Experimental study of (f722)7+ states in s-d shell nuclei

12~.F

Nuclear Physics A265 (1976) 220--252; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

EXPERIMENTAL

S T U D Y O F (f~)7 + S T A T E S I N s-d S H E L L N U C L E I t

R. M. DEL VECCHIO, R. T. KOUZES tt and R. SHERR Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08540 Received 27 November 1975 (Revised 22 March 1976) Abstract: The (u, d) reactions on 24, 26Mg' 2s. sosi ' 32.34S, 36.3s. hOAr and 42Ca at beam energies

from 34 to 40 MeV were performed to search for (f~2)7+ levels in order to confirm and extend systematics observed by previous investigators. Supplementary (SHe, p) experiments o n 2'tMg, 2sSi and 32S were also performed. DWBA calculations proved quite useful in interpreting the experimental results. In addition, the 2SMg(~, t) and 27Al(u, 3He) experiments were conducted to further distinguish the (f~2)7+ states from (dfffff)6- levels in ~6AI and 2SA1. The latter states are expected to be populated with significant strength in (u, d) as are (f~2)7+ states. A nearly linear dependence of the (u, d) Q-values to the 7 + states versus atomic mass holds for all the s-d shell nuclei studied here. This Q-value linearity ends as the f~ shell begins to fill although regularities are still apparent. A systematic and similar behavior of (d~2)3 + states was also observed.

E

NUCLEAR REACTIONS 24Mg, 2eMg, 2sSi, s°Si(~,, d), E----40 MeV; s2S, 34S, 42Ca(~,d), E = 34.8 MeV; 3eAr, 3SAr, 4°Ar(~,d), E = 34 MeV; 24Mg, 2sSi, 32S(aHe, p), E = 24.5 MeV; measured a(0). 26.2SA1, 30.32p, a4.36C1' 3s.4o.4~K, 4~Sc deduced levels, L, J, x. DWBA analysis. Resolution 40-110 keV. Enriched targets.

1. Introduction

The investigation o f multiplets in nuclei arising f r o m the i n t e r a c t i o n o f two particles in shells which are n o m i n a l l y e m p t y in the g r o u n d - s t a t e c o n f i g u r a t i o n is a n interesting b u t relatively u n e x p l o r e d field o f study. There are e x p e r i m e n t a l difficulties in identifying such states which generally occur at excitation energies where level densities due t o other states are quite high, T h e expected mixing with n e i g h b o u r i n g states c o u l d t h e n f r a c t i o n a t e the m u l t i p l e t a m o n g m a n y levels, greatly c o m p l i c a t i n g its identification. Nevertheless, the q u e s t i o n as t o the extent o f this f r a c t i o n a t i o n a n d the l o c a t i o n o f the energy c e n t r o i d s o f such states r e m a i n s quite interesting. The degree o f m i x i n g with o t h e r states is expected to be a s t r o n g f u n c t i o n o f the spin a n d parity. F o r instance ( l d ~ ) 5 + c o n f i g u r a t i o n states s h o u l d be relatively p u r e in d o u b l y o d d l p shell nuclei since 5 + states c a n n o t be s i m p l y c o n s t r u c t e d f r o m p shell nucleons. Similarly, t h e (1 f~)7 + states occurring in 2s- 1d shell nuclei are expected t o be relatively pure. I n fact, such stretched c o n f i g u r a t i o n states have been p r e v i o u s l y t Supported in part by the National Science Foundation Grant No. MPS71-03445. tt Present address: Indiana University, Department of Physics, Bloomington, Indiana 47401. 220

(f~2)7+ STATES

221

investigated with the (~, d) reaction in a systematic series of experiments ~-a). The kinematics of the (ct, d) reaction at cyclotron energies favor the formation of highspin states of the type under discussion. The investigations of refs. 1-3) in addition to establishing the utility of the (ct, d) reaction as a tool for studying such states, found that the Q-values for the formation of these (j2)Jm~ states followed a smooth and sometimes linear dependence on the mass of the residual nucleus. In an effort to extend the results of reK 2) on (f~)7 + states to other cases of interest and to provide additional evidence for previous 7 + assignments, we have performed the (u, d) reaction on 24Mg, 26Mg, 2sSi, 3°Si, 32S, 34S, 36AF, 3SAf, 4OAr and 42Ca. Some results of this work were presented elsewhere 4); in particular, the linear dependence of the (u, d) Q-value on nuclear mass was interpreted within the framework of the Bansal-French-Zamick weak coupling model 5, 6). This work supersedes ref. *) with regards to slight differences in excitation energies, !n addition, we per, formed the (3He, p) reaction on 24Mg, 2ssi and a2S in an effort to locate low-spin and/or even-spin members of the f~ multiplet. Although the (SHe, p) results did not provide convincing information on such states, they were a u seful supplement to the (~, d) work. A principle difficulty with the (3He, p) as well as the (x, d) experiments was the occurrence of pronounced contaminant peaks in the excitation region of interest from reactions on carbon and oxygen in the target backings or from oxidation and deposition on the target material itself. Previous spin assignments to these stretched configuration states have rested mainly on their high (~t, d) cross sections and on Q-value systematics. One objective of our study was to see whether DWBA calculations could be used to determine L-transfers to these states. The method of well matching, which has been useful in interpreting (d, ~t) angular distributions for heavier mass nuclei 7), was applied. A previous 24Mg(x, d)26A1 study s) utilizing well matching was quite encouraging. In order to be convincing, it was necessary to show that (x, d) angular distributions to states of known J= at lower excitation energy could be satisfactorily accounted for before attempting an analysis of unknown states. This necessitated an extensive amount of data analysis. Our results provide some guide to the selection of optical model parameters for use in s-d shell (=, d) or (d, ~) reactions. Such guides are necessary in view of the well known discrete and continuous parameter ambiguities encountered in fitting elastic scattering data. To further distinguish the (f~)7+ state from a (ld t lft)6- state expected at a comparable excitation energy in 26A1 and 2SA1,we performed the 25Mg(ct, 02 CA1 and the 27Al(~ ' aHe)2aA1 experiments. These are expected to populate the 6- state but not the 7 + state appreciably. Our results are consistent with previous suggestions 2, 9) as to which state is the 7 + state.

222

R.M. DEL VECCHIO et aL

2. Experimental procedure 2.1. THE (u, d) AND (3He, p) EXPERIMENTS The experiments were all performed with the Princeton azimuthally varying field cyclotron in our large (1.5 m diameter) scattering chamber. Details of the beam energies and targets employed are given in table 1. Reaction products were detected in a freon cooled A E - E silicon detector telescope followed by a third detector in anticoincidence. On-line particle identification was provided by a Sigma-2 computer using a range lookup table. Energy spectra corresponding to the particle type of interest were stored in 2048-channel arrays on magnetic tape for later analysis with the peak fitting routine A U T O F I T to). TABLE 1 Experimental details Target

Nominal thickness

Pressure (era Hg)

(~g/era2) 2tMg 2eMg 28Si 3°Si Cd3=S Cda4S 3eAr 3SAr hOAr

70 70 150 120 125 150

42Ca

200

10 9 10

Isotopic purity (~*)

E= (MeV)

E (3He) (MeV)

99.9 99.7 92.2 (nat) 95.5 95 (nat) 90 99.5 94 99.6 (nat) 94.4

39.95 39.95 40.06 40.06 34.82 34.82 34.3 34.0 34.3 34.8

24.47 24.47 24.47

Resolution FWHM (keV) (~, d) (aBe,p) 80 80 50 60 50 50 110 110 110 60

50 70 40

A detector kept at a fixed angle of either 30 ° or 45 ° monitored the elastic peak. Its use was principally diagnostic, in that the monitor to charge ratio served to indicate target deterioration. This ratio typically remained constant to within _+5 ~ for a given target so that relative cross sections could be reliably based on the integrated charge. For the gas targets, a 5 cm diameter gas cell with a 0.22 mg/cm 2 nickel entrance window and a 2 rag/era 2 H a v a r exit window for the direct beam were employed. Reaction products passed through a 1 rag/era 2 K a p t o n window and were doubly collimated before entering the detector array. This served to limit the region of the gas cell viewed by the detector. The experimental resolution achieved with this system was 110 keV F W H M due mostly to kinematic broadening, energy loss in the extended gas target, and straggling in the gas cell windows. Absolute cross sections were obtained from the gas pressure and the known geometry of the detector collimators and are expected to be accurate to + 20 ~o. Absolute cross sections for several of the solid target (~t, d) experiments were based on elastic scattering data taken simultaneously with the reaction data. F r o m published

