Study of isobaric analog ground states for several nuclei using the (3He, t) reaction

Nuclear Physics A163 (1911) 203 - 224; @ Not

North-Holland

Publishing

Co., Amsterdam

to be reproduced by photoprint or microfilm without written permission from the publisher

STUDY OF ISOBARIC ANALOG GROUND STATES FOR SEVERAL NUCLEI USING THE (3He, t) REACTION W. L. FADNER, Department

R. E. L. GREEN, S. I. HAYAKAWA

of Physics and Astrophysics,

University

t and J. J. KRAUSHAAR

of Colorado,

Boulder,

Colorado tt

and R. R. JOHNSON Department

of Physics,

University

of British

Columbia,

Vancouver

Received 28 September 1970 Abstract: The (3He, t) reaction was used at 37.7 MeV to investigate the isobaric analogs of the ground states of 46Ti , 54Fe 62Ni, 64Ni, 8gY, 90Zr, 91Zr, g2Zr and 93Nb. Differential cross sections were measured from 5” tb 50”. The angular distributions were described adequately by a macroscopic distorted wave theory. The effect of variations in the optical-model parameters and in the form of the interaction potential was investigated. The strengths of the interaction potentials extracted from the data are compared with values of the asymmetry potential obtained from a recent analysis of 3He and t elastic scattering data. Certain nucleus dependent effects are noted. Coulomb displacement energies were determined for nuclei with A around 90. NUCLEAR REACTIONS “‘Fe, 62Ni, 64Ni, 90&(JHe, t), E = 37.7 MeV; measured a(&, 0). *‘Y, 9o. ‘*Zr, 93Nb ground-state isobaric analogs deduced Coulomb energies. Enrichedtargets.

1. Introduction Quasi-elastic scattering using both the (p, n) and (3He, t) reactions has been the subject of a number of investigations. The (3He, t) reaction in particular is convenient experimentally for the study of the excitation of isobaric analog states. Following the work of Lane ‘) the macroscopic distorted wave analysis has used an interaction term that is directly related to the asymmetry term in the optical-model potential. In one study ‘) efforts were made to relate the strength and form of the interaction term for the (3He, t) reaction to the asymmetry term in the optical-model potentials used in the distorted wave analysis. In other recent studies 3p“) the macroscopic interaction term has been determined more phenomenologically from the (3He, t) reaction only. The present study is an attempt to extend the analysis for angular distributions of tritons for the isobaric analogs of the ground state of 46Ti, 54Fe, 62Ni, 64Ni and 9oZr. The work here was aided by the fact that an extensive optical-model analysis of 3He and triton elastic scattering data at approximately 20 MeV was completed by Urone et al. “). A self-consistent approach was made possible by the inclusion of 7 Presently at Foster Radiation Laboratory, McGill University, Montreal. tt Work supported in part by the US Atomic Energy Commission. 203

204

W. L. FADNER

et al.

an asymmetry term in the optical-model analysis with a strength that was consistent with the interaction needed to account for the (3He, t) data. The optical-model potentials so generated from the elastic scattering data were then used in the (3He, t) analysis. The dependence of the distorted wave analysis on the details of the optical-model parameters as well as on the form of the interaction potential were investigated. The strength of the imaginary part of the interaction potential was found to be relatively constant for the nine isotopes used as targets, but not consistent with the asymmetry potential strength from the independent optical-model analysis “) of elastic scattering data. 2. Experimmtal procedure Beams of 37.7 MeV 3He ions, extracted from the University of Colorado cyclotron, were used to bombard the ten self-supporting targets specified in table 1. Also shown in the table are the angular ranges over which cross-sectional data were taken for the nine

Fig. 1. Triton spectrum using E-AE particle identification. data (see table 1).

Tbii method was used for higher angle

nuclei: 46Ti, 54Fe, 62Ni, 64Ni, *‘Y, 9oZr, “Zr and 93Nb. Two methods of detecting the tritons were used. For higher lab angles (specifically, see table I), data were taken in a 92 cm scattering chamber, using a counter telescope consisting of a 171 or a 200 pm (depending on target) thick AE silicon surface-barrier detector and a 3000 pm thick E-AE lithium-drifted silicon detector, in conjunction with an ORTEC particle identifier and conventional electronics. A typical triton spectrum for this method is shown in fig. 1, for 89Y. The lower levels in *9Zr, at 0.0,0.58 and 1.45 MeV, as well as others, have been observed in a number of reactions. The isobaric analog ground state, which is of primary concern in this paper, was seen at an excitation

(‘He, t) REACTIONS

205

206

W. L. FADNER

energy

of 7.87kO.07

SO-120 keV FWHM

et

al.

MeV. The peak width of 100 keV seen here was typical obtained

with this experimental

method.

A Gaussian

of the fitting

program “) was used to extract the peak area and peak location for many of the runs discussed here. The higher angle data for the isobaric analog angular distributions of these nuclei have been incorporated in other studies 4, 7- ‘). To alleviate the problem of large numbers of 3He particles scattered into the detectors at low angles, a quadrupole lens spectrometer ’ “) was used for lab angles between 3” and 27.5” (specifically see table 1). This instrument is a charged-particle reaction products analyzing system which utilizes chromatic aberration in momentum in a high-quality quadrupole doublet magnet to achieve magnetic analysis of a scattered

1800

62Ni(3He t16*Cu =37.!i’MeV

t

v,

1600

e1

s ii!

+,.---

E3tie

,t

Lo = 100

.YD

N

+

/

/ 600-

/

/ ,’

4001 zoofl 0

l #mm

’ 570

** 4

*. %*

l _.

’ I 590

I 610

I 630

I 650

CHANNEL

I 670

I 690

:’

. ..* l * .‘._ *.

710

5. ‘._’

d 730

.*

l.

