- Email: [email protected]
Influence of velocity difference of correlated particles on the stopping power and vicinage effect
Nuclear Instruments and Methods in Physics Research B 267 (2009) 3175–3178
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Influence of velocity difference of correlated particles on the stopping power and vicinage effect Jean de Dieu Mugiraneza, Gaolong Zhang, Xiaoyun Le *, Liying Wang, Cuihua Rong, Ying Wang Department of Physics, Beihang University, Beijing 100191, China
a r t i c l e
i n f o
Article history: Available online 16 June 2009 Keywords: Correlated particles Velocity difference Stopping power Vicinage effect
a b s t r a c t We theoretically investigate the influence of the velocity difference of charged particles moving closely in a metallic target on the stopping power and vicinage function, as well as the energy loss of a pair of charged particles penetrating coherently in the target medium. These correlated charged particles are considered as a degenerate electron gas with appropriate density. Based on the dielectric formalism, the mathematical expressions for the stopping power, vicinage effects and related quantities for two correlated protons in an aluminium target are given. Plasmon-pole approximation without dispersion is used to describe the target response. We show that the energy loss is slightly enhanced with a small relative velocity difference. The interference between such protons exhibits a maximum at certain velocity, and exhibits an oscillating behaviour when product of the relative difference velocity and time is beyond a threshold value. The results can be applied to, for example, the energy deposition of high current density ion beams. Ó 2009 Published by Elsevier B.V.
1. Introduction Charged particles give rise to atom collision and electronic excitations in the target when penetrating the solid media. During the interaction, the energetic charged particles will lose energy in the materials. The lost energy along the unit length is defined as the stopping power or energy loss for energetic charged particles in a target material, which has been of great interest since the discovery of radioactivity by Marie Curie at the beginning of the last century. Early pioneering theoretical studies by Thomson and Bohr were followed by quantum theory of particle stopping developed by H. Bethe, F. Bloch, et al. More reliable calculations of energy loss by Lindhard [1] have been made through the linear response theory by modelling the solid target medium as a degenerate electron gas. Such model provides a comprehensive description of different excitation in medium and also shows the main features of the velocity dependence on the stopping power of metals. These features include proportionality with velocity in the adiabatic lowvelocity range, a threshold for plasmon excitation and maximum stopping power at intermediate velocities, and a decreasing behaviour at high energies, in agreement with the Bethe theory [2]. However, this model is not fit for low-velocity ranges since the strong interaction effects in this range cannot simply be described by linear or perturbative approximations [3]. Many other theories for
* Corresponding author. Tel./fax: +86 10 82317942. E-mail address: [email protected] (X. Le). 0168-583X/$ - see front matter Ó 2009 Published by Elsevier B.V. doi:10.1016/j.nimb.2009.06.071
different kinds of particles and energy ranges have been developed [4,5]. For high-velocity projectiles or clusters, the energy loss may be mainly due to collective and single particle excitation in the target medium. The correlated motion of the cluster and molecule constituents has been intensively studied and the induced vicinage effects have been satisfactorily evaluated up to the first-order perturbation theory within the dielectric formalism of the stopping power [6,7]. In these previous studies, it was assumed that the difference between components velocities of cluster or molecule is negligible and the cluster moves as a compact particle through the medium of the target. The objective of this paper is to study the influence of the difference between correlated particles velocity on the stopping power of correlated charged particles in a degenerate electron gas. The simplest and yet physically relevant way is via the stopping power of a dicluster, an ion pair of particles. Numerical expression of the stopping power for various diclusters in a degenerate electron gas is calculated within a linear-response theory and a simple plasmon-pole approximation (PPA). For the sake of mathematical simplicity, the vicinage force induced by one particle to another is extracted and compared with the stopping power for an independent projectile. The present study is of great interest since it gives additional knowledge on the complex problem of stopping power and can lead to application in energy deposition of high current density ion beams such as intense pulsed ion beam (IPIB). These beam pulses are like Gaussian and the beam particles are close enough to interfere with each other.
