Lattice:Simple cubic

SC or simple cubic is the simplest and most generic cubic lattice.

Canonical SC

Symmetry

• Crystal Family: Cubic
• Crystal System: Cubic
• Bravais Lattice: P (pcc)
• Crystal class: Hexoctahedral
• Point Group: m3m
• Space Group: Pm3m
• Particles per unit cell: $n=1$
• Volume of unit cell: $V_{d}=a^{3}$
• Dimensionality: $d=3$
• Projected d-dimensional volume: $v_{d}=a^{3}$
• Solid angle: $\Omega _{d}=4\pi$
• Nearest-neighbor distance: $d_{nn}=a$
• Assuming spherical particles of radius R:
  • Particle volume fraction: $\phi =4\pi R^{3}/\left(3a^{3}\right)$
  • Maximum volume fraction: $\phi _{max}=\pi /6\approx 0.524$ when $R=a/2$

Reciprocal-Space Peaks

peak    q value         h,k,l   m       f       intensity
1:      6.283185307180  1,0,0   6       1       6
2:      8.885765876317  1,1,0   12      1       12
3:      10.882796185405 1,1,1   8       1       8
4:      12.566370614359 2,0,0   6       1       6
5:      14.049629462081 2,1,0   24      1       24
6:      15.390597961942 2,1,1   24      1       24
7:      17.771531752633 2,2,0   12      1       12
8:      18.849555921539 3,0,0   30      1       30
9:      19.869176531592 3,1,0   24      1       24
10:     20.838968152189 3,1,1   24      1       24

Alternating Simple Cubic

A simple cubic lattice where the particle identity (in every direction) alternates between two possibilities. Thus this lattice has 2 distinct particles, in a 1:1 ratio. The unit cell may be thought of as having 8 particles (4 of each type).

Examples

Atomics

• Sodium chloride (NaCl) (a = 5.6402 Å)