Forward-to-backward asymmetry of the ( γ ,n) reaction in the energy...

PHYSICAL REVIEW C VOLUME 35, NUMBER 2 FEBRUARY 1987

Forward-to-backward asymmetry of the (y,n) reaction in the energy range 20—30 MeV

T. Murakami, I. Halpern, and D. W. StormNuclear Physics Laboratory, University of Washington, Seattle, Washington 98195

P. T. Debevec and L. J. MorfordNuclear Physics Laboratory, University of Illinois at Urbana Cham-paign, Urbana, Illinois 61801

S. A. WenderLos Alamos National Laboratory, Los Alarnos, New Mexico 87545

D. H. DowellBrookhaven National Laboratory, Upton, New Fork 11973

(Received 14 August 1986)

The forward-to-backward asymmetry of neutrons emitted in the (y,n) reactions on ""Pb and ""Cdtargets was measured for photons in the range of 20—30 MeV, where the isovector quadrupole giant

resonance is expected to lie. The asymmetry was observed to increase from small values (=0.2) tolarge ones (=0.6 and 0.8) for ""Cd and ""Pb, respectively. This phenomenon is interpreted as theinterference between E1 and E2 amplitudes. From an analysis of the asymmetry the excitation en-

ergies of the E2 isovector resonances were estimated to be 23.5+1.5 MeV and 26.5+1.5 MeV forthe Pb and Cd nuclei, respectively. The E2 isovector resonances are found to be considerably wider

than the E1 resonances.

I. INTRODUCTION

Compared with our knowledge of the isovector dipoleresonance (IVD), i.e., the well known El giant resonance,our knowledge of the isovector E2 resonance (IVQ) is verylimited. Most of the information about the IVQ has beenobtained from inelastic electron scattering. ' The shapesof (e, e') inelastic spectra and their dependence on scatter-ing angle show evidence for a concentration of E2strength at approximately 130A ' MeV for heavier nu-clei. For light nuclei the location tends to be somewhatlower. Upon dissecting these spectra into componentmultipoles, the authors concluded that the width of theIVQ resonance increases from = 5 to = 10 MeV as the nu-clear mass decreases and that the E2 strength concentra-tion corresponds to 70—100% of the T =1, E2 energy-weighted sum rule (EWSR) using the Cxoldhaber-Tellermodel. Because of the uncertain backgrounds and thestrengths due to other multipoles in the (e,e') spectra, it isimportant to study the IVQ and its parameters using anindependent method.

Evidence for the IVQ resonance has also been soughtusing the (p,p') reaction, the (n.+, tr ) reaction, th-e (y, y)reaction, and the (p,y) reaction. None of these hasmanaged to provide very definite information. In inelas-tic proton scattering, the dominance of the isoscalar termof the nuclear force obscures the excitation of the isovec-tor giant resonance even at high incident energies. Noevidence for IVQ strength was uncovered in a recentstudy of the (n. , m. ) reaction on Ca, Ni, and Zr tar-gets, although this reaction clearly excited the isovectormonopole and dipole resonances. According to a recent(y, y) reaction measurement no compact E2 strength is

observed in calcium between 19 and 51 MeV, nor in car-bon between 25 and 45 MeV. The sensitivities of the(n. , m. ) and the (y, y) reactions to the strength of the IVQare, however, so small that a substantial fraction of theIVQ sum rule would have to be concentrated within a nar-row energy interval for the IVQ to be seen.

Angular asymmetry studies using the (p,y) and (n,y) re-actions have the advantage over other reactions that theasymmetry depends on the amplitude of the E2 resonance(including its phase) rather than just on the intensity. It isdifficult to peel apart resonances on the basis of observedintensities whenever the resonances are wide. Because thephase of an amplitude changes most rapidly at resonance,it can provide valuable information to help pin down theresonance location and its other parameters. When the(p,y) reaction was used to study the IVQ, a large back-ground coming from direct E2 capture made identifica-tion of resonant E2 capture difficult. On the other hand,as has been pointed out earlier, There is essentially nodirect E2 amplitude in the (y,n) and (n,y) reactions, sincethe effective charge of a neutron undergoing a direct E2transition in a large nucleus is negligible. Thus, the (y,n)and (n,y) reactions are virtually background-free whenused to locate the IVQ through interference with the tailof the IVD.

Recently, Drake et al. have found striking evidence forthe existence and location of the IVQ in Pb, namely asharp rise at about 23 MeV in the (n,yo) forward-to-backward asymmetry. Subsequently, Bergqvist et al.have studied the IVQ in Ca using the same technique.The use of the (n,y) reaction for targets lighter than Pb isdifficult because the available fluxes of higher energymonoenergetic neutrons are small. We therefore decided

35 479 1987 The American Physical Society

480 T. MURAKAMI et al. 35

to use the inverse (y,n) reaction. In this paper we reportthe initial results of a survey of photoneutron asymmetriesat energies appropriate to the IVQ. In the next section wepresent the experimental details of the measurements, andin Sec. III we summarize the experimental results. A dis-cussion of these results in terms of both a simple schemat-ic model and the direct-semidirect (DSD) model appearsin Sec. IV.

II. EXPERIMENTAL PROCEDUREAND DATA REDUCTION

The experiment was performed using the continuouselectron beam from the University of Illinois MUSL-2electron microtron and its tagged photon facility. A 39.4MeV electron beam was focused onto a 1 cm aluminumconverter foil 0.13 mm thick to produce bremsstrahlung.To determine the energies of the photons produced, thepost-bremsstrahlung electrons were momentum-analyzedand detected in the focal plane of the analyzing magnet byan array of 32 plastic scintillators. The tagged brems-strahlung photons, after passing through a brass collima-tor, struck the photonuclear target located about 1.8 mdownstream from the converter. Fast neutrons ejectedfrom this target were detected by an NE-213 liquid scin-tillator, 30 cm in diameter and 5.1 cm thick. This scintil-lator was viewed by three photomultiplier tubes 12.7 cmin diameter. It was surrounded by borated paraffin andlead shielding to reduce the number of events comingfrom room background. To suppress the large photonyield from the targets, 5 cm of lead was placed just infront of the neutron detector. Neutron events were dis-tinguished from photon events by pulse-shape discrimina-tion. The neutron energies were determined by measuringthe time interval between the signals from the electroncounters and the neutron detector. The time resolution ofthe system was estimated to be about 2 ns. The usablecurrent of the primary electron beam was limited to a fewnA by the counting rate in the post-bremsstrahlung elec-tron counters. The allowable rate was typically =2& 10s 'counter '. In order to maintain a sufficiently highcounting rate for true events, it was necessary to use ashort flight path (1.5 m) to the neutron detector. Conse-quently, the neutron energy resolution was poor (1.6 MeVat E„=13MeV and 3.6 MeV at E„=23MeV). Thisresolution was not sufficient to distinguish neutron yieldsto different final states in the residual nucleus.

The main photonuclear targets and their thicknesseswere 11.1+0.2 g/cm natural lead and 32.8+0.4 g/cmnatural cadmium. To investigate a possible isotopicdependence of the reaction, a 5.4+0.1 g/cm enriched(88%) Pb target was also used. The target areas wereabout 15)&15 cm, large enough to intercept the wholephoton beam since the beam spot size was estimated to beless than 7 cm in diameter.

The cross sections of the (y,n) reaction on these targetswere measured for photon energies between 20 and 30MeV at 55' and 125. From these cross sections thefront-to-back asymmetry, defined as

a = [Y(55') —Y( 125 ) ]/[ Y( 55') + Y( 125')],was determined. The cross section at 90' was also mea-

sured for the natural lead target in the photon energyrange between 20 and 26 MeV. The attenuation of the in-cident photons in the target and of the photoneutrons inthe target and in the lead blocks just in front of the neu-tron detector were evaluated using the known photon ab-sorption cross sections' and the total cross sections ofneutrons predicted by a global optical potential. " Al-though the total attenuation was large (80—92%), thedifferences in attenuation at 55' and 125' were, in general,quite small. For example, for the natural lead runs thisdifference was calculated to be less than 4 5%. , with the125 target having the smaller overall attenuation.

