Equivalent Positions in 3D

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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
  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.

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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
  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,z½
  x,y,z y,x,z½

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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 0.7 0.8 0.1 0.8 0.8
0.1 0.2 0.9 0.8 0.3 0.9 0.8 0.7 0.2 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, 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 x y 0 x y 0 y
x ½ y x ½
_ _ _ _ _ _ _ _
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Table of Equivalent Positions
multiplicity
location of equivalent positions
lowest letter General Position a-j Special
Positions a position with highest symmetry
symmetry at that position 1 none
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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
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Using the Table of Equivalent Positions

• PdS has a tetragonal crystal structure with a b
  6.429? c 6.608? 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
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Using the Table of Equivalent Positions

• PdS has a tetragonal crystal structure with a b
  6.429? c 6.608? 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 6.429? c 6.608? 8S in 8k
0.19 0.32 0.23 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!
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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.
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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.
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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
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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?
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International Tables for Crystallography
P2221 x C2 y C2 z screw axis a ? b ? c
z
y
x
x
21 screw axis - to page 21 screw axis in page
_
z
y
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International Tables for Crystallography
P2221 x C2 y C2 z screw axis a ? b ? c
2-fold axis in page 2-fold axis - to page
_
z
y
x
x
21 screw axis - to page 21 screw axis in page
_
z
y
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International Tables for Crystallography
P2221 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
_
z
y
x
x
21 screw axis - to page 21 screw axis in page
_
z
y
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International Tables for Crystallography
P2221 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
_
z
y
x
x
21 screw axis - to page 21 screw axis in page
_
z
generated by
()
generated by ?
()
y
generated by ?
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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
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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
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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
inversion center ¼ above plane 21 screw axis -
to plane
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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


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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.
_ ,

_ ,
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International Tables for Crystallography
C2/m unit cell centered on C face
note multiplicity of 8, but only four coordinates
given.
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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.
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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 in special positions
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