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Wake potential and stopping power for a charged particle moving outside a nanosphere
Physics Letters A 376 (2012) 763–767
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Wake potential and stopping power for a charged particle moving outside a nanosphere Sheng-Bai An, Yan-Xia Wang, Yuan-Hong Song ∗ , You-Nian Wang School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Article history: Received 11 October 2011 Received in revised form 13 December 2011 Accepted 16 December 2011 Available online 21 December 2011 Communicated by R. Wu Keywords: Wake potential Stopping power Nanosphere
a b s t r a c t The interaction of a charged particle with a nanosphere is studied based on the dielectric response theory. We obtain the analytical expressions of the induced potential and stopping power, as the charged particle moving outside the nanosphere with a constant velocity. From our results, since the spherical shape limitation, the well-known V-shaped wake effect tracing the particle cannot be observed clearly no matter at the nanosphere surface or in the bulk. Besides, we also find that the particle can even gain energy from the electron polarization as the particle moves to the nanosphere at relatively low velocity. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Interactions of charged particles and matter have received increasing interest in the field of condensed-matter and surface physics over the past decades. Especially since the rapid development of semiconductor technology and new nanotechnology, the investigation of the nano-structured materials or electronics at nanoscale is well concerned. It has been demonstrated that the impact of charged particles on various surfaces can induce surface modification with nanometer dimensions [1]. And for nanostructured materials, suitable control of the properties and response of nanostructures can lead to new devices and technologies [2]. For instance, nanosphere has been successfully applied to fabricate patterns on substrates [3,4], by the method of ion beam sputtering or deposition. It is gradually known that, other than single molecules or infinite bulk materials, nano-structured materials can exhibit different physical and chemical properties, owing to the quantum size and shape effects on their electronic response and dynamics. Moreover, ion beam radiation can be a valid tool to probe the electronic excitation of nanomaterials. As a charged particle approaches solid surface with constant velocity, the electrons in the bulk will be excited and polarized, leading to the induced surface potential, i.e. the wake potential, which has been first studied by Neufeld and Ritchie [5]. Meanwhile, the induced electric field will then give rise to a force that acts on the moving charged particle and makes the particle get or lose energy, so-called the stopping power (i.e. the energy
*
Corresponding author. Tel.: +86 0411 84707939; fax: +86 0411 84707939. E-mail address: [email protected] (Y.-H. Song).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.12.029
lost per unit path length traveled by the charged particle). The well-known dielectric response theory has found wide applications in the investigation of ions and surface interaction [6–9]. Using the full random-phase approximation (RPA) for the bulk dielectric function [6], the calculation results assure us of the study of lowvelocity surface wake and show good agreement with those calculated by plasmon-pole approximation (PLA) in the high-velocity limit (v > v F , where v F is the Fermi velocity). Wang and Liu [7] use the local field correction (LFC) for the bulk dielectric function to calculate the induced potential of an ion moving slowly near a solid surface, with the effect of exchange-correlation interactions of electrons taken into account. It is found that the magnitude of the induced potential based on LFC is always larger than that from the RPA, and thus for low velocities, the exchange-correlation effects cannot be neglected. A detailed investigation of the wake potential has been made by Garcia and Echenique [8], in which the hydrodynamic approximation (HA) and plasmon-pole approximation (PLA) to (k, ω) are adopted. For high-velocity ions, it was shown that the asymptotic behavior of the wake potential is quite similar in both the HA and PLA formulation. The only difference between them is that the PLA includes the single particle excitations as well as the plasmon dispersion. So far, with technological development in the area of miniaturized devices, a wide field of studies of small systems with typical sizes ranging between the submicron and nanometer scale has been produced [10]. In addition, recent renewed interest in surface or interface plasmon has attracted more attention in the investigation of the nano-structured materials or electronics at the nanoscale [11,12]. Horing et al. [13] first investigated the energy loss of fast particles moving parallel to 2D electron gas in the framework of the RPA dielectric function within linear response
764
S.-B. An et al. / Physics Letters A 376 (2012) 763–767 m)! m where Alm (t ) = δ2 ((ll− P (cos θ0 ), and the spherical harmonic m +m)! l function can be written as,
Y lm (θ, ϕ ) = (cos mϕ cos mϕ0 + sin mϕ sin mϕ0 ) P lm (cos θ),
(3)
with δm = 2 for m = 0 and δm = 1 for m = 0. Since ϕ0 = 0, the spherical harmonic function can be written as Y lm (θ, ϕ ) = P lm (cos θ) cos mϕ . Meanwhile, in the spherical coordinate system, the induced potential due to the excitation of the electrons satisfies the Laplace’s equation and can be expanded as,
Fig. 1. Description of the system, for a charged particle Z 1 e moving outside the nanosphere with constant velocity v parallel to the axis z at a horizontal distance d in spherical coordinates systems (r , θ, ϕ ).
