Effects of density and parametrization on scattering observables

0.17 MB·English

by  M. Bhuyan

Effects of density and parametrization on scattering observables
0
1
0 M. Bhuyanaand S. K. Patrab
2
aSchool of Physics, Sambalpur University, Jyotivihar-768019, India
n
a bInstitute of Physics, Sachivalaya Marg, Bhubaneswar-751005, India
J

1 Wecalculate thedensitydistributionof protonsand neutronsfor40,42,44,48Caintheframe-work ofrelativistic
1
mean field (RMF) theory with NL3 and G2 parameter sets. The microscopic proton-nucleus optical potential

] for p+40Ca system is evaluted from Dirac NN-scattering amplitude and the density of the target nucleus using
h Relativistic-Love-FraneyandMcNeil-Ray-Wallaceparametrizations. Thenweestimatethescatteringobservables,
t
- such as elastic differential scattering cross-section, analysing power and the spin observables with relativistic
l impulse approximation. We compare the results with the experimental data for some selective cases and found
c
u that theuse of density as well as thescattering matrix parametrization is crucial for thetheoretical prediction.
n
[

1
v Explaining the nuclear structure by taking the successful NL3 [5] and advanced G2 [6] param-
9 toolofnuclearreactionisoneofthe mostcurious eter sets, respectively. As representative cases,
9 andchallengingsolutionforNuclearPhysicsboth we used these target densities folded with the
5
in theory and laboratory. So far the elastic scat- NN-aplitude of 1000 MeV energetic proton pro-
1
teringreactionofNeucleon-Nucleusismoreinter- jectile with Relativistic-Love-Franey (RLF) and
.
1
estingthanthatofNucleus-Nucleusatlaboratory McNeil-Ray-Wallace(MRW)parametrizations[7]
0
energy E ≃ 1000 MeV. The Neucleon-Nucleus for 40,42,44,48Ca in our calculations.
0 lab
1 interactionprovidesafruitfulsourcetodetermine The RMF and E-RMF theories are well docu-
: thenuclearstructureandaclearpathtowardthe mented [6,8,9] and for completeness we outline
v
formation of exotic nuclei in laboratory. One of here very briefly the formalisms for finite nu-
i
X the theoritical method to study such type of re- clei. TheenergydensityfunctionaloftheE-RMF
r action is the Relativistic Impulse Approximation model for finite nuclei is written as [10,11],
a
(RIA). In a wide range of energy interval, the
conventional impulse approximation [1,2] repro-
duces quantitatively the main features of quasi-
elastic scattering for medium mass nuclei [3,4]. E(r)= ϕ† −iα·∇+β(M −Φ)+W +
α
The observables of the elastic scattering reaction (
α
not only depend on the energy of the incident 1 X1+τ i 1
particle but also on the kinematic parameter as τ3R+ 3A− βα·(fv∇W + fρτ3∇
2 2 2M 2
wellasthe density discributions ofthe targetnu-
1
cleus. In the present letter, our motivation is to R+λ∇A)+ (β +β τ )∆A ϕ
2M2 s v 3 α
calculate the nucleon-nucleus elastic differential )
tsiceast,tleirkiengopctricoasls-psoectetinotnial((ddUΩσ) a)n,danoatlyhseirngqpuaownteir- + 1 + κ3 Φ + κ4 Φ2 m2sΦ2− ζ0 1 W4
opt 2 3! M 4! M2 g2 4!g2
(A ) and spin observables (Q−value) taking in- (cid:18) (cid:19) s v
y
putasrelativisticmeanfield(RMF)andrecently
1 Φ 1 Φ
proposed effective field theory motivated rela- + 2g2 1+α1M (∇Φ)2− 2g2 1+α2M
tivistic mean field (E-RMF) density. The RMF s (cid:18) (cid:19) v (cid:18) (cid:19)
1 Φ η Φ2 m 2 1
andE-RMFdensitiesareobtainedfromthemost (∇W)2 − 1+η + 2 v W2−
2 1M 2 M2 g 2 2g2
(cid:18) (cid:19) v ρ
1
