Dynamic screening and energy loss of antiprotons colliding with excited Al clusters
Natalia E. Koval, Daniel Sánchez-Portal, Andrey G. Borisov, Ricardo Díez Muiño
DDynamic screening and energy loss of antiprotonscolliding with excited Al clusters
Natalia E. Koval a, ∗ , Daniel S´anchez-Portal a,b , Andrey G. Borisov c , Ricardo D´ıezMui˜no a,b a Centro de F´ısica de Materiales CFM/MPC (CSIC-UPV/EHU),Paseo Manuel de Lardizabal 5, 20018 San Sebasti´an, Spain b Donostia International Physics Center DIPC,Paseo Manuel de Lardizabal 4, 20018 San Sebasti´an, Spain c Institut des Sciences Mol´eculaires d’Orsay, ISMO,Unit´e de Recherches CNRS-Universit´e Paris-Sud UMR 8214,Bˆatiment 351, Universit´e Paris-Sud, F-91405 Orsay Cedex, France
Abstract
We use time-dependent density functional theory to calculate the energy loss of anantiproton colliding with a small Al cluster previously excited. The velocity of theantiproton is such that non-linear effects in the electronic response of the Al clusterare relevant. We obtain that an antiproton penetrating an excited cluster transfers lessenergy to the cluster than an antiproton penetrating a ground state cluster. We quantifythis difference and analyze it in terms of the cluster excitation spectrum.
Keywords:
Energy loss, TDDFT, Metal cluster ∗ Corresponding author, tel: (+34) 943 01 87 59
Email addresses: natalia koval @ ehu.es (Natalia E. Koval), sqbsapod @ ehu.es (DanielS´anchez-Portal), andrei.borissov @ u-psud.fr (Andrey G. Borisov), rdm @ ehu.es (Ricardo D´ıezMui˜no) Preprint submitted to Nuclear Instruments and Methods B November 1, 2018 a r X i v : . [ phy s i c s . a t m - c l u s ] J a n . Introduction A charged particle moving across a metallic target is able to create electronic ex-citations in the medium at the expense of its own kinetic energy. Research on thisphenomenon has been broad in condensed matter physics and materials science becauseof its relevance in various fundamental and applied topics, such as radiation damage,medical physics, and ion sputtering.A key point in the theoretical analysis of the slowing down of charged particles inmetals is the intensity of the perturbation that the moving particle introduces in themedium. For a particle of charge Q moving with a velocity υ in a standard metal,the perturbation strength can be roughly characterized by the Sommerfeld parameter η = Q/υ [1]. If such ratio is small, η <<
1, linear theory is naturally applied andaccurate results for the particle energy loss are found. If η >> υ is much smallerthan the typical velocities of the electrons in the medium, various non-perturbativemethodologies have been successfully applied [2]. In between these two cases, in theregime of intermediate velocities, accurate descriptions of the energy loss process aremuch more involved because quasistatic or perturbative approximations break downeven for unit-charge projectiles. Only recently calculations based on time-dependentdensity functional theory (TDDFT) [3, 4, 5, 6, 7, 8, 9, 10] have shown its potential toclose this gap.The achievements of TDDFT in the non-linear description of electronic excitationspave the way to answer new and challenging questions in the field. In the traditionaldescription of energy loss processes, the target is always considered as initially in itsground state. However, the energy lost by a travelling charge in a metallic mediumshould be affected by the electronic state in which the target is. Otherwise said, if2lectronic excitations have been already created in the system, the electronic responseto the incident perturbation, and consequently the energy loss, will be different. In thiswork we try to quantify this difference for the particular case of a point charge crossingmetallic clusters of a few ˚A size.The experimental investigation of the excitation and ionization of neutral metalclusters by collision with positively or negatively charged particles has been intensive.In particular, the ionization of metal clusters by low energy singly and multiply chargedions and protons [11, 12] and the ionization of neutral metal clusters by slow electrons [13,14, 15, 16] have been studied. Description of such processes from the theoretical point ofview is incomplete and requires further investigation. In our work we study the collisionof an Al cluster with a slow negative point charge (an antiproton). The choice of theantiproton as a projectile allows us to avoid complications related to the electron captureby the cluster if the projectile is an electron and the electron capture by the projectile ifthe latter is an ion. Our goal is to identify the distinct effects that arise in the dynamicscreening and the projectile energy loss when the metallic target has been previouslyexcited by a preceding projectile. In spite of the fact that our model is simplified, theresults of our study can contribute to the understanding of the fundamentals of thedynamical processes during collision of charged particles with metallic clusters.We perform an explicit time propagation of the