theory. Recently, besides the dielectric response theory, the quantum hydrodynamic (QHD) theory was introduced to investigate the interaction of a charged particle with 2D and 3D electron gases [14–16], emphasizing on the quantum effects at the micro or nanoscale. However, by now the case of charged particle moving near to 2D or 3D electron gases has not received enough theoretical treatment. The purpose of this work is to present a numerical study of the interaction of the charged particle with a nanosphere. In order to get definite analytical expressions and calculated results of the induced potential and stopping power, the local frequencydependent (LFD) dielectric function [17] will be adopted to describe the bulk polarization of the nanosphere. The Letter is organized as follows: in Section 2 the theoretical model is described in detail, whereas simulation results and discussions are presented in Section 3. Conclusions are presented in Section 4. Gauss units will be adopted throughout the Letter.
(1 ) φind (r, θ, ϕ , t ) =
(k, ω) = 1 −
ω2p
and evaluate the analytical expression of the induced potential, where ω p = [4π n0 e 2 /me ]1/2 is the plasma frequency of the electron gas and γ is the damping factor. In the spherical coordinates system, the potential produced by the external charged particle outside the sphere r > a can be expressed by the series expansion in terms of the Legendre polynomials, as
φext (r, t ) =
⎧ ⎪ ⎨ Z 1 e l∞ =0 ⎪ ⎩ Z e ∞ 1 l =0
rl r0l+1 r0l
r l +1
P l (cos Θ)
(r < r0 ), (1)
P l (cos Θ)
(2 ) φind (r, θ, ϕ , t ) =
P l (cos Θ) =
l m =0
A lm (t )Y lm (θ,
ϕ ),
∞ l
(4)
C lm (t )r l Y lm (θ, ϕ )
for r < a.
(5)
In order to determine the coefficients B lm (t ) and C lm (t ) mentioned above, we take the following two boundary conditions into account. First, the electric potential remains continuous at the spherical surface r = a (1 ) (2 ) φext |r =a + φind r =a = φind r =a .
(6)
And, the normal direction of the electric displacement also keeps continuous, (2 ) ∂φind ∂ (1) . φext + φind = ( ω ) r = a ∂r ∂ r r =a
(7)
Moreover, a partial Fourier transform with respect to the time dependence is introduced here, as
G lm (t ) = −∞
dω
G lm (ω)e −i ωt ,
2π
(8)
where, G lm (t ) stands for B lm (t ) or C lm (t ) mentioned above. For the convenience, we take the transform of A lm (t ) as
Z 1e
Alm (t ) r0l+1 (t )
∞ = −∞
dω 2π
A lm (ω)e −i ωt .
(9)
Substituting Eqs. (1)–(5) into the boundary conditions in Eqs. (6) and (7), and combining them with Eqs. (8) and (9), the coefficients B lm (t ) and C lm (t ) can be obtained, as
B lm (ω) = C lm (ω) =
l[1 − (ω)] l (ω) + (l + 1) 2l + 1 l (ω) + (l + 1)
a2l+1 A lm (ω),
(10)
A lm (ω).
(11)
From the expressions given above, the analytical expression of the induced potential outside the sphere can be written as
∞
(1 )
φind (r, θ, ϕ , t ) = −∞
(2)
for r > a,
l =0 m =0
(r > r0 ),
where, Θ is the angle between r and r0 . Based on the additional formula, the Legendre function in Eq. (1) can be written as
Y lm (θ, ϕ )
and,
∞
(ω+i γ )ω to describe the response of the nanosphere
r l +1
l =0 m =0
2. Theoretical model We consider a point charge Z 1 e, moving outside a nanosphere of radius a, with constant velocity v parallel to the z axis, in a spherical coordinate system (r , θ, ϕ ), as shown in Fig. 1. The vacuum is in the region r > a, and the instantaneous position of the particle can be given by r0 = (r0 , θ0 , ϕ0 = 0). As the particle moves outside the nanosphere at the horizontal distance d, the electrons in the nanosphere with equilibrium density thus will be excited and polarized. The induced potential can be obtained by using the Poisson’s equation coupled with appropriate boundary conditions, in the framework of the dielectric function theory. In the present work, in order to get easily the distribution of the induced potential, we use the local frequency-dependent (LFD) dielectric function
l ∞ B lm (t )
∞ l dω B lm (ω)
2π
l =0 m =0
r l +1
Y lm (θ, ϕ )e −i ωt ,
(12)
while the potential inside the nanosphere sphere takes the form as, (2 )
∞
φind (r, θ, ϕ , t ) = −∞
∞ l dω
2π
l =0 m =0
C lm (ω)r l Y lm (θ, ϕ )e −i ωt .
