Why reciprocal lattice

The red square represents the translations of the smallest direct lattice produced by the periodic distributions of the small pieces of this mosaic. The yellow square represents another possible lattice, a bigger one, non primitive. Periodic stacking of balls, producing a 3-dimensional network direct lattice. The motif being repeated in the three directions of space is the contents of the small box with blue edges, the so called "unit cell".

Left: Elementary cell or unit cell defined by the 3 non-coplanar reticular translations cell axes or lattice axes Right: Crystal formation by stacking of many unit cells in 3 space directions. The structural motif shown in the left figure is repeated by a symmetry element symmetry operation , in this case a screw axis. Two-dimensional periodic distribution of one motif containing two objects a triangle and a circle. Position vector for any non-reticular point of a direct lattice.

Position vector for a non-reticular point black circle. Following with the argument given above, each motif in a repetitive distribution generates its own lattice, although all these lattices are identical red and blue. Of the two families of equivalent lattices shown red and blue we can choose only one of them, on the understanding that it also represents the remaining equivalent ones.

Note that the distance between the planes drawn on each lattice interplanar spacing is the same for the blue or red families. However, the family of red planes is separated from the family of blue planes by a distance that depends on the separation between the objects which produced the lattice.

This distance between the planes of different families can be called the geometric out-of-phase distance. Left: Family of reticular planes cutting the vertical axis of the cell in 2 parts and the horizontal axis in 1 part. These planes are parallel to the third reticular axis not shown in the figure. Right: Family of reticular planes cutting the vertical axis of the cell in 3 parts and the horizontal axis in 1 part.

The number of parts in which a family of planes cut the cell axes can be associated with a triplet of numbers that identify that family of planes. Brillouin Zones , an important tool in solid state physics, are also worked in reciprocal space. Previous Next Applications of reciprocal space Ewald sphere and Reciprocal space for single crystal and oriented samples Reciprocal space and the Ewald sphere have important implications for x-ray diffraction.

To observe more reflections one can: Rotate the reciprocal lattice i. Use a spread of wavelengths, i. This is described in Laue photographs. An interesting application is on-line assessment of orientation in single crystals such as turbine blades. Spread a reciprocal spot into a ring as used in powder diffraction. Reciprocal Space Maps This application for measuring the lattice parameters of a film as compared with those of the underlying substrate is shown in the following animation.

Indexing Assigning indices to the diffraction spots and working out the unit cell is done in reciprocal space. Brillouin Zones Brillouin Zones , an important tool in solid state physics, are also worked in reciprocal space.

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