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INFNFE-10-02

Fusion rate enhancement due to energy spread of colliding nuclei

G. Fiorentini(1,2) , C. Rolfs(3) , F.L. Villante(1,2) and B. Ricci

(1) Dipartimento (2) Istituto (3) (1,2)

arXiv:astro-ph/0210537v1 24 Oct 2002

di Fisica dell’Universit` a di Ferrara,I-44100 Ferrara, Italy Nazionale di Fisica Nucleare, Sezione di Ferrara, I-44100 Ferrara,Italy Institut f¨ ur Experimentalphysik III, Ruhr-Universit¨ at Bochum, Germany (February 2, 2008)

Abstract

Experimental results for sub-barrier nuclear fusion reactions show cross section enhancements with respect to bare nuclei which are generally larger than those expected according to electron screening calculations. We point out that energy spread of target or projectile nuclei is a mechanism which generally provides fusion enhancement. We present a general formula for calculating the enhancement factor and we provide quantitative estimate for e?ects due to thermal motion, vibrations inside atomic, molecular or crystal system, and due to ?nite beam energy width. All these e?ects are marginal at the energies which are presently measurable, however they have to be considered in future experiments at still lower energies. This study allows to exclude several effects as possible explanation of the observed anomalous fusion enhancements, which remain a mistery. I. INTRODUCTION

The chemical elements were created by nuclear fusion reactions in the hot interiors of remote and long-vanished stars over many billions of years [1]. Thus, nuclear reaction rates are at the heart of nuclear astrophysics: they in?uence sensitively the nucleosynthesis of the elements in the earliest stages of the universe and in all the objects formed thereafter, and they control the associated energy generation, neutrino luminosity and evolution of stars. A good knowledge of their rates is essential for understanding this broad picture. Nuclear reactions in static stellar burning phases occur at energies far below the Coulomb barrier. Due to the steep drop of the cross section σ (E ) at sub-barrier energies, it becomes increasingly di?cult to measure it as the energy E is lowered. Generally, stellar fusion rates are obtained by extrapolating laboratory data taken at energies signi?cantly larger than those relevant to stellar interiors. Obviously such an “extrapolation into the unknown” can lead to considerable uncertainty. In the last twenty years a signi?cant e?ort has been devoted to the experimental exploration of the lowest energies and new approaches have been developed so as to reduce the uncertainties in the extrapolations. In particular, the 1

installation of an accelerator facility in the underground laboratory at LNGS [2] has allowed the σ (E ) measurement of 3 He(3 He, 2p)4 He down to its solar Gamow peak, E0 ±?/2 = (21±5) keV [3] so that for this reaction no extrapolation is needed anymore. As experiments have moved well down into the sub-barrier region, the screening e?ect of atomic electrons has become relevant [4–6]. With respect to the bare nuclei case, the Coulomb repulsion is diminished, the tunneling distance Rt is reduced, and the fusion probability, which depends exponentially on Rt , is enhanced. The electron e?ect on the reaction can be seen as a transfer of energy U (the screening potential energy) from the electronic to the translational degrees of freedom. For each collision energy E , one has an e?ective energy Ee? = E + U and a cross section enhancement: f = σ (E + U )/σ (E ) (1)

The screening potential energy U is easily estimated in two limiting cases [5]: In the sudden limit, when the relative velocity vrel of the nuclei is larger than the typical electron velocity v0 = e2 /h ? : the electron wave function during the nuclear collision is frozen at the initial value Ψin and the energy transferred from electrons to the nuclei is thus Usu = Ψin |Z1 e2 /r1e |Ψin , (2)

where here and in the following the index 1 (2) denotes the projectile (target) nucleus and a sum over the electrons is understood. In the adiabatic limit, i.e. when vrel ? v0 : electrons follow adiabatically the nuclear motion and at any internuclear distance the electron wave function Ψad corresponds to an energy eigenstate calculated for ?xed nuclei. As the nuclei approach distances smaller than each atomic radius , Ψad tends to the united atom (i.e. with nuclear charge Z = Z1 + Z2 ) limit, Ψun . The kinetic energy gained by the colliding nuclei is thus Uad = ?in ? ?un , (3)

