International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 3, Issue 12, December 2014
ISSN 2319 - 4847
Wake Potential of penetrating Ionsin
Amorphous Carbon Target using Quantum
Harmonic Oscillator (QHO) Model
Nabil janan Al-Bahnam1, Khalid A. Ahmad2 and Abdulla Ahmad Rasheed3
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
Volume 3, Issue 12, December 2014
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 3, Issue 12, December 2014
ISSN 2319 - 4847
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
frame of the projectile (k)is the wave number and
and are cylindrical coordinates in the
.
P.M. Echenique , et.al. (1982) used interaction picture in the system wave function
equation [11],
Is the Hamiltonian operator in the interaction picture, the wake potential,
potential. Thus,
Obeys Schrödinger
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
function
.
: Function depends on cylindrical coordinate
.
Where the
and
positive zero of the ε
, wave number k, and real part of quantum Dielectric
is the step function,
denote the sine and cosine integrals, respectively[13, 14].
Volume 3, Issue 12, December 2014
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 3, Issue 12, December 2014
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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
.
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 3, Issue 12, December 2014
ISSN 2319 - 4847
3) The obtained results from Figure1 are agree with ideas found in Refs. [7, 8, 18].
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 3, Issue 12, December 2014
Volume 3, Issue 12, December 2014
ISSN 2319 - 4847
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 3, Issue 12, December 2014
ISSN 2319 - 4847
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
.
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