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J. de Dieu Mugiraneza et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3175–3178
2. Theoretical calculation of the stopping power Slowing down of ions in the target is related to the electronic structure and dynamics of the involved atoms, molecules, or a crystal. In the dielectric formalism the response of the target to an external perturbation is completely characterized by its dielectric function e(k,x), where k and x are the momentum and the energy transferred to the electronic excitation in the medium, respectively. Thus, the crucial factor in calculating and approximating the stopping power is the linear response function e(k,x) of the target material. Various models have been proposed for a wide range of materials and the dielectric function obtained by Lindhard [1] for a free-electron gas has been a basis of many applications in solid-states physics and particle–solid interaction phenomena. For a bunch of work on calculation of stopping power for clusters, ions and electron, other models have been used including semiconductor models, Mermin dielectric function [8] or extension of the Drude model [9]. Mermin dielectric function is the most accurate one, in which both outer-electron excitation and inner shell electron ionization are taken into account [10], but the function is tedious. To illustrate the vicinage effects between two particles moving at different velocity, we make some assumptions for simplicity. We calculate the effect between two aligned cluster-constituents moving at velocities greater than the Fermi velocity. We consider the simplest model of the dielectric function of a Jellium, a dense electron gas neutralized by a positive background, in which the lattice effect is included through effective mass of an electron. The model is a reasonable fit for target materials like Cu and Al. A plasmon-pole approximation to eðk; xÞ for an electron gas is used in order to obtain easily the analytical results. The medium is assumed to be non-dispersive and the relaxation rate of excitation 1=s is zero. In other words, the finite time of plasmon excitation is not considered. Basbas and Ritchie [11] employed a simplified form that exhibits collective and single-particle effects that can be expressed by:
1 Im eðk; wÞ " !# 2 p w2p hk ; Hðkc kÞdðw wp Þ þ Hðk kc Þd w ¼ 2 w 2m
1
0
pw2p 1 Im wdw ¼ : eðk; wÞ 2
ð1Þ
ð2Þ
During our calculation, atomic units (e = m = ⁄ = 1) are used unless stated otherwise. The real part of the energy loss function is calculated through Kramers-Kronig relation with the imaginary part:
Re
Z 1 1 1 1 1 1¼ P dx0 0 Im eðk; xÞ p 1 x x eðk; x0 Þ ¼
x2p 1 ; 2xk x xk
ð3Þ
with 2
xk ¼ Hðkc kÞxp þ Hðk kc Þ
k h : 2m
i
ð5Þ where r ij is the inter-ionic distance, vi and vj are velocities of the correlated particles, s is the relaxation time, which is related to the relaxation rate if excitation. A widely used representation of the charge density of the projectile is the statistical model of Brandt and Kitagawa (BK) [12]. The distribution of N electrons, which are bound at an isolated ion with the nuclear charge Z1, is modelled by a radically symmetric function Nð4pK2 rÞ1 expðr=KÞ, where r is the electron-nucleus distance. The ion size is defined by the parameter K with the value 2=3 K ¼ 0:48N , and its Fourier transform is expressed by: Z 1 N=7
qðBKÞ ðkÞ ¼ Z 1
ð4Þ
N 1 þ ðkKÞ2
ð6Þ
:
With the Eq. (3), the self-induced stopping power is approximated and an interference term is denoted as correlated stopping power between a di-cluster, which are:
Si
Iij ¼
where H(x) is the Heaviside unit-step function, wp is the plasma enhÞ1=2 allows the ergy of the electron gas and the choice kc ¼ ð2mwp = two functions in Eq. (1) to coincide at k ¼ kc in the k w plane. The first term describes the response due to non-dispersive plasmon excitation in the region of k < kc , while the second term describes free-electron recoil in the region of k > kc (single-particle excitations). The approximation function satisfies the sum rule for all values of k, i.e.:
Z
! ! Furthermore, in the region, h=2mv < jrij v ij j < v =xp , the two correlated ions are separated from comparison with the closest interaction with target electrons (electron-hole pairs excitation). They behave simultaneously as united ions with respect to the most distant collective interactions (resonant or plasmon excitations) with an adiabatic distance v/xp. We obtain the stopping power for a di-proton. The stopping power for a H+–H+ pair is given by: X x2p 2mv 2 i Sin ðv Þ ln hxp v2 i¼1;2 i v v Z Z kv i sin iv j xs X 2 dk 1 i ! ! x d x J ðkr Þ Im Iðr ij ; v ij Þ ¼ ; ij 0 v i v j p i¼1;2;i–j kv 2i 0 eðk; xÞ v xs
x2p
v 2i
ln
1
Z
pv 2i x2p Z
2mv 21 ; h xp
ð7Þ
v i v j Z kv i sin x s i vi dk sinðkrÞ 1 ; dx Im v i v j k kr eðk; xÞ 0 v s i
v i v j
dk sinðkrÞ sinð v i xk sÞ ¼ 2 ; v i v j v i kmin k kr v i s xk ( v v x2p sinð iv i j xp sÞ Z kc dk sinðkrÞ ¼ 2 v i v j kr vi kmin k v i sxp v v Z kmax dk sinðkrÞ sinð i j k2 sÞ) 2v i þ2v i : 4 kc rðv i v j Þk s kmax
ð8Þ
