UCSD NANO106 - 06 - Plane and Space Groups

Plane Groups and Spacegroups
Shyue Ping Ong
Department of NanoEngineering
University of California, San Diego

Readings
¡Chapter 10 of Structure of Materials
NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 6

Plane groups
NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 6

2D Crystallographic Point Groups

Principles of Derivation
¡ Point group + translations
¡ Every point group belongs to a crystal system. We combine all point groups
compatible with a crystal system with the corresponding 2D Bravais nets.
¡ Next, we try replacing mirror planes in the point group with glides (an
additional operation in 2D) and see if it generates new lattices.

¡We will now derive a few plane groups to
demonstrate the application of the principles. We
will focus on the oblique and rectangular nets to
demonstrate all principles, and one tetragonal net
for a more complex example. Not all plane groups
will be derived in lectures, but you are expected
to be able to derive all plane groups using the
same principles if given a net and a point group.

Derivation of Plane Group p2
¡Oblique net + 2 (C2)
Derivation
1. Put point group 2 (C2) at each lattice
point.
2. Consider a general motif.
3. 2 (C2) generates a motif rotated 180
deg about the rotation axis.
4. The two translation vectors generates
motifs at all lattice nodes.
5. By inspection, we see that an additional
2-fold rotation is implied from the
rotation and translation. (This is a
general principle that can be derived
mathematically).
6. Similarly, new rotation axes are
generated for other lattice translations.

Derivation of plane group pm
¡ Presence of additional mirror
plane implied by the presence of
mirror + translation

Derivation of plane group pg –
Replacing mirrors with glide planes
¡ Instead of the mirror in pm, let us now
try to add an axial glide plane (over
a/2) to the rectangular net.
¡ Again, we find that there is an
additional glide plane implied by the
combination of g with translation.

Derivation of plane group cm – Adding mirror (m) to
a centered rectangular net (oc)
¡ Let us now try to add a mirror to the
centered rectangular net
¡ This may seem similar to the pm plane
group, but note that there is an
additional lattice node in the center of
the net.
¡ Are there additional symmetries
implied by existing operations?
¡ Yes! There are additional glide planes!

What happens when we try to replace the
mirrors in cm with glide planes?
¡ Let’s go through the
exercise again.
¡ Does this look like a new
net?
¡ No! It’s simply cm after
you redefine the net basis
vectors!

p2mm – Adding 2mm to op
2mm

Adding m + g to op – p2mg
mg

Adding gg to op – p2gg
gg

A much more complicated example –
p4mm
4mm

The 17 Plane Groups

Space groups
NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 6

Space Group
¡ 32 crystallographic point groups + 14 3D Bravais lattices
¡ 1891 - First enumerated by Fedorov
¡ 2 omissions (I43d and Fdd2) and one duplication (Fmm2)
¡ 1891 - Independently enumerated by Schönflies
¡ 4 omissions and one duplication (P421m)
¡ 1892 - Correct list of 230 space groups was found by
Fedorov and Schönflies.
¡ Moral of the story: Enumerating the space groups
correctly is hard!

Approach
¡We are obviously not going to go through the
exercise of enumerating all 230 space groups.
Nor are you expected to memorize all the groups
and their symmetry operations.
¡The important thing is to demonstrate the
principles of derivation. After that, we will look at
examples in the International Tables of
Crystallography and learn how to find the
information when you actually need them.

Simple Example: mm2 + o lattices
¡Point Group mm2:
¡ Four operations (E, 2, m1, m2)
¡ Compatible with orthorhombic Bravais lattices (oP , oC,
oI, oF)

Pmm2 (oP + mm2)
¡Mirrors at t/2
are implied by
parallel
mirrors.

Cmm2 (oC + mm2)
Source: http://img.chem.ucl.ac.uk/sgp/

Imm2 (oI + mm2)

Fmm2 (oF + mm2)

Symmorphic Space Groups
¡ Symmorphic space group – Space group that does not contain screw
axes or glide planes in its symbol.
¡ Note that implied screw axes are fine, e.g., the Imm2 and Fmm2 that
we have just seen are also symmorphic.
¡ By combining point groups with Bravais lattices, we can obtain 61 of
the 73 symmorphic space groups. The additional symmorphic space
groups are obtained by:
¡ Considering additional orientations between the point groups and lattice
points, e.g., P¯42m (D12
d) and P¯4m2 (D52
d).
¡ For orthorhomic cells, we can position two-fold axes perpendicular or along
C plane.
¡ Trigonal point groups can be combined with rhombohedral lattice (rP) or
hexagonal primitive (hP) lattice, and in different orientations.

Non-symmorphic space groups (157)
¡ Obtained by replacing one or more of the symmetry
elements in the symmorphic point groups with screw axes
or glide planes.

Space Group Generators
¡Although some of the space groups have a high
order, the minimal number of generators required
to generate all 230 space groups is surprisingly
few.
¡ 14 fundamental symmetry matrices
¡ 11 translation magnitudes
¡ – highest symmetry space group with order
192, requires only 6 symmetry matrices
Fm3m

Space Groups Frequencies
http://www.bit.ly/sg_stats

Most common space groups

Chemistry Comparison
Oxides Sulfides
Any observations about the differences between the two?

