J. Plasma Fusion Res. SERIES, Vol.7 (2006) 150-153

Role of Excited States of Li Ions in the Stopping
Power of Molecular Hydrogen
MORIBAYASHI Kengo
Japan Atomic Energy Research Institute∗ , Kyoto 619-0215, Japan（Received: 5 October 2004 / Accepted: 21 Febrary 2006）

Abstract
We study the stopping power of H2 for Li ions and atoms. We consider the charges and electronic statesof Li for various densities of H2 . It is found that from the excited states of ions, the decay processes ofradiative transition and electron loss of the projectile play an important role in the low density and highdensity, respectively. This leads to the fact that the stopping power depends on the density.
Keywords:
heavy particle, density eﬀect, stopping power, charge number
1．Introduction
For stopping powers for ions, the charge numberof the ions plays an important role [1]. When ionspass through matter, the charge number changes as afunction of times. Therefore, the population of charge
numbers gives a significant contribution to the stopping
power.
When we treat protons as a projectile, the charge
numbers are determined by processes of charge transfer (Az+ + B → A(z−1)+ + B+ ), and electron loss ofthe projectile (Az+ + B → A(z+1)+ + B + e− ), whereAz+ ，B and e− are a heavy particle, a target molecule,and an electron, respectively, and z is the charge number of the particle [2]. Suppose that the cross sections of the electron loss of the projectile and chargetransfer are σELP and σCT , respectively, then, the ratesof the electron loss of the projectile and charge transfer become nt σELP vi and nt σCT vi , respectively, wherent and vi are density of the target and the velocity ofthe projectile, respectively. Then, by using a scaling
parameter r (= nt vi t), the rate equations [3,4], whichsolve the populations of protons (P p ) and hydrogenatoms (Ph ), become dP p /dr = σELP Ph − σCT P p anddPh /dr = −σELP Ph +σCT P p , where t is the time. Therefore, the populations become independent of nt and canbe expressed by analytical functions of σCT and σELP[2]. The stopping power, which depends on the charge
numbers [1], can also be given as a simple function ofnt . This comes from the fact that the charge transfer processes produce the ground state of hydrogen the mostoften.
For heavy particles, except for protons, charge trans-fer processes have often produced excited states of ionsor atoms. Then, the excited states also revert to stableones through not only electron loss of the projectile but
also radiative transition and auto-ionization processes.
After radiative transition processes occur, the ionizationenergy of the ions becomes larger, as a result, σ ELP decreases (see Fig. 1, the details will be discussed later).
Namely, after radiative transition processes, the electron loss of the projectile processes seldom occur, thatis, radiative transition processes play the role of retaining the charge number. Radiative transition probabili-Fig. 1 Measured electron loss of the projectile cross sections [8, 9] scaled by I 1.5/ Ne vs. ion energy forA
various projectile and a target ( H2 ), where IA andNe are the ionization energy and total number ofelectrons of the projectile.
C 2006 by The Japan Society of Plasma

Corresponding author’s e-mail: moribayashi.kengo@jaea.go.jp∗ Present: Japan Atomic Energy Agency

Science and Nuclear Fusion Research150

Role of Excited States of Li Ions in the Stopping Power of Molecular Hydrogen

He + H2 [9], He+ + H2 [9], Li+ + H2 [8], Li2+ + H2[8]. Namely, the cross section of the electron loss ofthe projection (σELP ) ∝ Ne /IA1.5 . Further, for ion energy (Ei ) lower than 300 keV/u, σELP ∝ Ei −0.6 . Figure 1 shows the cross sections scaled by using thesescaling from the experimental data [8,9]. For ion energies higher than 100 keV/u, the errors of the scalingare lower than 50 %. On the other hand, for energies
lower than 100 keV/u, the cross sections of the projectile of ions show a trend diﬀerent from those of neutral
atoms. Therefore, we employ the cross sections of Li
atoms and Li ions scaled from those of He atom andLi+(or2+) ions, respectively. For the radiative transition
probabilities and auto-ionization rates, we use atomic
data calculated by the Cowan code [10].
Suppose that ionization energies of the projectile of(z−1)+
and the target of B are IA and IB , respectively,Athen, energy loss by charge transfer processes is givenby1(1)∆E = me v2i + IB − IA2
where vi and me are the velocity of the projectile andthe mass of the electron, respectively. We assume thatthe charge transfer process occurs only where ∆E shows
the minimum value. This comes from the following
points. Janev et al. mentioned that the maximum ofσCT appears at vi ∼ (IB − IA )1/2 for high energy ions(vi ∼ 3(a.u.)) [11]. On the other hand, Ryufuku etal. assumed for low energy ions (vi  1(a.u.)), wherethe term (1/2me v2i ) is ignored, that charge transfer processes often take place at (I B − IA + (z − 1)/R) ∼ 0and σCT ∝ R2 , where R is the nuclear distance between
Az+ and B [12]. Namely, σCT increases as the value of(IB − IA ) becomes smaller. They also showed that σCTgiven by their assumption agrees well with some experimental results [12].

