Comparison of a deterministi rea tion model withan INC modelProdu tion of parti les from rea tions on light target nu leiH. DuarteCEA, DAM, DIF17/06/2015H. Duarte Comparison of a deterministi rea tion model with an INC model

Topi s3 Some Spe ities of the IntraNu lear Cas ade model3 Deterministi approa h for al ulation of double dierential ross se tion3 Results and omparison of non-lo al intera tion and zero-rangeapproximation3 Comparison with INC al ulation3 Summary and next developments

H. Duarte Comparison of a deterministi rea tion model with an INC model

Some Spe ities of the IntraNu lear Cas ade (INC) modelin summary INC is a as ade of in-medium NN ollisions• it needs of NN ross se tions (integrated and dierential)• it in ludes Pauli blo king of s attered nu leons(N1N2 → N3N4)• but INC al ulation is based on mean free path of movingnu leons in nu lear matter

⇒ then a part of the al ulations do not produ e outgoingnu leons : in-medium NN ollisions do not o ur due to Pauliblo king and lead to some transparen y of target nu lei.• this last point (transparen y of target nu lei) is parti ularlypenalizing for rea tions on light target nu lei (Z<10) and"high" omputation time is required to get enough statisti sfor double dierential ross se tions of outgoing nu leons.H. Duarte Comparison of a deterministi rea tion model with an INC model

Deterministi approa h for al ulation of double dierential ross se tionWe fo us here on the al ulation of the energy-angle distribution( d2σ

dEdΩ )lab of outgoing nu leons in the ontinuum after theintera tion of an in ident nu leon with the nu leons of a "dilute"system (light target nu lei).We sum all single intera tions that lead to the sele ted type τout ofoutgoing nu leon( d2σ

dEdΩ

)(τout)

lab=

Atarg+1∑

i

δ(τi − τout)(d2σ(in−med)

dEidΩi

)

labwhere (d2σ(in−med)

dEidΩi)lab is the double dierential ross se tion ofnu leon i after ollision Nproj + Ntarg −→ Ni + Ni+1H. Duarte Comparison of a deterministi rea tion model with an INC model

Cal ulation of (d2σ/dEdΩ)lab (2)Nproj is the in ident nu leon of energy-momentum(~Pproj , Eproj) in plane wave approximation.Ntarg is one of nu leons (neutron or proton) of the targetdes ribed by wave fun tion Φn,l(r2) (harmoni os illator wavefun tions that give realisti matter density ρ(r)) in the fermigas approximation :0 < Ptarg < PF (rtarg) where PF (rtarg) is the lo al Fermimomentum in the potential well Vneut/prot(rtarg) = Vτ (rtarg).(d2σ(in−med)

dEidΩi)lab in ludes Pauli blo king al ulation driven bythe ondition

PBlock(i, i+1) = [1−Heav(Pi−PF (ri))][1−Heav(Pi+1−PF (ri+1))]where PF (ri) is derived from eF (ri) = −Vτi(ri) − |ebind τi

|H. Duarte Comparison of a deterministi rea tion model with an INC model

Cal ulation of (d2σ/dEdΩ)lab (3)For a given position of Ntarg in phase-spa e (~rproj and(~Ptarg , Etarg) xed)

( d2σ

dEdΩ

)

lab(Nproj + Ntarg → Ni + Ni+1) =

Jcm→labdσ

dΩcm(Nproj + Ntarg → Ni + Ni+1)

Jcm→lab = (∂E∂Ω)lab

(∂E∂Ω)cmis the ja obian that transforms theelemental ell ∂Ecm∂Ωcm into ∂Elab∂Ωlab.this term is omputed by a Monte-Carlo method (as it isnaturally done in the INC model) for two reasons :H. Duarte Comparison of a deterministi rea tion model with an INC model

Cal ulation of (d2σ/dEdΩ)lab (4)1) The dependan e of Jcm→lab on ~β12 =~P1+~P2E1+E2

an not be easilyderived (where 1 and 2 are respe tively proj and targ in1 + 2 → 3 + 4 rea tion) apart the ase P2 = 0.in the general ase, ~β12 =