(f,2,2)7+ STATES

223

TABLE2 The (~t, d) optical parameters *) or-parameters

target

2't'26Mg 2 s. a oSi 32.3,tS 36.3s.,teAr , ,t2Ca

Elmbb) (MoV)

V (MeV)

ro (fln)

ao (fm)

W (MeV)

ri (fro)

at (fro)

z2/N ©) Notes

40.0 40.0 56.0 24.7

181.2 180.2 225.4 196.6

1.20 1.20 1.20 1.20

0.735 0.750 0.671 0.755

20.0 31.1 26.9 14.5

1.396 1.76 1.5 1.3

0.713 0.37 0,545 0.9

13.4 158.0 2.7 8.2

e) f) s) b)

Notes

d-parameters

target

24. 26Mg, 28. a°Si 32'34S a6'aa',t°Ar, ,t2Ca

El,b b) (MeV) 17.0 18.0 21.4

V (MeV) 82.5 86.41 88.5

ro (fro)

ao (fro)

4 FF_n (MeV)

rl (fro)

at (fm)

z2/N

1.20 1.215 1.2

0.75 0.513 0.644

31.2 43.6 33.9

0.84 1.66 1.361

1.32 0.596 0.875

a) d) 3.4

t) j) k)

~) All r, = 1.25 fin. b) Energy at which elastic data was taken. ! 0 Assumed error of 10 y. per point and used cross sections at angles from 5 ° to 120° in 5 ° steps. a) Original literature parameters used. e) Refit from elastic data of ref. xl) on 2*Mg. f) Refit from elastic data of ref. 11) on 2ssi. 8) Refit from "data" generated by g-parameters for 32S of ref. 3s). h) Data obtained from g-parameters of ref. as) on 4°Ca. t) Parameters from elastic scattering on 27A1 obtained by ref. 37). A spin-orbit potential was included: Va... = 5.63 MeV, rs... = 0.92 fro, a,.o. = 1.0 fro. J) Parameters from elastic scattering on a2S obtained by ref. 38). k) Refit from elastic data of ref. 39). A spin-orbit potential was included: Vs... = 6.51 MeV, rs.o. = 1.063 fro, as.o. = 0.721 fro.

40 M e V or-elastic d a t a 11) o n 2 * M g a n d 28Si, a b s o l u t e cross sections for 24Mg, 28Si(~, d ) were derived. T h e e s t i m a t e d a c c u r a c y is + 25 ~o. I n a d d i t i o n , optical p a r a m eters d e r i v e d f r o m a fit to t h e elastic d a t a o f ref. 11) v i a a c o m p u t e r s e a r c h r o u t i n e (table 2) were u s e d to p r e d i c t or-elastic cross sections for 2 6 M g a n d a°Si. These were t h e n a p p l i e d to the n o r m a l i z a t i o n o f the 2 6 M g ' 3 o s i ( ~ ' d ) data. I n view o f the r a t h e r p o o r fits o b t a i n e d to t h e elastic d a t a , these l a t t e r (~, d ) c r o s s - s e c t i o n scales c o u l d b e in error by an estimated + - loo 5o ~ . T h e fact t h a t we o b t a i n e d elastic d a t a at five o f six a n g l e s m a d e these n o r m a l i z a t i o n s less s u b j e c t to t h e details o f the elastic a n g u l a r distributions. • ......... E n e r g y c a l i b r a t i o n s were o b t a i n e d b y m e a n s o f the 12C(~t, d ) a n d 1 6 0 ( ~ , d ) rea c t i o n s to k n o w n states i n I * N a n d 18F since b o t h c a r b o n a n d o x y g e n were p r e s e n t in t h e b a c k i n g m a t e r i a l o r as c o n t a m i n a n t s o n all the solid targets, F o r the gas targets, air was i n t r o d u c e d i n t o the gas cell i n s e p a r a t e r u n s t o o b t a i n c a l i b r a t i o n p o i n t s . F o r

224

R.M. DEL VECCHIO et aL

p e a k s f r o m the r e a c t i o n o f interest g o o d internal consistency was o b t a i n e d for the e x c i t a t i o n energies m e a s u r e d a t different angles; t y p i c a l v a r i a t i o n s f r o m angle to angle were less t h a n -t-20 keV. H o w e v e r , w h e n excitation energies at low excitation c o u l d be c o m p a r e d with a c c u r a t e l y d e t e r m i n e d energies f r o m y-ray work, systematic e r r o r s b e c a m e a p p a r e n t . F o r m o s t cases where this c o u l d be studied, a linear r e l a t i o n between t h e c o r r e c t i o n n e e d e d a n d the excitation energy was f o u n d , a m o u n t i n g to, with t h e g r o u n d - s t a t e s e t t o 0, a n a d d i t i o n o f 10 to 12 keV p e r M e V o f e x c i t a t i o n t o o u r m e a s u r e d energies. This p r o c e d u r e i n t r o d u c e s a d d i t i o n a l u n c e r t a i n t y into o u r energies for new levels so t h a t a conservative estimate o f the overall e r r o r in these values is + 50 keV. TAnLE 3 2eAl levels seen in (~, d) or (SHe, p) Ref. is) Ex

(ct, d) this work

jr, T

Ez

(keV) 0

228 417 1058 1851 2070 2072 2365 3074-4-5 3159 3745 3745.4 5896 b) 6214 6265 6294 6453 6615 6738

L

J~

(keV) 5+

(j~b/sr) 4

0 +, 1 3+ 1+ 1+ 2 +, 1 1+ 3+ (2, 3) 2 +, 1 (2+) } 0 +, 1

(aHe, p) this work

~(20 °) ....

v' ©)

Ex (keV)

880

2

~/

1060

0

V/

240

L

jn

~(20o)c.m.

arbitrary units 4

x/

0 2 0+2 0+2

x/ ~/ x/ ~/

150 100 600 90 400

(a) 4

(3 +)

2+4 4 2 a)

360

5856

4

6257

(2)

6414 6586 6718

4

110 d) V' (3 +) ~/ }

180 160 290 370

(3-5) +

1170

}

6873 6950A:50 c) (6-)

6930

(5)

(4-6)-

1700

82704-50 ©) (7 +)

8220

(6)

(7 +)

1100

620 (3-5) +

(3)

6865

780 900 1200 ~ 1000 d)

7383

a)

8730 9355 10216

4, 1 4 4, 0q-2

~ 1400 (3-5) +

600 1200 1000

") Level excited but L-transfer not determined. b) Energies of levels from 5896 to 6873 keV are from ref. 16). c) Ref. ~). a) Angular distribution not shown. e) V' in this and subsequent tables means that our (at, d) L-transfers are consistent with known j n information.

(f.].2)7+ STATES

225

2.2. THE 2SMg(u, t)26Al AND 27Al(~t,aHe)2aAl EXPERIMENTS These experiments were performed with a 40 MeV u-beam in conjunction with the Princeton quadrupole-dipole-dipole-dipole ( Q D D D ) spectrograph. The focal plane detector was a 60 cm long gas proportional counter backed by a plastic scintillator. In order to rid the excitation region of interest of prominent and broadened peaks from reactions on carbon and oxygen, a spectrograph angle of 25 ° was chosen for both the 26A1 and the 2SAI study. Using targets of ~ 100 to 200/tg/cm 2 and a nearly full spectrograph aperature ( ~ 12 msr), the resolution of ~ 35 keV F W H M was considered adequate for our purposes. Calibration of the focal plane was provided by means of the 24Mg, 26Mg(~, d) reactions in addition to the 25Mg(0t, t) and 27A1(0~, 3He) reactions to states of known excitation energy. The excitation of the prominent states at 6930 and 8220 keV in 26A1and at 8611 and 9821 keV in 2SAI was observed in the (~, d) reactions (see tables 3 and 4), However, since the calibration peaks and the peaks of interest could not be seen with the same magnetic field setting we did not obtain significantly improved values for their excitation energies as compared with the determinations made in our large scattering chamber. In the 25Mg(0~, t)26A1 experiment, excitation of only the 6930 keV states was observed. Scarcely any (~, t) strength was seen within a region of _ 200 keV of the 8220 keV level. These results support an f~ configuration assignment to the 8220 keV level. In the 27Al(tx, 3He)2aA1 experiment, the 8611 keV level was excited with moderate strength while no evidence for excitation of the 9821 keV level was seen. Hence, the 9821 keV state is a strong candidate for the f~ configuration. TAaL~4 2aAI levels seen in (~, d) Ref. 15) E, (keV)

J", T

0 1014 1373 2202 2582:t: 3 5165 ") 6760-6835

3+ 3+ 1+ 1+

9800=[=50b) a) Ref. 18).