where ?in (?un ) is the electron energy of the isolated (united) atom in the corresponding states. We like to stress a few important features: i) Screening potential energies, which are in the range 10–100 eV, are de?nitely smaller than the practical collision energies (1–100 keV), nevertheless they can produce appreciable fusion enhancements due to the exponential dependence of the cross section. ii) In the adiabatic limit the electron energy assumes the lowest value consistent with quantum mechanics. Due to energy conservation, the energy transfer to the nuclear motion is thus maximal in this case (U < Uad ) and the observed cross section enhancement should not exceed that calculated by using the adiabatic potential: f ≤ fad = σ (E + Uad )/σ (E ) (4)

iii) The enhancement factors which have been measured are generally larger than expected. A summary of the available results is presented in Table I. The general trend is that the enhancement factors exceed the adiabatic limit. Recent measurement of d(d, p)3H with deuterium implanted in metals [7] have shown enhancements of the cross sections with 2

respect to the bare nuclei case by factors of order unity, whereas one expects a few percent e?ect. In other words, if one derives an “experimental” potential energy Uex from a ?t of experimental data according to eq. (1), the resulting values signi?cantly exceed the adiabatic limit Uad . In the case of deuterium implanted in metals, values as high as Uex ? 700 eV have been found [7], at least an order of magnitude larger than the expected atomic value Uad . Several theoretical investigations have resulted in a better understanding of small e?ects in low energy nuclear reactions, but have not provided an explanation of this puzzling picture. iv) Dynamical calculations of electron screening for ?nite values of the relative velocity show a smooth interpolation between the extreme adiabatic and sudden limits [8,9]. In fact, one cannot exceed the value obtained in the adiabatic approximation because the dynamical calculation includes atomic excitations which reduce the energy transferred from the electronic binding to the relative motion. v) The e?ects of vacuum polarization [10], relativity, Bremmstrahlung and atomic polarization [11] have been studied. Vacuum polarization becomes relevant when the minimal approach distance is close to the electron Compton wavelength but it has an anti-screening e?ect, corresponding to the fact that in QED the e?ective charge increases at short distances. All these e?ects cannot account for the anomalous enhancements. Although one cannot exclude some experimental e?ect, e.g. a (systematic) overestimate of the stopping power, the general trend is that most reactions exhibit an anomalous high enhanchement. Phenomenologically, this corresponds to an unexplained collision energy increase in the range of 100 eV. Actually, the anomalous experimental values Uex look too large to be related with atomic, molecular or crystal energies. Some other processes, involving the much smaller energies available in the target, should mimic the large experimental values of U. As an example, if the projectile approaches a target nucleus which is moving against it with energy E2 ? E , the collision energy is increased by an amount: U= 4m1 E E2 m1 + m2

1/2

.

(5)

For d + d reactions at (nominal) collision energy E = 10 keV, a target energy E2 = 0.5 eV is su?cient for producing U = 100 eV. Generally, one expects that opposite motions of the target nuclei are equally possible. Even in this case, however, the e?ect is not washed out: due to the strong nonlinearity of the fusion cross section the reaction probability is much larger for those nuclei which are moving against the projectile. In this spirit, we shall consider processes associated with the energy spread of the colliding nuclei. These processes generally lead to an enhancement of the fusion rate, for the reasons just outlined. In the next section we shall ?rst consider the thermal motion of the target nuclei. For this example, we shall derive an expression for the enhancement factor on physical grounds and then we shall outline the e?ects of an energy spread for the extraction of the astrophysical S-factor from experimental data. The treatment is generalized in sect. III and in sect. IV it is applied to study energy spreads due to motion of the nuclei inside atoms, molecules and crystals. Beam energy width and straggling are also considered. 3

In summary, all the e?ects turn out to be too tiny to explain the observed anomalous enhancements. Nevertheless, they have to be considered in analyzing the data, particularly in future experiments at still lower energies.