Tagging efficiencies were measured periodically byplacing a 25)&30 cm NaI detector directly in the photonbeam and counting the number of photons in coincidencewith the post-bremsstrahlung electrons. In order to makethis measurement, it was necessary to reduce the intensityof the electron beam by about 4 orders of magnitude.Since the tagging efficiencies were quite sensitive to theposition of the electron beam spot on the aluminum con-verter, the beam position was monitored by a pair of pro-file monitors every half-hour, and sidewise beam displace-ment was kept to less than +1 mm. This procedure en-sured that any variations of the tagging efficiencies duringa single measurement were less than 2%.

The pulse-height calibration for the neutron detectorwas obtained from the Compton edges of monenergetic yrays. These appear at 0.5 MeV (' Cs), 1.06 MeV ( Na),and 4.20 MeV (' C ). The threshold for the neutrondetector was set at 3.0 MeV (equivalent electron energy) toavoid a high accidental counting rate from low energyneutrons produced in the target by the great number ofuntagged photons. This threshold roughly corresponds to6.5 MeV in terms of neutron energy. The counting ratefor the remaining accidental coincidence events was es-timated from the number of the kinematically forbiddenevents in the time-of-flight spectra and was corrected for.A typical time-of-flight spectrum is shown in Fig. 1.

'Pb(y, n.)

E~=4.0 bleV g.s.

20— K~=6.5 hfeV

10—

0 100 200 300 400 500Channel

FICx. 1. A typical time-of-flight spectrum for ""Pb(y,n) atOl, b

——55'. The mean incident photon energy is 23.46 MeV. Thetime scale in this figure is about 0.5 ns/channel. The arrowsdenote typical energies relating to the neutron spectrum.

35 FORWARD-TO-BACKWARD ASYMMETRY OF THE (y, n). . . 481

The efficiency of the neutron detector as a function ofneutron energy was calculated for various thresholds by amodified version of the Monte Carlo code developed byStanton. ' The performance of the program was checkedby comparing the calculated efficiencies with those re-ported in Ref. 13 for an NE-213 liquid scintillator, 12 cmin diameter and 5.7 cm thick. The code reproduced theexperimental efficiencies within a few percent.

Overall uncertainties in the absolute cross section wereestimated to be +40%, and were mainly due to uncertain-ties in the neutron counter efficiency and in the attenua-tion of the y and neutron flux. The uncertainties in themeasured asymmetries (typically 10%) were much smallerthan those for the absolute cross sections because of thecancellations which occur between forward and backward

measurements.Possible spurious asymmetries of the system and sys-

tematic errors in the data reduction procedure were inves-tigated by measuring the known asymmetries of the'60(y, n0)' 0 reaction' in the gamma energy range of20—26 MeV. For this measurement it was necessary tolower the threshold for the neutron counter to 1.0 MeV interms of the effective electron energy. Our results turnedout to be in excellent agreement with those of Jury et al. '

Therefore the occurrence of any spurious asymmetrieswould appear to be negligible. This reaction also gave usgood calibrations of the time-of-flight spectra, since for' 0 the (y,n0) peak is well separated from the (y, n, ) peak.The uncertainty in the time calibration of the time-of-flight spectra was estimated to be less than +0.1 ns.

(o) 1.0

0.5

' Pb(y, n)

Ex( 2 MeV

(b) '0

0.5

"'Pb(q, n)

Ex ( 2 MeV

«)

«)

0.0Q)

1.0

0.5

Ex( 4 MeV«)

«)

«)

«)

0.0—Q

1.0

0.5

Ex ( 4 MeV

«)

0.0

20I

25E (MeV)

30

0.0

20I

25E (MeV)

30

0.5

II

'Cd(y, n)

Ex( 1 MeV

~ I

() «)

«)

0 0Q

1.0

«) () ()()~ 8 ~

I

I

Ex& 3MeV

0 ' 5«)

() «)

0.0

«)

«) ()«) «)

20 25Ez (MeV)

30

FIR. 2. AsymmetrY curves of the (y,n) reactions for ""Pb, Pb, and ""Cd targets for residual excitations indicated in the figures.The error bars in the figures reflect only the statistical uncertainties.

482 T. MURAKAMI et aL 35

III. EXPERIMENTAL RESULTS

The measured asymmetries integrated over an intervalof emitted neutron energies are shown in Fig. 2 as a func-tion of photon energy. The neutron energy interval interms of the excitation energy of the residual nucleus forthe most abundant isotope is indicated in the figures. Inthe case of ""Pb the isotope Pb has a 52% abundanceand the isotopes Pb and Pb each have abundances ofabout one-quarter. In the case of ""Cd the isotope " Cdis the most abundant (29%%uo), but the other isotopes""'"'" '" Cd also have fairly large abundances.Since Pb and " Cd do not have the lowest neutronbinding energies among the Pb and Cd isotopes, the low-excitation-energy end of the acceptance gate was set sothat the yields from the ground states of all isotopes wereincluded.

The asymmetries seem to have a slight dependence onthe energy range over which the neutron spectra were in-tegrated, but the statistical errors obscure this dependence.The data for the enriched Pb target agree within thestatistics with those for natural lead. This fact is con-sistent with the view that the asymmetry relates to collec-tive modes of nuclear excitation in which finer details ofnuclear structure are unimportant.

The asymmetry for ""Pb(y,n) changes more slowly withphoton energy than that observed earlier in the Pb(n, y )

8O

reaction. However, the two asymmetry curves are simi-lar in that the asymmetry changes from a small value to alarge one near the energy where the isovector E2 giantresonance is expected to lie. The asymmetry pattern for""Cd is similar to that for ""Pb, but the transition regionis about 4 MeV higher, consistent with the expected

dependence for the location of giant resonances inheavy nuclei.

The cross section ratios between the 90' measurement

1.0(o) . ;;

"'r b(q, n)55'

0.5

0.0 ~ ~

90

0.5C&(&

~-

b0.0

125'

0.5

0 0 i''t '' I '''

~

20

1.0

4&4@

25 30E (MeV)

I ~

and the average of the 55 and 125' measurements for the""Pb(y,n) reaction are shown in Fig. 3. Even though thedata do not have good statistical accuracy, these ratiosshow a definite dip in their values around Ez ——23 MeV.This suggests that the ratio of even to odd parity absorp-tion is increasing at that energy. This is consistent withascribing the observed asymmetry to the onset of the IVQ.The excitation functions of the cross section to the low ly-

I

- "Pb(y, n) 0.5

(b) " 'Cd(y, n)55

02 1

b'cf+0

0

0C)CD

00 102

CI& (&

(&

E„(2 Mev

«&C&

o.o

b0.5

125'

0Ex ( 4 MeV

I

20 25E (Me V)

30

FICx. 3. Ratios between the cross sections at 90 and the aver-age of those at 55 and 125 for the ""Pb(y,n) reaction. The resi-dual excitations are indicated in the figures.

0.020

I

25E (Me V)

30

FIG. 4. Cross sections of the (y, n) reactions for ""Pb and""Cd targets at the angles indicated in the figures. The residualexcitations are between 0 and 4 MeV for the ""Pb target and be-tween 0 and 3 MeV for the ""Cd target. The dotted lines in

panel (a) are the expected direct process contributions calculatedusing the DSD model (see Sec. IV C).


ing residual states do not, however, show a noticeable in-crease in the energy region where the asymmetry changesand where we expect the IVQ to lie. (See Fig. 4.)

