Charge evolution of swift heavy ions in fusion plasmas

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  • IL NUOVO CIMENTO VOL. 106A, N. 12 Dicembre 1993

    Charge Evolution of Swift Heavy Ions in Fusion Plasmas (*).

    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

    Laboratoire de Physique des Gaz et des Plasmas, Bat. 212, Universitd Paris XI 91405 Orsay, France

    (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

    (ricevuto il 25 Maggio 1993; approvato l'l Giugno 1993)

    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.

    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.

    1. - In t roduct ion .

    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

    (*) Paper presented at the International Symposium on Heavy Ion Inertial Fusion, Frascati, May 25-28, 1993.



    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.

    2 - Stopping and ion-beam-plasma experiments.

    Most of heavy-ion beam plasma experiments can be summarized as follow:

    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

    (1) AEp =f(Eo, qo, z, n, T)


    (2) qexit - - g(Eo, qo, z, n, T)


    with hEp = ISdx. The stopping power S can be written as 0

    () = - - - - 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

    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.

    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


    10 3 ~D



    I * I I I I


    i t6 32 48 64 80 96

    atomic number

    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).

    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].

    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

    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.

    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.


    3. - Charge state and atomic rates.

    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.

    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.

    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.

    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

    {) ~ i i i i i i ,

    9 u ~7"~ ion ionization ~ ..5 . . . . . . -

    m 7 - J '~ lectroni xg : ,, ~ ~n izat ion , ' "~:-. :

    5 I0 t ~ v , , ', J i i , -

    20 24 28 32 36 40 44 48 52 charge s ta te

    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.


    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

    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):

    All of the rates are nearly proportional to the density.

    Electron collision ionizations do not play a significant role.

    Free electron captures can only appear for highly ionized plasma.

    Ion-ion collision rates nearly scale as z 2 for not too high atomic number.

    The stopping rate has a little variation with density due to the plasmon energy in the Bethe formula.

    Ion-ion collisions are not sensitive to plasma ionization.

    Charge transfer decreases with plasma ionization so does the total recombination rate.

    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.

    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.

    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.

    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


    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.

    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 with z* is due to the fact that bound-bound charge transfer is much more effective than free electron recombination so it strongly depends on the target atomic number. For high-energy heavy ion, bound-bound charge transfer comes mainly from the strongly plasma bound electrons. In pure hydrogen case, recombination is proportional to the bound electron density unless a high degree of ionization allows for dielectronic or radiative effect. For other material, a large reduction of recombination rate is important only when K or L electrons are ionized and quite high temperatures are requested. The temperature effect on EPIP is then maximum for fully ionized target of small atomic number and is quite negligible for heavy material.

    3"3. Dynamical effect. - As shown in subsect. 3"1, a dynamical effect occurs when Rstp is larger than any atomic rate, and as can be seen from fig. 2, it occurs only if the recombination rate is reduced and if q is larger than or equal to Zdy. For not too high plasma atomic numbers, the ionization rate scales like z 2 and the charge transfer rate increases with z even faster, whereas the stopping rate, which depends mainly only on the free electron density, is quite insensitive of z. The dynamical effect is then favoured by small atomic numbers and in heavy material the moving ion charge is nearly equal to Zeq. In hydrogen targets experimental evidences of non-equilibrium charge are obtained even in cold material but more generally strong dynamical effects require fully ionized targets [11]. From fig. 2 one can see that for a fully ionized helium target a 4 MeV/u iodine beam incoming with a charge smaller than Zay = 38 ~11 quickly reach this charge, then as it slows down the ionization rate becomes smaller than the stopping rate so that the ionization state does not change until radiative or dielectronic recombination become higher than the stopping rate, that is for nearly stopped ions. In highly ionized targets, in the quite general case of an incoming charge lower than the equilibrium charge, the moving ion will then have a nearly constant charge on a large part of the range given by Zdy calculated for E = E(0). In the first part of the slowing-down process its charge is lower than Zeq while in the second part it becomes larger than Zeq. This frozen charge phenomenon greatly modifies the energy deposition profile. Whereas in cold target the swift heavy-ion energy profile is nearly fiat, in highly ionized targets it tends to the proton energy deposition profile with a Bragg peak at the end of the range [11].

    In fully ionized target the stopping rate slightly decreases with density due to the plasmon energy in the Bethe formula so that the dynamical effect is a little bit less effective in solid than in gas plasma.