II. THE EFFECT OF THERMAL MOTION OF TARGET NUCLEI

In this section we consider the e?ects of thermal motion of the target nuclei. We shall make several simpli?cations, in order to elucidate the main physical ingredients. In this way we shall derive a simple expression for the enhancement factor on physical grounds. Essentially, we shall concentrate on the exponential factor of the fusion cross section, neglecting the energy dependence of the pre-exponential factors, and we shall only consider the e?ect of the target motion in the direction of the incoming particle, neglecting the transverse motion. When these simpli?cations are removed the result is essentially con?rmed: see the more general treatment of sect. III. The fusion cross section at energies well below the Coulomb barrier is generally written as: σ= S (E ) V0 ? exp ?? E 2E/?

? ?

(6)

2 ?vrel is the collision energy, ? = m1 m2 /(m1 + m2 ) is the reduced mass, V0 = where E = 1 2 Z1 Z2 e2 /h ? and S (E ) is the astrophysical S-factor 1 . The cross section is more conveniently expressed in terms of the relative velocity of the colliding nuclei vrel :

σ (vrel ) =

V0 2S (vrel ) exp ? 2 ?vrel vrel

(7)

At energies well below the Coulomb barrier, vrel ? V0 , the main dependence is through the exponential factor, so we shall treat the pre-exponential term as a constant: σ (vrel ) ? B exp ? V0 vrel . (8)

We consider a projectile nucleus with ?xed velocity V impinging against a target where the nuclei have a thermal distribution of velocity. Since the target nucleus velocity v is generally much smaller than V = |V|, one can expand 1/vrel = 1/|V ? v| and retain the ?rst non vanishing term: σ ? B exp ? V0 V0 v ? 2 V V , (9)

where v is the target velocity projection over the V-direction.

1 For

convenience of the reader, we recall that v0 = e2 /? h and thus V0 = Z1 Z2 v0 .

4

The enhancement factor with respect to the ?xed target case, f = σ /σ (V ), is thus calculated by averaging exp(?V0 v /V 2 ) over the v distribution: ρ(v ) = where v 2 = kT /m2 . The integral f= 1 2π v 2

+∞ ?∞

1 2π v 2

1 v2 ? ? , exp ? 2 v2

? ?

?

?

(10)

v2 V0 v ? dv exp ?? 2 ? V 2 v2

(11)

is easily evaluated by using a (saddle point) trick similar to that used by Gamow for evaluating stellar burning rates. The product of the Gaussian and the exponential functions (Fig.1) results in an (approximately) Gaussian with the same width, centered at vG = ? v 2 V0 /V 2 , its height giving the enhancement factor: f = exp V02 v 2 2V 4 . (12)

Concerning this equation, which is the main result of the paper, several comments are needed: i) Since the term in parenthesis in eq. (12) is positive, one has f ≥ 1, i.e. the energy spread always results in a cross section enhancement. One cannot ignore the target velocity distribution for the calculation of the reaction yield since nuclei moving towards the projectile have a larger weight in the cross section. ii) The main contribution to the cross section comes from target nuclei with velocity close to vG . When V ≤ v 2 1/2 V0 , this velocity is larger than the typical thermal velocity v 2 1/2 . This result is equivalent to the Gamow peak energy in stars, which is signi?cantly m v 2 in higher than the thermal energy kT . In terms of the energy, by putting E2 = 1 2 2 G eq. (5), we see that the “most probable” collision energy is 2 : Emp = E + 2 m1 m1 + m2 V0 Et V , (13)

1 where Et = 2 m2 v 2 = 1 kT is the average thermal energy associated with the motion in the 2 collisional direction. iii) The energy dependence of eq. (12),

f = exp

1 m1 2 m1 + m2

Et E0 , E2

(14)

1 where E0 = 2 ?V02 , is di?erent from that resulting from electron screening f = exp(D/E 3/2 ).

2 The

most probable energy Emp has not to be confused with the e?ective energy Eef f .

