Harold T. Stokes and Branton J. Campbell, Brigham Young University,
Provo, Utah, USA
Sander van Smaalen, University of Bayreuth, Bayreuth, Germany
JJdW: A. Janner, T. Janssen and P. M. de Wolff, "Bravais Classes for Incommensurate Crystal Phases," Acta Cryst. A39, 658-666 (1983).
Our results agree with the tables of JJdW.
(1) The following two classes in our table are missing from the tables of JJdW:
3.23 P2/m(a1,b1,0)(1/2,0,g2)(0,1/2,g3)
3.24 P2/m(a1,b1,1/2)(1/2,0,g2)(0,1/2,g3)
We inserted these classes into the table immediately after JJdW 3.23.
(2) Four pairs of classes in JJdW's table are equivalent. We list below each pair, H1 and H2, followed by the affine transformation matrix S that takes one into the other, i.e., if h2 is an operator in H2, then the corresponding operator h1 in H2 is given by
h1=S*h2*S-1.
H1 = JJdW 3.7 P2/m(1/2,0,g1)(0,1/2,g2)(0,0,g3)
H2 = JJdW 3.8 P2/m(1/2,0,g1)(0,1/2,g2)(1/2,0,g3)
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 1 0 0
-1 0 0 1 0 1 0
0 0 0 0 0 0 1
H1 = JJdW 3.79 Pmmm(1/2,0,g1)(0,b2,g2)(0,-b2,g2)
H2 = JJdW 3.83 Pmmm(1/2,0,g1)(1/2,b2,g2)(1/2,-b2,g2)
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
-1 0 0 1 0 1 0
-1 0 0 1 1 0 0
0 0 0 0 0 0 1
H1 = JJdW 3.82 Pmmm(1/2,1/2,g1)(0,b2,g2)(0,-b2,g2)
H2 = JJdW 3.85 Pmmm(1/2,1/2,g1)(1/2,b2,g2)(1/2,-b2,g2)
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
-1 -1 0 1 0 1 0
-1 0 0 1 1 0 0
0 0 0 0 0 0 1
H1 = JJdW 3.92 Ammm(1/2,0,g1)(0,b2,g2)(0,-b2,g2)
H2 = JJdW 3.94 Ammm(1/2,0,g1)(1/2,b2,g2)(1/2,-b2,g2)
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
-1 0 0 1 0 1 0
-1 0 0 1 1 0 0
0 0 0 0 0 0 1
Because of these equivalences, we dropped JJdW 3.8, 3.83, 3.85, and 3.94 from our table.
(3) Because adding two new classes and removing four duplicate classes from the table forces a renumbering of the d=3 table, we took this opportunity to further modify the class order so that classes with the same Q vectors in the supercentered setting would be grouped together in the table. In contrast, JJdW tended to group classes according to the appearance of their q vectors in the setting of the basic space group. While Q-centric and q-centric orderings do not conflict for any of the d=1 or d=2 classes, they do occasionally conflict when d=3.
Example 1: for the d=3 classes based on 4/mmm, there are three
different forms of Q in the JJdW table:
(0,0,G1),(0,0,G2),(0,0,G3) in JJdW 3.159-161
(A1,0,0),(0,0,G2) in JJdW 3.162-165,167,169-170,173-174,177-179
(A1,A1,0),(0,0,G2) in JJdW 3.166,168,171-172,175-176,180-181
Example 2: for the d=3 classes based on -3m, there are three different
forms of Q in the JJdW table.
(0,0,G1),(0,0,G2),(0,0,G3) in JJdW 3.188-189
(A1,A1,0),(0,0,G2) in JJdW 3.190,193,196-197
(A1,0,0),(0,0,G2) in JJdW 3.191-192,194-195,198-199
In these cases, we can see that classes with the same form of Q get separated in the JJdW table. We reordered some of the d=3 classes so that classes with the same form of Q are grouped together.
(4) We fixed a typographical error: The 1/4 in JJdW 3.14 should be a 1/2.
(5) We interchanged the q vectors in JJdW 3.78-96, which changed the form of the Q vectors from (0,0,G1),(0,B2,0),(0,0,G2) to (0,B1,0),(0,0,G1),(0,0,G2), so that they would be more consistent with the Q vectors of JJdW 3.41-77, which have the form (0,B1,0),(0,0,G2),(0,0,G3).
(6) We also changed the notation for the following classes, to make it
more clear that each is a superposition of a d=2 class and a d=1
class.
JJdW 3.200 P6/m(a,b,g) became 3.198 P6/m(a1,b1,0)(-a1-b1,a1,0)(0,0,g2)
JJdW 3.202 P6/mmmm(a,0,g) became 3.200 P6/mmm(a1,0,0)(-a1,a1,0)(0,0,g2)
JJdW 3.203 P6/mmmm(a,a,g) became 3.201 P6/mmm(a1,a1,0)(-2a1,a1,0)(0,0,g2)