STATISTICAL EFFECTS IN THE INTERACTIONS OF COMPLEX NUCLEI

byDan Shapira

A Dissertation presented to the faculty of the Graduate School

ofYale University

in candidacy for the degree of Doctor of Philosophy

B.Sc. June 19 6 7 Israeli Institute of Technology, ISRAEL Ph.D. December 197^ Yale University

ABSTRACT

Heavy ion scattering and reaction excitation functions in many light nuclear systems have been characterized by marked energy dependence. This energy dependence has spanned a wide range of characteristic widths; on the one hard broad structure attributable to potential scattering has been observed and on the other much narrower structure suggestive of statistical fluctuation phenomena. In systems such as 1^0+1^0, 1^0+12C and ^2C+^2C structure of inter­mediate width appears which does not fall obviously into either of these categories and a number of novel mechan­isms have been suggested for its explanation.

This dissertation reports ©n a complete examination of this situation in the "^C+^C and ^0+^2C systems — both experimental and theoretical. Existing elastic scattering and newly measured reaction data have been sub­jected to a consistent statistical examination. Synthetic excitation techniques have been evolved to determine em­pirical corrections to the parameters extracted from such analyses to reflect the presence of non-statistical, e.g. potential scattering phenomena. Using these correc­tions, entirely consistent values of average widths, etc., have been obtained for all channels studied.

12 12In the C+ C system it has been found possible to encompass all the non-potential structure within the frame­work of the statistical models without requiring recourse to any of the more specialized mechanisms. This has also

-ip -| Cbeen the case in the C+ 0 system but with the strikingexception of sharp resonant structure which appears to

2 8correspond to a state at 36.5 MeV in Si having J=l4 and P=il00 keV. Considerable effort has been devoted to elucidating the structure of the state since it appears to open up an entirely new class of nuclear spectroscopy— both experimental and theoretical.

ACKNOWLEDGEMENTS

It is a pleasure to acknowledge the help and support of many people at the Wright Nuclear Structure Laboratory which made the execution of the work reported herein a more easy and pleasant task.

There are two people without whose help this work would have been impossible to complete. I am very grate­ful to my thesis advisor Professor D. A. Bromley who provided continuous guidance and cheerful encouragement throughout my research here and Professor R. G. Stokstad whose close support in the form of critical discussions and numerous work days devoted to this project have also been essential to its success.

I wish to thank all the people who have been involved in the acquisition and analysis of the data reported in this thesis. Among these are the many past and present members of the Yale heavy ion group, L. Chua, K. A. Erb,A. Gobbi, W. Reilly, M. W. Sachs, R. Wieland, C. Olmer and D. Hanson. The essential help provided by Dr. W. D. Callendar during the acquisition and analysis of the spectrograph data is gratefull acknowledged.

Special thanks go to M. W. Sachs who often served as

a patient mediator between the whims of the mindless com­puter and my baffled response.

Last but not least, I wish to thank all the members of the Wright Nuclear Structure Laboratory community, sci­entists, secretarial and professional staff for providing much and frequently needed help and support and for main­taining a stimulating and creative atmosphere which I thoroughly enjoyed and from which I very much benefitted.

Finally, appreciation is extended to Mrs. C. McMillen for fast and reliable typing of the thesis.

OUTLINE

I. Introduction1-1 General Introduction 1-2 Heavy Ion Elastic Scattering1-3 Heavy Ion Reactions1-4 Scope of Study

II. The Experimental Method II-l Introduction II-2 Data Acquisition Procedure II-3 Data Analysis Procedure

III. Method of Theoretical AnalysisIII-l IntroductionIII-2 Fluctuation AnalysisIII-3 Ericson's Model for Cross Section

FluctuationsIII-4 Finite Range of Data and Finite

Resolution EffectsIII-5 The Method of Synthetic Excitation

Functions.IV. The 12C+l2C System

IV-1 IntroductionFlue Data

IV-2 Fluctuation Analysis of 12C+^2C Elastic

IV-3 12C(12C,a)20Ne Data - Experimental Results and Fluctuation Analysis

IV-4 Comparison of Statistical Model to DataIV-5 Intermediate Structure - Its Possible

Coexistence With the Fluctuations in the Data

IV-6 Discussion - Validity of Fluctuation Analysis and Statistical Model Calcu­lations

V. The 12C+l60 SystemV-l The 12C(l60,a)2S/Ig Reaction and 12C+l60

Scattering - Experimental Results and Fluctuation Analysis

V-2 The Statistical Model - Comparison to Data

VI. Studies On the Resonance at E __.=19.7 MeV in theTo *17’ ' ' " "" " ' CulCt 0 SystemVI-1 IntroductionVI-2 Further Measurement on ResonanceVI- 3 The Structure and Formation Mechanism

of the ResonanceVI-4 Conclusion

VII. EpilogueVII-1 ExtensionsVII-2 Summary of Results and Conclusions

TABLE OF FIGURES

12 12Fig. 1-1 C+ C excitation curves

Fig. 1-2 ^ C + ^ C , ^ C + ^ O and ^ N + ^ N excitation curves

Fig. 1-3 Quasimolecular potential and double resonance.12 12Fig. 1-4 Double resonance mechanism predictions for C+ C

F ig .n - l Layout of scattering chamber

Fig. II-2 Detailed view of collimator+absorber+detector system

Fig. H-3 Diagram of data handling system

Fig. n-4 Typical spectra obtained at 2°16 12Fig. II—5 Absorber pressures needed for O and C

12Fig. H-6 Typical C recoil spectra

Fig. n - 7 Data analysis program

Fig. II-8 Monitor counter spectra

Fig. m-1 The Auto-correlation function12 12

Fig. IV -1 C+ C elastic excitation functionso 12 12

Fig. IV -2 90 C+ C elastic excitation function

Fig.IV -3 Dependence of average of normalized variance on aver-

-aging interval

Fig. IV -4 Distributions for R(P)°^S and widths on averaging interval

Fig.IV -5 Dependence of R(0)°^S and extracted widths on averaging

interval

Fig.IV-6 Same as fig. IV-5 but with gross structure included

Fig.IV-7 The Autocorrelation function of the reduced data24

Fig. IV-8 Dependence of width on excitation energy in Mg.

Fig. IV-9 Typical alpha particle spectrum12 12 20

Fig.IV-10 C( C ,a ) Ne excitation functions12 12

F ig ,IV - ll Partial cross sections for C+ C reactions

Fig.TV-12 Dependence of widths on excitation energy and mass

numbers

F ig .IV -13 Cross correlations for elastic data

Fig. IV -14 Cross correlations for reaction data

Fig.IV-16 Average compound cross sections^12 12

and Hauser Feshbach predictions for: C+ C elastic12 12

Fig.IV-17 same For: C+ C inelastic

Fig.IV-18 same for : Alpha particle Channels

Fig.IV-19 Results from analysis of synthetic excitation functions

with intermediate structure12 12

Fig.IV-20 Double resonance mechanism predictions for C+ C

inelastic scattering 16 16

Fig.IV-21 0+ O elastic excitation functions12 12

Fig.IV-22 Partial cross sections for C+ C.

Fig. IV-23 Effects of width to spacing ration on R(0 )°^S/ R(0)

Fig.IV-24 Effects of the number of open channels on the fluctuation

analysis 12 16

Fig. V - l C+ O elastic excitation functions12 16

Fig. V -2 Alpha particle spectra from C+ O reactions12 16

Fig. V-3 Cluster transfer in C+ O collisions19 1 94

Fig. V-5,6,7 C( 0 , a ) Mg* excitation functions

Fig. V-8 Comparison.with Hauser Feshbach predictions12 16F ig .V I-l C - O elastic angular distributions12 16Fig. VI-2 C+ O elastic and inelastic excitation functions

Fig. VI-3 Same as Fig. VI-212

Fig. VI-4 C recoil spectra on magnetic spectrograph plate.

Fig. VI-5 Detail of spectrum shown in fig. VI-4

Fig. VI-6 Expected angular distribution

Fig. VI-7 Setup for Gamma ray experiment12 16

Fig.VI-8 Gamma ray spectrum from C+ O

Fig. VI-9 6.1 MeV gamma ray yields

Fig. VI-10 Fit of gaussians to gamma ray spectrum

Fig. VI-11 Gamma ray excitation functions

Fig. VI-12 *2C (160 ,p )27Al excitation functions

Fig. VI-13 12C (160 ,p )27Al spectra12 16 27

Fig. V I -14 C( 0,p) A1 excitation functions12 1 fi

Fig. VI-15 Phase Shift analysis on the C+ O data1 fi 1 2 7

Fig. V I -16 Comparison of ( O, C) and( Li,t) spectra16 12

Fig. VI-17 0+ C elastic scattering and alpha transfer12

Fig.VI-18 C+a cluster wave function

Fig. V I -19 The exchange potential 12 16

Fig. VI-20 C+ O elastic excitation functions and optical

model predictions 12 16

Fig. VI-21 C+ O quasimolecular potential and the exchange

potential.

Fig. VI-22 Grazing pratial waves and anomalies observed in12 16

the C+ O system

TABLE OF TABLES

Table I

Table H

Table m

Table IV

Table V

Table VI

Table VE

Table Vm

Table IX

12 12C+ C elastic fluctuation analysis of:

12 12 20C( C, ) Ne fluctuation analysis of:

24Mg compound nucleus

level densities and potential parameters

Fluctuation analysis of theoretical predictions12 16 24

Fluctuation analysis of C( O, ) Mg data16 12 24

Fluctuation analysis of 0 ( C , ) Mg data28

Si compound nucleus

level densities and potential parameters

The resonance at 19.7 M e v Observations

The resonance at 19.7 M eV . . . . Partial width analysis

1

CHAPTER I

1-1 General IntroductionThe study of heavy-ion induced reactions has become

an important subfield of nuclear physics. Obviously the availability of high quality beams with well defined and variable energy spanning many ionic species is one reason for such increased activity in this field. A more fundamental reason, however, lies in the new scien­tific horizons opened by the use of heavy ions.

In a collision of heavy ions large linear and angular momenta are involved, and a large electric charge is brought close to the target. These unique features of the heavy ion collision permit the study of new physical phen­omena including the behaviour of quantum electrodynamics when eZ a-1 ( a is the fine structure constant 1/137), andthe properties of nuclear matter undergoing deformation and compression. New nuclear species can be formed in such collisions as a result of amalgamation or of many particle transfer. Entirely new regions of nuclear ex­citation can be studied which sre characterized by high angular momentum or which may have exotic configurations selectively populated by the transfer of nuclear clusters.

Curiosity, however, when satisfied exacts its price j A Pandora's box of problems is opened. How muchcan we. understand of the many interactions which may take place when two complex systems, which are themselves not well understood, collide?

Heavy ion collisions leading to systems in the 3-d shell at least have the advantage of much support­ing information already available on these systems.

This thesis is part of a broad program of study at the Wright Nuclear Structure Laboratory, undertaken to investigate the nature of heavy-ion interaction mechan­isms. A study of the elastic scattering was undertaken first (Ma 6 9,Re ?2) since an understandir^ of it is basic to any understanding of nuclear reactions. Further inter­esting features of heavy ion elastic scattering, our understanding of them and the problems posed will be discussed briefly in the following section.1-2 Heavy-ion Elastic Scattering

One of the most striking discoveries made in the early studies of heavy ion scattering was the observation of elastic cross sections which varied rapidly with energy(Br 60). This structure in the excitation functions,

12 1?which was particularly marked in systems such as C+ ~C, 12c+l6o an(j 16q+16q ^ag keen subject of continued

study since its discovery some 14 years ago. (St 73fSt 72) Even in the earlier elastic scattering measurements

several distinct classes of structure were identified.

12 12In the ~C+ C system, for example, it was found that at energies near and below the Coulomb barrier certain pro­nounced resonances were observed; these also appeared in all the reaction exit channels (A1 60)(see Fig. 1).They were first thought to reflect the formation of IP 12“C- C quasi-molecular configurations because of a large reduced width associated with the resonances ob­served in the elastic channel (A1 60,A1 63,Vo 60,Da 60,Im 68). Subsequent measurements at lower energies on thereaction channels revealed the existence of additional

12 12resonances in the C+ C system (Pa 69, Ma 73). Michaud and Vogt (Mi 72) then suggested that additional degreesof freedom, associated with the formation of cluster-like12 16C+3of and 0+2a "doorway'' configurations, were re­quired in order to explain the presence of these addition al resonances as well as other experimental features of ■^C+^C induced reactions.

Measurements at energies above the Coulomb barrier revealed fluctuating structure although no cross cor­relations between different reaction channels such asthose seen at lower energies were apparent. The struc-

12 12 20ture in the C( C,oi ) Ne reaction exhibits a character istic width of T « lOOkeV and was shown to be consistent with Ericson fluctuations arising from formation of

2 ixstrongly overlapping compound states in Mg (Bo 65,Bo 65a,Vo 64). Statistical analyses of the and^ 0 + ^ 0 elastic scattering (Bo 73) at bombarding energies

Fig. I - 1 Excitation functions for proton, neutron, alpha

12 12and gamma yields from the C+ C system

( Al-60)

CENTER OF MASS ENERGY IN M«v

just above the Coulomb barrier also supported the earlier conclusions that compound nucleus formation is an impor­tant, and probably the dominant, reaction mechanismproducing the fluctuating cross sections, Studies on the

1 1 fia -particle and elastic exit channels in the ' 0+ 0system resulted in similar conclusions (Sh 69). Recently,however, non-statistical behavior has been observed for

12 12 20the cross sections of the C( C, a ) Ne reaction lead-20ing to certain states in Ne (Mi 71).

With the availability of the first MP tandem acceler­ator, these studies were extended to several times the Coulomb barrier energy (Ma 69,Si 6?,Re 73). It immediate­ly became clear that yet a third type of structure of typical width in the range from 2 to 4 MeV was present. Detailed studies have shown that this gross structure corresponds to potential or shape elastic scattering and that it is reasonably well reproduced by appropriateoptical models (Re 73a,Go 73). Superposed on this broad

12 12structure, particularly in the cases of C+ C and12C+1 0, to a lesser extent in ^ 0 + ^ 0 and not at all

14 14for N+ N, one finds additional structure having char­acteristic widths significantly less than 1 MeV. (See Fig. 1-2)

The origin of this structure at the higher energies has remained an important and open question. Greiner and collaborators (Sc 70,Fi 72) have proposed a double resonance mechanism wherein one or both of the interacting

Fig. I - 2 90 deg. excitation functions for the systems

160+160, 12C+12Cand 14N+14N ( Re-73a)

d<r/d

ft (m

b/sr

)

Ecy(MeV)

nuclei are temporarily excited to one of their quantum states during the interaction. If the conditions for an orbiting resonance exist in the entrance channel (we have thevirtual resonance condition shown in Fig. T.-3a) and ifthe excitation of the nuclei (E*) drops the system intoa quasibound state of the interaction potential, such aprocess is expected to be enhanced. While the orbitingresonance produces the enhancement seen in the grossstructure, the coupling of the spin of the excited stateto the orbiting angular momentum produces the observedsplitting leading to the narrower structure. Fig. I-3bshows the predictions of this model calculated by Scheidet. al. (Sc 70) for the ^ 0 + ^ 0 system. Similar calcula-

IP 1 Ptions made for the 'C+ ~C system are shown in Fig. 1-4.It may be seen that this model does not account adequately for the narrow structure in the data. While one could possibly improve this calculation by coupling more excited states to the elastic channel it is also possible that the mechanisms advanced to explain the variation of the cross sections at lower energies, viz, alpha-cluster doorway states and statistical fluctuations also play a role in determining the structure observed at higher energies.1-3 Reaction of ][eavy Ions

The earlier studies of heavy ion reactions made on the systems discussed in the previous section, viz. the12'C(12C,a)20Ne, l60(l60, a )28Si and 12C(l60 , a )2%lg reac-

Fig. I - 3

16 16mechanism illustrated for the 0+ O system.

Quasimolecular potential and the double resonance

CROS

S SE

CTIO

N

cm (MeV)

POTE

NTI

AL(

MeV

)

. £=9 £=12(b) * *

2 0 -

15-

10 -

5 -

. VIRTUAL STATEm m \ \ \ \ \ \ e in

QUASI­BOUNDSTATE

^160+160 Elastic

6 8 10 12 14Rtf)

Fig. 1 - 4 Double resonance mechanism predictions for

12 12C+ C elastic excitation functions (solid line)

and elastic 1^C+1^C data .

c.m. ENERGY

[MeV]

CROSS SECTION [m b/sr]ELASTIC

CROSS SECTION

tions (Ba 65,Bo 65a,Vo 64) concentrated on the reaction channels leading to low lying states in the excited resi­dual nucleus and the dominant mechanism in these reactions was shown to be compoun nuclear, in accordance with subse­quent attribution of the structure observed in the elastic channel to the same mechanism (Bo 73). More recently studies were initiated on the same reaction channels leadingtrto higher excitation in the residual nucleus and with higher incident energies (Gr ?2,Mi 71a,Er 71). A sur­prisingly high selectivity in the population of levels in the residual nuclei was observed. Only a selected few states were strongly excited in a region of excitation energy where a dense continuum of states is known to exist. Subsequently it emerged that there is some connection between the presence of narrow structure in the elasticchannel and this remarkable selectivity. Absence of

14 14narrow structure in the elastic channel for the N+ N system, was correlated with the absence of such selectivity in the particle spectra for the reaction channels (Wi 73).

•j Z "IpThis selectivity in 0+ C reactions might be a result of heavy cluster transfer (e.g. involving alpha-clusters) or some other exotic intermediate process. Or it might be explained by a compound nuclear mechanism.

This interesting problem:has received considerable attention and a study of the nature of these selectively populated states was initiated in this laboratory and in others around the world (Fi 73»Go 71).

1-4 Scope of the Present StudyThe present study is concerned with the reaction

mechanism responsible for the phenomena we have just dis­cussed. The role of the statistical compound reaction mech anism in producing these effects is examined in particular

The experimental methods used for data acquisition and analysis are described in the next chapter. In Chap­ter III the theoretical approach and methods used are discussed. The use of excitation functions synthesized with the computer in this study and the rationale for using them are also discussed.

Our study then proceeds in three parts. We first12 12investigate the C+ C system where the non potential

structure in which we are interested is most pronounced in the elastic channel. For that purpose we have sup­plemented a large body of existing data (Re 73) with

12 12 20new measurements on the C( C, or ) Ne reaction. Itbecame evident from this study that a statistical com­pound reaction mechanism can account for all the struc­ture which is not attributable to potential scattering; furthermore it can reproduce the selectivity observed in some of the reaction channels.

TP TWe then turn to the C+' 0 channel wherein the12 12structure of interest is most marked save in the C+ C

system. We show that similar features observed in this system could stem from the same reaction mechanism and, thereby that the validity of the statistical compound

12 12reaction mechanism is not unique to the C+ C system.4 p 4 /In our study of the C+ 0 system, however, an un­

usually strong cross correlation between several outgoing channels was found, at an incident energy of 19.7 MeV in the center of mass. The narrow (non-potential) structure in the excitation functions for different channels showed behaviour very similar to that expected of a normal reson­ance. This unusual behaviour of the cross sections indi­cative of the decay of a single sharp level high in the

28continuum region in Si was entirely surprising and clearly merited more detailed study. Further measurements and theoretical studies were undertaken with the hope ofunderstanding the structure of this level and the mechan-

\ism by which it is populatedi these are described in Chapter VI, We believe that such sharp structure in the continuum may open an entirely new area, both experimental and theoretical, in nuclear spectroscopy.

It would clearly have been of great interest to extend our studies to the *^0+^0 systemi on the basis of all the evidence yet in hand (3c 70) it is not ruled out that this system may show significantly different behaviour and that additional non statistical mechanisms other than potential scattering (e.g. double resonance model pheno­mena) may be required to reproduce the observed data. Such an extension has been precluded by the unavailability of a body of data of comparable extent and quality to that available for the other systems. Measurement of these

■^0+^0 data, as would be necessary for such extension*would have required accelerator time far beyond thatavailable for the work reported herein.

In summary we have demonstrated that the structure 12 12m the C+ C system can be understood within the

framework of a statistical compound nuclear reaction model; we have also put the utilization of such models on a sounder basis through our synthetic excitation function studies and have demonstrated that proper statistical sampling of the data yields consistent sets of model parameters when applied to different reaction channels.

As a demonstration that this model's success is12 12not restricted to the C+ C system we have shown that

"L 6 12it has similar success in the " 0+ C system. In thecourse of our experimental studies on this latter systemwe have discovered an entirely new phenomenon - a sharp

28(400keV) state near 40 MeV of excitation in Si, This new phenomenon has been subjected to detailed study and various hypotheses concerning the structure and popula­tion of the very highly excited states have been developed.