5

iv) The resulting e?ects are anyhow extremely tiny. For example, for d + d collisions (V0 = e2 /h ? ) at E = 1 keV (V = 1/5 V0) and room temperature ( v 2 1/2 = 5 · 10?4 V0 ) one has f ? 1 ? 10?4 . A 10% enhancement would correspond to kT ? 30 eV. v) The same method can be extended to other motions of the target nuclei, provided that the velocity distribution is approximately Gaussian and if other interactions of the nuclei during the collision are neglected (sudden approximation). One has to replace v 2 in eq. (12) with the appropriate average velocity associated with the motion under investigation. Vibrations of the target nucleus inside a molecule or a crystal lattice can be treated in this way, since the vibrational times are much longer than the collision times. These and other similar e?ects will be discussed in sect. IV. vi) From the discussion presented above one gets an easy procedure to correct the experimental results for taking into account the e?ect of an energy spread. If the as1 trophysical S-factor has been measured at a nominal collision energy E = 2 ?V 2 , from Sexp = σexp E exp(V0 /V ), then the “true” S-factor is obtained as S = Sexp /f , where f is given by eq. (12) and the “true” energy is changed from E to Emp given in eq. (13) ( Fig. 2). In summary, the e?ect of the energy spread translates into both a cross section enhancement and an energy enhancement.

III. GENERAL TREATMENT

In this section we shall provide a more general discussion of the energy spread e?ects, which will substantially con?rm eq. (12) and which can be applied to a rather large class of processes. The main assumption is that the projectile motion is fast in comparison with the other motions, so that the sudden approximation can be used. Let us consider a projectile with velocity V impinging onto a thin target (density n and thickness L), where energy loss can be neglected. The interaction probability P is the product of the interaction probability per unit time p ˙ = n σvrel with the time spent in the target, L/V . The measured counting rate Λ = ?I p ˙, where I is the beam current and ? is the detector e?ciency, is thus: Λ= I?nL σvrel V . (15)

As in stars, the quantity which is physically relevant is thus σvrel , where the average has to be taken over the target nuclei velocity distribution. This distribution is due to the coupling with other degrees of freedom. Inside an atom (or a molecule, or a crystal) the nucleus is vibrating, its motion is altered by the arrival of the projectile nucleus and the calculation of the average is complicated in the general case. However, if the velocity V of the impinging particle is large in comparison with the velocity v of the target nucleus, the problem is simpli?ed. The target wave function does not have time for signi?cant evolution during the collision and it can be taken as that of the initial (unperturbed) state. This is the main content of the sudden approximation: the velocity distribution of the target nuclei ρ(v ) can be taken as the initial one ρin (v ) and one has to compute:

6

Λ=

I?nL V

d3 v ρin (v ) σ (vrel )vrel

.

(16)

By using eq.(7), one has thus to compute: Λ= I?nL V d3 v ρin (v ) V0 2S (vrel ) exp ? ?vrel vrel . (17)

We recall that S is a weakly varying function of energy, so that it can be taken out of the integral. 1 Since we are assuming V 2 ? v 2 , we expand the integrand g = vrel exp(?V0 /vrel ) in powers of v and keep the lowest order terms: Λ= 2SI?nL ?V 1 d3 v ρin (v ) gv=0 + vi (?i g )v=0 + vi vj (?i ?j g )v=0 2 . (18)

We shall consider distributions which are symmetrical for inversions and rotations around the collision axis V . In this case the term linear in v vanishes and the result is: Λ= V0 2SI?nL · exp ? 2 ?V V v 2 V02 V V {1 + 1?4 +2 4 2V V0 V0 2 2 2 V V v V } , ? + ⊥ 40 2V V0 V0 (19)

2

where the index (⊥) denotes the component of the velocity along (transverse to) the collision axis. The term in front of the curl bracket is the counting rate calculated neglecting the target energy spread. So, if we de?ne the enhancement factor f as the ratio of the measured counting rate Λ to the rate calculated for ?xed velocity ΛV , we have: f≡ we have now: v 2 V02 V V 1?4 +2 f ?1+ 4 2V V0 V0 2 2 2 v V V V + ⊥ 40 ? . 2V V0 V0 For a one dimensional motion (v⊥ = 0) it simpli?es to: f =1+ v 2 V02 V V 1?4 +2 4 2V V0 V0

2 2

Λ V0 = V exp ΛV V

d3 v ρin (v )

1 V0 exp ? vrel vrel

,

(20)

+

(21)

.

(22)

2 For the case of a spherically symmetrical distribution, v 2 = 1/2 v⊥ one gets:

7

f =1+

v 2 V02 V 1?2 4 2V V0

.

