Proceedings of the Third Conference on Reactions Between Complex Nuclei: Held at Asilomar (Pacific Grove, California) April 14–18, 1963 [Reprint 2019 ed.] 9780520316836

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PROCEEDINGS OF THE THIRD CONFERENCE ON REACTIONS BETWEEN COMPLEX NUCLEI

Proceedings of the Third Conference on

REACTIONS BETWEEN COMPLEX NUCLEI HELD AT ASILOMAR (PACIFIC GROVE, CALIFORNIA) APRIL 14-18, 1963

E D I T E D BY A. GHIORSO, R. M. DIAMOND, A N D H. E. C O N Z E T T

University of California Press Berkeley and Los Angeles

1963

UNIVERSITY OF CALIFORNIA PRESS Berkeley and Los Angeles, California C A M B R I D G E UNIVERSITY PRESS London, England

Library of Congress Catalog Card Number: 60-15752

Printed in the United States of America

FOREWORD The Third Conference on Reactions Between Complex Nuclei was held at Asilomar, in Pacific Grove, California, on 14-18 April 1963. The goals of the meetings were to provide information on the developments in the field since the Second Conference at Gatlinburg in 1960, to allow an informal exchange of ideas between scientists of the universities and laboratories participating in heavy-ion research, and to point to directions of fruitful research for the future. The Conference was truly international in its scope, with 30 representatives from Canada, Czechoslovakia, Denmark, England, France, Germany, India, Israel, Japan, The Netherlands, Poland, Russia, Sweden, and Switzerland, in addition to some 100 participants from the United States. The response to the initial request for contributions to the meeting was too good. Even after extending the meeting a half day and shortening the time allotted per paper, it was not possible to accept all of the papers for oral presentation or even for inclusion in the written proceedings. The organizing committee, composed of S. K. Allison, E. Almqvist, D. A. Bromley, H. E. Conzett, R. M. Diamond, A. Zucker, with A. Ghiorso as chairman, endeavored to select the papers so as to provide a representative cross section of the types of research and of the institutions involved. The diversity of topics covered testify to the wide applicability of heavy-ion beams to almost all fields of low-energy nuclear research as well as to inherently unique processes such as nuclear transfer reactions. This volume contains all the papers accepted for the proceedings with the discussions which followed those papers presented orally, and also the conference summary by J. O. Rasmussen. The papers vary greatly in style, length, and polish. Some are essentially finished pieces of experimental work, others are informal talks in the first person. The editors have not changed

them to a consistent style, although they have taken the liberty occasionally of making some changes for what they hope is the sake of clarity. In the interest of obtaining speedy publication, no chance was given to the authors (dispersed over a large part of the earth) to proofread their papers, so for any e r r o r s introduced, the editors must assume full responsibility. Some of the discussions had to be heavily edited or deleted because sentences were sometimes garbled in the tape script; we apologize if significant comments were thus removed from posterity's scrutiny. The task of the editors was greatly eased by the close to 100% yield of finished manuscripts handed in at the conference; for this remarkable feat we are greatly Indebted to all the authors and we desire to express our thanks. The very difficult job of transcribing and pre-editing the discussions promptly enough for them to be checked by individual speakers while still at Asilomar was accomplished by relays of Berkeley graduate students working long into the nights. We are very grateful to L. Altman, J. Clarkson, P. Croft, R. Kiefer, Y. E. Kim, M. Lederer, J. R. Nix, A. Pape, F. Plasil, K. Poggenburg, D. Reames, W. Simon, and B. Wllklns for performing with heroic zeal. We thank Cyrill Orly, Jerry Stivers, and their crew for arduous duties in connection with the technical operation and recording of the meetings. Special thanks must go to Judy Hartwell, Doral Buchholz, Nancy Schorn, and Lillian Mar. They not only did the extensive secretarial work behind the scenes but also noticeably boosted morale by their pleasant and charming presence at the social hours. For patiently and unobtrusively providing a wonderful conference locale, we thank Mrs. Philbrook and her Asilomar staff. Finally, for the difficult job of editorial supervision of the proceedings, we are indebted to Ernest Callenbach.

v

CONTENTS SESSION A—SCATTERING (A. Zucker, Chairman)

PAGE

A-l

HEAVY ION ELASTIC SCATTERING G. H. Rawitscher, J . S. Mcintosh, and J . A. Polak

3

A-2

OPTICAL MODEL DESCRIPTION OF LOW ENERGY COLLISIONS BETWEEN HEAVY IONS J . A. Kuehner and E. Almqvist

A-3

OPTICAL MODEL ANALYSIS OF HEAVY ION SCATTERING E. H. Auerbach and C. E. P o r t e r

19

A-4

PHASE SHIFT ANALYSIS OF HEAVY ION SCATTERING H. E. Conzett, A. Isoya, and E. Hadjlmichael

26

A-5

THE INELASTIC SCATTERING OF C 1 2 AND O 1 8 BY P b 2 0 8 K. H. Wang and J . A. Mclntyre

31

A-6

THE SCATTERING OF 168 MeV O 16 NUCLEI FROM C 1 2 G. T. Garvey and J . C. Hiebert

36

A-7

ELASTIC AND INELASTIC SCATTERING OF CARBON FROM CARBON R. H. Bassel, G. R. Satchler, and R. M. Drisko

45

A-8

ELASTIC SCATTERING AND REACTIONS OF LITHIUM IONS J . R. J . Bennett and I. S. Grant

50

A-9

INELASTIC SCATTERING OF 166 MeV O 1 8 IONS BY T a " A. Isoya, H. E. Conzett, E. Hadjlmichael, and E. Shield

54

11

A-10 NUCLEAR SCATTERING AND TWO-NUCLEON FORCES E. Schmid, K. Wildermuth, and Y. C. Tang

59

A-11 COMPLEX NUCLEUS ASPECTS OF ALPHA-PARTICLE SCATTERING R. H. Davis

61

A-12 THE QUASI-CLASSICAL ANALYSIS OF THE ELASTIC SCATTERING OF COMPLEX NUCLEI B. N. Kalinkin, T. P. Kochkina, B. I. Pustylnik SESSION B—DIRECT REACTIONS I (S. K. Allison, Chairman)

69 PAGE

B-l

SIMPLE INTERFERENCE CONCEPTS IN SCATTERING AND STRIPPING D. R. Inglis

81

B-2

PICKUP OF HEAVY CLUSTERS R. M. Drisko, G. R. Satchler, and R. H. Bassel

85

B-3

BORON-INDUCED TRANSFER REACTION STUDIES M. Sachs, C. Chasman, and D. A. Bromley

90

B-4

APPROXIMATIONS OF NUCLEON TRANSFER THEORY G. Breit

97 vii

B - 5 a NEUTRON TRANSFER IN THE N 1 4 (N 1 4 , N " ) N 1 5 REACTION L. C. Becker, F . C. Jobes, and J . A. Mclntyre

106

B-5b NEUTRON TRANSFER IN THE Pb 2 0 7 (N 14 , N13 ) P b 2 0 8 REACTION T. L. Watts and J . A. Mclntyre

110

B-6

SINGLE-NUCLEON TRANSFER EXCITATION FUNCTIONS K. S. Toth and E . Newman

114

B-7

A STUDY OF MULTINUCLEON STRIPPING REACTIONS IN BOMBARDMENT OF Ta WITH Ne 2 0 IONS W. Grochulski, T. Kwieciriska, Lian Go-chan, E. Lozyriski, J . Maly, L. K. Tarasov, V. V. Volkov

B-8

ANGULAR DISTRIBUTION OF THE TRANSFER REACTION PRODUCTS B. N. Kalinkin and J . Grabowski

B-9

SOME INVESTIGATIONS ON TWO-NEUTRON TRANSFER REACTIONS IN BOMBARDMENT OF ISOTOPES Z r 9 0 , Z r 9 2 , AND Zr 9 4 WITH N15 IONS L. Pomorskl, J . Tys, V. V. Volkov, and J . Wilczynski

SESSION C—DIRECT REACTIONS II (E. Almqvist, Chairman)

120

129

135 PAGE

C-l

ANGULAR DISTRIBUTION OF N15 PARTICLES FROM THE TRANSFER REACTION Al 2 7 (O 18 , N 15 ) S i 2 8 E. Newman, K. S. Toth, and A. Zucker

C-2

INDIRECT ASPECTS OF DIRECT INTERACTIONS K. R. Greider

148

C-3

MECHANISMS OF NUCLEAR REACTIONS BETWEEN LITHIUM NUCLEI S. K. Allison, M. Kamegai, and G. C. Morrison

155

143

C - 4 a ANGULAR DISTRIBUTIONS FROM Li 8 -INDUCED REACTIONS S. Takeda and R. Nakasima

159

C-4b DISTORTED WAVE ANALYSIS OF NUCLEAR TRANSFER REACTIONS T. Kammuri and R. Nakasima

163

C-5

NUCLEAR REACTION STUDIES WITH LITHIUM IONS ON B 1 0 AND B 1 1 TARGETS G. C. Morrison, N. H. Gale, M. Hussain, and G. Murray