IV. DISCUSSION OF RESULTS

This discussion is divided into four parts. It beginswith a general treatment of the relation between the mea-sured quantities (cross sections and asymmetries) and theodd and even parity amplitudes upon which these quanti-ties depend. The second part of the discussion is a semi-classical treatment of the asymmetry in which it is as-sumed, for simplicity, that the dependence of the asym-metry on photon energy is much the same for all residualstates. By expressing the odd and even amplitudes interms of the strength functions for nuclear excitations ofdifferent multipolarities, it becomes straightforward, us-ing this model, to estimate the sensitivities of the observedasymmetries and cross sections to the parameters whichcharacterize the relevant giant resonances. In the thirdpart of the discussion we go through the details of a cal-culation of the expected results using the so-called direct-semidirect (DSD) model for radiative capture reactions'along with the principle of detailed balance. The fourthand final part of the discussion compares the DSDanalysis of the resu1ts with that of the semiclassicaldescription.

A. The asymmetry and the cross sectionexpressed in terms of amplitudes

The data from our measurements provide three in-dependent pieces of information at each excitation energy:(1) a measured fore-aft asymmetry for neutrons whichleave the residual nucleus in its lowest few MeV of excita-tion, (2) some values for the ratio of the 90 yield for theseneutrons to their average yield at 55' and 125', and (3) ameasured total cross section for this group of neutrons.We consider emissions to a band of residual excitationsbecause the experimental neutron energy resolution wasinsufficient to distinguish between individual final statesand because it then makes more sense to average over abroad band of final states than to run a risk of biasedsampling over only a few final states in a narrow band.

We begin by deriving general expressions for the asym-metry and cross section for a typical residual neutron-holestate for which parity, spin and spin orientation are speci-fied. Let us assume, for the moment, that this state haseven parity. Since the target is an even-even nucleus, theintermediate state from which the neutron to this state isemitted is a superposition of both odd-parity (due mainlyto El excitation) and even-parity (due to E2 excitation)states. The amplitude for this neutron emission can there-fore be expressed as the sum for the emissions from theodd and even parity excitations, AJ+exp(i'+ )

+AJ exp(igj ). These two terms refer to a specific resi-dual state j, a specific incident photon energy, and theneutron-emission angle 55', our forward angle. The corre-sponding amplitude at 125, our backward angle, is there-fore AJ.+exp(i PJ+ )—AJ exp(i PJ ) since the odd parity neu-tron wave function changes sign (but not magnitude) be-

tween 55 and 125', whereas the even parity function doesnot change at all.

The quantities (A~ ) and (AJ+) at these angles providea measure of the contribution of state j to the angle-integrated odd and even parity cross sections, respectively,if the coefficients of P4(cosO) and higher terms in theLegendre-function expansion of the overall angular distri-bution are negligible. [We are using the fact thatP2(cos55 )=0.] Under this assumption the contributionto the angle-integrated cross section is therefore

crq——(Ai ) +(Ai+)

The front-to-back asymmetry aj can be expressed interms of the A's and p's for the residual statej as

2A+A—a = cos(d+. — . ) .

(A+) +(AJ. )

An important feature of this expression is that oj is sym-metric with respect to an interchange of + andThus, az has the same value for A&+/Az ——g as forAJ+/Aj ——X I. It depends on t-he ratio of odd to evenamplitudes, but not on which is the larger. This featureof Eq. (2) means that we are free to associate AJ+ with E2excitation and Az with E1 excitation in estimating a~, al-though (depending on the parity of the residual one-holestate) the correct association may actually be reversed.

The asymmetry, e„„&,observed in this experiment isthat for the sum of final states,

g[oi(55') —o~(125')]J

g[a J (55')+ crj(125')]J

gaj [o;(55')+crj(125')](3)

g[crj.(55 )+oj(125')]J

i.e., a„„]is the average az, where each state is weightedby the cross section to that state. As a result of theaforementioned symmetry of Eq. (2), with respect to theinterchange of + and —,one may associate the even par-ity emission amplitudes AJ+exp(i/~+) with the even (E2)excitations, although, for an odd parity j, the correspond-ing emission would actually be odd parity. Thus if the ex-citation amplitudes happen to be independent ofj, the a~.of (3) will be also, even though the emission amplitudeswill differ for j's of opposite parity.

In terms of the A's the observed cross section is

o«„~———,' g [o.i(55')+oi(125')]

J

=g [(A,+)'+(A,-)'] .J

B. A semiclassical approach

The relations of the cross sections and asymmetries tothe underlying odd and even parity amplitudes for indi-vidual residua1 states, as they have been described in Eqs.(1)—(4), are quite general. In this section we will take up a


semiclassical approximation to these relations. Theessence of the approximation will be the assumption thatthe A~.exp(ipj) are sufficiently the same for different j,but the same parity, to allow us to substitute for them asingle "typical" amplitude, A exp(i/ ) for the odd pari-ty excitations and A+exp(i/+) for the even ones. Thisassumption is suggested by the well-established similarityof giant resonances built upon different states. Thus inwhat follows the measured asymmetry will be expressed interms of A, A +, P, and P+ according to Eq. (2) ratherthan as a sum over the asymmetries for each of the finalsingle-hole states [Eq. (3)]. We must, however, alert thereader at this point that the DSD calculations (subsectionC) will suggest that the asymmetries az. are actually jdependent. Under these circumstances the energy depen-dence of a„„~can differ from that of the "typical" a~, aswe shall see in more detail below.

Our object here will be to deduce from the data whatwe can about the values of the A's and P's. In order toextract a value for the amplitude ratio A+/A (or its re-ciprocal) from the measured asymmetry, it is seen fromEq. (2) that one must provide an estimate for the phasedifference b,P between the odd and even parity excitations.For Pb such an estimate can be made with the least ambi-guity in the excitation energy range from 27 to 30 MeV.It is seen from Fig. 2(a) that the measured asymmetry inthis region has an average value of about 0.8, which isfairly close to the maximum possible value for an asym-metry, namely unity. It follows that cosh/ must lie be-tween 0.8 and 1. This, in turn, implies that the amplituderatio 3+/3 lies between —,

' and 2. The high value ofcosh/, i.e., the small difference between odd and evenparity phases, suggests that the region E~ =27—30 MeVlies above the even parity resonance since (I) this excita-tion region is definitely above the E1 resonance in Pb, and(2) the phase difference between two resonances tends tobe large in the energy range between the resonant energiesand small outside of it.

It is not as easy to deduce the amplitude ratios from theasymmetry data in Pb at the lower energies of Fig. 2(a) asit is at the higher energies, because there is then no way todetermine the value of cosh, P directly from our experi-ment. The reasonable requirement that cosh/ shouldchange smoothly with energy does not provide enough ofa constraint to be very helpful.

In order to extract more information from the measure-ments, it therefore becomes necessary to introduce somepreconceptions, some model-dependent assumptionswhether one uses the simple semiclassical model, the DSDmodel (Sec. IV C), or any other model. In the semiclassi-cal model, the odd and even amplitudes for the "typical"final state will be assigned on the basis of known or as-sumed properties of the nuclear giant resonances. Theessential elements in this picture are illustrated by theidealized excitation curves of Fig. 5. It is assumed herethat the photoemission proceeds through three resonances,the odd-parity E1 resonance and the even-parity E2 iso-scalar and isovector resonances. We will write the ampli-tudes for each resonance in Breit-Wigner (BW) form. Al-though the actual energy dependences are undoubtedlymore complicated than the BW dependences, this form

C,

0

OEV

CO

Qll/lO

(b)

4)~ -90—

-)8010 20

excitation energy (MeV)

FIG. 5. (a) Idealized giant resonance amplitudes for photonabsorption in Pb. (b) Idealized phases for the main odd andeven parity amplitudes for the (y,n) reaction.

ass (E2, isoscalar),~s+i I s/2

av~v= (E2, isovector),hv+)I v/2

where hD ——Ez —ED, As ——E& —Es, and Av ——Ez —Ev,and the energies E locate the centers of the respectiveresonances. The numerators, a, are taken to be positiveconstants. For a semiclassical model, this corresponds tothe correct relative phasing among the three multipolesfor the emission of neutrons. (For proton emission, thesign of the E2 isoscalar numerator would be reversed. ) Inthe DSD calculations the numerators will be complex.The imaginary parts are generally introduced to reflectcouplings to other modes. They will be found to effectthe values of the "best fit" resonance parameters.

