The Trojan Horse Method in nuclear astrophysics

ELSEVIER

Nuclear

Physics

A719

(2003)

99c-106~ www.elsevier.com/locate/npe

The Trojan

Horse Method

in nuclear

astrophysics

C. Spitaleri”,b, S. Cherubinic, A. Del Zoppob, A. Di Pietrob, P. Figuerab, M. Gulinoa,b, M. Lattuadab>d, D.Miljani?, A. Musumarra”‘b, M.G. Pellegriti”lb, R.G. Pizzone”lb*, C. Rolfsc, S. Romano”$b, S. Tudiscoalb >A Tuminoalb “Dipartimento di Metodologie Chimiche e Fisiche per I’Ingegneria, Universit& di Catania, Italy bINFN-Laboratori ‘Institut

Nazionali de1 Sud, Catania, Italy

fiir Physik mit Ionenstrahlen, Ruhr-Universitgt

Bochum, Germany

dDipartimento di Fisica ed Astronomia, Universit& cii Catania, Italy ?nstitut

Rudjer BoSkoviC, Zagreb, Croatia

The basic features of the Trojan Horse Method are discussed together with a review of recent applications, aimed to extract the bare astrophysical S(E)-factor for several twobody processes. In this framework information on electron screening potential U, was obtained from the comparison with direct experiments. 1. INTRODUCTION Understanding energy production in stars and nucleosynthesis processes requires the knowledge of a large number of charged particle reaction cross sections at interaction energies usually far below the Coulomb barrier [I-3]. For this reason accurate measurements of the relevant reaction cross sections (in this energy region o N pbtpb) are severely hampered. Therefore only in a very few favourable cases they have been measured within, or close to, the relevant Gamow energy window[4-61. Aiternatively the energy dependence of a(E) at stellar energies has been extrapolated from higher energies by using the definition of the S(E)-factor: g(E) = S(E)(l/E)eXp(-27rq)

(1)

where a(E) and S(E) are the cross section and the astrophysical S-factor, respectively and q is the Sommerfeld parameter. However such “extrapolation into the unknown” can lead to a considerable uncertainty as it was strikingly demonstrated in some cases [7,8]. Thus, different experimental approaches have been attempted to avoid these extrapolation procedures [I]. Improvements aimed to increase the detection statistics or to reduce the background were among the best proposed solutions: in this sense it is worth noticing the wide experimental work done in deep underground laboratories [9]. *Supported

by the C.S.F.N.S.M.,

03759474/03/$ - see front matter doi:10.1016/S0375-9474(03)00975-X

Catania,

Italy

0 2003 Elsevier

Science

B.V

All rights

reserved

lOOc

C. Spitalevi et al. /Nuclear

Physics A719 (2003) 99c-IO&

Performing experiments at astrophysical energies provides also the opportunity of a preliminary approach to the atomic physics effects. In fact target nuclei are in the form of neutral atoms or molecules and the surrounding electron clouds produce a screening field which reduces the bare nucleus-nucleus Coulomb barrier both in height and radial extension. This, in turn, leads to a larger value for the screened nucleus cross section, a,(E), than for the bare nucleus case, ah(E). This effect is usually described by introducing an enhancement factor flab(E) defined by [lo]:

where r;i, is the electron-screening potential energy. In the framework of direct measurement experiments, the screening energy U,, is determined by a comparison between the bare nucleus S(E)-factor, whose trend is extrapolated from higher energies, and the experimental data. Thus, although direct experiments are performed even down to the Gamow energy, extrapolation is needed. By comparing the experimental data with the extrapolation from higher energies, hints are drawn on the U, value. Experimental studies of reactions involving light nuclides [11,12,5] have shown that the expected enhancement of the cross section at low energies was in all cases significantly larger than what could be accounted for by available atomic-physics models. This aspect deserves special attention because one may have a chance to predict the effects of electron screening in an astrophysical plasma only if it is preliminarily understood under laboratory conditions. In order to overcome the experimental difficulties, arising from the small cross-sections involved and from the presence of electron screening, additional information related to these processesis required. In particular independent measurements of the electron screening potentials U, are needed; and in this context it can be helpful to develop indirect methods to measure cross sections at very low energies. Various indirect approaches, as the Asymptot,ic Normalization Coefficients (ANC) method [13,14], the Trojan Horse Method (THM) and the Coulomb dissociation have been applied over the years to reactions of astrophysical relevance whenever direct approaches alone could not provide definitive information. In this paper we will stress the role of the THM [15-271 as a complementary tool for studying reactions of astrophysical. interest. 2. THE PRINCIPLE

