Wake Potential of penetrating Ionsin Amorphous Carbon Target using Quantum Harmonic Oscillator (QHO) Model

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1 Wake Potential of penetrating Ionsin Amorphous Carbon Target using Quantum Harmonic Oscillator (QHO) Model Nabil janan Al-Bahnam 1, Khalid A. Ahmad 2 and Abdulla Ahmad Rasheed 3 1 Department of physics,college of Science for Women, Baghdad University, Iraq 2 Department of Physics, College of Science, AL-Mustansiriya University, Iraq 3 Department of Physics, College of Science, AL-Mustansiriya University, Iraq ABSTRACT The wake potential and induce forces for swift proton in an amorphous carbon target were studied using quantum harmonic oscillator (QHO) model with range of velocities. The wake results exhibited a damped oscillatory behavior in the longitudinal direction behind the projectile; the pattern of these oscillations decreases exponentially in the transverse direction. In addition, the wake potential extends slightly ahead of the projectile as well as it depends on the proton position and velocity. The effect of electron binding on the wake potential characterized by the electron density parameter, has been studied in quantum dielectric function formalism, the calculated wake potential, stopping and lateral force show that the results depend on the electron density parameter of the carbon target. Key word: wake potential, carbon,harmonic oscillator, quantum dielectric function 1. INTRODUCTION The subject of interactions of heavy charged particles and matter has received increasing interest in the field of condensed-matter and surface physics over the past years. Especially since the rapid development of semiconductor technology, new nanotechnology, irradiation of quantum dots, proton beam writing (PBW) and welding of nanotubes by ion beams or proton irradiation of carbon nanotubes (CNT). Material modification, induced by swift heavy ions (SHI), has been intensively studied in all kinds of materials, from insulators to superconductors and from crystalline to amorphous materials [1, 2]. Several attempts were made to incorporate quantum effects into the energy-loss problem in the 1920s. Henderson (1922) applied the concept of discrete energy levels to the problem by limiting the possible energy transfer to an atom from below by the ionization potential. In this manner he obtained an expression for the stopping power which is roughly one-half of the correct one (Henderson essentially ignored the distant collision contribution to the energy loss which accounts for the other one-half). The original classical result of Bohr was recreated in a quantum- mechanical treatment by Gaunt (1927), who treated the perturbation of an atom by the passage of a heavy charged particle moving with constant velocity. Bethe (1930) solved the problem quantum mechanically in the first Born approximation whereby the entire system [(charged particle) +atom] is considered within the framework of quantum theory; also A.Shinner and P.Sigmund (2012)have developedthe wake potential equation based on quantum harmonic oscillator.the significant difference between Bethe's approach and that of Bohr is the use by Bethe of momentum transfer rather than impact parameter to characterize collisions [3-6]. 2. Theoretical Model 2.1. Wake potential The wake potential induced by a swift proton has been studied theoretically for a random stopping medium consisting of quantal-harmonic-oscillator atoms. The primary purpose is to study the effect of atomic binding and electron densities on the wake potential and forces in a Fermi gas[7]. The Drude-Lorentz theory describes long-wavelength polarization phenomena and, therefore, breaks down near the trajectory, where the induced field is most pronounced. Unlike in the case of the classical oscillator a restriction to long wavelengths is not required. Within the Born approximation, the main part of the theory can be carried out analytically [8]. It is necessary, therefore, to utilize a description that describes variations over short wavelengths in a consistent manner. An appropriate tool here is the dielectric function based on the linear response of a 3D quantum oscillator, as derived in reference [7]based on reference [9]. For excited states, the approximation of the wake potential in terms of an anisotropic harmonic oscillator breaks down. For higher lying states, the system is a harmonic and non-separable. This problem provides therefore an Page 37

2 opportunity to study the quantum dynamics of a simple non-separable system with two degrees of freedom whose classical counterpart possesses a divided phase space of regular and chaotic motion. The Hamiltonian is given by [10], Where is the wake potentialwhich is approximation given by equation [10], The speed of the projectile is denoted by and its charge by and are cylindrical coordinates in the frame of the projectile (k)is the wave number and P.M. Echenique, et.al. (1982) used interaction picture in the system wave function equation [11],. Obeys Schrödinger Is the Hamiltonian operator in the interaction picture, the wake potential, potential. Thus, is the mean value of the scalar The ideas of the Fermi, Bohr,Lindhard A.Shiner and P.Sigmund used to express wake potentialin references [9, 12].The target atom is characterized by an assembly of spherical harmonic oscillators ; wake potential can be written as[8], Denote the positive zero of the ε. : Function depends on cylindrical coordinate, wave number k, and real part of quantum Dielectric function. Where the is the step function, and denote the sine and cosine integrals, respectively[13, 14]. Page 38

