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AT EXTREMES

Matter and Radiation at Extremes 1 (2016) 277-307 ^^^^^^^^^^^^^

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Review Article

Ion stopping in dense plasmas: A basic physics approach

Claude Deutsch*, Gilles Maynard

Laboratoire de Physique des Gaz et Plasmas, CNRS, Univ-Paris Sud, Université Paris-Saclay, 91405 Orsay, France

Received 1 June 2016; revised 21 September 2016; accepted 13 October 2016 Available online 2 December 2016

Abstract

We survey quite extensively the present research status of ion-stopping in dense plasmas of potential importance for initial confinement fusion (ICF) driven by intense and heavy ion beams, and also for warm dense matter (WDM). First, we put emphasis on every possible mechanism involved in the shaping of the ion projectile effective charge, while losing energy in a target plasma with classical ions and partially degenerate electrons.

Then, we switch to ion stopping by target bound electrons, taking detailed account of mean excitation energies. Free electron stopping has already been given a lot of attention in former works [C. Deutsch et al., Recent Res. Devel. Plasma 1 (2000) 1—23; Open Plasma Phys. J. 3 (2010) 88—115]. Then, we extend the usual standard stopping model (SSM) framework to nonlinear stopping including a treatment of the z3 Barkas effect and a confronting comparison of Bloch and Bohr Coulomb logarithms.

Finally, we document low velocity ion slowing down (LVISD) in single ion plasmas as well as in binary ionic mixtures (BIM), in connection with specific ICF fuels.

Copyright © 2016 Science and Technology Information Center, China Academy of Engineering Physics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

PACS Codes: 52.50.Gj; 34.50.Bw; 34.50.Fa; 34.80.Lx; 52.27.Gr; 52.27.cm

Keywords: Ion projectile effective charge; Mean excited energies; Nonlinear stopping; Binary ionic mixture

CrossMark

General considerations

Non-relativistic stopping of point like charges in a dense electron fluid has provided us with a fundamentally robust as well as highly versatile paradigm elaborated through a die-lectic formulation. So, it thus appears possible to envision now a higher level of complexity in our model approach. The conceptual framework previously developed could also allow for a quantitative treatment of additional basic features of ionplasma interaction. Those include in-flight balance of the projectile effective charge Zeff(Vp), stopping by electrons bound in the target, or the finite extension (non-punctuality) of the projectile ion electron distribution. The corresponding and suitably extended stopping theory could then be expected to be of quantitative accuracy in modeling the penetration of multi-

charged and non-relativistic ions in a small spherical pellet containing a thermonuclear deuterium + tritium (DT) fuel.

Through the remaining electrons bound to the target ions are now directly taking part in the stopping process of the incoming projectile.

These contentions may be given an immediate content by focusing our attention on a high temperature target. Then, the ion projectile-target electron coupling may be considered in the well-known Coulomb logarithm approximation. An obvious extension of the previous standard stopping model (SSM), may thus be written as

dE 4pN0e4

ATmeVp2

pZf Vp) (Zr - Z)ln Lb + ZG(V/Ve)ln LF

* Corresponding author.

E-mail address: claude.deutsch@u-psud.fr (C. Deutsch). Peer review under responsibility of Science and Technology Information Center, China Academy of Engineering Physics.

with G(X) / 1 for Vp [ Vthe, G(X) / 0 as (Vp/Vthe)3,

http://dx.doi.org/10.1016/j.mre.2016.11.004

2468-080X/Copyright © 2016 Science and Technology Information Center, China Academy of Engineering Physics. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Vhe = electron thermal velocity p = target density (g/cm3) N0 = Avogadro number

I = mean excitation energy of target bound electrons

with LB

2meVp2

: e p and Lf =

For instance, at ne z 1023 cm 3 and 100 eV one has

2 x 1016/s, Zupz 10 eV, 1D z 2 x 10"

cm, ne 1D z 1.

In Eq. (1), ZT denotes the target ions atomic number, Z their ionicity and AT, the corresponding atomic mass.

The first term on the right hand side (r.h.s) of Eq. (1) extends straightforwardly to electrons bound to partially ionized ions in plasma target, the usual Bethe expression, valid for an isolated ion. This pinpoints an obvious concern about the adaptability of the mean excitation potential I to plasma surroundings. Even more significant is the novel and conspicuous Zeff (V) behavior in a hot plasma target especially at low ZT. The enhanced projectile ionization in plasma (EPIP) cannot be extrapolated from the usual trends displayed by cold target homologues. It deserves specific developments considered at first, in the sequel.

The stopping expression (1) has been essentially introduced for illustrating a few specific behaviours of projectile ion stopping in a hot plasma medium. Nonetheless, it retains full validity in a high temperature and weakly coupled plasma target as long as the target ions concept remains operationally meaningfull, i.e. as long as its average extension is smaller than the plasma electron screening length 1D, i.e.

ac < 743r1=2 [eV]

ne '[cm^3]

When this is not the case, the difference between bound and free target electrons trends become rather blurred. Then, the atomic orbitals are likely to get delocalized around several ions. So, in the weakly coupled regime to which we restrict most of our present attention, the residual and direct ion—ion contribution in the r.h.s of Eq. (1) plays only a minor role (smaller than 5% in most situations of practical interest).

1. Projectile effective charge Zeff (Vp) in a plasma target

We recall the standard Betz expression for the projectile ion effective charge flowing in a neutral and cold gas target, which reads as [4].

Zeff (Vp) = Z{1 - 1 .034 x [ - (Vp/2 . 19 x 108 [cm/s])Z-0'688]}

in terms of the projectile atomic number Z and its instantaneous velocity Vp.

1.1. Hindered recombination

The projectile charge state, in a cold target, gets fixed through a balance between collisional electron losses and bound electron capture of the target atoms. Direct trapping of free electrons is much more problematic to achieve because the excess binding energy has to be evacuated through one of the three processes: (a) radiative recombination, (b) three-body recombination, or (c) dielectronic recombination. Therefore, one should expect different charge states of ions when they pass through plasma or cold targets, except at very high kinetic energy, where bound electron capture is also reduced due to momentum mismatch [5].

In contrast to atomic processes in ordinary plasmas, we emphasize here two specific features: (a) anisotropy of the electron velocity distribution in the ions rest frame, and (b) projectile collisions with target plasma ions are significant.

The dynamical distribution of projectile charge states through target is thus obtained with the coupled equation:

Ç = -P (4 + «0 + + Pj-K

where Pj = fraction of projectiles in charge state j, oR, «j = total rate coefficients of recombination and ionization for state charge j.

Given t-dependent outputs rely on charge state, Vp and target parameters. So, the corresponding t-dependent average charge state may be introduced in a standard stopping calculation.

Stopping due to free electrons is derived classically in high temperature plasmas. At each step, energy and velocity data are updated to advance the projectile, up to the following one.

In doing so, temperature effects have been investigated at length (Nardi-Zinamon [6]). Below a 200 eV electron target temperature, ionization of a single electron is the dominant process, mostly for light elements.

Nonetheless, multiple ionization can show up strongly and become the dominant effect especially for heavier projectiles and higher temperatures.

Moreover, kinematic constraints restrict 3-body recombination relative to the radiative one, in particular for highly charged and fast ions. In some cases, a significant dielectronic recombination can outnumber radiative recombination by far.

Finally, the projectile charge state dynamics gets mostly monitored through a balance at equilibrium between colli-sional ionization and radiative recombination.

At first sight, one seems to recover a statement for coronal equilibrium. However, the restricted electron distribution, in projectile frame, demands that excitation and recombination are performed by same electrons, although with standard Maxwell electrons, excitation arises from the high energy tail of the distribution, while recombination is due to the low velocity part of the bulk.

Typical atomic rates are given on Fig. 1 for ion Clq+ at 1.5 MeV/u.

(a) Electron-ion

Fig. 1. Ionization and one electron capture rates versus the projectile charge state for Clq+ at 1.5 MeV/u in gas and plasma target. Reproduced from M. Chabot, G. Maynard et al., Phys. Rev. E 51, 3504 (1995) with the permission of AIP Publishing.

Outputs of Eq. (5) for ion projectile energy 46 MeV/amu (cf 9 GeV Au ions) are given in Fig. 2(a) (electron-ion excitation) and Fig. 2(b) (ion—ion excitation) for a ZT = 10 ion in target. Such fast ions experience a nearly fully stripped charge state in every situation. Despite that the equilibrium charge is pretty close to the electron-ion case, thus fulfilling Bohr criterion. The time span for the stripping of the first electron is very short, and thus an even decreased initial stripping time does not matter very much. In order to provide a better understanding of the necessary equilibration time, we list in Table 1 the equilibration distance for a constant velocity ion in units of the cold range for the ions in Fig. 2(b).

The delay distances increases swiftly with projectile Z. For Au ions, it already amounts to a remarkable one-sixth of the total range.

Fig. 2 demonstrates that impinging projectiles are practically fully stripped at the target entrance. The calculations reported here refer to an electron density ne ~ 6 x 1023 cm~3 i.e. fully ionized aluminum close to solid density.

The stopping rate is roughly proportional to the electron density. Then, once the effective charge state is obtained, it will persist up to near the end of range. Indeed, estimations due to Nardi and Zinamon [6] (Fig. 3) yield a practically constant Zeff, far above cold target counterparts. However, both results converge at highest velocities.

Such an impressive plasma increase of Zeff arises from the hindered free electrons-ion projectile recombination, as compared to the much more favorable one-electron capture from bound electrons which accounts for a three orders of magnitude reduction.

As a result any surviving bound state in target is likely to overwhelm the recombination processes. This picture (Bell [7]) highlights the fact that the target ions with a smaller ionization degree and located close to the projectile trajectory should dominate recombination. Thus, the superposition of

ne=6x1023/cm3 " Eion=46 MeV/u Gold^-—^

- / Silver

- / / Copper / /^Calcium

——Silicon

10"3 10"' Time (ps)

(b) Ion-ion

ne=6x1023/cm3 Gold___—

~ Eion=46 MeV/u /

- Silver

^^tHKi i i......1 Copper

Calcium

Silicon

......1 ........I i .......1 ........

10"3 10" Time (ps)

Fig. 2. Time-dependent charge state for several ions with (a) electron collisions and (b) ion collisions for a target corresponding to fully stripped aluminum at solid density. Reproduced from D.S. Bailey, Y.T. Lee and R. More, J. Phys. (Paris) C8, 149 (1980) with the permission of EDP Sciences.

Table 1

Equilibration distance (in units of cold range). Calculated with the electron contribution to stripping (Fig. 2(a)) only. When one includes the ion contributions, it reduces substantially.

Ion type Equilibration distance

Gold ~0.16

Silver ~0.1

Copper ~0.025

Calcium ~0.01

Silicon ~0.005

Fig. 3. The charge state Zeff of a Xenon ion as a function of its energy as slowed in a fully ionized Carbon target. Reproduced from E. Nardi and Z. Zinamon, Phys. Rev. Lett. 49, 1251 (1982) with the permission of AIP Publishing.

accurate stopping estimates with a specific Zeff (Vp) in plasmas produces an enhanced range shortening.

1.2. Basic mechanisms

Total rates aR and appearing in Eq. (5) refer to sums over individual loss and capture rate processes ^ « with [8]

¿(Vp) = J dVfi (V, Vp)s(|Vp|)V

taken in a projectile moving frame, in terms of a shifted Maxwell distribution f for the plasma particles (mass m, number density n and temperature T) with thermal velocity V,

fi V, Vp

/ m \2pkBT

e 2pkgT

(V-Vp)2

and adequate cross-sections s,(| Vp|) for the projectile-plasma particles interaction. n and T designate plasma density and temperature, respectively.

The actual efficiency of a computer code calculation relies heavily on the approximate expressions used for Si(|Vp|). We detail below a few amongst the most significant. To keep presentation at a decent complexity level, one restricts first to a simple Slater model for the screened and hydrogenlike basic atomic quantities, such as excitation energies and oscillator strengths in plasma target. Nonetheless, the essential s,(|Vp|) behaviors featuring ionization and recombination are still given a proper perspective.

1.2.1. Ionization by plasma particles

First, we consider the one-electron excitation of projectile ions colliding with target free electrons and ions (denoted by A) respectively:

Xq+ + e/Xq+* + e',

Xq+ + Ap+(E)/Xq+* + Ap+ (E').

The above two processes are only significant for a deex-citation time which is long compared to intercollision time. Similarly, ionization through collisions with target free electrons and ions is given as:

Xq++ e/X(q+1)++ 2e, Xq++ Ayp+/X(q+1)++ Ajp_1)+, Xq+ + Ap+ /X(q+1)+ + Ap+ + e .

They feature the most important contribution to projectile ionization. aie refers to the electron reaction rate averaged over electron distribution and «j denotes a similar quantity for the sum of the following ion processes (see Eq. (6)). Autoionization matters when the projectile ion is doubly excited, with one electron ionized and the other one deexciting into a lower level.

One of the simplest and more robust modeling of projectile ionization by electrons and ions in the target is the well-known binary encounter model (BEM) introduced in a classical framework by Gryzinski [9]. According to this model, the cross-section for removing an electron in the n shell projectile ion cloud, by target ion impact is given as:

: 'YjNnSn = Nn

Here, Nn is the number of electrons in the nth shell, Un is their binding energy, and Vn = \J2Un/m their orbital velocity.

G is derived through the matching Vp = Vn for optimal ionization. It reads as:

G(V )=^ x

a + 3 (1 + b)ln(2 . 7 + V)

(1 - b)

1 - b(1+v2)

for V> 0.206,

G(V) = 4V4/15, for V< 0.206,

where a = V2/(1 + V2), b = 1/[4V(1 + V)].

According to its velocity Vp, the ion projectile is penetrating more or less deeply into a target atom or ion. Following Bell [7], one can approximate Z as the charge within a sphere of radius bn = \Jsn/p.

aBEM = VpnsBEM denotes Coulomb ionization rate. The upper dashed line in Fig. 4 shows the rate coefficient for the case of an iodine beam with Vp = 1.5ac and a = 1371036, fine structure constant in a hydrogen plasma with temperature T = 10 eV and ion density ni = 1017 cm~3. The relative fraction of bound electrons in such a plasma is no larger than 2 x 10~6.

The ionization edge (lower dashed curve in Fig. 4) documents a significant difference between ionization processes arising from target ions or target electrons.

