isobaric analog state excitation in proton radiative capture

l.E.7:

/

Nuclear Physics A346 (1980) 449 -460; @ N#r~h-~olI~nd P~~~shin~ Co., Amsrerdam

3.A Not to be reproduced by photoprint or microtilm without written permission from the publisher

ISOBARIC ANALOG STATE EXCITATION IN PROTON RADIATIVE CAPTURE

F. SAPORETTI

Comitalo Nazionale Energia Nucleare, CSR ‘E. Clemennrel’, Bologna, Ital_,

and

R. GUIDOnI

Facoltd di Ingegneria deN’ Universird, Bologna, Ita

Received 25 March 1980

(Revised 20 May 1980)

Abstract: The direct-semidirect model for proton radiative capture so far formulated is unable to describe the observed (p, y) excitation functions in the energy region where the isobaric analog resonances are located. To remove this diff%zulty, an extension of the model to include capture proceeding via isobaric analog state (IAS) excitation is here proposed. The calculated results are compared with the measured 90° excitation curve of the *‘*Pb(p, y,)Zo9Bi reaction and satisfactory agreement is achieved, A detailed analysis of the three mechanisms involved in the El dominant capture process is presented. In the case considered the model seems to provide a useful tool for study of the striking effects arising from IAS excitation.

1. Introduction

Since experimental investigation has revealed isobaric analog states (IAS) as sharp resonances in the nuclei excitation functions and shown that their existence in the heavy-element region is a quite extensive phenomenon, great attention has been focused on these showy effects. As a result, the lead-mass region has been investigated, in particular through high-resolution measurements 1,2) of the z”*P~p, y)‘09Bi reaction in and above the giant dipole resonance (GDR), where the excitation of IA% in *“Bi takes place; the nucleus 208Pb is particularly privileged in these studies by its structure (double-closed shell) and that of the nearest nuclei (e.g., purity of the low-lying single-neutron states in “‘Pb). The excitation curves provided by the experiment ‘) show a broad giant dipole resonance with sharp IAR’s at higher energies where the dipole excitation strength appears as a non-resonant background.

The need for a reaction model describing this physical phenomenon clearly stands out. In fact, the direct-semidirect (DSD) model for nucleon radiative capture 3,4) satisfactorily describes the basic nature of the reaction mechanism in several cases, and in particular, it supplies the correct gross structure of the excitation functions.

449

450 F. Saporetti, R. Guidotti / Isobaric ~a~og state

However, the model so far fo~ulated is unable to provide the details of the excitation pattern, and in particular, it fails to account for the sharp IAR’s.

Here we propose an extension of the DSD model to include capture proceeding through IAS excitation in addition to the collective electric multipole (A = 1,2,3) and magnetic dipole excitation modes of the target so far considered 5,6). Some first results, anticipated in a previous paper ‘), seem to indicate that this approach to the model is successful in providing a detailed description of the asymmetry observed in the y-ray angular distributions of the “*Pb(p, y) reaction.

First, we present the essential formalism of the physical IAS excitation process. Then, a comparison between the observed 90° excitation curve for transition to the proton 2fi final state in ‘09Bi and the predictions of the model is made in the 10-18 MeV energy region. Finally, the transition amplitudes of the El direct and semidirect (through GD and IA states) capture mechanisms and their interference are discussed in detail.

2. The model

Let us consider a medium or heavy target nucleus (A, N, Z) in the ground state

4 oo, characterized by the same quantum number for isospin and z-component: T = M, = To = $N-Z). In the target excitation

is assumed to be process the isobaric analog state

@AS = ‘*,

0

where T_ = )“‘t= 1 ~(2; the operator r’?, working ^ ^

(1)

in the isobaric space of the ith

nucleon, is defined by the matrix ( 21 i), while the proton and neutron states are represented by the vectors (y)i and (A), respectively. This operator changes (consis- tently with the Pauli principle) the ith neutron of the target into a proton, lowering the M, eigenvalue and thus producing the (To, To - 1) state.

