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Fragmentation of small van der Waals clusters in slow collisions with highly charged ions
Nuclear Instruments and Methods in Physics Research B 330 (2014) 61–65
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Fragmentation of small van der Waals clusters in slow collisions with highly charged ions Bilel Zarour ⇑ Department of Physics, Taibah University, P.O. Box 30080, Madinah 41477, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 24 February 2014 Received in revised form 22 March 2014 Accepted 27 March 2014 Available online 22 April 2014 Keywords: Classical Molecular Dynamics Charge transfer Fragmentation
a b s t r a c t We have studied the fragmentation of small argon clusters in slow collisions with highly charged projectiles. The studied systems are Xeqþ —ArN (N = 3, 5, 7, 10) for two impact velocities; v = 0.3 a.u. (for q ¼ 25) and v = 0.14 a.u. (for q = 5) which correspond to the impact energies 300 keV and 65 keV respectively. The theoretical method is based on the Classical Molecular Dynamics (CMD) which includes the Over-Barrier (OB) and tunneling (TU) mechanisms. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Atomic clusters are special cases of molecules. Due to their large internuclear distances the outer electrons stay localized at their nuclei in contrast to covalently bound molecules [1]. From the electronic point of view, van der Waals (VdW) clusters resemble very closely to separated neighboring atoms. Thus, the ion-cluster collisions involve necessarily a large number of electronic and nuclear degrees of freedom which implies that very different processes may occur simultaneously. For example, charge transfer may be strongly coupled with electron ionization and may lead to the fragmentation of the cluster. Interaction of VdW clusters with a variety of particles has already been reported, for instance with electrons [2] and strong laser pulses [3,4]. Collision with highly charged ions (HCI) exists only for VdW dimers [5–8]. The unique work which treats the reaction between HCI and VdW clusters has been presented by Tappe et al. [9]. They studied experimentally Xeq+–ArN collisions at impact velocities v = 0.14, 0.21, 0.25 and 0.3 a.u. for q = 5, 12, 17, and 25, respectively. Their experimental conditions led to the formation of an average cluster size hNi = 10 of argon atoms. On the other hand, highly charged ions have been used as projectiles in collisions with fullerenes [10,11], and metal clusters [12]. The formation of multiply charged atomic fragments has been observed in intense femtosecond laser-VdW clusters interaction [3,4] and in HCI–VdW clusters slow collisions [9]. In the last case, Ar7þ fragments have been observed. Such high charge states have not been ⇑ Tel.: +966 541119408. E-mail addresses: [email protected], [email protected] http://dx.doi.org/10.1016/j.nimb.2014.03.020 0168-583X/Ó 2014 Elsevier B.V. All rights reserved.
observed in ion-induced fragmentation of fullerenes or metal clusters [10–12]. To complete the experimental work of Tappe et al. [9] we present in this paper theoretical study of the interaction Xeqþ —ArN , where N = 3, 5, 7, 10, the projectile charges are q = 25 and 5. We have chosen to study theoretically these systems with the same impact energies used in the experiments of Tappe et al. [9]; 300 keV and 65 keV for q = 25 and q = 5, respectively, to have a direct comparison between the experimental data and the theoretical predictions. 2. Theory 2.1. Preparation of the cluster The positions of the cluster atoms should be determined before doing the simulation. For Van der Waals clusters we used the Lennard–Jones 12–6 potential [13,14]:
EðrÞ ¼ 4
" X r 12 i
r ij
6 #
r
rij
ð1Þ
where is the maximum depth of the potential well, and r ¼ r is where the potential changes sign from positive to negative. The Lennard–Jones potential is a function of the distance between the atoms. If the separation distance decreases below the equilibrium value the potential energy becomes increasingly positive. On the other hand, the potential energy is negative and approaches zero as the separation distance increases to infinity. For a distance slightly bigger than r, the potential energy reaches a minimum
62
B. Zarour / Nuclear Instruments and Methods in Physics Research B 330 (2014) 61–65
and at this point the system of atoms is more stable and remains in this orientation until an external force is applied and breaks this equilibrium. The molecular parameters used to form the argon clusters ArN (N 6 10) are r = 3.405 Å and =kB ¼ 119:8 K [13]. The energy minimization allows us to determine the coordinates of the cluster atoms. We give in Table 1 the smallest Ar–Ar distance and the radius of some small argon clusters. In fact, VdW clusters do not possess a spherical symmetry. For this reason we define the radius R0 of the cluster as the distance between the center of mass of the cluster and the farthest atom.