ties are independent of nt , on the other hand, the electron loss of the projectile processes, which increase thecharge number, are in proportion to nt as mentioned before. Therefore, it becomes more diﬃcult to solve therate equations. Resulting, the target density dependenceof the stopping powers for the heavy particles becomes
more complex.
In this paper, in order to study the eﬀect of the excited states of ions on stopping powers, we treat thepassing of Li ions through various densities of H2 . We
consider the ground state and the excited states (principal quantum number n = 1 − 3) of the Li atom andLi(1−3)+ ions. The rates of radiative transition processes
10from n = 2 to1 ( A2−1r ) are about 10 /s. On the otherhand, σELP of Li2+ ( n = 1) and vi at Ei of ∼ 100 keV
are about 10−17 cm2 and 108 cm/s, respectively. Since
the rate of electron loss is r ELP = nt σELP vi ∼ 10−9 nt ,it is comparable to the rate of the radiative transition at nt ∼ 1019 cm−3 . The densities we employ are rELP ), 1019 cm−3 ( A2−1
∼ rELP ), and1015 cm−3 ( A2−1rr
22−32−1
10 cm ( Ar  rELP ).
The eﬀect of target densities on charge transfer processes was already studied by Shevelko et al. [5]. They
solved cross sections of the charge transfer of O7+ +He as a function of densities. They considered excited states of O6+ produced through charge transfer
processes and employed the processes of charge transfer and electron loss of the projectile as one successive
process. On the other hand, we treat the two processes
separately and solve rate equations [3,4], which estimate the populations of the electronic states of the Liions or atom as a function of times. Further, we calculate the stopping power by using the populations.
2．Method of calculationsWe assume Li3+ ions with the energy of 1 MeV/u as
an initial projectile. Then, we calculate ion energies byusing stopping powers and the populations of the electronic states of the Li ions or atoms from the rate equations as a function of times. The stopping powers arecalculated by the same method given in Ref. [2].
Here the atomic data treated here and the production of the excited states of the Li ions and atom areshown. We have employed the measured cross sections of impact ionization [6], charge transfer [7], andelectron loss of the projectile [8], respectively. No paper giving cross sections of Li + H2 and of the excitedstates of Li ions has been found. Suppose that the total number of electrons and ionization energy are Neand IA , respectively, then, we derive the scaling forthe electron loss of the projectile from the measured
cross sections of H (1s) + H2 [9], H (2s) + H2 [9],3．Results and discussions
Figures 2 (a)–(c) show the stopping power as a function of ion energies for nt of 1015 , 1019 , and 1022 cm−3 ,respectively. It should be noted that the atomic andmolecular data for Li atom treated here may not bevalid at the density of 1022 cm−3 because the boundelectrons of the Li atom overlap with those of hydrogen molecules. However, we judge that we can satisfactorily discuss the density eﬀect. The reason will
be discussed later. The contribution of processes of the
impact ionization, charge transfer, and electron loss ofthe projectile to the stopping power is also shown. The
charge transfer and electron loss of the projectile play
an important role at the low energy region (less than afew 100 keV/u). Among Figs. 2 (a)–(c), we see five151

