Lattices and Structures

Bravais Lattices

There are other types of lattices, based on non primitive cells, which cannot be related to the previous ones.

In total, we have 14 types of lattices in 3D. These are called the Bravais Lattices

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In the previous cases, we have associated to each crystal system a primitve cell.

Each of these primitive cells defines a lattice type.

Let us exhamine these lattices in detail

No. Type Description

1 Primitive Lattice points on corners only. Symbol: P.

2 Face Centered Lattice points on corners as well as centered on faces. Symbols: A (bc faces); B (ac faces); C (ab faces).

3 All-Face Centered Lattice points on corners as well as in the centers of all faces. Symbol: F.

4 Body-Centered Lattice points on corners as well as in the center of the unit cell body. Symbol: I.

Four lattice centering types

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The 14 Bravais lattices

P

F

I

P

I

P

P

P

C

F

I

P

C

P 3

Crystal System & 14 Bravais Lattices

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Mystery of the missing entries in the Bravais List

A cell with two centered faces must be of type F (all face centered)

⁄ , , ⁄ , ⁄ , ⁄ ⁄ , ⁄ , ⁄ , ⁄ ,

(½ , ½, 0)

(0, ½ , ½)

(½ , 0, ½)

a

b

c

(0, ½ , ½)

(½ , 0, ½)

a

b

c

(½ , ½, 0)

Mystery of the missing entries in the Bravais List

A cell with is at the same time body centered and face centered can be always reduced to a face centered cell

, ⁄ , ⁄ ⁄ , ⁄ , ⁄ ⁄ , , ⁄ , , ⁄ , , ⁄ , ,

(0, ½ , ½)

(½ , 0, ½)

a

b

c

(½ , ½, ½)

(½ , 1, 1)

a

b

c

Mystery of the missing entries in the Bravais List!

1 Cubic Cube

Hence even though this lattice remains as it is it is called Simple Tetragonal(which is smaller in size)

Hence Cannot be called CubicBut then Cubic crystals need not have any 4-fold axes!!

(cubic lattices do need to have!)

What we chooseP I F C

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FCT = BCT

2 Tetragonal Square Prism (general height)

Mystery of the missing entries in the Bravais List! What we choose

P I F C

Smaller sized Body Centred Cell is chosen

Face Centred Tetragonal = Body Centred Tetragonal

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CCT = ST

2 Tetragonal Square Prism (general height)

Mystery of the missing entries in the Bravais List! What we choose

P I F C

Smaller sized Simple Cell is chosen

C Centred Tetragonal = Simple Tetragonal

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4 Hexagonal 120 Rhombic Prism

Mystery of the missing entries in the Bravais List!P I F C

Putting a lattice point at body centre destroys the 6-fold axis

Hence body centred hexagonal lattice NOT

possible

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Putting lattice points at face centresdestroys the 6-fold axis

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Putting a lattice point at face centredestroys the 6-fold axis

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& what we don’tMystery of the missing entries in the Bravais List!

As the FCC lattice has higher symmetry than the BCT cell(the one with higher symmetry is chosen)

Cubic F Tetragonal I(not chosen)

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Some example of the allowed and disallowed transfers

Crystal system Bravais lattices

1. Cubic P I F C

2. Tetragonal P I F

3. Orthorhombic P I F C

4. Hexagonal P

5. Trigonal P

6. Monoclinic P C

7. Triclinic P

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Space groups

We have seen so far the 32 crystallographic point groups: combinations of either simple rotations or inversion axes, compatible with the periodic nature of the lattice.

Combing these groups with the 14 Bravais lattices we obtain 73 space groups. These are called symmorphic space groups.

A crystallographic space group is the set of geometrical symmetry operations that take a three dimensional periodic object into itself.

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Crystal Class Bravais Lattices Point Groups

Triclinic P 1,1

Monoclinic P, C 2,m,2/m

Orthorhombic P, C, F, I 222,mm2,mmm

Trigonal P, R 3, 3,32, 3m, 3m

Hexagonal P 6, 6, 6 m⁄ , 622, 6mm, 62m, 6 mmm⁄

Tetragonal P, I 4, 4, 4 m⁄ , 422, 4mm, 42m, 4 mmm⁄

Cubic P, F, I 23,m3,43m,m3m

However, we can replace the proper or improper axes by screw axes of the same order and mirror planes by glide planes. Combining all possible symmetry elements with the 14 bravais lattices, we generate the 230 possible space groups.

Space groups

Important: when introducing symmetry elements with translational components, we do not have any more the restriction that all symmetry elements must meet in a point.

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Asymmetric unit

This is the smallest part of the unit cell which will generate the whole cell when applying to it the symmetry operations.

Examples of asymmetric units

Triclinic

P1 P

Monoclinic

P2 C2 C2/c

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Examples of asymmetric units

Orthorombic

P222 Immm

Tetragonal

P4 P

Fm m

Hexagonal

Cubic

Fm m

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Diagrams from International Table of Crystallography:

Triclinic system

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Diagrams from International Tables for Crystallography:

Triclinic system

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Diagrams from International Tables for Crystallography :

Monoclinic system

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Monoclinic system

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Orthorhombic system

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Tetragonal system

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Cubic system

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The 230 space groups

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Asymmetric units:

http://cci.lbl.gov/asu_gallery/

Space Group Diagrams and Tables:

http://img.chem.ucl.ac.uk/sgp/large/sgp.htm/

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The exact choice of the asymmetric unit is in reality arbitrary!

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Wigner-Seitz cells

The 14 Bravais Lattices are compatible with cells that are different from those conventionally associated with them.

