Table Of ContentMultiple s
attering e(cid:27)e
ts in quasi free s
attering from halo nu
lei: a test to
Distorted Wave Impulse Approximation
1,2 1 1 1 1
R. Crespo, A. Deltuva, E. Cravo, M. Rodríguez-Gallardo, and A. C. Fonse
a
1
Centro de Físi
a Nu
lear, Universidade de Lisboa,
Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
2
Departamento de Físi
a, Instituto Superior Té
ni
o, Taguspark,
Av. Prof. Cava
o Silva, Taguspark, 2780-990 Porto Salvo, Oeiras, Portugal
(Dated: February 4, 2008)
11
8 Full Faddeev-type
al
ulations are performed for Be breakup on proton target at 38.4, 100,
0
and 200 MeV/u in
ident energies. The
onvergen
e of the multiple s
attering expansion is inves-
0
tigated. The results are
ompared with those of other frameworks like Distorted Wave Impulse
2
Approximationthatare based on anin
omplete and trun
atedmultiple s
attering expansion.
n
a PACSnumbers: 24.50.+g
J
9
2 I. INTRODUCTION aim of the present work.
We
onsider here the quasi free s
attering of the A
] nu
leus by a proton target leading to the breakup rea
-
h Interest in quasi free breakup of a Halo nu
leus stems n−1
t mainly from the stru
ture information about the wave tion11p(A, A n)p. We take as an example the breakup
- of Be halo into a proton target at 38.4, 100, and 200
l fun
tionofthestru
kvalen
eneutron. Qualitatively,the
c quasifrees
atteringisapro
ess[1,2℄inwhi
haparti
le MeV/u. 11
u
Re
ently the elasti
s
attering of Be from a proton
n kno
ks a nu
leon out of the nu
leus (A) and no further
n−1 target was analysed using the Faddeev/AGS multiple
[ strongintera
tiono
ursbetweenthe(A )nu
leusand
s
attering approa
h [7℄. Its was shown that even at in-
the two outgoing parti
les.
1 termediate energiesof 200MeV/u oneneeds to takeinto
v Assumingn−th1atthenu
leus(A)
anbewelldes
ribedby a
ount 3rd order multiple s
attering
ontributions that
9 aninert(A )
oreandavalen
eneutron,thisrea
tion
arenotin
ludedintheGlaubers
atteringframework[8℄.
8
anbestudied usingthefew-bodyFaddeevorequivalent
In this work we brie(cid:29)y review the Faddeev/AGS mul-
4 Alt, Grassberger, and Sandhas (AGS) s
attering frame-
4 tiple s
attering framework in se
tion II. In se
tion III
work[3,4,5℄whi
h
anbeviewedasamultiples
attering
. wesumarizetheDWIAformalismandidentify theterms
1 seriesintermsofthetransitionamplitudesforea
hinter-
of the
orresponding multiple s
attering expansion. In
0 a
ting pair. Both equations are
onsistent with the
or-
se
tion IV we show the
al
ulated (cid:28)vefold breakup
ross
8 responding S
hrödinger equation and therefore provide
0 se
tion using the Faddeev/AGS equations and
ompare
anexa
tdes
riptionofthequantumthree-bodyproblem
: with the results of the DWIA type.
v but are more suitable for the numeri
al solution. The
i Faddeev equations areformulated for the
omponents of
X
the wave fun
tion, while the AGS equations are integral
II. THE FADDEEV/AGS EQUATIONS
r
a equations for the transition operatorsthat lead more di-
re
tly to the s
attering observables.
In this se
tion, for
ompleteness we brie(cid:29)y des
ribe
Traditionallythisrea
tionhasbeenanalysedusingthe
the Faddeev/AGS multiple s
attering framework whi
h
DistortedWaveImpulseApproximation(DWIA)s
atter-
treats all open
hannels in the same footing. Let us
ingframework[6℄. Inthisapproa
honeassumesthatthe
onsider 3 parti
les (1,2,3) intera
ting by means of two-
in
oming parti
le
ollides with thestru
k nu
leonasif it
body potentials. In this se
tion we use the odd man out
was free, and therefore the
ross se
tion for the quasi
notation appropriate for 3-body problems whi
h means,
free s
attering is given by the produ
t of the transition
for example, that the intera
tion between the pair (2,3)
v
amplitude betweenthe two
ollidingparti
lesmodulated 1
is denoted as . We assume that the system is non-
by the wave fun
tion of the nu
leon inside the nu
leus.
