hcp lattice is NOT a Bravais lattice. It can be constructed from a Bravais lattice with a basis containing two atoms. the packing efficiency is of course exactly the same as for the fcc structure (74 % of space occupied).

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44 Close-packed structures Close-packed structures are indeed found for inert solids and for metals. For metals, the conduction electrons are smeared out and directional bonding is not important. Close-packed structures have a big overlap of the wave functions. Most elements crystallize as hcp (36) or fcc (24).

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45 Close-packed structures: ionic materials In ionic materials, different considerations can be important (electrostatics, different size of ions) In NaCl the small Na are in interstitial positions of an fcc lattice formed by Cl ions (slightly pushed apart)

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46 Close-packed structures: ionic materials

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47 Non close-packed structures covalent materials (bond direction more important than packing) diamond (only 34 % packing) graphite

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48 Non close-packed structures molecular crystals (molecules are not round, complicated structures) α-Gallium, Iodine

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49 Crystal structure determination

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50 X-ray diffraction The atomic structure of crystals cannot be determined by optical microscopy because the wavelength of the photons is much too long (400 nm or so). So one might want to build an x-ray microscope but this does not work for very small wavelength because there are no suitable x-ray optical lenses. The idea is to use the diffraction of x-rays by a perfect crystal. Here: monochromatic x-rays, elastic scattering, kinematic approximation

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51 X-ray diffraction The crystals can be used to diffract X-rays (von Laue, 1912).

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52 The Bragg description (1913): specular reflection The Bragg condition for constructive interference holds for any number of layers, not only two (why?) and this only works for

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53 X-ray diffraction: the Bragg description The X-rays penetrate deeply and many layers contribute to the reflected intensity The diffracted peak intensities are therefore very sharp (in angle) The importance of the Miller indices becomes apparent. Crystal planes are essential The physics of the lattice planes is totally obscure!

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54 A little reminder about waves and diffraction λ = 1 Å hν = 12 keV complex notations (for real )

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55 X-ray diffraction, von Laue description incoming wave at r absolute E 0 of no interest outgoing wave in detector direction

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56 X-ray diffraction, von Laue description field at detector for one point and for the whole crystal

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57 X-ray diffraction, von Laue description so the measured intensity is with

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58 X-ray diffraction, von Laue description Volume V so the measured intensity is

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59 X-ray diffraction, von Laue description Volume V kk’ 55 with so the measured intensity is

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60 X-ray diffraction, von Laue description Volume V so the measured intensity is

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61 The reciprocal lattice for a given Bravais lattice the reciprocal lattice is defined as the set of vectors G for which or The reciprocal lattice is also a Bravais lattice

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62 The reciprocal lattice construction of the reciprocal lattice with this it is easy to see why a useful relation is

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63 The reciprocal lattice example 1: in two dimensions |a 1 |=a |a 2 |=b |b 2 |=2π/b |b 1 |=2π/a

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64 The reciprocal lattice if we have The vectors G of the reciprocal lattice give plane waves with the periodicity of the lattice. In this case G is the wave vector and 2π/|G| the wavelength. then we can write

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65 X-ray diffraction, von Laue description Volume V so the measured intensity is K=GK=G

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66 Lattice waves real spacereciprocal space a b 2π/a 2π/b (0,0)

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67 Lattice waves real spacereciprocal space a b 2π/a 2π/b (0,0)

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68 Lattice waves real spacereciprocal space a b 2π/a 2π/b (0,0)

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69 Lattice waves real spacereciprocal space a b 2π/a 2π/b (0,0)

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70 Lattice waves real spacereciprocal space a b 2π/a 2π/b (0,0)

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71 Lattice waves real spacereciprocal space a b 2π/a 2π/b (0,0)

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72 The reciprocal of the reciprocal lattice is again the real lattice |a 1 |=a |a 2 |=b |b 2 |=2π/b |b 1 |=2π/a

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73 The reciprocal lattice in 3D example 2: in three dimensions bcc and fcc lattice The fcc lattice is the reciprocal of the bcc lattice and vice versa.

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74 Other applications of the reciprocal lattice Greatly simplifies the description of lattice-periodic functions (charge density, one-electron potential...). 1D chain of atoms example of charge density 1 example of charge density 2

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75 Other applications of the reciprocal lattice example: charge density in the chain Fourier series alternatively

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76 Other applications of the reciprocal lattice 1D reciprocal lattice

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77 Other applications of the reciprocal lattice 1D 3D

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78 X-ray diffraction, von Laue description with use so the measured intensity is constructive interference when Laue condition

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79 X-ray diffraction, von Laue description with use so the measured intensity is for a specific spot

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80 X-ray diffraction, von Laue description Volume V so the measured intensity is

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81 X-ray diffraction, von Laue description Volume V so the measured intensity is K k k’ -k

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82 X-ray diffraction, von Laue description Volume V so the measured intensity is

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83 X-ray diffraction, von Laue description Volume V so the measured intensity is

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84 X-ray diffraction, von Laue description Volume V so the measured intensity is

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85 The Ewald construction Draw (cut through) the reciprocal lattice. Draw a k vector corresponding to the incoming x-rays which ends in a reciprocal lattice point. Draw a circle around the origin of the k vector. The Laue condition is fulfilled for all vectors k’ for which the circle hits a reciprocal lattice point. Laue conditionif G is a rec. lat. vec.

