Vibrations of a One-Dimensional Host-Guest System
Amelia Carolina Sparavigna
DOI: 10.4236/msa.2011.25041   PDF    HTML     5,723 Downloads   9,004 Views   Citations


A simple model shows how it is possible to create a gap in the vibrational spectrum of a one-dimensional lattice. The proposed model is a host-guest chain having, instead of point-like masses connected by spring, massive cages hosting particles inside. We imagine the cage as a rigid box containing a mass linked by a spring to the box inner wall. The presence of guests creates an energy gap in the dispersion of vibrational frequencies. The gap is about the internal resonance of the mass hidden in the cage. The model is proposed to help understanding the macroscopic behaviour of some phononic materials and the properties of materials with microscopic rattling modes.

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Sparavigna, A. (2011) Vibrations of a One-Dimensional Host-Guest System. Materials Sciences and Applications, 2, 314-318. doi: 10.4236/msa.2011.25041.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] R. M. Walser, “Metamaterials: An Introduction,” In: W. S. Weiglhofer and A. Lakhtakia, Eds. Introduction to Complex Mediums for Optics and Electromagnetics, SPIE Press, Bellingham, 2003, pp. 295-316. doi:10.1117/3.504610.ch13
[2] R. M. Walser, “Electromagnetic Metamaterials,” Proceeding SPIE Complex Mediums II beyond Linear Isotropic Dielectrics, San Diego, Vol. 4467, 2001, pp. 1-15.
[3] V. G. Veselago, “The Electrodynamics of Substance with Simultaneously Negative Values of ? and μ,” Soviet Physics Uspekhi, Vol. 10, No. 4, 1968, pp. 509-514. doi:10.1070/PU1968v010n04ABEH003699
[4] J. B. Pendry, A. J. Holden, D. J. Robbins and W. J. Stewart, “Magnetism from Conductors and Enhanced Nonlinear Phenomena,” IEEE Transactions on Microwave Theory and Techniques, Vol. 47, No. 11, 1999, pp. 2075-2084. doi:10.1109/22.798002
[5] A. Grbic and G. V. Eleftheriades, “Overcoming the Diffraction Limit with a Planar Left-Handed Transmission-Line Lens,” Physical Review Letters, Vol. 92, No. 11, 2004, 117403-1-117403-4.
[6] E. Yablonovitch, “How to be Truly Photonic,” Science, Vol. 289, No. 5479, 2000, pp. 557-559. doi:10.1126/science.289.5479.557
[7] A. Sparavigna, “Phonons in Conventional and Auxetic Honeycomb Lattices,” Physical Review B, Vol. 76, No. 13, 2007, pp. 1343021-1343021-6.
[8] K. E. Evans, M. A. Nkansah, I. J. Hutchinson and S.C. Rogers, “Molecular Network Design,” Nature, Vol. 353, No. 6340, 1991, pp. 124-125. doi:10.1038/353124a0
[9] A. Alderson, “A Triumph of Lateral Thought,” Chemistry & Industry, Vol. 17, May 1999, pp. 384-391.
[10] R. S. Lakes, “Foam Structures with a Negative Poisson’s ratio,” Science, Vol. 235, No. 4792, 1987, pp. 1038-1040. doi:10.1126/science.235.4792.1038
[11] C. P. Chen and R. S. Lakes, “Dynamic Wave Dispersion and Loss Properties of Conventional and Negative Poisson’s Ratio Polymeric Cellular Materials,” Cellular Polymers, Vol. 8, No. 5, 1989, pp. 343-359.
[12] C. W. Smith, J. N. Grima and K. E. Evans, “A Novel Mechanism for Generating Auxetic Behaviour in Reticulated Foams: Missing Rib Model,” Acta Materialia, Vol. 48, No. 17, 2000, pp. 4349-4356. doi:10.1016/S1359-6454(00)00269-X
[13] T. Suzuki and P. K. L. Yu, “Complex Elastic Wave Band Structures in Three-Dimensional Periodic Elastic Media,” Journal of the Mechanics and Physics of Solids, Vol. 46, No. 1, 1998, pp. 115-138. doi:10.1016/S0022-5096(97)00023-9
