1. Equivalent Positions in 3-D
Figure (a) shows equivalent points generated by a 4-fold rotation axis.
Start at x,y,z; a counterclockwise rotation of 90o generates a point with coordinates y,x,z; then x,y,z; and finally to y,x,z.
Because the points are related via a symmetry operation, they are equivalent.
What if the operator is a 42 screw axis (¼ rotation & translation by ½c)? Figure (b).
The equivalent positions will be: x,y,z; y,x,z+½; x,y,z; y,x,z+½ _
_ _ _ _
_ _ _

2. Equivalent Positions in 3-D
For the Space Group P42/m, in addition to the 42 screw axis there is a mirror plane perpendicular to that axis. So, for each of the four equivalent positions, there is an additional reflected point at –z.
So, if we know that there is an atom at x = 0.1, y = 0.2, & z = 0.3, there are seven additional (equivalent) atoms at:
note: 0.3 reflected yields -0.3, which is the same as 0.7.

3. Special Positions
Look at a particular point in P42/m, x,y,0, which lies on the mirror plane.
The eight equivalent points (using the general formula) would be:
obviously the four points on the right are identical to the four on the left.
so, there are only four distinct points.
Whenever the number of general equivalent points is reduced, usually because the point lies on a symmetry element, it is called a Special Position.
x y 0 x y 0
y x ½ y x ½
_ _
_ _ _ _
_ _

4. Table of Equivalent Positions
multiplicity
location of equivalent positions
e.g. if atoms are occupying position 2f, this means that there are two equivalent atoms at ½ ½ ¼ and ½ ½ ¾, and the symmetry at that position is 4.
lowest letter = General Position
a-j = Special Positions
a = position with highest symmetry
symmetry at that position
1 = none _

5. Using the Table of Equivalent Positions
PdS has a tetragonal crystal structure with a = b = 6.429Ǻ & c = 6.608Ǻ; its Space Group is P42/m, and there are 8Pd and 8S per unit cell. k 2e 2c 4j
the sulfurs occupy the General Position
there are 3 crystallographically different sets of Pd
2e = edges above corners
2c = edges along base; ½ up a face
4j = four interior positions
P42/m PdS
a = b = 6.429Ǻ & c = 6.608Ǻ
8S in 8k
2 Pd(1) in 2e
2Pd(2) in 2c
4Pd(3) in 4j with x = 0.48, y = 0.25
Note that all of the above information can be completely described by the following:
Note: Nothing at lattice points!

6. International Tables for Crystallography
primitive
inversion
of 230
o = symbol for inversion center.
location of symmetry elements in ab plane; may be other projections---here ac & bc planes.
location of equivalent points in the general position.
guide lines y x
+ ≡ general point within the unit cell.
means below this unit cell.*
, means the handedness has changed.
*if + = 0.1, 0.1, 0.1, then
= -0.1, -0.1, -0.1, or 0.9, 0.9, 0.9
within this unit cell.
location of the motif

7. International Tables for Crystallography
a c
d f
b a
e d
General Position
Special Positions b g a c
Not shown on right.
Why not?

8. International Tables for Crystallography
x: C2
y: C2
z: screw axis
a ≠ b ≠ c
C2 axis located ¼ up z-axis║ to y-axis.
2-fold axis in page
2-fold axis ┴ to page _ + y z x x
21 screw axis ┴ to page
21 screw axis in page _ z
generated by ()
generated by ↑ () y
generated by ↑

9. International Tables for Crystallography
C2/m
unit cell centered
on C face
mirror plane
in this plane
axial glide plane
in direction of arrow
located ¼ above plane
inversion center &
C2 rotation ┴ to plane
lattice pts on corners
lattice pt located ½ above this plane
inversion center
¼ above plane &
21 screw axis ┴ to plane
• • •
• • •
two atoms lie on top of each other in this projection.
_ , +
_ ,

10. International Tables for Crystallography
C2/m
unit cell centered
on C face
add these values to all coordinates; due to centered, rather than primitive unit cell.
note multiplicity of 8,
but only four coordinates given.

11. Pearson’s Crystallographic Data Index
Listing of many known crystals, sorted by Space Group.
TN690.P
a = triclinic
P = primitive
24 = 24 atoms/unit cell
4 atoms in formula
so, 6 units in cell
unit cell dimensions
most atoms in general
position, 2(i): x,y,z; x, y, z.
_ _ _
although two in special positions 24