PHYSICAL REVI E% C VOLUME 8, NUMBER 6 DECEMBER 1973

Proton Spectra from 54.8-MeV Alpha-Particle Reactions: Precompound Emission

A. Chevarier, N. Chevarier, A. Demeyer, Q. Hollinger, P. Pertosa, and Tran Minh DucInstitut de I'kysique Nucleaixe, Unieexsite de I.yon-l, 69622 Villeuxbanne, F'ranee

(Received 27 April 1973)

An experimental survey of 54, 8-MeV (n,p) proton spectra is undertaken for studying pre-equilibrium emission; such deexcitation yields on the average cross sections about 250 mb,the average value measured for all the investigated reactions. Different quantitative formula-tions are discussed in terms of exciton state mean lifetimes; the exciton and hybrid modelsare found to give a good. interpretation of the proton emission cross sections. A geometricaldistinction between the equilibrium and nonequilibrium processes may be attempted. Odd-eveneffects appear in (n, P) reactions by means of partial state density: Odd-Z targets require aninitial exciton number nr equal to five (3P-2n-Oh) and even-Z targets equal to four (2p-2n-0h).'/his result may be attributed to a pairing effect according to the odd-even character of resi-dual nuclei. .

I. INTRODUCTION

The statistical theory of the formation and de-excitation of the compound nucleus fails to des-cribe nuclear reations at excitation energies ofsome tens of MeV. The discrepancy is due toemission of more high-energy particles thanexpected from an evaporative process. Thisemission may be attributed to a. direct mechanismin the two-step model' used for interpreting high-energy reactions above 100-MeV incident particles.These models associate tw'o completely extremepoints of view. The particle emission in the faststep is assumed to occur during the primary bi-nary collisions of the incident nucleon with one ora very few additional nucleons of the target. Inthe second or slow stage, the emission is treatedas the degxcitation of a residual nucleus, in acomplete internal equilibrium and where the ex-citation energy is shared among all nucleons. Theattainment of the statistical equilibrium whichtakes place between these two extreme phases isgenerally neglected; such an assumption has beenshown to be valid. '

Recently, a slightly different approach formedium energy reactions initiated by Qriffinshas been proposed. The model considers the form-ation of the compound nucleus and the emission ofparticles during the establishment of the statisticalequilibrium. ' The compound nucleus can be ob-tained only by absorption of the incident particlethrough a sequence of intermediate precompoundnuclear states. Transitions among these states areassumed to proceed by successive binary intex-actions creating a particle-hole pair, the excitons,in a single-particle nuclear model. Each of theseparticle-hole intermediate states may decay byemitti. ng one of the pax ticles, or it may decay bya new residual interaction into another particle-

hole configuration and so on, up to the compoundstates which are linear combinations of numerousstates with a high exciton number. In order tomonitor the particle escape from the pre-equi-librium states, it is assumed that all transitionswhich take place ax'e governed by phase spaceavailable for these transitions, i.e. , by the den-sity of exciton states p(E, n): These functionscount the different mays in which the excitationenergy E may be partitioned among n excitons.

From the theorem of detailed balance, Clineand Blann' derive the probability of particleemission at each exciton state which can be writ-ten for proton emission as

2s+] p(U, w~ —1, w„, v~, v„)1T k » &P» &n» "P» ~n

where (2s+1) accounts for spin degeneracy, rn

and e are the reduced mass and kinetic energy ofthe emitted proton, and o,„„(e)is the inverse crosssection for the absorption by the residual nucleusof a proton with an energy e. p(E, w~, w„, v~, v„) and

p(U, w~ —1, w„, v~, v„) are the exciton state densities,respectively, for the initial nucleus at excitationenergy E and for the residual nucleus left atenergy U after one proton emission. The excitonstate density is written for two types of fermions:m~ and. m„are the numbers of protons and neutronsabove Fermi level; v~ and v„are the number ofproton and neutron holes.

The emission of a particle from an ji excitonstate can occur only dux'ing the mean lifetime ofthis state 7„. In addition, the probability of popu-lating a particular intermedi, ate state is not equalto one but is depleted by previous emission top(n, E) &1. Therefore the absolute emission from

a precompound state is

P3(e)de =P(n, E)W3(e)T„d~. (2)

where A.,(e) is, as stated by Blann, ' the escapeprobability of the nucleon with the energy ~, and

X„„(e)is the complementary probability that thisnucleon may collide with a nucleon of the targetand given an (n+2) state. The probability A.,(e)can be written out from the law of detailed bal-RnC 6

(2s+1)o „meC +3'-3

P

where g~ is the density of states inside the nucleusof the considered nucleon; with an energy &, itis taken equal to —,'g.

It was proposed by Blann' that A.„„(e)be ex-pressed as the ratio of the velocity of the nucleonto its mean free path A(e) in the nuclear matter:

(2e/m) ~'X„„(e)=

( )——.