It is important to appreciate that although each of theseresonances is characterized by three parameters (resonanceenergy, resonance strength, resonance width) the problemof explaining the observed asymmetry as a function ofphoton energy is not really a nine-parameter problem. Itis more nearly a four parameter problem. We are most in-terested in determining the relatively unknown parameters

has the advantage of being simple while still qualitativelyreproducing the main features of the amplitudes. Its usewill allow us to make some approximate deductions ofvalues of the E2 isovector resonance parameters thatmatch our observations and, perhaps more importantly,will allow us to examine the interplay between the variousamplitudes in a convenient and transparent way. The BWamplitudes can be written

aDAD —— (El),+iI /2


EP, I v and the ratio ai /aD which characterize the E2isovector resonance. The fourth somewhat uncertain pa-rameter is as/aD, the ratio of the E2 isoscalar to the Elamplitude. For the latter ratio it is the mean magnitudeof two amplitudes, (5) and (6), in the energy region of in-terest (Ez ——20—30 MeV) that mainly matters. This is be-cause the resonance energies for these excitations lie farbelow the region of interest. ' ' Their phases are there-fore relatively unambiguous and their amplitudes must bechanging relatively slowly with energy.

It is reasonable to wonder whether any model restrictedto the resonance amplitudes (5)—(7) is general enough.There are other possible amplitudes which are not includ-ed in these expressions:

1. Direct amplitudes

1.0) ~ (I

(&~~{)~C)~

0.020

I

25E (Me V)

30

FIG. 6. The asymmetry curve for the ""Pb(y,n) fitted by thesemiclassical model (see text) together with the experimentaldata. The residual excitations are between 0 and 4 MeV.

It should be recalled that any direct EA, amplitude(A, & 1) is negligible for (y, n) reactions because the neu-tron is uncharged. Any direct E1 amplitude would beclose in phase to the resonant E1 amplitude, and becausethe DSD calculations show that the direct E1 amplitudeis small and changing slowly with energy, we considerthat (5) can reasonably be considered to represent the sumof the direct and resonant E1 amplitudes,

2. Other multipole modes

In the same spirit one should probably think of the E1isovector and E2 isoscalar amplitudes as standing in forall odd- and even-parity excitations, respectively, whichare nonresonant in the 20—30 MeV range. There may be,for example, a small contribution from E3 (Ref. 15) aswell as some M1 contribution. The only amplitude whichwe assume to be resonant, between 20 and 30 MeV, andtherefore rapidly changing in phase and in magnitude inthat energy span, is the E2 isovector amplitude. The ob-served drop, in this energy interval, of the ratio of the 90'cross section to the average of those at 55 and 125' (Fig.3) shows that it is even rather than odd parity that isgrowing in, in support of this assumption.

3. Statistical emission

Any statistical decay would reduce the observed asym-metry. Indeed, no asymmetry at all is seen in Pb up to16 MeV. ' The contribution of statistical neutron emis-sion was estimated using the evaporation code PAcE (Ref.17) with standard parameters. At 20 MeV the cross sec-tion for such emission to the lowest 4 MeV of excitation iscalculated to be less than l%%uo of the observed cross sec-tion. Beyond this energy it falls off with increasing Er asexp ( E„/T),where T—is the nuclear temperature (about1 MeV in our energy range). The contribution of statisti-cal neutron emission should therefore be negligible at allenergies of this experiment, even considering the uncer-tainty in the evaporation calculation.

Our first application of the semi-classical model was tosee whether the amplitudes of the form in Eqs. (5)—(7),when used in Eq. (2), would give asymmetries close tothose observed when the resonance strengths were chosento be in accord with the energy-weighted sum rules. ' The

TABLE I. Giant resonance parameters used in the semiclas-sical model.

E1' E2 isoscalar E2 isovector'

Width (MeV)Energy (MeV)Strength a

4.013.5(1.0)

3.010.50.08

16.021.50.53

'From Ref. 19.bFrom Ref. 20.'From fit to asymmetry curve.Numerator in Eqs. (5)—(7).

numerators in the resonant amplitudes are actually pro-portional to the product of two factors. One factor is pro-portional to the square root of the sum-rule strength forphotoabsorption for the particular multipole. The secondfactor is the square root of the average branching ratio forneutron emission to one of the final states of this experi-ment. At the high energies involved, it is reasonable toassume that this average branching ratio is much the samefor all of the multipoles in Eqs. (5)—(7). This second fac-tor was consequently ignored in estimating the asym-metries since the asymmetries depend only upon ampli-tude ratios.

Thus the three parameters for the odd-parity amplitudewere assigned to be in accord with the accepted values forthe giant dipole resonance. ' Corresponding values forthe E2 isoscalar resonance were used for one of the twoeven-parity amplitudes. The parameters for the E2 iso-vector amplitude, the remaining contributor to even pari-ty, were chosen to be consistent with the E2 sum rule andto make the computed asymmetry curve match the ob-served one (Fig. 6). All of these parameters are listed inTable I.

It was found in this way that a rather good match tothe asymmetry curve could be obtained for values of theE2 isovector resonance energy and strength which wereclose to those expected and for a width that was a lot wid-er than expected. Although it was gratifying to find pa-rameters that gave a good fit to the asymmetry data, wemust not accept the implied E2 isovector parameterswithout further scrutiny, as we shall see below.

A number of additional exercises were carried out to ex-plore the sensitivity of the calculated asymmetry to the

T. MURAKAMI et al. 35

various resonance parameters. The results are shown inFig. 7.

It is seen that either an increase or a decrease in the E2isovector cross section (or amplitude) reduces the predict-ed asymmetry at the higher energies. The fact that theasymmetry is near its maximum possible value implies, aswe have already noted, that A+ must, for these energies,be comparab1e to A since the asymmetry peaks whenA+/A =1 [Eq. (2)]. Near this maximum, the asym-metry becomes relatively insensitive to the assumed mag-

1.0 I

Sensitivity to IVQ amplitude

0.5

0.01.0 l

.Sensitivity to ISQ amplitude

0.5

0.020

I

P.5E (Me V)

30

1.0-Sensitivity to DlQ location

0.5

0.0Q

1.0 ~ ~ ~

-Sensitivity to IVQ vridth

0.5

0.0P.O 25

E (MeV)30

FICx. 7. Sensitivity of the asymmetry to the values of the fourcritical resonance parameters defined in the text. The solidcurve in each figure is the fitted curve of Fig. 6. [Both curves inpanel (d) are for the same integrated strength for the IVQ reso-nance. ]

nitude of the E2 isovector cross section. The dashed anddotted-dashed solid curves in Fig. 7(a) correspond to E2cross sections which are four times higher and 4 timeslower than the best-fit curve.

The E2 isoscalar amplitude in the region of 20—30MeV is rather weak since the resonance lies near 10 MeVand is quite narrow. Nonetheless, because it is its ampli-tude rather than its intensity which matters, the isoscalarresonance plays a role. It slightly increases P+, the posi-tive parity phase. This tends to close the gap between P+and P and therefore to increase the asymmetry, as thecurve in Fig. 7(b) shows.

It is also seen from Figs. 7(b) and 7(c) that an increasein the assumed amount of E2 isoscalar resonance can beoffset by a shift of the E2 isovector resonance to a higherenergy. In particular a factor of 2 increase in the isoscalaramplitude from the originally assumed value can be com-pensated for by an upward shift in the isovector resonanceenergy of about an MeV.

There is a warmng in the foregoing result. Any effectwhich introduces an appreciable phase shift between nega-tive and positive parity amplitudes will shift the asym-metry curve. To provide a scale for these effects we notethat the doubling of the isoscalar amplitude in Fig. 7(b)changes P+ by values of only 3 —10' (between 20 and 30MeV). There can be sources of phase shift at least aslarge as this from effects which have been completely ig-nored in the semiclassical model. These are discussed inSec. IV D, where the results deduced using this model arecompared with those using the DSD model.

