Nuclear Physics A496 (1989) 305-332 North-Holland, Amsterdam

THE EFFECTIVE 3He--NUCLEON FORCE IN A M I C R O S C O P I C DWBA A P P R O A C H TO THE (3He, t) CHARGE-EXCHANGE REACTION

S.Y. VAN DER WERF, S. BRANDENBURG, P. GRASDIJK 1 and W.A. STERRENBURG 2

Kernfvsisch Versneller lnstituut, 9747 AA Groningen, The Netherlands

M.N. HARAKEH

Natuurkundig Laboratorium der Vrije Universiteit, Postbus 7161, 1007 MC Amsterdam, The Netherlands

M.B. GREENFIELD

The Florida A and M University, Tallahassee, FL, USA

B.A. BROWN

Qvclotron Laboratory, Michigan State University, East Lansing, Michigan 48823, USA

M. FUJIWARA

Research Center for Nuclear Physics, Osaka University, lbaraki, Osaka 567, Japan

Received 22 November 1988 Revised 20 January 1989

Abstract: The (3He, t) reaction has been studied on the targets 12C, 160, 2~Mg, 26Mg, 2SSi, 32S, 42Ca, 48Ca, 54Fe, 5SNi and *"Zr at incident energies 66-90 MeV. From angular distributions of transitions to selected states the parameters V~, V,~, V1~ and VLS ~ of an effective projectile-nucleon force parametrized with single Yukawa potentials have been obtained for this energy domain. Differences from the force proposed earlier by Schaeffer for a lower energy are found, notably in the tensor component.

NUCLEAR REACTIONS 12C, "O, 24Mg, 2('Mg, 2SSi, S2S, 42Ca, 4SCa, 54Fe, SSNi Eg°Zr(3He, t), E = 66-90 MeV; measured o-( Et, O,); DWBA analysis; deduced effective projec-

tile-nucleon force.

I. Introduction

T h e (3He, t) r e ac t i on m a y be u s e d to s tudy the s p e c t r o s c o p y o f nuc le i w h i c h are

diff icul t to r e a c h by o t h e r r eac t ions . It is an e s t ab l i shed a n d very su i t ab le r eac t ion

fo r exc i t i ng the i soba r i c a n a l o g state ( I A S ) o f the g r o u n d s ta te o f the t a rge t

n u c l e u s ~,2). D u e to t he la rge cross sec t ions a n d p e a k - t o - b a c k g r o u n d ra t ios , the

d e c a y p r o p e r t i e s o f the IAS m a y also be s t ud i ed 2 4).

The (3He, t) r e a c t i o n has a l so b e e n used in r ecen t years to s tudy c h a r g e e x c h a n g e

g ian t r e s o n a n c e s o f v a r i o u s mu l t i po l a r i t i e s . G a m o w - T e l l e r s t reng th has b e e n

Present address: PTT Telecommunicatie, Directoraat lnformatievoorziening en Automatisering, Groningen, The Netherlands.

2 Present address: Natuurkundig Laboratorium der Vrije Universiteit, Postbus 7161, 1007 MC Amsterdam, The Netherlands.

0375-9474/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

306 S.Y. van der Werf et aL / The effective 3He-nucleon force

observed in 9°Nb at E(3He) = 130 MeV by Galonsky et al. 5) and by Ovazza et al. 6) and in 48Sc at E ( a H e ) = 6 6 and 70 MeV, in 54Co at E (3He)= 70 MeV, and in 2o8 Bi

at E(3He) = 81 MeV by Gaarde et al. v-q). In the latter experiment also the proton

decay of the Gamow-Tel le r resonance was observed. Experiments at energies from E(3He) = 600 to 2300 MeV, performed at Saclay by

Bergqvist et al. lo), demonstrate that at these energies the (3He, t) reaction is as

selective for the giant Gamow-Tel le r strength as the (p, n) reaction at the same energy per nucleon.

Experiments at E(3He) = 200 MeV by Tabor et al. 11) and at E(3He) = 73-81 MeV by Sterrenburg et al. 12,13) and by van der Werf et aL t4), mostly on self-conjugate

nuclei, have shown that the (3He, t) reaction selectively excites charge-exchange A L = 1 resonances with both A S = O and 1.

An effective 3He-nucleon force, with which the (3He, t) reaction can be microscopi- cally described in DWBA as a one-step process, has been presented by Schaeffer 1_~) and has been applied to the 42Ca(3He, t)42Sc reaction at E ( 3 H e ) = 3 0 MeV. The

Schaeffer force has been used by Sterrenburg et al. 12,13) for the (3He, t) reactions on 12C and ~60 at E ( 3 H e ) = 81 MeV. Although this incident energy is almost three

times higher than that for which the force was first intended, it was yet found to

be successful in its quantitative description of the cross section. In the effective nucleon-nucleon force, appropriate for a microscopic description

of (p, p') and (p, n) reactions, the components are energy dependent, with the central V,~ term dominating for energies between E o = 100 and 200 MeV. One might expect the effective 3He-nucleon force to exhibit a corresponding energy dependence.

In refs. 12 14), which were initially focussed on the strength distributions of

charge-exchange giant resonances with L = 1 and J " = 1 (S = 0, 1 ) or J ~ = 0 , 2 (S = 1), the 12C, 160, 24"26Mg, 285i, 32S, 4°'42Ca and 58Ni(3He, t) reactions have been studied

at E (3He)= 73-81 MeV. These data, along with data from experiments on 48Ca and 54Fe at E ( 3 H e ) = 6 6 and 70MeV by Gaarde et al. 7,8) and on 9°Zr at E(3He) = 89.3 MeV by Fujiwara et al. 16), include many transitions to final states of simple

configurations which span a wide range of multipolarities. Transition amplitudes to these states or their analogs are well known, either from the (p,n), (p, p') or (e, e') reactions or from detailed shell-model calculations. In this paper data on these selected states are used as a basis to obtain an effective 3He-nucleon interaction appropriate for a microscopic DWBA analysis of the (3He, t) reaction for E(3He) =

22-30 MeV/nucleon. The form of this force will be taken as:

Ve~ = { V , Y ( r / Rc) + V,~,(~rl • ~2) Y ( r / Rc) + V , ~ ( L " S ) Y ( r / RL~)

-t- VTrr2S12 Y ( r / RT . ) } ( , r I • 72)

The Schaeffer force is) is of this form, apart from an overall multiplicative constant

with a value around 5. It has V, = - V~, = 1.32 MeV, VT, = --1.70 MeV, Rc = 1.415 fm and RT,=0 .878 fm, while the (L . S) term is absent. Since, however, the (L . S) term is known to dominate the excitation of high-spin natural-parity states 17,1s) in

S.Y. van der Werf et al. / The effective 3He-nucleon force 307

the (p, p') and (p, n) reactions, it should at least be considered as a possible

component in the 3He-nucleon force. The ranges that appear in the Yukawa functions have been taken so that they

reproduce the long range OPEP behaviour of the force. Components of shorter

range, arising from rho-meson or two-pion exchange on the level of the NN force, when averaged over the volume of the 3He particle, give rise to ranges close to the

OPEP values. Inclusion of exchange effects between the struck neutron and the nucleons that

constitute the 3He projectile does not alter the general form of the force, as has

been shown by Schaeffer ~9). We shall, in this paper, adhere to a direct one-step description of the data base

outlined in the introduction, restricting the radial dependence of each component in the force to a single Yukawa function with OPEP range and investigate its merits

and shortcomings. The (3He, t) reaction may have contributions from two-step processes such as

(3He, a ) (a, t) and (3He, d)(d, t) which, when calculated in second order DWBA,

have to be corrected for non-orthogonality terms 20.7). Two-step contributions may

affect the absolute values of the cross sections, but tend to have angular distributions with shapes similar to the corresponding one step process having the same angular- momentum transfer2~). The effect of two-step processes would therefore mainly

manifest itself as a renormalization of the effective interaction used in a one-step DWBA description. Since the effective force used herein for the 3He-neutron system has a bound state for S = T = 0, the two-step process (3He, a ) ( a , t) is effectively taken into account 22).