electronic state of the system usingTDDFT and evaluate the energy lost by the charge when crossing ground-state clusters.We compare this quantity with the amount of energy lost when the projectile crosses acluster excited from a previous collision. We show that the difference is appreciable andgive the explanation of this change as a consequence of the excited state of the clusteras well as of the emission of electronic charge from the excited cluster.Non-linear effects in the excitation of metal clusters have been previously analyzed3ith TDDFT. In particular, electron dynamics in clusters under intense laser fieldsare an active hot topic of research [17] because of the possibilities offered to exploreand control ultrafast processes. The resonance energy of collective excitations in thesesystems has been shown to depend on the intensity of the perturbation [18]. Here wefocus on a different type of external perturbation, namely, that derived from a pointCoulomb charge crossing the system. We will show, nevertheless, that similar shifts inthe position of the plasmon peaks are found.Hartree atomic units ( e = (cid:126) = m e = 1) will be used throughout this work unlessotherwise stated.
2. Methodology
Let us first define the system under study: We will focus on a negative point charge(an antiproton) crossing a metal cluster. Electron dynamics in metallic systems typicallylie in the femtosecond and subfemtosecond time scales [19, 20]. For this reason and forthe kind of processes that we study, we consider the cluster ion cores as frozen. Wefurther simplify the problem and use the spherical jellium model (JM) to represent thecluster. In the JM, the ions are substituted by an homogeneous background of positivecharge with density n +0 ( r ) = n ( r s )Θ( R cl − r ). Here R cl is the radius of the cluster, Θ( x )is the Heaviside step-function and n ( r s ) is the constant bulk density, which dependsonly on the Wigner-Seitz radius r s (1 /n = 4 πr s /
3) [21]. The number of electrons in aneutral cluster is N = ( R cl /r s ) .The ground state electronic density of the cluster n ( r ) is obtained using the Kohn-Sham (KS) scheme [22] of density functional theory (DFT) [23]. The ground state KSwave functions ϕ i ( r ) are expanded in the spherical harmonics basis set [24].4he evolution of the electronic density in the cluster in response to the field of themoving charge is calculated using TDDFT [25]. We propagate the ground state wavefunctions ϕ i ( r ,
0) = ϕ i ( r ) solving time-dependent KS equations:i ∂∂t ϕ i ( r , t ) = (cid:26) − ∇ + V eff ( r , t ) (cid:27) ϕ i ( r , t ) . (1)The effective potential includes four terms V eff ( r , t ) = V ext ( r , t ) + V H ( r , t ) + V xc ( r , t ) + V ¯ p ( r , t ), where V ext is the external potential created by the positive background. V H isthe Hartree potential created by the electronic density. V xc is the exchange-correlationpotential, calculated with the standard adiabatic local density approximation (ALDA)with the Perdew-Zunger parametrization of Ceperley-Alder exchange and correlationpotential [26]. Finally, V ¯ p ( r , t ) = − Q ¯ p (cid:112) ( z ¯ p ( t ) − z ) + ρ Θ( t ) is the potential created bythe antiproton and acting on the valence electrons of the cluster. We use cylindricalcoordinates ( ρ, z ) in the time-dependent calculations, which are more appropriate sincethe problem has axial symmetry. The origin of coordinates is located at the center of thecluster. The antiproton is represented by a negative point charge ( Q ¯ p = −
1) which moveswith constant velocity υ along the z -axis. At time t=0, the antiproton is located at adistance from the cluster (50 a.u.) far enough to avoid a significant interaction betweenthe projectile and the target. The time propagation of the electron wave function isperformed using the time-stepping algorithm: ϕ i ( ρ, z, t + dt ) = e − iH i dt ϕ i ( ρ, z, t ). Thesplit operator approximation is then used to separate the potential and kinetic energyterms in the e − iH i dt time propagator. The action of the kinetic energy operator iscalculated using Crank-Nicolson propagation scheme. A detailed description of thenumerical procedure can be found in Refs. [27, 28, 29].From the time-dependent KS orbitals we obtain the time-evolving electronic density5f the excited cluster n ( ρ, z, t ) = (cid:88) i ∈ occ | ϕ i ( ρ, z, t ) | . The force acting on the movingantiproton along the z -axis is obtained from the time-dependent electronic density andincludes the effect of the positive background: F z ( t ) = 2 π (cid:90) ρ dρ dz n ( ρ, z, t ) − n +0 ( ρ, z )[( z ¯ p ( t ) − z ) + ρ ] / [ z ¯ p ( t ) − z ] . (2)The energy loss is then obtained from the integral: E loss = − υ (cid:90) ∞ F z ( t ) dt. (3)We will study the energy loss in two different motion cycles. In the first cycle, theantiproton moves towards the cluster with a constant velocity υ , crosses it following asymmetry axis through the cluster center, and eventually moves away until it reaches aturning point arbitrarily defined. The turning point is at a distance from the cluster farenough not to have any residual interaction. The cluster is then left in an excited state.The electronic energy transferred to the cluster during the collision is calculated. In thesecond cycle, the antiproton turns back from the turning point and starts to approachthe excited cluster with the same constant velocity υ . In the second crossing of thecluster, the latter now in an excited state, energy is again transferred to the cluster. Wecalculate the energy lost in this second cycle and compare the obtained value with thatof the first cycle.
