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: ${\displaystyle n=1}$
• 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: ${\displaystyle d_{nn}=a}$
• Assuming spherical particles of radius R:
• Particle volume fraction: ${\displaystyle \phi =4\pi R^{3}/\left(3a^{3}\right)}$
• Maximum volume fraction: ${\displaystyle \phi _{max}=\pi /6\approx 0.524}$ when ${\displaystyle R=a/2}$

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).