The 58Ni(p, α) reaction mechanism through a study of the analyzing power

Nuclear Physics A499 (1989) 381-391 North-Holland. Amsterdam

THE

SXNi(p,cx) REACTION

MECHANISM THROUGH ANALYZING POWER R. BONETTI

A STUDY

OF THE

and F. CRESPI’

Istituto di Fisica Generule Applicata de/l’Uniuer.Gta’ di Milano, Milano, Ital) K.-I. KUBO

Departmentc$Physics, Tokyo Metropolitan

University,

Fukazawa,

Setagaya-ku,

Tokyo 158, Japan

Analyzing powers of the lXNi(p, 01) reaction at E,, = 22 and 72 MeV have been calculated for DWBA calculations transitions to the ground state ot . “Co and to excited states in the continuum. have been performed with both the a knock-out and triton pick-up models. The analyzing power at 72 MeV in the continuum was calculated with the statistical fnultistep direct theory. It is found that while the p.s. transition is dominated by the triton pick-up process, the analyzing power in the continuum can only be reproduced by an cy knock-out mechanism. Some general considerations on the interplay between these two mechanisms and on the consistency with results obtained in the past with semiclassical preequilibrium models are made.

Abstract:

1. Introduction It is really surprising that after so many years of investigation both experimentally and theoretically, a reaction such as the (p, cu), generally considered to be relatively simple, is yet far from being well understood. Three problems indeed remain to be solved: (i) The reaction mechanism responsible for exciting the low-lying state of residual nuclei, either the triton pickup or the a-knock out ‘). The two amplitudes must be coherently added up in principle. Such a work has been recently reported (ref. ‘)) by adopting the mixing coefficients which are adjusted to reproduce the observed differential cross sections. A unique set of the coefficients can reproduce the transition cross sections for several excited states quite well, but the magnitude of the coefficients needed is not well understood; (ii) The conflicting interpretations given for the two regions of the a-spectrum, the discrete one, where the a-particle angular distributions have been most frequently calculated within a triton pick-up model ‘), and the continuum one. Here, the preequilibrium particles have been traditionally calculated under a preformed cy knock-out hypothesis 3), although a few more recent calculations based on the multistep direct method “) have been performed with the pick-up model; (iii) The severe underprediction of the cross section given by DWBA calculations done with any of the two above models. It is found indeed that the calculated cross section is smaller by up to two or three orders of magnitude than the experimental Uh). one I Present address: AERITALIA, Nerviano, Italy. 0375-9474/89/Z-$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V

382

While

R. Bonetfi et al. / .“Ni( p, u)

the latter

problem

has been

recently

discussed

in detail

and a possible

solution has been suggested ‘), the present paper will address itself only to the other problems, and particularly to point (ii). To do this, we will adopt a somewhat unusual approach. First of all, both the resolved levels and the continuum part of the spectrum will be analyzed within the same framework. This is made possible by use of statistical method developed in the last few years to extend to the continuum the direct reaction methods traditionally used only for discrete-state transitions, therefore providing a unified description of the whole spectrum. These are referred to as the multistep direct-reaction (MSDR) method ‘) and the statistical multistep direct-emission (SMDE) theory “). Second, in the hope to discriminate better between the two competing processes, the comparison will be done on the basis of the cross section left-right asymmetry (also known as the analyzing power), due to the well-known sensitivity of spin-dependent data to the details of the reaction mechanisms. The reaction chosen in the 58Ni(p, a) induced by 72 MeV polarized protons, for which the analyzing powers have been measured both for the resolved g.s. transition ‘) and the continuum I”) at SIN in recent years. The g.s. transition by 22 MeV polarized protons is also calculated and compared with the experiment to see more generally the trend of the competition between pick-up and knock-out mechanisms.

2. The ground-state

transition

2.1. CALCULATIONS

Several reviews on the triton pick-up model have already been written ‘), therefore we will not discuss it here and will only give the technical aspects of the calculations, as the parameters used. On the other hand only few calculations on (p, a) reactions have been performed using the knock-out model ‘), due to the relatively recent availability of the appropriate DWBA code. In the knock-out model the target nucleus is considered to be made up of a core and an a-cluster in a bound state. The incoming proton interacts with this cluster via a residual effective two-body interaction; as a consequence the a-particle is emitted while the proton is captured in a shell-model state. The technique used in constructing such bound-state wave functions is the standard one so-called the separation energy method; the single (cluster) particle quantum numbers and the empirical separation energy are given, and the well depth is searched for. In the case of cluster particles (triton and a-particle) the radial and orbital quantum numbers are given by the harmonicoscillator energy-conservation rule. The DWBA calculations have been done with the help of the codes DWUCKS “) and TWOFNR ‘I) for triton pick-up and (Y knock-out processes, respectively. Both codes use the finite-range formalism in the computation of the DWBA transition amplitudes, and both use the cluster approximation in building the transfer form

383

R. Bonetti et al. / “Nit p, (Y1

factor. While DWUCKS can only treat transfer reactions such as pick-up and stripping, TWOFNR, by including the exchange term in the transition amplitude, is able to treat knock-out processes as well. We will now discuss the parameters used in the DWBA calculations within both models. To obtain a meaningful comparison, we have used the same parameters for both calculations whenever possible. We first point out that the difference in the two models lies in the form factor, the incoming and outgoing distorted waves being of course the same. The proton OM wave function was calculated using the potential of Sakaguchi ef al. “) which was derived for 6.5 MeV polarized protons on 5XNi, therefore quite close to the present case. For the a-particle OM potential we have chosen that of Weisser et al. 14) for 64.3 MeV particles on 5XNi. Other families of parameters such as the ones obtained from the global OM potentials of Becchetti-Greenlees for protons or the other sets of the a-particle OM potentials of the Perey compilation did not give such substantially different results that conflicts with the present conclusion. These parameters, together with those defining the proton, triton and cu-particle bound states are shown in table 1. The last quantity to be fixed is the transition potential: the proton-u effective interaction for the knock-out calculation and the proton-triton binding potential for the pick-up one. The former can be taken both from theoretical calculations 15) or from DWBA fits potential from such works of (Y inelastic-scattering reactions I”). The recommended is a gaussian with depth 37 MeV and range 2 fm. The latter is the same proton-triton binding potential taken from ref. “) to have a Woods-Saxon with geometical parameters ro= 1.488 fm (R = r,,t' ‘j) and a =0.144 fm; the 53.1 MeV was automatically searched for to reproduce the experimental binding energy in the a-particle.

as the shape depth proton

TAI~LI- 1

Potential

parameters

22 MeV P 0 72 MeV P g.s. and E,, = 65 MeV E,, = 60 MeV

t b.s. “) pt t b.s. “) a b.s. h, p b.s. h,

used in the present

VR

TK

52.1 206.X 33.654

1.17 1.41

158.9 173.1 93.1 53.1 133.6 52.4

depths

W,

f-L+\

a,,

I.705

0.75 0.52 0.708

2.14 25.8 II.34

1.32 1.41 1.052

0.58 0.52 0.X81

1.357

0.629

2 1.O 22.17

I .641 0.438

aIt

1.279 0.719 ‘1 ‘) ‘) <)

calculations,

1.5 1.488 1.2s 1.25

I.588

are in MeV, radii and ditfuseness

WS

separation

a,,

V

r,

,(>

I?

a, ,>

7.70

1.32

0.58

6.2

1.01

0.75

2.606

1.341

0.392

5.582

1.058

0.625

0.531

0.76 0.144 0.55 0.6

“) For pick-up calculations. “) For knock-out calculations. ‘) Searched for to obtain the experimental

TV”\

in fm

energies.

R. Bonetti et al. / SXNi( p, a)

384 2.2. RESULTS

The results of calculation for the g.s. (j = 3, I= 3) transition are shown in figs. 1 and 2. It is interesting to note that while the pick-up mechanism gives a reasonable account

to the analyzing

power

data, the knock-out

fails, in both 22 and 72 MeV

proton energies, as seen in fig. 2. The importance of pick-up mechanism in the analyzing powers is consistent with its predominance in the differential cross-section calculations as we can see in fig. 1. The ratio of the pick-up to the knock-out processes is indeed about a factor of 100 in the integrated cross sections. In the calculation of those cross sections, the spectroscopic amplitudes, S( cy = t x p) and S(58Ni = “5Co x t) for pick-up mechanism and S(SXNi = 54Fe x a) and S(“Co = 54Fe x p) for knock-out mechanism, are assumed to be unity. No any other renormalization has been made in fig. 1. In fig. 3 the results of analyzing-power calculation with the same I-transfer (I = 3) but different j-transfer (j = z or $) are shown. A striking difference between the two angular distributions exists in the knock-out mechanism.

k \ a i

1

-

% \ 4

72

lo-’

-

lo-=

IO-)1 -L O0

40~

800

120°

160’

Ocm

Fig. 1. Comparison of differential cross sections calculated for the “Co g.s. transition at E, = 22 and 72 MeV with the measurements. The two different curves show the results with the pick-up (full curve) and knock-out (dashed curve) mechanisms, respectively. The knock-out curves are multiplied by a factor of IO’. The experimental data are taken from Tagishi et al. in ref. *) for 22 MeV and from Dorninger et al. in ref. ? for 72 MeV.

385

R. Bonetti et al. / -5XNi( p, n)

72 MeV

Fig. 2. Comparison of analyzing powers calculated for the ‘Co g.s. transition at E, = 22 MeV (upper portion) and 72 MeV (lower portion) with the measurements. See also the caption in fig. 1.

3. The continuum 3.1. CALCULATIONS

As explained in sect. 1, our principal aim is to investigate the reaction mechanism of the (p, (w) reaction in the entire emission spectrum by using a unified description. We will use for the continuum the statistical multistep direct-emission (SMDE) theory of Feshbach, Kerman and Koonin “) which we have been using in recent years to interpret (p, n) [ref. IX)], (p, p’) [ref. ‘“)I and (LX,(w’) [ref. ‘(‘)I reactions characterized by forward-peaked angular distributions. While we refer the reader

to previous

papers

for the details

of the theory

and its

application, here we want only to stress that the SMDE theory (as the SMDR method of Tamura et al. ‘) allows calculation of the continuum cross section by ‘,“), the same DWBA formalism using, through certain statistical approximations as used in discrete calculations. In our case, the (p, cy) cross sections and analyzing powers computed by DWUCKS or TWOFNR for different combinations of incoming and outgoing energies, angular momentum transfer and shell-model-state configurations compatible with energy and angular momentum conservation are inserted in the code MUDIR. This code first averages the individual DWBA cross sections over the different ph configurations and L transfer according to the prescription of eqs.

R. Bonetti et al, / ““Ni( p, a)

386 I

1.0

i

,

I

/

1

/

/

f

g.s. 7/2-

22 MeV 0.5

c

I

58Ni(p.af55Co

.’

/

/-’

..,.

\ \

.,_ .._,. PU -

KO --_

-1 .I

i Fig. 3. Analyzing

power

calculations with the same I-transfer for both the pick-up and knock-out

(I = 3) but ditferent

j-transfer

(j = f or $1

mechanisms.

2.5 and 2.6 in ref. lx), subsequently calculates the analyzing powers using eq. 1 of ref. 19). This is accomplished by numerically integrating the left and right average cross sections over the intermediate angles and energies. It has been shown by explicit 7.‘8,‘y) that muhistep processes (i.e. the contributions due to collisions calculations following the first one) become important only for “large” energy transfer. In the particular case of the (p, N) reaction, the number of different routes the reaction can undergo when calculating the multistep components can be quite large compared with simpler reactions such as the (p, p’). For example, Tamura et af. in calculating the second step of their “Nb(p, cy) reaction up to -30 MeV energy transfer had to take into account processes as the (p--LY--cy’) and the (p-p’-cu)‘). An accurate calculation of the second and successive steps would therefore require an equally accurate calculation of the above (and other) processes and their relative weights. Such calculations would rapidly become impracticable with increasing number of steps; more impo~ant, any inaccuracy would make rather dif%icult extracting reliable information on the reaction mechanism. Therefore, we decided to limit the calculation of the analyzing power in the continuum to relatively small values of the energy transfer, -10 MeV, up to which

R. Bonetti

ef al. / “Ni( ,o, a)

the second (and higher) step contribution allowed us to approximate the second-step

