- Email: [email protected]
Nonlinear heavy-ion stopping in plasmas
Nuclear Instruments and Methods in Physics Research A 415 (1998) 680—686
Nonlinear heavy-ion stopping in plasmas Gu¨nter Zwicknagel* Laboratoire de Physique des Gaz et des Plasmas, BaL timent 212, Universite& Paris XI, F-91405 Orsay, France
Abstract In the context of ion-driven-inertial confinement fusion (ICF) the stopping power of ions in plasmas is crucial for the target design. For the stopping of heavy ions with high non-equilibrium charge states and nonlinear ion—target coupling, the theoretical description has to go beyond the standard approaches. To investigate nonlinear heavy-ion stopping in classical systems we performed both molecular dynamics (MD) simulations and Vlasov simulations. At nonlinear coupling we observed a dependence of the stopping power on the ion charge like Zx, x[1.5 and a screening length which now depends on Z and increases above the Debye length. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 34.50.Bw; 52.40.Mj Keywords: Stopping power; Nonlinear coupling; Nonideal plasmas
1. Introduction The experimental and theoretical determination of the energy loss of charged particles in matter has been an important issue of research throughout the whole century. One particular example is the stopping of heavy ions in plasmas recently discussed in context with inertial confinement fusion (ICF) and the electron cooling of heavy-ion beams in storage rings. There, the interaction of the ion beam with the target is of central importance. In particular the
* Correspondence address. Universita¨t Erlangen, Inst. fu¨r Theoretische Physik, Staudtstrasse 7, 91058 Erlangen, Germany. E-mail: [email protected].
target design in an ICF scenario requires a precise knowledge of the energy transfer rate. Here we investigate the nonlinear stopping power of heavy ions in classical free electron targets of density n and temperature ¹ and for non-relativistic projectile velocities as relevant for highly charged ions and fully ionized, dense plasmas where H" k ¹/E A1 and SgT"DZDe2/4pe +Sv TA1. Large B F 0 3 ratios H of the thermal energy k ¹ to the Fermi B energy E allow to neglect the Pauli principle and F to use classical Boltzmann statistics. At large Coulomb- or Bloch-parameters g"b /g the 0 3 ion—electron collisions can be described by classical trajectories while g@1 requires a quantum mechanical treatment. Here, b "DZDe2/4pe mv2 is the 0 0 3 classical collision diameter and g "+/mv where 3 3 v is the velocity of the relative motion and m the 3
0168-9002/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 0 4 4 8 - 3
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electron mass. Replacing v by the averaged relative 3 velocity Sv T yields the parameter SgT which char3 acterizes now the interaction of the ion with the target plasma. Thus, for HA1 and SgTA1 an entirely classical description is valid. The conventional descriptions of ion stopping presuppose that the ion represents a weak perturbation of the target, while nonlinear stopping is defined by a strong target perturbation. To quantify the strength of the perturbation caused by an ion of charge Z and velocity v, one can compare the potential energy for an electron in the field of the ion with the kinetic energy mv2/2 of the relative 3 motion. A weak perturbation is characterized by a potential energy which is smaller than the kinetic one. This definition of the ion—target coupling strength has been discussed in detail in Ref. [1] for any H, SgT. In the case of purely classical systems HA1, SgTA1 the derived criterion for a weak perturbation reads Sb T 2DZDe2 2J3DZDC3@2 0 " + @1, j 4pe mSv T2j 1#(v/v )3 5) 0 3
(1)
where Sb T is the classical collision diameter for the 0 averaged relative energy mSv T2 and j the velocity 3 dependent screening length. Here j was assumed as the Deybe length j"j "v /u (v2 "k ¹/m, D 5) 1 5) B u2"e2n/e m) at low projectile velocities and 1 0 j"v/u for high projectile velocities where Sv T 1 3 takes the values v and v, respectively. With 5) a simple interpolation between low and high projectile velocities v the product Sv T2j reads 3 Sv T2j+(v3 #v3)/u . Introducing this app3 5) 1 roximation together with the classical plasma parameter C"e2(4pn/3)1@3/4ne k ¹ yields the 0 B third term in Eq. (1), that is, the ion—target coupling strength is essentially given by the combination DZDC3@2, with weak coupling for DZDC3@2/(1#(v/v )3)@1 and nonlinear coupl5) ing for DZDC3@2/(1#(v/v )3)Z1. The plasma 5) parameter C alone represents the ratio of mean potential to thermal energy of the electrons and hence the coupling strength within the electron target. Ideal, collisionless targets correspond to C@1, nonideal ones to CZ0.1.