relativisti
and we write its total Hamiltonian as
Furthermoretherelativemotionoftheparti
lesintheen-
H =H0+Xvγ ,
tran
eand exit
hannelsisdes
ribed by distortedwaves. (1)
γ
Thisapproximationleads,inea
horder,toanin
omplete
and trun
ated multiple s
attering expansion. H0
with the kineti
energy operator and the intera
tion
v γ
Inordertoobtaina
urateinformationaboutthewave γ
for the pair . The Hamiltonian
an be rewritten as
fun
tion, one needs to investigate the e(cid:27)e
t of higher or- H =H +Vα ,
α
der res
attering between the two
oliding parti
les and (2)
the (An−1) nu
leus in the
al
ulated kno
kout observ- Hα α
where is the Hamiltonian for
hannel
ables. In addition, the
ontribution of the negle
ted
H =H +v ,
α 0 α
terms in ea
h order needs to be assessed. This is the (3)
2
Vα
and represents the sum of intera
tions external to
α
partition
+
Vα = Xvγ .
(4)
γ6=α
α = 0
The partition
orrVes0ponds to three free parti
les
in the
ontinuum where is the sum of all pair inter-
FIG.1: Singles
atteringdiagramsforbreakupintheFaddeev
a
tions. The total transition amplitude whi
h des
ribes
α β
s
attering framework.
thes
atteringfromtheinitialstate to the(cid:28)nalstate
isgiveninthepostform(seeRefs.[5,7℄foraderivation)
T+βα =Vβ + XT+βγG0tγ (5) waves
orresponding to the relative motion of three free
γ6=α parti
lesthatmaybeexpressedinanyoftherelativeJa-
G
o−b1i variables. In the latter
ase the
ontribution of the
0 term is zero.
with the transition amplitude
The solution of the Faddeev/AGS equations
an be
t =v +v G t
γ γ γ 0 γ (6) found by iteration leading to
U0α = Xtγδ¯γα+XtγXG0δ¯γξtξδ¯ξα
and the free resolvent
γ γ ξ
G =(E+i0−H )−1,
0 0 (7) + XtγXG0δ¯γξtξXG0δ¯ξηtηδ¯ηα
E γ ξ η
where is total energy of the three-parti
le system.
+ ··· ,
Equivalently in the prior form we write (13)
T−βα =Vα+XtγG0T−γα . (8) where the series is summed up by the Padé method [9℄.
γ6=β
The su
essive terms of this series
an be
onsidered
Uβα as (cid:28)rst order (single s
attering), se
ond order (double
Onethende(cid:28)nestheoperators whi
hareequivalent s
attering) and so on in the transition operators. The
to the transition amplitudes on the energy shell, su
h breakup series up to third order is represented diagra-
that mati
allyinFigs. 1-3wheretheupperparti
leistaken
Uβα = δ¯ G−1+Tβα as parti
le 1 s
attering from the bound state of the pair
βα α +
(23).
= δ¯ G−1+Tβα ,
βα β − (9) In our
al
ulations Eqs. (11 - 13) are solved exa
tly
afterpartialwavede
ompositionanddis
retizationofall
δ¯ =1−δ
βα βα momentumvariables. Sin
e,inadditiontothenu
learin-
where . From Eqs. (5-8)
Uβα =δ¯βαG−01+XUβγG0tγδ¯γα , (10) tinetrear
at
iotniobnebtweteweneeanlltthhereperpoatoirns,awnedin10
Blued,ewtehefoClloouwlotmhbe
γ te
hni
al developments implemented in Refs. [10, 11℄ for
α
proton-deuteron and -deuteron elasti
s
attering1a1nd
or breakup that were also used in Ref. [7℄ to study p- Be
Uβα =δ¯βαG−01+Xδ¯βγtγG0Uγα . (11) elasti
s
attering. ThetreatmentoftheCoulombintera
-
γ tionintheframeworkofthree-bodyFaddeev/AGSequa-
tionsisbasedonthemethodofs
reeningoftheCoulomb
intera
tion plus renormalization[10, 11℄; it leadsto fully
These are the well known three-body AGS equations [4℄
β =0
onverged results for the on-shell elasti
, transfer and
whi
h for breakup ( in the (cid:28)nal state) be
ome
breakup observables that are independent of the
hoi
e
U0α =G−01+XtγG0Uγα, (12) of s
reening radius usedRin≥th1e0
al
ulations, as long as
γ it is su(cid:30)
iently large ( fm for the observables
onsidered in the present work).