Electrons such as Vr < Vn do not have enough kinetic energy to ionize an nth shell bound electron. So, according to Lotz [10], ionization through free electron collisions features the cross section:

Fig. 4. Rates of electron capture and loss for a 1.5 MeV/u iodine beam in a 10 eV hydrogen plasma with ne = 1017 cm"3. The intersection between capture and loss is close to the equilibrium charge Zeq(Vp) at constant velocity (arrow). Atomic shells corresponding to charge states are given at the top. Reproduced from Th. Peter and J. Meyer-Ter-Vehn, Phys. Rev. A 43, 2015 (1992) with the permission of AIP Publishing.

Nnsn = 4 x 10-14 [(cm-eV)2] x^

■Kft)

q(Er - Un) (10)

with the step function 6(x) = (|x| + x)/2.

The relative energy between plasma electron and projectile ion is approximated by Er = (m/2)Vr2 x (m/2)(Vp + Vhe ), where Vthe is the thermal velocity of plasma electrons. The rate is ae x Vrnese. The edge at I24+ to I25+ in Fig. 4 pertains to the transition from the N shell to the M shell; the electrons do not have enough energy for M-shell ionization.

1.2.2. Radiative electron capture and three-body recombination

Radiative recombination:

Xq+ + e/X(q-1)+ + hv

has to be taken into account for highly ionized targets, and aREC is the corresponding reaction rate averaged over free-electron distribution.

Radiative electron capture (REC) is explained by the electronic transition probability A(n' / n) between major shells n' / n for the projectile (charge state Z) with energy E'n — En = Zu. In the quasiclassical approximation (Zel'do-vich and Raizer [11]),

A(n'/n) f

(1--LY

In2 n02)

The free-bound transition in the REC process can be derived from Eq. (11) through n' = ik, where k is the inverse velocity v of the photoelectron in ac units.

A(n'/n)dkf 3 /1 ^3

n2 + k2

A dk is proportional to the rate of REC into the nth projectile shell (snv)d3v = (snv)4pv2dv. One thus gets [12]:

Vo hvo g_

V f Vr2 n3

where:

A = 2.1 x 10"

hv = 2meVr2 =

^ hvo = Z2 [Ry],

g is the free-bound Gaunt factor, gz1.

The total rate SreC = VrnesREC is then determined by a sum from the ground state ng (Nng electrons) up to large quantum numbers. In the hydrogenlike approximation

«REC = 1 -

(1 - N1)

n=ng + 1

The quantity aREC shows up in Fig. 4 as a solid line.

At high density the probability for a simultaneous collision of the projectile ion with two electrons increases. One of these electrons is then captured by the projectile, while the other carries away the excess energy. The 3-body process thus reads as

a3BR = "

o5_2 10 73

2 p e Z ne

= 2.92 x 10-31Z3—

contrasted to radiative electron capture (Vr in ac, ne in cm 3):

aREC z a3BR

a.6 x 10

1.6 x 10

17 ZVr

Regarding to laboratory plasmas, radiative electron capture remains many orders of magnitude larger than three-body recombination. For 1.5 MeV/u ion projectiles (Vr = 7.75 ac) the two rates are equivalent when the density is close to solidstate one (Fig. 5).

1.2.3. Capture of bound target electrons

In partially ionized targets, a bound electron can jump from a target ion to the projectile (charge transfer (CT)). Oppenheimer and Brinckmann and Kramers [13] evaluated the cross section (OBK theory) for this process in first Born approximation for the Coulomb potential — e2/r and hydrogenlike wave functions (Laguerre polynomials). Using the cross section averaged over all initial (l, m) states and summing over all final (l, m) states,

en c (U o

bd 4d 4p U 3d 3p y 2p bsllsl J

[ ' 1.5 MeV/u I—>H gas ~

^^ZiVl) Ionization by '**•», Coulomb collisions —

Capture of bound electrons :-1-1- /"V ,l ,l ,l ,l ,l

Charge state

Fig. 5. Rates for 1.5 MeV/u iodine in cold hydrogen gas with ne = 1017 cm"3. Here the equilibrium charge is about 21 (arrow), thus considerably less than in the plasma case (Fig. 4). Reproduced from Th. Peter and J. Meyer-Ter-Vehn, Phys. Rev. A 43, 2015 (1992) with the permission of AIP Publishing.

SCT = 4.1 X 104^ ^NifleitX

(Ze2 )2 £f/2£f3/2£k

El + 2Ei(E + Et) + (Ei -Eff

where Ei and Ef are the binding energies of the electron >0 in initial and final states, respectively, Z is the charge state of the projectile Ek = mVp/2, and N the number of electrons in the shell of the target. aeik features a reduction factor (in the usual OBK theory aeik = 1). Generally 0.1 ( aeik ( 0.4.

Fig. 4 adds to all the other processes also the chargetransfer-rate coefficient in the 10-eV plasma that stems from the small part (2 x 10"6) of bound electrons. For comparison Fig. 5 also exhibits rate coefficients in a cold hydrogen gas of the same density 1017 cm"3. The total loss rate in both cases does not differ much from each other, but capture drops in the plasma case by about three orders of magnitude.

At large projectile velocities, one gets

«CT z «REC

< /1 Ef/2

Z2Ej5 Efrr e5=2Z :109-=rVs~ y 10 Z

with Ei in Ry and Vp in ac. In cold plasmas with many bound electrons OBK rate ranks many orders of magnitude above radiative recombination.

relationship for the Pn evolution in terms of loss and gain reaction rates as [14].

dPn dt

l(n, n)+Y^(2k2 - Pk)l(n, k)

+ (2n2 - Pn)

g(n, n) + Yl pkg(k; n)

which obviously extends Eq. (5) to arbitrary excited state n with every former basic mechanism included, dielectric recombination excluded.

When k S n, l(n, k) = g(n, k) represents the rate coefficient for one electron transfer from level n to level k, provided there is one electron in level n and one free place available in level k; g(n,n) and l(n,n) denote rate coefficients for recombination and ionization, respectively. l and g are computed owing to electron and ion ionization and excitation, spontaneous decay, bound—bound and radiative recombination. Eq. (19) has to be changed somewhat to include dielectronic recombination and autoionization process.

1.4. Dielectric recombination

In view of its rather involved behaviour, we postponed up to here a thorough presentation of the dielectric recombination (DR) effect on the projectile charge state evolution.

1.4.1. Basic mechanism

Dielectronic recombination:

Xq++ e/(X(q

>X(q-1)+ + hv

is a two-step process in which a free electron with kinetic energy Ek is captured by an ion X of charge Z and the excess energy is transferred to another electron already bound in shell i=(nili), which will be excited to shell j={jijlj). This

writes as

X(Z, i)+e(Ek, l±1) -4 X**(Z - 1; jnl).

The free electron will be captured into a highly excited level n [ 1, from where it will get autoionized by process Eq. (20) running from right to left (Auger effect with rate Aa). X** is then deexcited via a stabilizing radiative decay (with rate Ar) i.e.

1.3. Master equation

It looks now appropriate to systematize the overall effect of every basic mechanism on the inflight ion projectile charge state. During its penetration into a target, this charge state continuously fluctuates through gain and loss of electrons. It appears then convenient to focus attention towards the population Pn of a given excited state n of the ion projectile. Recalling that on a spherical electron shell n, there are 2n available locations, one can establish at once a masterlike

X**(Z - 1, jnl) / X*(Z - 1, inl) + Zu

The photon carries the energy Zu = Ej — Ei out. With its longer lifetime the higher excited electron remains after the i / j transition in the (n, l) level, and it finally cascades down to ground state. Here only the decay to state i is considered [8].

The basic DR features which includes resonance condition and fluorescence yield are pictured on Table 2.

Energy conservation demands Ek — En[ = Ej — Eh where Ek is the kinetic energy of the free electron, Ei the binding energy

of the initially bound electron (possibly in an inner shell), and Ej and Eni the energies of the excited states. We restrict to an ion at rest in target plasma. The number of DR processes per unit volume and time is n(X**)Ar/n(X)ne.

Detailed balance to the total loss of doubly excited ions n(X**)(Aa + Ar) is given by the loss nSaha(X**)Aa in a Saha equilibrium

-2(2l,), gj = 2(2lj + 1).

Ni and Nj are occupation numbers of shells in i and j of the ion X(Z, i). Hence the DR rate specified by the quantum numbers (i, j, ni) appears as:

aDR(i; j, ni)

n(X**)

_nSaha (X**);

Ar + Aa

where nSaha is given by

nsaha(X**) _ g(Z - 1,jnl±1) n(X)ne

_ Ar (j /j)Aa(jnl / iEk l± 1 )

Ar(j/j) +Aa(jnl/iEkl±1) g(Z - 1,jnl±1)

2g(Z, i)

2g(Z, i) (2pmkB T )3/2

-Ek/kBT. (23) (2pmkB T )3/2

-(Ej-Ei+EM)/kB T

Here, g(Z, i) and g(Z — 1, jni±1) are statistical weights of the states X(Z, i) and X**(Z — 1,jni±1), so that

g(Z; »>( ^ JIN ) > g(Z>fr

g(Z - ^K Ng- 0 U+02(21+1)

gi Ni + 1

Table 2

The dielectronic recombination mechanism.

STEP®: Radiationless Capture "—" (autoionization "•—") X(Z,i)+e(Ek, 1) -+X**(Z-\\j, ni + 1) STEP©: Stabilizing Radiative Transition

X**(Z-l;j, n-e± 1) — X*(Z-1; i, n-e ± 1)+hco

Ek + En ~ Ei - Ej

(Z\2 (Z + 1\2 (Z + In

Energy matching condition

"Branching Ratio"

(b) aDRKAa^

with ArAa/(Ar + Aa), branching ratio or fluorescence yield. The total rate of DR [15].

aDR = aDR(i, j, nl)

consists of a large sum over electronic states where (n, l ) denotes the states into which the plasma electron is captured with a simultaneous excitation of a projectile electron from state i to state j. Ek = mv2/2 is the initial kinetic energy of the recombining electron in the rest frame of the projectile. In the beam-plasma case, the projectile sees a Maxwellian electron distribution (6) shifted by the projectile velocity Vp. The corresponding rate

aDR(Vp) = J d3vf (v, Vp)vffDR(v),

is easily obtained from the normal plasma rate aDR (Vp = 0) if we take into account that the cross section aDR(Vp = 0) from

the work of Jacobs etal. [16]. Then one obtains 3/2 X g(j; ni)

aDR Vp =

4paBEH

2g(i) Aa + Ar + A;

:F(s, t),

with aB = Z2/me2 and EH = e2/2aB; g(j, nl) and g(i) are statistical weights. The l dependence of the Gaunt factor gl in Aa is normalized to ^ gl = 0.2. As a novel feature, autoionization of valence electrons (rate Av) in addition to auto-ionization of the captured electron (rate Aa) is found to be important in certain open-shell ions and has been included in Eq. (28).

The function

F(s, t) = {exp [ - (s - t)2] - exp [ - (s + t)2]}/4st in Eq. (28) results from the integral (27) with

2 _Ej Ei ^ Enl

and t2

2kBT .

In the limit Vp = 0, the pure plasma result with the Boltzmann factor F = exp[—(Ej — Ei + Enl)kBT] is recovered

from Eq. (28) and the rate is largest for high temperatures with Ej — Ei y |kB T. However, for fast ions in plasma of low temperature (Vp [ Vth), the resonance condition is best fulfilled for

-2Z yEj — Ei + Enl[kBT (syt[1),

the beam then probes individual resonance states as a function of beam velocity with a width kBT.

1.4.2. Balance equation

In order to include dielectronic recombination (DR), and the inverse autoionization process (AP) in the master expression (19), we reconsider [17].

+ A(P„ Pj; Pk) ) A^! , Pj+1, Pk+i)

where A (Pi, Pj, Pk) refers to a projectile with Pi, Pj, Pk electrons in levels i, j and k respectively (k, j > i). Reaction 1 pertains to DR and 2 to AP. In view of expression (19) one has for the rates t and t2:

tl = P,( 2j2 - Pj) (2k2 - Pk)d(i, j; k) and

t2 = PjPk (2i2 - Pi)a(i,j, k)

with d(i, j, k) and a(i, j, k) the rate coefficients for one electron or one free place in each level. t2 implies that Pj and Pk behave as independent quantities, which is roughly true in a high density plasma where numerous collisions can occur, but not in the low density case where the deexcitation time is now smaller than the collision time. In that case the approximation of independent populations leads to a very low and unphysical t2 as already mentioned in Ref. [18].

Adapting the DR expressions given in Section 1.5.1, to the present formulation we explain t through

15.28 x 1016EkuFijF(pU) ^ d(l; j; k) =-—-.-, / ' rz\ "

2k22j2k(DEflg(2k,JD^ 1=0 21 + 1

with DE = Ej — Ei (Ry), Fj is the dipole oscillator strength, u = Ek + Ej — Et (Ry) and l the azimuthal quantum number of level k. The function F is given by

1 + exp( a - S-+V-

1 + exp( a - g-p) }

with T denoting the temperature (Ry) and a the degeneracy of plasma free electrons (Vp in a.u.).

t2 is derived from micro-reversibility with the l dependence carefully included. When we use micro-reversibility we can start either from d(i, j, k) and calculate a(i, j, k), or start from

dl(i, j, k) at a given quantum number l which leads to al(i, j, k) and then sum over the number of electrons with quantum number l in level k, noted P(k, l), to finally reach a(i, j, k). The two calculations yield the same result only for a random distribution P(k, l) = Pk\(2l + 1)/2k2]. In that case one gets

a(i; j; k)

3.819 x 1016EkuFy 2j2k(AE)2lg(2kvf)

But if we consider a low density plasma with a collision time much longer than the deexcitation time, then one has to put P(k, l) = Pk and the rate coefficient gets multiplied by (1/ q) with

-1-/ 2l + 1/

(2l + 1)2

We thus make the approximation to use Eq. (34) for a random population of electrons that do not come from DR whereas the coefficient a/d provides the AP rate for electrons that result only from DR.

It must be emphasized here that DR and AP are two independent processes. In a high density plasma AP is the dominant effect because the excited states are populated mostly by collisions. In this case the overall effect of DR and AP is a loss of electrons.

1.5. Balance steady state

The above detailed atomic processes can suitably be given a perspective with the study of realistic ion-plasma interactions.

1.5.1. General

Projectile ionization stages get constantly modified through ion-electron and ion—ion collisions in the target. Most significant ones include collisions with free plasma particles, while autoionization also affects the projectile effective charge. Other mechanisms of noteworthy to be mentioned also include bound—bound charge transfer from plasma bound electrons, radiative transfer, while non 2-body recombinations (dielectronic and 3-body) provide only negligible contribution even at high densities and are not considered any further. Internal projectile charge state is also submitted to excitations caused by target particles and also to spontaneous projectiles radiative decay. Low density targets feature projectiles mostly in the ground state, which restricts considerations to total cross-sections.

