electronic stopping in astrophysical fusion reactions

b

with thections. Itexcitation

stigation

Physics Letters B 585 (2004) 35–41

www.elsevier.com/locate/physlet

Electronic stopping in astrophysical fusion reactions

C.A. Bertulani

National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA

Received 5 January 2004; accepted 29 January 2004

Editor: W. Haxton

Abstract

The stopping power of protons and deuterons in low energy collisions with helium gas targets is investigatednumerical solution of the time-dependent Schrödinger coupled-channels equations using molecular orbital wavefunis shown that at low projectile energies the energy loss is mainly due to nuclear stopping, charge exchange, and theof low energy levels in the target. The second and third mechanisms, called electronic stopping, dominate forElab � 200 eV.At lower energies it is also shown that a threshold effect is responsible for a quick drop of the energy loss. This invesheds more light on the long-standing electron screening problem in fusion reactions of astrophysical interest. 2004 Published by Elsevier B.V.

PACS: 26.20.+f; 34.50.Bw

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Nuclear fusion reactions proceed in stars attremely low energies, e.g., of the order of 10 keVour sun [1,2]. At such low energies it is extremely dficult to measure the cross sections for charged pcles at laboratory conditions due to the large Coulobarrier. Moreover, laboratory measurements of lowergy fusion reactions are strongly influenced bypresence of the atomic electrons. One has obseexperimentally a large discrepancy between theperimental data and the best models available to tthe screening effect by the electrons in the targetclei [3]. The screening effect arises because as thejectile nucleus penetrates the electronic cloud oftarget the electrons become more bound and thejectile energy increases by energy conservation. Sfusion cross sections increase strongly with the pro

E-mail address: [email protected] (C.A. Bertulani).

0370-2693/$ – see front matter 2004 Published by Elsevier B.V.doi:10.1016/j.physletb.2004.01.082

tile’s energy, this tiny amount of energy gain (of ordof 10–100 eV) leads to a large effect on the measucross sections. However, in order to explain the expimental data it is necessary an extra-amount of enabout twice the expected theoretical value [3].

In order to extract the fusion cross sections frexperiment one needs to correct for the energyin the target to assign the correct projectile enevalue for the reaction. The authors in Refs. [4,5] hashown that a possible solution to the long-standingcrepancy between theory and experiment for the retion 3He(d, p)4He could be obtained if the projectienergy loss by electronic excitations and chargechange with the target atoms would be smaller tpreviously assumed in the experimental data analyThere have been indeed a few experiments in whevidences of smaller than expected electronic stoppower were reported (see, e.g., Ref. [6]). Other retions of astrophysical interest (e.g., those listed in

http://www.elsevier.com/locate/physletb

36 C.A. Bertulani / Physics Letters B 585 (2004) 35–41

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Rolfs and collaborators [7,8]) should also be correcfor this effect. Whereas at higher energies the stoppis mainly due to the ionization of the target electroat the astrophysical energies it is mainly due to extations of the lowest levels, charge-exchange betwthe target and the projectile, and the nuclear stopppower.

In this Letter I address the problem of the stoppof very low energy ions in matter. I consider thsystems p+ 4He and d+ 3He, for which there areexperimental data available. A previous work [studied the energy loss of protons on hydrogentargets and showed that the stopping at veryproton energies is indeed smaller than what wouldexpected from extrapolations based on the Andeand Ziegler tables [10]. The case of helium targis more complicated due to the electron–electinteraction.

The present approach is based on the solutiothe time-dependent Schrödinger equation for the etron in a dynamical two-center field. The transitifrom the separated atoms (H+ + He) and the unitedatom (Li+) is obtained in the adiabatic approximtion, i.e., by assuming that the electronic motionfast compared to the nuclear separation motion sothe molecular orbitals (MO) are those for the distanR(t) between the nuclei. The atomic wavefunctioφµ = ∑

j cjµφSlatj , are constructed as a linear com

nation of Slater-type orbitals (STO) [11] of the forφSlatn = Nrn−1 exp(−ζ r)Ylm(r̂), where the Slater co

efficientsn andζ are chosen to best approximate texact atomic wavefunctions (see, e.g., Ref. [11]). Tmolecular orbital wavefunctions for the p+ He sys-tem, are obtained with theφµ’s chosen so that half othe STO’s are centered on the proton(A) and the otherhalf are centered on the helium nucleus(B). The to-tal wavefunction for the two-electron system is finawritten as a Slater determinant of the molecular orbwavefunctions,

(1)

ψe(r1, r2,R) = 1√2

∣∣∣∣φMO1 (1)α(1) φMO

2 (1)β(1)

φMO1 (2)α(2) φMO

2 (2)β(2)

∣∣∣∣ ,

whereα, β denote the spin state of the electron. Cofiguration interaction with double excitation configrations were included in the calculation, with the cefficientsn and the Slater parametersζ chosen in a

variational method to obtain the lowest energy staof the system.