750

NUMBER

Fig. 2. Triton spectrum using the quadrupole lens spectrometer. This method was used for low-angle data. The dotted line shows the transmission band of the spectrometer.

beam. The system is designed with very low intrinsic resolution (LIP/P x 10 o/,) since it is intended to be used principally as a magnetic rigidity band-pass filter, relying on a conventional solid-state detector for detailed energy analysis. Using a single detector for energy analysis, the field gradient in the quadrupole magnets is sufficient for M/Z’ particle identification of light charged reaction products. Separation is achieved for deuterons, tritons and 3He ions, but protons and alpha particles, while separated from the others, are not separated from each other. Particles of magnetic rigidity differing from that of the focussed (completely transmitted) value are transmitted through the system with efficiencies which decrease as their difference from the focussed-value rigidity increases. The particles enter the spectrometer through an aperture-baffle combination at the quadrupole entrance. The fraction of these which are transmitted to the detector is indicated by the dashed curve in fig. 2 as a function of channel number (energy) for the mass-3 charge-l passing band. In these runs the passing band was centered on the isobaric analog ground state (i.g.s.) of the spectrum. The triton spectrum shown in

207

(3He, t) REACTIONS

this figure is for 62Ni at 10”. The transmission falls off on either side of the i.g.s., although it is essentially constant at 100 y0 near the i.g.s. The angular resolution for the instrument was estimated to be approximately + 1.0”. In addition, there was possible angular position uncertainty of up to 0.2”. Experimental angular distributions for 54Fe, 62Ni, 64Ni and 90Zr are shown in figs. 3 and 4. The two different methods of detecting tritons are represented by different symbols. The error bars shown on the angular distributions are relative only and are essentially determined by statistics. The absolute cross sections have an additional uncertainty of about 10 %. 3. Theoretical analysis In the optical model proposed by Lane ‘), the optical potential was generalized to include an isospin term: U(r) = U,(r)+

%(r)(t

- q4

(1)

where t and Tare the isospin operators for the projectile and the nucleus, respectively. Following this generalization, elastic scattering would be described by a matrix element using the U,(r) term plus the diagonal term of U,(r)(t - T)/A. Extensive studies il) have been made with nucleons that related the asymmetry term of the optical-model potential to the strength of the isospin interaction needed to account for quasi-elastic scattering. Similar studies for the mass-three projectiles are lacking except for the work of Drisko et al. ‘) on 64Ni and 9oZr. Generally, optical-model potentials derived for mass-three particles have not included an asymmetry term, although the analysis of elastic scattering triton data by Flynn ef al. 12) indicated the need for such a term in the imaginary part of the potential. The 3He and triton opticalmodel parameters found by Urone et al. ‘) were parameterized to include a complex isospin term. As expected, the magnitude of the term was found to be small compared to isospin independent terms. In the work reported here, optical-model parameters of both types (with and without isospin dependent terms) were used to calculate the incoming 3He and outgoing triton distorted wave functions. Further information on the optical-model parameters used is included in the next section. It has been found 2-4) that the off-diagonal terms of the t * T matrix will lead to a reasonably good description of charge exchange reactions leading to the isobaric analog of the ground state. This is often termed a macroscopic description. In this paper, the computer program DWUCK 13) was used for the macroscopic distorted wave calculation for the (3He, t) reaction. The transition amplitude was calculated using the equation Tfi = $ (iTo)*/

dr$-‘(r)

U,(r)x$+‘(r).

(2)

The terms xi+)(r) and xi-)( r ) are the initial and final distorted waves, computed from the 3He and triton optical-model parameters, respectively. The isospin of the target

208

W. L. FADNER

et al.

nucleus is To. The factor ($T,,)*/A comes from the off-diagonal term of the (f - T)U,/A matrix in eq. (1) between the target (I’,, T,) and the final analog state (T,, To - 1). Assuming that eq. (1) is valid, the Wi(r) potential found in eq. (2) should be identical to the isospin dependent potential in the optical parameters for elastic scattering. 3eyond this expected relationship, there does not seem to be any conclusive physical argument as to the form or value of the U, term. Since the isospin dependent effect is rather weak in elastic’ scattering, the optical-model fit to the elastic data does not provide very strong evidence as to the form or magnitude of U, . The form of the interaction potential, U, , is taken to be

U-) = - Vly(l +eX)-‘+

dX(l +eX’)-‘1 [ V1, d

- [iW,,(l +eX’)-‘]+iW,,

-J& (1 +eX’)-‘,

with x = (r-G-J*)/ae,

x’ = (r-r&4+)/&

where the subscript 1 refers to interaction potentials, and the v and s refer to volume and surface terms respectively. In practice, most calculations have been done using those terms which have led to the best description of the (3He, t) cross-section data. For example, it was found by Wesolowski et al. “) that the excitation of the isobaric analog ground state with the (3He, t) reaction on titanium nuclei was well described by a macroscopic analysis using V,, = Wls = 108 MeV + and V1, = WI, = 0. In the case of nickel nuclei, Kunz et al. “) found that values of V,, = W,, = 54 MeV and V,, = W,, = 0 were satisfactory. In both cases, the surface imaginary term (WI,) was found to be the most important. Wesolowski ef al. “) found that the calculations were very insensitive to the strength and shape of the real part of the potential. Setting the real part equal to zero or to a volume shape (Vi”) with conventional strength produced little change in the differential cross section for the titanium nuclei they examined. This was also found to be true in the analysis reported here, except that a very large real term did in certain cases cause an appreciable change in the shape of the macroscopic angular distribution. The differences between the more correct coupled-channels calculation and the distorted wave calculation have been explored previously 3, *). It was found that the angular distributions that resulted from the distorted wave calculation were almost identical in shape but shifted overall by a few degrees with respect to the coupled channel calculation in the angular range of interest. These results were considered as providing justification for the use of distorted-wave calculations in this study. t There is considerable variation in the literature with respect to parameterizing the Wr term [see eqs. (1) and (3) J. The parameterization in this paper is the same as in refs. I* 4*5). The extracted values of W, obtained in ref. 3, must be multiplied by 4 to be consistent with,our notation, while the values of Wr obtained in refs. *s r6) must be multiplied by 16 for comparison. All values quoted in this paper have been multiplied by the appropriate factor.