387

is expected contribution

to be rather small ‘). This of the SMDE chain with

the (p, p’) process alone. Moreover, the (p, p’) left and right DWBA cross sections (which act as the “source” term for the 2-step process, see eq. 22 of ref. ‘“) were taken, for the sake of saving computer time, from our previous work on ‘TXNi(p, p’) at 65 MeV [ref. I’)], thus ignoring the small difference in incident energy in respect to the present case. In such a way we calculated the 5XNi(p, cy) analyzing powers for two energy bins, 65 and 60 MeV, corresponding to excitation energies of -4.5 and 9.5 MeV. Due to large uncertainties in the calculation of the magnitude of the continuum cross section which, in addition to the problems typical of any DWBA calculation outlined in sect. 1, include also those of a-hole level density and cy preformation factors, we decided to consider only the analyzing power data of ref. I(‘). In this case indeed one has to deal with a ratio of cross sections, therefore avoiding the above problems.

“Ni

(6,~f’~Co

E,,= 72 MeV

-0.6 ;

_o.2

t

-0.6

(!+2)

‘~.__~~~“KO f I

tl’d_L_u___ 10

30

50

70

90

“O

9,

m. @leg)

Fig. 4. Comparision of analyzing power measured for two energy bins in the continuum with calculations based on the triton pick-up (full line) and a-knock-out models. In the latter case, calculations have been done with the first step only (1. dot-dashed line) and with the fir\t + second step (1+2. dashed line) of the SMDE chain.

388

R. Bonetti et al. / 58Ni( p, a)

As in the case of the g.s. transition, process,

using both the pick-up

the calculations

and the knock-out

were done,

models.

for the (p, a)

The parameters

in the

DWBA calculations are those discussed in sect. 2 and shown in table 1. In particular, for the calculations at E,, = 65 MeV we used the same OM potential as for the g.s. transition, while for those at E, = 60 MeV we used the parameters of Madland et al. “). The DWBA calculations on “Ni(p, p’) and the relevant parameters have been discussed

in previous

work I’)).

3.2. RESULTS

The results are shown in fig. 4. Contrary to what happens for the g.s. transition, the triton pick-up now does not fit at all the experimental data. The knock-out model on the other hand does reasonably at E,, = 65 MeV, well at E, = 60 MeV and even better after the second step is included. The fact that the second step gives only a small correction at these outgoing energies justifies its neglection in the case of the pick-up calculation.

4. Discussion To summarize the results described in previous sections, the triton pick-up process is active in the g.s. transition while the knock-out dominates for transitions to highly excited states in the continuum. The very different predictions given by the two models leave no doubt about the results of the fits. Indeed, the advantage in using the analyzing power to test different direct-reaction models results, clearly when one compares the sign and the rather structured behaviour of the experimental data analyzed here with the different calculations. This point is very useful when the angular differential cross section has such a smooth and structureless shape as the cases happen to be at incident energies higher than those considered here (72 MeV, ref. I”)) and in heavier target nuclei (Dragtin et al. in ref. “)). Theoretically, the analyzing power predicted by the two models are expected to be, and indeed are, very different, because of the different form factors. The analyzing powers calculated for the g.s. transitions at 22 and 72 MeV are found to show a clear j-dependence over the wide scattering angular range, where j is the value of transferred total angular momentum. This kind of feature was pointed out in the triton pick-up cases for the forward angles by several authors ‘) in the past. The present knock-out calculations, however, give more complete j-dependence as shown in fig. 3; j = z analyzing-power distribution is almost out-ofphase in all the angles in comparison with the j = $- angular distribution. It is interesting to see the interplay between the two reaction mechanisms by looking carefully at how the experimental analyzing power changes in shape in going from the discrete state to the continuum. Here at 72 MeV, it assumes at forward angles the largely negative values typical of the knock-out process (see fig. 2). With

