Commensurate Double-wall Nanotubes: Symmetry and Phonons E. Dobardži´ c, I. Miloševi´ c, B. Nikoli´ c, T. Vukovi´ c and M. Damnjanovi´ c Faculty of Physics, University of Belgrade, P. O. Box 368, Belgrade 11001, Serbia and Montenegro Abstract. For translationally periodic double-wall carbon nanotubes stable configurations and full symmetry groups are determined. Using this, the phonon dispersions and eigenvectors are calculated and assigned by the complete set of conserved quantum numbers. Breathing-like modes, friction modes and acoustic modes are studied. The latter are found to be linear in wave vector k. Recently, double-wall carbon nanotubes (DWCNs) have been generated by coales- cence of C 60 molecules encapsulated into the single-wall CNs [1] and by chemical vapor deposition [2]. Resonant Raman measurements have been used to determine the inner and outer diameter of the DWCNs [3]. Theoretically, stable structures of DWCNs have been calculated [4] and their breathing-like phonon modes have been studied [5]. How- ever, up to the knowledge of the authors, no systematic theoretical study of the DWCNs stable configurations, symmetry and lattice dynamics has been reported yet. In this con- tribution we find the stable configurations and symmetry groups of the commensurate (i.e. translationally periodic) DWCNs and calculate their phonon dispersions and atomic displacements. DWCN W@W is a pair of co-axial single-layer tubes: the inner W= n 1 n 2 and the outer W n 1 n 2 . Such a tube is commensurate (CDWCN) if the ratio aa of the layers’ periods is rational. We study CDWCNs with inner diameters 2.8Å D 50Å. Assuming that the inter-layer distance is 3.44 0.2Å, there are 318 of them (roughly 0.5% of all DWCNs with such diameters). Frequently, in 178 cases, the walls have the same or opposite chiral angle, Tab. 1; such tubes we denote as ˆ n 1 ˆ n 2 n@n and ˆ n 2 ˆ n 1 n@n , respectively, where (ˆ n i n i n n i n , i 1 2). Particularly, here are 60 zig-zag ZZ n 10 n@ n 9 , and 35 armchair, AA n 11 n@ n 5 tubes. Here we sketch the results of the extensive symmetry based research of these CD- WCNs. This includes determination of the stable configurations, full symmetry analy- sis, phonon frequencies and displacements and their assignment by conserved quantum numbers. To this end we use the POLSym code [6], recently applied to SWCNs [7]. Symmetry group of CDWCN is the intersection of the groups of its layers [8]. It depends on the relative position of the walls, which is completely determined by the angle of rotation Φ and length of translation Z (around and along the tube axis, respectively) which are to be performed on x- to match x -axis (here x- and x -axis pass through the centers of carbon hexagons of W and W ; layers z- and z -axis coincide). The stable configuration is found numerically, as the minimum of the W-W interac- tion potential V Φ Z 1 2 ∑ vr α r α (summation over all atoms α of W and α of W ) © 2003 American Institute of Physics 0-7354-0154-3/03/$20.00 Electronic Properties of Novel Materials, edited by H. Kuzmany, J. Fink, M. Mehring, and S. Roth CP685, Molecular Nanostructures: XVII Int'l. Winterschool/Euroconference on 340