Analysis of the Ratio of (p;pn) to (p;2p)
Reation Cross Setions
V.B. Shostak 1
, G.P. Palkin 1
,N.I. Woloshin 1
, V.P. Likhahev 2
,
J.D.T. Arruda-Neto 2
, M.T.F. daCruz 2
, and M.N. Martins 2
1
Institutefor NulearResearh,Kiev,Ukraine
2
LaboratoriodoAeleradorLinear,
Institutode Fsia,
Universidadede S~aoPaulo,
CaixaPostal66318,05315-970, S~aoPaulo,SP, Brasil
Reeivedon7November,2000
The ratio of the dierential ross setions for the reations 7
Li(p;2p) 6
He and 7
Li(p;pn) 6
Li, at
inident protonenergyof 70MeV, wereanalyzedinthe frameworkof DWT approah. FSIand
o-shellontributionswerefatorizedandomparedwithappropriateexperimentaldata.
I Introdution
The quasi-elasti knokout of nuleons by protons,
(p;pn) and (p;2p) reations,are thesoureof several,
relatively independent piees of information: reation
mehanism,two-partilefores, interationsin the
ini-tialandnalstates,propertiesofthestatesofthe
resid-ualnulearsystems,andthenulearwavefuntionsof
theknokedoutpartiles.
Forinitial proton energies(E
o
)below400MeV,
the single-partile harater of the proton-nuleon
in-teration starts to be signiantly distorted, even for
lightnulei,asaresultofvariouseets. Amongthem
are o-shell eets, when the two-partile sattering
is aeted by the presene of other partiles in the
medium. Other kindsof distortionare onnetedwith
theinterationoftheprotonwiththenuleusinits
ini-tialstate(beforethequasi-elastisattering),andwith
theinterationoftheoutgoingnuleonswiththe
resid-ual nuleus in the nal state (FSI eets). At these
energies,theagreementbetweenthealulatedand
ex-perimentalrosssetionsmerelyindiatesthatthewave
funtions of theintranulear nuleon, thetwo-partile
potential,andthedistortionoftheinidentprotonwave
funtion havebeenorretlydetermined.
AtE
o
<100 MeV,distortion eets are so strong
that theydeterminetheharaterof thereation. For
thisreason,thestudyofknokoutreationsinduedby
protons at these energiesdeals only with thereation
mehanismandwiththeorretestimateofthe
distor-tionsinvolved.
In Refs. [1-3℄ various phenomenologial
nuleon-that,while givingasatisfatorydesriptionofthefree
ppsattering,theydonotdesribeadequatelytheross
setionsof (p;2p)reationsdueto o-shell eets. At
low to intermediate energies, when the inoming
par-tile energy is of the same order of magnitude of the
separation energy, B, of the outgoing partile, the
(p;2p)dierentialrosssetionsdependstronglyonthe
o-shell eets present on nuleon-nuleon sattering
withinthenulearmedium.
O-shell phenomena are inherent to all
many-partile proesses, when the sattering in the
two-partile system is distorted by other partiles in the
medium. Only when B =0and the kinetienergy of
the reoil nuleus an be negleted, the relative
mo-menta in the initial and nal hannels are equal and
the two-partile amplitude will be determined on the
massshell.
The o-shell properties of the two-partile
ampli-tude,whihould,inpriniple,beobtainedfrom
knok-outreations,an be importantin the solutionof the
inverse problem (deriving the two-partile amplitude
fromdataonnuleon-nuleonsattering). Information
aboutthe o-shell behaviorof thetwo-partile
ampli-tude in knokout reationswould eliminate the
ambi-guity of the problem, sine even if a omplete set of
data on nuleon-nuleon sattering was available, an
innitenumberofphase-equivalentpotentialsouldbe
onstruted,givingthesametwo-partileamplitudeon
the massshell but diering o it. Unfortunately,
o-shell properties are not easy to obtain, sine in most
experimentsit is impossible to disentanglethem from
reations are not sensitive to the o-shell behavior
of the two-partile amplitude at energies above 300
MeV. Therefore projetiles above this energy are
re-ommended for the study of the intranulear nuleon
wavefuntions. Forprojetileenergiesbelow200MeV,
the (p;2p) ross setion beomes sensitive to the
o-shellbehaviorofthenuleon-nuleoninteration,
mak-ingthisenergyregionthemostonvenientforthestudy
ofo-shelleets.
