2. -x - y ½+z - ( ½+y ½-x ¼+z) -½-x-y -½+x-y ¼
Symmetry operator 2 -Symmetry operator 4
-x y ½+z
- ( ½+y ½-x ¼+z)
-½-x-y -½+x-y ¼

3. -x - y ½+z - ( ½+y ½-x ¼+z) -½-x-y -½+x-y ¼
Symmetry operator 2 -Symmetry operator 4
-x y ½+z
- ( ½+y ½-x ¼+z)
-½-x-y -½+x-y ¼
Plug in u.
u=-½-x-y
0.18=-½-x-y
0.68=-x-y
Plug in v.
v=-½+x-y
0.22=-½+x-y
0.72=x-y
Add two equations and solve for y.
+(0.72= x-y)
1.40=-2y
-0.70=y
Plug y into first equation and solve for x.
0.68=-x-(-0.70)
0.02=x

4. -x - y ½+z - ( ½+y ½-x ¼+z) -½-x-y -½+x-y ¼
Symmetry operator 2 -Symmetry operator 4
-x y ½+z
- ( ½+y ½-x ¼+z)
-½-x-y -½+x-y ¼
Plug in u.
u=-½-x-y
0.18=-½-x-y
0.68=-x-y
Plug in v.
v=-½+x-y
0.22=-½+x-y
0.72=x-y
Add two equations and solve for y.
+(0.72= x-y)
1.40=-2y
-0.70=y
Plug y into first equation and solve for x.
0.68=-x-(-0.70)
0.02=x

5. -x - y ½+z - ( ½+y ½-x ¼+z) -½-x-y -½+x-y ¼
Symmetry operator 2 -Symmetry operator 4
-x y ½+z
- ( ½+y ½-x ¼+z)
-½-x-y -½+x-y ¼
Plug in u.
u=-½-x-y
0.18=-½-x-y
0.68=-x-y
Plug in v.
v=-½+x-y
0.22=-½+x-y
0.72=x-y
Add two equations and solve for y.
+(0.72= x-y)
1.40=-2y
-0.70=y
Plug y into first equation and solve for x.
0.68=-x-(-0.70)
0.02=x

6. -x - y ½+z - ( ½+y ½-x ¼+z) -½-x-y -½+x-y ¼
Symmetry operator 2 -Symmetry operator 4
-x y ½+z
- ( ½+y ½-x ¼+z)
-½-x-y -½+x-y ¼
Plug in u.
u=-½-x-y
0.18=-½-x-y
0.68=-x-y
Plug in v.
v=-½+x-y
0.22=-½+x-y
0.72=x-y
Add two equations and solve for y.
+(0.72= x-y)
1.40=-2y
-0.70=y
Plug y into first equation and solve for x.
0.68=-x-(-0.70)
0.02=x

8. ½-y ½+x ¾+z - ( -y -x ½-z) ½ ½+2x ¼+2z Symmetry operator 3
Plug in v.
v= ½+2x
0.48= ½+2x
-0.02=2x
-0.01=x
Plug in w.
w= ¼+2z
0.24= ¼+2z
-0.01=2z
-0.005=z

9. ½-y ½+x ¾+z - ( -y -x ½-z) ½ ½+2x ¼+2z Symmetry operator 3
Plug in v.
v= ½+2x
0.46= ½+2x
-0.04=2x
-0.02=x
Plug in w.
w= ¼+2z
0.24= ¼+2z
-0.01=2z
-0.005=z

10. ½-y ½+x ¾+z - ( -y -x ½-z) ½ ½+2x ¼+2z Symmetry operator 3
Plug in v.
v= ½+2x
0.46= ½+2x
-0.04=2x
-0.02=x
Plug in w.
w= ¼+2z
0.24= ¼+2z
-0.01=2z
-0.005=z

11. Xstep4=-0.02 ystep4= ?.?? zstep4=-0.005
From step 3
Xstep3= ystep3= zstep3=?.???
From step 4
Xstep4= ystep4= ?.?? zstep4=-0.005
Clearly, Xstep3 does not equal Xstep4 .
Use a Cheshire symmetry operator that transforms xstep3= 0.02 into xstep4=
For example, let’s use:
-x, -y, z
And apply it to all coordinates in step 3.
xstep3-transformed = - (+0.02) = -0.02
ystep3-transformed = - (- 0.70) = +0.70
Now xstep3-transformed = xstep4
And ystep3 has been transformed to a reference frame consistent with x and z from step 4. So we arrive at the following self-consistent x,y,z:
Xstep4=-0.02, ystep3-transformed=0.70, zstep4=-0.005
Or simply, x=-0.02, y=0.70, z=-0.005
The x, y coordinate in step 3 describes one of the heavy atom positions in the unit cell. The x, z coordinate in step 4 describes a symmetry related copy. We can’t combine these coordinates directly. They don’t describe the same atom. Perhaps they even
referred to different origins.
How can we transform x, y from step 3 so it describes
the same atom as x and z in step 4?

