Screw axis

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Crystallography

  In crystallography, a screw axis is a symmetry operation describing how a combination of rotation about an axis and a translation parallel to that axis leaves a crystal unchanged.

Screw axes are noted by a number, n, where the angle of rotation is 360°/n. The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2₁ is a 180° (two-fold) rotation followed by a translation of 1/2 of the lattice vector. 3₁ is a 120° (three-fold) rotation followed by a translation of 1/3 of the lattice vector. The possible screw axes are 2₁, 3₁, 4₁, 4₂, 6₁, 6₂, and 6₃, and the enantiomorphous 3₂, 4₃, 6₄, and 6₅.

Mathematics

A screw operation is the combination of a rotation by some angle φ about an axis (called the screw axis), combined with a translation by some distance d along the axis. A positive rotation direction usually means one that corresponds to the translation direction by the right-hand rule. Except for φ = 180°, we have to distinguish a screw operation from its mirror image. Unlike for rotations, a righthand and lefthand screw operation even generate different groups.

The combination of a rotation about an axis and a translation in a perpendicular direction is a rotation about a parallel axis. However, a screw operation with a nonzero translation vector along the axis cannot be reduced like that. Thus the effect of a rotation combined with any translation is a screw operation in the general sense, with as special cases a pure translation. a pure rotation, and the identity. Together these are all the direct isometries in 3D.

Screw axis symmetry is invariance under a screw operation.

If φ = 360°/n for some positive integer n, then screw axis symmetry implies translational symmetry with a translation vector which is n times that of the screw operation.

Applicable for space groups is a rotation by 360°/n about an axis, combined with a translation along the axis by a multiple of the distance of the translational symmetry, divided by n. This multiple is indicated by a subscript. So, 6₃ is a rotation of 60° combined with a translation of 1/2 of the lattice vector, implying that there is also 3-fold rotational symmetry about this axis. The possibilities are 2₁, 3₁, 4₁, 4₂, 6₁, 6₂, and 6₃, and the enantiomorphous 3₂, 4₃, 6₄, and 6₅.

Continuous case

A non-discrete screw axis isometry group contains all combinations of a rotation about some axis and a proportional translation along the axis (in rifling, the constant of proportionality is called the twist rate); in general this is combined with k-fold rotational isometries about the same axis (k ≥ 1); the set of images of a point under the isometries is a k-fold helix; in addition there may be a 2-fold rotation about a perpendicularly intersecting axis, and hence a k-fold helix of such axes.

Motion

The motion of a rigid body may be the combination of rotation about an axis (the screw axis) and a translation along that axis. This screw move is characterized by the velocity vector for the translation and the angular velocity vector in the same or opposite direction. If these two vectors are constant and along one of the principal axes of the body, no external forces are needed for this motion (moving and spinning).As an example, if gravity and drag are ignored, this is the motion of a bullet fired from a rifled gun.

See also

• Helical symmetryhttp://lexicon.server.privat/lexikon/e/Symmetry#Helical_symmetry
• Screw theory
• Space group
• Corkscrew (roller coaster element)

References

• Walter Borchardt-Ott (1995). Crystallography. Springer-Verlag. ISBN 3-540-59478-7.