From Wikipedia,
the free encyclopedia
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 vecior 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). Apart from gravity
and drag, this is e.g. the motion
of a
bullet fired from a
rifled
gun.
See also
References
