Rest of the Point Groups (continued)

3 Individually: +z +y 1 = rotate 180º A’→B’ and A→B 1 2 = rotate 180º A→B’ and A’ →B

3 = rotate 180º A→A’ and B→B’ 2 +x by doing the 3 x→y→z 2-fold rotations in conjunction: 1 2 3 “No mirrors & no inversion” Label the equipoints: These are 6 more of the 32 point groups to total 23 so far.

+y

Above plane +x Below plane

Out- In-plane plane diads “No mirrors & no inversion” diad “next page” Class19/1 Recall Stereographic projections (Case study of how structure determines properties)

[001] stereographic projection of cubic Microcrystalline Fe one nanoindent crystal sample taken with UNT’s Environmental-SEM (Quanta) with electron backscatter diffraction (EBSD) detector

•Hardness and Elastic Modulus vary from grain to grain which exhibit different crystallographic orientation

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. Class19/2 EBSD Grain/Phase Orientation Mapping

2 Ni-Ti containing intermetallic phases

Ni3Ti pole figures NiTi pole figures

Hexagonal Cubic

Ni3Ti NiTi (D024 structure) (B2 structure/CsCl) {0001}//{110} and also <11-20>//<111> directions Class19/3 Rest of the Point Groups (continued)

and

These are 6 more of the 32 point groups to total =222 =422 =622 =2 =4 2 29 so far. mmm mmm mmm m 3 m 3 m

These are last 3 of the 2 32 point groups. =3 m

Class19/4 Point Group Nomenclature

•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): =[0001] =[2110] =[0110]

•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>. Class19/5 Cubic: 1st <100>, Ex.: Heusler Str. is m3m 2nd <111>, =4/m 3 2/m 3rd <110>

m Ga Mn 3 m Six of them Ni along face diagonals, e.g. [110]

6

Three of them bisect Recall 3 cubic has cube, e.g. Recall 3: •fcc metals, rocksalt 9 mirror (200) [010] Four of them along body also have this point Class19/6 planes: + [001] diagonals, e.g. [111] (space) group Cubic: 1st <100>, Ex.: Zincblende is 43m 2nd <111>, 3rd <110>

Three of them along 3 Four of them along body m Six of them along face 4 [100],[010],[001] diagonals, e.g. [111]: diagonals, e.g. [110]:

Class19/7 The 32 Point Groups

Class19/8 The 32 Point Groups

Point groups: representation of the ways that a macroscopic symmetry element can be self-consistently arranged around a single, immobile geometric point.

Centric (centro- symmetric) have center of symmetry (i)

Cubic →(21) →Laue class/ group (11) 3-D: http://neon.mems.cmu.edu/degraef/pg/pg_gif.html Class19/9 Missing -43m

Class19/10 Descent in Symmetry for the 32 Point Groups

Class19/11 Determining the Crystallographic Point Group of an Object 1.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