Using these conditions and variation method, oobtains the following Hatree–Fock equation:F ⊗C = O ⊗ C ⊗ E, whereF is the “Fock” matrix

Fµν =Hµν

+∑λσ

Pλσ

[〈µν| 1

r12|λσ 〉 − 1

2〈µλ| 1

r12|νσ 〉

],

(2)Pλσ = 2occ∑i=1

cλicσ i,

in which “occ” refers to the occupied moleculorbital,

(3)

Hµν =∫ ∫

φµ(1)

[−1

2∇2 −

∑L=A,B

1

r1L

]φν(1) dτ1,

is the one-electron integral and

〈µν| 1

r12|λσ 〉

(4)=∫ ∫

φµ(1)φν(1)1

r12φλ(2)φσ (2) dτ1dτ2,

are the two-electron integrals. TheC matrix is thecoefficient matrixcµν and O is the overlap matrix〈φµ(1)|φν(1)〉. E is a diagonal matrix with eacdiagonal element corresponding to the energy ofassociated molecular orbital. Solving the HartreFock equations one obtains the coefficientscij whichgive the proper linear combination of atomic orbitato form the molecular orbital. The energy of thmolecular orbitals are then given by

(5)

E(R) =∑µν

PµνHµν + 1

2

∑µνλσ

PµνPλσ

[〈µν| 1

r12|λσ 〉

− 1

2〈µλ| 1

r12|νσ 〉

].

Table 1 shows the states involved in the calculatwhere it is shown how the states in the separahydrogen and helium atoms become molecular stin the united atom system. For large distances betwthe nuclei,R > 15 a.u. (1 a.u. of length= 0.53 Å)the energy levels for the 1s, 2s, and 2p statesH and He are reproduced to within 2% and 4%the spectroscopic data, respectively. The energie

C.A. Bertulani / Physics Letters B 585 (2004) 35–41 37

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Table 1Lowest states in the p+ He molecule

Separated atom United ato

H+ + He(1s2) 0ΣH(1s)+ He+(1s) 1ΣH+(1s)+ He(1s2s) 2ΣH(n= 2)+ He+(1s) 1ΠH(n= 2)+ He+(1s) 3ΣH(n= 2)+ He+(1s) 4ΣH+ + He(1s1p) 5ΣH+ + He(1s1p) 2Π

Fig. 1. Adiabatic energies (1 a.u. of energy= 27.2 eV, 1 a.u.of length= 0.53 Å) for the electronic orbitals for the (H–He)+system as a function of the internuclear separation. As the aapproach each other slowly curves of same symmetry repelother. A transition between states s and s′ can occur in a slowcollision. In a fast collision a diabatic transition, with the stacrossing each other, will occur. This is shown in the inset.

these states are shown in Fig. 1 as a function ofinternuclear distanceR.

At very low proton energies (Ep � 10 keV) it is fairto assume that only the low-lying states are involvin the electronic dynamics. Only for bombardienergies larger than 25 keV the proton velocity wbe comparable to the electron velocity,ve � αc. Thus,the evolution of the system is almost adiabaticEp � 10 keV. Also shown in Fig. 1 (inset) are thintersection points of the states with same symmeIn a fast collision these states would cross (diab

collisions), whereas in collisions at very low energ(adiabatic collisions) they obey the von NeumanWigner non-crossing rule.

In the dynamical case the full time-dependwavefunction for the system can be expanded in teof two-center states,ψn(r1, r2, t), given by Eq. (1),with expansion coefficientsan(t). It is further assumedthat the nuclei follow a classical straight-line trajectodetermined by an impact parameterb, so that thetime dependence of the molecular wavefunctiondetermined by the conditionR = √

b2 + v2t2, wherev is the collision velocity. The dynamical evolutioof the H+ He system is calculated using the saapproach as described in Ref. [9]. We solve the selinear coupled equations

(6)iS · dAdt

= M · A,

where the column matrixA represents the timedependent expansion coefficients,S is the overlap matrix with the elementsSij = 〈ψi |ψj 〉 and M is thecoupling matrix with the elementsMij = 〈ψi |Hel −i∂/∂t|ψj 〉, whereHel is the electronic HamiltonianThe solutions are obtained starting from initial intenuclear distance of 15 a.u. for the incoming trajectand stopped at the same value for the outgoing tratory. The probability for the capture in the protonobtained by a projection of the final wavefunction inthe wavefunctions of the 1s, 2s and 2p states of thedrogen atom

(7)Pexch=∣∣∣∣∑m

am(∞)〈ψH |ψm(∞)〉∣∣∣∣2

.