(3He,t)REACTIONS

209

As discussed in previous work 3), it was expected that the macroscopic formalism for the (3He, t) isobaric analog reaction would be relatively insensitive to differences in the optical-model parameters used to calculate the incoming and outgoing distorted waves, as long as those parameters were a good fit to the appropriate elastic data. This is because the reaction being studied is essentially quasi-elastic in nature. Analyses were begun by setting V,, = WI, = 54 MeV and VI, = WI, = 0, as in ref. 4), using the unbracketed terms of eq. (3). Subsequently, several other variations in the form of the interaction potential were tried, including the bracketed terms in eq. (3). In all important cases, values for W,,, the dominant term, were extracted from the no~alization of the theoretical angular ~stributions to the data. The results are presented later. For purposes of comparison, analyses using a microscopic model are also reported. The details of this type of interaction have been discussed elsewhere ‘). A simple shell model was assumed for the present study, a Yukawa interaction occurring between the projectile and one of the nucleons in the outer shell. The orbitals assumed for these nucleons were f* for 54Fe, f3: for 62Ni and 64Ni, and gp for 90Zr. Wave functions for the bound state nucleons were generated from a Woods-Saxon potential with a radius of 1.25 A” fm and diffuseness parameter of 0.65 fm. (The proton potential also included the potential due to a uniformly charge sphere of radius 1.25 A” fm.) Distorted waves for the incoming 3He and outgoing triton particles were calculated using optical parameter sets as discussed in the section on results. A range parameter of 1.0 fm-l was used in the Yukawa potential. All calculations assumed a spinindependent interaction. The computer code DWUCK 13) was used for these calculations. 4. optical-model parameters Using the suggestion of Lane ‘) and Drisko et csl. “) for the generalization of the optical model, the potentials for elastic scattering of 3He and tritons should be the diagonal elements of eq. (1): U(GHt elastic= u~(r)-~s~~(r)~

U(r), etastic= u*(r)+~su*(~),

(4)

where & = (N-Z)/A and where U,(r) is the same potential as in eqs. (2) and (3). For the optical-model parameters used here, the U,(r) term was expanded into a volume real term plus an imaginary term of either volume or surface type (de~nding upon the optical-model set). The parameters due to Urone et al. ‘) included isospin dependent terms. Then U(r) for the optical-model parameters can be parameterized: U(r),,,

= - Voy(l +e”)-l++EV;,(l 4

[

W,,(1+e*)‘-(W,,iSrW;,)~,

i-e”)-’ (1 +exY]

f V&9

(5)

210

W. L. FADNER

et al.

WhHX x =

(Y-roAQzo,

x' =

(r - t-i A+)/ah .

The W;, and Vi, term will have opposite signs for 3He and triton particles, the upper sign applying for 3He. The W;, term in eq. (5) should, in the macroscopic theory used, be equal in form and magnitude to the interaction potential, U,, in eqs. (1) and (2). In fact this is not quite the case. In several of the optical-model parameter sets used, the isospin term was purely surface imaginary; whereas in the interaction potential it has been found that a complex form fit the data better, as discussed previously. This was not an important inconsistency since the surface imaginary has previously been found to be the predominant term, as mentioned before. Since the isospin terms are small in the optical-model potential, the effect on the other optical parameters is relatively small. (Actually, as found in this investigation, either a surface imaginary or a volume imaginary term in the interaction potential may serve as the predominant term.) TABLE 2

A summary of features of optical-model

parameter

sets that were used in DWBA calculations

Optical-model fit to 37.7 MeV 58Ni(3He, 3He) data. No isospin dependent term. Triton Set I ref. 14) parameters were assumed to be equal to 3He parameters. The same parameters were used for all nuclei studied. Sets II-V Optical-model fit to 21 MeV (3He, 3He) and 20 MeV (t, t) data for seven nuclei (including ref. 5, 54Fe, 62Ni, 64Ni, vOZr). These parameters contain isospin dependent terms. Set II

Fixed V’r, = w’,, = 46 MeV, r0 = 1.14 fm, and r’ c = 1.600 fm. Made a four-parameter (VO”, a, I~o,, a’) search to fit the 3He and t data for all seven nuclei simultaneously.

Set III

Initial I’,, = 170 MeV. Fixed a = 0.718 fm (previously determined to be essentially constant at this value for 170 MeV families). Set rc, = 1.14 fm, for a five-parameter (V0”> WO”, W’,,, r’c, a’) search to fit the “He and t elastic data for all seven nuclei simultaneously. Resulting w’i, = 141.6 MeV. Then for each nucleus separately, rc was searched to fit the appropriate 3He and t data.

Set IV

Same as III except initial Voy = 130 MeV, and a = 0.760 fm, (previously determined to be essentially constant at this value for 130 MeV families). Resulting w’,, = 115.0 MeV.

Set V

Initial W’,,,

V,, = 170 MeV. Fixed r0 = 1.14 fm. Used a five-parameter search (V,,, Wo,, r ‘o, a’) to fit the aHe and t elastic data for each nucleus. This set has a large surface

imaginary term, rather than a volume imaginary. Set VI

Fixed r0 = 1.14 fm, r ‘a = 1.600 fm. Fixed v’r, = w’r, = 46.0 MeV in a four-parameter (Vu,, a, WO,, a’) search to fit the 3He and t elastic data for 62Ni and 64Ni. Fixed I”,, = 414.0 MeV and W 1s = 46.0 MeV for a four-parameter ( Vov, a, W,,, a’) search to fit 3He and t elastic data for 54Fe and 90Zr.

The magnitude of the W;, term in the optical-model parameters and the important or WI, term in the interaction potential, eq. (3), should be essentially equal for a consistent formalism, and furthermore they should be approximately constant for different nuclei. The extent to which this was true is discussed in sect. 6. WI,

211

(‘He, t) REACTIONS

Table 2 summarizes the features of the optical-model parameter sets that were used in the macroscopic distorted wave calculations in this paper. Set I is due to Gibson et al. 14). It includes no isospin dependent term. The other sets were taken from the 4001

\

,

I

I

400,

I I

I

-

\

(

I

I

I

I

I

I

200

te,tl54Co I00

R

il II

trl

-u

500

200 100 50

20 IO 5 0

IO

20

30 @CM.,

40

deg.

@C M., deg.

Fig. 3. Data and theoretical curves of differential cross section for the isobaric analog ground state of four nuclei. The data points represented with an open box were taken with the quadrupole spectrometer. Those with the solid circle were taken with E--dE particle identiiication. Three different sets of optical parameters were used to generate these distorted wave macroscopic curves, using interaction potential type A. W:,

which was searched upon along with the other optical parameters in the fit to the 3He and triton elastic data. Sets II and VI contain isospin terms Vi, and W;, which were fixed at values determined to be reasonable from the present (3He, t) work. Set II has V;‘” equal to W;,, in accordance with the type of interaction potential found by several work

of Urone

et al. “). Sets III, IV and V contain

an isospin

term

212

W. L. FADNER

et al.

investigators ’ - “) to b e useful in fitting the (3He, t) data. Set VI has a very large Y;, term for 54Fe and 90Zr only, in keeping with the interaction potentials required by these nuclei according to this investigation.

