ON LINE EDUCATION ON CRYSTALLOGRAPHY. I-EXTINCTIONS

Last Update 31/10/1998

em PortuguÍs

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.

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rotation (radians) p 2p/3 2p/3 p/2 p/2 p/2 p/3 p/3 p/3 p/3 p/3
simbol 21 31 32 41 42 43 61 62 63 64 65
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)2B5O8(OH)2Cl 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

Federal University of Vicosa