168

C-6

GAMMA-RAY STUDIES OF B " , N 1 4 , AND F 1 9 INDUCED TRANSFER REACTIONS G. Hortig, H. Werner, and W. Gentner

178

C-7

TWO PARAMETER ANALYSIS OF LITHIUM REACTIONS R. R. Carlson and E. Norbeck

181

C-8

THE B 1 0 (d, Li 6 ) Li 6 REACTION AT DEUTERON ENERGIES FROM 8 TO 13.5 MeV D. S. Gemmell, J . R. Ersldne, and J . P. Schiffer

186

C-9

DISSOCIATION REACTIONS IN LIGHT NUCLEI

191

R. Ollerhead, C. Chasman, and D. A. Bromley SESSION D—COMPOUND SYSTEMS I (D. A. Bromley, Chairman)

PAGE

D-l

1 2 (C 12 , a) Ne 2 0 REACTION THE J. C Borggreen, B. Elbek, R. B. Leachmann, M. C. Olesen, and N. O. R. Poulsen

201

D-2

HIGH SPIN RESONANCES IN C 1 2 (C 12 , a) Ne 2 0 E. Almqvist, J . A. Kuehner, E. Vogt, D. McPherson, and J . D. Prentice

207

D-3

FLUCTUATIONS IN THE C 1 2 (O 18 , a) Mg 2 4 REACTION M. L. Halbert, F . E. Durham, C. D. Moak, and A. Zucker

213

viii

D-4

A SPONTANEOUSLY FISSIONING ISOMER PRODUCED IN REACTIONS WITH CHARGED PARTICLES G. N. Flerov, S. M. Polikanov, V. L. Mikheev, V. P. Perelygin, and A. A. Pleve

219

D-5

ANGULAR-MOMENTUM E F F E C T S IN COMPOUND-NUCLEUS REACTIONS M. L. Halbert and F. E. Durham

223

D-6

MOMENTUM TRANSFER IN HEAVY ION INDUCED REACTIONS T. Sikkeland and V. E. Viola, J r .

232

D-7

TOTAL REACTION CROSS SECTIONS FOR HEAVY IONS B. Wilkins and G. Igo

241

D-8

PROBABILITY FOR ISOMERISM FROM THE STATISTICAL NUCLEAR MODEL V. M. Strutinski

246

D-9

HIGH ANGULAR MOMENTUM CUTOFF IN NUCLEAR LEVEL DENSITIES

248

D. W. Lang SESSION E - C O U L O M B EXCITATION (R. M. Diamond, Chairman)

PAGE

E-l

ONK. THE THEORY OF MULTIPLE COULOMB EXCITATION WITH HEAVY IONS Alder

253

E-2

THIRD ORDER REORIENTATION E F F E C T IN HEAVY ION COULOMB EXCITATION D. L. Lin and J . F. Masso

267

E-3

SECOND-ORDER E F F E C T S IN THE HEAVY ION COULOMB EXCITATION OF Se 7 6 AND Se 78 A. C. Douglas, W. Bygrave, D. Eccleshall, and M. J . L. Yates

274

PRECISION MEASUREMENTS OF EXCITATION FUNCTIONS FOR THE COULOMB EXCITATION OF THE 2+ and 4+ ROTATIONAL LEVELS IN Sm 152 G. Goldring, J . de Boer, and H. Winkler

278

E-4

E-5

MULTIPLE COULOMB EXCITATION IN DEFORMED NUCLEI H. LQtken and A. Winther

285

E-6

COULOMB EXCITATION OF VIBRATIONAL STATES IN DEFORMED NUCLEI Y. Yoshizawa, B. Elbek, B. Herskind, and M. C. Olesen

289

E-7

HIGHER ORDER PERTURBATIONS IN NUCLEAR COLLECTIVE MOTION J. S. Greenberg, D. A. Bromley, G. C. Seaman, and E. V. Bishop

295

E-8

HEAVY ION COULOMB EXCITATION OF DEFORMED NUCLEI F. S. Stephens, B. Elbek, and R. M. Diamond

303

E-9

SYMMETRIZATION IN COULOMB EXCITATION CALCULATIONS P. J . B r u s s a a r d and L. C. Biedenharn

311

E-10 MULTIPLE COULOMB EXCITATION OF ROTATIONAL LEVELS IN EVEN-EVEN NUCLEI J. de Boer, G. Goldring, and H. Winkler SESSION F-COMPOUND SYSTEMS II (H. E. Conzett, Chairman) F-l

EQUILIBRIUM SHAPES OF A ROTATING CHARGED DROP AND CONSEQUENCES FOR HEAVY ION INDUCED NUCLEAR REACTIONS S. Cohen, F. P l a s i l , and W. J . Swiatecki

317 PAGE 325

F-2

TERNARY FISSION OF HEAVY COMPOUND NUCLEI P. B. P r i c e , R. L. F l e i s c h e r , R. M. Walker, and E. L. Hubbard

332

F-3

NEUTRON SPECTRA FROM HEAVY ION BOMBARDMENT OF GOLD W. G. Simon

338

ix

F-4

E F F E C T OF ANGULAR MOMENTUM ON NEUTRON EMISSION FROM Tb AND Dy COMPOUND NUCLEI G. N. Simonoff and J . M. Alexander

345

F-5

RECOIL RANGE STUDIES OF NUCLEAR REACTIONS INDUCED BY HEAVY IONS M. Kaplan and R. D. Fink

353

F-6

INTERACTION MECHANISMS OF 200 MeV Ne 2 0 WITH Ag AND Br NUCLEI R. Pfohl, Ch. Winter-Gegauff, and J . P. Lonchamp

360

F-7

ESTIMATES OF FISSION-FRAGMENT KINETIC-ENERGY DISTRIBUTIONS ON THE BASIS OF THE LIQUID-DROP MODEL J . R. Nix

366

F-8

EMISSION OF CHARGED PARTICLES IN HEAVY ION INTERACTIONS D. V. Reames

374

F-9

STATES OF THE COMPOUND NUCLEUS WITH HIGH ANGULAR MOMENTUM D. Sperber

379

F-10 RECOIL RANGE STRAGGLING OF HEAVY ION REACTION PRODUCTS IN HELIUM J . Gilat and J . M. Alexander

387

F - l l DETERMINATION OF SOME FRAGMENT YIELDS FROM FISSION OF HEAVY NUCLEI BY MULTICHARGED IONS I. Z v a r a SESSION G-SPECTROSCOPY (G. Breit, Chairman)

389 PAGE

G-l

NEW REGIONS OF NUCLEAR DEFORMATION R. N. Chanda, J . E. Clarkson, and R. K. Sheline

G-2

CHARGED DISTRIBUTIONS OF PRODUCTS OF REACTIONS BETWEEN COMPLEX NUCLEI N. H. Steiger

407

G-3

LIFETIMES OF LEVELS IN N e 2 1 R. D. Bent, J . E. Evans, G. C. Morrison, and I. J . Van Heerden

417

G-4

METASTABLE STATES IN THE LIGHT THALLIUM ISOTOPES PRODUCED IN HEAVY ION REACTIONS R. M. Diamond and F. S. Stephens

422

CONCERNING THE PRODUCTION OF DELAYED PROTON EMITTERS IN REACTIONS BETWEEN COMPLEX NUCLEI V. I. Goldanskii

428

SEARCH FOR PROTON EMITTERS AMONG THE PRODUCTS OF HEAVY ION INDUCED REACTIONS V. A. Karnaukhov, G. M. Ter-Akopian, and V. G. Subbotin

434

G-5

G-6

x

397

CONFERENCE SUMMARY J . O. Rasmussen

439

CONFERENCE PARTICIPANTS

452

A-l

HEAVY ION ELASTIC SCATTERING G. H. Rawitscher, J . S. Mcintosh, and J . A. Polak* Yale University New Haven, Connecticut The theoretical calculation of the long-range part of the interaction of elastic heavy ion scattering, previously reported, has been systematized and detailed comparison with measurements on C - N, O - C, and N - N from 8 to 20 MeV c e n t e r - o f - m a s s energy will be presented. Deviations from the p r e dictions of this simplified theoretical model a r e particularly noticeable in the c a s e of C - O, and the possibility of attributing them to resonances will be briefly discussed. Comparison with experiment involves also short-range interactions treated either by means of the nucleus-nucleus optical model (OM) or by an ingoing wave boundary (IWB) condition used at an empirically adjusted internuclear distance r = R » 1 . 2 ( A 1 1 / 3 + A 2 1 / 3 ) without absorption for r > R assuming the JWKB approximation for r ^ R . The parabolic b a r r i e r approximation provides an analytical approximation for the IWB. The near equivalence of IWB and OM indicates that the OM mainly supplies a representation of absorption in close collisions to be combined with the distant collision theoretically estimated potential effect.