The range of our interest is between kmin=xp/v and kmax=2mv/⁄. 3. Results and discussion Fig. 1 illustrates the dependence of interference term of the stopping power on the particle velocity with rij = 1 and v2 = 5. The term, obtained from Eq. (8), not only includes the inter-ionic distance rij of the correlated particles, but also features the difference between particle velocities vij. The plasma frequency xp = 0.55 a.u was chosen, corresponding to the plasma frequency of the electron gas of Al and the first peak of the energy loss function of Ti, respectively. The individual stopping Sp is compared to the interference term Iij at different relaxation times with s = 1, 2 and 5. As shown in Fig. 1, the interference term Iij(r,v,t) exhibits a similar behaviour as the stopping power of a proton Sp but is less than the magnitude of Sp. Above the threshold, the interference term
J. de Dieu Mugiraneza et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3175–3178
Fig. 1. The self-induced term and interference term of the stopping power as function of velocity calculated by Eq. (8) at different relaxation times.
goes through a pronounced maximum and then falls off with velocity increasing. Since the characteristic wave number is between kmin = xp/v and kmax = 2mv/⁄, the number of contributing k diminishes with the increase of velocity for all the terms. Roughly speaking, half of all possible excitations will interfere constructively as compared with the individual stopping term Sp. In Fig. 2, the interference term Iij(rij,vij,s) decreases as the time elapses and the cutoff velocity shifts to higher velocities. The three (two of them shown in Fig. 2) curves intersect at v = 5 for the second particle, in which the interference term between two correlated cluster or molecule components that moves at the same velocity is reduced to a much lower value. The peculiarity of the velocity-dependent interference term is that an oscillating behaviour occurs at small velocity range as compared with that of the neighbouring charged particle. With the time increase, the cutoff velocity of larger constructive interference shifts closely to that of the neighbouring ion. This phenomenon has not been observed for correlated particles moving at the same velocity. Fig. 3 shows that the cutoff velocity for the constructive interaction does not change for the correlated particles with a fixed inter-ionic distance. As illustrated in Fig. 4, the interaction decreases with inter-particle distance increasing. Another feature of the interference between the moving correlated particles lies in the ratio of the interference term to the self-induced stopping power, R ¼ Iij ðrij ; v ij ; sÞ=Si ðv Þ. When the two protons have the same velocity, the vicinage effect reduces to the one of dicluster or diatomic molecule. For velocities around v2, a slight enhancement of stop-
Fig. 2. Comparison among the proton stopping power SP, the interference term Iij(rij) for a pair of correlated ions separated from a distance r = 1 and the velocity dependant Iij(r,v,s) with r = 1 and t = 6 in a medium of wp = 0.55.
3177
Fig. 3. The proton stopping power and velocity-independent inter-ference term with different inter-ionic distance r = 5 and 3.
Fig. 4. Vicinage effect (ratio R = Iij/S0) as function of time for correlated pair of protons, which corresponds to the familiar vicinage for a di-cluster moving at velocity. The other curves show the vicinage of correlated protons with one velocity at v2 = 5 and another velocity at different times with s = 0.5, 2 and 5. Inter-ionic distance is at r = 1and the medium plasma is xp = 0.55 a.u. in above calculation.
ping power is observed. It symmetrically increases with the increase of the difference of velocities and reaches a maximum at a certain velocity, then drops. The velocity corresponding to the peak changes with the interaction time. Namely, it should satisfy certain relation with the relaxation time of the medium. In Fig. 4, we can easily find that the range of velocities with interfering constructively becomes narrow with the increase of time. At very short time the energy loss enhancement becomes larger than that in the lower velocity range. For the same particle velocities, the expression reduces to that of molecules and ion cluster. The enhancement of energy loss due to difference between velocities of correlated particles is very small as compared with the self-induced stopping power. A narrow range of velocity in a given short interaction time is observed. The enhancement or decrease in energy loss due to this effect can be negligible for thermal and mechanical application because slight amount of energy loss will not affect the thermal related excitation such as phonons, electron–hole and other phenomena requiring higher energy. However, for the low energy excitation like spin excitation, its contribution cannot be neglected. At present, its use is challenging because of the strict condition needed for the ion beam and pulse. More theoretical and experimental studies on this aspect may lead to a better understanding of interaction of ion beam such as IPIB with the materials.
