A relativistic model for α-nucleus elastic scattering

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 589 (1995) 307-319

A relativistic model for a-nucleus elastic scattering S. A i t - T a h a r a, J.S. A 1 - K h a l i l i b, Y. N e d j a d i b a Physics Department, Nuclear Physics Laboratory, 1 Keble Road, Oxford OX1 3RIt, UK b Department of Physics, University of Surrey, Guildford, Surrey GU2 5XIt, UK Received 1 March 1995; revised 17 March 1995

Abstract

It is the purpose of this paper to present the first relativistic description of alpha-nucleus elastic scattering using an optical model with the Duffin-Kemmer-Petiau (DKP) equation as the applicable wave equation. The DKP optical model potentials that are used are obtained by a folding procedure involving phenomenological Lorentz scalar and vector Dirac nucleon-nucleus potentials. In order to probe the DKP description of the alpha-nucleus interaction, we consider the elastic a-scattering on a range of nuclei over the 100 < E < 200 MeV energy range. The effective potential has, in its Sch~dinger equivalent form, a linear energy dependence and differs from the form factors commonly used. We find that this model gives a good description of the elastic scattering data and reproduces the diffraction pattern and the exponential fall-off of the cross section beyond the rainbow region. The predictive power of the model is emphasized noting that all the energy dependence is inherently built into the model and no fits were thus necessary when going from one energy to the next. A comparison of the DKP model with non-relativistic models is also carried out highlighting the advantages of the relativistic approach over the non-relativistic one.

1. Introduction

Extensive studies of nucleon-nucleus elastic scattering in recent years have shown that Dirac equation-based approaches often give a better description of the data than the usual Schrrdinger equation framework [ 1-4]. This has raised the issue of the relevance of this improvement when the scattering involves spinless or spin-1 projectiles. It is the purpose of this paper to present the first relativistic description of or+nucleus elastic scattering using the Duffin-Kemmer-Petiau (DKP) equation [5] as the applicable wave equation. The DKP equation is a first-order relativistic equation for scalar and vector bosons [ 6 - 8 ] . The spin-1 form of the DKP equation has been applied to 0375-9474/95/$09.50 (~ 1995 Elsevier Science B.V. All rights reserved

SSDIO375-9474(95)OO128-X

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the study of deuteron-nucleus elastic scattering by Kozack et al. [8]. In their work, the single folding model is used to determine the DKP deuteron-nucleus potentials in terms of the nucleon-nucleus scalar and vector potentials obtained from a Dirac equation approach. The DKP equation for a given deuteron-nucleus interaction is then approximately reduced to a Schr~dinger-like equation with some effective potential. This can then be solved to obtain the elastic scattering cross section. The approach used here to construct the alpha-nucleus interaction follows the same path. A set of Lorentz scalar and vector local DKP potentials for the alpha-nucleus interaction are obtained by folding global Lorentz scalar and vector Dirac nucleonnucleus potentials, evaluated at one quarter of the incident lab energy of the a-particle, over the a-particle's matter density distribution. As will be shown later, it is a salient feature of this model that, even when local and energy-independent interactions are used, the DKP-based Schr~dinger equivalent equation contains a potential with an explicit linear energy dependence as well as a Darwin potential that reflects some non-locality effects. In order to probe the DKP description of the alpha-nucleus interaction, we study the elastic scattering of a-particles from a number of target nuclei over the 100 < E < 200 MeV energy range where much experimental data are available. At low energy the scattering is mainly defined by the nuclear sizes and there exists an ambiguity in determining the optical potential: potentials belonging to different families give equivalent description of the elastic data while at the same time yielding different wave functions and S matrix elements. In contrast, in the intermediate-energy region considered in the present calculations, the diffraction region in the elastic data is followed by a refractive angular pattern due to the deep penetration of the a-particle into the interior radial region owing to the weak absorptive role of the optical potential. This feature in the data helps to discriminate between different models as it is sensitive to the details of the real potential over a wide radial range extending from the surface towards very small radii [9]. This intermediate-energy a+nucleus scattering has been the focus of considerable attention using non-relativistic optical model and microscopic calculations [ 10-13]. In non-relativistic models, relativistic kinematics are sometimes included but it will be interesting to investigate how a fully relativistic treatment affects the predicted observables. The nuclear rainbow scattering and the refractive fall-off the cross section which are stringent constraints on the alpha-nuclear interactions have been used to eliminate ambiguities in the potential shapes and distinguish between alternative non-relativistic microscopic calculations. Such data has also been used recently to distinguish between different NN interactions within the double folding model with G-matrix elements based on the Reid or Paris potentials [ 11 ]. In order to identify what features distinguish our relativistic model from non-relativistic calculations, we carry out a comparison between the DKP model and two non-relativistic semi-microscopic optical models. The first contrasting non-relativistic optical potential is obtained by starting from the same Lorentz scalar and vector Dirac potentials for the nucleon-nucleus interactions as are use in the relativistic approach, but now the

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Schr6dinger reduction is made directly from the Dirac equation to obtain effective proton- and neutron-nucleus potentials. These are then folded over the a-density as before. In the second non-relativistic approach, the alpha-nucleus potential is obtained by starting from phenomenological central proton- and neutron-nucleus optical potentials, obtained using the Becchetti and Greenlees global parameterization [ 14], which are then folded over the a-density. Furthermore, we also compare our results with those obtained with a global optical potential [ 10] and give a qualitative comparison with the recent results of the double folding model. Apart from the motivation for nuclear reaction studies, these calculations have also other practical implications. For instance, Khoa and von Oertzen used the results of their optical model calculations to deduce Hartree-Fock values for the nuclear incompressibility K [ 11 ]. The importance of such potentials for nuclear astrophysics (through extension to low energies) has also been studied extensively in recent years by Auce et al. [ 15,16]. The structure of the paper is as follows: in the next section we discuss the DKP equation. In Section 3 we describe how the various optical potentials are evaluated. In Section 4, we present the results of applying these potentials to a-particle scattering from 12C and 4°Ca at 104 MeV, from 58Ni at 139 MeV and from 4°Ca and 9°Zr at 141 MeV and discuss the limitations of the present calculations and future work to improve them. The conclusions are presented in Section 5.

2. The D K P nuclear scalar-vector modal

A distinguishing feature of a relativistic nuclear optical model is the Lorentz transformation character of its assumed interaction. Given that for nucleon-nucleus scattering the Lorentz scalar-vector combination was shown to provide the best description, we restrict ourselves here to this model. The DKP equation that describes the scattering of an a-particle in the static, local and spherically symmetric alpha-nucleus potentials Us and Uv is written (h = c = 1) (b. p + n o + Us +/3°Uv) ~b(R) = t S ° E , ~ ( R ) .

(1)

In this frame, the potentials Us and Uv are Lorentz scalar and time-like vector, respectively, and R is the position vector of the centre of mass of the a-particle with respect to the target. In Eq. (1), E,~ is the energy of the a-particle of rest mass m,~, p its momentum while b and /3° are the internal variables known as DKP matrices, which satisfy the commutation relation fl~fl~fla + flafluflu = g~Ufla + gj,aflu

(/.t = 0, 1,2, 3).

(2)

In the spin-0 representation, the DKP matrices are /30=

0

'

-P~r

( i = 1,2,3)

(3)

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with 6, (), 0 as 2 x 2, 2 × 3, 3 × 3 zero matrices, respectively and

o:(010) '

0

i :)

0

0

3(:0 l)

0

0

0

' '

(4)

while the dynamical state ~O is a five-component spinor. We define ( ~upper "] ~t ( r ) = k, i¢lower J

with

°) 0upper -- ( ~A0

and

~/,,ower-

(a) A2 A3

,

(5)

in which case the DKP equation can now be expressed as

(ma + Us)~o = (Ea - Uv)A0 + V . A, (m,~ + Us)Ao = (E,~ - Uv)~o,

(6)

(m~ + Us)A = ~7~o, where A is the vector (AI,A2,A3). In order to identify the pieces of the usual non-relativistic optical model potential, we reduce Eq. (6) to a Schrbdinger-like equation, by eliminating Ao and A in favor of ~, so that

-2m~V~--~V~o+ [ 2 m ~ ( U s + - ~ U v ) +U2-U2v]~o=(E~-m2)~o,

(7)

where m~* is a local effective mass (m~, = m~+Us). This reduction is exact and we refer to this equation as Schrbdinger-like since ~odoes not satisfy the usual orthonormalization requirements and the energy term in the fight-hand side involves a relativistic expression for the energy. What is readily noticeable about Eq. (7) is that the "hamiltonian" is non-local and involves a linearly energy-dependent central potential. The DKP optical potentials are static and local so that the non-locality and the energy dependence arise naturally from the relativistic description. In other words, the DKP description of the alpha-nucleus interaction appears, in the Schrbdinger language, as a non-local parameterization of the optical potential which includes an explicitly energy-dependent term. While it remains to be investigated whether the amount of energy dependence and non-locality inherently built in the DKP approach is the required one, this at least suggests that the relativistic optical potential may be less energy dependent and non-local than the non-relativistic one. The wave function ~o(R) is then written as the product of the exact local equivalent wave function q~(R) and a damping factor P(R) in analogy to the damping effect relation associated with the non-relativistic Schrbdinger equation for a potential with a non-locality of the Perey-Buck type [ 17]. The function P(R) satisfies the relations:

~TP(R______~) _ 1 dUs P(R) 2(m,~ + Us) dR

and

lim P(R) = 1. R--oo

(8)

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Eq. (7) can then be reformulated as the second-order equation Iv 2 +

2 _ 2m~Uefr(R)] ~b(R)

-

=

0

(9)

where 1

Ucff(R) = Us + E~Uv + ~m~ (Us2 - U~) + enarwin. rn,~

(10)

The main term in the effective central potential is (Us + E,~Uv/ma); the empirical real parts of Us and Uv are very large and of opposite signs, typically Us '~ - 1 . 4 GeV and Ov ~ 1.1 GeV at the centre of the nucleus and as in the Dirac case they cancel out and give a potential of reasonable strength. The factor E~/ma then makes the cancellation even stronger. The central potential contains also quadratic scalar and vector terms of opposite signs similarly to those obtained with the reduction of the Dirac equation in the case of proton scattering. Such terms appear to be indispensable for the description of proton scattering as well as for deuteron scattering and are responsible for the "wine bottle" shapes of the potentials [8]. The monopole Coulomb potential, Uc, is also included by adding it to Uv in the above equation. This gives rise to a nuclear-Coulomb cross term that may sometimes play an important role. The Coulomb potential used in the calculations is obtained from a uniform spherical distribution or from two-parameter Fermi distribution. The Darwin term is defined as UDarwin- 2m~ with

2 ,Y(r) =

dR

1 dUs (m,~ + Us) dR"

(12)

The Darwin potential which simulates the non-locality is essentially a surface contribution. This potential involves various derivative forms of the scalar potential and leads to oscillations of very small amplitudes in the surface region of the central potential. Although it has been included in the calculations, its contribution is almost negligible. Note that the DKP-based effective potential in Eq. (10) has the same form as the Diracbased one [ 18] except for the spin-orbit term absent here and the functional form of the Darwin potential - the latter involves both the scalar and vector potentials in the Dirac case. The total energy of the a-particle in the c.m. frame is defined by E , ~ = ( S + m , ~2 - m2a)/2X/~,

(13)

The total invariant energy, v/-S, can be defined in terms of the kinetic energy of the in the lab frame as S = (ma + ma) 2 + 2maTaLab.

(14)

To take into account the small corrections arising from the recoil of the target, we considered two prescriptions that give approximately the same effect. The first

3i2

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prescription follows that of Craten and Alstine [ 19] and makes use of a fictitious particle of relative motion. The second one [20] defines recoil factors As and Av by which the respective potentials are multiplied:

as

=

mA/V/-S,

Av

=

EA/X/~,

(lS)

where mA is the mass of the target, EA = x/~ - E,~ and x/S is the total invariant energy of a+nucleus system. The Coulomb potential is also be multiplied by ,~v. We find that the first method gives better agreement with the experimental data and is therefore used in the present model.

3. The potentials Within the relativistic folding model used here, the DKP vector and scalar potentials are obtained by folding the nucleon-nucleus Dirac vector and scalar potentials (Vv, Vs) over the a-particle density distribution. The same approach was adopted for deuteron scattering calculations [7,8]. The notation used here will be that alpha potentials are denoted by a "U" and nucleon potentials by a "V". We write

Us.v(TaLab, R) =4

d r p a ( r ) V s , v ( ~1TLab . a , I R - rl),

(16)

where T~Lab is the kinetic energy of the a-particle in the lab frame and r is the position vector of the nucleons in the a with respect to its centre of mass. The inputs Vs and Vv refer to the nuclear Lorentz scalar and time-like vector potentials, respectively, determined by fitting nucleon+nucleus scattering observables at one quarter of the alpha kinetic energy. They are parameterized

Vj (r) = Vjf(r, rjR, aiR) + i W j f ( r , rji, aji),

(17)

in which j =S,V and where the form factors f ( r , rj, a) are two-parameter Fermi distributions,

f ( r , rj, aj) = [ 1 + e x p ( ( r - r j A l / a ) / a j ) ] -1

(18)

The parameters we use were obtained from the global Dirac optical potential fit of Hama et al. [ 18] to available nucleon scattering data. They obtained a number of sets of global prescriptions giving fits to the data for a range of nuclei and energies. Here, we used exclusively one set which included energies from 21 to 1040 MeV and targets of mass range 12-208. The matter point density distribution of the a-particle, p~, is taken to have a simple gaussian form,

pa(r) = po e - ~ r2, with fl = 0.744 fm -2 based on a RMS point radius for the a-particle of 1.42 fm.

