Hamiltonian of Acoustic Phonons in Inhomogeneous Solids

Affiliation(s)

Department of Physics, Faculty of Education, Catholic University, Ruzomberok, Slovak Republic.

Department of Physics, Faculty of Education, Catholic University, Ruzomberok, Slovak Republic.

ABSTRACT

Theoretical solid-state physicists formulate their models usually in the form of a Hamiltonian. In quantum mechanics, the Hamilton operator (Hamiltonian) is of fundamental importance in most formulations of quantum theory. Mentioned operator corresponds to the total energy of the system and its spectrum determines the set of possible outcomes when one measures the total energy. Interpretation of results obtained by the applying of models based on the Hamiltonian indicates very specific mechanisms of some observed phenomena that are not fully consistent with the experience. Such approach may occasionally lead to surprises when obtained results are confronted with expectations. The aim of this work is to find Hamilton operator of acoustic phonons in inhomogeneous solids. The transport of energy in the vibrating crystal consisting of ions whose properties differ over long distances is described in the work. We modeled crystal lattice by 1D “inhomogeneous” ionic chain vibrating by acoustic frequencies and found the Hamiltonian of such system in the second quantization. The influence of long-distance inhomogeneities on the acoustic phonons quantum states can be discussed on basis of our results.

Cite this paper

S. Minarik and V. Labas, "Hamiltonian of Acoustic Phonons in Inhomogeneous Solids,"*Journal of Modern Physics*, Vol. 4 No. 3, 2013, pp. 373-379. doi: 10.4236/jmp.2013.43052.

S. Minarik and V. Labas, "Hamiltonian of Acoustic Phonons in Inhomogeneous Solids,"

References

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[3] R. Fahrenbacher, “Coupling of Optical Phonons in One-Dimensional t-J Model: Effect of Superconducting Fluctuation and Phase Separation,” Physical Review Letters, Vol. 77, No. 11, 1996, pp. 2288-2291. doi:10.1103/PhysRevLett.77.2288

[4] V. M. Kuznetsov and V. I. Khromov, “Fractal Representation of the Debye Theory for Studying the Heat Capacity of Macro- and Nanostructures,” Technical Physics, Vol. 53, No. 11, 2008, pp. 1401-1406.

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[6] D. C. Mattis and W. D. Langer, Physical Review Letters, Vol. 25, No. 6, 1970.

[7] S. Tajima and J. Schützmann, “Role of Phonons in Electronic and Spin Excitations of High-Tc Superconductors,” Physica B: Condensed Matter, Vol. 219-220, 1996, pp. 128-131. doi:10.1016/0921-4526(95)00672-9

[8] B. W. Faughnan, M. W. P. Strandberg, Journal of Physics and Chemistry of Solids, Vol. 19, No. 1-2, 1961, pp. 155-166.

[9] E. B. Magnusson, H. Flayac, G. Malpuech and I. A. Shelykh, “Role of Phonons in Josephson Oscillations of Excitonic and Polaritonic Condensates,” Phyical. Review B, Vol. 82, No. 19, 2010, Article ID: 195312. doi:10.1103/PhysRevB.82.195312

[10] P. D. Bogdanoff and B. Fultz, “The Role of Phonons in the Thermodynamics of the Martensitic Transformation in NiTi,” Philisophical Magazine B, Vol. 81, No. 3, 2001, pp. 299-311. doi:10.1080/13642810108221985

[11] M. A. Stroscio, Yu. M. Sirenko, S. Yu and K. W. Kim, “Acoustic Phonon Quantization in Buried Waveguides and Resonators,” Journal of Physics: Condensed Matter, Vol. 8, No. 13, 1996, p. 2143. doi:10.1088/0953-8984/8/13/006

[12] M. A. Stroscio, et al., In: P. Bhattacharya, Eds., Properties of III-V Quantum Wells and Superlattices, INSPEC, London, 1996, p. 194.

[13] B. A. Auld, “Acoustic Fields and Waves,” Wiley, New York, 1973.