(23)

This equation can be easily compared with the result of the previous section concerning the thermal energy e?ect. By expanding eq. (12) one gets: f =1+ 1 v 2 V02 2 V4 . (24)

This is the same as eq. (23) apart for the last term which is negligible at small velocities, since it is a higher order contribution in V /V0. Note that this last term arises from the variation of the pre-exponential factor 1/vrel , which was neglected in the simpli?ed treatment of sect. II. Clearly this term, once averaged over the target distribution, is smaller than 1/V and therefore it provides a reduction of the rate, as implied by the negative coe?cient in eq.(23). The previous results have been obtained by neglecting higher order terms in the expansion of g . Their contribution is suppressed by a factor v 2 V02 /V 4 .Thus the previous results are not valid for V ? v 2 1/2 V0 , as can be simply understood. In this case, one cannot expand the integrand function g (v ), since it changes faster than the distribution function ρ(v ) over a large range of target velocities. More precisely, the decrease of ρ(v ) is counter-balanced by the increase of g (v ) in a velocity range which is typically larger than the average target velocity dispersion v 2 1/2 . As a consequence, the tails of the distribution function ρ(v ) give a relevant contribution to the counting rate, leading to an increase of the factor f with respect to the simple estimate eq.(23). It is di?cult to obtain a general expression for f in this low velocity regime. The factor f depends, in fact, on the shape of the distribution function. In the case of a gaussian distribution function, ρ(v ) ∝ exp(?v 2 /2 v 2 ), one can use the Gamow “trick” described in the previous section which leads to eq.(12). For distribution functions which decrease more slowly with v one expects larger e?ects. In order to have, however, a general result for the low velocity ( v 2 < V 2 < v 2 1/2 V0 ) behaviour of f , we note that, being the counting rate Λ an increasing function of the projectile velocity V , one has: Λ ≥ Λ(0) ≡ I?nL A V d3 v ρin (v ) V0 1 exp ? v v (25)

This means that the enhancement factor f should be larger than: f0 = V exp V0 V d3 v ρin (v ) V0 1 exp ? v v . (26)

IV. APPLICATIONS

The method developed in the previous sections, summarized in eq. (12) or in the more accurate eq. (21), can be applied to several motions of the target nuclei (vibrations inside 8

an atomic, molecular or crystal system), provided that interactions with other degrees of freedom during the collision can be neglected. Simply, one has to compute the value of v 2 which is appropriate to the system under consideration. Also, the treatment can be easily extended to the e?ect of beam energy width and straggling.

A. Nuclear motion inside the atom

Very much as the motion of a star in the sky is a?ected by the presence of planets around it, the nucleus inside an atom is vibrating around the center of mass of the atomic system. The nuclear momentum distribution P (p) is immediately determined from that of the atomic electrons Pe (pe ) by requiring that the total momentum of the atom vanishes in the center of mass (p = ?pe ), where pe is the (total) momentum carried by the electron(s), i.e. P (p) = Pe (?pe ) and the initial nuclear velocity distribution ρin (v ) is immediately determined from v = p/m2 , where m2 is the target nucleus mass. For the case of Hydrogen (isotope) in the ground state, the atomic electron momentum distribution is: 8 (me v0 )5 P e (p e ) = 2 2 (27) 2 4 π (pe + m2 e v0 ) so that the nucleus velocity distribution is: ρin (v ) = 8 u5 0 4 π 2 (v 2 + u 2 0) , (28)

where u0 = (me /m2 )v0 = (me /m2 )e2 /h ? , is the typical velocity associated with the target nuclear motion. In practice, this is de?nitely smaller than the collision velocity V , so that the sudden approximation holds and the results of the previous section can be applied. One can easily evaluate that: v2 = 1 2 1 me u = 3 0 3 m2

2 2 v0

,

(29)

so that for Hydrogen-Hydrogen (or deuterium-deuterium) collisions, for which V0 = Z1 Z2 e2 /h ? = v0 , by using eq. (23), one obtains for the enhancement factor : fat = 1 + 1 6 me m2

2

V0 V

4

1?2

V V0

.