~P1+~P2E1+E2

is not along ~P1 and the al ulation of ja obian an not be done easily (analyti alderivation and "double value" problem when the velo ity ofthe enter-of-mass system is greater than the velo ity ofparti les in the .m.s : 2 dierent kineti energies in laboratoryframe are al ulated at 1 emission angle θlab)⇒ Monte-Carlo integration an do it expli itly without di ulty(but leads to statisti al u tuation).2) we are also interested in a deterministi method that an beeasily "translated" into a pure Monte-Carlo approa h.H. Duarte Comparison of a deterministi rea tion model with an INC model

Cal ulation of (d2σ/dEdΩ)lab (5)The total in-medium double-dierential ross se tion in thelaboratory frame of nu leon i (=3,4) after N1=projN2=targ ollision omes from the integration on ~r2 and ~P2 :(d2σ(in−med)

dEidΩi)lab =

∫

∞

0dr2r

22Φ

2n,l(r2)

∫ PF (r2)

0dp2p

22

∫

Ωp2

dΩp2

PBlock(i, i + 1)Jcm→labdσ

dΩcmwhere dσdΩcm

is the dierential NN ross se tion we used in our INCmodel BRIC (H. Duarte Phys. Rev. C 75, 024611 (2007)) that wastted our experimental data.H. Duarte Comparison of a deterministi rea tion model with an INC model

Comparison of non-lo al intera tion and zero-rangeapproximation2 approximations are presented here (with 1=proj, 2=targ) :zero-range approximation : δ(~r1 − ~r2)

∫

∞

0dr2r

22Φ

2n,l(r2)

∫ PF (r2)

0dp2p

22

∫

Ωp2

dΩp2non-lo al intera tion : |~r1 − ~r2| < r(cut)12 =

√

σ12(~P1, ~P2)/π

∫

∞

0dr2r

22Φ

2n,l(r2)

∫ PF (r2)

0dp2p

22

∫

Ωp2

dΩp2

∫ r(cut)12

0

∫

Ω12

d3r12

H. Duarte Comparison of a deterministi rea tion model with an INC model

(d2σ/dEndΩn)lab vs En for 7Li(p, xn) at 186 MeVzero-range approximation and non-lo al al ulation7Li (p,xn) X (Ep = 186 MeV)

θlab=0o θlab=5o

θlab=10o θlab=15o

θlab=20o θlab=24.5o

Eneut(lab) (MeV)

d2 σ/dΩ

dE

(m

b M

eV-1

sr-1

)

7Li (p,xn) X (Ep = 186 MeV)

θlab=29.4o θlab=34.4o

θlab=39.4o θlab=44.4o

Eneut(lab) (MeV)d2 σ/dΩ

dE

(m

b M

eV-1

sr-1

)zero-range : δ(r1-r2)

non-local : |r1-r2| < ∆rcut

* data of L. Wang et al. Phys. Rev. C 50, 2438 (1994)distribution at 0o is peaked for the al ulation with zero-rangeapproximation H. Duarte Comparison of a deterministi rea tion model with an INC model

(d2σ/dEndΩn)lab vs En for 6Li(p, xn) at 186 MeVzero-range approximation and non-lo al al ulation6Li (p,xn) X (Ep = 186 MeV)

θlab=0o θlab=5o

θlab=10o θlab=15o

θlab=20o θlab=24.5o

Eneut(lab) (MeV)

d2 σ/dΩ

dE

(m

b M

eV-1

sr-1

)

6Li (p,xn) X (Ep = 186 MeV)

θlab=29.4o θlab=34.4o

θlab=39.4o θlab=44.4o

Eneut(lab) (MeV)d2 σ/dΩ

dE

(m

b M

eV-1

sr-1

)zero-range : δ(r1-r2)

non-local : |r1-r2| < ∆rcut

* data of L. Wang et al. Phys. Rev. C 50, 2438 (1994)distribution at 0o is peaked for zero-range approximation as forp + 7Li rea tion due to the lo al Pauli blo kingH. Duarte Comparison of a deterministi rea tion model with an INC model