The experimental procedure is described in sect. 2. The elastic scattering of 3He

and the optical-model potentials derived from that data are discussed in sect. 3.

In sect. 4 the different components of the 3He-nucleon force are obtained. The

VT component is adjusted to fit the isobaric analog states of the target ground state. The tensor force dominates the transitions to high-spin states of unnatural parity. Therefore the VrT term is derived from transitions to stretched states with one- component single-particle-hole amplitudes and their relevant quenching factors. After V~ and VTT have been fixed, the central V,~ component may be determined from fits to several states of low spin and unnatural parity. The feasibility of including

a spin-orbit component in the effective interaction is investigated. In sect. 5 the resulting force is applied to the excitations of low-lying states in

26A1 and 2Sp observed in the reactions 26Mg(3He, t)26A1 and 28Si(3He, t)ZSP. Sect. 6 contains a discussion and conclusions.

2. Experimental procedure

Momentum analyzed beams of 3He particles from the KVI AVF cyclotron were used to bombard 12C, ~60, 24Mg, 26Mg, 288i, 32S, 4°Ca, 42Ca and 5SNi targets. The

308 S.Y. van der Werf et al . / The effective ~He-nucleon Jbrce

thicknesses, composi t ions and the ~He bombard ing energies for these targets are given in table 1.

The incident energies were chosen such that outgoing tritons from the (3He, t) reaction could still be bent in the Q M G / 2 magnet ic spectrograph 23,24). The detector

system consisted o f two, 120 cm long, gas-filled drift chambers provided with a

resistive wire 25). They were placed 10 cm one behind the other, such that the angle

o f incidence could be determined from the positions measured in both detectors. A plastic scintillator behind the two drift chambers provides energy and time signals which are used for particle identification.

TABLE 1

Target thicknesses, compositions, and bombarding energies

Target Thickness Bombarding (ixg/cm 2) Composition energy (MeV)

~2C 460 self-supp. 81 160 160 (160) + 720 (Ta) Ta205 81 160 250 (mylar) mylar 73

2aMg 350 self-supp. 81 26Mg 650 self-supp. 73 .... t Si 350 self-supp. 81 ~2S 890 (S)+800 (Au) Au-S-Au (sandwich) 73

42Ca 1000 self-supp. 73 aSCa 700 self-supp. 66 4~Ca 350 on 51 ~g/cm2 C 66 S~Ni 800 self-supp. 73

An overall energy resolution of about 50 keV was obtained in all experiments.

Angular distributions o f outgoing tritons were measured from 0 ° to 30 °. For the 0 °

measurement the beam was s topped in a Faraday cup inside the first dipole magnet o f the spectrograph.

The 73 MeV experiment on mylar, per formed in order to obtain data on the 0 ,

1 , 2 , 3- quadruple t in 16F free from background , was done with a 48 cm multiwire

drift chamber 26). During this experiment the oxygen content of the target was

moni tored by measuring elastic scattering at a fixed angle with a moni tor consisting of a z l E - E solid state telescope.

In all runs the vertical acceptance angle was 6 ° . The horizontal acceptance angle was 2 ° for the two smallest non-zero angles and was increased to 6 ° for angles beyond 15 °. At 0 ° spectrograph angle different horizontal acceptances were used

and in order to obtain differential cross sections at different scattering angles below 3 °"

For all targets the absolute cross-section scale was established by compar ison with forward-angle elastic scattering o f 3He off the same target as used for s tudying the (3He, t) reaction.

S. Y. van der Wer f et al. / The effective 3He-nucleon Jbrce 309

For the 26Mg, 28Si, 32S, 42Ca and 58Ni targets, elastic scattering of 3He has been

measured from 4 ° to 20 ° in steps of 2 ° and from 20 ° to 60 ° in steps of 2.5 ° . In these runs the vertical acceptance angle was 5.5 ° and horizontal acceptance angles was

1 ° for angles less than 20 ° and 2 ° for angles larger than 20 °.

3. Elastic 3He scattering and optical-model potentials

Optical model parameters for 3He particles o n 26Mg, 28Si, 32S, 42Ca and 5SNi are

presented in this section. For ~2C optical-model parameters at E (3He)=81 MeV have been obtained by Tanabe et al. 27). These parameters have a real volume and an imaginary surface term. The parameters for 160 have been extracted ~3) from

those of ~2C by using the systematics on A-dependence as given by Trost et al. 28).

Optical-model parameters for 9°Zr used by Fujiwara et al. ~6) also have a real volume and an imaginary surface term. Optical potentials for 3He o n 4SCa and 54Fe a r e

taken from the work of Gaarde et al. 7.8). These have real and imaginary parts which

are both of the volume type. Usually triton optical-model parameters are assumed to be similar or equal to

the 3He parameters. As will be discussed in sect. 4.1, they can be obtained from the

angular distributions of charge-exchange elastic scattering, i.e. the transition to the isobaric analog state (IAS). Optical-model parameters are generally obtained from

fitting the elastic-scattering cross section either with DWBA or CCBA, in the latter case taking into account the cross sections for excited states. In the coupled channels approach, however, not only the coupling to excited states of the target but also to their analogs in the final nucleus should be included on the same footing. Since the analysis of (3He, t) cross sections in this work will be based on a one-step DWBA

description, it is appropriate to use optical-model parameters that are obtained from

fitting the elastic scattering only. Two sets of 3He optical-model parameters have been found, both of which are

based on the geometry of Gibson 29), that fit the elastic-scattering data. They both

have real and imaginary volume terms, but differ essentially in the depth of the real potential. They are listed in tables 2 and 3 with their corresponding fits given in fig.

TABLE 2

3He optical-model parameters, shallower depth set, with rv = 1.14fro, rw = 1.60 fm and r c = 1.25 fm fixed

Nucleus V o (MeV) W, (MeV) av (fm) aw (fm) X2/(N- m)

26Mg 112.0 19.4 0.85 0.83 120 2~Si 110.0 18.0 0.80 0.80 40 32S 117.3 18.8 0.89 0.89 200

42Ca 117.6 16.5 0.82 0.94 184 5~Ni 117.6 16.8 0.83 0.81 80

310 S.Y. van der Wer f et a l . / The effective ~He-nucleon force

TABLE 3

3He optical-model parameters, deeper set, with r v - 1.14 fro, r w = - 1.60 fm, av - 0 . 7 2 fm, aw - 0.81 fm and

r e= 1.40 fm

Nucleus V~ (MeV) W~ (MeV) X2/(N m)

-'6Mg 187.6 19.9 76 2sSi 186.9 17.7 154 ~-'S 179.8 18.8 260

42Ca 177.3 19.1 184 ~SNi 170.7 17.6 207

1. For the shallower set of table 2 the ranges were fixed in the search and the depths and diffusenesses were fitted. For the deeper set of table 3 only the depths of the real and imaginary terms were left free. The shallower set which in most cases gives a better fit has been used for the DWBA calculations described in later sections.

Table 4 lists the optical-model parameters used in this analysis but based on elastic-scattering data from other work. Optical-model parameters consisting of real

volume and surface imaginary terms, based on the systematics of Trost et al. ~-~), have also been tried but generally did not fit the (3He, t) differential cross sections of the transitions to the IAS.