3. Results and discussion
We have chosen a small Al ( r s = 2.07) cluster with N =18 electrons and with radius R cl = 5.43 a.u. ( ≈ .
29 nm). In all the calculations shown in this article, the projectilevelocity is υ = 0.5 a.u. The ALDA-TDDFT method used here predicts very well the6nergy loss of antiprotons in Al targets. The method gives very good agreement withmeasurements in Al bulk for antiproton velocities up to 1.8 a.u. Above this velocity theexcitations of the inner shells in Al start to contribute to the energy loss and resultsdeviate from the experimental ones [4].The antiproton starts its motion at time t = 0 from the position z = −
50 a.u.After the first collision, the projectile continues to move until t = 1000 a.u. At thistime the second cycle starts and the antiproton takes the way back to collide againwith the cluster. We call τ to the time interval between both collisions. In both cycleswe calculate the force F z ( t ) experienced by the projectile due to the interaction withthe cluster through Eq. 2. From F z ( t ) we obtain the value of the energy lost by theantiproton E loss by means of Eq. 3. In addition, we consider three other different timespots for the second cycle to start: 1003.5, 1005, and 1010 a.u. The purpose of usingdifferent time delays is to check the sensitivity of the final result to the dynamics followedby the electron density in the cluster excited state. With our choice of time delays, theantiproton reaches the excited cluster respectively ∆ τ = 7, 10 and 20 a.u. of time laterthan in the reference calculation. Depending on the value of ∆ τ , the antiproton willstart to cross the surface of the excited cluster meeting a minimum or a maximum inthe electronic density oscillations, or an intermediate state. The density oscillations willbe discussed later. The results for the energy loss are summarized in Table I.The first interesting conclusion that can be extracted from the results of Table Iis that the energy loss of the antiproton crossing the excited cluster is consistentlylower than the corresponding value for the antiproton colliding with the cluster in theground state. There are two reasons for the decrease of the energy loss. One reason isthat, after the first collision, the cluster is emitting an amount of electronic charge thatroughly corresponds to one electron. This means that during the second collision the7ntiproton is interacting with a smaller amount of electronic charge and therefore losesless energy. In order to check the relevance of the change in the electronic charge ofthe cluster, we performed an additional calculation, namely that of the energy loss in apositively charged cluster which contains 17 electrons and remains in its ground state.The obtained value of 0.8211 a.u. is lower than the value of the energy loss in the neutralcluster, which is given in the Table I and is equal to 0.8527 a.u. The difference betweenthese two results is around 4%. However, the difference is not as big as in the case oftime delays ∆ τ = 0 and 20 a.u. given in the table, which is up to 11% of the value ofthe energy loss in the first collision. This allows us to conclude that the emission of oneelectron from the cluster only partially explains the observed decrease of energy loss inthe second collision.Another reason for the lowering of the energy loss is that the cluster is excited afterthe first collision with the antiproton. Namely, the electronic density of the cluster isstarting to oscillate in time with a given frequency. As we mentioned before, dependingon the value of the time delay ∆ τ , in the second collision the antiproton meets differentstates of the electronic density oscillations at the surface of the cluster. In what followswe are going to analyze the difference in the energy loss between two collisions dependingon the time delay of the second collision.For different time delays ∆ τ of the second collision, the value of the energy lossslightly varies. In order to illustrate this, we show the difference in the force betweenthe first and the second collision ∆ F z = F stz − F ndz for different values of ∆ τ . Here F stz is the force felt by the moving charge colliding with the non-excited metal cluster,which is equal for all τ ; F ndz is the force felt by the antiproton colliding with the excitedcluster. ∆ F z is shown in Fig. 1 as a function of the antiproton position. In this figure,large negative values of z ap indicate positions of the