Resonant charge-exchange in atomic collisionsfirst observed by Everhart and collaborators [12].these experiments it was determined that the exchaprobability in homonuclear atomic collisions osclates with the incoming energy for a collision withgiven impact parameter, or scattering angle. Thisinterpreted [13] as due to transitions between degeate states of the system at large internuclear separdistance. In the simplest situation of a p+H the degen-erate states are the symmetric and antisymmetric sobtained from the linear combination of the (H+H)and (HH+) wavefunctions. This effect was studiedRef. [14], where a relation between the damping ofoscillatory behavior of the exchange probabilities athe Landau–Zener effect was established. The p+ H


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Fig. 2. Probability of charge exchange in the collision p+ 4Heshowing the resonant behavior as a function of the impact paramand for proton energyEp = 10 keV.

collisions at small energies was recently studiedRef. [9] and the oscillatory effect was shown to belated to the Sommerfeld quantization rule for the ingral from t = −∞ to t = ∞ of the energy differencebetween the symmetric and antisymmetric state.electron tunnels back and forth between the protile and the target during the ingoing and the outgoparts of the trajectory. When the interaction time isexact multiple of the oscillation time, a minimumthe exchange probability occurs.

A similar situation occurs for p+ He collisions, asshown in Fig. 2 for the electron capture probabilby the proton at 10 keV bombarding energy. Theoscillations are due to the electron exchange betwthe ground state of the hydrogen and the first excstate in He(1s2s). But, in contrast to the H+H system,the oscillations are strongly damped. Following twork of Lichten [14] we interpret this damping effeas due to the interference between the participstates and a band of states of average energy〈Ea〉 andwidth 2Γ , as seen in Fig. 1. The important regiowhere the diabatic level cross occurs is shown in Fiinside the encircled areas. The damping mechanis best understood using the Landau–Zener theorylevel crossing. At the crossing there is a particuprobability (1− P ) of an adiabatic transition wherP is given by the Landau–Zener formula

(8)Pexch= exp

[2πH 2

ss′v(d/dR)(Es −Es′)

],

Fig. 3. Charge-exchange cross sections for the p+ 4He as a functionof the proton energy. The solid line was obtained by solvingcoupled-channels Eq. (6), and using Eq. (7) for the exchaprobability. The experimental data are from Ref. [15].

wherev is the collision velocity andHss′ is the off-diagonal matrix element connecting states s and′.The oscillatory behavior shown in Fig. 2 is duto the many level transitions at the crossing, etime governed by the probability of Eq. (8). Thinterference with the neighboring states introducedamping in the charge exchange probability, i.e.,

Pexch(b, t)� cos2( 〈Ea〉b

v

)exp

[−2πΓ 2b

v〈Ea〉],

where〈Ea〉 � 1. a.u. is the average separation enebetween the 0Σ level and the bunch of higher-enerlevels shown in Fig. 1. The exponential damping facagrees with the numerical calculations if one uΓ � 5 eV, which agrees with the energy intervalthe band of states shown in Fig. 1.

The total cross section for charge exchangecalculated from

σ = 2π∫

Pexchb db.

The numerical results for the p+ He system is shownin Fig. 3 as a function of the proton energy. The soline is the result of using the coupled-channels Eqsand (7) for the exchange probability. The experimtal data are from Ref. [15]. We observe that the callation reproduces the trend of the experimental dBut, whereas the maximum of the cross sectionEp � 20 keV is rather well described, the calcutions underestimate the cross sections at smallerergies. The low energy slope of the cross sectio


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Fig. 4. Stopping cross section for proton incident on gas4He targets,as a function of the proton energy. The experimental data areRefs. [15,16].

nonetheless well reproduced. At energies higher tthe Bragg peak (Ep � 20 keV) the numerical resultshould not be trusted as the adiabatic approximafor the molecular orbitals and also the inclusiononly the lowest energy levels are not adequate (conuum states should also be included). ForEp → 0, thecharge exchange cross section must go to zero sthe higher binding of the electrons in He preventscapture by the incident proton in an extreme adiabcollision. This feature is correctly reproduced by tnumerical calculations.

In Fig. 4 we show the stopping cross sectionthe proton. The experimental data are from Refs. [16]. The stopping cross section is defined asS =∑

i /Eiσi , where /Ei is the energy loss of thprojectile in a process denoted byi. The stoppingpower,SP = dE/dx, the energy loss per unit lengof the target material, is related to the stopping crsection byS = SP/N , whereN is the atomic densityof the material. In the charge exchange mechanone of the electrons in He is transferred to incomproton and the energy loss by the proton is givby /E = mev2/2, where v is the proton velocityAssuming that there are few free electrons inmaterial (e.g., in the helium gas) only one mostopping mechanism at very low energies shouldconsidered: the nuclear stopping power,Sn. This issimply the elastic scattering of the projectile o

the target nuclei. The projectile energy is partiatransferred to the recoil energy of the target atoThe stopping cross section for this mechanismbeen extensively discussed in Ref. [17]. The tostopping power is given byS = Sexch + Sn. In unitsof 10−5 eV cm2 the nuclear stopping for the p+ 4Hesystem atEp < 30 keV is given by

(9)Sn = S0ln(1+ 1.1383ε)

(ε + 0.01321ε0.21226+ 0.19593ε0.5),

whereS0 = 0.779 andε = 5.99Ep, with the protonincident energyEp given in keV.