2. Standard approaches There are two conventional approaches to the stopping power, the dielectric linear response formalism and the binary collision treatment. In the dielectric description the stopping power dE/ds is calculated as the force between the ion and the polarization cloud created by the ion and can be expressed in terms of the dielectric function e(k, u) [2] dE j j2 D "Z2J3C3@2 D ds k ¹ 2p2 B 1 k ) K ] d3k Im . (2) e(k, k ) ) k2 k:km For classical systems the dielectric function e can be written in terms of the Fried—Conte plasma dispersion function [3] and represents the limit +P0 of the RPA dielectric function for any degeneracy of the electron target [4]. The classical linear response description, however, cannot treat close ion—electron collisions. To correct for this, a cutoff k is m introduced to incorporate a Bloch-correction for the stopping power which accounts for the contribution of close collisions in an approximative manner for DZDC3@2/(1#(v/v )3)@1. For classical 5) ion—electron collisions with g'1 this cutoff is given by k +1/Sb T"4pe mSv T2/DZDe2" m 0 0 3 (1#v2/v2 )/J3DZDC3@2j [1]. For a discussion of 5) D more sophisticated corrections on the linear response stopping power see Ref. [5]. In the binary collision approach the stopping power is obtained as the average over the momentum transfer in isolated collisions between the ion and the target electrons. In terms of the transport cross-section p for the ion—electron collisions, 53 the stopping power dE/ds on heavy ions (i.e. in the limit of an infinite projectile mass) reads [6,7]
P
C
dE j 1 D" ds k ¹ 4pJ3C3@2 B
A
GP
D
d3v 3 (2p)3@2v3 5)
B
H
( #)2 v K ) p (v ) 3 3 53 3 , (3) ]exp ! 3 2v2 v v j2 5) D 5) 5) for an electron target with a Maxwell velocity distribution. To account for the screening of the ion
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potential in the electron target an effective ion— electron interaction » must be used for calculatI% ing the transport cross-section p . This screened 53 interaction » is approximated by a Debye potenI% tial
A B
Ze2 r » (r)" exp ! . (4) I% 4pe r j 0 Investigating the binary collision approach not only for weak coupling but also for nonlinear one requires to evaluate the transport cross-section for arbitrary strengths of the ion—electron coupling. Thus p is determined from the scattering phase 53 shifts d for potential (4), i.e., l 1 = p (v )J + (l#1)sin2[d (v )!d (v )], (5) 53 3 l 3 l`1 3 v2 3 l/0 in the limit gA1. For this evaluation of the transport cross-section, expression (3) for the stopping power can be derived as well starting from a quantum Boltzmann equation where the binary collisions are described in terms of the T-matrix calculated for the Debye potential (4) [8,9].
3. Numerical simulations To determine the stopping power on a heavy ion moving through a target plasma of classical electrons two different descriptions of the ion—target system are considered and treated by two different numerical schemes. In the molecular dynamics (MD) simulations the classical equations of motion are solved for a system of point like particles where the projectile—electron and the electron—electron interactions are given by the bare Coulomb potential. In the second approach the target is approximated as a charged fluid. This allows to describe the ion—target system by the nonlinear Vlasov—Poisson equation which is solved numerically by test-particle simulations. There a smooth phase-space density is approximated by splitting real electrons into many (here 350) charged testparticles. Completely analogous to the well-known standard particle-in-cell (PIC) schemes [10] the test-particles act as sources for a mean field and the evolution of the phase-space density of the system
is provided by the motion of the test-particles in this self consistent mean field and, in our case, in the bare Coulomb field of the projectile-ion. In the Vlasov description the interaction between the electrons is given solely by the mean field while all electrons interact directly in the MD simulation. Hence, the MD simulations can also treat nonideal electron targets (CZ0.1) where electron—electron correlations become important whereas the Vlasov—Poisson simulations are restricted to ideal, collisionless plasmas (C@1). Since both schemes can be applied, however, to strong nonlinear ion—target coupling (DZDC3@2/(1#(v/v )3)Z1), 5) a comparison of the results of both methods reveals the influence of correlations between the electrons, more general, it allows to separate effects due to nonlinearity from those due to nonideality. Both numerical schemes use a elementary cubic simulation volume of typically 500 electrons or the corresponding number of test-particles with a Maxwell velocity distribution. The elementary simulation box is continued periodically in all directions. This cuts off contributions to the stopping power coming from the long range part of the interaction larger than the box length ¸. These are, however, only important in the high-velocity regime when the screening length will exceed the box size ¸, but they are negligible in the low velocity region as has been tested by varying the particle number at constant density. Thus the finite box size hardly affects the investigation of the nonlinear regime at low velocities or the comparison between MD and Vlasov—Poisson simulations which are performed using the same ¸. For further details on the applied simulation techniques see Refs. [11—13]. In Fig. 1 we compare the MD and Vlasov simulation results with the standard approaches. To take into account the finite box size in the theoretical descriptions a lower cutoff k "2p/¸ was .*/ introduced in the linear response description (2) which excludes the long wavelengths with k4 2p/¸. For the binary collision stopping power (3), p was evaluated using in Eq. (4) a screening length 53 j"j (1#v2/v2 )1@2 for j(¸/4 and j"¸/4 D 5) otherwise. As stated previously this modifies the corresponding stopping powers only at high velocities. The errorbars of the MD simulation reflect averaging over several runs with different initial
G. Zwicknagel/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 680—686
configurations. The test-particle simulations show less fluctuations and the errorbars lie within the size of the symbols (stars). For the upper example in Fig. 1 we have DZDC3@2"0.36 and the nonlinear