From the above relations, the proton spectrumis given by the following expression given byalarm' and Harp and Miller' with a different deriva-tion:

, p(U, v3 —1, 1T„, V3, V„) 3 g (e)P(E~ &py &,~ &pi ~, ) iW.,(e)+A,„„(e)'

In this expression referred to Rs hybrid model,r„ is dependent only on the energy of the particleand is quite independent of the n exciton numberand independent also of the excitation energy.From another point of view closer to the statisti-

Two different approaches have been adoptedfor evaluating T„. As pointed out previously, anlntermedlRte stRte may decRy by emitting pRx'tlclesor by decaying to an n +2 exciton state with a two-body interaction creating one additional particle-hole pair. One has to neglect the probability fortransitions leading configurations where the exci-ton number decreases to n —2 or stays unchanged. '

7„may be defined rather as the mean time duringwhich the particle of kinetic energy & to be emittedcan be considered as one of the n excitons of the

precompound state. It is only during this timethat the particle emission is governed by thephase space specified in the expression (1) whereone proton possesses an energy c and the otherexcitons share the residual energy. So z„ isexpressed as'

1

Z, (e) +Z„„(e)'

cal theory, the mean lifetime 7.„must accountfor all the competing emissions and internal tran-sitions which participate in the decaying of then exciton state. This formulation, referred toas exciton model, ' expresses g„as

Z, (E,n)+X„„(E,n) '

wh~~e, following Gadioli. ,'" z, (E, n) is defined

quite differently than previously as the sum overall particle emissions, mainly neutrons and

protons:

A.,(E, n) = g W„'(e)de .t -—neutron,

proton

A.„,,(E,n) is the probability of transition fromn exciton state to (n+2) state As. the matrixelements responsible for the transitions areunknown, ' the different empirical approaches' ' ' "are based upon nucleon-nucleon scattering crosssections inside the nuclear matter similar to thehybrid model.

The pre-equilibrium model has been employedfor analysis of both excitation functions and parti-cle spectra. " The purpose of this paper is toinvestigate the different theoretical formulationsanalyzing spectral data which are still scarcelyfound in the literature. The selected experiment,a survey of 54.8-MeV (o., p) reactions, is especial-ly indicated since no data are available except fora single result at 59 MeV, "which extends theanalyses performed by Griffin, "byline and alarm, 'and Blann and Mignerey" of results at 30 and42 MeV' "with the pre-equj. librjum model.

II. EXPERIMENTAL PROCEDURE

The experiments were performed on the LyonUniversity synchrocyclotron with 54.8-MeV nparticles impinging on self-supported foils of51V 54, 56, 57pe 59co»tCu 66gn 93Nb natA~

y p

g'5In '6'Ta and '97Au. The thickness of the tar-gets varies from 3 to 10 mg/cm'. Beam intensityis 5 nA on the average. The protons are detectedby Si detectoxs. In order to stop 45-MeV protonscompletely, a thickness greater than 10 mm isrequired, so we have used R transverse field15-mm-thick Si-Li junction" mounted on a rota-table arm and cooled by a liquid nitrogen cryostat.To identify the proton among the other chargedparticles, the method of Goulding et al."has beenemployed. The domain of emission energy thatthe reported experiments intend to cover extendsfrom 5 MeV, the evaporation region, to 45 MeV,the maximum proton energy to be detected. Con-sequently, a classical two-detector telescopecannot cover this whole energy range with a good

XROTON SPECTRA FROM 54. 8-MeV ALXHA-PARTICI. E. . .

identification, "so it was necessary to use athree-detector telescope. The first two detectorsare transmission surface barrier Si, and typical-ly 100 and 1000 p. m thick; disposed in front ofthe 15-mm detector they are thermically isolatedfrom it. The final resolution of the telescopeattains 200 keP. The low-energy protons areanalyzed by the first two detectors, since theydo not reach the last one. The higher-energyprotons are detected by the entire telescope: Thetwo signals delivered by the transmission junc-tions are added to supply the energy loss LEinformation. The electronic system for the treat-ment and selection of the proton signals is de-scribed elsewhere. " With this procedure, the

energy loss information ~E is always large enough,around 10% of the total energy of the particle, and

the condition for obtaining a good identificationsignal is fulfilled for the whole proton energy dis-tribution. Therefore it was possible to recordthe spectrum in a single measurement. A sampleof measurements is shown in Fig. 1 for 'Nb as atarget.

10

The data are then corrected for energy absorp-tion in the target and expressed in the center-of-mass system. When running the experiments,contaminations with light elements are carefullyavoided because of their high (n, p) cross sections.This problem has been discussed at length byWest. " %'e have made a few measurements, atthe same incident energy of 42 Me& as West'sexperiments, and no disagreement was seenbetween the two different sets of values. " Finally,the errors affecting the absolute value for thecross sections are estimated to be lower than30%.

III. DATA REDUCTION

The differential cross sections for proton emis-sion, as indicated in Fig. 2 concerning reactionswith "Co hand "'Ta, have been accumulated atangles from 20 to 150, at 15 intervals forwardand 30' intervals backward. From the angulardistribution obtained with these data, the integra-tion over all angles yields the total absolutespectrum; these results will appear in Fig. 11.

