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Fusion enhancement due to energy spread of colliding nuclei*

Fusion enhancement due to energy spread of colliding nuclei*. 1. Motivation: anomalous electron screening or what else ? 2. Energy spread Þ fusion enhancement 3. Calculate fusion enhancement due to thermal motion of target atoms 4. Generalize to similar processes: - Lattice vibration

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Fusion enhancement due to energy spread of colliding nuclei*

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  1. Fusion enhancement due to energy spread of colliding nuclei* 1. Motivation: anomalous electron screening or what else ? 2. Energy spread Þ fusion enhancement 3. Calculate fusion enhancement due to thermal motion of target atoms 4. Generalize to similar processes: - Lattice vibration - Beam energy width - Energy straggling 5. So far, so what? * Work in progress by B Ricci, F Villante and G Fiorentini

  2. Anomalous electron screening or what else? • At the lowest measured energies, fusion rates are found generally larger than expected, i.e. the enhancement factor • f= smeas/sBare is: • f > fad • where fad is the maximal screening effect consistent with QM • [In the adiabatic limit energy transfer from electrons to colliding nuclei is maximal] . • Measured values are typically: • (f-1) » (10 - 20) % for p+d, d+d, d+3He in gas target • (f-1) » 100% for d+d with d inplanted in metals • Are we seeing anomalous screening or something else?

  3. V V+vt V-vt Energy spread implies fusion enhancement • Consider an example: • -The projectile has fixed V • -The target has a velocity distribution • due (e.g.) to thermal motion: • P(v) » exp[-v2 / 2vt2] • -Can one neglect the target velocity distribution assuming that “it is zero on average”, i.e.: • <s> =s(V) ??? • NO:Due to strong velocity dependence, • approaching particles have larger • weight than the receding ones. • Þ i f=<s>/s(V) >1

  4. s Maxwell vmax v Calculation of the effect Generally one has to calculate: f=<s>/s(V) >1 Where < > is the average over the velocity distribution: f = òdv exp [-vo/ïV-vï -v2/ 2vt2] / exp[- vo/V] òdv exp [-v2/ 2vt2] The calculation can be easily done by a “Gamow trick”. The result is*: f= exp [ vo vt2 / 2V4] where V=proj. vel., vt= target av. thermal vel. ,vo=2pZ1Z2ac *We are assuming S=const; for S=S(E) see later…The result is to the leading order in v/V

  5. f= exp [ vo 2vt2 / 2V4] V=proj. vel., vt= target av. thermal vel. vo=2pZ1Z2ac Remarks • 1. f> 1 i.e. always enhancement • Gamow-like peak; largest contribution from target nuclei with: • v = vmax= vo vt2 / V2 • f is strongly energy dependent, f » exp (-k/ E2). • En. Dependence different fromscreening: • fsc» exp (-k/ E3/2) • Effect is small in present conditions: • (f-1) » 7 10-4 for d+d at (c.m.) E = 2 keV • It would be significant at extremely low energies: • (f-1) » 10% for d+d at E = 0.2 keV

  6. s Maxwell vmax v Effective energy enhancement • If one takes into account that • S=S(E) an additional effect arises. • The “Gamow peak”means that the • S factor is measured for an • effective velocity • Veff = V(1+ vovt2/ V3) • Equivalently, the effective c.m. energy Eeff is larger • than E=1/2 mV2: • Eeff =E (1 + vovt2 (m/2E)3/2]2. • The effect can thus be interpreted as an enhancement • of the effective collision energy. • Really a very tiny effect: 10-5 for d+d at ECM = 2 keV S

  7. Sexp(E) S(Eeff) E Eeff Correction of S • If Sexp has been measured at a nominal energy E=1/2 m V2, from: • Sexp(E) = sexp E(V) exp (vo/V) • in order to obtain the true S factor one has to: • - Change to the effective energy • E--> Eeff • -Apply a renormalization factor: • S(Eeff) = Sexp (E) Eexp/E exp [-vo 2vt2 / 2V(E)4]