InRefs. [4-6℄itwasshownthatusualexperimental
setups are unable to separate FSI and o-shell eets
on theexperimental stage. This situation preludes a
systematistudyofo-shelleets,sineitisevidently
neessarytoseparatethedistortionsandtheo-shell
ef-fetsalreadyattheexperimentstage[7℄. Nevertheless,
asshowninRefs. [8-10℄,someindependentonlusions
aboutthebehaviorofo-shell andFSIorretionsan
bedrawnfromtheanalysisoftheratiooftheross
se-tionsfor (p;pn) and(p;2p) reations,aomplishedin
theframeworkofthedistorted-waveapproximationfor
non-loalrealistit-matrix(DWTA)approah.
Inthis work wepresentan analysis of the ratioof
theross setionsfor the(p;pn) and(p;2p) reations,
toshowthat itispossibletofatorize theFSI and
o-shell ontributions, studying the behavior of eah of
theseontributionsversustheseparationenergy.
II Theoretial Model
The rosssetion for the 7
Li(p;pn) and 7
Li(p;2p)
re-ation was alulated in the distorted-wave
approxi-mation for non-loal realisti t-matrix (DWTA). This
method was developed by MCarthy and o-workers
[11,12℄andthenimprovedfortheaseofarbitrary
ge-ometry, eliminating ambiguities in parametersand
in-ludinganindiretproess[13-15℄. Theindiretproess
orrespondstothereleaseofanintranulearnuleonas
aresult of theinteration of theinident proton with
theresidualnuleus[16℄.
MCarthyabandonedthezero-rangeapproximation
and derivedatheory of aneetive non-loal realisti
t-matrix suitable for desribing the diret (p;2p) and
(p;pn) knokoutreations. Theenergy-dependent
o-shellnuleon-nuleonmatrixt(01;01;e)isanexat
so-lutionoftheLippmann-Shwingerequationwitha
sep-arablenonloaltwo-nuleonpotential. Here0;1;0,and
1are generalizedoordinates of the nuleons(spatial,
spin, and isospin oordinates). A separable non-loal
potentialwithaGaussianformfatorwasproposed[11℄
to alulate the radialomponent of the two-nuleon
t-matrix. The orresponding parameters were
deter-mined from the phase shifts for elasti p-p and p-n
sattering over the energy range 0-350 MeV. The s,
p, and d waveswere taken into onsideration. In the
presentstudyweusedasimilar potential, onstruted
byLevshinet. al. [17℄,withatensorinteration. That
potentialgivesagooddesriptionoftheenergy
depen-dene of the phase shifts when s, p, d, and f waves,
inhannelswithisospinsT =0and1and
orrespond-ingmixingparameters,aretakenintoaountoverthe
energyrangefrom 0to 500MeV. Italsodesribesthe
singletandtripletsatteringlengthsandeetiveradii.
In general, the quantity e, the relative energy of the
nuleon-nuleoninteration, hasnot beendened
rig-orously for a desription of quasielasti proesses. In
the three-partile problem at hand, the range of this
unertaintyissetbyasumoftwoquantities: the
sepa-rationenergyof thenuleonwhihis knokedoutand
theenergyofthereoilnuleus.
Havinga nonloal t-matrix, MCarthyinluded in
the matrix element the oordinates of all partiles.
Thenthematrixelementforatransitionfrom the
ini-tial state i (the system onsisting of A nuleons and
theinidentproton)tothenal statef (reoilnuleus
andtwoemittednuleons)forthemehanismofdiret
quasifreeknokoutis
T
fi =C
T
i N
i
TfNf 1
2
P
m C
J
i M
i
J
f M
f jm
R
[ ( )
~
k
1 (~r
0 )
( )
~
k
2 (~r
1 )S
1 (0)S
2 (0)t
1 (1)t
2 (1)℄
t(01;01;e) h
(+)
~
k0 (~r
0
~r1
A )
i
S
0 (0)t
1 (0)
jm (1)d~r
0 d~r
1 d~r
0 d~r
1
(1)
d
Herethesubsripts0,1,2speifytheinident
par-tile and the two emitted nuleons, respetively; C
are vetor-addition oeÆients (the rst isospin
oef-ients);S
andt
arethespinandisospinwave
fun-tions of the partiles; and are the projetions of
the spin and the isospin; j;m;J
f ;M
f ;J
i
and M
i are
theangularmomentaandrespetiveprojetionsforthe
knokedoutnuleon,fortheore,andfortheinitial
nu-leus,respetively.