2 M. Bhuyan and S.K. Patra
(∇R)2− 1 1+η Φ m2ρR2 NN-scatteringoperaterF canbewritteninterms
2 ρM g2 of five complex functions (the five terms involves
(cid:18) (cid:19) ρ
1 1 1 in the proton-proton pp and neutron-neutron pn
− (∇A)2+ A∆W + A∆R, (1) scattering) as follows:
2e2 3g g g g
γ v γ ρ
F(∐,E)=FS +FVγµ γ +FPSγ5 γ5
where the index α runs over all occupied states (0) (1)µ (0) (1)
ϕα(r) of the positive energy spectrum, Φ ≡ +FTσµνσ +FAγ5 γµ γ5 γ , (2)
g φ (r), W ≡ g V (r), R ≡ g b (r) and A ≡ (0) (1)µν (0) (0) (1) (1)µ
s 0 v 0 ρ 0
eA0(r). Thetermswithgγ,λ,βsandβv takecare where (0)and(1)arethe incidentandstrucknu-
of the effects related with the electromagnetic cleons respectively. The amplitude for each FL
structure of the pion and the nucleon (see Ref. is a complex function of the Lorentz invariants T
[11]). The energy density contains tensor cou- andSwithE =E andqisthefourmomentum.
lab
plings,andscalar-vectorandvector-vectormeson We recommend the redearsfor detail expressions
interactions, in addition to the standard scalar to Refs. [14,15,16,17,18,19,20,21,22]. Then the
selfinteractionsκ3andκ4. Thus,theE-RMFfor- Dirac optical potential Uopt(q,E) can be written
malism can be interpreted as a covariant formu- as,
lation of density functional theory as it contains
all the higher order terms in the Lagrangian,ob- −4πip A
tained by expanding it in powers of the meson Uopt(q,E)= hψ| expiq.x(n)F(q,E;n)|ψi, (3)
M
fields. The terms in the Lagrangian are kept fi- nX=1
nite byadjustingtheparameters. Furtherinsight
where F is the scattering operator, p is the mo-
into the concepts of the E-RMF model can be
mentum of the projectile in the nucleon-nucleus
obtained from Ref. [11]. It may be noted that
center of mass frame, |ψi is the nuclear ground
the standard RMF Lagrangian is obtained from
state wave function for A-particle. Finally using
that of the E-RMF by ignoring the vector-vector
the Numerov algorithm the obtained wave func-
and scalar-vector cross interactions, and hence
tionismatchedwiththecoulombscatteringsolu-
does not need a separate discussion. In each of
tion for a boundary condition at r → ∞ and we
the two formalisms (E-RMF and RMF), the set
getthescatteringobservablesfromthescattering
of coupled equations are solved numerically by a
amplitude, which are defined as:
self-consistent iteration method and the baryon,
scalar,isovector,proton,neutronandtensorden- dσ
≡|A(θ)|2+|B(θ)|2 (4)
sities are calculated. dΩ
Thenumericalprocedureofcalculationandthe 2Re[A∗(θ)B(θ)]
A ≡ (5)
detailedequationsforthegroundstateproperties y dσ/dΩ
offinitenuclei,wereferethereadertoRefs. [9,8]. and
The densities obtained from RMF (NL3) [5] and 2Im[A(θ)B∗(θ)]
E-RMF(G2)[6]areusedforfoldingwiththeNN- Q≡ . (6)
dσ/dΩ
sacttering amplitude at E = 1000MeV, which
lab
gives the proton-nucleus complex optical poten- Now we present our calculated results of neu-
tial for RMF and E-RMF formalisms. RIA in- trons and protons density distribution obtained
volvesmainlytwosteps[12,13]ofcalculationsfor from the RMF and E-RMF formalisms [8]. Then
the evaluation of the NN-scattering amplitude. weevaluatethescatteringobservablesusingthese
In this case, five Lorentz covariant function [7] densities in the relativistic impulse approxima-
multiply with the so calledFermi invariantDirac tion, which involves the following two steps:
matrix (NN-scattering amplitudes). This NN- in the first step we generate the complex NN-