CHAPTER 2

II-l IntroductionThe experimental methods employed in the different

studies are outlined in sect. II-2, A description of the data analysis follows in sect. II-3.

Extensive use has been made of the modern IBM 3^0/44 computer system at the laboratory. The use of the key­board and light pen facilities at data analysis time provided for a high degree of user interaction with the analysis program while at the same time maintaining a high level of automation and speed so essential in the reduction of the large amount of experimental data ac­cumulated and the extensive statistical calculations required,

II-2 Experimental ProcedureThe present study focuses on the variation of heavy

ion cross sections with incident energy. Correspondingly, most of the measurements undertaken involved the accumula­tion of excitation functions of the same interacting system leading to many outgoing channels. The typical period, or width, of the structure being studied is several hundred keV, hence variability, reproducibility and definition (resolution) of the beam energy to better than 100 keV are

essential.

The Yale MP Tandem Van-do-'Trnn r nccnlnrntnr is nor-:t suitable for such studies, providing the experimenter withmonoenergetic beams whose energy can be varied easily and• 12 1 (5with precision. The C and 0 beams used here were ex­

tracted as negative ions or compounds from a Penning type source (He 69). They were then accelerated toward a posi­tive higjivoltage terminal where they were stripped oftheir electrons by allowing them to pass through a thin 12C foil or a gas stripper canal and were then further accelerated back to ground potential. Final momentum selection was accomplished with a 90° analyzing magnet. Thin targets with 10-30 t ig /cm - typical thickness were used in the experiments in order to keep the energy losses in the target to a v?l ue less than the characteristic widths

of the structure being investigated. The resulting experi­mental energy resolution in the center of mass system was better than 50 keV in most experiments.

The physical layout of the scattering chamber is shown in Fig. IT.-l, All the detectors used were Si surface barrier detectors. AE and E denote a thin transmission mounted and a thick (5mm) Si(Li) detector, respectively. Reaction products were detected and identified at extreme forward angles. This angle of observation had the advan­tages of simplifying the fluctuation analysis of the data, to be described later, and also enabled us to perform a sensitive check of fore-aft reaction asymmetry by a simple

Fig. I I - Layout of scattering chamber used in small

angle measurements of excitation functions.

\

SCATTERING CHAMBER LAYOUT

RECOIL COUNTER

MONITOR COUNTER

exchange of beam and target. Fig. II-2 is a more detaileddiagram of the collimator, absorber and detector system.

The beam is carefully focussed and collimated beforeit is scattered from the target. A tantalum beam stop 1.5mmthick and 2.5mm in diameter was placed before the entrance

12window of the absorber, preventing C buildup which might otherwise occur on this window and the entry of direct beam into the gas cell which would cause flooding of the counter with recoiling protons.. The projectiles scattered at for­ward angles of 2°, were prevented from reaching the de­tector by this absorber while the lighter products were transmitted. These were identified with the E-E tele­scope system and their summed spectra recorded.

A rough monitoring of the beam was provided by measure­ment of the beam intercepted by the tantalum beam stop.More reliable monitoring, however, was provided by record­ing the number of projectiles scattered by a thin gold layer deposited on the target into a monitor counter at an angle - ^ (Fig. TI-1). The recoil counter shown atposition © was used for coincidence gating of scattering

17 12by C which allowed us to monitor the "C content in the12target and thereby check for possible 'C buildup. Such

2buildup was kept below 20$ for runs with a thin 20^g/cm 12C target, by placing a liquid-Nitrogen cooled metal plate near the target.

Since the experimental program was aimed also at un­derstanding the reaction mechanism which selectively popu­lates high excited states in the residual nucleus, good

Fig. n - 2 Detailed view fo collimation + absorber +

detector system.

50— I100 mm

SC A L E

resolution for the outgoing alpha-particles was desirable. The contribution to resolution made by the detector and pulso handling system was approximately 70 keV, the target thickness contributed a few tens keV, but the main contribU' ti.on,~250 keV, was caused by straggling of the a-particles in the absorber when Ni foils were used.

The amount of straggling in an absorber is given by (Co 6 6 ) t

where z = projectiles' average chargeZ = target's charge (absorber),A = absorber's mass numberX - absorber's thickness in mg/cm'N^,y = Avogadro's number

n - la) Os: 29.6 z [ ? £X(mg/cm2)

An efficient absorber, therefore, must have the character­istics that only a small amount of material ( x) will be needed to stop the beam, and thus the straggling will be re­duced. We have used hydrogen gas for the stopping material as it is the most effecient material available for such use. Thedetectors were mounted inside the pressurized hydrogen gascell and a thin 2mg/cm" Havar entrance window was used to admit the particles and contain the gas. The overall en­ergy resolution was thereby improved to better than 150 keV.

The logic of the signal processing system is shown in Fig. II-3. Summed E + AE pulses were fed into different analyzers. The particle identifier pulses were used for routing and sorting the different particle spectra into either different analyzers or different memory subgroups of a single analyzer. Fig. II-4 presents two typical spectra obtained with a 1mm thick E detector and a 5mm Si(Li) detector (E). Part (a) shows a singles AE spectrum and part (b) shows a summed (E + AE) proton spectrum gated by the AExE particle identifier, The low energy cutoff in the proton spectrum corresponds to those protons which were stopped in the AE counter and were therefore not recorded as coincidences. After correction for the energy lost in the absorber, this threshold corresponds to protons of 12 MeV at the target. The proton yield is about 1% or less of the total, particle yield. Since, after alpha particles, the protons are the most intense light charged particle particle group observed in these experiments (^2C+^2C and ^2C+^0), it is safe to assume that the singles spectra shown in Fig. II-4a contain mainly a- particles, especially if channels larger than ~ 350 are considered. Consequently singles measurements were made whenever a-particle groups were studied. The hydrogen absorber pressure was always kept at the minimum levelneeded to stop the beam particles. Fig, II-5 shows a

1_ 2 16graph giving the gas pressure needed to stop C and 0 projectiles as a function of their energy. The Hydrogen

Fig. II - 3 Diagram of data handling system

AMP ■ DELAY-

-•PREANP-TS.CA.-

AE-»|PREAMP

AMP

AMP

IV

PI.(E*AE) AMP S.CA.

AMP

TS.CA -

DELAY

MONITORPREAMP

RECOIL(PREAMP

AMP DELAT

TS.CA.

LINEARGATE

TAG.

AMP * DELAY

TS.CA ILMEARGATE

GATEa

DELAY *

TO ANALYZER

SUB GROUP 2

SUB GROUP 3

SUB GROUP 4

Fig. II - 4 Typical spectra obtained at 2 deg. from

1 2 _ 12_C+ C reactions,

COUN

TS

CHANNEL NUMBER

Fig. n -

projectiles given as a function of particle's

energy.

Absorber pressures needed to stop C and O

I

PRES

SURE

DI

ST.

(Atm

.-cm

)

ENERGY (MeV)

PRES

SURE

(A

tm)

15

absorber thickness needed, in excess of the 2mg/cm" Havar window materia.! , was calculated using stopping power data from tables of Northcliffe & Schilling (No 70) and using the ideal gas law the hydrogen pressure needed to pro­duce the appropriate absorber thickness over a path length of 115mm (see Fig, II-2) was calculated.

The ability to vary the absorber thickness (by changing the pressure) without breaking the target chamber vacuum was especially valuable for the ^2C ("^0, "^0 )^2C elasticand inelastic scattering measurements. The absorber pres-

12sure was lowered sufficiently to allow C recoiling nuclei through but prevented the passage of ^ 0 projectiles scat­tered to forward angles. A thin AE detector was used

12which completely stopped the C ions while the alpha particles and protons deposited very little energy. In this way cross sections for elastic and inelastic scatter­ing were measured at a center of mass angle of ~1?7° .

12Fig, II-6 shows a typical C recoil spectrum.

II-3 Analysis and Reduction of Experimental DataIn a typical experiment particle spectra were accumula­

ted at approximately one hundred different incident energies and each spectrum contained many peaks representing transi­tions to discrete final states. At each energy an associated monitor spectrum was also recorded for normalization purposes. Since a large number of such experiments were involved an interactive analysis program was written to analyze these

2

12Fig. II - 6 Typical C recoil spectrum obtained at 2 deg.

COUN

TS

200 300 400CHANNEL NUMBER

data making use of the function keyboard, display and light pen facilities of the IBP.'! 3^0/44 computer system at the laboratory. A schematic outline of the program operation is shown in Fig. II-7, Spectra, previously recorded on tape are read in and displayed. Using information on the calibration of the analyzer, the absorber thicknesses in­volved, the beam energy and reaction involved, expected positions of particle groups populating known levels in the residual nucleus are indicated by display markers super­posed upon the displayed spectra. In this way spectral regions of interest can be quickly and reliably identified. The peaks of interest are then selected by the light pen (up to six at a time) and their areas and exact positionsare extracted by fitting Gaussians riding on a linear or

IP X6quadratic background. An example from the “C+ 0 systemis shown in the inset on Fig. II-7. All the extracted peak positions and areas are stored for each spectrum - the yield under the peak in the monitor spectrum correspond­ing to scattering from the thin gold target layer is re­corded last and is used to obtain a relative normalization for all the peaks in the spectra. A program is then called by the operator which transforms the channel position and area information into excitation energy and absolute cross sections respectively. This final information, together with proper identification of the group is stored on tape and printed out for each spectrum. At each point in the analysis a summary excitation function can be obtained for

Fig. n - 7 Schematic diagram of data analysis program

operation.

T A P E -I

O D

opERAT0R

SPECTRUM I.O.

(R U N * ANAL.)

INPUT INFO

CALIBRATION

BEAM ENERGY

ABSORBERS

FITTIN G ROUTINE

STORAGE FOR

FITTING ROUTINE

UP TO 1 0 0 PEAK S

PE R S PE C T

CALIBRATION S

NORMALIZATION

(A B SO L . C R O SS SECTIONS)

P R IN T E R

SEGMENT OF Q PARTICLE SPECTRUM

INFO. F L O W -O N E WAY

INFO. FLOW IN BOTH

DIRECTIONS

any state by skimming through all the spectra already analyzed and recorded on the storage tape.

Absolute Normalization!As already mentioned relative normalization was

achieved using the yield of beam particles scattered into the monitor counter by a thin gold layer which was evapo­rated onto the target. At the energies used here, the

1 f\ IP'scattering of the 0 or C projectiles from gold is pure Rutherford in nature,

The reaction and Rutherford yield are given by H - 2 ) a) -- - ------

v w S ? d 0r

b)

YR - NAuVd-fi> %Jtv

Yr,Y^ - are yields - number of particlesare the solid angles of the 0° and Monitor detectors, respectively

= # of target nuclei per unit target areaN^u = # of gold nuclei per unit target areaN, = # of beam particles incident on target

The reaction cross section is then given by:

II-3)/ d a ) = / \ / d_c_)

3 f?r YR ( N t d0r d p R

the only unknown quantity which cannot be calculated is N^u/N .. This ratio can, however, be easily found by per­forming a scattering experiment using the same target material and very low beam energies such that both scatter­ing by~the~ target atoms and by the gold atoms will be pure Rutherford. All the cross sections involved are thus pre­viously known or measured quantities. Using the measured yield? one can then obtain the ratio The monitorspectra can be quite complicated sometimes as shown in Fig. II-8 for a SiO target. The coincidence counter is then used to identify the peaks in the spectrum. The absolute cross sections deduced are accurate to within±20$ in general, and most of the uncertainty arises from 12C buildup on the target during the experiment,

Energy CalibrationsPositions of groups populating known states in the

spectra were identified by means of comparison with pre­viously measured spectra e.g. magnetic spectrograph data of Middleton et. al. (Mi 71a). The exact positions of several known states was fed to the program together with kinematic and absorber data and were then used to produce a linear energy calibration. Gain shifts in the analog

Fig. I I - Monitor counter spectrum with and without

recoil coincidence gating.

CO

UN

TS

CHANNEL NUMBER

pulse handling system were detected and corrected for by monitoring the position of one or two reference peaks generated by a pulser. The accuracy of the calibration used is limited by our ability to determine accurately the centroids of the peaks used in the calibration, Ey using strong groups corresponding to the excitation of known single states in the residual nucleus, the calibra­tions obtained were accurate to ~20 keV, The deter­mination of excitation energy for the alpha particle groups identified was limited to a ~70 keV accuracy mainly by the available resolution in the alpha particle spectrum.

CHAPTER 3 Method of Theoretical Analysis

III-l Introduction <This study is concerned with the variation of the

IP \ 2 16"0+ C and C+ 0 cross sections as a function of bom­barding energy. The investigation is aimed at finding out which one or ones of several existing interaction models can best explain this variation.

One possible approach is to calculate the reaction or elastic cross section using a certain model and then see if the observed energy dependence can be reproduced. An improved fit to the data can sometimes be achieved by adjusting parameters within these models. An example ofsuch theoretical analyses is the treatment of the correlated

12_ 12_ . 12_ ,16- . , resonances appearing m the C+ C and. C+ 0 systems atand below the coulomb barrier (A1 60,A1 63»Da 60fIm 68,Mi 72,Vo 6 0). The gross structure appearing in heavy *ionscattering is well reproduced by potential scattering oroptical models, A similar attempt, made for the narrowstructure appearing at energies high above the barrier,would seem to represent an impossible task; this structureis abundant, has no apparent regularities, and an unwieldy

number of adjustable parameters would be required for a detailed description.

An alternative approach is to compare the model- based predictions for the average characteristics, e.g. strength and width, of the structure, with the average width and strength of the structure deduced from the measur­ed cross sections. This method is usually applied in con­junction with the statistical model of compound nuclear reactions but can also be applied, at least in principle, in other cases (Ja 73). The latter method is applied throughout most of the present study. For such studies a large enough body of data must be collected so that the average characteristics of the structure can be obtained with some degree of confidence.

III-2 Fluctuation AnalysisThe mathematical tool used in studying the average

features of the cross section's energy dependence is the autocorrelation function which is defined by*

I I I - l ____________________________________

This is a mathematical entity related to the correlation coefficient - which for two random variables X and Y is defined as (Ea 72)

22

IIT-2.(X -u ) (Y - u )

» < X ' Y > ° g g yx y

Ux = X Expectation value of X

and: a =(X- 4V )2 Variance of XX X°y =<*- V 2

This correlation function takes on values between -1 and 1 and vanishes when X and Y are independent variables.

The meaning and use of the autocorrelation function is obvious. Fig. III-l illustrates this for two extreme cases. The energy displacement K (case 1) shown is a typical width measure, for E- -Eg < ^ A(E^) and A (Eg) are not likelyto differ in sign, but for larger displacements A(E^) and A(Eg) become two independent random variables. Case 2

shows the dependence of R ( c ) on obtained if a (E) were a periodic function. Needless to say such extremes as case 1 and case 2 are not found in measured cross sections. Furthermore, the absolute vanishing of R(0 obtained from randomly varying cross sections can be obtained only if true expectation values can be calculated. For that end we would need, however, an infinite sample of data!

Usually the uncertainty involved, or allowed, in calculating the correlation of two random variables, using only a finite sample, will depend on the parent distribu­tion to which the variables X and Y belong. In the case of A(E^) and A(Eg) (cf. Fig, III-l) it will be the same

Fig. DI - 1 The auto-correlation function - illustrated.

CASE IA(E|) independent of A(E2)

if |Er E2|>k

R(0)=Average normalized variance

€

(b)

R(e)

(C)

R(e)CASE 2 A(E) - Periodic

23

distribution.The exact shape of R(c) for e^K and the uncertain­

ties that must bn allowed when determining R(0 from a finite sample of data depend then on the statistical properties of nuclear reaction cross sections ,These properties, in turn, can be determined only within the framework of a model for the reaction.

Such a model was first introduced by Ericson (Er 6 3,Er 6 5) and is based on the same assumptions of randomness for scattering amplitudes as those made in the statistical model of Hauser and Feshbach (Ha 52).

Ericson's theory for fluctuations in nuclear cross sections will be discussed briefly in the next section.The problems in extending these results to the case of heavy ion reactions and the ensuing limitations will also be described.

III-3 Ericson*s Model for Cross Section FluctuationsThe form of R(e) has been specified by Ericson (Er 6 3)

within the framework of a compound nuclear reaction model which describes the fluctuations arising in the measured cross sections. In particular he has shown that

p2 III-3 aR(c) = -x t R(°)

r + c2

! - y d2 1 1 1 -3 bR(°) = — ---

eff

and

I H i ) ,III-4

P(x) = r~r exp 1 y .Id i - yt i

where R(0) is the average normalized variance, T is the average width of levels populated in the compound nucleus, and is the ratio of the average direct component of the reaction cross section to the average of the measured, cross section. Nepf is the effective number of independent chan­nels contributing to the observed cross section and is equal to or less than the number of different positive spin projections, Eq. (4) gives the distribution of the fluc­tuating cross sections, x=a/a, for the case of I here denotes the modified Bessel function of zero order.

Ericson's model and its assumptions may be summarized as follows: The scattering matrix which is generally usedto describe transitions from an initial state a with total angular momentum J to a final state o/' and J can be cast in the form

o

III-5

where it is assumed thati) 5*^, - is approximately energy independent, such

that the changed in over an energy range ofthe order of many can be neglected.

25

ii) The resonances in the compound system overlapstrongly, i.e. T /DJ » 1 where isthe average level width in the compound system and J JD = ^r' " -e average spacing betweenlevels of the same spin and parity,

iii) The quantities g ^ are in general complex con­stants whose phases vary randomly with respect to the channel a and the level index X .

iv) The distribution of the level widths has a small dispersion, i.e. ^

According to Ericson the relation r^= T T ? remainsA a Aavalid in the region of overlapping levels and the partialwidths I\J have the same distribution as that observed Aafor isolated neutron resonances. Assumption (iv) thenholds for the case of many open channels. Furthermore, for

flthe case of strong fluctuations, i.e. if < a > is comparable to the non-fluctuating component of a , the complex quanti­ties g*^ may be identified with the partial width amplitudes y ^ and S^a, becomes , the matrix element for director potential scattering,

Moldauer (Mo 64) studied the fluctuations in nuclear reaction cross sections for arbitrary and D^. He con­firmed Ericson's results in the limit of large r/D and evaluated corrections to Eqs. III-3 a and b which were shown to arise from partial width fluctuations and reson- ance-resonance interference phenomena. Similar corrections also appear in the formulae for the average cross sections

and will be discussed later.

III-4 Effects of Finite Data Sample and Finite Experi­mental Energy Resolution

Since we deal with random variables it is clear that analyzing an excitation function over a finite energy range will almost certainly not yield R(e)=0 for large e even if the fluctuations are random. As noted in sect. III-l knowledge of the statistical properties of a (E)(or A(e )), however, will enable us to estimate the uncer­tainty that must be involved when analyzing a finite random sample.

These finite-range-of-data (F.R.D.) effects were discussed by several authors (Ha 6 7,Sh 6 9,Vo 64,Wo 6 6),An analytic expression for the shift and uncertainty in the value of R(0) obtained from finite samples was derived by Dallimore and Hall (Da 66), Their procedure will be followed here but will be generalized to the case where a non fluctuating component is also, and simultaneously, present in the data (y^O).a (E) will be defined to be the true average, or the expectation value of a(E). <a(E)> will then be definedto denote an experimental local average. Otherwise the notation of sections III 2 and 3 is retained.

It is clear that, in general, «j(E)> 4 a (E) and that the two operations a(E) and <®(E)> do not commute. If unweighted averages are used, as is the case here, these two operations obviously commute. Also o(E) is a

27

constant and clearly:

( <o (E ) > - o (E ) ) 2 =■ < a (E )2 - O(E) 2 > ■ < o (E )2fl l^ R ( f ) >

using Eq. III-3 have

a. r2( <o(E)> - 5717)2 = a 2 lim • 2X g >eff r +e

2 ad f 1 1. ^ r2a t:— i-j=, lim E. — ---------------- —eff ^ V Ej T + ( E . - E )

1 J

here the assumption that oq and a r e constant withinthe data range AE was made. Such an assumption is valid when the fluctuations are very narrow and a small enough range AE is sufficient for the analysis. At high excita­tion energies, however, the characteristic width of the fluctuation is several hundreds keV and the data range that has to be analyzed is at least several MeV, Over such large ranges (given by l-yd ’) may change appreciably.Nevertheless, if the plausible assumption that the fast fluctuations are independent of the slow variation of yd with energy is made one can factorise

“a r2 > <Neff r2 + d

into ot „ 2

<$ c ~ > < — r — 2 *eff r + C Z

28

For r/D » 1 the summation over i and j is replaced by an .integration resulting im

(<o(E)>-a(E))2= a(E)2- ^ aeff

wherei 2 1 2a = — arctg (m)-- log (l + m )m ° ' 2 em

and

m = —=- is the sample size.