4. D W B A calculations

Distorted-wave Born approximation (DWBA) calculations, using the programs DWBA82 of Raynal 3o) and DW81 by Comfort 3~), have been fitted to the experi-

mental angular distributions. Both programs are updated versions of the original program DWBA70 by Schaeffer and Rayna132). With the recoil correction according to the prescription of Blok and Kunz 33) the programs give identical results. The faster DWBA82 code was used except in those cases which give rise to particle unbound states for which DW81 has the appropriate option.

Single-particle wave functions were generated from Woods-Saxon potentials with

reduced radius ro-- 1.25 fm and diffuseness a = 0.60 fm, a reduced Coulomb radius rc = 1.3 fm and a Thomas spin-orbit strength a = 25. The single-particle energies were determined according to the separation energy method, i.e. for the neutron hole orbitals they were taken as B .E . ( j , )= Sn(g.s.)-Ex(.jn) where Sn(g.s.) is the neutron separation energy of the target ground state and E(j ,) is the centroid excitation energy of the jn orbital in the ( N - 1, Z ) nucleus. Correspondingly the proton single-particle energies were taken as B . E . ( j p ) = S p ( g . s . ) + E~(jp) where Sp(g.s.) is the proton separation energy of the final ( N - 1, Z + 1) nucleus and Ex(jp) the excitation energy of the final state excited by the (3He, t) reaction.

The components of the effective 3He-nucleon force are obtained from different classes of excitations using the following procedure:

S.Y. van der Werf et al . / The effective 3He-nucleon force

lO5r ~ ~ i 7 F ] r 1 - -

1 0 ~ p ~ ]He etastic scattering

103 t

W ~A E,H~73~ /

:°o: , II It, zBS]

: ;,, °b lOS~ - ~ ~ ~ 4

i \ ~'~ 32 S

~ 7 ~ ~ 1~ ~2Ca

~o~ t111 ~,=~ev 103 5 ~ ~" ' --.

10 2 ~ / ~ I,'

,o, ] A

'°° II ~a

20 ~,0 60 80 @~m (deg I

311

Fig. 1. 3He e las t ic different ial cross sect ions. The sol id curves are for the sha l low po ten t ia l set of table 2 and the da shed curves for the deepe r set of table 3.

312 S.Y. van der Werf et aL / The e[/ective ~He-nucleon lbrce

TABLE 4

Other 3He optical-model parameters used in this work

Nucleus V~ W o Wt) r~ r~ rw~ ) a v au, aw.f) re Ref. (MeV) (MeV) (MeV) (fm) (fro) (fro) (fro) (fro) (fm) (fro)

12C 118.2 14.3 1.00 1.17 0.80 0.81 1.25 27)

~'O 119.1 17.5 1.00 1.17 0.80 0.81 1.25 ~) 24Mg 112.0 19.4 - 1.14 1.60 0.85 0.83 1.25 2~,) 4~Ca 114.2 18.4 1.14 1.62 0.84 0.79 1.30 7) 54Fe 114.2 18.4 - 1.14 1.62 0.84 0.79 1.30 s) '~Zr 154.6 24.2 1.22 1.13 0.68 0.94 1.30 ~')

(i) V~ can be obtained, in the absence of the (L. S) term, from the transitions

to isobaric analog states. The central o-r term and the tensor term give no contribu- tions here.

(ii) From (p, p') and (p, n) reactions the real (L. S) term is known to dominate the cross sections for transitions to high-spin natural-parity states. The even-J states

in 48Sc and 9°Nb are suitable for determining this component. For these states, the

central o-r term and the tensor term do not contribute but the central r term does.

(iii) The tensor term completely dominates the cross sections of high-spin states

of unnatural parity. Quenching factors for the underlying single-particle transitions

may be taken from (p, n) reaction data on the same transitions or from the (p, p') or (e, e') reaction to the analog states.

(iv) After the other components are fixed, the central err force is determined from

a number of selected unnatural-parity states that are especially sensitive to it. These are the transitions to the 0 state in " F and a number of 1 ~ states in various nuclei.

The wave functions used in this work are those of Cohen and Kurath 34) for ~2N.

For ~6F we use a wave function close to that of Gillet and Vinh Mau 35), adjusted

to give the right ft value for the decay of its analog, the 16N(0 ) state (see appendix). For 26A1, 2Sp, 32CI we use the wave functions by Brown and Wildenthal 36) and for 42Sc, 4SCa, 5SCu and 9 ° N b w e take pure particle-hole configurations.

4.1. TRANSITIONS TO THE IAS: THE V, PART OF THE EFFECTIVE INTERACTION

Only the central VT and the VLsT components of the effective force contribute to the excitation of the isobaric analog state of the target. The optical-model parameters of the triton are usually taken equal or close to the 3He parameters at the same

energy. In the absence of the Coulomb energy this would be rigorously true. For the excitation of the IAS, the exit channel, triton plus IAS, is obtained from the entrance channel, 3He plus target ground state, by rotation in isospin space. The

triton optical-model parameters are, however, not exactly the same as those for 3He

due to Coulomb energy differences. Phenomenological optical-model parameters

S. Y. van der Werf et al. / The effective 3He-nucleon force 313

from Perey and Perey 37) contain a ( N - Z ) / A term which makes the 3He potentials

deeper than the triton potentials. First an analysis with only the V, component of the effective force, using the

OPEP range R~ = 1.415 fm like in the Schaeffer force is presented. It will be shown

in sect. 4.2 that the L S r force gives a contribution to the angular distribution that

is out of phase with that of the central V, force. Even a small L S r force causes a

rapid deterioration of the fit. Therefore, in this section the IAS transitions will be analyzed with V, only and the need of including a LS'c component in the force will

be discussed further on. In order to avoid proliferation of free parameters, triton optical-model parameters

are set equal to 3He parameters with real and imaginary depths scaled by a common factor k. The IAS transitions on the targets 26Mg, 42Ca, 48Ca, 5SNi and 9°Zr are used

to determine k. The angular distributions are shown in fig. 2. The calculated angular

distributions have been averaged over the opening angle of the spectrograph. Since the cross section at forward angles is evidently insensitive to k, the search

for V, and k may be decoupled. In table 5 the resulting values for V, are given.

These values are found to be relatively independent of the target mass and bombard-

ing energy range of this work. The mean value, V, = 6.65 MeV, is adopted.

The effect of the optical-model scale factor k on the shape of the angular

distributions is seen from fig. 2. The value of k was varied in steps of 0.05. The best

overal value for this set of nuclei was found to be k = 0.85. This value will be used in the following for all triton optical-model potentials including those not determined

directly from fitting to an IAS transition. The resulting triton parameters used in

this work are given in table 6.

4.2. NATURAL PARITY HIGH-SPIN STATES: THE VIs , PART OF THE EFFECTIVE

INTERACTION

It is known that in (p, p') and (p, n) reactions the cross sections of transitions to

natural-parity high-spin states are dominated by the real spin-orbit term in the

effective interaction. In this section the relevance of such a term in the (3He, t)

reaction is discussed. The spin-orbit term in the effective NN interaction has the form

VLs,( r) = V~.s~ Y ( r / RLsT)( L " S)( 'r , • "r2) .