The width of the E2 isovector resonance which wasneeded to obtain the fit in Fig. 6 is enormously large com-pared to that of the dipole isovector resonance (Table I).Even though the E2 isovector width is certainly expectedto be larger than the E1 width since both the escape andspreading widths increase with the energy of a giant reso-nance, an increase as large as that in Table I is unexpect-ed. Figure 7(d) shows that a significantly narrower as-sumed width will not fit the data.

One might wonder whether the difficulty here stemsfrom having assumed a wrong energy dependence for A +

or A, but it is easy to demonstrate that even largechanges in the relative sizes of the A's lead to only smalleffects on the asymmetry curve. For example, if the evenparity cross section is held fixed and the odd parity crosssection is doubled at 20 MeV and then allowed to changegradually until it is only half of the best-fit cross sectionat 30 MeV, the asymmetries remain within 0.03 of thosein the "best-fit" curve throughout the energy range. Theeffect is only slightly larger if it is the odd parity ampli-tude that is kept fixed and the even parity cross section ischanged by the factor of 4 between 20 and 30 MeV. Thereason for this remarkable insensitivity of the asymmetryto the cross sections is that at the higher energies whereX=A+/A approaches unity, the asymmetry becomesstationary as a function of g. At lower energies where 7and the asymmetry are both small, even large percentagechanges in the asymmetry correspond to small absolutechanges. As a result, the overall shape of the curve ishardly affected.

A comparison of Figs. 7(a) and 7(d) suggests that if one

35 FORWARD- TO-BACKWARD ASYMMETRY OF THE (y, n). . . 487

combines a narrow IVQ width with a reduction of theIVQ strength from the sum-rule value, the resulting asym-metry would possibly give as good a fit to the data as thatobtained with the full sum-rule strength and the 16 MeVwidth (the IVQ parameters in Fig. 6). This is indeed thecase, as the asymmetry curve for half the sum rulestrength and a width of 12 MeV is virtually indistinguish-able from the curve in Fig. 6. It becomes difficult, how-ever, to fit the data for widths smaller than 12 MeV.Thus the analysis of the data using the semiclassicalmodel suggests that the IVQ resonance is substantiallywider than the IVD resonance. A possible reason thatthis model gives so large a width is discussed in Sec. IV Dbelow (after we have had a chance to examine the corre-sponding analysis using the DSD model).

We now turn briefly to the second major quantity mea-sured in this experiment, the cross section for the (y,n) re-action (to the particular residual-energy interval) as afunction of photon energy. In contrast to the fore-aftasymmetry, the cross section depends on the A's and notat all on the P's. Thus the fitting of the asymmetries andof the cross section data are essentially decoupled.

The cross section calculated from the amplitudes inTable I [see Eq. (I)] is found to decrease by a factor of 3between 21 and 29 MeV, whereas the experimental crosssection falls by more nearly a factor of 6 in the same ener-gy interval. This sort of discrepancy is not altogethersurprising since the branching ratios for neutron decay tothe low lying levels in Pb have not been included in thesemiclassical calculation. Although, as we have noted,these ratios should play only a minor role in the asym-metries, they enter directly in the estimates of the crosssections. Since they necessarily fall off as the energy goesup and new emission channels open, one must expect theactual cross section to fall off faster than that of a calcu-lation where the effect of branching was neglected.

No attempt has been made to use the semiclassicalmodel to fit the asymmetries or cross sections for our oth-er target, ""Cd. The observed cross section in ""Cd fallsoff at about the same rate between 21 and 29 MeV as itdoes in ""Pb. The asymmetry curve in Cd is shifted tohigher photon energy by an amount consistent with anA ' dependence for the location of the isovector reso-nance and it seems to level off at a somewhat lower valuefor the maximum asymmetry. This should be checked byextending the measurements for both Pb and Cd to higherenergies.

C. The DSD model

The semiclassical model which we have used up to thispoint to discuss the data does not include a number ofconsiderations which affect the magnitudes and phases ofthe interfering amplitudes in the (y,n) reaction. These arepresumably included in the so-called direct-semidirect(DSD) model, ' which, as its name implies, includes directamplitudes as well as the resonant (semidirect) ampli-tudes, (5)—(7), of the semiclassical model. The DSDmodel also treats individual final states one at a time. Tocompare the results of DSD calculations with our data,we must sum the calculations over the final states whichlie in the residual energy interval.

The differential cross sections for the (y,n) reactionwere obtained using detailed balance from calculationsmade in the inverse direction; i.e., we calculated the ma-trix elements for neutron capture by a target which is inan excited single-hole state. In order to apply the DSDmodel to the case of targets with spin, e.g., to targets in anexcited state, we followed the prescription of Snover inassuming that the semidirect process depends only on thefinal state parentage to the target ground state plus a sin-gle nucleon. The details of this treatment can be found inRef. 23. Beside this assumption, it was supposed that theoptical potential which is used for the ground state calcu-lation also describes the scattering of neutrons by the ex-cited nucleus, and that this optical potential is indepen-dent of the spin of the target nucleus. The bound stateneutron wave function was calculated using the samegeometrical parameters as those for the real part of theoptical potential. The depth of the real part of this poten-tial was adjusted to reproduce the binding energy of theneutron. The strength of the spin-orbit force in the boundstate problem was the same as in the optical potential. Incarrying out our numerical calculations, we used two setsof global optical potentials, one due to Rosen et aI. andthe other due to Becchetti and Greenlees. " Our purposewas to see the sensitivity of the calculations to opticalmodel prescriptions. The explicit values of the optical po-tential parameters which we used are given in Refs. 24and 11.

According to the DSD model, the transition amplitudefor the neutron capture reaction on spin-zero targets forelectric radiation of rnultipolarity L, circular polarizationp (=+1), and spin projection s (with the quantizationaxis in the projectile direction) can be written as

T =0, 1E ELy + l I LT /2

Here, X & &2, is the neutron scattering wave function ob-tained from the optical model, and N„I& is the single par-ticle state wave function with quantum numbers n, I, j,and m into which the neutron is captured. ' The vec-tors r and r~ denote coordinates for the neutron and pho-ton, respectively. The spectroscopic factor for the finalstate is denoted C SI&. E is the total energy of the system,

I

while EL& and I Lz- are the position and width of the gi-ant resonance. The subscript T represents the isospinstate. For the quadrupole excitations, the isoscalar (ISQ)resonance must be included as well as the isovector (IVQ)resonance, since the isoscalar quadrupole resonance wasfound to affect the fore-aft asymmetry. The termdL~(r~, r) is the single-particle multipole operator for mul-


ULpT(ry, r) =y CLp(k )eL hl T(r)DpM'(ry) YIM (r),with

(10)

(k)2+Wc

V

1/2

1/22m.(2L+ 1)(L+1)

X L1

(2L+ 1)!!

and

—m n

m„+m~Zte . (12)

Here, k is the photon wave number and V is the volumeenclosed by the surface on which periodic boundary con-ditions are imposed. The mass of the neutron and target,and the charge of the target, are denoted by m„,m„andZ„respectively. We chose the following radial forms forhLT(r):

3% A' &ii 1 I if . dg(r)A m„E~~ (y2) 4 dr

(13)

a' P2ohqo(r) =2Z r

mn E2o

vo df(r)r dr

(14)

h (y) 10~~ (" ~„2 f( ) PTm„E~~ (r') 4

(15)

These radial forms are essentially the same as those ofRef. 28, but we explicitly include the possibility that onlya fraction PLT of the EWSR is exhausted. Here, N, Z,

tipolarity L. The factor ULpT(ry, r) is the incident neutrontarget-nucleus vibration coupling interaction, i.e., theform factor for the inelastic excitation of the collectiveelectric multipole state by the incident neutron. It shouldbe noted that, in order to obtain the correct resonanceshape for heavy nuclei in capture reactions, it is usuallynecessary to introduce a complex form factor for theparticle-vibration coupling, with a relatively large imagi-nary strength.

