Inclined scanning, auxiliary tables

Kopskyacute, V.; Litvin, D. B.

doi:10.1107/97809553602060000652

International
Tables for
Crystallography
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

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International Tables for Crystallography (2006). Vol. E. ch. 5.2, p. 407 | 1 | 2 |

Section 5.2.4.5.2. Inclined scanning, auxiliary tables

V. Kopský^a ^* and D. B. Litvin^b

^a Department of Physics, University of the South Pacific, Suva, Fiji, and Institute of Physics, The Academy of Sciences of the Czech Republic, Na Slovance 2, PO Box 24, 180 40 Prague 8, Czech Republic, and ^bDepartment of Physics, Penn State Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA
Correspondence e-mail: kopsky@fzu.cz

5.2.4.5.2. Inclined scanning, auxiliary tables

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There are no orientation orbits with variable parameter and hence no auxiliary tables to the Laue class $[{\bar 3}]$ ( $[C_{3i}]$ ).

Orientation orbit $[(mn\,{\overline {m+n}}\,0)]$ : This orbit appears in groups of the Laue class [6/m] ( $[C_{6h}]$ ), where it contains the three orientations ( $[mn\,{\overline {m+n}}\,0]$ ), ( $[\,{\overline {m+n}}\,mn0]$ ) and ( $[n\,{\overline {m+n}}\,m0]$ ); further, it appears in groups of the Laue class [6/mmm] ( $[D_{6h}]$ ), where it contains six orientations – to the three orientations we add their images generated by auxiliary axes or planes, which are the orientations ( $[nm\,{\overline {m+n}}\,0]$ ), ( $[\,{\overline {m+n}}\,nm0]$ ) and ( $[m\,{\overline {m+n}}\,n0]$ ). The choice of basis vectors for the scanning group of the first orientation ( $[mn\,{\overline {m+n}}\,0]$ ) is: $[{\bf a}' = {\bf c}]$ , $[{\bf b}' = n{\bf a} - m{\bf b}]$ and $[{\bf d} = p{\bf a} + q{\bf b}]$ ; as always in monoclinic/inclined scanning, the bases for other orientations are obtained by rotations around the principal axis [Laue class [6/m] ( $[C_{6h}]$ )] and by reflections in auxiliary planes [Laue class [6/mmm] ( $[D_{6h}]$ )], so that the scanning groups and the scanning are expressed by identical symbols in their respective bases.

For the particular values [m = 0] , [n = 1] or [m = -1] , [n = 2] , the orientation orbit turns into a special orbit $[(01{\bar 1}0)]$ or $[({\bar 1}2{\bar 1}0)]$ with fixed parameters, respectively, for which the scanning group and hence the scanning is orthorhombic.

Orientation orbits $[(0h{\bar h}l)]$ and $[({\bar h}2h{\bar h}l)]$ : These two orbits include those orientations which contain the secondary or tertiary directions of the hexagonal system. Both orbits exist in the Laue classes $[{\bar 3}m]$ ( $[D_{3d}]$ ) and [6/mmm] ( $[D_{6h}]$ ); the orbit ( $[0h{\bar h}l]$ ) appears in the arithmetic classes [321P] , [3m1P] , $[{\bar 3}m1P]$ and [32R] , [3mR] , $[{\bar 3}mR]$ , where it contains further the two orientations ( $[{\bar h}0hl]$ ) and ( $[h{\bar h}0l]$ ); the orbit ( $[{\bar h}2h{\bar h}l]$ ) appears in the arithmetic classes [312P] , [31mP] and $[{\bar 3}1mP]$ , where it contains the two other orientations ( $[{\overline {hh}}2hl]$ ), $[(2h{\overline {hh}}l]$ ): both orbits appear in all groups of the Laue class [6/mmm] ( $[D_{6h}]$ ) where they contain additional triplets of orientations: $[(0h{\overline {hl}})]$ , $[({\bar h}0h{\bar l})]$ and $[(h{\bar h}0{\bar l})]$ in the first case and $[({\bar h}2h{\overline {hl}})]$ , ( $[{\overline {hh}}2h{\bar l}]$ ) and $[(2h{\overline {hhl}})]$ in the second case.

Transformation of Bravais–Miller indices : hexagonal axes . The orientations $[(0h{\bar h}l)]$ are specified by Bravais–Miller indices with reference to the hexagonal basis (a, b, c) through integers h, l. To find their Miller indices [(mn0)] with reference to auxiliary bases $[({\widehat {\bf a}}, {\widehat {\bf b}}, {\widehat {\bf c}})]$ , we consider a vector $[{\bf w} = {\bf u}+{\bf v} \approx [l({\bf a}+2{\bf b}) -2h{\bf c}]]$ as shown in Fig. 5.2.4.7. This vector is proportional to a vector $[{\bf b}']$ , which is used as a vector of the conventional basis $[({\bf a}', {\bf b}', {\bf d})]$ of the scanning group in both centring types P and R. Vector $[{\bf b}']$ is defined as $[{\bf b}' = n{\widehat {\bf a}}-m{\widehat {\bf b}}]$ , where $[{\widehat {\bf a}} = {\bf a}+2{\bf b}]$ for both the centring types P and R, while $[{\widehat {\bf b}} = {\bf c}]$ for the centring type P and $[{\widehat {\bf b}} = {\bf c}_{r}]$ for the centring type R.