The present report concerns itself with the elastic scattering of heavy ions at energies close to the Coulomb barrier. A preliminary account of this work was presented at the 1960 Gatlinburg meeting! and elsewhere^ where the theoretical procedures were outlined. The method has since then been systematized and comparison with experiment is now carried out for the nuclear systems N - N, N - C and O - C. The basis of the approach consists in considering only collisions where the nuclei "do not touch," and in addition, are going by each other slowly enough so that the collision can be considered as a succession of static confrontations of the two nuclei taken at various separation distances. When the two nuclei are very far apart they interact mainly via the Coulomb force. But since the nuclear surfaces are fuzzy, nucleons from either nucleus can penetrate into the region occupied by the other nucleus, and an interaction due to nuclear forces takes place. The effect is to change the binding energy of each nucleón to its original nucleus slightly, which when summed over all available nucleons, is interpreted as anucleus-nucleus interaction energy. The input information required for calculating this change in binding energy is (1) a nucleon-nucleus optical model potential extrapolated to negative nucleón energy, (2) the probability density of finding each nucleón at the nuclear surface, and (3) the binding energies of the nucleons in each nucleus. The first piece of information is *Work supported by the U. S. Atomic Energy Commission and by the U.S. Army R e s e a r c h Office (Durham).

ELASTIC

SCATTERING

Scattering

Angle

(0CM)

Fig. 1. N 1 4 — N 1 4 and N * 4 — C 1 2 elastic scattering near 10 MeV.

3

SCATTERING

20

bid

40

1.0 \ .8 bla .6 TJ T3 '

60

80 100 120 140 Scontrine Anglt (8Cy>

.4

.2 .10 — • 13.67 Me» Kuehner & Almqviat

.08 .06 .04

° 13.7 Mev Quinton H . a !

The other curve corresponds to a potential about 5 times larger, which should illustrate the importance of the normalization of the cross sections. None of the theoretical curves have the "wiggles" found in the experimental cross sections. This is particularly the case for the O - C system, as shown in Figure 2. 13.6 MeV is an interesting energy because here the resonances first pointed out by the Chalk River group in 1960, and interpretedS as possibly due to molecule formation, do definitely occur. Two sets of data are shown, measured at slightly different energies. The open circles are measurements by A. R. Quinton and G. P. Lawrence which are preliminary and will be remeasured, but were kindly supplied for the purpose of this talk. The same O - C potential is used for all three theoretical curves in the figure. It is surprising that at very much higher energies, where the adiabatic approximations are presumably not applicable, agreement with experiment is again obtained with the same parameters which fitted the low-energy experiments. The comparison is shown in Figure 3. From these potentials one can now deduce the

.02

0

20

40 60 80 Scattering Angle (®CM)

• Williomt

100

S Stcigtrt " Theory

Fig. 2. C 1 2 —O 1 6 elastic scattering, 8 to 14 MeV.

known within certain limits from experiments on nucleon-nucleus scattering, the second piece is similar to a reduced width and is a free parameter in our theory. The value of this parameter, where averaged over the various nucleons which participate in the interaction of the nuclei, will be denoted by X. It enters only as a multiplicative constant in front of the whole expression for the change in binding energy and can therefore be easily changed. The dependence of the long-range part of the resulting nucleus-nucleus potential on the internuclear distance r is fixed by items (1) and (3) mentioned above, and is found to be of an approximately exponential form. This potential, when used in the calculation of differential cross sections in a way to be discussed further on, yields information on the parameters X. The first few slides show the results obtained, and the remainder of the talk will be devoted to discussing in some detail the procedures used in obtaining these results. Figure 1 shows the comparison of the differential cross sections with experiment for N - N and N - C. The same potential which gave the fit at 9 MeV for N C gives rise to the curve labeled .145 at 9.92 MeV. 4

— Smith & Steigert

—

Theory

4

8

Scattering

6

7

Internuclear

12

Angle

8

9

16

(0cm)

Distance

10

20

II

(Fermi»)

Fig. 3. O 1 6 —C 1 2 andC 1 2 —N 1 4 elastic scattering near 63 MeV.

A - l Rawitscher, Mcintosh, and Polak

values of the parameters X. One way to present these values is in units of what one would expect if a single-particle, shell-like description of the nucléons were applicable. This value of X is denoted by the subscript s.p., and the results may be summarized thus: REDUCED WIDTH PARAMETERS THEORY NUCLEUS O c N

EXPERIMENT x sp

SYSTEM

41.0 F" 3 48.0 F" 3 6.9 F - 3

O -C C - N N - N

Vxsp (.16) .14 .17

REDUCED WIDTH PARAMETERS The ratios x / x S p are all found to be approximately equal to 1/6. The two labels on the theoretical curves shown in Figure 1 indicate the values of X / A s p employed. The remainder of this talk will be spent on describing the procedure used to incorporate the tail of the potential into a nucleus-nucleus Schr5dinger equation. One possibility is to arbitrarily extrapolate a potential V from the tail region into the origin of r and to add an imaginary potential W to allow for the existence of absorption into the inelastic channels. This is the usual optical model procedure. Then, by changing the values of the potentials for the small values of r, one empirically verifies that the elastic scattering cross sections are essentially independent of the potentials at small internuclear distances.I, 4 The question then is: for what values of r does the potential have to be known with precision? This question can be examined by means of the JWKB approximation^ and it is found that for the O, C, and N cases examined above, it is in the barrier region formed by the Coulomb and nuclear potentials where the nuclear potential matters most. The argument is as follows. In the optical model procedure the wave function is taken to have a two-body character for all values of r, including the situation in which the nuclei overlap completely. In the decomposition into partial waves, for each value of the angular momentum quantum number L, an effective potential is defined as V e ff = V N UCL + V C OUL + V CENTRIFUGAL as in Figure 4. There is a valley region in V e ff, where the kinetic energy is positive. The region where the effective potential has a positive hump is summar-

0

2

4

Internucleor

6

8

10

Distance

12

14

(Fermis)

Fig. 4. Effective potential, V e f f .

ily to be called the barrier region. It is the more pronounced, the larger the Coulomb interaction. For heavy ions, the effective (or local) wavelength is small compared to nuclear distances and the JWKB approximation is likely to be valid, particularly in the valley region. In the JWKB form the wave function has two terms, which can be interpreted as the ingoing and the outgoing parts of the wave:

JWKB = L l A L e

B

Le

'

kjr) = |£(E-V-iW) BT 2 i J r r kT dr outgoing _ L L ingoing ÂTe 5

SCATTERING Ingoing wave boundary condition: -t IW

= k

R• 5 F o R. 4 F • R « 2.5 F • I.W.B. A

•

/ X dr

8 6

L

The effective wave number times 2 it is kL(r). It is complex in the presence of an imaginary potential and gives rise to a real exponential factor in the wave function. If the imaginary potential is sufficiently deep in the valley region, and the valley region contains sufficiently many wave lengths, then the outgoing portion of the wave disappears relatively to the ingoing one, and the wave function is given by the expression on the last line in the slide. In this case the logarithmic derivative of the wave function is dependent only on the local value of the potential, and changes in the potentials to the left of where the outgoing part of the wave function disappears do not affect the cross sections.

A

A

(r - Rl/D l+e

Si • CM

_• M

o

5

•

I

•

•

A

e

1.0

•

A

.8

e a

2.0

.6 .4

•

8 * 8 A4 f « 0

1.5

2

r

4 6 8 10 12 Angular Momentum Number

.2

14 L —

16

18

Fig. 6. Phase shifts obtained from IWB and optical model calculations.

1.0

0.9

0

2

4 Angular

6

8 Momentum

I

10

L

12

14

L—-

Fig. 5. Logarithmic derivatives calculated by the IWB method compared with those from optical model calculations. 6

One of the procedures utilized in the calculation of the cross sections shown before consisted in assuming the wave function to be of the ingoing form, as shown in the last line above, at a point in the valley region corresponding to an internuclear separation distance as large as possible but yet compatible with the validity of the JWKB approximation. This distance, denoted by R, corresponds to roughly 1.25 fermis times ( a / / j + A 2 l / S ) . The advantage of this procedure is that it provides a very simple absorption mechanism. No particular form of the nuclear potential needs to be specified to the inside of R, the introduction of a specific imaginary potential can be avoided, and in addition the effect of disregarding the many-body character of the nucleus-nucleus wave function at distances beyond R may not be as serious as could be the case if all radial distances were involved.6 The method just described will be labelled by the letters IWB, 7 in order to distinguish it from the usual optical model procedure also employed. To illustrate

A - l Rawitscher, Mcintosh, and Polak

the relationship of this procedure to the usual optical model method, a numerical example has been calculated employing the potential shown in Figure 4. First, logarithmic derivatives are calculated at r = 7.1 fermis, which lies in the valley region, by both the optical model and the IWB procedures. Figure 5 illustrates that there is good agreement between the two, showing that the JWKB approximation and the neglect of the outgoing part of the wave leads, in this case, to results nearly equivalent to those obtained by the optical model procedure. The disagreement beyond L = 9 is due to the failure of the JWKB approximation at 7.1 fermis. At smaller values of r the failure of the JWKB occurs for larger values of L. In optical model calculations the corresponding phase shifts are then sufficiently small so that the effect on the cross section is negligible in the cases investigated. Fig-

i.o

1

w

1 \l

1

.9 .8

w0 1+ e

.7 .6 o

u.

