Wake potential in interaction of charged particles with Carbon nanotubes

8/19/2019 Wake potential in interaction of charged particles with Carbon nanotubes

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International Journal of Application or Innovation in Engineering& Management (IJAIEM)Web Site: www.ijaiem.org Email: [email protected]

Volume 5, Issue 2, February 2016 ISSN 2319 - 4847

Volume 5, Issue 2, February 2016 Page 11

ABSTRACTThe 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) andK m(x), with

m=0,±1, ±2, ±3,…….[11]:

, for ρa(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]:

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


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(3c)

dna

(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: ω p is the plasma frequency and γ the damping constant.

For γ


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due x x

x xiu )(

0

1)(

,

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),

mmmm

m

m

mmmm I Ak k I k

k k I Akak k I

)(

)()()()( 000

Thus ,

mmmmmmmmmmmm I Ak k k k I k k I Ak k k I )()()()()( 000 mmmmmmmmmmm k k k I k k k I I k k k I A )()()()()( 000

then :

)()()(

)](1

),( 00

k I I k k k I

k k

k A mmmmm

mmm

(18a)

where,

(18b)

and by the same way obtain :

where,

(19)

Using Fourier transforming Eq. (12) after inserting from Eq. (18b) , the induced potential inside the cylinder is

given as follows :


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(20)

Integrating Eq. (20) over ω using Fourier transformation and Euler's )sincos(

iei

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 ofthe 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 ω p/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) areincreases 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 φ ind

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 φ ind 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 ω p =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=

(w p/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 ω p =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

ω p =0.5 and ɤ=0.05 usingLor entz 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 ω p =0.5 and ɤ=0.05 usingLor entz dielectric function.

REFERENCES

[1]. R. H. Ritchie, Phys. Rev. 114, 644 (1959).

[2]. R.J.Warmack,et al., Phys. Rev.B 29, 114, 4375 (1984).

[3]. N.A. Arista and M. A. Fuentes, Phys. Rev.B 63, 165401 (2001).

[4]. S.Iijima.Nature (London) 354 (1991).[5]. Zhu Z Y ,et al., Proc. SPIE 597413(2005).

[6]. T.W. Ebbesen and P.M.Ajayan, Nature (London) 358,220 (1992).

[7]. N.A. Arista, Phys. Rev.A 64, 032901(2001).

[8]. ThomasStoïckli ,et al. PHILOSOPHICAL MAGAZINE B79 , 10( 1999).

[9]. N.A. Arista, Nuci. Instr. And Math. In Phys. Res. B 182(2001) 109-115.

[10]. F. J. Garcia de Abajo and P. M. Echenique , , Phys. Rev. B 48,18, (1993).

[11]. D. Jackson , Classical Elctrodynamics (Wiley, New York (1975).

[12].Seham Z. Abbas ,“wake potential and energy loss for charged particles in nanotubes”, PhD thesis , to be

submitted to college of science, Al –mustansirya University.

[13]. J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 28, 8 (1954).

[14]. F. J. Garcia de Abajo and P. M. Echenique , , Nuci. Instr. And Math. In Phys. Res. B 48, (1990 )25-28.

[15]. M. Abramowitz and I. A.Stegun,’Hand book of Mathematical Functions (Dover,New York, 1975).[16]. D. Penn, Phys. Rev. 128, 2093 (1962).

[17]. J. P. Walter and M.L. Cohen, Phys. Rev. B5, 3101 (1972).

[18]. I. Abril, R. Garcia-Molina, C.D. Denton,F. J. Pe´rez, and N. R Arista, Phys. Rev. A 58, 357 (1998).

[19]. A. Schinner and P. Sigmund, Eur. J. Phys. 66, 1-11 (2012).

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Wake potential in interaction of charged particles with Carbon nanotubes

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