The only other part of the effective interaction contributing to the excitation of natural-parity high-spin states, is the central isovector part, V~. Fig. 3 shows the differential cross sections for even-spin states in 4SSc and 9°Nb. The fits, indicated

by the dashed curves are made with only the VT term in the interaction, fixed to the best values found for the IAS (7.25 MeV for 4SSc and 5.7 MeV for 9°Nb) as

derived in the previous section. For the IAS even a small admixture of a L S r term causes a deterioration of the

fits. The solid lines in fig. 3, which represent calculations in which a term with

314 S.Y. van der Werf et al. / The effective ~He-nucleon force

10 ~

~SEa (3He,f) 48Sc (IAS) l ~ i t~ Ex=6.67 MeV

I01 _~ i0 0 I t ~ E3He=66 HeY I I I

26Hg (3He,f) 26AI (IAS) ~ I Ex=0.228 HeY ] i ~ k

O E3H._: 73 HeY | I " . IO ~ IO I !f\/~, \~

-.~ I 0 - I I 0 °

-~ i' "~", " ~ J SSNi(3He,f}SSEu (IAS)

-o f~,~ 3He= 73 HeY

io' io-2 I.,'! ; \-;', [ I I /

,o ~'/ E,=O.OOO .eV t ,~o~ ~ ,, ~t ~P~ 9°Zr (3He,f) 9°Nb (IAS)~ I

I I E3He:73 HeY ~ ' ~ E×:5.008 HeY t ,., E3He=893 HeY

I i I _, ~ 1 , , I

• I ~xl

_ ,, ~ 1 0 - 2 ~ io ',

I0 L i l i I i L I 10 - 3 . . . . . . J . . . . . . . . . . . . . . . ] .....

0 20 40 60 0 20 40 60 8cr n (deg)

Fig. 2. Differential cross sections of the (3He, t) reaction exciting the IAS. For 3He the shallow optical- model parameter set of table 2 has been used. Calculations are shown for triton parameters of the same geometry as for 3He with the same potential depths (k = 1, dashed curves) and with depths scaled down by a factor k = 0.85 (solid curves). These calculations were averaged over the finite solid angle of the

spectrograph.

S.Y. van der Werf et al . / The effective ~He-nucleon force

TABLE 5

V~ determined from IAS transitions (with k = 0.85)

Nucleus E× E(3He) V~ Configuration (MeV) (MeV) (MeV)

26A1 0.228 73 6.93 Brown-Wildentha136)

42Sc 0.000 73 6.50 idem. 4XSc 6.67 66 7.25 idem. 58Cu 0.203 73 6.50 idem. ~°Nb 5.008 89.3 5.70 (g9)2, g9/2)

315

TABLE 6

Triton optical-model parameters

Nucleus V° W~ W D rv rw rwD av aw awD rc (MeV) (MeV) (MeV) (fm) (fro) (fm) (fro) (fm) (fm) (fm)

IZN 101.8 - 14.0 1.00 1.17 0.80 0.81 1.25 16F 100.5 - 12.2 1.00 - 1.17 0.80 0.81 1.25

26AI 98.2 16.5 1,14 1.60 0.849 0.832 1.25

2~p 93.5 15,3 1,14 1.60 0.800 0.799 1.25 32CI 99.7 16.0 1.14 1.60 0.886 0.893 1.25

42Sc 99,9 14.0 1.14 1.60 0.817 0.935 1.25

4~Sc 100.5 12.5 1.14 1.62 0.842 0,793 1.30

5SCu 99.9 14.3 1.14 1.60 0.831 0.808 1.25 9"Nb 131.4 - 20,6 1,218 - 1.128 0.677 0.939 1.30

VLS~ = 4 MeV and RLS~ = 1.2 fm has been added to the V~ term illustrate this effect. Since the angular distributions for V~ alone and for VLS, alone are out o f phase with each other, negative or imaginary values for VLs~ of the same magnitude do

wash out the steep minima in a similar way. As is seen from fig. 3, the higher the spin of the final state, the more the calculations

including only V, underestimate the cross sections suggesting an increasing domin- ance o f the VLs~ term. Though the shape of the cross sections is not very sensitive to the range of this component a value o f RLs, = 1.2 fm is found to reproduce best

the experimental shapes for the highest spins. Because the VLS, term dominates for all transitions but the IAS, interference

between this term and V~ is relatively small and thus the fits are basically insensitive

to whether VLS, is chosen real or imaginary. The calculations shown in fig. 3 are restricted to positive real values o f VLST with a range of 1.2 fm. Only for the transition to the 2 + state in 48Sc a fit with a negative value is shown in addition.

The VLs~ strengths needed to reproduce the absolute differential cross sections, keeping the value o f V, fixed, are given in table 7. It is found that they increase rapidly with spin. An increase with spin of the normalization, needed to fit the cross sections o f transitions to natural-parity states was reported by Schaeffer 15) in an

316 S. Y. van der Werf et at / The effective +He-nucleon Jbrce

~SCa (3He,t) L'BSc (3He,t) 9°N b E3He=66HeV I ' I E 3 H e : 8 9 . 3 M e V ~ + 0 4 ~

lC ~ ~ 6 " , gs 8" gs

10 2 10

,0 J+,! \ i 10 : T - ,

1/ x x ~_ ] ] / \ ~ I ~ x ~

• " " - . . . . i 6 + " ' ' " - +] 10- G. 104- - . E - 0 1 2 2 H e V

- O ~ *

- / t "q

i " - O ' ~ i ~ E~=1.143 MeV • S,*

t,+, t' f ~ , • - _ [ f \ " . ~ '~0 " ~ E x : O 3 2 8 M e V +

10 o 0* AS 0 1 I ' L ~ 0', IAS --

10 -2 I I I I ~ \ / I 10 ' ~ , h I \ , ] L / I 10 30 50 10 30 50

Ocm ( d e g . )

Fig. 3. Differential cross sections for transitions to natural-parity states in 4SSc [ref. 7)] and 9°Nb [ref. n6)]. The dashed curves represent calculations without a spin-orbit component in the effective force and V~ equal to the value derived from the IAS transition (7.25 MeV for 4~Sc and 5.70 MeV for 911Nb, The sol id

curves have been obtained with an addit ional positive component VLS ~ and the dotted curve for the 2 + state in 4SSc is obtained for a negative V~.s,. Corresponding values are given in table 7.

S.Y. van der Werf et al./ The effective 3He-nucleon force

TABLE 7

VLS . s t r eng ths , u sed to ca l cu l a t e the c ross sec t ions o f e v e n - J n a t u r a l - p a r i t y s ta tes in 48Sc a n d 9°Nb ~)

317

N u c l e u s J ~ Ex V~s, (MeV) (MeV)

48Sc 6 + 0 .000 50.0

4 + 0.252 26.0

2 + 1.413 16.0

( IAS) 0 + 6.67 4.0

9°Nb 8 + 0.000 34.0

6 + 0.122 20.0

4 + 0.328 8.0

( IAS) 0 + 5.008 4.0

( a n d - 8 . 0 )

~') Wi th V, = 7.25 MeV fo r 488c a n d 5.7 MeV for 9°Nb.

analysis of 30 MeV data with V, only. The addition of a VLs, term would seem the logical solution to cure this effect. However, we find that this cannot be achieved

with a unique value of VLs,. We shall come back to this as yet unsolved problem

in sect. 6.

4.3. T H E V;~ T P A R T O F T H E E F F E C T I V E I N T E R A C T I O N

The tensor strength may most appropriately be deduced from stretched spin states of unnatural parity. Such states have rather simple wave functions and within a

restricted basis space have few, and sometimes only one, configurations. Surveys of stretched-spin states excited via the (p, n), (p, p') and (e, e') reactions

have been given by Lindgren and Petrovich 3~) and by Anderson et a139). One

generally finds that the spectroscopic amplitudes of these unnatural stretched-spin states are reduced with respect to shell-model estimates. Quenching factors are almost the same for the three reactions. This quenching may be due to fragmentation of the main shell-model configuration which may be more substantial than obtained within the limited basis space used. Alternatively Papanicolas 40) has suggested that the observed quenching would result from a partial filling of the shells below the Fermi surface, their depletion being caused by short-range and tensor correlations.

For our purpose the origin of the observed quenching is irrelevant. The important thing is that an analysis in terms of an (effective) projectile-nucleon interaction, based upon a single shell-model configuration, is adequate and that the amplitudes of these configurations may be obtained from the (p, n), (p, p') or (e, e') reactions.