\ \ \\ \\ \\ \\ 0 \ »

.2

SENSI1 IVITY

to w

ure 6 shows the comparison between the phase shifts obtained by means of the two methods. The imaginary potential was set equal to zero everywhere for the IWB method. The various optical model results correspond to different amounts of imaginary potential in the barrier region. Two of the imaginary potentials employed are shown in Figure 4. They correspond to the parameters R equal to 2.5 F and 4 F. The nuclear phase shifts R I " are K T + i K T , and the combinations, e Li

2K

d

and e

2

k

L

LI

- 2 KL

sin

, have been plotted versus L in

Li

Figure 6. The squares, open circles, and triangles represent optical model results, with progressively decreased amounts of imaginary potential W present in the barrier region. As W is removed from the barrier region, the points shift smoothly towards the values obtained by the IWB method (black dots) until nearly perfect equivalence between the two methods is obtained. The half-shaded circles indicate that the open circles and the black dots coincide. As W is removed still further, the values of the phase shifts (squares) begin to jump around the IWB values and wiggles start appearing in the cross sections. This is illustrated in Figure 7 where ratios to Rutherford cross sections are shown versus center-of-mass angle. The dotted curve corresponds to the smallest amount of W in the barrier region. It oscillates around a smooth curve given by the IWB® method, which however is not shown so as not to clutter up the figure. The IWB procedure was utilized in the preparation of the theoretical curves presented in the first few figures, since then only the parameters describing the tail of the potential need to be specified. The value of the radius R, at which the ingoing wave boundary condition is imposed, does not affect the cross section, as long as R is contained in the val-

Y : V ( r ) = ( V * s p ) A E ( r ) A A s p is a f r e e p a r a m e t e r A E ( r ) is c a l c u l a t e d f r o m theory

- < * vnN

V


COMPARISON WITH EXPERIMENT: V = V Q e - r / a 1 1

1 1

i

30 60 Scattering Angle Fig. 7. Effect on differential cross c;hange in W.

i 90 (0CM)

120

section due to

SYSTEM

V 0 , MeV

a, fermis

O - C C - N N - N

(.108 x 108) .194 x 108) .150 x 107)

.472 .452 .543 7

SCATTERING ley region. The theoretical result for the tall of the potential energy 1s well represented by an exponential function of distance - V 0 exp (- r/a), and the values of V 0 and a are listed above. The parameter a is given by the theory and the values of V 0 shown here give rise to the values of X /x sp shown in the table above. The IWB procedure is of course very schematic, and is introduced so as to obtain an initially simple method to be used in connection with the theoretical long-range part of the potential. The fact that the "wiggles" in the cross section are not fitted by the calculated values is an indication of the complexity of the actual situation. The remainder of this talk will be devoted to describing an approximation, which although not very precise, gives insight into the dependence of the nuclear phase shifts on the angular momentum quantum number L.9 It is assumed that only the ingoing wave part of the wave function exists in the valley region of V e ff, and the penetration through and reflection from the hump in the barrier region is obtained by approximating the hump by means of Inverted parabolas. Hill and Wheeler 10 used this approach in 1953 for the calculation of barrier penetrabilities in fission. The solution of the SchrOdinger equation in the presence of parabolic potentials is given exactly in terms of Weber's parabolic cylinder functions, but for realistic calVC + * 2 L ( L + l ) / 2 ^ r a

culations, difficulties do occur. A minor one is that the barrier is not symmetric around its maximum so that two parabolas of different curvature, one at each slide, give a better fit to V e ff. Figure 8 shows a fit employing two such parabolas. The Weber functions can lie continued from the left side towards the right without any complications. A more serious difficulty consists in continuing the wave function from the parabola region out to the region where the nuclear potential is negligible. One can derive an approximation to the parabolic cylinder function which converges faster than the usual series expansion around the origin, and, by matching the resulting values to numerically obtained Coulomb functions at a convenient point, quite reasonable values for the nuclear phase shifts can be obtained. Another much simpler method consists in utilizing the asymptotic form of the parabolic cylinder function and continuing it to large values of r into a combination of Coulomb functions by means of the JWKB approximation. This procedure is applicable only for a small but important range of angular momentum values for which the top of the Veff barrier grazes the energy. The imaginary part of the nuclear phase shift is then expressible in terms of gamma functions and exponential functions involving the parabola parameters, and the real part involves some JWKB integrals as well. This procedure was applied for the numerical example which has been shown in the last few figures, and the result is contained in the following table.

Comparison of the Parabolic Barrier approximation (P.B.) with numerically computed I.W.B. results

» 1 P.B L = 7 9 11 13

4

5

6

7

8

9

10

lnt«rnuel*ar Distanc* (Permit)

Fig. 8. Two-parabola fit to V e ff.

II

12

e - » L sin 2 K^

e NUMERICAL

.86 .34 .029 .0014

.856 .356 .023 .0005

P.B.

NUMERICAL

-.03 .22 .11 .04

+ .136 +.262 +.090 +.019

The first column gives the barrier transmission coefficients which for the IWB condition are related to the imaginary parts of the phase shifts as indicated. The second column involves the real phase shift. The columns labelled "numerical"

A - l Rawitscher, Mcintosh, and Polak

were obtained by employing the ingoing wave boundary condition, numerical integration across the barrier, and matching to numerical Coulomb wave functions. The comparison shows reasonable agreement, indicating that the general trend of the phase shifts with L can be obtained in this way. It is recognized that the applicability of the IWB procedure for the values of L which are important in explaining the main features of the experimental results has not been established in terms of general principles but depends on particular combinations of magnitudes of Coulomb barriers, the real internuclear potential and the amount of empirically required absorption all combining to provide a not too unreasonable range of values of the boundary condition radius at which the JWKB approximation is good. It is also realized that the IWB procedure as used implies a special L dependence of the logarithmic derivative at the boundary radius which has not been justified from a dynamical viewpoint and that arguments for it are mainly the simplicity of the procedure combined with adequacy for representing the principal features of the data. On the other hand, the customary employment of the optical model also lacks a theoretical justification regarding concrete correlation with nuclear properties. ACKNOWLEDGMENT The authors wish to express their gratitude to Professor G. Breit for many helpful discussions on the work. REFERENCES 1. Proceedings ofthe Second Conference on Reactions Between Complex Nuclei, eds. A. Zucker, F. T. Howard, and E. C. Halbert (New York: John Wiley and Sons, Inc., 1960). 2. J. S. Mcintosh, S. C. Park, G. H. Rawitscher, Bull. Amer. Phys. Soc. 6, 296 (1961) and Bull. Amer. Phys. Soc. 8, 61 (1963). 3. D. A. Bromley, J. A. Kuehner, E. Almqvist, Phys. Rev. 123, 878 (1961). 4. J. A. Kuehner and E. Almqvist, this conference; G. Igo, Phys. Rev. Letters 1, 167 (1958) and Phys. Rev. 115, 1665 (1959). 5. N. Austern, Annals of Physics 15, 229 (1961) has employed the JWKB approximation in providing an interpretation of the angular momentum dependence of phase shifts. 6. The boundary condition distance R mentioned in the text is equal to 6F for the C - O system. At this distance the fraction of the charge of the carbon nucleus contained within a certain volume V of the oxygen nucleus, neglecting distortion, is less than 1%. Volume V is such that it contains 80% of the oxygen charge. The nuclear interactions at this distance are already quite large so that the

many body admixture to the wave function is not necessarily small, 7. Ingoing wave boundary conditions are of course well known in the literature. The names of H. Feshbach and V. F. Weisskopf, Phys. Rev. 76, 1550 (1949) are associated to it in connection with reactions involving neutrons. Their method was applied to heavy ion interactions for the first time by R. L. Becker and M. E. Ebel. R. L. Becker, Yale Dissertation 1957, Part II (unpublished). 8. Whenever wiggles occur in the optical model cross sections, where the IWB results do not show any, then this is an indication that changes of the potentials in the valley region will affect the cross sections. 9. The help received from Mr. V. H. Mesch and Miss J. Gibson in performing the calculations required for this discussion is gratefully acknowledged. 10. D. L. Hill and J. A . Wheeler, Phys. Rev. 89, 1102 (1953) p. 1141; see also K. W. Ford, D. Hill, M. Wakano, and J. A. Wheeler, Annals of Physics 7, 239 (1959).

