InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 334-344
## Section 2.5.3.3.6. Examples of space-group determination M. Tanaka
^{f} |

A simple example of point-group determination has already been given for Si in Section 2.5.3.2.5. In this section, two examples of space-group determination for rutile and samarium selenide are described, in which the point-group determination still accounts for an important part. The examples look to be a little sophisticated but are a good exercise for those who want to acquire experience in CBED space-group determination. The present determination is carried out by assuming the lattice parameters to be known.

*Rutile (TiO _{2})*. The space group of rutile is well known to be

*P*4

_{2}/

*mnm*. The lattice parameters are

*a*=

*b*= 0.459 nm and

*c*= 0.296 nm. Fig. 2.5.3.15(

*a*) shows a CBED pattern taken with the [001] incidence at an accelerating voltage of 80 kV. Since no fine HOLZ lines appear in all the discs, projection diffraction groups (column VI of Table 2.5.3.3) have to be applied to explain this pattern. The projection (proj.) WP shows symmetry 4

*mm*. The projection diffraction group is found to be 4

*mm*1

_{R}from Table 2.5.3.3. Thus, possible diffraction groups are 4

*m*

_{R}

*m*

_{R}, 4

*mm*, 4

_{R}

*mm*

_{R}and 4

*mm*1

_{R}. Another CBED pattern at a second crystal orientation needs to be taken because Fig. 2.5.3.15(

*a*) shows only projection symmetry. Figs. 2.5.3.15(

*b*) and (

*c*) show CBED patterns taken with the [101] incidence at an accelerating voltage of 100 kV. In Fig. 2.5.3.15(

*b*), which is the central part of Fig. 2.5.3.15(

*c*), no HOLZ lines are seen. The symmetries of the projection BP and projection WP are both 2

*mm*. The projection diffraction group of the pattern is 2

*mm*1

_{R}. The WP of Fig. 2.5.3.15

*(c*) is seen to have one mirror symmetry

*m*. The diffraction groups which satisfy symmetry

*m*are

*m*,

*m*1

_{R}and 2

_{R}

*mm*

_{R}. Among these diffraction groups, the diffraction group whose projection becomes 2

*mm*1

_{R}is only diffraction group 2

_{R}

*mm*

_{R}. By consulting Fig. 2.5.3.4, diffraction group 2

_{R}

*mm*

_{R }obtained from Figs. 2.5.3.15(

*b*) and (

*c*) and one diffraction group 4

*mm*1

_{R}among diffraction groups 4

*m*

_{R}

*m*

_{R}, 4

*mm*, 4

_{R}

*mm*

_{R}and 4

*mm*1

_{R}obtained from Fig. 2.5.3.15(

*a*) commonly satisfy point group 4/

*mmm*. Thus, the point group of rutile is determined to be 4/

*mmm*.

Fig. 2.5.3.15(*d*) shows an ordinary diffraction pattern taken with the [001] incidence at an accelerating voltage of 80 kV. With the help of the lattice parameters and the camera length, the indices of the reflections are given as shown in the figure. There are no kinematically forbidden reflections. Thus, the lattice type is determined to be primitive *P*.

Possible space groups which satisfy point group 4/*mmm* and primitive lattice type *P* are those of Nos. 123–138 in Table 2.5.3.9. In Fig. 2.5.3.15(*a*), the dynamical extinction line *A*_{2} is seen in the 100 disc and in the equivalent 010 disc. By consulting Table 2.5.3.9, four space groups *P*4/*mbm*, *P*4/*mnc*, *P*4_{2}/*mbc* and *P*4_{2}/*mnm* are selected. In Fig. 2.5.3.15(*b*), the dynamical extinction line *A*_{2} is seen in the 010 disc but not in the disc. Two space groups *P*4/*mnc* and *P*4_{2}/*mnm* are selected from the four. To distinguish the two space groups, it is found from Table 2.5.3.9 that a CBED pattern taken with the [110] electron incidence should be examined. Fig. 2.5.3.15(*e*) shows a CBED pattern taken with the [110] incidence at 100 kV, where the 001 reflection is exactly excited. The reflections are kinematically allowed for space group *P*4_{2}/*mnm* but kinematically forbidden for *P*4/*mnc*. Since in the case of *P*4/*mnc*, no *Umweganregung* (multiple scattering) paths to the 001 reflection exist in the zeroth-order Laue zone, only the intensities of HOLZ lines, which are caused by *Umweganregung* *via* HOLZ reflections, are expected to appear in the 001 disc. If such *Umweganregung* is not practically excited, the 001 reflection must have no intensity. However, strong intensity produced by two-dimensional interaction is seen in the 001 disc of Fig. 2.5.3.15(*e*). This indicates that the reflection is an allowed reflection. Therefore, the space group of rutile is determined to be *P*4_{2}/*mnm*, which agrees with the space group already known.

