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!

High-energy proton-induced nuclear reactions 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.

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

High-energy nuclear reactions: Models 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 Empirical parametrizations looked more promising at that time

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

Early attempts for empirical parametrizations 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

Proton- vs. heavy-ion induced reactions Proton- and heavy-ion induced reactions give very similar isotope distributions: Proton- vs. heavy-ion induced reactions 28 GeV p+238U 8 GeV 48Ca+Be Na 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!

History of EPAX Versions 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

Ingredients of EPAX Y = YA • n • exp(R|Zprob-Z|Un,p) 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

Heavy-ion-induced fragmentation cross-section datasets System Energy (A MeV) Laboratory Remarks 40Ar + C 200 LBL pioneering work 48Ca + Be 212 -"- 58Ni + Be 500 GSI p-rich, very small xsects 86Kr + Be large fluctuations 129Xe + Al 700 208Pb + Cu 750 36Ar + Be 1050 p-rich only 112Sn + Be 1000 124Xe + Pb Pb target! 136Xe + Pb 40,48Ca + Be 140 MSU new dataset 58,64Ni + Be 136Xe + Be n-rich, p-removal 64 MSU/RIKEN very low energy! EPAX 1 EPAX 2 new

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

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

Centroid Zprob(A) Zprob(A) = Zß(A)+ Δ(A) + corr(A,Ap) corr Δ 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) ??

Width parameter R(A) Y ~ exp(R|Zp-Z|Un,p) For A=Ap, the 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)

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)

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

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

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

Status and outlook work in progress at 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

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