DISCUSSION PORTER: You have a parameter, lambda, which should let you adjust the fit you made, but for nitrogen the calculations didn't seem to close in on the data. RAWITSCHER: Yes, perhaps it might be interesting to point out that the curve which we have shown for the nitrogen-nitrogen, was the ratio to Coulomb. Usually it is shown as the cross section directly, where the agreement, of course, looks much nicer. It follows the general trend quite nicely, but it is certainly true that this one parameter which we are giving ourselves is not capable of completely fitting the wiggles, and this is similar to the lack of agreement to the wiggles in the case of even nitrogen on carbon. There is a certain amount of the wiggling which we do not fit, which is certainly the case also in the oxygen-carbon system. For the nitrogen-nitrogen system I do not think that one should say that it is lack of proper interference with the Mott part of the scattering, but it is probably a nuclear effect treated too schematically. PORTER: I think there are other people that have remarked in the other abstracts that the fit is not terribly sensitive to model parameters. I'm surprised at the sensitivity of the calculations you have made. I think there is something to be learned from that. RAWITSCHER: We could say that the calculations that we have done represent a rather crude method which points out only certain aspects to which we may add other effects. In this case one can see they are not there. 9

SCATTERING PORTER: I'm curious to know what the qualitative thing is that prevents the match. RAWITSCHER: We have speculated on it, but I would hesitate to say. BREIT: I think one thing that may have something to do with it is interaction associated with absorption. At reasonably large distances there seems to be some evidence for such a phenomenon in nucleon transfer. The fact that in the paper we heard, there is essentially one parameter, is helpful in getting a satisfactory fit. On the other hand, it would be surprising if one could get a complete fit with just one parameter. One might point out that in this matter with ingoing wave boundary conditions, one depends on special circumstances for heavy ion reactions which are used in this paper. This boundary condition differs from the usual ones in that it applies only if the JWKB approximation applies. This shows again that one is dealing with the simple way of making fits. Furthermore, the static potential used for large distances cannot really be expected to be correct. It is only an approximation so there are a lot of reasons why the method used should not give a fit in detail. PORTER: Do you think there might be a dynamical effect? BREIT: I think that for large distances there will be a dynamic effect. I think that for nuclear distances there are absorption effects which the IWB cannot be expected to take care of. BROMLEY: In a recent issue of the Philosophical Magazine, Hodgson of Oxford has a paper in which he suggests that the optical model fits, particularly f o r nucleon-nucleus, deuteronnucleus, and alpha-nucleus situations, are relatively insensitive as far as fitting experimental data is concerned providing that you take the imaginary part of the potential to have a larger radius than the real part. I think that your treat-

10

ment would say just the opposite. Would you attribute this to a difference between nucleusnucleus and nucleon-nucleus potentials or would you disagree with Hodgson? RAWITSCHER: Yes, this is a very interesting point. We have not investigated the nucleusnucleus case in detail and cannot determine whether one can rule out or really needs an imaginary part of the potential which extends as far out as the barrier region; this is something which should be investigated. I think that it is interesting that the simple model which we have been using succeeds without having to assume a complex part of the potential in this part of the barrier region. We did actually introduce some, experimentally, to see what would happen, and it does not seems to be demanded by the data. I would further comment that from our simple way of doing things, we do not get wiggles at these lower energies. If we were using the optical model entirely, we could reduce the imaginary potential in the interior or valley region and then we would get the wiggles. But the fact that we didn't get the wiggles in the ingoing wave boundary case is an indication that the outgoing wave in the optical model calculation is still there and not negligible compared with the ingoing waves which means that the wiggles produced depend sensitively on the assumptions made on the inside region of the nucleus-nucleus interaction. They are not a diffraction effect. ZUCKER: What are these lambdas? Are they the same ones as in the transfer reaction theory? RAWITSCHER: Basically they should be related to the lambdas in the transfer reactions if both theories were perfectly complete. We actually tried to establish a relation between the two lambdas. Only the nitrogen-nitrogen case was available for the comparison. The two parameters agreed within a factor of ten, and this may not really be so bad.

A - 2

OPTICAL MODEL DESCRIPTION OF LOW ENERGY COLLISIONS BETWEEN HEAVY IONS J. A. Kuehner and E. Almqvist Chalk River Laboratories Chalk River, Ontario, Canada An optical model has been used to describe the elastic scattering of O i e + C 1 2 , N 1 4 + O 1 2 , and N 1 4 + Be 9 for energies near and above the Coulomb barrier. Using a Woods-Saxton form for both the real and imaginary potentials good agreement with the experimental data is obtained. Quantitative differences between O 1 6 + C 1 2 , which exhibits well-developed diffraction structure, and the other two systems, which exhibit l e s s pronounced diffraction structure, are well accounted for by the model and are reflected mainly by the size of the imaginary potential, a small imaginary potential being associated with the largeamplitude diffraction oscillations. The parameters determined by fitting the elastic scattering data yield reaction cross sections in agreement with measured data. Although the model gives a good description of the data there are difficulties in the physical interpretation which will be discussed. This arises from the deep interpénétration of the colliding ions implied by the model; for O1® + C* 2 , which requires a small absorption in order to fit the large diffraction oscillations, the mean free path inside the potential is ~6fm. Such a deep interpénétration does not seem physically realistic, yet, within the framework of the model it appears to be a necessary condition for producing the observed amplitude of diffraction oscillations.

The advent of the tandem accelerator, a few years ago, for the first time made possible precise studies of heavy-ion elastic scattering in the energy range near the top of the Coulomb barrier and such studiesl,2,3 have revealed some very interesting new features. In some cases, sharp resonances are observed, e.g., for C12 + C12 scattering, in others a broad diffraction-like structure is seen, and in other cases only a smooth featureless energy dependence is found. Thus the measurements appear to exhibit, in varying degrees, properties of three types. These are 1. Smooth (e.g., N14 + Be9) 2. Diffraction (e.g., O1» + C " ) 3. Resonances (e.g., C12 + C12 ) It is important to gain an understanding of why different nuclear pairs produce such widely differing results. Figure 1 contains angular distributions for 0 1 9 + C l z scattering on the left and for N14 + Be9 scattering on the right. The quantity plotted here and in subsequent figures is the observed scattering cross section divided by the Rutherford cross section A diffraction-like structure is quite marked f o r O i a + C 12 scattering while a much smoother

angular dependence is exhibited by the N14 + Be9 scattering. One must ask why these systems are so different in their behavior. It is primarily this question with which this paper deals. Figure 2 illustrates the energy dependence of the scattering cross section for the same two cases. The N14 + Be9 scattering has a smooth, rather featureless, energy dependence. On the other hand the O16 + C 12 scattering exhibits sharp resonant structure ( r ~ 200 KeV) as well as suggesting a broader structure. An attempt has been made to account for the average features of the scattering with an optical model, in which the incident wave is scattered by a complex potential. Since this model cannot be expected to account for the sharp resonant structure, which appears at energies well above the Coulomb barrier in the O i a + C 12 case, only low energy data were considered initially. Note that the angular distributions of Figure 1 correspond to energies near the "break" from Rutherford scattering and are well separated from the resonant region. In addition, low energy data were considered initially since one might expect the optical model to be most successful in describing the behavior for distant collisions in which, a priori, only interpénétration of the low density outermost nu-

ll

SCATTERING

)

•

•

i

N + Be9

o ce o

5 MeiV •••• » •

0-6

0 01 o

bltì bici

-ora

0-2

0-6

•••

0-4

6 MeV ••• ••

b|CS

•o I'd

•

0-2

20

40

60

80

CENTER-0F-MASS

100

120

20

SCATTERING ANGLE

40

60

CENTER-0F-MASS

80

100

120

SCATTERING

140

ANGLE

Fig. 1. Angular distributions for elastic scattering of N 1 4 by Be 9 at the left and of O 16 by C 1 2 at the right. The ordinate is the observed scattering cross section divided by the Rutherford cross section.

—

„16 „ 12 U + L 90°

—

—

.. 14 N + B< (30" 10

•

\\ —

0-8

0-6

•

• •• •

—

t

/ r

••

••

o-oe 006

_ 8

9

IO

II

CENTER-0F-MASS

12 ENERGY

13

14

IN MeV

15 CENTER-0F-MASS

ENERGY IN MeV

Fig. 2. Energy dependence for elastic scattering at 90° in the center-of-mass system of O 16 by C 1 2 at the left and of N 1 5 by Be 9 at the right. The ordinate is the observed scattering cross section divided by the Rutherford cross section. 12

A-2 Kuehner and Almqvist

6 e io IN FERUIS Fig. 3. The functional form of the potential is shown inset at the left and graphs of several potentials are shown at the right. The curve at the left is X 2 when attempting to fit the N 1 4 + C 1 2 angular distribution data when the parameters of the potential are changed as described in the text.

cleons was expected. In fact we shall see that the optical model fits obtained imply greater interpenetration than seems physically reasonable. Figure 3 contains the potential we have used in the optical model. Calculations were carried out on a Datatron computer using an existing programme. 4 The shape of the potential is illustrated at the right of Figure 3. A logarithmic scale is used to emphasize the tail region. Figure 3, on the left, shows what happens to x 2 when attempting to fit the N14 + C 12 data (see Figure 5) when the par a m e t e r s of the potential are changed in such a way that the tail region is unaffected (illustrated at the right of Figure 3). In the comparison W is changed also, keeping the ratio of W to V constant. The r e sults are quite insensitive to the central region of the potential. A similar insensitivity to the central region has been demonstrated5 for alpha particle scattering. This result allows one to choose any value of V provided that it is sufficiently large; e.g., in Figure 3 any | V | > 10 MeV gives a good fit. In what follows V has been fixed at -50 MeV. Figure 4 demonstrates the effect of varying W. The data points are those for O18 + C 11 at 10 MeV. As you can see, a small W produces large diffraction oscillations, while a larger Wproduces smalle r diffraction oscillations. This is just the difference that we want to be able to describe. The fits obtained to the low energy angular distributions are shown in Figure 5. These are for, left to right in Figure 5, O19 + C 1 2 , N14 + C12 , and N14 + Be 9 . The values of W required to fit the varying sizes of diffraction oscillations in the three systems are -2 MeV (O1* + C 1 2 ), - 4 MeV (N14 + C " ), and -10 MeV (N14 + Be 9 ). It is mainly this one difference, the size of W, which distinguishes

the different cases. The remaining parameters, r 0 and a, while not affecting the size of the oscillations affect the positions of the peaks and valleys. In fact, it is possible to cause these peaks and valleys to move one way or the other until a new and approximately equally good fit is obtained. In Figure 6 optical model curves are compared with the elastic scattering energy distributions for O19 + C 12 and N14 + Be 9 . The three curves in the O n + C l * case correspond to three possible fits obtained to the low energy angular distributions as discussed above. They lie essentially on top of each other in the region from 8 to 10 MeV where the angular distributions were fitted. For the higher energies these different cases give somewhat different results, possible allowing a choice between them to be made. The optical model predicts a broad dip at about 13 MeV, consistent with a possible trend of the data if one averages over narrow resonance structure. In the N14 + Be 9 data the energy dependence is smooth and it is possible to obtain a very good fit. The different curves, for different values of W, give an indication of the degree to which this parameter is determined. The curve with W= -10 MeV corresponds to the fit shown for the low energy angular distributions of Figure 5. It becomes apparent from the curves shown in Figures 4,5, and 6 and is born out by detailed inspec-