Hence the uncertainty in determining the average cross section from a finite data sample is:

III-6

I a a, <o(E) > = a(E) l

V eff

Since R(c) is a quadratic function of o(E) the un­certainty in <a(E)> will result in a shift in the value of R(c) in addition to the uncertainty involved. This shift is defined by:

y(f=°) S 5T“ -R(€=0) eff By R(0 is meant an autocorrelation function calculated over a finite sample. y(e=0), then, is the shift from the

-ad"true" expectation value •eff

We have themy(e=0) <a>2 ~ < a>2 - <a2 - <a>2 >

eff

assuming uncorrelated fluctuations in y and a

one obtainsia ,+ N

. ^ 2 d eff 2 2y (e=P) <a > = — =---- <ct > - <a >

eff

using eq. III-6 we obtainIII-7

a a N + a / d eff d

y (c=0) = N t t N « + a ,eff eff d

The uncertainty in the value of R(0) can be evaluated similarly

1 1 1 -8 2 2 2 2-.2 S<Neffiaod> arctg(m) - 2a V e f f ^ d * +24a “d^ett^d*ad ' 2 3 2 2 2 q<Ne«f+a“d) Neff +6a“dNeft + 3 <a“d> Neff + 24,1 “d b

"a" has been defined previously and 2b = - 8 a rc tg (T ) - - j loge (1 + — )

m mfor yd=0 Eqs, III-7 and III-8 reduce to the corresponding expressions in ref. Da 66.

For large values of the sample size, m, one obtains

a -* — and b-* —m m

and the shift and uncertainty in R(0) reduce to the ap­proximate expressions in refs. Ha 6 7# Vo 64, and Sh 6 9.

For the uncertainty and shift in the values ex­tracted for r , the average width, the approximate expressions in ref. Sh 69 are used, i.e.

30

III-9 robsr’true

111-10

The finite, and sometimes poor, experimental energy resolution, 6E, has an effect on the fluctuation analysis which has been treated by several authors for the case of 6E<r (La 65,Wo 66), The resolution effect was expressedin a series expansion in 6E/T the first term of which is

2of order (6E/D,Knowledge of the functional form of the energy spread­

ing functions, and the expected form of R(f) - the auto­correlation function obtained from data with arbitrary width and energy resolution -6 .

For a rectangular resolution function one has (Bi 68)

if we put a= 6 /r then

R 6(r) = [R(0)/a2 ] [g(l+a) + g(l-a) - 2g(l) ]R.(0) = [R(0)/a2] [ g(a) + g(-a) -2g(0) ] o

R fi(c) is known (experimental value) to solve for Ti

III-llaRg(c) = R(0) (^) { g < ^ ) + e ( ^ A ) + g ( 5

whereI I I - l l b

g ( £ ) = £ arctg(£) - ilo g e (1 + C2)

R 6( D g ( ! + £ ) + g ( l - - | > - 2 g (l )

All the expressions obtained or listed in sec, III-3»4 depend upon tho assumptions listed in sec, III-3 on which Ericson's model for fluctuating cross sections is based.The allowed uncertainties con provide a useful consistency check on results obtained in fluctuation analyses. Widths extracted, for example, from several excitation functions decaying from tho came energy and angular momentum region in the compound nucleus, populated by the same incoming channel, should not, in general, deviate from their common average much more than the allowed deviation based on the sample size analyzed. The values of y^, - direct to total cross section ratio, extracted by the two methods distri­bution (eq. J.II-4) and autocorrelation (eq. III-3 ) should also be close, or identical within the allowed uncertain­ties. Consistency checks such as these were repeatedly ap­plied but will be mentioned hereafter only when some new insights are gained through them,

III-5 Synthetic Excitation FunctionsIt is clear that the experimental date to be con­

sidered here do not satisfy a.ll the criteria mentioned in sect. III-3. In particular the presence of gross structure in the data arising from shape elastic or po­tential scattering (Go ?la,Go 7 3 ) implies that (Eq. T.II-8) has a strong energy dependence when a=a' .This specific energy dependence must be removed from the data before a proper statistical analysis can be attempted.

Also, it is not obvious whether assumption (li) is satis­fied since for the heavy ion coJlisions considered here, high angular momentum states in the compound ore popul.at.od which may lie .in the vicinity of the Yrast lie where their density is rather low. Finally, the presence of structure of intermediate width, arising from intermediate inter­action mechanisms, could be reflected in a large range of widths in violation of assumption (iv) of sect. IIT-3 .All these effects will be examined quantitatively when the fluctuation analysis of these data are discussed (Chapter IV and V). The method used in these studies and its phil­osophy will be described briefly in the following paragraphs.

According to Ericson's model the fluctuations in the

observed cross sections rresult from the random superposition of a large number of S-matrix pole terms which arise from the contribution of many overlapping levels in the compound

nucleus.There is no simple a priori way of evaluating the ef­

fect of non-satisfaction of the assumptions underlying that model, some of which are clearly violated as were listed above. A most straightforward way of evaluating such effects is by direct simulation of this process under different conditions and examine the effect of deviations from the ideal situation postulated.

The simulation is done with the aid of the computer.A summation of S-matrix pole terms chosen at random is per­formed, Nuclear models provide us with the statistics of

nuclear levels i.e. the distributions of their widths, spacing and reduced widths (Po 56,Wo 5 2, Fr 6?) nil of which can be verified from lower energy studies of isolated resonances. These distributions can be used in the gen­eration procedure in order to simulate what is thought to be the actual process of the reaction.

The synthetic cross section is given by (Wo 66, Da 66, Ho 69)111-12

aJ (E) = — (2J +lj I J ,Fv 12a a '( ) k2 (21 +1) (21 +1) ' a a ' ( ) I

a.

whereJ SaX gXa,a . m - j

E - V ? rx

i,I-are spins of target and projectile and J = total angular momentum

andtgJ v - are random numbers normally distributed about aXzero.

The level spacings = E +1 - E^ obeys the Wigner dis­tribution (Wi 52, Fr 62) and the width are random^y

' 2dispersed around an average value - T according to a X distribution of n-degrees of freedom where n is the number of open channels.

The above statistics are consistent with Ericson’s fluctuation model wherein the quantities represent

the partial width amplitudes. For T fD>i , or many open channels, the dispersion in the decreases and may be assumed. Moldauer (Mo 6 7,MS 68) has pointed out, however, that the distributions obeyed by the resonance parameters in the region of isolated resonances, and which are used above, do not necessarily pertain in the region where the resonances overlap, and has demonstrated, for the case of strong absorption, that a wide distribution of

widths may be appropriate even when many channels are open.12 12For the case of the C+ C system where we shall carry

out a more comprehensive analysis, a large part of the total absorption is due to direct processes and not to compound formation (Fr 71,Ri 7 2 ,Wi 73), thus we expect the distributions of widths and pole strengths to behave according to Ericson's model limit for many open channels.

The synthetic excitation functions generated will beused to study the effect of such situations as small T/Dratios or small numbers of open channels. Ey superposingthe fluctuations on different types of smooth background,and mixing them with structure of different width, effectsresulting from presence of gross or intermediate structurein the data can also be examined. Such studies will be

12 12very useful when analyzing the ' C- 'C elastic data and their results will be described in the next chapter where these data are also discussed.

One important feature of all such synthetic excita­tion function studies is the fact that on each occasion

a finite random sample is analyzed; thus in order to obtain reliable estimates for the average outcome of such analyses as well as the uncertainty .involved, many random samples pertaining to the same situation (5., e, average width and strength) must be generated and analyzed. Nature can only provide a limited number of independent samples, but a computer has unlimited capacity for that; we have exploited this capacity in our studies, For each situation which

we have examined many synthetic excitation functions were generated with independent sets of random number inputs.The outcome of the analyses of such a set is averaged and used as an estimator for the average resulting quantity while the standard deviation from this average is used as an estimate of the uncertainty associated with deducing the average from the study of a single sample of finite size.

CHAPTER 4

IV-1 Introduction 1? 12The C+ C system is investigated with the aim of

understanding the origin of the narrow structure in theelastic excitation function as well as the selectivity in

12 12some of the reaction channels. The C+ C elastic data,taken in this laboratory by W. Reilly and his collaborators

19 ip 20and shown on Fig. IV-1, and new "C( C, )' Ne* reaction data obtained in the present studies are examined and analyzed. Predictions of the statistical model and the double resonance model are then compared to the results of our fluctuation analyses. Finally, the validity of applying the fluctuation analysis to such heavy ion data is checked.

IV-2 Fluctuation Analysis of the ^ C + ^ C Elastic Data Fig. IV-?a displays the ^ C + ^ C differential cross

section for elastic scattering at 90° in the center of mass over an energy range Ecm=13.5 to 37,5 MeV (Re 73).The data points are at intervals of 0,125 MeV, a spacing which is also comparable to the experimental energy resolu­tion. Also shown is a running average of the cross section

12 12Fig. IV - 1 C+ C elastic excitation functions ( Re-73)

\

d<x/dft

(m

b/sr

E cm(MeV)

The full curve is a running average of the data

taken over an interval £ -2 .5 MeV

h) The "reduced" data, i.e . excitation functions

devided by its average.

12 12Fig. IV - 2 a/ 90 deg. elastic excitation function for C+ C

determined with an averaging interval A =2,5 MeV which shows clearly the modulating gross structure which must he removed. The method used here, first suggested by Pappllardo (Pa i-), is to divide the experimental cross section by this emoirical average. Fig. IV-2b shov.'s the reduced data obtained in the way for A =2.5 ft.eV. In principle the size of the averaging interval A is chosen such that the resulting average contains all or most of the broad structure which we wish to remove and very little of the fluctuating component. Thus we would choose T « A « r g E where r and Tgg represent the characteristic widths of the narrow and the gross structure, respectively. For a case in which r ot?>20r , we would then expect that the

OI L

value of R(0), obtained from data which have been reducedwith an averaging interval A , would increase as afunction of A until A~10T , At this point R(0) shouldvary little with increasing A and this "plateau region"should extend until A ~rgT7. Pig. IV-3 shows R(0)obs(the superscript "obs" denotes that these are observedval\ies of R(G) obtained directly from the reduced datawith no corrections applied for averaging by A ). Onlya suggestion of a plateau is apparent for the data at70°, 80°, and 90°, while the 50° and 60° data give no

12 12indication of a plateau. This shows that for C+ C elastic scattering, at these energies, the difference in widths between the narrow and the broad structure is not sufficiently large to permit a complete separation of the

RATIO

do

/d£2

(mb/

sr)

Fig. IV - 3 Dependence of the average normalized variance

R(0)°^8, on the averaging interval. A .

A(MeV)

38

two .in the autoenrrel ati on function.T’he dependence of R(0)° s on A for the ?0°,80°, end

90° date indicates that £ should be chosen somewhere in the range of 1,5^ 3 MeV but that even within this limit there is a large variation in the values of R(0)°^S, In order to obtain true values of R(0) and T , which are by definition independent of A , ?t is necessary to know, for

r\ V\oa given value of A » the extent to which P.(0) and T

are affected by the. fact that the reduced data do not contain all the fluctuating strength and, moreover, still retain some of the gross structure. This was studied quantitatively as discussed y'-o'?': by generating synthetic excitation functions whose true statistical characteristics (R(0),T , r/D etc.) are known, a-priori, performing the eu.tocorrelatian analysis and then comparing the resulting values of R(0)°^s and r0^53 with the "true" values (see sec. III-4),

Fig, J.V-4 shows the results of such studies made on a sample of ~ 300 fluctuating excitation functions with E(0)=1 a constant width r=400 keV and r/D«20, Parts a and b of the figure show the distr.ibu.tion of T 0^ and R(0)°^s obtained after a. conventional autocorrelation analysis of these 3^9 random samples. As discussed previously (Chapter IIT, sec, 4) the average and the standard deviation of these distribution give us the average values and the un­certainties for R(0) and r which one might expect from analyzing a data, sample having the same size as that of

Fig. IV - 4 Typical distributions obtained for R(0)°^S and

_obsr from the study of a set of 300 synthetic

excitation functions, all generated with the same

input parameters ( r=0.4 Mev, R(0)=1.0 ) but

with different sets of random numbers. Note

the shift in the distributions when A is varied.

NUM

BER

OF

OCC

URR

ENCE

S

r 0BS(keV) R(0)obs

the synthetic excitation functions analyzed. Parts c and

d of the same figure show what values of r°^'J and R(0)°^S

are expected when the analysis is done with the sliding,

average method using a A ~ 1 ,5 !V*eV averaging interval, a

large shift to lower average values is evident. It

should be borne in mind that these were "pure" synthetic

excitation functions with only one constant width input

and no gross structure yet the dependence of the ex­tracted values of R(0)°^>‘J> and r° J ’ on is marked. The

full dependence of R(0)o'k,J ar.d p ° ^ s 0n the averaging

interval-£ is shown in Fig. TV-5 a and b. Similar stud­ies were performed with known fluctuations superposed on known gross structure and the results are shown on Fig.

IV-6 a and b. The open circles display the results of analyzing excitation functions having equally strong back-

lground but with no gross structure in them. The grossstructure used here was the actual gross structure ap-

12 12peering in the 'C- ' C data and determined with a 2,5 MeV running average of the 90° excitation function. The results from this study show that because of the gross structure, the optimum value for A to be used in the analysis is 1,5 MeV. Put et al.(Pu 68) have noted that the corrections to be applied to R(C)°^S and r°^s can be quite large even if A is as l^rge as 10T . This is also the case in the present work.

Examples of R(c) and P(X) obtained from the analysis of the reduced data are given in Figs. IV-7 a and b, re-

Fig. IV - 5 Dependence of r ° * >Sand R(0)°^S on the value

of £ used in the analysis of the same excitation

functions. The variation with £ becomes small

for values beyond £ 20 r.

OBS

r^

Sfk

eV

)

A (IN UNITS OF D

Fig. IV - 6 R(0)°bs and r ° bs versus A for excitation functions

with and without gross structure in the background

At small values of A the gross structure has

very little effect.

r0BS

(keV

)

40

spocti.vf.ly, for the excitation ■f'unctionr measured at. 90°.To Fig . TV-7;i t the osc i l l a M onr: in R(e ) f o r ( > ? T are

as soc i a ted with the f i n i t e range of' data a v a i l a b l e f o r

analysis. The value of T obtained from Eq. TTI-3a isr 0^ J - 1 9 1 keV, The value of yd obtained from Eq, IIJ.-3b

with Naj. =l is Yd°^S=0.9 3 2. (^eff ^ en^ ca^ y unityfor elastic scattering of spinless particles.) Fig. IV-7bcompares the distribution of cross sections for the reduceddata with Eq, ITI-4 for a best-f i t-value of ydol:s=0,956, -in close agreement with the above value of 0.932 obtainedfrom the autocorrelation analysis (Eq. III-3b),

The results of the autocorrelation analysis for allscattering angles are given in table I, Columns ? and 3

give the values of R(0)°^s and y^°^S a.s obtained fromEqs, IIT-1 and III-3i respectively, and column 4 listsfor comoari.son the value of Vj0l3° obtained from Eo . ITI-4.v 0

The values in columns 3 and 4 are in reasonable agreement. The fifth column gives the value of y^°^S obtained from Eq. III-3 ond corrected for the bias introduced by the finite range of data. This correction has been discussed in see. III-4, The error in y^ given in column 5 is based on the sample size (column 11) which is defined by n= A E A r where AE is the energy range of the excitation function. The value of y^ presented in column. 6 includes the correction for the effect of the averaging interval For a -1.5 MeV, it may be seen that this correction is

sizeable: the compound nuclear contribution to the reaction

Fig. IV - 7 a) The autocorrelation function R (c)°^S for the reduced

elastic scattering data at 90° c, m,

b) The distribution of cross sections. The full curve is

obtained from Eq. Ill - 4 with y °^ S adjusted to 0.956.

2The value of \ per degree of freedom is 0.9.

I2C+,2C ELASTIC SCATTERING

€(MeV)

2.0 3.0cr/<c7>

TABLE I

A=l.5 MeV

c.m.Angle R(0)obs

obsyd yd

— obs (keV

f FRD (keV) r rP.c. Sample

Size

r >

oo1f> 0.051 0.984s) 0.974b) 0.965*0.007°) 0 945+0‘012d) -0.023 160°) 171*22^ 330±48d) 306*37 23

60° 0.057 0.970 0.971 0.970*0.007 0 938+0*014 ° *938-0.026 157 160*22 327*47 341*40 23

70° 0.078 0.967 0.960 0.957*0.008 0 914+°*020 -0.037 178 191*24 367*53 341*40 21

80° 0.176 0.923 0.908 0.901*0.018 0 794+0*049 -0.101 178 191*24 367*53 341*40 21

90° 0.134 0.956 0.932 0.926*0.018 0.848^?*938 -0.070 191 205*26 393*57 371*45 21

a) Using the autocorrelation method (eq. 3)

b) Using the distribution of cross sections (eq. 4)

c) The value of y°^S from eq. (3), corrected only for the finite range of data (FRD)

d) Includes correction for FRD and the effect of the averaging interval A= 1.5 MeV

®) Using the autocorrelation method (eq. 2)

*) The value of T °bS from eq. (2), corrected only for the FRD

is changed by a factor of two. The larger error given in

column 6 reflects the uncertainty in the size of this

correction combined with the effect of the finite range

of data. The compann^ or f ’ct,io.ting frs'ct^n of the

cross ructions reaches a maximum of 20% at 80° and 90°.

As is expected, the ratio of the average fluctuating part of the cross section to the total cross section,(l-yr-j), decreases at the more forward angles where the potential scattering is much larger.

The values of r °^J obtained from Ea, IIT.-2. are given in column 7. When correcti.ons for the bias accom­panying the finite range of data are applied, the values in column 8 are obtained. As is the case with R(0) the correction associated with the averaging interval A =1.5 heV is much larger, again by about a factor of two. The final value is given in column 9. It is interesting to compare these values of T with those obtained from the peak counting method introduced by Brink and Stephen (Er 6 3) (column 10), In this case the relation r=0.50/N v.'as used, when N is the average number of peaks per r.eV. mho agreement between the values of T given in columns 9 ar.d 10 serves to further validate the use of synthetic excitation functions to determine the effects of using reduced data in our analyses,

It is apparent from an inspection of Fig. III-l and 2 that the typical width of the fluctuations is larger in the portion of the excitation function at higher energies

41

than in that at lower energi.es, A quantitative study was therefore undertaken to determine the dependence of the average width op the excitation energy in the compound nucleus. Fig. TIT-fl shows values of T obtained from analyzing 10 MeV-wide subintervals and averaging over the five scattering angles. The abscissa is the value

MeV and A=24, The horizontal bars denote the extent of the 10 MeV-wide subintervals. Within the larger errors on r (associated with the smaller sample size of the subinterval) the logarithm, of T is seen to increase

obtained at each angle were assumed to be statistically independent. The validity of this assumption depends, however, on the magnitude of the difference between the two angles at which the excitation functions were measured.

The differential cross section for scattering reactions proceeding via levels of spin j in the compound nucleus can be written as:

where E* is the excitation energy in' Mg in

analysis the values of T

Fig. IV - 8 The observed dependence of T on/A/E* (filled circles).

A and E* are the mass (a. m. u .) and excitation energy

24(MeV) in Mg, respectively. The theoretical value of

r obtained from Eq. IV - 1.

AVER

AGE

WID

TH

(keV

)

1000900800

700

600

500

400

300

200

074 0.76 0.78 0.80 0.82 0.84 0.86 0.88

^ S T e *

100

AVERAGE WIDTHS IN 24Mg5 ANGLE AVERAGE •

,2C(l2C,a)20Ne

r THE0RY. 1 J

¥o

A

e

▲

oA□

_ i

.AA

The first sum, inside the squared brackets, takes account of the different ways in which the channel spin S ® I + i (I,i=target and projectile spins) can combine with the orbital angular momentum to form the compound spin J, The primed variables are for the exit channels.The energy dependent part is a random amplitude and for a fixed angle (9,<p) the different can be regardedas weighting factors for the different random amplitudes at each energy - E. As long as 9 and 0’ do not differ suf­ficiently for these weighting factors to change the differ­ential cross sections measured at 6 and 0' will be correlat­ed. A large angular difference, 0-0', will change the weighting factors of all these random amplitudes and cor­relation is then unlikely. A measure for such angular dif­ference, the so-called coherence angle, is then given by the angular interval over which Y^O.p) changes appreciably.