According to the formalism of ref. 4), the interaction for short-range two-body forces between the incom~g particle (here considered as a proton) and the ith target nucleon is given in the form

vi = [PO0 + P’O(z 1 zJ+ PO’@ * a,)$ P”(a. a&. ri)]d(r- ri), (2)

where r, tr and r denote the position, spin and isospin of the projectile and vi, bi and ri refer to the target nucleons. The quantities PO’, PIO, Pol and PI1 specify the strengths of the different interaction terms. Calculating the matrix element between the target ground state and the excited state (l), namely

A

WASI c ql#o& i=l

(3)

F. Sapdretti, R. Guidotti 1 Isobaric analog state

the effective coupling interaction for excitation of the IAS is derived as

%s PO

= 2n P(r)z+ T-,

451

(4)

where r, is represented by the matrix (i A); the factor PAS(r)r+ changes the free incident proton into a bound-state neutron; the operator T_, acting on the target nucleus, excites the IAS as previously mentioned.

In eq. (4) the radial function hiAS is defined by

(##Pl i r(“Cqr- ?$r21#,,>

h’AS(r) = i=l

WASIT k&J ’ (5)

so, by neglecting the non-diagonal terms and using the commutation relation [r+, r-1 = z3, we obtain

thus

h’As(r) = f

<&),I $ q&r- r,)l#,,> i=l

2%

9 (6)

hiAS = & rP,wP,(r)l7 (7)

where p,(r) and p,(r) are the neutron and proton densities. In the usual approximation (p, -p,)/(p, + p,) = (N- Z)/A, the radial function can be finally obtained, in a Woods-Saxon form, as

/$==(r) = $ jj+ [l+e~p(~)~l, 03)

where in the Fermi distribution

3A P+ =-------

&R3

is the mean nuclear density. By treating the coupling hamiltonian H&- as a perturbation we can calculate the

transition amplitude Qr,, IAs for the El proton radiative capture proceeding via IAS. Indeed, taking account of the fact that this intermediate state is characterized by a fairly well defined resonance energy ,?ZiM and a finite decay width PAS, we can write

(9)

where a’ is the incident proton energy and Hla the electric dipole operator of the

452 F. Saporetti, R. Guidorti / Isobaric analog stare

particle-target system. The initial, intermediate and final state wave functions are, respectively, given by

Fi = 1 i” exp (ia,.)[4n(21’+ l)]f(l’~om’lj’m’)~j,,.(8, ~)@$+#J,,~, f'j'

!f$f = (pj,,.(e,~)Ujlll,j.(r)~'AS.

'Y, = @j,(', 4)"$.(r)$oo*

(10)

(11)

(12)

Here cI, is the Coulomb phase shift and Qj,,, are the spin-angular wave functions. The labels (n) and (p) distinguish the neutron from the proton radial wave functions. In eq. (9) since the binding energy difference between the states 400 and +IAS is given by the Coulomb energy, the intermediate-state resonance energy is

2 E'AS R = Es&,+ ; g,

c

(13)

where the first term represents the bound-state neutron energy and Rc is the Coulomb radius of the nucleus. In eq. (9) the matrix element of Hi,, for the IAS excitation has the form

(W~lH;,sl6,) = (2nT$P” C i”(21’+ l)(l’~m’lj’m’) I'j

The matrix element of sir, describing the El decay of the intermediate state, that is a neutron transition from the level (n’l’j’m’) to the level (nljm), may be written as

where Z@) = - MZ/A is the dipole neutron effective charge with M the reduced mass. In El proton capture, the mechanism proceeding via IAS excitation interferes

with the direct process and the mechanism proceeding via the giant dipole state @DS, the latter characterized by the excitation energy ~~~~~ and the decay width r’m. Therefore, the transition amplitude for El proton radiative capture is expressed as a sum of the contributions of the three capture processes, viz.