compared to a random number n. If this probability is greater than n (0 6 n 6 1) the electron is released. Thus, a new electron is ‘‘born’’ at the position of the ion ‘‘mother’’ with a kinetic energy equal to the difference between the binding energy Eb and the coulomb barrier V(r), thus the local momentum is given by:
2.2. The collision dynamics
In the framework of the Classical Molecular Dynamics (CMD), we can suppose two steps to describe the collision; firstly, the projectile interacts with the closest argon atom Arð1Þ leading to the creation of an ion Arð1Þkþ . From the classical Over-Barrier (COB) model [17] the critical distance Rm , at which the projectile with a charge q can remove the mth electron from an atomic target, can pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffipffiffiffiffiffi be given by Rm ¼ ½2 q m þ 1 m þ m=IEm , where IEm is the ionization energy of the mth electron. This gives for the first electron the value R1 = 19.00 a.u. when the projectile is a Xe25þ . If we take into account the radius of the molecular target as it is defined in Table 1, then for the Xe25+–Ar10 system, the first electron will be removed for a critical projectile-target distance equal to 28.7 a.u. In our calculations, we have found that the first electron is removed for a distance RCMD-TU = 31.0 a.u. when tunneling is 1 included and RCMD-OB = 23.0 a.u. without tunneling. In the second 1 step of the collision dynamics, the coulomb interaction of the charged argon atom Arð1Þkþ with its neighbors is taken into account. Thus, the outermost electrons of the surrounding argon atoms will feel an average field created by both the projectile and the ‘‘already’’ charged argon ion. We can generalize this image and say that during the collision process, the charged argon ions will help the projectile to remove electrons from the other argon atoms of the ArN molecule. For very close collisions, the projectile Xe25þ can remove easily more than 10 electrons from the first argon atom Arð1Þ . In fact, the Xe25þ needs a distance of 1.3 a.u. to remove 15 electrons from an argon atom. If we suppose that an Arð1Þ10þ is formed, in this case, the first four removal distances for this ion are R1,2,3,4 12.65, 10.33, 8.55, 6.63 a.u. Only the last distance is lower than the smallest Ar–Ar distance in the Ar10 molecule (see Table 1). This shows the important role of the interactions between the argon ions inside the cluster during the collision process. In fact, the importance of tunneling mechanism for molecular targets is quite different from the atomic case. In a previous work Zarour and Saalmann showed for ion-atom slow collisions [15] that only the first electron is removed by tunneling. The inner electrons of the atomic target are mainly removed by the Over-Barrier mechanism. The reason is trivial; in order to remove electrons by the Over-Barrier mechanism the projectile needs to be inside a sphere centred on the atomic target with radius given by the expression of Bárány et al. [17]. Outside of this sphere, removal of electrons becomes possible only when tunneling is included. For molecular targets and especially for VdW clusters where the equilibrium distances between the atoms are large, tunneling plays an important role during the whole collision process. It was shown by Zarour in a recent paper for slow HCI–argon dimers interactions [16] that for very large distant collisions, when tunneling is included the probability to remove only one electron from the dimer is almost equal to zero. This means that the projectile removes electrons from the molecular target by tunneling, and also the highly charged ion of the dimer interacts with the lower charged one and removes one electron or more by tunneling. Thus, tunneling mechanism has to be taken into account for molecular targets because its importance in the interactions between the atoms of the molecule has been demonstrated.
We start by briefly describing the Classical Molecular Dynamics (CMD). This method has been already used to treat ion–atom [15] and HCI–Ar2 [16] slow collisions. In few words This technique is based on three points: (i) the target electrons are not explicitly treated, (ii) all particles are classically described, and (iii) only coulomb-type interactions are involved in the collision. This is defined for a pair of particles with charges q1 and q2 and a distance r 12 as:
q1 q2 ffi W ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 212 þ a
ð2Þ
where a ¼ 0:2 is a constant which prevents the unphysical instability of the classical simulations. The ArN cluster is considered as a system of N argon atoms; each argon atom is defined by a position and the binding energies of the its 18 electrons. Thus, the initial conditions for the collision are specified for the projectile by its initial distance from the origin, its velocity and its impact parameter b defined with respect to the center of mass of the ArN. At the beginning of the collision, the projectile is placed far from the target such as there is no coulomb interaction with the target electrons. During the propagation of the projectile and for each time step Dt, the coulomb barrier seen by the outermost target electron is calculated and compared to its shifted binding energy Eb . To remove an electron, we distinguish two cases: (i) When the binding energy Eb is bigger than the top of the barrier. The probability that the considered electron is removed in the framework of the Classical Molecular Dynamics with the Over-Barrier mechanism (CMD-OB) is
pCMD-OB ¼
Dt Tb
ð3Þ
where T b is the classical period of the electron. (ii) If not, we calculate the tunnel probability [15]
pCMD-TU ¼
Z b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dt exp 2 dr 2½VðrÞ Eb Tb a
ð4Þ
V(r) is the coulomb potential seen by the considered electron and deduced from (2) [for more details see [15,16]], a and b are the turning points defined by V(r) = Eb. In the two cases the probability is Table 1 The minimum Ar–Ar distances and the radius R0 of some small atomic argon clusters. N
Minimum Ar–Ar distance (a.u.)
Cluster radius R0 (a.u.)
2 3 4 5 6 7 8 9 10
7.21404 7.21402 7.21364 7.19714 7.17671 7.16724 7.14406 7.09521 7.11015
3.60702 4.16505 4.41773 5.86988 6.58639 6.14560 7.87217 7.32441 9.70360
!
k¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2ðEb VðrÞÞ u
ð5Þ
!
where u is the direction of the coulomb field seen by the electron.
