•A basic knowledge of crystallography is essentialfor solid state physicists;

–- to specify any crystal structure and-–- to classify the solids into different types according

to the symmetries they possess.

•Symmetry of a crystal can have a profoundinfluence on its properties.

•We will concern in this course with solids withsimple structures.

ELEMENTARYCRYSTALLOGRAPHY

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CRYSTAL LATTICE

What is crystal (space) lattice?In crystallography, only the geometrical properties of thecrystal are of interest, therefore one replaces each atom bya geometrical point located at the equilibrium position ofthat atom.

Platinum Platinum surface Crystal lattice andstructure of Platinum(scanning tunneling microscope)

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•- An infinite array ofpoints in space,

•- Each point hasidenticalsurroundings to allothers.

•- Arrays arearranged exactly ina periodic manner.

Crystal Lattice

α

a

bCB ED

O A

y

x

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Crystal Structure•Crystal structure can be obtained by attaching atoms,

groups of atoms or molecules which are called basis(motif) to the lattice sides of the lattice point.

Crystal Structure = Crystal Lattice + Basis

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A two-dimensional Bravais lattice withdifferent choices for the basis

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E

HO A

CB

Fb G

D

x

y

a

α

a

bCB ED

O A

y

x

b) Crystal lattice obtained byidentifying all the atoms in (a)

a) Situation of atoms at thecorners of regular hexagons

Basis A group of atoms which describe crystal structureA group of atoms which describe crystal structure

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Crystal structure

•- Don't mix up atoms withlattice points

•- Lattice points areinfinitesimal points in space

•- Lattice points do notnecessarily lie at thecentre of atoms

Crystal Structure = Crystal Lattice + Basis

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Crystal Structure 8

Crystal Lattice

Bravais Lattice (BL) Non-Bravais Lattice (non-BL)

All atoms are of the same kind All lattice points are equivalent

Atoms can be of different kind Some lattice points are not

equivalentA combination of two or more BL

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Types Of Crystal Lattices

is an infinite array of discrete points withBravais lattice1)an arrangement and orientation that appears exactlythe same, from whichever of the points the array isviewed. Lattice is invariant under a translation

Nb film

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Types Of Crystal Lattices

•- The red side has a neighbour to itsimmediate left, the blue one insteadhas a neighbour to its right.

•- Red (and blue) sides are equivalentand have the same appearance

•-Red and blue sides are not equivalent.Same appearance can be obtainedrotating blue side 180º.

2)2) NonNon--Bravais LatticeBravais LatticeNot only the arrangement but also the orientation mustappear exactly the same from every point in a bravais lattice.

Honeycomb

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Translational Lattice Vectors – 2D

A space lattice is a set of points suchthat a translation from any point inthe lattice by a vector;

Rn = n1 a + n2 blocates an exactly equivalent point,i.e. a point with the same environmentas P . This is translational symmetry.The vectors a, b are known as latticevectors and (n1, n2) is a pair ofintegers whose valuesdepend on the lattice point.

P

Point D(n1, n2) = (0,2)Point F (n1, n2) = (0,-1)

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•bandtwo vectors aThe-form a set of lattice

vectors for the lattice.•- The choice of lattice

vectors is not unique. - Thusone could equally well takethe vectors a and b’ as a latticevectors.

Lattice Vectors – 2D

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Lattice Vectors – 3DAn ideal three dimensional crystal is described by 3fundamental translation vectors a, b and c. If there isa lattice point represented by the position vector r,there is then also a lattice point represented by theposition vector where u, v and w are arbitraryintegers.

r’ = r + u a + v b + w c (1)

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Five Bravais Lattices in 2D

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Unit Cell in 2D

•The smallest component of the crystal (group of atoms, ions ormolecules), which when stacked together with pure

translational repetition reproduces the whole crystal.

Sa

b

S

S

S

S

S

S

S

S

S

S

S

S

S

S

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Unit Cell in 2D

•The smallest component of the crystal (group ofwhen stackedatoms, ions or molecules), which

with pure translational repetitiontogetherreproduces the whole crystal.

S

S

The choice ofunit cell

is not unique.

a

Sb

S

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2D Unit Cell example -(NaCl)

We define lattice points ; these are points with identicalenvironments

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Choice of origin is arbitrary - lattice pointsunit cell size shouldbut-need not be atoms

.always be the same

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This is also a unit cell -it doesn’t matter if you start from Na or Cl

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- or if you don’t start from an atom

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even though they are all theNOT a unit cellThis issame - empty space is not allowed!

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In 2D, this IS a unit cellIn 3D, it is NOT

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Q : Why can't the blue triangle be a unit cell?

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