After some Racah algebra, we obtain

Q,, = e 1 C(j'm'; p)(FJY+FyDS+F”1,9), .r

1

(17)

F. Saporerti, R. Guidotti / Isobaric analog state 453

where

C( Pm’; 11) = (- ly’-+i”[(2ll+ 1)(2j’+ l)]+(l’~m’ljm’)(jlm’~~jm),

9: = (fj+ - $1 10)$p) exp (ia,.)W~, (20)

where the radial integrals are

al

g!*S = I’ s z$(r)u$(r)r3dr

0 s mU~l,j(r)h’AS(r)~~~,(~)r2dr, (21)

0

s m

.@?DS - J’ -

u’“!(r)hcDs(r)~‘“!( ) ‘d nlJ

,‘J< r r I, (22)

0

(23)

and Z(P) = MN/A is the dipole proton effective charge, E$ the final proton energy and hGDs(r) the radial form function already the subject of extended attention in several papers [see, e.g., refs. 4, 8* ‘)I. In eq. (19) the transition matrix element between the ground and theGDS may be estimated by sum rule or by direct use of the measured reduced electric transition strength B(E1).

We can now calculate the integrated El cross section for proton radiative capture to the single-particle bound state (I, j), namely

and the differential cross section

(24)

(25)

where k’ and ky are the incident and photon wave numbers, XIP = [A@+ l)]-*LYAP

454 F. Saporetti, R. Cuidotti 1 isobaric analog state

are the vector spherical harmonics, and 1 denotes the multipolarities of the capture participating in the reaction process. In the present paper we consider only the dipole (,? = 1) and quadrupole (A = 2) captures. The transition amplitude Q2,, for the E2 direct and semidirect processes going through the isoscalar (T = 0) and isovector (7” = 1) giant quadrupole states is defined in ref. 5).

Finally, expanding eq. (25) in Legendre polynomials, we find

(27)

where E is the incident proton energy in the laboratory system and A, are coefficients related to the multipolarities involved in the emission processes; in particular, A,, is the integrated cross section divided by 411.

3. Analysis of the results

Experimental data to test the model proposed here are taken from ref. “) which provides the 90” excitation curves from high-resolution measurements of proton radiative capture on ‘**Pb filling the single-proton final states in zOgBi. The observed excitation functions cover the energy interval 8.8-18 MeV, thus also the region where the analog resonances in *“Bi are present. The predictions of the model are in this paper compared with the 90° curve of the first excited state 2f+ mainly for two reasons: first, this state is experimentally better resolved with respect to the other final proton states “); second, the gross structure of the observed excitation shape is fairly well reproduced by the DSD capture model in the whole energy region of interest when the appropriate potential is used in optical-model calculations lo). Therefore, according to a shell-model description, the excitation of three single-neutron IAS’s is involved here, i.e., the 2g,, 3d, and 2% resonances.

The comparison between the calculated 900 differential cross section and the experimental points of ref. ‘) for the 208Pb(p, y,)“‘Bi is presented in fig. 1. The dashed curve represents the result given by the DSD model when only the El (direct and semidirect via GDS) and E2 (direct, isoscalar and isovector) processes are taken into account; the continuous line is calculated by including the excitation of the IAS’s in the model.

In the energy interval 14-18 MeV the dashed curve, essentially the GDR component, appears as a background smoothly decreasing with the proton incident energy. Calculations regarding the dashed curve are carried out here according to ref. lo) where the Van Oers et al. 11) optical-model potential is used to generate the initial scattering wave functions of the incident proton. It must be noted that this potential was determined from proton elastic scattering measurements at the single incident energy E = 16 MeV, that is exactly in the middle of the lb18 MeV energy interval

F. Saporetti, R. Guidotti / Isobaric analog state 455

10 -

T

3 c 0 iY 0 5-

. .*

.

0 I I I I I

10 12 14 16 16

Ebw

Fig. 1. Comparison between the measured ‘) and calculated 90° cross sections for the z”*Pb(p, y,)‘09Bi reaction. Dashed curve: El (direct + semidirect via GDS) + E2 (direct + isoscalar and isovector semidirect) capture processes only. Continuous curve: El + E2 + El (semidirect via IAS) capture processes.

The location of the single-neutron resonances involved in the reaction is indicated by arrows.

which is of interest here. This fact, of course, offers a better description of the incident proton wave functions in the considered energy region. The more representative parameters used here in calculations for the dominant giant dipole process are, however, reported, for greater convenience, in table 1.

