Threefold Screw Axes (3n)
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The symbols above show threefold screw axes perpendicular to the
plane of the screen, i.e parallel to Z.
A threefold screw axis is equivalent to an anticlockwise rotation of 120°
about a line followed by a translation parallel to that line.
The translation is either 1/3 or 2/3 of the repeat distance in the lattice
along that direction for three-one and three-two screw axes, respectively.
A threefold rotation about the c-axis, i.e. about
the line 0,0,z, followed by a translation of 1/3 c
along it, will have
the corresponding symmetry operator -y,x-y,1/3+z.
In contrast to a two-one screw axis,
three-one and three-two axes exhibit handedness corresponding to
right-handed and left-handed helices, respectively.
They have the written symbols "31" or
In space group diagrams for cubic symmetry,
all the threefold screw axes are inclined
at an angle of 54.736° to the XY plane. The symbol for these are
shown above, the dot indicating the exact position of intersection with
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Birkbeck College, University of London.