1. Optical and Electromechanical Properties of Carbon Nanotubes via a Two-Field Elastic Description Cristiano Nisoli Vincent H. Crespi Penn State University

2. Elasticity of Continua: Basics elastic displacement field strain tensor Elastic free energy…...in harmonic limit, lower derivatives… …for isotropic media. Linear spectra No Brillouin zone No acoustic modes L. D. Landau, E. M. Lifshitz “Theory of Elasticity” Pergamon Press Oxford (1986)

3. Two-Field Description: In plane Elastic free energy… …direct terms … …cross term. Direct term is isotropic. (accounts for NNNs)

4. Cross Term Coupling between sub-lattice strains Cost for displacing sub lattices apart Internal displacement to strain coupling Similar formalism can be introduce for out-of-plane deformation fields.

5. Uniform Deformations

6. Phonons The equations of motion: Admit analytical solution Predict a Brillouin zone Predict optical modes

7. Carbon Nanotubes. Raman Modes Coupling between pure breathing and optical graphite-like mode** One- field result * Eigenvectors can be obtained: in plane displacement, B- mode *** * ** ***

8. Electro Mechanical Effects Hopping Integrals Softening of the Longitudinal optical modes in metallic CNTs * Band Gap *

9. Phonon Softening Softening of B modes in metallic tubes Softening of speed of sound of Twist mode in metallic tubes Softening in Zig-Zag

10. Self-Trapped Electrons. Correction of Yang formula * * More general solutions, not angularly invariant can be found…

11. 2-Field formalism explains many observed features  Analytical results for complete phonon spectra.  Coupling between Raman modes seen in DFT.  Eigenvectors previously observed in DFT.  Phonon softening in metallic CNTs, observed in DFT.  Framework for electromechanical effects.  Correction terms for accepted formulas.