200 100 50

23 '

IO

f 5

5

t I

z c

PARSE11

---PAR.SETlt%f INT.TYPE

F’

:

ti i

i:.-fs\i ii hi

2

.a

50

ec

M,

60

70

0

IO

20

30

40

50

60

eC.t,t,deg.

deg.

Fig 4. Data and theoretical curves of differential cross sections for the isobaric ground state of four nuclei. The solid and dashed curves represent distorted wave macroscopic curves for two optical parameter sets using interaction potentials having very large surface real terms. The dotted curve represents a microscopic distorted wave analysis.

5. Results The theoretical angular distributions are compared with the data in figs. 3, 4 and 5 for the four nuclei 54Fe, (j2Ni, 64Ni and “Zr. In the angular range where data were taken from both methods of measurement (E--BE particle identi~cation in addition to the quadrupole lens spectrometer), the values of the cross sections for

213

(aHe, t) REACTIONS

the two methods are seen to be in good agreement. The exceptions at 18.5” for J4Fe and at 18.5” and 24” for ?Ii are not surprising in that these points occur at a location of large slope in the differential cross section curves. Since the two methods of

-ii

2

> -is c D 1000 $ 500

5 0

,o

20

30

40

%.A, deg.

50

60

70

0

IO

20

30

40

50

60

70

kM., deg.

Fig. 5. Data and theoretical Curves of differential cross sections for the isobaric ground state of four nuclei. The solid and dashed curves represent distorted wave macroscopic curves using optical parameter set V and interactions A and Al respectively. The dotted curve represents a microscopic distorted wave analysis using optical-model parameter set V.

me~urement have different angular resolutions (+F for E- AE, rt:1” for quadrupole spectrometer), and since small discrepancies may be expected in the absolute values of the laboratory angles, the difference in cross section at these angles is not unreasonable. The differences in the results of the two methods at 16” and 29” for 62Ni are not so easily explained. It is possible, however, that normalization difficulties due to secondary electrons scattering into the Faraday cup of the quadru-

214

5%‘.L. FADNER

et ai.

pole spectrometer may account for the discrepancy at these two points. This difficulty was corrected in the data taken for the other nuclei. The experimental angular distributions had peaks in the cross sections occurring at about 15”, 25” and 40” for all nuclei. It is worth noting, however, that in the cases of 54Fe and 9oZr, the first p eak is reduced in height with respect to the second and third peaks, while in 62Ni and 64Ni, the second peak is reduced with respect to the first and third. It would be desirable for any theoretical description to account for these features of the data. The distorted wave angular distributions resulting from the use of the first four optical-model parameter sets were very similar. These results are summarized in figs. 3 and 4. In fig. 3, theoretical angular distributions are shown for three of the optical parameter sets examined, using the macroscopic theory with interaction type A (v,v = w,,, V1, = WI, = 0), as used by Kunz et al. “). Variations in shape were minor. There was some difference in the extracted values of the imaginary interaction potential WI,, as shown in table 4 and fig. 6. Good fits for the first two peaks in the angular distribution of 62Ni and (j4Ni were obtained using the first six parameter sets, although the theoretical predictions were somewhat low at the third peak. The first two peaks in the angular distribution of 54Fe and 9Zr were not as well described by the theory. According to the data, the magnitudes of the first two peaks were about equal, whereas the theoretical ratio of first to second peak height was typically about 2. Some ambiguity in the extracted asymmetry potential resulted in these cases due to the inability to simultaneously fit the first and second peaks. Several other variations were made in the theoretical calculations using the first four sets of optical-model parameters, through changing the form and magnitude of the interaction potential terms as summarized in table 4. Some of the angular distributions are shown in fig. 4. The solid and dashed curves show the effect of introducing a very large volume term, I/,, , into the interaction potential of the macroscopic analysis for two of the optical parameter sets. The ratio of V1,/W,, in the interaction potential was set in the range of 5.0 to 10.3, depending on the opticalmodel parameter set used. It is evident that the resulting theoretical angular distributions describe all three peaks in the 54Fe and “Zr data rather well. Furthermore, these calculations are still relatively insensitive to the optical-model parameter set used. The fits to the 62Ni and 64Ni data are not good, however, and furthermore these angular distributions are very sensitive to the different optical-model parameter sets. These features in the theoretical curves occurred only when there was a large real volume term in addition to the usual imaginary surface or volume term (interaction types B, B’, F, F’). These results can be understood for the most part by noting that the imaginary part of the potential has a far larger radius than the real part (approximately 1.6 A’ fm, compared to 1.1 AS fm). The larger radius makes the imaginary terms more important in the interaction, and therefore when the real and imaginary well depths are about equal, the real terms have little effect, as discussed in previous work 3*4). If

215

(3He, t) REACTIONS

a large real volume term is introduced, then more of the interaction can proceed through this term, changing the shape of the angular distribution. However, the imaginary term is still more important, especially in determining the magnitude of the theoretical cross sections. These considerations are further demonstrated in ths results using o$ical-model parameter set V. This set is unusual in that a larg: surface imaginary t:rm, WOS, was used with the volume imaginary term, W,, assumed zero. As can be seen in table 3, this resulted in an imaginary radius, ri, which was only slightly larger than TABLE