(30)

This is an extremely tiny correction, since one has fat ? 1 ? 2 · 10?5 for a d-d collision at E = 1 keV energy . √ In the low energy regime, i.e. when V ≤ u0 V0 = (me /m2 )1/2 v0 , the previous estimate has to be corrected to take into account the contribution of the tails of the distribution function. By using eq.(26) we can easily estimate: (fat )0 ? 32 · 5! V π V0 u0 V0

5

exp(

V0 ) V

(31)

In Fig. 3 we compare the approximate expressions with the numerical evaluation of eq. (21). In the whole range a good approximation to the full numerical calculation is provided by f = fat + (fat )0 . 9

B. Molecular vibrations

Let us consider, as an example, reactions involving a deuterium nucleus bound in a D2 molecule. The target nucleus is vibrating, the vibration energy in the ground state being Evib = 0.19 eV . This energy is shared between the two nuclei and between potential and kinetic energy, so that the average kinetic energy of each nucleus is 1 m v 2 vib = 1/4Evib. 2 d 2 The target nucleus velocity, v 2 vib ? 10?6 v0 , is much smaller than the projectile velocity so that the sudden approximation applies again. By using eq. (12) and assuming a random orientation of the molecular axis , v 2 = 1/3 v 2 vib , we get: fmol = exp v 2 vib V02 6V 4 . (32)

This corresponds to a 10?4 correction at E=1 keV. Conversely, an enhancement correction of 10% would correspond to Evib ? 200 eV.

C. Local vibrations in a crystal lattice

When a deuterium nucleus is implanted in a crystal, it generally occupies an interstitial site where it performs local vibrations. The vibration energy Ecr depends on the host lattice, being typically in the range of 0.1 eV, very similar to the molecular vibration scale. E?ects associated with vibrations in the crystal are thus similar to those calculated for the D2 molecule: fcr ? fmol . (33)

D. Finite beam width and straggling

In an ideal accelerator all projectiles have the same energy Elab . Actually, due to several physical processes (voltage ?uctuations, di?erent orbits...) the beam will have a ?nite energy width ?. As an example, in the LUNA accelerator one has ? ? 10 eV. Furthermore, when the beam passes through the target, ?uctuations in the energy loss will produce an enlargement of the energy width (straggling). Thus, even neglecting the target motion, there is a collision energy spread. The beam energy distribution, P (E ) ? exp ? gives a velocity ditribution with: v

2

′

(E ? Elab )2 2?2

′

(34)

=

?2 md Elab

(35)

10

By using eq. (21) the enhancement factor is thus 3 : f = exp V02 ?2 6 m2 1V (36)

E?ects are very small in the case of LUNA: for d+d at E = 1 keV and ? = 10 eV one has f ? 1 ? 2 · 10?5 . The e?ect behaves quadratically with ? and it can be signi?cant if momentum resolution is worse. Conversely, an enhancement correction of 10% corresponds to ? ? 250 eV.

E. Polynomial velocity distributions

One could suspect that velocity distributions of di?erent shape can provide enhancements signi?cantly larger than the tiny e?ects which we have found so far. In this spirit, let us consider the case of a polynomial velocity distribution, ρ(v ) = A (v 2 + B 2 )n (37)

where the slowly decreasing tail should provide a signi?cant enhancement. Clearly the more favourable cases correspond to small values of n. The requirement that v 2 is ?nite implies n ≥ 3, so we consider n = 3 in order to maximize the tail e?ect. The normalized distribution is in this case: ρ(v ) = v 2 3/2 4 π 2 · 33/2 [v 2 + (1/3) v 2 ]3 V0 V V V0 (38)

The low energy enhancement factor f0 of eq. (26) becomes now: f0 ? 16 · 3! π · 33/2 v 2 /V02

3/2

exp

(39)

In order to have f0 ? 1.1 for d+d collisions at E = 1 keV one needs v 2 ? 3 · 10?2 V02 , which corresponds to an average energy in the range of 1 keV, well above the physical scale of the process.

V. CONCLUDING REMARKS

We summarize the main points of this paper:

the sake of precision, the counting rate is now: Λ = ?InL σ beam . This is di?erent from eq. (15). A calculation of the average, similar to that presented in sect. III, yields the same expression as in eq.(19) for the leading term in V /V0 and di?erent numerical coe?cients for the higher order (negligible) terms.