Comparison with INC al ulationthis method is faster than an INC al ulation :(several minutes ompared to several hours on one CPU withbetter statisti al results : bin width in energy and angle are 1MeV and 1o ompared to 2.5 MeV and 3o for INC al ulation)

⇒ in one hand, se ond and higher-order are missing in thisapproa h while they are impli itly in luded in INC al ulation.⇒ on the other hand, optimization of parameters an be doneeasier

En=0,1,.. : energy levels of inner nu leons of target nu leusr(cut)12 : range of intera tion (sharp ut approximation).

H. Duarte Comparison of a deterministi rea tion model with an INC model

(d2σ/dEndΩn)lab vs En for 6Li(p, xn) at 186 MeVpartial de omposition and omparison with INC6Li (p,xn) X (Ep = 186 MeV)

θlab=0o θlab=5o

θlab=10o θlab=15o

θlab=20o θlab=24.5o

Eneut(lab) (MeV)

d2 σ/dΩ

dE

(m

b M

eV-1

sr-1

)

6Li (p,xn) X (Ep = 186 MeV)

θlab=29.4o θlab=34.4o

θlab=39.4o θlab=44.4o

Eneut(lab) (MeV)d2 σ/dΩ

dE

(m

b M

eV-1

sr-1

)Φn=0

Φn=1

total

INC (BRIC)

* data of L. Wang et al. Phys. Rev. C 50, 2438 (1994)H. Duarte Comparison of a deterministi rea tion model with an INC model

(d2σ/dEndΩn)lab vs En for 7Li(p, xn) at 186 MeVpartial de omposition and omparison with INC7Li (p,xn) X (Ep = 186 MeV)

θlab=0o θlab=5o

θlab=10o θlab=15o

θlab=20o θlab=24.5o

Eneut(lab) (MeV)

d2 σ/dΩ

dE

(m

b M

eV-1

sr-1

)

7Li (p,xn) X (Ep = 186 MeV)

θlab=29.4o θlab=34.4o

θlab=39.4o θlab=44.4o

Eneut(lab) (MeV)d2 σ/dΩ

dE

(m

b M

eV-1

sr-1

)Φn=0

Φn=1

total

INC (BRIC)

* data of L. Wang et al. Phys. Rev. C 50, 2438 (1994)H. Duarte Comparison of a deterministi rea tion model with an INC model

Summary• Preliminary results of this semi-deterministi method areen ouraging when one ompares the results and the time omputations of this method and of INC model for the samenu lear rea tions on light nu lei.• Cal ulation of d2σ

dEdΩ , σdE and σ

dΩ are urrently available for 2type of outgoing parti les (proton and neutron)• and 2-parti les orrelations d3σ

dEdΩ3dΩ4 an be obtained.

H. Duarte Comparison of a deterministi rea tion model with an INC model

Next developments (1)• the urrent method is limited to light target nu lei.• se ond order al ulation (intera tion of parti les 3 and 4 withother nu leons of target nu lei) is ne essary in order to

⋄ al ulate in-medium orre tions,⋄ get upto 22 outgoing parti les,⋄ and in lude heavier nu lei than Li.

• for rea tions on medium target nu lei, higher order al ulationswould be ne essary (in some interative pro edure if possible)H. Duarte Comparison of a deterministi rea tion model with an INC model

Next developments (2)• extension to higher in ident energy will require pion andresonan e degree of freedom but should be more or lessstraight forward as long as the rea tion is on light nu lei (1storder approximation).• another interesting development would be the "translation" ofthis deterministi method into a pure Monte-Carlo al ulation

⇒ to repla e (a part of) our INC al ulation of nu learrea tions on light target nu lei (too high omputation timedue to mean free path al ulation and transparen y of lighttarget nu lei)H. Duarte Comparison of a deterministi rea tion model with an INC model