B. Zarour / Nuclear Instruments and Methods in Physics Research B 330 (2014) 61–65
3. Results and discussion The theoretical method used in this study allows us to follow the dynamics of the collision with time. Thus, Fig. 1 shows the evolution with time of some dynamical quantities; the projectile and target charges and the number of removed electrons for the Xe25þ —Ar10 system. Based on the values of the impact parameter b, three cases can be distinguished; frontal collision which corresponds to b < R0 . Here, the projectile passes through the cluster and almost all the cluster components will feel the field generated by the incoming highly charged projectile. If b ’ R0 , the projectile mainly interacts with the cluster’s surface atoms. The last case is defined by b > R0 and corresponds to distant collision. We have to mention here that the direction of the projectile is randomly chosen in this study. The upper panel of Fig. 1 shows the case of a frontal collision when tunneling is included. t = 0 fs corresponds to a minimum projectile–target distance. The collision time depends on the initial charge of the projectile, its impact velocity and the size of the target. For the collisinal system Xe25þ —Ar10 ; the first electron is removed for a projectile-target distance RCMD-OB = 23.0 a.u. and RCMD-TU = 31.0 a.u. when tunneling is 1 1 included. The speed of the projectile in this case is v = 0.3 a.u. We can conclude that the maximum interaction time for this system is 4.3 fs when only Over-Barrier mechanism is taken into account and 5.8 fs with tunneling. Now if we consider all the studied systems, the collision time will vary from its minimum value 3.93 fs (5.53 fs when tunneling is included) for Xe25þ —Ar3 to its maximum value 5.42 fs (6.82 fs with tunneling) for Xe5þ —Ar10 . The interaction time decreases with increasing b. The oscillations in the projectile and target charges in Fig. 1 show the molecular character of the removed electrons. For frontal collisions, the field generated by the incoming highly charged ion is so huge that all the removed electrons are released during the collision time. For distant collisions, removal process requires longer time. In lower panel of Fig. 1, we can see that about 45 electrons have been removed from the atoms of the Ar10 cluster at the end of the collision time. At this moment, the distance between the projectile and the center of mass of the cluster is about 26.6 a.u. This means that the influence of the projectile on the cluster atoms is very weak. We have seen that the first electron is removed from the molecular target for a distance Xe25þ —Ar10 equal to 23.0 a.u. and 31.0 a.u. for CMD-OB and CMD-TU mechanisms respectively. For b = 15 a.u. and at
Fig. 1. The evolution with time of the projectile charge (dot lines), the number of removed electrons (dashed lines) and the target charge (solid lines) for the system Xe25þ —Ar10 . The upper panel gives the case b = 0 a.u. and lower panel gives the case b = 15 a.u. The collision time for each case is indicated.
63
t = 0 fs the distances between the Xe25þ and the 10 argon atoms are 8.32, 9.36, 11.11, 14.19, 14.60, 17.49, 17.71, 20.08, 21.46 and 23.24 a.u. from the closest to the farthest atom. From the classical Over-Barrier model [17], the number of removed electrons from these atoms are 6, 5, 3, 2, 2, 1, 1, 0, 0 and 0 respectively. The large distances between the projectile and the last three argon atoms do not allow to remove their outermost electrons. By a simple sum we can see that the model developed by Bárány et al. [17] predicts the removal of 20 electrons from the Ar10 cluster in this case. Since in our Classical Molecular Dynamics model, the influence of all charged particles is taken into account in the simulation, the top of the barrier is directly affected by them. Consequently, the distances at which the inner electrons are removed are slightly bigger than the ones given by the model of Bárány et al. [17]. Our CMD-OB calculations show that at the end of the collision time (t 2 fs), 25 electrons are already removed from the cluster. When tunneling is included this number becomes 45 electrons. When the projectile leaves the collision zone, the cluster charge distribution is not uniform. The argon atoms which were close to the projectile trajectory are highly charged, the other constituents of the cluster are lowly charged. The equilibration mechanism continues beyond the collision time. Electrons are removed from the lower charged argon ions by the highly charged ones. Most of these removed electrons are recaptured by the argon ions. We can remark that the global target charge does not change dramatically even when the electron removal process continues. As a consequence of the high charge of the argon atoms inside the cluster, the coulomb fragmentation is a dominant process. This is shown by the evolution of the cluster radii with time in Fig. 2. We suppose that the cluster is completely destroyed if its radius is multiplied by a factor of 2. As it was predicted by the measurements of Tappe et al. [9], we can see from the figure that all the studied clusters do not fragment during the collision time, which varies from 3.93 fs to 6.82 fs for our different systems. When the cluster radius is multiplied by 2, the projectile is far from the interaction area. In fact, only for very slow projectiles, the fragmentation time becomes of the order of the collision time [16]. The panel (a) of Fig. 2 shows that the fragmentation time increases with increasing N. Thus, for the same projectile conditions; the initial projectile charge, its speed and the impact parameter value, Ar3 and Ar10 need 16 fs and 24 fs to fragment, respectively. This is due
Fig. 2. Evolution with time of the cluster radii defined by R=R0 for the reactions Xeqþ —ArN . In panel (a), q = 25. Solid, dashed, dot and small dashed lines for N = 10, 7, 5 and 3 respectively. In panel (b), N ¼ 10, Solid and dashed lines represent the cases b = 0 a.u. and b = 15 a.u., respectively. In panel (c), solid and dashed lines for q ¼ 25 and 5 respectively. The impact parameter is fixed at b = 0 a.u.