The contribution of the IAS excitation to the El capture is represented by eqs. (18) and (21). The continuum neutron wave functions @“) are calculated with the optical potential of ref. r4). The neutron bound-state wave functions u(“) are then obtained by a Woods-Saxon form potential and radius parameters taken from refs. ls* 16);

the well depths, used as free parameters, are adjusted to give the experimental binding energies. The coupling strength of the IAS excitation process is treated as a free parameter and adjusted in order to approximately fit the well-pronounced 2g, isobaric analog resonance. According to this procedure, we have derived the value Plop+ = 45 MeV as the IAS coupling strength. Such a value is typical for heavy

TABLE 1

Giant dipole resonance parameters considered in calculations

hwGos (MeV) rGns (MeV) &El) (fm2) V, (MeV) WI (MeV)

13.42 “) 4.05 “) 65 “) 40 Y 80 ‘)

‘) Taken from refs. I23 13). b, See ref. to).

456 F. Saporetti, R. Guidotti / Isobaric analog state

TABLE 2

*09Bi* single-neutron IAR parameters *‘) used in calculations

IAS EAAs (MeV) Yla (MeV) ~,!?j,~, (MeV)

2g9,, 14.92 0.25 3.94 3d 5/z 16.50 0.31 2.37 2&,, 17.43 0.29 1.44

nuclei in DSD mode1 analysis with real volume coupling [see, e.g., ref. I’)]. Finally, the single-neutron analog resonance parameters used in the calculations are taken from ref. is) and summarized in table 2.

As can be seen from fig. 1, inclusion of the IAS excitation in the DSD model produces a remarkable improvement in the agreement between theory and experiment in the 14-18 MeV energy region. However, the observed analog resonances exhibit at their basis an asymmetry (due to interference with the background component) which is not reproduced by the corresponding calculated excitation curve. Moreover, the predicted 3d, IAR peak is lower than that suggested by experiment. Attempts are made to remove these discrepancies, e.g. by changing the parameters generating the wave functions; nevertheless, the discrepancies remain. It is worth noting that, as pointed out in ref. ‘), the g, + & transition (involving spin-flip) is not observed in the experimental excitation curve; the absence of any 2g, EAS contribution in the calculated curve is clearly ascribable to the corresponding Clebsch-Gordan coefficient in eq. (18) which damps this excitation.

Considering eq. (27) and the properties of the Legendre polynomials, the differential cross section at 90° can be briefly expressed as

AccordingIy, we will confine ourselves in what follows to investigator the Ai’ and A;’ coefficients; we note, indeed, that only in the latter coefficients does the IAS excitation mechanism act, while the E2 contributions simply represent a background component not affected by IAR’s.

On the other hand, we observe that the A:’ quantity can be written as

(29)

This means that a detailed investigation of this coefficient requires, in turn, a more elaborate analysis of the single transition amplitudes described by eqs. (18~20). So, in fig. 2 we have separately represented the calculated energy behaviour of the real and imaginary parts of the amplitudes 9; (dotted line), gy (dashed line) and

F. Saporetti, R. Guidotti / Isobaric analog state 451

j’: 5/2 0.8

0.6

b) / I

I

0.6 i’:

0.4

9/2 0.6

- 02 a) I

10 14 ’ ’ w 18 10

E(MeV)

Fig. 2. Real and imaginary parts of the El amplitudes F_j” (dotted line), 9:‘s (dashed line) and 9jbs (continuous line) defined in eqs. (l&(20) for the quantum numbers involved :/ = p. 3, :.

9:” (continuous line) for the involved initial quantum numbers j’ = ?j, 4, z. This figure offers a clear means to understand the energy dependence of the At’ coefficient reported for the 10-18 MeV energy interval in fig. 3. Here, to show the weight of the different El capture mechanisms, the contributions to the A:’ coefficient are plotted by subsequently adding to the contribution of the direct process (At: dotted curve) first that of the semidirect process (AZ+ A, GDS. dot-dashed curve), then that of the . IAS excitation(A~+A~DS+A~AS: continuous curve). If we compare figs. 2a, 2b and 2c, we observe that the greatest contribution to the GDR excitation curve (dot-dashed line, fig. 3) arises from the transition initiated by the d, state. Moreover, we note for this state a destructive interference between the imaginary components of the direct and semidirect (via GDS) processes above 12 MeV; so, we learn that it is precisely this event which allows the calculated 90° cross-section curve (dashed line, fig. 1) to tit the rapid decrease with incident energy of the experimental points.

