The 31P(3He, p)33S reaction

Nuclear Physics Al62 (1911) 593-604; Not

@ North-Holland Publishing Co., Amsterdam

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

THE 31P(3He, P)~~SRJ3ACTION A. GRAUE,

J. R. LIEN, L. RASMUSSEN and G. E. SANDVIK University of Bergen, Bergen, Norway and

E. R. COSMAN Physics Department and Laboratory for Nuclear Science t Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Received 23 January 1970 (Revised 7 November 1970) Abstract: The 3’P(3He, P)~~S reaction has been studied at 12.0 MeV incident energy. High-resolution magnetic analysis of the reaction protons enabled 36 levels to be identified up to & = 8.3 MeV in 33S. Detailed angular distributions were extracted for most of the transitions. DWBA analyses using the reaction formalisms of Glendenning and Bayman were compared and the predictions of the theoretical wave functions of Glaudemans were tested. Presence of L = 0, 1, 2 and 3 stripping amplitudes was tentatively inferred for several levels and consequent spin assignments suggested. Three analogs of 33P levels were seen, and a brief discussion of Coulomb and symmetry energies is given. E

NUCLEAR

I

REACTIONS: 3’P(3He, p) E = 12.0 MeV, measured o(E,, O), Q. 33S deduced levels, L, J, n, isobaric analogs.

1

1. Introduction

The 31P(3He, P)~~S reaction is studied at a bombarding energy of 12.0 MeV. From this two-nucleon stripping reaction, it is possible to learn about the 2p-lh character of the levels of 33S relative to 32S and thus co m p lement the large body of existing data on this nucleus. In addition, because of the special coherences involved in this reaction it can provide a sensitive test for theoretical calculations made for the target and final nucleus. The highresolution of the magnetic spectrographused to detect the reaction protons enabled most of the levels in 33S known up to the time of the 1967 compilation by Endt and van der Leun ‘) to be identified here. The experimental results are described in sect. 2, including energy levels and detailed angular distributions. Several DWBA analyses of the strongest transitions involving different calculational procedures are given in sect. 3. As input we have taken theoretical wave functions of Glaudemans both with and without correlations in the ‘*Si core in an attempt to assess the importance of such effects in explaining the observed cross sections. In sect. 4, a qualitative t This work has been supported in part through funds provided by AEC Contract AT(30-l)-2098. 593

A. GRAUE et al.

594

discussion is given of other aspects of the data including identification ofL = 0, 1,2 and 3 transfers, analog states, and a comparison to the previous knowledge of the 33S level scheme. 2. Experimental procedure and results Ions of 3He+ + from the MIT-ONR Van de Graaff accelerator were used in this experiment to bombard a Zn,P, target with 51 pg/cm2 of 31P. The target was prepared by vacuum evaporation of Zn3Pz onto a x 5 yg/cm2 backing of formvar. The EXClTATlON ENERGY (t&V) 80 I

60 I

40 1

/0.0

20 I

31P(3He,p)33S LAB ANGLE=15* E,= 12.0 t&V SPECTROGRAPH FIELU =123089 kG

1

200

I ___A 100

0

15

20

25

30

35

40

I.5

so

55

60

d

65

DISTANCE ALONG PLATE (cm)

Fig. 1. The 8,,, = 15” proton spectrum from the 31P{3He,p)33S reaction measured in the MIT multiple-gap spectrograph. The numbers over the proton groups correspond to the scheme adopted in table 1. The weaker peaks not labelled result mainly from 6“*s6Zn(3He, P)~~*~*G~ reactions.

reaction protons were energy analysed in the MIT multiple-gap spectrograph 2), and the target thickness was determined by a 4 MeV 3He’ Rutherford scattering measurement. The reaction protons were registered at 24 different angles simultaneously on 50 pm nuclear emulsions. After a 2000 /.E exposure of the target to the beam, the emulsions were developed and scanned under a microscope. Aluminium degrading foils over the emulsions during exposure enabled the scanners to discriminate visually the proton tracks from those of other particles that emanated from the target. In fig. 1 is shown a typical spectrum of protons from the 31P(3He, p)s3S reaction measured at &,, = 22.5”. The energy resolution is about 30 keV. Table 1 summarises the excitation energies for the 33S states labelled in fig. 1. They are based on a magnetic-field calculation derived from the positions of known (3He, d) and (3He, p) contaminant reaction groups present on the emulsions and should be good to about + 10 keV. In several cases, the data indicate a ciustering of unresolved levels; this is

Spectroscopic

TABLE 1 information from the present investigation of the 31P(3He, p)% reaction and from previous studies of the 32S(d, P)~~S and 34S(3He, a)33S reactions 31P(3He, P)~~S

Level no.