Uγα
where isobtainedfromthesolutionofEq.(11)with
α,β,γ =(1,2,3)
U. βTαhes
atteringamplitudesarethema-
trix elements of
al
ulated between initial and (cid:28)nal III. THE DWIA/PWIA
states that are eigenstates of the
orresponding
hannel
H (H ) A(a,ab)B
α β
Hamiltonian with the same energy eigenvalue. Let us
onsider the rea
tion where an in
i-
a
For elasti
s
attering and transfer those states are made dent parti
le kno
ks out a nu
leon or a bound
lus-
α(β) b A
upbytheboundstatewavefun
tionforpair timesa ter in the target nu
leus resulting in three parti
les
α(β) α(β) (a,b,B)
relativeplanewavebetweenparti
le andpair . in the (cid:28)nal state. This rea
tion has been ana-
For breakup the (cid:28)nal state is a produ
t of two plane lyzed within the DWIA as des
ribed for example in the
3
V V
aB bB
The potentials and are taken to be the opti
al
a+B b+B
potentials whi
hdes
ribethe and s
attering
+ + ǫ ǫ
aB bB
atenergies and respe
tively. Thereforeonewrites
T ∼hη(−)η(−)|t |φ η(+)i.
AB aB bB ab Bb aA (19)
If we now introdu
e plane waves in the exit
hannel
FIG. 2: Double s
attering diagrams for breakup in the Fad- T ∼hχ χ |(1+t G )(1+t G )t |φ η(+)i,
deevs
attering framework. AB aB bB aB 0 bB 0 ab Bb aA (20)
we then obtain the trun
ated multiple s
attering series
t +t G t +t G t +t G t G t .
+ + ab aB 0 ab bB 0 ab aB 0 bB 0 ab (21)
The (cid:28)rst order term is the single s
attering represented
diagramati
allyin Fig. 4. This is the only term retained
+ + in Plane Wave Impulse Approximation (PWIA)
al
u-
lations. The se
ond and third order terms in Eq. (21)
represented in Fig. 5 are due to distortion in the exit
hannel. The relative importan
e of these terms to the
+ +
s
attering
anbe assessed by performing Faddeev multi-
ple s
attering
al
ulations.
Thedistortionintheentran
e
hannelismoredi(cid:30)
ult
to bridge with few-body
al
ulations. In order to esti-
matethis distortionin the
ase ofa lightemited parti
le
FIG.3: Triples
atteringdiagramsforbreakupintheFaddeev one
ould say for instan
e, that
s
attering framework.
η(+) ∼(1+G t )χ .
aA 0 aB aA (22)
workof Chantand Roos[6℄. Thisis an approximatefor- Thisleadstoadditionalmultiples
attering
ontributions
malism and its validity in the appli
ation to radioa
tive t G t +t G t G t +t G t G t +O(t4).
ab 0 aB aB 0 ab 0 aB bB 0 ab 0 aB
beam studies needs to be assessed. In the work of ref-
eren
e [6℄ it is assumed that the proje
tile stru
ks the (23)
eje
ted parti
le freely and that the transition amplitude
is given by These terms are represented in Fig. 6. We note that the
T =hη(−)|t |φ η(+)i, distortion on the entran
e and exit
hannels
ontribute
AB Bab ab Bb aA (14) to a trun
ated and in
omplete multiple s
attering series
φ (Bb) at ea
h order.