For onveniene in the alulation of
multidimen-sionalintegrals,MCarthyproposed theuseof an
an-alyti form to represent the distorted wave funtions
refration,absorptionandfousing,andhavean
analyt-ialrepresentationsimilartotheeikonalapproximation
[13℄:
(+)
~
k
(~r)=e
~
k R
N
e i(+i)
~
k~r
1+Fe (~r R
^
k ) 2
S 2
( )
~
k (~r)=
h
(+)
~
k (~r)
i
(2)
where +i = D is the omplex refrative index of
theoptialmodel. Thequantitykplaystheroleofa
modiedwavenumber,anddeterminesthedamping.
F;R andS arethefousingparameters. R
N
ishosen
tobeequaltothesumofthehargeradiiofthenuleus
andtheproton.
In ouralulations the DWF parameterswere
un-ambiguouslyhosenfromtherequirementof: 1)a
quan-titatively orretdesription of theexperimental data
for the elasti (dierential and integrated),
el ;
rea-tion,
r
, andtotal,
tot
, rosssetions forthe
intera-tionoftheproton(neutron)withtheorresponding
nu-leiintheentraneandexithannels[10℄and;2)
agree-ment betweenthe DWF and theexatwavefuntion,
obtained by numerial integration of the Shrodinger
equationinarangeomparablewiththesizeofthe
nu-leus[14℄. Sinetheexperimentalresultswereobtained
fortwosetsofkinematisparameters,orrespondingto
thefollowingaverageenergiesofthenalnuleons:
hE
1
i=22MeV
hE
2
i=22MeVfor1s
hE
2
i=40MeVfor1p
hE
1
i=30MeV
hE
2
i=14MeVfor1s
hE
2
i=30MeVfor1p
(3)
and the parameters of the DWF were determined for
these averageenergies.
The obtained DWF parameters reprodue the
ex-perimentalrosssetions
el ;
r ,and
tot
[10℄with
a-uraybetterthan10%.
Thesingle-partile bound statewavefuntion
jm
wasalulatedfora Woods-Saxon potential[18℄, with
parameters hosenfrom theorret desription of the
bindingenergiesandelastiformfators.
Inourase,theeetivequasi-two-partilet-matrix
is theoherentsumof twotermswhose squared
mod-uli determine the ross setions of elasti p-p and
p-ore sattering [16℄. The transition from the matrix
t(01;01;e) to the quasielasti matrix is made by
ex-panding the t-matrix in partial waves and separating
the relativeand theenterof massoordinates. After
this, the 12-fold integral (1) redues to a 9-fold
inte-gral, and it an be alulated analytially by virtue
of the exponential representation (expansion with
re-spet to a Gaussian funtion) and the useof
generat-angular-momentum eigenfuntions. After all the
ma-nipulations, all the integrals in (1) beome
exponen-tials,theargumentof theprodut ofthe exponentials
beingaquadratiformthatistransformedtoasumof
squares.
III Experimental Proedure
Therststudyofthe 7
Li(p;pn)and 7
Li(p;2p)reations
wasreportedin[10℄.
Theexperiment wasarried outusing the70-MeV
proton beam from the U-240isohronous ylotron of
the Institute for Nulear Researh of the Ukrainian
Aademy of Sienes. The experimental faility was
desribedelsewhere[8,10℄.
Protonsandneutronsweredetetedinoplanar
ge-ometrybytwospetrometers,loatedonoppositesides
of the initial proton beam trajetory axis. A
mag-netispetrometer,basedonatwo-quadrupoleand
one-dipole optis, was used for themomentum analysis of
thesatteredprotons,ataxedangle(
p
1 =45
0
)with
respetto the initialbeamaxis. Momentum analyzed
protonsweredetetedinthefoalplanebyan8-hannel
sintillationounterwithmomentumaeptaneof3%.
Energies of the seondary protons (E
p2
) and
neu-trons (E
n
) were determined, at denite angles
p
2 ,
by the time-of-ight spetrometer (TFS). The TFS
onsisted of ve sintillation-ounter telesopes,
posi-tioneduniformlyalongairulararoveringtherange
45 0
69 0
, instepsof6 0
.