64
B. Zarour / Nuclear Instruments and Methods in Physics Research B 330 (2014) 61–65
to the screening effect of the removed electrons which decreases the coulomb repulsion between the cluster argon ions for larger clusters. For example, the radius of trimer Ar3 is only increased by a factor of 0.1 when the projectile Xe25þ leaves the interaction area. However, the fragmentation process for this system is faster than the other systems, 11 fs after the collision, the radius of the trimer is multiplied by the factor 2. The destruction of Ar10 requires 20 fs for frontal collisions. When we change the initial projectile conditions, the fragmentation time is directly affected. Thus for lower charged projectiles or more distant collisions the fragmentation becomes slower as it is showed in the panels (b) and (c) of Fig. 2. To understand the charge distribution over the cluster fragments, we present in Fig. 3 the average highest and lowest fragment charges for the system Xe25þ —Ar10 . We can see that the formation of Ar7þ is possible with a large probability. For frontal collisions ðb < R0 ), all the cluster constituents are charged and the destruction of the cluster is faster. For distant collisions, as some argon atoms remain neutral, the fragmentation is slower. This figure shows clearly the zone between 23.0 a.u. and 31.0 a.u. where the Over-Barrier mechanism is forbidden because of the potential barrier and electron removal is possible only by tunneling. In Fig. 4, we show for frontal collisions the number of fragments as a function of their charge. Two systems are presented; Xe25þ —Ar10 and Xe25þ —Ar2 . The last one is extracted from Ref. [16]. The information that we can obtain from this figure are:
Fig. 4. The number of fragments (in arbitrary units) as function of their charge for two systems: Xe25þ —Ar2 (right panel) and Xe25þ —Ar10 (left panel). For the two cases filled and empty circles design CMD-TU and CMD-OB mechanisms respectively.
The non-uniform distribution of the charges over the cluster fragments. The fragments Ar7þ can have as origin the cluster Ar10 with a significant probability. The fragments Arrþ with r 8 have as origin smaller clusters with significant probability. The most abundant fragment ions are Ar4þ , Ar5þ and Ar6þ for the two systems, in addition to Ar7þ for the case of the dimer. Further insight for the influence of the cluster size on the electron removal process is evident from Fig. 5, for frontal collisions. The evolution of the average charge per atom defined by the PN s PN q i q i ¼ Nj¼1 i;j is the average charge expression Q atom ¼ Ni¼1 , where q s of the ith argon ion. N s is the number of simulations. We include the atomic case for comparison. The average charge for Xe25þ —Ar2 is extracted form Ref. [16]. The figure shows a decrease of the average charge per atom with increasing N. This is due to what we have called the equilibra-
Fig. 5. The evolution with the cluster size N of the average charge per atom for Xeqþ —ArN , solid circles and solid squares for q ¼ 25 and q ¼ 5 respectively.
tion process. When the projectile leaves the collision area, the molecular target is characterized by two characteristics: The cluster is still intact. The charge is not uniformly distributed over all constituents of the cluster. Consequently, the highly charged argon ions have enough time to remove more electrons from the lower and neutral argon atoms. Most of these removed electrons are recaptured by the argon ions and do not leave the cluster. When the size of the cluster decreases, the removed electrons are more likely to leave the target and captured either by the highly charged projectile or completely ionized in the continuum. A similar behavior of the average charge per atom with the cluster size was reported in a theoretical study of strong laser pulses-argon clusters interaction [3]. It was shown that the clusters were less effectively ionized at high fields than atoms. 4. Conclusion
25þ
Fig. 3. The average highest and lowest fragment charges for the system Xe —Ar10 as function of the impact parameter b. Dashed and solid lines for CMD-OB and CMD-TU mechanisms respectively.
In conclusion, we presented a theoretical model to explain the fragmentation mechanism of VdW clusters which were ionized
B. Zarour / Nuclear Instruments and Methods in Physics Research B 330 (2014) 61–65
by HCI. We confirm the experimental results of Tappe et al. [9]. The large equilibrium distances between the atoms of VdW clusters do not allow a total charge equilibration over all the fragments. The formation of highly charged atomic fragment ions Arrþ is possible, up to r ¼ 7 when the target is Ar10 and up to r ¼ 9 for smaller argon clusters. We pointed out the evolution of the fragments charge with the cluster size; the charge per atom decreases with increasing N.
References [1] B. Ulrich et al., Phys. Rev. A 82 (2010) 013412. [2] T. Pflüger et al., Phys. Rev. Lett 107 (2011) 22321.