E, (keV)

(6-) (7 +)

8611 9821

(or, d) - this work L jr 2 2 0+2 0+ 2 4 (5) (3)

V/ x/ X/ ~/ (3-5) + X/

(6)

(7 +)

a(20 o).... Q~b/sr) 150 110 100 130 100 400 220 340 700

b) Ref. 2).

3. DWBA analysis 3.1. THE (~, d) CALCULATIONS Distorted wave Born approximation (DWBA) calculations were performed with the two-nucleon transfer option of the computer code D W U C K 12). Parameter ambiguites in the elastic channels were resolved by means of the well matching

226

R.M. DEL VECCHIO et aL

criterion 7). This procedure constrains the real well depths and the real well geometric parameters to physically reasonable values, i.e. values which are expected on the basis of folding calculations. While some of the deuteron parameters found in the literature satisfied this criterion, most of the u-particle parameters for the nuclei of interest did not. Hence, new parameters were obtained with an optical model search routine using either the literature elastic cross-section data when available or "elastic data" generated with the published best fit optical parameters. Table 2 summarizes the final optical parameters used in the (~, d) calculations. Although the x2/N value is rather high for some of the fits, the positions of the maxima and minima in the elastic data were usually well reproduced. No excessive effort was made to optimize the fits to the elastic data while keeping within the well matching criterion. The geometric parameters for the Woods-Saxon potential well of the bound state particles were r o = 1.25 fm and ao = 0.75 fm, and the well depth was adjusted to bind the nucleons at ½x (Ithe deuteron separation energy[ +2.225 MeV). The DWBA curves used for specific levels were obtained from calculations which came within a few MeV of satisfying this binding energy prescription. In addition, a spinorbit potential of the Thomas type with the parameter 2 = 25 was employed. Simple shell-model configurations were chosen for the single-particle states. Further, all calculations were performed with a zero-range interaction although standard nonlocality parameters were used for the incident and outgoing projectiles 12).~ 3.2. THE (aHe,p) CALCULATIONS Similar procedures were followed for the (3He, p) DWBA calculations. The 3He optical parameters used for all three (3He, p) reactions were 13): 1/° = 177 MeV, r o = 1.138fm, a o = 0 . 7 2 4 f m , W = 2 1 . 2 4 MeV, r i = 1.602 fm, a i = 0 . 7 6 9 fm, P's.o. = 5.0 MeV, rs.o. = 1.138 fm, as.o. = 0.724 fm. The proton parameters were calculated from a general formula in ref. 14). This particular combination of 3He parameters (including a prescription for changing W with bombarding energy according to dW/dE = 0.5) and proton parameters were found to work well in other s-d shell calculations 13). Other 3He and proton parameters which weie more realistic in terms of fitting elastic data for nuclei reasonably close to those studied here were tried. Similar acceptable fits to the data were obtained with combinations which came close to satisfying the well matching criterion. It appears, as in the case of (~, d) and in view of the reasonable success in fitting the data of this study, that the requirement of well matching is perhaps more important than obtaining a detailed fit to the elastic data.

4. Experimental results and comparison with other work 4.1. GENERAL DISCUSSION The primary experimental information is presented in the form of graphs and tables in this section. Comparison with previous work and points requiring further

(f;2)7+ STATES

227

clarification are discussed below. Systematic trends in the data and general aspects of the results are presented in sect. 5. Figs. la and b show sample spectra of the (~, d) experiments and fig. 2 shows representative (aHe, p) spectra. The prominent impurity peaks, present for the solid targets, are cross-hatched. These occasionally overlapped peaks of interest creating gaps in the angular distributions. Some peaks were probably missed in regions of 300

I eSMg (a,d) ;~ltAI 20 Di[GRI[ES £e " 39.95 IdeV

14ug(e,jl2eAi Is Dir OREirS ira • 39,9'J IdeV

t N |

|.o

6:o 4~o irXCITATION (Me'V)

o~o irXCITATION ( MeV)

30$1 ( a , d ) ~ p 2 0 DIrGR£ES ire • 40.Oir MeV

~SSi(j,d)T'Op 20 DEORf£O ir a • 40.06 MeV

zo¢ L"

¢n

e~O

~C

g

'o

7ZO

a'.o EXCITATION (MeV)

sZo

t;o

J:O

,io ~o EXCITATION (MeV)

s.o

32 S (a,d)34CI 25 DI:'OREES E4 • 34.02MeV

eO

S4S{a,d)3eCI ;~S DEOREES [ g • 34.82 MeV

~'o

60

4:o

£0 EXCITATION (MeV)

EXCrTATIOR (MEW)

Fig. la.

,:o

!i

g

COUNTS

[

COUNTS I

I

lllll

t~

,

~t~ COUNTS

g

~,

COUNTS

~

,~

..~

g~

~g g~ -q

~

. 3 ~

3?53

-3d~8

~.

~. r~

g

'~

,89~

.<


P~ 7o 7a OIHDD~IA "[~[~ " ~ "~I

gZ:~:

(f~T~)7+ S T A T E S

I

229

2 4 Mg (sl,.le, p ) :)SAI

20

DEGREES

E3He "24.47 MBV Z~500

-

. . . CHANNELNUMBER

z~oo

.

. 30oo

~ ~

V~./~,,

zssi(SHe.p)~:)p 20 DEGREES ESHo'24:ATMoV

1 ql-

ap

A ° e

'

30~)0

g

)

A

i

CHANNEL NUMBER

3i0o

too

~S(SJ.le.p) 34CI 30 DEGREES E3He- 24.47 MeV

o

Z1500

'

3~O

i

CHANNEL NUMBER

m

35oo

Fig. 2. The (aHe, p) spectra obtained with solid state detectors. See fig. l caption.

excitation where impurity peaks were numerous. The resolution also precluded the separation of doublets less than about 40 keV apart for the solid targets and less than about 80 keV apart for the gas targets. Angular distributions for the (~, d) experiments are shown in figs. 3-12 andthosc for the (aHe, p) experiments in figs. 13 and 14. Angular distributions for the a2S(aHe, p)a4Cl experiment are not given because only four angles were taken and statistics

230

R.M. DEI.,VECCHIO et

aL

were poor. Error bars on the experimental points reflect statistical and background subtraction errors only. The DWBA curves were adjusted by eye to produce a best fit. In cases were the spin and parity, J~, are known from previous work, the angular momentum (L) transfer(s) are determined. When two L-values can contribute, a best fit with only one L was preferred unless a noticeable improvement could be achieved with two. A knowledge of the parity of a level is very useful since it sometimes happens, for example, that an incoherent sum of two even L-transfer curves closely resembles the intermediate odd L-transfer curve. Since DWBA fits to levels of known J~ were generally acceptable, attempts were made to fit unknown levels. One of the difficulties, especially for the high-spin states of interest here, is the rather structureless nature of the angular distributions. Nevertheless, for many of the unknown levels studied, reasonably convincing fits were obtained. Information on the states observed in this study is summarized intables 3-12. The most general reference to previous work cited is the compilation of Endt and Van der Leun 1s). Energies are taken from this compilation, unless superseded by later work or determined in the present study. Errors in excitation energies taken from ref. 15) that are on the order of or less than 1 keV are not given. Our energies, as mentioned previously, can only be considered accurate to +50 keY. Uncertain L-transfers are enclosed in parentheses or several alternatives given. Blanks in the L-columns mean that no convincing DWBA fit was obtained; lack of data points, a possible doublet, a structureless angular distribution, etc. could be responsible. When our L-transfers are consistent with previous J~ determinations a check is placed in the J~ column referring to this work. Otherwise the J~ or J~ limits determined here are given. The differential cross section at 20 ° in the c.m. is also given in the tables. This number was obtained by a smooth extrapolation from neighbouring experimental points, guided by the DWBA prediction when necessary. Absolute cross-section errors were discussed in sect. 2. Isospins, T, are only listed when they are known to be one unit more than that of the target. One possible reason for the differences in levels excited by (g, d) versus (3He, p) is that the former reaction excites only T = Tta~ct states while the latter can excite both T = T,.rgct or T = T,-r,et + 1 levels, Since our primary interest was in studying the strongly excited states, often statistics on the weaker states were poor so that their study was omitted. 4.2. THE 24Mg(c¢, d)26A1 AND "4Mg(3He, p)26Al REACTIONS