This point of view is illustrated on Fig. 6 with atomic collision rates for 4 MeV/u Iq+ ions in helium plasma with n = 4 x 1017 cm—3 and T = 4.4 eV. These parameters give rise to a plasma ionization degree z = 0.5.

From Figs. 1 and 6 the equilibrium charge state Zeq is obtained when ionization rate matches the recombination one. Ionization rates arise through ion collisions (Fig. 6) while recombination rate arises from charge transfer yielding Zeq = 31. Looking now at the rate teq for this q = Zeq 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/req.

Atomic rates (Fig. 6) are given as frequencies for projectiles gains and losses of electrons, thus establishing a rate for a relative charge variation 1/q. In the same vein, one gets a stopping rate Rstp for a relative energy variation of DE/E = 1/q and Rstp = qSV/E which is also reported in Fig. 6 in terms of the stopping power S = —dE/dx.

Aside from density dependence, every mechanism involved in the evolution of the ion projectile charge while crossing in plasma is given on Fig. 6.

As an example, we now consider an iodine ion with a given ionization state q. For instance, let us take an initial value q = 24. Fig. 6 tells us that this ion should be ionized by collisions with plasma ions until reaching Zeq = 31 with a time teq = 5 ns.

Then, the atomic rate is 10 times above the stopping rate, and the q increases from 24 up to 31 preventing the projectile from slowing down, since the target length required from reaching equilibrium amounts to 13.8 at the given density.

Removing charge transfer, the projectile ion charge increases up to q = 38 through ion collisions when ionization rate balances the stopping rate. The given ionization stage defines the dynamical charge Zdy.

Energy changes more swiftly than the charge state during the slowing down process, when q > Zdy. Then, the projectile charge stays unchanged until energy decays sufficiently with a recombination rate >Rstp.

These considerations thus allow us to foresee from Fig. 6 the evolution of the projectile charge state in terms of target parameters through the behaviour of most significant atomic rates.

E : i 1 1 1 1 1 1 : o^orno transfer___— ■ j

\ / \ / A . ' \ i i A III! Ion ionization \ •N

: » \ V

• % - ^ > — N * - > \ \ Dielectronic+ »

— ______' auloionization > — • \ v J

y Ws rvX

• » Electron ionization 1 J \ \

: 1 _ it 1 V — 1 • \ \:

^ 1 1 11 1 I« < * -1 » i i i 1 1 ! 1

Charge state

Fig. 6. Reaction rates for electron capture (solid line) and loss (dashed lines) together with stopping rate (dotted curve) of 4 MeV/u Iq+ in half-ionized helium plasma at T = 4.4 eV and n = 4 x 1017 cm"3 as a function of projectile charge state q. [G. Maynard et al., Nuo. Cim. 106 A, 1825 (1993)] with kind permission of Societa Italiana di Fisica, copyright 2016.

Overall low-density plasma trends then feature:

1. All of the rates are almost directly proportional to the density.

2. Electron collision ionizations do not play a significant role.

3. Free electron capture can only appear for highly colli-sional plasma.

4. Ion-ion collision rates nearly scale as Z2 for not too high atomic number. The stopping rates experience little variations with density due to plasmon energy in the Bethe formula. Ion-ion collisions are insensitive to plasma ionization.

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

At first sight, modifying Z/ZT has not much bearing on curve slopes. A trend can be predicted by moving up or down (in Fig. 6) the three curves of ion collision ionization rate, stopping rate and bound-bound charge transfer rate. The plasma atomic number has a large influence on ionization and charge transfer probability, while plasma ionization modifies the stopping rate and charge transfer. In an energetic ionplasma encounter ion-ion collisions are more relevant than the electron-ion ones, at variance with collisions between plasma particles, which rules out collisional radiative models in the present situation.

1.5.2. Temperature effects

An obvious temperature signature stems from the quadrature of electron-ion collisions cross-section over the plasma electron velocity distribution in the moving projectile frame. As in the stopping expression, temperature leaves an immediate imprint for Vp < Vthe. However, electron ionization stays below ion ionization and free electron recombination remains under the stopping rate. So, the temperature does not directly influence the projectile charge dynamics. Nonetheless, charge transfer relies on plasma excitation and ionization Z/ZT. Ion-ion ionization hardly changes with increasing Z/ZT, the main effect of which is to lower the charge transfer rate, yielding (Fig. 6) enhanced Zeq and bigger teq. Zeq is only obtained at a large enough plasma linear density. When Z/ZT keeps rising up, charge transfer rate falls below the stopping rate and the projectile charge now relies on initial q value. When q < Zdy, the projectile charge attains q = Zdy with decreasing energy, while if q > Zdy energy decays even faster with nearly unmodified projectile charge state. Z induced recombination arises from enhanced bound-bound charge transfer monitored by the strongly bound electrons. In hydrogen target, recombination is usually proportional to bound electron density while a large ionization will allow dielectronic and radiation processes to occur. In other targets, significant reduction of recombination only occurs for K and L ionized electrons and also at a large temperature. The influence on the enhanced projectile ionization in plasma (EPIP) reaches its top value in fully ionized targets with a small ZT and shows us as negligible at a high ZT [20].

1.5.3. Dynamical effect

As seen previously, a dynamical imprint appears for Rstp bigger than any atomic state. Fig. 6 demonstrates it appearing at reduced recombination with q > Zdy. At not too big ZT, the ionization rate is proportional to ZT and the charge transfer rises even swifter with ZT, with a nearly ZT-independent stopping rate. Dynamical effect is then triggered by small atomic numbers. In heavy targets, the projectile charge remains close to Zeq. Cold hydrogen targets document nonequilibrium charges while strong dynamical effects demand full target ionization. Moreover, in a fully ionized He target, a 4 MeV/u iodine beam with an initial charge below Zdy = 58, will rapidly attain it. As ion projectile slows down, ionization turns smaller than stopping.

As a result, ionization remains constant until the radiative or dielectronic recombination rate is above the stopping rate, at a very low projectile velocity.

In a mundane situation featuring large target ionization and an initial ion projectile charge state <Zeq, the projectile will be keeping a nearly constant charge ~Zdy on most of its trajectory for E = E(0), the initial energy.

The slowing down process is initiated with a charge <Zeq while it turns >Zeq further on. This fixed charge process largely influences the energy deposition profile. Although nearly flat, in cold targets, the swift heavy-ion energy profile becomes proton-like, featuring a Bragg peak near the end of range as it is the case in strongly ionized plasmas targets. In the latter case, stopping slightly decreases with density (plasmon energy in Bethe expression) so that solid targets exhibit less dynamical effects than plasma ones.

Dielectronic and autoionization (DA) processes account for electron losses except for a resonance at q ~ 38. DA documents autoionization after a dielectronic recombination and also arises from vacancies out of plasma ion collisions. Autoionization rate ~z2 (z denoting degree) can rank high in large Zp targets.

1.5.4. Density effects

Density highlights a rather tricky case featuring pretty large uncertainties due to the use of partial cross-sections. We shall mainly focus on the density impact on ion projectile excitation. Every atomic rate is proportional to target density at the exclusion of radiative decay and autoionization. In the presence of an increasing density, the incoming ion has no time to relax to ground state between two collisions in a row. Then, a few collisions later, it can pile up enough excitation energy available for Auger emission at surface exit. The main density effects are: excitation collision effect, partial recombination effect and vacancy production effect. Obviously, excitation arises because excited states are easier to ionize than ground state. With the excitation cross-section ~ Zp; such an effect appears first in a heavy material. Excitation collisions are documented by the Betz—Grodzins model [19] for a projectile endowed with a number of weakly excited electrons [19]. A small enhancement will secure ionization with a large Auger emission taking place at surface exit.

Recombining electrons on high lying levels may trigger ionization in lieu of transfer. Partial recombination on excited

levels rises with decreasing z. The recombination reduction via partial charge transfer remains small for heavy targets and features a dominant density effect at low Zp. For instance, in a hydrogen target with n = 1020 cm—3, the given diminution rises Zeq from 24 to 29 for a 1 MeV/u iodine beam. Low Zp targets favor the Bohr-Lindhard model highlighting a large density enhancement for the projectile in plasma and some Auger electrons at target exit.

On the other hand, low Zp targets stress a density dependence through vacancies in ion projectile inner shell. Then, electrons are allowed to recombine on high-lying levels but are not allowed to do so in a vacancies produced by ion collisions. The charge then increases with density in relation to excitation rates higher than radiative decay rates and also the vacancies number increases with density. This effect seems to be the largest for carbon targets.

Finally, it should be noted that systematic discrepancy between q exit and q outside prevents a straightforward experimental vindication of density effect on heavy ion charge state within target. Nonetheless, recent stopping power measured data demonstrate a 15% increase for the heavy ion effective charge in solid when contrasted to the equivalent cold gas.

1.6. Inflight effective charge (hydrogen target)

Now we can integrate all those atomic rates into a complete calculation of the ion projectile velocity-dependent effective charge Zeff (Vp). We focus attention on IN+ in view of its high resonant contribution to dielectronic recombination. So, it can provide a convincing benchmark for optimizing the impact of this effect on inflight projectile charge state.

1.6.1. Solution of the rate equation neglecting dielectronic recombination

The total rates for ionization and recombination without dielectronic recombination thus read as:

a (Z / Z + 1 ) = Sbem (Z) Vpn + ffe (Z) VX a(Z/Z - 1) = Oct(Z) Vpn + oRec + Obr(Z) Vrn

which allow to solve the rate equations for 1.5 MeV/u iodine ions incident on 10 eV hydrogen with a density of 1017 cm—3. Fig. 7, displays the charge-state evolution starting from Zeff = 0. For Vp = 10ac we find Eeq,hot [ Eeq,cold. Only at very high velocities is Zeqhot < Zeqcold because according to Eq. (18), the cross section for charge transfer drops faster at high velocity than the cross section for high radiative capture.

Fig. 7 also reveals that the equilibrium areal density / nedl (integrated from Z = 0 to Z = Zeq — 1) increases by two orders of magnitude. Thus, in order to observe the different behaviors of the effective charges in plasma and cold gas in a laboratory experiment, it matters to have a plasma with high enough areal density.

1.6.2. Dielectronic recombination included [21]

Again we consider the interaction of iodine ion at 1.5 MeV/ u with a fully ionized hydrogen plasma at T = 10 eV and

11 11 11 || 11 11 11 11 1.5 MeV/u l->H(10" cm"3) ч ч 4 1 ■Zeff=4 0.2

7=10eV / Betz 22.5

Ss T=0 I I I—I I I I I I I

¡nedl (cm

Fig. 7. Time evolution of the effective charge of an iodine beam with 1.5 MeV/ u in hydrogen neglecting stopping-power effects. The beam ions have charge zero initially. The dash-dotted line gives the semiempirical value of Betz [Rev. Mod. Phys. 44, 465 (1972)] for cold gas.

ne = 1017 cm—3. Now the evaluation of the inflight projectile ion charge state may include simultaneously a DR contribution and the stopping effect, as well (cf. Fig. 8). When the energy loss is not taken into account, Zeff(Vp) is strongly dependent on the DR rate. This behavior is well documented on Fig. 8 and Table 2 by the constant Vp charge states initialized at Zeff = 20. Asymptotically, i.e. near equilibrium, they climb up to 29 and 41 respectively, with and without DR included. Now, comparing distances for a complete stop, with and without DR included, one gets a ratio R = 1.17 for initial charge state 20 and R = 1.16 for 32, respectively when stopping is neglected. It has to be appreciated that those R values result only in a 16% DR modification for the complete stopping process. This is 6 times smaller than the ratio 2 of stopping powers pertaining to equilibrium charges Zeq(Vp) at constant velocity. When stopping is included, the ion projectile is brought to rest at the end of 8 m penetration in the given target plasma. On the other hand, Zeq(Vp) is obtained only after 40-m penetration depth, when DR is not taken into account. Fig. 8 also displays corresponding evolutions for an initial projectile charge of 32, in between the above two equilibrium values (29 and 41). The

Fig. 8. Variations of projectile ion Iq+ (initial energy 1.5 MeV/u) charge states in a hydrogen plasma as a function of penetration depth. Reproduced from G. Maynard and C. Deutsch, Phys. Scr. 48, 471 (1993) with the permission of IOP Publishing.

main conclusion to be drawn from this detailed investigation is that the realistic interplay of dielectronic recombination and stopping power, likely to be met in an experimental situation, yields a dynamical picture with a rather modest DR impact on the projectile ion effective charge. This statement may be taken as a robust one because we have essentially considered Iq+ projectile which features one of the highest resonance capabilities to accommodate the DR process [22].

1.7. Inflight effective charge (non-hydrogenic target)

A hydrogen plasma target provides easily a high density of free electrons. However, in order to make contact with realistic stopping plasmas of interest for inertial confinement fusion (ICF) through intense ion beams, one has to consider also high Z and low Z targets of nonhydrogenic materials such as Au, Pb, C, Al, etc.

The corresponding dense and partially ionized plasmas always retain a large amount of bound electrons which contribute significantly to the stopping and also to charge exchange induced electron transfer between incoming partially ionized projectile and target ions in the vicinity of its path. Figs. 9 and 10 display reaction rates weighted by density of excited target bound states for the interaction of Cu9+ ions with a hot (Te = 250 eV) Al plasma with a free electron density ne = 1018 cm"3. Fig. 9 features variations in terms of the projectile charge q at 0.781 MeV/u kinetic energy. In Fig. 10 the rate variations are given in terms of projectile Cu7+ energy [20].

According to the discussion in Section 1.5 dielectronic recombination may be given for its reaction rate a constant average value aDR = 10"11 cm"3-s_1.

This value has been identified as an upper bound (Nardi and Zinamon [6]). Chen's investigations [21] show that aDR remains dependent on target temperature.

The reported variations remain moderate, in particular for a projectile ionization smaller than 10. In order to estimate the final output of those projectile charge variations, we used a parameter study demonstrating that increasing aDR by an order of magnitude or reducing it by the same amount does not change at all the results displayed in Figs. 9 and 10 as well as in experimental cases considered below. In a partially ionized plasma target, the projectile charge state is mostly reached with collisions and charge transfer to target ions. Other free-electron contributions such as dielectronic recombination or three-body recombination are kept three orders of magnitude below the charge transfer once there is 0.5% of bound electrons in the target.