OF THE METHOD

2.1. Quasi-free mechanism The basic idea of the THM [28] re1’ies on the assumption that a three-body reaction A(B, cd)S can proceed via a quasi free reaction (QFR) mechanism that is dominant under particular kinematical conditions. In these conditions, the reaction A(B,cd)S is considered to be described by a polar diagram, as in fig. 1 (pseudo-Feynman diagram), where only the first term of the Feynman series is retained. The target nucleus .4 is assumed to break-up into the clusters 2 and S; S is then considered to be a spectator of the B+x -+ d+c reaction, where c and d are the outgoing particles. In this pict~ure, the cross section of the three body reaction can be factorized

C. Spitaleri et al. /Nuclear

Physics A719 (2003) 99c-106~

1Olc

into two terms corresponding to the two vertices of the diagram (fig. 1). It should be outlined that in order to apply the THM a suitable three-body reaction and proper kinematical (angular region and energy in the c.m.) conditions should be found. Generally in a A(B, cd)S three body reaction the same final state can be reached through reaction mechanisms other than the QF mechanism e.g. via sequential decay (SD) or direct breakup. For this reason a preliminary study of the kinematical conditions is necessary in order to select the energy regions where the expected QF mechanism contribution is separated from the other contributions.

5

A

Figure 1. Pseudo-Feynmann diagram for the polar approximation.

2.2.

Three-body

cross

section:

the

impulse

approximation

In order to describe the QF process a simple model based on the Plane-Wave Impulse Approximation (PWIA) can be used. In this framework the QF triple-differential cross section is given by: d30 dEdRldRz

Here, KF is a kinematical factor; i@(ps)/2 is the momentum distribution of the particle J: inside the nucleus A, da/d0 is the two-body off-energy-shell differential cross section of the virtual B(z, c)d reaction at a given center-of-mass angle ecln and at the energy EC,, given by Em = Ec-d - Q where EC, is defined in the so-called post-collision prescription [31], Q is the two-body Q-value of the reaction x+ B + c+d and &-d is the relative energy between the outgoing particles c and d. iQ(p3) I2 can be calculated using well-known parameterizations in terms of Hulthen, Hankel or Eckart wave functions depending on the nucleus A considered. A more sophisticated approach using a modified Plane Wave Born Approximation which accounts for both Coulomb effects in the two body entrance channel and off-energy-shell effects was also developed [29,30].

C. Spitaleri et al. /Nuclear

102c

Physics A719 (2003) 99c-106~

In this approach the differential two body cross section of eq. 3 is expressed by:

where doi/dil represents the on-shell two body cross section in partial wave 1, Cl is a constant and Pi is the penetrability factor which compensates for the Coulomb suppression of dal/dS2 at low energies. If ]@(ps)I2 is known and KF is calculated, it is possible to derive (da/da) from a measurement of d3a/dEldf&dRz by using eq. 3. 2.3.

Validity

test

for the

Pole

Approximation

above

the

Coulomb

barrier

.4n experimental way of testing the basic assumptions in the polar description of the QF reaction was originally proposed [32] for high-energy single-pion exchange reactions and later extended [33] to some non-relativistic cases involving non zero spin particles. It has been suggested [34-361 that the energy behaviour o(Ecm)lnd of the virtual twobody reaction, extracted from the QF process according to Eq. 3, could be compared with the corresponding two-body a(Ecm)oir measured in a direct way. If ~(&Tn)lnd ranges within an energy region above the Coulomb barrier, EC, we should obtain the same trend observed in the direct data for both the excitation function and the angular distribution. It was experimentally shown that, indeed, in this energy region (EC, > EC) the energy dependence of the virtual cross section g(Ecm)lnd agrees fairly well with the known free excitation functions in the case of the 7Li(p, o)4Ne[22], %i(d; a)“He[20], 6Li(p, a)3He, [23,37] and “%‘(a, a)12C [18] reactions. The fact that the absolute values of the cross section calculated in the framework of any approximation are not reliable, makes it necessary to normalize the extracted QF two-body cross section to the directly measured one in a suitable energy region. It has to be stressed, however, that a good agreement between the two trends (direct and indirect cross section) is a necessary condition (test of applicability of the approximation) to be fulfilled before extracting the astrophysical S(E) factor by means of the THM. A further point is that all the excitation functions are obtained by evaluating the two body cross section energies using the post collision prescription. 3. FROM 3.1.