3178
J. de Dieu Mugiraneza et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3175–3178
4. Conclusions
Acknowledgment
We theoretically investigate the influence of the velocity difference of charged particles moving closely in a metallic target on the stopping power and vicinage function, as well as the energy loss of a pair of charged particles penetrating coherently in the target medium. Using the dielectric formalism, the mathematical expressions have been given for the stopping power of correlated charged particles penetrating the medium at different velocities. When aligned projectile protons move at high velocities and target medium is not dispersive but the corresponding dielectric function exhibits both single and collective excitation, the interaction between such protons in their vicinities is weakened with the increase of time. Such interaction depends on both absolute and relative velocity difference. It is observed that energy loss is slightly enhanced with a small relative velocity difference. The interference between such protons exhibits a maximum at certain velocity, and exhibits an oscillating behaviour when product of the relative difference velocity and time is beyond a threshold value. This study provides additional knowledge on the complex problem of stopping power and can apply for the energy deposition of high current density ion beams such as intense pulsed ion beam.
This work is supported by the National Natural Science Foundation of China under Grant No. 60572177. References [1] J. Lindhard, K. Dan, Vidensk. Selsk. Mat. Fys. Medd. 28 (1954) 1. [2] M.A. Kumakhov, F.F. Komarov, Energy Loss and Ion Ranges in Solids, Gordon and Breach, New York, 1981. [3] P.M. Echenique, M.E. Uranga, in: A. Gras-Martí, H.M. Urbassek, N.R. Arista, F. Flores (Eds.), Interaction of Charged Particles with Solids and Surfaces, Plenum, New York, 1991. [4] P. Sigmund, Phys. Rev. A 26 (1982) 2497. [5] J. Calera Rubio, A. Gras-Martı´, N.R. Arista, in: R.A. Baragiola (Ed.), Ionization of Solids by Heavy Particles, Plenum, New York, 1993, Nucl. Instrum. Methods Phys. Res. B 93 (1994) 137. [6] W. Brandt, A. Ratkowsky, et al., Phys. Rev. Lett. 33 (1974) 1325. [7] Nestor R. Arista, Phys. Rev. B 18 (1978) 1. [8] N.D. Mermin, Phys. Rev. B 1 (1970) 2362. [9] J.C. Ashley, J. Phys.: Condens. Matter. 3 (1991) 2741. [10] S. Heredia-Avalos, J.C. Moreno-Marín, I. Abril, R. Garcia-Molina, Nucl. Instr. and Meth. B 230 (2005) 118. [11] G. Basbas, R.H. Ritchie, Phys. Rev. A 25 (1982) 1943. [12] W. Brandt, M. Kitagawa, Phys. Rev. B 25 (1982) 5631.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Influence of velocity difference of correlated particles on the stopping power and vicinage effect Jean de Dieu Mugiraneza, Gaolong Zhang, Xiaoyun Le *, Liying Wang, Cuihua Rong, Ying Wang Department of Physics, Beihang University, Beijing 100191, China
a r t i c l e
i n f o
Article history: Available online 16 June 2009 Keywords: Correlated particles Velocity difference Stopping power Vicinage effect
a b s t r a c t We theoretically investigate the influence of the velocity difference of charged particles moving closely in a metallic target on the stopping power and vicinage function, as well as the energy loss of a pair of charged particles penetrating coherently in the target medium. These correlated charged particles are considered as a degenerate electron gas with appropriate density. Based on the dielectric formalism, the mathematical expressions for the stopping power, vicinage effects and related quantities for two correlated protons in an aluminium target are given. Plasmon-pole approximation without dispersion is used to describe the target response. We show that the energy loss is slightly enhanced with a small relative velocity difference. The interference between such protons exhibits a maximum at certain velocity, and exhibits an oscillating behaviour when product of the relative difference velocity and time is beyond a threshold value. The results can be applied to, for example, the energy deposition of high current density ion beams. Ó 2009 Published by Elsevier B.V.