In Fig. 3 spectra at three selected angles, 30,90 and 150', are presented for several targets

I I

Cg 70

V

" 10''

10

eg

Og

110

10

I10

0

10 i

70 20

oy

781

4

30 90 150

de~ ~

150 +,10

10

50 100I

750

C hannet number

100 70

I~ I

20 30 40 30 90 150

FIG. 1. The differential experimental spectra for the+Nb(m, P) reaction at several angles. The arrows indi-cate the connection of the spectra collected with two andthree detectors, respectively.

e(MeV) 0FIG. 2. The differential cross sections for the ~~Co

and ~Ta{e,p) reactions with angular distributions forthree kinetic energy regions.

2158 A. CHEQARIER et al.

with A values covering the mass table. For anyof the studied reactions no peculiar structure isnoticeable in the energy spectrum or in the angulardistribution energy. The common behavior can becharacterized by a variation of emission crosssection with a wide maximum near the Coulombbarrier and then decaying nearly exponentiallyand monotonously toward high energy. Thisdescription is not true for heavy-mass targetswhere the evaporation peak is very depressed.Indeed, this low-energy portion of spectra classi-cally exhibits every feature of a compound-nucleusprocess: an angular distribution, flat or rathersymmetrical around 90', and cross sections de-creasing with increasing mass values.

On the other side, toward larger energy, the

angular distribution is peaked forward with nomarked 3 dependence of the cross section. Actual-ly the spectra are almost parallel and nearlymerged for most of the reactions. These featuresmay be understood intuitively with a fast mech-anism in which a small number of nucleons, iden-tical for any target, participates in sharing theexcitation energy, the remaining nucleons formingan inert core.

10

10

21

(

eq

"U

10

As for the intermediate kinetic energy region,the angular distribution, while still remainingpeaked forward, rises at backward angles and itmay appear to be flat for angles greater than 120'.This trend leads several authors" "" to attributethe whole emission at very large angles, 150,to the evaporation process. However, the valuesof the level density parameter a extracted fromsuch analyses are found to be very low and aboveall, independent of mass number, but they de-crease with the excitation energy. These conclu-sions are quite inconsistent with the Fermi-gasmodel. Another more realistic point of view"is to consider that these misleading conclusionsare caused by particles emitted in excess of thecross section predicted by the statistical theory.So, some pre-equilibrium emission has to be takeninto consideration. In evaporation calculations weare led to use level density parameter values ex-tracted from analyses of low-energy reaction and

classically equal to ~. Proceeding in this man-ner, we have fitted the calculated compound-nu-cleus emission shape with the experimental rear-angle spectrum. The normalization region isclearly the low-energy portion of the spectra;the high-energy part which is not interpreted willbe attributed to noncompound processes (Fig. 4).%e have thus obtained from the total integratedspectrum approximate values of cross sections forthe compound and the so-called precompound inter-actions (Table I). It can be noticed that nonequilib-rium processes are increased by a factor of 2 or3 when the incident energy of n particles in-creases from 42 to 54.8 MeV. This fact fullyjustifies the reported experiments and furnishesa good experimental test of the pre-equilibrium

10

10

0.1

0.01

Q2 s 150

l

0 20 30

VInNbZnAgFe0ioAu

e (MeV )

FIG. 3. The differential cross sections at 30, 90, and150' angles for (n, p) reactions with different targetsalong the mass table.

10

70 20 10 20

& (MeV)

FIG. 4. The 120 cross sections for 5~Co(m, P) spec-trum at 42- and 64.8-MeU bombarding energies: Thedashed lines are evaporation spectrum shapes calculatedwith a = 7.9 MeU and fitted with experimental data atthe lowest-energy region of emission. The hatched partis due to noncompound events.

PROTON SPECTRA FROM 54.8-MeV ALPHA-PARTICLE. . . 2159

theory. Figure 5(a) shows indeed that at 54.8MeV the proton energy spectrum covers a widerrange than data with lower incident energies, sothe confrontation with theoretical predictions ismore compelling. Moreover, the experimentaldata for precompound emission are also extractedmore precisely. As explained in Fig. 5(b), wherethe spectra are plotted against residual energycalculated for emission of the first proton, theinterference with the equilibrium process is muchmore appreciable for lower incident energies.

IV. ANALYSIS AND DISCUSSION

In the expression for the probability of pre-compound emission, two factors play a fundamen-tal part. The first parameter to be specified isthe exciton state density. The basic assumption ofprecompound formalism is that the initial excita-tion energy is shared only among a small numberof degrees of freedom, the excitons, and, hence,the statistics of partition and particle emissionare governed by the particle-hole state density.Secondly, as discussed in Sec. I, the importanceof the transition rates between intermediate stateshas been pointed out in relating the mean lifetimesof these states in order to obtain absolute valuesof the cross sections.

p(w~ —1, w„, v~, v„, U) according to

p (11~ —1, 7t„, v~, v„, p)go. „7

Two different procedures for conducting theanalysis can be followed: One is to fix a priorithe initial exciton number n~ from argumentsconcerning the reaction mechanisms, """theother mechod, which we shall adopt, is to extractthis parameter from the spectral data.

