Abstract
Line groups are introduced as symmetry groups of the system periodic in a single direction, with periodicity being not restricted to the translational one. Their structure is a weak direct product of the intrinsic symmetry of monomer and the group of generalized translations, arranging these monomers along the direction of periodicity. Continuously many of these groups are classified into 13 infinite families. Only 75 of the line groups are subgroups of the space groups and they are known as rod groups.
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Notes
- 1.
Capital A denotes the translational period of the commensurate helical group, only. The period of the total group is denoted as a.
- 2.
This does not mean that the elements of \({{\boldsymbol P}} \) and \({{\boldsymbol Z}}\) commute, but only that for each P and t there is a choice of P′ and t′ such that \(PZ^t=Z^{t'}P'\).
- 3.
Essentially, we exploited here nonuniqueness of the helical subgroup of the first family line groups, which will be discussed in Sect. 2.2.2.
- 4.
Strictly, number of elements \(R_\phi\) is countable; however, the obtained angles ϕ are dense in the interval \([0,2\pi)\), and for physical applications, when the involved quantities are continuous functions on ϕ, there is no difference between such countable groups and mentioned continuous ones.
- 5.
This type of coset decomposition is applied in crystallography for derivation of the space groups, and it is reflected in the international notation of these groups (see Sect. 2.2.3). The (complicated) mathematical construction used for this purpose is known as extension from the translational subgroup by the isogonal group. It gives [4] classification of the commensurate line groups only.
- 6.
Also, one can find the minimal Q s not less than one; in the above-mentioned cases, it is \(Q_{\mathrm{min}}=nQ'/(n+nQ'-Q')\) (for \(n\leq Q'\)) and \(Q_{\mathrm{min}}=nQ'/(n+Q'[(nQ'-n)/Q'])\) (for \(n\geq Q'\)).
- 7.
The solvability, i.e., commensurability, has been already provided by rationality of Q.
- 8.
Note that r and p, respectively, correspond to the minimal (possibly not pure) rotation and translation involved: r is (convention C0) chosen such that \(C^r_q\) is the minimal rotation mapping initial monomer to the monomer at this minimal height f.
- 9.
For the families 2 and 9 S 2n can be used instead of C n to get the general forms \(\ell_{ts}=(I|a)^tS^s_{2n}\) and \(\ell_{tsp}=(I|a)^tS^s_{2n}{{\sigma}_\text{v}}^p\), respectively (this reduces the number of generators to 2 and 3).
References
M. Damnjanović, M. Vujičić, Phys. Rev. B 25, 6987 (1982) 3
T. Janssen, Crystallographic groups (North-Holland, Amsterdam, 1973)
L. Jansen, M. Boon, Theory of Finite Groups. Applications in Physics (North-Holland, Amsterdam, 1967)
M. Vujičić, I. Božović, F. Herbut, J. Phys. A 10, 1271 (1977)
V. Kopsky, D. Litvin, Subperiodic Groups, International Tables for Crystallography, vol. E (Kluwer, Dordrecht, 2003)
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Damnjanović, M., Milošsević, I. (2010). Line Groups Structure. In: Line Groups in Physics. Lecture Notes in Physics, vol 801. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11172-3_2
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