For angular differences larger than 0-0’ ~ l/t, where I - is the highest orbital angular momentum involved (l ~kR) , correlation between the differential cross sections at 0 and 0' is unlikely. Similar values for the coherence angle were obtained by Brink et al.(Br 64) who considered specific models for the reaction (black nucleus, surface emission) and obtained an anlytio expression describing the angular cross correlation. For the scattering problem considered here (kR)“1 ~6° hence a difference of 10° between the measured excitation functions is sufficient for the as­sumed statistical independence. Another exception which must be considered is the possibility that

one partial wave amplitude dominates the cross section. For such case since S=S’ =0 (spinless particles) we have l = V = J and only on Y™ determines the cross section, then cross correlations would prevail between the differ­ential cross sections measured at any two angles. We shall demonstrate, however, in the following section that this is not the case for the fluctuations observed in this system.

IV-3 The 12C(12C,a)20Ne ReactionExperimental Results And Fluctuation Analyses

IP IP 20The 'C( 'C, a ) Ne reaction was investigated fortwo reasons. First, the study of exit channels otherthan elastic scattering provides important additionalinformation on the reaction mechanism. The statisticalmodel predicts the average behaviour of all energeticallyallowed channels, and each additional measurement providesanother independent determination of the statistical

20properties of the coumpound nucleus. The a+ Ne channels also have the advantage of being strongly populated in this reaction. Further, any non-statistical effects arising from possible alpha particle intermediate sturc- ture would be expected to be prominent in the alpha particle channel. This is also a channel where there is a strong selective population of highly excited states in the residual nucleus.

Absolute cross sections were measured for the12 12 20C( C, a) Ne reaction at 50 keV intervals over the

energy range E =16-21 MeV using the 0(5+) "beam from the Yale MP tandem accelerator. The target consisted of a 20 jjg/cm" natural carbon foil nnd resulted in an energy resolution in the incident channel of ~ 5 0 keV c.m. The alpha particles were detected at an average angle of 3°c.m» which facilitates the fluctuation analysis by eliminating any uncertainty in the value of N The experimentalsetup and methods of analysis have already beer, discvsse^ in Chapter II.

A typical, calibrated, alpha particle spectrum ob­tained in these studies is shown on Fig. IV-9. The energyresolution for the alpha particles was ~150 keV, States

20in ' Ne as high as 15 MeV in excitation are seen to bestrongly populated.

For the purpose of fluctuation analyses, however,only six excitation functions were chosen. These wereexcitation functions leading to well resolved and known

2 0(Mi 71a) final states in ' Ne which were also populated strongly throughout the incident energy region studied. We also avoided analyzing states at high excitation because we consider our energy resolution too poor to be able to re­solve, with confidence, states in a region of excitation

20in Ne where the expected density of states is ~ JO states per MeV. Even small contributions to a strongly excited state from neighboring states can have large effects on the results of a fluctuation analysis of this state’s excitation function. Gross sections extracted for levels

12

Fig. IV - 9 Typical a particle spectrum from the

12 12 20 C( C ,a ) Ne reaction.

COUN

TS

500

375

250

125

j L

l2C(l2C,a)20Ne E |2 C=40.50MeV

8.48 (784,2+)

(743,2+)

(5.78. D

r(5.62,3')

(4.25,4+)(l.63,2+)

(g.s.,0+)

JL80 160 240 320 400 480 560

CHANNEL NUMBER640 720 800 880

12 12 20Fig. IV - 10 Excitation functions for C ( C , a ) Ne*. Also

shown is a r unning average of the data taken with

A 2. 0 MeV

20

16

12

8

4

<12

9

6

3

<8

6

4

2

•

• • ••“ C C ^ C / x r 'N e

A U=?°• •

# •0________ _

• 4•• m

A.* Eexc=9.02

1• —

•*•

>

• • * pm 0?

•••••%

' s " v •> Eexc=^® ^

C \•

• •S'S*»

•• 1

<i1

9

••

A>

•• -

••

_4__j/P —• V -— ^ * »v

•• •

•F = i^co

•••#

••>

Cexc

k

J.Vb -

r ••* •

•••• •

• • y

•

y > 7 -16 17 1B 19 20 2

with excitation energies and spins (MeV,J ) of (0,0,0+), (1.63,2+), (4.25, 4+), (5 .6 3, 3"), (7.83, ?+), and (9.02,4+) are shown in Fig, IV-10, The absolute cross sections are accurate to ±20%.

The statistical analyses of the above experimental results proceeded in the same manner as outlined in sec, IV-2 for the case of elastic scattering. The observa­tion of the alpha particles at angles close to 0° ensures that only m=0 magnetic substates make appreciable con­tributions to the cross section and, hence, N „=1 remains’ ei 1a good approximation. The data were first divided by a running average obtained with A=1.5 MeV in order to reduce the effects of,possible gross structure. The averaged cross sections, shown as full curves in Fig. IV-10, sug­gest that some modulating structure is present.

The results of the analyses performed on the reduced data are listed in Table II. A comparison of yd with

^d0133 an(* ^ r °^S reveals the large correctionsassociated with analyzing reduced data and, particularly in this case, over a very limited range of energy. The value of r obtained when the six individual values are averaged is 263±42 keV, This compares favorably with the average value 225±49 keV obtained from counting maxima,An analysis of the elastic scattering data over the same range of bombarding energy yields a value of T =277±68 keV. These widths are comparable even though the a -particle data were obtained with an experimental resolution which

TABLE H

A = 1.5 MeV

Eexc

R(0)obsobsd Y d

— obs., __ T (keV) r r P .c .

0.0 0.306 0.83 °.60+“™-0.60148 281 ± 70 227 ± 49

1.63 0.338 0.81 136 258 ± 68 227 ± 49

4.25 0.169 0.91 175 333 ± 85 208 ± 45

5.62 0.323 0.82 0 57+0*23 -0.57

124 236 ± 62 233 ± 51

7.83 0.232 0.88 0 72+° - 14 -0.24

109 207 ± 55 227 ± 49

9.02 0*210 0.89 0.75+® *^-0.20140 266 ± 70 227 ± 49

Sample size ~ 7

47

war; less than one half that used for the elastic scattering. This demonstrates that in this energy region there is no experimental structure finer than the 750 keV structure observed here.

A. remarkable feature of the present results is the large direct or non-fluctuating component in the cross sections: y^ i greater than 0.5 in most cases. This

i

result has also been obtained by Greenwood et al. (Gr 73)and is in contrast to the results obtained for this reac-

7 Qtion at lower bombarding energies ’ where the fluctuations in the data were consistent with 7^=0.

IV Comparison of the Statistical Model Predictions With the Experimental ResultsIndependent estimates of the average width, the

distribution of the cross sections and the magnitude oftheir fluctuations, obtained from the statistical modelof nuclei and nuclear reactions, will be compared withthe experimental results.

>

A. Comparison of Widths The average width of the levels of a given angular

momentum J and energy E in the compound nucleus must be independent of the exit channel for which this width is deduced. This is the case for the channels studied here.The six reaction channels listed in Table II yield an average coherence width of r =263±42 keV. The elastic data at five different angles yield, when analyzed over

the same excitation energy range in ~’Mg(30s3Se <35 MeV),an average coherence v/idth of 277+68 keV, Hauser-Feshhachcalculations (Sec, IV-C) of th^ dependence of the cnsm

1? 20sections for elastic and for 'C( C,a )' Ne scattering or, the total angular momentum suggest that the various angular momenta make similar contributions to the observed width and that J=12 is the dominat spin. Fig, IV-11 shows a plot of crj as calculated for the elastic, inelastic and alpha particle channels at an average bombarding energy of 18.5 MeV c.m. The rapid increase in the calculated cross section with increasing angular momentum illustrates the sensitive dependence of the cross section on the Yrast angular momentum cut-off. The partial cross section for J=12 is larger than that for J=10 by more than an order of magnitude, in the case of elastic scattering.

The dispersion in the widths in Table II extracted for individual channels from the average width most prob­ably reflects the finite range of data which has been analyzed to obtain these results. The results shown in Table I were obtained from an analysis of a larger sample of data and indeed the individual widths for each angle shown in Table I do show smaller deviations from their common average of 377 keV.

The different values for the average widths given in Tables I and II (377 keV and 263 MeV, respectively) reflect the fact that the average width varies with excitation energy and that different ranges of excitation energy have

2 h.

Fig IV - 11 Partial cross sections at E 18.5 MeV calculatedc.m.

with a Hauser - Feshbach statistical model. At this

energy, the maximum angular momentum allowed by

our angular momentum cut off parametrization (Ba 74)

is J 12.

PARTIAL CROSS SECTIONS AT E tm = 18.5 MeV

l2C(l2C,l2C)l2Cl2C(l2C,l2C)l2C* (4.43 MeV. 2 + )

a l2C(l2C,a)20Ne(gs.)

* l2C(l2C,a)20Ne (1.63 MeV. 2+)

been studied in obtaining these averages, In Tables I and?LII, the corresponding- ranges of excitation energy in ~ Mg

were 30 to 35 MeV and 27.5 51.5 MeV, respectively.A theoretical expression for the energy dependence

of the average width can be obtained from simple considera­tions on the assumption that the compound nucleus is in thermal equilibrium (Er 66). The average width depends on the temperature, T, and threshold energy for particleemission, W, and is given by T ~ exp(-W/kT), Since the

-1/2temperature varies as E*, we expect log T~(E*) 7 ' where6

E* is the excitation energy. Fig, IV-7 shows that the present results for the energy dependence of T are in agreement with this simple model. If one takes into account the variation of temperature with mass number and neglects the variations of pairing energies and particle emission thresholds from nucleus to nucleus, it follows that r =C exp {-a \/a/E* ) where A is the mass of the nucleus and C and a are constants. Fig, IV-12 presents the logarithms of the average experimentally determined widths taken from a recent compilation (Mi 70) plotted

a least-squares fit to these data (filled circles only)

tained in the present work. The excellent agreement with the values predicted by this semi-empirical evaluation of coherence widths obtained for other nuclei strongly suggests that the origin of narrow structure in the

for various nuclei. The straight line is

24and the open circles represent the widths for Mg ob-

Fig IV - 12 The dependence of experimental widths on/A/E*

( see Fig. IV -8 caption). The straight line fit to

the data pdlnts has the same slope as the line shown

in Fig. IV-8. Only the data shown as filled circles

have been included in the least square fit.

elastic scattering.The above comparison neglects the dependence ob the

average v/idth on angular momentum. Thus, we would notexpect, a priori, a coherence v/idth deduced from an analy-

2 3 2 0sis of the Na(p, oc ) ' Ne reaction to agree with the value1? 12 20obtained from the "C( C, a ) ' Ne reaction because of the

different angular momenta in the entrance channel, hencein the compound system. Estimates of this angular momentum

2^dependence for widths in Mg have been ma.de using the statistical model relation

IV-1)

< d > ~ c ’l 1

(c " denotes summation over all the open exit channels)and a Fermi gas model to compute the density of levels(DJ)”■*■. The procedure for the evaluation of the number

Jc11of open channels (J T ) and the level density will be described later in this section. The results for T 1' are shown in Fig. IV-7 for spins J=8, 10, 12, and 14. The theoretical values compare favorably with the experimental values over the energy range studied. The angular momentum dependence is not pronounced, a result which has been noted by several workers (Er 66), and this probably accounts for the general agreement shown in Fig. IV-11,

12 p2 12 I*5C( ~C, C) ‘C excitation functions is indeed, compound

Wh.ile the agreement between the theoretical and the experimental widths shown in Fig. IV-7 is good, the un-

• Q _ "1certainty in the level density parameter " a " ( f = 0 , l A MeVwas used in the Fermi-gas formula to obtain D' ) places a ±50$ confidence limit on the theoretical values of r^.Thus the agreement between the widths predicted on a seni-empirica.1 basis and the measured ones shown in Fig. IV-12provides the stronger argument in favor of the compoundorigin of the fluctuating structure,

IV-4B Comparison of Cross Section DistributionsThe statistical model predicts that the fluctuations

in the cross sections should be distributed in the form specified by Eq. IV-3. Fig. IV-6 shows that the agreement obtained for 0 cm=90° is quite good when is adjustedto the value 0.956. Similar agreement was found at the other four angles, and the values of obtained fromthis analysis agreed well with those from the autocorrela­tion analysis (see Table 1).

The fluctuations in the cross sections for excitation functions at angles differing by more than the coherence angle 0_~l/kR should be uncorrelated (Er 64) provided thatv

several partial waves contribute to the cross section.Since the data are at 10° intervals and 0 ~6° for thecpresent case no strong correlations would be expected on this basis. However, Fig. IV-10 shows that the cross section can be dominated by the contribution from a single partial wave at some energies, which might place this

assumed independence in question. The degree of independ­ence may be tested by considering the fractional deviationsof the cross sections from their average values at each

5 a i<E>angle. Fig, IV-13a shows the quantity D(E)= T I — tet; 1|i = 1 a i<E )

where o^(E) is the cross section at the i angle at energy E (Ma 72). Although no strong cross correlations are apparent in this figure, the statistical independence of the fluctuations in the five elastic excitation functions can be tested more quantitatively by comparing predictions for the distribution of D(E), based on such an independence hypothesis, to the observed distribution. The distribution of D(E) can be predicted in a combinatorial calculation since the distribution of the fluctuations at each angle is known (Eq, IV-4) and the data at each angle are assumed to be statistically independent. In Fig. IV-13b we present the experimental end the calculated distributions of D(E) as a histogram and a full curve, respectively; the good agreement shown here does not suggest any significant angular cross correlations. Similar results were obtained from a cross correlation analysis of the six alpha particle channels listed in Table II, and are shown in Fig. IV-14,The lack of cross correlations in the data and the predicted domination of the cross section by a single partial wave at some energies (see Fig. IV-10) is most probably ex­plained in the following way. In the theoretical expres­sion for a fluctuating differential cross section, ampli­tudes for the individual partial waves are added coherently.

Fig. IV - 13 a) The cross correlation function D(E) for elastic

scattering at five angles.

b ) The distribution of D(E). The calculated distribution

(continuous curve) is obtained by assuming statistical

independence of the fluctuations in the five individual

channels.

MAGNITUDE

OF D

NUMBER OF OCCURRENCES ui o oi 8 roOl

mo

£

D(E)

Fig. IV - 14 a) The cross correlation function D(E) for five outgoing

V 1 , . ., 12_ .12_ .20.. *channels in the C ( C .a ) Ne* reaction

b) The distribution of D(E). The calculated distribution

(continuous curve) is obtained by assuming statistical

independence of the fluctuations in the five individual

excitation functions.

MAGNITUDE

OF

NUMBER OF OCCURRENCES_ _ roo oo ro o) o

DISTRIBUTION OF

D(E)

D(E)

In terms of amplitudes, the contributions of the adjacent partial waves becomes relatively larger than would be indicated by Fig. IV-11, which gives angle-integrated cross sections. These contributions in amplitude may well be sufficient to remove the angular cross correlation which would be obtained if only one partial wave ampli­tude was present.

IV-^C Compe.ri.snn of Average Magnitudes of FluctuationsIn this subsection we compare the predictions of the

Hauser-Feshbach statistical model with the experimentally deduced fraction of the cross section which proceeds via non-direct reaction mechanisms. The average fluctua­ting cross section is related to the average normalized variance by Eq. IV-3b and the relation <af^> = <o>(l-y,;j) where <a> and are the averages of the measuredcross section and its fluctuating component, respectively. The magnitude of yd is assumed constant over the entire range of bombarding energy, while the average value of the cross section, <a>» is determined at each energy by a 2.5 MeV-wide running average, (The value of y^ is, in fact, energy dependent as is the case with T, However, analysis of subintervals shows this dependence to be small and imprecisely determined so that we believe that we are justified in using a constant value in this analysis.

The Hauser-Feshbach theory has had considerable success recently in predicting absolute cross sections for compound

nuclear reactions in this nass region and well above the Coulomb barrier (Ha ?4,Gr, 7 2 , ?o 74), Earlier studies (ho 6 3 1 Do 65-', Vo 64, Ha 6 7) also emphasized the promise of this theory for estimating compound nuclear cross sections for heavy ion reactions. We refer here to the full expression for the theoretical cross section as given by Hauser and Feshbach (?e 60) and not to an approximate form of the theory as developed by Eberhard et al. (Eb 6 9) which is also used frequently (Em 73) and which is usually normalized to the experimental data at some point. A discussion of the difficulties presented by ihis particular approximation in evaluating the number of open channels in conjunction with heavy ion reactions has been given by Greenv/ood et al. (Gr 72)

Our calculations were performed with the computer code STATIS (St 72a) which evaluated the following ex­pression for the differential cross section for scattering from channel a to channel a* ,I V - 2 )

d § 0) Wf t Ws WD £ 2 j f ir(2I+l)(2i+l) ^ V a ) ^ V l ' f r T (c»a c", i"

Z (tJtJ;SL) Z(t'JUJ;S'L) (-)S_S’ P^cosQ)

Except for the first three factors in Eq. IV-2, the notation used in conjunction v.’ith the optical model trans­mission coefficient T , and angular momentum recou.pling coefficients Z is standard (Vo 64, Fe 60) and self ex­planatory. The denominator T T (clf) includes a

d ' , l " 1summation over all possible outgoing channels. Low lying

states of known spin in the residual nuclei were summed explicitly and the following level density expression derived by Lang (La 6 3) was used for higher excitation energies in these nuclei:

I V - 3 ) 2p(E,J) = — i<2- +1i--- 5t exp[2 aU ] exp [-(J+ P - ]

12a4 (U+t) (2ct) 2ctwhere U = E*- A E* = excitation energy

A = pairing energy correctionThe spin cutoff parameter a is given by:

2 , 2 o = «9t / ftwhere t is the nuclear temperature defined by

2U=at -tand «9 is the moment of inertia (we assume it to be the rigid body value under the present circumstances)

«9 = |mR2(l +0.315 +0.4482) R = r5 OThe summation J, ir in Eq, IV-2 is over all the spins (and parities) of states that are populated in the com­pound nucleus. This sum runs from J=0 to an upper limit determined by the following constraints.1, The highest angular momentum state that can be sup­

ported at an excitation energy - E* by a nucleus is limited, If wo consider rotational excitation the limit is determined by the maximum rotrtional energy available for-excitation. This so called Yrast limit has a.n effect on the outgoing channel, limiting the number of outgoing channels.

2, The more stringent limit on the population of high spin states in the compound is of dynamical origin.The combined centrifugal and coulomb repulsion may prevent the fusion of the colliding nuclei, this angular momentum limit has been discussed by several authors and is usually the more stringent limit or. the highest spins achievable in compound nucleus formation by heavy ions (Ea 74, V.7i 73).The factors V/g, and in eq. IV-2 are associated

with the symmetrization of the scattering amplitude re­quired by the presence of identical particles, the in­ability of the detector to distinguish the two reaction products (direct and recoil) in the exit channel, and the fluctuation width correction, respectively. The presence of identical bosons in the entrance channel requires W =2 in all calculations described here and. limits thebsummation in Eq, (7) to even values of J and posi-

12tive parity. The identity of C ions in the exit channelfor elastic scattering obviously requires that W^=2for this case. Although the detection systems used byWieland et al. (Wi 7 3 ) and Em ling et al. (Err. 73) separateelastic and inelastic scattering, they do not distinguish,

] 2for the case of inelastic scattering, which ' 'C ion is in the excited state and which is in the ground state. Thus Wq=2 for inelastic scattering and Wg=l for the 12C(12C, a )2 reaction. The origin and value of will be discussedbelow.