Theightpathsfor protonsand neutronswere 3.4
mor5.7m. Eahtelesopeonsistedoftwoplasti
sin-tillators(NE102A),5-and200-mmthik,respetively,
oupledtophotomultiplier tubesPM-36. Betweenthe
sintillatorswasplaedaleadabsorber8-mm thik to
avoidhargedpartilesarrivingattheseond
sintilla-tor. The rstsintillator detetspratially only
pro-tons,sineitsneutrondetetioneÆienyisverysmall
(0.3%). Theseondsintillatordetetsonlyneutrons
(with about 10%eÆieny, see below), sinethe
pro-tons were stopped at the absorber. Signals from the
sintillators were used for timing purposes. The
ener-giesoftheseondarypartilesweredeterminedfromthe
diereneinighttimebetweenthemandthesattered
protons deteted by the magneti spetrometer. The
timeresolution,assoiatedmainlywiththedimensions
of the plasti sintillators, was about 5 ns, produing
a7-8MeVenergyresolutionforthedetetedpartiles.
To keep the bakground within aeptable levels, the
sintillatorswereshielded byleadandparaÆn.
spetrometers) were reorded by the aquisition
sys-temin event-by-eventmode. An o-line analysis ode
allowed theseletionof dierent spetrafrom the raw
data, to alulate the momentum of the reoiling
nu-leus,k
A 1
,and theseparationenergy,B
p orB
n ,and
rearrangethespetrumversusnewvariables.
Forthe analysispresentedin this work onlyevents
orresponding to symmetri oplanar geometry were
hosen:
1 =
2 = 45
0
. These events, for eah
sat-tered proton energy, E
p
1
, were arranged in spetra as
funtionof theseparationenergy.
The pn- and pp-oinidene spetra asfuntion of
B
n and B
p
, respetively, were deomposed on partial
ontributions from quasi-free knokout of 1s and 1p
shells. Thedeompositionwasbasedonaleast-squares
toftwoGaussianstotheexperimentalspetra,leaving
twofreeparametersforeahpeak: heightandFWHM.
Peak positions were xed aordingto the separation
energies,obtainedin ref. [19℄ atE
0
=1GeV.
The dierential ross setions for 7
Li(p;pn) and
7
Li(p;2p) reations,for 1sand 1p shells, versus(E
p ),
obtained as a result of the deomposition proedure,
areshownin Figs. 1and2bytheopenirles.
σ
Ω
Ω
µ
Figure 1. Dierential ross setions for the reation
7
Li(p;pn) 6
Li at Eo = 70 MeV, for 1p and 1s shells,
ver-susEp. CirlesrepresentexperimentaldatafromRef. [10℄.
SolidurvesorrespondtoaalulationwiththeDWF
pa-rameters for 6
Li in the standard sheme. Dashed urves
representresults oftheDWTAalulation fora
hypothet-ialFSI,withmodiedDWFparametersof 4
He (seetext
fordetails).
IV Analysis of the ratio of
(p;pn) to (p;2p) ross setions
Theanalysis oftheratioof(p;pn) to(p;2p)ross
se-tionsisespeiallyinterestingbeause:
the ratio of (p;pn) to (p;2p) ross setions,
ob-tained simultaneously and under idential
kine-matial onditions are free of systemati errors,
allowingadiretomparisonofthesevalueswith
otherdata;
usingthis ratioitispossibletoseparate theFSI
ando-shellontributionsandtostudy them
in-dependently. Moreover, sine the DWTA
ap-proah isbasedontherealistiparameterization
oftheo-shellandFSIontributionsthroughthe
use of experimental ross setions, there is no
otherwaytoontrolthem inthealulations.
σ
Ω
Ω
µ
Figure 2. Dierential ross setions for the reation
7
Li(p;2p) 6
HeatEo=70MeV,for1sand1pshells. Cirles
represent experimental data from Ref. [10℄. Solid urves
orrespondtoalulations withmodiedDWFparameters
of 4
He. Thedashedurvesrepresentresults oftheDWTA
alulation for ahypothetialFSI,withparameters of 6
Li
(seetextfordetails).