3 For

11

i) Energy spread is a mechanism which generally provides fusion enhancement. ii) We have found a general expression for calculating the enhancement factor f : Z1 Z2 e2 f = exp ? h ?

?

2

v 2? 2V 4

?

(40)

iii) We have provided quantitative estimates for the enhancement e?ects. For a d+d collision one has: thermal motion: f ? 1 ? 10?4 (E/1keV)?2 vibrational motion: f ? 1 ? (10?5 – 10?4 ) (E/1keV)?2 beam width: f ? 1 ? 10?5 (E/1keV)?3 iv) All these e?ects are marginal at the energies which are presently measurable, however they have to be considered in future experiments at still lower energies. v) This study allows to exclude several e?ects as possible explanation of the observed anomalous fusion enhancements, which remain a mistery.

12

REFERENCES

[1] C.E. Rolfs and W.S. Rodney, “Cauldrons in the Cosmos”, The University of Chicago Press, Chicago 1988. [2] G.Fiorentini, R.W. Kavanagh and C.E. Rolfs, Z.Phys. A 350 (1995)289 [3] R. Bonetti et al. (LUNA coll.), Phys. Rev. Lett. 82 (1999) 5205 [4] H.J. Assenbaum , K. Langanke and C.E. Rolfs, Z. Phys. A 327 (1987) 461. [5] L. Bracci et al. Nucl. Phys A 513 (1990) 316. [6] F. Strieder et al., Naturwissenschaften 88 (2001) 461. [7] F. Raiola et al., Eur. Phys. J. A 13 (2002) 377 and Phys. Lett. B (in press). [8] L. Bracci, G. Fiorentini and G. Mezzorani Proc. of TAUP 1989, Aquila, Italy, Sep 25-28. [9] T.D. Shoppa et al., Phys. Rev C 48 (1993) 837. [10] S. Degl’Innocenti and G. Fiorentini, Astron. Astrophys. 284 (1994) 300. [11] A.B. Balantekin, C.A. Bertulani and M.S. Hussein, Nucl.Phys. A 627 (1997) 324. [12] U. Greife et al. Z. Phys. A 351 (1995) 107 [13] M. Aliotta et al. Nucl.Phys. A 690 (2001) 790. [14] M. Junker et al., Phys. Rev. C 57 (1998) 2700. [15] S. Engstler et al. Z. Phys. A 342 (1992) 471. [16] A. Musumarra et al. Phys. Rev. C 64 (2001) 068801. [17] M. Lattuada et al., Ap. J. 562 (2001) 1076. [18] D. Zahnow et al., Z. Phys. A 359 (1997) 211. [19] C. Angulo et al., Z. Phys. A 345 (1993) 231.

13

TABLES

TABLE I. Summary of results for electron screening e?ects Reaction d(d,p)t 3 He(d,p)4 He d(3 He,p)4 He 3 He(3 He,2p)4 He 3 He(3 He,2p)4 He 6 Li(p,α)3 He 6 Li(d,α)4 He 7 Li(p,α)4 He 9 Be(p,d)8 Be 11 B(p,α)8 Be

a

Uex (eV) 25 ± 5 b 219 ± 7 109 ± 9 294 ± 47 432 ± 29 470 ± 150 320 ± 50 330 ± 40 900 ± 50 430 ± 90

Uad a (eV) 28.5 114 102 240 240 184 184 184 262 346

Ref. [12] [13] [13] [3] [14] [15] [16] [17] [18] [19]

Values calculated for atomic target, following [5]. It is assumed that at fusion hydrogen projectiles are charged or neutral with equal probability. Helium projectiles are assumed to be He+ (He) with 20% (80%) probability. b This value results from gaseous target. Much larger values have been found when deuterium is implanted in metals [7].

14

FIGURES

FIG. 1. A sketch of the contribution to the averaged cross section. ρ(v ) is de?ned in eq.(10) and ? = exp(?v0 v /V 2 ).

15

FIG. 2. Extraction of the S -factor from experimental data.

16

FIG. 3. Fusion enhancement due to nuclear motion inside a H atom. we present the numerical evaluation of eq.(21) (full line), the approximations of eq.(12) (dotdashed) and of eq.(23) (dotted), the low velocity limit eq.(31) (dashed).

17

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