Charge transfer

xq+ + лр+ +

with the rate aqq_b features the most important recombination process in neutral gases, solids, or partially ionized plasmas. Nonetheless, in a fully ionized medium, this contribution nearly vanishes. Such a behavior accounts for a drastic reduction of recombination in plasma. Then the result for the

Fig. 9. Reaction rates weighted by density of excited target bound states, versus charge state q, encountered in the interaction of 50 MeV Cu ions with Al plasmas (Te = 250 eV, ne = 1018 cm—3; curves labelling: 1: ne «rec; 2: n(12+)«qq—1; 3: ne «dr; 4: n(12+) «j 5: n(9+)«y; 6: n(9+)aqq—1; 7: ne «e; 8: n(10+)aij; 9: n(10+)aqq—1; 10: n(11+)aqq—1; 11: n(11+)aii. Reproduced from C. Couillaud et al., Phys. Rev. E 49, 1545 (1994) with the permission of AIP Publishing.

Fig. 10. Reaction rates in the interaction of the 50 MeV Cu7+ ions with Al plasmas of Fig. 9 in terms of projectile energy. Other unspecified quantities are as in Fig. 9. Reproduced from C. Couillaud et al., Phys. Rev. E 49, 1545 (1994) with the permission of AIP Publishing.

projectile charge is expected to keep same plateau value over most of its velocity range.

Decay by spontaneous emission

Xq(E)/Xq+(E') + hv

appears non-negligible for highly excited projectiles, while three-body recombination is ruled out because at high projectile velocity and average electron density, conservation of momentum cancels three-body reactions.

In particular, momenta of the two recombining electrons have to compensate each other at the ion location. Such an occurrence decreases very rapidly with increasing projectile velocity (Bailey, Lee and More) [5] (Section 1.2).

As shown above, charge transfer features the most efficient capture process, so a higher projectile charge is expected in a fully ionized plasma than in the equivalent cold gas. Also, for a partially ionized plasma retaining a large number of bound electrons, the projectile equilibrium charge, reached when electron losses balance gains, is seen at higher projectile velocity that in totally stripped plasmas.

In Fig. 9, we picture the above considered reaction rates versus charge state q for the 50-MeV Cu ions interacting with a strongly ionized Al plasma where Te = 250 eV and ne = 1018 cm"3. This target is assumed in a coronal regime with Aln+ ions and n = 9 - 12.

Excepted the q-independent dielectronic recombination, all other mechanisms are strongly affected by the varying projectile charge state. On the other hand, when that projectile charge state (see Fig. 10) is kept fixed, the corresponding rates remain nearly constant in terms of incoming kinetic energy.

A lot of efforts has been devoted to asserting the actual significance and quantitative relevance of dielectronic recombination on the projectile ion inflight charge state Zeff (Vp), in a strongly ionized target plasma. As a result, the initial picture set forward in Section 1.2 highlighting higher charge states than in equivalent cold gas emerges strengthened by this thorough examination. Hindered recombination actually appears as the trade mark of plasma conditioned projectile effective charge.

Those considerations allow to extract easily the most significant pieces of information encapsulated in Fig. 9. This is performed on Fig. 11. We again consider a Cu9+ ion projectile with 50 MeV total kinetic energy, interacting with strongly ionized plasma targets with ion density = 1018 cm—3. We restrict to ionizing collisions with target ions and free electrons, as well as to radiative electron capture and 3-body recombination. Projectile excitation is neglected in those simulations.

Fully ionized targets (Fig. 11(a) and (b)) at high temperature exhibit very similar trends, although the first is aluminum and the second carbon. Then, the absence of charge transfer

ln(1 )

Enfnln(En)

Fig. 11. Reaction rates for 50 MeV Cu ions versus charge state q in the interaction with a plasma (ion density = 1018 cm—3). (a) Al at 300 eV (fully ionized), (b) C at 100 eV (fully ionized), and (c) Al at 100 eV.

between target ions and projectile can be observed at top efficiency. However, lowering by a factor of 3 the Al plasma temperature, makes to appear a dominant contribution of recombination through charge transfer. The target ions are then mostly heliumlike with two bound electrons.

where fn is the dipole oscillator strength for the nth energy level En, nT is the ion density in the target. Up to now, most of measurements of the ion stopping have been restricted to cold solid or gaseous targets. In order to optimize the efficiency of a pellet compression, however, an accurate estimation of the stopping parameters in dense and hot plasmas is required. The target material is likely to be heated up to temperatures of 50—200 eV, before fuel ignition starts up in a direct drive approach. Under these conditions, such tamper elements as Pb will be only partially ionized. Consequently, the stopping by bound electrons is no longer negligible and thus, we have to anticipate an energy range of a given ion beam in a cold matter and in an equivalent plasma for the same line density nl (l is the linear projectile range).

The central quantity of present concern is the mean excitation I.

We are thus obviously required to extend to nonhydrogenic cold targets, the usual knowledge already used for neutral atoms and hydrogenic ions. Furthermore, we shall compute the corresponding I in hot and dense plasmas of atomic charge ZT.

Postponing the presentation of low Vp behavior we restrict presently to the standard Bethe-Bohr-Bloch regime with 2meVp/I > 1, i.e

MV2 >— 5 ,15Zr[eV]

where ZT now denotes the average nuclear charge in target. Here, we put the main emphasis on theoretical methods which are able to deliver accurate mean excitation energies based on a clear understanding, so that comparison of cold and hot stopping, respectively, for a given ion beam is easily achieved.

This explains why we pay attention to a variational approach yielding upper and lower bounds with a narrow gap in between. Other methods including a cubic spline interpolation, a pseudo-analytic expression for I, and the local plasma approximation (LPA) are also considered.

They all agree in showing that high-temperature plasmas (kBT > 10 eV) have significantly smaller mean excitation energies than their cold target homologue.

2.1. Variational method

2. Stopping by bound electrons

Now, we switch attention to the first term in the right hand side under brackets in Eq. (1). It pertains to the stopping contribution arising from the remaining electrons bound in the partially ionized plasma target. It thus reads as

2.1.1. Sum rules

Starting with a formal approach initiated by Pekeris [23] and Dalgarno [24] and subsequently developed by many authors [25], we consider the sum rules (—1 < m < 2):

S(m) = I>nE

>nE0n,

dE_ 4p [Zee (Vp) e2]'

me Vp2

-nT ( ZT

Z ïln^p

(36) L(m) = 5>nE0>|E0n|,

in terms of the mean excitation energy

in terms of optical oscillator strengths (q / 0 limit),

f0n = 2Er(n 0) Ia-u-]>

and excitation energy E0n for transitions {0} / {n}ina given atom (ion) with N bound electrons, and nuclear charge Z. Eqs. (38) and (39) interpolate for any m 2 [—1, 2], the usual physical quantities specified for integer m. For the sake of simplicity, we restrict our presentation to a one-electron model so that a given subshell {a} may be indexed either through classical {n, l} or relativistic {n, l, j} one-electron quantum numbers. We thus specify Eqs. (38) and (39).

Sa(m) ^ ^.faa' Eaa'

La(m) = Xf*' E^, ln|Eaa|.

Noticing that an optical transition to an already filled subshell has to be forbidden, we introduce the more realistic sums

sa(m)=x)(1 - ha fa Ema,

La(m) = — ha')faa'Eaa'ln|Eaa' |;

(41') (42 )

in terms of h« = g«/d«, g« is the number of electrons {«'} while da, refers to level degeneracy, i.e., 2(21 + 1) for a = {n, l} and (2j + 1) for {n, l, j}. The overall quantities corresponding to Eqs. (38) and (39) thus read:

sp(m) = E gaSa(m);

LP(m) = X ^(m),

SP(2k)=S(2k); k < 1; SP(2k + 1) <S(2k + 1), k < 1.

2.1.2. Upper and lower bounds

We now consider a generalized mean excitation energy under the form:

ln[i(m)]

■sP(m)

sP(m) sP(m) dm

The physical I is retrieved in the m = 0 limit. Now, we turn to the variational procedure itself as considered by Shimamura and Watanabe [26] for an atom in its ground state.

Upper and lower bounds for ln[I(m)] are then deduced from the basic atomic quantities (in a.u.):

Sa(-1)=2(a|r2|a) = |ra,

Sa(0) = 1 ,

Sa(1)=|(a|(P2/2)|a) = 3

Sa(2)=3(a

4pZ d(r)--— [rV (r)]

(48b) (48c)

The operators:

Pa = X) V1 — ha | a'><«' |

a' < a

D = V2z = V2(q-r/r) (49)

allow us to rephrase Eqs. (41) and (42) under the form (a'<a): S«(m) = <a|DPa(H — Ea)m+1PaD|a>, (50)

Ma(m) = <a|DPa(H — E«)m+:ln(H — E«)P« D|a>, (51)

featuring upper and lower bounds straightforwardly deduced from convexity considerations.

Their obvious numerical interest arises from the sharp max-min interval, which allows us to well approximate ln(I) by the bounds average ln<I> the simplest case of a basis reduced to its first vector u\. This procedure thus yields

Sa(m - 1)

Ma(m) , < 77 < ln Sa(m)

Sa(m +1)'

2.1.3. Analytic formula

For an ion (atom) in its ground state, Eq. (52) reduces to

SP(m -1).

SP(m +1)

because there is no superelastic transition. SP(m) fulfills

sp(0) = S(0), Sp(1)<S(1), Sp(—1)<S(—1).

In the m = 0 limit one gets:

LS(-1)J

< ln(I) < ln

Taking the bounds average leads to

ln(I) = 2ln

which compares very well with the Hartree-Fock-Slater (HFS) results given by Dehmer, Inokuti and Saxon [27] (DIS).

Estimate (56) is the basis for variational results plotted in Figs. 4—6 together with the HFS ones. Neglecting correlations between bound electrons allows us to use (a0 = Bohr radius)

(48a) S(-1) =

e 2 2 a0

S(1)=3*0,

which, when introduced into Eq. (55) yield the explicit form:

(I) = i FT0 [a • u •]

with rg = <r2)0.

This novel approximation already provides a very good estimate for I. The practical interest of Eq. (58) would remain rather limited if one could only proceed to a sophisticated evaluation for K0 and r§. Hopefully, these quantities are easily deduced from electron-ion pseudo-analytic effective interactions which enables us to compute these parameters with the same accuracy.

As experienced earlier, we find it convenient to explain these effective interactions in the GSZ form [28]

?M = -(— ) U(r),

(59b) (59c)

Q(r)=[H (er/d — 1) + 1] 1 .

H and d denote two adjustable parameters fitted on the lowest bound state energies.

For our present goals, the quantity of interest is the electron density p(r) fulfilling the Poisson equation,

V24(r) = —4pp(r),

, 2r(r) H r/dH(er/d + 1) — 1

= — re/d—^--

[H(er/d - 1) + 1f From Eq. (60) one readily obtains

r24pr2r(r)dr :

r2d4 )dr

± (2r34)R - 6 J r24(r)dr

through an integration by parts, so (R / to),

Nr0 = -6 J dr-r2ç(r) = rG F (a), a = 1 - 1/H,

where rG = 6Nd2 /H and

TO _,n

F(*) = J2-

n=0 (n ± 1)

with 0<a < 1.

The average kinetic energy K0 is reached through a constrained virial relationship:

Zj - N ± 1\/N - 1

H(er'd - 1) + 1_

11 Z - N 6 ± ÜH

4pr2p(r)dr

Finally, putting Eqs. (61) and (62) into Eq. (57) yields the Garbet expression [29]

NL(H\2[1 — (i) (1 + 2H)]

F (1 — H) "

If we now specify the above results to the neutral atom limit (ZT = N) and recall the empirical estimate

Iy 10• 32 [eV],

one expects a behavior (I) y ZT, arising from the GSZ scaling law [28]

h = Zß, ß = 0 • 5 • d

This result is indeed confirmed when the Thomas-Fermi estimates NK0 - ZT/3 and Nr2 - zJ/2 are introduced into Eq. (58). The corresponding (I) falls within 6% of DIS ones [27] for ZT > 28 (see Fig. 12). Putting Eq. (65) into Eq. (63) yields

ZtN [1 - (i) (1 ±H)]

6d F( 1 - H) '

with the atomic limit (ZT = N )

2_ Z2 9 + a

6d 12F(a) In the TF limit (Z/œ), one has

a = 1, F(a)=—,

which when combined to the most likely estimated d/a0 — 0 . 7, provides

(I) Zj

y 9 •4462 [eV]

in fair agreement with Eq. (64). The main interest of Eq. (63) lies in its flexibility. It applies to any ionicity (see Fig. 13) and to arbitrarily partially stripped ions in plasmas.

Fig. 12. Mean excitation energies for neutral species (1 < Zr < 40). Reproduced from X. Garbet et al., J. Appl. Phys. 61, 907 (1987) with the permission of AIP Publishing.

Ind.) =dm ln[s«(m)]

and the interpolation formula

s. (0)

5 — 2m 7 — 2m

where a, b, c are another set of disposable parameters. Setting

Ta(m) = — ln m

ln(I.) becomes

ln(4)=-ra (—l)+_ra(l)—3_r„(2).

2.3. Ions in plasma

Fig. 13. Mean excitation energies for isolated ion Aln+ according to ionicity. Reproduced from X. Garbet et al., J. Appl. Phys. 61, 907 (1987) with the permission of AIP Publishing.

2.2. Cubic spline method

Here we essentially show how some variational results can improve a spline interpolation method initially worked out by Peek et al. [30]. For this purpose, let us reconsider Eq. (47) with the P superscript deleted, under the form

ln(i)= d— ln[Sa(m)]

which suggests a numerical interpolation by considering real m values, provided the integer data m = — 1, 0, 1 and 2 are known. Arguing that S(m) diverges logarithmically at m = 2.5, Dalgarno [24] has considered the formula

S(m) = [a + bm + cm2 + dm ln(2.5 — m)] ms(0),

where a, b, c, d are suitable constants prescribed by specific constraints. These techniques have been further [31] taken up by specifying Eq. (46) to a subshell a, with

2.3.1. Average atom model

With the obvious intention of extending the scope of the above methods to the plasma case, we implement a scheme proposed by Rozsnayi [32]. It usually leads to results close to those of the Liberman [33] average atom model.

This is essentially an algorithm delivering electronic levels and equation of state of atoms embedded in matter with arbitrary density and temperature.

The self-consistent field treatment starts with the relativistic Thomas-Fermi-Dirac (TFD) model in the iterative procedure.

The Fermi statistics and the central field approximation are maintained, giving an average atom representation. It is generally believed that in the region of high temperature and/ or pressure, the TFD model gives a reasonably accurate electron potential. This potential, in turn, can be used as a basis to treat the atom in a more quantum-mechanical way by solving the single-electron wave equation for the bound states.