(19)

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313

In order to explore the new features that may distinguish this relativistic model from non-relativistic calculations, we seek to compare the DKP approach with two non-relativistic folding models. In the first non-relativistic approach, the alpha-nucleus potential is obtained using the same folding procedure as in Eq. (15) but where the nucleon-nucleus potential is now an effective potential (Ven) derived from Vs and Vv. At the level of nucleon+nucleus scattering, the Dirac equation is (20)

[or . p + f l ( m i + Vs(ri)) + Vv(ri) +SipVc(ri)]d/Dime(ri) =EiODirac(ri),

where mi is the nucleon rest mass and Ei is the total energy of the nucleon in the nucleon-nucleus centre of mass (c.m.) frame, can be exactly reduced to the Schr0dinger equivalent equation Dime(ri) = 0. [V/2 + E~ -- m 2i -- 2mi(Vieff(ri) +SipVP(ri))] ~upper

(21)

The V/ff (i =n,p) comprise both central and spin-orbit components. In this work, however, we neglect the spin-orbit potential so that

viff(r) = Vs + Ei lg~l + 1...~( V~ - V2v) - ._~..1wp/,/p + 2Vv)6ip + eiVDarwin, (22) mi

2mi

2mi "C ~ "C

mi

while

I d (r2dB~ 2Br 2 dr \ dr J '

(23)

B ( r ) = ( Ei + mi + Us - Uv - 6ipV~) / ( Ei + mi).

(24)

3

VDarwin(r) = ~

(dB~ 2 \ dr]

where

Note that the proton and the neutron are taken to be of equal mass. Thus

UNRFl(T~b,R)=2/drp~(r)[vP'l'rL~blRk.~f~,~.~,

-- rl) + ,~ff,~,~"~'''rL~blR--rl) . ,

) (25)

The second contrasting non-relativistic optical potential is obtained directly by folding the central parts of the p- and n-nucleus optical potentials over the a-density. Here the nucleon optical potentials are taken to be complex Woods-Saxon with the global Becchetti-Greenlees parameterization [ 14]. Thus

UNRF2(Tyb,R)

=2f

drp~,(r) (VaP.6.(tTLab ~-~ , IR - rl) + V,nn. c . , (l'rLab ~-~ ,IR-rl)).

(26) Both UNRFl and UNRF2 then replace Uefrin Eq. (9) to calculate the corresponding elastic amplitudes.

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i

:

w

-50

-100

............................. ,'/ :;Phenom W S

...........

-150

,

,

;~ -I0

~

-20

-30

/°

a 2

a 4

I 6

: 8

10

R (fm) Fig. 1. The effective relativistic potential (solid line) compared with the Woods-Saxon potential (dotted line) and with the non-relativistic folding potentials NRF1 and NRF2 for the a + 58Ni system at 139 MeV. 4. Results

In this section we present that results o f the D K P optical model calculations for a - n u c l e u s scattering in the 100-200 MeV energy region. We compare these results to the data and to other theoretical calculations based on non-relativistic models. From the outset, we stress that we are do not have in mind the optimization o f the fits to the data. We are more concerned about assessing the predictive power o f the D K P based model. As described below, only the strength of the potential is normalized by a factor NR which is kept at roughly the same value for all nuclei and energies. Such a normalizing factor is also essential in folding model calculations and was found indispensable in relativistic calculations [2]. It is however not surprising that we need such a factor as the present model can only be viewed as a first-order calculation in which other effects are not included. Therefore (besides the above-mentioned factor), once the Dirac scalar and vector potentials are obtained for a given system and are directly used to obtain the effective normalized D K P potential, no parameters are varied to adjust the fits to the

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315

102 -o

+~2C

atE~=

104MeV

10

/

........

.....

-1

lO

_ _

DKP

-2

lO

.

.

-3 10 0

.

.

.

.

.

.

.