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[3] U. Saalmann, Ch. Siedschlag, J.M. Rost, J. Phys. B At. Mol. Opt. Phys. 39 (2006) R39–R77. [4] T. Ditmire et al., Phys. Rev. Lett 78 (1997) 2732. [5] J. Matsumoto et al., Phys. Rev. Lett 105 (2010) 263202. [6] T. Ohyama-Yamaguchi, A. Ichimura, Phys. Scr. T144 (2011) 014028. [7] T. Ohyama-Yamaguchi, A. Ichimura, Nucl. Intrum. Meth. B 205 (2003) 620. [8] J. Matsumoto et al., Phys. Rev. Lett. 105 (2010) 263202. [9] W. Tappe, R. Flesch, E. Rühl, R. Hoekstra, T. Schlathölter, Phys. Rev. Lett 88 (2002) 143401. [10] B. Walch, C.L. Cocke, R. Vlpel, E. Salzborn, Phys. Rev. Lett. 72 (1994) 1439. [11] S. martin, L. Chen, A. denis, J. Desesquelles, Phys. Rev. A 57 (1998) 4518. [12] L. Plagne, C. Guest, Phys. Rev. A 59 (1999) 4461. [13] R. Guido Della Valle, E. Venuti, Phys. Rev. B 58 (1998) 206. [14] D.J. Wales, J.P.K. Doye, J. Phys. Chem. A 101 (1997) 5111. [15] B. Zarour, U. Saalmann, Nucl. Instrum. Meth. B 205 (2003) 610. [16] B. Zarour, Nucl. Intrum. Meth. B 320 (2014) 12. [17] A. Bárány et al., Nucl. Instrum. Meth. B 9 (1985) 397.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Fragmentation of small van der Waals clusters in slow collisions with highly charged ions Bilel Zarour ⇑ Department of Physics, Taibah University, P.O. Box 30080, Madinah 41477, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 24 February 2014 Received in revised form 22 March 2014 Accepted 27 March 2014 Available online 22 April 2014 Keywords: Classical Molecular Dynamics Charge transfer Fragmentation
a b s t r a c t We have studied the fragmentation of small argon clusters in slow collisions with highly charged projectiles. The studied systems are Xeqþ —ArN (N = 3, 5, 7, 10) for two impact velocities; v = 0.3 a.u. (for q ¼ 25) and v = 0.14 a.u. (for q = 5) which correspond to the impact energies 300 keV and 65 keV respectively. The theoretical method is based on the Classical Molecular Dynamics (CMD) which includes the Over-Barrier (OB) and tunneling (TU) mechanisms. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Atomic clusters are special cases of molecules. Due to their large internuclear distances the outer electrons stay localized at their nuclei in contrast to covalently bound molecules [1]. From the electronic point of view, van der Waals (VdW) clusters resemble very closely to separated neighboring atoms. Thus, the ion-cluster collisions involve necessarily a large number of electronic and nuclear degrees of freedom which implies that very different processes may occur simultaneously. For example, charge transfer may be strongly coupled with electron ionization and may lead to the fragmentation of the cluster. Interaction of VdW clusters with a variety of particles has already been reported, for instance with electrons [2] and strong laser pulses [3,4]. Collision with highly charged ions (HCI) exists only for VdW dimers [5–8]. The unique work which treats the reaction between HCI and VdW clusters has been presented by Tappe et al. [9]. They studied experimentally Xeq+–ArN collisions at impact velocities v = 0.14, 0.21, 0.25 and 0.3 a.u. for q = 5, 12, 17, and 25, respectively. Their experimental conditions led to the formation of an average cluster size hNi = 10 of argon atoms. On the other hand, highly charged ions have been used as projectiles in collisions with fullerenes [10,11], and metal clusters [12]. The formation of multiply charged atomic fragments has been observed in intense femtosecond laser-VdW clusters interaction [3,4] and in HCI–VdW clusters slow collisions [9]. In the last case, Ar7þ fragments have been observed. Such high charge states have not been ⇑ Tel.: +966 541119408. E-mail addresses: [email protected], [email protected] http://dx.doi.org/10.1016/j.nimb.2014.03.020 0168-583X/Ó 2014 Elsevier B.V. All rights reserved.
observed in ion-induced fragmentation of fullerenes or metal clusters [10–12]. To complete the experimental work of Tappe et al. [9] we present in this paper theoretical study of the interaction Xeqþ —ArN , where N = 3, 5, 7, 10, the projectile charges are q = 25 and 5. We have chosen to study theoretically these systems with the same impact energies used in the experiments of Tappe et al. [9]; 300 keV and 65 keV for q = 25 and q = 5, respectively, to have a direct comparison between the experimental data and the theoretical predictions. 2. Theory 2.1. Preparation of the cluster The positions of the cluster atoms should be determined before doing the simulation. For Van der Waals clusters we used the Lennard–Jones 12–6 potential [13,14]:
EðrÞ ¼ 4
" X r 12 i
r ij
6 #
r
rij
ð1Þ
where is the maximum depth of the potential well, and r ¼ r is where the potential changes sign from positive to negative. The Lennard–Jones potential is a function of the distance between the atoms. If the separation distance decreases below the equilibrium value the potential energy becomes increasingly positive. On the other hand, the potential energy is negative and approaches zero as the separation distance increases to infinity. For a distance slightly bigger than r, the potential energy reaches a minimum
62
B. Zarour / Nuclear Instruments and Methods in Physics Research B 330 (2014) 61–65
and at this point the system of atoms is more stable and remains in this orientation until an external force is applied and breaks this equilibrium. The molecular parameters used to form the argon clusters ArN (N 6 10) are r = 3.405 Å and =kB ¼ 119:8 K [13]. The energy minimization allows us to determine the coordinates of the cluster atoms. We give in Table 1 the smallest Ar–Ar distance and the radius of some small argon clusters. In fact, VdW clusters do not possess a spherical symmetry. For this reason we define the radius R0 of the cluster as the distance between the center of mass of the cluster and the farthest atom.
