Example: for 2,3,4 and 6 fold axes, rotational symmetry are allowed while for 5, 7 and higher fold axes rotational symmetry are not allowed. why?
Nowadays it cannot be said that 5, 8, 10, 12 or n-fold rotational symmetry is not allowed in crystals since the discovery of quasicrystal has shown nature is more complex than our original understanding of crystals.
Being this said, in general, 3D periodic crystals (not quasicrystals which may be periodic in 3+n dimensional space) only show 1, 2, 3, 4 or 6 fold symmetry because a periodic arrangements of lattice points (or atoms or molecules) cannot be made to have other symmetry.
To understand this you may think about the Bravais lattices that define the 7 periodic crystal systems. A simplification of the reasoning can be made in 2D. If you want to completely cover a plane with equal tiles arranged in such a way that two tiles can be related by a translation made of a linear combination of the lengths of the sides of the tiles then the only tiles that can be used are: paralelogram, rectangle, general rhombus (with angles alpha/180-alpha) all with 2-fold symmetry, square with 4-fold symmetry, 60/120 rhombus with 3-fold symmetry and regular hexagon with 6-fold symmetry (although the smallest quadrilatera inside this hexagon is a 60/120 rhombus). In all the cases the quadrilatera put together around a vertex of any of them have their angles add 360. For the paralelogram and general rhombus two angles are alpha and the other two are 180-alpha, therefore alpha+alpha+(180-alpha)+(180-alpha)=360. For the square and the rectangle the four angles are 90 and for the 60/120 rhombus alpha=60.
If you try to cover the whole plane with regular pentagons you'll find that voids will arise every time three pentagons meet at a corner since the internal angles of the rhombus are 108 degrees so 3 of them add up to 324 degrees, not enough to fill the space around the point. If you add a 4th pentagon you go over 360 so no way. The same with regular heptagons, octagons, and general n-agons where the problem is similar to pentagons.
This implies that periodic arrangements can only be made that they display 1, 2, 3, 4 and 6 fold rotational symmetry and the large (by far) majority of crystals are of the periodic kind so it is said that crystals can only show these kinds of rotational symmetry. Even in many cases aperiodic crystals are such that have a periodic arrangement in one or two directions that define its shape and lack of periodicity in the remaining ones.
This question you make is much larger than you may think, since the finding of quasicrystals obliged crystallographers to change the definitio of crystals from periodically arranged solid bodies to bodies that display a unique diffraction pattern with sharp peaks, that includes both periodic and aperiodic crystals such as quasicrystals.
I'd suggest you check a good book on fundamental crystallography for a better explanation of this important issue.
BTW a nice article in wikipedia devoted to tiling of the plane with regular polygons (see http://en.wikipedia.org/wiki/Tiling_by_regular_polygons) is illuminating if you manage to see that no matter the symmetry and number of specific polygons that are used in the tiling, the whole pattern always has 1, 2, 3, 4 or 6-fold symmetry but none a different one.
Hope this helps,
best of luck
Leo
Nowadays it cannot be said that 5, 8, 10, 12 or n-fold rotational symmetry is not allowed in crystals since the discovery of quasicrystal has shown nature is more complex than our original understanding of crystals.
Being this said, in general, 3D periodic crystals (not quasicrystals which may be periodic in 3+n dimensional space) only show 1, 2, 3, 4 or 6 fold symmetry because a periodic arrangements of lattice points (or atoms or molecules) cannot be made to have other symmetry.
To understand this you may think about the Bravais lattices that define the 7 periodic crystal systems. A simplification of the reasoning can be made in 2D. If you want to completely cover a plane with equal tiles arranged in such a way that two tiles can be related by a translation made of a linear combination of the lengths of the sides of the tiles then the only tiles that can be used are: paralelogram, rectangle, general rhombus (with angles alpha/180-alpha) all with 2-fold symmetry, square with 4-fold symmetry, 60/120 rhombus with 3-fold symmetry and regular hexagon with 6-fold symmetry (although the smallest quadrilatera inside this hexagon is a 60/120 rhombus). In all the cases the quadrilatera put together around a vertex of any of them have their angles add 360. For the paralelogram and general rhombus two angles are alpha and the other two are 180-alpha, therefore alpha+alpha+(180-alpha)+(180-alpha)=360. For the square and the rectangle the four angles are 90 and for the 60/120 rhombus alpha=60.
If you try to cover the whole plane with regular pentagons you'll find that voids will arise every time three pentagons meet at a corner since the internal angles of the rhombus are 108 degrees so 3 of them add up to 324 degrees, not enough to fill the space around the point. If you add a 4th pentagon you go over 360 so no way. The same with regular heptagons, octagons, and general n-agons where the problem is similar to pentagons.
This implies that periodic arrangements can only be made that they display 1, 2, 3, 4 and 6 fold rotational symmetry and the large (by far) majority of crystals are of the periodic kind so it is said that crystals can only show these kinds of rotational symmetry. Even in many cases aperiodic crystals are such that have a periodic arrangement in one or two directions that define its shape and lack of periodicity in the remaining ones.
This question you make is much larger than you may think, since the finding of quasicrystals obliged crystallographers to change the definitio of crystals from periodically arranged solid bodies to bodies that display a unique diffraction pattern with sharp peaks, that includes both periodic and aperiodic crystals such as quasicrystals.
I'd suggest you check a good book on fundamental crystallography for a better explanation of this important issue.
BTW a nice article in wikipedia devoted to tiling of the plane with regular polygons (see http://en.wikipedia.org/wiki/Tiling_by_regular_polygons) is illuminating if you manage to see that no matter the symmetry and number of specific polygons that are used in the tiling, the whole pattern always has 1, 2, 3, 4 or 6-fold symmetry but none a different one.
Hope this helps,
best of luck
Leo
Leopoldo's answer is excellent and I can add little, except to amplify the point that this is not a property of crystals, but more generally of symmetry itself. The 230 space groups represent the ONLY ways of creating 3D periodicity. Crystals are - by the simplest definition which precludes the quasicrystals mentioned above - periodic arrangements of atoms in space, and therefore must fall into one of the space groups.
@Leopoldo Suescun
Thanks a lots for the detail description. it helps a lot in my exam. also thanks @Arun Kumar and Philip Howie
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Materials Science
- Materials