Apart from parallelepipeds, there are other classes of polyhedra which can fill up the space by translation

One of them is obtained by the Dirichlet construction:

1. Connect a lattice point with its nearest neighbors 2. Trace through the mid points of the segments the planes perpendicular

to them.3. The intersecting planes delimit a region of space known as the Wigner-

Seitz cells

Examples in 2D....

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Construction of a Wigner-Seitz cell in the case of a face-centered cubic lattice

Crystal lattice filled with the repetition of a Wigner-Seitz cell

As we shall see later, the Wigner-Seitz construction delimits a region in the reciprocal space known as the first Brilllouin Zone.

Construction of a Wigner-Seitz cell in the case of a body-centered cubic lattice

Examples in 3D....

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Closed-packed structures

• There are an infinite number of ways to organize spheres to maximize the packing fraction.

There are different ways you can pack spheres together. This shows two ways, one by putting the spheres in an ABAB… arrangement, the other with ACAC…. (or any combination of the two works)

The centresof spheres at A, B, and C positions

Overview of some common crystal structures

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The FCC and hexagonal closed-packed structures (HCP) are formed from packing in different ways. FCC (sometimes called the cubic closed-packed structure, or CCP) has the stacking arrangement of ABCABCABC… HCP has the arrangement ABABAB….

HCP

ABABsequence

FCC(CCP)

(looking along [111]

direction

ABCABCsequence

[1 1 1][0 0 1]

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Cubic clos packing (fcc) packing

An enormous number of solids crystallize in the face-centered structure, with an atom at each lattice site. In the cell, the direction in which the ABC staking can be seen is the 111 direction.

An example is represented by gold.

Cube side a = 4.08 Å

Space group: Fm3m (number 225)

The fcc structure is a Bravais lattice

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Hexagonal close packing (same packing density of cubic close packing)

Also a great number of solids crystallize in the hexagonal close packed structure, with an atom at each lattice site. In the cell, the direction in which the AB staking can be seen is the 001 direction.

This is described by an hexagonal unit cell

a= 2.51 Å, c= 4.07 Å

Space group: P63/mmc (number 194)

Comparison between AB and ABC stackingThe hcp structure is not a Bravais lattice!

An example is represented by metallic Co

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Diamond structure

The diamond structure consists of two interpenetrating face-centered cubic (fcc) Bravais lattices, displaced along the body diagonal of the cubic cell by one quarter of the length of the diagonal

The diamond lattice is not a bravais lattice!

a


Space group: Fd3m (number 227)

Also Si, Ge and -Sn crystallize in the diamond structure

The two sublattices are colored differently, although all atoms refer to carbon atoms

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Similarly to the case of diamond, a honeycomb net, like the case of a single graphene sheet, is not a Bravais lattice, in the sense that each C atom cannot be considered as a point of a 2D lattice. Instead, we can describe it as two interpenetrated hexagonal lattices.

The two sublattices are colored differently, although all atoms refer to carbon atoms

Honeycomb net (graphene, boron nitride)

The same structure of two interpenetrating sublattices can describe a single sheet of boron nitride (BN). In this case one sublattice is formed by B atoms, the other by N atoms.

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Zinc-blende structure

Strongly related to the diamond structure is the zinc-blende structure, named after the mineral sphalerite (ZnS), which consists of two interpenetrating face-centered cubic Bravais lattices, displaced along the body diagonal of the cubic cell by one quarter of the length of the diagonal

The diamond lattice is not a bravais lattice!

a


Space group: F43m (number 216)

Many semiconductors (GaAs, CdTe, CdSe, etc) crystallize in the zinc-blende structure

In this case, the two sublattices, colored differently, are made of different atoms: Zn and S

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Wurtzite structure

The wurtzite structure, named after the mineral wurtzite (Zn, Fe)S, consists of two interpenetrating hcp lattices.

The wurtzite lattice is obviously not a bravais lattice!

a = 3.25 Å, c = 5.21 Å

Space group: P63mc (number 186)

Many compounds (AgI, ZnO, CdS, CdSe, α-SiC, GaN, AlN, BN, etc.) crystallize in the wurtzite structure

In this case too, the two sublattices, colored differently, are made of different atoms

The wurtzite structure is non-centrosymmetric (i.e., lacks inversion symmetry). Due to this, wurtzite crystals can (and generally do) have properties such as piezoelectricity and pyroelectricity, which centrosymmetric crystals lack

c axis

One lattice, the blue one in the figure, is shifted by - 3 8⁄ 0. 375 along the c direction

Example: ZnO

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A comparison between wurtzite and zincblende

d d

d is smaller in wurtzite than in zinc-blende

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Fluorite structure

The antifluorite structure is identical except that the anion and cation coordinations are reversed.

The fluorite structure can be generated by starting with a FCC lattice of cations (A) and filling all of the tetrahedral holes with anions (X).

Antifluorite structure

Cubic Cu2Se for example has antifluorite crystal structure

It is convenient to see this structure in terms of coordination polyhedra

Comparing Zinc Blende and Antifluorite structures

Zinc Blende

The sphalerite structure can also be seen as derived from the antifluorite structure, by removing ½ of the cations from the tetrahedral sites

Antifluorite

Rock salt (NaCl) structure

• The NaCl structure is fcc• The basis consists of one Na atom and one Cl atom,

separated by one-half of the body diagonal of a unit cube

• There are four units of NaCl in each unit cube• Each atom has 6 nearest neighbors of the opposite kind

Another example of rocksalt structure is represented by NiO

Also, it is most convenient to see this structure in terms of coordination polyhedra

The rocksalt structure is then represented by edge-sharing NaCl6 octahedra 42

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Lattices and Structures