Plane Groups and SpacegroupsShyue Ping OngDepartment of NanoEngineeringUniversity of California, San Diego

Readings¡Chapter 10 of Structure of Materials

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

Plane groups

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

2D Crystallographic Point Groups

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

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.

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

¡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.

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

Derivation of Plane Group p2¡Oblique net + 2 (C2)

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

Derivation1. 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

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

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.

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

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!

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

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!

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

p2mm – Adding 2mm to op

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

2mm

Adding m + g to op – p2mg

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

mg

Adding gg to op – p2gg

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

gg

A much more complicated example –p4mm

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

4mm

The 17 Plane Groups

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

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!

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

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.

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

Simple Example: mm2 + o lattices

¡Point Group mm2:¡ Four operations (E, 2, m1, m2)¡ Compatible with orthorhombic Bravais lattices (oP , oC,

oI, oF)

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

Pmm2 (oP + mm2)

¡Mirrors at t/2 are implied by parallel mirrors.

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

Cmm2 (oC + mm2)

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

Source: http://img.chem.ucl.ac.uk/sgp/

Imm2 (oI + mm2)

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

Source: http://img.chem.ucl.ac.uk/sgp/

Fmm2 (oF + mm2)

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

Source: http://img.chem.ucl.ac.uk/sgp/

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 (D12d) and P¯4m2 (D52d).¡ 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.

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

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.

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

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

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

Fm3m

Space Groups Frequencies

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

http://www.bit.ly/sg_stats

Most common space groups

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

Chemistry Comparison

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

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

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

Simple example: mmm¡ Using the generator matrices, we can now generate the 8

symmetry operations in this point group.

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

1 0 00 1 00 0 −1

"

#

$$$

%

&

'''=m1

1 0 00 −1 00 0 1

"

#

$$$

%

&

'''=m2

−1 0 00 1 00 0 1

"

#

$$$

%

&

'''=m3

−1 0 00 1 00 0 −1

"

#

$$$

%

&

'''=m1 ⋅m3 = 2y

−1 0 00 −1 00 0 1

"

#

$$$

%

&

'''=m1 ⋅m2 = 2z

1 0 00 −1 00 0 −1

"

#

$$$

%

&

'''=m2 ⋅m3 = 2x

−1 0 00 −1 00 0 −1

"

#

$$$

%

&

'''=m1 ⋅m2 ⋅m3 = i

1 0 00 1 00 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 im2 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%20Group.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_symmetry/

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

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).

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

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

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

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.

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

Example 1 contd.

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

Species Wyckoff Symbol x y z

Ba2+ 0 0 0

Ti4+ 2b 0 0.51

O2- 2a 0.49

O2- 0.5 0.253 0.237

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

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 Wycoffsymbols of Zn and S?

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

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