The (p, n) [ref. 39)], (e, e') [refs. 41-43)] and (p, p') [refs. 44-46)] data either on the

same stretched-spin unnatural-parity transitions or their analogs for which we measured (3He, t) angular distributions are shown in table 8. Analyses of these transitions 38.39) have been done with the use of harmonic oscillator single-particle wave functions and the effective NN force of Love and Franey 47).

318 S. Y. van der Wer[ et aL / The effective ~He-nucleon Jbrce

TABLE 8

Strength factors S(p, n), S(p, p') and S(e, e') for some unnatural-parity stretched-spin states ~') and values of V~.

E x Nucleus (MeV) I n, T Transition h) S2(p, n) ~) S2(p, p') ~) N2(e, e') d) $2 Vt, adopt. (MeV)

l~'F/~'O 6.372 4 , 1 (lds/2 , lp3/z) 0.51 0.52 0.64 0.52 5.6 24AI/24Mg 5.47 6 , 1 (lf7/2, lds/2) 0.38 0.41 0.38 6.5

2~'A] 0.000 5 ~ , 0 ( ld5/2, ld~/2) 0.56 0.56 6.5 2~p/2SSi 4.95 6 , 1 (1f7/2, ld~/2) 0.32 0.35 0.40 0.35 6.9

4~Sc 1.096 7 +, 3 ( 1['7/2, 1 f7/2) 0.62 0.62 9.8 54Cu 0.198 7+,0 ( lf7/2, 1 f7/2) 0.61 0.61 7.4 '~SY 9 +, 4 ( lgu/2, lgw2) 0.50

'~°Nb 0.82 9 +, 4 ( lg,)/2, lgw2) (0.50) (6.6)

") The (p, n) data are mostly from the Kent State group 3,~). The data for the (e, e') reaction are from refs. 4t 43). Those for (p, p') are from refs. 47 4,).

~' The single-particle wave functions were obtained from a Woods-Saxon potential with geometry ro - 1.25 fm, a - 0.60 fm and r C - 1.25 fm. A Thomas spin-orbit strength A 25 was used. Binding energies were determined from separation energies. The 24Mg ground state has been taken as a (ld~/~) s configur- ation and that of 2~Mg as (ld~/z) u~. For the other targets closed-shell configurations have been taken.

") The calculations were done with the program DWS1 [ref. 3~)] using the updated NN force of Franey and Love 4s).

d) The calculation were done with the programs WSAXM and HEIMAG 4,,).

W e h a v e r e p e a t e d the ana lys i s us ing a W o o d s - S a x o n po t en t i a l o f the s a m e

g e o m e t r y as u sed t h r o u g h o u t this w o r k to g e n e r a t e the s ing le -pa r t i c l e w a v e func t ions .

Fo r the (p, n) and (p, p ') r eac t i ons we used the u p d a t e d ef fec t ive n u c l e o n - n u c l e o n

fo rce o f F r a n e y and Love 48). The resu l t ing q u e n c h i n g fac tors h a v e b e e n l is ted in

t ab le 8. We use the s q u a r e roo t o f the i r a d o p t e d ave rages as r e d u c t i o n fac tors o f

the s ing le -pa r t i c l e t r an s i t i on a m p l i t u d e s in o u r ana lys i s o f the (3He, t) r eac t ion .

The t e n s o r i n t e r a c t i o n has the fo rm:

VTT(r) = VrTr 2 Y ( r / R )S ,2 ( "c l • 72 ) .

F o r the t r ans i t i ons c o n s i d e r e d here the VT fo rce g ives no c o n t r i b u t i o n . The V,~

fo rce is o f l i t t le i n f luence a n d inc rea s ing ly less i m p o r t a n t , the h i g h e r the sp in o f the

final s tate. T h e final va lues o b t a i n e d for Vr, resul t f r o m an ana lys i s in w h i c h a v a l u e

o f V,T, = - 3 . 0 M e V has b e e n a d o p t e d . Th is v a l u e fo l l ows f r o m the ana lys i s o f a

n u m b e r o f t r ans i t ions to s tates o f low spin a n d u n n a t u r a l pa r i ty (see sect. 4.4 be low) .

T h e s h a p e s o f the d i f fe ren t ia l cross sec t ions o f these s tates are not ve ry sens i t ive

to the r ange o f the t e n s o r t e rm. We t h e r e f o r e a d o p t the v a l u e as g iven by Schaef fer ,

R = 0.878 fro, w h i c h was c h o s e n so as to r e s e m b l e at la rge d i s t ances m o s t c lose ly

the t e n s o r pa r t o f the O P E P force , f o l d e d o v e r the 3He v o l u m e .

Fits to a n g u l a r d i s t r i b u t i o n s o f the t r ans i t i ons to all s t r e t ched spin s tates l is ted in

t ab le 8 are s h o w n in fig. 4. T h e va lues d e d u c e d fo r Vr~ fo r these d i f fe ren t t r ans i t ions

are g iven in the last c o l u m n o f tab le 8. T h e r e is a g o o d overa l l cons i s t ency . O n l y

the v a l u e for the 7 + s ta te in 4SSc is s o m e w h a t high. Th is 7 + s ta te is, h o w e v e r , no t

S.Y. van der Werf et al. I The effective 3He-nucleon force 319

IO°L , T T ~ ~ 10]~ ~ l ~ ~ ~

i i £ = 6.372 HeY 4

10°[

L ~,SCa (3He, t ) ~,8Sc ]

i ) / \ _

!

i

4

i / ' '~\ i0_31 ," ~,

~'o-'i 1,. '",, \ t ~°- J i O 7"

\ 6 ] ~ E , = 0 198 MeV 0-'~ Z6Mg (3He, f ) z AI ] ...,, '

,o - , r r , , ' ~ , ~,o-, ,- j L I / ,, \ ~

l_.,7 ',,..-. " ~ t_i,i ",,,-,,,,. t 10 10 ~

9°Zr (3He, f) 9°Nb ] <

104! 6- : I0"2 ,iEx:O.82 HeY i

, ~ = 1,95 M~V i

I0- ~/ L i "\ J L J 0 20 LO 60 0 20 4~',, 60

O~m(deg )

Fig. 4. Differential cross sections for transit ions to stretched states of unnatural parity. The dashed curves are for a central force V,,~ = -3 .0 MeV and the solid curves with an additional tensor force. Quenching

factors on the single-particle ampli tudes and the resulting tensor strengths VTT are given in table 8.

320 S.Y. van der Wef t et al . / The effective 3He-nucleon force

resolved experimental ly f rom the much more intense 2 + state and could only be

analyzed for a few angles. The error bars on these points in fig. 4 do not reflect any possible systematic errors in the peak shape analysis. Also the value for V~ was

found somewhat high for 4SCa (table 5), indicating that the absolute cross section

may be overest imated by some 20% in this experiment.

Giving less weight to the transition to the 4~8c(7+) state, we adopt a value o f

VT, = -6 .5 MeV from the ensemble o f data on unnatural-par i ty stretched-spin states. The sign does not follow from the analysis in this section, since the tensor force

dominates the V,~, componen t completely. As will be shown in sect. 4.4, the negative

sign for VT~, consistent with Schaeffer ~5), is largely determined by the transition to the 0 state in ~6F.

The absolute value o f VT, derived here, is significantly smaller than in the Schaeffer

force, where it was 30% larger than V~. In Schaeffer 's analysis the quenching of

stretched-spin states was not taken into account , as this effect was not known at that time. Would it have been included, the value of Vr, would have to be taken

almost twice as large as that o f VT. We conclude that the tensor force is significantly

weaker at 22-30 MeV/nuc leon incident energy than at 10 MeV/nuc leon for which

the Schaeffer force was derived. On the other hand the value o f VT has not changed significantly.