amplitudes are folded with the target densities interaction from the Lorentz invariant matrix
of protons and neutrons to produced a first or- FL(q,E) as defined in Eq. (2). Then the in-
dercomplexopticalpotentialU . Theinvariant teraction is folded with the ground state target
opt
Effects of density and parametrization on scattering observables 3
nuclear density for both the RLF and MRW pa- In Fig. 1, the protons and neutrons density
rameters[7]separatelyandobtainedthenucleon- distribution for 40Ca using NL3 and G2 parame-
nucleus complex optical potential U (q,E) for ter sets (upper panel) and the optical potential
opt
the parametrisations. It is to be noted that pair- obtained with RLF and MRW parametrisation
inginteractionistakencareusingthePauliblock- for p+40 Ca at 1000 MeV proton energy (lower
ing approximation. In the second step, we solve panel) are shown. From the figure, it is noticed
the wavefunction of the scattering state utilising that, there is no significant difference in desities
the opticalpotential preparedin the first step by for RMF and E-RMF parameter sets. However,
well known Numerov algorithm [23]. The result a careful inspection shows a small enhancement
approxumatedwith the non-relativistic Coulomb in central density (0-1.6 fm) for NL3 set. On the
scattering for a longer rangeof radialcomponent otherhand the densities obtained from G2 elon-
which results the scattering amplitude and other gated to a larger distance towards the tail part
observables [24]. In thr present paper we calcu- of the density distribution. As the optical poten-
late the density distribution of protons and neu- tial is a complex function which constitute both
trons for 40,42,44,48Ca in NL3 and G2 parameter real and imaginary part for both scalar and vec-
sets. From the density we evalute the optical po- tor, we have displyed those values in the lower
tentialandotherscatteringobservablesandsome panel of Fig. 1. Unlike to the (upper panel) of
representativecasesarepresentedinFigures1−3. protons and neutrons density distribution, here
we find a large difference of U (q,E) between
opt
theRLFandMRWparametrisation. Further,the
U (q,E)valueofeitherRLForMRWdifferssig-
opt
nificantly depending on the NL3 or G2 force pa-
rameters. That means, the optical potential not
0.1
-3m) 00..0068 PNPrreooutttoornno n((GN (G2L)23)) oonflNyLse3nosirtiGve2topaRrLamFeotrerMsReWts.bIuntvaelsstoigtaotitnhgeuthsee
ρ (f 0.04 Neutron (NL3) figureitisclearthat,the extrimummagnitudeof
0.02 realandimaginarypartofthescalarpotentialare
0 -442.2 and 113.6 MeV for RLF (G2) and -372.4
0 1 2 3 4 5 6 7 8
400 and 109.1 MeV for RLF (NL3). The same values
Re S (NL3)
V) 200 (a) (b) Im S (NL3) fortheMRWparametrisationare-219.8and32.8
e Re V (NL3) MeVwithG2and-175.1and33.2MeVwithNL3
M Im V (NL3)
V(central -2000 RLF MRW RIRImmee SSVV ((((GGGG2222)))) sv-1ea7tlsu9.e.2sIfnMorecVaresaefloorafnRvdLeicFmtoa(rgGipn2o)atreaynntpdiaar2l,t7s9tah.2reeae3nx6dt1r.e-31m6au4nm.d8
-400
MeVforRLF(NL3)butwithMRWparametrisa-
0 2 4 6 8 0 2 4 6 8 10
tiontheseareappearedat128.1and-87.4MeVin
r (fm)
G2 and 99.2 and -76.6 MeV in NL3. From these
large variation in magnitude of scalar and vector
potentials,itisclearthatthepredictedresultsnot
Figure 1. (upper panel): The neutrons and only depend onthe input targetdensity, but also
protons density distribution for 40Ca with NL3 highlysensitivewiththekinematicofthereaction