The basic assumption is that during an elementary stopping process, the target ion is nearly instantaneously neutralized by plasma fluctuations. This hypothesis is particularly relevant to matter treated by intense ion beams, in view of the relatively long pulse time (~20 — 30 ns) allowing the target species to be considered in local thermodynamic equilibrium (LTE) with comparable electron and ion temperature T and number density n.

The numerical procedure is initialized by taking each R0 = (3/4pn)1/3. According to Rozsnayi [32] the HFS scheme is implemented as follows.

1. Bound electrons are consider independent. They are supposed to be modeled in spherically symmetric and self-consistent potential Veff (r) (see Eq. (73) below), the same for all electrons. However, exchange and correlation are still retained in Veff (r). The total electron density:

P(r)=Pb(r) + Pf(r); fulfills

4py dr-r2p(r) = ZT,

, . ^ — Z „ , . Z (71) V0(r)= — U(r)=RZ0

Ro 1 r n Ro

with rb(r) in terms of single-electron eigenquantities

fe, J).

2. The average atom assumption allows us to replace the various excitation states in target with those of a fictitious atom with noninteger occupation numbers for excited orbitals. Vf (r) is taken constant within each subshell.

3. Conceptually and numerically speaking it is most convenient to derive (ei, Ji) from Dirac equation, which proves especially convenient to tackling the eigenvalue problem through two coupled first-order equations.

We do not intend to emphasize relativistic and spin effects of increasing relevance for heavy elements. However, this could be easily done within the present framework. The bound electron radial density rb(r)

4pPb(r) = X J?'+ 1) ,

lif exP Enj - m)\ +1

with total electron energy

E'nlj = Enlj + Veff (r), and

Veff (r) = Vcb (r) + Vexch (r) + Vcor(r),

' + B'r)

a sum of Coloumb, exchange, and correlation potentials is readily expressed in terms of radial Dirac wave functions deduced from

Veff (r)-Enlj

Veff(r) -Enlj\ cs K r

where K = -s(j + l/2), c = 137 .037 (a .u .), s = ±1. 4. The free electron density is taken to be Thomas-Fermi like

V? 3=2 f x1=2dx

P.f (r)=p3/ïT J f [m-Veff (r)1°

Veil (r) eXPi X KbT f

with the boundary condition Veff(R0) = 0. En[ is devided from the Wigner-Seitz condition J(R0) = 0.

5. The number Z(n, T) of free electrons per nuclei in target may be initialized with a Thomas-Fermi approximation.

6. It appears convenient to initialize Veff (r) with (see Eq. (59))

a superposition of GSZ (bound) and ion-sphere potential (free), allowing for a self-consistent resolution of the TF model. (£,-, J) are also initialized by their WKB values.

Table 3

Average Al atom in plasma with T = 100 eV (3.657 a.u.) at the three densities (a) r = 0.1 p0, (b) r = p0, and (c) p = 10p0, with p = 2.699 g-cm"3, solid density. Occupation numbers (usually noninteger) and eigenvalues are plotted for the successive (orbitals) subshells (n, l, j ). Variational outputs for Sp(0), Sp(1), and Sp(2) together with mini-max for Lp(0) and Lp(1) are given, with the corresponding K0 and (r2)0 required to implement GSZ expression (58). Chemical potential m, bound electrons number N, and free electron number Z(N + Z = ZT) are also displayed.

(a) p = 0.1p0, R0 = 6.44 a.u.

Orbitals Electrons occupation number Hamiltonian eigenvalue

1s2 2.000 -69.453 Sp(-1) = 1. 947

2s^ 0.391 -12.307 Sp(0) = 4. 056

2p2 0.331 -11.560 Sp(1) = 2. 391 x 102

2p2 0.657 -11.528 ^(2) = 1 .275 x 105

3s2 0.042 -3.3995 K0 = 46.81

3p2, 0.039 -3.1300 (r2>0 = 0. 733

3p2 0.078 -3.1209

3d| 0.072 -2.8112

3d252 0.108 -2.8044

4s212 0.021 -0.8836

4p2 0.021 -0.7737

4p2 0.042 -0.7707

4d2 0.040 -0.6494

4d252 0.060 -0.6161

4f7 0.079 -0.5872

4f252 0.058 -0.4926

5s1 0.017 -0.0755

m = -1.751; N = 4.06; Z = 8.94; 14.83 < Lp(0) < 15.08; 2.529 x 103 < Lp(1) < 2.718 x 103

(b) p = po, Ro = 2.99 a.u.

Orbitals Electrons occupation number Hamiltonian eigenvalue

1s2 2s212 2p2 2p2 3s2 3p22 3P2 3d2 2.000 0.792 0.667 1.327 0.154 0.290 0.140 0.407 -63.951 -8.3632 -7.3733 -7.3450 0.7896 -0.5509 -0.4078 -0.2823 Sp(-1) = 3 . 004 Sp(0) = 5 . 777 ^(1) = 2. 515 x 102 Sp(2) = 1 .342 x 105 K0 = 35.90 r2>0 = 0 828

m = -9.916; N = 5.78; Z = 7.22; 18.29 < Lp(0) < 19.31; 2.589 x 103 < Lp(1) < 2.825 : x 103

(c) p = 10po, R0 = 1.39 a.u.

Orbitals Electrons occupation number Hamiltonian eigenvalue

1s2 2.000 -57.335 Sp(-1) = 1.139

2s212 1.252 -3.4574 Sp(0) = 6. 528

2p22 2.214 -2.3550 Sp(1) = 2.507 x 102

2p2 1.062 -2.0194 Sp(2) = 1 .272 x 105 K0 = 31.86 r2>0 = 0 340

m = -1.565; N = 6.53; Z = 6.47 20.32 < Lp(0) < 22.20; 2.536 x 103

< Lp(1) < 2.814 x 103

7. Once convergence is reached, these eigenquantities are self-consistently determined for a given set (n,T) of plasma conditions, together with definitive expressions for Veff, m(n, T) and ne (see Table 3 for illustration).

2.3.2. Mean excitation energies in plasmas

To illustrate the whole procedure, let us compare <I> in Al plasma and for an isolated Al3+ ion, respectively, according to Eq. (56). In the first option, we take KBT = 100 eV and n = 10 solid density, so that (in a.u.)

K0 = 22.42, r02 = 0.493, Z = 3

leads to <I> = 2.255 slightly overshooting the more accurate (variational) data 2.16. The corresponding isolated ion results are

24.79, r2 = 0.341; (/) = 2.49

in lieu of the exact 2.48.

On Figs. 14 and 15, we have also reported data due to the local plasma approximation (LPA) [34]. The LPA consists of averaging over density of the inhomogeneous fluid of bound electrons around a target ion.

Despite its obvious plasma connotation, this method provides results among the least accurate, according to (g = 2),

ln(/)= / ln[gZWp(r)]Pb(r)dr

up(r)=4ppb(r)e2/me.

We recall that accuracy of these theoretical results in cold matter is evaluated with respect to those of DIS [27] which are ab initio HFS.

The present bleak performances of the LPA even at room temperature, are not completely unexpected [35] in view of the basic difficulties encountered when one tries to derive this scheme from the first principles, i.e, starting with the standard definition of the oscillator strength in terms of dipole matrix elements and carrying out a systematic deduction. This point and many others concerning the LPA have recently been critically reviewed by Johnson and Inokuti [35].

In this connection, it should be appreciated that our present results reinforce the paradoxical status of the LPA because it is supposed to be precisely at its best in yielding accurate I values. Nonetheless, we have described three other methods which are easily accurate in cold matter, where contact can be performed with the most accurate DIS ones [27].

The above atomic methods detailed in Sections 2.1 and 2.2 for isolated species are thus renormalized with plasma effects according to Section 2.3.

Their respective ln I (I in a.u.) are contrasted in Figs. 7 and 8 for Al at various densities and Fe. All the results agree in showing

/(Zr,n,T> 10 [eV]) </[Zr,Z(n, T)]

Fig. 14. Mean excitation energies for Al in a hot plasma at (a) solid density, (b) 0.1 x solid density, (c) 10 x solid density. Reproduced fromX. Garbet et al., J. Appl. Phys. 61, 907 (1987) with the permission of AIP Publishing.

Up to a temperature of 10 eV, it is acceptable to deduce / from a Saha determination of ionization Z in target. However, at a higher temperature, occupation of higher subshells has to be taken into account, and their contribution to stopping is larger than that arising from more tightly bound electrons in

Fig. 15. Mean excitation energies in Fe plasma at solid density, as a function of temperature. Reproduced from X. Garbet et al., J. Appl. Phys. 61, 907 (1987) with the permission of AIP Publishing.

lower subshells. The spline interpolation adequately improved with variational inputs seems to provide fair data. As stated, the LPA produces the least accurate results.

At this juncture, it has to be appreciated that the accuracy of the variational results is not restricted by the numerical procedure. For instance, the mesh used for computing the basis wave functions may be taken as densely as required. The variational accuracy is mostly limited by neglecting of any electron exchange between neighboring ions. As shown in Tables 3 and 4 variational results can be given with controlled accuracy, when compared to those of other methods [36]. Finally, we display in Table 4 corresponding variational results for an average Al (Tables 3 and 4) atom in dense hot matter. As stressed above, the excited subshells have a noninteger occupation number.

The present calculations clearly exhibit the temperature limit above which the isolated ion approximation implemented with Saha distribution of excited states, has to be questioned. Through the AAM the plasma effects result in a larger population of excited bound orbitals, which respond more efficiently to the projectile field than corresponding ground state [37,38].

Table 4

ln(1) (I in a.u.) for isolated ion Al"+. Maximum discrepancy for the variational results are given within parentheses.

Ionicity McGuire [36 GSZ [Eq. (58)] Variational method Mehlhorn [3-

0 1.51 1.51 1.61 (7.0%) 1.52

1 1.77 1.91 1.87 (4.5%) 1.65

2 2.11 2.13 2.16 (4.2%) 1.85

3 2.44 2.39 2.47 (3.5%) 2.10

4 2.48 2.53 2.59 (3.0%) 2.15

5 2.66 2.58 2.72 (2.7%) 2.20

6 2.82 2.75 2.87 (3.2%) 2.28

7 2.97 2.88 3.04 (3.2%) 2.37

8 2.16 3.04 3.24 (3.0%) 2.50

9 3.40 3.26 3.51 (2.9%) 2.71

10 3.82 3.68 3.15

11 4.47 4.10

The results add a novel and significant contribution to the shortening of the projectile range in target, in addition to those due to partial degeneracy of the electron fluid and enhanced projectile charge in plasma [37] already considered previously.

3. Nonlinear stopping

Up to now, we restricted our investigations to the standard stopping model (SSM) framework and to a first term ~ Ze2ff(Vp) in the interaction expansion. Nonetheless, for completeness and above all to secure the quantitative pertinence of the above results, it appears compulsory to pay detailed attention to the higher and so-called nonlinear corrections to the usual Bohr-Bethe quadratic dependence in the projectile charge of the stopping power expression. A lot of attention has already been given to those terms for the usual case of a gaseous target [39]. Our main concern here is to adapt these results to a noneutral target by revisiting the corresponding formalisms.

In order to focus attention on the stopping mechanisms, we take it here for granted that the ion projectile keeps a fixed inflight charge Zp. We therefore pay a first attention to the so-called Barkas contribution [39] Z3.

It is also the first nonlinear stopping contribution which now depends on the Zp sign.

3.1. Barkas effect

3.1.1. Classical formulation

The theoretical framework considered up to now is essentially based on the first Born approximation in which the stopping power for a particle of velocity Vp energy, and charge number Zp ^ Vp/V0, where V0 = e2/Z is proportional to (Zpe)2. This section extends the stopping-power theory to the term y (Zpe)2 in an impulse approximation, i.e., the classical equivalent of the second Born approximation. It accounts for the observed differences between the electromagnetic stopping power of a particle and its antiparticle, and also for differences between the stopping powers of a particles and protons which, so far, have remained unexplained.

Consider first an electron bound harmonically with a frequency u to the nucleus of the atom in a target with atoms of atomic number ZT and density nT. A heavy projectile of charge Zpe approaches the oscillator with non relativistic velocity Vp at an impact parameter b relative to atom nucleus. f(t,b) denotes the classically force on the electron set up by the projectile at time t. The displacement of the electron at time t from its equilibrium position (at the origin with zero velocity at t = —to), Z(t), is then obtained through Z + u2Z = f, which we write in the form [39].

?(0 = Re

— if (t, b)e-lu(t-t0 W meu J

At time t, the electron gained the energy 1me(Z2 + u2Z2). Thanks to Eq. (79), this energy is given by

Wr(t, b)=2m

J f (t, b)e-iut' dt

Ion projectile

M, z„

Thus the energy lost by the incident particle per unit distance dR, i.e., the stopping power of the fraction FM of target electrons bound with a frequency u, appears as:

~~dx) = 2pnTZrFu

bdbWS(œ, b)

Bound electron

.....U^

+ bdbWr (to, b)

where Wr is the energy transferred to resonance excitations of the electrons and WS is the energy transferred in close single collisions. au advocates a lower limit of the impact parameter with the electrons seen as harmonically bound. For b < au the electrons are treated as unbound. If the incident particle moves along the line x = — b in the +y direction, the force (see Fig. 16) on the electron can be written with Eq. (79) as:

f (t, b)

[b + Kx(t)]i - [Vpt - Zy(Q]?

{[b + Zx(t)}2 + [Vp t - Zy(t)]2}3/2

where i and j are unit vectors in the +x and +y directions. For small electron displacements such that Ç(i)/(b2 + Vpi2)1/2^1

and retaining first-order terms, one gets:

f (t, b) =fo(t, b) + Af (t, b)

where f0(t, b) denotes the standard expression: bi — Vpt]

f 0 (t, b)

b2 + (Vpi) The new term Df is explained as:

Fig. 16. Schematic of the ion projectile with an extended charge distribution interacting with electrons bound to the partially stripped target. The impact parameter b is assumed larger than the average projectile diameter.

volume so that the electrons responding with frequency u remain bound. This restricts au to values larger than the respective shell radius. For smaller impact parameters the momentum transfer becomes so large that the electrons turn quasi-free and their contribution to the Zp effect becomes small, because the cross section for Rutherford scattering with free electrons is exactly proportional to Zp2. Such a division into glancing collisions and close collisions with large momentum transfers is in agreement with Bohr's early semi-classical treatment of stopping [40].