Asarststep,wedene theratioofthe(p;pn) to
(p;2p)reationrosssetions,reduedtoanequal
num-berof protons,N `
p
,and neutrons,N `
n
,inagivenshell,
`, and normalized to the elementary proton-neutron,
d(p;n),andproton-proton,d(p;p),elastisattering
dierentialrosssetionsforthesameenergiesand
an-gles,as:
< = d
3
(p;pn) exp
d(pn)N `
,
d 3
(p;2p) exp
d(pp)N `
where d 3 (p;2p) exp and d 3 (p;pn) exp
are the
experi-mental ross setions fot the (p;pn) and (p;2p)
rea-tions,respetively.
InRef. [19℄ it wasshownthat, at suÆientlyhigh
E
o
,where theimpulseapproximationisvalid, <is
de-nedbythesingle-partileboundstatewavefuntions
forprotonsandneutrons. Intheaseswheretheproton
and neutron distributions areidential, < =1.
Devi-ations of < from unity at high energies are onneted
with dierenesbetween theRMSradii ofthe nulear
shells for protons (r p
rms
) and neutrons (r n
rms
) [20,21℄.
This dependene isstrong< / (r n r ms ) 4 (r p r ms ) 4
[19℄,andsmall
dierenes in r
rms
anresult in largedeviations of <
from unity.
Table1showsthevaluesof<obtainedfromthedata
for light nulei at E
o
= 70 MeV and 1 GeV,the last
valuesbeingassoiatedwiththesingle-partile
meha-nism. AtE
o
=70MeV,<presentsastrongdeviation
from the single-partilemehanismvalue. This
devia-tionshouldbeonnetedwithFSIando-shelleets.
TableI-Experimentalvaluesof<(seetext)for
severalnuleiat 70MeVand1GeV.
<
E
o
(MeV) 70 1000
Nuleus Shell
4
He 1s 2.1(2)
b)
6
Li 1p 3.5(4)
) 0.95(8) d) 1s 3.6(4) ) 1.03(10) d) 7
Li 1p 2.3(4)
) 1.05(4) d) 1s 2.3(4) ) 1.08(15) d) 9 Be 1p a) 3.9(1.5) f) 1.58(5) d) 1s 3.9(1.2) f) 0.97(12) d) a) B n
=18,1MeV, b) [13℄, ) [22,23℄, d) [19℄, e) [10℄, f) [8℄.
In order to separate FSI and o-shell eets in
the ross setion ratio it is neessary to alulate the
(p;2p)and(p;pn)rosssetionswithahypothetial
-nalstateinteration(d 3 (p;2p) hyp andd 3 (p;pn) hyp ).
This is done using, for the DWF in the exit hannel,
the set of parameters obtained for an isotone of the
residual nuleus, insteadof theset parametersfor the
residual nuleus itself, keeping unhanged all another
DWTA parameters (o-shell nuleon-nuleon
intera-tion t-matrix, and DWF parameters for the entrane
hannel).
Then the ratio in eq. (2) ould bepresented asa
produtoftwofators:
< = d 3 (p;pn) exp d 3 (p;pn) hyp d 3 (p;pn) hyp d(pp)N ` p d 3 (p;2p) exp d(pn)N ` n (5) < = d 3 (p;2p) hyp d 3 (p;2p) exp d 3 (p;pn) exp d(pp)N ` p d 3 (p;2p) hyp d(pn)N ` n : (6)
The denition of the hypothetial ross setions
d 3 (p;2p) hyp and d 3 (p;pn) hyp
implies that the rst
fatorsin eqs. (3)and(4),aretheratios:
< FSI 1 = d 3 (p;pn) hyp d 3 (p;pn) exp (7) and < FSI 2 = d 3 (p;2p) hyp d 3 (p;2p) exp ; (8)
whih haraterize the relative ontribution of
the distortions aused by FSI eets, sine
in both ratios d 3 (p;pn) exp =d 3 (p;pn) hyp , and d 3 (p;2p) exp =d 3 (p;2p) hyp
, there are no hanges
on-netedwith possibledierenesin o-shelleets, but
onlyhanges onnetedwith thedierene in theexit
hannels. IfFSIeetswerenegligible(oridentialfor
that pair of isotones)this ratio should be equal to 1.