When we focus our attention on the IAS excitation mechanism in fig. 2, it is clear that the main contribution comes from the 2g, IAS. Its amplitude at 14.9 MeV rises

4% F. Sagoretti, R. Guidotti / Isobaric analog state

29 Q- % 3d 5/2 2Q7/2

1 1 I

Fig. 3. Coefficient A:’ versus incident proton energy. Dottedcurve: direct process only. Dot-dashed curve: direct + semidirect via GDS capture processes. Continuous curve: direct + semidirect via GDS and I AS capture processes. The arrows indicate the energy location of the single-neutron analog resonances

involved.

well above the direct and GDS components, thus no appreciable interference is present. On the contrary, the 3dt IAS excitation mechanism (fig. 2b) gives, at 16.48 MeV, a contribution comparable with those provided by the other El processes. This leads to a strong constructive interference among the three El mechanisms in the real part of the ,FFLt amplitude, while in the corresponding imaginary part the direct and GDS contributions cancel each other. This interference plays an important role in support of the strength of the 3d, resonance in the A:’ curve of fig. 3. Indeed, the 972, peak values are about half those of Yyzt (see figs. 2a and 2b) and the (2j’+ 1) coefftcient of eq. (29) favours the j’ = $ state with respect to]’ = s, which means that one expects a ratio A&’ = $)/A&j = 5) % 7. In spite of this, the 2g IAR peak is

.p only twice the 3d, IAR peak as appears in fig. 3, due to the constructive interference in the fly!+ real part. Finally, we note that no contribution from the 2gI IAS appears at 17.43 MeV in the -4:’ curve (fig. 3), as expected on the basis of the same considera- tions previously made concerning the 90° cross section.

Let us now consider the calculated energy dependence of the A:’ coefficient shown in fig. 4. The meaning of the curves is the same as in fig. 3. We can see from the figure that the IAR mechanism is present in the coefficient with a strong contribution from the 2gt IAS at 14.92 MeV. Such a contribution, according to eq. (28), adds construc- tively to the A:’ one to create the 900 cross-section curve reported in fig_ 1. However, the 2% contribution is the sole effect produced by the IAS excitation on the 90* cross section through the A, coefficient; indeed, no appreciable presence of the 3d, and 2g, IAR’s exists in the A, pattern.

Therefore, from analysis of the calculated results it seems that the 2g% IAR is

F. Saporetti, R. Guidatti / Isobaric analog state 459

I I I I I

lo 12 14 16 E(hW)

‘2

,.*

16

Fig. 4. Coefficient A:’ versus incident proton energy. See fig. 3 caption for the meaning of the curves.

better investigated through the 90” differential cross section with respect to the integrated cross section; this is due to the coherent sum of effects in the A, and A, coefficients. On the contrary, no preference exists between the two cross-section data as regards the 3d, IAR study; indeed, the peak at 16.48 MeV arises exclusively from the A, contribution. Finally, information on the presence of a 2g+ IAR at 17.43 MeV, not producing effects on the A, and A, coefficients, can be drawn only through a possible interference with the excitation of the isovector giant quadrupole resonance (located at ~2 MeV above the considered IAR and having a 5 MeV width); this brings out the need to investigate cross sections at angles different from 90° in order to reveal some possible 2g, IAS excitation effects.

In summary, the advent of high-resolution experiments eliminates the possibility of describing the observed (p, y) reaction data by the DSD model so far formulated in the energy region where IAR are located. The natural way to remove this di~culty is to include the EAS excitation in the model. The present paper represents an attempt to realize this new approach of the DSD model. The comparison between theory and experiment for the 208Pb(p, y,)20gBi reaction shows satisfactory agreement, though some discrepancy still remains. A detailed model analysis supplies indications on the most suitable angles for investigating the different IAR’s excited in the reaction process.

The authors wish to thank Dr. F. Fabbri for programming the required compu- tations.

460 F. Saporetti, R. Guidotti f Zsobnric analog state

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isobaric analog state excitation in proton radiative capture

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