Bb
where is the bound state wave fun
tion, and
η(+) η(−) We
on
entrate on the study of the multiple s
atter-
aA and Babdes
ribetherelativemotionoftheparti
les ing expansion. In addition, we want to investigate if the
intheentran
eaVndexit
hannels,respe
tively,negle
ting single s
attering term be
omes the dominant
ontribu-
ab
ηth(+e)intera
tion . Therefore in the entran
e
hannel tion astheenergyin
reases. Itwasshownin theworkof
aA satis(cid:28)es Chant and Roos [6℄ that,
ontrary to what is expe
ted,
(T +V −V −ǫ )η(+) =0, the DWIA results do not
onverge to the PWIA results
aA aA ab aA aA (15) as the proje
tile energy in
reases. The importan
e of
ǫ T in
luding the distortionin
omparisonwith the full Fad-
aA aA
where is the relative kineti
energy and the rel- deev multiple s
attering series is also estimated in the
ative kineti
energVy op−eraVtor.∼InVsta(n~rdar)d DWIA
al
u- present work. We refer as DWIA-full a multiple s
at-
aA ab aB aA
lations one takes . For the exit tering
al
ulation that in
ludes the multiple s
attering
hannel one assumes as aηg(−oo)d approximation to write terms in Eqs. (21) and (23). The
al
ulations based on
the 3-body wave fun
tion Bab as a fa
torized produ
t these equationsfollowexa
tly the samepro
edureasex-
η(−) ∼η(−)η(−), plainedbeforeforthesolutionoftheFaddeev/AGSequa-
Bab aB bB (16) tions(11)through(13), both in termsofinitialand (cid:28)nal
states as well as treatment of Coulomb, so that one may
where
obtain a
onsistent
omparison between di(cid:27)erent terms
(TaB +VaB −ǫaB)ηa(+B) =0, (17) in the multiple s
attering representation of DWIA and
Faddeev/AGS amplitudes.
and Wenotethatfurtherapproximationsareusuallymade
(T +V −ǫ )η(+) =0. in standard appli
ations of the DWIA when evaluating
bB bB bB bB (18) the transition amplitude, namely:
4
1.80×104
2) 38.4 MeV θn=15o SDSS
V sr θ10Be=0o MS
e
FfrIaGm.ew4:orSki.ngle s
attering diagram in the DWIA s
attering mb/M 1.20×104 θn=30o
S ( θn=45o
d
2
Ω
d 6.00×103
1
+ Ω
d
σ/
5
d
0.00
0 30 60 90 120
S (MeV)
FIG.5: Diagramsduetothedistortionintheexit
hannelin
theDWIAs
attering framework.
F11IG. 7: (Color online) Cross se
tion for the breakup
Be(p,pn) at 38.4 MeV.
•
The potential approximationin the entran
e
han-
VaA−Vab ∼VaB(~raA) 8.00×103
nel .
• 2) 100 MeV θn=15o SDSS
The fa
torizationapproximation, whi
h is only ex- sr MS
a
t in PWIA. eV 6.00×103 θ10Be=0o
M
• On-shell approximation of the transition ampli- mb/ θ=30o
tude. dS (24.00×103 θn=45o n
Ω
We referto this asDWIA-standard. An attempt to esti- d
1
mate the e(cid:27)e
t of su
h approximations was done for ex- Ωd 2.00×103
ample in the work of Ref. [12℄ for (p,2p) rea
tions. The σ/
5
Faddeev
al
ulationsandtheDWIA-fulldonotmakeuse d
ofsu
happroximationsinthesummationofthemultiple 0.00
50 100 150 200 250 300
s
attering expansion, and therefore we are not address- S (MeV)
ingherethevalidityofsu
happroximationswhi
harere-
sponsibleforadditionalshort
omingsofDWIA-standard
vis-a-vis the exa
t
al
ulations. F11IG. 8: (Color online) Cross se
tion for the breakup
Be(p,pn) at 100 MeV.
IV. RESULTS
asde1s0
ribed in [7℄. Forthe potential betweenthe proton
We now pro
eed to solve Faddeev/AGS equations (11 and Be
ore we use a phemenologi
al opti
al model
-1113) and to
al
ulate the observables for the proton- with parameters taken from the Watson global opti
al
Be breakup. As mentioned above the treatment of the potential parametrization[7, 15℄. The energy dependent
Coulomb intera
tion, as well as the
al
ulational te
h- parameters of the opti
al potent1ia1l are taken at the pro-
nique, is taken over from Refs. [10, 11℄. 10 ton laboratory energy of the p- Be rea
tion in inverse
1F0irstwedes
ribethepairintera
tionsp-n, p- Beand kinemati
s.
n- Be. Forp-nwetaketherealisti
nu
leon-nu
leonCD Inthe solutionof the Faddeevequationswe in
lude n-
Bonnpotential[1130℄. Theintera
tionbetweenthevalen
e pℓ pa≤rti8al w1a0ves up to rℓel≤ati3ve total a10ngular momLe≤ntu1m0
np
neutronand the Be
oreisassumedto beL-dependent , n- Be up to , and p- Be up to .