*Samarium selenide (Sm _{3}Se_{4})*. Sm

_{3}Se

_{4}has the Th

_{3}P

_{4}structure type with space group at high temperatures. The lattice parameters are

*a*=

*b*=

*c*= 0.8885 nm. It was expected that Sm

_{3}Se

_{4}would transform to an ordered state of electrons with two valences of +2 and +3 around 150 K. The determination of the space group of the material was conducted at 100 K and room temperature. The space groups at both temperatures were determined by CBED to be the same. The following experiments were performed at 100 K.

Fig. 2.5.3.16(*a*) shows a CBED pattern taken with the [111] incidence at 80 kV, which clearly shows the first-order-Laue-zone reflections. The symmetry of the WP is seen to be 3*m* with the help of the enlarged insets. Possible diffraction groups are 3*m*, 3*m*1_{R} and 6_{R}*mm*_{R} from Table 2.5.3.3. Fig. 2.5.3.16(*b*), which is the central part of Fig. 2.5.3.16(*a*), shows projection symmetry 3*m*, indicating that the projection diffraction group is 3*m*1_{R}. Among the three groups 3*m*, 3*m*1_{R} and 6_{R}*mm*_{R}, diffraction groups for which the projection diffraction group is 3*m*1_{R} are 3*m* and 3*m*1_{R}. Possible point groups are found to be 3*m*, and from Fig. 2.5.3.4. Fig. 2.5.3.16(*c*) shows a CBED pattern taken with the [100] incidence at 80 kV. The WP is seen to have symmetry 2*mm*. Allowed diffraction groups are 2*mm*, 2*mm*1_{R} and 4_{R}*mm*_{R}. Fig. 2.5.3.16(*d*), which is the central part of Fig. 2.5.3.16(*c*), shows projection WP symmetry 4*mm*, indicating that the projection diffraction group is 4*mm*1_{R}. The diffraction group among the three groups 2*mm*, 2*mm*1_{R} and 4_{R}*mm*_{R} whose projection diffraction group is 4*mm*1_{R} is 4_{R}*mm*_{R}. Possible point groups are found to be and from Fig. 2.5.3.4. Thus, the point group which satisfies the results obtained at the two crystal orientations is .

Fig. 2.5.3.16(*e*) shows an ordinary diffraction pattern taken with the [100] incidence at 80 kV. With the help of the lattice parameters and the camera length, the indices of the reflections are given as shown in the figure. The reflections 0*kl* (*k* + *l* = 2*n* + 1) are found to be kinematically forbidden. Thus, the lattice type is determined to be *I*.

The space groups having point group and lattice type *I* are and from Table 2.5.3.9. Fig. 2.5.3.16(*d*) shows dynamical extinction lines *A*_{2} in the 033 disc and equivalent discs (also broad lines *A*_{2 }in the 011 discs). Since the former space group does not give any dynamical extinction lines, the space group is determined to be . For confirmation, a CBED pattern which contains the second-order-Laue-zone reflections was taken (Fig. 2.5.3.16*f*). Dynamical extinction lines *A* are seen in the 2,22,22 disc and the equivalent discs. This result also identifies the space group to be not but with the aid of Table 2.5.3.12.