Crystallographic Orbit
¡ The crystallographic orbit of a symmetry group is the set of all points
that are symmetrically equivalent to a point.
¡ For a general position with coordinates (x, y, z), the # of points in an
orbit = Order of group
¡ For an higher symmetry position, the # of points in the orbit < Order of
group

Simple example: mmm
¡ Using the generator matrices, we can now generate the 8
symmetry operations in this point group.
1 0 0
0 1 0
0 0 −1
"
#
$
$
$
%
&
'
'
'
= m1
1 0 0
0 −1 0
0 0 1
"
#
$
$
$
%
&
'
'
'
= m2
−1 0 0
0 1 0
0 0 1
"
#
$
$
$
%
&
'
'
'
= m3
−1 0 0
0 1 0
0 0 −1
"
#
$
$
$
%
&
'
'
'
= m1 ⋅m3 = 2y
−1 0 0
0 −1 0
0 0 1
"
#
$
$
$
%
&
'
'
'
= m1 ⋅m2 = 2z
1 0 0
0 −1 0
0 0 −1
"
#
$
$
$
%
&
'
'
'
= m2 ⋅m3 = 2x
−1 0 0
0 −1 0
0 0 −1
"
#
$
$
$
%
&
'
'
'
= m1 ⋅m2 ⋅m3 = i
1 0 0
0 1 0
0 0 1
"
#
$
$
$
%
&
'
'
'
= m1 ⋅m2 ⋅m3 ⋅m1 ⋅m2 ⋅m3 = i⋅i = E
E i m1 m2 m3 2x 2y 2z
E E i m1 m2 m3 2x 2y 2z
i i E 2z 2y 2x m3 m2 m1
m1 m1 2z E 2x 2y m2 m3 i
m2 m2 2y 2x E 2z m1 i m2
m3 m3 2x 2y 2z E i m1 m3
2x 2x m3 m2 m1 i E 2z 2y
2y 2y m2 m3 i m1 2z E 2x
2z 2z m1 i m2 m3 2y 2x E
http://nbviewer.ipython.org/github/materialsvirtuallab/nano106/blob/master/lectures/lecture_4_point_
group_symmetry/Symmetry%20Computations%20on%20mmm%20%28D_2h%29%20Point%20Gro
up.ipynb

Simple example: mmm, contd
¡Orbit of General position (x, y, z)
¡ [-x y z] [x -y z] [x y -z] [x y z] [-x -y z] [-x y -z] [x -y -z] [-x -
y –z]
¡Ipython notebook for Oh point group
¡ http://nbviewer.ipython.org/github/materialsvirtuallab/nan
o106/tree/master/lectures/lecture_4_point_group_symm
etry/

Special Positions
¡ For general positions (x,y,z), the orbit always has the same number of
points as the order of the point group.
¡ But for positions that lie on a particular symmetry element, the orbit
will contain fewer number of points than the order of the point group.
¡ Continuing the mmm point group example, what happens when we
consider a point that lie on the 2-fold rotation axis parallel to the c-
direction, i.e., (0, 0, z)?
¡ Continuing the analysis, we find that there are only two unique points
(0, 0, z) and (0, 0, -z) [several operations map this point to the same
point].
¡ Such positions are known as special positions, and they have higher
symmetry that of the general position with point group 1 (C1).

The International Tables for
Crystallography
¡Please refer to your handouts.
¡Online version of IUCR
¡ http://it.iucr.org/Ab/contents/
¡A more user-friendly version
¡ http://img.chem.ucl.ac.uk/sgp/large/sgp.htm

Example 1
¡ One of the high-temperature polymorphs of a compound containing
Ba, Ti and O has the spacegroup Amm2 (38). Please answer the
following questions:
¡ What is the crystal system and point group associated with this space
group?
¡ Describe the symmetry operations in this space group (you need to state
the symmetry operation and the position of the axes, particularly if it is not
at the origin, e.g., X-fold rotation axis passing through (x,y,z) parallel to b-
direction.).
¡ Write down all the 4x4 matrices for the symmetry operations for the (0, 0, 0)
set for this space group.
¡ The table below provides partially completed information on the location of
all sites in the structure. Fill in all missing fields, shaded in light grey.
¡ Determine the formula of the compound and calculate how many atoms are
present in the unit cell.

Example 1 contd.
Species
Wyckoff
Symbol
x y z
Ba
2+
0 0 0
Ti
4+
2b 0 0.51
O
2-
2a 0.49
O
2-
0.5 0.253 0.237

Example 2
¡The crystal structure of the
wurtzite form of ZnS is
shown below. It has
spacegroup P63mc. The
fractional coordinates of one
of the Zn and S atoms are
(1/3, 2/3, 0) and (1/3, 2/3,
0.3748) respectively.
Determine the
crystallographic orbit for Zn
and S. What are the Wycoff
symbols of Zn and S?

UCSD NANO106 - 06 - Plane and Space Groups

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UCSD NANO106 - 06 - Plane and Space Groups