In the conventional long wavelength approximation thequantities dLp(ry, r) and ULpT(ry, r) can be written as

drp(ry, r) =g CLp(k)et r DpM'(ry) YLM(r) (9)

and

and A denote the neutron, charge, and mass numbers ofthe target, respectively, and (r ) is the mean of the radialcoordinate of a nucleon raised to the A, power. The func-tions f(r) and g (r) are the Woods-Saxon form factors forthe real and imaginary parts of the optical potential,respectively, and a is the diffuseness in the function g (r).

Since the photon energies involved in this experimentwere quite high, we were concerned that the long wave-length approximation might not be applicable. We there-fore made some calculations replacing the ei r factor inthe form factors dip(ry, r) and uLpr(ry, r) with

(L+1)k~ &R(16)

where R = —m„r/(m„+m,). This factor is a more accu-rate expression for the electric multipole moment. Thosecalculations showed that this modification only changedthe cross section by 7% or less around Ez ——30 MeV.There were no noticeable changes in the asymmetrycurves. Nevertheless, we use the full form rather than thelong wavelength form throughout this paper.

Before applying this formalism to the present (y, n) ex-periment, we calculated the asymmetry for the capture re-action Pb(n, yo) in order to compare it with the availabledata. The spectroscopic factor of the Pb ground statewas set equal to 1, and the (r ) and (r ) values, 30.5 fmand 1161 fm, for the Pb, were taken from electronscattering data. For the coupling strength parametersV& and O'I we used the values suggested by Potokaret al. for the IVD, namely V~ ——75 MeV and 8'& ——125MeV for the Rosen potential, and V& ——60 MeV and8'~ ——120 MeV for the Becchetti-Greenlees potential. Theisoscalar coupling constant Vo was assumed to be —50MeV. The resonance parameters used for the calculationare given in Table II. They are essentially empiricalvalues, ' except for the parameters for the IVQ, whichwe adjusted to give the best simultaneous reproduction ofthe asymmetry curves for both the Pb(n, yo) and""Pb(y,n) reactions. The cross sections of the Pb(n, yo)reaction in the energy range E„=6—15 MeV (Ref. 30) arewell reproduced by a DSD calculation, as Potokar et aI.have already reported. The asymmetry curve measured byDrake et al. is also reasonably well explained (Fig. 8),but it is seen that at the high energy end of the calculationthe two optical potentials predict quite different results.The rapid rise of the asymmetry clearly comes from theinterference between the IVD and the IVQ. Without theIVQ, we can get only a slow and small increase in theasymmetry. The inclusion of the ISQ seems to be essen-tial to explain the absence of a negative asymmetry belowthe IVQ resonance, as has been already pointed out byLongo et al. '

TABLE II. Cxiant resonance parameters used in the DSD calculation. '

Target

PbCd

E13.4215.8

4.056.3

(%%uo)

117106

Ev

23.5026.50

I v

6.007.00

&2i(%)

4050

Es

10.5013.20

I s

2.803.30

'Energies in MeV.


.0 I I II

I I I II

I I I II

I I I II

I I I I

"'Pb(n, q, )(a)

0.5

1.0 I

'Pb(7, n)

Ex ( 4 hf.eV

I I I I I I I I$.0 s i c i s

"'Pb(n, q,)(b)

I I I I I I I I I II I I I I I I I I 1 I

0.020

I

25E (Me V)

30

0.5

. . . I. . . , I. . . , I. . . , I

10 15 20 25E (MeV)

30

FIG. 9. Asymmetry curves for the ""Pb(y,n) reaction predict-ed by the DSD model with the resonance parameters (Table II)based on available data and the assumption EJvQ —23.5 MeVand I 1vQ ——6 MeV shown together with the experimental data.Solid and dashed curves are the results with the optical poten-tials from Refs. 24 and 11, respectively.

FIG. 8 ~ Comparison of the DSD model prediction for the'Pb(n, yo) reaction to the experimental data (from Ref. 8). (a)

The solid and dashed curves are predictions of the DSD modelincluding the ISQ, IVD, and IVQ, and using the optical poten-tials from Refs. 24 and 11, respectively. In panel (b) the solidand dashed curves are the DSD model prediction including onlythe IVD and IVQ, and the dotted and dotted-dashed curves arethe predictions including only the ISQ and IVD. The resonanceparameters are listed in Table II.

TABLE III. Neutron hole states in Pb. '

E„(MeV)

0.00.5720.8991.6292.3343.415

'From Ref. 32.

Spin

3p 1/2

2fsn3P3/21 l13/2

2fvn1 h9/2

C S(J.

2.36.27.0

12.07.36.9

Although the (y,n) target in the experiment was ""Pb,the calculations were carried out for only Pb since theisotope dependence of the (y,n) reaction at these energiesis small. Candidates for the final states of the Pb(y, n)reaction are the single-hole states, which are listed inTable III. The spectroscopic factors for these states aretaken from Ref. 32. They are all reasonably close to max-imum possible values. The ( r ) and ( r ) values for

Pb were 30.4 fm and 1155 fm, respectively, again ob-tained from electron scattering data. All other parame-ters were taken to be the same as in the calculation of

Pb(n, yo).In Fig. 4(a) we compare the calculated cross section for

only a direct (nonresonant) process (dotted line) with themeasured one for ""Pb. It is clear that the direct processalone gives a serious underestimate of the observed cross

I I

0

M

b

+0

1—b

0C)Cb

b'aCQ

I

. "r b(q, n)

Ex & 4 hIeVI

20I

25E (MeV)

30

FIG. 10. Measured and predicted cross section ratios between90' and the average of those at 55 and 125' for the ""Pb(y,n) re-action. Solid and dashed curves are the results with parametersreferred to in Fig. 9.

section. To account for the latter it is necessary to involvea semidirect, i.e., a resonant process.

The results of the full DSD calculation summed overseveral final states are shown along with the data in Figs.9—11. For reference the asymmetry curves and the exci-tation functions of the cross sections for each of these fi-nal states are shown in Figs. 12(a) and 12(b). It is seenthat there is an appreciable final state dependence espe-cially for final states which have angular momenta closeto the grazing angular momentum (=7ssi). The generalbehavior of the calculated averaged asymmetries (weight-ed by the cross section to each individual state) and crosssection ratios seem to reproduce the trend of the datareasonably well. Above Ez ——26 MeV there are, however,rather large discrepancies between the DSD predictionand the data for the asymmetry; i.e., the measured asym-metry keeps increasing, while the calculated asymmetrybegins to decrease as Ez increases. The inclusion of an


1.0

0.5

0.0

0.5

"'r b(7, n)55

I I

where VJ is the occupation number. Again, we treat onlythe most abundant isotope, i.e., " Cd, in the model calcu-lation. The V~ values and the relative single quasiparticleenergies Ej for the " Cd nucleus were obtained from Ref.34. The absolute locations of the neutron single quasipar-ticle states were determined by assuming that the 3s&&2state is at E„("Cd)=214 keV, which is close to thecenter of gravity excitation energy for j= —, states excitedby the " Cd(d, t)" Cd reaction. The unknown V~ valuefor the 1g7/2 state in " Cd was assumed to be the same asin " Cd. These values are summarized in Table IV. Theresonance parameters ' used for the calculation aregiven in Table II. Since the ISQ in the " Cd is not well

0.0 I . ~ . ~ I ~. . . I

I I I125'

0.51.0

0.5 3 415h

0.020 25

E (Me V)30

0.0

2.334f

1 62S

FIG. 11. Predicted cross section for the ""Pb(y,n) reaction at55, 90', and 125 together with the data. Solid and dashedcurves are the results with the parameters referred to in Fig. 9.The residual excitations are between 0 and 4 MeV.

