1. Effects Of Distortion On Trojan Horse Applications Rosario Gianluca Pizzone INFN – Laboratori Nazionali del Sud Catania

2. The Trojan Horse Method Indirect Methods can improve Nuclear Astrophysics results. Among them the Trojan Horse Method (THM). (C. Spitaleri’s talk)It allows the study of reactions of astrophysical interest like x(A,C)c at energies as low as the astrophysical ones after selection of an appropriate a(A,Cs)c reaction, induced at energies greater than the Coulomb barrier in quasi free conditions. (C. Spitaleri’s talk) For QF processes the binary cross section, as a function of the three body one is: a quasi free break-up s A x virtual reaction in nuclear field A + x  c + C c C Measured 3-body cross section 2-body cross section What is  (p s )?

3. The nucleus a can be brought into nuclear field of nucleus A and the cluster x induces the reaction A + x  C + c Coulomb effects and electron screening are negligible (C. Spitaleri’s talk) The incoming energy E A of the incident particle is larger than the Coulomb barrier energy (E AB ) Coul. Bar. E A > (E Aa ) Coulomb Barrier (This means that A and x have a non-negligible probability to be very close) x A C c a S

4. Outlook Spectator momentum distribution in TH nucleus is necessary for THM application TH nuclei (table) show a strong cluster configuration In recent years the impulse distribution of spectator inside TH nucleus has been extensively studied for 6 Li as a function of the transferred momentum ( Pizzone et al. 2005); Goal of the present work: to evaluate the momentum distribution distortion as a function of the transferred momentum also for the other nuclei used as TH nuclei Knowing that:

5.  p s ): the momentum distribution 6 Li d    It represents the momentum distribution of the cluster s inside the Trojan Horse nucleus es. 6 Li or 2 H; A very well known case: 6 Li=  +d (Pizzone et al, PRC 058801 2005) Step 1 to evaluate the distortion effects is the study of the  -d momentum distribution for 6 Li

6. Momentum distribution If the reaction proceeds via a QF mechanism the process is a direct one and the impulse distribution of the spectator should be identical to the one it has inside TH nucleus Since : Then if d  /d  ~ constant (small relative energy interval)  QF mechanism is present and can be separated from other contributions.

7. Hulthen function: Standard parameters a=0.2317 fm -1 b=1.202 fm -1 Only one variable in the fit, the normalization constant Momentum distribution Dots: experimental momentum distribution (n in deuteron case) Red: DWBA calculation Black line: PWIA If -30<ps<30 MeV/c they almost agree (typical ranges for THM application, according to Shapiro prescriptions)

8. Momentum distribution for 6 Li By increasing the transferred momentum the momentum distribution widens up Low transferred momentum higher transferred momentum B

9. FWHM vs. transferred momentum variation: 6 Li Step 2 investigate how momentum distribution shape changes with transferred momentum (reaction 6 Li( 6 Li,  ) 4 He); Correlation FWHM – transferred momentum 6 Li( 6 Li,  ) 4 He at different beam energies (2.1 – 44 MeV)  6 Li( 3 He,  p) 4 He at different beam energies (5-6 MeV)  Data from Barbarino et al. PRC 1980 f 0 =73 MeV/c; q 0 =324MeV/c

10. Is this behaviour independent from Trojan Horse Nucleus?? Deuterium Helium 3 Beryllium 9 Tritium

11. FWHM vs. transferred momentum variation: 2 H With growing transferred momentum the distribution width get closer to its asymptotic D=p+n

12. Additional study: momentum distribution for n in deuteron Experimental points are compared with theoretical prediction (Lamia et al., PRC, 2012) D wave contributing up to 4%.

13. Remarks We see therefore that for decreasing transferred momenta q t distortions in the momentum distribution of n inside d shows up. This works also for other possible TH nuclei used in several experiments (Pizzone et al 2009)

14. FWHM vs. transferred momentum variation: 3 He 3 He=p+d

15. FWHM vs. transferred momentum variation: 9 Be 9 Be= a + 5 He

16. FWHM vs. transferred momentum variation: tritium t=d+n

17. Parameter f 0, q 0 and k=(2  E b ) 1/2 for examined nuclei Correlation between k and q 0

18. Conclusions In all cases when transferred momentum is much higher than k the asymptotic value of the momentum distribution width is reached. Otherwise a narrowing shows up.That is a clear signature that distortion effects show up as soon as one get far from an “ideal” quasi free situation. One can take into account these effects by adopting in THM applications the effective momentum distribution instead of the asymptotic one. q t >>k q t ~k

19. Remarks: Distortions in THM Distortions accounted for in THM by inserting in the experimental momentum distribution width (in general different from the asymptotic one). Two possible effects: -Impact of altered width in TH nucleus momentum distributions -Impact of assuming only s-wave contributes to the d wave function (e.g. for deuteron)

20. The S(E) factor for the 6 Li(d,a) 4 He reaction has been extracted for different values of the momentum distribution FWHM (Pizzone et al. 2005). FWHM = 70 MeV/c FWHM = 60 MeV/c FWHM = 50 MeV/c No remarkable change (within experimental errors) differences around 5% 1

21. The variation of the extracted S(E) factor for taking into account the d wave instead of the s wave alone in the n-p intercluster motion in d turned out to be negligible within experimental errors: In both the examined cases the discrepancy is less than 0.5% 2

22. Bibliography: