Alpha–Ni optical model potentials

s ofon andxperi-using ah pro-tegralson thee rea-range

leons,nsid-

Nuclear Physics A 762 (2005) 50–81

Alpha–Ni optical model potentials

M.M. Billah a, M.N.A. Abdullaha,∗, S.K. Dasb, M.A. Uddina,A.K. Basaka,∗, I. Reichsteinc, H.M. Sen Guptad, F.B. Malike,f

a Department of Physics, University of Rajshahi, Rajshahi 6205, Bangladeshb Department of Physics, Shahjalal University of Science and Technology, Sylhet, Bangladesh

c School of Computer Science, Carleton University, Ottawa, Ontario K1S 5B6, Canadad Department of Physics, University of Dhaka, Dhaka 1000, Bangladesh

e Department of Physics, Southern Illinois University, Carbondale, IL 62901, USAf Department of Physics, Washington University, St. Louis, MO 63130, USA

Received 8 July 2005; accepted 27 July 2005

Available online 18 August 2005

Abstract

The present work reports the analyses of the experimental differential cross-sections ofα elas-tic scattering on58,60,62,64Ni, over a wide range of incident energies, in terms of four typeoptical potentials, namely shallow (molecular), deep non-monotonic, squared Woods–Saxsemi-microscopic folding. All the four potentials produce a reasonable description of the emental data. The potential parameters, calculated from the energy density functional theoryrealistic two-nucleon interaction, resemble closely the molecular potential parameters, whicduce the best description of the experimental data for the four isotopes. The volume inand the energy variation of the parameters indicate the effect of the shell-model structurepotentials. The folding potentials, without any need for renormalization, are found to describsonably well the elastic scattering cross-section data for the four isotopes within the energyconsidered. In conformity with the previous observation on Ca isotopes, the number of nuc4Aα = 49, existing inα-like clusters in the target nucleus, is the same for the four isotopes, coered herein. 2005 Elsevier B.V. All rights reserved.

* Corresponding authors.E-mail addresses: [email protected] (M.N.A. Abdullah), [email protected] (A.K. Basak).

0375-9474/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysa.2005.07.020

M.M. Billah et al. / Nuclear Physics A 762 (2005) 50–81 51

ofd

with

re aslent lo-d [17].

[18–l part

ndthesed a-on over

-

on-als in,

thecross-

theoods–cycross-

at theyedmagi-tes thec calcu-

PACS: 25.55.Ci; 24.10.Ht; 21.60.Gx

Keywords: α–58,60,62,64Ni optical potentials; Elastic scattering;α-clustering; Non-monotonic; SquareWoods–Saxon and folded potentials

1. Introduction

Anomalous large angle scattering (ALAS), observed in the elastic scatteringα-particles [1–7] as well as the non-elastic processes involvingα-particles [5–9] has triggerea considerable interest since the mid-seventies [10–13]. ALAS in theα elastic scatteringhas been accounted for by a number ofα–nucleus potentials.

Two simple local potentials have been proposed to explain ALAS. The first onea squared Woods–Saxon (SWS) geometry with the real volume integralJR/(4A) ≈300 MeV fm3, advocated by Michel and his collaborators [13–16], is referred to heMichel potential. The SWS potential shape has a close resemblance with the equivacal potential obtained in the microscopic analysis using the resonating group methoThe second one, which has its roots in the energy-density functional (EDF) formalism21] is a molecular type of complex potential [23–25] with a repulsive core in its reaandJR/(4A) ≈ 100 MeV fm3.

Microscopic double folding [7,26] of the effective nucleon–nucleon(N–N) and semi-microscopic single folding of eitherα–N or α–α potential [27] interaction have been fouto give satisfactory account of ALAS in theα elastic scattering. In both the cases,folding potential needs renormalization. Very recently, Abdullah et al. [28,29] proposingle-folding potential based on the combined distributions ofα-like clusters and unclustered nucleons in the target. The resulting potential does not need any renormalizatia wide range of energies.

ALAS in the α-particle elastic scattering on24Mg and28,30Si [24] as well as on calcium isotopes [30]; the inelastic scattering ofα-particles by24Mg and 28Si [31]; the27Al(α, t)28Si reaction [32]; and the27Al(α,d)29Si reaction [25] data have been reasably accounted for by using complex non-monotonic molecular and Michel potentitheα-channel. However, the experimental data of the28,29,30Si(α,d)30,31,32P reactions [3334] and28Si(α,p)31P reaction [35] are better described by the molecular potential inα-channel, where the Michel potential underestimates the absolute magnitude ofsections by several orders. Although ALAS is the feature observed in theα-interactionwith the light and light-medium targets, no significant ALAS effects can be found incase of heavier targets, namely the Ni-isotopes [36]. Cowley et al. [37] used the WSaxon (WS) form to obtainα–58,60,62,64Ni elastic scattering fits. They found a discrepanat back angles between the optical model (OM) calculations and the experimentalsections even atEα = 32.3 MeV, which increases with the target mass.

The calculations of Singh et al. [38] and Budzanowski et al. [39], however, show thα–Ni potential is different from the WS form [38,39]. Budzanowski et al. [11] emplothe squared WS (SWS) form factors for the real, volume imaginary and derivative inary potentials in the OM and coupled channel (CC) analyses. The SWS form mitigadiscrepancy between the calculations and the data at backward angles. Microscopi

52 M.M. Billah et al. / Nuclear Physics A 762 (2005) 50–81

s

lar po-3],

ignif-to

rved byf,.

emi-

e the

ulsion.ent

itycom-

aussian

osed

lations using the double folding potential forα–58Ni can also give excellent descriptionof the data at back angles but the potential needs renormalization [26].

The purpose of the present paper is to investigate how the non-monotonic molecutential with its origin from the EDF formalism, the SWS potential of the form in [11–1and the single-folded potential from the model of [28,29] work for theα elastic scatteringon the even isotopes of Ni in the f–p shell, where ALAS does not seem to play a sicant role [36]. While theα–nucleus potential with the WS form factor is inadequatereproduce the absolute magnitude of the three-nucleon transfer reactions as obseKajihara et al. [40] in the60Ni(α,p)63Cu reaction, the above mentioned three types oα–Ni potential will, in a future paper, be examined with the available (α,p) reaction data [4142] on the Ni isotopes in an attempt to search for a global form of theα–nucleus potential

2. Alpha–nucleus potentials

The parameters of the non-monotonic (molecular), SWS (Michel) and folding (smicroscopic)α–Ni potential are parameterized in the following sub-sections.

2.1. Non-monotonic potential

The analytical form of the non-monotonic (molecular) potential is taken to havfollowing forms [19,23] for the real,Vm(R), and imaginary,Wm(R), parts:

Vm(R) = −V0

[1+ exp

(R − R0

a0

)]−1

+ V1 exp

[−

(R − D1

R1

)2]+ VC(R), (1)

Wm(R) = −W0 exp

[−

(R

RW

)2]− WS exp

[−

(R − DS

RS

)2], (2)

with the Coulomb potentialVC(R), of a uniformly charged sphere with the radiusRC ,given by

VC(R) =

[Z1Z2e

2

2RC

][3− R2

R2C

]for R � RC,

Z1Z2e2

Rfor R > RC.

(3)

The real part is non-monotonic due to the second term in (1) with a short-range repThe Coulomb radiusRC , in the molecular potential is the sum of the radii of the incidα-particle and the target nucleus and is given byRC = RαC + r0A

1/3, wherer0 = 1.35 fm,A is the target mass number andRαC is linked with the relatively large range of the densdistribution of alpha, as observed in [35]. The imaginary part in (2) is assumed to beposed of a volume term with a Gaussian shape and a surface term with a shifted Ggeometry.

2.2. Squared Woods–Saxon potential

The SWS (Michel) potential including the Coulomb term is considered to be compof the following forms [13] of the real,VM(R), and imaginary,WM(R), parts:


by

uding

e

o

eterstheb

ks ofrole,43].

VM(R) = −V0

[1+ exp

(R − R0

2a0

)]−2

+ VC(R), (4)

WM(R) = −W0

[1+ exp

(R − RW

2aW

)]−2

, (5)

with the Coulomb potentialVC(R) again given by (3) and the Coulomb radius givenRC = rCA1/3 with rC = 1.30 fm.

2.3. Folding potential

Assuming the density distributionsρα andρN , respectively, of theα-like clusters andunclustered nucleons in the target, the folding potential for the real part [28,29], inclagain the Coulomb potentialVC(R), is given by

VF (R) =∫

ρα(rα)Vαα

(|R − rα|)d3rα +∫

ρN(rN)VαN

(|R − rN |)d3rN

+ VC(R). (6)

In (6), ρα andρN satisfy the following normalization integral:∫ρα(rα) d3rα +

∫ρN(rN)d3rN = 4Aα + AN = A. (7)

HereAα andAN denote, respectively, the number ofα-like clusters and the number of thunclustered nucleons in the target.

The α–α potentialVαα and theα–nucleon potentialVαN in Eq. (6) are considered thave the forms:

Vαα(r) = VR exp(−µ2

Rr2) − VA exp(−µ2

Ar2), (8)

VαN(r) = −V0 exp(−K2r2). (9)

In (8), VA and VR are the attractive and repulsive depths with the range paramµA andµR , respectively. In (9),V0 andK are, respectively, the depth and range ofα–nucleon potential. The Coulomb potentialVC is again given by (3) and the Coulomradius isRC = rCA1/3 with rC = 1.30 fm.

A phenomenological Gaussian form is taken for the imaginary part with the form:

W(R) = −W0 exp

(− R2

R2W

). (10)

3. Energy density functional method

The shallow non-monotonic (molecular) potential is embedded in the early worBlock and Malik [18,19], who recognized that this potential is a manifestation of thedue to the Pauli exclusion principle and is obtainable from the EDF formalism [22


ution

nucleonservedergy inensity

on

to theene-within

tance

e,

de-of the

According to EDF [20], the energy of a system of fermions for a given density distribρ(r) can be expressed as

E =∫

ε[ρ(r)

]d3r, (11)

where

ε[ρ(r)

] = 0.3

(h̄2

2M

)(3π2

2

)2/3[(1− x)5/3 + (1+ x)5/3]ρ5/3 + ν(ρ, x)ρ

+ e

2ΦC(r)ρP − 0.739e2ρ

4/3P +

(h̄2

8M

)η(∇ρ)2. (12)

Herex is the neutron excess in the target nucleus,M is the nucleon mass, andη is the freeparameter adjusted to reproduce the nuclear mass. The average potential seen by ahas been calculated from the realistic two-nucleon potential that accounts for the obdeuteron properties and two-nucleon scattering data up to the pion-production enthe Brueckner–Hartree–Fock approximation [44] theory for nuclear matter. The ddependence is given by

ν(ρ, x) = b1(1+ a1x

2)ρ + b2(1+ a2x

2)ρ4/3 + b3(1+ a3x

2)ρ5/3, (13)

with a1 = −0.200, a2 = 0.316 anda3 = 1.646, b1 = −741.28, b2 = 1179.89 andb3 =−467.54. The Coulomb potentialΦC in (12) is related to the proton density distributiρP by

ΦC = e

∫ρP (r)|r − r′|d

3r′. (14)

The fourth and last terms in (12) denote, respectively, the exchange correctionCoulomb potentialΦC and correction to the first and second terms due to inhomogity in density distribution. The observed nuclear masses [45] can be reproduced1.5% using the observed density distributions [46] takingη = 8.0 [21].

The potentialV (R) between the projectile alpha and the target at a separation disof R is given by

V (R) = E[ρ(r,R)

] − Eα

[ρα(r,R)

] − ET

[ρT (r,R)

], (15)

whereρ is the density function of the composite system.ρα andρT are, respectively, thdensity functions for the projectileα and the target atR = ∞. In the sudden approximationthe density distribution function is given by

ρ(r) = ρα(r) + ρT (r). (16)

For the alpha particle the density distribution function is given in [47] by

ρα(r) = 4

(γ

π

)3/2

exp(−γ r2), (17)

with γ = 0.45–0.50 [23]. For Ni-isotopes, the density distribution function is taken,pending on the availability of the parameters values in de Vries et al. [46], as onefollowing forms:

(i) ρ(r) = ρ0

[1+ exp

(r − c

)]−1

(18)
z


n04.0V

;and

e

2]. Forbeeng

cor-ted

gener-rame-

set

in the two-parameter Fermi (2pF) model,

(ii) ρ(r) = ρ0

(1+ w

r2

c2

)[1+ exp

(r − c

z

)]−1

, (19)

in the three-parameter Fermi (3pF) model, and

(iii) ρ(r) = ρ0

(1+ w

r2

c2

)[1+ exp

(r2 − c2

z2

)]−1

, (20)

in the three-parameter Gaussian (3pG) model.

4. Analysis

Theα elastic scattering data are taken from the following sources:

(i) The α + 58Ni elastic scattering data forEα = 18.0, 21.0 and 24.1 MeV are takefrom [36]; 23.4 and 166.0 MeV from [38]; 26.5, 29.0, 32.3, 34.0, 38.0, 58.0, 1and 139.0 MeV from [11]; 42.0 MeV from [48]; 50.2 MeV from [49]; and 60.0 Mefrom [50].

(ii) The α + 60Ni elastic scattering data forEα = 15.0 MeV are taken from [51]18.0 MeV from [36,51]; 21.0, 24.1 and 27.0 MeV from [36]; 29.0, 32.3, 34.0104 MeV from [11]; and 60.0 MeV from [50].

(iii) The α + 62,64Ni elastic scattering data forEα = 18.0, 21.0, 24.1 and 27.0 MeV artaken from [36]; 32.3 MeV from [37]; and 60.0 MeV from [50].

The experimental data have been analyzed using the optical model code SCAT2 [5fitting the angular distributions, the chi-square minimization code MINUIT [53] hasused in conjunction with SCAT2. The fitting parameters are obtained by minimizinχ2

defined by

χ2 = 1

N

∑i

[σexp(θi) − σth(θi)

�σexp(θi)

]2

. (21)

Hereσexp(θi) and�σexp(θi) are, respectively, the experimental cross-section and theresponding error at the scattering angleθi . σth(θi) is the calculated cross-section generafrom the optical potential.N is the number of data points for a given incident energy.

4.1. Shallow non-monotonic (molecular) α–nucleus potential

The starting parameters for the shallow non-monotonic (molecular) potential areated by the EDF calculations. The calculations are performed using the following pater values:

(a) The width parameterγ in (17), for the alpha density distribution function, has beento γ = 0.45, leading to the binding energy BE= 20 MeV for the alpha particle.


g

t

ly,

s withing

he

ental

with

reEDFr

F-2o thers–6. At-factortermfound

heused.in Ta-F and

calcu-.rs (dot-effect ofdeter-ectednergywors-

stment

Table 1The parameters of the equivalent 2pF density distribution functions in (18) withz adjusted to generate the bindinenergies close to the experimental values.c andz are in fm,ρ0 in fm−3 and the energies in MeV

Target Parameters of 2pF density function Binding energy

c z ρ0 Calculation Experimen58Ni 4.154 0.5531 0.1644 506.37 506.4560Ni 4.200 0.5597 0.1645 526.83 526.8362Ni 4.262 0.5464 0.1645 549.14 545.2564Ni 4.212 0.5804 0.1722 561.99 561.74

(b) The density distribution functions for58Ni and 60Ni are then chosen, respectiveas 3pF in (19) and 3pG in (20), and those for62,64Ni, as 2pF given in (18). All thedensity distribution functions have been transformed to the equivalent 2pF onethe parameterz adjusted within the error bars quoted in [46] to obtain the bindenergies (BE) for58Ni, 60Ni, 62Ni and 64Ni close to their experimental values. Tparametersc and z of the 2pF distribution functions, noted in Table 1, yield BE=506.37, 526.83, 549.14 and 561.99 MeV, which compare favorably to the experimvalues 506.45, 526.83, 545.25 and 561.74 MeV, respectively, for58Ni, 60Ni, 62Ni and64Ni. Fig. 1 shows the comparison between the values of the density functionsthe parameters of [46] and those to reproduce BE of the four Ni-isotopes.

The values of the nuclear potentials for theα–Ni interaction, calculated from EDF, agiven in solid circles in Fig. 2. The broken and solid curves are the analytical fits to thepotentials in terms of the parameters in Eq. (1) withD1 = 0.0, resulting in the parameteset EDF-1 and withD1 �= 0.0 leading to EDF-2, respectively, in Table 2. EDF-1 and EDdiffer mainly in the central region of the target nuclei. Figs. 3 and 4 show the fits tα elastic scattering cross-section data on58.60,62.64Ni using the real potential parameteof set-1 and set-2 given in Tables 2–6, and those of the imaginary part in Tables 3the lower incident energies, the surface absorption with the shifted Gaussian formgiven by the second term in Eq. (2) is found to work better than the volume imaginarywith the Gaussian shape. At the higher incident energies, the volume absorption isto be important and applied to58Ni for Eα � 58 MeV and to60Ni at Eα = 104 MeV, inaddition to the surface imaginary part. For the62,64Ni targets, the data are restricted to tmaximum incident energy of 60 MeV and hence no volume imaginary part has been

The fit-parameters of set-1 and set-2 for the real part (even rows) are comparedble 2 with the corresponding EDF-parameters (odd rows). The closeness of the EDfit-parameters for all the isotopes of Ni substantiates the importance of the EDFlations in predicting the parameters of the shallow non-monotonicα–nucleus potentialFurthermore, the proximity of the calculated cross-sections using the set-1 parameteted curves) and those using the set-2 parameters (solid curves) suggests that thethe potential shape in the central region of the target nucleus is not that significant inmining the angular distribution of cross-sections at low incident energies. This is expsince the potential determined from the EDF calculations is valid near zero incident eand the scattering is dominated by the nuclear potential at the nuclear surface. Theening effects of using the unshifted core are seen to be mitigated by the slight adju


2pF

arerenceialt alpha

ints, asts

-nergiese po-he real

the

Fig. 1. Density distribution function values in solid circles using (a) 3pF function for58Ni, (b) 3pG for 60Ni,(c) 2pF for62Ni and (d) 2pF for64Ni from de Vries et al. [46]. The solid curves are the densities using theparameters in Table 1 to reproduce the experimental BE values.

of the parameters other than the shifting parameterD1. However, the effect of the nucleinterior becomes significant at higher incident energies as can be seen from the diffin the predictions (Figs. 3 and 4) for theα + 58,60Ni elastic scattering using the potentinvolving Gaussian and shifted Gaussian repulsive core at and beyond the incidenenergy of 60 MeV. In particular, the fold at around the scattering angleθcm = 100◦ in theangular distribution ofα + 58Ni at 60 MeV can be generated only by the solid curveFig. 3, resulting from the potential set-2. Moreover, the overall fits for the four targedictated by the totalχ2-value,χ2

T = ∑i,j (χ

2)ij with i running over all the energy poinandj referring to the targets, improves fromχ2

T = 1560.7 for set-1 toχ2T = 1133.1 for

set-2. The potential set-1 with one parameter (withD1 = 0.0) less is obviously more advantageous. It is noticeable from Tables 3 and 4 that the experimental data at ebeyondEα = 50.2 MeV need changes in the parameter-values for the real part of thtential. This is a consequence of the dispersion relation [54–56] existing between tand imaginary parts of the potential.

In order to generate a more consistent potential parameter set withD1 = 0.0 for theshallow non-monotonicα–Ni potential, all the parameters of set-1 are freed, keeping


hedlsive core

Thising fitshough

hinse

et-2soveringhe realthe–

r

Fig. 2. Parametrization of theα–58,60,62,64Ni potentials from the EDF calculations (solid circles). The dasand solid curves denote the real potentials using, respectively, the Gaussian and shifted Gaussian repuwith the EDF-1 and EDF-2 parameters in Table 2.

volume integralJR/(4A) nearly same, to achieve better fits to the experimental data.procedure results in the parameter set-3, listed in Tables 7 and 8. The correspondare shown in the dotted curves in Figs. 5 and 6 and solid curves in Figs. 7 and 8. Altthe set-3 parameters fail to reproduce the fold near the scattering angle of 100◦ in theangular distribution for theα + 58Ni case atEα = 60 MeV, the overall fits are better witχ2

T = 1401.8, compared toχ2T = 1560.7 for set-1. Moreover, the domain of energy-b

for each of the targets, with the constancy of theV0, R0, a0, V1 andR1 parameters, is morexpanded for set-3 than that for set-1 or set-2. For example, while set-1 for58Ni needsdifferent values ofV1 at almost all energy points in the interval 58.0–166 MeV and srequires two sets of theV0, R0 anda0 values in addition to the differentV1 values, set-3 haone set of parameter values in the same interval leading to only two parameter sets cthe entire 18.0–166.0 MeV range. Likewise, set-3 has only one set of parameters for tpart of theα–60,62,64Ni potential covering the whole range of incident energies. Whilevolume integralsJR/(4A) varies from 122.1 to 120.0 MeV fm3 in the energy range 18.0166.0 MeV for58Ni, and the volume integralJR/(4A) = 101.3, 103.5 and 109.2 MeV fm3

are obtained, respectively, for60Ni in the energy range 15.0–104.0 MeV, and62Ni and64Nithe energy range 18.0–60.0 MeV. The volume integrals for58Ni are higher than those fothe other isotopes of Ni.


r

-et-4)

Table 2Comparison of the potential calculated from EDF and the real part of the non-monotonicα–58,60,62,64Ni poten-tials determined from the fits shown in Figs. 3 and 4. The shift parameterD1 is zero for set-1 and non-zero foset-2.V0 andV1 are in MeV;R0, R1, RC , a0 andD1, in fm; andJR/(4A), in MeV fm3

Target Set V0 R0 a0 V1 R1 D1 RC JR/(4A)

58Ni EDF-1 27.52 6.00 0.668 46.40 3.20 0.00 8.50 84.0Set-1 33.00 6.00 0.668 23.00 3.20 0.00 8.70 126.4EDF-2 28.50 5.92 0.668 20.00 2.30 2.15 8.50 88.50Set-2 36.00 5.92 0.668 15.00 2.30 2.15 8.70 128.1

60Ni EDF-1 36.31 5.949 0.738 49.18 3.945 0.00 8.80 83.6Set-1 37.50 5.949 0.738 28.00 3.945 0.00 8.80 118.9EDF-2 38.77 5.891 0.706 29.88 2.860 2.217 8.80 88.50Set-2 44.75 5.891 0.706 28.00 2.860 2.217 8.80 117.2

62Ni EDF-1 33.66 6.050 0.699 46.93 3.902 0.00 8.90 79.9Set-1 36.00 6.050 0.699 25.00 3.902 0.00 8.90 119.1EDF-2 36.47 5.978 0.672 27.47 2.754 2.378 8.90 85.00Set-2 47.30 5.978 0.672 35.50 2.754 2.378 8.90 110.5

64Ni EDF-1 39.32 5.968 0.758 55.66 3.990 0.00 9.00 81.6Set-1 40.00 5.968 0.758 34.50 3.990 0.00 9.00 113.6EDF-2 42.71 5.916 0.728 35.41 3.000 2.110 9.00 86.8Set-2 52.50 5.916 0.728 37.50 3.000 2.110 9.00 120.2

Table 3Non-monotonic set-1 potential parameters for58Ni. The energy independent parametersV0, R0, a0, R1, D1 =0.0 andRC are same as in Table 2.Eα , V1, W0 andWS are in MeV.RS , DS andRW are in fm. The volumeintegralsJR/(4A) andJI /(4A) are in MeV fm3

Target Eα V1 DS RS W0 RW WS JR/(4A) JI /(4A) χ2

58Ni 18.0 23.0 5.55 1.53 0.00 4.00 6.00 126.4 28.2 7.5121.0 6.00 28.2 14.823.4 6.35 29.8 6.8324.1 6.60 31.0 9.6126.5 7.20 33.8 112.229.0 7.30 34.3 9.6532.3 7.50 35.2 14.234.0 7.50 35.2 14.838.0 7.50 35.2 18.742.0 8.00 37.6 17.050.2 8.50 39.9 17.258.0 70.0 1.0 12.0 89.4 57.9 30.160.0 12.0 57.9 25.8

104.0 87.0 8.0 10.0 76.0 59.3 19.5139.0 117.0 9.0 10.0 52.4 60.8 54.1166.0 119.0 9.5 10.0 50.9 61.6 12.0

4.2. Deep non-monotonic α–nucleus potential

The present work investigates the existence of deep non-monotonicα potentials for58,60,62,64Ni, with parameters defined in Eqs. (1)–(2) withD1 = 0.0 fm having volume integralsJR/(4A) ≈ 300 MeV fm3. Tables 9 and 10 list the best-fit potential parameters (s


e

Table 4Same as Table 3 for60,62,64Ni. The Coulomb radiiRC are same as those in Table 2

Target Eα V1 DS RS W0 RW WS JR/(4A) JI /(4A) χ2

60Ni 15.0 28.0 5.55 1.53 0.00 4.00 5.00 118.9 22.7 10.618.0 5.50 25.0 8.8821.0 6.00 27.2 8.3324.1 7.00 31.8 59.827.0 7.50 34.1 47.229.0 8.00 36.3 52.832.3 8.00 36.3 13.834.0 8.00 36.3 20.260.0 72.0 10.0 56.2 45.4 387.5

104.0 96.0 6.0 11.0 22.0 58.9 77.962Ni 18.0 25.00 5.55 1.53 0.0 – 7.50 119.1 33.0 3.80

21.0 7.50 33.0 7.2824.1 7.70 33.8 167.427.0 8.50 37.4 24.132.3 10.0 43.9 19.360.0 73.75 14.0 54.0 61.5 36.8

64Ni 18.0 34.50 5.55 1.55 0.0 – 8.00 113.6 34.5 3.6621.0 8.00 34.5 6.3124.1 8.00 34.5 135.027.0 9.25 39.9 39.932.3 12.0 51.8 31.260.0 80.00 16.0 50.7 69.1 15.0

Table 5Non-monotonic set-2 potential parameters for58Ni. The parametersRC andRW are kept same as in Table 2. Thshift parameterDS = 5.55 fm is held constant for all energies.Eα , V0, V1, W0 andWS are in MeV.R0, a0, R1,D1 andRS are in fm. The volume integralsJR/(4A) andJI /(4A) are in MeV fm3

Target Eα V0 R0 a0 V1 R1 D1 RS W0 WS JR/4A JI /4A χ2

58Ni 18.0 36.0 5.92 0.668 15.0 2.30 2.15 1.53 0.0 5.50 128.1 25.8 8.6721.0 5.90 27.7 15.523.4 6.20 29.1 8.1924.1 6.40 30.1 12.126.5 7.00 32.9 146.129.0 7.00 32.9 8.3332.3 7.50 35.2 11.734.0 7.50 35.2 12.538.0 7.50 35.2 15.542.0 8.00 37.6 14.550.2 9.00 42.3 17.358.0 33.5 6.0 0.770 59.5 2.10 1.90 1.5 14.0 83.2 68.1 8.1060.0 61.0 2.0 14.0 81.5 68.8 9.21

104.0 42.0 0.590 87.0 9.0 14.0 79.0 79.6 11.6139.0 99.0 13.0 13.0 65.1 81.0 33.4166.0 119.0 16.5 12.0 64.0 81.7 12.8


Table 6Same as Table 5 for60,62,64Ni. The shift parameters areDS = 5.20,5.25 and 5.30 fm, respectively, for60,62,64Ni.The Coulomb radiiRC are same as those in Table 2

Target Eα V0 R0 a0 V1 R1 D1 RS W0 WS JR/4A JI /4A χ2

60Ni 15.0 44.75 5.891 0.706 28.0 2.86 2.217 1.53 0.00 5.00 117.2 22.7 9.9018.0 5.00 22.7 8.5821.0 6.00 27.2 9.6124.1 8.00 36.3 65.927.0 8.00 36.3 38.229.0 8.00 36.3 60.832.3 8.00 36.3 12.134.0 8.00 36.3 18.360.0 38.50 5.800 0.920 74.5 3.30 1.250 15.5 28.1 70.4 73.2

104.0 87.5 2.0 15.0 4.50 71.1 7.7062Ni 18.0 47.30 5.978 0.672 35.50 2.754 2.378 1.53 0.0 5.50 110.5 24.2 6.40

21.0 5.50 24.2 9.8124.1 7.50 33.0 248.427.0 8.30 36.5 27.232.3 9.90 43.5 20.960.0 43.50 5.850 0.870 80.00 3.400 1.000 0.0 17.0 74.7 48.2 10.24

64Ni 18.0 52.50 5.916 0.728 37.50 3.00 2.111 1.55 0.0 7.00 120.2 30.2 5.1821.0 7.20 31.1 6.5624.1 8.50 36.7 84.427.0 9.50 41.0 20.432.3 10.5 45.3 29.160.0 44.0 5.850 0.830 81.00 3.40 1.000 17.5 75.5 44.3 14.8

Table 7Non-monotonic set-3 potential parameters for58Ni. The parameters,D1 = 0.0, R1 = 4.0 andRW = 4.0, RS =1.40,RC = 8.7 andDS = 5.20 fm, are used at all energies.Eα , V0, V1, W0 andWS are in MeV.R0 anda0 arein fm. The volume integralsJR/(4A) andJI /(4A) are in MeV fm3

Target Eα V0 R0 a0 V1 W0 WS JR/(4A) JI /(4A) χ2

58Ni 18.0 28.5 6.40 0.56 15.0 0.00 6.00 122.1 22.6 8.8221.0 6.70 25.2 23.023.4 7.80 29.4 7.7624.1 8.50 32.0 13.826.5 10.0 37.7 85.529.0 11.0 41.4 13.132.3 12.5 47.1 51.634.0 13.1 49.3 26.938.0 14.3 53.9 15.542.0 16.1 60.6 19.950.2 17.0 67.0 22.058.0 32.5 6.50 0.52 33.4 11.5 17.4 120.0 83.2 40.260.0 12.5 17.0 83.2 56.1

104.0 22.0 16.5 95.9 15.0139.0 26.5 14.8 96.5 34.4166.0 30.3 13.4 97.0 20.5


nd set-2

olidbetter

t of

Fig. 3. Experimental differential cross-sections (solid circles) for theα + 58Ni elastic scattering at differenincident energies are compared to the predictions using the set-1 parameters (dotted curves) in Table 3parameters (solid curves) of the non-monotonic potentials in Table 5.

along with theχ2 values at different incident energies for the fits, which are shown in scurves in Figs. 5 and 6. The deep non-monotonic set-4 potential produces a tangiblyfit to the α + 58Ni data at 60 MeV (Fig. 5) and an overall superior fit withχ2

T = 1234.0to that withχ2 = 1401.8 for set-3. The performance of set-4 compares closely to th

ta

a
T


ntal
Fig. 4. Same as in Fig. 3 for theα+60,62,64Ni elastic scattering with solid circles and triangles as the experimedata. The set-1 parameters are now in Table 4 and set-2 in Table 6.


e

Surves.fit is-set-3

et-2

Table 8Same as Table 7 for60,62,64Ni. The parameters,D1 = 0.0, R1 = 4.0 andRW = 4.0 fm, are same for the thretargets. The range parameters of the surface imaginary part areRS = 1.40 fm for 60,62Ni andRS = 1.47 fm for64Ni. The corresponding shift parameters areDS = 5.20 fm for 60Ni andDS = 5.25 and 5.30 fm, respectivelyfor 62,64Ni. The Coulomb radii areRC = 8.8, 8.9 and 9.0, respectively, for60,62,64Ni

Target Eα V0 R0 a0 V1 W0 WS JR/(4A) JI /(4A) χ2

60Ni 15.0 29.0 6.50 0.59 33.0 0.00 7.00 101.3 25.5 5.0018.0 7.50 27.3 5.7221.0 9.50 34.6 36.924.1 11.0 41.9 111.727.0 12.0 43.7 27.029.0 12.5 45.5 106.632.3 15.0 54.6 31.634.0 15.0 54.6 24.960.0 18.0 65.5 26.4

104.0 15.0 16.0 80.5 8.1862Ni 18.0 30.0 6.50 0.60 33.0 0.0 7.50 103.5 26.9 8.07

21.0 9.50 34.1 10.124.1 11.9 42.7 171.027.0 12.0 43.0 28.732.3 15.5 55.6 65.860.0 21.0 75.4 11.1

64Ni 18.0 31.5 6.50 0.64 33.0 0.0 5.75 109.2 21.5 9.0421.0 9.00 33.6 8.8524.1 10.0 37.3 179.327.0 12.5 46.6 25.532.3 17.0 63.4 35.460.0 21.5 80.2 10.9

set-2 withχ2T = 1133.1. The volume integralsJR/(4A) vary from 363.9 to 311.4 MeV fm3

in the energy range 18.0–166.0 MeV for58Ni, and 316.2 to 310.5 MeV fm3 in the en-ergy range 15.0–104.0 MeV for60Ni. For the62,64Ni isotopes, the volume integrals aJR/(4A) = 312.4 and 278.4 MeV fm3 in the energy range 18.0–60.0 MeV for62Ni and64Ni, respectively. The volume integralsJR/(4A) are distinctly higher for58Ni and lowerfor 64Ni than those for60,62Ni.

4.3. Squared Woods–Saxon (SWS) α–nucleus potential

Results of analyses of theα + 58,60,62,64Ni elastic scattering data in terms of the SW(Michel) optical potential parameters in Eqs. (4) and (5) have been shown in dottedin Figs. 7 and 8 with the potential parameters and theχ2-values noted in Tables 11 and 1Although the fits at the higher incident energies are satisfactory and, in particular, thbetter for theα + 58Ni scattering atEα = 60 MeV (Fig. 7) than that produced by the nomonotonic potentials, the fits are inferior at the lower energies to the non-monotonicpotential. This is reflected in the totalχ2 valueχ2

T = 2782.2 which is much higher than thvaluesχ2

T = 1560.7, 1133.1 and 1401.8, respectively, for the non-monotonic set-1,and set-3 potentials. The volume integrals vary fromJR/(4A) = 373.3 to 312.5 MeV fm3

in the energy range 18.0–166.0 MeV for58Ni, 353.9 to 303.4 MeV fm3 in the energy range

e

,

r

c2en

es


Table 9Non-monotonic set-4 (deep) potential parameters for58Ni. The parameters,D1 = 0.0, RW = 4.0, RC = 8.7 fm,are used for all incident energies. No surface imaginary potential has been employed.Eα , V0, V1 andW0 are inMeV. R0, a0 andR1 are in fm. The volume integralsJR/(4A) andJI /(4A) are in MeV fm3

Target Eα V0 R0 a0 V1 R1 W0 JR/(4A) JI /(4A) χ2

58Ni 18.0 180.0 5.20 0.66 148.0 3.60 29.0 363.9 44.6 8.2221.0 31.0 47.6 10.123.4 33.0 50.7 4.5324.1 36.8 56.5 7.3626.5 39.7 61.0 13.329.0 39.7 61.0 14.932.3 170.0 45.0 339.3 69.1 142.634.0 45.0 69.1 69.138.0 48.0 73.7 21.142.0 48.0 73.7 9.5850.2 50.0 76.8 71.058.0 116.0 0.70 32.0 67.0 311.4 102.9 5.0160.0 67.0 102.9 14.5

104.0 90.0 138.3 7.69139.0 125.0 192.0 141.5166.0 130.0 200.0 19.5

Table 10Same as Table 9 for60,62,64Ni. The Coulomb radii areRC = 8.8, 8.9 and 9.0 fm, respectively, for60,62,64Ni

Target Eα V0 R0 a0 V1 R1 W0 JR/(4A) JI /(4A) χ2

60Ni 15.0 111.8 5.30 0.66 15.4 3.75 22.0 316.2 32.7 11.018.0 23.0 34.2 8.8921.0 30.0 44.6 18.024.1 34.5 51.2 34.527.0 38.0 56.4 34.529.0 41.0 60.9 29.632.3 45.0 66.8 85.534.0 45.5 67.6 13.160.0 115.8 0.62 25.4 65.5 310.5 97.3 45.7

104.0 84.0 124.7 7.1262Ni 18.0 126.6 5.30 0.69 50.0 3.75 23.0 312.4 33.1 8.33

21.0 28.0 40.2 14.724.1 34.0 48.9 50.727.0 38.0 54.6 63.732.3 51.0 73.3 49.360.0 86.0 123.6 25.7

64Ni 18.0 128.2 5.20 0.70 60.0 3.75 22.0 278.4 30.6 16.321.0 30.0 41.8 23.624.1 35.0 48.7 23.827.0 40.0 55.7 34.432.3 45.0 62.7 66.660.0 80.0 111.4 8.97


Table 11SWS potential parameters for58Ni. Eα , V0 andW0 are in MeV.R0, a0, RW andaW are in fm. The volumeintegralsJR/(4A) andJI /(4A) are in MeV fm3. RC is given byRC = 1.3A1/3

Target Eα V0 R0 a0 W0 RW aW JR/(4A) JI /(4A) χ2

58Ni 18.0 142.0 5.85 0.52 12.50 6.20 0.52 371.3 39.2 11.621.0 138.0 15.0 360.9 47.0 29.023.4 137.9 15.9 360.6 49.8 21.924.1 137.8 16.0 360.4 50.1 90.326.5 137.7 17.0 360.1 53.3 244.529.0 135.0 18.0 353.0 56.4 21.232.3 133.0 20.0 347.8 62.7 45.934.0 132.9 20.1 347.5 63.0 13.238.0 130.0 22.0 340.0 69.0 17.342.0 128.0 24.0 334.7 75.2 11.350.2 124.9 25.0 326.6 78.3 48.158.0 124.8 32.0 326.4 100.3 25.060.0 124.0 32.2 324.3 100.9 42.5

104.0 122.0 40.0 319.0 125.3 10.5139.0 120.0 47.0 313.8 147.3 38.9166.0 119.5 47.1 312.5 147.6 12.8

Table 12Same as Table 11 for60,62,64Ni

Target Eα V0 R0 a0 W0 RW aW JR/(4A) JI /(4A) χ2

60Ni 15.0 140.0 5.85 0.52 10.0 6.20 0.52 353.9 30.3 7.3618.0 139.0 12.0 351.4 36.3 22.321.0 138.9 15.0 351.1 45.4 27.124.1 137.0 17.0 346.3 51.5 308.627.0 136.0 17.5 343.8 53.0 164.829.0 135.0 21.0 341.3 63.6 31.032.3 134.0 22.0 338.7 66.6 92.234.0 133.5 22.1 337.5 67.0 16.760.0 130.0 35.0 328.6 106.0 54.1

104.0 120.0 41.0 303.4 124.2 11.662Ni 18.0 145.0 5.85 0.53 15.0 6.20 0.53 354.1 43.9 8.32

21.0 141.0 16.0 344.3 46.8 12.124.1 139.0 18.0 339.4 52.6 609.927.0 137.0 18.5 334.5 54.1 70.632.3 133.0 23.0 324.8 67.3 108.560.0 123.0 40.0 300.3 117.0 12.2

64Ni 18.0 148.0 5.85 0.54 14.0 6.20 0.54 349.5 39.6 25.621.0 143.0 16.0 337.7 45.2 26.124.1 140.0 18.0 330.6 50.9 218.427.0 137.0 19.0 323.5 53.7 136.332.3 134.0 24.0 316.4 67.9 116.360.0 124.0 41.5 292.8 117.3 18.19


turves) in

Vnce

Fig. 5. Experimental differential cross-sections (solid circles) for theα + 58Ni elastic scattering at differenincident energies are compared to the predictions using the non-monotonic set-3 parameters (dotted cTable 7 and set-4 parameters (solid curves) in Table 9.

15.0–104.0 MeV for60Ni, 354.1 to 300.3 MeV fm3 in the energy range 18.0–60.0 Mefor 62Ni, and 349.5 to 292.8 MeV fm3 for 64Ni in the same energy range. In consonawith the shallow and deep non-monotonic potentials, the volume integralsJR/(4A) aredistinctly higher for58Ni than those for the other three isotopes.


id
Fig. 6. Same as in Fig. 5 for (a)α + 60Ni, (b) α + 62Ni and (c)α + 64Ni elastic scattering with the data in solcircles and triangles with the potential parameters in Tables 8 and 10.


WSand 14.

Fig. 7. Same as in Fig. 5 for theα + 58Ni elastic scattering using the non-monotonic set-3 (solid curves), S(dotted curves) and folding (dashed curves) potential parameters, respectively, noted in Tables 7, 11, 13

4.4. Folding α–nucleus potential

In the analyses, the parameter valuesVA = 122.62 MeV andµA = 0.469 fm−1 inEq. (8) taken from Buck et al. [57], andV0 = 47.3 MeV andK = 0.435 fm−1 from Sacket al. [58] have been kept fixed. The Coulomb radius parameterrC = 1.30 fm has also


id
Fig. 8. Same as in Fig. 7 for (a)α + 60Ni, (b) α + 62Ni and (c)α + 64Ni elastic scattering with the data in solcircles and triangles and the potential parameters in Tables 8, 12, 13 and 15.


for then

the

thethe

topesisotopes

. The-rokenverall

Table 13Energy independent parameters and the deduced results for the number of nucleons 4Aα in α-like clusters andAN in unclustered configuration.ρ0α andρ0N are in fm−3, RW , cα , cN andz are in fm.VA = 122.62 MeV,µA = 0.469 fm−1, V0 = 47.3 MeV, K = 0.435 fm−1 andrC = 1.3 fm. µR is in fm−1

Target ρ0α ρ0N RW w cα cN µR z 4Aα AN

58Ni 0.03574 0.06496 3.80 −0.131 4.309 3.10 0.533 0.517 49.0 9.060Ni 0.03599 0.09325 3.80 −0.267 4.489 3.10 0.535 0.537 49.0 11.062Ni 0.03501 0.07890 4.00 −0.209 4.443 3.40 0.530 0.539 49.0 13.064Ni 0.03402 0.13280 4.00 −0.228 4.521 3.00 0.530 0.528 49.0 15.0

Table 14The energy dependent parameters of the folding potential for58Ni. The depth parametersVR andW0 are in MeV,and volume integralsJR/(4A) andJI /(4A), in MeV fm3 at different incident energies

Target Eα VR W0 JR/(4A) JI /(4A) χ2

58Ni 18.0 56.0 25.0 350.2 31.5 14.421.0 65.0 34.0 333.4 42.9 12.823.4 80.5 34.1 304.5 43.0 34.424.1 81.0 36.0 303.6 45.4 24.326.5 81.5 40.0 298.0 50.5 49.129.0 85.0 43.0 296.1 54.2 19.532.3 87.0 46.0 292.4 58.0 119.834.0 87.1 49.0 292.2 61.8 30.838.0 95.0 60.0 277.4 75.7 13.942.0 98.0 63.0 271.9 79.5 20.250.2 105.0 65.0 258.8 82.0 24.158.0 110.0 75.0 249.5 94.6 26.560.0 110.2 75.2 249.0 94.9 32.3

104.0 110.3 82.0 248.9 103.4 9.82139.0 110.4 82.1 248.7 103.6 29.3166.0 110.5 82.2 248.5 103.7 7.47

been taken to be the same for all the four targets. The density distribution functionsα-like clusters and the unclustered nucleons in each of the58,60,62,64Ni targets have beeassumed to have the 3pF functional form given in Eq. (19). We attach the subscriptsα andN to the radius parameterc to denotecα andcN as the corresponding parameters forα-density and nucleon-density distributions in the target nucleus. Similarly,ρ0α andρ0N

refer to the central values for theα and nucleonic density distributions. The values ofother two parameters, namelyw andz in Eq. (19) have been assumed the same forα and nucleonic distributions. The values of the parametersw, cα andz, from de Vrieset al. [46], which are noted in Table 13, have been left unaltered for each of the isoduring the analyses. The energy independent adjustable parameters for each of theareρ0N , ρ0α , µR (the range parameter for the repulsive part) andcN for the real part ofthe potential, andRW in Eq. (10). These parameter values are also listed in Table 13energy dependent best-fit parametersVR in (8) andW0 in (10) along with the volume integrals and theχ2-values are recorded in Tables 14 and 15. The fits are displayed in bcurves in Figs. 7 and 8. Apart from the few cases at lower incident energies, the ofits are satisfactory with the totalχ2-valueχ2 = 1207.1, which is less thanχ2 = 1401.8
T T


er, the

ntial.there

and 15

n-depthls theenergy

-range

Table 15Same as Table 14 for60,62,64Ni

Target Eα VR W0 JR/(4A) JI /(4A) χ2

60Ni 15.0 40.0 20.0 393.7 24.4 12.618.0 45.0 27.0 384.8 32.9 28.321.0 48.0 32.0 379.4 39.0 50.924.1 58.0 42.0 361.6 51.2 25.027.0 65.0 42.1 349.1 51.3 119.629.0 67.1 47.0 345.3 57.3 58.932.3 68.0 50.0 343.7 61.0 81.334.0 70.0 52.0 340.2 63.4 32.460.0 100.0 88.0 286.6 107.3 12.1

104.0 110.0 95.0 268.8 115.9 4.8262Ni 18.0 64.0 25.0 360.6 34.3 11.7

21.0 65.0 32.0 358.8 43.9 11.524.1 70.0 37.5 350.0 51.5 35.727.0 78.0 38.0 335.8 52.1 29.932.3 80.0 48.0 332.2 65.9 61.960.0 95.0 80.0 305.6 109.8 15.8

64Ni 18.0 61.0 28.0 378.0 37.2 9.3421.0 62.0 35.0 376.3 46.5 12.524.1 63.0 38.0 374.6 50.5 14.027.0 64.0 43.0 372.8 57.2 21.332.3 68.0 50.0 366.0 66.5 55.760.0 82.0 80.0 342.0 106.3 33.1

and 2782.2, respectively, for the non-monotonic set-3 and SWS potentials. Howevfold near the scattering angle of 100◦ in the angular distribution of theα + 58Ni scatter-ing atEα = 60 MeV (Fig. 7) could not be reproduced by the generated folding poteThe remarkable result emerging from the analyses with the folding potential is thatis consistency in the deduced number of nucleons 4Aα = 49 (Table 13) forming theα-likeclusters in all the four isotopes, the number of unclustered nucleons being 9, 11, 13in 58Ni, 60Ni, 62Ni and64Ni, respectively. The volume integralsJR/(4A) varies from 350.2to 248.5 MeV fm3 in the energy range 18.0–166.0 MeV for58Ni, 393.7 to 268.8 MeV fm3

in the energy range 15.0–104.0 MeV for60Ni, and 360.6 to 305.6 MeV fm3 for 62Ni and378.0 to 342.0 MeV fm3 for 64Ni in the common energy range of 18.0–60.0 MeV.

4.5. Energy dependence of the depth of the real and imaginary parts of potentials

The parameters of the real attractive depthV0 of the non-monotonic potentials are eergy independent within the energy region considered in the work. The repulsiveV1 varies discretely within the energy-bins. Hence for the non-monotonic potentiaanalytical form of the energy dependence for the real part is not investigated. Thedependences of the depthV0 for the SWS potential, the repulsive depthVR for the foldingpotential and the imaginary depths have been investigated within the common energyfor the four isotopes.


d

10,and 17.the

at forrbital

Fig. 9. Energy dependence of the depthV0 (Table 16) of the real part of the SWS potential for (a)58Ni, (b) 60Ni,(c) 62Ni and (d)64Ni.

4.5.1. Energy dependence of the depth of the real part of potentialsTo study the energy dependent parameters in the real depthV0 of the SWS potential an

the repulsive depthVR of the folding potential, the following relations are employed:

V0 = V00(1+ A1Eα + A2E

2α + A3E

3α

), (22)

VR = VR0(1+ B1Eα + B2E

2α + B3E

3α

). (23)

The analytical fits to theV0 andVR values are displayed, respectively, in Figs. 9 andwhile the energy dependent parameters of Eqs. (22) and (23) are listed in Tables 16The energy variation curve forV0 is similar for the four isotopes, as also reflected inparameter values given in Table 16. But the curve pattern ofVR for 64Ni, apart from thedifferent values of the parameters in Table 17, bears a different curvature from ththe other isotopes. This may be linked to either a few energy points or different oconfiguration of the nucleons.


al

g

Fig. 10. Same as in Fig. 9 for the repulsive depthVR (Table 17) of the real part of the folding potential.

Table 16Parameters associated with the energy dependence of the depthV0 of the real part of the SWS (Michel) potentifor 58,60,62,64Ni

Target V00 (MeV) A1 (MeV−1) A2 (MeV−2) A3 (MeV−3)58Ni 157.20 −0.006281 0.0000459 0.060Ni 148.11 −0.003907 0.0000310 0.062Ni 165.29 −0.008147 0.0000647 0.064Ni 171.44 −0.009382 0.0000796 0.0

Table 17Parameters associated with the energy dependence of the depthVR of the repulsive real part of the foldinpotential for58,60,62,64Ni

Target VR0 (MeV) B1 (MeV−1) B2 (MeV−2) B3 (MeV−3)58Ni 8.4219 0.4567 −0.006031 0.000029560Ni 3.8577 0.7117 −0.006568 0.000002762Ni 29.1637 0.07505 −0.0006239 0.064Ni 53.2572 0.007127 0.0000314 0.0


al

e non-:

ab-es. Atr

factor

catter-

shownre

(25)

of thel ande oneallow

Table 18Parameters associated with the energy dependence of the depthWS of the surface imaginary andW0 of thevolume imaginary parts of the non-monotonic set-3 potential for58,60,62,64Ni

Target WS0 (MeV) W00 (MeV) C1 (MeV−1) C2 (MeV−2) C3 (MeV−3)58Ni 0.09055 – 1.4770 0.14840 −0.002060Ni 0.1065 – 3.2967 0.06735 −0.0012662Ni 0.1094 – 2.3742 0.1156 −0.00170364Ni 0.1981 – −0.9001 0.1820 −0.00228558Ni – −4.814 −0.06802 0.000147 0.0

Table 19Parameters associated with the energy dependence of the depthW0 of the imaginary part of the SWS potentifor 58,60,62,64Ni

Target W00 (MeV) C1 (MeV−1) C2 (MeV−2) C3 (MeV−3)58Ni 3.8455 0.1319 −0.0001943 0.060Ni 3.8370 0.1302 0.0001600 0.062Ni 4.9237 0.1022 0.0002756 0.064Ni 4.9181 0.0929 0.0005061 0.0

4.5.2. Energy dependence of the depth of the imaginary parts of potentialsThe energy dependence of the depth parameters in the imaginary part of th

monotonic set-3, SWS and folding potential is sought through the following relations

WS = WS0(1+ C1Eα + C2E

2α + C3E

3α

), (24)

W0 = W00(1+ C1Eα + C2E

2α + C3E

3α

). (25)

Eq. (24) involvingWS applies to the non-monotonic potentials, where purely surfacesorption with the shifted Gaussian geometry is used at the lower incident energihigher energies, the volume absorption involvingW0 is found to play a significant role. Fothe SWS and folding potential only the volume absorption with the Gaussian form-has been applied.

