# $P 4_{2} 22$ tetragonal group choice of origin for symmetry of special projections

I was looking at the description of the tetragonal group $$P 4_{2} 2 2$$ (No 93 in the International Tables of Crystallography) and there is one aspect that I do not understand. Namely, what is the convention and implications of choosing the origin point? For example, for the two fold rotation along the [110] plane, the origin is choosen to be $$(0, 0, \frac{1}{4})$$ and thus we should have the following transformation: $$x \rightarrow y, y \rightarrow x, z \rightarrow -z + \frac{1}{2}.$$

Why the origin is not set at $$(0, 0, 0)$$ and then one would have: $$x \rightarrow y, y \rightarrow x, z \rightarrow -z?$$

• Be aware that the Chemistry SE site policy prefers ( in contrary to some other SE sites ) plain text titles, as formatting affects indexing/searching and displaying titles in Q lists in some browsers. Commented Mar 26, 2021 at 13:01
• @Poutnik Thanks! I will have that in mind next time. Commented Mar 26, 2021 at 13:13
• You can choose the origin once only, not separately for all the different symmetry elements. So in order to keep one operation simple (like having an axis going through the origin), some other operation will be a bit less simple (like having an axis that does not go through the origin).
– Karsten
Commented Mar 29, 2021 at 0:56
• @KarstenTheis I think Piotr referred to the origin on the collection of planes of the special projections [110], not the space group origin. That planar origin depends on which exact plane is being considered, hence why it is (x,x,1/4). A different planar origin for these planes may be chosen just as well as p2mm provides multiple options, although none of these options includes (0,0,0). Commented Mar 29, 2021 at 19:46

As stated in the International Tables of Crystallography vol A, as a rule the origin is locatedat a centre of symmetry or at a point of high site symmetry. Anyway, the choice of such a coordinate system is not mandatory since in principle a crystal structure can be referred to any coordinate system. In particular, all centrosymmetric space groups are described with an inversion centre as origin, whereas for noncentrosymmetric space groups, the origin is at a point of highest site symmetry.

In the space group n°93 the origin is at the intersection of 3 different 2-fold rotation axes (site 2a: 0,0,0), not at 0,0,1/4. I think this you're confused on this point, since the origin is always at 0,0,0 by definition.

From the trasnformation that you have assumed (y,x,-z+1/2), I guess you are referring to the operation number (7), that is a 2-fold rotation along the crystallographic [x,x,1/4] axis (not a plane! rotations are along an axis). As stated above, you can choose another coordinate system with a new origin shifted in respect the standard one, but any case the new origin will have again 0,0,0 coordinates in the new coordinate system. If you shift the origin, the coordinates of all the symmetry operations will be obviously redefined in the new coordinate system, since they are not independently distributed in space.

• Maybe I was not very precise. I did not mean the general origin, but the origin mentioned in the special symmetry of projections point, where for the projection along [110] it is mentioned that the origin is at [x, x, 1/4] Commented Mar 28, 2021 at 20:04
• OK, I was misled by the meaning that you attributed to the term 'origin'. An origin is always a point (with coordinates 0,0,0,), it cannot be an axis (such as [x,x,1/4]). The axis coordinates are defined in respect of the selected origin. Commented Mar 29, 2021 at 7:13
• Look at that entry in the table, you will see that they define multiple origins :) Commented Mar 31, 2021 at 8:23
• I guess that you mean with 'multiple origin' the origins specified for the symmetry of special projections. Anyway the location of the origin of the plane group is always with respect to the unit cell of the space group (whose origin is at 0,0,0) ;-) Commented Apr 1, 2021 at 9:49

The space group $$P4_222$$ has some inherent symmetry on special projections beyond its eight symmetry equivalents. The International Tables for Crystallography, Volume A (DOI: 10.1107/97809553602060000114) notes that planes normal to each of the three projections for tetragonal space groups, namely $$[100]$$, $$[001]$$ and $$[110]$$, having their own symmetry in fact. Here's the relevant section for $$P4_222$$ from the International Tables:

It looks like you are referring to the planes associated with the projection $$[110]$$. The planes normal to [110] have symmetry $$p2mm$$ of their own as well as their own origin at $$x,x,\frac{1}{4}$$, although only one of these planes has its origin at $$0,0,\frac{1}{4}$$, and of course none of these planes has its origin at $$0,0,0$$. Here is how the collection of origins at $$x,x,\frac{1}{4}$$ looks like:

Lets take an example of one such plane, say $$(110)$$ normal to the projection $$[110]$$. Defining that plane is $$x+y=1$$. Here is how this plane looks like:

Applying the relationship $$y=1-x$$ to the $$P4_222$$ symmetry equivalents, the planar symmetry $$p2mm$$ reveals itself in the following: $$(x,\bar x,\tfrac{1}{4}), (\bar x,x,\tfrac{1}{4}), (x,x,\tfrac{3}{4}), (\bar x,\bar x,\tfrac{3}{4}),$$ $$(\bar x,\bar x,\tfrac{3}{4}), (x,x,\tfrac{3}{4}), (\bar x,x,\tfrac{1}{4}), (x,\bar x,\tfrac{1}{4})$$

For this one plane $$(110)$$ the origin at $$(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{4})$$.

The analysis may be generalized for the rest of the infinite collection of planes normal to the projection $$[110]$$, each having its origin at $$x,x,\tfrac{1}{4}$$ as noted above.

As for the origin for the space group $$P4_222$$ it is at $$0,0,0$$.

• That was the explanation I was looking for. Thanks a lot! Commented Mar 28, 2021 at 20:01
• @z1273 I don't understand why you define (x,x,1/4) a collection of origins. This coordinate triplet simply indicates the location and orientation of the symmetry element which corresponds to the symmetry operation. Moreover, I dont' understand how the plane (110) can have the origin at (1/2,1/2,1/4). In your figure the Miller indexing clearly refers to an origin at the intersection of the x,y,z axes, that is at (0,0,0). Commented Mar 30, 2021 at 9:45
• @gryphys I added the relevant section of the ITC-A about the Special Projections for P 4sub2 2 2. Each of the planes normal to the projection [110] has the Rectangular p2mm symmetry (Plane group No.6 of the ITC, found on p.97 of 2006 edition) and has its origin of symmetry where that plane crosses the directional vector (x,x,1/4). Although by definition the plane group p2mm has its origin at (0,0), the relevant 2-fold axis in P 4sub2 2 2 crosses the z axis at ¼, not at 0, hence the plane origin at (x,x,1/4). Commented Mar 30, 2021 at 12:37