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<ul><li><p>IL NUOVO CIMENTO VOL. 106A, N. 12 Dicembre 1993 </p><p>Charge Evolution of Swift Heavy Ions in Fusion Plasmas (*). </p><p>G. MAYNARD(l), W. ANDRt~ (3), M. CHABOT(2), C. DEUTSCH (1), C. FLEURIER(2), D. GARDt~S (2), D. HONG(3), J. KIENER (2), M. POUEY(l) and K. WOHRER (4) (1) Groupe de Recherche 918 du Centre National de Recherche Scientifique </p><p>Laboratoire de Physique des Gaz et des Plasmas, Bat. 212, Universitd Paris XI 91405 Orsay, France </p><p>(2) Institut de Physique Nucldaire - Orsay, France (3) Groupe de Recherche sur l'Energdtique des Milieux Ionisgs - Orlgans, France (a) Groupe de Physique du Solide - Paris VI, France </p><p>(ricevuto il 25 Maggio 1993; approvato l'l Giugno 1993) </p><p>Summary. -- Influence of target temperature, density and atomic number on the charge state of s~f-t heavy ions interacting with hot and dense plasma is considered. Hydrogen targets exhibit strong temperature effects and large non-equilibrium charges, whereas interactions with heavy material are more sensitive to the density effect. Our new average correlated hydrogenic atom model (ACHAM) is presented. It enables us to cover the whole range of target densities of interest for swift heavy- ion-plasma interaction experiments. </p><p>PACS 28.50.Re - Fusion reactors and thermonuclear power studies. PACS 52.40.Mj - Particle beam interactions in plasma (including intense charged- particle beams). PACS 52.20.Hv - Atomic molecular ion, and heavy-particle collisions. PACS 52.70.Nc - Particle measurements. </p><p>1. - In t roduct ion . </p><p>In the case of direct or indirect heavy-ion inertial fusion, a precise knowledge of the beam energy deposition profile is needed especially near its maximum, i.e. for a few MeV/u energy. Even though current densities up to 10 kAcm -2 are required for fusion, the heavy-ion beam in the dense target can be seen as a collection of individual uncorrelated particles [1]. As a first step one is thus entitled to replace a complex interaction of intense ion beams with a given target by that of a dilute beam out of a standard accelerating structure. The plasma target may thus be fired independently of the ion beam. Such experiments were already performed at Orsay[1] and at </p><p>(*) Paper presented at the International Symposium on Heavy Ion Inertial Fusion, Frascati, May 25-28, 1993. </p><p>1825 </p></li><li><p>1826 G. MAYNARD, W. ANDRt~, M. CHABOT, C. DEUTSCH, ETC. </p><p>Darmstadt[2] using Ohmic heating discharge in a low-pressure hydrogen column. These experiments were primarily destined to confirm the so-called enhanced plasma stopping (EPS) as predicted by the standard stopping model (SSM)[1, 2]. They were then upgraded allowing higher free electron densities, using the Z-pinch effect, and measurements of the enhanced projectile ionization in plasma (EPIP) of the heavy ion beam at the exit of the discharge. This paper is devoted to the relation between EPS and EPIP and to the influence of the plasma density, atomic number and temperature on the projectile ionization state. In sect. 2 basic features of heavy-ion beam plasma interaction experiments are presented together with the SSM. The various atomic rate coefficients are detailed in sect. 3 with the general trends of the EPIP. Two examples are analyzed in sect. 4. Our new theoretical model is presented in sect. 5 and sect. 6 summarizes our results. Atomic units (h = e = me = I) are used throughout except when otherwise specified. This paper restricts to heavy ion beams in the MeV/u energy range interacting with plasma of maximum temperature of a few hundred eV. </p><p>2 - Stopping and ion-beam-plasma experiments. </p><p>Most of heavy-ion beam plasma experiments can be summarized as follow: </p><p>An initial ion beam of atomic number Z, atomic mass M, kinetic energy per nucleon Eo and charge state qo enters first in a sheath region (region A), then flows through a nearly homogenous plasma of atomic number z, ionization z *, total electron density n (free and bound), temperature T and length D. q(x) and E(x) are the mean ionization state and energy of the ion beam at a distance x from the entrance surface of the plasma with the notation qexit and Eex i t for q(D) and E(D). At the exit surface the moving ions encounter a second sheath region (region B) before they are analyzed in a magnet spectrometer where their average charge qoutside and average energy Eouts id e are measured. One then wants to deduce from these experiments the two functions f and g defined by </p><p>(1) AEp =f(Eo, qo, z, n, T) </p><p>and </p><p>(2) qexit - - g(Eo, qo, z, n, T) </p><p>D </p><p>with hEp = ISdx. The stopping power S can be written as 0 </p><p>() = - - - - L b L=Lo+ q + q 2 dx V 2 z z ' vL1 ~ L2. In eq. (3), V is the projectile velocity, L f and L b are stopping numbers of free and </p><p>bound electrons and are expressed as the sum of three functions L0, L1 and L2. Equation (3) is the most straightforward extension to a plasma target of the Bohr-Bethe-Bloch-Barkas stopping in cold matter. </p><p>L0 in eq. (3) gives the so-called dominant term, at high