For the non-monotonic set-3 potential, the parameters of Eq. (24) for the elastic sing on58,60,62,64Ni in the energy range up to 60 MeV and those of (25) for the case of58Niin the energy interval 58.0–166.0 MeV are given in Table 18. The parameter fits arein Figs. 11 and 12. The parameters concerningW0 for the SWS and folding potentials alisted, respectively, in Tables 19 and 20 while the analytical fits in terms of the relationare displayed in Figs. 13 and 14.

5. Discussions and conclusions

The present work reports the results of analyses on the experimental valuesα + 58,60,62,64Ni elastic scattering cross-sections in terms of three phenomenologicaone semi-microscopic optical potentials. The phenomenological potentials includmonotonic type with the SWS form-factor and two non-monotonic potentials, sh


l.

based

entialse pa-own in

of theEDF

Fig. 11. Same as in Fig. 9 for the surface imaginary depthWS (Table 18) of the non-monotonic set-3 potentia

Table 20Same as in Table 19 for the Folding potential

Target W00 (MeV) C1 (MeV−1) C2 (MeV−2) C3 (MeV−3)58Ni 3.2318 0.4721 −0.00154 0.060Ni 3.2247 0.4473 −0.0000902 0.062Ni 3.1729 0.4389 −0.0005632 0.064Ni 3.5586 0.4392 −0.001336 0.0

(molecular) and deep. The semi-microscopic potential, used in the present work, ison the folding model of Abdullah et al. [28,29]. Both visually and in terms ofχ2, set-2gives the best overall fits, although the deep non-monotonic, SWS and folding potwhich have been propounded to account for the ALAS effect seem to work well. Thrameters of the real part of the molecular potentials set-1 and set-2, providing fits shFigs. 3 and 4, are close to the EDF-1 withD1 = 0.0 and EDF-2 withD1 �= 0.0 parameters(Table 2), respectively, for all the four isotopes. This suggests that the successesmolecular potentials, including those reported in [25,30–33], are due to their origin in


ial

ch is a

meterslyticalnuclearere aremeters,lid nearnuclear

reducebeen

aset-2wider

ng one

0

ofof the

logi-

forms

Fig. 12. Same as in Fig. 11 for the volume imaginary depthW0 (Table 18) of the non-monotonic set-3 potentfor 58Ni.

which determines the potential from the realistic two-nucleon interaction and as sumicroscopic method to obtain the potential in a fundamental way.

The fits to the experimental data of the four targets using the set-1 and set-2 parain Figs. 3 and 4, particularly at lower incident energies, and the comparison of the anashapes of the EDF-1 and EDF-2 suggest that the effect of the potential shape in theinterior on the cross-sections is not significant. However, at the higher energies, thtangible differences in the predicted cross-sections due to the set-1 and set-2 paraas can be seen in Figs. 3 and 4. This is not surprising as the EDF calculations are vazero energy and at higher energies the partial waves can penetrate more into theinterior.

The fits to four different isotopes over a range of incident energies are expected toboth continuous [59] and discrete [60] ambiguities. The latter type of ambiguities hasobserved by Cowley et al. [37] in their analysis ofα + 58,60,62,64Ni elastic scattering datin the narrow energy range 24.3–32.3 MeV using the WS potential. Although thepotential is adjudged as the best performer, the non-monotonic set-3 potential withenergy-bins in terms of the constancy of the parameters of the real part and haviparameter (D1 = 0.0) less than set-2 is likely to have less ambiguities.

A close inspection of Figs. 5–8 reveals that at the scattering angles beyond 15◦ theexperimental cross-sections of58Ni at the incident energies 18 and 21 MeV, those of60Niat 15–21 MeV and those of62Ni at 18 and 21 MeV could not be reproduced by anythe four types of potentials considered herein. It is possible that this is a signaturepresence of the resonance process contributing to the scattering.

The volume integralJR/(4A) values in the cases of the three types of phenomenocal potentials, namely SWS, and shallow and deep non-monotonic suggest that58Ni, withhigher volume integrals, behaves differently from the other even isotopes. This conto the findings of Budzanowski et al. [11] who obtained higher volume integrals for58Ni


egraleffect.the

ture.g, have

entted

ntialom

hatdoesnge of

Fig. 13. Same as in Fig. 9 for the volume imaginary depthW0 (Table 19) of the SWS potential.

than those of60Ni, although in their global analysis they parametrized for the same intvalues for both the isotopes. It is possible that the difference stems from the shellIn the case of58Ni the orbital 2p1/2 does not participate, while it does so possibly forother isotopes. In the case of the folding potential, however,58Ni has the lowest volumeintegral for the real part of the potential, showing another face of its distinctive feaNevertheless, all the deep potentials, namely, non-monotonic set-4, SWS and foldincomparable volume integral valuesJR/(4A) = 363.9, 371.3 and 350.2 MeV fm3, respec-tively, at Eα = 18 MeV for 58Ni. The shell-effect is also visible in the energy dependcurve ofVR for 64Ni (Fig. 10) with the curvature distinctively different from that associawith the other isotopes.

In consonance with the work of Abdullah et al. [28,29], the present folding poterequires a repulsive part of theα–α potential along with the attractive parameters frBuck et al. [57], as given in (8). However, the range parameter for the former partµR =0.530–0.535 fm−1 (Table 8) are slightly higher thanµR = 0.50 fm−1 used in generatingthe folding potential for16O [28] and40,44,48Ca [29]. The present study also confirms tthe folding model of Abdullah et al. [28,29] can generate an optical potential whichnot need any renormalization in accounting for the elastic scattering over a wide ra


in theered

gs ofall

oepkesentationon,vegral toffect

s for

y

Fig. 14. Same as in Fig. 9 for the volume imaginary depthW0 (Table 20) of the folding potential.

incident energies. Moreover, the model yields the number of nucleons that existsα-like clusters as 4Aα = 49, which is the same for the four even isotopes of Ni, considherein. This result conforms to the constancy in the number of theα-like clusters in theeven isotopes of Ca, as observed in [29]. Moreover, in conformity with the findin[28,29], the radiicN of the density distribution function of the unclustered nucleons inthe four Ni-isotopes are found to be smaller than the corresponding radiicα for theα-likeclusters. This feature also confirms the observation of Brink and Castro [61] and R[62] that theα-particle formation is energetically favored in the surface region. The prefolding procedure, which is valid over a wide energy range, is much simpler in applicthan that prescribed in [26], where the foldedα–nucleus potential needs renormalizatiand that in [64], where the potential calculation involves a density-dependent effectiNNinteraction, an energy dependent factor with a parameter in it and the exchange inteyield the potential renormalization-free, in addition to the inclusion of the dispersion ein the latter two works. All the three folding potentials have similar volume integralthe real part.

The present investigation presents four types ofα–58,60,62,64Ni potential. These manext be examined in the non-elastic processes including the challenging (α,p) reaction.


arch for

encee au-Tariq

l. Phys.

574

ski,

owski,Rev.

. C 28

toms

F.B.

F.B.

.M.

The latter processes, as noted by Satchler [63], provide a more stringent test in a sethe nature of theα–nucleus potential and the effort to find its global characteristics.

Acknowledgements

This work is partly supported by the grant INT-0209584 of the US National SciFoundation and a grant from the University Grants Commission of Bangladesh. Ththors thankfully acknowledge the grants. The authors are also thankful to Dr. A.S.B.of Rajshahi University for his help in preparation of the manuscript.

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Alpha–Ni optical model potentials

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