The partial state density for a uniform-spacingmodel of two fermion gases is given by"

gn-1p( p» » vt» v»& @) &gp &g»

~~ 1 n'„I v~ t v„ I in —1q l

1OO

10

1

A. Partial State Density 0.1

As will be stated quantitatively later, the de-excitation of the first intermediate state, charac-terized by the exciton number n, , from which theparticle emission may occur, makes the mostimportant contribution. So it can be recalledfrom expressions (1) and (2) that the shape of theproton pre-equilibrium spectrum is then propor-tional to the density of the particular (n, —1) state

0.01—

10 20 30 40e (MeV)

54 MeVCompound Precompound

Target cross section cross sectionnuclei (mb) (mb)

42 MeVCompound Precompound

cross section cross sectiol(mb) (mb)

TABLE I. Integrated cross sections for compound andnoncompound processes in proton emission with 54- and42-MeV e-particle reactions.

10

1

b)

30

"v56Fe"Co

10501265

980

275235220 881 a 89 a

90b

0.1

30 10 0U (MeV)

Zn92NbnatA

'"Sn

1190700500650

'See Ref. 17.b See Ref. 21.

230240250255

335

315

85

125

FIG. 5. Integrated cross sections for 93Mb(n, P) re-actions at different bombarding energies. Data at 30MeV are due to Swenson and Gruhn (Ref. 16), 42 MeVdue to W'est (Ref. 17). Spectra in (a) are plotted againstkinetic energy of protons and in (b) against residualenergy U =E —e -8&.

2160 A. CHEVARIER et al.

where g~ and g„are constant single-particle leveldensity of the proton gas and neutron gas, takenboth equal to —,'g.

It is easily seen by means of the above relationsthat the spectrum shape is a function of U"I ',so the plot of the logarithm of (do/de)/eo, „„v„versus the logarithm of U is linear with a slope

qua t nIIn this analysis we first have to examine the

expression of the residual energy: U=E -B~—&.

B~ is the proton binding energy. For the leveldensity calculations in the statistical model one

usually adds a correcting term according to theodd-even character of the nucleus. Then, assome authors" propose, the question of correctingthe energy for pairing effects in the exciton statedensity expression can be examined. As pre-compound emissions populate low-energy residualstates, this problem may be of some importance.

A second correction. is due to the Pauli principlewhich tends to reduce the exciton state density;

Williams proposes to shift the energy by a negativeterm".

In (n, p) reactions, n, seems to describe 4p-Oh

or 5p-Oh configurations, i.e., with m =2 or 3,m„=2, and v~= v„=0, so with g~=g„=A/26. 6; thesefactors 8 are typically equal (in MeV) to 78/Aor 120/A.

Therefore, we have expressed the residualenergy without. correction, then with an exclusionprinciple correction and with a pairing term.We have also used the different expressions ofthe mean lifetime: T„constant with e followingWilliams' or exciton formulation' "or varyingwith e in the hybrid model. ' A third methodderived from a direct mechanism point of viewaccording to Griffin" has also been used by plotting(do/de)/E

100

10

10 2 10

U(MeV)20

FIG. 6. Different plots for the determination of the intial configuration number nI in the 88Zn(n, P) and 59Co(n, p) re-actions. The curves a correspond to the exciton or Williams formulation, b to the hybrid model, and c to the Griffinexpression. The symbols are as follows: ~: denotes plots for uncorrected U; ~: for Pauli-principle-shifted U; 0:forpairing-shifted U; and 6:for back-shifted U. The angle of emission is 30'. The parameters s are the slope valuesobtained from the analysis with unshifted energies.

PROTON SPECTRA FROM 54. 8-MeV ALPHA-PARTICLE. . .

Figure 6 displays plots from "Co(o,, p)e2Ni and"Zn(o. ,p)"Ga spectra at a 30' angle. As empha-sized by the results, when the exciton state densi-ty expression from the constant spacing model laused the analysis seems to be more satisfactorywith the uncorrected residual energy. In thiscase indeed, the expected linearity extends fromthe lowest excitation energies up to about 20 MeV,where contributions from higher exciton statesappear significant. With shifted energy, especiallyby pairing effect, the departure of the plot from asingle straight line is obvious and the analysisgives two slopes [the residual nuclei selected inthis example are even-even ("Ni), so 5= 3.0 MeVand odd-even ("Ga) so 5 =1.5 MeV"]. For thehigher residual energy region, the slope valueis the same as given by the analysis with unshifted

U; for the lower-energy region, it is roughly oneunit smaller, and this applies for both nuclei.

The different approaches for evaluating T„donot modify the previous remarks and they yieldin fact the same n, values. This conclusion under-lines the importance of exciton state density in theshape of the distribution of the kinetic energy.Acually, the slope analysis is basically dependenton the accuracy of the functional form (8) obtained

10

0, 1

I I 1

00o~ 0

~00

Oo~e« inv. ~ p~ Q

0~ p0~ 0

07

59 62~ 27CO {a,P) Ni

54,56, 57 57, 59160F~( p)

0

U(Mev)5 10 2O

40I

30 20 10

from the constant spacing model. This point hasbeen discussed by Bohning" and Williams, "whohave concluded that the analytical form is a quitepoor approximation only when U is small. Thecomparison made by Williams, Mignerey, andBlann29 of the approximation (8) with some directnumerical values computed for the density ofstates generated by combinations of Nilsson orbit-als yields fairly large discrepancies only for lowU and near the closed shells. On the contrary,from rea) levels of a few nuclei, Grimes et al."have found the expression in U" ' to be a goodapproximation even at low excitation energy. At