antiproton before each collision with8he cluster. The total force during the first collision F stz is shown in the inset of Fig. 1as a function of the projectile position. The two strong features in F stz correspond tothe antiproton crossing the cluster surface. Away from the cluster, the antiproton isattracted by the induced dipole. Inside the cluster, the electronic density rearranges inorder to screen the strong perturbation created by the moving antiproton. The forceinside the cluster oscillates about a mean value that roughly corresponds to the effectivestopping power for this particular velocity of the projectile ( υ = 0.5 a.u.) [5]. The curvesin the main panel of Fig. 1 show how the force felt by the antiproton changes dependingon the time at which the second collision starts.The fact that the energy loss is different at different time delays of the second collisioncan be also analyzed looking at the total energy of the cluster. The energy is shown inFig. 2 as a function of the antiproton position. From Fig. 2 we can see that the totalenergy of the cluster is increased by the collision. This increase in energy is the valueof the energy transferred by the antiproton to the cluster or, in other words, the energylost by the antiproton. We can see as well that, in all cases, the energy loss after thesecond collision is lower than after the first collision. The curves for ∆ τ =0 and for∆ τ =20 a.u. are similar. This is consistent with the values of the energy loss given inTable I for these two cases. We can also see the longer range of the cluster-antiprotoninteraction during the second collision. This is due to the net positive charge of theexcited cluster.The dependence of the energy loss on the time delay between collisions can be under-stood by looking at the time evolution of the induced electronic density. Figure 3a showsthe change in electronic density ∆ n ( z, ρ = 0 , t ) = [ n ( z, ρ = 0 , t ) − n ( z, ρ = 0 , t = 0)]inside the cluster in units of the background density n , along the z − axis and as a func-tion of time. The results are shown for the calculation with ∆ τ =0. The time interval in9ig. 3a is chosen to include the moment at which the second collision of the antiprotonwith the cluster takes place. In the figure, the projectile moves from the right to theleft. The second collision starts at t ≈ τ , the impact of the incoming antiproton with the previously excited clustercan bump into a minimum or a maximum of the electronic density oscillations. In thefirst calculation (∆ τ = 0) and when the time delay is ∆ τ = 20, the antiproton starts tocross the excited cluster when there is a maximum in the electronic density oscillationsat the surface of the cluster (the change in density in Fig. 3a is positive). In the case of∆ τ = 10 a.u. the second crossing finds a minimum of the electronic density oscillationsat the cluster surface. The time delay ∆ τ = 7 a.u. is chosen to have a case in which thesecond crossing falls neither on the minimum nor on the maximum of the change of theelectronic density, but in-between these two situations. Depending on this circumstance,the value of the energy lost by the antiproton varies.From Fig. 3a, we can also see that the minima and maxima in the induced electronicdensity become more pronounced after the second collision, indicating that the clusteris further excited by the second collision. This can also be seen in Fig. 3b where thechange in density is shown as a function of time for the particular value of z = 4a.u., marked with a dashed line in Fig. 3a. The amplitude of the electronic densityoscillations increases after the second collision. This is also observed in Fig. 3c andFig. 3d where we illustrate the density distribution in the cluster before ( t = 1812 a.u.)and after ( t = 1992 a.u.) the second collision. The change in density is plotted in a10lane in ( ρ, z ) coordinates with the center of the cluster at ( ρ = 0 , z =0). The negativeand positive peaks in the right panel (Fig. 3d) are much more intense than in the leftpanel (Fig. 3c). These pronounced oscillations show that, after the second collision, theoscillations of the induced electronic density are stronger. The excitation created by thesecond antiproton enhances that created during the first collision. However, the similardistribution of the induced charge seems to indicate that similar electronic modes areexcited in both events.In order to calculate the frequency of the density oscillations we perform a Fourieranalysis of the time evolution of the dipole moment P ( t ) → P ( ω ) created by the elec-tronic density in the excited cluster. The Fourier transform is done for two differentcases: a) after the single collision and without including the second collision, and b)after the two collisions. In this analysis we use the time evolution of the dipole during ∼ | P ( ω ) | are shown in Fig. 4. Two peaksare shown by arrows at frequencies ω = 0.261 and 0.284 a.u. (corresponding periods ofthe plasmon oscillations are T ≈ n as ω p = (cid:112) πn / r s = 2 .
