International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 5, Issue 2, February 2016
ISSN 2319 - 4847
Wake potential in interaction of charged
particles withCarbon nanotubes
*Khalid A. Ahmad, **Seham Z. Abbas
*Department. of physics , college of science, Al –mustansirya University. Iraq
** Minicitiry of sciences & technology,Baghdad, Iraq
ABSTRACT
The interaction of charged particles with nanotubes is described on the basis of the dielectric formalism, starting with simple
dielectric models, like Random Phase Approximation (RPA)and Lorentz models.The response of the medium is characterized
in terms of the wake effects, calculating the main effects on the moving particles, including induced field and induced potential
in 2D and 3D of cylindrical coordinates. The wake results exhibits a damped oscillatory behavior in the longitudinal direction
behind the projectile (proton), the pattern of these oscillations decreases exponentially in the transversal direction. Good
agreement is achieved with previous work .
Keywords: wake potential, dielectric function ,carbon nanotube
1. INTRODUCTION
The interaction of charged particles with surface modes in cylindrical channels in solids is a subject of interest for
current studies of electron and ion interaction with microchannels, capillaries, and nanotubes in various
materials[1,2,3].Ever since the discovery of carbon nanotubes (rolled-up cylinders of graphene) by Iijima in (1991)[4],
many efforts have been made both theoretically and experimentally to look into various aspects of nanotubes owing to
their outstanding properties and great expectations for their applications,such as in semiconducting devices and
biomedical. In addition,the progressive technology of ordering and straightening of carbon nanotube arrays makes it
become a probable candidate for channeling application [5].
The production of nanotubes of graphite orfullerenes has been reported, and there are already electron-spectroscopy
experiments, and studies of particle channeling in these structures [6]. There is growing interest in studying the
interaction of charged particles and the formation of hollow atoms in micro capillaries and nanotubes. Hence, the study
of plasmon excitation in these systems is a subject of great current interest [7, 8, 9].
In present work the interaction between nonrelativistic charged particles and surface modes of a cylindrical cavity using
classical mechanism formulas, the Random Phase Approximation (RPA)andLorntz models of dielectric function have
been used to find the induced potential of a material having more complicated dielectric properties than the simple
resonance.
2. DIELECTRIC MODELS
2.1Plasma resonance model (Random Phase Approximation (RPA))
Let us suppose acharge Ze is moving uniformly with trajectory parallel to the channel axis z with velocity v and with
instantaneous coordinates (ρ₀,φ₀,vt)[3,10] .The electrostatic modes of a cylindrical cavity of radius a in a solid can be
determined by the solutions of the Laplaceequation , in terms of cylindrical Bessel functions Im(x) andKm(x), with
m=0,±1, ±2, ±3,…….[11]:
, for ρ<a(1)
, for ρ>a(2)
where (ρ ,φ,z) are cylindrical coordinates and k is a wave vector along the axial channel direction denoted by z.To
obtaining the relation between the coefficients Am and Bm, and the frequencies of the modes ωk,m=ωm(k), use the
boundary conditionsas follows[12]:
(i)
(3a)
(ii)
(3b)
where (ω) is the dielectric function of the medium. This yields [12]:
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Volume 5, Issue 2, February 2016
ISSN 2319 - 4847
(3c)
adn
(3d)
(3e)
where
and
Equation (3e) gives the dispersion relation of the modes, ω=ωm(k), and it may be solved for each material using the
appropriate expression for
. In particular, the simple approximate to the dielectric function around the plasma
resonance is given by[13,14]:
(4)
Where: ωp is the plasma frequency and γ the damping constant.
For γ<ωp, the frequency of the modes ωk,m=ωm(k) according to Eqs. (3e,4) isin the form:
Thus,
(5)
where
(6)
Since
is positive quantity therefore,
Using the Wronskian property [15]:
0.
