The Harmonic Approximation in Heavy-Ion Reaction Study

Godwin Joseph Ibeh^1*, Elijah Dika Mshelia²

¹Department of Physics, Nigerian Defence Academy, Kaduna, Nigeria.
²Department of Physics, University of Abuja, Abuja, Nigeria.

DOI: 10.4236/am.2015.611161                

Abstract

The derivation of the harmonic approximation of the Hamiltonian of a model of coupled three-dimensional harmonic oscillator is presented. It is shown how the splitting of the total Hamiltonian into the intrinsic and collective Hamiltonians leads to the description of the mechanism for energy dissipation in physical systems.

Keywords

Harmonic Approximation, Energy Dissipation, Coupled Oscillators, Heavy-Ions, Dinuclear System, Cluster Model

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Diaz-Torres, A., Hinde, D.J., Dasgupta, M., Milburn, G.J. and Tostevin, J.A. (2008) Dissipative Quantum Dynamics in Low-Energy Collisions of Complex Nuclei. Physical Review C, 78, Article ID: 064604. http://dx.doi.org/10.1103/physrevc.78.064604
[2]	Diaz-Torres, A., Hinde, D.J., Dasgupta, M., Milburn, G.J. and Tostevin, J.A. (2009) Coupled-Channels Approach for Dissipative Quantum Dynamics in Near-Barrier Collisions. International Conference on New Aspects of Heavy-Ion Barrier, Chicago. http://dx.doi.org/10.1063/1.3108859
[3]	Swiatecki, W.J. and Bjфrnholm, S. (1972) Fission and Fusion Dynamics. Physics Reports, 4, 326-342. http://dx.doi.org/10.1016/0370-1573(72)90003-8
[4]	Nörenberg, W. and Weidenmüller, H.A. (1976) Introduction to the Theory of Heavy-Ion Collisions. Lecture Notes in Physics 5, Springer-Verlag, Berlin.
[5]	Hasse, R.W. (1978) Approaches to Nuclear Friction. Reports on Progress in Physics, 41, 1027-1101. http://dx.doi.org/10.1088/0034-4885/41/7/002
[6]	Weidenmüller, H.A. (1980) Transport Theories of Heavy-Ion Reactions. Progress in Particle and Nuclear Physics, 3, 49. http://dx.doi.org/10.1016/0146-6410(80)90030-7
[7]	Mshelia, E.D. (1997) Nuclear Science and Technology in Human Progress. (Unpublished University Inaugural Lecture). Abubakar Tafawa Balewa University, Bauchi.
[8]	Aritomo, Y., Hagino, K., Nishio, K. and Chiba, S. (2012) Dynamical Approach to Heavy-Ion Induced Fission Using Actinide Target Nuclei at Energies around the Coulomb Barrier. Physical Review C, 85, Article ID: 044614. http://dx.doi.org/10.1103/PhysRevC.85.044614
[9]	Liu, Z.H. and Bao, J.D. (2013) Possibility to Produce Element 120 in the 54Cr+248Cm Hot Fusion Reaction. Physical Review C, 87, Article ID: 0344616. http://dx.doi.org/10.1103/PhysRevC.87.034616
[10]	Gontchar, I.I., Bhattacharya, R. and Chushnyakova, M.V. (2014) Quantitative Analysis of Precise Heavy-Ion Fusion Data at Above-Barrier Energies Using Skyrme-Hartree-Fock Nuclear Densities. Physical Review C, 89, Article ID: 034601. http://dx.doi.org/10.1103/PhysRevC.89.034601
[11]	Wen, K., Sakata, F., Li, Z.-X., Wu, X., Zhang, Y. and Zhou, S. (2014) Energy Dependence of the Nucleus-Nucleus Potential and the Friction Parameter in Fusion Reactions. Physical Review C, 90, Article ID: 054613. http://dx.doi.org/10.1103/PhysRevC.90.054613
[12]	Mshelia, E.D., Scheid, W. and Greiner, W. (1975) Theory of Energy Dissipation in Heavy-Ion Reactions. Il Nuovo Cimento A, 30, 589-608. http://dx.doi.org/10.1007/BF02730488
[13]	Mshelia, E.D., Haln, D. and Scheid, W. (1981) Energy Dissipation in a Model of Coupled Oscillators. Il Nuovo Cimento A, 61, 28-55. http://dx.doi.org/10.1007/BF02776606
[14]	Mshelia, E.D. and Ngadda, Y.H. (1989) A Stimulation for Energy Dissipation in Nuclear Reactions. Journal of Physics G: Nuclear and Particle Physics, 15, 1281-1290. http://dx.doi.org/10.1088/0954-3899/15/8/023
[15]	Ibeh, G.J. and Mshelia, E.D. (2014) Energy Dissipation in a Model of Coupled Three-Dimensional Harmonic Oscillator. Far East Journal of Mathematical Sciences, 88, 107-136.
[16]	Corben, H.C. and Stehle, P. (1960) Classical Mechanics. John Wiley, New York, 113-131, 364-372.
[17]	Wells, D.A. (1967) Theory and Problems of Lagrangian Dynamics. Schaum Publication, New York.