    3"4. Dielectronic and autoionization influence. - A few comments have to be added on dielectronic and autoionization processes (DA) because our results as presented in fig. 2 greatly differ from those of ref.[9] where dielectronic recombination rates are analyzed in detail. In fig. 2 one can see that DA gives a loss of electrons except for some resonance at q = 38 while in ref. [9] DA produces a larger contribution and is never negative. The reason for this discrepancy arises from the fact that we have included in DA not only autoionization coming after a dielectronic recombination process but also autoionization processes coming from vacancy production by plasma ion collisions. This autoionization rate is proportional to the square of z and it can reach high values in case of heavy material.

    3"5. Density effects. - Density effect is the most complicated case and great uncertainties remain mainly because now one has to introduce partial cross-section. We will comment only briefly here density effect on the projectile excitation state, density screening is discussed in ref. [9].

    All atomic rates describes in sect. 2 are proportional to the target density except for radiative decay and autoionization. As the density increases, the moving ion has not enough time between two successive collisions to decay to the ground state, and so, after several collisions it gets some excitation energy which is responsible for Auger emission at the exit surface. There are mainly three density effects : excitation collision effect, partial recombination effect and vacancy production effect.

    Excitation effects come from the fact that excited states are more easily ionized than ground state. Due to the z 2 scaling of the excitation cross-section this effect appears sooner for heavy material. It can be shown that excitation collisions lead to the so-called Betz and Grodzins model [12] which states that the projectile has many excited electrons each of them with small excitation energy. Ionization has a small enhancement but a lot of Auger emission can occur at the exit surface.

    Electron which recombines in high excited level can lead to ionization instead of radiative decay, the recombination rate is then reduced. For bound-bound charge transfer, partial recombination in excited shell increases when z decreases. The lowering of recombination rate due to partial charge transfer is small in heavy materials but yields the dominant density effect for small atomic number. We found that in hydrogen target at density n = 102~ -8 this recombination reduction increases the equilibrium charge of a 1 MeV/u iodine beam from 24 to 29. In this case of small atomic number target, we are close to the Bohr and Lindhard model [13] which predicts a large density enhancement of the charge inside the target and few Auger electrons at the exit surface.

    In light material another density effect, which is often neglected, is due to vacancies in the projectile inner shell. For small plasma atomic number, not only the electrons can recombine in highly excited states but also they cannot recombine in vacancies created by ion collisions. The charge will then increase with density because, as the excitation rates become larger than the radiative decay rates, the number of vacancies increases with density. We found that this effect is maximum for carbon target [14].

    Due to the already mentioned discrepancy between qexit and qoutside, there is no direct experimental measurement of the density effect on the heavy-ion charge state inside target. However, using stopping power measurement, experiments showed heavy-ion effective charges in solid 15% larger than in gas [5, 15].


    4. - Examples of charge evolution in plasma.

    4"1. Hydrogen target. - In fig. 3, 4 results for incoming 4 MeV/u 127 + on hydrogen gas discharge are given for density n = nl = 4.10 '7 cm -s and n = n2 = 102~ cm -3 at T = 0 and T = 10 eV. Comparing T = 0 and T = 10 eV at n = n2 we recover the large predicted EPS and EPIP. The stopping range is reduced by a factor of two and the energy profile shows a Bragg peak at T = 10 eV. At T = 10 eV, there is first a transient zone between 4 and 3 MeV/u, where the projectile charge changes from its initial value to the so-called plateau value, and then the charge remains unchanged during the slowing down. This transient zone increases with Z, E(0) and 1/z . As was stated the charge plateau value q = 38 is close to the Zdy value calculated at E = = 4 MeV/u. Due to the dynamical effect, the maximum ionization state of the beam does not vary very much with temperature, at T = 0 we have qm~ = 36 and at T = 10 eV qm~x = 38, fully ionized helium target shows up as a better stripper than hydrogen.

    In fig. 3 we have also reported the iodine charge state for cold low-density gas n = = 4-10 '7 cm -3 and one can observe, as stated in sect. 3, that the density effect is already appreciable for n = 10 ~~ cm -8 .