1.0

0.5

o.eSS0.572 fs/s

0.000 P 1/'a

Ez spin symbol

C Stj ——VJ (2j+1), (17)

isovector E3 giant resonance which is predicted aroundE =33 MeV with a strength of the 58% of the energy-weighted sum rule does not improve this discrepancy atall. Perhaps the easiest way to obtain better agreementwould be to introduce extra even-parity strength aboveEz ——26 MeV or to widen the isovector E2 resonance inorder to give larger even-parity strength on the high ener-

gy tail of the IVQ.The energy dependence of the cross section forPb()/, n) presented another problem. The parameters

(Table II) for the best overall fit to ()/, n) and to (n, yo) onPb put the IVQ width at 6 MeV and the fraction of the

sum rule strength in the resonance at 40%. Any increasein strength or decrease in width enlarges the bump (seenat 55 in the calculated cross section) which appearsaround the center of the IVQ resonance (Fig. 11). Sinceno such bump is observed in the experimental data, thesedata require an IVQ strength distribution that is morespread out than that deduced from an analysis of inelasticelectron scattering.

For the ""Cd(y,n) reaction the DSD calculation of theasymmetry is more complicated than for Pb. The distri-bution of single particle strength among the low-lyingstates in Cd is less obvious since Cd is far from a closedshell. According to the quasiparticle model, for an even-even parent nucleus in its J=0 ground state, the popula-tion of single-hole states with angular momentum j isgiven by the spectroscopic factor

0.0

15I. . . . I

20 25

E» (MeV)30

1.0 I

"'r b(», n, )'90'

(b) .

0.5

0.0

0.5

o.o15 80 25 30

E» (MeV)

FIG. 12. Final state dependence (a) of the asymmetry and (bjof the cross section at 90 for the Pb(y, n;) reaction calculatedwith the DSD model. Top and bottom parts correspond to theresults with the optical potentials from Refs. 24 and 11, respec-tively, and the resonance parameters referred to in Fig. 9. Solid,dashed, dotted, long dashed, fine dotted, and dotted-dashedcurves correspond to the 3p~/2, 2f5/z, 3p3/p ii~3/p 2f7/2 and1h9/2 states, respectively.


State

2d ]/22d 5/2

1g7/21h

& &/2

EJ (keV)

214445700480180

pT21

0.270.250.780.840.35

TABLE IV. Occupation number V~ and single quasiparticleenergy EJ for" Cd.

count for the observed Cd asymmetry curve. Thestrength of the IVQ resonance required for a best fit in Cdis again about half the sum-rule strength (see Figs. 13 and14). The IVQ strength appears to be more widely distri-buted than the analysis of the (e,e') data suggests.

D. Comparison of the two models

In the earlier parts of this section we have examinedour experimental results (especially those for ""Pb) using

0.5

.zlEtC d ( )85% of EWSR

known, we adopted the empirical resonance parametersfor " Sn. For (r ) and (r ) values for " Cd we used21.4 fm and 577.8 fm, respectively. Since the couplingstrengths V& and 8'& for ""Cd have not been investigatedpreviously, we treat them the free parameters adjusted tofit the measured absolute crosses sections while keepingthe ratio between them the same as for the ""Pb(y,n) reac-tion. The final values for ""Cd are then V~ ——54 MeV andW& ——91 MeV for the Rosen potential, and V~ ——41 MeVand 8'& ——82 MeV for the Becchetti-Greenlees potential.In Fig. 13 the result for the asymmetry is compared withthe experimental data integrated over the range of residualexcitation energies up to 3 MeV. This range is probablylarge enough to include all fragmented quasiparticlestates. The position and the width of the IVQ wereroughly adjusted to get fair agreement with the data.

Despite the crudeness of the DSD calculations, we find,as we did for Pb, that the calculation can qualitatively ac-

1.0

0.5

0.0

b0.5

0.0

1.0

20I

25E (MeV)

" 'Cd(y, n)55'

50

30

"cd(y, n. )55'

0.01.0

(p 0.5

I

I

50% of EWSR

0.5

0.04g~ y

I

I

0,01.0 I

I

25% of EWSR

b0.5

0.5

0.0

20I

25E, (Mev)

30

FIG. 13. Asymmetry curves for the ""Cd(y,n) reactionpredicted by the DSD model together with the experimentaldata. The residual excitations are between 0 and 3 MeV. Solidand dashed curves are the results with the optical potentialsfrom Refs. 24 and 11, respectively. Resonance parameters arelisted in Table II.

3000

20 25E (Me V)

FIG. 14. Cross section for the ""Cd(y,n) reaction predictedby the DSD model together with the experimental data. Theresidual excitations are between 0 and 3 MeV. (a) and (b) are theresults with the optical potentials from Refs. 24 and 11, respec-tively. Solid, dashed, and dotted-dashed curves correspond to85%, 50%, and 25% of the EWSR strength, respectively. Theother resonance parameters are listed in Table II. Dotted curvesare the calculated direct process contributions.

492 T. MURAKAMI et al.

two theoretical models to determine the values of the pa-rameters which characterize the IVQ resonance.

Asymmetry curves

It was found that although the measured asymmetriescould be qualitatively reproduced with either a veryschematic semiclassical model or with the direct-semidirect model, the sets of resonance parameters de-duced from the two models disagreed. The semiclassicalmodel gave a lower energy and a much wider resonance.Some of this disagreement can be accounted for by a closeexamination of the two models.

a. The location of the IVQ resonance The. DSD modelplaces the IVQ resonance at 23.5 MeV, whereas the semi-classical fit puts it at 21.5 MeV. In both models it is theshape and location of the asymmetry curve that is used toassign the IVQ resonance energy. We have seen that theasymmetry depends mainly on the phase difference be-tween interfering positive and negative parity amplitudesrather than on the magnitudes of these amplitudes. TheDSD calculation includes three contributions to the phaseof these amplitudes which do not appear in our semiclas-sical calculation.

(i) First, the optical potential introduces a phase intothe (y,n) matrix element through the distortions of theoutgoing neutron wave function. The phase differenceP+ —P due to the real part of the attractive nuclear po-tential is negative and it increases with l (see the Appen-dix). Since P+ —P is already negative (without the opti-cal model correction) because the El resonance lies belowthe E2 resonance, the correction adds to the phase differ-ence (see Fig. 5). This additional phase difference has theeffect of moving the asymmetry curve to higher energy.The magnitude of this shift, due to real part of the poten-tial, increases with l and can be observed in the DSD cal-culations to states of different angular momentum (Fig.12). The phase difference due to the imaginary part of thepotential is found to be relatively small. The mean shiftof the asymmetry curves due to the optical potential(averaged over l) is upward in photon energy. To fit thedata would therefore require a lower rather than a higherresonance energy in the DSD calculation than in the semi-classical calculation. This shift is therefore in the wrongdirection to explain the discrepancy between the resonanceenergy assignments in the two treatments. There musttherefore be additional differences between these treat-ments which move the DSD asymmetry curves down inenergy.

(ii) The second effect in the DSD calculation which isnot present in the semiclassical formulation is the in-clusion of a direct component as well as the resonant com-ponent for the odd parity excitation. The addition of anonresonant component makes P less negative and in-creases the magnitude of the phase difference P+ —PThis is in the same direction as the preceding effect but ismuch smaller.

(iii) The third phase-dependent difference between theDSD and semiclassical treatments is that the numeratorof the resonant term is taken to be complex rather than

real in the former in order to represent coupling to othermodes. The resulting phase shift of the amplitude comesfrom the surface region of the nucleus because of thecharacter of the form factor [Eqs. (13) and (15)] and islarger for E2 than El excitation because of the differencein weightings (r vs r) for these excitations. The effect isto reduce the magnitude of P+ —P . This effect resultsin a downward energy shift of the calculated asymmetrycurves and is in the right direction to explain why the Ezvalue deduced from the DSD calculation lies higher thanthat from the Breit-Wigner semiclassical treatment. ADSD calculation using a reduced value for the imaginarycoupling shows that the use of a complex numerator doesindeed move the asymmetry curves down by a few MeVand that this shift is rather independent of the I value.

