EPAX : an Empirical PArametrization of fragmentation CROSS sections

Klaus Sümmerer, GSI Darmstadt (Germany)

2. The EPAX formula

Ingredients: Parameters and their mass dependence

3. Attempts to derive a new set of parameters

Measured data vs. EPAX predictions

work in progress!

1. Introduction:

High-energy proton-induced reactions

History of empirical parametrizations

Two-step models of high-energy reactions

High-energy proton-induced nuclear reactions

Some early high-energy proton accelerators:Facility Energy From yearBevatron (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.: -spectroscopy with or without chemical separations,

Production cross sections and (some) kinematics for suitable radioactive

isotopesImportant 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

Energy-independence of cross sections

Bell-shaped Z-distributions for constant A

p+Au

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.Physical models: Two-step approach

Step 1:•Intranuclear-cascade models or•Abrasion models

Step 2: evaporation calculation

not very accurate in the1970's and 1980's

Empirical parametrizations looked more promisingat that time

High-energy nuclear reactions: Two-step models

after intra-nuclear cascade after evaporation

400 A MeV 20Ne + 197Au

slope:~ Zp/Np

Zprob(A) line

β-stability line

Early attempts for empirical parametrizations

Proton-induced reactions:

• Silberberg-Tsao parametrization

Mainly used for cosmic-ray purposes: Collisions of light (<Fe) nuclei with

H 2

Not useful for heavier targets or projectiles.

• Rudstam parametrization (from 1966)

• Rudstam parametrization was

later

extended and modified

Proton- vs. heavy-ion induced reactions

Proton- and heavy-ion inducedreactions give very similarisotope distributions: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-stabilityThe widths depend mainly n fragment mass.The fragments reflect the proton/neutron-excess of the projectile

28 GeV p+238U

8 GeV 48Ca+Be

Na

Target fragmentation: GeV p + Ap A Projectile fragmentation: GeV/nucleon Ap + p Aare equivalent!

History of EPAX Versions

EPAX Version 1: Phys. Rev. C 42, 1990

based on p+A cross sections; Bevalac heavy-ion data for 40 Ar+C,

48 Ca+Be

First parametrization of "memory effect"

EPAX Version 2: Phys. Rev. C 61, 2000

only high-energy heavy-ion data (E/A > 200

MeV)

Bevalac: 40 Ar+C, 48 Ca,

GSI/FRS: 58 Ni,86 Kr+Be, 129 Xe+Al, 208 Pb+Cu

Main problem of EPAX Version 1:

strong overprediction of p-removal cross

sections

Ingredients of EPAX

ZprobEPAX uses a modified "Rudstam

formula":

Y = Y A • n • exp(R|Z prob-Z|Un,p )

Y A = S • P • exp(P(A p-A)) R UpUn

exp.slope

n-rich p-rich

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

500 A MeV 58Ni + Be A=50

Z

(b

arn)

Mass yield YA:

exponential slope

slope parameter depends on Ap

Heavy-ion-induced fragmentation cross-section datasets

System Energy (A MeV) Laboratory Remarks

40Ar + C 200 LBL pioneering work48Ca + Be 212 LBL -"-58Ni + Be 500 GSI p-rich, very small xsects86Kr + Be 500 GSI large fluctuations129Xe + Al 700 GSI

208Pb + Cu 750 GSI36Ar + Be 1050 GSI p-rich only

112Sn + Be 1000 GSI p-rich only124Xe + Pb 1000 GSI Pb target!136Xe + Pb 1000 GSI Pb target!

40,48Ca + Be 140 MSU new dataset58,64Ni + Be 140 MSU -"-136Xe + Be 1000 GSI n-rich, p-removal86Kr + Be 64 MSU/RIKEN very low energy!

new

EPA

X 1

EPA

X 2

Attempts to improve EPAX

2006: GSI

First attempt to modify EPAX parameters compared to Version 2Slightly better fits, but no drastic improvementProblems occur with 124Xe+Pb

2009: Santiago de Compostela

Second attempt to modify EPAX parametersInclude new datasets from MSU at 140 A MeV

the following examples date from this attempt

Most probable fragments – Zprob(A)

line of ß-stability

136Xe

124Xe

ch

arg

e n

um

be

r Z

loci of largest cross sections, Zprob(A)

evaporation-residue corridor

Zprob(A) can be expressed relative tothe line of -stability:

Z prob(A) = Z ß(A)+ Δ(A) + corr(A,Ap)

depends on n(p)-excess of

projectile

"memory effect":fragments "remember" then(p)-excess of the projectile

Centroid Zprob(A)

line of beta stability

residue corridor

Z-

units

rela

tive

diffe

renc

e to

co

rrid

or

n-rich projectile (136Xe)

n-deficient projectile (124Xe) ??

ß-stable projectiles (129Xe, 208Pb)

Δ

corr Z prob(A) = Z ß(A)+ Δ(A) +

corr(A,Ap)

For A=Ap, Zprob =Zp !

Width parameter R(A)

witd

h pa

ram

eter

R

n-deficient projectile (124Xe) ??

ß-stable projectiles (40Ar,129Xe,208Pb)

n-rich projectiles (86Kr, 136Xe)

Y ~ exp(R|Z p-Z|

Un,p )

For A=Ap, the width must shrink!

A

1 A GeV 36Ar+Be

neutron-deficientfragments only!

new version 3.02

old version 2.1

data:M.Caamano et al.NP A733, 187 (2004)

electromagneticdissociation

1 A GeV 136Xe+Pb

bad fit!

bad fit!

data:D. Henzlova et al.Phys. Rev. C 78, 044616 (2008)

Zσ

(b)

1 A GeV 112Sn+Be

neutron-deficientfragments only!

new version 3.02

old version 2.1

data:A. Stolz et al.PR C 65, 064603 (2002)

50Sn

49In

48Cd

47Ag

46Pd

45Rh

44Ru

43Tc

Proton-removal channels?

good!

bad!

1 A GeV 136Xe+Be

data:J.Benlliure et al.Phys. Rev. C 78, 054605 (2008)

Status and outlook New EPAX fits to old and new data sets give satisfactory results

Parameter dependences of Y A(A) and R(A) yield slightly better

quality than EPAX Version 2

Z prob(A)-dependence for 124 Xe+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:

work in progress at

Mass Yield YA(Ap)

slope parameter P ofmass yield depends on size of projectile:

exponential slope for0.50 < A/Ap < 0.90

new: S ~ S0 . (Ap2/3 + At

2/3) Y(A,Ap) = S P exp(-P(Ap-A))