Moreover,diretly from thedenition of d 3 (p;2p) hyp andd 3 (p;pn) hyp
,followsthattheratios:
< off shell 1 = d 3 (p;pn) hyp d(pp)N ` p d 3 (p;2p) exp d(pn)N ` n (9) and < off shell 2 = d 3 (p;pn) exp d(pp)N ` p d 3 (p;2p) hyp d(pn)N ` n (10)
haraterizetherelativeontributionofo-shelleets
only, sinein d 3 (p;pn) exp and d 3 (p;2p) exp wehave
thesameentraneandexithannelsasind 3 (p;pn) hyp and d 3 (p;2p) hyp
,respetively, the only hange being
onnetedwiththediereneino-shell eetsdue to
thediereneinthepandnseparationenergies. Ifthe
o-shell eets were negligible (or idential for those
isotones) these ratios should be equal to (r n r ms ) 4 (r p r ms ) 4 . In
partiularfor the1p and 1sshells in 7
Li,these ratios
shouldbeequalto1[19℄ forE
o
=1GeV(Table1).
Summarizing, the hoie of the hypothetial nal
statesisruledbytheneedtoseparateo-shellandFSI
eets. So,ifwewantto studyo-shell eets, we
ar-tiiallyhoosearealistinalstateofoneofthe
rea-tionstomakethenalstatesofboth(p;2p)and(p;pn)
reations exatly the same, so that the deviations of
<
off shell
from unity will be due to o-shell eets.
Ontheotherhand,ifwewanttostudyFSIeets, we
artiiallyhoosearealistinalstateinordertomake
thesamereationproduedierentnalstates,likein
eqns. (7)and(8).
Usingtheexperimental andhypothetialross
se-tions,itwaspossibleto alulate< FSI and< off shell for 7 Li(p;pn) 6 Li and 7 Li(p;2p) 6
HereationsatE
TableII -Valuesof<
off shell
,< FSI
,and<(seetext) forseveralnuleiat 70MeV.
Nuleus
7
Li
4
He
9
Be
Shell 1p 1s 1s 1p 1s
B (MeV) 9 25 21 18 28
<
off shell
1
= (p;pn)
hy p
(pp)e`N `
p
(p;2p) exp
(pn)
e` N
`
n
1.14(16) d)
0.68(5) d)
0.57(5) a)
-
-<
off shell
2
= (p;pn)
exp
(pp)e`N `
p
(p;2p) hy p
(pn)
e` N
`
n
1.15(5) )
0.65(7) )
0.58(5) a)
[0.86(8)℄ 0.68(8) b)
< FSI
1 =
(p;pn) exp
(p;pn) hy p
2.0(1) d)
3.3(2) d)
3.6(1) a)
-
-< FSI
1 =
(p;2p) hy p
(p;2p) exp
1.7(3) )
3.1(2) )
3.6(1) a)
3.0(8) b)
6.0(14) b)
<= p;pn)
exp
(pp)
e` N
`
p
p;2p) exp
(pn)e`N `
n
2.3(4) )
2.2(3) )
2.1(2) a)
3.9(15) b)
3.9(12) b)
< FSI
1
<
off shell
1
= < t
1
2.3(3) 2.1(2) 2.1(2) -
-< FSI
2
<
off shell
2
= < t
1
2.0(3) 2.0(2) 2.1(2) 4.1(12) 3.6(8)
a)
Referene[9,13℄. b)
Referene [2℄. )
Referene[10,22℄. d)
This work.
V Disussion and onlusions
Results of standard DWTA alulations for the
rea-tion 7
Li(p;pn) 6
Li are shown in Fig. 1, for both 1s
and 1p shells, by the solid urves, whih orrespond
to the oherent sum of the diret and indiret
meh-anisms. DWTA alulationsquantitatively reprodue
the experimental data for both shells within absolute
unertainties(15%).
For the reation 7
Li(p;2p) 6
He it is impossible to
arry out the DWTA alulations using the standard
alulationsheme usedin thease of 4
He(p;pn) 3
He,
4
He(p;2p) 3
H[13℄and 7
Li(p;pn) 6
Li[10℄,sinethedata
of elasti and total ross setions for the reations
(p; 6
He)and(n; 6
He)donotexist. Toirumventthis
problemweusedadierentDWTAalulationsheme,
desribedin [13℄,andalulatedtheDWF parameters
usingtherosssetionsforahypothetialresidual
nu-leus(a neighboringstable isotope,forwhihdata
ex-ist) and then tted the fousing parameter R . Using
this proedure it was possibleto dedue the RMS
ra-diusof 8
Be[8℄.
