6 Crystal Bonding Was hält einen Kristall zusammen? Requirements:Positive ions as far as possible from each otherValence electrons as far as possible from each otherValence electrons as close as possible to positive ionsRequirements a)-c) reduce the potential energy,these should be fulfilled without enlarging the kinetic energy to much.In quantum mechanics confinement of electron increases their kinetic energy

9 COVALENT BONDINGEach electron in a shared pair is attracted to both nuclei involved in the bond. The approach, electron overlap, and attraction can be visualized as shown in the following figure representing the nuclei and electrons in a hydrogen molecule.Attraction cannot be explained with classical physics: quantum mechanical effectCalculation of exchange interaction (overlap of electron oritals)Compare with bonding of Hydrogen molecule: spins must be antiparallelee

10 COVALENT BONDINGPropertyExplanationMelting pointand boiling pointVery high melting points because each atom is bound by strong covalent bonds. Many covalent bonds must be broken if the solid is to be melted and a large amount of thermal energy is required for this.ElectricalconductivityPoor conductors because electrons are held either on the atoms or within covalent bonds. They cannot move through the lattice.HardnessThey are hard because the atoms are strongly bound in the lattice, and are not easily displaced.BrittlenessCovalent network substances are brittle.If sufficient force is applied to a crystal, covalent bond are broken as the lattice is distorted. Shattering occurs rather than deformation of a shape.

12 Crystalline SolidCrystalline Solid is the solid form of a substance in which the atoms or molecules are arranged in a definite, repeating pattern in three dimension.

13 Crystalline SolidSingle crystal has an atomic structure that repeats periodically across its whole volume. Even at infinite length scales, each atom is related to every other equivalent atom in the structure by translational symmetrySingle PyriteCrystalAmorphousSolidSingle Crystal

14 Polycrystalline SolidPolycrystal is a material made up of an aggregate of many small single crystals (also called crystallites or grains).The grains are usually 100 nm microns in diameter. Polycrystals with grains that are <10 nm in diameter are called nanocrystallinePolycrystallinePyrite form(Grain)PolycrystalEinkristalline Bereiche, von Korngrenzen getrennt, beliebig zueinander orientiert

15 POLYCRYSTALLINE MATERIALS“Nuclei” form during solidification, each of which grows into crystals

16 POLYCRYSTALS • Most engineering materials are polycrystals. 1 mmAdapted from Fig. K, color inset pages of Callister 6e.(Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)1 mm• Nb-Hf-W plate with an electron beam weld.• Each "grain" is a single crystal.• If crystals are randomly oriented,overall component properties are not directional.• Crystal sizes typ. range from 1 nm to 2 cm(i.e., from a few to millions of atomic layers).

23 Kristallgitter Don't mix up atoms with lattice pointsLattice points are infinitesimal points in spaceLattice points do not necessarily lie at the centre of atomsCrystal Structure = Crystal Lattice Basis

24 Lattice Vectors – 2DThe two vectors a and b form a set of lattice vectors for the lattice.The choice of lattice vectors is not unique. Thus one could equally well take the vectors a and b’ as a lattice vectors.The lattice vectors define the unit cell.b`aCrystal Structure

25 NP = Non-Primitive Unit CellThe primitive unit cell must have only one lattice point.There can be different choices for lattice vectors , but the volumes of these primitive cells are all the same.P = Primitive Unit CellNP = Non-Primitive Unit CellCrystal Structure

29 The volume enclosed is called as aWigner-Seitz MethodA simply way to find the primitivecell which is called Wigner-Seitzcell can be done as follows;Choose a lattice point.Draw lines to connect these lattice point to its neighbours.At the mid-point and normal to these lines draw new lines.The volume enclosed is called as aWigner-Seitz cell.Crystal Structure

32 14 BRAVAIS LATTICES AND THE SEVEN CRYSTAL SYSTEMCRYSTAL STRUCTURES IN 3D14 BRAVAIS LATTICES AND THE SEVEN CRYSTAL SYSTEMThere are only seven different shapes of unit cell which can be stacked together to completely fill all space (in 3 dimensions) without overlapping. This gives the seven crystal systems, in which all crystal structures can be classified.Cubic Crystal System (SC, BCC,FCC)Hexagonal Crystal System (S)Triclinic Crystal System (S)Monoclinic Crystal System (S, Base-C)Orthorhombic Crystal System (S, Base-C, BC, FC)Tetragonal Crystal System (S, BC)Trigonal (Rhombohedral) Crystal System (S)