57

In h.is treatment of fluctuating cross sections Moldauer (lio 64) obtained the following expression for the average compound cross section, similar in appearance to the Hauser-Peshbach formula, but valid for arbitrary values of T and D,IV-4

it , _9Xa8Xa'_ „ rwaa,<9Xa><9Xa> , ,aa'= . T t < - T >X - M aa'}— twfl. “ J57>------ M aa'J

CCCt*The fluctuation width correction VJ arises from re­ft

placing the first term in Eq. IV-4 by <®\a><®Xtf>/ <®\>2 ff i i2where <0Xo>= DxN x l*Xo(l . and 8^= T 8Xo and g is a

term arising from resonance-resonance interference effects. (N^- is a volume integral of the wave function for the resonant state.) Assuming the amplitudes g^a for differ­ent channels to be uncorrelatei, Moldauer showed that, in the limit of large T/D or many competing direct transi­tions, M . becomes small and <0N >N may be replaced by aa1 Xa X J Jthe transmission coefficient for compound nucleus formation. Furthermore, the fluctuation width correction W in the

AM Ilimit of large T/D becomes =1 + a 6 , with l^aS!X4/ uuand, in the presence of many competing direct transitions, a^l. Since in the present study r/D as given by Eq. IV-1 is usually greater than 5 and there is significant competi­tion from direct reactions, the resonance-resonance inter-

tttt*ference term has been neglected and has been setequal to 1 + 6^,

The transmission coefficients were calculated with the optical model code APACUS-2 (Au 64), Standard opti­cal potential parameters were used for protons and neu­trons (Pe 6 3). For the and a+'^Ne channelsoptical potentials taken from coupled channel fits to the elastic and inelastic scattering in these channels (Ri 72, Fr 71) were used. In addition the higher partial waves near to and including the grazing partial wave were excluded at higher energies by the introduction of a Yrast limit on the total angular momentum in the compound nucleus. This effectively set T^=0 for the partial waves that are expected to contribute mainly to direct reactsons. The requirement that the transmission coefficient used in Eq. IV-? should correspond only to compound nuclear ab­sorption is thus approximately met by the above choices of optical potentials for a+2( Ne and and byincluding angular momentum cut-off in the entrance channel. The parameters determining this cut-off (see Table 1) resulted in angular momentum limits of J=l4 andJ=l6 at energies below E “*21,5 WeV and E “* 281-leV,0 cm cmrespectively. The parameters are similar to those used in an analysis of ^2C+^N reactions (Ha ?4) and the resulting cut-off-values agree closely with the limiting angular momenta in the entrance channel for fusion of12 12C+ C predicted by a semi-classical model recently suggested by Eass (Ba 73).

Table III lists all the parameters used in the optic?.! model calculations, in the level, density formula and in the Yrast cut-off calculations, A discussion of the sensitivity of the calculated cross sections to the various input parameters is given in Ref. Ha 7^ ^rd applies here as well. The very shallow imaginary potential depths for

12 1? 2o .the C+ "C and Ne+a channels arise from the inclusion of coupled channel effects in their derivation from a com­parison to experimental data (Ri 72,Fr 71). The weak

1? 12.absorption of the ~C+ C potential results in some degree of transparency even for 1=0 partial waves. This un­physical result for the lower partial waves is not of real consequence for the present comparison because these low partial waves make a negligible contribution to the predicted cros~ sections (see Fig.IV-11). The important feature is that compound nuclear absorption be reduced for the near-grazing partial waves and reducing the magnitude of V/ is one way of parametrizing this effect.

Uncertainties in the above parameters and in the value of (especially at energies where T/D is not very large)lead us to expect the absolute normalization of the theoretical compound nuclear cross sections to be accurate to about a factor of two. Figs. IV-16 and IV-17 show the calculated compound nuclear and experimental fluc­tuation cross sections for the elastic channel and sixreaction channels. The agreement for elastic scattering

12 17 20and for several C( “C,a) Ne channels is good. In some

Optical Model Parameters

Channel V_ .(MeV) VT (MeV) R . a , R a. Ref.Real' Imag real Real Imag imag

TABLE m ( 24Mg C .N :)

2%e+0i 50.0 2.0a) 4.94 0.59 4.94 0.46 F r 71

16oVfe 14.0 0.4+0.15Ea* 6.10 0.49 6.10 0.49

12c+12c 14.0 0.82a) 6.18 0.35 6.41 0.56 Ri 72

23Na+p 56.-0.55E 13.5b) 3.56 0.65 3.56 0.47 Pe 63

23Mgf f(\ 48.2-0.3E 11.5b) 3.56 0.65 3.56 0.47 Pe 63

a) Volume absorption

b) Surface absorption

Level Density Parameters ( Ha 74, Fa 68 )

24Mg 20„Ne 16o 12c 23 23

Na( Mg)

a / A 0.140 0.152 0„152 0.152 0.167

%5.13 5.13 5.13 5.13 2.67

The spin cut-off factor is evaluated using the rigid body moment of inertia

2 2 %«9=— m R where R = r A and r =1.50 fm.

5 o o

cross section ( energy-averaged over 2.5MeV) and

corresponding Hauser - Feshbach predictions. The

indicated errors reflect the uncertainty in the value of

y^ obtained in a fluvtuation analysis. The discontinuities

at the arrows* positions are caused by the sharp angular

momentum cutoff applied which allows the values of

J 14 and J 16 to contribute beginning at E ~21C » I D •

and 28 MeV, respectively.

12 12Fig. IV - 16 The average fluctuating component of the C- C elastic

do-/

dil(m

b/sr

)

Fig. IV - 17 The average fluctuating component of cross sections

12 12 20 for the C ( C ,a ) Ne* reaction and corresponding

Hauser - Feshbach predictions.

1 2 1 2 2 0cases, however, the predicted "C( 'C,a)'' Ne cross sections are higher than the experimental cross section (by a fac­tor of 2 at most). Disoontirnn tier, jn the calculated cross sections in Fig, IV-16 occur at a bombarding energy where an additional partial wave is suddenly allowed by the sharp Yrast cut off. A modulating structure in add tion tc this, in the calculated cross sections, and which is also apparent in Fig, IV-16, arises from the use of very weakly absorbing potentials which permit "shape resonances" in the variation of with energy.

It should be emphasized again that these Hauser- Feshbach predictions are absolute and not normalized to the average fluctuating cross section which was extracted from these data. Thus, for elastic scattering, the amount of compound scattering which is theoretically predicted is consistent with the average size of the fluctuations pre­sent in the data,

Fluctuating structure in the excitation functions for 12 12C+ C inelastic scattering has also been observed. The measurements of Emling et al. (Em 73) were carried out in sufficiently fine steps to permit an autocorrelation analysis, and they obtained a value of R( 0)ol:>s=0.257 * 0, 061 for the 120(12C,12C)12C* (4.43) reaction at 6cm=90° over the energy range E =10-18 MeV using a running average ofurnA =2.1 MeV. If we correct their value of R(0)°^s for the finite range of data and for the effect of averaging with A =2.1 MeV, a value of R(0)=0,290*q * results, Emling

et al. use ^ ^ = 2 which then yields a value of y ^ O .65+q*Jg

The average fluctuating cross section in the region Ecm=l3-18MeV is thus<a^>« 0.7^q*2 rcb/sr, and the correspondingHauser-Feshbach prediction for this energy region is 0.6 mb/sOn the assumption that y^ does not change appreciably with

12 1 ?energy we have used the same yd value for C+ 'C inelastic cross sections measured by R. VM eland et al. (Wi 7 3 ) over the energy range Ecm=20-30 MeV. Fig, IV-18 shows the cross sections and the corresponding Hauser-Feshbach prediction for ;{thisr r e g i o n e m ^ t ,.ejcc e 1 1 ent and the sta­tistical model therefore accounts for the magnitude of the

j'*'. ■ v" ' • ' v -sr.; >. -.he rr.< a f»u '» . * •' -1- '

fluctuations in the inelastic as well as in the elastic. . j i i hr,r) i.-vc tr.r U'.< Hurt.mim/ -

scattering,M.'cUon ar* r<v:»u< !' th* •. 1.. • , '■In summary, the width, strength and cross correlations

predicted b/“ e' y^a^ist*icainmo"d‘eis of'’nuclei and of nu­clear reaction agree very well with the characteristics of the narrow structure appearing in the measured crosssections for the elastic, inelastic and reaction channels

12 1?in the C+ ~C system, It seems reasonable to conclude on this basis that the origin of this observed fluctuating structure is statistical compound nucleus formation.

IV-5 Intermediate StructureIn the preceding section we have shown that the fluctuating structure with a characteristic width of ~ 300 keV originates with statistical compound nucleus formation. It is also well known (Lo 72,Ho 66) that cross sections exhibiting

Fig. IV - 18 C - C inelastic cross sections and Hauser - Feshbach

predictions (heavy line). The measured cross section (Wi-73)

(light line) and the average fluctuating component of the cross

section (points) are related by the value of which is based

on the value given by Emling (Em-74).

12 12

dcr/

dlM

mb/

sr)

10

0.1

c.m. (MeV)

statistical fluctuations may, with reasonable probability, contain occasional individual peaks in an excitation func­tion with widths or spacings two or even three times larger than the characteristic width. The only way to positively identify structures of non-statistical origin in the pre­sence of strong fluctuations is to observe structure which falls outside the range of the statistical model, i.e. a correlation or fluctuation which, statistically, is veryunlikely. Although the present data indicate no such cor-

20relations in the elastic scattering or in thea + Ne reac­

tion channels, Scheid et a l . (Ja 73) have suggested that

this test may be insufficient to establish the presence of intermediate structure in this case. In particular, they

have performed a cross correlation analysis on theoretical

excitation functions of Fink et a l . in which the structure

necessarily arises solely from known direct and intermediate

mechanisms. They report no apparent correlations and that

the ''compound" component of the cross section, deduced by

applying an autocorrelation analysis to the results of their original calculation, was not unlike the value of (1-y^1 ) deduced from the experimental data (Em 73). They conclude,therefore, that the results of an autocorrelation analysis

12 12for 'C+ C do not provide a reliable indicator of the re­action mechanism (Ja 73).

It is clear that an autocorrelation analysis, i.e. the extraction of R(e) from the experimental data, determines only the average width and strength of the component of

63

tho cross section which vari.es with energy; by itself it provides no indication of the physical origin of this energy variation. It i.3 only through, a comparison of R(0 with independent theoretical predict.ions (such as is made in Sec. IV) that information on the origin of the fluctuating structure is gained.

We have also performed a statistical analysis of these sane theoretical cross sections. All the structure in these calculated cross sections stems from an intermediate process, the double resonance mechanism a detailed descrip­tion of which was given in Sec. 1-2. Since the calculation was done by coupling in only states at low excitation (4,43 MeV) in the inelastic channel the comparison of the calcu­lated cross section with the data should not be carried beyond 20 MeV center of mass incident energy, and was thus limited to that region. Several comments in addition to those given in Ref. Ja 73 are in order. Our analysis was performed in a manner analogous to that discussed above in our analysis of the data; the slow energy variation was removed by deviding out a running average of the calcu­lated cross section obtained with A=2.5 MeV (Sc 73).The results of such an analysis are given in Table IV.We obtain values of R(0)obs comparable to those reported by Jansen and Scheid, From peak counting we obtained r~800 keV for the characteristic width of the structure in their theoretical calculations. Since these theoretical data are to be analyzed in exactly the same manner as were the

12 12Fluctuation Analysis of Theoretical C - C Data

A = 2.5 MeV

TABLE IV

R(0)°bs a) R(0) h)_obs a

(keV)_obs \x

(keV) 1_ P i C . c.

(keV) }

50° 0.10 0 34+0*20 3 -0.180 81+ ° * 10 0,81-0.13 241 830

60° 0.13 0 42+ ° * 25 ' -0.22

0 76+0-13 -0.49 357 750

70° 0.22 0 7fi+(0*24> -0.40

0 49+ ° * 31 -0,49 316 940

oo00 0.24 0 82+ (° * 18) -0.43 335 750

90° 0.28 0 93+(0*07) -0.49

316 750

a) Using the autocorrelation method (Eq. (3)).

k) Includes corrections for F. R .D . and the effect of an averaging interval of A= 2.5 MeV.

c) Using the peak counting method.

experimental data, this large value of T requires that very large corrections must be appljed to account for the effect of averaging with A=2.5 MeV, Such corrections result in values of R(0) which yield a rather small direct component (consistent with 0 for the ?0°, 80° and 90° theo­retical cross sections). Such large "compound" cross sections and large widths are not indicated by the statisti­cal analysis of the experimental data and also could not be accounted for within the framework of any reasonable statistical models of nuclear structure and reactions.Thus a self-consistent statistical analysis of these theoretical cross sections (treated as statistical fluc­tuations) yields results which are at variance with both the experimental data and statistical models. Two related questions present themselves at this point:

i) V.'hat would be the effect on the present statistical analysis of the experimental data if there were a signifi­cant component of intermediate structure of the type predicted by Fink et al. (Fi 72) present in the experimental data?

ii) Could the present statistical analysis give any positive indication of such structure if it were present in the experimental data, given the presence of statistical fluctuations with T~300 keV,

The answers to these questions again may be obtained from a study of synthetic excitation functions. Synthetic excitation functions were generated which contained both

a 3°0 keV and an 800 keV fluctuating component. These components were generated with equal intensities, i.e.

O V) dequal average cross sections. Values of R(0) 0 andr°K ’ obtained from an analysis of these excitation functions wore then compared .wi th those..obtained from excitation functions containing only the 300 keV com­ponent and are shown in Fig, IV-19. The results of the comparison depend on the value of A used in the re­spective analyses since the 800 keV structure is attenu­ated more than the J 0 0 keV structure, particularly for smaller values of A . We find that r obs increases by 15/S when the 800 keV structure is added and when A =1.5 MeV. If A=?,5 MeV, this increase is 22.%. Given the uncertainties in the values of r 0^’" and R(0)obs ex­tracted from the experimental data, such changes in r ° bs or in R(0)obs are not sufficient to seriously affect the analysis of the~300 keV strucutre. Similarly, the present analysis is not sufficiently sensitive to either detect the presence, or demonstrate the absence, of 800 keV structure at a level of intensity equal to that of the ~ 300 keV fluctuating component. It is thuspossible that some of the structure in the excitation

12 12function for the elastic scattering of ' C+ C originates with the virtual excitation of quasibound states (Sc 70,Fi 72). However, the positive evidence for the effects of this reaction mechanism, which has been sought in this study and in -studies of the inelastic scattering

Fig. IV - 19 The dependence of r ° bs and R(p)obs on the averaging

interval A from the analysis of synthetic excitation

functions with and without intermediate structure mixed

in.

R(0)obs

r0

SS(ke

V)

/

Fig. IV - 20 Comparison of the double resonance mechanism's

predictions for 12C (12C ,12C )12C(2+ ,4.43 MeV) to

the experimental data.

/

dcr/

dU(m

b/sr

)

INELASTIC I2C [I2C,I2C*(2+)] l2C 90°

0.01 IM25

c . m ^ e V )

structure observed.ip the elastic,scattering, It wouldbe interesting with regard to the question of intermediatestructure to have a nuclear reaction in which the couplingto excited states of the projectile and target were strong,and the compound elastic scattering were weak. In thiscase, compound nuclear fluctuations would not hinder theverification of the mechanism suggested by Fink et al. (Fi

Although no evidence for non-statistical behavior hasbeen observed in the experimental data studied here,Cosman et al. (Bi 74) have observed an anomaly in thecross section for ^2C(^2C,p)2^Na at E =19.3 MeV, Thecmorigin of this structure is not known and the absence ofany corresponding anomaly in the elastic channel and in

20the a + Ne channels investigated here is noteworthy.The a-cluster ''doorway’’ state model may also predict

resonant structure in the excitation function. The differ­ence between such a picture and the statistical compound nucleus may not be that great if the a-cluster model con­tains sufficient degrees of freedom and the intermediate levels overlap strongly. Clustering effects may also explain the large direct components observed in some of the ^2C(^C,a)2^Ne reaction channels studied.

Finally, there has been considerable discussion in ■T_6the literature on the possible existence, in ' 0+ 0

scattering, of two types of structure in addition to the gross structure (Ma 69,Sc 70X^he argument for two types of structure was based on a fluctuation analysis by Shaw

et al.(Sh 6 9) which yielded 80 keV at an average c,m.energy of 18.5 I'ieV, and on the observation of broader structure, 200-300 keV wide, at higher bombarding energies (see Fig. IV-21 taken from Maher et al. (Ma 69a). It should be noted, however, that much of this variation in width could be accounted for in the framework of the statistical model of nuclei. Included in Fig. IV-12 as filled squares are the values of T obtained by Shaw et al. (Sh 69) and values of r obtained by counting the maxima in Fig. IV-2), The latter widths are in the range 20 0 -300 keV, even though the data were taken with 25 keV c.m. resolution and the agreement with the semi- empirical predictions of the compound nuclear fluctuation widths is good. The verification of an intermediate mechanism in this system would require a simultaneous observation of both narrow and intermediate width struc­ture over the same range of excitation energy and a demon­stration that the wider peaks associated with the inter­mediate width structure could not result, with reasonable probability, from the narrower statistical fluctuations.It has not been possible within the time available for the present studies to carry out such experimental studies.

IV-5 DiscussionIn the previous sections Ericson’s formalism has

been used in relating the results of the fluctuation analysis to the nuclear reaction mechanism (y i and nuclear

16 16Fig. IV - 21 0 - 0 elastic scattering data (Ma-69a)

dcr/

doO

nb/s

tero

d)

E CM. ( M e V )

structure ( T) information,

Fd g, TV-77 shows the contributions o^ severat part:a1 wriV'T to the cross sections for the olcst.ic inelastic mid

tv/o of the react i r n chp.nnei s studied, Jt " s clear f.hn"hthe vpi id 1+y of Ericsor’s r’or,’i''i i sm is "hv no nears certain

hero since th.n main contribution to the cross scot" on come,

fron high partial waves and r^/E^ may net be largo enough

in. such esses, and the number of open channels becomesC'TTS n 1

The consequence of email r/P ratios ir: the formalism

outlined in. Chapter III are twofold.o., In the mathematical derive tiers used there, r / D » l

1-./-,,-. V/]0 ^ r *1 r-. ■* ” r f t p r O r,,T r ' ’

rue] cun '~c;:l" oo'7 b” an integral,

b, The small number of open channels results in a large

dispersion of the level, widths, , in the compound

nucleus and r. * T is no longer a safe assumption.A

Poth effects have been studied iuantitatively using

synthetic excitation functions. To study the first effect synthetic excitation functions with values of r/D as low o.s 2 were generated and analyzed. Mo effect on the widths extractd was observed and the effect of extracting R(0)

from "data" where r/D is small is shown in Pig. IV-23.It is shown here that for small r / D the values extracted for R(0) tend to increase.

It should be noticed, however, that when the da+a are

analyzed, with the running average method these effects on

12 12Fig. IV - 22 Partial cross sections for four outgoing C - C

channels calculated with a Hauser - Feshbach

statistical model, over the energy range 16 < E <21 MeV.c.m.

(MeV)

PARTIAL CROSS SECTIO N S - C T^m b)

Fig. IV - 23 Dependence of the correction factor R(0)°^8/ R(0) on

the value of IV D used to generate the excitation functions.

This dependence is shown for different values of the

averaging interval, A, used in the analysis.

JS/ - v s . r/oAVERAGING IN iu .. USED IN ANALYSIS

Fig. IV - 24 Dependence of the results obtained in an autocorrelation

analysis on the number of open channels i. e. the

distribution of the the level widths in the compound

around their mean.

(A©>DSq0j: sao

SYNTHETIC EXCITATION FUNCTION ANA

15 20 25 30 35AON UNITS OF D

R(0) tend to decrease in magnitude.Tho effect of many open channels wos studied hv re­

placing tho constant width T used in tho polo termsO(eq. 111-13) with T^-s picked from a \ distribution

centered around T . Fig. IV-24 shows the effect of havingonly 20 open channels - this corresponds to T/D values ofapproximately 3. From our Hauser-Feshbach calculationsfor the cross sections with limiting angluar momentumin the entrance channel we saw that.the lowest valuesoccuring for I/D were not less than~4 and usually muchhigher. We have also used a small running average, A~ 5 » in our fluctuation analysis. By inspection ofFigs, IV-23 and 24 it appears that the analysis performedwas valid and that it yields correct values for R(0); thechanges in R(0) due to small T/D ratios or few openchannels, are much smaller than the uncertainties allowedbecause of the finite range of data.

It may thus be concluded that the statistical modelfor compound reactions accounts very well for the narrow

12 12structure observed in the ' C+ C elastic scattering excitation functions. The rather large compound elastic cross section needed to account for this structure are provided for through the population of high angular momentum states in the compound nucleus, which then have a limited number of channels open for their decay. Likewise theselectivity in populating some highly excited states in20 1? 12 20Ne via the "C( C,a) Ne reaction is accounted for at

12 12 20 Fig. IV - 25 C( C, a ) Ne* excitation functions

in.'it. in M.-irt by the .largo compound crocs rootionc pro-20die-tod for transition:; ’lo high spin states :in No, Re-

20ofirtly the spine of come of the high lying states in ' Me

populated in this reaction wore measured (F i 7 4 ) and most of them indeed proved to have high spins. Fig, IV-25 shows the absolute cross section measured for some of

these, states (uncertainty ± 40%) and the Hauser-Feshbach

predictions fco tbe-o er-oss sections based on the spin

values shown in the figure, .A s has been noted before

while r.ot all of the cross section is accounted for by the

compound nucleus reaction meciruv] s m , it does appear to be responsible for a substantial fraction of that observed.