Equation (86) then reads as:

/d^ = 4pnTZTUFu(Zpe2)37 (W { dxj u m2Vp5 Up'

Df (t, b)

b2 + (Vpi) + 3bVptZy(t) + fb2

f du . . f cos uv r/ i ) rr/ \ I(X)= - K1 (u) dv-^ (v2 - 2) Fx(u, v)

(-2b2 + Vp¥) Zx(t) 1 'J u21 1 _{ (1 + v2)5/2U J

2Vp2t2

3bVptZy (t) Ky(t)] ]} .

Solving Eq. (83) iteratively and, replacing f by f0 when calculating Z(t) from Eq. (79), we insert the Fourier transforms f0u and AfU into Wr(TO,b), yielding Eq. (80) the stopping-power correction due to Af, i.e

3vF2(u, v)] + K0(u) dv

(1 - 2v2)F - 2(u, v)

sin uv

(1 + v2)

[3vF1 (u, v)

2pnTZTFu

b db(Ref 0u$ReDf u

The functions Fj and F2 are defined by

sin[u(v — y)]

+ Imf0u$ImAfu) . (86)

The dipole approximation implied in Eq. (86) limits its validity range to distances from nucleus outside the atomic

F(u,v)= J dy

f ysin F2(u, v)= dy —-1

. ,2) 3/2

1 + y 2 3

ysin[u(v - y)}

Recalling that g(w) is related to the electron distribution p(r) in a target atom (ion) by

Kv is the modified Bessel function of the second kind of order v. For X = waw/Vp^1, the function I(f) can be approached by I(X) z — 2p ln(X) + A, where A z —2.4 is a constant. In this

case the function I(f) vanishes approximately as ZTg(U)= d3rp(r)d[w0(r) — w]

I(?)z( |f)e—2x.

The fraction F(u) of oscillators responding in the frequency range between u and u + du is given by g(u)du, where g(u) is the differential oscillator strength normalized such that J0°° g(u)du = 1.

As long as the above result is restricted to large b values [1, the corresponding quantum-mechanical derivation would yield to an identical expression (88). Then, the corresponding stopping power correction reads as

Uo(r) =

8pp(r)e

we are now entitled to estimate the Zp3 stopping power correction

Z„3/d = (Z^3

m2Vp p

(Zpe2)3 / Mu)^

dEv\ _ 4pWpZi,z

dx I = (Zp"

/ 2\ 3 I I

(Zpe ) dug(u)u л—

3.1.2. Estimation of equation (88)

The intricate expression (88) may be drastically simplified [41] by invoking an instructive connection with the quadrupole contribution to the electron broadening of neutral lines in Stark broadening theory [43]. Quadratures very similar to

r ~izx r

MmAz, Z )= / dx

(1 + x2)m

(1 + y2)v

to the enlarged stopping expression

= 2Zp/d + ZpJd ,

where the customary lowest-order energy loss (Bohr-Bethe-Bloch) is expressed as twice a long-distance contribution.

Toward that goal we use again the Thomas-Fermi like-GSZ-expression

r(r)_ H H (er|d + ^ - 1

N d2'~ [H(er/d — 1) + 1]3'

Upon introducing a dimensionless ratio V = Vp/\[ZTV0 with V0 = e 2/Z, the Zp2 -term becomes

have already been reduced to a sum of products of modified Bessel function.

Actually, all the quadratures contributing to I(X) may be derived from Mm v(z, z') and its derivatives with respect to the arguments.

This allows one, after lengthy manipulations, to reduce Eq. (88) to the much simpler expression

H) = XKi(X)[Ko (X)/o(X)-Ki(X)/i(X)]

+ J duK1(2u)K1(u)I1 (u),

with Iv as the modified Bessel function of the 1st kind.

Id =7й

d3rp(r)?K! (X)Ko (X),

? =<r^, C = = 0.307 [Mey.cm2.g-1]

y mer2 mec2

N0 being the Avogadro number.

The rel finally by:

The relative magnitude of the Zp (Barkas) term is measured

F(V ) = VZV 2 J 21 d

• For X < 1, we extend a previous limit with

I (?)« — —ln(X) — 2.417 — 2pf{ [ln(X)]2 + 1,14ln(X) — 0.33},

• For large arguments, we get a decay

1 / \ 3/2

I(?)« e2x?/œ, (92)

differing slightly from the Ashley et al. [42,43] expression.

displayed in Table 5 for four typical atoms and small and large V values respectively. F(V) is typically 15 up to 30 percent of the Zp2 -term. Jd is obviously a non-negligible quantity. It appears convenient to put Eq. (94) under the form:

Jd = C Ж 1/d3rr(r)I(X)X2-

In contradiction to the Lenz-Jensen expression which does not discriminate the atomic structure, GSZ is more selective because it is built upon accurate experimental data for lowest bound states.

Table 5

Jd and F(V) data.

Condition V Jd F(V)

0.5 0.22196 0.327

1 0.29387 0.371

Ne: Z = 10 1.5 0.27711 0.366

d = 0.427 9 0.05799 0.179

9.5 0.05412 0.173

10 0.05064 0.167

0.5 0.21793 0.215

1 0.25206 0.257

Ca: Z = 20 1.5 0.23594 0.267

d = 0.880 9 0.058472 0.170

9.5 0.054863 0.166

10 0.051600 0.162

0.5 0.20626 0.205

1 0.24476 0.254

Hg: Z = 80 1.5 0.23434 0.271

d = 0.620 4 0.14005 0.251

4.5 0.12717 0.244

5 0.11601 0.236

0.5 0.19718 0.178

1 0.2337 0.231

Th: Z = 90 1.5 0.22616 0.252

d = 0.980 4 0.13992 0.246

4.5 0.12757 0.240

5 0.11678 0.234

3.2. Bloch term

Within the SSM framework outlined in Section 2.2, it is now straight-forward to complete the stopping expression with Barkas ~ Zp and Bloch terms as well. In cold gas, those latter are no longer negligible when the Born parameter ZpVo/(V2 + VL)1/2 is comparable to 1. V0 = 1 in atomic units. Vth is the thermal velocity of plasma electrons. Retaining also the very small target-ion contribution, one thus gets a more accurate stopping expression

4pNofVZD2 ATmeVp2

DZt •

Z —i

_ F Zt Z g

+zd—

+ z 2V0 f

H ZD v2f

vd2) + mzTLD

,J2me<

and up = \J4pe2ne/me, the plasma frequency of the target free electrons.

( Vhe) is the usual average in terms of Fermi functions. f(Vp2) accounts for the standard Bloch correction, which bridges a gap between Bohr semi-classical expression and Bethe quantum-mechanical one. The last term (me/Ar) ZTLp accounts for inelastic encounters between ion projectiles and target ions. It is nonnegligible only for Vp < Vthe, or at a very high plasma temperature.

The third term within brackets in the right-hand side (r.h.s.) of Eq. (98) is the Barkas contribution [44] ~ Zp. It pertains only to

bound electrons and thus vanishes identically in a fully ionized target. In this case, the Born parameter becomes ZpV0/Vp.

For Vp [ Vthe, Lp/Ap and LBB/Ab (cf. Eq. (2)). The relative importance of the four last terms in the rhs of Eq. (98) is evaluated in Table 6 for a target electron density of subsequent experimental interest. Obviously, the Barkas and Bloch corrections to the main Bethe contribution remain rather weak for all values of the Born parameter ZpV0/Vp.

In the sequel Eq. (98) will be often referred to in the canonical form

dx kV2

Lo + ZpLi + f[Z

Bohr, Bethe and Bloch stopping formulae differ essentially from each other in their respective implementation of a classical description combined with the Born approximation. Before attempting a quantitative comparison [39] we consider anew the projectile high velocity expression

Z2u2e2 Zp upe

: V2 D

where u2 = 4pNee2/me is the target bulk plasma frequency, with L intrinsically given by Eq. (101) within the selected slowing down expression. To render more transparent the given comparison we consider a fiducial "atom" built on a single harmonically bound electron. It is then easily seen that I = Zu, with u, electron circular frequency.

O - ind - b2> - b2;

O - 1'n(1 - b2> -2H j(1)

1.123mv3

|Zp|e2u

(102a)

(102b)

(102c)

j(x) denotes the digamma function, logarithmic derivative of G(x) and a

137.036

denotes the fine structure constant.

Relativistic extensions of these expressions feature similarly but are not identical. For instance, half of Bethe rela-tivistic correction arises from distant interactions and one-half stems from close ones. Bohr and Bloch relativistic modifications to distant collisions yield back Bethe's i.e.

-ln(1 - b2)-b2/2.

Moreover Bethe's relativistic corrections to close interactions apply only within the first Born approximation while similar Bohr and Bloch contributions are not restricted in the same way which leads to a confusing confrontation. Noticing that these relativistic extensions amount only to -7% for b = 0.5, thus we neglect them in the sequel.

Table 6

Relative importance of Bloch and Barkas terms for cold-gas and plasma target. The target has a linear density of free electron nel = 1.5 G. Maynard et al., Phys. Rev. 146, 5101 (1992)].

ZpVo Vp

Bethe L0

Barkas

1019 cm [D. Gardes,

Ion E (MeV/amu) Gas Plasmas Gas Plasmas Gas Plasmas Gas Plasmas

C4+ 2 0.56 0.62 5.63 12.48 0.08 0 -0.3 -0.35

S7+ 1 1.75 2.07 4.94 11.79 0.35 0 -1.16 -1.32

1.5 1.43 1.69 5.34 12.2 0.23 0 -0.98 -1.13

2 1.34 1.47 5.63 12.48 0.18 0 -0.85 -1.00

Br6+ 0.93 2.79 3.45 4.86 11.72 0.64 0 -1.61 -1.82

Nonrelativistic expressions for ¿Bethe, ¿Bloch, and ¿Bohr are given on Fig. 17 in terms of as a function of ln(b) at hu = 100 eV for Zp = 1, 10, and 92. Let us remark that ¿Bloch = ¿Bethe for Zp = 1 and also that LBloch ~ ¿Bohr from

b / 0.5 while LBloch /¿Bohr when b / 0.05.

Despite that ¿Bethe = ¿Bohr at Zp = 10, when b / 0.13, both quantities differ by 5% from ¿Bloch. This highlights a case when a classical result gives back a first-order quantum-mechanical one. Nonetheless both ¿ values turn negative for small b's demonstrating their non-validity for low velocities.

Now it is appropriate to comment on the physics motivating the distinct ¿ behaviors on Fig. 17. Let us first notice that the first Born approximation documents any lowest-order quantum-mechanical one. It is used by Bethe for any kind of collisions while Bloch restricts the use of a first order approximation to distant collisions taken in a dipole approximation. In the non-relativistic case, Bloch considers close interactions with the exact asymptotic's of the Coulomb scattering amplitude altogether with an adequate form of the electron wave packet structure in the c.m. frame allowing a finite and lateral extent of the projectile scattering center in given frame.

Non-relativistic Bloch comes close to non-relativistic Bethe when |Zp|a/b^1(^1) from Born (classical) validity. That

Fig. 17. Nonrelativistic forms of Eqs. (102a), (102b) and (102c). Note that LBethe is independent of Zp and that L^hr = ¿Bloch for Zp = 92, LBethe = ¿Bohr for Zp = 1 and that LBloch / LBohr at small velocities and LBloch / LBethe at large velocities for Zp = 10 (after Ref. [39]). Reproduced from S.P. Ahlen, Rev. Mod. Phys 52, 121 (1980) with the permission of AIP Publishing.

requirement features the valid approach to close-collision slowing down. It is more difficult to pinpoint distant interactions because the dipole approximation interferes with Born's.

Nonetheless, as Bloch and Bohr use same assumptions for distant collisions while Bloch's treatment retrieves Bohr's even at relativistic energies, any discrepancies between the three approaches should imply close and/or intermediate interactions.

A straightforward contention of this view, due to Bohr, runs as follows: we are allowed to introduce an intermediate impact parameter b so that collisions fulfilling b > bj can be seen as electromagnetic excitations of charged harmonic oscillators in a homogeneous electric field arising from the passing projectile, while collisions with b < bj can be seen as free-electron scattering in momentum frame center.

When b < b1, Bohr exact expression is retrieved through bj = Zpe2/mv2g with v, relative projectile-electron velocity and g = (1 — b2) — 1/2 while restricting to "distant-collision" energy loss.

This is achieved by restricting the impact parameter so that the energy transfer is given as 1/b2.

However, when the electron de Broglie wavelength > Zpe2/mvg in c.m. frame, it is expected to replace the classical minimum impact parameter which accounts for the elimination of close collisions energy transfer with electron wave packet smearing.

As a result, when Zpa/b < 1, quantum-mechanical bi's lead to Bethe expression while Zpa/b yields back Bohr one. Those conditions feature accurately the validity of first Born and classical approximations, respectively.

3.3. ¿ow velocity stopping

Here, one focuses attention on the slow velocity regime, where the projectile is still transferring a significant amount of energy to the target inner shells. A typical example is afforded by proton beams with initial energy (at pellet entrance) in the MeV range. Then, the range of present concern lies around E/ A ~ 100 keV, where E denotes the projectile kinetic energy and A, its atomic mass.

Keeping constant effective charge Zp, present considerations are easily adapted to heavy ion driven fusion, because recent numerical simulations show an enhanced energy transfer between projectiles and target.

To investigate stopping at low velocity, we have to extend the conceptual framework beyond the standard expression (37).

More specifically, it remains to pay a due attention to projectile energy range (M = Amp, mp being the proton mass)

e (25 keV)

105 [eV] 1

<-MVp < 5.15—Zt [eV]

2 p me

for which stopping is expected to advocate a frictionlike expression

with a Brownianlike constant a to be fixed in the sequel. Approximation (104) is fulfilled by RPA free electrons in agreement with L ~ aVp ([1]). Then, the stopping process is viewed as a kind of Brownian motion with the projectile, assaulted by many tiny particles (the stopping electrons). Up to now, we advocated a Zp^ZT regime, which is to be contrasted to the case ZpyZT considered in the literature (e.g. for instance the so-called LSS model [45,46] through a quasi-molecular approach.

Eq. (104) may be derived in the impulse approximation (IA) when the Born validity criterion,

Z e2 ZVn

V0 = average bound target electron velocity, is fulfilled (by close collisions, for instance). Then, the stopping contribution of bound subshell I(=n, l) is explained as

4pnT Vdx/ n

dq dfni(q, e') q dE'

in terms of the generalized oscillator strength (GOS) density dl labelled by excitation energies E and the impulse Zq transferred from projectile to target electron. nT.

