We present the full in-plane phonon dispersion of graphite obtained from inelastic x-ray scattering, including the optical and acoustic branches, as well as the mid-frequency range between the K and M points in the Brillouin zone, where experimental data have been unavailable so far. The existence of a Kohn anomaly at the K point is further supported. We fit a fifth-nearest neighbour force-constants model to the experimental data, making improved force-constants calculations of the phonon dispersion in both graphite and carbon nanotubes available.

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We study the breathing-like phonon modes of double-walled carbon nanotubes in a simple analytical model by considering the tube walls as coupled oscillators. The force constant of the oscillator coupling is found to be proportional to the inner tube diameter. Thus, only for small-diameter tubes, the shift of the breathing-like phonon mode frequencies (relative to the radial breathing modes of the isolated layers) scales with the tube diameter D. For tubes with larger diameter the in-phase breathing-like phonon mode frequency is inversely proportional to the diameter (like the radial breathing modes of the single wall carbon nanotubes), while the out-of-phase mode approaches asymptotically, as 1/D 2 , the graphite B 2g phonon mode frequency.Due to the remarkable physical properties and potential applications carbon nanotubes (CNTs) have been investigated intensively since their discovery by Iijima [1]. In particular, the vibrational properties of single-wall carbon nanotubes (SWCNTs) [2] have been studied extensively by many research groups [3][4][5]. More recently, experimental studies of double-walled CNTs have appeared [6,7] focussing on the role of the wall-to-wall interaction for the so-called radial breathing mode of the tubes. For isolated single-walled tubes this frequency is generally taken to be a good indicator of the tube diameter D. For double-walled tubes the simple 1/D dependence of the frequency is modified by the presence of the additional wall-to-wall interaction. Popov and Henrard [8] studied the breathing-like phonon modes (BLMs) of double-and multi-walled CNTs within a valence-force field model and a continuum model, showing clearly the effect of this additional interaction. Here we present a very simple, analytically solvable model of two coupled oscillators by which we are able to explain the main features of the BLMs of double-walled carbon nanotubes (DWCNTs) as predicted in Ref. [8]. In particular, we derive simple expressions in the small and large-diameter limit from which one may estimate the diameter of inner and outer tubes from experimental Raman frequencies.A DWCNT consists of two co-axial SWCNTs, the walls of which are assumed to be at a distance d/2 = 3.44 Å and the diameters of which are D 1 = D and D 2 = D + d. The frequency of the radial breathing mode (RBM) of isolated SWCNT scales inversely with the tube diameter (ω = α/D) as was originally