12. Xstep4=-0.02 ystep4= ?.?? zstep4=-0.005
From step 3
Xstep3= ystep3= zstep3=?.???
From step 4
Xstep4= ystep4= ?.?? zstep4=-0.005
Clearly, Xstep3 does not equal Xstep4 .
Use a Cheshire symmetry operator that transforms xstep3= 0.02 into xstep4=
For example, let’s use:
-x, -y, z
And apply it to all coordinates in step 3.
xstep3-transformed = - (+0.02) = -0.02
ystep3-transformed = - (- 0.70) = +0.70
Now xstep3-transformed = xstep4
And ystep3 has been transformed to a reference frame consistent with x and z from step 4. So we arrive at the following self-consistent x,y,z:
Xstep4=-0.02, ystep3-transformed=0.70, zstep4=-0.005
Or simply, x=-0.02, y=0.70, z=-0.005
Cheshire Symmetry Operators for space group P43212
X, Y, Z
-X, -Y, Z
-Y, X, 1/4+Z
Y, -X, 1/4+Z
Y, X, -Z
-Y, -X, -Z
X, -Y, 1/4-Z
-X, Y, 1/4-Z
1/2+X, 1/2+Y, Z
1/2-X, 1/2-Y, Z
1/2-Y, 1/2+X, 1/4+Z
1/2+Y, 1/2-X, 1/4+Z
1/2+Y, 1/2+X, -Z
1/2-Y, 1/2-X, -Z
1/2+X, 1/2-Y, 1/4-Z
1/2-X, 1/2+Y, 1/4-Z
X, Y, 1/2+Z
-X, -Y, 1/2+Z
-Y, X, 3/4+Z
Y, -X, 3/4+Z
Y, X, 1/2-Z
-Y, -X, 1/2-Z
X, -Y, 3/4-Z
-X, Y, 3/4-Z
1/2+X, 1/2+Y, 1/2+Z
1/2-X, 1/2-Y, 1/2+Z
1/2-Y, 1/2+X, 3/4+Z
1/2+Y, 1/2-X, 3/4+Z
1/2+Y, 1/2+X, 1/2-Z
1/2-Y, 1/2-X, 1/2-Z
1/2+X, 1/2-Y, 3/4-Z
1/2-X, 1/2+Y, 3/4-Z

13. Xstep4=-0.02 ystep4= ?.?? zstep4=-0.005
From step 3
Xstep3= ystep3= zstep3=?.???
From step 4
Xstep4= ystep4= ?.?? zstep4=-0.005
Clearly, Xstep3 does not equal Xstep4 .
Use a Cheshire symmetry operator that transforms xstep3= 0.02 into xstep4=
For example, let’s use:
-x, -y, z
And apply it to all coordinates in step 3.
xstep3-transformed = - (+0.02) = -0.02
ystep3-transformed = - (- 0.70) = +0.70
Now xstep3-transformed = xstep4
And ystep3 has been transformed to a reference frame consistent with x and z from step 4. So we arrive at the following self-consistent x,y,z:
Xstep4=-0.02, ystep3-transformed=0.70, zstep4=-0.005
Or simply, x=-0.02, y=0.70, z=-0.005
Cheshire Symmetry Operators for space group P43212
X, Y, Z
-X, -Y, Z
-Y, X, 1/4+Z
Y, -X, 1/4+Z
Y, X, -Z
-Y, -X, -Z
X, -Y, 1/4-Z
-X, Y, 1/4-Z
1/2+X, 1/2+Y, Z
1/2-X, 1/2-Y, Z
1/2-Y, 1/2+X, 1/4+Z
1/2+Y, 1/2-X, 1/4+Z
1/2+Y, 1/2+X, -Z
1/2-Y, 1/2-X, -Z
1/2+X, 1/2-Y, 1/4-Z
1/2-X, 1/2+Y, 1/4-Z
X, Y, 1/2+Z
-X, -Y, 1/2+Z
-Y, X, 3/4+Z
Y, -X, 3/4+Z
Y, X, 1/2-Z
-Y, -X, 1/2-Z
X, -Y, 3/4-Z
-X, Y, 3/4-Z
1/2+X, 1/2+Y, 1/2+Z
1/2-X, 1/2-Y, 1/2+Z
1/2-Y, 1/2+X, 3/4+Z
1/2+Y, 1/2-X, 3/4+Z
1/2+Y, 1/2+X, 1/2-Z
1/2-Y, 1/2-X, 1/2-Z
1/2+X, 1/2-Y, 3/4-Z
1/2-X, 1/2+Y, 3/4-Z