QUASI

Experimental

FREE

REACTION

TO

THE

TROJAN

HORSE

METHOD

conditions

If the bombarding energy E, is chosen high enough to overcome the Coulomb barrier in the entrance channel of the three-body reaction, the Coulomb effects as well as, a fortiori, the electron screening effect can be assumed negligible. The nucleus B enters the nuclear interaction region and the cluster x induces the reaction B + z + c + d. If the binding energy of the particle II: inside A compensates for the initial projectile velocity, the two-body reaction is induced at very low (even vanishing) IC- B relative energy, so as to match the relevant astrophysical energy region, which is spanned due to the Fermi motion of .2:inside A [28]. In this way it is possible to extract the two-body cross section from eq. 3 which already accounts for the penetrability effects, affecting the direct data below the Coulomb barrier. Thus the comparison between direct and indirect data can be performed down to the low energy region.

C. Spitaleri et al. /Nuclear 3.2.

Comparison between Coulomb barrier and

direct and S(E)-factor

Physics A719 (2003) 99c-IO& indirect

excitation

functions

103c

below

the

In all the cases here considered after normalization the behaviour of the indirectly extracted excitation function is similar to the direct one in the full range investigated except below 100 keV where the electron screening effect is no longer negligible in the direct data, as shown in figures 2a-4a. The astrophysical S(E)-factors, extracted for the 7Li(p, a)4Ne [22], ‘Li(d, cr)4He [20] and ‘Li(p, a)3He [23,37] reactions in the astrophysical energy range, are shown in figures lb-3b together with direct data from [la].

Table 1 Electron screening potential (adiabatic limit U,=lSS eV) and S(0) coefficient measured for different reactions via direct experiments and THM [20,22,23,37] Reaction S(0)TWIMeV$ Uf [evl WY ur* 380&250 17.4 6Li(d, a)4He 340f50 16.9icO.5 ‘Li(p, ol)4He 330 * 40 300 f160 0.059 0.055*0.003 6Li(p, a)3He -400 44oZt150 2.86 3Zto.3

At lower energies THM data agree with the previous extrapolation as well as the values of lJ,, but with a better accuracy level. Both the significant discrepancy between U, and Uad and the isotopic independence of electron screening are thus confirmed. The results indicate that this discrepancy is not easy to be explained only in the area of nuclear physics and thus most likely its origin should lie in the area of atomic physics. These results are summed up in table 1.

20

0.05

0.1

0.‘15

0.2

E

0.25

(MeV)

0.3

J

I

lo-’

lo-’

E (Me”;

Figure 2. Cross section (left) and astrophysical j(E)-factor (right) for the reaction 7Li(p, a)4He as obtained by THM (full dots) [22] and direct measurements (open dots) [12]. The dashed line represents a polynomial fit to the indirect S(E) while the solid line is a fit to direct data to obtain the U, value.

et al. /Nuclear Physics A719 (2003) 99c--106~

C. Spitaleri

Figure 3. Cross section (left) and astrophysical S(E)-factor (right) for the reaction %(JI, ol)3Ne as obtained by THM (full dots) [23,37] and direct measurements (open symbols) [la]. The dashed line represents a polynomial fit to the indirect, S(E) while the solid line is a fit to direct data to obtain the U, value.

,--. -$. E a

IO?

--

10

1 -1 2 ‘“*

-‘IO &g lo;;-lo-'-O

* A

present Elwyn

work 1977

10

-r 10 10 lo

-6

5

-7 i

0 a

E

0++4

E

(keV)

Figure 4. Cross section (left) and astrophysical S(E)-factor (right) for the reaction FLi(d, a)4He as obtained by THM (full dots) [20] and direct measurements (open symbols) [12]. The solid line represents a polynomial fit to the indirect S(E) while t,he dotted line is a fit to direct data to obtain the U, value.

4. CONCLUSIONS

The present paper reports on the basic features of the THM together with a general review OFrecent applications to astrophysically relevant reactions. interesting results have been achieved and the THM has proven to be a powerful tool for measuring sub-Coulomb nuclear reaction cross sections. However, a lot remains to be done in the future to achieve

C. Spitaleri et al. /Nuclear Physics A719 (2003) 99,.-1Ohc

1OSC

reliable information for many key reactions and processes, especially for the effects of the electron screening in fusion reactions where new theoretical developments are strongly needed to meet, progresses in the experimental field. REFERENCES 1. 2. 3. 4. 3. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

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