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86 Why does the Bragg condition appear so much simpler? Laue condition automatically fulfilled parallel to the surface (choosing specular reflection) 1D reciprocal lattice in this direction: define vector d connecting the planes kk’

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87 Relation to lattice planes / Miller indices The vector is the normal vector to the lattice planes with Miller indices (m,n,o)

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88 x-ray diffraction in practice

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89 X-ray diffraction in practice image source: wikipedia

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90 typical result

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91 typical result (with filter)

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92 This is all good and fine but one needs to have the crystal aligned EXACTLY such that the condition of specular reflection is fulfilled. This is not so simple because the reflections are very narrow. One trick is to use a powder of small crystals instead of one big crystal. Due to the disorder in the powder, the specular reflection is always realized for some crystals. Another trick is to use non- monochromatic (white) x-rays to increase the chances of constructive interference (see next slide)

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93 Laue Method Using white x-rays in transmission or reflection. Obtain the symmetry of the crystal along a certain axis.

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94 advanced X-ray diffraction The position of the spots gives information about the reciprocal lattice and thus the Bravais lattice. An intensity analysis can give information about the basis. Even the structure of a very complicated basis can be determined (proteins...) every spot crystallize protein remember

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95 advanced x-ray sources: synchrotron radiation A highly collimated and monochromatic beam is needed for protein crystallography. This can only be provided by a synchrotron radiation source. SPring-8ASTRID

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96 What is in the basis? with (sum over the j atoms in the unit cell, model this) we have N unit cells in the crystal

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97 The holy grail of x-ray diffraction Is it possible to determine the structure of a molecule (protein, virus....) by taking a diffraction pattern created from a single, intense shot of x-rays? pulse: 25 fs, 4x10 13 Wcm -2 from FLASH in Hamburg Chapman et al. Nature Physics (2006) Fig. 1 from Chapman et al., Nature Physics 2, 839 (2006)

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98 The holy grail of x-ray diffraction Is it possible to determine the structure of a molecule (protein, virus....) by taking a diffraction pattern created from a single, intense shot of x-rays? pulse: 25 fs, 4x10 13 Wcm -2 from FLASH in Hamburg, heats sample to 60.000 K (and hence destroys it) Fig. 2a from Chapman et al., Nature Physics 2, 839 (2006)

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99 The holy grail of x-ray diffraction Is it possible to determine the structure of a molecule (protein, virus....) by taking a diffraction pattern created from a single, intense shot of x-rays? pulse: 25 fs, 4x10 13 Wcm -2 from FLASH in Hamburg, heats sample to 60.000 K (and hence destroys it) Fig. 3 from Chapman et al., Nature Physics 2, 839 (2006)

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100 Inelastic scattering Gain information about possible excitations in the crystal (mostly lattice vibrations). Not discussed here.

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101 Other scattering methods (other than x-rays) electrons (below) neutrons

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102 Electron microscopes / electron diffraction Electrons can also have a de Broglie wavelength similar to the lattice constant in crystals. For electrons we get This gives a wavelength of 5 Å for an energy of 6 eV.

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103 Electron microscopes For high energies, we should get sufficiently good resolution to “see” atoms. But in practice, the resolution is limited to 5 Å or so. For a transmission electron microscope (TEM) the samples have to be very thin. insert TEM picture here

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104 Electron diffraction We can get also electron diffraction patterns from a crystal which is quite similar to x-ray diffraction patterns (Davisson and Germer, Nobel 1927, wave nature of the electron) insert LEED picture

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105 Electron microscopes In a scanning tunnelling microscope (STM) a sharp tip is scanned across the sample surface. Atomic resolution is achieved (Binnig and Rohrer, Nobel prize in 1986) insert STM principle and picture

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106 Quasicrystals (1984) Sharp 5-fold diffraction spots are observed, indicative of high crystalline order... but a Bravais lattice can only contain 2, 3, 4 and 6-fold symmetric axis, long-range order is impossible 5-fold symmetry Laue image insert picture Nobel Prize in Chemistry (2011) to Dan Shechtman

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107 Penrose tilings (1974) Tilings which fill 2D space without translational symmetry but local 5-fold symmetry. Why does this give rise to such sharp spots (indicative of long range order)? insert pictures green mosque in Bursa (Turkey) from 1424, quasicrystal in 1453

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