compared to a random number n. If this probability is greater than n (0 6 n 6 1) the electron is released. Thus, a new electron is ‘‘born’’ at the position of the ion ‘‘mother’’ with a kinetic energy equal to the difference between the binding energy Eb and the coulomb barrier V(r), thus the local momentum is given by:
2.2. The collision dynamics
In the framework of the Classical Molecular Dynamics (CMD), we can suppose two steps to describe the collision; firstly, the projectile interacts with the closest argon atom Arð1Þ leading to the creation of an ion Arð1Þkþ . From the classical Over-Barrier (COB) model [17] the critical distance Rm , at which the projectile with a charge q can remove the mth electron from an atomic target, can pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffipffiffiffiffiffi be given by Rm ¼ ½2 q m þ 1 m þ m=IEm , where IEm is the ionization energy of the mth electron. This gives for the first electron the value R1 = 19.00 a.u. when the projectile is a Xe25þ . If we take into account the radius of the molecular target as it is defined in Table 1, then for the Xe25+–Ar10 system, the first electron will be removed for a critical projectile-target distance equal to 28.7 a.u. In our calculations, we have found that the first electron is removed for a distance RCMD-TU = 31.0 a.u. when tunneling is 1 included and RCMD-OB = 23.0 a.u. without tunneling. In the second 1 step of the collision dynamics, the coulomb interaction of the charged argon atom Arð1Þkþ with its neighbors is taken into account. Thus, the outermost electrons of the surrounding argon atoms will feel an average field created by both the projectile and the ‘‘already’’ charged argon ion. We can generalize this image and say that during the collision process, the charged argon ions will help the projectile to remove electrons from the other argon atoms of the ArN molecule. For very close collisions, the projectile Xe25þ can remove easily more than 10 electrons from the first argon atom Arð1Þ . In fact, the Xe25þ needs a distance of 1.3 a.u. to remove 15 electrons from an argon atom. If we suppose that an Arð1Þ10þ is formed, in this case, the first four removal distances for this ion are R1,2,3,4 12.65, 10.33, 8.55, 6.63 a.u. Only the last distance is lower than the smallest Ar–Ar distance in the Ar10 molecule (see Table 1). This shows the important role of the interactions between the argon ions inside the cluster during the collision process. In fact, the importance of tunneling mechanism for molecular targets is quite different from the atomic case. In a previous work Zarour and Saalmann showed for ion-atom slow collisions [15] that only the first electron is removed by tunneling. The inner electrons of the atomic target are mainly removed by the Over-Barrier mechanism. The reason is trivial; in order to remove electrons by the Over-Barrier mechanism the projectile needs to be inside a sphere centred on the atomic target with radius given by the expression of Bárány et al. [17]. Outside of this sphere, removal of electrons becomes possible only when tunneling is included. For molecular targets and especially for VdW clusters where the equilibrium distances between the atoms are large, tunneling plays an important role during the whole collision process. It was shown by Zarour in a recent paper for slow HCI–argon dimers interactions [16] that for very large distant collisions, when tunneling is included the probability to remove only one electron from the dimer is almost equal to zero. This means that the projectile removes electrons from the molecular target by tunneling, and also the highly charged ion of the dimer interacts with the lower charged one and removes one electron or more by tunneling. Thus, tunneling mechanism has to be taken into account for molecular targets because its importance in the interactions between the atoms of the molecule has been demonstrated.
We start by briefly describing the Classical Molecular Dynamics (CMD). This method has been already used to treat ion–atom [15] and HCI–Ar2 [16] slow collisions. In few words This technique is based on three points: (i) the target electrons are not explicitly treated, (ii) all particles are classically described, and (iii) only coulomb-type interactions are involved in the collision. This is defined for a pair of particles with charges q1 and q2 and a distance r 12 as:
q1 q2 ffi W ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 212 þ a
ð2Þ
where a ¼ 0:2 is a constant which prevents the unphysical instability of the classical simulations. The ArN cluster is considered as a system of N argon atoms; each argon atom is defined by a position and the binding energies of the its 18 electrons. Thus, the initial conditions for the collision are specified for the projectile by its initial distance from the origin, its velocity and its impact parameter b defined with respect to the center of mass of the ArN. At the beginning of the collision, the projectile is placed far from the target such as there is no coulomb interaction with the target electrons. During the propagation of the projectile and for each time step Dt, the coulomb barrier seen by the outermost target electron is calculated and compared to its shifted binding energy Eb . To remove an electron, we distinguish two cases: (i) When the binding energy Eb is bigger than the top of the barrier. The probability that the considered electron is removed in the framework of the Classical Molecular Dynamics with the Over-Barrier mechanism (CMD-OB) is
pCMD-OB ¼
Dt Tb
ð3Þ
where T b is the classical period of the electron. (ii) If not, we calculate the tunnel probability [15]
pCMD-TU ¼
Z b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dt exp 2 dr 2½VðrÞ Eb Tb a
ð4Þ
V(r) is the coulomb potential seen by the considered electron and deduced from (2) [for more details see [15,16]], a and b are the turning points defined by V(r) = Eb. In the two cases the probability is Table 1 The minimum Ar–Ar distances and the radius R0 of some small atomic argon clusters. N
Minimum Ar–Ar distance (a.u.)
