Dissemination of IT for the Promotion of Materials Science (DoITPoMS)


Reciprocal space

The animation below shows the relationship between the real lattice and the reciprocal lattice. Note that this 2D representation omits the c* vector, but that it follows the same rules as a* and b*.

The key things to note are that:

  • The reciprocal lattice has reciprocal vectors a* and b*, separated by the angle γ*.
  •  a* is perpendicular to the (100) planes, and equal in magnitude to the inverse of d100.
  •  Similarly, b* is perpendicular to the (010) planes and equal in magnitude to the inverse of d010.
  • γ and γ* will sum to 180º.

Due to the linear relationship between planes (for example, d200 = ½ d100 ), a periodic lattice is generated. In general, the periodicity in the reciprocal lattice is given by

 \({\rho _{hkl}}^ *\) = \(\frac{1}{{{d_{hkl}}}}\)

In vector form, the general reciprocal lattice vector for the (h k l) plane is given by

\({s_{hkl}}\) = \(\frac{{{{\rm{n}}_{hkl}}}}{{{d_{hkl}}}}\)

where nhkl is the unit vector normal to the (h k l) planes.

This concept can be applied to crystals, to generate a reciprocal lattice of the crystal lattice. The units in reciprocal space are Å-1 or nm-1