Within the impulse approximation (IA), df/dE is expressed in terms of the p-space target wave functions cnl(p), retaining their unperturbed labelling (n, l ). After several transformation, Eq. (106) may be given the form (104), with

3Ei \pn

showing that, the best accuracy can be expected for eigen-quantities of the stopping electron. However, the most serious drawback is the insufficiently accurate Ej for the highest subshells which provide the largest stopping. Generally speaking, weakly bound electrons seem more efficient stoppers than more tightly bound ones. The JA results compare pretty well (Fig. 18) with the Andersen-Ziegler compilation [47] for low velocity stopping of protons in cold argon gas.

The extrapolations from Bethe formula stand down by more than a factor of 2.

VD(a.u.)

Fig. 18. Impulse approximation (JA) results compared with experimental values and Bethe formula for stopping by neutral argon. Pergamon refers to Ref. [47].

4. Low velocity ion stopping in binary ionic mixtures (BIM) [48]

A specific interest devoted to low velocity ion slowing down (LVJSD) in dense multicomponent plasmas has recently emerged in conjunction with a few other topics featuring concerns for inertially confined fusion (JCF) operated through super intense and femtosecond laser produced ion beams. As expected, this trend got initiated with the routine production of proton beams with a kinetic energy up to 100 MeV and also highlighting a very small transverse emittance. An intense activity is thus presently deployed to extending the proton performances up to fully ionized carbon ions (C6+) and heavier ions whenever possible. Moreover, the quantitative control of the standard fusion reaction based on the deuteriumtritium (DT) fuel sustained by the resulting a particles (He2+) requires a thorough investigation of an a LVSJD near the end of range.

This explains that we pay a certain attention to proton and a stopping in dense DT-mixtures with a variable relative composition with density range 1023 < ne [cm-3] < 1026 and thermal velocity range 10 < T [eV] < 20 000. An adequate and fine tuning out of the relevant LVJSD is thus expected for the beam-target interaction involved in the fast ignition scenario (FJS).

Jt is also worthwhile to mention the strong relevance of LVJSD to many astrophysics issues, and also in the production through heavy ion beams stopped near Bragg peak in thin targets, of the so-called warm dense matter (WDM) i.e strongly coupled plasmas with a density close to the solid one and a few eV temperature [49].

4.1. Basic formulation

Basically, LVISD features canonically as dE

-dE = AVp,

for a typical energy/nucleon <100 keV/nucleon.

The pertaining dielectronic picture includes partially degenerate electrons and classical ion species (Z,, M,) as well. The ion component highlights a weakly coupled BIM. Then, it appears appropriate to introduce the overall dielectronic function

e(k, u) ^

W(W) + W W

+ W [ VM~2 W

with the usual Fried-Conte dispersion function W (Im ? > 0)

w (Z):

__ lim

V2P v/0+

and X(?) = ReW(?) , Y(?) = ImW(?).

Extending the standard one-component stopping expression

k +1 f „ ,3 f

mY (mVp)

[k2 + x(mVp)] + Y2(MVp) (111)

with Z = Zeff/Nd, where Zeff denotes the projectile effective charge at velocity Vp, ND = nelD in terms of target electron density and corresponding Deb ye length. In the sequel, Vp will be scaled by Vthe = \JkBT/me, thermal electron velocity with T, thermalized target temperature. In Eq. (111) we select kmax by considering quantum diffraction effects when kBT^1Ry so that

kmax= min

mVp2 + Vi) 2m^Vp + V2

vp2+A^^D"

where Vp is now dimensionless on the second line on the right-hand-side, a = 1/137.036 is the fine structure constant, and c, light velocity.

The specification of Eq. (111) to BIM stopping requires the relative ion concentration of species 1, i.e.

N1 + N2

in terms of ion number N, with i that BIM densities

1, 2, in target plasma, so

n1 = ' n2

ne ( 1 - a —

Z = Z1a + Z2(1 - a) (114)

are straightforwardly expressed in terms of electron density ne.

Then, we can estimate the stopping contributions of every target component: electron 0, ion 1 and ion 2 as follows:

mY Ov)

D(k , mv) '

dE1 dx

mY (VM1mv )

D(K, mv) '

(115a)

(115b)

Fig. 19. Proton stopping in H+-D+ BIM in terms of Vp/Vthe-ne = 1023 cm"3, a

denotes the proton concentration in BIM T = ... 10 eV,--100 eV,--1000 eV.

Red pertains to e-stopping, green to H+-stopping and blue to D+-stopping. Reproduced from B. Tashev, et al., Phys. Plasmas 15,102701 (2008) with the permission of AIP Publishing.

n„ a

dE=cj dk-e f dmmx^l, (115c)

dx 2 J J * d(k, mv) ' ^ ;

D(k; mv) = [k2 + X(mv) + x( pMffim v) + x (pMjffi mv)]2

+ [Y (mv) + Y (VM/v) + Y (vMmv)]2, (116)

0 a 2 i4 ,

4p2nelD

C1 = C2 =

CoZ4a CoZ4(1 - a)

altogether with

kmaxo = Min

134p(v2 + 2) kmax1 ne ^D

3 v2 + 2

14.78nelDj

e D\ kKT

kmax1 / Z kmax2 = ~zTl~a

(117a)

(117b) (117c)

Expressions (115) are in kBT/1D, with the Debye screening length lD respectively adapted to electron 0, ion 1 and ion 2. Vp is in Vthe.

Fig. 20. Alpha particle stopping in a Tokamak-like D-T BIM with ne = 10 cm- and T = 10 keV. a denotes D+-concentration. Red refers to estopping, green to D+-stopping and blue to T+-stopping. Reproduced from B. Tashev et al., Phys. Plasmas 15, 102701 (2008) with the permission of AIP Publishing.

Fig. 21. Proton stopping in a DT-mixture. ne = 1025 cm-3, T = ... 0.5 keV,--

1 keV,--2 keV. Red refers to e-stopping, green to D+-stopping and blue to

T+-stopping. Reproduced from B. Tashev et al., Phys. Plasmas 15, 102701 (2008) with the permission of AIP Publishing.

4.2. Main stopping trends

4.2.1. Basic feature

We detail now the LVISD specificities in the DT-BIM in the velocity range Vthi < Vp < Vthe. Stressing the usual target parameters, density, temperature, and the charge and mass (Mj , Zj) of ion species i, we notice that a pertinent bookkeeping of stopping modes has to be performed for a projectile stopping power measured in kBT/1D, in terms of the target equilibrium temperature T and screening Debye length 1D, respectively, featuring the electron contribution (e-stopping) and the ion contribution i = 1, 2. Moreover, we shall frequently reduce the projectile effective charge Zeff to that of the proton, because the kmax [Eq. (117)] Zeff dependence remains very weak, so the usual quadratic dependence Ze2ff of the stopping patterns shows up as fulfilled in the whole space parameter of present interest.

We first focus on deuterium-tritium (DT) BIM of thermonuclear fusion interest, with a strong emphasis on superdense DT target considered for the so-called fast ignition scenario (FIS) in inertial fusion energy (IFE).

Stressing the stopping of the thermonuclear produced a particles, we remark that as soon as the target temperature T lies above 1 keV, the corresponding a particles velocity Vp should remain below the target thermal electron velocity Vthe. So the crucial self-sustained target ignition through He2+ stopping in DT-targets is strongly dependent on LVISD mechanisms considered here.

On the other hand, proton stopping remains a major issue for assessing the feasibility of FIS/IFE in view of the recently advocated potential of very intense proton beams produced in the irradiation of thin solid foils by PW-lasers. The given

proton beams show a very low transverse emittance, which are easier to handle than the relativistic electron beams (REB) initially selected out.

The very fine tuning of the tiny hot spots produced in the bombardment of DT targets, demands that the proton end of range be carefully monitored through LVISD.

Denoting a as the relative proportion of ion species A in the so-called AB-BIM and expliciting in Fig. 19 the a-dependence of the stopping profiles (SP) in terms of dimensionless projectile velocity 0^ Vp/Vthe^1, for proton projectiles losing their energy in a H+—D+ BIM at solid density ne = 1023 cm-3 and T = 10—1000 eV, BIM stopping appears essentially restricted to the narrow range 0^Vp/Vthe^0.2.

Blue qualifies projectile stopping through the D+-ionic component, green pertains to the target H+-component, and red highlights the usual e-stopping treated classically in view of T^10eV > TF = 7.85 eV. In this Vp range, e-stopping remains secondary to i-stopping.

Lower T-values would emphasize Z S 0 effects due to electron partial degeneracy through T-dependent electron-ion and electron-electron effective interactions.

The code of colors runs as follows: Black for blue lines, dark gray for red lines, and light gray for green lines. The most conspicuous feature highlighted by the a-dependence of Fig. 19 is obviously the quasi symmetry a^1 — a due to Z: = Z2 and M2/Mj = 1.5.

Such a behavior is also fulfilled by a D-T BIM exposed to a He2+ beam at a much lower density ne = 1014 cm-3, qualifying an MFE-like plasma at T = 10 keV. The given quasi symmetry is also updated in both situations by the nearly equal ion stopping contributions at a = 0.5.

Fig. 22. Alpha particle projectile in a Deuterium-Tritium BIM with ne = 1023 cm"3 and different temperatures with D+ relative proportion. (a) a = 0.001, (b) a = 0.1, (c) a = 0.5 and (d) a = 0.9. Red refers to electron stopping, green to D+-stopping while blue denotes T+-stopping; Dots refer to T = 10 eV, other lines with the same color denote T = 100 eV, 2 keV and 5 keV from top to bottom. Reproduced from B. Tashev, et al., Phys. Plasmas 15, 102701 (2008) with the permission of AIP Publishing.

Let us remark that the sum of the highest ordinates of the ion SP keeps a value constant when a varies, at fixed temperature. All those trends are shared by proton stopping in a superdense mixture (Fig. 22 of FJS concern with ne = 1025 cm-3 and T = 0.5 — 2 keV. Jt is also a remarkable fact that target ion SP peaks always appear at very low Vp/ Vthe < 0.05, in the three cases (Figs. 19 and 22) herein considered.

4.2.2. T-dependence

FJS ignition performances strongly rely on the the compressed DT-fuel initial temperature. Jn this regard we show on Figs. 23 and 24 the stopping profiles (SP) on a logarithmic scale for alpha particle slowing down at 1023 cm-3 and 1026 cm-3, respectively.

With this change of scale, the e-stopping does not appear negligible and fine structures at the intersection of e-stopping and i-stopping are clearly seen, a denotes deuterium relative concentration and the quasi-symmetric features of target ions SP look even enhanced contrasted to their linear counterparts.

4.2.3. Critical velocity Vp,crit

Above displayed stopping data make it clear that target estopping contribution matches the sum of remaining i-pro-jectile stopping contributions for a critical projectile velocity Vp,crit such that Vp crjt/Vthe ~ 0.15 — 0.2, when the given BJM are not too asymmetric in charge and mass.

This process is illustrated in Fig. 24 through proton stopping down in a H+ — He2+ BJM with 1012^ne [cm-3]^1018 and 0^T [eV]^103. Fig. 24(a) highlights a rather flat three-dimensional snapshot for a a = 0.5 BJM vindicating 0.1^Vp ; crit/Vthe^0.13. However, separate variations in T [Fig. 24(b)] and density [Fig. 24(c)] document a strong a-dependence for Vp,crit. Jt is a remarkable fact that these Vp,crit estimates fall in quite a close agreement with those derived from an independent Fokker-Planck analysis, relying on the ratio of projectile kinetic energy to that of target electrons.

Thus we get

2 P p.crit

1m v2 :

2 e * the

C2/3 :im V2

2 e ' the

3p1=VMr

in terms of the BJM quantities Z = aZ\ + (1 - a)Z2 and M = aMj + (1 - a)M2. Jn the present case, the above ratio becomes

Fig. 23. He2+ projectile in a deuterium-tritium mixture where ne = 1026 cm-3 and T = . _ 1 keV,--2 keV,--5 keV with a deuterium relative concentration. Color attribution are the same as in Fig. 22. Reproduced from B. Tashev et al., Phys. Plasmas 15, 102701 (2008) with the permission of AIP Publishing.

use in Tokamak physics can be retrieved for alpha particle stopping, as well.

4.2.4. Straggling

Within the rather high temperature range of present interest, one can use the high-T straggling approximation,

Q2 = 2kB TS

V^s0.12 Vthe

in rather good agreement to that displayed on the Fig. 24(a) vertical scale. A similar quantitative matching between the present dielectric approach and the collisional one of current

where S is the total stopping including electron and ion contributions. Fig. 25 illustrates the case of proton projectiles stopped in a DT-equimixture (a = 0.5) with ne = 1018 cm-3 and T = 500 eV. It unambiguously contrasts the usual bell-shaped curve featuring e-stopping at any Vp, to the highly peaked BIM contribution restricted to Vp/Vthe < 0.2.

16 14 12 10

0.1600L lgn=18

J 0.1400 "

I ----«=0.1

* 0.1200^^ [ "

_ A_L_I_I_I_I_L

v 200 400 600 800 1000 Te (eV)

12 14 16 18 Ig n

Fig. 24. Critical proton velocity Vp,crit in a = 0.5 H+—He2+ BIM. (a) 3D view in terms of ne and T, (b) cross-section at ne = 1018 cm-3 for a = 0 and 0.5, (c) cross-section at T = 200 eV for a = 0.1 and 0.5. Reproduced from B. Tashev et al., Phys. Plasmas 15, 102701 (2008) with the permission of AIP Publishing.

5. Conclusion and outlook

The present review of ion stopping in dense plasmas appears mostly geared toward theoretical and simulation features. However, corresponding experimental aspects already

__ 3.0

VpIVlhe

Fig. 25. High temperature straggling for proton projectiles in a DT-BIM with a = 0.5, ne = 1018 cm-3 and T = 500 eV. Reproduced from B. Tashev et al., Phys. Plasmas 15, 102701 (2008) with the permission of AIP Publishing.

previously referred to [2] demand also a timely updating motivated by very recent and significantly promising developments.

In this regard, the potentialities afforded by intense ion beams to emulate high energy density physics have just been presented by Sharkov et al. [54] which details, for instance, the specific ion-plasma target designs such as HIHEX and PRIOR envisioned for isochoric plasma heating and proton radiography.

Pertaining facilities include FAIR at GSI — Darmstadt (presently in construction) as well the HIAF combination in China headed by IMP-Lanzhou.

Typically, these setups would secure intense and highly charged (Z ~ 33) uranium beams with up to 4.4 x 109 particles/bunch in 100—125 ns.