(See figs. 3 and 13 and table 3.) Statistics in the (6, d) experiment were such that only a few of the stronger states could be studied. Many more were seen in a previous (6, d) study 2). Our suggested (3+), T = 0 assignment for the 3074 keV state is new and is implied by both the (u, d) and (3He, p) data. The (u, d) study at 50 MeV of ref. 8) only looked at states at low excitation. Our L-transfers agree with theirs for the first three states listed in table 3. They observe L = 2 transfer to the 3074 level which is consistent with our L = 4 transfer if J~ = 3 +. Our DWBA curves for

(f.t.2)7+ STATES

io 4

231

26Mg (=,d)28AI Ea • 39.95 MeV

to-'

Z4Mg'(a,d) 26AI Ea = 39.95 MeV

klV

0

tO~~" 103 ~

L,2

0 keV

102 / ~

103

I0~. klV

Lo2

_

417 keV 3+L=2

~ L'O*2

10 ~

÷ I0 -I

1851 keV

tOI

"~

L,O~'2

i

I0 ~

* ,Q :~" 102

3074 keV (5*) % - 4

'~

~

2582keY

L

si.s k*v

L°4

104 "~

6930 keV •

I03

L-5

~

I0|

• "~

104

I03

~

8220 keV (7 ÷) 0

F

10;

• 8611key * *

lO=~ ~

1o~ ;~ 2'o A ,'o 5; 6'o 7'o 8b 8~m. Fig. 3. The (a, d) angular distributions. The curves are DWBA calculations, arbitrarily nor= realized and adjusted by eye to achieve a best fit to the data. The distributions are labelled by the excitation energy in keV, the ~r~ when known, and the L-transfer value o f the curve drawn.

9112!keV

'°°~,, id

J.;

i ..,<-,.. I

I

I

T

I

I

I

f

!

0 tO 20 3040 506070 ~

~.m. Fig. 4. See caption to fig. 3.

232

R . M . DEL VECCHIO et aL

the 6930 and 8220 keV levels were only suggestive of the L-transfers indicated in table 3. However, no other lower L-transfer would have fit the 8220 keV level as well as L = 6. This, together with its strength, suggests an (f~)7 + conflgnration, in agreement with previous work 2,9). The 6930 keV level has been proposed as the (d~, fi)6- level by previous investigators 2. 9). Our (u, d)results together with the 25Mg(u, t) results discussed in sect. 2 are in accord with this proposal. Our (aHe, p) L-transfers agree quite well with previous (aHe, p) results at 18 MeV bombarding energy 16). However, the energies of our levels above 5 MeV of excitation appear to be systematically about 30--40 keV lower than those given in ref. 16) (the correspondences shown are also based on cross-section comparisons). Our (3He, p) L-transfers to these upper states are all new. The 6930 and 8220 keV states, strongly excited in (~, d), are apparently only weakly excited in (aHe, p). However, several interesting rather strong levels appear in this excitation region and even higher. 4.3. THE 26Mg(Qt,d)2SAl REACTION

(See fig. 4 and table 4 for the experimental information.) Our (~, d) L-transfers to states below 5 MeV agree with those previously obtained in a (SHe, p) study of this nucleus 17). A spin of 4 + was suggested for the 2582 keV state by ref. I7) which is consistent with our findings. A (6-) assignment for the 5165 keV level has been rather convincingly argued by ref. la) based in part on considering analog states in 28Si. On the other hand, the argument for the (6-) assignment strongly depends on a 5assignment for the 4033 keV state in 2SA1. This latter J~, however, conflicts with a fairly convincing L = 3 transfer in (aI-Ie, p) reported in ref. 17). The strong L = (5) transition for the 5165 keV level which we observe in (ct,d) argues for ahigh spin, 5or 6-. The 8611 keV level has not been previously reported. By analogy with 26A1 it could be a (d t, f~)6- state in view of the 27Al(ct, 3He) results discussed in sect. 2. Its (~, d) angular distribution is quite structureless. We support an (f~)7 + assignment for the 9821keV level made previously 2), based on the (u, d) strength, the shape of the angular distribution, and the (~t, SHe) results discussed in sect. 2. 4.4. THE 2sSi(~', d)S°P AND 2sSi(SHe, p)SOp REACTIONS

(Refer to figs. 5 and 14 and table 5.) Only a few new J~ suggestions could be made in SOp. Acceptable DWBA fits were achieved for a number of levels, using allowed Ltransfers. The (SHe, p) L = (0 + 2) transfer to the 1454 keV level conflicts with the 2 + assignment from the literature. Our (SHe, p) work on this nucleus agrees with a recent study 19) at 28 MeV which also observes L = (0+2) transfer to the 5412 keV state. Our excitation energy for the presumed (f~)7 + state at 7231 keV disagrees with ref. 2) by 200 keV which is outside the errors. An energy of 7110 keV is given for this state in ref. s) which is only ~ 100 keV removed from ours and within the combined errors. We observe an interesting

233

(fg,2)7 + STATES

102~-

0 keY 2 a S i ( =,d) ~Op

E= =40.06 MeV

L=2

IOI

I0 I

102

42340 keV

,o, -

1454 keV 2+ L=2

io~

,

I01 _-

" \ ~

t

102 ::k

19733+keY L=4

ioI

t

~..

~

4921 + keV f4945

÷

+

~

,oa~-

I/~

\.

oW ~c~.-

~,,~.

2538k~v

'L=3

6496 keY

• I01 _--

'

~ '

~

TL=4!

~" ~

7231 keV ( 7+)

28319+keY L=2

i0 = -

io 2

I0~

4143 keV

o

, I

~o

I

I

I

i

20 30 4o 50 Oc.m.

I

I

I

6o 7o 8o

i001 o

L 6~

I I I 1 I I I I ~o 20 3o 40 5o 60 7o so

8c.m. Fig. 5. See caption to fig. 3.

R. M. DEL VECCHIO et aL

234

TABLE 5 aop states seen in (u, d) or (SHe, p) Ref. xs) E~ (keY)

J=, T

0 677 709 1454 1973 2538 2839 b) 2937 3019 3927 4144 4230 4342 4625 4921 ± 3 4945~5 5412±10 5883 c) 6O89 6476 6661 6865 7O40 7180 7030 d) 7289

1+ 0 +, 1 1+ 2+ 3+ 3+ 1+ 2+,1 1+

Ex (keY)

2 0 2 4 2 2

24-

3 3 3, 2

33- r) 1

(7 +)

(u, d) this work L jn ~(20 °).... ~b/sr)

¢ V: ,/ ~,: ~: v:

x/ v:

")

6496

(3, 4)

7231

(6)

7392

(6)

32 14 13 60 35 30

55 100 15

~ 30

(0+2)

¢

320

(0+2)

¢

650

(0+2) 2 (2)

~/

95 380 260

") ") 2 1+3 3+5

}

50

(7 + )

(3He, p) this work L j~t 0"(20°) .... arbitrary units

E~ (keY)

V

< v:

200 4O0 180 1300 800

")

150

")

160

5885 6065 6438 6629 6849 7014 7149

0+2 ") ") ") 2 (2) (0+2) ")

7287

")

")

7628 7972 8092 8610

4,1 a) ") ")

820 2OOO

250

I+

26O

d)

1200 400 200O 700 3OO 50O

110

b)

")

a) L-transfer not determined. b) Probably a 3 + state according to ref. 19). c) Excitation energies for this and remaining states in this column except the 7030 keV level are from ref. 19). d) Ref. z). c) Insufficient data for ~r(20°)c.m. determination. The 8092 and 8610 keV levels are fairly strong in (3He, p). r) Refs. 4o.41). state at 7392 k e V , r a t h e r s t r o n g l y e x c i t e d in (ct, d ) a n d n o t p r e v i o u s l y r e p o r t e d . Its a n g u l a r d i s t r i b u t i o n is n e a r l y i d e n t i c a l t o t h a t o f t h e 7231 k e V level. T h e L = (6) t r a n s f e r suggests J ~ = 5 +, 6 + o r 7 +. A h i g h - s p i n n e g a t i v e p a r i t y a s s i g n m e n t c a n n o t be r u l e d o u t , h o w e v e r , in v i e w o f t h e s t r u c t u r e l e s s a n g u l a r d i s t r i b u t i o n .