40 Richtungen im KristallBenutze das Triplett mit den kleinsten möglichen ganzen ZahlenIn real space, a crystal is made of a periodic lattice. A unit cell in a real lattice is defined by three unit cell vectors: a, b, c, which should be non-coplanar.Any lattice point in real space can be represented by a lattice vector, r = ua + vb + wc, where u, v and w are the components of the direction index [uvw]. Beachte: Eckige Klammern für Kristallrichtungen

42 CRYSTALLOGRAPHIC PLANESCrystallographic planes specified by 3 Miller indices as (hkl)Procedure for determining h,k and l: (ein Beispiel)If plane passes through origin, translateplane or choose new originDetermine intercepts of planes on eachof the axes in terms of unit cell edge lengths(lattice parameters) (½ ¼ ½).Note: if plane has no intercept to an axis (i.e., it isparallel to that axis), intercept is infinityDetermine reciprocal of the three intercepts (2 4 2)If necessary, multiply these three numbers by acommon factor which converts all the reciprocals to small integers (1 2 1)The three indices are not separated by commas and are enclosed in curved brackets: (hkl) (121)If any of the indices is negative, a bar is placed in top of that index

43 Take the reciprocals of the fractional interceptsCrystal Planes: Miller indicesMiller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice and are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes.Identify the intercepts on the x- , y- and z- axesx = a (at the point (a,0,0) )parallel to the y- and z-axesIntercepts : a , , Specify the intercepts in fractional co-ordinatesIn the case of a cubic unit cell each co-ordinate will simply be divided by the cubic cell constant aa/a , /a, /a i.e. 1 ,  , Take the reciprocals of the fractional interceptsyieldingMiller Indices : (100)

46 Das reziproke Gitter ghkl = hg1 + kg2 + lg3The reciprocal space lattice is a set of imaginary points constructed in such a way that the direction of a vector from one point to another coincides with the direction of a normal to the real space planes and the separation of those points (absolute value of the vector) is equal to the reciprocal of the real interplanar distance multiplied by 2p.ghkl = 2p / dhklPhysikalischer HintergrundBei elastischer Streuung an einer Kristallebene (hkl) beträgt der Impulsübertrag auf die gestreuten Teichen genau ghkl, wobei ghkl der reziproke Gittervektorsenkrecht zur (hkl)-Ebene ist.ghkl = hg1 + kg2 + lg3

53 Reciprocal Lattice in 3Dwhere nx, ny, and nz = +/- 1,2,3…kxky2p/LEach dot represents a point of the reciprocal latticeEach reciprocal lattice point represents a crystal plane in real spaceThe volume of the unit cell of the reciprocal lattice is (2p/L)3 = 8p3/V

54 Reciprocal Lattice (Zusammenfassung)The reciprocal lattice vector is given by:ghkl = ha* + kb* + lc* where h, k, l are integer numbers and correspond to the Miller indices of the plane (hkl).Physical meaning of the reciprocal space vector:- The direction of ghkl is perpendicular to the set of atomic planes that intercept the real space lattice at (a/h, b/k, c/l).- The magnitude of ghkl is 2p/dhkl, where dhkl is the spacing between lattice planes with the Miller indices (hkl).The components h, k and l of the reciprocal vector ghkl in reciprocal space give the plane index (hkl).The reciprocal vector ghkl in the reciprocal lattice is always perpendicular to the plane (hkl) in the real lattice.

57 {kx= ±π/a, ky= ±π/a, kz= ±π/a}Brillouin ZonePoints of symmetry on the Brillouin zone are given particular importance especially when determining the bandstructure of the material.Points of high-symmetry on the Brillouin zone have specific importance. Perhaps the most important, at least for optoelectronic devices, is at k = 0 which is known as the gamma point. As you might expect, it is given the symbol, Γ .Symmetry PointkΓkx= 0, ky= 0, kz= 0X{ki= ±2π /a, kj= 0, kk= 0}L{kx= ±π/a, ky= ±π/a, kz= ±π/a}Brillouin zone of a fcc lattice