14. Xstep4=-0.02 ystep4= ?.?? zstep4=-0.005
From step 3
Xstep3= ystep3= zstep3=?.???
From step 4
Xstep4= ystep4= ?.?? zstep4=-0.005
Clearly, Xstep3 does not equal Xstep4 .
Use a Cheshire symmetry operator that transforms xstep3= 0.02 into xstep4=
For example, let’s use:
-x, -y, z
And apply it to all coordinates in step 3.
xstep3-transformed = - (+0.02) = -0.02
ystep3-transformed = - (- 0.70) = +0.70
Now xstep3-transformed = xstep4
And ystep3 has been transformed to a reference frame consistent with x and z from step 4. So we arrive at the following self-consistent x,y,z:
Xstep4=-0.02, ystep3-transformed=0.70, zstep4=-0.005
Or simply, x=-0.02, y=0.70, z=-0.005
Cheshire Symmetry Operators for space group P43212
X, Y, Z
-X, -Y, Z
-Y, X, 1/4+Z
Y, -X, 1/4+Z
Y, X, -Z
-Y, -X, -Z
X, -Y, 1/4-Z
-X, Y, 1/4-Z
1/2+X, 1/2+Y, Z
1/2-X, 1/2-Y, Z
1/2-Y, 1/2+X, 1/4+Z
1/2+Y, 1/2-X, 1/4+Z
1/2+Y, 1/2+X, -Z
1/2-Y, 1/2-X, -Z
1/2+X, 1/2-Y, 1/4-Z
1/2-X, 1/2+Y, 1/4-Z
X, Y, 1/2+Z
-X, -Y, 1/2+Z
-Y, X, 3/4+Z
Y, -X, 3/4+Z
Y, X, 1/2-Z
-Y, -X, 1/2-Z
X, -Y, 3/4-Z
-X, Y, 3/4-Z
1/2+X, 1/2+Y, 1/2+Z
1/2-X, 1/2-Y, 1/2+Z
1/2-Y, 1/2+X, 3/4+Z
1/2+Y, 1/2-X, 3/4+Z
1/2+Y, 1/2+X, 1/2-Z
1/2-Y, 1/2-X, 1/2-Z
1/2+X, 1/2-Y, 3/4-Z
1/2-X, 1/2+Y, 3/4-Z

15. Use x,y,z to predict the position of a non-Harker Patterson peak
x,y,z vs. –x,y,z ambiguity remains
In other words x=-0.02, y=0.70, z= or
x=+0.02, y=0.70, z= could be correct.
Both satisfy the difference vector equations for Harker sections
Only one is correct. 50/50 chance
Predict the position of a non Harker peak.
Use symop1-symop5
Plug in x,y,z solve for u,v,w.
Plug in –x,y,z solve for u,v,w
I have a non-Harker peak at u=0.28 v=0.28, w=0.0
The position of the non-Harker peak will be predicted by the correct heavy atom coordinate.

16. x y z -( y x -z) x-y -x+y 2z symmetry operator 1 -symmetry operator 5
u v w
First, plug in x=-0.02, y=0.70, z=-0.005
u=x-y = =-0.72
v=-x+y= = 0.72
w=2z=2*(-0.005)=-0.01
The numerical value of these co-ordinates falls outside the section we have drawn. Lets transform this uvw by Patterson symmetry v,-u,-w.
-0.72,0.72,-0.01 becomes
-0.72,-0.72,0.01 then add 1 to u and v
0.28, 0.28, 0.01 This corresponds to the peak shown u=0.28, v=0.28, w=0.01
Thus, x=-0.02, y=0.70, z= is correct. Hurray! We are finished!
In the case that the above test failed, we would change the sign of x.