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stopping regime Eq. (1) is not yet reached. Here, the numerical results and the theoretical descriptions agree quite well. For increasing parameters DZDC3@2"1.99 (centre) and 11.2 (bottom) the nonlinear regime appears at low and intermediate velocities and considerable deviations between the simulation data and the standard descriptions occur. The linear response description with Bloch correction, Eq. (2), differs earlier and stronger from the numerical results compared to the binary collision description, Eq. (3), which remains closer to the Vlasov simulations (stars). The strong ion—electron interactions which become increasingly important in the nonlinear regime are well described in the binary collision picture but are treated in a very crude approximation in the linear response approach. When comparing on the other hand the binary collision results and the Vlasov results with the MD simulations results (triangles) a deviation at low velocities shows up which increases with the nonideality of the target C. This must be ascribed to the different treatments of the electron—electron interaction. The Vlasov—Poisson equation describes ideal target plasmas C@1 and depends only on the coupling parameter ZC3@2. In the MD simulations correlations between the electrons are included. They depend separately on both parameters ZC3@2 and C. The differences between the MD and Vlasov data thus document the influence of the nonideality of the target on the stopping power. The nonlinear ion—target coupling, on the other hand, is responsible for the strong deviations between the linear response description (dashed curves) and the simulations. As the strongest nonlinear effects occur for low ion velocities, we now look in more detail on this regime where the stopping power is linear in v. This allows to discuss it in terms of a dimensionless friction coef-
b Fig. 1. Normalized stopping power dE/ds/Z2 in units of 31@2C3@2k ¹/j as function of the ion velocity v in B D v "(k ¹/m)1@2 for an ion of charge Z"10 in electron plasmas 5) B with C"0.11, (top) 0.34 (centre) and 1.08 (bottom): MD simulations (n), with typical size of errorbars as indicated right top in each case, Vlasov simulations (*), the linear response description, Eq. (2), (dashed curve) and the binary collision treatment, Eq. (3), (dash-dotted).
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Fig. 2. Dimensionless friction coefficient R(Z, C) (6) as function of ZC3@2. MD simulation results for various ion charges Z and targets with C"0.11 (L), C"0.34 (n), C"1.08 (e) and C"3.41 (]) together with the results of Vlasov simulations (*), the linear response description, Eq. (2), (dashed) and the binary collision treatment, Eq. (3), (dash-dotted). For comparison, the dotted line shows a pure Z2C3 dependence.
ficient R(Z, C), defined through
AB
dE j J3 v D "! R(Z, C), v@v , 5) ds k ¹ C3@2 v B 5)
(6)
which depends separately on C and Z in the MD simulations and is function only of the combination ZC3@2 in all other discussed treatments. In Fig. 2 the simulation results for R as function of ZC3@2 are compared to the standard descriptions. Here, the variation of R(Z, C) with ZC3@2, assuming a fixed C, yields the charge dependence of the stopping power at low velocities. Obviously, the friction coefficient increases slower than Z2 (dotted line): for ZC3@2[0.1 with the usual weak coupling Z2 ln(const/Z) behavior, but for 0.1(ZC3@2[10 with a Z-dependence of approximately RJZx, x+1.5, and for still stronger nonlinear coupling with Zx, x(1.5. Such deviations of the charge dependence from the usual weak coupling behavior
have been also observed experimentally in context of cooling force measurements in heavy-ion storage rings. For a comparison of these experimental data with our simulation results, see Refs. [13,14]. This change of the charge dependence observed in the Vlasov and the MD simulations is the essential feature of nonlinear stopping. A closer look to Fig. 2 reveals the additional influence of the nonideality of the target on R(Z, C) by the separate dependence of the MD results on C and ZC3@2 as documented through the different offset for different C in contrast to the Vlasov results (stars) which line up with a pure ZC3@2 dependence. Comparing the simulation results with the theoretical approaches shows, as in Fig. 1, that the linear response description (2) fails grossly for nonlinear coupling. The binary collision description (3) agrees generally better with the simulation results, but also considerably underestimates R for large ZC3@2'10. But why do the binary collision results and the Vlasov simulations deviate at large ZC3@2? In the low-velocity regime dynamical polarization effects play no role and in both approaches the stopping power is determined by the movement of the electrons in an effective ion potential. Since the ion—electron interaction can be treated for arbitrary strength in the Vlasov simulations as well as in the binary collision approach based on the transport cross-section (5), one may conclude that the Debye potential (4) with j"j used in the binary D collision calculation does not approximate sufficiently well the selfconsistent potential of the Vlasov description. The effective potential » (r)"Ze2 exp(!r/j )/4pe r, however, was obI% D 0 tained from a linear theory and is not designed for a nonlinear situation. To investigate this point, we performed simulations where the ions are at rest and extracted the total electrical potential about the ion. The functional form of this total potential turns out to be rather close to Eq. (4), i.e. Jexp(!r/j)/r, however, with a screening length j which differs from j . From a fit of the functional D form exp(!r/j)/r to the “measured” total potential we obtain the (effective) screening lengths shown in Fig. 3. The rather large errorbars are here mainly due to the not perfect match of the real potential to exp(!r/j)/r. The found screening length j increases with the coupling strength and now
G. Zwicknagel/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 680—686
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Fig. 3. Static screening length j in units of the Debye length j as function of ZC3@2. Results of MD simulations for different D ion charges Z and electron plasmas C"0.34 (n) and C"1.08 (e) and of Vlasov simulations (*). The typical size of the errorbars for both MD and Vlasov data is given left top.
