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
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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:
0.1 0.2 0.3 0.1 0.2 0.70.8 0.1 0.8 0.8 0.1 0.20.9 0.8 0.3 0.9 0.8 0.70.2 0.9 0.8 0.2 0.9 0.2
note: 0.3 reflected
yields -0.3, which is the same as 0.7.
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 0y x ½ y x ½x y 0 x y 0y x ½ y x ½
_ __ _ _ _ _ _
Table of Equivalent Positions
multiplicity
lowest letter = General Positiona-j = Special Positionsa = position with highest symmetry
symmetry at that position1 = none
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.
_
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.
the sulfurs occupy the General Positionthere are 3 crystallographically different
sets of Pd
k
2e2c4j
2e = edges above corners2c = edges along base; ½ up a face4j = four interior positions
Note that all of the above information can be completely described by the following:
P42/m PdSa = b = 6.429Ǻ & c = 6.608Ǻ8S in 8k 0.19 0.32 0.23
2 Pd(1) in 2e2Pd(2) in 2c4Pd(3) in 4j with x = 0.48, y = 0.25
Note: Nothing at lattice points!
International Tables for Crystallography
primitiveinversion
of 230
location of symmetry elements in ab plane; may be other projections---here ac & bc planes.
o = symbol for inversion center.
location of equivalent points in the general position.
yx
guide lines
+ ≡ 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
International Tables for Crystallography
General Position
Special Positions
a c
d f
b ga c
b a
e d
Not shown on right.Why not?
International Tables for CrystallographyP2221x: C2
y: C2
z: screw axisa ≠ b ≠ c
2-fold axis in page
2-fold axis ┴ to page
C2 axis located ¼ up z-axis║ to y-axis.
x
x
y
y
z
z
21 screw axis ┴ to page
21 screw axis in page
_
+
+ + _
generated by ↑
generated by
()
() generated by ↑
International Tables for CrystallographyC2/munit cell centeredon C face
mirror planein this plane
axial glide planein direction of arrowlocated ¼ above plane
inversion center &C2 rotation ┴ to plane
inversion center ¼ above plane &21 screw axis ┴ to plane
• • •
• • •
lattice pts on cornerslattice pt located ½ above this planetwo atoms lie on top of each other in this projection.
_ ,
_ ,
+
International Tables for CrystallographyC2/munit cell centeredon C face
note multiplicity of 8,but only four coordinates given.
add these values to all coordinates; due to centered, rather than primitive unit cell.
Pearson’s Crystallographic Data Index
• Listing of many known crystals, sorted by Space Group.• TN690.P361 1991
a = triclinicP = primitive24 = 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