Then,
(7)
Where x=ka and
2.2. Lorentz model
Apply the similar consideration to the case of Lorentz model of the dielectric function [16,17]:
(8)
With model parameters ω0 and ω1. For γ→0, the frequency of the modes ωk,m=ωm(k)and according to Eqs. (3c,8):
(9)
3. INTERACTION WITH AN EXTERNAL POINT CHARGE
3.1-Induced field
The Coulomb potential of the external charge may be expanded in terms of cylindrical functions as follow[11]:
(10)
where ρ<and ρ>inside and outside the cavity respectively .
Using the Fourier transform [15],
the relation
cos[k ( z  vt)]  [eik ( zvt)  eik ( zvt) ] / 2
and according to property of delta function[15]:
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Volume 5, Issue 2, February 2016
1
 ( x  x ) 


ISSN 2319 - 4847
iu ( x  x  )
e
du ,
0
the electrostatic potential,
(11)
3.2. Induced potential
The induced potential inside and outside the cavity may be expanded also in terms of the regular solutions [11] :
(1) at ρ < a,
(12)
so that the total potential forρ< a becomes,
(13)
(2) at ρ > a,
It is suitable to expand the total (external plus induced) potential as follows:
(14)
The coefficients Amand Bmin Eqs. (12) and (14) will be determined from the boundary conditionsgiven in Eqs. 3a, 3b
atρ = a:
(15)
(16)
where the primes denote the derivatives with respect to the variable ρ . From equations 11-14 ,
(17)
Solving these two Eqs. for Am(k,ω) and Bm(k,ω),
I m ( k 0 ) k m ( ka )  Am I m ( k 0 ) 
km
I m (k 0 )k m  Am I m 
k m  ( )
Thus ,
 (  ) I m ( k  0 ) k m k m  A m  ( ) I m ( k  0 ) k m  I m ( k  0 ) k m k m  k m A m I m
A m  ( ) I m ( k  0 ) k m  k m I m   I m ( k  0 ) k m k m   ( ) I m ( k  0 ) k m k m
then :
Am (k , ) 
1 
 (  )] k m k m 
 (  ) I m ( k  0 ) k m  k m I m

I m (k 0 )
(18a)
where,
(18b)
and by the same way obtain :
where,
(19)
Using Fourier transforming Eq. (12) after inserting
given as follows :
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from Eq. (18b) , the induced potential inside the cylinder is
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(20)
Integrating Eq. (20) over ω using Fourier transformation and Euler's (e i  cos   i sin  ) equation, one can get:
(21)
where we have used the property:
(22)
and the frequency is now given by ω=kv .
4. RESULTS AND DISCUSSION
The calculation of the induced (wake) potential is according to Eq. 21 using RPA andLorentz dielectric function taking
in the consideration the special cases of velocity, and will use atomic unit (a.u) in present work.Fig.1show the values of
the induced potential calculated at ρ= (0.1,0.5,2,4 and 6), for proton moving with velocityv=10and ρ0=5 in cylindrical
cavity of carbon nanotubesas a function of coordinate Z using RPA dielectric function . At ρ0→0, φ→φ0 and Z→0, the
induced potential
is given as follows:
With this approximation,
as shown in Fig. 1 ,which at ρ > 0 for 0˂Z˂0 the induced potential is
running according to modified Bessel function Im(kρ0) and Im(kρ),remember that cos(-x)=cos(x) and sin(-x)= ̶
sin(x).The shape of the induced potential derived from Eq. 21 which exhibits a damped oscillatory behavior in the
longitudinal direction behind the projectile (proton) , the pattern of these oscillations decreases exponentially in the
transversal direction. This result agreement with a previous work [18] .
Figure 1The induced potential of proton as a function of the coordinate Z at ρ0 = 5, a=10 with radial
coordinates ρ = ( 0.1,0.5,2,4 and 6) using RPA dielectric function.
Figs.2 (a,b,c,d,e) show the values of the induced potential calculated at ρ= (0.1,0.5,1,6 and10), for proton moving with
velocity v=10, ρ0=5 and for different electron density in terms of ωp/ω0 in cylindrical cavity of carbon nanotubes as a
function of coordinate Z usingLorentz dielectric function .The value of the induced potential in figs. 2(a,b,c,d,e) are
increases when ρ decreases and the shape of the induced potential which exhibits a damped oscillatory behavior in the
longitudinal direction behind the projectile [18,19].