Fig. 4. Dimensionless friction coefficient R(Z, C) (6) as function of ZC3@2. MD (now with error bars) and Vlasov results as in Fig. 2. The curves are obtained from the binary collision description, Eq. (3), using different screening lengths j in the potential Eq. (4). The dash-dotted curve corresponds to j"j , the D solid one to j"j (1#0.09(ZC3@2)0.6). D
depends on the ion charge in contrast to the linear regime where j"j independent of Z. For the D variation of j with the coupling parameter we obtain from the data in Fig. 3 as a rough approximation j"j (1#0.09(ZC3@2)0.6). Taking this D screening length as an input in Eqs. (4) and (5) clearly improves the agreement of R predicted by the binary collision approximation Eq. (3) with the simulation data, as shown in Fig. 4. This demonstrates that both the close and strong ion—electron collisions and the collective nonlinear screening must be taken into account for a description of the nonlinear stopping power. Here, the previously discussed simple modification of the screening length might be sufficient at low velocities but much more sophisticated theoretical approaches are required for increasing velocities when nonlinear dynamical polarization effects will contribute to the stopping power (see Fig. 1).
4. Summary We have presented MD and Vlasov (test-particle) simulation results on the energy loss of heavy ions in classical electron plasmas. At nonlinear ion—target coupling (DZDC3@2/(1#(v/v )3)Z1) we 5) observed considerable deviations from the standard approaches for the stopping power as the dielectric linear response description and the binary collision approximation. The main features of nonlinear stopping are (1) the stopping power depends on the ion charge like Zx with x[1.5 and clearly deviates from the Z2 ln(const/Z) scaling of the standard theories at weak coupling and (2) this change in the Z-scaling is connected to nonlinear screening characterized by a Z-dependent screening length j(Z)5j . The additional influence of D the nonideality of the target (at CZ0.1) has been documented by comparing the results of MD simulations which include electron—electron correlations with those of Vlasov simulations which are based on a mean-field treatment.
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Acknowledgements G.Z. acknowledges clarifying and illuminating discussions with C. Deutsch, G. Maynard, P.-G. Reinhard and C. Toepffer and a grant from the “Deutsche Forschungsgemeinschaft” (DFG).
References [1] G. Zwicknagel, C. Deutsch, Phys. Rev. E 56 (1997) 970. [2] A.I. Akhiezer, I.A. Akhiezer, R.V. Polovin, A.G. Sitenko, K.N. Stepanov, Plasma Electrodynamics, vol. 2, Pergamon Press, Oxford, 1975. [3] B.D. Fried, S.D. Conte, The Plasma Dispersion Function, Academic Press, New York, (1961). [4] C. Gouedard, C. Deutsch, J. Math. Phys. 19 (1978) 32.
[5] G. Maynard et al., Nucl. Instr. and Meth. A 415 (1998) 687. [6] L. Bo¨nig, K. Scho¨nhammer, Phys. Rev. B 39 (1989) 7413. [7] L. de Ferrariis, N.R. Arista, Phys. Rev. A 29 (1984) 2145. [8] D.O. Gericke, M. Schlanges, W.D. Kraeft, Phys. Lett. A 222 (1996) 241. [9] K. Morawetz, G. Ro¨pke, Phys. Rev. E 54 (1996) 4134. [10] R.W. Hockney, J.W. Eastwood, Computer Simulations Using Particles, McGrawHill, New York, 1981. [11] G. Zwicknagel, C. Toepffer, P.-G. Reinhard, Hyperfine Interactions 99 (1996) 285. [12] G. Zwicknagel, C. Toepffer, P.-G. Reinhard, Fusion Eng. Des. 32—33 (1996) 523. [13] G. Zwicknagel, Q. Spreiter, M. Miller, C. Toepffer, P.-G. Reinhard, in: D.M. Maletic´, A.G. Ruggiero (Eds.), Crystalline beams and related issues, World Scientific, Singapore, 1996, p. 185. [14] G. Zwicknagel, Q. Spreiter, C. Toepffer, Hyperfine Interactions 108 (1997) 131.