CHAPTER VThe 12C+*^0 System and the Statistical Model

In this chapter we report on measurements and on a12 16statistical analysis done on C+ 0 induced reactions,

It was of particular interest to see whether the impor­tant role the statistical reaction mechanism plays inproducing the structure and selectivity observed in the 12 12C+ C system was restricted to that system,

12 16The C+ 0 system was chosen since it has similar12 12experimental features to those of interest in the C+ C

system. Strong fluctuations superposed on gross structure were observed in the elastic channel (Pig, V-l) in addi­tion to strong selectivity in the alpha-particle channels. Measurements carried out here and at other laboratories(Go 71,Fi 73) have also shown that the states selectively

12 16 2 4populated in C( 0,a) Mg reaction have high spins (Js6).

V-l The *2C(^0,a)2^Mg and Reaction and ScatteringCross Sections— A Fluctuation AnalysisWe have measured excitation functions for the reaction

12C(*^0,a)2Slg* over the energy range 18 MeV to 21 MeV in the center of mass, in 60 keV steps and at extreme for­ward angle ( 0 ^ ~ 3 0), The apparatus and experimental

12 16C - O elastic excitation functions (S i-71)

A* •!L» 'XM• • . A • •

j i i i i i i i i i i i i— i— i— l I l8 12 16 20 24

I 1 1 ' I 1 1 1 | 1 1 1 | ' r

i . *- tf •

• •/ > . / V120

° c ° 0V o * 0 °0 o ° <

° -

L \ 130°

i • i *+/* ♦ * *. *• -♦ ♦

10

10

*140 •o • S0A O ®5 °n O OO 0)0

o o© n „

%• >150 - 4 v / H * f -A.- *. •/

C. M. ENERGY (MeV)

methods outlined in chapter II were used again. The samereaction channels were also measured over the same energyrange but at extreme back-angles ( 177° ). This wasdone by interchanging beam and target and scaling the pro-

12jectile energies. Natural C and SiO targets with areal2 2 densities of 20 yg/cm and 30 Pg/cm were used, resulting

in an energy resolution of approximately 100 keV in thecenter of masB. The bombarding energy ranges were 32^E12c-37MeV and 42<E16 <49 MeV. The resolution in the spectrumwas about 150 keV. Two spectra obtained at close center ofmass energies are shown in fig. V-2. The resolution ofthe a-spectrum and uncertainties in determining the energyloss for a-particles in the different absorbers force usto put a 70 keV uncertainty limit on the excitation ener-

24gies in Mg for states which are shown on fig. V-2,In general the states populated in one spectrum show up alsoin the other onei a noteworthy exception, however, is a

24group of states in Mg around 21 MeV in excitation which12is strongly excited in the C target experiment but which

does not show up in the other. This assymmetry persisted over an incident energy range larger than 1 MeV which is much larger than the fluctuation period found in the exci­tation functions analyzed. It is hoped that the comparison of the average cross sections, to a given state, obtained at such forward and backward angles, could yield information on the reaction mechanism feeding this channel. Pronounced assymmetry might indicate, for example, that a direct heavy

12 16 24, Fig. V - 2 Typical alpha particle spectra from the C ( O .a ) Mg*

reaction at ~ 2 ° and 178° laboratory angle.

COU

NTS

CHANNEL NUMBER

cluster transfer contributes to the observed cross section. Fig, V-3 illustrates in a simple way why one might expect assymmetry in the observed cross section if large clusters are transferred directly in this reaction. It might be ex­pected that such direct processes would dominate in grazing collisions, i.e. such processes are enhanced for large im­pact parameter scattering which could result in strong forward peaking. Such a simple picture is admittedly very crude} it reflects, however, our early hypotheses that the states involved might have strong cluster structure in­volving direct transfer of four or eight nucleons.

Figures V-4,5 and 6 present absolute cross sectionsoh

f o r excitation functions leading to 15 states in Mg overthe energy range studied. It must be realized that thesame problems that were mentioned in the analysis of the 12 12 20C( C,a) Ne reaction also exist here. Precise energy calibration and good alpha-spectra resolution are needed for the conclusive identification of states populated at such high excitation energies. Some of the "states", whose excitation functions were studied here have already been identified as multiplets (Gr 72,Fi 73)I e.g. 'the states at Eexc=l4*14 MeV (5’,8+), 15.15 MeV (4+,7“), 16.29 MeV (8+ ,9"»10+) and 16.55 MeV (8+ ,9‘).

Because of carbon buildup on the SiO target, which12 12 20results in many intense a-groups from the C( 0,0;) Ne

reaction some of which were identified in Fig. V-2, SiO

Fig. V - 3 Cluster transfer in C + collisions

Denotes transfer of three a ' s , o r C Observed a comes from 0 16

Denotes transfer of two a ' s Observed a comes from C12

12

Mg* excitation functions

o+*C\J

a*5

OJU+o

CMIc<oT

js /q u j ^ (u p z -o p )

2IO98

7

6

9

4

32

I0

6

9

4

3

2

I0

9

4

3

2I

0 16 + C12 — ► M g24*+ a

®LAB “ 2°E „ c in Mg24

32

I0

8

7

654

32

I0

94

32I

0

4

32I0

0 16 ♦ C12— - M g24* 4- a® L * B " 2 °

c'2 BEAM-c.m. MeV

O16 BEAM

75

targets were changed frequently during the experiment. Re­sulting normalization problems limit the accuracy of the extracted cross sections shown here to ±30$.

Quite strong fluctuations are readily apparent in the data. In most excitation functions, however, the fluctuations are superposed on some wider structure, 0.5-1 MeV wide, which is indicated in the figures with a heavy line.

A fluctuation analysis was performed on these data using the same procedures outlined before in section IV-2,The cross sections were divided by a running average of the data and only the ratio is analyzed. Tables V and VI show the results of the fluctuation analyses. The sample size -"n", averaging interval -"A" and a first estimate of the coherence width from peak counting (Br 64) -"T" are also shown. Using these values for A, T and n we were able to determine the combined correction factor arising from the effects of averaging and finite sample size. These corrections were then applied to R(0)ots and rots and the resulting values are given in column 5 and column 6 of the tables, followed by the quantity yd. The quantity given in column 8 is the average of the experimental cross section taken over the energy range studied. The average width, r , as calculated from the 11 cases analyzed, isshown with the standard deviation calculated for this sample.

1 2 16A fluctuation analysis performed on elastic C+ 0 cross sections measured at an angle of ~178°, (of. Fig. VI- )

TABLE V

*2C(16Ot a )24Mg* Fluctuation Analysis

Eexc J* R(0)°bs I**8 (keV) m T ( keV ) yd<a>

exp

6.01 4+ 0.103 47 0.64 ±0 .36 183 ± 53 1.13

7.35 2+ 0.076 41 0.47 ±0 .27 160 ± 46 0.54

7.56 l ” 0.082 38 0.51 ±0 .29 148 ± 43„ +0.18

-0.310.53

8.11 6+ 0.109 49 0.68 ±0 .38 191 ± 55 0 57+0’ 27 ° * 5 -0.572.26

9.27 2+ 0.074 45 0.46 ±0 .26 176 ± 51 0 73+° a 6 * -0.20 1.68

9.52 4+ 0.123 64 0.77 ±0 .43 250 ± 73 0 48+0’ 33 -0.48

1.60

13.45 6+ 0.038 29 0.237±0.13 113 ± 33 0 87+0* °7 ° -0.09

1.57

13.86 6+ 0.050 30 0.31 ±0 .18 117 ± 34+0.10

° - 83-0.111.32

14.14 8+ 0.040 28 0.25 ±0 .14 109 ± 32 0 87+0#07 0 -0.09

1.99

15.15 7“ 0.040 29 0.25 ±0 .14 113 ± 33 0 87+0#07 -0.092.00

16.29 8+ 0.080 49 0.50 ±0 .28 191 ± 55 0 71+3a7 * -0.24

2.37

I

Averaging Interval - A = 0.4 MeV

Sample Size - ( m = 13’) n = 13/IT

T (width) from peak counting ■ 200 keV

Average width - T = 160 ± 45 keV

TABLE VI

* 60 ( * 2C, a ) 24Mg Fluctuation Analysis

Eexc J* R (0 )°bs r °bs

(keV) R(0) T ( keV)yd

< a > €

6.01 4+ 0.142 48 0.89 ± 0.50 187 ± 54A „ +0.44

-0.340.64

7.35 2+ 0.035 42 0.22 ± 0.12 161 ± 47 n Q +0.07 -0. 07

0.69

7.56 l ” 0.094 58 0.59 ± 0.33 226 ± 66 « .,+0-22 -0. 35

0.47

8.11+

6 0.196 58 1.22 ± 0.69 226 ± 66 0.0 +0-68 0.80

9.27+

2 0.032 52 0.20 ± 0.11 203 ± 59 1.64

9.52+

4 0.139 53 0.87 ± 0.49 207 ± 60 1.09

13.45+

6 0.047 55 0.29 ± 0.17 215 ± 62 „ o,+0.09 -0.11

1.83

13.86+

6 0.066 55 0.41 ± 0.23 215 ± 62 2.05

14.14+

8 0.080 67 0.50 ± 0.28 261 ± 76 3.31

15.15 7~ 0.089 56 0.56 ± 0.31 218 ± 63n +0.20

-0.312.54

16.29+

8 0.135 51 0.84 ± 0.47 199 ± 58 -_+0.39

-0.402.53

Averaging Interval - A = 0.4 MeV

Sample Size - n = m/n = 13/tt

Width - r from peak counting = 200 keV

Average Width - T = 211 ± 25 keV

76

over the same energy range, resulted in an average co­herence width of r -220 ±56 keV and y(jas0*69+Q*2 * In following section we examine the narrow structure in the data by comparing statistical model predictions to the strength deduced from the data. The problem of the inter­mediate structure is taken up later.

V-2 Comparison of the Statistical Model to the Experimental ResultsWe first note that the average coherence widths of the

fluctuations observed in the excitation functions from the 12C(l6o,a)24Mg and l6o(l2c,a)2i,Mg experiments as well as from the elastic scattering data are all the same within their expected uncertainty limits. When 11 channels were averaged (Tables V and VI) the standard deviation from the average also had the magnitude of the uncertainty expected because of the finite range of data. When a straight-forward fluctuation analysis of the same data was attempted, ignor­ing the possible existence of intermediate structure, re­sults that are not consistent with the statistical model were obtained. Some unusually large widths were obtained, and the standard deviation from the average calculated for the same 11 channels was larger than the average itself. This could well be a signature for some unusual, or non- statistical, underlying intermediate structure in these excitation functions.

The semi-empirical prediction for the average fluctua­tion width expected at these energies, based on the

statistical model and obtained from Fig. IV-12 (A=28,E*»36 MeV) is 220 keV. This is in good agreement with the average width of 197±29 keV observed in the data.

Finally, the magnitudes of the fluctuating cross sections and the compound cross sections calculated with a Hauser-Feshbach formula (Eqs. IV-2,3) are compared in Fig. V-7, Table VII lists the channels included in the calculation, the optical model parameters used for calcu­lating the transmission coefficients and the level densities and pairing corrections used in Eq. IV-3.

In Fig. V-7 the central bar represents the Hauser- Feshbach prediction and the bars flanking it represent the average measured cross sections. Shaded parts on these bars delineate the limits that could correspond to compound or fluctuating part of the cross sections as calculated with a rather large uncertainty from yd. The deutron channels was excluded from the calculations shown here, its inclusion would lower the predicted cross sections by about 5$. The wide bars on the R.H. side are obtained by summing the experimental and the predicted cross sections separately and dividing by the number of cases. They represent, then, the average measured and deduced fluc­tuating cross sections and the average cross section predicted for these 11 channels. Caution should be exer­cised in assessing the significance of the agreement. In the case of some doublets, for example, cross sections were calculated only to the strongest member but the compound

77

Optical model parameters

Channel V (MoV) V T (MeV) It, , a , H a Ref.Heal Imag Real Iteal Imag hnag

28TABLE VH ( Si C .N . )

12c+16o 17.0 0.8+0. 2Ea) 6.49 0.49 6.10 0.15 Go 71

24Mgk* 54.4 9.8 a) 4.90 0.53 4.90 0.53 Sa 65

27A1+ p 52.2-0.3E 11.5b) 3.75 0.65 3.75 0.47 Pe 63

27Si+n 48.8-0.3E 11.5b) 3.75 0.65 3.75 0.47 Pe 63

8Be+2QNe 14.0 0.4+0. 15Ea) 6.10 0.49 6.10 0.49 W i 74

26A l+d 117. 18.9 a) 3.10 0.86 4.70 0.54

a) Volume absorption

b) Surface absorption

Level Density Parameters ( La 63, Fa ,68, W i 74 )

16rx 24__ O Mg 27Al

27Si

20Ne 26a i

28_.Si

a / A 0.136 0.149 0.137 0.137 0.152 0.152 0.116

5.13 5.13 1.80 2.09 5.13 0.00 3.89

The spin cut-off factor is evaluated using the rigid body moment of inertia

Fig. V - 7 Comparison of Hauser - Feshbach calculations to

12 16 24the observed C ( O .a ) Mg* cross sections.

The shaded area on each bar represents the limits

of the fluctuating part of the cross sections as obtained

from a fluctuation analysis.

cfo-

/d&

(mb/

sr)

,60 beam ,2C beam

^ C + ^ O — a +^4 M g *HAUSER - FESHBACH - v s - MEASURED COMPOUND

CROSS SECTION

Average Cross Section (14.14,8+)

(8.11,6+)(6.01,4+)

fll!v ;«/ t oI

II

(735,2*)(756, n

M ii

(13.86.6*)(9.27 2+) (13.45,6*)

(9.52,4*)

il |;li'

(15.15.7')(16.29,8*)

J j |r

iV * /j

AVERAGE

fraction, obtained in the fluctuation analysis of the excitation function of this doublet, is also underesti­mated. Emphasis should be put on the low lying states which are well resolved, i.e. (6.01,4+), (7.35»2+) and (8.11,6+) and perhaps on the overall agreement as seen from the average compiled for these 11 cases. On the whole it follows that the Hauser-Feshbach predictions tend to overestimate slightly the experimentally deduced average fluctuating cross sections (by less than a factor of ~2). It is also noteworthy that, to the extent that we are sure that the same state was identified at 0° and 180°, no strong asymmetry exists and the compound cross sections at forward and backward angles do overlap in magnitude for all cases.

A similar comparison can be made between the calcu­lated compound elastic cross section and the average fluc­tuating cross section deduced from the data. Using the same parameters as were used for the a-channels and taking the fluctuation width correction to be Waa, = l +6aa* as in the C+ C case we obtain a jjp-lO mb/str for the pre­dicted cross section. This agrees quite well with theaverage fluctuating cross section in the data which is

where yd was ob- given byi tained in our

<a>c = <o>exp<l-ydM mb M r fluctuationanaly-(Sect. V-l).It thus can be concluded, that the selectivity ob-

served in the C( 0,a) Mg reaction, as well as the

structure observed in the measured elastic excitation function are accounted for by a statistical compound nuclear reaction mechanism.

There remains one feature, however, in the reaction data which is not reconciled with the statistical model, namely, the broader structure appearing in many of the alpha channel excitation functions which exhibits a correlated minimum in most of the channels observed at 19.7 MeV incident energy. This phenomenon is discussed in the following chapter.

CHAPTER 6

The ^2C+^0 System - The Resonance at E. _ =19.7 MeV ------------ «... — ---------------------- c ,m,--------

VI-1 IntroductionIn the previous chapter it was shown that essentially

12 16all of the structure in the C+ 0 elastic excitationfunction which does not stem from potential scatteringcan be accounted for by statistical compound reactioncontributions. Furthermore, the selectivity observed in

12 1 6 2 Llthe C( 0,a) Mg reactions populating highly excited2Astates in Mg as well as the structure in the excitation

functions leading to those 3tates could be accounted forby the same mechanism. There is one already mentionedexception, however, namely the correlated minima observed,in many of the alpha particle channels studied, at anincident energy of E -19.7 MeV. The observation of ac «in •correlated resonance in the elastic channel (Si 71) atthe same incident energy was clearly associated with thiscorrelated minimum. It should be noted also that the sameelastic data show correlated structure at E„ „ =13.7 MeVc.m.which coincides with the energy at which Halbert et al.(Ha 6 7) have observed a non-statistical fluctuation in

their eariier studies of the 12C(1^0,o,)2^Mg* reactions.Angular distributions measured on and around the 19.7

MeV resonance (Si 71) bear evidence that at the resonance energy the decay into the elastic channel is governed by the J=L=1^ partial wave. Fig. VI-1 taken from Ref. Ma 72 shows the measured elastic angular distribution and the angular distributions calculated with a P^(cos0) Breit Wigner amplitude added coherently to the optical model background.

Without detailed knowledge of the structure of the state involved it has been convenient in our subsequent work to refer to it as a specific resonance corresponding to a J=l4 state at 36.5 MeV of excitation in 2®Si, having a total decay width of ^00 keV. Such a strong isolated state, being excited so high in the continuum is quite un­precedented in nuclear physics and therefore merited much more detailed study. It was clear from the outset that if such resonances indeed are a feature of the extreme con­tinuum region ( 5 times the nuclear separation energy) they would open up an entirely new class of nuclear spectro­scopy - both theoretical and experimental.

A more detailed experimental study of this resonance was thus initiated. Results of this study are reported in the following section. Measurements were also undertaken at other laboratories (Ma 72,Co 72) and the information obtained from these studies will also be included herein

12 12Fig. VI -1 C - C elastic angular distributions at three

energies in the vicinity of the resonance.

dcr/d uj ( m b / s r )

82

since it will be of importance in discussing some of the models which have been advanced for this resonance and its population.

VI-2 The Resonance At 19.7 MeV - Further Measurements One interesting feature of that resonance was its '

enhanced prominence at the more backward angles measured as far as 160° in the center of mass. As a first step we therefore extended the elastic scattering measure­ments to the most extreme backward angles and also studied some of the inelastic channels at this same angle.

The 12C(1^0,160*)12C differential cross sections atan angle of 0 - 1 7 7 ° in the center of mass were measured cmat energies around the resonance and were also extended

2to higher incident energies. A 20 ng/cm natural Carbon16 1 2 target was bombarded with 0 and the recoiling C nuclei

were detected at 2°. The apparatus and method used are12described in sect. II-2 where we also showed a C recoil

spectrum obtained in such an experiment carried out, how-oever, with a 100 ug/cm thick carbon target (Fig, II-5).

The resolution in the spectrum shown is about 250 keV.While the resolution is slightly better at the lower energies around the resonance where we used a thinner target, the poor resolution still left unresolved the two particle groups corresponding to the 6,05 and 6,13 MeV states in ^0. The measured excitation functions are shown in Figs. VI-2 and VI-3. Strong fluctuations are present in

12 16Fig. VI - 2 C - O elastic and inelastic scattering data

Thin target (20jig/cm2) data.

100

50

05

0

10

5

0

5

05

018 19 20 21

— Ec.m.(MeV)

Fig. VI - 3 C - O elastic and inelastic excitation functions

2Thick target data(I00 pg/cm )

40

20

0

10

5

0

10

5

0

10

5

0

5

0

Ec.m.(MeV)

the elastic scattering data, which, in general, can be accounted for by statistical compound contributions (see sect. V-l,2), The resonance seen in the elastic data persists also at these backward angles and in some of the inelastic data. In Fig. VI-3 data at higher energies are shown. At least two more "resonances" appear, the maxima at E =22.3 and 23.9 MeV are correlated in the elastic and in two of the inelastic channels. Correlated structure at 22.1 MeV has also been observed in some of the a-particle channels by a group at Saclay (Ch 73). The structure at 22.1 MeV is almost 1 MeV wide and appears to be an unre- solved doublet. We have used a 100 ug/cm carbon target in these measurements and measurements with a thinner target are required to establish whether this wide cor­related bump in the data correspond to a single or several closely spaced resonances.

One attractive model which has been suggested for themechanism underlying that resonance (St 72b\ postulates

12 12the formation of a C-a- C quasimolecule in which an1 2alpha-particle is shared by the two C cores and binds

them into this molecular complex. The large alpha cluster16 12 7parentage of the 4p-4h states in 0 observed in C( Li,t)

12 7studies (Pa 72), and the similarity between C('Li,t) and 12C(lb0,12C) reaction spectra at higher energies (cf.Fig. VI-16), showing theoL-transfer nature of this16 12( 0, C) reactiow were suggestive of such a model.