56

26

3.110

0.1—

66 69Z (,p) 6

63 66Cu (u, p) Zn

I I I I0 Q~0

~0 0'V~

~o~ Q

~ 0~ 0

~ 0~ 0

~ 0~ 0

02.7 0

ep~ o

~ 0

5927

I

30I

20

U(MeV)5 10 20I

40I

10

63,65

29

66

30

5 10 50 5 10 50

U ( MeV)

10 I

0.1

QQI oo~ Q~ ~ Q

e 0e 0

3.3

U(MeV)2 5 10

116 119Sn (u, p) 51Sb

0~ p 115 118

0 ~ In (a, p) Sn.~ 0

00

o0

~ 000

FIG. 7. Plots versus U for (n, P) spectra with somecouples of neighboring targets: evidence of an energygap between the excitation of even-even nuclei and odd-odd or odd-A nuclei. The slope analyses are also re-called.

I

40I

30 20 10

U (McV)

FIG. 8. Results of slope analysis for all studied (e, p)reactions. Plots at 30' angle with the exciton method.

2162 A. CHEVARIER et al, .

any rate, for the present (u, p) reactions, verylow-lying states are not excited (see Figs. 6-8).By considering the region of residual energyanalyzed here, and as the majority of the studiednuclei are far from closed shells, the use of theuniform-spacing model approximation seems tobe quite justified.

The results of the slope analysis with unshiftedenergies and by means of the expression (8) forexciton state density show evidence for an odd-even effect: As listed in Fig. 7 and Table II, odd-Z targets require &&1=5 and even Z, nI =4. Untilnow, most of the (u, p) reactions have been in-vestigated for odd-Z nuclei and precompoundanalyses actually yielded n, = 5.' A few resultswith even Z have reported n, =4 for 30-MeV datawjth Nj and ' Sn and 42 MeV wjth Nj and"'Sn." In this work this last trend is firmlyestablished for """Fe, 66Zn, and '"Sn targets.

This odd-even effect is manifested by the inter-mediary of the residual energy. As shown indetail by Fig. 8, for a nearly identical kineticspectrum of protons the energy of the levels ex-cited in (u, p) emission is much lower for odd-gand odd-odd residual nuclei ('"Sb, "'"'"Co) thanfor even-even nuclei ("Ni, '"Sn). The differencemay exceed 3 MeV. The Q values cannot explainthe result, since for the reactions with the three""'"Fe isotopes the Q values differ up to 2 MeV,and nevertheless the plots of the spectra againstU are merged and interpreted by the same nrnumber. In other terms, the even-even nucleirequire higher energy to be excited. This gap isquite consistent with a pairing effect. The spectraldata for these nuclei are then interpreted by largern, values. The same trend may be observed in(u, n) reactions": The '7Fe(u, n), "7Sn(u, n), and'"Sn(u, n) lead to even-even nuclei and are fitted,respectively, by nr

——5, 5, and 4, while the (u, n)reactions leading to different parity nuclei areinterpreted by n1=3 and 4. This odd-even featureis not so obvious in ('He, n) reactions" and itcannot be observed in ('He, p) or (p, n) reactionswhere even-even nuclei can be reached from onlyodd-odd targets. For '(p, n) reactions, odd-eveneffects have been demonstrated; however, nogeneral trend is observed. '

Now, if some pairing correction has to beintroduced for evaluating the residual energy, onemust first examine the form of the intermediatestate density to be used in connection with shiftedenergy. The effect of pairing inclusion in theexciton state density has been investigated byGrimes et al.";with shifted energies, the resultsfrom a ¹ilsson calculation" are fitted by a, powerone unit larger (or even greater near the closedshells) than expression (8), i.e. , U" instead of

U" '. According to that result, the slope of theplots with pairing correction 'is now n~ —1. Theanalysis with pairing term will not change the

n~ value if we take into account only the lowerresidual energy region; otherwise, for the higherregion, a number larger by one unit is obtained.However it remains that such a conclusion givingtwo exciton numbers differing by one unit andcharacterizing a same reaction cannot be easilyexplained by assuming two-body internal transi-tions. The calculated contribution of precom-pound emission of a second proton for this emis-sion energy range is too small in comparison withthe first proton emission to be taken into account.

It may be noted that in analyses of nuclear re-actions with the statistical theory another alterna-tive for correcting energy is the back-shiftedFermi-gas model; the zero of the energy scaleis taken now for even-even nuclei. " Unhappily,no expression of the exciton density state hasbeen investigated for this procedure. Thus, theapproximation (8) has to be assumed to remainvalid. Then the analysis of the (u, p) reactionswith an odd-Z target is identical to the previousone with uncorrected energy, since the residualnucleus is even even. For even-Z targets, theanalysis with the residual energy shifted towardshigher energy will raise the nI value: n~ = 5 forodd-even residual nucleus (see Fig. 6). The odd-odd residual nuclei would require even a highernr, for example, with the "Fe(u, p)~Co reaction.