1. Introduction Charged particles give rise to atom collision and electronic excitations in the target when penetrating the solid media. During the interaction, the energetic charged particles will lose energy in the materials. The lost energy along the unit length is defined as the stopping power or energy loss for energetic charged particles in a target material, which has been of great interest since the discovery of radioactivity by Marie Curie at the beginning of the last century. Early pioneering theoretical studies by Thomson and Bohr were followed by quantum theory of particle stopping developed by H. Bethe, F. Bloch, et al. More reliable calculations of energy loss by Lindhard [1] have been made through the linear response theory by modelling the solid target medium as a degenerate electron gas. Such model provides a comprehensive description of different excitation in medium and also shows the main features of the velocity dependence on the stopping power of metals. These features include proportionality with velocity in the adiabatic lowvelocity range, a threshold for plasmon excitation and maximum stopping power at intermediate velocities, and a decreasing behaviour at high energies, in agreement with the Bethe theory [2]. However, this model is not fit for low-velocity ranges since the strong interaction effects in this range cannot simply be described by linear or perturbative approximations [3]. Many other theories for
* Corresponding author. Tel./fax: +86 10 82317942. E-mail address: [email protected] (X. Le). 0168-583X/$ - see front matter Ó 2009 Published by Elsevier B.V. doi:10.1016/j.nimb.2009.06.071
different kinds of particles and energy ranges have been developed [4,5]. For high-velocity projectiles or clusters, the energy loss may be mainly due to collective and single particle excitation in the target medium. The correlated motion of the cluster and molecule constituents has been intensively studied and the induced vicinage effects have been satisfactorily evaluated up to the first-order perturbation theory within the dielectric formalism of the stopping power [6,7]. In these previous studies, it was assumed that the difference between components velocities of cluster or molecule is negligible and the cluster moves as a compact particle through the medium of the target. The objective of this paper is to study the influence of the difference between correlated particles velocity on the stopping power of correlated charged particles in a degenerate electron gas. The simplest and yet physically relevant way is via the stopping power of a dicluster, an ion pair of particles. Numerical expression of the stopping power for various diclusters in a degenerate electron gas is calculated within a linear-response theory and a simple plasmon-pole approximation (PPA). For the sake of mathematical simplicity, the vicinage force induced by one particle to another is extracted and compared with the stopping power for an independent projectile. The present study is of great interest since it gives additional knowledge on the complex problem of stopping power and can lead to application in energy deposition of high current density ion beams such as intense pulsed ion beam (IPIB). These beam pulses are like Gaussian and the beam particles are close enough to interfere with each other.
3176
J. de Dieu Mugiraneza et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3175–3178
2. Theoretical calculation of the stopping power Slowing down of ions in the target is related to the electronic structure and dynamics of the involved atoms, molecules, or a crystal. In the dielectric formalism the response of the target to an external perturbation is completely characterized by its dielectric function e(k,x), where k and x are the momentum and the energy transferred to the electronic excitation in the medium, respectively. Thus, the crucial factor in calculating and approximating the stopping power is the linear response function e(k,x) of the target material. Various models have been proposed for a wide range of materials and the dielectric function obtained by Lindhard [1] for a free-electron gas has been a basis of many applications in solid-states physics and particle–solid interaction phenomena. For a bunch of work on calculation of stopping power for clusters, ions and electron, other models have been used including semiconductor models, Mermin dielectric function [8] or extension of the Drude model [9]. Mermin dielectric function is the most accurate one, in which both outer-electron excitation and inner shell electron ionization are taken into account [10], but the function is tedious. To illustrate the vicinage effects between two particles moving at