(f.1.2)7 + STATES

235

3°Si (=,d)3zp E,, -40.06 MeV

,7~,~,v

,o, ~ I0

,o,

5077 keV

([.-I,,)-

I0 I

I0

,

I0' .

I ~

I01

I00'

V< t

I0

~

~

I0 I

5835 + key 5858

~t

i02"

.

i0(;

,°'f

Io'

6140 keV

~t

tt

,o,~-

--~ %;~,v L-3

:ili I0°L

I0 j

f

I

~

~'?" "~,'~. 3"""

101

e 4' (, 4, ~, 4696 keY

7420 keV (7*) L'6

I0"

I01

I01

o,E,' Z030' ~ ' 5060'~o~o

o ~ ; o ' ~040' ~ o '6070'~o

ec.m. Fig. 6. See caption to fig. 3.

8c.m,

236

R. M. DEL VECCHIO et

al.

TAnI~. 6 a2p states seen in (~, d)

Ref. Is) Ez

(ct, d) this work jn, T

(keY)

F_~

L

J~

(keV)

1755 2175 3264 3443 4007 42804-6

3+ 3+ 242-

(/~h/sr) 4

V

54

2 3 3 3

x/ v/ ~/ x/

25 7o 7o 3o 90 80 55 17

2, 3

4696 50774-7 55094-2 58354-8 58584-8 65304-20

(2-4)(0-2)-

(1) (1)

(2-) x/

5849 6140

} (2--4)-

6880 7420

o(20°)c.m.

(3) (6) (6)

~/ (7 +)

5O 25 30 180 380

4.5. THE a°Si(~, d)a2P REACTION (See fig. 6 and tables 6.) This (u, d) reaction has not been previously reported. As table 6 indicates, we suggest a ( 2 - ) assignment for the 5077 keV state. Our (at, d) Ltransfers are consistent with known J~ assignments to the low-lying states. At higher excitation, we again, as in the 3op case, see two reasonably strong states proceeding b y L = (6)transfer. The choice of which of these is the 7 + state was made on the basis of the cross section although they both are possibly 7 + states. 4.6. THE a2S(% d)a4Cl AND a2S(aHe, p)a4Cl REACTIONS (Refer to fig. 7 and table 7 for a summary of the data.) Two strong states have again been observed in (~t, d) proceeding by L = 6 transfers. The L = 6 fits are quite convincing (fig. 7) and differ noticeably from the L = 5 transfer to the known 3633 keV 5 - state. The stronger state at 5230 keV is suggested as a 7 + state in agreement with ref. 9), however, both are possibly 7 + states. 4.7. THE 34S(ct, d)a6Cl REACTION (See fig. 8 and table 8 for a presentation of the data.) Statistics on this reaction were rather poor. An L = 1 transfer to the 5000 keV level is quite convincing and provides a narrower limitation on the J'~ possibilities than previously given. The 5303 keV level is, on the basis of strength and L = (6) transfer, suggested as a (7 +) state. 4.8. THE 36Ar(ct, d)aSK REACTION (Fig. 9 and table 9 present our experimental results on 3SK.) A comparison with relevant states from a recent extensive study of this nucleus 2o) is given in table 9. The

(f~=)7* STATES I03

.

237

32S(a,d ) 34CI Err =34.82 MeV

102

~,

147 keV

I01

34S (a,d) 36CI

102

•

Ea=34.82 MeV 101

789

"~d~'

t ~--'

461.i.keV

keV

3+

I0'

+ 27 222- keV

u) c

t ÷t

i0 0

t

tt

2517 key

102

5-

$

io ~

~*

3653 keY

4÷+

,+

c

•-

'

I0 I

i01

,o, 4075 keV

.~,

::5

:=

I01

r

~

5000 keV (0-3 )I

I0 I

I0 io z

keY

5303

102` 5285 keY (7"")

o 4

I01

i

i

i

I0 20 30

40

a

i

!

I

!

50 60 70 80 90

8c.m.

Fig. 7. S¢¢ caption to fig. 3.

,°,I ioOl 0

I

|

!

f

I

1

•

|

1

i

I0 20 30 40 50 60 70 80 90

8c m.

Fig. 8. S¢¢ caption to fig. 3.

7 + assignment to the 3460 keV level is consistent with the work of ref. 2 o). Additional evidence for a 7 + assignments comes from ref. zl, 2z) as well as f r o m the strong L -- 6 transfer observed here. In view of the high level density, our 3665 keV level m a y not correspond to the

238

R. M. DEL VECCHIO et aL TABLE 7 34C1 states seen in (ec, d) or (aHe, lO)

Ref. xs) Ez (keY) 0 146 461 2158 2722 3128"4-2 33834-2 3633 4075 4137 44164-2

jTr, T

Ex (keY)

0 +, 1 3+ 1+

(u, d) this work L jr 0"(20°).... arbitrary units (2)

x/

9

2

~/

8

(3)

~/

9

5 (5)

V' ~/

25 14

(3He, p) this work Ez (keY)

L

0"(200 ) . . . .

arbitrary units 21 4 20 b)

2 +, 1 21+ 2 +, 1 54-

25 10 6 18 2O

(1-3)-

4790 c) 5230 c)

jtt

32

(4-) (7*)

4789 5283

6 6

(5-7) + (7 +)

18 35

b)

4674 4790 5292 6319 6724

95 15 ~) 26

") No L-transfers were determined in the (3He, p) work. b) Insutticient data to determine 0.(20°)c.m.. c) Ref. 9). TABLE8 3eCl states seen in (u, d) Ref. 1s) Ex

(u, d) this work J"

(keV) 789 2517-4-5 5000± 8

Ez

L

~r~r

(keV)

0 ( 2 0 °) . . . .

arbitrary units

3+

3

5- a) (0-3)5303

1 (6)

(0-2)(7 +)

9 25 ,~ 50

") Ref. 42). 3668 k e y level o f ref. zo). T h e L = 4 transfer a n d (~t, d ) strength suggests a 5 + level w h i c h is expected in this r e g i o n o f excitation f r o m a s b e l l - m o d f l c a l c u l a t i o n 2o). T h e shell-model s t u d y o f ref. 2o) f u r t h e r predicts a n o t h e r 7 + state a t a b o u t 5 M e V o f excitation. A s t r o n g level a t 5280 keV was first o b s e r v e d in a 4°C.a(d, ~ ) a s K experiment, p r o c e e d i n g b y L = 6 transfer a n d given a 7 + a s s i g n m e n t 23). W e observe a n L = 6 t r a n s f e r t o a level a t 5127 keV. This disagrees in energy w i t h the 5280 keV level o f ref. 23) sufficiently outside t h e c o m b i n e d e r r o r s t h a t it m a y be a different level.

(f;2)7 + S T A T E S 36Ar(,,,d) ZmK E,, -34.3 MeV

239 1031

i0 • keV (5*) L.4

L,2 1

0

2

~

I02

i011

~. 459 keV i+

•

~lAr~l,d) 4o K Ee ,~N..O MeV

.•-••.•,3665

'°=~'~,.

o~ev

L

i0t - " ~

,

800_keV

I0 I lOt i

jo ~

t

102 See

** (,

4' $

• "

t

3737

keV

I02

I01

,o,

~. ,-~

f [ ~,

i0~

1698 keV ft. L.O

IOz 3 9 6 5 keY

lot

I01

t

t

o! . " "~

I0~

• ,•,

2291 keV

%,

.t

2613 + keV

~L

,, #3

,o,r \

t t +t ..~ ~\\

2070 keY

!