.

.

.

NRF1 NRF2

, , , I . . . . I . . . . ) . . . . I . . . . I . . . . I,,~I,,~,I,=,, 10 20 30 40 50 60 70 80 90

0(deg) Fig. 2. Angular distributions for a + 12C at 104 MeV. The results of the relativistic DKP effective potential (solid line) and the potentials NRFI (dash) and NRF2 (dot-dashed) are compared with the data from Ref. [21].

data for all target nuclei and energies. The model is in this sense parameter free apart from the inputs inherent to the Dirac potentials which are themselves based on both theoretical and phenomenological relativistic models. First we start by presenting the effective potentials obtained for the a+58Ni system at 139 MeV laboratory energy. In Fig. 1, the DKP effective potential is compared with the NRF1, NRF2 potentials and with the phenomenological Woods-Saxon potential. The DKP real potential (solid line) is found to be deeper than the other potentials over the entire radial region. This part is found to be deeper than the global potential of Nolte et al. [ 10] for all the cases we looked at. The DKP absorptive part is also distinct from the usual WS form factor exhibiting a pronounced surface peaked behaviour. The real part of the effective potential seems to be very close to the potential recently obtained from double folding G-matrix calculations using the M3Y-Reid potential (DDM3Y1). It has only been possible to make a qualitative comparison with this latter as we do not have the actual values of the DDM3Y1 potential. However, the results of Ref. [ 11 ] for 4°Ca at 104 MeV show that as expected the scattering is sensitive to very small radii and the potential is unambiguously determined down to around 2 fm. This potential has a depth of 150 MeV with an uncertainty of about 30 MeV. Our real potential appears to have the same radial behaviour and has a depth of 135 MeV at the centre in agreement with the above values. In contrast, the imaginary part of our potential is markedly different from its DDM3Y1 counterpart. This latter is of WS shape of depth around 20 MeV at the centre whereas our potential is surface peaked at around 4 fm with a depth of about 20 MeV at this point. It is also weaker at the centre having a strength of about

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1051

.~

,,,

4F

(o) a+ CootE.=141UeV

10 2 ~,

~,

10

(c) a +'*'Zr ot E,, = 141MeV

-~ /

1~ '~ , , ~ , 1 , 0

10

20

30

40

,,I,,,,I,,,,~, 50

60

70

,,I

80

O( eg) Fig. 3. Experimental angular distributions for (a) a+4°Ca at 141 MeV (data from Ref. [91 ), (b) a+SSNi at 139 MeV (data from Ref. [22] ), and (c) a+9°Zr at 141 MeV (data from Ref. [9] ) compared with the predictions of the effective DKP potential.

2 MeV. Moreover, judging from the figures our absorptive potential is steeper in the radial region from 4-8 fermis. In Fig. 2 are shown the cross sections for a-particle scattering from 12C at an energy of T ~ b = 104 MeV. The potentials used for these cross sections are not renormalized. The solid curve gives the DKP results while the two non-relativistic (NRF1 and NRF2) results are indicated by the dashed and the dot-dashed curves, respectively. The DKP results give clearly better fits. Below 30 ° , the diffraction pattern is accurately described by the DKP model whereas the NRF1 and NRb2 do not fit the data as well. Beyond this angle, all three calculations fail to predict the experimental data [21]. In the following, the results of the DKP based model with a normalization factor are presented. All the cross sections shown in Figs. 3 and 4 are as a ratio to the Rutherford cross section with the ratio being one at O = 0 (the scales are chosen for convenience of display). In Figs. 3a-c, the results of the elastic angular distributions are presented for the following systems: (a) ot+4°Ca at 141 MeV, (b) ot+58Ni at 139 MeV, (c) ot+9°Zr at 141 MeV. Given that no serious attempt to optimize the fit closely to the data was made, the overall reasonable agreement with the data is quite striking and a number of points need to be noted. The real potential yielding these results has been obtained with a normalization factor NR --~ 0.75. For 4°Ca, the data at 141 MeV [9]

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317

10 4

o o

10 3 10 2 10

I

(o) a +12C otF'~= 104MeV

1

1(31 16 2 (b)~ +~CoatE~= 104MeV 16 4 ,~,1, , , I J L , ~ l , ~ , , I , , ~ l k , , , I , , , ~ l , ~ , , 10 20 30 40 50 60 70 o

80

O(deg) Fig. 4. Same as Fig. 3 for (a) tr+12C at 104 MeV (data from Ref. [21]), and (b) a+4°Ca at 104 MeV (data from Ref. [ 13] ).

