KevinYager: Created page with "'''FCC''' or '''face-centered cubic''' is a cubic lattice where the symmetry involves having additional atoms/particles sitting on the faces of the conceptua..."

2016-11-13T22:53:04Z

Created page with "'''FCC''' or '''face-centered cubic''' is a cubic lattice where the symmetry involves having additional atoms/particles sitting on the faces of the conceptua..."

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'''FCC''' or '''face-centered cubic''' is a [[Lattice:Cubic|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
* [http://en.wikipedia.org/wiki/Lattice_system Crystal System]: [[Lattice:Cubic|Cubic]]
* [http://en.wikipedia.org/wiki/Bravais_lattice#Bravais_lattices_in_3_dimensions Bravais Lattice]: F (fcc)
* Crystal class: Hexoctahedral
* [http://en.wikipedia.org/wiki/Crystallographic_point_group Point Group]: m3m
* Space Group: Fm3m
* Particles per unit cell: <math>n=4</math>
* Volume of unit cell: <math>V_d=a^3</math>
* Dimensionality: <math>d=3</math>
* Projected ''d''-dimensional volume: <math>v_d=a^3</math>
* Solid angle: <math>\Omega_d=4\pi</math>
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{2}a/2</math>
* Assuming spherical particles of radius ''R'':
** Particle volume fraction: <math>\phi=16 \pi R^3/\left(3a^3\right)</math>
** Maximum volume fraction: <math>\phi_{max}=\pi\sqrt{2}/6\approx0.740</math> when <math>R=a/(2\sqrt{2})</math>

===Structure===

One can also consider the FCC to be a [[FCC_to_BCC|distorted BCC]]:

[[Image:Fcc to bcc025.png|400px]]

===Reciprocal-Space Peaks===
* Allowed reflections:
*: <math>f_{hkl} = \left\{ \begin{array}{c l} 4 & \mathrm{for} \,\, \left( h \and k \and l \right) = \mathrm{even} \\ 4 & \mathrm{for} \,\, \left( h \and k \and l \right) = \mathrm{odd} \end{array} \right. </math>
* Peak multiplicities:
*:: <math>m_{h00}=6</math>
*:: <math>m_{hh0}=12</math>
*:: <math>m_{hhh}=8</math>
*:: <math>m_{hk0}=24</math>
*:: <math>m_{hhk}=24</math>
*:: <math>m_{hkl}=48</math>
* Peak positions:
*: <math>q_{hkl}=\frac{2\pi}{a}\sqrt{h^2+k^2+l^2}</math>
*: For ''a'' = 1.0:

<pre>
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
</pre>
[[Image:Lattice_peaks-FCC_07.png|450px]]

==Face-centered Four-particle==
A lattice where the unit cell has four distinct atoms/particles, arranged in an FCC-like way. The lattice has '''[[Lattice:Simple Cubic|simple cubic]]''' symmetry.
===Symmetry===
* Crystal Family: Cubic
* [http://en.wikipedia.org/wiki/Lattice_system Crystal System]: [[Lattice:Cubic|Cubic]]
* [http://en.wikipedia.org/wiki/Bravais_lattice#Bravais_lattices_in_3_dimensions Bravais Lattice]: P (bcc)
* Crystal class: Hexoctahedral
* [http://en.wikipedia.org/wiki/Crystallographic_point_group Point Group]: m3m
* Space Group: Pm3m
* Particles per unit cell: <math>n=4</math> (distinct)
* Volume of unit cell: <math>V_d=a^3</math>
* Dimensionality: <math>d=3</math>
* Projected ''d''-dimensional volume: <math>v_d=a^3</math>
* Solid angle: <math>\Omega_d=4\pi</math>
* Nearest-neighbor distance: <math>d_{nn}=\sqrt{2}a/2</math>

===Examples===
====Atomics====
* [http://en.wikipedia.org/wiki/Sodium_chloride NaCl] (two distinct atoms) (''a'' = 5.64 Å)

Lattice:FCC - Revision history

KevinYager: Created page with "'''FCC''' or '''face-centered cubic''' is a cubic lattice where the symmetry involves having additional atoms/particles sitting on the faces of the conceptua..."