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

DoITPoMS Teaching & Learning Packages Reciprocal Space Reciprocal space
Reciprocal Space AimsBefore you startIntroductionReciprocal spaceMathematical representation of reciprocal latticeEwald sphereApplications of reciprocal spaceSummaryQuestionsGoing further
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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

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