Rest of the Point Groups (continued)

Class19/1

These are 6 more of the 32 point groups to total 23 so far.

3

2

1+y+z

+x

1

= rotate 180º A→A’ and B→B’

2

= rotate 180º A’→B’ and A→B

3

= rotate 180º A→B’ and A’ →B

+y

+x

x→y→z

1 2 3

by doing the 3

2-fold rotations

in conjunction:

Label the

equipoints:

In-plane

diads

Out-

plane

diad

“No mirrors & no inversion”

Individually:

“next

page”

“No mirrors & no inversion”

Above planeBelow plane

Recall Stereographic projections (Case study of how structure determines properties)

Figure above. Inverse Pole figure

for all Berkovich nanoindentations

(technique to measure hardness,

H, and elastic modulus, E). Black

spots on the stereographic triangle

represent various indentation H and

E values for each grain.

•Hardness and Elastic Modulus

vary from grain to grain which

exhibit different crystallographic

orientation

Microcrystalline Fe

sample taken with UNT’s

Environmental-SEM (Quanta)

with electron backscatter

diffraction (EBSD) detector

one nanoindent

Class19/2

[001] stereographic

projection of cubic

crystal

HexagonalNi3Ti

(D024 structure)

CubicNiTi

(B2 structure/CsCl)

EBSD Grain/Phase Orientation Mapping

Class19/3

2 Ni-Ti containing intermetallic phases

{0001}//{110} and also <11-20>//<111> directions

Ni3Ti pole figures NiTi pole figures

Rest of the Point Groups (continued)

These are 6 more of the

32 point groups to total

29 so far.

These are last 3 of the

32 point groups.

Class19/4

and

=222mmm

=422mmm

=622mmm

=2m

=4m

2m33

2m=3

Point Group Nomenclature

Class19/5

•m is used in preference to 2

•a mirror plane normal to a symmetry axis is indicated by X/m where X is 2, 3, 4 or 6

•where there are two distinct sets of mirrors parallel to a symmetry axes mm is used

•up to three symbols or combination

of symbols can be used to describe a

point group, e.g. 3m, 23, 432 and

6/mmm are all point groups.

•The order is extremely important,

follow (and know) this table (from DeGraef):

•Orthorhombic system: the 3 symbols refer to 3 mutually perpendicular x,y, and z axes, in that order

•All tetragonal groups have a 4 or 4 rotation axis in the z-direction and this is listed first. The

second component refers to symmetry around the mutually perpendicular x and y axes, and the

third component refers to the directions in the x-y plane that bisect the x and y axes.

•Trigonal (3 or 3) or hexagonal (6 or 6) rotation axes first, the second symbol describes the

symmetry around the equivalent directions (either 60° or 120° apart) in the plane perpendicular to

the 3, 3, 6 or 6 axis. A third component in hexagonal system refers to directions that bisect the

angles between the axes specified by the second symbol.

•If there is a 3 in the second position it is cubic (4 rotation triads along <111>, the body diagonals),

first symbol refers to the cube axis <100> and the third to face diagonals <110>.

=[2110]=[0001] =[0110]

3

6

Cubic: 1st <100>,

2nd <111>,

3rd <110>

Recall

cubic has

9 mirror

planes:

m

Three of

them bisect

cube, e.g.

(200) [010]

+ [001]

3

Four of them along body

diagonals, e.g. [111]

Recall 3:

Six of them

along face

diagonals,

e.g. [110]

•fcc metals, rocksalt

also have this point

(space) groupClass19/6

m

Ex.: Heusler Str. is m3m

=4/m 3 2/m

GaMn

Ni

4 3

Cubic: 1st <100>,

2nd <111>,

3rd <110>

mThree of them along

[100],[010],[001]

Four of them along body

diagonals, e.g. [111]:Six of them along face

diagonals, e.g. [110]:

Ex.: Zincblende is 43m

Class19/7

The 32 Point Groups

Class19/8

3-D:

http://neon.mems.cmu.edu/degraef/pg/pg_gif.html

Point groups: representation of the ways that a macroscopic

symmetry element can be self-consistently arranged around a single,

immobile geometric point.

The 32 Point Groups

Class19/9

Cubic→Laue class/

group (11)

Centric (centro-

symmetric) have

center of symmetry (i)

→(21)

Class19/10

Missing -43m

Descent in Symmetry

for the 32 Point Groups

Class19/11

Determining the Crystallographic

Point Group of an Object1.First, check for the highest or lowest symmetry groups.

2. If no translational symmetry axes exist, the group is 1, 1, or m.

3. On the other hand, if the 4 triad axes characteristic of cubic point groups are present,

then the object has the symmetry of one of the cubic point groups.

4. To determine which of the 5 cubic groups it is, systematically search for the additional

elements.

5. If it is neither cubic nor triclinic, then search for axis of highest rotational symmetry.

6. If it is a rotation hexad, it is hexagonal.

7. If it has a single triad or tetrad axis, then it is trigonal or tetragonal, respectively.

Remember the simultaneous occurrence of these two elements would imply a

cubic group.

8. If the object’s highest symmetry element is a diad and there are two mutually

perpendicular elements, it is orthorhombic.

9. If not, it is monoclinic.

10. In each case, once the crystal system has been identified, the presence or absence

of perpendicular diads and mirrors tells you which group it is.

Note: Point group symmetry can have a profound affect on physical properties of

crystals, e.g. the absence of an inversion center is an essential requirement for

piezoelectricity: crystals which polarize simultaneously below a critical temperature are

called pyroelectric, if polarization is reversible the crystal is ferroelectric. Class19/12