The present discussion shows the importanceof evaluating the energy in the exciton state den-sity expression. More information about thisproblem is highly desirable. For the study here,we may conclude that no pairing correction of the

Reaction

Odd-even characterof the residual

nucleusSlope

S

5iV(e,p)'4Fe(o, ,p)58Fe(o. ,p)"Fe(e,p)"Co(n,p)"65Cu(o. ,p)66Zn(o. ,p )~~Nb(n, p )iQ7,109Ag(y p)ii~In(d, p )'"Sn(a p)18iTB (~ p)i97Au(o. ,p )

even-evenodd-evenodd-evenodd-odd

even-eveneven-even

odd-eveneven-eveneven-eveneven-even

odd-eveneven-eveneven-even

3,31.91.91.92.72.72.33.5331.93.12.8

544455

555

55

TABLE II. Results of the determination of the parame-ter nr for 54.8-MeV (e,p) reactions. The even-evenresidual nuclei require the nr value to be one unit largerthan the other nuclei.

PROTON SPECTRA FROM 54. 8-MeV ALPHA-PARTICLE. . . 2163

energy seems to be appropriate for interpreting54.8-MeV (o., p) reactions. This conclusion isconsistent with the next quantitative analysis ofthe cross section. At the present stage of thequestion, we prefer to state that pairing effectsare manifested through the difference of n~ valuesgiven by the analysis with unshifted energies ac-cording to the parity of the proton number of thetarget. A point of view from the reaction mecha-nism has been adopted by Griffin' '' for discuss-ing the odd-even effects in precompound reactions.%'ith such an analysis, it can be said that for(a, p} reactions with an odd proton number target,the extra proton participates with the four nucleonsof the diluted incident n particle for dissipatingthe excitation energy. Thus, the initial configura-tion with n1=5 is a 3p-2n-Oh state and with n~=4,a 2p-2n-Oh state.

B. Cross Sections

model. "'6 InitiaQy, Blann and Mignerey" havetaken the reciprocal of A.„„asthe exciton statelifetimes 7„ to be used then for calculating absoluteexciton model spectra. " Characteristic magni-tudes for a„+,(e) are (2.5x10") s ' and (4.5 x10")s ' according to e =20 and 40 MeV. It is usefulto compare these results wwith calculations givenby Harp and Miller' from the resolution of the~3.ster equations for the equilibration processinside a two-fermion gas by nucleon-nucleon col-lisions. , Values extracted in this way are depen-dent on n and E. For 18-MeV protons bombarding"'Ta, these authors have found A.„„(E,n) equalapproximately to (2 x10") s ' for n =4. Anotherestimation, formally similar, due to Braga-Marcazzan et al. ,

"yields values of the same orderof magnitude. With the exciton model formulation,we shall use for g„(n, E) values derived fromA„„(I,E}estimated by Birattari et al." accordingto the relation:

This last section will deal with the comparisonbetween the experimental integrated cross sec-tions and the different calculations of pre-equilib-rium theory. Such formulations require the esti-mation of the transition probability g„.

In the expression for A.„„(e)of the hybrid model, 'Blann uses values computed by Kikuchi and Kawai'4for evaluating the mean free path of nucleons in-side the nuclear matter. These values have beenfound consistent" with results from the optical

700

~„„(n,E}=X„„(I,E) 2

102

10

93Nb (u, p)

10 10

70

0.1

10 20 40

e (MeV)

FIG. 9. The different calculated contributions for theproton emission induced by 54.8-MeV 0,' particles on93Mb. The curve ( ) is the total theoretical spectrum;the curve (——) is the precompound contribution forthe initial configuration: nl = 3p-2g-Oh; the curve(- ————-) is the total precompound emission; thecurve (———) is the evaporation spectrum from thefinal nuclei after precompound emission; and the curve(—~ —- ) is the evaporation spectrum from the compoundnucleus at equilibrium when no precompound emissionhas occurred.

20 30 40

e &MeV)

FIG. 10. The comparison of the experimental crosssections (0) for the 54.8-MeV ~3Mb(n, p) reaction withthe results of the different calculations. The curve( — ) represents the hybrid model (Ref. 7); the curve(———) represents the exciton model (Ref. 9); thecurve (------) represents the Griffin's model {Ref.24); and the smooth averaging of the l sum in this lastformulation is represented by the curve (-"~ —~ ~ ). Allthe calculations have been performed for nz = 5 and witha priori fixed parameters.

2164 A. CHEVARIER et al.

For instance, with 9'Nb(o. , p) reactions at 54.8MeV the numerical value with nz = 5 is (2 x 10")s-'.