[18]	Marion, J.B. and Thornton, S.T. (1995) Classical Dynamics of Particles and Systems. 4th Edition, Saunders College Publications, New York.
[19]	Goldstein, H., Poole, C. and Safko, J. (2002) Classical Mechanics. 3rd Edition, Addison-Wesley, San Francisco, 238-258.
[20]	Wilson Jr., B., Decius, J.C. and Paul, C.C. (1980) Molecular Vibrations. Dover Publications, Inc., New York.
[21]	Desloge, E.A. (1982) Classical Mechanics. Volume II, Wiley-Interscience, New York, 665-707.
[22]	Mshelia, E.D. (1995) Method of Normal Coordinates in the Formulation of a System with Dissipation: The Harmonic Oscillator. Il Nuovo Cimento A, 108, 709-721. http://dx.doi.org/10.1007/BF02813376
[23]	Schiff, L.I. (1987) Quantum Mechanics. 3rd Edition, McGraw-Hill, New York, 50-52.
[24]	Greiner, W. (1994) Quantum Mechanics: An Introduction. 3rd Edition, Springer-Verlag, New York, 97-105.
[25]	Courant, R. and Hilbert, D. (1953) Methods of Mathematical Physics. Interscience Publishers, New York, 424.
[26]	Bromley, D.A. (1978) Nuclear Molecules. Scientific American, 239, 58-68. http://dx.doi.org/10.1038/scientificamerican1278-58
[27]	Volkov, V.V. (1978) Deep Inelastic Transfer Reaction—The New Type of Reactions between Complex Nuclei. Physics Reports, 44, 93-157. http://dx.doi.org/10.1016/0370-1573(78)90200-4
[28]	Iachello, F. and Jackson, A.D. (1982) A Phenomenological Approach to α-Clustering in Heavy Nuclei. Physics Letters B, 108, 151-154. http://dx.doi.org/10.1016/0370-2693(82)91162-5
[29]	Greiner, W., Park, J.Y. and Scheid, W. (1995) Nuclear Molecules. World Scientific, Singapore.
[30]	Buck, B., Merchant, A.C. and Perez, S.M. (1998) Systematic Study of Exotic Clustering in Even-Even Actinide Nuclei. Physical Review C, 58, 2049-2060. http://dx.doi.org/10.1103/physrevc.58.2049
[31]	Buck, B., Merchant, A.C. and Perez, S.M. (1999) Cluster Structure and Gamma Transitions in Actinides. Physical Review C, 59, 750-754. http://dx.doi.org/10.1103/PhysRevC.59.750
[32]	Adamian, G.G., Antonenko, N.V. and Scheid, W. (2000) Isotopic Dependence of Fusion Cross Sections in Reactions with Heavy Nuclei. Nuclear Physics A, 678, 24-38. http://dx.doi.org/10.1016/s0375-9474(00)00317-1
[33]	Mshelia, E.D. and Scheid, W. (2004) Collective Dynamics of a Dinuclear System. The European Physical Journal A, 20, 251-254. http://dx.doi.org/10.1140/epja/i2003-10102-7
[34]	Li, W., Nan, W., Fei, J., Hushan, X., Wei, Z., Li, Q.F., et al. (2006) Particle Transfer and Fusion Cross-Section for Super-Heavy Nuclei in Dinuclear System. Journal of Physics G, 32, 1143-1155. http://dx.doi.org/10.1088/0954-3899/32/8/006
[35]	Kalandarov, S.A., Adamian, G.G., Antonenko, N.V. and Scheid, W. (2011) Role of Angular Momentum in the Production of Complex Fragments in Fusion and Quasifission Reactions. Physical Review C, 83, Article ID: 054611. http://dx.doi.org/10.1103/PhysRevC.83.054611
[36]	Kalandarov, S.A., Adamian, G.G, Antonenko, N.V., Scheid, W., Heinz, S., Comas, V., Hofman, S., Khuyagbaatar, J., Ackermann, D., Heredia, J., HeBberger, F.P., Kindler, B., Lommel, B. and Mann, R. (2011) Emission of Cluster with Z > 2 from Excited Actinide Nuclei. Physical Review C, 84, Article ID: 054607. http://dx.doi.org/10.1103/PhysRevC.84.054607
[37]	Kalandarov, S.A., Adamian, G.G., Antonenko, N.V., Scheid, W. and Wieleczko, J.P. (2011) Role of the Entrance Channel in the Production of Complex Fragments in Fusion-Fission and Quasifission Reactions in the Framework of the Dinuclear System Model. Physical Review C, 84, Article ID: 064601. http://dx.doi.org/10.1103/PhysRevC.84.064601
[38]	Bansal, M., Sahila, C. and Rajk, G. (2012) Dynamical Cluster-Decay Model Using Various Formulations of a Proximity Potential for Compact Non-Coplanar Nuclei: Application to the 64Ni+100Mo Reaction. Physical Review C, 86, Article ID: 034604. http://dx.doi.org/10.1103/physrevc.86.034604
[39]	Adamian, G.G., Antonenko, N.V. and Lenske, H. (2015) Role of the Neck Degree of Freedom in Cold Fusion Reactions. Physical Review C, 91, Article ID: 054602. http://dx.doi.org/10.1103/PhysRevC.91.054602