Optical-model

3

parameters used in the DWBA

calculations

-

Set I

170.6

0

1.14

0.712

18.50

0

0

1.600

0.829

173.7

46.0

1.14

0.726

16.68

0

46.0

1.600

0.826

Set III

3He t 3He t 3He t 3He t

174.3 174.3 174.3 174.3 174.3 174.3 174.3 174.3

0 0 0 0 0 0 0 0

1.111 1.112 1.175 1.142 1.168 1.150 1.111 1.146

0.718 0.718 0.718 0.718 0.718 0.718 0.718 0.718

19.46 19.46 19.46 19.46 19.46 19.46 19.46 19.46

0 0 0 0 0 0 0 0

141.6 141.6 141.6 141.6 141.6 141.6 141.6 141.6

1.522 1.522 1.522 1.522 1.522 1.522 1.522 1.522

0.836 0.836 0.836 0.836 0.836 0.836 0.836 0.836

Set IV

3He

128.6 128.6 128.6 128.6 128.6 128.6 128.6 128.6

0 0 0 0 0 0 0 0

1.109 1.107 1.178 1.147 1.169 1.157 1.100 1.168

0.760 0.760 0.760 0.760 0.760 0.760 0.760 0.760

15.98 15.98 15.98 15.98 15.98 15.98 15.98 15.98

0 0 0 0 0 0 0 0

115.0 115.0 115.0 115.0 115.0 115.0 115.0 115.0

1.629 1.629 1.629 1.629 1.629 1.629 1.629 1.629

0.812 0.812 0.812 0.812 0.812 0.812 0.812 0.812

-48.0 -48.0 521.6 653.6

1.174 1.171 1.174 1.160

0.807 0.785 0.771 0.791

46.0 46.0 46.0 46.0

1.600 1.600 1.600 1.600

0.826 0.738 0.738 0.826

Set II

3He t 3He 3He Set V

3He 3He 3He 3He

and and and and

t t t t

167.5 173.4 173.0 174.0

0 0 0 0

1.14 1.14 1.14 1.14

0.722 0.757 0.764 0.739

Set VI

‘He sHe 3He 3He

and and and and

t t t t

174.0 171.8 171.8 174.0

414.0 46.0 46.0 414.0

1.14 1.14 1.14 1.14

0.690 0.768 0.768 0.690

0 0 0 0 17.45 18.53 18.53 17.45

109.5 129.6 136.3 141.6 0 0 0 0

These terms were combined according to eq. (5) to give the total optical potential. The potential depths are in MeV and the geometric parameters in fm.

the real radius ro. When this same geometry is used in the interaction potential has been the case in all results reported here previously), then one expects that decrease of the imaginary radius with respect to the real radius would decrease relative effect of the imaginary terms in the interaction potential with respect to

(as the the the

216

W. L. FADNER

et al.

real terms. This should have two effects: (i) to greatly increase the extracted value of the imaginary term W,, or W,, needed to fit the data, and (ii) to increase the relative effect of the real terms, similar to the effect seen using interaction types B and F with the other optical parameters. These effects are essentially what is seen in the theoretical angular distributions using optical parameter set V, as evidenced by the solid curves in fig. 5, and the appropriate value of WI, found in table 4. The theoretical angular

Interaction Interaction type

A A’ Al

B B’ C D E F F’ G H I

TABLE4 potentials used, with resulting Wi valucc

Int. potentials used V I”

VI,

Wl”

WI,

Approximate shape

Average extracted opt. set I

opt. set11

opt. set111

opt. setIV

46&6

44f5

48*6

40*5

33h5

% 36

+; 00 0 Wl w1 same as type A, but opt. set I geometry for int. potential RWI 5Wi 0 0

0 0 WI RWl

0 0 0 0

W, w, Wl W,

RW1 5w,

0 0 0

WI Wi WI

0 0 0

Wl RWl 0

WI WI 0

0 0 w,

0 0 0

Approx. avg. extracted WI for each opt. set Value of W’l used in opt. set

35&6 z 51 37*5 = 42 32f2 x 44

39 0

37*5 44+7 33*4 42&6

opt. set V 24lzt37’) % 230 b, 42f 5

37*5

35*5 42&7 32&5 4618 39.4 46

Wi

39 141.6

z 45

332*53

40 115.2

287

All values are in MeV. The ratio R = Vo,/ W,, and depends upon the optical parameter set, being in the range of S-10.3. Errors shown reflect only the uncertainty in fitting the theoretical angular distributions to the data. Where a z sign appears with the extracted value of WI, fewer than three nuclei were examined. Shape a is typical of macroscopic interactions which proceed largely through the surface imaginary term WI,; this type of shape has been reported in several previous papers. Shape b is considerably different, giving a good description of the (3He, t) data for 54Fe and “Zr, a poor description for the other targets. “) Shape b for opt. set V. ‘) Used only with S4Fe.

distribution for 54Fe shown by the solid curve in fig. 5 is a slightly modified interaction potential, A’, which gave a somewhat better fit than A. To further explore the effect of the interaction potential geometry, interaction type Al was tried with optical parameter set V. This interaction potential uses the geometry of optical parameter set I (r. = 1.14, a = 0.712, ri = 1.60, a’ = 0.829). With this more typical geometry for the interaction potential only, it became apparent that optical parameter set V was really quite similar to the others. It can be seen that the dashed curve in fig. 5 is very

(3He, t) REACTIONS

217

similar to all of the curves in fig. 3. The data for 62Ni and ‘j4Ni are fit acceptably well and for 54Fe and ‘*Zr not so well. Using the hybrid optical-model parameters, set VI, (and the appropriate hybrid interaction potentials: type A for 62Ni and 64Ni, type B for 54Fe and ‘*Zr), good fits of the theoretical macroscopic angular distribution to the data were obtained. That is, the result of imposing the best interaction potential upon the optical-model parameters is to slightly improve the (3He, t) fit over that obtained using one of the other optical-model parameter sets, but the same hybrid interaction potentials just mentioned. The above results show that for the wide range of optical-model parameter sets examined, good fits to the experimental data could be obtained only by using different interaction potentials for 54Fe and gOZr than for 62Ni and 64Ni. Interaction potentials that proceed primarily through the surface imaginary term (types A and C are typical of these and have been employed in previous work), have typical theoretical angular distributions with an approximately constant slope of the peaks of the distribution, as can be seen for all of the curves of fig. 3 and the dashed curve of fig. 5. The titanium nuclei ‘9’) follow this pattern fairly closely, as does 26Mg at 16 MeV [ref. ‘“)I. For other cases that have been studied ls - “) it has been found that the theoretical slope is approximately correct for the first two peaks, but that the third peak is of the same magnitude as the second. This was also the case for 62Ni and 64Ni examined here. The effect seen here for 54Fe and “Zr, in which the first two peaks are of the same magnitude, is apparently not common. Furthermore, the pattern of the angular distribution for 54Fe changes considerably when the energy is reduced to 30 MeV [ref. 16)]. A ch an ge in the patterns of the angular distribution of the analog ground state of 48Ca at 30.2 MeV versus 18.2 MeV has also been observed 16). These differences in shape of the angular distribution are not adequately explained at present. The dotted curves in figs. 4 and 5 show the calculated differential cross sections for a microscopic formulation using optical parameters III and V respectively. Opticalmodel parameter sets I, II, IV and VI were also investigated using the microscopic model but the results are not shown. All of these optical-model parameter sets gave similar results. The results were more strongly dependent upon the optical parameter set for 62Ni and 64Ni than for 54Fe and ‘*Zr. None of these microscopic-model calculations provided as good a description of the data as did the better macroscopic calculations. In summary, the above results indicate that the inclusion of isospin dependent terms in the optical-model parameters does not cause a significant difference in the theoretical (3He, t) angular distributions with either the macroscopic or microscopic model. Furthermore, the 130 MeV family of optical-model parameters gave results similar to the 170 MeV family. The macroscopic formulation was sensitive to the interaction potential used, and these results indicate that for certain nuclei a different asymmetry term is required than for others.