Nonlinear heavy-ion stopping in plasmas Gu¨nter Zwicknagel* Laboratoire de Physique des Gaz et des Plasmas, BaL timent 212, Universite& Paris XI, F-91405 Orsay, France
Abstract In the context of ion-driven-inertial confinement fusion (ICF) the stopping power of ions in plasmas is crucial for the target design. For the stopping of heavy ions with high non-equilibrium charge states and nonlinear ion—target coupling, the theoretical description has to go beyond the standard approaches. To investigate nonlinear heavy-ion stopping in classical systems we performed both molecular dynamics (MD) simulations and Vlasov simulations. At nonlinear coupling we observed a dependence of the stopping power on the ion charge like Zx, x[1.5 and a screening length which now depends on Z and increases above the Debye length. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 34.50.Bw; 52.40.Mj Keywords: Stopping power; Nonlinear coupling; Nonideal plasmas
1. Introduction The experimental and theoretical determination of the energy loss of charged particles in matter has been an important issue of research throughout the whole century. One particular example is the stopping of heavy ions in plasmas recently discussed in context with inertial confinement fusion (ICF) and the electron cooling of heavy-ion beams in storage rings. There, the interaction of the ion beam with the target is of central importance. In particular the
* Correspondence address. Universita¨t Erlangen, Inst. fu¨r Theoretische Physik, Staudtstrasse 7, 91058 Erlangen, Germany. E-mail: [email protected].
target design in an ICF scenario requires a precise knowledge of the energy transfer rate. Here we investigate the nonlinear stopping power of heavy ions in classical free electron targets of density n and temperature ¹ and for non-relativistic projectile velocities as relevant for highly charged ions and fully ionized, dense plasmas where H" k ¹/E A1 and SgT"DZDe2/4pe +Sv TA1. Large B F 0 3 ratios H of the thermal energy k ¹ to the Fermi B energy E allow to neglect the Pauli principle and F to use classical Boltzmann statistics. At large Coulomb- or Bloch-parameters g"b /g the 0 3 ion—electron collisions can be described by classical trajectories while g@1 requires a quantum mechanical treatment. Here, b "DZDe2/4pe mv2 is the 0 0 3 classical collision diameter and g "+/mv where 3 3 v is the velocity of the relative motion and m the 3
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electron mass. Replacing v by the averaged relative 3 velocity Sv T yields the parameter SgT which char3 acterizes now the interaction of the ion with the target plasma. Thus, for HA1 and SgTA1 an entirely classical description is valid. The conventional descriptions of ion stopping presuppose that the ion represents a weak perturbation of the target, while nonlinear stopping is defined by a strong target perturbation. To quantify the strength of the perturbation caused by an ion of charge Z and velocity v, one can compare the potential energy for an electron in the field of the ion with the kinetic energy mv2/2 of the relative 3 motion. A weak perturbation is characterized by a potential energy which is smaller than the kinetic one. This definition of the ion—target coupling strength has been discussed in detail in Ref. [1] for any H, SgT. In the case of purely classical systems HA1, SgTA1 the derived criterion for a weak perturbation reads Sb T 2DZDe2 2J3DZDC3@2 0 " + @1, j 4pe mSv T2j 1#(v/v )3 5) 0 3
(1)
where Sb T is the classical collision diameter for the 0 averaged relative energy mSv T2 and j the velocity 3 dependent screening length. Here j was assumed as the Deybe length j"j "v /u (v2 "k ¹/m, D 5) 1 5) B u2"e2n/e m) at low projectile velocities and 1 0 j"v/u for high projectile velocities where Sv T 1 3 takes the values v and v, respectively. With 5) a simple interpolation between low and high projectile velocities v the product Sv T2j reads 3 Sv T2j+(v3 #v3)/u . Introducing this app3 5) 1 roximation together with the classical plasma parameter C"e2(4pn/3)1@3/4ne k ¹ yields the 0 B third term in Eq. (1), that is, the ion—target coupling strength is essentially given by the combination DZDC3@2, with weak coupling for DZDC3@2/(1#(v/v )3)@1 and nonlinear coupl5) ing for DZDC3@2/(1#(v/v )3)Z1. The plasma 5) parameter C alone represents the ratio of mean potential to thermal energy of the electrons and hence the coupling strength within the electron target. Ideal, collisionless targets correspond to C@1, nonideal ones to CZ0.1.