  8. Generalization to similar processes • There are two ingredients in the calculation: • s» exp [ - vo/Vrel (Vrel=relative velocity) • P(v) » exp [-v2/ 2vt2] • The same scheme can be used for other processes, which produce a (Gaussian) velocity spread of target and/or projectile nuclei. • One only has to re-interpret vt2, by introducing a suitable (vt2)eff f= exp [ vo 2 (vt2 )eff/ 2V4]

  9. Vibrational effects d • Consider the target nucleus • (e.g. d) inplanted in a crystal*). • The typical vibrational energies are • Evib =(0.1-1) eV • Since the collision time is short compared to the vibrational period, one can use the sudden approximation for the target nucleus motion. • In the harmonic oscillator approximation, one has: • <Ekin>= 1/2 Evib • This means <Ekin > = 1/2 (KT)effÞ (Vt2)eff = Evib /md *)Similar considerations hold for molecular vibrations

  10. Enhancement due Vibrational effects d This gives for the enhancement factor: f= exp [ vo 2Evib/ 2 md V4] • Resulting effects are small: • (f-1) » 3 (10-3 - 10-2 ) for d+d at ECM = 2 keV • They can become significant at smaller energy. • The value observed for d in some metals, f-1 » 100%, • would correspond to Evib» 10 eV

  11. D EL E Beam energy width • The produced beam is not really monochromatic: • P(E) » exp [-(E-EL)2/2D2beam] • For LUNA, Dbeam» 10 eV. • This can be transformed into an approximately gaussian • velocity distribution with: (Vt2)eff = D2beam/ 2mp EL (mp= projectile mass)

  12. D EL E Enhancement due toBeam energy width The enhancement factor is thus: f= exp [ vo 2D2beam/ 2 m2p V6] • Effects are very small in the LUNA condition: • (f-1) » 3 10-4 for d+d at ECM = 2 keV and Dbeam» 10 eV • The effect behaves quadratically with Dbeam and it • can be significant if momentum resolution is worse.

  13. D EL E Check • As a check, one can show that the same result can be obtained by integrating s directly over the energy distribution : • P(E) » exp [-(E-EL)2/2D2beam] • By using the saddle point method and assuming Dbeam << EL • one finds: f =< s >/ s(EL)= exp( vo2D2beam / 2mp2V6) • This is the same result as before.

  14. E Energy loss and straggling • Due to atomic collisions in the target one has • - Energy loss, Elost • - Energy Straggling, Dstra • If e is the energy lost in each of N • collisions • Elost» N e ; Dstra» (N)1/2 e Þ Dstra =(Eloste)1/2 • For Elost» 1KeV, e» 10 eV Dstra» 100 eV. • One has the same formula as before, however with Dstra >>Dbeam: f =< s >/ s(EL)= exp( vo2 Eloste / 2mp2V6)

  15. E Competition between energy loss and straggling • Consider particles entering the target with • kin. energy Ein. As they advance their kin. • energies are decreased. • When the average kin. energy is EL=1/2 mpV2 the correct weight to the cross section is: • 1/V2 exp (vo/V) exp( vo2 Eloste / 2mp2V6) • The last term is due to straggling. It is: • -negliglible at EL» Ein ( since Elost=0), where most fusions occur. • -large at small energies, when fusion is anyhow suppressed. • For this reason we expect the effect is not important...

  16. So far, so what ? • Energy spread is a mechanism which provides fusion enhancement • We have found a general expression for calculating the enhancement f= < s >/ s due to a gaussian spread: f= exp [ vo 2 (vt2 )eff/ 2V4] • Quantitative estimates for d+d at Ecm= 2KeV: • thermal (f-1) » 10-3 • vibrational (f-1) » 3 10-3 -3 10-2 • beam (f-1) » 3 10-4 • No explanation for anomalous screening found; actually we can exclude several potential candidates.

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