The exchange of an alpha particle can be initiated through

83

84

the entrance channel by direct dissociation of *^0 in its ground state or by a two step mechanism wherein is first excited to one or more of its 4p-4-h states and then breaks

1 Oup into ano(+ C configuration. The question of which of the two states (6 .0 5,0+ ) or (6.13,3”) resonates in theinelastic excitation, or whether both do, is of great im-pnj-tj.'nue in eva'iir*t > r..- i,ho vnlf’H y of any such model.

? opi t i r i p v i n t p t 1 t " . } . i f ? c I?; t e i r i ^ i'A

would p r e f e r ^ * ■ h- st.-ite wh :<-h

has a large ap-ah < -p -.. i r-i mac i:vi

J " state with 5 is >-.• ••»>- • • -» • ,t'*p . K « . » •mentally this rai :• < - of f/-e difI i no !in resolving these two c.ioaoly spaced states (65 keV apart).

1 o *1 ZA high resolution measurement of C+ 0 elastic

scattering was carried out at the resonance energy using the Yale Multigap magnetic spectrograph (Ko 70), The

pexperiment consisted of bombarding a very thin, 5^.g/cm , carbon foil with a 4-5.96 MeV *^0(7+) beam. After a 30 hour exposure the photographic plates recording the reaction pro­ducts were removed, developed and scanned. The recoiling 12C nuclei were identified by the length of their tracksin the emulsion. Plates were positioned at forward angles,corresponding to large center of mass scattering angles

12when C recoils are observed. The plate at 3 3/4-° had toomuch background from small angle *^0 scattering and could

12not be counted. Fig. VI-4- shows the C recoil spectrum atA IP7 1/2 lab. The C recoil peak at the position cor-

Fig. VI - 4 C recoil spectra on magnetic spectrograph plate.

16Several scattered O peaks are shown too.

12

DIS

TA

NC

E

ALO

NG

F

OC

AL

PL

AN

E

(cm

)

COUNTS PER 0.5 mm

85

responding to 6.13 MeV excitation of ^ 0 has a full width at half maximum of ~70 keV. There are very few counts at the position corresponding to the 6.05 MeV excitation of *^0. Fig. VI-5 shows a detail of the spectrum of Fig. VI-4. By fitting two gaussian peaks to the spectrum in the region corresponding to 6.05 and 6.13 MeV doublet, limits on the ratio of the yields were obtained for the two states and are indicated in the figure. More than 90$ of the yield on resonance comes from the (6.13,3”) level excitation, at the angle measured. Fig. VI-6 shows the expected angular distribution for the decay of a J=14 compound nucleus to the 0+ and 3” levels. The measurements were done at an angle that would permit theobservation of the 0+ yield, were it there.

12 16Gamma rays emitted from C+ 0 collisions were studied next. The 3“ state in *^0 at 6.13 MeV can decay, to the ground state by emitting a 6.13 MeV E3 gamma-ray. The 0+ state at 6.05 MeV can decay only by pair produc­tion and there is a negligibly small probability that all of the energy carried by the pair will be deposited in the same crystal. Subsequently, observation of y-rays at energies corresponding to 6.1 MeV can be attributed to the 6 . 1 3 decay to the ground state.

Fig. VI-7 shows a diagram of the experimental ap­paratus* the Faraday cage was charged to 5000 Volts in order to prevent secondary electrons scattered from the target and collimators from reaching it. A ty­pical 1024 channely-ray spectra is shown in Fig. VI-8.

Fig. VI - 5 Detail of spectrum shown in previous figure.

CO

UN

TS

PER

0.5

mm

D I S T A N C E A L O N G F O C A L P L A N E ( c m )

Fig. VI - 6 Angular distribution expected from the decay of

the J a 14 resonance.

W id

)

CENTER OF MASS ANGLE

Fig. VI - 7 Set up of the y -ray experiment

TO ANALYZER

LAMP

TO FARADAY CUP

X-RAY EXPERIMENT

VOLTAGE DIVIDER

PM. TUBE

Nal(TI) (5"x5")

86

The energy calibration was obtained by using the y-rays from standard calibration sources* (1 ^Cs, 0,662 MeV), (88Y, 0.898, 1.836 and 2 .7 3 k MeV), (6°Co, 1.173 and 1.33 MeV) and (se-Po» ^.^3 MeV) y-ray sources were used. The strong transition seen in the spectrum around 1.37 MeVand 1 , 6 3 MeV are the 2+— *0+ (ground state) transitions

2 k 2 0 j.in Mg and Ne, respectively. These 2+ states can befed in numerous ways and the normalized yields for their decays should vary slowly with energy, likewise the total y -ray yield should show a very slow and smooth energy dependence. That this is the case was verified by study­ing these 1.37 and I . 6 3 MeV transitions and the total -ray yield normalized to the B.C.I. (Beam Current Integrator)counts, A slow rise was observed in the normalized total

12yield that could be mainly attributed to C buildup on the target of approximetly 20# the original target thick­ness ( a 20 fig/cm natural carbon foil was used). We then used the total y-ray yield in a spectrum to normalize the yields under individual peaks in that spectrum, thusautomatically correcting for any target thickness changes

12due to C buildup and also for dead time losses. An examination of the spectra in Fig. VI-8 shows hardly any peak corresponding to yields under the 6,13 MeV y-ray photopeak position. An expanded detail of this region in Fig. VI-8 is shown in Fig. VI-9 after summation of the counts in every k adjacent channels. The position of the 6.13 MeV photopeak and the 1st and 2nd escape peaks are

Fig. VI - 8 Typical gamma ray spectrum obtained from O

2 12bombardment of 20fig/cm thick C target.

16

Fig. VI - 9 6.1 MeV gamma ray yield compared at energies

on and off the resonance.

COUN

TS

PER

CHAN

NEL

/

CHANNEL NUMBER

indicated on the figure. It can further be seen from Fig. VI-9 that the 6,13 MeV y-ray yield is enhanced on resonance relative to the background in comparison to the yields off resonance. Yields under the photopeak were extracted using the same fitting program described before (sect. II-3) but in order to minimize uncertainties due to the large background several constraints were imposed on the fits. The triplet of photopeak and 1st escape and 2nd escape peaks were fit together. A fit at the reson­ance energy was carried out first (Fig, VI-10) and the relative position, areas and width of all three peaks were constrained in all the subsequent fits, done at different energies, to the values thus extracted. The bombarding energy dependence of the normalized yield under the photo­peak is shown in Fig. VI-11,

In order to obtain the absolute cross section the target thickness was measured by studying the range of alpha particles from a Po source in a combined target and air layer, and was determined to be 20 yg/cm+8 jig/cm . The flux of beam particles was calculated using the amount of charge collected in the beam current integrator. The formulae used are shown on the next page.

Eeff was taken from tables in ref. Ma 68. Since the outgoing ^0*(6,13 MeV) is probably unaligned, the cor­rection for non-isotropy Cy was taken to be unity.

Fig. VI - 10 The fit of the three gamma ray peaks associated

with the gamm ray of 6.1 MeV, taken on resonance

and used as reference.

COUN

TS

PER

CHAN

NEL

4000

o '6 + c 12Ec.m. = 19.7 MeV

GAMM A-RAY S INGLES••

3 0 0 0

2nd escape Is* escape

5.11r 5.62~ r

6.13 MeV

2000

1000

• • / \ * . •

* r w v \-***>\ V \/ v

^ \ < T"-v / \ \

Ol— 100

JL120 140 160

CHANNEL NUMBER

1 ••••••••••••

180

Fig. VI - 11 Gamma ray excitation functions in the region of the resonance.

( only normalized yield is shown here)

COUNTS IN PEAKT OT A L COUNTS

N

VI-l Y = (cy A^ab Eeff Nbeam Ntarget 5 ,Y = Observed y-ray yielda = Cross section for 1^0*(6,13) excitationC = Angular correlation co-efficientAO= Solid angle of acceptance for Nal crystal

E = Photopeak detection efficiency for 6.1 MeV -rays in a 5" x 5" Nal(Tl) crystal

2target ” Number of target particles per unit area (cm )Nbeam = Numt)er of beam particles that fell on target =

_ B.C.I. counts ♦ B.C.I. (units) unit charge * Zeff

Zeff w effec'fcive charge of projectiles reaching the B.C.I. (Beam Current Integrator)An additional correction factor Ay >1 was applied to

take account of the attenuation of y-rays in the Aluminumabsorber which was put in front of the crystal to absorbhard X-rays emitted. The resultant relation for theabsolute cross sections was thenVT -2 _ Yield

a B. C. I (counts)x From the two experiments described here two facts

become clear. The 6.13 MeV, 3" state in *^0 resonatesand from the spectrograph data one can see that for thescattering angle studied at least 90$ of the resonant yieldcan be attributed to the 6.13 MeV state.

Malmin and Paul (Ma 73) obtained a more stringentupper limit of 1*200 for the 0+ to 3" cross sections ratio.They measured total cross sections for the 12C(1^0,1^0* 6,05))inelastic scattering by observing the e+ - e" pairs produced

89

in the decay of the 0+ state at 6.05 MeV to the ground state in 1 0.

It has been suggested at this point (Ma 72,St 72)thata double resonance mechanism could be responsible for this

12resonance. In such a framework the coupling of the C and 160 excitations to the elastic channel would produce thestructure seen in the elastic and inelastic channels in amanner described already in section 1-2. Such a mechanismmay explain the correlated enhancements in the (3”, 6.13 MeV)inelastic channel, and perhaps even the 6.0 MeV energydifference between the two resonances at 19.7 and 13.7 MeV,but it remains to be seen how well a calculation of thetype Scheid et al. (Sc 69) and Fink et al, (Fi 72) havecarried out for the *^0-*^0 and *2C-*2C systems fares in

12 16explaining the phenomena observed in C- 0 scattering.One would be especially hard pressed to explain, within

the framework of any of the two models already suggestedfor the resonance, the surprising enhancement observed atthe resonance energy in some of the proton outgoing channels.

Cosman et al. (Co 72) have measured 12C(1^0,p)2^A1*proton excitation functions and found two protons groups

27corresponding to 15.53 and 15.83 MeV excitation in Alwhich resonate strongly at E =19.7 MeV. The excitationcmfunctions leading to these two states obtained at two angles are shown in Fig, VI-12 which is taken from their work (Co 72), A finer structure is observed for the reson­ance, which is not apparent from other data (shown in parts

12 16Fig. VI - 12 a) Cross correlation function for elastic C+ O data

( Ma-72 ).

12 16b) . C+ O inelastic scattering excitation functions.

12 16 27c) C ( O.p) A l* excitation functions (Co 72).

a and b). Using the experimental system which has been described in Chapter II, we have studied the *2C(*^0,p)2^A1 reaction at 2° outgoing proton's angle (i.e. 3° in the center of mass). By interchanging projectile and target the same reaction was studied at 177° in the center of mass. Since the projectile and target are both spin zero particles there is only one angular momentum in the entrance channel that can feed a certain compound state of spin J, the decay of the J=l4 resonance should then be symmetric around 90°. (This is essentially the requirement l^+lg+I^ Even imposed on the L values appearing in the general expan­sion for the unaveraged compound reaction cross section in terms of the legendre plynomials P^cosS)). The three spectra shown in Fig, VI-13 demonstrate how strong is the enhancement at the resonance energy of the two protongroups corresponding to 15.5 MeV and 15.8 MeV excitation

27in 'Al, The spectrum which was measured off the reson­ance energy shows no selective population of any state in 27Al, this may be attributed, in part, to the rather poor resolution, approximately 150 keV F.W.H.M., in the proton spectra. At the resonance energy, however, these statesshow up very strongly at both measured angles. Excitation

/

functions for these states are shown in Fig. VI-14. The finer structure of the resonance, which was observed by Cosman et al. at 15° and 23°, persists also at the other angles measured here.

The salient results of the experimental work carried

spectra at ~ 2 ° and 178° laboratory

COUN

TS

CHANNEL NUMBER

12 16 27Fig. VI - 14 C ( O, p ) Al proton excitation functions in

the energy region of the resonance.

dcr

/d^(

mb

/sr)

out on this resonance at Brookhaven National Laboratory, Argonne National Laboratory and Yale, to mention only a few laboratories, are summarized in Table VIII. A number of the theoretical ideas pertaining to this resonance will be discussed in some detail in the following section,

VI-3 The Resonance at E^ _ =19.7 MeV in the System" C « lu #" ' ' ”-Models Suggested For Its Formation and Decay-Inspection of the elastic angular distributions

measured at the Argonne National Laboratory (Ma 72) on resonance, shows that the decay into the elastic channel is governed by an L=14 partial wave, permitting a spin assignment of J=l4 for the resonance.

In principle a more exact procedure would consist of a phase shift analysis performed on the excitation functions and angular distributions in the region of the resonance. Such a procedure was attempted (Sh 72) using the three angular distributions and two excitation functions shown in Fig. VI-15, A Breit Wigner resonance term was super­posed on an optical model background. The results, as can be seen from Fig. VI-15, are rather discouraging; the reason for this may lie in the appreciable compound contributions to the elastic cross section forming the background. With such analyses one could not venture beyond confirming the J=l4 spin assignment. This in­ability to carry out a phase shift analysis also precludes a rigorous partial width analysis on the decay channels of this resonance. Estimates for the partial widths were

TABLE VIII

Summary of Measurements Carried Out on the Resonance at E_m=19.7 MeV in the 12C+l60 System

1. Elastic Channel! (Si 71,Ma 72)a. Resonance pronounced at backward anglesb. Angular distribution indicates J=l4

2. Inelastic Channels! (St 72,Sh 72,St 73»Ma 72)a. Many studied but only (6.13,3”) l80*

coupled to resonance.

3. Alpha-particle Channels. (St 72,Br 71)b. Correlated minimum appears at E =19.7

MeV at 0 = 3 ° , 177° and 12° cm cm

k. Proton Channel! (Co 72)a. Two states in 2^A1* at Ex=15.53 and 15.83

are coupled to resonance, p-angles observed 8 =2®, 1 5 °, 22.5°, 178°

Q5. Deutron Channel, Be channel: no evidence yet for for resonance

Fig. VI - 15 Results of Phase shift analysis for elastic data

( Anti resonance is a resonance with the phase

varying in an opposite direction to the usual)

9 0 °

177°

\19.5 MeV

19.8 MeV

20.3 MeV '■

CI2 - o 16 PHASE SHIFT ANALYSIS

L= 12 BW-Resonance L=I4 BW-Resonance L= 12 Antiresonance

obtained, however, by taking the apparent 400 keV width to be an estimate for the total width. Cross sections for the resonance in the various channels studied were obtained by first using differential cross sections to normalize anticipated angular distributions which were then integrated to obtain the total cross section. For the inelastic cross sections we also had the absolute cross sections determined from our y -ray experiment.

As mentioned before, the amount of compound elastic background in the elastic cross sections at 177° c.m. is comparable to the optical model background. Since the resonance also looks very prominent the background was neglected altogether. The resonance is thus described by a single level formula, (La 58) for the elastic as well as reaction channels.

3 2 J rXc’>(EJ +AJ . E)2+i(rJ,2

The contributions of distant levels are also ignored i.e. the background R-matrix, RQf is taken to be zero. In this limitVI-4a i i 2

Xc " c yXc

VI-4b =t kR

C F 2 + G2 R=R,1 1 c

2 ewhere y^c - is the reduced width, -is the penetrability, L-the angular momentum, F^ and G^ are the regular and ir­regular coulomb wave functions.

93

= f2'i+lV{'li+iy wher*e 1 and i are the target's andprojectile's spins respectively,

j Ir = .T I\ - is the partial width and the sum isAC *{/, S ACtaken over all t,s such that -t+s=J, s = channel spin,r J = T r J - is the total width.A c Xc

For the elastic channel l=i=0 and only one l value is permitted; for the reaction channels with non-zero channel spin the lowest t-value possible is taken. Table IX summarizes the results of this analysis. The uncer­tainties in the partial widths reflect the uncertainty in the magnitude of the extracted cross section; other groups' data were also used and are referenced in the table. The errors arising from neglecting the background and the contribution of distant levels to the R-matrix were not considered here. The reduced widths were calcu­lated using an interaction radius-R shown in the table; choice of different radii may change the penetrabilities drastically, For the proton channel with 2^A1* excited to 15 MeV the maximum orbital angular momentum is 3 and thus the lowest possible spin that can be populated in 2?A1 is 21/2, As shown in Table IX the reduced width for the proton channel is only a small part of_ the single particle width. For the elastic and inelastic channels, on the other hand, considerable configuration overlap is required to account for the strength of the resonance in these channels. If optical model transmission factors were used instead of the coulomb penetrabilities somewhat

28Compound Nucleus: Si Excitation Energy: 36.5 Mev Spin(J ): 14 _____ — _ c

TABLE IX

OutgoingChannel

12C+160 (g .s . ,g .s . ) 12C+160 (g .s. ,6.1) p +2?A1 (15.53 Mev) 27 * p+ Al (15.83 Mev)

expa 25 ±6 mb 30 ± 10 mb 10±3 mb ( Co 72 ) 6± 2 mb ( Co 72 )

Partial Width 92 ± 23 keV 94 ± 35 keV 31 ± 9 keV 18 ± 6 keV

S= Maximum Channel spin 0

13 1 0

I

I 19 1 10 1 11

' |

1 I

9 ' 10 | 11 1 1

L= Lowest ang. momentum 14

111 1 14

i5 ! 4 1 3

1

1 1 5 | 4 | 3

. 1

P Penetrabilities0.842 0.218 0.0048

1

10.085| 0.417 | 1.140

1 1 I

1 1 0.095 1 0.450 1 1.312

1 1 ■ |

k b / ' 16.50 13.70

1 i

3.60

1 1

3.50

Reduced Width 0.054

1i

0.216 i 0.050**)i •>

0. 183 , 0.037 1 0.0141 1

0.095 ' 0.020 ! 0.007(Mev-fm) I 1

1 11 1 1 1

Wigner Limit 0.217 0.217 2.230 2.230(Mev-fm)

* * ) The cross section ratio a (6. 05, 0+) / a (6 .13, 3 ) = 1 / 200 was assumed ( Ma 73 )

different values are obtained but the general trends mentioned above persist.

The above partial width analysis should not be con­sidered as a rigorous structure analysis of the resonance

28in Si, but it serves to indicate which of the resonating exit channels does require considerable overlap of that exit channel configuration with the structure of the reson­ance, Especially noteworthy is the fact that the decay of

p Qthe J=14 resonance at 36.5 MeV excitation energy in Siwould proceed into p+2^Al* with the observed strength oncea spin 21/2 level in 2^A1 can be reached, even thoughconfiguration overlap of this compound state with that of

27the p+ 'Al exit channel is extremely small. As a resultit is not necessary that the mechanism suggested for thisresonance account for an overlap of the state populated

27with the p+ Al channel configuration in spite of theobserved enhancement in cross section observed for the

27 *p+ 'Al channels. Another observation that follows fromTable IX is the large difference in penetrabilities forthe transitions to the 0+ and 3" states.

Although these observations do not provide any evidencein favor of the double resonance mechanism or the forma-

12 12tion of a C molecule they do not necessarilyrule out these models.

The possible role of an a-exchange between the two12C cores as an underlying mechanism for that resonance was studied first. The main arguments in favor of an

a-particle exchange mechanism hinge on the following facts* The resonance is induced by the grazing partial wave which

1 ? 1/) ip 1 c.usually participates in direct reactions. C( 0, C) 0nis a "good" a-transfer reaction* our comparison of ('Li,t)

1 f\ 1 P 1 fi(Pa 72) and ( 0, C) spectra from reactions on 0 athigher energies show a striking similarity as can be seen in Fig. VI-16* especially one should note the strong population of the 4+ member of the Ap-4h rotational band in at 10.96 MeV by both reactions. Finally, the enhancement in the average cross section at backward angle, could be accounted for by elastic a-exchange,(Oe 70) as illustrated in Fig. VI-17. An especially attractive formalism to treat such resonant particle exchange is the theory of Linear Combination of Nuclear Orbitals first suggested by W. Von Oertzen (Oe 70). The elastic scattering and the elastic transfer are treated in a simple two state approximation. The nuclear states describing the transferred particle, or cluster, as bound to one or the other cluster, are combined in a symmetric and an antisymmetric combination. Such formalism incor­porates the treatment of multiple exchange of the trans­ferred particle between the cores and when the relative velocity of the colliding ions becomes comparable to the ortibal velocity of the exchanged particles resonant en­hancement occurs. Such conditions could materialize in heavy ion scattering because of the high angular momentum and coulomb barrier which act to cancel much of the

16 12 7Fig. V I - 16 Comparison of ( O, C) and ( Li, t ) spectra

12from C target bombardment.