The total proton spectrum from (u, p) reactionsis computed according to the following scheme:(i) First, the precompound emission probabilityis computed from the composite system (N, Z)with energy E for only the first nucleon with n

varying from nz to n~ =~gE by a step of 2.(ii) Then, for the residual nuclei (N —1, 2) and

(N, Z —1) with energy P populated by this nucleonpre-equilibrium emission, an evaporation calcula-tion for nucleons and z particles is applied.(iii) Finally, the last stage computes the de-excitation of the compound nucleus at equilibrium

which is obtained with a probability equal to

neutrons ni-2protons

The sum of these contributions multiplied bythe cross section of capture of the n particleobtained with an optical-model code" yields thetotal proton spectrum. A sample calculation isgiven in Fig. 9 where the pre-equilibrium con-tribution is computed according to the hybridmodel. In this example, the contribution of thefirst precompound emission is also plotted inorder to show its dominance. The importance of

100 100

10

~ 1002

to

10

100

100 100

10 10 oi

100

10— 10—

100—

10 10

0.1 0.1

I

10

I I l

20 30 40

(MeV)

10 20 30 40

(MeV)

FIG. 11. Comparison of the experimental data (0) with the hybrid model (Ref. 7) results for all the studied 54.8-MeV(0.', p) reactions. Each spectrum has its cross-section scale shifted from the preceding one by two decades. The calcu-lations were performed with unshifted energies and with the nucleon mean free paths given by Kikuchi and Kawai {Ref.84). The odd-Z target reactions are interpreted by an initial exciton state el =5 (curves ), and even Z by nz =4(curves ————).

PROTON SPECTRA FROM 54.8-MeV ALPHA-PARTICLE. . . 2165

the deexcitation of the residual nuclei (N, Z —1, U)and (N —1,Z, U) should be noted also [stage (ii)]:It contributes about 20% of the total evaporationprocess.

The compound-nucleus calculations were per-formed classically with Fermi-gas parameters:g = and level density of the exponential formU ' exp2~aU, with pairing correction.

Figure 10 compares an experimental "Nb(cv. , p)spectrum with calculations of exciton and hybridmodels. The different parameters have been al-ready specified; as given by the slope analysis,the n~ configuration is built with Sp-2n-Oh. Forenergy calculations only, the exclusion principlecorrection is added without pairing term.

