Lattice:Hexagonal

Hexagonal is a general class of lattice symmetries (i.e. how unit cells can be arranged in space).

Reciprocal-space Peaks

• Forbidden reflections, when both:
  • $h+2k=3n$
  • $l$ odd
• Peak positions:

  $q_{hkl}=2\pi \left({\frac {(h^{2}+hk+k^{2})^{?}}{a^{2}}}+{\frac {l^{2}}{c^{2}}}\right)^{1/2}$

  For a = b = 1.0, c = 1.0:

peak    q value         h,k,l   m       f       intensity       intensity_scaled
1:      0.126933036509  1,0,0   2       1       2       0.192150
2:      0.146569645595  1,0,0   6       1       6       0.499222
3:      0.193893415997  1,1,0   12      1       12      0.754752
4:      0.253866073017  2,1,0   8       1       8       0.384301
5:      0.283830898224  2,1,1   12      1       12      0.515594
6:      0.293139291189  2,0,0   18      1       18      0.748832
7:      0.319441136669  2,1,0   12      1       12      0.458117
8:      0.359020843488  2,2,1   12      1       12      0.407613
9:      0.380799109526  3,0,0   2       1       2       0.064050
10:     0.387786831994  3,1,0   24      1       24      0.754752

Canonical 2D

A canonical 2D hexagonal lattice, such as formed by hexgonally-packed cylinders (where the c direction is conceptually infinite) has peak positions in the ratio:

• $1:{\sqrt {3}}:2:{\sqrt {7}}:{\sqrt {9}}:{\sqrt {12}}:{\sqrt {13}}$

See Also

• Hexagonal close-packed (HCP)
• Hexagonal diamond
  • Wurtzite
• AlB₂