~

2,543 keV

4345keV ,it t

I01

t t

I0 t lot

\ tO:

5127 keV

L -3

IO;

j / ~ ,o' r

3094 keY

\,

L.4

I01 I0 I

I0 o

t 5313 keV

h I0" /

3753 keY L- 4

io z et

#

$ $

tel

t

m~

3908 keV

I01

I0 !

eeee I0=

ee e e

t 20 t ~ 4050 ~ t ~o o ,o

8c.m.

e~o~O

0 ~ 2 ' 0 ' '3' 0 4 0 5 0 6 0'7-0' ~ 0 ; 0 8cm

Fig. 9. See caption to fig. 3.

40 50 60 ecru.

Fig. 10. See caption to fig. 3.

240

R . M . D E L V E C C H I O et al. TABLE 9 aSK states seen in (~, d) Ref. 2o)

E, (keV)

(u, d) this work L jn

J" (keV)

0 459 1698 2613 2646 2870 3460-4-6 3668

3+ 1+

(2)

1+

0

(3)(4) 2(7 + ) (1 +, 3)

5280 ")

2621 3445 3665 3737 3965 4345 5127

a(20 °) . . . . (pb/sr)

v/

4o

~/

4O 25

3, 2

50

3 6 4

~/ (7 + ) (5 + )

6

(5-7) +

35 700 400 320 30 70 110

7+ 5313

60

") Ref. 2a). TAm~ 10 4 ° K states seen in (~, d) Ref. ~s)

F~

(~, d) this w o r k

j,r

(keY) 0 800 892 2070

2291 2543 2787

F~

L

jn

(3) (1) 5 3 3 6

v/ v~ v/ v/ v/ (7 + )

(keY) 425-

34(3, ~) 7 + ") 3 + ')

4 3094 3445 3753 3908

4 4 4

~/ (3-5) + (3-5) + (3-5) +

a(20 °) . . . . O,b/sr) 60

7o 200 45

65 1000 700 150 120 330 300

") Ref. 2,).

The L-transfers in the 80 MeV (d, ~) study of ref. 23) were extracted for the ground state, the 1698 keV state and the 5280 keV state. The first two are observed to proceed by L = 4 and L = 2 predominately, i.e. the larger of the two possible L-transfers, while our (~, d) L-transfers at 34 MeV proceed by the smaller L-value. 4.9. T H E aSAr(~, d ) 4 ° K R E A C T I O N

(See fig. l0 and table 10.) Good (~, d) DWBA fits were obtained for a number of levels in 4°K and new J~ limitations placed on several levels. The 2543 keV state is, in

(f~2)7+ STATES

241

I0s 4°Ar (=od)42K E= =34.3 MeV 102 0 keV

f + +

io

Ioz kev (3)-

42Ca [cl,d)44Sc Ec~='~4.8MeV

F \r

,

77~eV

. .

~7o,.~

IOo '°'~,x_

Z~

i01[ I0f

1170keY

I0

"" ~

L=6 ~

IOI .~ =L

L ~ ~ L "

1534keV 4

I0

iO~ - -

1950 key

,of

IOs. iO=' •

~.t

2315 keY

fA

lolL ~

10 2

•¢

;=616keV (Z-5)-

IOI

I01

¢

,i.+

0

0c.m. Fig. ] L S¢¢ caption to fig, 3,

8c.m. Fig. 12. See caption to fig. 3.

R. M. DEL VECCHIO e t al.

242

TABLE 11 4~K states seen in (c~, d) Ref. ts)

E, (keY)

dn

0 107 700.-4-2

1948 a)

Ex (keY)

2(3)(5)1170 1534 1950 2315 2829

(7 +)

(~t, d) this work j,r L

1

~/

5

v:

4 6 4 4

(3-5) + (7 + ) (3-5) + (3-5) +

a

o (20) .... ~b/sr)

32 15 150 50 300 700 350 190

") Ref. 2s). TABLE 12 "4Sc states seen in (~, d) Ref. 24) E~ (keV) 765 969 1052 1186 1532 1684 2618

(~, d) this work d~

3+ 7+ 5+ 3+ 5+ 5(2-5)-

E. (keV)

L

J~

tT(20°)©.=. ~b/sr)

773 970 1052 1186 1533 1683 2616

2 6 4 2 4 5

~/ V/ V/ ~/ ~/ V'

10 400 240 12 70 15 100

line with previous arguments, a strong possibility for a (f~)7 + state, lit has recently been given a 7 + assignment 2,)]. The 3094 keV state is suggested as a 5 + level, predominantly of the f~ configurataon. Other possibilities for this (f~)5 + state are certainly present. Alternatively this level may be mixed with a number of others and the (x, d) strength fractionated. 4.10. THE a°Ar(oc, d)42K REACTION

(Fig. 11 andtable ll present the experimental information.) Our results o n 42K are consistent with previous work. The 1950 keV level has been previously given a (7 +) assignment 2, 23, 25). Only one strong level, proceeding by L = 6, was seen in the **Ca(d, ex)42K reaction 23). The only other (d, u) angular distribution given for this reaction was to the ground state, which proceeded by L = 3, the larger of the possible L-transfers. This is in contrastto our (~, d) L = 1 transfer, but is like the3SK situation mentioned previously. Several almost equally likely candidates for an (f~)5 + state are present.

(f.I_=)7+ STATES

243

24Mg (3He, p)26AI E3He • 24.5 MeV 103L

~

,o,

i05 .

0 keV

~..

5856 keV

~:'~ I0 2

,03\ ~

2~.v

103`

,o, .ii

I0 |

6257 keV

"

~

ION

"% " . ~

io 3,

ION

•

6586 keV

102

• ~ z

,,

'°'I

I0z

~ ,

f~

6718 keY

n.I,-

<[

-

102

236?.,keV •

•

3i

4

io i: ~,,.. _

3074 keY (3+) L-4

t %-'-,,,--~,*'~

I0:

• , 7383 keY

I0 2

=03.

=03

8730 keV "q"

~',,,

L-4

IOZ

,,'"% 102 ~

3159k,V

-; 103

LM,

9355 keY

I0 3

•"•

I

3745 * keY 3745.5 •oo

•

i02

10216keV

I0

102 l

I

I

I

I

I

I

0 1020304050607O80 0c.m.

i

•t

; -_J_.-J._.

0 1020304050607080 ~c.m.

Fig. 13. The 24Mg(3He, p) angular distributions. See figure caption for the (0c, d) spectra (fig. 3).

244

R . M . DEL VECCHIO et al.

28Si(SHe, p)30p ESHe "34.5 MeV i0 ,~ .

los I

~.

~

0 keV i" . "~-o-2

"\

•" ~

,o~-

o, \

.~..

6.~7.v

~- .~.~ ,o~ .~-'°=r.'>':~.

t

-

¢J) I.-

t~ ~j

I-W

v,

iiI

-

~...~.o.~ ~.o

• f\'~

".

,97p.v

.4~.v

.

I0 !

•

•

'~,o~

•

0

1%.~c'X 54,2.v

,o,t v ~

. 30119

e e

1

4921 key 3-

•

.

~,.o.2

\.. ~

I0~

'

1454key

io ~L

Io

I" I" ,o'~

IOa; . keY

103 ~

~

~7014keV •

• e

2

~

,o~~..

I0 "~

7149 keV •

~,~.-~

• e

103

, o ~

4234ok.

" . •

,o~

28.v

i0 ~, 102~

"

,

~'L" I

~ * • 4625 keV

' ~ o~o~.~ ~;o~o, #c.m.

7972 I

i

I

i

I

keY

I

o ~b 2o 3o 4o r,o ~ 7o ~o

Oc.m.

Fig. 14. The 2sSi(SHe, p) angular distributions. See figure caption for the (,% d) spectra (fig. 3).

(f~z)7+ STATES

245

4.11. THE 42Ca(~,d)*4Sc REACTION (Refer to fig. 12 and table 12.) We performed this reaction in order to determine the (x, d) L-transfer to the known 7 + state empirically so that exclusive reliance on DWBA, with its various uncertainties, could be avoidedl Since the beam energy was the same as that used for the argon (u, d) studies and since the masses and Q-values are nearly the same, we anticipated similar angular distribution shapes for a given Ltransfer. In fact the same DWBA curves were used to fit the 42Ca(u, d) and the 3s, ,OAr(u ' d) reactions. Only for a6Ar(~, d) was the mass difference taken into account. The quality of the DWBA fits can be judged from fig. 12. The relevant results of this experiment are comparedin table 12 with previous information found in ref. 26). The 46Ti(d, ~)L-transfers measured 26) at 19 MeV bombarding energy agree with those observed here in our 34 MeV (~, d) experiment. This should be compared with the 80 MeV (d, ~) findings discussed previously (subsects. 4.8 and 4.10). A comparison of the angular distributions to the selected 7 + states in the potassium isotopes with that to the 4"Sc 7 + state is shown in fig. 15. The empirical *2Ca(~, d) shape to the "4Sc 7 + state (solid line) has been shifted upward to pass through the argon (~, d) angular distributions (dashed lines). The close similarity of the shapes is apparent.