W. L. FADNER

218

al.

et

Well depths for the asymmetry potential

6.

The value of W,, or I+‘,, was in each case extracted from the (3He, t) data by adjusting the normalization of the theoretical distributions to fit the data. One of the goals of this investigation was to determine if the values of W;, from the optimal-modei parameters of Urone et al. “) were consistent with the values of W,, or W,, extracted from the (3He, t) data, and, if possible, to provide feedback for possible correction to the W,f, value in the optical-model parameters. The extracted values for the imaginary isospin term W, ( W,, or W,,) are summarized in table 4 and fig. 6. For a given set of optical parameters and a given type of interaction potential, these are roughly constant; i.e. independent of (N-2)/A. This tends to verify that the isospin dependency contained in eqs. (2) and (4) is approximately correct at the 37.7 MeV energy used in this analysis. I

“%i

5’Fe

I

I

I

I

tt

I-A

X5 - HYBRID -_-___.c 1 I

0.02

I

I

004

I-F I

I

0.06

I

I

0.06

I

I1

I

0 IO

0.12

I

0.14

(N-2)/A

Fig. 6. Extracted interaction potential for a macroscopic distorted wave analysis as a function of (N-2)/,4. The optical parameter set used is given by the Roman numeral, followed by the symbol for the type of interaction potential used.

The approximate average for the extracted value of W, for five sets of optical parameters (sets I-IV and VI) was about 40 MeV, slightly lower than the value of 54 MeV which had previously “) been determined for the nickel isotopes at 37.5 MeV incident energy. Both of these values are considerably lower than the values ranging from 80-140 MeV found by other investigators 2~3~15)at 25 and 30 MeV incident energy. There are apparently two factors contributing to this difference. First, the extracted value of I+‘, has some sensitivity to the geometrical parameters used in the interaction. This is discussed in ref. “). A second factor appears to be an energy dependence on the part of the magnitude of the experimental cross section. Recent data for 62Ni indicates an energy dependence in the cross section beyond that predicted

219

(3He, t) REACTIONS

by the DWBA analysis. Wesolowski et al. “) obtained cross-section data for 46Ti at 24.6 MeV which had a first peak magnitude of about 108 pb/sr as compared to the value of about 72 Bb/sr found at 37.5 MeV [ref. ‘)I. The difference in shape of the experimental angular distribution at different energies for both 54Fe and 48Ca, mentioned previously, also indicates an energy dependence in the interaction. The above two factors seem to be sufficient to account for the difference in W, values obtained in the quoted papers. In summary, comparisons in the magnitude of W, are not necessarily valid when different geometries are used in the interaction potential. Furthermore, there is an indication of an energy dependence in the magnitude of W, which requires further study. Optical-model parameter sets I and II gave approximately equal results for the extracted value of the imagina~ interaction term, W, . For interaction potentials, such as A, C, E, G, I, which do not have a large volume term, the extracted value for W, was approximately 43; quite consistent with the value of 46 imposed upon optical set II. For interactions with a large volume term, the extracted WI was about 35. Optical-model parameter set III was obtained ‘) by including a search on the W;, value, and the result was a value of 141.6 MeV. This is not in good agreement with the average extracted value of 39 MeV using this parameter set in the DWBA analysis. Optical-model parameter set IV was obtained in the same way as set III, but for the 130 MeV rather than the 170 MeV family of parameters. This resulted in a W; of 115 MeV, also considerably higher than the approximate average extracted value of 40 MeV. Since the isospin dependent term is small compared to the other optical-model parameters, and since the parameters of Urone et af. “) were obtained from elastic data taken at about 20 MeV, these results are not surprising. It is also reasonable that the resulting W;, of optical parameter set III agrees quite well with those obtained by Drisko et al. ‘), since both groups used the same 20 MeV triton data and an rh of about 1.5 fm. Both experiments at 20 MeV yield a value of W ;, which is a factor of about 3 higher than the value of WI, extracted from t3He, t) data at 37.7 MeV incident energy. TABLE 5

Extracted potentials Optical parameter set I

II III VI

( W,) obtained from the hybrid interaction

46Ti

s4Fe

44 *5 43.5k2.5 50 *5 54.5*3

33.5i2 38 &2 35 f3 29.5 + 1

9oZr

Avg.

44 &3 42.5f1.5

36.512 39.5*3.5

42 f2.5 51.5*3.5

36 &2 29 k2.5

41.5*3 42.3+3 43.5k3.5 44.6*2.5

62Ni

64Ni

47 12 48 rt3 55 rfi3 58.5&3

For parameter sets I, II and VI, a type A inte~ction was used for 46Ti, “%i and 64Ni, and a type B interaction for 54Fe and *@Zr. For parameter set II, a type A interaction was used for 46Ti, 6ZNi and 64Ni, and a type B’ interaction for s4Fe and 90Zr. These hybrid interactions resulted in better fits of the macroscopic angular distributions to the (sHe, t) data for all nuclei than did any other type of interaction.

220

W. L. FADNER

ef al.