QI6+CI2 a

10 MeV

su.

tt UJ r t-

2-

a.

blcs 1-0 0-8

0-6

W = -5

a: x 0-4 ui bid

Tj|"0

V = -50 r0= 1-265 a = 0-39

0-2

01 20

40

60

CENTER-OF-MASS

80

100

SCATTERING

120

ANGLE

Fig. 4. Optical model curves with three different values of the imaginary potential depth, W, are compared with the data for O 1 " + C 1 2 scattering at 10 MeV. 13

SCATTERING

1-0 0-8

0«

0-4

9-0

02

MeV


AI27 ELA£l58MeV

: I

:

0 \ " .2 n 1 1111 1 1V-«" i i i i i 0 2 4 6 8 10 12 14 16 18 20 22 24 ecm.( DEGREES)

Fig. 5. Calculations using the parameters of Figure 2 for the data of Mclntyre et al. 5 Although the agreement for Aul97 (Ol6, ol6) Aul97 isnotso close in the region of the oscillations, the low angular resolution of the experiment did not seem to warrant a more thorough analysis. Agreement between data and computations is apparently worst for forward angles in the scattering of 0 1( > by Al27. probably some adjustments in radii would alleviate this discrepancy.

in Figure 5. The radius seems to be correct, as evidenced by the position of the drop from unity. The oscillating region for gold is not too well reproduced. For Ni and Al, the small oscillations present in sharp cutoff calculations appear in the optical model calculations as well. As is the case for gold, the fits at larger angles are better than at smaller angles. The question whether volume absorption is the correct description for the imaginary part of the potential does arise. For neutrons, at least, Bjork-

A-3 Auerbach and Porter I I II

I I

TRANSMISSION

Table 2 COEFFICIENT «s

I

1.0 0.9

0.8 0.7

0.6

-VOLUME -SURFACE

ABSORPTION ABSORPTION

0.5 0.4

a

0.3

0.2 0. I 0

I I I I I I I I I 20

60

V 120

140

Fig. 6. Absorption coefficient T [ as a function of f for the potential of Figure 2 corresponding to the calculations of Figure 3. A best fit "surface-imaginary" computation is also shown indicating a slightly greater transparency of the center of the interaction region and a sharper surface region.

lund and Fernbach? found a surface imaginary term gave good fits for data over abroad range of target nuclei; other investigators have followed suit. We examined the rather restricted case of a gaussian surface imaginary potential of width 1 fermi (Bjorklund and Fernbach used 0.98 f) at the same radius as was necessary to fit the data with a volume imaginary potential. We then obtained a "best fit," varying only the depths V and W. The correspondence to the data was not as good as for the volume case; in particular, the position of the last (large) oscillation is not reproduced correctly. This is hardly conclusive; further investigation is warranted. It may of course be argued that the incident particle sees only the tail of the potential and therefore the form in the interior is not relevant. As we saw in Figure 2, there is a valley in the real part of the potential located not too far inside the nuclear radius; any particle getting through the outer region of potential, comes under its influence. How great that influence will be depends on whether there is also strong absorption at that radius. As a measure of the relative effects of volume and surface absorption, we calculated the absorption (sometimes called, "transmission") coefficients as a function of i . We find (Figure 6) that the surface absorption potential, is slightly more transparent to incident particles of small 4 . In the vicinity of the cutoff, surface absorption cuts off more sharply than volume. This may be what causes

Reaction

a r (barns)

Reaction

crr (barns)

Q16 + Au197 O18 + Au187 C u + Au187 N " + Au1"7

2.31 a 2.36 b 2.19 2.29

Ne20 + Au187 O18 + Bi209 O18 + Ni O18 + Al27

2.41 2.29 2.07

Volume imaginary.

1.82

b Surface imaginary.

the oscillating region to fit the data badly in the surface absorption case. We have also computed the reaction cross sections corresponding to Figures 3-6. They are given in Table 2. m . DISCUSSION The optical model may, of course, be viewed as a smoothed Akhiezer-Pomeranchuk- Blair model, or vice versa. However, the explicit introduction of an optical potential with its parameters leads to radii, in particular, which are more compatible with those obtained for optical potentials used in nucleon-nucleus scattering. In general, the APBmodel radii tend to be rather large. The real parts of the potentials which fit the data well do not seem to be much larger in strength than that experienced by a single nucleon; they c e r tainly are not 16 times the nucleon-nucleus potentials; it appears, rather, that the collision of two complex nuclei is somewhat like a single nucleon experiencing a complex nucleus. Perhaps the interaction "valence" of two complex nuclei is of the order of unity. Since the original calculations were made, it has been found that if only Reynolds data for O18 on Au197 is considered, a W of 32 MeV gives a significantly better fit than that shown in Figure 3. This W is of the same order of magnitude as V. There are slight differences between the various versions of the optical model that can be considered. It will be interesting to see to what extent a static optical potential continues to reproduce the interaction of heavy ions. There are small oscillations present in the computed elastic angular distributions for lighter targets (see Figure 5) which are not demanded by the data and which may be related to finer features of the static potential model that must really be described by a more r e fined dynamical picture, e.g., the small valley in the total real potential (Figure 2) may not be so realistic dynamically. 23

SCATTERING In addition, the data at back angles would be useful in stimulating further work on models for scattering. Since the ratio to Rutherford scattering there is down by four or five orders of magnitude, this is not easy experimentally; however, due to the heavy cancellations in the partial wave series at backward angles, this is expected to be a rather sensitive test of any models which are proposed. A helpful by-product will quite likely be a more precise statement of the techniques of the semiclassical analysis of scattering data. It is also possible that the information obtained from these results may be turned around and applied to the calculation of fission cross sections. We have shown that an optical model analysis of heavy ion scattering is both feasible and realistic. Continued analyses along these lines, in concert with experiments designed to test the detailed predictions of the model, may well provide abetter understanding of the interactions between complex nuclei in addition to providing a more fruitful approach to the relationship between nucleon-nucleon or nucleon-nucleus interactions and heavy ion interactions.

made accurate measurements of oxygen on lead and nitrogen on lead which he has been fitting with a partial wave analysis to within a few percent and he finds that he does not need four parameters, but can do a pretty good job with three. AUERBACH: Yes, it seems that there is a redundancy in the size. The parameters R, a paired together could certainly be reduced to one parameter.

REFERENCES 1. A. Akhiezer and I. Pomeranchuk, J. Phys. (USSR) 9, 471 (1945). 2. J.S. Blair, Phys. Rev. 95, 1218 (1954). 3. N.S. Wall, J.R. Rees, and K.W. Ford, Phys. Rev. 97, 726 (1955). 4. E. H. Auerbach, BNL Internal Report (unpublished). 5. J.A. Mclntyre, S.D. Baker, and T.L. Watts, Phys. Rev. 1 1 6 , 1 2 1 2 (1959). 6. H.L.Reynolds, E. Goldberg,and D.D. Kerlee, Phys. Rev. 119, 2009 (1960). 7. F. BjorklundandS. Fernbach, Phys. Rev. 109, 1295 (1958).

