EPAX : an Empirical PArametrization of fragmentation CROSS sections

1. EPAX : an Empirical PArametrization of fragmentation CROSS sections Klaus Sümmerer, GSI Darmstadt (Germany) • Introduction: • High-energy proton-induced reactions • History of empirical parametrizations • Two-step models of high-energy reactions • The EPAX formula • Ingredients: Parameters and their mass dependence • Attempts to derive a new set of parameters • Measured data vs. EPAX predictions work in progress!

2. Some early high-energy proton accelerators: Facility Energy From year Bevatron (Berkeley) 6 GeV 1954...... AGS (Brookhaven) 11 GeV 1960...... Fermilab (Chicago) >300 GeV 1967...... They were also used to bombard various stable target materials. These targets were analyzed with radiochemical methods, i.e.: g-spectroscopy with or without chemical separations,  Production cross sections and (some) kinematics for suitable radioactive isotopes High-energy proton-induced nuclear reactions • Important findings: • Reaction products are practically at rest in the target. • Above 3-10 GeV, the cross sections do not change any more. • At high energies, the mass yields show an exponential slope. • The Z-distributions for each fragment mass exhibit a "bell" shape.

3. Mass yields: exponential slope High-energy proton-induced nuclear reactions: Isobaric cross sections Bell-shaped Z-distributions for constant A Energy-independence of cross sections p+Au

4. At GeV energies, a nucleon can be regarded as a classical particle • Nucleon-nucleon collisions can be treated classically using measured free nucleon-nucleon cross sections (intra-nuclear cascade). • In these collisions, very little transverse momentum is exchanged. • After the cascade, the residual nucleus is highly excited. • Heavy-ion projectiles can be treated as a bag of individual nucleons. High-energy nuclear reactions: Models • Physical models: Two-step approach • Step 1: • Intranuclear-cascade models or • Abrasion models • Step 2: evaporation calculation not very accurate in the 1970's and 1980's looked more promising at that time Empirical parametrizations

5. High-energy nuclear reactions: Two-step models after intra-nuclear cascade after evaporation slope: ~ Zp/Np Zprob(A) line β-stability line 400 A MeV 20Ne + 197Au

6. Proton-induced reactions: • Silberberg-Tsao parametrization • Mainly used for cosmic-ray purposes: Collisions of light (<Fe) nuclei with H2 • Not useful for heavier targets or projectiles. • Rudstam parametrization (from 1966) Early attempts for empirical parametrizations • Rudstam parametrization was later • extended and modified

7. Proton- and heavy-ion induced reactions give very similar isotope distributions: Proton- vs. heavy-ion induced reactions Na 8 GeV 48Ca+Be 28 GeV p+238U • Important observations: • The "bell" slopes are asymmetric! • The peaks of the distributions seem to follow a universal "corridor" located on the p-rich side of the valley of β-stability • The widths depend mainly n fragment mass. • The fragments reflect the proton/neutron-excess of the projectile Target fragmentation:GeV p + Ap A Projectile fragmentation: GeV/nucleon Ap + p  A are equivalent!

8. EPAX Version 1: Phys. Rev. C 42, 1990 based on p+A cross sections; Bevalac heavy-ion data for 40Ar+C, 48Ca+Be First parametrization of "memory effect" History of EPAX Versions • Main problem of EPAX Version 1: • strong overprediction of p-removal cross sections EPAX Version 2: Phys. Rev. C 61, 2000 only high-energy heavy-ion data (E/A > 200 MeV) Bevalac: 40Ar+C, 48Ca, GSI/FRS: 58Ni,86Kr+Be, 129Xe+Al, 208Pb+Cu

9. EPAX uses a modified "Rudstam formula": Y = YA • n • exp(R|Zprob-Z|Un,p) YA = S • P • exp(P(Ap-A)) Ingredients of EPAX Zprob s (barn) Un Up R • Zprob(A) and R(A) are fragment-mass dependent • Un=1.65 and Up=2.1 ("Gauss") can be fixed • For very small cross-sections of very p-rich fragments, the "Gauss" curve turns into an exponential exp.slope • Mass yield YA: • exponential slope • slope parameter depends on Ap 500 A MeV 58Ni + Be  A=50 n-rich p-rich Z

10. Heavy-ion-induced fragmentation cross-section datasets EPAX 1 EPAX 2 new

11. Attempts to improve EPAX 2006: GSI First attempt to modify EPAX parameters compared to Version 2 Slightly better fits, but no drastic improvement Problems occur with 124Xe+Pb the following examples date from this attempt 2009: Santiago de Compostela Second attempt to modify EPAX parameters Include new datasets from MSU at 140 A MeV

12. Most probable fragments – Zprob(A) 124Xe line of ß-stability Zprob(A) = Zß(A)+ Δ(A) + corr(A,Ap) 136Xe Zprob(A) can be expressed relative to the line of b-stability: depends on n(p)-excess of projectile charge number Z "memory effect": fragments "remember" the n(p)-excess of the projectile evaporation-residue corridor loci of largest cross sections, Zprob(A)

13. Centroid Zprob(A) Zprob(A) = Zß(A)+ Δ(A) + corr(A,Ap) residue corridor Z-units corr line of beta stability Δ For A=Ap, Zprob =Zp ! relative difference to corridor n-rich projectile (136Xe) ß-stable projectiles (129Xe, 208Pb) n-deficient projectile (124Xe) ??

14. Width parameter R(A) Y ~ exp(R|Zp-Z|Un,p) witdh parameter R n-deficient projectile (124Xe) ?? For A=Ap, the width must shrink! ß-stable projectiles (40Ar,129Xe,208Pb) A n-rich projectiles (86Kr, 136Xe)

15. 1 A GeV 36Ar+Be neutron-deficient fragments only! new version 3.02 old version 2.1 data: M.Caamano et al. NP A733, 187 (2004)

16. electromagnetic dissociation σ(b) 1 A GeV 136Xe+Pb Z bad fit! bad fit! data: D. Henzlova et al. Phys. Rev. C 78, 044616 (2008)

17. 1 A GeV 112Sn+Be 49In 50Sn neutron-deficient fragments only! new version 3.02 48Cd 47Ag old version 2.1 45Rh 46Pd data: A. Stolz et al. PR C 65, 064603 (2002) 43Tc 44Ru

18. Proton-removal channels? 1 A GeV 136Xe+Be bad! good! data: J.Benlliure et al. Phys. Rev. C 78, 054605 (2008)

19. New EPAX fits to old and new data sets give satisfactory results • Parameter dependences of YA(A) and R(A) yield slightly better quality than EPAX Version 2 • Zprob(A)-dependence for 124Xe+Pb is difficult to describe with the current parameterization • Problem with p-removal cross sections less severe, shifted to larger Z • There is still much room for improvement....therefore: Status and outlook work in progress at

20. Mass Yield YA(Ap) exponential slope for 0.50 < A/Ap < 0.90 slope parameter P of mass yield depends on size of projectile: new: S ~ S0 . (Ap2/3 + At2/3) Y(A,Ap) = S P exp(-P(Ap-A))