07) this value is ω p = 0 .
4. Conclusions
In summary, we have calculated the energy loss of an antiproton colliding with asmall Al cluster, both when the cluster is in the ground state and when the cluster is inan excited electronic state. We have shown that the antiproton loses less energy whenpenetrating a cluster previously excited. The lowering of the energy loss is related notonly to the fact that the cluster is transferred to an excited state, but also to the factthat the cluster loses one electron during the first collision with the antiproton.We have also shown that the projectile creates a plasmon in the cluster and that theplasmon peak shifts to higher frequencies in the second collision. This corresponds tothe observed shorter period and larger amplitude of the electron density oscillations inthe cluster after the second collision of the antiproton with the cluster. The shift of theplasmon peak to higher frequencies is partially due to the emission of one electron fromthe cluster, which thus becomes positively charged.Our work is another example of how TDDFT in the time domain is an extremelyuseful tool to study electron dynamics in finite-size objects, as well as to analyze theenergy loss processes of charges interacting with condensed matter.12 cknowledgements
NEK acknowledges support from the CSIC JAE-predoc program, co-financed bythe European Science Foundation. We also acknowledge the support of the BasqueDepartamento de Educaci´on and the UPV/EHU (Grant No. IT-366-07), the SpanishMinisterio de Econom´ıa y Competitividad (Grant No. FIS2010-19609-CO2-02) andthe ETORTEK program funded by the Basque Departamento de Industria and theDiputaci´on Foral de Gipuzkoa. 13 eferences [1] I. Nagy, A. Arnau, and P.M. Echenique, Screening and stopping of charged particlesin an electron gas, Phys. Rev. B 48 (1993) 5650.[2] P.M. Echenique, R.M. Nieminen, J.C. Ashley, and R.H. Ritchie, Nonlinear stoppingpower of an electron gas for slow ions, Phys. Rev. A 33 (1986) 897.[3] R. Baer and N. Siam, Real-time study of the adiabatic energy loss in an atomiccollision with a metal cluster, J. Chem. Phys. 121 (2004) 6341.[4] M. Quijada, A.G. Borisov, I. Nagy, R. D´ıez Mui˜no, and P.M. Echenique, Time-dependent density-functional calculation of the stopping power for protons andantiprotons in metals, Phys. Rev. A 75 (2007) 042902.[5] M. Quijada, A.G. Borisov, R. D´ıez Mui˜no, Time-dependent density functional cal-culation of the energy loss of antiprotons colliding with metallic nanoshells, Phys.Stat. Sol. 205 (2008) 1312–1316.[6] A.V. Krasheninnikov, Y. Miyamoto, and D. Tom´anek, Role of electronic excitationsin ion collisions with carbon nanostructures, Phys. Rev. Lett. 99 (2007) 016104.[7] J.M. Pruneda, D. S´anchez-Portal, A. Arnau, J.I. Juaristi, and E. Artacho, Elec-tronic stopping power in LiF from first principles, Phys. Rev. Lett. 99 (2007) 235501.[8] A.A. Correa, J. Kohanoff, E. Artacho, D. Sanchez-Portal, and A. Caro, Nonadia-batic forces in ion-solid interactions: the initial stages of radiation damage, Phys.Rev. Lett. 108 (2012) 213201.[9] M.A. Zeb, J. Kohanoff, D. S´anchez-Portal, A. Arnau, J.I. Juaristi, and E. Artacho,14lectronic stopping power in gold: The role of d electrons and the H/He anomaly,Phys. Rev. Lett. 108 (2012) 225504.[10] A. Castro, M. Isla, J. I. Martinez, and J. A. Alonso, Scattering of a proton withthe Li4 cluster: Non-adiabatic molecular dynamics description based on time-dependent density-functional theory, Chem. Phys. 399 (2012) 130.[11] F. Chandezon, C. Guet, B.A. Huber, D. Jalabert, M. Maurel, E. Monnand, C. Ris-tori, and J.C. Rocco, Critical sizes against Coulomb dissociation of highly chargedsodium clusters obtained by ion impact, Phys. Rev. Lett. 74 (1995) 3784.[12] J. Daligault, F. Chandezon, C. Guet, B.A. Huber, and S. Tomita, Energy transferin collisions of metal clusters with multiply charged ions, Phys. Rev. A 66 (2002)033205.[13] V. Kasperovich, G. Tikhonov, K. Wong, P. Brockhaus, and V.V. Kresin, Polariza-tion forces in collisions between low-energy electrons and sodium clusters, Phys.Rev. A 60 (1999) 3071.[14] V. Kasperovich, K. Wong, G. Tikhonov, and V.V. Kresin, Electron capture by theimage charge of a metal nanoparticle, Phys. Rev. Lett. 85 (2000) 2729.[15] V. Kasperovich, G. Tikhonov, K. Wong, and V.V. Kresin, Negative-ion formationin collisions of low-energy electrons with neutral sodium clusters, Phys. Rev. A 62(2000) 063201.[16] A. Halder, A. Liang, Ch. Yin and V.V. Kresin, Double and triple ionization of silverclusters by electron impact, J. Phys.: Condens. Matter 24 (2012) 104009.1517] Th. Fennel, K.-H. Meiwes-Broer, J. Tiggesb¨aumker, P.-G. Reinhard, P. M. Dinh,and E. Suraud, Laser-driven nonlinear cluster dynamics, Rev. Mod. Phys. 82 (2010)1793.[18] F. Calvayrac, P.-G. Reinhard, and E. Suraud, Nonlinear plasmon response in highlyexcited metallic clusters, Phys. Rev. B 52 (1995) 17056-17059.[19] R. D´ıez Mui˜no, D. S´anchez-Portal, V.M. Silkin, E.V. Chulkov, and P.M. Echenique,P. Natl. Acad. Sci. USA 108 (2011) 971.[20] A.G. Borisov, D. S´anchez-Portal, R. D´ıez Mui˜no, P.M. Echenique, Building up thescreening below the femtosecond scale, Chem. Phys. Lett. 387 (2004) 95–100.[21] N.W. Ashcroft, N. David Mermin, Solid State Physics, Philadelphia, Harcourt Col-lege Publishers, 1976, Ch. 1, pp. 4-5.[22] W. Kohn and L.J. Sham, Self-consistent equations including exchange and correla-tion effects, Phys. Rev. 140 (1965) A1133–A1138.[23] P. Hohenberg and W Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (1964)B864–B871.[24] W. Ekardt, Work function of small metal particles: Self-consistent spherical jellium-background model, Phys. Rev. B 29 (1984) 1558.[25] E. Runge and E.K.U. Gros, Density-functional theory for time-dependent systems,Phys. Rev. Lett. 52 (1984) 997.[26] J.P Perdew, A. Zunger, Self-interaction correction to density-functional approxi-mations for many-electron systems, Phys. Rev. B 23 (1981) 5048.1627] A.G. Borisov, J.I. Juaristi, R. D´ıez Mui˜no, D. S´anchez-Portal, P.M. Echenique,Quantum-size effects in the energy loss of charged particles interacting with a con-fined two-dimensional electron gas, Phys. Rev. A 73 (2006) 012901.[28] A.G. Borisov, J.P. Gauyacq, S.V. Shabanov, Wave packet propagation study of thecharge transfer interaction in the F − –Cu(111) and –Ag(111) systems, Surf. Sci. 487(2001) 243–257.[29] E.V. Chulkov, A.G. Borisov, J.P. Gauyacq, D. S´anchez-Portal, V.M. Silkin, V.P.Zhukov, and P.M. Echenique, Electronic excitations in metals and at metal surfaces,Chem. Rev. 106 (2006) 4160–4206.