R. Boneiti

increasing

excitation

energy

et al. / “Ni( p, a)

the analyzing

power

flattens

389

and approaches

the zero

value I”) as a consequence of the progressively increasing importance of multistep components (see two cases in fig. “)). This fact produces a progressive loss of memory of the initial polarization and would therefore make it difficult to extract information on the reaction mechanism, thus justifying the choice of relatively high a-particle energy values for our comparisons. An illuminating interpretation of the results outlined above can be obtained within a classical collision model. Due to the large mass difference between the incoming proton and the a-particle a complete energy transfer between such particles as a consequence of their interaction would not be possible, classically speaking. The proton would be left with some energy and would therefore occupy not the lowest possible state but some excited state. Therefore, even in presence of preformed cY-particles, an cy knock-out process will be unable to populate the g.s. or any low-lying levels of residual nucleus and will only populate (or preferentially populate) the highly excited levels in the continuum. Further understanding in terms of differences in the nuclear associated with the present reactions is as follows. The knock-out be approximately expressed as

Tko-

i

wave functions amplitude may

dr,, [x)rm’*(r,r)cp,v(r,)l dr,[cp~(r,)x~‘(r,)V,,(r,,,)l, i

(1)

where cp and x represent the bound- and the scattering-particle’s wave functions, respectively. Here we assumed that the nuclear surface takes place a predominant role in the transition so that two space-coordinate vectors in cp and x in each square bracket in eq. (1) is the same. Hence now the amplitude has overlap form of the bound- and the scattering-state wave functions and tends to zero in a long-range interaction limit. On the other hand,

the pick-up

amplitude

is approximated

as

where

4 and, for simplicity

=

dr,,

cP$(r,JVJr,,J ,

we used

x)l’(r~~)-x~,~‘(r,,)xb, ‘(rrr), From eqs. (1) and (2), it is then easily seen that the pick-up amplitude has less orthogonal property compared to the knock-out amplitude. In other words, only a very small cross section results in the knock-out calculations for the bound state transitions.

390

R. Bonerrieral./ =Ni(p, a)

In the continuum state transitions, however, the proton state in eq. (1) is considered as in the scattering state providing now a better overlap. Numerical calculations of sample transitions at E = 60 MeV with the I = 1 proton unbound wave function show indeed an increase of the knock-out cross section ranging between 8.1 and 72.9 depending on the particular ph configuration. Furthermore, the a-clustering amplitude is enhanced in the continuum since in a neutron-excess nucleus the two protons and two neutrons can be taken from the same shell orbit configurations in the transitions leaving the residual nucleus in highly excited states. This effect, which could be quantitatively estimated only with the help of a fully microscopic form factor is, for example, responsible for the prevailing a-clustering properties of light nuclei (IhO, ‘“Ne, 24Mg, . . .). These two features change the roles of pick-up and knock-out contributions in the cross section calculations from the bound-state cases. We would like now to compare our result with those obtained by other authors. MSDR calculations have been performed by the Austin group on several (p, (.y) reactions at 65 and 72 MeV [refs. ‘,“‘)] by using a triton pick-up model. Although the fits were generally successful, there are several discrepancies, particularly with analyzing power data for light nuclei I’). This led Lewandowski et al. ‘“) to criticize the triton pick-up model and to consider, as alternative process, a quasifree pscattering. This is, of course, consistent with our result for continuum. Recently, Gadioli et al. 2’) made a detailed comparison between the predictions given by the pick-up and knock-out models for the shapes of the energy spectra of (n, a) reactions at 14-18 MeV. Their calculations show that the knock-out model, while doing reasonably in the highly excited continuum, greatly underestimates the cross section for transition to g.s. and levels in its immediate vicinity which, on the other hand, are nicely reproduced by the pick-up model. This is also consistent with our results.

5. Conclusion

To summarize, taken altogether, these revisitations of the (p, a) reaction with the help of the modern technologies settled for direct and multistep direct reactions, support both the results obtained for the continuum with the preequilibrium exciton models in the 1970s [ref. “)I and also the work which has been traditionally done on the discrete-state angular distributions with the DWBA. There is, therefore, no conflict between the pick-up and knock-out mechanisms, which can be contemporary present in a given (p, a) reaction with different weights in the different parts of the energy spectrum. We are indebted to R. Wagner and H. Mueller for supplying us with their experimental data, and particularly for the unpublished E, = 65 MeV data, and for stimulating discussions. The help with the TWOFNR code of M. Igarashi is gratefully acknowledged.

R.

Bonetti

et al. / 5HNi(p, n)

391

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