The extracted values of WI for optical-model parameter set V were very large, as expected, due to the small value of the imaginary well radius, rd. The average extracted value of WI for the hybrid optical-model parameter set, set VI, is about equal to the value imposed on the set (see tables 5 and 3). The extracted values of WI given in table 4 were arrived at by averaging the values for all nuclei using the same optical parameter sets and interaction potentials for all nuclei. However, the discussion in the previous section indicates that different interaction potentials might be required for the different nuclei. That is, for best fit of the macroscopic theoretical angular distributions to the data, we would require an interaction like type A (that is, A, C, E, or some similar interaction) for 46Ti, 62Ni and 64Ni and an interaction like type B or F for 54Fe and 9‘Zr. Under these conditions, the extracted WI could be better determined for 54Fe and 90Zr. However, when interaction types are mixed in this way, the isospin dependency for different nuclei is more complicated than the form given in eq. (1). The form and magnitude of U1 are not constant for this hybrid interaction, and therefore it would not be expected that the imaginary part, WI, would be constant in magnitude. Nevertheless, it is interesting to examine how nearly constant WI remains: this is presumably a rough measure of the extent to which WI is still the most important term. The resulting values of WI for this hybrid type interaction are shown in table 5 for several optical-model parameter sets, including set VI, the hybrid optical-model set. As expected there is a general tendency for the values of WI for 54Fe and 90Zr to be lower than those for the other nuclei. These results indicate that the best fit to the data does not yield truly constant values for WI, for the nuclei examined. Yet to a fair approximation (x 25 %) WI is constant for all the nuclei studied using the interaction potential which fits each nucleus best. 7. Information on nuclear shape Some fairly simple arguments and reasonable approximations, summarized by Satchler i I), lead to the conclusion that the imaginary asymmetry potential WI should be proportional to the difference in neutron and proton densities:

JJW = Wh(f-)-~,(r))~ where K1 is a proportionality constant. To the extent that the macroscopic analysis is valid, the comparison of this theory to the data for the (p, n) and (jHe, t) quasielastic scattering data should be useful for studying the form of WI(r) and hence the difference in neutron and proton densities in the nuclei. Investigations ‘*) have been carried out to deduce nuclear shape in a more indirect way using optical-model fits to (p, p) and (n, n) data. The fact that the surface imaginary term has been found to be the most important term in describing the analog ground state angular distribution in many earlier studies, as discussed previously, tends to support the contention that the neutron excess in most nuclei is surface peaked. According to this analysis, 62Ni and 64Ni would seem

(3He, t) REACTIONS

221

to follow that pattern. However, the large volume terms required in this study for 54Fe and 90Zr, in addition to the normally expected surface imaginary terms, tend to indicate that the neutron excess for these nuclei is not as strongly surface peaked as other nuclei which have been examined, and that the neutron density is somewhat greater than the proton density in the interior of the nucleus. In order to gain further information on relative neutron and proton densities, an investigation has been initiated using the difference between two Woods-Saxon wells as the interaction term, U, , in a macroscopic analysis. These two wells are intended to represent the potential due to neutrons and protons respectively. It can easily be shown that the difference between two Woods-Saxon wells can lead to a wide variety of shapes (as a function of r), including those which are primarily surface peaked, those which are volume shapes, and combinations of volume and surface. Furthermore it is possible in this way to generate shapes which are different from those obtained through a combination of a Woods-Saxon and a derivative Woods-Saxon well. In the preliminary investigation to date, the restriction has been made that the volume integral of the radial function for the two Woods-Saxon wells be proportional to the masses of the total neutrons and protons in the nucleus, respectively. The radii and diffusenesses reported by Greenlees et al. 1“> were used as a starting point for the wells, after applying a factor to account for the greater effective diameter of the ‘He particle compared to the proton. Several macroscopic calculations were then made using the DWUCK program and varying values of radius and diffuseness for the two wells (with the volume limitation as discussed above). The results, although very preliminary, are encouraging. On those nuclei for which this method was tried, it was possible to fit the (3He, t) data as well as any of the methods described previously. These results were not strongly sensitive to the optical-model parameter set used. (This is undoubtedly because the effective interaction potential radius, r,, , was chosen independently of the optical-model parameter set used.) The difference between the two wells could be used as a purely imaginary interaction potential, with no need for the additional real potential, which has been found necessary in previous macroscopic formulations. Finally, one combination of two Woods-Saxon wells in a macroscopic formulation has an angular distribution which fit the 54Fe data at 37.7 MeV, and in addition there was a tendency for the calculated cross section to change in the proper way, qualitatively, to assume the quite different shape of the already observed data at 30 MeV, as discussed under the section on results. Therefore it is thought that this general method of formulating an interaction potential may lead to an internally consistent description of elastic and quasi-elastic scattering. Furthermore, (3He, t) interactions, being very sensitive to the isospin dependent terms, may lead to a two-well interaction potential which, through the ~cros~opic formulation, will describe the (3He, t) isobaric analog ground state data at different energies. If this can be accompfished, then the results may lead to useful information on the relative neutron and proton densities in the nucleus. Further work is presently underway on this point,

222

W. L. FADNER

et al.

8. Coulomb excitation energies for the isobaric analog ground state In addition to the determination of angular distributions for several nuclei, the Coulomb excitation energies for the isobaric analog ground states of “Y, 9oZr, g2Zr and 93Nb were extracted from the spectra taken at 25”. The value of Coulomb excitation for ‘IZr was taken to be equal to 11.83_+0.03 MeV, as given by Long et al. I’). This value is the average of the excitation energy determined using the (d, p) reaction and of that using mass data. Differences between the excitation energy of 91Zr and that of “Y , 9oZr, 92Zr and Q3Nb were each determined by measuring the appropriate channel differences, using the known excitation energy of the isobaric analog ground state of QIZr to establish the energy per channel. Monitoring of the

Fig. 7. Coulomb displacement energy for isobaric analog ground states of several nuclei with A = 90.