[30] John A. Blackman (ed.), Metallic Nanoparticles, in: Prasanta Misra (ed.), Thebook series, Handbook of metal physics, Elsevier B.V., 2009, p. 202.[31] V. Kresin, Static electric polarizabilities and collective resonance frequencies ofsmall metal clusters, Phys. Rev. B 39 (1989-I) 3042.[32] F. Calvayrac, P.-G. Reinhard, E. Suraud, and C.A. Ullrich, Nonlinear electrondynamics in metal clusters, Phys. Rep. 337 (2000) 493.[33] Walt A. de Heer, The physics of simple metal clusters: experimental aspects andsimple models, Reviews of Modern Physics 65 (1993) No. 3.[34] Matthias Brack, The physics of simple metal clusters: self-consistent jellium modeland semiclassical approaches, Reviews of Modern Physics 65 (1993) No. 3.17able I: Energy loss E loss (in a.u.) of an antiproton crossing the spherical Al cluster inground and excited states.1 st collision 2 nd collision 2 nd collision, 2 nd collision, 2 nd collision,∆ τ = 7 au ∆ τ = 10 au ∆ τ = 20 au E loss F z , for different time delays τ between collisions as a function ofthe projectile position z ap . Inset: total force during the first collision F stz , as a functionof the projectile position. Dashed lines show the borders of the cluster ( R cl = 5.43 a.u.).All quantities are in a.u.Figure 2. Total energy of the cluster E tot for different collisions and for differenttime delays ∆ τ , as a function of the antiproton position z ap . All the energy curvescorresponding to the second crossing are referred to the value of the energy prior to thefirst crossing when antiproton is far from the cluster. Dashed lines show the borders ofthe cluster ( R cl = 5.43 a.u.). All quantities are in a.u.Figure 3. (a) Time evolution of the induced electronic density inside the cluster alongthe z − axis ( ρ = 0 .
02 a.u.) including the time at which the antiproton crosses the excitedcluster. The color code shows the change in density [ n ( z, ρ = 0 , t ) − n ( z, ρ = 0 , t = 0)] inunits of the background density n . The dashed line in panel (a) indicates the position z = 4 a.u. for which, in panel (b), we show the change in density as a function of time.Dashed line in panel (b) indicates the moment when the second collision starts. (c) and(d) show the change in the electronic density [ n ( r , t ) − n ( r , ρ, z ) coordinates at times t = 1812 a.u. and t = 1992 a.u. respectively.Figure 4. Dipole power spectra | P ( ω ) | (arbitrary units) for the excited cluster afterone collision (red solid line) and after two collisions (black dashed line). Frequency isshown in a.u. 19 ∆ F z ( a . u . ) z ap (a.u.) ∆τ =0 ∆τ =7 ∆τ =10 ∆τ =20 -0.3-0.2-0.1 0 0.1 0.2 0.3 -30 -20 -10 0 10 20 30 Figure 120 E t o t ( a . u . ) z ap (a.u.)1 st crossing 2 nd crossing 2 nd , ∆τ =7 2 nd , ∆τ =10 2 nd , ∆τ =20 Figure 221
10 -5 0 5 10z (a.u.) 1800 1850 1900 1950 2000 2050 2100 ti m e ( a . u . ) -1-0.8-0.6-0.4-0.2 0 0.2 0.4 (a) -1-0.8-0.6-0.4-0.2 0 0.2 0.4 1600 1800 2000 2200time (a.u.)c) d)z=4 a.u. (b) -10 -5 0 5 10z (a.u.)-10-5 0 5 10 ρ ( a . u . ) -0.01-0.005 0 0.005 (c) -10 -5 0 5 10z (a.u.)-10-5 0 5 10 ρ ( a . u . ) -0.01-0.005 0 0.005 (d) Figure 322 | P ( ω ) | ( a r b it r a r y un it s ) ω (a.u.) 1 st crossing 2 nd crossingcrossing