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, pp. 404-405 | 1 | 2 |

Section 5.2.4.4.2. 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

5.2.4.4.2. Inclined scanning, auxiliary tables

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Orientation orbits [(mn0)] occur in groups of both tetragonal Laue classes [4/m] ( $[C_{4h}]$ ) and [4/mmm] ( $[D_{4h}]$ ). Orientation orbits [(0mn)] occur only in groups of the Laue class [4/mmm] ( $[D_{4h}]$ ).

Orientation orbits [(mn0)] : These orbits contain two orientations, namely ( [mn0] ) and ( $[{\bar n}m0]$ ) in groups of the geometric classes [4] ( $[C_{4}]$ ), $[{\bar 4}]$ ( $[S_{4}]$ ) and [4/m] ( $[C_{4h}]$ ) which belong to the Laue class [4/m] ( $[C_{4h}]$ ), and four orientations, namely ( [mn0] ), ( $[{\bar n}m0]$ ), ( $[{\bar m}n0]$ ) and ( [nm0] ) in groups of the geometric classes [422] ( $[D_{4}]$ ), [4mm] ( $[C_{4v}]$ ), $[{\bar 4}2m]$ ( $[D_{2d}]$ ) and [4/mmm] ( $[D_{4h}]$ ) which belong to the Laue class [4/mmm] ( $[D_{4h}]$ ).

For special values [m = 1] and [n = 0] , the orbit contains only two orientations (100) and (010) which form an orbit with fixed parameters with an orthorhombic scanning group for groups of the Laue class [4/mmm] ( $[D_{4h}]$ ). For groups of the Laue class [4/m] ( $[C_{4h}]$ ) these two orientations represent just one particular case of the orbit ( [mn0] ). Analogously, the orbit with two orientations (110) and ( $[1{\bar 1}0]$ ) for groups of the Laue class [4/mmm] ( $[D_{4h}]$ ) is an orbit with fixed parameters [m = 1] , [n = 1] while for groups of the Laue class [4/m] ( $[C_{4h}]$ ) it is a particular case of the orbits ( [mn0] ).

There are no other special orbits with variable parameter in groups of the Laue class [4/m] ( $[C_{4h}]$ ). Auxiliary bases are defined by one table common for both centring types P and I.

Auxiliary bases for this orbit are also common for both centring types in groups of the Laue class [4/mmm] ( $[D_{4h}]$ ) and they are given in the tables of orientation orbits for both types.

Orientation orbits [(0mn)] : These orbits, consisting of orientations ( [0mn] ), ( $[0{\bar m}n]$ ), ( [m0n] ) and ( $[m0{\bar n}]$ ), appear only in groups of the Laue class [4/mmm] ( $[D_{4h}]$ ). The first two orientations contain the vector a, the other two contain the vector b, scanning groups are monoclinic with unique axes along vectors a and b, respectively, for the first and second pair of orientations; the scanning is inclined because the vectors a and b lie in the respective orientations. To primitive and centred lattices of the scanned groups there correspond primitive and centred lattices of the scanning groups, respectively, which is reflected in the reference tables.

Auxiliary bases for this orbit are common for both centring types in groups of the Laue class [4/mmm] ( $[D_{4h}]$ ) and they are given in tables of orientation orbits for both types.

For special values of parameters, the orbit coincides either with the orbit [(100)] , [(010)] or with the orbit [(110)] , $[(1{\bar 1}0)]$ .

Orientation orbits [(hhl)] : These orbits, consisting of orientations ( [hhl] ), ( $[{\overline {hh}}l]$ ), ( $[h{\bar h}l]$ ) and ( $[{\bar h}hl]$ ), appear again only in groups of the Laue class [4/mmm] ( $[D_{4h}]$ ). The first two orientations contain the vector $[({\bf a}-{\bf b})]$ , the other two contain the vector $[({\bf a} +{\bf b})]$ , scanning groups are monoclinic with unique axes along these vectors $[({\bf a}-{\bf b})]$ and $[({\bf a} + {\bf b})]$ , respectively, for the first and second pair of orientations; the scanning is again inclined because the vectors $[({\bf a}-{\bf b})]$ and $[({\bf a}+{\bf b})]$ lie in the respective orientations.

The auxiliary bases for the monoclinic scanning groups in the case of a primitive (P) tetragonal lattice are chosen as $[{\widehat {\bf a}} = {\bf a} + {\bf b}, \; \; {\widehat {\bf b}} = {\bf c} \; \; {\rm and} \; \; {\widehat {\bf c}} = {\bf a} - {\bf b} \eqno(5.2.4.4)]$ for the first pair of orientations and as $[{\widehat {\bf a}} = {\bf b} - {\bf a}, \; \; {\widehat {\bf b}} = {\bf c} \; \; {\rm and} \; \; {\widehat {\bf c}} = {\bf a} + {\bf b} \eqno(5.2.4.5)]$ for the second pair of orientations.

The auxiliary bases for the monoclinic scanning groups in the case of an I-centred tetragonal lattice are chosen as $[{\widehat {\bf a}} = ({\bf a}+{\bf b}+{\bf c})/2, \; \; {\widehat {\bf b}} = {\bf c}\; \; {\rm and} \; \; {\widehat {\bf c}} = ({\bf a} - {\bf b}) \eqno(5.2.4.6)]$ for the first pair of orientations and as $[{\widehat {\bf a}} = ({\bf b}-{\bf a}+{\bf c})/2, \; \; {\widehat {\bf b}} = {\bf c} \; \; {\rm and} \; \; {\widehat {\bf c}} = ({\bf a} + {\bf b}) \eqno(5.2.4.7)]$ for the second pair of orientations.

A vector parallel with planes of orientation [(hhl)] and orthogonal to $[{\bf a}-{\bf b}]$ is a multiple of $[2h{\bf c}-l({\bf a}+{\bf b}). \eqno(5.2.4.8)]$ In terms of Miller indices [(mn0)] with reference to the first auxiliary basis for a P-centred lattice, such a vector is a multiple of $[m{\bf c}-n({\bf a}+{\bf b}) \eqno(5.2.4.9)]$ and in terms of Miller indices [(mn0)] with reference to the first auxiliary basis for an I-centred lattice, it is a multiple of $[(2m-n){\bf c}-n({\bf a}+{\bf b}). \eqno(5.2.4.10)]$ Therefore, for a P-centred lattice, the pair of numbers [(m,n)] must be proportional to the pair [(2h,l)] . Since Miller indices must be relatively prime, we get [n = l] , [m = 2h] if l is odd and [n = l/2] , [m = h] if l is even.

For an I-centred lattice, the pair of numbers [(2m-n,n)] must be proportional to the pair [(2h,l)] and hence the pair [(2m,n)] must be proportional to the pair [(2h+l,l)] . If l is odd, then [2h+l] is also odd and we put [m = 2h+l] , so that [n = 2l] . If l is even, we put [n = l] and [m = h+l/2] .

These relations are printed in the last rows across the tables of orientation orbits within the block for orbit [(hhl)] .

References

International Tables for Crystallography (2006). Vol. E. ch. 5.2, pp. 404-405