For comparison, it was useful to perform acalculation according to an expression by Lee andGriffin'4 which we recall here without further dis-cussion:

~~~ P(w~ —1, m„, v~, v„,U) 2l +1 2m

where (fj', ,') is a ratio where the numerator is theaverage square matrix element of transition be-tween two states, the initial one is bound, and thefinal one has a proton in continuum with orbitalangular momentum E. The denominator corres-ponds to internal transitions between bound states.

This ratio is approximated as:

(a, ') =-,'r(n, r)'[f, (&,r)]'+[q, (k,r)]',where y is the nuclear radius equal to i.2A. '~' and

k, the proton wave number. j, and g, are thespherical Bessel functions. The summation isover all the l removed by the emission: Thistruncation with integer value of E is responsiblefor the discontinuity in calculated values whichcan be removed by a smooth averaging of the lsum

As predicted by our previous discussion, thedifferent theoretical shapes are quite concordantwith the experimental spectrum. The last formula-tion is somewhat higher than the two others whichare in quantitative agreement with the experiment.The fit is equally good for the evaporation andpre-equilibrium components of the spectra. Inthe estimation of v„ the leading term is g, » andsince for this quantity two approaches use valuesof the same order of magnitude, it is not sur-prising that both calculated cross sections arenearly identical. The values employed here forA,„„arequite realistic and correspond to meanfree paths computed by Kikuchi and Kawai. "Thus, in the following series of Fig. 11 we com-

100 100—

10

0.1

10 20 30 40

e (MeV)

FIG. 12. The Fe(n, p) Co cross sections (the ezper-imental points are denoted by open circles, 0) arebetter interpreted by a hybrid calculation with nl ——4 andunshifted energy (curve -) than with nz =5 and back-shifted energy (curve ———). The result of a hybridcalculation with ni =4 and back-shifted energy (curve—.—) is also shown.

0.1

10 20 30

oll

40 50

( MeV)

FIG. 13. Same as Fig. 10 for Fe(u, p) reaction at 5gMeV, except that the calculations are carried with nz =4.The experimental data are due to Bertrand and Peele(Ref. 13).

A. CHEVARIER et al.

pare the experimental spectra with only hybridformulation, assuming that the exciton modelleads to the same conclusion. The results of thisinterpretation of the absolute cross sections areconsistent with the conclusion of the last sectionanalysis of the spectrum shape. The whole setof the reported (a, p) reacti'ons are well inter-preted by expressing the energy without pairingor back-shift correction and by taking n, =5 or 4according to the odd-even parity of the targetatomic number. Another alternative for theseconditions may be possible: Hybrid calculationswith nl =4 can also interpret the (u, p) reactionswith odd-Z targets when the energy is shifted bya pairing correction and when an exciton statedensity of the form U" ' is used. If a U" form inthis case seems to be more appropriate as found

by Grimes et al. ,"a more appropriate value for

nl will be 5. However, such absolute cross-sec-tion calculations do not follow the conclusion ofthe previous slope analysis which yields moreacutely the nz value. The back-shifted energyprocedure in the cross-section computation leadsalso to some inconsistency with the slope analy-sis; for instance, the S7Fe(o., p)"Co is not fittedby a calculation with n~ =5 or more as suggestedby the shape analysis. Actually, this reactionis quite well interpreted by n, =4 when the energyis uncorrected (see Fig. 12).

With (o., p) reactions at 54.8 MeV or higherenergy [see Fig. 13 for the interpretation of

0.1

0.01,- 10 20l

10

10

0.1

0.01I

10

20I

5

l

{bj

tT Ill b

g 1

00

I

20

I

20

ho

10

30

l

30

40

Q 210

E

10

1

0 Nb

10

0. 1

i'044 o

0.01

30 l ~n &ol5 10

10I

20l l

30 40e (MeV)

10I

30 40

e (MeV)

FIG. 14. Same as Fig. 10 for 93Mb(o,', p) reaction at 42MeV. The experimental data are due to West (Hef. 17).A hybrid calculation performed with a value of meanfree path 2 times larger than that calculated in Ref. 34gives a better fit (curve ———).

FIG. 15. (a)—(c) The results for the geometry-dependent hybrid formulation (Ref. 38) for a few 54.8-MeV (n, P) spectra. In addition to the calculated curve( -) of the cross sections, the partial capture crosssection (curve ) and the precompound proton emis-sion for each partial zone (hatched surface) are a1so indi-cated in the inserted figure, versus the impact para-meter.

PROTON SPECTRA FROM 54.8-MeV ALPHA-PARTICLE. . . 2167

"Fe(u, p) data of Bertrand and Peele" at 59 MeV]the experimental results are in agreement withthe theoretical predictions. However, signifi-cant discrepancies appear in proton-induced re-actions, "in (u, n) reactions, "or in (u, p) reac-tions themselves at lower energies, as for in-stance in 42-MeV "Nb(u, p) reaction, the analysisof which is shown in Fig. 14. The disagreementmay be due at low excitation energy to an under-estimated evaporation calculation when no angularmoment effect is added. A more probable reasonis that no geometrical factor is included in theprecompound closed formulation even though itis a surface interaction. " In fact, mean-free-path values larger than preceding ones, generallyby a, factor of 2, give better agreement for (u, n)reactions" or 42-MeV (u, p) reactions (see Fig.14).

The hybrid model allows inclusion of the re-action geometry dependence. " Thus, the con-sideration of nuclear matter distribution in thecalculation of the internal transitions enhancesthe mean-free-path values at the diffuse edge ofthe nucleus. It gives then v„values larger fornucleon escape. A second effect accompanyingthe nucleon density distribution is a geometricaldependence of the g values and the exciton statedensity. " However, the g variation according tothe nuclear radius" leads to unrealistic valuesunexplained if we consider that g is a property ofthe potential created by all the nucleons, and,besides, the importance of this effect in crosssection is weak. The exciton state density in thephilosophy of our analysis is a result extractedfrom the experimental data and in fact the nlnumbers used (2p or Sp-2n-Oh) implicitly assumethat holes do not participate as a degree of freedomin sharing the excitation energy. This is consis-tent with a relative difficulty of creating very deephole states.

Consequently, we shall use the following expres-sion for the different partial zones:

where wX'(2l+1)T, is the capture cross sectionof the incident particle with an impact parameterequal to LX, and P~(e, l) has a dependence on theFermi-Thomas density distribution according to

p(wp —1, wp, vp, &„,U) A. (e, l)

with

X„„(e)n+2( ) l +e(R-Rq)/

g, is equal to 1.074' ' fm and diffuseness a =0.55

2 1~.(e, f) = g (28+1)Tf,

0

where l' is the angular momentum of emittedproton and T~ the corresponding transmission co-efficient.

The precompound emission has been computedwith the above expressions for the n, =4 or 5exciton state only; the deexcitation of this firststate is quite predominant. Results which appearin Figs. 15(a)-15(c) for reactions with "2ln, 9'Nb,

and "'Ta are in good agreement with experiment.In these figures, we can see that the major

amount of precompound emission can be localizedby this treatment in the nucleus periphery. On

the contrary the central trajectories lead to thecompound nucleus almost without any pre-equilib-rium emission; this conclusion is consistent withresults of the master equation for equilibriumnuclear relaxation. ' Another distinction betweenthe two processes can be pointed out in this geome-try-dependent formulation. For precompoundemission the times used for relating r„rae &(2x10 ") s; these values are acceptable for inter-mediate states since they are a little longer thanthe transit time of a nucleon through the nucleus.The 7„values for central impacts are considerablyshorter, about a few times 10 "s; the reason isnaturally the dense nuclear matter involved inthe calculations. If we suppose that the equilib-rium is attained within a few collisions (2), theequilibrium time assumed by the model is typically&10 "s. The long lifetime compound nucleus isthen obtained in a very short time. This idea isconsistent with the statistical theory.

V. CONCLUSION

The pre-equilibrium formulations have beenshown to interpret with a fairly good success54.8-MeV (u, p) reactions either in the analysisof the proton spectrum shapes or in the analysisof the absolute cross sections; some insight intothe reaction mechanism may also be consideredby a geometry-dependent model. Both analyseshave also demonstrated that precompound emissionmay be a possible way to investigate by the inter-mediary of exciton state density gross nuclearstructure effects related to odd-even characterof nucleus and pairing energy.

ACKNOWLEDGMENTS

We would like to thank Professor M. Blann forhaving communicated to us some of his computerprograms. We also would like to thank ProfessorE. Gadioli, Professor M. Lefort, Professor T.Magda, and Professor J. M. Miller for somevaluable and stimulating discuss|ons.

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