Role of Excited States of Li Ions in the Stopping Power of Molecular Hydrogenpeaks, which we call them Peak I to V (shown in Figs.
2), respectively. Peak IV comes from the relationshipbetween ionization and charge transfer. Namely, thestopping powers of charge transfer and impact ionization increase and decrease, respectively. Peak V corresponds to the maximum value of charge transfer process. The heights of Peak II and III depend on the density, where the peaks become higher as the density increases.
Firstly, we will show that the production of Peak II
and III is caused from the peaks shown in Fig.3. Figure 3 shows the averaged ion charge number (Zav ) asa function of ion energies for nt of 1015 , 1019 , and1022 cm−3 . In this figure, we mark the corresponding
energy to Peak II and III whose average charge numbers are also on the top of the peaks. As seen in Ref.
[6], the impact ionization cross sections increase as thecharge number becomes larger. The increase of Zav corresponds to the increase of impact ionization cross sections, furthermore, to the increase of the stopping powers. Namely, the peaks in Fig. 3 bring about Peak IIand III. As mentioned before, the atomic and molecular data for Li atoms may not be valid for the densityof 1022 cm−3 . However, the Peaks II and III appear atZav ∼ 1.5 and Zav ∼ 1, respectively. Namely, the atomicand molecular data for Li atoms have a little eﬀect on
the production of these peaks. This is the reason why it
is judged that we could satisfactory discuss the densityeﬀect here.

Fig. 2 Stopping powers vs. ion energies for target densities of (a) 1015 cm–3 , (b) 1019 cm–3 , and (c)1022 cm–3 . The contribution of processes of the
impact ionization, charge transfer, and electron
loss of the projectile to the stopping power is alsoshown.

Fig. 3 Averaged charge number vs. ion energies fortarget densities of 1015 cm–3 , 1019 cm–3 , and1022 cm–3 .

152

Role of Excited States of Li Ions in the Stopping Power of Molecular Hydrogen

Next we show the reason why the peaks exist in Fig.
3 only at high densities. In the energy region higher thanthat Peak II, the charge transfer processes produce theground state of the projectile ion. However, near Peak IIand III, the excitation states of n = 2 and 3, are formed,
respectively because the energy of the term of 1/2me vionin eq. (1) becomes comparable to the bound energies ofn = 2 and 3. As mentioned before, the rates of theelectron loss of the projectile for n = 2 and 3 becomemuch larger than those for n = 1. At high density, theprocesses of the electron loss of the projectile occur justafter charge transfer processes. Namely, it looks as ifcharge transfer processes seldom occurred and for theexisting electrons in the projectile ions, processes of theelectron loss of the projectile continue to occur. This
gives rise to the increase of the charge number and thepeaks in Fig. 3. On the other hand, in the lower density
region, the decay of the excited state to the ground stateis faster than the processes of the electron loss of theprojectile. Namely, only processes of the electron lossof the projectile from n = 1 appear. As a result, nopeaks are seen in Fig. 3 for low density.
In this paper, we show the reason why the density
eﬀect appears. The eﬀect comes from the fact that thecharge number of the projectile ions becomes larger as
the density increases. This result is consistent with thatgiven in Ref. [5], where the charge transfer cross sections becomes smaller as the target density increases.
Our model given may not be suﬃcient in the intermediate ion energy region. However, in this region, we showthat the electronic states must be treated in the calculation of the stopping power because some excited statesof ions are definitely produced.

tion cross sections increase according to charge number. For projectile ions with larger atomic number, theion energy, which produces the excited states of ions,increases because the ionization energy of the groundstate becomes larger. Namely, the electronic states may
be more important even in the higher energy region.
We expect that the density eﬀect and the peaks such
as Peak I-V in Fig. 2 (c) shown here can be measured.
This measurement may give us some new informationsuch as the density of targets and ion energy there. The
information of the density may be also useful for plasma
diagnosis.
Acknowledgements
We wish to thank Profs. M. Inokuchi, T. Kato, andT. Tajima, Drs. J. Koga, M. Yamagiwa, A. Nagashima,and T. Kimura for their useful discussions.
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4．Conclusion
We study the eﬀect of the density of the target for thestopping power of H2 for Li ions. We have found thatthe stopping power increases as the density becomeslarger when charge transfer processes produce theexcited states of the projectile ions. The electron loss ofthe projectile ions and the decay processes of radiativetransition occur more often in the higher and lower
energy region, respectively. As a result, in the higher
density, the charge number becomes larger. This gives
rise to larger stopping power because impact ioniza-153