Crystallographic structure Physical vs Chemical bonding in solids

Transcription

1 Crystallographic structure Physical vs Chemical bonding in solids Inert gas and molecular crystals: Van der Waals forces (physics) Water and organic chemistry H bonds (physics) Quartz crystal SiO 2 : covalent bonds (chemistry) Stronger bonds

2 Ionic bond : NaCl Neutral building block + - fcc corresponds to packing in an as close packed as possible structure For most solids bonds are a partly partly ionic and partly covalent CsCl bcc structure (bonds are more directional)

3 Chemical bonding in solids Covalent bond Bonding and antibonding levels form due to orbital hybridization. The bond is stable if only (or mostly) bonding orbitals are filled

4 Hybridization Goup IV elements Diamond Structure sp 3 Graphite and graphene sp 2

5 Interaction beyond first nearest neighbors Metallic bond For Ni bonds occurs through s orbitals Fcc structure If directional d orbitals matter bcc or hcp structures (e.g. Fe and W)

6 3D Crystals a 1 T a 2 R R =R+n 1 a 1 + n 2 a 2 + n 3 a 3 T=n 1 a 1 + n 2 a 2 + n 3 a 3 R n 1, n 2, n 3 arbitrary integers a 1, a 2, a 3 fundamental translation vectors The set of points R defines a lattice while spanning all over n i

7 3D Crystals and Lattice R =R+n 1 a 1 + n 2 a 2 + n 3 a 3 T=n 1 a 1 + n 2 a 2 + n 3 a 3 n 1, n 2, n 3 arbitrary integers a 1, a 2, a 3 fundamental translation vectors The set of points R defines a lattice while spanning over all n i The atomic arrangement looks the same in every respect (including orientation) when viewed from any point R of the lattice The lattice is the regular periodic arrays of points in space The crystal structure is formed when a basis is attached identically to every lattice point

8 3D Crystals and Translation Operation The lattice and the translation vectors a 1, a 2, a 3 are said to be primitive if with a suitable choice of the integers n 1, n 2, n 3 any two points R and R always satisfy R =R+n 1 a 1 + n 2 a 2 + n 3 a 3 In this case the vectors a 1, a 2, a 3 are primitive translation vectors and no cell of smaller volume can serve as a building block for the crystal structure A lattice translation operation is defined as the displacement of a crystal by a crystal translation vector T=n 1 a 1 + n 2 a 2 + n 3 a 3

9 3D Crystals Often, though not always, primitive translation vectors a 1, a 2, a 3 are used to define the crystal axes. More than one lattice is always possible for any given structure More than one set of axes is always possible for a given lattice The basis is chosen with an arbitrary choice.

10 Primitive Cell and Axes a 1 a 1 a 1 a 2 a 2 a 1 a 2 a 2 All pairs of vectors a 1, a 2 are translation vectors, but (a 1, a 2 ) is not a primitive translation vector pair All other pairs can be taken as primitive translation vectors and the related unit volume is the same

11 3D Crystals and Symmetry Operations A symmetry operation of a crystal carries the crystal structure into itself Translation operations (T=n 1 a 1 + n 2 a 2 + n 3 a 3 ) are symmetry operations Rotations, reflections and inversions can be symmetry operations, called point symmetry operations Compound operations can be symmetry operations The collection of symmetry operations is a group The simultaneous fulfillment of the translation operations with the point group symmetry operations leads to 14 special lattice types (Bravais lattices)

12 3D Crystals x x x x x x x x x a 1 T a 2 R R Point Operations A rotation of π radians around any point marked x is a symmetry operation since it carries the crystal structure into itself

13 Possible symmetry operations: Rotations Reflections Inversions Combinations of them Group of the octahedron Cubic symmetry Group of the tetrahedron Different symmetry operations are possible!

