**ON LINE EDUCATION ON CRYSTALLOGRAPHY. I-EXTINCTIONS**

Last Update 31/10/1998

Crystals are able to destroy energy. It is not fiction. This
can be observed during a diffraction experiment. Notice that
energy can destroy energy. For this to occur just add two
parallel light beams with the same wavelength but with a phase
difference of p radians, this is named
destructive interference. Electrons on symmetric distributed
atoms in crystals cause x-ray diffraction. If the atomic
distribution in the crystal is of low symmetry, than no energy
destruction will be possible or no radiation extinction will be
observed during diffraction. On the other hand, if the crystal
material is distributed inside the space limits defined by the
observed sample in such arrangement that a rotation followed by a
displacement symmetry operator cyclically determines a new
location equivalent to the previous one, some specific diffracted
beams will be destroyed. This destruction was explained above as
destructive interference. As the materials in those crystals are
distributed according to symmetry laws, the resulting extinctions
will naturally accord to symmetry laws as well. Several symmetry
elements perform the mentioned kind of operation, they are named
screw axes. They can be of four different rotation angles,
actually with rotations of p, 2p/3, p/2 and p/3 radians, with their characteristic
displacements. The following table shows the symbols for each
screw axis and the respective fractional displacements along the
unit cell edge **a**, (with dimension=a) if the axis is
directed parallel to it.

.

rotation (radians) | p | 2p/3 | 2p/3 | p/2 | p/2 | p/2 | p/3 | p/3 | p/3 | p/3 | p/3 |

simbol | 2_{1} |
3_{1} |
3_{2} |
4_{1} |
4_{2} |
4_{3} |
6_{1} |
6_{2} |
6_{3} |
6_{4} |
6_{5} |

displacement | a/2 | a/3 | 2a/3 | a/4 | a/2 | 3a/4 | a/6 | a/3 | a/2 | 2a/3 | 5a/6 |

Otherwise, if the screw axis is parallel to the unit cell edge
**b**, then the displacement will be a fraction of the b
parameter, and if parallel to **c**, the displacement will be
a fraction of c.

Another set of symmetry elements (or operators) is the glide planes. They combine
mirror reflection and translation. There are five different glide planes: a,
b, c, n and d. The a-glide plane parallel to **b** will reflect and displace
each point a distance of a/2 along the direction of **a**. In a similar way,
b-glide plane and c-glide plane execute a translation of b/2 and c/2 respectively,
combined with the reflection. Next, the n-glide plane perpendicular to **a**
will reflect and combine a translation of a/2 + b/2. A d-glide plane will reflect
and translate a/4+b/4. Other symmetry elements can be exhibited by a crystal,
as simple rotation axes and reflexion plane, including the center of symmetry
but these operators do not cause extinction. Finally extinctions can result
from the interaction of radiation with nonprimitive lattices A, B and C which
contain 2 lattice points as so as the body-centered lattice I and F lattice
with 4 lattice points. If a single crystal of the mineral Hilgardite, strontian,
(Ca,Sr)_{2}B_{5}O_{8}(OH)_{2}Cl which belongs
to the low-symmetry space group P1 of the triclinic crystal system is properly
irradiated with x-rays during a diffraction experiment, no extinction will be
observed. In the figure the non-extinction graphics are drawn initially, in
the three perpendicular zero layer planes hk0, h0L and 0kL and the corresponding
three first layer planes hk1, h1L and 1kL. The graphics are projections over
square grids, just to observe extinctions; they do not display other parameters.
If you click the mouse button when the cursor is pointing over a symmetry element
indicated on the buttons, it will display the corresponding extinctions by blanks
and the diffracted reflexions by black points. The "clean extinctions"
button cleans the system and shows the primitive non-extinction case. The zero
order reflexion is always missing in the figure, representing its difficult
experimental observation.

Reference

Buerger, M.J.(l966) X-ray Crystallography. New York: John Willey & Sons, p.83

Prof. Dr. Roberto Andrea Mueller