Screw Symmetry
III. Helical Screw Axes 
Helical Screw Axes
The twoone axis illustrated previously is not the only type of screw axis that can be found within a crystal structure. A threefold rotation axis, for example, may be combined with either a translation of 1/3 or 2/3 of the repeat distance along the axis direction. These two types of screw axis are referred to as "threeone" and "threetwo" axes, and have the crystallographic notation 3_{1} and 3_{2}, respectively.
In contrast to twoone screw axes, threeone and threetwo axes possess
a sense of handedness. Looking down the threeone axis, then the rotation
is seen as 120° anticlockwise and the translation is 1/3 of the
repeat distance (A to D) up the axis (as shown by the black arrows).
This is illustrated in the figure below where the length of the horizontal black
arrow corresponds to a translation length of 1/3 of the repeat distance
along the axis direction.
Starting with the red atom (A) on the far left of the diagram,
you should be able
to follow its movement along the axis as a result of repeated action
of the threeone screw symmetry operator (A to B to C to D to E to F to G).
The movement corresponds
to that of a righthanded helical screw as shown by the blue ribbon.
Note that it takes three times
3_{1} to reach the middle (and equivalent) lattice position (D).
3_{1}  
View perpendicular to axis  View down axis 
3_{2}  
View perpendicular to axis  View down axis 
The majority of the table is reference material.
In general, there exist crystallographic screw axes of the generic type n sub m, where n is the order of rotation, i.e. 2, 3, 4, or 6, m/n is the translation along the axis, and m < n, as shown in the table below:
Order  Screw axes 

2  2_{1} 
3  3_{1}, 3_{2} 
4  4_{1}, 4_{2}, 4_{3} 
6  6_{1}, 6_{2}, 6_{3}, 6_{4}, 6_{5} 
© Copyright 19952006.
Birkbeck College, University of London.

Author(s):
Jeremy Karl Cockcroft Huub Driessen 