2. Standard approaches There are two conventional approaches to the stopping power, the dielectric linear response formalism and the binary collision treatment. In the dielectric description the stopping power dE/ds is calculated as the force between the ion and the polarization cloud created by the ion and can be expressed in terms of the dielectric function e(k, u) [2] dE j j2 D "Z2J3C3@2 D ds k ¹ 2p2 B 1 k ) K ] d3k Im . (2) e(k, k ) ) k2 k:km For classical systems the dielectric function e can be written in terms of the Fried—Conte plasma dispersion function [3] and represents the limit +P0 of the RPA dielectric function for any degeneracy of the electron target [4]. The classical linear response description, however, cannot treat close ion—electron collisions. To correct for this, a cutoff k is m introduced to incorporate a Bloch-correction for the stopping power which accounts for the contribution of close collisions in an approximative manner for DZDC3@2/(1#(v/v )3)@1. For classical 5) ion—electron collisions with g'1 this cutoff is given by k +1/Sb T"4pe mSv T2/DZDe2" m 0 0 3 (1#v2/v2 )/J3DZDC3@2j [1]. For a discussion of 5) D more sophisticated corrections on the linear response stopping power see Ref. [5]. In the binary collision approach the stopping power is obtained as the average over the momentum transfer in isolated collisions between the ion and the target electrons. In terms of the transport cross-section p for the ion—electron collisions, 53 the stopping power dE/ds on heavy ions (i.e. in the limit of an infinite projectile mass) reads [6,7]
P
C
dE j 1 D" ds k ¹ 4pJ3C3@2 B
A
GP
D
d3v 3 (2p)3@2v3 5)
B
H
( #)2 v K ) p (v ) 3 3 53 3 , (3) ]exp ! 3 2v2 v v j2 5) D 5) 5) for an electron target with a Maxwell velocity distribution. To account for the screening of the ion
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potential in the electron target an effective ion— electron interaction » must be used for calculatI% ing the transport cross-section p . This screened 53 interaction » is approximated by a Debye potenI% tial
A B
Ze2 r » (r)" exp ! . (4) I% 4pe r j 0 Investigating the binary collision approach not only for weak coupling but also for nonlinear one requires to evaluate the transport cross-section for arbitrary strengths of the ion—electron coupling. Thus p is determined from the scattering phase 53 shifts d for potential (4), i.e., l 1 = p (v )J + (l#1)sin2[d (v )!d (v )], (5) 53 3 l 3 l`1 3 v2 3 l/0 in the limit gA1. For this evaluation of the transport cross-section, expression (3) for the stopping power can be derived as well starting from a quantum Boltzmann equation where the binary collisions are described in terms of the T-matrix calculated for the Debye potential (4) [8,9].
3. Numerical simulations To determine the stopping power on a heavy ion moving through a target plasma of classical electrons two different descriptions of the ion—target system are considered and treated by two different numerical schemes. In the molecular dynamics (MD) simulations the classical equations of motion are solved for a system of point like particles where the projectile—electron and the electron—electron interactions are given by the bare Coulomb potential. In the second approach the target is approximated as a charged fluid. This allows to describe the ion—target system by the nonlinear Vlasov—Poisson equation which is solved numerically by test-particle simulations. There a smooth phase-space density is approximated by splitting real electrons into many (here 350) charged testparticles. Completely analogous to the well-known standard particle-in-cell (PIC) schemes [10] the test-particles act as sources for a mean field and the evolution of the phase-space density of the system
is provided by the motion of the test-particles in this self consistent mean field and, in our case, in the bare Coulomb field of the projectile-ion. In the Vlasov description the interaction between the electrons is given solely by the mean field while all electrons interact directly in the MD simulation. Hence, the MD simulations can also treat nonideal electron targets (CZ0.1) where electron—electron correlations become important whereas the Vlasov—Poisson simulations are restricted to ideal, collisionless plasmas (C@1). Since both schemes can be applied, however, to strong nonlinear ion—target coupling (DZDC3@2/(1#(v/v )3)Z1), 5) a comparison of the results of both methods reveals the influence of correlations between the electrons, more general, it allows to separate effects due to nonlinearity from those due to nonideality. Both numerical schemes use a elementary cubic simulation volume of typically 500 electrons or the corresponding number of test-particles with a Maxwell velocity distribution. The elementary simulation box is continued periodically in all directions. This cuts off contributions to the stopping power coming from the long range part of the interaction larger than the box length ¸. These are, however, only important in the high-velocity regime when the screening length will exceed the box size ¸, but they are negligible in the low velocity region as has been tested by varying the particle number at constant density. Thus the finite box size hardly affects the investigation of the nonlinear regime at low velocities or the comparison between MD and Vlasov—Poisson simulations which are performed using the same ¸. For further details on the applied simulation techniques see Refs. [11—13]. In Fig. 1 we compare the MD and Vlasov simulation results with the standard approaches. To take into account the finite box size in the theoretical descriptions a lower cutoff k "2p/¸ was .*/ introduced in the linear response description (2) which excludes the long wavelengths with k4 2p/¸. For the binary collision stopping power (3), p was evaluated using in Eq. (4) a screening length 53 j"j (1#v2/v2 )1@2 for j(¸/4 and j"¸/4 D 5) otherwise. As stated previously this modifies the corresponding stopping powers only at high velocities. The errorbars of the MD simulation reflect averaging over several runs with different initial
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configurations. The test-particle simulations show less fluctuations and the errorbars lie within the size of the symbols (stars). For the upper example in Fig. 1 we have DZDC3@2"0.36 and the nonlinear