DISCUSSION DAVIS: Wildermuth's remarks pertained to an a particle in infinite nuclear matter and this is certainly a different situation than the one which obtains in the scattering of carbon on oxygen, carbon, or gold. Consequently, the overlap of the particles, taking a semiclassical view, would be the key for the multiplicative factor for the nucleon-nucleus optical model potential. AUERBACH: Surely, at first count you would not expect to go anywhere near 16 times, but you might expect a factor of 2 or 3. Apparently it is not required. It may work but it is not required. McINTYRE: We have continued on our previous measurements and Mr. S. D. Baker at Yale has 24

HIW YORK TIMIS DAILY AVI* AG IS SO COMBINED STOCKS

A-3 Auerbach and Porter McINTYRE: I would like to present here some data which have not yet appeared In the literature and have been called to my attention by Mr. Baker. I have left a slide at the desk for insertion at this point. These data look very similar to those which we have already seen. [ Much laughter] AUERBACH: If there is a reporter here from the Wall Street Journal, he might report that the model fits the stock market tool RAWITSCHER: I would like to come back to the comparison of the nucleon-nucleus with the nucleus-nucleus optical model. The comparison between the two must be done with a certain amount of care because the nucleus-nucleus optical model that one needs extends to much larger distances than the nucleon-nucleus. It is certainly true that the nucleon-nucleus enters into the production of the nucleus-nucleus, but in the way we are trying to calculate this, there enters a lot of complexity especially at these large interaction distances. In addition, perhaps inside the interaction region the analogy fails. AUERBACH: We would certainly hope that the optical model analysis would tie in with proposed theories as to what is going on. We are not say-

ing that the nucleon-nucleus interaction defines the nucleus-nucleus interaction. RAWITSCHER: We tried in a preliminary way to use our exponential potential which, this time, was not calculated from any theory. With an incoming wave boundary condition and an exponential tail for oxygen on lead, we find that we get reasonable agreement but a certain amount of surface absorption is called for in this case. Nevertheless, it is never a clear-cut case because one can always change the real potential to compensate for it. RASMUSSEN: You have shown the transmission coefficient from the optical model calculation as a function of i . Can you make a comment about the real part of the phase shift? AUERBACH: I can say nothing about the real part of the phase shift in the interior region because we would then try to take the arc tangent of the machine round-off error. In the boundary region it seems to follow the type of pattern that was shown in the a paper by Elsberg and Porter. This is the real part of the 6 1 plotted against £ as Is shown for the a case but we would be able to make no comment for the lower portion of the tail because the calculations would not justify making any statement.

25

A-4

PHASE SHIFT ANALYSIS OF HEAVY ION SCATTERING H. E. Conzett, A. Isoya,* and E. Hadjimichael Lawrence Radiation Laboratory University of California Berkeley, California A comparison is made between (complex) phase-shift analysis and optical model analysis of elastic scattering data. In the former treatment, one works directly with the scattering amplitude by parameterizing the partial wave phase shifts. The optional model achieves the same result by working through the intermediary of a parameterized complex potential. The equivalence of the two procedures is shown. The heretofore somewhat arbitrary phase-shift parameterization is altered to a form based on theoretical considerations. Several advantages of phase-shift analyses of heavy ion scattering data are pointed out. The apparent 'discrepancy* between "interaction-radii" as measured by heavy ion scattering and those determined by optical model analyses is explained.

In this paper we want to emphasize the equivalence between phase-shift and optical model analyses of the elastic scattering of strongly absorbedl particles, such as heavy ions. Then, we will point out some advantages in using the phase-shift analyses. We include here all analyses2 in which the partialwave (complex) phase shifts are explicit parameters in the calculation and are not adjusted through an Intermediary optical potential. The differential cross-section for elastic scattering is |f(0)|2

o(e) =

with the scattering amplitude given by

1 ( 0 )

=

f

J

O

)

+

e 2 i o r i P„ (cose)

^

t

f =0

C 2 Î + 1 )

(>-v 2ia

fÄ*« • 1 U O oo 1 *

250 PULSE

HEIGHT

Fig. 1. Pulse-height distribution at 0(lab) = 35.6 for C 1 2 bombardingPb^Oo. The Q-values indicated represent the excitation energies of the nuclei. The contribution to Q from mass changes in the reaction has not been included here.

of the projectiles corresponds then to that of 44 MeV alpha particles. The experimental technique is straightforward. A solid state counter is used to detect the nuclei scattered from the Pb 208 target. The pulse height spectrum is analyzed by a 400 channel analyzer. A typical spectrum for the scattering of C12 by Pb208 is shown in Figure 1. Four peaks are evident, the elastic scattering peak and three other peaks corresponding to energy losses of 2.7 + 0.5 MeV, 4.5 + 0.5 MeV, and 6.1 + 0.5 MeV. A 3-mil mylar absorber was then placed between the target and the counter and the pulse height spectrum taken again.

The result of this measurement is shown in Figure 2. Again four peaks occur, the elastic peak and three other peaks corresponding to energy losses at the counter of 3, 5.2, and 9.3 MeV. From a calibration curve measured experimentally, ? the energy losses of C12 nuclei at the target which correspond to the counter peaks can be determined. These energy losses at the target are also indicated in Figure 2 by the arrows and are 2.7 + 0.5 MeV, 4.5 + 0.5 MeV, and 8.0 + 0.5 MeV. The first two energy losses agree with the values obtained without the degrader, the 8.0 MeV loss is in disagreement with the 6.5 MeV loss obtained without the degrader. The 2.7 MeV and 4.5 MeV peaks therefore are produced by inelastically scattered C12 nuclei, while the 6.5 - 8.0 MeV peak is not. However, if the 6.5 - 8.0 MeV peak is assumed to be produced by C13 nuclei, then the energy at the counter after 3 mils of mylar degradation would

14.71 14.08 13.34 12.71 II .83V

in MeV

M - A

(1+)

/ , ' A

(I-)

10.84 WWW, (I-) 10.10 WWW, pertinent results for the present analysis are the 0--I0 57 i expression for the elastic scattering 419 'A " : / \ / \ 1 T (Ko Ro ' ) 1 : (5) Ji ' \ ' \ \ 55= M k r e 1 / \l o o 1 \M 1 t // . 1 , 1 , 11 ,. 1 . 1 . 1 , 1 ! 1 . 1 . 1 .ii . 1 . 1 .\i 1 . 1 . 1 .11 . and the expression for the inelastic scattering between the ground state of an even-even nucleus to Fig. 9. The inelastic scattering cross section for the an excited state of spin and parity 2+

;

mutual excitation of the 6.14 MeV state in O 1 6 and the 4.43 MeV State in C 1 2 . The dashed line represents the Born approximation cross section fit with Rq = 6.85 f, V 0 = 4.46 MeV, B 2 = .136, and C3 = 2700 MeV. (The experimental cross section is too small by ~20%. Proper normalization is shown in Figure 5.)

tual excitation of the 2+ state in C12 and the 3- in O18 is the following. da dQ K

R

5

417

_7 hWj 4 7T 2CS

o1l-l(

K

T

R

o)-

2 fi V R s o o

( ,

-

l ) i

l(

^ C2 (23 i,000) f K

T

R

o)i'

Ik r 2 ! 2 L o oj

jVk r ov 0 0 J 4

3J, a (K_ R_

6

Ê

4tt

(6)

The J (KR6)'s in this case are the regular cylindrical Bessel functions. Thus if the same target is bombarded by different projectiles and or different energies over a region where the approximation used to derive the above results are equally valid, and the resulting experimentally observed cross

(4)

C (23 i ,000) is a vector addition coefficient as defined by Rose. This particular form will vanish unless t equals an odd number, 1 < I < 5. Thus for K T Ro » 1 the even order spherical Bessel functions will dominate the cross section and it will be out of phase with the elastic scattering as it is observed to be. All the parameters in Equation (4) have been determined by fitting equations 1, 2, and 3 to the experimental results. The results obtained using the previously mentioned parameter values are shown along with the experimental points in Figure 9. The agreement is seen to be fairly good with regard to phase and over-all magnitude, and as the calculation predicts the differential cross section changes less steeply with angle then does the excitation of a single nucleus. These fits are shown merely for illustration purposes. Any truly serious attempt to get detailed agreement with the experiment and subsequently 42

dQ

c^-c12

Il

EcmM«V 72 0 1385 63 4 1.385 62.7 1.37 33 6 1 33

I I !I II

>0«J8 Fig. 10. Comparison of elastic scattering of different projectiles from C 1 2 within the framework of the Blair adiabatic diffraction scattering model. The data used were obtained from references 7, 11, and the present paper.

A-6 Garvey and Hiebert 1

1 1 1 r-

1 I

I

1 1 1 1 1 1 r^ — -—

gamma-ray lifetimes.) Using 150 MeV protons, D. J . R o w e l 6 t a l . found the above ratio to be .24 + .04. A comparison using this data yields .21 + .05. It is interesting to note that the relative excitation probability for the protons and the heavy ions has very nearly the same value. The difference from the electromagnetic transition probability presumably comes about because the overlap integrals emphasize different regions in the nuclear volume. e

o"-c" c"-c" . -c"

REFERENCES

Pig. 11. Comparison of the inelastic scattering to the first excited state of C 1 2 within the framework of the Blair adiabatic diffraction scattering model. The data used were obtained from references 7, 11, and the present paper.

sections are divided by (KQ ROZ) SO as to remove the momentum and radius effects, plotting the results against KQRQS should yield similar results f o r a l l c a s e s . 14

Figures 10 and 11 show the results for the elastic scattering from C12 and the inelastic scattering to the 2+, 4.43 MeV state inC 12 respectively. In the elastic scattering (Figure 10) the heavy ion results cluster rather nicely but at larger momentum transfer the a scattering deviates rather markedly. This deviation would be even more marked if Iho 1