Then, the given and possible beam—target configurations appear very well suited to document heating of warm dense matter (WDM) plasmas with T ~ 1 — 10 eV in a volume ~ mm3 much larger than the biggest ones affordable on the most performing laser beam lines (LCLS at SLAC for instance) while featuring only small density gradients at plasma edges.

Specific energy deposit is thus expected up to 100 MJ/g enabling the investigation of hot plasma targets with T up to 1 keV.

Other significant highlights concern the very first and accurate determination of light ion (proton and a particle) stopping in the vicinity of the Bragg peak (Vp ~ Vthe) by Frenje et al. [55] using SiO2 thin capsules filled with D-3HeHe gas in same proportion. Strength and location of the Bragg peak are demonstrated strongly dependent on density (~1023 cm-3) and temperature (T ~ 0.4 — 0.5 keV) variations in plasmas of current inertial fusion interest.

The same MIT-LLE team headed by Petrasso [56] in collaboration with others also carefully investigated proton stopping in WDM targets with T ~ 32 eV at moderate Coulomb coupling (0.3) and moderate electron degeneracy (T/TF ~ 2).

In this regard, it is worthwhile to notice that such investigations open the door to a systematic inquiry of nonrel-ativistic ion stopping in arbitrary degenerate electron targets [1,57]. Moreover, an intriguing and promising possibility for measuring stopping and straggling of very light ions such as protons and a particles got very recently implemented [58] through the simultaneous use of the same high power laser to ignite a target plasma in a gas jet and also to produce the interacting ion beam out of the usual TNSA (target normal sheath acceleration) mechanism.

For that purpose, two arms on the setups ELFIE (Ecole Polytechnique) and TITAN (Lawrence Livermore) have been respectively selected to produce a short TNSA pulse (10 J, 400 fs) and a long pulse (60 J, 500 ps). The latter one being devided in two-halves enabling target firing and also a probing beam (~30 mJ, 400 fs).

This novel metrological approach could allow to probe very conveniently different target plasmas (H, He, N, Ar, Xe ...) at high temperature (100 < Te [eV] < 1000) with a target density ne ~ 1020 cm-3 while using a well energy—calibrated impacting ion beam through a carefully designed velocity

selector out of the initially and laser produced broadband beam. Then, the corresponding ion-plasma interactions stand at variance to the former accelerator-based ones [2,3,50-53] restricted to moderate temperatures (1 < Te [eV] < 10) [2,3].

References

[1] C. Deutsch, G. Maynard, Stopping of pointlike charges in a dense electron fluid, Recent Res. Devel. Palsmas 1 (2000) 1—23.

[2] C. Deutsch, G. Maynard, M. Chabot, D. Gardes, S. Della-Negra, et al., ion Stopping in dense plasma target for high energy density physics, Open Plasma Phys. J. 3 (2010) 88—115.

[3] R.O. Bangerter, Targets for heavy ion fusion, Fus. Technol. 13 (1988) 348—355;

Ion beam interaction with ICF targets, Las. Int. Rel. Plasma Phen. 6 (1984) 1013—1027.

[4] H.D. Betz, Ion projectile effective charge in, Appl. At. Coll. Phys. 4 (1984) 1—42.

[5] D.S. Bailey, Y.T. Lee, R.M. More, Ab initio calculations of the charge state of a fast heavy ion stopping in a finite temperature target, J. Phys. (Paris) 44 (C8) (1983) 149—158.

[6] E. Nardi, Z. Zinamon, Charge state and slowing of fast in plasmas, Phys. Rev. Lett. 49 (1982) 1251—1254;

G. Maynard, C. Deutsch, Interaction of heavy ions with gas and plasma targets, J. Phys. (Paris) 49 (C7) (1988) 89—97.

[7] G.I. Bell, The capture and loss of electrons by fission fragments, Phys. Rev. 90 (1953) 548—557.

[8] Th. Peter, J. Meyer-Ter-Vehn, Energy loss of heavy ions in dense plasma. II Nonequilibrium charge states and stopping powers, Phys. Rev. A 43 (1991) 2015—2030.

[9] M. Gryzinski, Two-particle collisions. I. General relations for collisions in the laboratory system, Phys. Rev. 138, A (1965) 305—321;

M. Gryzinski, Two-particle collisions. II. Coulomb collisions in the laboratory system of coordinates, Phys. Rev. 138, A (1965) 322—325; J.H. McGuire, P. Richard, Procedure for computing cross sections for single and multiple ionization of atoms in the binary-encounter approximation by the impact of heavy particles, Phys. Rev. A8 (1973) 1374—1384.

[10] W. Lotz, An empirical formula for the electron-impact ionization cross-section, Zeitschrift für Physik A Hadrons and Nuclei 206 (1967) 205—211.

[11] Ya. B. Zel'Dovich, Yu.P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena vol. 1, Academic Press, New-York, 1966, p. 406.

[12] D.H. Menzel, The physical processes in gaseous nebulae 1. Absorption and emission radiation, Astrophys. J. 85 (1937) 330—339;

L. Spitzer, The temperature of interstellar matter 1., Astrophys. J. 107 (1948) 6—33.

[13] J.R. Oppenheimer, On the quantum theory of the capture of electrons, Phys. Rev. 31 (1928) 349—356;

H.C. Brinkman, H.A. Kramers, The theory of the collection of electrons by means of a-particles, Proc. Ned. Akad. Wet. 33 (1930) 973—984.

[14] G. Maynard, W. Andre, M. Chabot, C. Deutsch, C. Fleurier, et al., Charge evolution of swift heavy ions in fusion plasmas, Nuo. Cim. 106A (1993) 1825—1993.

[15] Th. Peter, R. Arnold, J. Meyer-Ter-Vehn, Influence of dielectronic recombination on fast ion charge states in a plasma, Phys. Rev. Lett. 57 (1986) 1859—1862.

[16] V.L. Jacobs, J. Davis, P.C. Kepple, P. Blaha, The influence of auto-ionization accompanied by excitation on dielectronic recombination and ionization equilibrium, Astrophys. J. 211 (1977) 605—616.

[17] G. Maynard, C. Deutsch, Dynamical effect and dielectronic recombination of fast heavy ions in a plasma, Phys. Scr. 48 (1993) 471—473.

[18] G. Maynard, C. Deutsch, P. Fromy, K. Katsonis, Atomic physics for inertial fusion using average correlated hydrogenic atom model, Laser Part. Beams 13 (1995) 271—279.

[19] H.D. Betz, L. Grodzins, Charge states and excitation of fast heavy ions passing trhough solids : A new model for the density effect, Phys. Rev. Lett. 25 (1970) 211—214.

[20] C. Couillaud, R. Deicas, Ph. Nardin, M.A. Beuve, J.M. Guihaume, et al., Ionization and stopping of heavy ions in dense laser-ablated plasmas, Phys. Rev. E49 (1994) 1545—1562.

[21] M.H. Chen, Relativistic intermediate-coupling calculations of dielectronic-recombination coefficients for the Li isoelectronic sequence, Phys. Rev. A44 (1991) 4215—4223.

[22] Th Peter, R. Arnold, J. Meyer-Ter-Vehn, Private Communication, 1985.

[23] C.L. Pekeris, 11S and 23S states of helium, Phys. Rev. 115 (1959) 1216—1221.

[24] A. Dalgarno, The stopping powers of atoms,, Proc. Phys. Soc. 76 (1960) 422.

[25] U. Fano, J.W. Cooper, Spectral distribution of atomic oscillator strengths, Rev. Mod. 40 (1968) 441—507;

I. Shimamura, T. Watanabe, Bounds on mean excitation energies in terms of oscillator-strength moments, Phys. Rev. A 23 (1981) 2914—2915.

[26] I. Shimamura, T. Watanabe, Bounds on mean excitation Energies, lamb-shift, stopping power, straggling and grazzing collision of high charged particles, J. Phys. Soc. Jap. 34 (1973) 483.

[27] J.L. Dehmer, M. Inokuti, R.P. Saxon, Systematics of moments of dipole oscillator-strength distributions for atoms of the first and second rows, Phys. Rev. A12 (1975) 102—121.

[28] A.E.S. Green, D.L. Sellin, A.S. Zachor, Analytic independent-particle model for atoms, Phys. Rev. 184 (1969) 1—9;

R. Garvey, C.H. Jackman, A.E.S. Green, Independent-particle model potentials for atoms and ions with 36<Z<54 and a modified Thomas-Fermi atomic energy formula, Phys. Rev. A12 (1975) 1144—1152.

[29] X. Garbet, C. Deutsch, G. Maynard, Mean excitation energies for ions in gases and plasmas, J. Appl. Phys. 61 (1987) 907—916.

[30] J.H. Peek, L.C. Pitchford, E.J. Shipsey, Methods for predicting stopping and straggling mean excitation energies, Phys. Rev. A29 (1984) 1096—1102.

[31] J. Bell, D.R.B. Bish, P.E. Gill, Separate subshell contribution to the stopping power of rare gases, J. Phys. B5 (1972) 476—482.

[32] B.F. Rozsnayi, Relativistic Hartree-Fock-Slater calculations for arbitrary temperature and matter density, Phys. Rev. A5 (1972) 1137—1149.

[33] D.A. Liberman, D.T. Cromes, J.T. Waber, Average atom models in dense plasmas, Compt. Phys. Comm. 2 (1971) 107.

[34] J. Lindhard, M. Scharff, Energy loss in matter by fast particles of low charge, Dan. Vidensk. Selsk. Math. Fys. Medd. 27 (15) (1953) 1—31.

[35] R.E. Johnson, M. Inokuti, The local plasma approximation to the oscillation strength spectrum: How good is it and why? Comm. At. Mol. Phys. 14 (1983) 19—31.

[36] T.A. Mehlhorn, A finite material temperature model for ion energy deposition in ion-driven inertial confinement fusion targets, J. Appl. Phys. 52 (1981) 65226532.

[37] C. Deutsch, G. Maynard, H. Minoo, Ion stopping in dense plasmas, Las. Int. Rel. Plasma P. H. 6 (1984) 1029—1048;

C. Deutsch, Inertial confinement fusion driven by intense ion Beams, Ann. Phys. Fr. 11 (1986) 1 — 111.

[38] F.I. McGuire, J.M. Peek, C.L. Pitchford, Proton stopping power of aluminium ions, Phys. Rev. A26 (1982) 1318—1325.

[39] S.P. Ahlen, Theoretical and experimental aspects of the energy loss of relativistic heavily ionizing particles, Rev. Mod. Phys. 52 (1980) 121 —173.

[40] N. Bohr, The penetration of atomic particles through matter, Kgl. Dan. Vidensk. Selsk. Mat. Phys. Medd. 18 (1948) N8.

[41] G. Maynard, C. Deutsch, The Barkas effect or Zp-contributions to stopping of swift charged particles, J. Phys. (Paris) 43 L (1982) 223—227.

[42] J.D. Jackson, R.L. McCarthy, Zp connection to energy loss and range, Phys. Rev. B6 (1972) 4131—4141;

J.C. Ashley, R.H. Ritchie, W. Brandt, Zp effect is stopping power of matter for charged particles, Phys. Rev. B5 (1972), 2393—2347; K.W. Hill, E. Merzbacher, Zp effect in stopping power of matter for charged particles, Phys. Rev. A9 (1974) 156—165.

[43] C. Deutsch, S. Klarsfeld, Quadrupole electron broadening of neutral lines in a plasma, Phys. Rev. A7 (1973) 2081—2086.

[44] J. Lindhard, The Barkas effect or ZpZP corrections to stopping of swift charged, Nucl. Instr. Meth. 132 (1976) 1—5.

[45] J. Lindhard, M. Scharff, H.E. Schioett, Range concepts and heavy ion ranges. Notes on atomic collisions II, Kgl. Dan. Vidensk. Selbsh. Mat. Fys. Medd. 33 (39) (1963) 1—42.

[46] X. Garbet, Charged Particle Stopping by Bound Electrons in a Plasma, Report L.P. N vol. 209, Universite Paris XI, Orsay, 1984.

[47] H.H. Andersen, J.F. Ziegler, Stopping and Ranges of Ions in Matter, Pergamon, New-York, 1977.

[48] B. Tashev, F. Baimbetov, C. Deutsch, P. Fromy, Low velocity ion stopping in binary mixtures, Phys. Plasmas 15 (102701) (2008) 1—10.

[49] See for instance A. Ortner, A. Franck, A. Blazevic, M. Roth, Role of charge transfer in heavy-ion-beam-plasma interactions at intermediate energies, Phys. Rev. E91 (23104) (2015) 1—7.

[50] C. Deutsch, R. Popoff, Low ion-velocity slowing down in a strongly magnatized plasma target, Phys. Rev. E78 (56405) (2008) 1—8.

[51] C. Deutsch, N.A. Tahir, Fragmentation and stopping of heavy cluster ions in a lithium target-Application to target implosion, Phys. Fluids B4 (1992) 3735—3746;

N.A. Tahir, H.J. Lutz, O. Geb, J.A. Maruhn, C. Deutsch, et al., Inertial

confinement fusion using hohlramm radiation generated by heavy ion cluster, Phys. Plasmas 4 (3) (1997) 796—816.

[52] Z. Wang, Bin He, Z.G. Fu, P. Zhang, Energy relaxation of multi-MeV protons travelling in compressed DT + Be plasmas, Phys. Plasmas 21 (72703) (2014) 1—4.

[53] H.B. Nersisyan, C. Deutsch, Stopping of ions in a plasma irradiated by an intense laser field, Laser Part. Beams 29 (2011) 389—397.

[54] B. Yu Sharkov, D.H.H. Hoffmann, A.A. Golubev, Y.T. Zhao, High energy density physics with intense ion beams, Matter Radiat. Extrem. 1 (2016) 28—47.

[55] J.A. Frenje, P.E. Grabowski, C.K. Li, F.H. Seguin, A.B. Zylstra, et al., Measurements of ion stopping around the Bragg peak in high-energy-density plasmas, Phys. Rev. Lett 115 (2015) 205001—205005.

[56] A.B. Zylstra, J.A. Frenje, P.E. Grabowski, C.K. Li, G.W. Collins, et al., Measurement of charged-particle stopping in warm dense plasma, Phys. Rev. Lett 114 (2015) 215002—215006.

[57] G. Maynard, C. Deutsch, Energy loss and straggling at any velocity of swift ions in dense matter, Phys. Rev. A26 (1982) 665—668;

Born RPA approximation for ion stopping in arbitrary degenerate electron fluid, J. de Physique 46 (1985) 1113—1123.

[58] S.N. Chen, C. Deutsch, S. Atzeni, M. Gauthier, J. Fuchs, EPS — 2015 Lisboa and to be published.