# Lattice:Diamond

The diamond lattice, which consists of tetrahedrally-arranged atoms/particles, has simple cubic symmetry.

Along an appropriate axis, the lattice has six-fold symmetry (hexagonal).

## Contents

## Canonical Diamond

A canonical diamond lattice (single atom/particle type arranged as shown above) has symmetry *Fd3m*. The atoms occupy the positions (0,0,0;1/4,1/4,1/4)+face-centering.

### Symmetry

- Crystal Family: Cubic
- Crystal System: Cubic
- Bravais Lattice: F (fcc)
- Crystal class: Hexoctahedral
- Point Group: d3m
- Space Group: Fd3m
- Particles per unit cell:
- Volume of unit cell:
- Dimensionality:
- Projected
*d*-dimensional volume: - Solid angle:
- Nearest-neighbor distance:
- Assuming spherical particles of radius
*R*:- Particle volume fraction:
- Maximum volume fraction: when

### Structure

The lattice may be thought of as two interpenetrating FCC lattices.

### Particle Positions

There are 18 positions. In total there are 8 particles in the unit cell:

### Examples

#### Elemental

- 6. Carbon (C) (
*a*= 3.567 Å) - 14. Silicon (Si) (
*a*= 5.431 Å) - 32. Germanium (Ge) (
*a*= 5.657 Å) - 50. Gray Tin (Sn) (
*a*= 6.491 Å)

#### Atomic

- TBD

#### Nano

- Nanoparticle superlattice
- Liu, W.; Tagawa, M.; Xin, H.L.; Wang, T.; Emamy, H. Li, H.; Yager, K.G.; Starr, F.W.; Tkachenko, A.V.; Gang, O. Diamond family of nanoparticle superlattices
*Science***2016**, 351, 582–586. doi: 10.1126/science.aad2080

- Liu, W.; Tagawa, M.; Xin, H.L.; Wang, T.; Emamy, H. Li, H.; Yager, K.G.; Starr, F.W.; Tkachenko, A.V.; Gang, O. Diamond family of nanoparticle superlattices

## Diamond-like Two-particle

Also known as **zincblende**. This is effectively

### Examples

#### Atomics

- Gallium arsenide (GaAs)
- Zinc sulfide (ZnS), a.k.a. zincblende
- Indium phosphide (InP)
- Cubic silicon carbide (CSi)
- Cubic gallium nitride (GaN)

## Double-filled Diamond-like Two-particle

The diamond lattice includes 8 "tetrahedral holes", with only 4 occupied in a 'normal' diamond structure. A two-particle lattice can be formed by filling all 8 internal holes with the 2nd particle-type, in which case the particles exist in a 1:2 ratio. This is often called a **CaF _{2}** lattice.

### Particle Positions

There are 22 positions. In total there are 12 particles in the unit cell:

### Neighbors

There are 12 particles in the unit cell: 4 of type A, and 8 of type B. The type A particles each connect to 8 particles of type B. The type B particles each connect (tetrahedrally) to 4 particles of type A.

The lattice may also be thought of as having cubes of type B with a connected type A particle at the center. However the extended lattice is not BCC-like, since only every second "BCC cube" has a central particle of type A.

### Examples

#### Atomics

- Calcium fluoride (CaF
_{2}) (*a*= 5.46 Å)

## Double Inter-penetrating Diamond Two-Particle

This lattice is formed from two diamond lattices that inter-penetrate. It is often called the **NaTl** lattice.

### Particle Positions

There are 36 positions. In total there are 16 particles in the unit cell, 8 of each type.

#### Particle Type A

#### Particle Type B

### Examples

#### Atomics

- Sodium Thallium crystal (NaTl)

#### Nanoparticles

- Au NPs and proteins: Petr Cigler, Abigail K. R. Lytton-Jean, Daniel G. Anderson, M. G. Finn & Sung Yong Park DNA-controlled assembly of a NaTl lattice structure from gold nanoparticles and protein nanoparticles Nature Materials 9, 918–922 (2010) doi:10.1038/nmat2877

## Cristobalite

Cristobalite is a diamond-like lattice formed in silicon dioxide (SiO_{2}). The Si atoms bond tetradrally, each to four O atoms. Each O atom is bonded to 2 Si atoms, effectively acting as a bridge.

### Structure

Can be thought of as a diamond lattice, where the tetrahedral species sit on the positions of the canonical diamond sites, and the two-bonded species sites at the midway point of each bond.

The tetrahedral species sit on the 14 "FCC" sites, plus 4 "internal" diamond sites. The two-bonded species sit on the 16 internal bonds in the unit cell. The tetrahedral species sit at a distance from one another of:

- Next-nearest-neighbor distance:

The different species thus have a distance of half that:

- Nearest-neighbor distance:

### Particle Positions

There are 34 positions (24 in unit cell)

#### Particle Type A (bond tetrahedrally)

There are 18 positions. In total there are 8 particles in the unit cell:

#### Particle Type B (two-fold bonded)

There are 16 positions (all 16 in the unit cell):

### Examples

#### Atomics

- Cristobalite form of silicon dioxide (SiO
_{2})