/ \dT21

w a s

s h o w n

1

K R o

2

o

- 1 1 1 Fig-

observed/ V ' Rutherford ure 11 the C12 - C12 cross section should be halved to take into account that either C12 nucleus can be excited. Thus the heavy ion results compare well again and the a results are high. It is difficult to assign the origin of these descrepancies however, further plots of this type may show a systematic behavior that will make the understanding more obvious. Another interesting point arising from this work involves a comparison of the magnitudes of the cross sections for the excitations of the first 2+ states in O18 and C 1 2 . If the relative magnitude of the cross sections were determined by a matrix element identical in form to the electromagnetic transition p r o b a b i l i t y 1 5 i t would be expected that d a (2+ 6.92 MeV O18) KO . d g (2+ 4.43 M e V O " r ° - 5 2 ± ( U 5 - ( T h e E e n " ergy dependence has been removed from the

1. E. Rostand N. Austern, Phys. Rev. 120, 1375 (1960). 2. N. Austern, p. 323, Proceedings of the Kingston Conference, ed. D.A. Bromley and E.W. Vogt, Toronto: University of Toronto Press, 1960. 3. R.H. Lemmer, A. de ShalitandN.S. Wall, Phys. Rev. 124, 1155 (1961). 4. G.T. Pinkston and G.R. Satchler, p. 394, Proceedings of the Kingston Conference, op. cit. 5. H.W. Kendall and Jan Oeser (to be published). 6. N. Austern, R.M. Drisko, E. RostandC.R. Satchler, Phys. Rev. 128, 733 (1962). 7. D.J. Williams and F.E. Steigert, Nuclear Phys. 30, 373 (1962). K. H. Wang, S.D. Baker and J . A. Mclntyre, Phys. Rev. 127, 187 (1962). 8. G . T . Garvey, A.M. Smith, J . C. Hiebert and F . E . Steigert, Phys. Rev. Letters 8, 25 (1962). 9. G.T. Garvey, Proceedings of the Padua Conference (to be published). G.T. Garvey, A.M. SmithandJ.C. Hiebert (to be published in Phys. Rev.). 10. W.W. Eidson and J.G. Cramer (to be published). 11. University of Washington, Cyclotron report 1962. 12. J . S . Blair, Phys. Rev. 115, 928 (1959). 13. A.I. Yavin and G.W. Farwell, Nuclear Phys. 12 1, (1959). 14. J . S . Blair, p. 824, Proceedings of the Kingston Conference, op. cit. 15. F. Ajzinberg-SeloveandT. Lauritzen,Nuclear Phys. 11, 1 (1959). 16. D.J. Rowe.A.B. Clegg, G.L. Salmmand P.S. Fischer (to be published).

DISCUSSION McINTYRE: In the introductory remarks you mentioned that the excitation was independent of projectile. I just wanted to say that we excited the level in Pb208 with carbon, but not with oxygen. GARVEY: Yes, and it is the only example of this that I have seen. It may be just the large Coulomb barrier coming into play. RASMUSSEN: With respect to that I wonder if you have calculated what the grazing velocities of the oxygen and the carbon were in the two cases? 43

SCATTERING McINTYRE: They were both the same; 10-MeV per nucléon in both cases. RASMUSSEN: When they are grazing, they are slowed down by the Coulomb potential. I wondered if there was a difference in the velocity at the grazing of the surface. McINTYRE: No, I don't think so. The Coulomb barrier ZZ1 e 2 / R is proportional to the Z and hence also to the A of the projectile. The bombarding energy is also proportional to the A of the projectile. The kinetic energy of the projec-

44

tile when grazing the nucleus is thus also proportional to the A of the projectile so that the velocity of the projectile when grazing the nucleus is independent of A. PORTER: How significant were the differences between those curves? Were there any differences within experimental e r r o r s ? GARVEY: I would say they were outside of experimental errors. I am sure it could be corrected by adding a Coulomb term and smoothing off the absorption.

A-7

ELASTIC AND INELASTIC SCATTERING OF CARBON FROM CARBON R. H. Bassel, G. R. Satchler, and R. M. Drisko* Oak Ridge National Laboratory t Oak Ridge, Tennessee The measurements of Wang et al. and Garvey et al. on the scattering of 127- MeV carbon ions from carbon show, in addition to elastic scattering, strong single-excitation of the 4.43-MeV (2+) state in one of the nuclei, the single-excitation of the 14-MeV (4+) state, and the "mutual" excitation of both nuclei to the first excited state. We, assuming the two-body effects to be predominant, have analyzed the data for the single excitations in terms of the collective model (1) in distorted wave approximation and (2) in strong coupling approximation where the rotational model is assumed. Good agreement was found between theory and experiment, both in absolute magnitude and in shape of the angular distributions.

Recent analyses of the inelastic scattering of nucleonsl and alpha particles2,3,4,5,6 from nuclei have shown the validity of the collective model' in explaining these phenomena. On this model the initial and final nuclear states are very similar, differing only in the degree of shape oscillation or rotation, and the elastic and Inelastic scattering are simply related.2 It is of interest to inquire whether this simple model, which invokes only the two-body aspects of the problem, can account for similar reactions in the collision between two complex nuclei. The recent measurements of Wang et al.% and Garvey and his collaborators^ provide a sensitive test of these ideas. These experimentalists, using 127-MeV carbon ions, have observed the elastic scattering of carbon from carbon, the excitation of the 4.43 MeV (2+) and 14 MeV (4+) states in one of the carbon nuclei, and the mutual excitation ofboth carbon nuclei to the 4.43 MeV state. Theory To specialize the analyses of these data to the collective model, the interaction is derived from a deformed (nonspherical) potential well. It is assumed that the potential depends on the distance from the "surface" (r-R). The familiar Saxon well is of this form. The deformation is introduced by allowing the radius p a r a m e t e r to be angle-

•On leave from the University of Pittsburgh, t Operated for the USAEC by Union Carbide Corporation.

dependent, i.e., R = R G) 1 - Z |

%

|V4,

+ R (i) + 2

%

Y

kq

1

+

(i)

Y k q (0J.0O

-Z|%|a/4'r « i.«5)] = R 0

+ 6R

where 6', are the polar angles relative to the body-fixed axes, the o^q are the collective coordinates, and we have allowed for the fact that either or both of the nuclei can be excited. A Taylor expansion of the potential about R = RQ yields the series U(r-R)

=U(r-Ro)-6R^U(r-Ro)

J j t i i #2 dr 2

V

V(r-R) o)

Of these terms U(r - RQ) is spherical and leads to elastic scattering; the second term is linear in the collective coSrdinates and, taken once, leads to excitation of the 2+ state in one or the other of the nuclei. The excitation of the 4+ and the mutual excitation of both nuclei can come about in two ways, quadratic and bilinear terms in the second-order interaction can directly excite either the 4+ state or both of the nuclei. These reactions can also proceed via an intermediate state arising from the 45

SCATTERING first-order interaction taken twice. Previous studies of the excitation of the 4+ two-phonon state in Ni 18 have shown these two mechanisms to be comparable in magnitude.5,6 With these definitions of the interaction, we analyze the data using distorted wave methods. If the coupling between the ground and first excited states is weak, the distorted wave Born approximation (DWBA) should give a reasonable description of this reaction. In this approximation the transition is between elastic scattering states, the distorted waves usually being generated from an optical potential which reproduces the observed elastic scattering at the proper energy. On the other hand, if the coupling is strong, explicit account of the strong interaction between the states is afforded by simultaneous solution of coupled SchrOdinger equations. 1,5 The analysis of the excitation of the 4+ state and the mutual excitation process is somewhat more complicated. The angular distributions of the reaction products arising from these states8,9 clearly violate the Blair phase rule2 for even parity transitions, suggesting that both direct and two-step excitation are important. For the mutual excitation process a further complication is added by the fact UNCLASSIFIED ORNL-LR-DWG 78345

f,

exp

-2 /

£

R

k

(p)dp

where p i s the classical turning point. BREIT: My comment about the first paper on this program was really referring to this particular situation; the employment of the classical mechanical approximation. When one speaks about ingoing and outgoing waves, under special circumstances, like the progress of a wave at large distances, those concepts have a definite meaning. But at small distances, you would have to determine the position of the particle and its velocity at the surface which contradicts the uncertainty relation. It is for this reason that the two methods are not really equivalent. They would be equivalent if one were dealing with a plane wave problem for then the wave would be definitely propagated inward.

53

A-9

INELASTIC SCATTERING OF 166-MEV O 1 * IONS BY Ta 181 A. Isoya,* H. E. Conzett, E. Hadjimichael, and E. Shield Lawrence Radiation Laboratory University of California Berkeley, California The relative yields of inelastic and elastic scattering of 166-MeV O 1 6 ions by Ta1®1 have been measured at several forward scattering angles. The inelastic events were identified by coincident detection of known energy (136 and 166 KeV) de-excitation 7-rays. The objects of the experiment were twofold: first, to determine the previously unresolved inelastic contribution to our elastic differential cross section measurements, and second, to compare this determination with that calculated from Coulomb excitation theory at scattering angles where the latter is applicable—that is, at forward angles where the (classical) trajectory of the O ion does not pass through the nucleus even though the incident ion energy is well above the Coulomb barrier. Inelastic excitation of the 136-KeVand 303-KeV levels of Ta 1 8 1 predominated, and the ratio of inelastic to elastic scattering yield agreed with that calculated within an experimental uncertainty o f ± (