    In Ohmic plasma heating discharge impurities can be desorbed either from the wall or from electrodes. In a heavy-ion fusion device the heavy ion beam energy absorber can also be a mixed compound with a few percent of heavy material mixed for example with lithium. We have reported in fig. 3, 4 results for 2.5% of A1 in hydrogen gas at the same free electron density n = n2. If one supposes an additivity rule for the stopping power, the plasma compound stopping power has to be slightly larger than in pure hydrogen case due to the bound aluminium electrons. That is what we find at the very beginning of the stopping but during the slowing down, the stopping with A1 impurities is greatly reduced due to the much lower charge in gas

    40 400







    / / *

    / * J *

    j s

    / /

    .~ 300


    -v 100

    I i I I ~ I I I . 0 2 3 4 0 10

    energy (MeV/u) Fig. 3. Fig. 4.

    , I J I J I J I

    2 4' 6 8 distance (cm)

    Fig. 3. - Iodine ion charge state evolution vs. energy during the slowing-down process in fully ionized hydrogen target ( - - - T= 10eV, n=n2; - - T=O, n=n2; . . . . . T=O, n=n l ) and in a compound of hydrogen and 2.5% of aluminium ions ( * T = 10 eV, n = n2). Initial energy is 4MeV/u and initial charge is 27. Total electron densities are nl=4-1017cm -s and n2=10~~ -s.

    Fig. 4. - Stopping power vs. range for the same cases as in fig. 3 with one electron density, n = n 2 = 1020 am -s.


    compound as shown in fig. 3. Impurities have also a large influence on the heavy-ion beam radiation emission. We found that 2.5% of aluminium increases the total energy radiated by the projectile during the slowing down by a factor of 5.

    0.1% of A1 in H have still some influence on the stopping of our beam. Highly accurate diagnostics of impurities is then crucial for experiments in hydrogen plasma [16].

    4"2. Dense a lumin ium target. - Here we show an example close to heavy-ion fusion direct experiments with a 30MeV/uBi beam incoming on an aluminium target. In fig. 5, 6 the projectile charge state and stopping power during the slowing down are reported for solid target at normal density ~ = ; 0 at T = 0 and T = 100 eV and also for a reduced density p =~:0/50 at T= 100eV. One can see in fig. 6 a moderate EPS due to the plasma ionization. The larger stopping at lower density is due to a larger ionization z* = 9.85 at 6 -- ~0/50 and z* = 5.4 at p = Po at T = 100 eV, and also to a lower plasma frequency in the Bethe formula. The projectile charge state evolution is reported for the three cases on fig. 5. At the solid density there is no visible change in the charge state with temperature and no evidence for non- equilibrium charge state. Large temperature effects on the charge state require ionization of the K-shell plasma bound electron. In aluminium, at solid density, K- shell electrons begin to be ionized at ~ 300 eV. The frontier between what we have called small and large atomic numbers is then close to 10. We therefore conclude that there is no temperature effect and no dynamical effect in heavy material. There remains only some density dependence, in our example and starting

    55 r

    70 , , i i I i i i , I i i i i I [ i i i I i i i

    65 ]~,

    60 , "


    50 ," / /

    ~ /

    45 /

    . /

    4( i i ' i I ' i i i L ' i i i I I i i i I ' J I 5 10 15 20 25 30

    120t,,,, ' ' ' ' ' ' ' ' ' ' ' ' ' * ' ' ' ' ' ' ' '

    110 f "'\ ,


    30 , L t I i J t t I i i i i J i i i L I i i i 5 10 15 20 25 30

    energy (MeV/u) energy (MeV/u)

    Fig. 5. Fig. 6.

    Fig. 5. - Ionization state of Bi ions on their path through dense aluminium plasma for two temperatures and two densities: - - ~ = Po, T = 0; . . . . . ~: = ;o/50, T = 100eV; * p = ~o, T = 100 eV. Po is the normal solid density. Initial energy is 30 MeV/u. All initial charges up to 60 give the same curve.

    Fig. 6. - Stopping power of Bi ions vs. energy for the same cases as in fig. 5.


    at normal density, projectile charge and stopping power have a reduction of 10 and 20 percent when the density is divided by 50.

    5. - Theoret i ca l mode l .