(13)
S.-B. An et al. / Physics Letters A 376 (2012) 763–767
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Fig. 2. The surface induced potential (normalized by φ0 = e /2π a B ) dependent on the angle θ(π ∼ 0) for different projectile speeds v 0.6v B (a), and v 0.6v B (b) as ϕ = 0. In both of two figures, the black dot is the position of the charged particle.
The stopping power, which is determined by the retarding force acting on the moving charged particle, can be given in spherical coordinate system, as follows, (1 )
S = Z 1e
v · ∇φind v
r =r (t )
0
∂ sin θ ∂ (1 ) φind = Z 1 e cos θ − . ∂r r ∂θ r =r0 (t )
(14)
For the sake of numerical calculation, we introduce dimensionless variables v → v / v B , ω → ω/ω p , l → l/a B , t → t ω p , γ → γ /ω p , where l stands for any quantities of length, ω p = [4π n0 e 2 /me ]1/2 is the plasma frequency in the sphere of uniform density, a B = h¯ 2 /me e 2 is the Bohr radius, v B = e 2 /¯h is the Bohr velocity, and the dimensionless parameter r s = [3/(4π n0 a3B )]1/3 which is introduced to represent the electron gas density in equilibrium. 3. Numerical results In the following calculations, we take the charged particle as a proton (i.e. Z 1 = 1). The spherical radius a = 20a B , dimensionless parameter r s = 3.0, and damping factor γ = 0.002ω p are fixed. In Fig. 2, we calculate the distribution of the induced potential [normalized by φ0 = e /2π a B ] at the spherical surface as a function of the angle θ(π ∼ 0) for different particle speeds when ϕ = ϕ0 = 0. The horizontal distance between the particle and the z axis is d = 22a B and z0 = vt = 10a B is taken as the particle instantaneous position. In Fig. 2(a) in which the particle speeds are less than 0.6v B , the oscillating wake effects appear behind the particle, and the oscillation becomes more obvious with the increasing particle speeds. But in high speed region v 0.6v B in Fig. 2(b), the oscillation becomes less evident and the oscillating magnitude is getting smaller, indicating less electrons can respond to the polarization effect as the particle speed increases further. Especially for the case of v = 2.0v B , no more positive oscillating amplitude can be observed within the valid size. As we are considering electron excitation at a spherical surface, the induced potential approaches to zero much rapidly at the opposite direction of the particle motion, compared with that at a plane surface. The two-dimensional distribution of induced potential is shown, at the surface of the nanosphere in Fig. 3(a), and at r = 7a/8 inside the nanosphere in Fig. 3(b). In both of the figures, we take the particle speed v = 1.0v B , and the particle instantaneous position (r0 , θ0 , ϕ0 ) = (22a B , π /2, 0). From the figures, one can see that, the oscillating amplitudes of the induced potential rapidly decrease and quickly vanish as the azimuthal angle ϕ = 0, showing the electron excitation mainly occurs in the region around ϕ = 0,
Fig. 3. The surface induced potential (a) and the induced potential at r = 7a/8 (b), with the angle θ(π ∼ 0) and ϕ (−π ∼ π ), as well as the particle speed v = 1.0v B , and the particle instantaneous position (r0 , θ0 , ϕ0 ) = (22a B , π /2, 0).
within a shorter distance to the particle. Thus, we can see that the wake potential could not propagate far away from the projectile like those in the two-dimensional infinite plane [14,18] because of the limitation of the spherical structure. More spherical shape effects can be noticed in the following discussions. With the same particle speed v = 1.0v B and particle instantaneous position (r0 , θ0 , ϕ0 ) = (22a B , π /2, 0), we plot the spatial distribution of the induced potential in the sphere at the cross section x = 19.5a B , 10a B and x = 0 in Fig. 4(a)–(c). Similar calculation results are shown in Fig. 4(d)–(f), for the particle instantaneous position d = 22a B and z0 = vt = 10a B . In Figs. 4(a) and 4(d), the induced potential attains much larger maximum absolute values as the cross section is closer to the charged particle, at x = 19.5a B , in which the radius of the circle is 4.4a B . Since the size and shape effects, we can easily notice in all the six figures that, the ordinary wake potential with V-shaped cone structures behind the particle cannot be found clearly. But the oscillating property of the induced potential can still be observed. Fig. 5 shows the stopping power (normalized by S 0 = ( Z 1 e /a B )2 /2π ) as a function of the particle velocity, for different particle positions. The curves of the stopping power with the charged particle at the position z0 = 0 but different horizontal distance d, are shown in Fig. 5(a). It is easily to see that, the values of the stopping power increase when the particle stays closer to the spherical surface, because of strong polarization effect from the electron excitation. For the case of d = 22a B , the peak value locates approximately at v = 0.6v B , in agreement with the results in Fig. 2, in which the oscillating profiles attain amplitude maximum at v = 0.6v B . Figs. 5(b) and 5(c) display similar stopping power results, but for different z0 from −20a B to 15a B , at the fixed d = 22a B . We notice that, the values of the stopping power increase as the particle approaches to the nanosphere but decrease as the particle moves away, parallel with the z axis
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S.-B. An et al. / Physics Letters A 376 (2012) 763–767
Fig. 4. The induced potential distribution at the cross sections in the nanosphere, at x = 19.5a B , 10a B and x = 0, with the particle speed v = 1.0v B and the horizontal distance d = 22a B for z0 = 0 (a)–(c), and z0 = 10a B (d)–(f).