different velocity, we make some assumptions for simplicity. We calculate the effect between two aligned cluster-constituents moving at velocities greater than the Fermi velocity. We consider the simplest model of the dielectric function of a Jellium, a dense electron gas neutralized by a positive background, in which the lattice effect is included through effective mass of an electron. The model is a reasonable fit for target materials like Cu and Al. A plasmon-pole approximation to eðk; xÞ for an electron gas is used in order to obtain easily the analytical results. The medium is assumed to be non-dispersive and the relaxation rate of excitation 1=s is zero. In other words, the finite time of plasmon excitation is not considered. Basbas and Ritchie [11] employed a simplified form that exhibits collective and single-particle effects that can be expressed by:
1 Im eðk; wÞ " !# 2 p w2p hk ; Hðkc kÞdðw wp Þ þ Hðk kc Þd w ¼ 2 w 2m
1
0
pw2p 1 Im wdw ¼ : eðk; wÞ 2
ð1Þ
ð2Þ
During our calculation, atomic units (e = m = ⁄ = 1) are used unless stated otherwise. The real part of the energy loss function is calculated through Kramers-Kronig relation with the imaginary part:
Re
Z 1 1 1 1 1 1¼ P dx0 0 Im eðk; xÞ p 1 x x eðk; x0 Þ ¼
x2p 1 ; 2xk x xk
ð3Þ
with 2
xk ¼ Hðkc kÞxp þ Hðk kc Þ
k h : 2m
i
ð5Þ where r ij is the inter-ionic distance, vi and vj are velocities of the correlated particles, s is the relaxation time, which is related to the relaxation rate if excitation. A widely used representation of the charge density of the projectile is the statistical model of Brandt and Kitagawa (BK) [12]. The distribution of N electrons, which are bound at an isolated ion with the nuclear charge Z1, is modelled by a radically symmetric function Nð4pK2 rÞ1 expðr=KÞ, where r is the electron-nucleus distance. The ion size is defined by the parameter K with the value 2=3 K ¼ 0:48N , and its Fourier transform is expressed by: Z 1 N=7
qðBKÞ ðkÞ ¼ Z 1
ð4Þ
N 1 þ ðkKÞ2
ð6Þ
:
With the Eq. (3), the self-induced stopping power is approximated and an interference term is denoted as correlated stopping power between a di-cluster, which are:
Si
Iij ¼
where H(x) is the Heaviside unit-step function, wp is the plasma enhÞ1=2 allows the ergy of the electron gas and the choice kc ¼ ð2mwp = two functions in Eq. (1) to coincide at k ¼ kc in the k w plane. The first term describes the response due to non-dispersive plasmon excitation in the region of k < kc , while the second term describes free-electron recoil in the region of k > kc (single-particle excitations). The approximation function satisfies the sum rule for all values of k, i.e.:
Z
! ! Furthermore, in the region, h=2mv < jrij v ij j < v =xp , the two correlated ions are separated from comparison with the closest interaction with target electrons (electron-hole pairs excitation). They behave simultaneously as united ions with respect to the most distant collective interactions (resonant or plasmon excitations) with an adiabatic distance v/xp. We obtain the stopping power for a di-proton. The stopping power for a H+–H+ pair is given by: X x2p 2mv 2 i Sin ðv Þ ln hxp v2 i¼1;2 i v v Z Z kv i sin iv j xs X 2 dk 1 i ! ! x d x J ðkr Þ Im Iðr ij ; v ij Þ ¼ ; ij 0 v i v j p i¼1;2;i–j kv 2i 0 eðk; xÞ v xs
x2p
v 2i
ln
1
Z
pv 2i x2p Z
2mv 21 ; h xp
ð7Þ
v i v j Z kv i sin x s i vi dk sinðkrÞ 1 ; dx Im v i v j k kr eðk; xÞ 0 v s i
v i v j
dk sinðkrÞ sinð v i xk sÞ ¼ 2 ; v i v j v i kmin k kr v i s xk ( v v x2p sinð iv i j xp sÞ Z kc dk sinðkrÞ ¼ 2 v i v j kr vi kmin k v i sxp v v Z kmax dk sinðkrÞ sinð i j k2 sÞ) 2v i þ2v i : 4 kc rðv i v j Þk s kmax
ð8Þ
The range of our interest is between kmin=xp/v and kmax=2mv/⁄. 3. Results and discussion Fig. 1 illustrates the dependence of interference term of the stopping power on the particle velocity with rij = 1 and v2 = 5. The term, obtained from Eq. (8), not only includes the inter-ionic distance rij of the correlated particles, but also features the difference between particle velocities vij. The plasma frequency xp = 0.55 a.u was chosen, corresponding to the plasma frequency of the electron gas of Al and the first peak of the energy loss function of Ti, respectively. The individual stopping Sp is compared to the interference term Iij at different relaxation times with s = 1, 2 and 5. As shown in Fig. 1, the interference term Iij(r,v,t) exhibits a similar behaviour as the stopping power of a proton Sp but is less than the magnitude of Sp. Above the threshold, the interference term
J. de Dieu Mugiraneza et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3175–3178
Fig. 1. The self-induced term and interference term of the stopping power as function of velocity calculated by Eq. (8) at different relaxation times.