I\» .& <J) OB COUNTS

16 12Fig. VI - 17 0+ C elastic scattering and alpha transfer.

do-dft

0 16 + C 12

16

1 2

96

incoming kinetic energy. Some of the mathematical detail and the approximations involved in such treatments are described in appendix-A. In what follows the highlights of this formalism are listed. The final states of the system describing the elastic scattering and transfer are combined into symmetric and antisymmetric states,VI"5 A + (X + B ) (el.) h x *(B)

A + C =( A + X ) + B (tr .) s Xa b * ( a )

x(AB) describes relative motion and >3>(A), 'I'(B) describestates with X bound to A or to B respectively.

% , a = T { * { A )t

The total wave function of the system beingV = X * + X *Aa a As s

Xa „-are relative motion wave function, symmetric and9|aantisymmetric with respect to core exchange.

As long as the energy is not too high above the barrier, and the nuclear cores A and B do not approach each other and are not excited or changed in any way,If describes all the states the system can be in. TheSchrodinger equation for ^ is given by H ^ = E'J'where (HAB=Relative motion of cores) and goingthrough some algebra, which is described in Appendix A, one obtains the following Schrodinger equation for the relative motion of the cores.VI-6 ( A,ti + V AT, + J ) x — E x AB AB s,a As,a cm As,a

Jg a is an effective exchange potential given by

VI-6a j = - (~)/tjs ,a 1 ± ( - ) l S

wherei j = Jdr *(A) VfiX *(B)and s = Jdr tf(A) tf(B)more detailed expressions for J and S can be found in the appendix. VfiX is the potential binding the ex­changed particle to one of the cores. At infinite sep­aration (R -» ® ) the exchange term J vanishes and X „ 0S f clbecome degenerate states.

In studying the effects of this process on theelastic scattering the assumption is now made that xdescribes also the relative motion in the elastic channel,i.e. AB-»AC we than solve eq. VI-6 to obtain the elasticscattering amplitudes S(R) and J(R) were evaluated byexact numerical integration. For V£X a Woods Saxon 12C+ a. potential obtained in elastic scattering studiesby Brady et al.(Br ) was used. The real well parametersof this potential were V=66 MeV, r0=1.8f, and a=s0.353f, howeverthe depth has been varied in order to obtain the desired

12number of nodes for the bound state C+a radial wave function, but the surface was kept the same during this variation of depth by readjusting the radius and dif­fuseness of the well. The number of nodes in the radial 12C+a wave function was obtained from cluster nodel consideration. Energy conservation is applied to the cluster state and four single particle states in Har­monic Oscillator potentials

97

98

VI-8 [2(n.-l) + l i ] = 2 ( n^- 1 ) + L

ni’ = ra(*ical quantum number and orbital angular momentum of particles in IP# shell1 2n , L = radical quantum number of a-particle in C-a

potential, and a- C relative orbital angular momentum

n.=l 1^=1 for IP# nucleons. L=0 since for lb0 (ground state) 12J=0 and C and a-particle also have spin zero ground states.

We get then na = 3. In the above notation (Sh 6 3) n a is the number of nodes excluding the one at zero but including that at infinites.

12Fig. VI-18 shows the a- C potential used and theresulting wave function. The Harmonic Oscilla or wavefunction with the same tail is also shown for comparison

12(broken line). This wave function and that same C-a potential were now used to evaluate the exchange integral J(R) shown in Fig, VI-19 where it is compared to the exchange potential obtained with the Buttle Goldfarb ap­proximation (Bu 66, Bu 68) previously used (Oe 70). Com­parison is meaningful only for radii greater than 7 fermiwhere the L.C.N.O. approximation is applicable. Fig. VI-20

12 16shows C- 0 elastic scattering data measured at ArgonneNational Laboratory (Si 71) and Yale and the calculatedcross sections for it. The optical potential was chosento fit the data at angles below 100°c.m. and the exchangepotential was then coupled in to reproduce the magnitudeof the cross section at backward angles. A spectroscopic

16 l ?factor of S=0,7 for the < 0 | C®a > overlap was

4

Fig. VI - 18 C + a cluster wave function ( radial dependence )

12and C +0!potential well.

12

Fig. VI - 19 The dependence of the exchange potential on the

relative core distance, also the one obtained with

the Buttle - Goldfarb approximation is shown for

comparison.

8 10 12 RADIUS (FERMI)

14

12 16Fig. VI - 20 C + O elastic excitation functions and Optical

Model predictions with exchange included.

mb

/sr

OPT. MODEL + E X C H A N G E

VRE =7.53 + 0.4 xE VIM = -16.5 +2.04xE

RR =6.43 fm AR =0.45 fm R I =3.9 I fm A l =0.80 fm

S = 0 .7

■c.m.

needed to produce enough strength in J(R) (see Appendix A) to raise the elastic cross section at backward angles by the missing order of magnitude. Fig. VI-21 shows the ex­change potential used (bottom part) and also illustratesthe proposed mechanism for the formation of the resonance.

12 12A C-a- C molecule formation is proposed for which an ex­changed particle serves as a binding agent. In orderfor such an exchange process to resonate the relative velocity of the two cores must be comparable to the orbi­tal velocity of the exchanged particle ( ). Such conditions could exist in the entrance channel when the coulomb and centrifugal barriers balance out most of the kinetic energy, and for E « 20 MeV the L=l4 grazing partial wave providesc«iu *a sufficiently high barrier.

For a semiclassical description the resonant transfertime for the exchanged particle (Oe 70) is given byVI-9 r =“A y tr J (R)

With the strength postulated for the exchange potential the life-time of this molecule should correspond to about 600 keV (depending on the distance R at which J(R) is evaluated), which is comparable to the magnitude of the observed width.

There are several problems with this approach. A1 12rather large spectroscopic factor for ( 0 | C ® a ) (all

in ground state) is needed to reproduce experimental obser­vation, Theoretical estimates of this spectroscopic factor (Dr ?4,Ic 73) are considerably smaller than the 0.7 figure

12 16Fig. VI - 21 c - O quasimolecular potential and the exchange

potential as a function of the relative core distance.

\

V(R)cl2-o16(MeV)

J(R)(MeV)

C ' 2 - 0 16EXCHANGE POTENTIAL

CORE-CORE DISTANCE (fm)

IOC

used here.A smaller exchange potential and subsequently a smaller

spectroscopic factor will be needed if we allow also othermechanisms to account for the rise in the average elasticcross sections at backward angles. We have shown, for example,that about 40% of the cross section at 177° can be attributed,on the average, to compound nucleus mechanism contributions.

12Some additional ^-dependence in the C- 0 optical po­tential of the type suggested by Robson may be present(Ch 70). Another problem lies in the supposition that 16 12( 0, C) reactions are "good" a-transfer reactions, some

of the evidence for this hypothesis has been questioned recently (Ro 72). The decay pattern of the resonance with the small reduced width in the (0+,6 .0 5) channel can hardly support such mechanism, the much larger spectro­scopic factor, or reduced width to single particle widthratio, for the (3”» 6 .1 3 ) decay is rather disturbingsince the (0+, 6.05) state has a much larger parentage to

12the C+a channel.The strong population of the (3”i6.13) channel via

the decay of the resonance on the other hand is suggestive of a double resonance mechanism (Ma ?2,St 73a) as the mechanism underlying the resonance.

An alternative approach, very different from the various intermediate process mechanisms, was suggested by Moldauer (Mo 6?). In this approach the intermediate reson­ant structure, amidst uncorrelated narrower fluctuations, such

101

as seen by Halbert et al. in the *2C(*^0, a)2Sflg reaction E =13.7 MeV and the resonance reported here at 19.7 MeV

V |ZU|

may be phenomena within the realm of a statistical compound nuclear reaction theory. In his so-called R-matrix approach to statistical compound reactions, (Mo 6 7,6 8) Moldauer show­ed for the case of strong compound nuclear absorption, the distribution of the strengths of the many Breit Wigner pole terms comprising the compound scattering amplitude may have an abnormally large dispersion. By abnormally large is meant a distribution of pole strengths which is much wider than that of reduced widths amplitudes that is usually assumed in the standard statistical model (Er 6 3).As a result there is an enhanced probability that one single pole term may become very large and dominate the scattering amplitude. Such an effect would simulate the population of an isolated resonance in that it would be correlated in several exit channels and would show good spin and parity, but the background, for example, is not the smooth optical model background which is dominant in the case of isolated resonances. Moldauer also showed that such large pole strength magnitudes occur preferentially in conjunction with larger-than-average widths. It should be noted that both of these features characterize the19.7 MeV resonance. Also the 13.7 MeV anomaly seen by Halbert et al. has a width larger than the average coherence width deduced for this region. Occurrences of such quasi” resonances cannot be predicted, only the probability of

their appearance can be assessed. According to Moldauer's statistical R-matrix theory such phenomena occur more fre­quently than expected from the standard statistical models. If a large enough sample of data is studied their frequency of occurrence can be estimated from experiment and the theoretical predictions compared to it. Another trend that exists for these quasiresonances is that to the extent thatItheir spins can be identified the spin values measured would most probably be in the vicinity of the grazing angular momentum value. The grazing partial waves popu­late the highest spin states in the compound system; these states find the least number of open channels for decay and have the widest distributions of pole strength. It would be interesting in this connection to determine thespins of the two anomalies observed at higher energies in

1 ? 16the C+ 0 system at E =23.9 MeV. The resonance atc.m.19.7 MeV has J=14, the one observed by Halbert at 13.7 MeV is assumed to take the grazing partial wave value, i.e.J=10 — 11. The ones at higher energies should also have a grazing partial wave spin value. This has not been verified yet. Fig. VI-22 shows two possible angular mo- mentum-vs,-energy lines on which such resonances may lie, so far only the J=14 one was verified. It is interesting to note, however, that the highest partial wave absorbed at the 4 points on this figure is 10 or 11 for the lowest one and 14, 15 or 16 for the higher ones.

12 16Fig. VI - 22 Position of the anomalies observed in the C+ O

system as a function of grazing partial wave magnitude.

Except in one case (19. 7 Mev ) none have their spins

determined yet.

28Si has been collected. Several hypotheses have been ad­vanced for its structure and for the mechanism underlying its formation. From the results of a phase shift analysis of the elastic data it became clear that while this resonance has sharp angular momentum (J=14) its background cannot be parametrized as a simple optical model amplitude. The width of the resonance is considerably larger than the average width at this excitation energy , as obtained from a fluc­tuation analysis of the data. The statistical R-matrix theory of Moldauer, which in essence is the correct treat­ment of fluctuating compound cross sections appropriate for the continuum excitation energy region, can account for the appearance of such a resonance, and some of the other ano­malies seen in that system, with considerable probability.The strengths of the pole terms in Moldauer's formalism, how­ever, are related in a non linear way to the reduced width amplitudes describing the overlap of this compound state

with the various exit channels and the fact remains that the enhancement seen in the elastic and inelastic channel arising from that J=l4 dominant pole-term does imply a large magnitude (residue) of this pole term, which is related in turn to the reduced widths for these channels. There may be one or more states in the compound nucleus at this excitation energy having large overlap with the elastic and inelastic channels and that are responsible

VI-4 DiscussionA large body of experimental data on this resonance in

for the extra strength of that pole term to which we are sensitive in our measurement.

It must be emphasized that although the availabledata do not permit the identification of the 19.7 MeVanomaly as a true- or a quasi-resonance, the differencebetween the two is much less than might appear to be thecase at first sight. For practical purposes of spectro­scopy the difference may be more in mathematical detail than substance; if the Moldauer explanation holds, the dominance of one amplitude may reflect the dominance of several reduced widths or the coherent contribution of several reduced width amplitudes (the amplitude's strength in the exact R-matrix theory is related to the reduced width amplitudes for the different channels). In either case, if such structures are found at other energies and in other systems--as is indeed already the case (Bi 740, they form the basis for a totally new spectroscopy— both experimental and theoretical.

A test of this statistical explanation vs. nuclear structure explanations for these intermediate resonances will have to come from systematics. If simple systematics emerge; then it is nuclear structure. Otherwise Moldauer's explanation is preferred because it is simpler.

EPILOGUE

VII-1 ExtensionsWhile the work presented here answered some of the

questions arising in the study of Heavy Ion scattering it left some unresolved and posed new questions. The few that immediately come to mind arei1. The origin of the narrow (or intermediate) structure

in the ^ 0 + ^ 0 system. This problem has been alludedto in the section where the intermediate structure in 12 12C+ C was discussed, but a careful study remains to be done. The strength of the orbiting resonances, as evidenced in the marked gross structure, and the smaller compound elastic contributions expected in this system make it a better candidate for inter­mediate structure generated by a double resonance

12 12mechanism than the C+ C system studied here in detail.

2. In connection to the phenomena of the unusual anomaly1 12observed in 0+ C reactions an estimate of the

probability of its occurrence could be made within the R-Matrix statistical model. The prescription to do that has already been given by Moldauer (Mo 6?,68)

but a systematic calculation remains to be done. In parallel an extensive search for similar resonance phenomena should be carried out. The spins of the "resonances" at 1 3 .7 * 2 2 . 3 and 2 3 . 9 could be studied and some systematic trends looked for. Such course is indeed pursued by various groups at Stony Brook New York, the Australian National University and other places (Ma 73,Ma 74,Br 73a), Search for similar phenomena in other systems is also being carried out (Co 74).

VI-2 Conclusion12 12A careful study of the energy dependence of C+ C

induced reactions and scattering cross sections has de­monstrated that all the structure appearing in the elastic excitation functions which does not stem from potential scattering can be fully accounted for by statistical com­pound reaction contributions. Likewise, the observed

12 12 20selectivity in the C( C,a) Ne can be accounted for by such a reaction model. In this study predictions of the statistical model for the strength and width of the structure were shown to account for the structure ob­served in the elastic channel and for the selective popu­lation of high spin states at high excitation energiesin 2®Ne in the *2C(*2C,a)2^Ne reaction.

12 16A study performed on the C+ AO0 system showed that12 12this situation is not unique to the C+ C system, but

can also account for similar features occurring in the 42C+*^0 system.

An extensive study carried out with synthetic exci­tation functions also enabled us to ascertain the validity of the methods of statistical analysis and the statistical model calculations used herein for the heavy ion systems studied.

Appendix A

Linear^om bm ation^ofJNuclear^^rbi^ls^^heo^^HigU^Ms^

( Refs. Oe71, Bo71 )

In the scattering problem A ( C , C ) A

where C = B + X and A = B

the two processes A ( B+X , B ) A+X

and A ( B+X , B+X ) A interfere

The theory of linear combination of nuclear orbitals (L. C. N. O .) treats

this problem under the following assumptions:

1) No channel other then the elastic transfer is coupled to the entrance

channel -» Two state approximation.

2) The relative motion of the two cores is slow ■+ Adiabaticity.

3) "No recoil", the relative motion of the two cores A and B is described

by the same radius vector which describes the motion of A and C.

Using assumption l:The system can be described completely by the

following two states: a) X bound to A -* (A)

b) X bound to B -* (B)

an orthogonal combination of these two states is given by:

* + = (tf (A) +* (B))/Aand _ = ( * (A) - V (B) )//2

Three interaction regions can be specified:

Region III: Large separation, X is tied to core A or B. The wave functions

(A) and ^ (B) are Shell or Cluster Model wave functions.

Region II: Intermediate separation, the cores are unperturbed but X is

now shared by the two cores. L. C. N. O. region.

Region I: Small separation, energy of relative motion high above the

barrier. Core structure dissolved, Two Center Shell model region.

In region m and are degenerate: Energy eigenvalues = E ( » ) .+ - BIn region II degeneracy is removed by the exchange interaction, the energy

is now a function of the distance between the two cores:

A I

Relative to the energy at infinite separation the energy can be written as:

E(R) = Eb (R) - Eb ( » )

This R dependent "Exchange Energy" term will affect the relative motion

of the two cores. In the following this term will be derived assuming

that energies in the entrance channel are low enough and the L. C .N . O. theory

is applicable ( region II )

The Hamiltonian of the system A + X + B is given, in region II, by:

» H = HAB + Ho = <&AB + VAB> + <A X + VXA + VBX + HA + HB + HX>

H . _ = Core - core Hamiltonian ABV . _ = Core - core interaction ABH = Static Hamiltonian

oV , V - are the A -X and B -X binding potentials.AA d A

If we denote by ^ the internal wave functions describing A, B and XA B X

and by E. ’ ’ their internal energies (All are assumed in ground state)

we have: TT _ _,A _H. ^ . = E. . and the same for B and X.

A A 1 A

The total wave function which is a solution of H ^ = Ety is separable

In region III = XAB (R) ^ (B)

whereABXAB AB XAB

^ (AX + VBX + HA + HB + HX} * (B) = ( Ef + + Ef + (B)

The explicit form of 'I' (B) is then given by:

^A,mA ^C,mC 4a) (B) = * A ® V C ( C = B+X )

a.. T jC ,mC _ _ _ .. . ,. . T jB ’ mB . j.m.-t4b) *C ' ^ m B ( , BmB ' jm | )CmC) ^B BJ3

4c) * B m’t ' U ,iXX X

4d) <PB k , *r BX* Yl lt>BX>

A 2

A3

^ i m ] ’ •*2m 2 ^ 3 m3 iS an£u*a r momentum coupling

cool'fjclont describing tho coupling'

of J1 and J2 In fo rm j

c - orbital wave function of X in C.J5bound wave function of X in C.

S^. - spectroscopic factor for ( C | B+X ) separation

The system A+X+B can be described as a combination of the two product

wave functions Y AT ^ (B ) and Y . ^ f A ) where:AAB AAB

5a) (B) = S<) <*:«'>^“x lim I > *X 3

i . i . n . V j i r t i r " x , 4,«*A B X * 8

and

5c) >K A )= r S m' jV | j m‘ ) «fn, m'jj m'A> 'V ’ >A’ mA t jB 'mB , jX 'mX J .\ u '

* A *B X B

The notation used in Eq. 4 is

i he total energy ot the system is given by

6) E (R) = < ^ | H° ]**> . o <>F±[>F±>

the denominator is different from zero only if

m =m\ , m =m ' and m = m' holdsA A B B X X

In the following derivations we further assume that no transfer of

angular momenta is taking place between the relative core motion

to the core - cluster binding states i. e.

A 4

\j = j and if no spin flip is further assumed then 1 = 1 and p = p

it follows then that the expression for 'I' (A) is the same as the one

for 'I' (B) except for the orbital part describing the binding of X to

the cores A or B.

To obtain the energy of the system in states and 'I' we evaluate

of (A) and (B) as given in the beginning of this section.

After some straightforward algebra one obtains:

K = ^ [< ^ (A )| Vb x | * (A )> + < * (B )| v a x | * (B ) > ]

| [< * (A ) | v a x | * (B )> + < * (B )| v b x | * (A ) > ]

and S = <>F (A )|>K B)>

The explicit form of these integrals is:

9) K(R) = Jd? * (A) V BX * (A) =

the expression 6 by substituting the expressions for in terms

o i i i B

and

7) E± = EA + EB + EX + E (oo) + K ± J 1 ± S

8) E(R, = Eb (R) - Eb M =

where:

10) J(R) = Jd? * (A) VBX * (B) =

= ES^ (-e,M,jxmx |jm)2 (jAmA,jm|jcmc )2 Jd?$^M$ -r ) VgX( r )< ^ ( r )

11) S(R) = J dr (A) * (B ) =

A5

The orthogonal wave functions defined before do not incorporate yet

the symmetry properties required. Since the two cores are identical the

wave functions must be symmetric under core exchange, this symmetry

operation amounts to an inversion of the wave function of the exchanged

particle - X in C with respect to the center of mass of the two cores.

The result of such inversion would be reflected in a (-1 ) factor in the

expression for <£ and consequently for 'I' _ . We can thus defineA, B A, Ba symmetric and an antisymmetric wave functions for the system

A+X+B:

12) * S = O ( A ) + ( - ) ^ ^ (B ) ] //2

* a = [ * ( A ) - ( - ) S ( B ) ] / /2

and in this case the exchange energy is given by:

l 3 > e S ’ 1 | K ) = t i h ^

(+) for the symmetric case (s)

(- ) for the antisymmetric case (a)

Since in regions II and III the cores are far enough apart for the

integral K to vanish ( or contribute very little to the total energy)

we take K(R) = 0 in expression 13 and this results in the exchange

integral J (R) used in the text.S ) SL

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