direction. However, in these calculation results, the particle attains its maximum stopping power values at z0 = 4a B , instead of z0 = 0 at which the particle is nearest to the sphere surface. Meanwhile, the peak value location tends to shift to lower speed region slightly as the particle gets close to the nanosphere in Fig. 5(b), but shifts back to higher speed region as the particle moves far away in Fig. 5(c). Similar peak position shift can be also noticed from Fig. 5(a). As we know the LFD dielectric function adopted in this work is a crude approximation, these stopping power results can only show the properties of those from the collective electron excitation. In our future work, we will study similar issues in the framework of quantum hydrodynamic (QHD) theory, in which the single particle excitation will be included by introducing a binary collision term. What attracts us is the negative stopping power that appears in Fig. 5(b), as the incident velocity of the particle is less than 1v B and the position z0 0. To figure out much clearly what happens, we plot curves of the stopping power dependent on the particle position z0 for three different particle speed v, in Fig. 6. In the figure, one can see that, as the particle moves slowly towards the nanosphere, especially for the case of v = 0.4v B , the value of the stopping power becomes negative, suggesting that the particle can even gain energy from the electron polarization at
Fig. 5. Stopping power S (normalized by S 0 = ( Z 1 e /a B )2 /2π ) versus the particle speed v (normalized by v B ), for different horizontal distance d with certain z0 = 0 (a), and for different z0 with d = 22a B in (b) and (c).
Fig. 6. Stopping power S dependent on z0 for different particle speeds v, with d = 22a B .
S.-B. An et al. / Physics Letters A 376 (2012) 763–767
767
the nanosphere surface. And then the stopping power increases and becomes positive as the particle moves on. As the particle speed increases, however, this special phenomenon cannot reproduce any more, as shown for the case v = 2.0v B . Similar results have been obtained in our former work [12]. In our opinion, the reason why the particle can gain energy as the particle approaches and then lose energy when the particle moves away is that the special spherical structure, which is different from the plane electron gas. And thus, the above description can also explain the sudden dive as v 0.5v B in the stopping power curves in Fig. 5(b) and Fig. 5(c).
will be extended based on a more reasonable model by means of quantum hydrodynamic (QHD) theory, to study the quantum effects in the interaction of charged particle with the nanosphere.
4. Summary
References
In summary, we have investigated numerically the interaction of a charged particle with a nanosphere, based on the dielectric response theory. The analytical derivations of the induced potential and stopping power are put forward, as the charged particle moves outside the nanosphere with constant velocity parallel to the z axis. It can be seen from our calculation, the oscillating wake effects occur behind the particle opposite to the particle motion at the spherical surface, as we know from former studies. However, since the limitation of the spherical structure, the surface wake potential could not propagate long distance behind the particle. And the typical V-shaped cone structures of the induced potential cannot be obtained clearly, no matter at the sphere surface or in the bulk. In addition, we analyze the dependence of the stopping power on the charged particle position, and notice that negative stopping power can be achieved as the particle approaches to the nanosphere slowly. Since the LFD dielectric function adopted in this Letter is a relatively crude approximation, our present work
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 10975028), the National Natural Basic Research Program of China (Grant No. 2010CB832901) and Program for New Century Excellent Talents in University (NCET-080073).