‘He elastic peak during each data run provided a check on any shift of cyclotron beam energy or gain changes in the electronics. Shifts of beam energy and gain changes were negligible. The value of the Coulomb excitation energy for each particular nucleus was then determined by adding the energy difference thus obtained to the assumed value for ‘IZr. The entire process was done twice on different runs using different cyclotron tunes, and bombarding the isotopes in a different chronological order, in order to check on any unexpected experimentai errors. The values obtained with the two different runs were well within the estimated error bars (for each nucleus). Results are displayed in fig. 7, along with lines which are extrapolations from leastsquares fits done by Long et al. ’ “) on other nuclei. A general tendency of nuclei near this value of Z/A* to have Coulomb displacement energies lower than the extrapolated

(3He, t) REACTIONS

223

line was typical in the data quoted by Long. The values determined for the Coulomb excitation energies were: for *‘Y, 11.64kO.05 MeV; for “Zr, 11.65kO.05 MeV; for ‘OZr, 11.87+_0.05 MeV; and for 93Nb, 11.93+0.05 MeV. 9. Conclusions The differential cross sections for the isobaric analogs of the ground states of 62Ni and 64Ni in the (3He, t) reaction at 37.5 MeV incident energy are well described by a macroscopic distorted wave analysis which proceeds primarily through a surface imaginary interaction term of about 40-50 MeV strength, to a large extent independent of the optical parameter set used. The i.g.s. of 54Fe and 90Zr in the (3He, t) reaction at 31.7 MeV incident energy show the distinct feature of having the first two peaks of the differential cross section about equal in magnitude. This feature is not well described by a macroscopic analysis which proceeds primarily through a surface imaginary interaction term. However, when the volume real term in the interaction potential is made more effective, either through using a very large volume real term or through decreasing the imaginary radius, the fits of the macroscopic angular distributions to the 54Fe and 90Zr (3He, t) data are considerably improved, while those to the 62Ni and 64Ni (3He, t) data are degraded. For any given optical parameter set, it is apparent from the above results that good fits to all four nuclei can only be obtained by requiring different types of interaction potentials for different nuclei. Since these results are valid for several different types of parameter sets, it would seem unlikely that the fitting of the first two peaks could be resolved by further study of new optical-model fits to elastic data. Thus considerable doubt is cast on the validity of attempts to determine optical-model parameters which account for isospin dependency using only terms directly proportional to E = (N-2)/A. Certain nuclei (such as 54Fe and 90Zr) deviate sufficiently from this dependence to require an extra, nucleus dependent, and possibly energy dependent, factor. The factor would probably be a function of the radius and would not necessarily use the same geometry as the opticalmodel potential. The new optical parameter sets examined here described the i.g.s. data as well as previously used sets, but did not give any significant improvements in fits. The macroscopic analysis was not very sensitive to optical parameters when conventional interaction potentials were used. The microscopic distorted wave analysis was highly sensitive to optical parameters and gave a poor fit in all cases, even though an isospin dependent term was present in several of the optical parameters tried. Therefore the i.g.s. would not seem to provide a very stringent test for the relative merit of various optical parameter sets, except to provide information on the isospin terms which must be imposed upon the optical models, as discussed above. A better method to obtain consistency would be to use the interaction potential shapes and magnitudes determined from the (3He, t) data to impose isospin dependent terms upon the opticalmodel parameters, and then vary other terms in the optical-model parameters to fit

224

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et al.

the (3He, 3He) and (t, t) data. The new parameters thus determined could be reexamined using the macroscopic fit to the (3He, t) data in an iterative procedure. Apparently the isospin dependent terms must be individually examined for each nucleus in this way. The authors are pleased to acknowledge many helpful discussions with Professors P. D. Kunz, B. W. Ridley and E. Rost. The close cooperation of Dr. P. P. Urone in supplying optical parameters and running various optical-model searches for us is greatly appreciated. The help of Messrs. L. C. Farwell and H. H. Chang in analyzing data and helping with the calculations is gratefully acknowledged. References 1) A. M. Lane, Phys. Rev. Lett. 8 (1962) 171 2) R. M. Drisko, P. G. Roos and R. H. Bassel, Proc. Int. Conf. on nuclear structure,Tokyo, Japan , 1967, Supplement to J. Phys. Sot. Japan 24 (1968) 347 3) J. J. Wesolowski, E. H. Schwartz, P. G. Roos and C. A. Ludemann, Phys. Rev. 169 (1968) 878 4) P. D. Kunz, E. Rost, G. D. Jones, R. R. Johnson and S. I. Hayakawa, Phys. Rev. 185 (1969) 1528 5) I’. P. Urone, L. W. Put, H. H. Chang and B. W. Ridley, Nucl. Phys. Al63 (1971) 225; P. P. Urone, Ph.D. Thesis, University of Colorado (1970) unpublished 6) Computer program written by D. Zurstadt 7) S. 1. Hayakawa, J. J. Kraushaar, P. D. Kunz and E. Rost, Phys. Lett. 29B (1969) 327 8) S. I. Hayakawa, W. L. Fadner, J. J. Kraushaar and E. Rost, Nucl. Phys. Al39 (1969) 465 9) W. L. Fadner, L. C. Farwell, R. E. L. Green, S. I. Hayakawa and J. J. Kraushaar, to be published 10) R. E. L. Green and D. A. Lind, Bull. Am. Phys. Sot. 14 (1969) 1242 11) G. R. Satchler, Isospin dependence of optical-model potentials, in Isospin in nuclear physics, ed. D. H. Wilkinson (North-Holland, Amsterdam, 1970) ch. 9 12) E. R. Flynn, D. D. Armstrong, J. G. Beery and A. G. Blair, Phys. Rev. 182 (1969) 1113 13) Computer program written by P. D. Kunz 14) E. F. Gibson, B. W. Ridley, J. J. Kraushaar, M. E. Rickey and R. H. Bassell, Phys. Rev. 155 (1967) 1194 15) H. H. Duhm, K. Peterseim, R. Seehars, R. Finlay and C. Detraz, Nucl. Phys. A151 (1970) 579 16) G. Bruge, A. Bussiere, H. Faraggi, P. Kossanyi-Demay, J. M. Loiseaux, P. Roussel and L. Valentin, Nucl. Phys. Al29 (1969) 417 17) P. Kossanyi-Demay, P. Roussel, H. Faraggi and R. Schaeffer, Nucl. Phys. A148 (1970) 181 18) G. W. Greenlees, G. J. Pyle and Y. C. Tang, Phys. Rev. 171 (1968) 1115; G. J. Pyle and G. W. Greenlees, Phys. Rev. 181 (1969) 1444; ’ G. W. Greenlees, W. Makofske and G. J. Pyle, Phys. Rev. Cl (1970) 1145 19) D. D. Long, P. Richard, C. F. Moore and J. D. Fox, Phys. Rev. 149 (1966) 906