The Reciprocal Lattice


In the introduction to crystal symmetry I have shown that a crystal consists of a periodic arrangement of the unit cell (filled with the motif and its symmetry generated equivalents), into a lattice. In the same fashion we can define the reciprocal lattice, whose lattice dimensions are reciprocal to the original cell (and correspond to the reflection positions) and whose 'size' (the intensity of the reflection) corresponds to the contents of the unit cell. The following picture will make this clear.

Each of the lattice points corresponds to the diffraction from a periodic set of specific crystal lattice planes defined by the index triple hkl. The dimensions of the reciprocal lattice are reciprocally related to the real lattice. In the case of the orthorhombic system I have drawn, the relations are simple: c* = 1/c etc., but in a generic oblique system the relation is more complicated. The length of a reciprocal lattice vector d(hkl)* (from origin to reciprocal lattice point h,k,l) again corresponds to the reciprocal distance d(hkl)of the crystal lattice planes with this index. In our simple case, for 001 this is just the cell dimension c for d(001) or 1/2 c for 002 etc.(d(001)*=1/c, thus d=c).

 PROGRAM - Click here to go to the reciprocal lattice program

Resolution revisited

The vector d(hkl) also determines the location of the diffraction spot in the diffraction image. The diffraction angle at which we observe the reflection is given by Bragg's formula sin(theta)=lambda/2d. The higher the index of a reciprocal lattice point, the larger the diffraction angle will be. We have already seen that the larger the diffraction angle, the higher the resolution, i.e. the finer the detail we can observe in the reconstruction of the crystal structure. This can be easily understood now : we need to 'slice' the crystal fine enough (i.e, small d(hkl) = high indices hkl) to have enough information contained in our diffraction pattern to reconstruct details. In a similar way we needed to make the FT grid (slices) fine enough to resolve details in the Fourier synthesis. A Fourier map from atomic resolution data still looks bad when you use too coarse a FT grid.

In the next chapter we will use the reciprocal lattice and the Ewald construction to visualize some important concepts in data collection.

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Last revised Dezember 27, 2009 01:40