Three-body total angular momentum is in
luded up to
100.
As a (cid:28)rst result we show indF5iσg/s.dΩ7(cid:21)9dΩthdeS(cid:28)vefold dif-
1 2
ferentialbreakup
rossse
tion versusthe
+ + S E /A =
lab
ar
length at 38.4 MeV, 100 MeV, and 200
MeV respe
tively. The kinemati
al
on(cid:28)gurations are
(θ ,φ )
i i
hara
terized by the polar and azimuthal angles
of the two dete
ted parti
les labeled 110and 2 in Fig. 10.
We assume those parti
les to be the Be
ore (1) and
FIG.6: Diagramsduetothedistortioninthein
oming
han- the neutron (2). We take thr(eθe ,qφua)si=fre(e0◦s,
0a◦t)tering
nel in the DWIAs
attering framework. 1 1
on(cid:28)gurations: (
onf 1) with and
5
2MeV sr) 65..0000××110033 20θ100B eM=0eoV θn=15o SDMSSS 2eV sr) 1.80×104 38θ1.04Be =M0oeV θn=15o SSSS ((nnpp )+ pcore)
ΩΩddS (mb/12432...000000×××111000333 θn=45o θn=30o ΩddS (mb/M1216..2000××110043 θn=45oθn=30o
d Ω
5σ/ 1.00×103 σ/d
d 5
d
0.00
200 250 300 350 400 450 500 0.00
0 30 60 90 120 150
S (MeV)
S (MeV)
F11IG. 9: (Color online) Cross se
tion for the breakup
F11IG. 11: (Color online) Cross se
tion for the breakup
Be(p,pn) at 200 MeV.
Be(p,pn) at 38.4 MeV.
y
8.00×103
θ=15o
2sr) 100 MeV n SSSS ((nnpp )+ pcore)
p ϕ12 pp eV 6.00×103 θ10Be=0o
1 M
ϕ1 ϕ2 mb/ θn=30o
θ1 p2 x S ( 4.00×103 θn=45o
θ2 Ωd2
d
1
Ωd 2.00×103
z σ/
p 5d
3
0.00
50 100 150 200 250 300
S (MeV)
FIG. 10: (Color online) Kinemati
angles for breakup.
F11IG. 12: (Color online) Cross se
tion for the breakup
(θ ,φ ) = (45◦,180◦) (θ ,φ ) = (0◦,0◦)
2 2 1 1 Be(p,pn) at 100 MeV.
(θ ,φ ) = (30◦,;1(8
0o◦n)f 2) with (θ ,φ ) =
2 2 1 1
a(0n◦d,0◦) (θ ,φ ) = (15;◦,(1
8o0n◦f)3) with S
2 2
and . The ar
length
E
i
is related to the energies of the two deteS
ted par- produ
es well the full
al
ulation at 200 MeV/u energy.
S = dS
ti
les in the standard way [14℄ as R0 with ThereforetheFaddeev/AGSmultiples
atteringseriesfor
dS = (dE2 +dE2)1/2
1 2 . In ea
h
on(cid:28)guration, the solid breakup has
onverged at se
ond order level at this en-
ergy.
lines represent the full multiple s
attering (MS) results
obtained from the solution of Faddeev/AGS equations. Nowwe
ompareFaddeev/AGSandDWIA-fullresults
The single s
atteringapproximation(SS) represented by for the breakup observables. We begin by analyzing
thedashedlinesis
al
ulatedtakingintoa
ountthepro- the single s
attering term in Figs. 11(cid:21)13. The breakup
ton s
attering from the stru
k nu
leon and the s
atter-
rossse
tion
al
ulatedtakingintoa
ountbothproton-
ing from the valen
e
ore as represented by diagrams in neutron and proton-
ore
ontributions is represented by
Fig. 1. As follows from Fig. 7 the SS aproximationover- the solid line. The
ir
les
orrespond to the PWIA, i.e.,
estimates the exa
t result in any
on(cid:28)guration at 38.4 without the proton-
ores
attering
ontribution. At 38.4
MeV/u. At 200 MeV/u, the SS already provides a good MeV/u this
ontribution is very small in the
on(cid:28)gura-
representation of the full
al
ulation in the
ase of
onf tions1and2,butissizablewhentheneutronisemmited
1, but in
reasingly overestimates MS in
onf 2 or
onf 3 in the forwardregion. Therefore, even at thesingle s
at-
where the neutron is emitted at smaller angles. Thus, teringlevel,
al
ulationsbe
omeinna
urateinsome
on-
ontrary to what is predi
ted by Chant and Roos [6℄, if (cid:28)gurations at low energies. At high energies the proton-
the neutron is not dete
ted at very forward angles, the
ores
attering
ontribution isnegligiblein all
on(cid:28)gura-
single s
attering approa
hes the full
al
ulation as the tions.
