Hi all
For various reasons, I'm writing some code to generate the general
positions of the space groups. I'm trying to follow the implementation of
Shmueli in their 1984 paper and ITB chapter[1]
My generated positions are fine for all except about 10 settings, and I
think that I might have an issue with how I'm generating the various
matrices which have improper rotations with translation vectors.
I've deduced that a matrix, P, with translation vector, t, multiplied by Q
and u, can be represented as (P,t)*(Q,u) = (PQ, Pu + t), and so I assume
that (P,t)^2 is just (PP, Pt+t).
Following equation 12 from the 1984 paper, I can see that, (for example)
from space group P4(1)32, one of the generating matrices is (4C, 393), and
if I follow the above schema, I can generate (4C, 393), (4C, 393)^2, (4C,
393)^3, and (4C, 393)^4==(1A,000) (the identity).
{(-y+1/4, x+3/4, z+1/4), (-x+1/2, -y, -z+1/2), (y+1/4, -x+1/4, z+3/4),
(x,y,z)} (see the end for matrices)
I can't get the same cycle when I start with an improper rotation with a
non-zero translation vector.
For example:
I3Q999 from Fd-3c. -> Improper rotation of matrix 3Q (z,x,y) with a
translation vector of (3/4,3/4,3/4)
>From Table 4 in the 1984 paper, I know that this is (-z+3/4, -x+3/4,
-y+3/4), so the "improper" rotation only affects the signs of the matrix,
not the translation
But I think I'm missing out something fundamental, as (I3Q999)^3 !=
(P1A000)
eg: {(-z+3/4, -x+3/4, -y+3/4), (y, z, x), (-x+3/4, -y+3/4, -z+3/4)}
I3Q999 = (P,t)
[ 0 0 -1 | 3/4 ]
[ -1 0 0 | 3/4 ]
[ 0 -1 0 | 3/4 ]
(I3Q999)^2 = I3Q999 * I3Q999 = (P,t)*(P,t) = (PP, Pt + t)
[ 0 1 0 | 0 ]
[ 0 0 1 | 0 ]
[ 1 0 0 | 0 ]
(I3Q999)^3 = I3Q999 * I3Q999 * I3Q999 = (P,t)*(PP, Pt + t) = (PPP, PPt + Pt
+ t)
[ -1 0 0 | 3/4 ] [ 1 0 0 | 0 ]
[ 0 -1 0 | 3/4 ] != [ 0 1 0 | 0 ]
[ 0 0 -1 | 3/4 ] [ 0 0 1 | 0 ]
There aren't any examples of improper transformations in the papers, I
don't know enough group theory to get enough out of the rest of the
international tables, and I don't know the notation of Zachariasen [2].
Can anybody help point me in the right direction?
Thanks
Matthew Rowles
[1] Acta Cryst (1984), A40, 567-571 and International Tables, Vol. B,
Chapter 1.4
[2] Theory of X-Ray Diffraction in Crystals (1967), Ch. 2.
P4C393 = (P,t)
[ 0 -1 0 | 1/4 ]
[ 1 0 0 | 3/4 ]
[ 0 0 1 | 1/4 ]
(P4C393)^2 = (P,t)*(P,t) = (PP, Pt + t)
[-1 0 0 | 1/2 ]
[ 0 -1 0 | 0 ]
[ 0 0 1 | 1/2 ]
(P4C393)^3 = (P,t)*(PP, Pt + t) = (PPP, PPt + Pt + t)
[ 0 1 0 | 1/4 ]
[ -1 0 0 | 1/4 ]
[ 0 0 1 | 3/4 ]
(P4C393)^4 = (P,t)*(PPP, PPt + Pt + t) = (PPPP, PPPt + PPt + Pt + t)
[ 1 0 0 | 0 ]
[ 0 1 0 | 0 ]
[ 0 0 1 | 0 ]
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