14 The Seven crystal systems: a) Cubic (simple cubic, body centered cubic, face centered cubic) b) Tetragonal, cubic symmetry is reduced by pulling on two opposite bases one obtains a square base and a non equivalent height or c-axis (simple and body centered tetragonal) Distortion of fcc and bcc lattices leads to the same tetragonal symmetry

15 The Seven crystal systems: c) Orthorombic, the square faces are deformed into mutially perpendicular rectangles: i) Simple orthorombic ii) Base centered orthorombic iii) Body centered orthormbic iv)face centered orthorombic

16 The Seven crystal systems: d) Monoclinic, the rectangular face othogonal to the c axis is distorted into a rhombus: simple base centered

17 The Seven crystal systems: e) Triclinic, the c axis is tilted and no longer perpendicular to the other two. The crystal has only inversion symmetry f) Trigonal or rhombohedral, this lattice is obtained by stretching a cube along a body diagonal. The lattice vectors have identical length and make equal angles with one another g) Hexagonal, it is the symmetry group of a right prism with a hexagon at the base

18 3D Bravais Lattices There are 14 special lattice types (Bravais lattices) in 3D space (hkl) Indices of a plane {hkl} Planes equivalent by symmetry [uvw] Indices of a direction Combining point symmetry and translations one gets 230 possible 3D lattices

19 Classification of the point groups

20 Combining rotational and translational symmetry: Bravais lattices (basis of spherical symmetry) Crystal lattices (basis of arbitrary symmetry) Number of point groups 7 32 Number of space groups 14 Bravais lattices 230 space groups C cyclic; D dihedral; S Spiegel (mirror) The subscripts h, v and d stand for horizontal, vertical and diagonal mirror planes

21 3D Crystals and Primitive Lattice Cell The unit volume defined by the primitive a 1, a 2, a 3 axes is called primitive cell A unit cell will fill all space by the repetition of suitable translation operations A primitive cell is a minimum-volume cell There are many possible choices for the primitive axes and for the cell for a given lattice The number of atoms in a primitive cell or primitive basis is always the same for a given crystal structure

22 Compact structures: fcc ABCABC hcp ABABAB

23 fcc, different possible choices of unit vectors The primitive cell has one fourth of the volume of the conventional unit cell Packing fraction 0,74 The primitive choice is complicated, better working with a non-primitive unit cell like the face centered cube

24 bcc Packing fraction 0,68 Different choices are possible for the unit cell. The primitive cell has half the volume of the conventional unit cell Elements crystallizing in bcc structure have mostly directional bonds determined by d states. However, also alkali metals are bcc, the reason being that the bcc lattice optimizes the superposition of the orbitals corresponding to the 2 nd and 3 rd nearest neighbors as illustrated in the figure

25

26 Fcc and bcc lattices are Bravais lattices since, even if they have four and two atoms in the unit cell the surroundings look alike when viewed from any lattice point

27 Hexagonal layer : graphene Layered crystals: graphite hcp structure: ABAB packing More complicated ABCABABCAB packing possible, too

28 Lattice with two atom basis: the diamond structure Packing fraction = 0,34 It consists of two identical fcc lattices displaced by (¼¼¼) In total we therefore have 8 atoms in the unit cell. It is not a Bravais lattice since the surroundings do not look the same from the two atoms of the basis.

29 Primitive Cell and Basis There is always one lattice point per primitive cell The basis associated with a primitive cell is called primitive basis No basis contains fewer atoms than a primitive basis Wigner-Seitz cell: defined as follows 1) Draw lines to connect a given lattice point to all nearby points 2) Draw new lines at the midpoint and normal to the above lines 3) The smallest volume enclosed in this way is the Wigner-Seitz primitive cell

30 Wigner Seitz cells bcc fcc Complicated geometry, used rarely

31 Compounds NaCl structure Two fcc lattices shifted by (½,0,0) filled by two different atoms 4 atoms per unit cell

32 CsCl structure Two simple cubic lattices consisting of different atoms 2 atoms per unit cell

33 zincblende structure Like diamond but with two different atomic species, 4 atoms per unit cell

34 NaCl CsCl Fluorite CaF 2 Perovskite BaTiO 3 Laves phase Cu 2 Mg A15 or β tungsten structure Nb 3 Sn