The dashed line in Fig. 4 gives the energy trafer by means of nuclear stopping, while the solid lshows the results for the electronic stopping mecnism, i.e., due to charge-exchangeand excitation inhelium target. We see that the nuclear stopping donates at the lowest energies, while the electronic sping is larger for proton energies greater than 200We do not consider the change of the charge sof the protons as they penetrate the target mateThe exchange mechanism transforms the protonsH atoms. These again interact with the target atoThey can loose their electron again by transferrinback to a bound state in the target.

At very low energies the only possibility that thelectron is captured by the proton is if there istransition 1s2(1S0) → 1s2s(3S) in the helium targetOnly in this case the energy of one of the electroin helium roughly matches the electronic energythe ground state in H. This resonant transfer effis responsible for the large capture cross sectiWhen this transition is not possible the electrons preto stay in the helium target, as the energy ofwhole system is lowest in this case. Another possmechanism for the stopping is the excitation ofhelium atom by the transition 1s2(1S0) → 1s2s(3S).Thus, there must be a direct relationship betweenenergy transfer to the transition 1s2(1S0) → 1s2s(3S)and the minimum projectile energy which enabelectronic changes. Ref. [18] reported for the first tithis effect, named by threshold energy, which canunderstood as follows. The momentum transfer inprojectile-target collision,/q , is related to the energtransfer to the electrons by/q = /E/v, where v isthe projectile velocity. In order that this momentutransfer absorbed by the electron, induces an atotransition, it is necessary that̄h2/q2/2me ∼ /E.


ntion

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HY-

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ett.

Fig. 5. Energy loss of deuterons in3He gas as a function of deuteroenergy. Data are from Ref. [18]. The solid curve is the calculafor the electronic stopping power, while the dashed curve showsnuclear stopping.

Solving these equations for the projectile energyfinds

(10)Ethresp ∼ mp

4me/E.

This is the threshold energy for atomic excitatioand/or charge exchange. If the projectile energysmaller than this value, no stopping should occThe energy for transition 1s2(1S0) → 1s2s(3S) in Heis /E = 18.7 eV. Thus, for p+ He collisions, thethreshold energy isEthres

p ∼ 9 keV. This roughlyagrees with the numerical calculations presentedFig. 4 (solid curve).

Fig. 5 shows the energy loss of deuterons in3Hegas as a function of deuteron energy. The data are fRef. [18]. The solid curve is the numerical calculatifor the electronic stopping power, while the dashcurve shows the nuclear stopping. For this systhe coefficients in Eq. (9) areS0 = 1.557 andε =4.491Ed, respectively. As discussed in Ref. [18] tthreshold deuteron energy in this reaction is oforder of 18 keV, which agrees with the estimate baon Eq. (10). However, the numerical calculatiobased on the electronic stopping (solid curve of Figindicate a lower threshold energy for this systeNonetheless, the agreement with the experimentalis very good forEd > 20 keV. The threshold effecis one more indication that the extrapolationS ∼ v,

based on the Andersen–Ziegler tables is not applicto very low energies.

The steep rise of the fusion cross sectionsastrophysical energies amplifies all effects leadinga slight modification of the projectile energy [19]. Tresults presented in this article show that the stoppmechanism does not follow a universal pattern forsystems. The threshold effect reported in Ref. [is indeed responsible for a rapid decrease ofelectronic stopping at low energies. It will occwhenever the charge-exchange mechanism andexcitation of the first electronic state in the targinvolve approximately the same energy. However,drop of the electronic stopping is not as sharpexpected from the simple classical arguments givenEq. (10).

The experiments on astrophysical fusion reactihave shown that the screening effect is much larthan expected by theory. The solution to this problmight be indeed the smaller stopping power, duea steeper slope at low energies induced, e.g., bythreshold mechanism. This calls for improved thretical studies of the energy loss of ions at extremlow energies of and for their independent experimeverification. The present situation is highly disturbibecause if we cannot explain the laboratory screeneffect, most likely we cannot explain it in stellar envronments.

Acknowledgements

I would like to express my gratitude to D.Tde Paula and I. Ivanov for helping me with tprogramming during the earlier stages of this woThis work was supported by the National ScienFoundation under Grants Nos. PHY-007091 and P00-70818.

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