velocity it reduces to the Bethe limit Lo = ln(2V2/u) with the electron resonance energy ~ given by the plasmon energy ~op for free electrons (~Op = 1.36-10-S(n(101Scm-S)) ~ and by the </p></li><li><p>CHARGE EVOLUTION OF SWIFT HEAVY IONS IN FUSION PLASMAS 1827 </p><p>10 3 ~D </p><p>102 </p><p>0 </p><p>I * I I I I </p><p>/// </p><p>i t6 32 48 64 80 96 </p><p>atomic number </p><p>Fig. 1. - Square of [] effective, A exit and * outside charge of 4 MeV/u heavy ions penetrating a carbon target as a function of projectile atomic number Z. Experimental Zef comes from the tabulation of ref. [7] and experimental qoutside ( - - - ) from the tabulation of ref. [6]. Our results are theoretical calculations made using ACHAM (see sect. 5). </p><p>mean excitation energy / for bound ones (i = z/2 for neutral atom). For Eo of a few MeV/u, V 10 and one obtains f b ~" Lo/Lo ~ 2 for a hydrogen target. This EPS close to 2 has been checked out in Ohmic heating discharges [1,2]. </p><p>The stopping number L in eq. (3) is ~t ten as an expansion in the Born parameter q/V. L1 and L2 are corrections to the linear stopping theory which states S ~ q2. L1 (Barkas term) comes from distant collisions and L2 (Bloch term) arises from close ones. For collisions with bound electrons L2 is connected to ionization and L1 to excitation cross-section. It is well known that the effective charge for ionization is smaller than q [3] so that L2 is negative, on the other hand, second-order perturbation theory shows that L1 increases S[4]. Defining the effective charge Zef by Z[f (ql)/S(q]) = Z[f (q2)/S(q2) non-linearities contributions to the stopping can thus be quantified using the ratio Rnl - - (q /Zef ) 2 9 </p><p>Even in cold target there are few direct experimental measurements of Rnl for heavy-ion beams in the MeV/u energy regime and most of them are for low-pressure target gas: Geissel and his collaborators[5] found a maximum R,j of 1.32 for 1.4 MeV/u ions in argon gas. In dense target experiments, one of the most important difficulties is due to the unknown discrepancy between qe~t and qoutside due to Auger emission at the exit surface. Results for 4 MeV/u heavy ions in carbon are reported in fig. 1 where parameterization of experimental data for qo~t~de[6] and Zef[7] is shown together with our theoretical results. One gets for uranium a maximum ratio (qoutside/Zef) 2= 1.73 whereas our theoretical Rnl is only 1.35. </p><p>In cold-target cases, if the Born parameter q/V is large the moving ion still retains many bound electrons, so that atomic collision processes, like charge transfer, and projectile structure can have some influence in the stopping mechanism. One advantage of highly ionized plasma target experiments is to yield higher ionized heavy projectiles providing a larger Born parameter together with a reduced charge transfer rate so that heavy-ion beam-plasma interaction experiments producing qe~t and AEp give very good checks of stopping power theories. </p></li><li><p>1828 G. MAYNARD, W. ANDRE, M. CHABOT, C. DEUTSCH, ETC. </p><p>3. - Charge state and atomic rates. </p><p>3"1. General description. - As the projectile passes through the plasma target its ionization state changes owing to collisions with plasma ions and electrons. The projectile is ionized by plasma ions collisions, free electrons collisions and autoionization processes, it can also gain electrons by bound-bound charge transfer from plasma bound electrons, by radiative transfer, dielectronic recombination and three-body recombination R3 of free electrons. Due to angular integration of free electrons velocities in the projectile reference frame, R3 always gives a negligible contribution even at high densities [8] and will not be considered here. The internal projectile state is also varying due to plasma ions and electrons excitations and to spontaneous radiative decay. </p><p>In low-density target, beam ions are mainly in their ground state and so only total cross-sections have to be considered. As a pedagogic example we show in fig. 2 atomic collision rates for 4 MeV/u I q + ions in helium plasma with n = 4.1017 cm-3 and T = = 4.4 eV giving a plasma ionization z * = 0.5. Similar examples for H target can be found in ref. [9,10]. Using fig. 2 one can define the equih'brium charge state Zeq as the charge state where the ionization rate is equal to the recombination rate. In our case the ionization rate comes from ion collisions and the recombination rate is due to charge transfer and we get Zeq = 31. Looking now at the rate Zeq for this charge q = Z~q we can have an estimate of the necessary time for the ion to reach its equilibrium charge, keeping its velocity constant, by defining the equilibrium time teq = 1/Z~q. </p><p>Atomic rates in fig. 2 are equal to the frequency at which one projectile gains or loses one electron, so that they give the rate for a relative charge variation of 1/q. In the same manner, we define a corresponding stopping rate Rstp for a relative energy variation of ~/E = l/q: Rst p = qSV/E which is also reported in fig. 2. </p><p>Except for the density influence, all of the basic features of the evolution of