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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 90ogenerates 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.

_

__ _

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Figure (a) shows equivalent points generated by a 4-fold rotation axis.

Start at x,y,z; a counterclockwise rotation of 90ogenerates 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 42screw axis ( rotation &

translation by c)? Figure (b). The equivalent positions will be: x,y,z; y,x,z+;

x,y,z; y,x,z+

_

__ _

_

_ _ _

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For the Space Group P42/m, in addition to the 42screw axis there is a mirrorplane perpendicular to that axis. So, for each of the four equivalent positions,there is an additional reflected point atz.

So, if we know that there is an atom at x = 0.1, y = 0.2, & z = 0.3, there areseven 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.

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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, usuallybecause the point lies ona symmetry element, it is called a Special Position.

x y 0 x y 0y x y x

x y 0 x y 0y x y x

_ __ _ _ _

_ _

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Table of Equivalent Positions

multiplicity

lowest letter = General Position

a-j = Special Positions

a = position with highest symmetry

symmetry at thatposition

1 = none

location of

equivalent

positions

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Table of Equivalent Positions

multiplicity

lowest letter = General Position

a-j = Special Positions

a = position with highest symmetry

symmetry at thatposition

1 = none

location of

equivalent

positions

e.g. if atoms are occupying

position 2f, this means that

there are two equivalent

atoms at and ,and the symmetry at that

position is 4._

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Using the Table of Equivalent Positions

PdS has a tetragonal crystal structure with a = b = 6.429 & c = 6.608; itsSpace Group is P42/m, and there are 8Pd and 8S per unit cell.

the sulfurs occupy

the General Position

there are 3

crystallographically

different sets of Pd

k

2e2c4j

2e = edges above corners

2c = edges along base; up a face4j = four interior positions

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Using the Table of Equivalent Positions

PdS has a tetragonal crystal structure with a = b = 6.429 & c = 6.608; itsSpace Group is P42/m, and there are 8Pd and 8S per unit cell.

the sulfurs occupy

the General Position

there are 3

crystallographically

different sets of Pd

k

2e2c4j

2e = edges above corners

2c = edges along base; up a face4j = four interior positions

Note that allof the above information can becompletely described by the following:

P42/m PdS

a = b = 6.429 & c = 6.6088S in 8k 0.19 0.32 0.23

2 Pd(1) in 2e

2Pd(2) in 2c4Pd(3) in 4j with x = 0.48, y = 0.25

Note: Nothing at lattice points!

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International Tables for Crystallography

primitive

inversion of 230

location of

symmetry elementsin ab plane; may be

other projections---

here ac & bc planes.

o = symbol for

inversion center.

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primitive

inversion of 230

location of




o = symbol for

inversion center.

location of equivalent

points in the generalposition.y

x

guide lines

+ general point withinthe unit cell.

- means belowthis 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

withinthis unit cell.

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primitive

inversion of 230

location of




o = symbol for

inversion center.

location of equivalent

points in the generalposition.y

x

guide lines

+ general point withinthe unit cell.

- means belowthis 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

withinthis unit cell.

location of the motif

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General Position

Special Positions

a c

d f

b

g

ac

b a

e d

Not shown on right.

Why not?

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International Tables for CrystallographyP2221x: C2y: C2z: screw axisa b c

x

x

y

y

z

z

21screw axis to page

21screw axis in page+ +

_

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2-fold axis in page

2-fold axis to pagex

x

y

y

z

z


21screw axis in page

_

+

+ +

_

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2-fold axis in page

2-fold axis to page

C2axis located up

z-axis to y-axis.

x

x

y

y

z

z



_

+

+ +

_

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2-fold axis in page

2-fold axis to page

C2axis located up

z-axis to y-axis.

x

x

y

y

z

z



_

+

+ +

_

generated by

generated by

)

) generated by

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International Tables for CrystallographyC2/munit cell centeredon C face

mirror planein this planeaxial glide plane

in direction of arrowlocated above plane

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inversion center &C2rotation to plane

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inversion center above plane &21screw axis to plane

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lattice pts on cornerslattice pt located above this plane

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lattice pts on cornerslattice pt located above this plane

two atoms lie on topof each other in this

projection.

_ ,

_ ,

+

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note multiplicity of 8,

but only four coordinates given.

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note multiplicity of 8,

but only four coordinates given.

add these values to allcoordinates; due tocentered, rather thanprimitive unit cell.

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Pearsons Crystallographic Data Index

Listing of many known crystals, sorted by Space Group. TN690.P361 1991

a = triclinic

P = primitive

24 = 24 atoms/unit cell

4 atoms in formula

so, 6 units in cell

unit cell dimensions

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Pearsons Crystallographic Data Index

Listing of many known crystals, sorted by Space Group. TN690.P361 1991

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 inspecial positions

24

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