4.4. THE V,r, COMPONENT OF THE INTERACTION

The range of the central cT-componen t o f the effective force derives from the

O P E P tail of the underlying N N force and is taken as R = 1.415 fm. The contr ibut ion

of this cry" componen t is most prominent in the transitions to states o f low spin and

unnatural parity. For those V~ does not contribute and also the contr ibut ion o f V~s~

is negligible, even if it is given rather large values. The tensor force does, however,

contribute strongly.

The most obvious transitions that could serve to pin down the o-T componen t of the effective interaction are those to 1 + states for which the Gamow-Te l l e r matrix

element is large. Such transitions are known to be domina ted by the o-r force in

(p, n) reactions at high energies.

In fig. 5 angular distributions of transitions to some 1 + states are shown. For these transitions, listed in table 9, wave functions from detailed shell-model calculations

were adopted where possible and the degree to which these wave functions do reproduce the available experimental Gamow-Te l l e r strengths is indicated as a check.

One finds that the tensor force dominates the cross sections already at the maximum of the L = 2 transfer and beyond. Only in the region of L = 0 transfer at very small forward angles the cross sections are found to be sensitive to the value

adopted for V,,~. The latter was chosen to be negative in the Schaeffer force and equal in magni tude to V~ so that the central force would be active only in the S = 0

S.Y. van der Wer f et aL / The effective ~He-nucleon fi)rce 321

12C(3He,f) I2N _ 1o I ' 32S(3He't) 32Cll I

o E3He = 81 HeY I \'-,"~! I~, E3He= 73 MeV 1 0 " .

10'~sj~ 26Mg (3He,f)26A[ ~ 10 ] " , [ ~ E3He=73 MeV J ~ 1" I

. t :1 l

E 0 ~ I I"

• 3 I0-2~ - 17 ~.,,O x0C - - I0- ' - - I ~ , l ~ j "

x\

- 28Si (3He ' f) ZBp _ b~ SBNi (3He,f) S8Cu 10 -1 10 ' ~ E3He = 7 3 1 " MeV i ~-x E 3 H e : 81 MeV

F g.s ° Ex=1313 HeY

10 -2 f \

• . ~,~ ~.., ~o 22

10-" 1; I 30 S;0 10 310 ~ 50 0 (deg)

Fig. 5. Differential cross sections For transitions to some 1 + states with fits For effective Force components Vr, - -6.5 MeV, RT. --0.878 fm and V,,, = -1.0 MeV (solid curves), -3 .0 MeV (dash-dotted curves) and -5 .0 MeV (dashed curves), all with R,~, = 1.415 fm. The calculated cross sections have been scaled by

factors indicated in the figure.

322 S.Y. van der Werf et al./ The effective ~He-nucleon force

TABLE 9

Properties of some selected transitions to 1 + states

E(3He) E x Nucleus (MeV) ( M e V ) wavefunction crexp/cr~,l ~ B(GT)~xp ") B(GT)~,~

12N 81 g . s . Cohen-Kurath 34) 1.0 0.90 b) 0.98 26A1 73 1.06 Brown-Wildenthal 36) 1.0 1.12 ") 1.91

73 1.85 idem. 0.4 0.55 ~) 0.87 28p 81 1.31 idem. 0.2 32C1 73 g.s. idem. 2.5 0.0022 d) 5.0 X 10 5

73 1.17 idem. 1.4 42Sc 73 0.61 (| f7/2) 2 1.0 2.67 ~) 2.57 58Cu 73 g.s. (lf7/2) 2 0.22 0.15 1) 2.57

~') Derived from (CA/Cv) 2 ]M(GT)]2=6170/ft with CA/Cv = --1.251 [50]. b) f t value from F.A. Ajzenberg-Selove and C. Langell Busch 51). ") From the /3 + decay of 26Si. f t values from Brown and Wildentha1152). d) f t value from ref. 52). e) From the /3+ decay of 42Ti. f l values from Endt and Van der Leun 53). r) f t value from Peker 54).

channel . This is reasonable since at lower relative energy the 3He-n system has a

b o u n d state, the 4He g round state, only for S = 0. Adopt ing V~ = 6.65 MeV, as found

from our above analysis and in agreement with the lower energy data from which

the Schaeffer force was obta ined, one finds that a value of a round -4 .5 MeV is

required for V~ to reproduce the 4He ground-s ta te b ind ing energy with a reasonable

trial wave funct ion.

In fig. 5 fits are shown for Vr, = -6 .5 MeV, fixed from the analysis in sect. 4.3 of

high-spin stretched states, and values - 1 , - 3 and - 5 MeV for V~. Most fits are of

good qual i ty and the ensemble of the angula r dis t r ibut ions seems to favour a value

of V~T a round - 3 MeV. In the angular d is t r ibut ions for ~2N(g.s.) and :6Al(g.s.) the

L = 0 m a x i m u m is conspicuous ly absent. Attempts to reproduce these angular

d is t r ibut ions by adopt ing a cry" force or a tensor force with two ranges (not shown)

brought no improvement . This suggests a dependence on nuclear-s t ructure details

that cannot be reproduced with the present form of the effective force. At this stage

we conclude that V~T does not make an impor tan t cont r ibu t ion to 1 + transit ions. A

compromise value of a r o u n d - 3 MeV gives the best overall agreement with the data.

The fit with V, TT = - - 5 MeV overshoots the data at small angles for all angular

distr ibutions. The V~T term thus turns out to be clearly smaller than V~ at energies

of 25 MeV/nuc l eon , unl ike for the Schaeffer force at 10 MeV/nuc leon .

The only case where the cross section scales up or down with V,~ over the full

angular range and not just for forward angles is offered by the ~60(3He, t)16F

t ransi t ion to the 0- g round state. The wave funct ion of this state is given, in a 1 h w

basis space, by x/1 - a21pl/12, S1/2)+ celp3/t2, d3/2). The T D A calculat ion of Gillet and

Vinh Mau 35) gives c~ = 0.055 and a similar ca lcula t ion with the interact ion of Clark

S.Y. van der Werf et al . / The efJective ~He-nucleon force 323

and Elliot 55) gives c~ = 0.069 [refs. ,3.~4)]. The cross section calculated for the (3He, t)

reaction is very sensitive to the amplitude of the [P3/'2, d3/2) component. The beta decay rate of the 0 state in ]6N at 0.1201 MeV, which is the analog of

the 0- state in 16F, t o the '60 ground state has been measured by Palffy et al. 56)

and by Gagliardi et al. 57). As pointed out by these authors, this beta decay rate

may be sensitive to meson-exchange currents and to the induced pseudo-scalar

interaction. Although one should be aware of this, we attempt here an analysis in terms of an effective, purely nucleonic, form factor in a 1 hw particle-hole basis which will make it tractable for our analysis in terms of an effective projectile-nucleon

force. In order to reproduce the experimental f t value of this first forbidden beta decay, one has to adopt a value ~ = 0.125 for t h e Ipf/12, d3/2) component. The details

of this calculation are given in the appendix. Data on the (p, n) reaction for this transition at E v = 35 MeV have been reported

by Orihara et al. 58). An analysis in terms of the M3Y interaction as given by Bertsch et al. 59) and assuming a pure ]pi/'~, s,/2) amplitude gives a cross section that

overshoots the data by a factor 2. The analog transition has been studied in (p, p') at Ep = 65 MeV by Hosono et

aL 6o). Using the same version of the M3Y force they find the experimental cross

section reduced by a factor 0.48 relative to a calculation with a pure ]p,/~ ~, s,/2)

amplitude. Using the same M3Y force, but the two-component amplitude given above with

c~ = 0.125 as found from the nuclear beta decay we find a ratio o -~p /~ ,~ of 1.3 for

10-

f_.

