Threefold Rotation with Cartesian Coordinates

reference What follows is a proof of the symmetry operator for a threefold rotation in a Cartesian coordinate system for those who like mathematics: this page is very much supplementary course material.

Since z coordinate is unaffected by the rotation, one needs to consider only the effect on the (xy) coordinates in the XY plane as shown in the figure below:

It is simplest to work in polar coordinates, so the initial coordinates (xy) coordinates can be written as:

(rcosθ, rsinθ)

After rotation by 120°, the new coordinates can be written simply using polar coordinates as:

(rcos(θ+120°), rsin(θ+120°))

Expanding the double-angle sine and cosine terms gives:

(rcosθcos120° − rsinθsin120°, rsinθcos120° + rsin120°cosθ)

Substituting for rcosθ (= x) and rsinθ (= y) gives:

(xcos120° − ysin120°, ycos120° + xsin120°)

Finally, substituting values for sin120° (= √3/2) and cos120° (= −1/2), and reordering gives:

(−x/2 − y√3/2, x√3/2 − y/2)

quod erat demonstrandum

In addition, the symmetry operator for a threefold rotation may be represented in matrix form, both for Cartesian:

( cos120° −sin120°  0 ) ( x )
sin120° cos120° 0 y
0 0 1 z

and for crystallographic coordinate systems:

( 0 −1  0 ) ( x )
1 −1  0 y
0 0 1 z

An extended form of the latter (to include translational components) is used by virtually all crystallographic computer software.


Back
© Copyright 2002-2006.  Birkbeck College, University of London. Author(s): Jeremy Karl Cockcroft