Inert gas and molecular crystals: Van der Waals forces (physics)

Crystallographic structure Physical vs Chemical bonding in solids

Water and organic chemistry H bonds (physics)

Quartz crystal SiO2: covalent bonds (chemistry)

Stronger bonds

Ionic bond : NaCl

For most solids bonds are a partly partly ionic and partly covalent

- +

fcc corresponds to packing in an “as close packed as possible” structure - 12 nearest neighbors

CsCl bcc structure - 8 nearest neighbors (bonds are more directional)

Neutral building block

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

Hybridization Group IV elements

Diamond Structure sp3

Graphite and graphene sp2

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)

Interaction beyond first nearest neighbors

a2

a1

R

R’

T

R’=R+n1a1+ n2a2+ n3a3

T=n1a1+ n2a2+ n3a3

n1, n2, n3 arbitrary integers a1, a2, a3 fundamental translation vectors

The set of points R’ defines a lattice while spanning all over ni

3D Crystals

R’=R+n1a1+ n2a2+ n3a3

T=n1a1+ n2a2+ n3a3 n1, n2, n3 arbitrary integers a1, a2, a3 fundamental translation vectors

The set of points R’ defines a lattice while spanning over all ni 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

3D Crystals and Lattice

The lattice and the translation vectors a1, a2, a3 are said to be primitive if with a suitable choice of the integers n1, n2, n3 any two points R and R’ always satisfy R’=R+n1a1+ n2a2+ n3a3 In this case the vectors a1, a2, a3 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=n1a1+ n2a2+ n3a3

3D Crystals and Translation Operation

Often, though not always, primitive translation vectors a1, a2, a3 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.

3D Crystals

a2’’’’

a2’

a1’

a2’’

a1’’

a1’’’’

a1’’’

a2’’’

All pairs of vectors a1, a2 are translation vectors, but

(a1’’’’ , a2’’’’) 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

Primitive Cell and Axes

3D Crystals and Symmetry Operations

A symmetry operation of a crystal carries the crystal structure into itself

Translation operations (T=n1a1+ n2a2+ n3a3) 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)

a2

a1

R

R’

T

x

x x x

x x

x x x

A rotation of π radians around any point marked x is a symmetry operation since it carries the crystal structure into itself

Point Operations

3D Crystals

Group of the octahedron Cubic symmetry

Possible symmetry operations: Rotations Reflections Inversions Combinations of them: glide operation

Group of the tetrahedron

Different symmetry operations are possible!

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

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

The Seven crystal systems: d) Monoclinic , the rectangular face othogonal to

the c axis is distorted into a rhombus: simple base centered

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

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

Classification of the point groups

Number of point groups

Number of space groups

Bra

vais

la

ttic

es

(bas

is o

f sp

her

ical

sym

met

ry)

Cry

stal

la

ttic

es

(bas

is o

f ar

bit

rary

sym

met

ry)

7 32

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

Combining rotational and translational symmetry:

3D Crystals and Primitive Lattice Cell The unit volume defined by the primitive a1, a2, a3 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

Compact structures: fcc ABCABC hcp ABABAB

fcc, different possible choices of unit vectors

The primitive choice is complicated, better working with a non-primitive unit cell like the face centered cube

The primitive cell has one fourth of the volume of the conventional unit cell

Packing fraction 0,74

bcc

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 2nd and 3rd nearest neighbors as illustrated in the figure

Packing fraction 0,68

Different choices are possible for the unit cell. The primitive cell has half the volume of the conventional unit cell

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

hcp structure: ABAB packing

Hexagonal layer : graphene Layered crystals: graphite

More complicated ABCABABCAB packing possible, too, e.g. Nd

Packing fraction = 0,34

Lattice with two atom basis: the diamond structure

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.

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

Wigner Seitz cells

bcc

fcc

Complicated geometry, used rarely

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

Compounds

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

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

NaCl CsCl

Fluorite CaF2

Perovskite BaTiO3

Laves phase Cu2Mg A15 or

β tungsten structure Nb3Sn