Lattice:FCC

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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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=4}
  • Volume of unit cell: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_d=a^3}
  • Dimensionality: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=3}
  • Projected d-dimensional volume: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_d=a^3}
  • Solid angle: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_d=4\pi}
  • Nearest-neighbor distance: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_{nn}=\sqrt{2}a/2}
  • Assuming spherical particles of radius R:
    • Particle volume fraction: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi=16 \pi R^3/\left(3a^3\right)}
    • Maximum volume fraction: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_{max}=\pi\sqrt{2}/6\approx0.740} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R=a/(2\sqrt{2})}

Structure

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

Fcc to bcc025.png


Reciprocal-Space Peaks

  • Allowed reflections:
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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. }
  • Peak multiplicities:
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{h00}=6}
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{hh0}=12}
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{hhh}=8}
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{hk0}=24}
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{hhk}=24}
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{hkl}=48}
  • Peak positions:
    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

Lattice peaks-FCC 07.png

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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=4} (distinct)
  • Volume of unit cell: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_d=a^3}
  • Dimensionality: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=3}
  • Projected d-dimensional volume: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_d=a^3}
  • Solid angle: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_d=4\pi}
  • Nearest-neighbor distance: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_{nn}=\sqrt{2}a/2}

Examples

Atomics

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