[1] S. Facsko, et al., J. Phys. Condens. Matter 21 (2009) 224012. [2] Joshua Jortner, C.N.R. Rao, Pure Appl. Chem. 74 (2002) 1491. [3] Z.C. Zhou, X.S. Zhao, X.T. Zeng, Surface and Coatings Technology 198 (2005) 178. [4] C.W. Kuo, J.Y. Shiu, P. Chen, Chem. Mater. 15 (2003) 2917. [5] J. Neufeld, R.H. Ritchie, Phys. Rev. 98 (1955) 1632; J. Neufeld, R.H. Ritchie, Phys. Rev. 99 (1955) 1125. [6] F.J. Garcia de Abajo, P.M. Echenique, Phys. Rev. B 48 (1993) 13399. [7] Y.N. Wang, W.K. Liu, Chem. Phys. Lett. 254 (1996) 122. [8] F.J. Garcia de Abajo, P.M. Echenique, Phys. Rev. B 46 (1992) 2663. [9] H.G. Eriksson, et al., Phys. Rev. B 31 (1985) 843. [10] B. Satpati, B.N. Dev, Nanotechnology 16 (2005) 572. [11] J.M. Pitarke, et al., Rep. Prog. Phys. 70 (2007) 1. [12] C.Z. Li, Y.H. Song, Y.N. Wang, Nucl. Instr. Meth. B 267 (2009) 3129. [13] N.J.M. Horing, H.C. Tso, G. Gumbs, Phys. Rev. B 36 (1987) 1588. [14] C.Z. Li, Y.H. Song, Y.N. Wang, Chin. Phys. Lett. 25 (2008) 2981. [15] C.Z. Li, Y.H. Song, Y.N. Wang, Phys. Lett. A 372 (2008) 4500. [16] C.Z. Li, Y.H. Song, Y.N. Wang, Phys. Rev. A 79 (2009) 062903. [17] Y.N. Wang, X.L. Deng, T.C. Ma, Nucl. Instr. Meth. B 135 (1998) 164. [18] K. Jiang, L.J. Hou, Y.N. Wang, Chin. Phys. Lett. 22 (2005) 1713.
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Wake potential and stopping power for a charged particle moving outside a nanosphere Sheng-Bai An, Yan-Xia Wang, Yuan-Hong Song ∗ , You-Nian Wang School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Article history: Received 11 October 2011 Received in revised form 13 December 2011 Accepted 16 December 2011 Available online 21 December 2011 Communicated by R. Wu Keywords: Wake potential Stopping power Nanosphere
a b s t r a c t The interaction of a charged particle with a nanosphere is studied based on the dielectric response theory. We obtain the analytical expressions of the induced potential and stopping power, as the charged particle moving outside the nanosphere with a constant velocity. From our results, since the spherical shape limitation, the well-known V-shaped wake effect tracing the particle cannot be observed clearly no matter at the nanosphere surface or in the bulk. Besides, we also find that the particle can even gain energy from the electron polarization as the particle moves to the nanosphere at relatively low velocity. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Interactions of charged particles and matter have received increasing interest in the field of condensed-matter and surface physics over the past decades. Especially since the rapid development of semiconductor technology and new nanotechnology, the investigation of the nano-structured materials or electronics at nanoscale is well concerned. It has been demonstrated that the impact of charged particles on various surfaces can induce surface modification with nanometer dimensions [1]. And for nanostructured materials, suitable control of the properties and response of nanostructures can lead to new devices and technologies [2]. For instance, nanosphere has been successfully applied to fabricate patterns on substrates [3,4], by the method of ion beam sputtering or deposition. It is gradually known that, other than single molecules or infinite bulk materials, nano-structured materials can exhibit different physical and chemical properties, owing to the quantum size and shape effects on their electronic response and dynamics. Moreover, ion beam radiation can be a valid tool to probe the electronic excitation of nanomaterials. As a charged particle approaches solid surface with constant velocity, the electrons in the bulk will be excited and polarized, leading to the induced surface potential, i.e. the wake potential, which has been first studied by Neufeld and Ritchie [5]. Meanwhile, the induced electric field will then give rise to a force that acts on the moving charged particle and makes the particle get or lose energy, so-called the stopping power (i.e. the energy
*
Corresponding author. Tel.: +86 0411 84707939; fax: +86 0411 84707939. E-mail address: [email protected] (Y.-H. Song).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.12.029
lost per unit path length traveled by the charged particle). The well-known dielectric response theory has found wide applications in the investigation of ions and surface interaction [6–9]. Using the full random-phase approximation (RPA) for the bulk dielectric function [6], the calculation results assure us of the study of lowvelocity surface wake and show good agreement with those calculated by plasmon-pole approximation (PLA) in the high-velocity limit (v > v F , where v F is the Fermi velocity). Wang and Liu [7] use the local field correction (LFC) for the bulk dielectric function to calculate the induced potential of an ion moving slowly near a solid surface, with the effect of exchange-correlation interactions of electrons taken into account. It is found that the magnitude of the induced potential based on LFC is always larger than that from the RPA, and thus for low velocities, the exchange-correlation effects cannot be neglected. A detailed investigation of the wake potential has been made by Garcia and Echenique [8], in which the hydrodynamic approximation (HA) and plasmon-pole approximation (PLA) to (k, ω) are adopted. For high-velocity ions, it was shown that the asymptotic behavior of the wake potential is quite similar in both the HA and PLA formulation. The only difference between them is that the PLA includes the single particle excitations as well as the plasmon dispersion. So far, with technological development in the area of miniaturized devices, a wide field of studies of small systems with typical sizes ranging between the submicron and nanometer scale has been produced [10]. In addition, recent renewed interest in surface or interface plasmon has attracted more attention in the investigation of the nano-structured materials or electronics at the nanoscale [11,12]. Horing et al. [13] first investigated the energy loss of fast particles moving parallel to 2D electron gas in the framework of the RPA dielectric function within linear response
764
S.-B. An et al. / Physics Letters A 376 (2012) 763–767 m)! m where Alm (t ) = δ2 ((ll− P (cos θ0 ), and the spherical harmonic m +m)! l function can be written as,
Y lm (θ, ϕ ) = (cos mϕ cos mϕ0 + sin mϕ sin mϕ0 ) P lm (cos θ),
(3)
with δm = 2 for m = 0 and δm = 1 for m = 0. Since ϕ0 = 0, the spherical harmonic function can be written as Y lm (θ, ϕ ) = P lm (cos θ) cos mϕ . Meanwhile, in the spherical coordinate system, the induced potential due to the excitation of the electrons satisfies the Laplace’s equation and can be expanded as,
Fig. 1. Description of the system, for a charged particle Z 1 e moving outside the nanosphere with constant velocity v parallel to the axis z at a horizontal distance d in spherical coordinates systems (r , θ, ϕ ).