and G2 parameter sets. (lower panel) (a) the dynamics. A further analysis of the results for
Dirac optical potential for p +40 Ca system us- theopticalpotentialwithNL3andG2,itsuggest
ing RMF (NL3) and E-RMF (G2) densities with thattheUoptvalueextendsforalargerdistancein
RLF parametrisation, (b) same as (a), but for NL3thanG2. Forexample,withRLFthecentral
MRWparametrisation. Theprojectileprotonwith partofUopt with G2is moreexpandedthanwith
E =1000 MeV is taken. NL3 and ended at r ∼ 6fm, whereas the optical
lab
potential persists till r ∼ 8fm in NL3. Similar
4 M. Bhuyan and S.K. Patra
situation is also valid in MRW parametrisation. tion, it is clearly seen that dσ with NL3 desity is
dΩ
This nature of the potential suggests the appli- moreclosertoexperimentaldatawhichinsistnot
cability of NL3 over G2 force parameter. This is only the importance of parametrization (RLF or
because in case of NL3 the soft-core interaction MRW) but also to choose proper density input
between the projectile and the target nucleon is for the reaction dynamics. Analysing the elastic
more effective. differential cross-section along the isotopic chain
ofCafromA=40to48,thecalculatedresultsim-
provewith increasingmass number of the target.
104 RLF(G2)
MRW
102 ERXLFP(.NL3) 1
MRW
mb/sr)1100-02 40Ca 42Ca 00..68 (a) (b)
Ω (104 0.4
σ/d 0.2
d102 0
100 44Ca 48Ca -0.2 Ay Q-Value
10-2 -0.4
-0.6 NL3 (RLF)
0 10 20 30 0 10 20 30 G2 (RLF)
θ (deg) -0.8 NL3 (MRW)
cm G2 (MRW)
0 10 20 30 0 10 20 30 40
θ (deg)
c.m.
Figure2. The elastic differential scatteringcross-
section (dσ) as a function of scattering angle
dΩ
θcm(deg) for 40,42,44,48Ca using both RLF and Figure 3. (a) The calculated values of analysing
MRW parametrisations. The value of ddΩσ is power Ay as a function of scattering an-
shown for RMF (NL3) and E-RMF (G2) densi- gle θ (deg) for 40Ca (b) The spin observ-
cm
ties. able Q−value as a function of scattering angle
θ (deg) for 40Ca. In both (a) and (b), the RLF
cm
and MRW parametrisations are used with RMF
(NL3) and E-RMF (G2) densities.
In Fig. 2., we have plotted the elastic scatter-
ing cross-section of the proton with 40,42,44,48Ca
atlaboratoryenergyE =1000MeVusingboth
lab
densities obtained in the NL3 and G2 parameter The analysing power for p +40 Ca composite
sets with RLF and MRW parametrisations. The system is calculated from the general formulae
experimentaldata [25]are alsogivenfor compar- given in eqns. (4) and (5) and are shown in Fig.
ison. It is reportedin Refs. [7,26] the superiority 3withRLFandMRW.TheA andQ−valuesob-
y
of RLF over MRW for lower energy (E ≤ 400 tained by NL3 and G2 sets almost matches with
lab
MeV), howeverthe MRW shows better results at eachotherbothinRLFandMRW.Butifwecom-
energy E >400 MeV. In the present case, our paretheresultswithRLFandMRWitdifferssig-
lab
incident energy is 1000 MeV which matches bet- nificantly. Again,we geta smalloscillationofA
y
ter(MRW)withexperimentalvalues. Thisiscon- and Q in G2 set with increasing scattering angle
sistent with the optical potential also (see Fig. θ0 for RLF which does not appear in NL3 set.
c.m.
1). From the differential cross-section for both There is a rotation of Q−value from positive to
NL3 and G2 densities with MRW parametriza- negative direction when we calculate with MRW
Effects of density and parametrization on scattering observables 5
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