Within the HINDAS Collaboration
J. CUGNON 1
, TH.AOUST 2
, P. HENROTTE, B. VAN DENBOSSCHE
UniversityofLiege,Physi sDepartmentB5
B-4000SartTilmanLiege1,Belgium
A.BOUDARD, S.LERAY and C. VOLANT
DAPNIA/SPhN,CEA/Sa lay,F-91191Gif-sur-YvetteCedex,Fran e
Abstra t. The HINDAS ollaboration put forward the improvement of the intranu lear as ade
(INC)plus evaporationmodel for spallationrea tions in the 200 MeV to 2GeV range. This e ort
has led to the onstru tion of the re ent version INCL4 of the Liege INC model, whi h is brie y
des ribed here. Current developments of the modelare alsopresented.
1 Introdu tion
One of the tasks of HINDAS was \to develop a theoreti al model for p-indu ed rea tions in the
200 MeV to 2 GeV range, to be validated on ben hmark experiments, and to be in luded in the
parti le transport ode HERMES". The hoi e of the HINDAS ollaboration turned towards the
modelresultingfromthe ouplingof the Liege intranu lear as ade modelINCL 1 3)
and the
Karl-Heinz S hmidt evaporation- ssionmodelABLA 4)
. The present paperreports onthe developments
of the INCL model during the period of the HINDAS program.
Prior toHINDAS,the INCL2 1)
andINCL3 2)
versions ofINCLwereable togivegoodresults for
neutron doubledi erential ross-se tions, but su ered ofsome short omings: (i) they failedonthe
quasi-elasti peak,(ii)residue ross-se tions losetothe targetmasswere underestimated and(iii)
the statisti al implementation of the Pauli blo king led to unphysi al results. We explain below
how these short omings are ured in the new version INCL4 3)
developed within the frame of the
HINDAS program. We will omment on some sele ted important results and say a few words on
the developments of the INCL modeloutside HINDAS.
2 Des ription of INCL4
Let us rst des ribe the main features of the INCL model. Its basi premise is the des ription
of the nu leon-nu leus intera tion as a sequen e of binary ollisions(and de ays) well separated in
spa e-time. Infa t,alltheparti lesarefollowedintime,moveonstraight-linetraje toriesuntiltwo
ofthem rea htheir minimumdistan e of approa h(when ollisionisde ided or noton aminimum
distan e of approa h riterion), or until a parti le hits the surfa e (when it an be transmitted or
re e ted), afterwhi hstraight-line motionis resumed, and so on. Collisionsare subje ted toPauli
blo king,sto hasti ally,a ordingtoaprobabilitylinkedwithphasespa eo upan iesevaluatedon
the neighbourhoodof the andidate nal states of the parti les. Cross-se tions are supposed to be
asinfreespa e. Pionand degrees offreedomare introdu ed. The targetissupposed too upy a
spherewithasharp surfa e,inwhi hnu leonsundergo the e e tof a onstantattra tivepotential.
Animportantaspe tofthe INCLmodelisthe\self- onsistent"determinationofthestoppingtime,
atwhi hanevaporationmodelisto be ranked: the average evolutionof many physi alquantities
exhibits a hange of rate roughly at the same time, whi h is hosen as the stoppingtime.
1
e-mail:J.Cugnonulg.a .be
2
1. Introdu tion of a smooth surfa e. Target nu leons are positioned at random a ording to
a Saxon-Woods distribution for the density (r), ut at R
max = R
0
+8a. Their momentum is
randomlygenerated in a sphere of radius p
F
; r p orrelations are introdu edin order to a ount
for the fa t that fast nu leons travel farther out than slow nu leons. Nu leons with momentum
between pand p+dpare assumed to ontributeto the density (r)by the layer
Æ(r)=((R (p)) (R (p+dp)))
H
(R (p) r); (1)
where
H
isthe Heavisidefun tion. This de nes the orrelating, monotonouslyin reasing, fun tion
R (p). This pres riptionamountstotake rst momentumpatrandomand thenpositionatrandom
inasphereofradiusR (p). Furthermore,parti lesofmomentumparefeelingapotentialof onstant
depthV
0
andofradiusR (p). ItisshowninRef. 3)
thatthisleadstoatotalstabilizationofthetarget
(in terms of r and p distributions) in absen e of ollisions. This pro edure is basi ally equivalent
to putting parti les in a Saxon-Woods potential well, but keeps the simpli ity of the straight-line
motion,whi hallows topropagate parti lesin a singletime step between ollisions.