goes through a pronounced maximum and then falls off with velocity increasing. Since the characteristic wave number is between kmin = xp/v and kmax = 2mv/⁄, the number of contributing k diminishes with the increase of velocity for all the terms. Roughly speaking, half of all possible excitations will interfere constructively as compared with the individual stopping term Sp. In Fig. 2, the interference term Iij(rij,vij,s) decreases as the time elapses and the cutoff velocity shifts to higher velocities. The three (two of them shown in Fig. 2) curves intersect at v = 5 for the second particle, in which the interference term between two correlated cluster or molecule components that moves at the same velocity is reduced to a much lower value. The peculiarity of the velocity-dependent interference term is that an oscillating behaviour occurs at small velocity range as compared with that of the neighbouring charged particle. With the time increase, the cutoff velocity of larger constructive interference shifts closely to that of the neighbouring ion. This phenomenon has not been observed for correlated particles moving at the same velocity. Fig. 3 shows that the cutoff velocity for the constructive interaction does not change for the correlated particles with a fixed inter-ionic distance. As illustrated in Fig. 4, the interaction decreases with inter-particle distance increasing. Another feature of the interference between the moving correlated particles lies in the ratio of the interference term to the self-induced stopping power, R ¼ Iij ðrij ; v ij ; sÞ=Si ðv Þ. When the two protons have the same velocity, the vicinage effect reduces to the one of dicluster or diatomic molecule. For velocities around v2, a slight enhancement of stop-
Fig. 2. Comparison among the proton stopping power SP, the interference term Iij(rij) for a pair of correlated ions separated from a distance r = 1 and the velocity dependant Iij(r,v,s) with r = 1 and t = 6 in a medium of wp = 0.55.
3177
Fig. 3. The proton stopping power and velocity-independent inter-ference term with different inter-ionic distance r = 5 and 3.
Fig. 4. Vicinage effect (ratio R = Iij/S0) as function of time for correlated pair of protons, which corresponds to the familiar vicinage for a di-cluster moving at velocity. The other curves show the vicinage of correlated protons with one velocity at v2 = 5 and another velocity at different times with s = 0.5, 2 and 5. Inter-ionic distance is at r = 1and the medium plasma is xp = 0.55 a.u. in above calculation.
ping power is observed. It symmetrically increases with the increase of the difference of velocities and reaches a maximum at a certain velocity, then drops. The velocity corresponding to the peak changes with the interaction time. Namely, it should satisfy certain relation with the relaxation time of the medium. In Fig. 4, we can easily find that the range of velocities with interfering constructively becomes narrow with the increase of time. At very short time the energy loss enhancement becomes larger than that in the lower velocity range. For the same particle velocities, the expression reduces to that of molecules and ion cluster. The enhancement of energy loss due to difference between velocities of correlated particles is very small as compared with the self-induced stopping power. A narrow range of velocity in a given short interaction time is observed. The enhancement or decrease in energy loss due to this effect can be negligible for thermal and mechanical application because slight amount of energy loss will not affect the thermal related excitation such as phonons, electron–hole and other phenomena requiring higher energy. However, for the low energy excitation like spin excitation, its contribution cannot be neglected. At present, its use is challenging because of the strict condition needed for the ion beam and pulse. More theoretical and experimental studies on this aspect may lead to a better understanding of interaction of ion beam such as IPIB with the materials.
3178
J. de Dieu Mugiraneza et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3175–3178
4. Conclusions
Acknowledgment
We theoretically investigate the influence of the velocity difference of charged particles moving closely in a metallic target on the stopping power and vicinage function, as well as the energy loss of a pair of charged particles penetrating coherently in the target medium. Using the dielectric formalism, the mathematical expressions have been given for the stopping power of correlated charged particles penetrating the medium at different velocities. When aligned projectile protons move at high velocities and target medium is not dispersive but the corresponding dielectric function exhibits both single and collective excitation, the interaction between such protons in their vicinities is weakened with the increase of time. Such interaction depends on both absolute and relative velocity difference. It is observed that energy loss is slightly enhanced with a small relative velocity difference. The interference between such protons exhibits a maximum at certain velocity, and exhibits an oscillating behaviour when product of the relative difference velocity and time is beyond a threshold value. This study provides additional knowledge on the complex problem of stopping power and can apply for the energy deposition of high current density ion beams such as intense pulsed ion beam.
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