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stopping regime Eq. (1) is not yet reached. Here, the numerical results and the theoretical descriptions agree quite well. For increasing parameters DZDC3@2"1.99 (centre) and 11.2 (bottom) the nonlinear regime appears at low and intermediate velocities and considerable deviations between the simulation data and the standard descriptions occur. The linear response description with Bloch correction, Eq. (2), differs earlier and stronger from the numerical results compared to the binary collision description, Eq. (3), which remains closer to the Vlasov simulations (stars). The strong ion—electron interactions which become increasingly important in the nonlinear regime are well described in the binary collision picture but are treated in a very crude approximation in the linear response approach. When comparing on the other hand the binary collision results and the Vlasov results with the MD simulations results (triangles) a deviation at low velocities shows up which increases with the nonideality of the target C. This must be ascribed to the different treatments of the electron—electron interaction. The Vlasov—Poisson equation describes ideal target plasmas C@1 and depends only on the coupling parameter ZC3@2. In the MD simulations correlations between the electrons are included. They depend separately on both parameters ZC3@2 and C. The differences between the MD and Vlasov data thus document the influence of the nonideality of the target on the stopping power. The nonlinear ion—target coupling, on the other hand, is responsible for the strong deviations between the linear response description (dashed curves) and the simulations. As the strongest nonlinear effects occur for low ion velocities, we now look in more detail on this regime where the stopping power is linear in v. This allows to discuss it in terms of a dimensionless friction coef-
b Fig. 1. Normalized stopping power dE/ds/Z2 in units of 31@2C3@2k ¹/j as function of the ion velocity v in B D v "(k ¹/m)1@2 for an ion of charge Z"10 in electron plasmas 5) B with C"0.11, (top) 0.34 (centre) and 1.08 (bottom): MD simulations (n), with typical size of errorbars as indicated right top in each case, Vlasov simulations (*), the linear response description, Eq. (2), (dashed curve) and the binary collision treatment, Eq. (3), (dash-dotted).
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Fig. 2. Dimensionless friction coefficient R(Z, C) (6) as function of ZC3@2. MD simulation results for various ion charges Z and targets with C"0.11 (L), C"0.34 (n), C"1.08 (e) and C"3.41 (]) together with the results of Vlasov simulations (*), the linear response description, Eq. (2), (dashed) and the binary collision treatment, Eq. (3), (dash-dotted). For comparison, the dotted line shows a pure Z2C3 dependence.
ficient R(Z, C), defined through
AB
dE j J3 v D "! R(Z, C), v@v , 5) ds k ¹ C3@2 v B 5)
(6)
which depends separately on C and Z in the MD simulations and is function only of the combination ZC3@2 in all other discussed treatments. In Fig. 2 the simulation results for R as function of ZC3@2 are compared to the standard descriptions. Here, the variation of R(Z, C) with ZC3@2, assuming a fixed C, yields the charge dependence of the stopping power at low velocities. Obviously, the friction coefficient increases slower than Z2 (dotted line): for ZC3@2[0.1 with the usual weak coupling Z2 ln(const/Z) behavior, but for 0.1(ZC3@2[10 with a Z-dependence of approximately RJZx, x+1.5, and for still stronger nonlinear coupling with Zx, x(1.5. Such deviations of the charge dependence from the usual weak coupling behavior
have been also observed experimentally in context of cooling force measurements in heavy-ion storage rings. For a comparison of these experimental data with our simulation results, see Refs. [13,14]. This change of the charge dependence observed in the Vlasov and the MD simulations is the essential feature of nonlinear stopping. A closer look to Fig. 2 reveals the additional influence of the nonideality of the target on R(Z, C) by the separate dependence of the MD results on C and ZC3@2 as documented through the different offset for different C in contrast to the Vlasov results (stars) which line up with a pure ZC3@2 dependence. Comparing the simulation results with the theoretical approaches shows, as in Fig. 1, that the linear response description (2) fails grossly for nonlinear coupling. The binary collision description (3) agrees generally better with the simulation results, but also considerably underestimates R for large ZC3@2'10. But why do the binary collision results and the Vlasov simulations deviate at large ZC3@2? In the low-velocity regime dynamical polarization effects play no role and in both approaches the stopping power is determined by the movement of the electrons in an effective ion potential. Since the ion—electron interaction can be treated for arbitrary strength in the Vlasov simulations as well as in the binary collision approach based on the transport cross-section (5), one may conclude that the Debye potential (4) with j"j used in the binary D collision calculation does not approximate sufficiently well the selfconsistent potential of the Vlasov description. The effective potential » (r)"Ze2 exp(!r/j )/4pe r, however, was obI% D 0 tained from a linear theory and is not designed for a nonlinear situation. To investigate this point, we performed simulations where the ions are at rest and extracted the total electrical potential about the ion. The functional form of this total potential turns out to be rather close to Eq. (4), i.e. Jexp(!r/j)/r, however, with a screening length j which differs from j . From a fit of the functional D form exp(!r/j)/r to the “measured” total potential we obtain the (effective) screening lengths shown in Fig. 3. The rather large errorbars are here mainly due to the not perfect match of the real potential to exp(!r/j)/r. The found screening length j increases with the coupling strength and now
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Fig. 3. Static screening length j in units of the Debye length j as function of ZC3@2. Results of MD simulations for different D ion charges Z and electron plasmas C"0.34 (n) and C"1.08 (e) and of Vlasov simulations (*). The typical size of the errorbars for both MD and Vlasov data is given left top.