theory. Recently, besides the dielectric response theory, the quantum hydrodynamic (QHD) theory was introduced to investigate the interaction of a charged particle with 2D and 3D electron gases [14–16], emphasizing on the quantum effects at the micro or nanoscale. However, by now the case of charged particle moving near to 2D or 3D electron gases has not received enough theoretical treatment. The purpose of this work is to present a numerical study of the interaction of the charged particle with a nanosphere. In order to get definite analytical expressions and calculated results of the induced potential and stopping power, the local frequencydependent (LFD) dielectric function [17] will be adopted to describe the bulk polarization of the nanosphere. The Letter is organized as follows: in Section 2 the theoretical model is described in detail, whereas simulation results and discussions are presented in Section 3. Conclusions are presented in Section 4. Gauss units will be adopted throughout the Letter.
(1 ) φind (r, θ, ϕ , t ) =
(k, ω) = 1 −
ω2p
and evaluate the analytical expression of the induced potential, where ω p = [4π n0 e 2 /me ]1/2 is the plasma frequency of the electron gas and γ is the damping factor. In the spherical coordinates system, the potential produced by the external charged particle outside the sphere r > a can be expressed by the series expansion in terms of the Legendre polynomials, as
φext (r, t ) =
⎧ ⎪ ⎨ Z 1 e l∞ =0 ⎪ ⎩ Z e ∞ 1 l =0
rl r0l+1 r0l
r l +1
P l (cos Θ)
(r < r0 ), (1)
P l (cos Θ)
(2 ) φind (r, θ, ϕ , t ) =
P l (cos Θ) =
l m =0
A lm (t )Y lm (θ,
ϕ ),
∞ l
(4)
C lm (t )r l Y lm (θ, ϕ )
for r < a.
(5)
In order to determine the coefficients B lm (t ) and C lm (t ) mentioned above, we take the following two boundary conditions into account. First, the electric potential remains continuous at the spherical surface r = a (1 ) (2 ) φext |r =a + φind r =a = φind r =a .
(6)
And, the normal direction of the electric displacement also keeps continuous, (2 ) ∂φind ∂ (1) . φext + φind = ( ω ) r = a ∂r ∂ r r =a
(7)
Moreover, a partial Fourier transform with respect to the time dependence is introduced here, as
G lm (t ) = −∞
dω
G lm (ω)e −i ωt ,
2π
(8)
where, G lm (t ) stands for B lm (t ) or C lm (t ) mentioned above. For the convenience, we take the transform of A lm (t ) as
Z 1e
Alm (t ) r0l+1 (t )
∞ = −∞
dω 2π
A lm (ω)e −i ωt .
(9)
Substituting Eqs. (1)–(5) into the boundary conditions in Eqs. (6) and (7), and combining them with Eqs. (8) and (9), the coefficients B lm (t ) and C lm (t ) can be obtained, as
B lm (ω) = C lm (ω) =
l[1 − (ω)] l (ω) + (l + 1) 2l + 1 l (ω) + (l + 1)
a2l+1 A lm (ω),
(10)
A lm (ω).