heavy- ion charge in plasma can be understood from fig. 2. To find what can happen to the iodine ion at a given ionization state q, one has just to look at the largest atomic rate for this charge. Let us take an initial charge q = 24. From fig. 2 we see that our ion </p><p>{) ~ i i i i i i , </p><p>9 u ~7"~ ion ionization ~ ..5 . . . . . . - </p><p>m 7 - J '~ lectroni xg : ,, ~ ~n izat ion , ' "~:-. : </p><p>5 I0 t ~ v , , ', J i i , - </p><p>20 24 28 32 36 40 44 48 52 charge s ta te </p><p>Fig. 2. - Atomic rates for electron capture (solid line) and loss (dashed lines) together with stopping rate (dotted curve) of 4 MeV/u lq in half-ionized helium plasma of T = 4.4 eV and n = 4.1017 cm -a as a function of projectile charge state q. </p></li><li><p>CHARGE EVOLUTION OF SWIFT HEAVY IONS IN FUSION PLASMAS 1829 </p><p>will be ionized by plasma ions collisions until it reaches the equilibrium value Zeq = 31 at a time teq = 5 ns. In this case the atomic rate is ten times larger than the stopping rate so that from q = 24 to q = 31 the projectile has not enough time to slow down and the necessary target length to reach equilibrium is then 13.8 cm at this density. If we take now the same example but without charge transfer, the moving ion charge increases with ion collisions until it reaches the charge q = 38 where the ionization rate is equal to the stopping rate. We define this ionization state as the dynamical charge Zdy. If q is larger or equal to Zay energy will change more rapidly than charge state during the slowing down. The projectile will remain in a nearly frozen charge until the energy becomes low enough for the recombination rate to be larger than Rstp 9 </p><p>We are now able to predict from fig. 2 the influence of the target atomic number and ionization on the evolution of the moving ion charge state considering the variation of the most important atomic rates. The general guidelines will be (low-density plasma case): </p><p>All of the rates are nearly proportional to the density. </p><p>Electron collision ionizations do not play a significant role. </p><p>Free electron captures can only appear for highly ionized plasma. </p><p>Ion-ion collision rates nearly scale as z 2 for not too high atomic number. </p><p>The stopping rate has a little variation with density due to the plasmon energy in the Bethe formula. </p><p>Ion-ion collisions are not sensitive to plasma ionization. </p><p>Charge transfer decreases with plasma ionization so does the total recombination rate. </p><p>As a first approximation, we can suppose that the curve slopes do not change very much when varying one parameter like atomic number or z*/z. </p><p>To get a simpler view we can say that T, z and z*/z influences can be predicted moving up or down in fig. 2 only the three curves of ion collision ionization rate, stopping rate and bound-bound charge transfer rate. Plasma atomic number has large influence on ionization and charge transfer curves, while plasma ionization modifies stopping rate and charge transfer. </p><p>Before going to temperature effect in more detail, it has to be stressed out that in swift heavy-ion-plasma interaction, ion-ion collisions are generally much more effective than ion-electron collisions, in contradistinction with collision rates inside a plasma where the electron-ion collisions are dominant, so that plasma collisional radiative models results are not useful for our case. </p><p>3"2. Temperature effects. - A direct temperature effect comes from the integration of the electron-ion collision cross-section over the plasma electron velocity distribution function in the reference frame of the moving projectile. Temperature then, like in the stopping formula, can have some direct influence only if the ion velocity is equal or smaller than the plasma mean free electron velocity. But even in this case, electron ionization is always smaller than ion ionization and free electron recombination is smaller than the stopping rate. One then can state that temperature </p></li><li><p>1830 G. MAYNARD, W. ANDRI~, M. CHABOT, C. DEUTSCH, ETC. </p><p>has no direct effect on the charge state evolution. An indirect temperature effect can be seen first through the plasma state: charge transfer is modified by the plasma excited state, and secondly through the plasma ionization z*/z. </p><p>When increasing z*/z, ion-ion ionization remains almost constant. Changes in the plasma bound electron screening can only be seen for high atomic number and very high plasma temperature. The main effect of an increase in z*/z is to reduce the charge transfer rate. One then gets in fig. 2 an enhanced Zeq but also a larger teq, the target linear density has to be large enough to reach Zeq. If z*/z increases again, the charge transfer rate curve can be under the stopping rate curve and the charge evolution now depends on the initial q value. For q lower than Zdy the moving ion charge will reach q = Zay and then energy will decrease, while for q larger than Zdy the energy will decrease quickly so that the projectile ionization state remains almost unchanged. Reduction of recombination rate wi...</p></li></ul>