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