(11)
From the expressions given above, the analytical expression of the induced potential outside the sphere can be written as
∞
(1 )
φind (r, θ, ϕ , t ) = −∞
(2)
for r > a,
l =0 m =0
(r > r0 ),
where, Θ is the angle between r and r0 . Based on the additional formula, the Legendre function in Eq. (1) can be written as
Y lm (θ, ϕ )
and,
∞
(ω+i γ )ω to describe the response of the nanosphere
r l +1
l =0 m =0
2. Theoretical model We consider a point charge Z 1 e, moving outside a nanosphere of radius a, with constant velocity v parallel to the z axis, in a spherical coordinate system (r , θ, ϕ ), as shown in Fig. 1. The vacuum is in the region r > a, and the instantaneous position of the particle can be given by r0 = (r0 , θ0 , ϕ0 = 0). As the particle moves outside the nanosphere at the horizontal distance d, the electrons in the nanosphere with equilibrium density thus will be excited and polarized. The induced potential can be obtained by using the Poisson’s equation coupled with appropriate boundary conditions, in the framework of the dielectric function theory. In the present work, in order to get easily the distribution of the induced potential, we use the local frequency-dependent (LFD) dielectric function
l ∞ B lm (t )
∞ l dω B lm (ω)
2π
l =0 m =0
r l +1
Y lm (θ, ϕ )e −i ωt ,
(12)
while the potential inside the nanosphere sphere takes the form as, (2 )
∞
φind (r, θ, ϕ , t ) = −∞
∞ l dω
2π
l =0 m =0
C lm (ω)r l Y lm (θ, ϕ )e −i ωt .
(13)
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Fig. 2. The surface induced potential (normalized by φ0 = e /2π a B ) dependent on the angle θ(π ∼ 0) for different projectile speeds v 0.6v B (a), and v 0.6v B (b) as ϕ = 0. In both of two figures, the black dot is the position of the charged particle.
The stopping power, which is determined by the retarding force acting on the moving charged particle, can be given in spherical coordinate system, as follows, (1 )
S = Z 1e
v · ∇φind v
r =r (t )
0
∂ sin θ ∂ (1 ) φind = Z 1 e cos θ − . ∂r r ∂θ r =r0 (t )
(14)
For the sake of numerical calculation, we introduce dimensionless variables v → v / v B , ω → ω/ω p , l → l/a B , t → t ω p , γ → γ /ω p , where l stands for any quantities of length, ω p = [4π n0 e 2 /me ]1/2 is the plasma frequency in the sphere of uniform density, a B = h¯ 2 /me e 2 is the Bohr radius, v B = e 2 /¯h is the Bohr velocity, and the dimensionless parameter r s = [3/(4π n0 a3B )]1/3 which is introduced to represent the electron gas density in equilibrium. 3. Numerical results In the following calculations, we take the charged particle as a proton (i.e. Z 1 = 1). The spherical radius a = 20a B , dimensionless parameter r s = 3.0, and damping factor γ = 0.002ω p are fixed. In Fig. 2, we calculate the distribution of the induced potential [normalized by φ0 = e /2π a B ] at the spherical surface as a function of the angle θ(π ∼ 0) for different particle speeds when ϕ = ϕ0 = 0. The horizontal distance between the particle and the z axis is d = 22a B and z0 = vt = 10a B is taken as the particle instantaneous position. In Fig. 2(a) in which the particle speeds are less than 0.6v B , the oscillating wake effects appear behind the particle, and the oscillation becomes more obvious with the increasing particle speeds. But in high speed region v 0.6v B in Fig. 2(b), the oscillation becomes less evident and the oscillating magnitude is getting smaller, indicating less electrons can respond to the polarization effect as the particle speed increases further. Especially for the case of v = 2.0v B , no more positive oscillating amplitude can be observed within the valid size. As we are considering electron excitation at a spherical surface, the induced potential approaches to zero much rapidly at the opposite direction of the particle motion, compared with that at a plane surface. The two-dimensional distribution of induced potential is shown, at the surface of the nanosphere in Fig. 3(a), and at r = 7a/8 inside the nanosphere in Fig. 3(b). In both of the figures, we take the particle speed v = 1.0v B , and the particle instantaneous position (r0 , θ0 , ϕ0 ) = (22a B , π /2, 0). From the figures, one can see that, the oscillating amplitudes of the induced potential rapidly decrease and quickly vanish as the azimuthal angle ϕ = 0, showing the electron excitation mainly occurs in the region around ϕ = 0,
Fig. 3. The surface induced potential (a) and the induced potential at r = 7a/8 (b), with the angle θ(π ∼ 0) and ϕ (−π ∼ π ), as well as the particle speed v = 1.0v B , and the particle instantaneous position (r0 , θ0 , ϕ0 ) = (22a B , π /2, 0).