2. Consistent dynami al Pauli blo king. The ombination of the statisti al generation of the
initial state and the statisti al implementation of the Pauli blo king leads to unphysi al e e ts.
Be ause of u tuations the phase spa e o upan y f (measured by ounting parti les in a small
phasespa e volume) an be smallerthan one, even inthe initialstate. As a onsequen e, even the
rst ollision made by the in oming parti le may lead to a de rease of the energy of the olliding
target parti le, if the foreseen position of the latter in phase spa e is just in a region where f <1.
In su h a ase, the target ex itation energy may be ome negative. In most events this de ien y
is ured by subsequent ollisions. To remove this undesirable e e t we do not allow ollisions
whi h ould lead to a negative ex itation energy of the urrent Fermi sea, i.e. the energy of all
parti les with momentum below p
F
annot be smaller than the ground state of the urrent Fermi
sea, de ned as the energy of the original Fermi sea minus the separation energy of the nu leons
whi h havees aped from it.
3. In ident light lusters. In ident light lusters, up to 4
He, an now be a ommodated by the
INCL4 ode.
4. Improved pion dynami s. The dynami al pi ture of pion produ tion has been improved
on some points. In parti ular, a orre ted detailed balan e formula, whi h a ounts for the nite
lifetimeof the resonan es, is used todetermine the N!NN ross-se tion.
5. Remnant angular momentum. This quantity is provided as an output of the ode. It is
al ulated as the di eren e between the initial angular momentum and the angular momentum
arriedby the eje tiles.
6. Spe tatorsandparti ipants. Spe tatorsaremoving,whi hgeneratesaphysi alrearrangement
of the density, but are not allowed to ollideamong themselves.
Thenumeri al odeINCL4isbasi allyaparameter-free ode. Inputdataare takenfrom
experi-ment(targetradius, ross-se tions, et )or xed on eforall(e.g. Fermimomenta). Te hni allyonly
two parameters (potential depthand stopping time)are leftfree, althoughthey annotreasonably
hanged mu hfrom their optionalvalues.
InRef. 3)
INCL4predi tionsareshowntobequitesu essfulwhen omparedwithalargebodyof
experimentaldatain ludingtotal ross-se tions, neutronandprotondi erential ross-se tions,
neu-tronand hargedparti lemultipli ities,residuemassand hargedistributions,isotopi distributions
The amplitude of the quasi-elasti peakboth in n and p spe tra is well des ribed by INCL4. This
is basi ally due to the introdu tion of a smooth nu lear surfa e, whi h enhan es the rate of single
s atteringevents. However, if thepositionofthe peakiswellreprodu edinpspe tra(see Fig.1),it
islo atedata toohigh energy for n spe tra. This probably re e ts the e e ts of olle tivedegrees
of freedom, whi h show up oherently in single s attering events, but are wiped out in multiple
s attering events, whi hare dominated by in oherent e e ts.
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σ
/d
Ω
dT (
µ
b/sr MeV)
T (MeV)
p(500 MeV)+Ni
θ
= 65
θ
= 90
θ
= 120
θ
= 160
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p(600 MeV)+Ta
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= 30
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= 60
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= 90
θ
= 120
θ
= 150
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Figure1: Comparisonof theINCL4+ABLApredi tionswithexperimentalprotondoubledi erential
ross-se tions for the indi ated systems. Data are from Refs. 5 7)
. Adapted from Ref. 8)
Theresiduemassspe traare onsiderablyimprovedinINCL4(seeRef. 3)
),espe iallyforresidues
withmass lose tothe targetmass. This is alsodue tothe introdu tionofa smoothsurfa e, whi h
enhan estherateofeventswithasmallex itationenergy. The ssionyieldisusuallywelldes ribed,
a result whi h is mainlyattributable to the use of the ssion model ontained in ABLA, but also
tothe ex itationdistribution generated by INCL4.