5. Discussion and conclusions

The (~, d) normalization constant, N, for the case of a spin 0 target is defined by the formula 12, i3) dcr dQexp

3 N(d'~ \ d ~ ] DWUCK

N being sensitive to the optical parameters employed, whether zero-range or finite-range calculations are performed, etc. For the series of (~, d) experiments performed here the value of N obtained for the selected (f~)7 + states was 284 (26A1), 141(28A1), 64(3°P), 107(32P), 631(38K), mlll(*°K), 1196('2K) and 889('*Sc). The spectroscopic amplitudes for the last two cases were calculated under the assumption that the target nucleus contains two f~ neutrons. The average N is 550 which is fairly close to a typical value obtained in a previous 2*Mg(0c, d)26A1 study which employed shellmodel wave functions for the low-lying states 8). The variation in N from nucleus to nucleus observed here is unsatisfactory. A similarly large variation was found in the 2*Mg (~, d) study of ref. 8). The situation would be improved somewhat if the two strong L = 6 states observed in sop and 32p were both 7 + states (see discussion below). Perhaps the lack of good DWBA fits for the (7 +) states in the lighter mass nuclei and experimental normalization errors are responsible for much of the variation in N. Optical parameter effects should not be too pronounced since parameters within the same family were chosen for all the nuclei studied.

246

R . M . DEL VECCHIO e t al. I0 == E

E] 5 . 4 4 M e V 3 6 A r ( a , d ) 3 a K x 2 0 2.55 MeV 3 S A r ( - , d ) 4 0 K X 1.95 MeV 4 0 A r ( a , d ) 4 2 K A 0.97 MeV42Ce(a, d)44Sc

0

,o\\

k £

I .8

X__x~-x---x..?\.,.,'q'\ ~

.6

\

.(3

E

.4

\X

X\ \ \

Z

_o It.) =,, o3 (/3

N N o X \ El

.2

n.-

I0-I .o8

-J

.06

X

I--z .04 LU n-" I.l.I 14_ " .02

i0 -~'

O.

"~E)----(~ ... 0

_

\ \ x... X

0

7 + STATES 34. MeV a

I

I

I

I0.

20.

:50.

I

40.

I

I

50.

60.

70.

C E N T E R OF MASS A N G L E , DEGREES

Fig. 15. The (g, d) angular distributions of proposed 7 + states in doubly odd potassium isotopes compared to the 4aCa(~, d) angular distribution to the known 7 + level in "4Sc. The solid curve is an empirical fit to the 44Sc distribution and the dashed curves are the same curve shifted upward to best pass through the potassium distributions.

A plot of the (~, d) Q-value for formation of the selected (f~)7 + states verses the atomic mass of the residual nucleus is shown in fig. 16. The linear relation first established by ref. 2) is seen to hold with the inclusion of new data points at A = 32, 36, 38 and 40. A possible interpretation of this linear relationship in terms of the BansalFrench-Zamick weak-coupling model has been given elsewhere 4). It would be quite interesting to continue searching from 7 + states in lighter mass nuclei, e.g. 22Na, ISF, etc. Although the 2°Ne(~t, d)22Na, 160(~, d)XSF, and 12C(u, d)14N reactions have been performed [ref. 3), ref. 27) and ref. 2s), respectively], no convincing candidate for a 7 + state was seen. A (d~)5 + State dominates the (~t, d) spectra in this region of the s-d shell. Possible candidates for the 7 + state which have reason-

(fk2)7 +

STATES

247

- Q ( a , d ) for 7 + STATES

28--

X. \ %

24-

\

\

\

\

0 N=Z= odd x N= Z÷2= odd % \ N

22--

\

\

\

%

\ \ 26AI

~>

~E = 20"1o

i

16--

x"~

14--

I0

I 20

I 30

I 40

A Fig. 16. Plot of the negative of the (et, d) Q-value to 7 + states in T = 0 and T = 1 s-d shell nuclei. Here, A is the mass number of the residual nucleus. able (~, d) strength and Q-values which lie nearly along the line of fig. 16 are the 9.99 or 9.36 MeV states in 22Na and the 7.65 or 6.76 MeV states in 18F. A level at 9.312 MeV seenin I°B(160, ~) 22Na has been proposed as a (7 +) state based on a HauserFeshbach analysis 29) and possibly is identical with the 9.36 MeV (~, d) state. Table 13 lists the excitation energies of states of assumed simple configurations derived from this and previous work. In fig. 17, the binding energies of the 7 + states are plotted verses atomic mass of the residual nucleus. The binding energy, B ~ , is taken relative to the removal of a neutron and proton (chosen positive for bound states) and is related to the (~, d) Q-value by B,,~, = Q(~, d) -I-26.072. The B~, binding energies of the ground states of various nuclei of interest are given in the last column of table 13. Also plotted in fig. 17 are data for 7 + states of T = 0 and 1 nuclei continued into the f~ shell. References to this data are given besides the excitation energies listed in table 13. A break in the binding energy curve occurs when more than two particles are in the f~ shell. The (f~)7 + states in T = 0 nuclei follow a different B~, versus A curve f r o m those in T = 1 nuclei. The differences in these curves, however, are not so clear if only s-d shell nuclei are considered.

248

R . M . D E L V E C C H I O et aL

TABLE 13 Excitation energies (keV) of states assumed to have relatively pure T = 0 two-particle configurations in T = 0 and T = 1 nuclei Nucleus

14N lSF 2ZNa 26A1 2SAI 3op 32p 3'tel 3eCl aSK 4°K "2Sc 44Sc 4eV "sV ~°Mn 52Mn 54Co a) f) J) k)

(fgz)7 +

8220 9821 7231 7420 5283 5303 3460 2543 618 °) 969 1590 b) 1255 ©) 1030 d) 870 1) 199 d)

(d~2)3 +

3074 1973 1755 147 789 0

(d~2)5 + 8963 1122 1528 0

(d~fg.)5 -

f) ') h) 1)

(dtf.].)6 -

Bxv(g.s. ) J)

15100 k) 9440 ~) 7460 t) 6930 5165 m)

12497 9751 13502 13638 15996 14073 15233 13785 14953 13922 14180 12622 14637 14884 15705 15170 15810 15030

4696 3633 2517 892

Ref. 43). b) Pet'. 4,). ©) Ref. 4s). d) Ref. 4e). *) Ref. "~). Refs. 2s.,s). s) Refs. 27.,9). h) Ref. 3). l) Ref. 2). Based on (x, d) Q-value from ref. so). Ref. 2s). 1) Ref. 2). m) Ref. Is).

A similar phenomenon was first observed for (d~)5 + states in ref. 1). A plot of B~v veisus A for these states is also shown in fig. 17. References to the data points can be found in table 13 beside the listed excitation energies. It seemed worthwhile to attempt a similar plot for (d~)3 + states. The criteria used to select such states were (a) a reasonably large (~, d) cross section and (b) they should proceed primarily by L = 4 transfer in (~, d) or (3He, p). The excitation energies of the states selected are given in table 13 and the B~, curve is included in fig. 17. Of the states chosen on the basis of this work, the above criteria were not rigorously met. The 1973 keV state in 3Op, while populated by L = 4 in (0c, d) appears to favor L = 2 in (3He, p). The 147 keV level in 34C1 is reached by a tentative L = 2 in (~, d). The (~, d) L-transfer to the 789 keV level in 36C1 was not obtained and the 3SK ground state is reached by a tentative L = 2 transfer. The 1755 keV level in 32p was observed to be excited about three times more strongly than other nearby 3 + states in a 3°Si(3He, p)32p experiment 3o). Moreover, the data was consistent with dominant d~ transfer predicted by structure calculations. The B~, curve for the possible (d~)3 + gtates shown in fig. 17 is similar to the (f~)7 + B ~ curve. A dependence of B ~ on T = 0 or 1 also seems to be present as in the 7 +

(f.}2)7 + S T A T E S

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t .,,

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nn 5

q ,0

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~ I

i

.