beam energy is in
reased. The double s
attering (DS) Sin
e, as shown in Figs. 7(cid:21)9, the DS results are very
approximation,representedbythe
ir
les,and
al
ulated
losetothefullMSresults,we
omparenextinFigs.14(cid:21)
taking into a
ount the diagrams of Figs. 1 and 2, re- 16 the predi
tions of Faddeev/AGS and the DWIA-full
6
6.00×103 8.00×103
θ=15o θ=30o
2b/MeV sr) 54..0000××110033 2θ0100Be =M0oeV θ=30no SSSS ((nnpp )+ pcore) 2mb/MeV sr) 6.00×103 3θ810.B4e= M0oeV θn=45o n θn=SDD1SWW5+oIIDAAS --F ddaiidssd eeexxeiivtt+in
dS (m 3.00×103 θ=45o n ΩdS (24.00×103
Ω2 n d
Ωdd12.00×103 σΩ/d12.00×103
σ/ 1.00×103 5d
5
d
0.00
0 30 60 90 120
0.00 S (MeV)
200 250 300 350 400 450 500
S (MeV)
F11IG. 14: (Color online) Cross se
tion for the breakup
F11IG. 13: (Color online) Cross se
tion for the breakup Be(p,pn) at 38.4 MeV.
Be(p,pn) at 200 MeV.
6.00×103
θ=15o
2sr) 5.00×103 100 MeV n SDDSWW+IIDAAS --F ddaiidssd eeexxeiivtt+income
approa
h in
luding in both
al
ulations the
orrespond- mb/MeV 4.00×103 θ10Be=0o θn=30o
ing multiple s
attering terms up to se
ond order. As S ( 3.00×103 θn=45o
d
explainedbeforetheDWIA-fullapproa
hprovidesanin- Ω2
omplete series in all orders, thought in (cid:28)rst order the Ωd12.00×103
e(cid:27)e
ts of the missing term are negleagible at high ener- d
gies as shown in Figs. 11(cid:21)13. 5σd/ 1.00×103
0.00
The Faddeev/AGS (solid line) results in
lude all the 50 100 150 200 250 300
S (MeV)
single and double s
attering terms shown in Figs 1(cid:21)2.
TheDWIA-fullassumesthatthe
ontributionofthes
at-
tering between the proton and the neutron is dominant F11IG. 15: (Color online) Cross se
tion for the breakup
and therefore the single s
attering
ont1r0ibution due to Be(p,pn) at 100 MeV.
the s
attering of the proton from the Be
ore is ne-
gle
ted. The
ir
les in
lude the (cid:28)rst and se
ond order
diagramsoriginatedfromthedistortionsintheexit
han- V. CONCLUSIONS
nel representedin Fig. 5, and the dashed
urvein
ludes,
11
in addition, the se
ond order diagrams originated from We have
al
ulated the breakup
ross se
tion for Be
the distortion in the in
oming
hannel represented in breakupfromprotonsat38.4,100and200MeV/u. Quasi
Fig. 6. The DWIA-full results to se
ond order (dashed frees
attering
onditionsare
onsideredin di(cid:27)erent
on-
line) provideapoorapproximationto the Faddeev/AGS (cid:28)gurations for the emitted neutron. The Faddeev/AGS
at38.4MeVinany
on(cid:28)guration. Astheenergyin
reases
DWIA-full approa
hes the exa
t result. However one 5.00×103
should keep in mind that DWIA-standard
al
ulations θ=15o
assume further approximations as dis
ussed in Se
t. III. 2eV sr) 4.00×103 2θ0100B eM=0eoV n SDDSWW+IIDAAS --F ddaiidssd eeexxeiivtt+income
M
In Figs. 9 and 11(cid:21)13 we show that in the high en-
ergyregimePWIAapproa
hestheexa
tresultatleastin mb/ 3.00×103 θn=30o
esonmeregysurietgaibmlee,birneaokrduepr
toong(cid:28)egtuararteiloinabs.leTphreerdeif
otrioe,ninoftthhies ΩdS (22.00×103 θn=45o
d
breakupobservablesinquasifrees
attering,weadvo
ate Ω1
that a better approa
h is to work with the PWIA s
at- σ/d 1.00×103
tering framework within suitably
hosen
on(cid:28)gurations 5d
rather than in DWIA where un
ontrolable approxima- 0.00
250 300 350 400 450 500
tionsaremade. Onthe
ontrary,in PWIA thefa
toriza- S (MeV)
tion is exa
t and the transition amplitude for the s
at-
tering between the in
ident and kno
ked parti
le is on
theenergyshellintheweakbindinglimitofthekno
ked F11IG. 16: (Color online) Cross se
tion for the breakup
Be(p,pn) at 200 MeV.