Inthisalulationsheme,asarststep,weinlude
in the alulation of the dierential ross setions for
the 7
Li(p;2p) 6
Hereationtheinterationeetsofthe
outgoingnuleonswith ahypothetialresidualnuleus
in the nal state (FSI eets) using the ross setion
of 4
He instead of 6
He, and then t the fousing
pa-rameterR . Theresultsof suh alulationsareshown
in Fig. 2by thesolid lines. This alulationgivesan
forboth1sand 1pshells,whihhavedierentrelative
ontributionsfromFSI ando-shelleets.
To evaluate the dierene in FSI eets when we
have 6
Li or 6
He in the exit hannel, weanalulate
the 7
Li(p;2p) ross setion with a hypothetial nal
stateinteration, usingrosssetions for 6
Li in the
-nalstate,insteadfor 6
He. Theresultsofsuh
alula-tions are shown in Fig. 2by the dashed urves. The
urvesobtainedusingthehypothetialnalstate
inter-ationinthealulationpresentthesameshapeas,but
overestimatetheexperimentaldata.
The results of suh hypothetial ross setion
al-ulations (FSI for 6
He instead 6
Li) areshown in Fig.
1 by the dashed lines. It is lear that the
hypothet-ial ross setion strongly underestimates the
experi-mental resultsfor three oftheexperimental data sets,
reproduingoneof them (1p;E
p
30 MeV). This
sit-uation is expeted, sine the region around E
p 30
MeVorrespondstotheminimumofboththe
momen-tum transferandthe 1pproton rosssetion[24℄,and
theFSI ontributionshould llthisminimum,sineit
tendstoinreasetherosssetionaroundtheminimum
andtodereaseitotherwise. Figures1and2showthat
thehangefrom 6
Lito 6
Hein thenalstateinreases
the FSI, and, sine the RMS radius of 6
He is bigger
thanthatof 6
Li,andallotherDWFparametersarethe
same,oneanonludethataninreaseintheRMS
ra-diusoftheresidualnuleustendstoinreasetheFSI,as
itwasshownin[19-21℄. NotethattheDWFissensitive
ouldusefortheneighboring nuleithesamevaluesof
other mirosopiparameters,hangingonlythe RMS
radius.
Based on the results of the DWTA alulations,
whihorretlyreproduetheexperimental(p;2p)and
(p;pn)rosssetions,weanestimatetheorderof
mag-nitude of the FSI and o-shell eets in the ratio of
theserosssetionsandstudytheirbehaviorasa
fun-tionoftheseparationenergy(Table2).
Table2 showsthat < FSI
and<
off shell
, obtained
at E
o
= 70 MeV, signiantly dier from those
ob-tained for E
o
= 1 GeV. This means that the (p;pn)
to (p;2p)ratios at E
o
=70 MeV ontain information
abouto-shellandFSI eets.
Thevaluesobtainedfor thepairs < FSI
1
and < FSI
2 ,
and <
off shell
1
and <
off shell
2
, are ompatible within
unertainties,althoughalulatedusingompletely
in-dependent experimental ross setions. This suggests
that the ross setions are ompatible and that the
DWTA adequately desribes o-shell and FSI eets.
In Table 2 only the datum for the 1p shell of 9
Be
orresponds to a ase where there is a dierent
spa-tial distribution of protons and neutrons in the shell
(
p (r)6=
n
(r)). Tokeeptheompatibilitywithother
alulations,the valueof<
off shell
2
forthe 1pshellof
9
Be wasorretedbyafator(r n
rms )
4
=(r p
rms )
4
[19℄,to
aountforthediereneinprotonandneutron
distri-butions. TheorretedvalueispresentedinTable2in
squarebrakets.
<
off shell
dereases as a funtion of the
separa-tion energy and nulear number, while < FSI
presents
the opposite tendeny. Table 2 also presents
prod-uts of the ratios: < t
1 = <
FSI
1
<
off shell
1
and
< t
2 = <
FSI
2
<
off shell
2
. These produts are
om-patible with the experimental valuesof <. This
indi-atesthat,withintheahievedauray,the
ontribu-tion of two-stage proesses [11,25℄ are not notieable
in the dierential ross setions of 7
Li(p;2p) 6
He and
7
Li(p;pn) 6
Li reationsatE
o
=70MeV.
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