Cluster radius R0 (a.u.)
2 3 4 5 6 7 8 9 10
7.21404 7.21402 7.21364 7.19714 7.17671 7.16724 7.14406 7.09521 7.11015
3.60702 4.16505 4.41773 5.86988 6.58639 6.14560 7.87217 7.32441 9.70360
!
k¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2ðEb VðrÞÞ u
ð5Þ
!
where u is the direction of the coulomb field seen by the electron.
B. Zarour / Nuclear Instruments and Methods in Physics Research B 330 (2014) 61–65
3. Results and discussion The theoretical method used in this study allows us to follow the dynamics of the collision with time. Thus, Fig. 1 shows the evolution with time of some dynamical quantities; the projectile and target charges and the number of removed electrons for the Xe25þ —Ar10 system. Based on the values of the impact parameter b, three cases can be distinguished; frontal collision which corresponds to b < R0 . Here, the projectile passes through the cluster and almost all the cluster components will feel the field generated by the incoming highly charged projectile. If b ’ R0 , the projectile mainly interacts with the cluster’s surface atoms. The last case is defined by b > R0 and corresponds to distant collision. We have to mention here that the direction of the projectile is randomly chosen in this study. The upper panel of Fig. 1 shows the case of a frontal collision when tunneling is included. t = 0 fs corresponds to a minimum projectile–target distance. The collision time depends on the initial charge of the projectile, its impact velocity and the size of the target. For the collisinal system Xe25þ —Ar10 ; the first electron is removed for a projectile-target distance RCMD-OB = 23.0 a.u. and RCMD-TU = 31.0 a.u. when tunneling is 1 1 included. The speed of the projectile in this case is v = 0.3 a.u. We can conclude that the maximum interaction time for this system is 4.3 fs when only Over-Barrier mechanism is taken into account and 5.8 fs with tunneling. Now if we consider all the studied systems, the collision time will vary from its minimum value 3.93 fs (5.53 fs when tunneling is included) for Xe25þ —Ar3 to its maximum value 5.42 fs (6.82 fs with tunneling) for Xe5þ —Ar10 . The interaction time decreases with increasing b. The oscillations in the projectile and target charges in Fig. 1 show the molecular character of the removed electrons. For frontal collisions, the field generated by the incoming highly charged ion is so huge that all the removed electrons are released during the collision time. For distant collisions, removal process requires longer time. In lower panel of Fig. 1, we can see that about 45 electrons have been removed from the atoms of the Ar10 cluster at the end of the collision time. At this moment, the distance between the projectile and the center of mass of the cluster is about 26.6 a.u. This means that the influence of the projectile on the cluster atoms is very weak. We have seen that the first electron is removed from the molecular target for a distance Xe25þ —Ar10 equal to 23.0 a.u. and 31.0 a.u. for CMD-OB and CMD-TU mechanisms respectively. For b = 15 a.u. and at
Fig. 1. The evolution with time of the projectile charge (dot lines), the number of removed electrons (dashed lines) and the target charge (solid lines) for the system Xe25þ —Ar10 . The upper panel gives the case b = 0 a.u. and lower panel gives the case b = 15 a.u. The collision time for each case is indicated.
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t = 0 fs the distances between the Xe25þ and the 10 argon atoms are 8.32, 9.36, 11.11, 14.19, 14.60, 17.49, 17.71, 20.08, 21.46 and 23.24 a.u. from the closest to the farthest atom. From the classical Over-Barrier model [17], the number of removed electrons from these atoms are 6, 5, 3, 2, 2, 1, 1, 0, 0 and 0 respectively. The large distances between the projectile and the last three argon atoms do not allow to remove their outermost electrons. By a simple sum we can see that the model developed by Bárány et al. [17] predicts the removal of 20 electrons from the Ar10 cluster in this case. Since in our Classical Molecular Dynamics model, the influence of all charged particles is taken into account in the simulation, the top of the barrier is directly affected by them. Consequently, the distances at which the inner electrons are removed are slightly bigger than the ones given by the model of Bárány et al. [17]. Our CMD-OB calculations show that at the end of the collision time (t 2 fs), 25 electrons are already removed from the cluster. When tunneling is included this number becomes 45 electrons. When the projectile leaves the collision zone, the cluster charge distribution is not uniform. The argon atoms which were close to the projectile trajectory are highly charged, the other constituents of the cluster are lowly charged. The equilibration mechanism continues beyond the collision time. Electrons are removed from the lower charged argon ions by the highly charged ones. Most of these removed electrons are recaptured by the argon ions. We can remark that the global target charge does not change dramatically even when the electron removal process continues. As a consequence of the high charge of the argon atoms inside the cluster, the coulomb fragmentation is a dominant process. This is shown by the evolution of the cluster radii with time in Fig. 2. We suppose that the cluster is completely destroyed if its radius is multiplied by a factor of 2. As it was predicted by the measurements of Tappe et al. [9], we can see from the figure that all the studied clusters do not fragment during the collision time, which varies from 3.93 fs to 6.82 fs for our different systems. When the cluster radius is multiplied by 2, the projectile is far from the interaction area. In fact, only for very slow projectiles, the fragmentation time becomes of the order of the collision time [16]. The panel (a) of Fig. 2 shows that the fragmentation time increases with increasing N. Thus, for the same projectile conditions; the initial projectile charge, its speed and the impact parameter value, Ar3 and Ar10 need 16 fs and 24 fs to fragment, respectively. This is due
Fig. 2. Evolution with time of the cluster radii defined by R=R0 for the reactions Xeqþ —ArN . In panel (a), q = 25. Solid, dashed, dot and small dashed lines for N = 10, 7, 5 and 3 respectively. In panel (b), N ¼ 10, Solid and dashed lines represent the cases b = 0 a.u. and b = 15 a.u., respectively. In panel (c), solid and dashed lines for q ¼ 25 and 5 respectively. The impact parameter is fixed at b = 0 a.u.