10 -2 E qZ3

'o q [3

10 -3

10 -~.

I I

' 16 0 (3He,f) 16 F ~.. E3He=73 HeY

%

~.f~ ° O- g ' s "

L'

I I I 10 30 50 @cm

f I I I 160 (3He,t)16F l O - ' E3He=81 HeY

-"~ \ o- ~\~ g.s.

10 -2 _-- il

lO_B[-~ i ' !~- ~

-- v,~,, , \ \ 1

10 -~ I I \ k l t 10 30 50

(deg.) Fig. 6. Differential cross sections of the ~O(3He, t) '6F(0 ) transition at E ( 3 H e ) - 7 3 and 81 MeV. The fits represent calculations with the same effective forces as in fig. 5: V;~ - 6.5 MeV, R;~ = 0.878 fm and V,,, T = 1.0 MeV (solid curves), -3 .0 MeV (dash-dotted curves) and -5 .0 MeV (dashed curves), all with

R~,~ - 1.415 fm.

324 S.Y. van der Wer.f et at / The effective ~He-nucleon force

the (p, n) data and 0.8 for the (p, p') data. In the following we adopt this transition ampli tude for the analysis o f our (3He, t) data.

In fig. 6 the angular distribution o f the 160(3He, t)16F(0 ) transition is shown for

73 and 81 MeV incident energy. Fits are shown for VT~ = -6 .5 MeV fixed as above,

and for values of - 1 , - 3 and - 5 MeV for V,T~. The contributions o f the components

V~ and VLST are identically zero for this transition. Unlike for the 1 + states, discussed

above, both VT~ and V,T, contribute with comparable weight to the first maximum (L = 1), resulting in magni tudes o f the cross section that vary strongly with V, TT.

Unfortunately, the shapes of the calculated curves fall off much steeper than the experimental cross sections. The data favour a value o f V,,~ : - 3 to - 5 MeV. A

value o f - 1 MeV underest imates the data at all angles forward of 20 °.

Combin ing the result on the ~6F(0 ) state with those on the 1 + states, we adopt V,,~ = - 3 MeV as the best value, with an uncertainty o f about 1 MeV.

5. Application of the effective force to the reactions 26Mg(3He, t)26Al and 28Si(3He, t)28p

An effective 3He-n interaction for the (3He, t) reaction at 22-30 MeV/nuc leon and

having the general form given in the introduct ion has been analyzed in the previous

section. From different classes o f excitations we arrived at the potential strengths

VT = 6.65 MeV and V~T = - 3 . 0 MeV. For both these central potentials the range o f

the associated Yukawa potential was taken as Rc = 1.415 fro. The tensor force was found to have a strength VT~ = --6.5 MeV and its radial dependence was taken as

r 2 Y ( r / R T T ) with RT~ = 0.878 fro. The presence o f a spin-orbi t type potential seems

required for transitions to high-spin states o f natural parity, but needs strengths that

vary strongly with the spin o f the final state. For IAS transitions it is found not to

be needed and for transitions to states o f unnatural parity it does not or in some cases only weakly contribute.

In this section we analyze transitions to low-lying states in ~6A1 and 2Sp that were

not included in the analysis presented in sect. 4. The LST componen t is left out f rom the effective force. We use transition ampli tudes of Brown and Wildenthal 36).

In figs. 7 and 8, angular distributions for transitions to states in 26A1 and 2Sp are

shown, respectively, and in tables 10 and 11 the spins and excitation energies o f the final states are listed, together with the model states with which they have been identified and the normalizat ion factors that are required.

Our part icular interest is in the overall degree to which the above effective force is able to reproduce the cross sections quantitatively. It can be seen that the normalizat ion factors scatter a round unity for states of unnatural parity. In particular, the cross sections for states o f spin 3 + agree well. This spin-parity class was not included in the analysis o f sect. 4.

Normal iza t ion factors for 2 + states are generally larger. This is the same effect

that was observed in sect. 2 where we found that transitions to states o f natural

S.Y. van der Werf et al. / The effective 3He-nucleon force 325

10 0 1 I I I I

Z6Mg (3He,i.) Z6A[ _ E3He=73 MeV "

I0-I ~/ ~ 3" / ~ • E,,=0.~-17 HeY "

• O i l

10-1l--.

Ex=1.759 MeV

100

, ~ e 3" E,=2.365 MeV

10_ t • •

I0-~1 J = J i I 0 20 z~O 6

10 -1 I I I I I - -

Ex:2.739 HeY 10-z

10 -3 -

10-

) a O

• 2* 10" 13 MeV

100I

lO ~ 2"; T=I ~ e o e Ex=3.160 HeY

10" - ~ ° ° °

10-~1 / [ t I [ I 0 20 40 60

Qcm (deg.)

Fig. 7. Different ia l cross sec t ions of 26Mg(3He, t)2~Al for some low-ly ing states. The fits are for the effective force descr ibed in the text.

326 S. Y. van der Werf et al. / The effective ~He-nucleon force

10 0

10 -1

10-;

10-'

"C

I0- ;

10

10-:

.C

10 -I

10 -2

10 -3

I I I •

28Si (~He,t) 28p E3He =81 HeV

1.131

" ~ E x = 1.516 HeY

20 40

- - 10 - '

10-:

10 o

10 ~

10-2

10-

10-

~ 10-1

I I - - I I ]

• 1 + .569 HeY

i.+2 ÷ .... ~ . -~ Ex=2.'O HeY

-. ~ ~\ SUM

\

'.....f 2 +

1"~2.216 MeV

60 0 20 40 Oc~ (deg.l

Fig. 8. D i f f e r e n t i a l c r o s s s e c t i o n s of 2SSi (~He, t) 2sp f o r s o m e l o w - l y i n g s t a t e s . The fits a r e f o r t h e e f f e c t i v e

f o r c e d e s c r i b e d i n t h e t e x t .

parity seem to require a spin-orbit force. An alternative possibility is that the transition amplitudes for the 2 + states might be enhanced by core polarization effects. This effect has been studied by Brown 63) for sd shell nuclei. He finds enhanced E2 transition rates for isoscalar but not for isovector amplitudes. It seems therefore unlikely that core polarization would enhance the isovector charge- exchange amplitudes.

With the exception of the 2 + states the overall trend is that absolute cross sections are reproduced quite well.

S.Y. van der Werf et al. / The effective ~ He- nucleon force

TABLE 10

States in 26A1 excited in the 26Mg(3He, t)26Al reaction at E(3He) = 73 MeV

327

Model wave E~ (MeV) ~') I ", T function b) N ~)

0.417 3 +, 0 3BW01 0.71 1.759 2 +, 0 2BW01 4.6 2.365 3 t, 0 3BW02 1.95 2.545 3 ÷, 0 3BW03 1.21 2.739 1 ~, 0 1BW04 1.74 2.913 2 ÷, 0 2BW03 3.84 3.160 2 ÷, 1 2BW11 1.34 3.403 5 ÷, 0 5BW02 1.16

'~) Excitation energies from Boerma et al. ~). b) IBWTn stands for the nth model wave function of

Brown and Wildenthal with spin I and isospin T.

TABLE 11

States in 2~p excited in the 28Si(3He, t)28P reaction at E(3He) = 81 MeV

Model wave E X (MeV)~') I =, T function b) N ")

0.000 3 +, 1 3BWI1 1.1 0.106 2 +, 1 2BWll 1.4 1.131 3 + , 1 3BW12 0.8 1.516 2 ~, 1 2BW12 1.0 1.569 1 ~, 1 1BW12 1.5 2.10 1 +, 1 1BWI3 1.1

2 t , I 2BW13 2.8 2.216 4 ~, 1 4BWI1 1.5 2.483 5 +, 1 5BW11 0.9

~') Excitation energies from Ramstein et al. 62), b) IBWTn stands for the nth model wave function of

Brown and Wildenthal with spin I and isospin T. ~) N=v'o'x~,/o- I ~.