3 2.2. Induce Forces in Quantum Harmonic Oscillator A swift projectile channeling through a solid target interacts with the target electrons and nuclei, which reduces gradually its energy, and affects its direction of motion as well as its charge state. When a projectile moves inside a target, it can vary its charge state by exchanging (capturing or losing) electrons with the target, reaching an equilibrium charge state after a few femtoseconds. The equations based on the Born approximation, the quantum dielectric formalism provides the following expressions for the stopping force, and the lateral force, of a material for a projectile with mass m, and Charge Ze[15, 16]. Stopping and lateral forces modeled from potential equation (5), are given by, 2.3. Quantum Dielectric Function (Quantum Oscillator) To treat the complex many-body problem of the interaction of charged particles with a degenerate electron gas, Lindhard and Hubbard have developed independently a quantum theory of the dielectric constant of such a system. In their treatments, the electric field is assumed to be classically described, although the electron motion in the gas is treated by the quantum perturbation theory[17]. The dielectric function of a medium composed of 3D quantum oscillators distributed at random in space has been Suggested by A.Belkasem and P.Sigmund[9], 3. Numerical Results and Discussion From 3D plotting wake potential figures, one can see and conclude the following: 1) The amplitude of the oscillations increases with increasing electron density, while the wavelength decreases,because of the influence of electron densities and electron binding of the electron gas of the solid target atoms. This effect depends on the values of each of the plasma frequency and the resonance frequency. 2) Oscillations of induce potential increases with decreasing electron density parameter ratio. i.e in the high electron densities a gradual transition is seen from an oscillation governed by the plasma frequency. More rapid oscillation this occurs in the case of. Page 39

4 ISSN ) The obtained results from Figure1 are agree with ideas found in Refs. [7, 8, 18]. Page 40

5 ISSN Page 41

6 4.CONCLUSION We use quantum harmonic oscillator (QHO) model, to obtain wake potential and induce forces (stopping and lateral force) to the proton moving with velocities in amorphous carbon. The general shape of wake potential derived in equation (5) shows in the longitudinal direction behind the target position, the oscillatory behavior is more pronounced for quantum gas. From our work we observe that the amplitude of the oscillations increases to the induce forces and the oscillation potential decreases with increasing electron density parameter in free electron metal, where the ratio is normally. REFERENCES [1] A. V. Krasheninnikov and K. Nordlund, Nucl. Instr. and Meth. B 216, (2004). [2] A. S. El-Said, Nucl. Instr. and Meth. B 282, (2012). [3] S. P. Ahlen, Rev. Mod. Phys. 52, (1980). [4] N. Bohr and E. London, J. Sci 25, 10 (1913). [5] J.A.Grant, Proc. Cambridge Philos.Soc 23, 732 (1927). [6] H. Bethe, Ann.Physik 397, (1930). [7] A. Schinner and P. Sigmund, Eur. J. Phys. 66, 1-11 (2012). [8] P. Sigmund, Particle Penetration and Radiation Effects,Springer, Switzerland, (2014). [9] A. Belkacem and P. Sigmund, Nucl. Instr. and Meth. B 48, (1990). [10] J. Mueller, J. Burgdoerfer, and D. W. Noid, "Semiclassical eigenenergies in the wake of fast ions in solids," presented at the Mini workshop on quantum chaos, Treiste-Italy (1990). [11] P. M. Echenique, J. C. Ashley, and R. H. Ritchie, Eur. J. Phys. 3, 25 (1982). [12] P. Sigmund and A. Schinner, Eur. Phys. J. D 61, (2011). [13] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions,dover New York, (1972). [14] F. W. Olver, NIST handbook of mathematical functions,cambridge University Press, (2010). [15] Gustavo Garc ıa G omez-tejedor and D. M. C. Fuss, Radiation Damage in Biomolecular Systems, 1st ed.springer Dordrecht Heidelberg, London New York (2012). [16] R. G.-M. Isabel Abril, P. d. Vera, I. Kyriakou, D. Emfietzoglou, C. Champion, C. C. Montanari, et al., Advances in Quantum Chemistry. vol. Volume 65, B. Dževad, Ed., ed Elsevier: Academic Press, (2013). [17] R. H. Ritchie, Phys. Rev. 114, (1959). [18] I. Abril, R. Garcia-Molina, C. D. Denton, F. J. Pérez-Pérez, and N. R. Arista, Phys. Rev. A 58, (1998). Page 42