Fig. 4. Dimensionless friction coefficient R(Z, C) (6) as function of ZC3@2. MD (now with error bars) and Vlasov results as in Fig. 2. The curves are obtained from the binary collision description, Eq. (3), using different screening lengths j in the potential Eq. (4). The dash-dotted curve corresponds to j"j , the D solid one to j"j (1#0.09(ZC3@2)0.6). D
depends on the ion charge in contrast to the linear regime where j"j independent of Z. For the D variation of j with the coupling parameter we obtain from the data in Fig. 3 as a rough approximation j"j (1#0.09(ZC3@2)0.6). Taking this D screening length as an input in Eqs. (4) and (5) clearly improves the agreement of R predicted by the binary collision approximation Eq. (3) with the simulation data, as shown in Fig. 4. This demonstrates that both the close and strong ion—electron collisions and the collective nonlinear screening must be taken into account for a description of the nonlinear stopping power. Here, the previously discussed simple modification of the screening length might be sufficient at low velocities but much more sophisticated theoretical approaches are required for increasing velocities when nonlinear dynamical polarization effects will contribute to the stopping power (see Fig. 1).
4. Summary We have presented MD and Vlasov (test-particle) simulation results on the energy loss of heavy ions in classical electron plasmas. At nonlinear ion—target coupling (DZDC3@2/(1#(v/v )3)Z1) we 5) observed considerable deviations from the standard approaches for the stopping power as the dielectric linear response description and the binary collision approximation. The main features of nonlinear stopping are (1) the stopping power depends on the ion charge like Zx with x[1.5 and clearly deviates from the Z2 ln(const/Z) scaling of the standard theories at weak coupling and (2) this change in the Z-scaling is connected to nonlinear screening characterized by a Z-dependent screening length j(Z)5j . The additional influence of D the nonideality of the target (at CZ0.1) has been documented by comparing the results of MD simulations which include electron—electron correlations with those of Vlasov simulations which are based on a mean-field treatment.
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Acknowledgements G.Z. acknowledges clarifying and illuminating discussions with C. Deutsch, G. Maynard, P.-G. Reinhard and C. Toepffer and a grant from the “Deutsche Forschungsgemeinschaft” (DFG).
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[5] G. Maynard et al., Nucl. Instr. and Meth. A 415 (1998) 687. [6] L. Bo¨nig, K. Scho¨nhammer, Phys. Rev. B 39 (1989) 7413. [7] L. de Ferrariis, N.R. Arista, Phys. Rev. A 29 (1984) 2145. [8] D.O. Gericke, M. Schlanges, W.D. Kraeft, Phys. Lett. A 222 (1996) 241. [9] K. Morawetz, G. Ro¨pke, Phys. Rev. E 54 (1996) 4134. [10] R.W. Hockney, J.W. Eastwood, Computer Simulations Using Particles, McGrawHill, New York, 1981. [11] G. Zwicknagel, C. Toepffer, P.-G. Reinhard, Hyperfine Interactions 99 (1996) 285. [12] G. Zwicknagel, C. Toepffer, P.-G. Reinhard, Fusion Eng. Des. 32—33 (1996) 523. [13] G. Zwicknagel, Q. Spreiter, M. Miller, C. Toepffer, P.-G. Reinhard, in: D.M. Maletic´, A.G. Ruggiero (Eds.), Crystalline beams and related issues, World Scientific, Singapore, 1996, p. 185. [14] G. Zwicknagel, Q. Spreiter, C. Toepffer, Hyperfine Interactions 108 (1997) 131.