~---~lr-

. ~

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A

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of

/

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{j2),me •

/'dr"

249

..- -0~0~"

/..R

STA TFS CORE

o 7÷, T--1 CORE • 5÷, T-O CORE Zx 3÷, T=I CORE • 5*, T=O CORE

I I I I [ I 18 26

~ I

I I I I I I I I I I I I I 34 42 50 58 A Fig. 17. Plot of the binding energy relative to the removal of a neutron and proton from possible (j2)./mz states in nuclei in the s-d and neighboring shells. T i e cores remaining after this two-particle r e m o v a l h a v e isospins of T = 0 o r I. Here, A is the m a s s n u m b e r o f the nucleus i n w h i c h the two-particle state occurs. T h e lines are d r a w n t o guide the eye. T h e a r r o w s indicate where the linear p o r t i o n o f the curve is expected to end.

case. While a pure (d~)3 + configuration for these states should n o t be taken t o o literally, they appear to form a generic family of states. Whatever the forces responsible for mixing them with other states and perturbing their energies, they seem to operate in a smooth and regular way from nucleus to nucleus. While the Bansal-French-Zamick weak-coupling model s' 6) can account for the linear dependence of the (~, d) Q-value (or B , ) on atomic number for the (f~)7 + states in s-d shell nuclei #), the behavior of Q(~, d) or B,, as a function of atomic number for 7 + states in f i shell nuclei requires a more detailed shell-model description since an (f~)7 + configuration is present. Similar remarks apply to the 5 + and 3 + binding energies plotted in fig. 17. The expected linear portion of these latter curves is not as apparent due to a lack of data on (d~)5 + and (d~)3 + states in p-shell nuclei. Although the information is scarcer, some attempt was made to locate (dt, f l ) 6 - and (dp fi)5- states. The energies of the states selected appear in table 13 and an appropriate reference is given except when the assignment was discussed previously or its selection deduced from the data presented here. The (~, d) angular distribution for the 4696 keV level in 32p is consistent with L - 5 but could not be determined due to a lack of points. It appears as a moderately strong level in the (3He, p) work of ref. ao) (at 4.66 MeV). A 5- assignment to the 2517 keV level in 36C1 has not been firmly established. When more information becomes available on such 5- and 6- states, a search for systematic trends would be interesting.

250

R. M. DEL VECCHIO et aL

The other strong L = 6 (x, d) levels found in aop at 7392 keV, in 32p at 6880 keV, and in 34C1 at 4789 keV, although not plotted, have B=, values which fall very close to the 7 + curve of fig. 17. These could be (f~ fi)6 + or (f~)5 + levels but it is reasonable to expect such states to have binding energies a few MeV less than those of (f~)7 + states in the same nucleus. The (f~)5 + states are expected to proceed predominantly by L = 4 transfer in (x, d). This was observed in the *2Ca(x, d)*4Sc experiment discussed in subsect. 4.11. A more likely possibility is that these are also 7 + levels, although a splitting of the (f~)7 + strength is rather unexpected. If another 7 + states is present at nearly the energy of the (f~)7 + state, they would mix to some extent. The degree of mixing would depend on the structure of the two states and the strength of the interaction which mixes them. The (~, d) reaction then populates such states in proportion to the amount of (f~)7 + admixture. Assuming that such two-state mixing is occurring in 3Op, 32p and 34C1, one can calculate an interaction matrix element from the observed energies and (c~, d) cross sections. One obtains a mixing matrix element of 74 keV for aOp, 252 keV for a2p and 229 keV for 34C1. Such small matrix elements are expected for connecting states of rather different composition. As a check on the reasonableness of the selection of two-particle configuration states shown in fig. 17, we extracted empirical two-particle interaction energies, As was demonstrated previously for (d~)5 +, (f~)7 + and (g~)9 + configuration states 3), a smooth behavior from nucleus to nucleus is expected. Following refs. at, a2, 3) the (j2)j configuration two-particle interaction energy relative to a J = 0, T = 0 core is given by

= B (j) + 8 , U ) - B ,(je)j, where B~(j) and B~(j) are the proton and neutron binding energies in single-particle states j relative to the core and B~,(j2)J is the proton-neutron binding energy in the state (j2)j relative to the same core. With the convention that the B-values are positive for bound states, the sign of E above is negative for attractive interaction energies. For (Jr J2) d, T = 0 configuration states relative to a J = 0. T = 0 core, the B~(j) and By(j) in the above formula should be replaced by ½(B~(jl) +B~(j2)) and ½(B,(jl) +B~(j2)), respectively. The choice of the relevant single-particle energies is made in table 14, based largely on the compilation of Endt and Van der Leun 1s). The fiction that all the singleparticle strength is concentrated in one state was assumed. Moderate fractionation of this strength would not affect our qualitative arguments. Table 14 gives the empirical (f~)7 +, (d~)3 + and (d~, f~)5- interaction energies for T = 0 nuclei. For T = 1 cores, the interaction energies can be obtained by making suitable assumptions 3). The (f~)7 + interaction energies were given previously *). They are quite close to the results of ref. a) (except for aSK which was not studied). The (d~)3 + and (d~ f~)5energies in a4CI agree with ref, 33) while the remaining ones are new. The relatively smooth behavior of these two-particle energies suggests that the configuration assign-

(f.~2)7 + STATES

251

men~s are basically correct. To what extent these regularities are dependent on a bias present in selecting the appropriate levels is difficult to assess. Additional independent determinations of spin-parity assignments would be helpful. No strong candidate for (f~)5 + states emerged from this study, except possibly in the potassium isotopes. DWBA calculations predict the (f~)5 + cross sections to be about a factor of five lower than th~ (f~)7 + cross sections at the beam energies used TABLE 14 Single-particle excitation energies (keV) and experimental T = 0 neutron-proton interaction energies (keV) for assumed simple configurations in T ~ 0 nuclei

Core

f~z~

f~v

d~

d~

E(f~.2)7 +

E(dg,2)3 +

3696 3447 2688 1379 0

3968 3623 2934 1611 0

945 1384 0 0

975 1273 0 0

--3481 --2691 --3203 --2807 --2557

--2883 --3536 --2717 --3277

E(d~q)5-

nucleus ~4Mg 2aSi a2S a6Ar 4°Ca

--2042

here. If the 3665 k e y level in 38K is indeed the (f~)5 + level, then the f~ multiplet as observed in 42Sc is significantly altered upon removal of a neutron and proton from the s-d shell. Possible (f~)0 + and (f~)l + configurations for levels in a s k at 3045 and 3342 keV, respectively, given by ref. 2o) [the calculated wave functions are ~ 70 (d~"4) (f~)]together with the S + possibility mentioned above suggests that the energy spacings of the multiplet members are contracting in 3SK relative to ~2Sc. Considerable mixing with other states might be expected for low-spin members of the f¢ multiplet. In a 32S(t, p)34S experiment, however, plausible candidates were observed for (f~) 0 +, 2 +, 4 + T = 1 states which preserved the 42Ca spacing for these levels rather well 34). Further, ref. 4) suggested the beginning of a possible (f~)0 + sequence of levels in s-d shell nuclei based on available (t, p) data. The existence of relatively pure f~ multiplets in s-d shell nuclei would be very difficult to verify experimentally. However, such information would be useful in studying the systematics of wave function mixing or the influence of the core on the effective two-particle interaction. We wish to thank E. Marmer for help with the data taking and analysis and D. Rohrlich for help with the data analysis. We are grateful to T. J. Bowles for much assistance with the experiments using the QDDD. References I) 2) 3) 4) 5)

B. E. C. R. R.

O. Harvey, J. Cemey, R. H. Pehl and E. Rivet, Nucl. Phys. 39 (1962) 160 Rivet, R. H. Pehl, J. Corney and B. O. Harvey, Phys. Roy. 14I (1966) 1021 C. Lu, M. S. Zisman and B. G. Harvey, Phys. Roy. 186 (1969) I086 Sherr, R. Kouzes and R. Del Vecchio, Phys. Lett. 5213 (1974) 401 K. Bansal and J. B. French, Phys. Lett. I I (1964) 145

252 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49) 50)

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