    5"1. Average atom model. - Our main goal is to describe heavy-ion-target interaction for densities from 1017 cm -3 up to the solid density 10 ~ cm -8. In solid target we have already seen that projectiles can have many excited electrons so that the number of atomic states to be taken into account is extremely large. As in EOS calculations of dense plasma the most useful approximation is to use the average atom model (AAM) : Instead of dealing with all the ionic species, an average configuration has to be determined and the only variables are the non-integer electron populations p on each atomic shell. When going to non-equilibrium situations the atomic model has to be simplified and the most commonly used approximation is the average hydrogenic atom model (AHAM) with screening constants. Non-equilibrium charge states of heavy ions in dense plasma have already been determined using AHAM [6, 8,14] with the screening constants of More[17]. /-splitting was not included in More constants. It has been introduced by Perrot [18] and then used in opacity calculations [19]. l-splitting has some influence only for not too high excited states, we have then introduced /-splitting in our model for s, p and d states. Our model therefore take into account metastable states which can play some role in low- density target. Numbering the atomic level from ls with increasing energy order (ls, 2s, 2p, 38, 3p...) we define Pi, hi and g~ as the electron population, hole population and degeneracy of the level i. The heavy-ion beam charge evolution in a plasma is determined by the relations:

    (4a) q = Z- ~p~, . . . . i

    dt ih Gi

    + dielectronic + autoionization rate,

    (4c) Pi + hi = gi 9

    In eqs. (4) I and R are the rate coefficients for ionization and recombination while Gi stands for the rate coefficient of one electron transfer from level i to level j, the dielectronic rate is of the form Pi hi hk D~ k for the excitation of one electron from level i to level j and the recombination of a free electron on level k, in the same manner for the reverse reaction, the autoionization rate is written as hipjp~Ajk. All the rate coefficients in eqs. (4) are calculated according to the AHAM energy and effective charge atomic level.

    In AHAM, only elementary processes have to be determined through eqs. (4). There is no approximation to perform about branching ratios. The situation is then much easier here than in full configurations model like in ref. [9]. Unfortunately, as it stands, eqs. (4) can only be used for high-density targets. All of the electron and hole population products entering into the rate coefficients are derived by assuming that hole and electron populations of different levels are independent quantities. Let us suppose that a dielectronic recombination occurs which transfers an electron from


    level i to level j and a second one in level k, AHAM supposes that there are many other processes before an autoionization from levels j, k to level i could happen. In fact, at low density, AHAM gives a nearly zero autoionization rate and greatly reduces de-excitation rate in inner shell vacancies. The only way to consider low density targets is then to take into account electron-hole and electron-electron-hole correlation and we have found that the most economical way to do this is to introduce explicitly the electron-hole and electron-electron-hole population created by excitation and dielectronic recombination. We have called the new model the average correlated hydrogenic atom model (ACHAM) which will be now detailed.

    5"2. Average correlated hydrogenic atom model. - In ACHAM we introduce the new populations pO, hio, hpij and hpPijk; pO and h ~ are independent electron and hole populations first created by direct ionization or recombination, hpij are correlated electron-hole population with one electron on level j and one hole on level i first created by excitation collisions and hpp~jk are correlated electron-electron-hole population with one hole on level i and one electron on each of the levels j and k coming from dielectronic recombination. As was shown in subsect. 5"1, electron-hole correlation is important only for de-excitation in inner shells so that for hpij we take i < j and i ~< 12 (i = 12 is the 5d level) and for hpPijk we take i < j ~< k and i ~< 12. In terms of our new populations, p and h are given by

    (5b) j i k>~j i


    5"3. Atomic-rate expressions. - In AHAM and ACHAM one needs analytical expressions for all of the atomic-collision cross-sections in terms of energy and effective charge of each projectile atomic shell. Many formulae have been published (see ref.[9]) and can be used, but, and especially when one needs partial cross-sections, there remain large uncertainties. In the particular case of heavy- ion-heavy-ion collisions at MeV/u energies, most of the analytical expressions are quite out of range. One therefore needs new experimental or theoretical atomic data to make accurate descriptions of heavy-ion interaction in heavy material like gold or lead. Hydrogen is here again a special case, because nearly all of the necessary cross- sections are known with good accuracy. Theory can thus be quantitatively checked against experimental results.

    6. - Conc lus ion .

    Looking at temperature and density effects on heavy-ion beam charge states in dense plasma we have shown that the target atomic number has a large influence. A pure hydrogen target is actually a particular case which allows for strong temperature, density and dynamical effects. In that case, however, impurities and the non-ionized sheath region at the exit surface can greatly modify the experimental results and highly accurate diagnostics of the target, included sheath regions, are thus needed. In large atomic-number targets, the moving ion charge is nearly independent of plasma temperature and stays at its equilibrium value. Density in this case is the most important parameter. A new theoretical model has been presented, still much simpler than the multiconfiguration model. It allows to check out theory against experiments in a large density domain.


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