Figure 5.2.4.7 | top | pdf |

Illustration of the transformation of Bravais–Miller indices in a hexagonal basis to Bravais indices in an auxiliary basis.

The proportionality relations therefore read for the centring type P $[l({\bf a}+2{\bf b})-2h{\bf c} \approx n({\bf a}+2{\bf b})-m{\bf c},\eqno(5.2.4.13)]$ from which we express n, m through h, l as follows: $[l \hbox{ odd } \Rightarrow n = l, m = 2h;\quad l \hbox{ even }\Rightarrow n = l/2, m = h.]$

In the case of the centring type R, we have $[\eqalign{{\bf b}' &= n({\bf a}+2{\bf b})-m(-{\bf a}-2{\bf b}+{\bf c})/3 \cr&= (n+m/3)({\bf a}+2{\bf b})-m{\bf c}/3,}]$ so that the proportionality relation reads $[l({\bf a}+2{\bf b})-2h{\bf c} \approx (n+{{m}/{3}})({\bf a}+2{\bf b}) - ({{m}/{3}}){\bf c}.\eqno(5.2.4.14)]$ Comparing the coefficients, we obtain that the pair [(n,m)] must be proportional to the pair [(l-2h,6h)] , from which we express n, m through h, l as follows: $[l \hbox{ odd }\Rightarrow n = l-2h, m = 6h;\,\,\,\, l \hbox{ even }\Rightarrow n = l/2-h, m = 3h.]$

For the orientation orbit $[({\bar h}2h{\bar h}l)]$ , we obtain the proportionality relation by comparing the proportional vectors $[{\bf b}'=]$ $[n{\widehat {\bf a}}-m{\widehat {\bf b}}=]$ $[n{\bf b}-m{\bf c}]$ and $[l{\bf b}-2h{\bf c}]$ , which leads again to the relations $[l \hbox{ odd }\Rightarrow n = l, m = 2h;\quad l \hbox{ even }\Rightarrow n = l/2, m = h.]$

The relations between indices h, l and m, n are, as usual, recorded under each orbit in a row across the table.

The orientation orbits $[(0h{\bar h}l)]$ and $[({\bar h}2h{\bar h}l)]$ turn into the special orbits $[(01{\bar 1}0)]$ and $[({\bar 1}2{\bar 1}0)]$ with fixed parameter for the special values [h = 1] , [l = 0] , and their symmetry increases to orthorhombic for groups of the Laue class [6/mmm] ( $[D_{6h}]$ ). In groups of the Laue class $[{\bar 3}m]$ ( $[D_{3d}]$ ), the symmetry of these orbits remains monoclinic but the scanning changes from monoclinic/inclined to monoclinic/orthogonal.

Rhombohedral axes . Auxiliary tables for the five group types with a rhombohedral lattice are given in a compact manner for all three arithmetic classes. Neither auxiliary nor conventional (in the sense of the convention for scanning groups, see Section 5.2.2.3) bases of scanning groups change. The orientations of the orbit are expressed by Bravais–Miller indices in the hexagonal basis and these are transformed to Miller indices [(mn0)] with reference to the auxiliary basis as shown above. In the rhombohedral basis, we describe orientations of the orbit by Miller indices [(hhl)] . The integers h, l here are considered independently of the same letters in Bravais–Miller indices. To transform them into Miller indices with reference to the auxiliary basis, we take into account that the vector $[{\bf w}]$ from Fig. 5.2.4.7 is proportional to $[l({\bf a}_{r}+{\bf b}_{r})-2h{\bf c}_{r}]$ as well as to $[n({\bf a}_{r}+{\bf b}_{r}+{\bf c}_{r})-m{\bf c}_{r}=]$ $[n({\bf a}_{r}+{\bf b}_{r})+(n-m){\bf c}_{r}]$ . Comparing coefficients at $[({\bf a}_{r}+{\bf b}_{r})]$ and $[{\bf c}_{r}]$ we obtain $[l \hbox{ odd }\Rightarrow n = l, m = 2h+l;\quad l \hbox{ even }\Rightarrow n = l/2, m = h+l/2.]$

The reference table is given as a common table for consideration in hexagonal or rhombohedral axes. It is also common for all five group types with rhombohedral lattice for which this type of orientation orbit occurs.

References

International Tables for Crystallography (2006). Vol. E. ch. 5.2, p. 407