parti
le.
7
formalismisusedtoshowthatattheseenergiesthemul-
ontrolableapproximationssu
hasthefa
torizationand
tiple s
attering expansion has
onverged at the se
ond on-shellapproximations,oneadvo
atesthatatthesehigh
order. Wehavealsoshownthatathighin
identenergies energies it is adviseable to use the exa
t approa
h or in-
the single s
attering (PWIA) approa
hes the full
al
u- stead the PWIA in suitable
on(cid:28)gurations.
lation in some suitable
on(cid:28)gurations, namely when the A
knowledgements: The work of R. Crespo is
neutron is not emitted at forward angles. We have also supported by the Fundação para a Ciên
ia e Te
nologia
shownthattheDWIA-fulls
atteringframeworkprovides (FCT), through grant No. POCTI/FNU/43421/2001.
a poorapproximationof the Faddeev/AGSformalism at A. Deltuva is supported by the FCT grant
low energies, but approa
hes it at high energies. How- SFRH/BPD/34628/2007 and all other authors by
ever sin
e standard appli
ations make further use of un- the FCT grant POCTI/ISFL/2/275.
[1℄ G. Ja
ob, TH. A.J. Maris, Rev. Mod. Phys. 38, 121 and I.J. Thompson,Phys. Rev. C 75, 024608 (2007).
(1966). [9℄ G.A.BakerandJ.L.Gammeleds.,The PadéApproxima-
[2℄ G.Ja
ob,TH.A.J.Maris,Rev.Mod.Phys.45,6(1973). tioninTheoreti
al Physi
s(A
ademi
,NewYork,1970).
[3℄ L.D.Faddeev,Zh.Eksp.Theor.Fiz.39,1459(1960)[Sov. [10℄ A.Deltuva,A.C. Fonse
a, andP.U.Sauer,Phys.Rev.C
Phys.JETP 12, 1014 (1961)℄. 71,054005(2005);72,054004(2005);73,057001(2005).
[4℄ E.O. Alt, P. Grassberger, and W. Sandhas, Nu
l. Phys. [11℄ A. Deltuva, Phys. Rev. C 74, 064001 (2006).
B 2, 167 (1967). [12℄ N. Kanayama, Y. Kudo, H. Tsunoda, and T. Wakasugi,
[5℄ W. Glö
kle, The Quantum Me
hani
al Few-Body Prob- Progress of Theoreti
al Phys83, 540 (1990).
lem (Springer-Verlag, Berlin/Heidelberg, 1983). [13℄ R. Ma
hleidt, Phys. Rev. C 63, 024001 (2001).
[6℄ N.S. Chant and P.G. Roos, Phys.Rev. C 15, 57 (1977). [14℄ K. Chmielewski et al., Phys.Rev. C 67, 014002 (2003).
[7℄ R.Crespo,E.Cravo,A.Deltuva,M.Rodríguez-Gallardo, [15℄ B.A.Watson,P.P.Singh,andR.ESegel,Phys.Rev.182,
and A.C. Fonse
a, Phys.Rev. C 76, 014620 (2007). 978 (1969).
[8℄ J.S. Al-Khalili, R. Crespo, R.C. Johnson, A.M. Moro,