We would like to omment on the re oil energy of the residues. Its value is overwhelmingly
determined by the as ade stage. It is well des ribed by INCL4, as depi ted by Fig.2. More
importantly, longitudinal re oil velo ities have been measured. It is found 9)
that their mean value
for a given mass loss in reases linearly with this mass loss and that their varian e is also a linear
fun tion of the mass loss (at least for not too large mass losses). Therefore, varian e and mean
value are proportional. This is akinto the ngerprintof a randomwalk pro ess, whi h an readily
be identi ed with the sequen e of su essive ollisions: ea h ollision produ es another re oiling
parti ipantwhi hultimatelytransfersitsmomentumtotheremnant. The observed proportionality
thus provides us with a justi ation of the basi assumption of the INC approa h, namely the
independen e of the su essive ollisions.
Finally,wesay afewwordsaboutthestoppingtime. We he ked that,by hangingthestopping
timebyafewfm/ 's,theresultsarenotmodi edsensitively. Of ourse,ifitis hangedmoresizably,
resultsmay hangedrasti ally. Inparti ular,ifitisdiminishedbymorethan,say,10fm/ 's,neutron
spe tra anbesubstantiallydepletedinthe regionbetween20and50MeV.Itlooksasourstopping
time has just the value whi h makes the introdu tionof an intermediate so- alled pre-equilibrium
stage 10)
0
2
4
6
8
10
12
0
10
20
30
40
50
60
70
∆
A
<
E
rec
>
(MeV)
Model
Experiment
Figure 2: Comparison of the INCL4+ABLA predi tions (triangles) with experimental average
residue re oil energy( ir les), as a fun tion of the mass loss. Data are from Ref. 9)
. Adapted from
Ref. 3)
evaporationpart, see Fig. 3). Perhaps itis appropriateto say that our as ade modelexhibits two
important features: itis able to des ribe the features usually attributed to pre-equilibriummodels
(ex ept for luster emission, but see below the extensions of our model) and it pi ks up (by the
'self- onsistent' determination of the stopping time) the time at whi h the system is suÆ iently
randomizedfor evolvingfurther by evaporation.
4 Developments outside HINDAS
4.1 Cluster emission
Sin e this topi is largely overed by A. Boudard 12)
in his ontribution at this meeting, we just
explain brie y the model used to a omodate luster emission during the as ade stage. When a
parti le is he ked for leaving the nu leus (and provided it has full lled the test for transmission
through the Coulomb barrier), it is veri ed whether it an drag along a luster, in whi h it is
embedded. A luster isde ned asa groupof nu leons losetoea h otherin phasespa e. A tually,
the andidate lusteris onstru ted, startingfromthe onsideredparti le,by ndingase ond,then
athird, et , nu leon satisfying the following ondition
r i;[i 1℄ p i;[i 1℄ P 0 (2)
on the Ja obian oordinates of the i-th nu leon, i.e. the relative oordinates of this nu leon with
respe t to the subgroup onstituted of the rst [i 1℄ nu leons. A luster is emitted if its
(ki-neti +potential) energy is suÆ ient to give an asymptoti bound luster with positive kineti
en-ergyand if itsu eeds the test for transmission through the Coulomb barrier. Of ourse, energy is
onserved duringthe emissionof the luster.
Clusters up to 4
He are onsidered. Ifalarge luster isbuilt and if itdoes not su eed the tests
for emission,the next largest luster in the buildingpro ess isthen tested for emission,and soon.
Inother words, weadoptthe following hierar hy: 4
He> 3
He;t>d>n;p. Finally,asnu leonsare
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/d
Ω
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n
(mb/sr MeV)
T
n
(MeV)
p (1200 MeV) + Pb
0
0
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0
(10
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0
(10
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0
(10
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0
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(10
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0
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0
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0
(10
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)
145
0
(10
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)
160
0
(10
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)
Figure3: Comparison of the INCL4+ABLA predi tionswith experimentalneutron double di er
en-tial ross-se tions for the indi ated system. Data are from Refs. 11)
. Adapted from Ref. 8)
small,we get itba k by asmalldistan e dbeforebuildingthe luster. In summary, this modelis a
sortofsurfa e oales en emodel,relying onthe instantaneousphase-spa eo upan y,dynami ally
generated by the INC itself. It is quitedi erent frompure (momentumspa e) oales en e.