6. Discussion and conclusions

T h e e f f e c t i v e f o r c e a p p r o p r i a t e f o r a o n e - s t e p D W B A d e s c r i p t i o n o f t h e ( 3 H e , t)

r e a c t i o n a r o u n d 25 M e V / n u c l e o n i n c i d e n t e n e r g y , e s t a b l i s h e d i n s e c t . 4, m a y b e

e x p r e s s e d a s :

Veer = { V T Y ( r / R c ) + V,,T(~r, • or2) Y ( r / R c ) + VLs~(L" S ) Y ( r / RLs~)

+ V T ~ r 2 S , 2 Y ( r / R T ~ ) } ( X , " "r2),

w i t h V~ = 6 .65 M e V , V,,T = - -3 .0 M e V , Tr , = - 6 . 5 M e V a n d r a n g e s Rc = 1 .415 f m a n d

328 S.Y. van der Werf et aL / The effective ~He-nucleon jorce

RrT ---- 0.878 fm. No unique value for VLs, can be found for any reasonable choice

of the range RLs~ and this term has been left out in the application to the low-lying states of 2~AI and 28p.

The aim of our analysis was to investigate to what extent this effective projectile-

nucleon force for the (3He, t) reaction at incident energies of about E(3He)=

22-30 MeV/nucleon, is different from the interaction proposed by Schaeffer for

bombarding energies of about E(3He)= 10 MeV/nucleon. We have chosen the Schaeffer interaction as a starting point and have obtained new values for the

strengths of the different terms. In addition a spin-orbit force has been studied.

The result of fits to a large body of data may be summarized as follows: Calcula-

tions of transitions to the IAS states give consistently good fits from which an

average value of VT = 6.65 MeV results. This is essentially the same value as in the

Schaeffer force.

The o-r component of the force contributes only little to the cross sections of most transitions. From the forward angle data on several 1 + transitions and the

transition to the 0 state in ~6F a value of V,r~=-3.0 MeV emerges as the best

compromise. This is about half the Schaeffer value and this difference may well be

due to an energy dependence of this term. This dependence on energy differs from

the trend in the NN force where the V~ component increases relative to VT over this energy range 4~).

From a large number of transitions to stretched states of unnatural parity the value of the strength of the tensor term is found to be Vr~ = -6.5 MeV, with relatively

little spreading. Use has been made of quenching factors of single particle-hole

amplitudes as found from the analysis of the (p, n) data on the same transitions

and the (p, p') and (e, e') reactions exciting their analogs. Taking into account that

the Schaeffer force was derived for pure, unquenched configurations, one is lead

to the conclusion that at 22-30 MeV/nucleon the tensor force is only half that of

the Schaeffer value. Whereas the 1AS transitions basically require the absence of a spin-orbit term in

the interaction, such a component seems necessary to explain the cross sections of

transitions to high-spin states of natural parity. The value required for VLs~ must, however, be drastically normalized for each spin value. It appears that the above

functional form of the force does not allow a consistent analysis of notably these high-spin natural-parity states, despite the fact that for the (p, p') and (p, n) reactions

it is precisely the spin-orbit force that dominates. Two-step contributions are expected to diminish with increasing spin and cannot

account for the increasing degree in which the data tend to overshoot the calculation for fixed parameter values of the force. Even if the direct reaction approach may

take care of two-step effects via a renormalization of the strength parameters the explicit study of two-step contributions remains a necessity. Such studies have been reported by Gaarde et al. 7) for some transitions to states in 4~Sc at 66 and 70 MeV incident energy and by Clarke ~4), also for states in 4SSc for polarized 3He particles

of 33 MeV.

K Y. van der Wer fe t al . / The effective *He-nucleonforce 329

The choice of a one-step analysis in the present work has been made for reasons of consistency in analysis and in order to assess its merits when used for a large

data base on transitions of various multipolarities. In view of the remaining dis- crepancies, notably in the excitation of natural-parity states it may be interesting to investigate the possibility of other possible force components, not included in the above form of the force, such as for example a (L . L) term. The available computer programs do not at present allow for such a contribution.

An application of the force, derived in this work, to the low-lying states in 26A1

and 2Sp shows a very satisfactory quantitative agreement.

It is a pleasure to acknowledge the participation in different experiments of A.G. Drentje, H. Riezebos, W. Segeth, A. van der Woude, C. Gaarde, J.S. Larsen and S.K.B. Hesmondhalgh.

This work was performed as part of the research program of the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Neder landse Organisatie voor Wetenschappelijk Onderzoek" (NWO). It was sup- ported in part by the US National Science Foundation Grant PHY87-14432.

Appendix

THE 16N(0 ) t~ , i~,O(g.s.) TRANSITION

In this appendix we obtain the transition amplitude within a 1 hto particle-hole basis that can account for the observed decay rate of the 16N(0 - ) s t a t e to the ground state of ~60. This transition amplitude is used in sect. 4.4 in the analysis of the t60(3He, t)16F reaction populating the 0- ground state, which is the isobaric analog state of the 0 state at 0.1201 MeV in 16N.

The relation between the beta-decay shape factor and the f t value for a 0 ~ 0 + transition is given by 65):

CA 2 ~ a Z = ~ for e .

The Dirac equation for a spherical scalar potential reads [66], with h = c = l,

W t h = i y s ( t r • r) -~ - - + V ( r ) + f l M t h . r r

The large and the small radial wave functions, G~ and FK, with K = j + ½ f o r j = l--~ and K = - ( j + ½ ) for j = l+½ are related through the coupled equations:

d G~(r) K + I G K ( r ) + [ W + M - V ( r ) ] F ~ ( r ) ,

d r r

d F ~ ( r ) K - 1 - F ~ ( r ) - [ W - M - V ( r ) ] G ~ ( r ) .

d r r

330 £ Y. van der Werf et aL / The effective 3He-nucleon force

By replacing [ W + M - V(r)] by 2M in the upper equation, the two equations are decoupled and yield:

d2GK(r) 2 d G K ( r ) l ( l + l ) d r 2 ~ G K ( r ) + 2 M [ W - M - V ( r ) ] G ~ ( r ) = O r dr r 2

dr r '

showing that G~(r) is a solution of the Schr6dinger equation and that F~(r) can

be obtained from GK (r) through the second equation. The most general wave function for the ~6N(0 ) state is, in a 1 ha) space,

I '6N(0-)) = ,/1 - c~ 2(p,/~, s,/2) + ~ (p 3/'2, d3/2)

The single-particle wave functions have been generated from a Woods-Saxon potential with the same geometry as used throughout this work: ro = 1.25 fm, a--- 0.60 fm and a Thomas spin-orbit strength of a = 25.

An L S coupling scheme has been used, with radial wave functions positive near the origin. The single-particle transition matrix elements are given by:

<K,'If~'.,IIKi) = i f [ G ~ F ~ , - O~F~]r 2 dr with K,-= -K i , ./i

(Kr][i(er" r)l[Ki): i[4"a-(2ji + ' ) (2 j r+ 1) ] l /2 /~ / \jr.

x<lll<0½) f d r r 3 G~,GK~. d

'i) <z,-II Y, llO Ji

With the above wave function we obtain for the shape factor:

S¢, °)= [0.21541 - c~2- 0.78S a ] .

Experimental values for S] °l are: (1.42+0.33) x 10 : by Palffy et al. 56) and (1.35± 0.20) × 10 -2 by Gagliardi et al. 57), indicating a (p3/l, d3/2) admixture in the 16N(0 )

wave function with an amplitude a-~0.125.

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