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B. Zarour / Nuclear Instruments and Methods in Physics Research B 330 (2014) 61–65
to the screening effect of the removed electrons which decreases the coulomb repulsion between the cluster argon ions for larger clusters. For example, the radius of trimer Ar3 is only increased by a factor of 0.1 when the projectile Xe25þ leaves the interaction area. However, the fragmentation process for this system is faster than the other systems, 11 fs after the collision, the radius of the trimer is multiplied by the factor 2. The destruction of Ar10 requires 20 fs for frontal collisions. When we change the initial projectile conditions, the fragmentation time is directly affected. Thus for lower charged projectiles or more distant collisions the fragmentation becomes slower as it is showed in the panels (b) and (c) of Fig. 2. To understand the charge distribution over the cluster fragments, we present in Fig. 3 the average highest and lowest fragment charges for the system Xe25þ —Ar10 . We can see that the formation of Ar7þ is possible with a large probability. For frontal collisions ðb < R0 ), all the cluster constituents are charged and the destruction of the cluster is faster. For distant collisions, as some argon atoms remain neutral, the fragmentation is slower. This figure shows clearly the zone between 23.0 a.u. and 31.0 a.u. where the Over-Barrier mechanism is forbidden because of the potential barrier and electron removal is possible only by tunneling. In Fig. 4, we show for frontal collisions the number of fragments as a function of their charge. Two systems are presented; Xe25þ —Ar10 and Xe25þ —Ar2 . The last one is extracted from Ref. [16]. The information that we can obtain from this figure are:
Fig. 4. The number of fragments (in arbitrary units) as function of their charge for two systems: Xe25þ —Ar2 (right panel) and Xe25þ —Ar10 (left panel). For the two cases filled and empty circles design CMD-TU and CMD-OB mechanisms respectively.
The non-uniform distribution of the charges over the cluster fragments. The fragments Ar7þ can have as origin the cluster Ar10 with a significant probability. The fragments Arrþ with r 8 have as origin smaller clusters with significant probability. The most abundant fragment ions are Ar4þ , Ar5þ and Ar6þ for the two systems, in addition to Ar7þ for the case of the dimer. Further insight for the influence of the cluster size on the electron removal process is evident from Fig. 5, for frontal collisions. The evolution of the average charge per atom defined by the PN s PN q i q i ¼ Nj¼1 i;j is the average charge expression Q atom ¼ Ni¼1 , where q s of the ith argon ion. N s is the number of simulations. We include the atomic case for comparison. The average charge for Xe25þ —Ar2 is extracted form Ref. [16]. The figure shows a decrease of the average charge per atom with increasing N. This is due to what we have called the equilibra-
Fig. 5. The evolution with the cluster size N of the average charge per atom for Xeqþ —ArN , solid circles and solid squares for q ¼ 25 and q ¼ 5 respectively.
tion process. When the projectile leaves the collision area, the molecular target is characterized by two characteristics: The cluster is still intact. The charge is not uniformly distributed over all constituents of the cluster. Consequently, the highly charged argon ions have enough time to remove more electrons from the lower and neutral argon atoms. Most of these removed electrons are recaptured by the argon ions and do not leave the cluster. When the size of the cluster decreases, the removed electrons are more likely to leave the target and captured either by the highly charged projectile or completely ionized in the continuum. A similar behavior of the average charge per atom with the cluster size was reported in a theoretical study of strong laser pulses-argon clusters interaction [3]. It was shown that the clusters were less effectively ionized at high fields than atoms. 4. Conclusion
25þ
Fig. 3. The average highest and lowest fragment charges for the system Xe —Ar10 as function of the impact parameter b. Dashed and solid lines for CMD-OB and CMD-TU mechanisms respectively.
In conclusion, we presented a theoretical model to explain the fragmentation mechanism of VdW clusters which were ionized
B. Zarour / Nuclear Instruments and Methods in Physics Research B 330 (2014) 61–65
by HCI. We confirm the experimental results of Tappe et al. [9]. The large equilibrium distances between the atoms of VdW clusters do not allow a total charge equilibration over all the fragments. The formation of highly charged atomic fragment ions Arrþ is possible, up to r ¼ 7 when the target is Ar10 and up to r ¼ 9 for smaller argon clusters. We pointed out the evolution of the fragments charge with the cluster size; the charge per atom decreases with increasing N.
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