We did not try to t the parameters P
0
and d. The adopted values are P
0
=387 MeV/ and
d=2.6fm. They allowustogiveareasonablea ountof lusterprodu tion ross-se tions, asshown
inRef. 12)
.
4.2 Low-energy limit of validity of INCL
Itis ommonlystatedthat separabilityof the su essive nu leon-nu leon ollisions annotbe
guar-anteed for in ident energies below 200 MeV. O asionally INC models have been used at lower
energy, with some su ess. In a re ent work 13)
, the validity of INC has been systemati ally tested
below200 MeV, justby omparing the INCLpredi tions with the existingdata for n andp double
di erential ross-se tions. The on lusionsofthisinvestigationare: (i)totalrea tion ross-se tionis
underestimatedunder100MeV;(ii)predi tedenergyandangulardistributions,downto40-50MeV
in ident energy, are generally well des ribed, provided a stri t Pauli blo king (instead of the
sta-tisti alimplementation)is adopted; (iii) better resultsare obtained when Pauliblo king isstri tly
enfor ed for the rst ollision and statisti ally for the su eeding ones; (iv) in many ases, INCL3
gives as good results (even, sometimes, better) as multi-step dire t models. The latter, somewhat
surprising, result presumably arises from the fa t that, for multiple ollisions, what really matters
isthe average energy-momentum ow, whi h an bedes ribed by mean ( lassi al)traje toriesand
The isospin dependen e of the mean eld is introdu ed by adopting di erent Fermi momenta k i
F ,
i=n;pand by requiringthe Fermi energies to be equaltominus the separation energies S
i : (hk i F ) 2 2M V i = S i ; (3) whereV i
isthe potentialdepth. Thefollowingenergydependen e V
i = i i E isintrodu ed,with
parameters taken from the opti al potential phenomenology 14)
(see Ref. 15)
for details). The main
e e tofthesedependen esistodrivethequasi-elasti peaktowardslowerenergiesinneutronspe tra
atsmallangles,partly uringoneoftheshort omingsofINCL.Thise e thasbeen on rmedbyan
analyti al model assuming single s attering 16)
, whi h is a good approximation in the quasi-elasti
region.
5 Con lusions
In the frame of the HINDAS ollaboration, the Liege INCL model has rea hed a high level of
predi tability in the 200 MeV to 2 GeV range. It has been onfronted su essfully with a large
body of experimental data, espe ially with those obtained by the HINDAS groups. This new
version (INCL4) has been introdu ed or willsoon be introdu ed inthe transport odes HERMES,
LAHET3andMCNPX.TheINCLmodelisunder onstantimprovement. Weonlymentionedthree
of them, related to luster emission, extension to low energy and detailed properties of the mean
eld.
A knowledgments.
ThedevelopmentoftheINCL4model,thebuildingoftheINCL4 ode,itsin lusionintheHERMES
transport ode and some further studieshave been performed inthe frameof the HINDAS
ollab-oration (European Union Contra t N Æ
FIKW WCT-2000-00031). We are grateful to our HINDAS
olleagues for helpful dis ussions and for havingparti ipated tovarioustests of INCL4.
Referen es.
1. J. Cugnon, C. Volant and S.Vuillier, Nu l. Phys. A620(1997) 475.
2. J. Benlliure etal, Nu l. Phys. A700(2002) 469.
3. A. Boudard et al,Phys. Rev. C66(2002) 044615.
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5. R. E. Chrienet al,Phys. Rev. C21(1980) 1014.
6. K. R.Cordell etal, Nu l. Phys. A352(1981) 485.
7. G. Roy etal, Phys. Rev. C23(1981) 1671.
8. A. Boudard et al, ontributiontothe A App03Conferen e, San Diego,2003.
9. T. Enqvistet al,Nu l. Phys. A686(2001) 481.
10. K.K. Gudima,S. G.Mashnik and V.D. Toneev, Nu l. Phys. A401 (1983)329.
11. S.Leray etal,Phys. Rev. C65 (2002) 044621.
12. A. Boudard etal, ontribution tothis meeting.
13. J. Cugnon and P.Henrotte, Eur. Phys. J. A16 (2003) 393.
14. P. E. Hodgson, \The Nu leon Opti alPotential", World S ienti , Singapore, 1994.
15. T. Aoust and J. Cugnon, Preprint University of Liege, September 2003.