is very well described by the present calculations over the entire angular range with the rapid decrease at large angles as expected. The oscillations are all well predicted except the oscillation at around 24 ° where the data is slightly underestimated. For 5SNi at 139 MeV [22] the agreement is also very good. The mismatched oscillation at around 26 ° is also underestimated but in phase with the data. Finally, for 9°Zr at 141 MeV [9], the agreement with the data is even better and all the essential features are well described down to some details. Here again we find a small angular region, now located at around 34 ° where the data is higher than the calculations although the oscillation is given at the correct position. In Figs. 4a-b, the results for (a) ot+12C at 104 MeV, (b) a+4°Ca at 104 MeV are shown. The angular distribution for 12C is well reproduced including the large angular region with the exponential fall-off. We note however that the model gives values lower than the data at around 18° although the oscillation is predicted at the right position. This feature may not be purely accidental as it seems to be common in all the cases we have looked at. For 4°Ca, the data at 104 MeV [ 13] is very well described by the present model. Here also the region where there is a slight mismatch between model predictions and the data is a small angular region 30-45 °. Since no serious attempts to optimize the fits to the data were made, these results are at the least very encouraging. The fact that roughly the same value of NR is used for all nuclei and energies may suggest that there is an underlying reason for this value. A similar value for this normalization factor was found to be necessary in the past in relativistic calculations [2]. Furthermore, this is in contrast with the normalization

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factor that is generally used in folding model calculations where the values obtained for NR by fitting the data are sometimes very different for different nuclei or for different energies and for different model calculations. Returning to Figs. 3 and 4, we see that the main point where the present model slightly disagrees with the data is for all cases an angular region encompassing the one oscillation just below the maximum plateau is reached. In this region, although the calculations predict the oscillation at the right angle, they yield a cross section lower than the data. This does not represent a major problem for the model as no optimization of the fits was carried out. Nevertheless, it is interesting to note that this feature is quite common to all the cases looked at here and persists for other nuclei including up to 2°8pb. Moreover, because it occurs in the same angular region and is shifted gradually to larger angles as the target mass increases, it is possible that there is a common cause for it, i.e. a feature built in the resulting DKP potentials. This is presently being looked at. A priori, it seems likely that the culprit in this case is the imaginary potential which is different from the usual forms assumed in non-relativistic calculation. Some adjustments of this part of the potential could lead to improved results.

5. Conclusions We have developed a relativistic model for a-nucleus scattering based on the DKP equation for spin-0 particles. The resulting effective potential that enters the Schrrdingerlike equation is found to involve terms similar to those that are present in the counterpart equation derived in Dirac phenomenology studies including the explicitly energydependent term and the non-local Darwin type term. We have used this model to calculate the angular distributions of a-particles elastically scattered from a number of target nuclei at energies where the data is expected to be sensitive to the real potentials at very small radii due to the effects of the refractive (rainbow) scattering. Our effective potential is constructed through a folding procedure from the well tested Dirac scalar and vector potentials. The effective potential is then normalized by a given factor and no further parameters were varied in order to optimize the fit to the experimental elastic scattering data. The present results show that the theoretical model gives a good description of the data. This is very encouraging as it emphasizes the predictive power of the model. The essential merits of the model thus far is that it does not seem to require any adjustments of parameters as the energy or the target nucleus is varied. The dependence on target mass stems directly from the one built in the scalar and vector Dirac potentials. The energy dependence has its origins both in the explicit linear energy dependence of the effective potential and in the energy dependence implicit in the Dirac input potentials. The model at the moment represents a first-order description of the elastic scattering of a-particles. Several improvements are therefore potentially possible. These will centre around the assumptions inherent in the model and the basic ingredients that enter the calculations. Among these, the folding procedure and the Dirac global potentials.

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Regarding these latter, Hama et al. [ 18] provide a number of possible sets that fit the proton scattering equally well and we have used only one of these so far. Adopting other sets of Dirac potentials together with the possible future updating of these potentials will no doubt reinforce this approach. The present calculations show that the DKP based optical model developed here for a scattering possesses clear advantages over non-relativistic models and thus reinforces the consistency of the relativistic description of nuclear scattering phenomena.

Acknowledgements Support for EPSRC grants GR/G/53556 and GR/G/53648 is acknowledged. One of us (S.A.) acknowledges with thanks the hospitality of the Physics Department and in particular the Nuclear Theory Group at the University of Oxford.

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