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Fig.3 (a,b,c,e) show the three dimensions 3D surface induced potential for a proton with different velocitiesv=1,3,5and
10 in a channel of radius a=10 using RPA dielectric function. From these figs. one can see the oscillation in φind
increases as a coordinate Z increases in negative values and also these figs. show the wake behind the proton i.e (ρ˃a) .
Fig.4(a,b,c,e) show the 3D surface induced potential for a proton with velocity v=5 in a channel of different radii a=1,
5, 10 and 20using RPA dielectric function. These figs. show the values of the induced potential behind the proton. As it
may be observed the cusp behavior at ρ=a is due to the accumulation of electronic charge at the boundary.
Fig.5 (a,b,c,e) show the 3D surface induced potential for a proton with velocity v=5 in a channel of radius a=10 at
different ρ0 =1, 5, 10 and 20using RPA dielectric function. These figs. show the increasing of oscillation in φind as a
function of Z when ρ0 increases and also these figs. show the wake behind the proton i.e (ρ˃a) .
But when using the Lorentz dielectric function and the calculations were done for a medium with ωp =0.5., γ=0.05, as
shown in figs. 6,7,8 one can see:
Fig.6 (a,b,c,e) show the 3D surface induced potential for a proton with different velocity v=1,3,5and 10 in a channel of
radius a=10. These figs. show, when the value of the proton s velocity increases, the oscillation in φind decreases as a
coordinate Z increases in negative , and also the amplitude of the oscillation increases as ρ increases .
Fig.7 (a,b,c,e) show the 3D surface induced potential for a proton with velocity v=5 in a channel of different radii a=1,
5, 10 and 20. These figs. show the values of the induced potential behind the proton. As it may be observed, the
potential has a ‘‘normal’’ wake behavior outside the cavity (ρ˃a) but becomes flat inside it.
Fig.8 (a,b,c,e) show the 3D surface induced potential for a proton with velocity v=5 in a channel of radius a=10 at
different ρ0 =1, 5, 10 and 20 .These figs. show the decreasing of oscillation in φind as a coordinate Z increasing in
negative when ρ0 increases and also these figs. show the wake behind the proton i.e (ρ˃a) .These results agreement
with previous work [7,19].
In the present work a program SihmVind.for [12] is written in Fortran 90 using Gaussian method in the calculation of
numerical integrations.
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Figure.2. The induced potential of proton as a function of the coordinate Z at ρ0 = 5 with radial coordinates ρ =
(0.1,0.5,1,6 and 10) , and for ωp/ω0=(10 ,5,2,1,0.2and 0.1) using Lorentz dielectric function ,where wp/w0=
(wp/w0)2.
Figure. 3Theinduced potential of a proton with deferent velocities v= (1,3,5 and 10) in a channel of radius a=10 for
CNTs using RPA dielectric function at ρ0=5 .
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Figure. 4 The induced potential of a proton with velocity v= 5 in a channel of radius a= (1,5,10 and 20) for CNTs
using RPA dielectric function at ρ0=5.
Figure. 5The induced potential of a proton with velocity v =5 in a channel of radius a=10 for CNTs at ρ0 =(1,5,10 and
20) using RPA dielectric function.
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Figure. 6. The induced potential of a proton with deferent velocities v= (1,3,5 and 10 ) in a channel of radius a=10 for
CNTs with ωp =0.5 and ɤ =0.05 using Lorentz dielectric function.
Figure. 7The induced potential of a proton with velocities v =5 in a channel of radii a=(1,5,10 and 20) for CNTs with
ωp =0.5 and ɤ=0.05 usingLorentz dielectric function.
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Figure. 8 Calculation of the induced potential for a proton with velocity v =5 in a channel of radius a=10 for CNTs at
ρ0 =(1,5,10,20) with ωp =0.5 and ɤ=0.05 usingLorentz dielectric function.
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