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).

**Exercise 1**: show that in the orthorhombic case the generic equation for d reduces to

d(hkl) = 1/sqrt((h/a)^{2}+ (k/b)^{2}+ (l/c)^{2}). Eeeeek, math!**Exercise 2**: The following program calculates reciprocal cell parameters for a given cell, transformation matrices and the d-spacing of a specified reflection.Run the program with default values.

Notice the simple relation between the direct and the reciprocal lattice in the orthonormal case. Make beta now larger than 90 deg (monoclinic). What happens to the reciprocal cell constants? Why is b* the only one still having the simple relationship b* = 1/b ? (Hint : look at the formula). Also notice that for the same reason d(001) is not equal to 1/c any more.

Set beta back to default value of 90.0. Look at the 4 matrices listed in the printout. The first is the cell's

*metric tensor*, used for fast coordinate system independent calculation of d between atoms in the cell (the SEXIE program was initially designed to do just that). It should be easy to figure out what its components are.

Application of the*reduced orthogonalization matrix**orthogonalization matrix*to the lower left is expanded with the unit cell dimensions, and transforms fractional coordinates into orthonormal Cartesian coordinates in units of Angstrom, which are what is listed in a pdb file (Cartesian coordinates are convenient to use in visualization programs, because no cell information is needed to draw the molecule). The*deorthogonalization matrix*does exactly the opposite : it transforms Cartesian in to fractional coordinates. It is listed in the SCALE records in the pdb files. In oblique systems the coordinates must be transformed into the crystallographic (fractional coordinate) reference system before any symmetry operators can be applied to a molecule!Change back to an oblique system and check how the transformation matrices change.

** PROGRAM -
Click here to go to the reciprocal lattice program**

The vector * d*(hkl) also
determines the location of the diffraction spot in the
diffraction image. The diffraction

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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by Bernhard Rupp. ****
Last revised
Dezember 27, 2009 01:40**