# Lattice:FCC

FCC or face-centered cubic is a cubic lattice where the symmetry involves having additional atoms/particles sitting on the faces of the conceptual unit cell.

## Canonical FCC

### Symmetry

• Crystal Family: Cubic
• Crystal System: Cubic
• Bravais Lattice: F (fcc)
• Crystal class: Hexoctahedral
• Point Group: m3m
• Space Group: Fm3m
• Particles per unit cell: ${\displaystyle n=4}$
• Volume of unit cell: ${\displaystyle V_{d}=a^{3}}$
• Dimensionality: ${\displaystyle d=3}$
• Projected d-dimensional volume: ${\displaystyle v_{d}=a^{3}}$
• Solid angle: ${\displaystyle \Omega _{d}=4\pi }$
• Nearest-neighbor distance: ${\displaystyle d_{nn}={\sqrt {2}}a/2}$
• Assuming spherical particles of radius R:
• Particle volume fraction: ${\displaystyle \phi =16\pi R^{3}/\left(3a^{3}\right)}$
• Maximum volume fraction: ${\displaystyle \phi _{max}=\pi {\sqrt {2}}/6\approx 0.740}$ when ${\displaystyle R=a/(2{\sqrt {2}})}$

### Structure

One can also consider the FCC to be a distorted BCC:

### Reciprocal-Space Peaks

• Allowed reflections:
${\displaystyle f_{hkl}=\left\{{\begin{array}{c l}4&\mathrm {for} \,\,\left(h\land k\land l\right)=\mathrm {even} \\4&\mathrm {for} \,\,\left(h\land k\land l\right)=\mathrm {odd} \end{array}}\right.}$
• Peak multiplicities:
${\displaystyle m_{h00}=6}$
${\displaystyle m_{hh0}=12}$
${\displaystyle m_{hhh}=8}$
${\displaystyle m_{hk0}=24}$
${\displaystyle m_{hhk}=24}$
${\displaystyle m_{hkl}=48}$
• Peak positions:
${\displaystyle q_{hkl}={\frac {2\pi }{a}}{\sqrt {h^{2}+k^{2}+l^{2}}}}$
For a = 1.0:
peak    q value         h,k,l   m       f       intensity
1:      10.882796185405 1,1,1   8       4       128
2:      12.566370614359 2,0,0   6       4       96
3:      17.771531752633 2,2,0   12      4       192
4:      20.838968152189 3,1,1   24      4       384
5:      21.765592370811 2,2,2   8       4       128
6:      25.132741228718 4,0,0   6       4       96
7:      27.387769797535 3,3,1   24      4       384
8:      28.099258924163 4,2,0   24      4       384
9:      30.781195923885 4,2,2   24      4       384
10:     32.648388556216 5,1,1   32      4       512
11:     35.543063505267 4,4,0   12      4       192
12:     37.171825569274 5,3,1   48      4       768
13:     37.699111843078 6,0,0   30      4       480
14:     39.738353063184 6,2,0   24      4       384
15:     41.201601388628 5,3,3   24      4       384
16:     41.677936304377 6,2,2   24      4       384
17:     43.531184741621 4,4,4   8       4       128
18:     44.870918174495 5,5,1   24      4       384
19:     45.308693596556 6,4,0   24      4       384
20:     47.019053434156 6,4,2   48      4       768
21:     48.262062105313 5,5,3   24      4       384
22:     51.812473373661 6,4,4   24      4       384
23:     53.314595257900 6,6,0   12      4       192
24:     54.413980927027 5,5,5   8       4       128
25:     54.775539595071 6,6,2   24      4       384
26:     58.941502773372 6,6,4   24      4       384
27:     65.296777112432 6,6,6   8       4       128


## Face-centered Four-particle

A lattice where the unit cell has four distinct atoms/particles, arranged in an FCC-like way. The lattice has simple cubic symmetry.

### Symmetry

• Crystal Family: Cubic
• Crystal System: Cubic
• Bravais Lattice: P (bcc)
• Crystal class: Hexoctahedral
• Point Group: m3m
• Space Group: Pm3m
• Particles per unit cell: ${\displaystyle n=4}$ (distinct)
• Volume of unit cell: ${\displaystyle V_{d}=a^{3}}$
• Dimensionality: ${\displaystyle d=3}$
• Projected d-dimensional volume: ${\displaystyle v_{d}=a^{3}}$
• Solid angle: ${\displaystyle \Omega _{d}=4\pi }$
• Nearest-neighbor distance: ${\displaystyle d_{nn}={\sqrt {2}}a/2}$

### Examples

#### Atomics

• NaCl (two distinct atoms) (a = 5.64 Å)