within a shorter distance to the particle. Thus, we can see that the wake potential could not propagate far away from the projectile like those in the two-dimensional infinite plane [14,18] because of the limitation of the spherical structure. More spherical shape effects can be noticed in the following discussions. With the same particle speed v = 1.0v B and particle instantaneous position (r0 , θ0 , ϕ0 ) = (22a B , π /2, 0), we plot the spatial distribution of the induced potential in the sphere at the cross section x = 19.5a B , 10a B and x = 0 in Fig. 4(a)–(c). Similar calculation results are shown in Fig. 4(d)–(f), for the particle instantaneous position d = 22a B and z0 = vt = 10a B . In Figs. 4(a) and 4(d), the induced potential attains much larger maximum absolute values as the cross section is closer to the charged particle, at x = 19.5a B , in which the radius of the circle is 4.4a B . Since the size and shape effects, we can easily notice in all the six figures that, the ordinary wake potential with V-shaped cone structures behind the particle cannot be found clearly. But the oscillating property of the induced potential can still be observed. Fig. 5 shows the stopping power (normalized by S 0 = ( Z 1 e /a B )2 /2π ) as a function of the particle velocity, for different particle positions. The curves of the stopping power with the charged particle at the position z0 = 0 but different horizontal distance d, are shown in Fig. 5(a). It is easily to see that, the values of the stopping power increase when the particle stays closer to the spherical surface, because of strong polarization effect from the electron excitation. For the case of d = 22a B , the peak value locates approximately at v = 0.6v B , in agreement with the results in Fig. 2, in which the oscillating profiles attain amplitude maximum at v = 0.6v B . Figs. 5(b) and 5(c) display similar stopping power results, but for different z0 from −20a B to 15a B , at the fixed d = 22a B . We notice that, the values of the stopping power increase as the particle approaches to the nanosphere but decrease as the particle moves away, parallel with the z axis
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Fig. 4. The induced potential distribution at the cross sections in the nanosphere, at x = 19.5a B , 10a B and x = 0, with the particle speed v = 1.0v B and the horizontal distance d = 22a B for z0 = 0 (a)–(c), and z0 = 10a B (d)–(f).
direction. However, in these calculation results, the particle attains its maximum stopping power values at z0 = 4a B , instead of z0 = 0 at which the particle is nearest to the sphere surface. Meanwhile, the peak value location tends to shift to lower speed region slightly as the particle gets close to the nanosphere in Fig. 5(b), but shifts back to higher speed region as the particle moves far away in Fig. 5(c). Similar peak position shift can be also noticed from Fig. 5(a). As we know the LFD dielectric function adopted in this work is a crude approximation, these stopping power results can only show the properties of those from the collective electron excitation. In our future work, we will study similar issues in the framework of quantum hydrodynamic (QHD) theory, in which the single particle excitation will be included by introducing a binary collision term. What attracts us is the negative stopping power that appears in Fig. 5(b), as the incident velocity of the particle is less than 1v B and the position z0 0. To figure out much clearly what happens, we plot curves of the stopping power dependent on the particle position z0 for three different particle speed v, in Fig. 6. In the figure, one can see that, as the particle moves slowly towards the nanosphere, especially for the case of v = 0.4v B , the value of the stopping power becomes negative, suggesting that the particle can even gain energy from the electron polarization at
Fig. 5. Stopping power S (normalized by S 0 = ( Z 1 e /a B )2 /2π ) versus the particle speed v (normalized by v B ), for different horizontal distance d with certain z0 = 0 (a), and for different z0 with d = 22a B in (b) and (c).
Fig. 6. Stopping power S dependent on z0 for different particle speeds v, with d = 22a B .
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the nanosphere surface. And then the stopping power increases and becomes positive as the particle moves on. As the particle speed increases, however, this special phenomenon cannot reproduce any more, as shown for the case v = 2.0v B . Similar results have been obtained in our former work [12]. In our opinion, the reason why the particle can gain energy as the particle approaches and then lose energy when the particle moves away is that the special spherical structure, which is different from the plane electron gas. And thus, the above description can also explain the sudden dive as v 0.5v B in the stopping power curves in Fig. 5(b) and Fig. 5(c).
will be extended based on a more reasonable model by means of quantum hydrodynamic (QHD) theory, to study the quantum effects in the interaction of charged particle with the nanosphere.
4. Summary
References
In summary, we have investigated numerically the interaction of a charged particle with a nanosphere, based on the dielectric response theory. The analytical derivations of the induced potential and stopping power are put forward, as the charged particle moves outside the nanosphere with constant velocity parallel to the z axis. It can be seen from our calculation, the oscillating wake effects occur behind the particle opposite to the particle motion at the spherical surface, as we know from former studies. However, since the limitation of the spherical structure, the surface wake potential could not propagate long distance behind the particle. And the typical V-shaped cone structures of the induced potential cannot be obtained clearly, no matter at the sphere surface or in the bulk. In addition, we analyze the dependence of the stopping power on the charged particle position, and notice that negative stopping power can be achieved as the particle approaches to the nanosphere slowly. Since the LFD dielectric function adopted in this Letter is a relatively crude approximation, our present work
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 10975028), the National Natural Basic Research Program of China (Grant No. 2010CB832901) and Program for New Century Excellent Talents in University (NCET-080073).
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