[14] M. Kafesaki, M. M. Sigalas and N. Garcìa, “Frequency Modulation in the Transmittivity of Wave Guides in Elastic-Wave Band-Gap Materials,” Physical Review Letters, Vol. 85, No. 19, 2000, pp. 4044-4047. doi:10.1103/PhysRevLett.85.4044
[15] Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chang and P. Sheng, “Locally Resonant Sonic Materials,” Science, Vol. 289, No. 5485, 2000, pp. 1734-1736. doi:10.1126/science.289.5485.1734
[16] G. W. Milton, “New Metamaterials with Macroscopic Behavior outside that of Continuum Elastodynamics,” New Journal of Physics, Vol. 9, No. 10, 2007, pp. 359-1- 359-13.
[17] G. W. Milton and J. R. Willis, “On Modi?cations of Newton’s Second Law and Linear Continuum Elastodynamics,” Proceedings of the Royal Society A, Vol. 463, No. 2079, 2007, pp. 855-880.
[18] D. M. Rowe, “Review, Thermoelectric Waste Heat Recovery as a Renewable Energy Source,” International Journal of Innovations in Energy Systems and Power, Vol. 1, No. 1, November 2006, pp. 13-23.
[19] A. Sparavigna, “Influence of Isotope Scattering on the Thermal Conductivity of Diamond,” Physical Review B, Vol. 65, No. 6, 2002, pp. 064305-1-064305-5. doi:10.1103/PhysRevB.65.064305
[20] A. Sparavigna, “Lattice Thermal Conductivity in Cubic Silicon Carbide,” Physical Review B, Vol. 66, No. 17, 2002, pp. 174301-1-174301-5. doi:10.1103/PhysRevB.66.174301
[21] M. Omini and A. Sparavigna, “Role of Grain Boundaries as Phonon Diffraction Gratings in the Theory of Thermal Conductivity,” Physical Review B, Vol. 61, No. 10, 2000, pp. 6677-6688. doi:10.1103/PhysRevB.61.6677
[22] M. Gao and D. M. Rowe, “A serious Limitation to the Phonon Glass Electron Crystal (PGEC) Approach to Improved Thermoelectric Materials,” Journal of Materials Science Letters, Vol. 18, No. 16, 1999, pp. 1305-1306.
[23] G. A. Slack, “Design Concepts for Improved Thermoelectric Materials,” Materials Research Society Symposium Proceedings, Vol. 478, 1997, pp. 47-54.
[24] J. P. Fleurial, T. Caillat and A. Borshchevsky, “Skutterudites: An Update,” Proceedings ICT '97 XVI International Conference on Thermoelectrics, Dresden, 1997, pp. 1-11.
[25] G. S. Nolas, “Semiconductor Clathrates: A PGEC System with Potential for Thermoelectric Applications,” Materials Research Symposium Proceedings, Vol. 545, 1999, pp. 435-442.
[26] J. S. Tse and M. A. White, “Origin of Glassy Crystalline Behavior in the Thermal Properties of Clathrate Hydrates: A Thermal Conductivity Study of Tetrahydrofuran Hydrate,” Journal of Physical Chemistry, Vol. 92, No. 17, 1988, pp. 5006-5011. doi:10.1021/j100328a036
[27] J. S. Tse, V. P. Shpakov, V. V. Murashov and V. R. Belosludov, “The Low Frequency Vibrations in Clathrate Hydrates,” Journal of Chemical Physics, Vol. 107, No. 21, 1997, pp. 9271-9275. doi:10.1063/1.475218
[28] J. Baumert, C. Gutt, V. P. Shpakov, J. S. Tse, M. Krisch, M. Müller, H. Requardt, D. D. Klug, S. Janssen and W. Press, “Lattice Dynamics of Methane and Xenon Hydrate: Observation of Symmetry-Avoided Crossing by Experiment and Theory,” Physical Review B, Vol. 68, No. 17, 2003, pp. 174301-1-174301-4. doi:10.1103/PhysRevB.68.174301
[29] M. Christensen, A. B. Abrahamsen, N. B. Christensen, F. Juranyi, N. H. Andersen, K. Lefmann, J. Andreasson, C. R. H. Bahl and B. B. Iversen, “Avoided Crossing of Rattler Modes in Thermoelectric Materials,” Nature Materials, Vol. 7, No. 10, 2008, pp. 811-815. doi:10.1038/nmat2273

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