It is unfortunate that the assignment of the IVQ reso-nance energy Ez depends as sensitively as it appears to onhigher order effects in the excitation of the collectivemode. This is particularly so with regard to the effect ofthe assumed imaginary part of the resonant form factorsince there is not yet available a convincing derivation ofits form or magnitude.

b. The uridth of the IVQ resonance The sem. iclassicaltreatment clearly overestimates the width of the IVQ reso-nance because it tacitly assumes that the energy depen-dence of the asymmetry curves is independent of I. Wehave seen that due to the attractive potential for neutrons,the DSD-calculated asymmetry curves for final singlehole states of different I are displaced from one another.Our own DSD calculations (Fig. 12) and those of Longoet al. ' show an appreciable difference in the computedasymmetry curves for different final hole states. Since themeasured asymmetry is actually an average over suchasymmetry curves [Eq. (3)], it is clear that if these curveshappen to be displaced from one another in excitation en-

ergy, their sum will give rise to an asymmetry curve withsmaller slope than that of any of the individual curves.This would then be taken to correspond to a larger E2 iso-vector width than would match any of the individualcurves of Fig. 12.

It would be useful to confirm the strong l dependenceof the calculated curves by measuring the asymmetries toindividual final states rather than to a superposition ofstates as we have had to do in this experiment. Thesharper rise that was observed in the asymmetry of theearlier Pb(n, y) study (compared with that of thepresent data) might be accounted for by the fact that asingle final state was involved in that experiment.

In any event, one must. assume at this point that theIVQ resonance width is closer to that deduced throughthe DSD calculation than it is to that from the semiclassi-cal model.

c. The strength of the IVQ resonance. Neither modelcan be trusted for a reliable estimate of the observedstrength of the IVQ resonance. There is no way to judgefrom the excitation functions how much of the cross sec-tion is due to the IVQ and how much to other excitations.The asymmetry curves are not much more helpful thanthe cross-section curves. The observed asymmetry is onlyweakly dependent on the ratio A+/2 and there is,

35 FORWARD- TO-BACK%'ARD ASYMMETRY OF THE (y, n). . . 493

moreover, considerable uncertainty about the contribu-tions to these amplitudes from the various excitationmodes. There is also uncertainty in the relative amountsof direct and collective El amplitudes.

2. The energy dependence of the (y, n) cross section

We close this section with a comment on the calculatedand observed (y,n) cross sections as a function of photonenergy. It has already been remarked that this cross sec-tion in the semiclassical calculation is expected to fall offmore slowly with energy than the actual cross sectionsince the (energy-dependent) neutron branching ratio wasomitted in that calculation. Figures 11 and 14 show thatwe have a similar discrepancy between the DSD calcula-tion and experiment, although here the branching ratio ispresumably being taken into account.

%'e have already remarked about the uncertainties inthe magnitudes of A+ and 3 . These uncertainties areeven more serious for estimates of the cross sections thanthey are for estimates of the asymmetries.

V. SUMMARY

The forward-to-backward asymmetry of the (y, n) reac-tion to a band of low-lying states has been measured for""Pb and ""Cd in the photon energy range from 20 to 30MeV. The asymmetries were found to increase fromsmall values (=0.2) to large ones (=0.6—0.8) for the twotargets. This asymmetry is attributed to an interferencebetween an isovector quadrupole collective resonance andan odd parity background which is due mainly to the tailof the isovector dipole resonance. Direct-semidirect cal-culations were carried out for Pb and Cd to determine abest set of parameters for the IVQ resonance by fitting tothe observed asymmetries and cross sections. Moderatelygood fits were obtained for assumed resonances at23.5+ 1.5 MeV (with width 6 MeV) for Pb, and at26.5+1.5 MeV (width 7 MeV) for Cd. The strengths ofthese resonances were however only 40% and 50% of therespective sum rule strengths suggesting that the E2 iso-vector strength is more broadly distributed than the as-signed widths would indicate. It should be emphasizedthat the present evaluations of the resonance width reflectthe large dependence of the asymmetry upon the l of theresidual state which the DSD calculation predicts. Itwould be most useful to confirm this prediction by inves-tigating the I dependence experimentally.

The main advantage of the present method for studyingthe IVQ resonance arises from the fact that in the IVQresonance region the odd and even parity amplitudes hap-pen, by chance, to be nearly equal. This near equalitymakes the fore-aft asymmetry insensitive to the precise ra-tio of the amplitude magnitudes. The asymmetry then de-pends mainly on the phase difference between the ampli-tudes and since one of the amplitudes is going throughresonance, this phase difference is changing rapidly. Onewould think that it would therefore be fairly easy to ex-tract the location and the width of the resonance from theasymmetry data. Although this is to a large extent a

reasonable assessment, we have seen that in dealing withreaction amplitudes rather than just with the intensitiesone can be very sensitive to the phases of these ampli-tudes. Unfortunately, some of the important phase fac-tors in the present experiment are relatively uncertain.Although they may not be directly related to the giant res-onances, they nevertheless limit the certainty with whichthe resonance parameters can be extracted from the datadespite the unambiguous and even dramatic asymmetriesthat one observes.

This work was supported in part by the U.S. Depart-ment of Energy and the National Science Foundation(under Grant No. NSF PHYS 83-11717).

APPENDIX: THE DEPENDENCEOF THE ASYMMETRY ON ANGULAR MOMENTUM

The DSD calculations of Longo and Saporetti, ' as wellas our own (see Fig. 12), suggest a definite dependence ofthe asymmetry on the angular momentum of the residualsingle-hole state. The larger this angular momentum, thesmaller the asymmetry. The main source of this depen-dence appears to be the phase-shift difference between theeven- and odd-parity emissions arising from the attractionbetween the emitted neutron and the residual nucleus.

When the DSD calculation for the (y,n) reaction is car-ried out in terms of the inverse reaction (see Sec. IV) theincoming neutron wave is distorted by the attractive po-tential. The distortion introduces a phase factor exp(i5t)into each partial wave, where 5~ is the asymptotic phaseshift due to the nuclear potential for the lth radial wave.If the residual hole state in the (y,n) reaction is labeledl —1, then the main odd and even parity scattering statesfor the neutron correspond to angular momenta l andI+ 1, respectively. A phase-shift difference,:—5I+~ —6I, occurs between scattering waves of neighbor-ing I. To the extent that the main contributions to thematrix elements for the two interfering amplitudes comefrom the nuclear surface region where the neutron wave

R

FIG. 15. A diagram to show how one can estimate the differ-ence in phase shifts for neutrons of consecutive I values and thesame energy, E, in a square well of radius R and depth Vo. Thecurves show the centrifugal potentials up to the classical turningpoints when the well is present (unprimed r's) and before it isturned on (primed r's).


functions have already accumulated almost their fullasymptotic phase shift, it is seen that AI 1 should give areasonable approximation to the phase difference betweeneven and odd amplitudes which arises from the nuclearattraction.

It has been emphasized in this paper that the asym-metries are particularly sensitive to the phase differencebetween interfering odd and even amplitudes and muchless sensitive to the magnitude ratio of these amplitudes.We have therefore carried out a very simple exercise toobtain an approximate estimate for b ~ 1 as a function ofangular momentum l.

The phase shift for a given partial wave due an attrac-tive potential can be approximated by the difference be-tween the integrals f dr lk over the classically allowed re-

gion when the potential is present and when it is absent.For a neutron of orbital momentum 1 (we are ignoring thespin), in a three-dimensional square well of radius R thisdifference is easily shown to be

2 1/2Rl. — —1

trI—arccos —.R

R1/2

rI—arccos—R

where rr and r~' are the classical distances of closest ap-proach with and without the well (Fig. 15) for the lth par-tial wave of the neutron.

With reasonable square-well parameters for Pb andwith a neutron energy appropriate to this experiment, wefind that A~ 1 is negative and that its magnitude becomesrapidly larger as l increases. For impact parameterswhich correspond to the edge of the Pb nucleus, thisphase difference reaches about —30. The magnitude ofthe l dependence found in the DSD calculations is quali-tatively in accord with that suggested by the square-wellexercise.

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