Crystallographic restriction theorem proof

  • 3-fold Rotation Axis - Objects that repeat themselves upon rotation of 120o are said to have a 3-fold axis of rotational symmetry (360/120 =3), and they will repeat 3 times in a 360o rotation.
    A filled triangle is used to symbolize the location of 3-fold rotation axis.

  • What is 5 fold symmetry?

    More precisely the term “having fivefold symmetries” means that their symmetry groups contain rotations through angles 2kπ/5 (k = 1, 2, 3 and 4)..

  • What permissible rotational symmetries in crystals are limited to 1 2 3 4 and 6 fold?

    In radians, the only allowed rotations consistent with lattice periodicity are given by 2π/n, where n = 1, 2, 3, 4, 6.
    This corresponds to 1-, 2-, 3-, 4-, and 6-fold symmetry, respectively, and therefore excludes the possibility of 5-fold or greater than 6-fold symmetry..

  • Why 5 fold symmetry does not exist?

    Because if you try to pack molecules with a 5 fold symmetry you cannot, at the same, conserve the repetition of the molecules in all directions of space.
    The basis of a crystal being that molecules are regularly located with the same distance in all directions of the space cannot be maintained with a 5 fold symmetry..

  • In radians, the only allowed rotations consistent with lattice periodicity are given by 2π/n, where n = 1, 2, 3, 4, 6.
    This corresponds to 1-, 2-, 3-, 4-, and 6-fold symmetry, respectively, and therefore excludes the possibility of 5-fold or greater than 6-fold symmetry.
  • Prior to 1991 crystals were defined to be solids having only 2-, 3-, 4- and 6-fold rotational symmetry because only these rotational symmetries have the required translational periodicity to build the long-range order of a crystalline solid.
If it exists, then we can take every other edge displacement and (head-to-tail) assemble a 5-point star, with the last edge returning to the starting point. The vertices of such a star are again vertices of a regular pentagon with 5-fold symmetry, but about 60% smaller than the original. Thus the theorem is proved.
That proof assumes the existence of some rotational axis plus the crystal itself being provided by some 3D lattice. Therefore some plane orthogonal to the axis, when containing at least a single lattice point, then shall contain a whole subdimensional 2D lattice.

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