[14] K. Kneipp, L. T. Perelman, H. Kneipp, V. Backman, A. Jorio, G. Dresselhaus and M. S. Dresselhaus, “Coupling and Scattering Power Exchange between Phonon Modes Observed in Surface-Enhanced Raman Spectra of Single-Wall Carbon Nanotubes on Silver Colloidal Clusters,” Physical Review B, Vol. 63, No. 19, 2001, Article ID: 193411. doi:10.1103/PhysRevB.63.193411

[15] D. J. Safarik, R. B. Schwarz and M. F. Hundley, “Similarities in the Cp/T3 Peaks in Amorphous and Crystalline Metals,” Physical Review Letters, Vol. 96, No. 19, 2006, Article ID: 195902. doi:10.1103/PhysRevLett.96.195902

[16] P. Giudicelli and N. Bernhoeft, Europhysics Letters, Vol. 67, 2004, pp. 117-122.

[1] V. M. Axt and A. Stahl, “Influence of Phonon Bath on the Hierarchy of Electronic Densities in a Optically Excited Semiconductor,” Physical Review B, Vol. 53, No. 7244, 1996.

[2] T. Hotta and Y. Takada, “Effect of Electron Correlation on Phonons in a Strongly Coupled Electron-Phonon System,” Physical Review B, Vol. 56, No. 21, 1997, pp. 13916-13926. doi:10.1103/PhysRevB.56.13916

[3] R. Fahrenbacher, “Coupling of Optical Phonons in One-Dimensional t-J Model: Effect of Superconducting Fluctuation and Phase Separation,” Physical Review Letters, Vol. 77, No. 11, 1996, pp. 2288-2291. doi:10.1103/PhysRevLett.77.2288

[4] V. M. Kuznetsov and V. I. Khromov, “Fractal Representation of the Debye Theory for Studying the Heat Capacity of Macro- and Nanostructures,” Technical Physics, Vol. 53, No. 11, 2008, pp. 1401-1406.

[5] W. C. Walker and D. M. Roesslev, “Phonon Induced Splitting of Exciton Lines in MgO and BeO,” Physical Review Letters, Vol. 20, No. 16, 1968, pp. 847-848. doi:10.1103/PhysRevLett.20.847

[6] D. C. Mattis and W. D. Langer, Physical Review Letters, Vol. 25, No. 6, 1970.

[7] S. Tajima and J. Schützmann, “Role of Phonons in Electronic and Spin Excitations of High-Tc Superconductors,” Physica B: Condensed Matter, Vol. 219-220, 1996, pp. 128-131. doi:10.1016/0921-4526(95)00672-9

[8] B. W. Faughnan, M. W. P. Strandberg, Journal of Physics and Chemistry of Solids, Vol. 19, No. 1-2, 1961, pp. 155-166.

[9] E. B. Magnusson, H. Flayac, G. Malpuech and I. A. Shelykh, “Role of Phonons in Josephson Oscillations of Excitonic and Polaritonic Condensates,” Phyical. Review B, Vol. 82, No. 19, 2010, Article ID: 195312. doi:10.1103/PhysRevB.82.195312

[10] P. D. Bogdanoff and B. Fultz, “The Role of Phonons in the Thermodynamics of the Martensitic Transformation in NiTi,” Philisophical Magazine B, Vol. 81, No. 3, 2001, pp. 299-311. doi:10.1080/13642810108221985

[11] M. A. Stroscio, Yu. M. Sirenko, S. Yu and K. W. Kim, “Acoustic Phonon Quantization in Buried Waveguides and Resonators,” Journal of Physics: Condensed Matter, Vol. 8, No. 13, 1996, p. 2143. doi:10.1088/0953-8984/8/13/006

[12] M. A. Stroscio, et al., In: P. Bhattacharya, Eds., Properties of III-V Quantum Wells and Superlattices, INSPEC, London, 1996, p. 194.

[13] B. A. Auld, “Acoustic Fields and Waves,” Wiley, New York, 1973.

[14] K. Kneipp, L. T. Perelman, H. Kneipp, V. Backman, A. Jorio, G. Dresselhaus and M. S. Dresselhaus, “Coupling and Scattering Power Exchange between Phonon Modes Observed in Surface-Enhanced Raman Spectra of Single-Wall Carbon Nanotubes on Silver Colloidal Clusters,” Physical Review B, Vol. 63, No. 19, 2001, Article ID: 193411. doi:10.1103/PhysRevB.63.193411

[15] D. J. Safarik, R. B. Schwarz and M. F. Hundley, “Similarities in the Cp/T3 Peaks in Amorphous and Crystalline Metals,” Physical Review Letters, Vol. 96, No. 19, 2006, Article ID: 195902. doi:10.1103/PhysRevLett.96.195902

[16] P. Giudicelli and N. Bernhoeft, Europhysics Letters, Vol. 67, 2004, pp. 117-122.