E. (MeV) 0

0.840 1.966 2.314 2.868 2.934 6 I 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

2.974 3.223 3.832 3.939 4.059 4.146 4.214 4.377 4.432 4.746 4.933 ‘) 5.281 5.487 5.607 5.723 5.913 6.370 6.512 6.684 r) 6.900 6.973 7.032 ‘) 7.163 r) 7.346 ‘) 7.463 ‘) 7.567 ‘)

32 33 34 35

7.902 ‘) 8.013 8.107 8.329 all +0.012

~W~)nmx (mb/sr)

34S(3He, a)33S ‘)

32S(d, P)~~S “)

00 “) (degrees)

L

0.078 0.193 0.018 0,062 0.017

22.5 7.5 22.5 7.5 30

2 (i) 0+2 (2)

0.071

31.5

(3)

1. ‘) (h&)

0.207 0.019 0.054 0.198 0.052 0.161

15 30 30 1.5 15 7.5

0.078 0.100 0.188 1.48

7.5 15 7.5 7.5

0

0.201

7.5

1

(23) 2 0+2 (1) 1

1

0.209

30

0.170 0.131 0.140

22.5 22.5 22.5

0.294 0.199 0.238

30 22.5 22.5

(2) (2)

1.03 0.231

7.5 15

0

0.450

7.5

(2)

2 2

0

0.839 1.965 2.314 2.869 2.936 2.971 3.222 3.832 3.935 4.049 4.145 4.211 4.377 4.425 4.147 4.919 5.287 5.479 5.613 5.711 5.915 6.360 6.513 6.676 6.903 6.965 7.037 7.164 7.353 7.452

Ex (MeV)

L

0

2 (i) (2) (2) 3

0.842 1.964 2.316 2.870

0

2.950 ‘)

3

3.225 3.837 3.935

1 2 ‘) 2

d)

2 2

Jr= ‘)

;: :: (h, 8) + ?I-

1 (4) 1

4.920 5.285 r) 5.419

1

(:, 0

5.720 6.361

2

6.900

(2)

(%+,Tf=B 9

7.348 7.450 all *O.OlO to 0.020

(3)

(&+, T,=+) “)

7.892 8.015 all kO.006

(!z,3)’

“) Ref. I*).

“1 Refs. lg. z” ). ‘1 13~is the angle at which the measured cross section is a maximum.

9 Bracketed Z,,values are from unpublished papers [see ref. ‘)I. ‘) Compilation

of the results from the present work and those of several others I* r* -“).

9 Probably doublet, Two closely separated levels were observed in the ‘*S(d, P)~~S reaction for these cases. The listed E, in column 6 is always the lower level. For this level Z, = 2 is given in ref. 20) while a tentative 2. = (3) is suggested in ref. 19). Spin and parity of assumed 33P analog level ‘I).

1;

596

A. GRAUE

et al.

indicated in the table. The ground state Q-value was determined to be Q0 = 9.787 f 0.015 MeV. Fig. 2 displays the angular distributions for 12 of the 36 33S transitions seen here. The data points at backward angles were not extracted, because most of the yields were extremely low there, and the important spectroscopic information is contained in the very forward-angle cross sections. In the figure we have included curves from DWBA calculations that will be discussed below. 3. DWBA analysis As seen in fig. 2, several angular distributions show a strong diffraction structure, indicative of a direct stripping mechanism. The specific details of the yields and stripping patterns are known to depend sensitively on the coherent superposition of the several scattering amplitudes that might contribute to populating a given final state “). Thus, a qu~ti~tive analysis requires some knowledge of the initial and final wave functions involved. In this section we perform such a quantitative analysis using two sets of wave functions calculated by Glaudemans et al., the first set “) determined by assuming the *aSi core to be inert, and the second set “) derived by including the interactions with Id% core particles. In addition, we have made our stripping calculations with two different computer codes, one with the Glendenning formalism “), and the other with the Bayman formalism 6), and with different stripping interactions to determine the sensitivity of the results to these approaches. The objective is to provide a test for the wave functions of Glaudemans, at least for the strongest transitions, and to see to what extent detailed correlation effects can be detected from our 31P(3He, P)~~S scattering cross sections. Assuming a zero-range, spin4sospin independent interaction between the outgoing proton and the stripped nucleons, and no spin-orbit coupling in the optical potential, the (3He, p) cross section can be written as an incoherent sum over L, S, J and T of the transferred pair

The quantum numbers Jo, To and Jr, Tf are for the initial and final nucleus respectively. The factor A involves the strength of the stripping interaction and the normalization factor for the internal 3He wave function and it is not yet known theoretically. In the reaction function CT &B), which corresponds to the one used in single-nucleon transfer reactions, we use the modified form factor of Glendenning “)

where tiRL is the radial part of the harmonic

oscillator wave function describing

31P(3He,p)%

597

the c.m. of the captured pair. As emphasized by Drisko and Rybicki ‘) it is important that the wave function has the asymptotic behaviour of a finite well. Therefore the h.o. wave function for each N is matched to a Hankel function corresponding to the 1

Level No.0 ErO.oMAeV _--__ 6 _~-_ c

:

Level No.2 E.-l.966MeV -A ____- 6 -__.- c

10-l

E

LeveiNo.6.6

i

I-----

LevelNo.26

LevelNo.20 ?I:"'"

1

Level No.16 Es.933 MeV L'l

Er7.163 MeV L=2

MeV

1

t

0

30

60

90

0

30

60

90

0

30

60

90

0

30

60

90

B,,,(degrees) Fig. 2. Angular distribution for some of the observed transitions in the 31P(3He, P)~~S reaction. For level nos. 0, 1, 2, 3,4, 18 and 25 detailed DWBA calculations using available shell-model wave functions were performed (see text). Curves labelled A are resulting from the Glendenning procedure with dq correlations in the initial and final state wave functions 5), whereas curves labelled B result from the same procedure but assuming a closed dt shell 4). The Bayman procedure for the reaction analysis with d+ correlated wave functions resulted in the curves labelled C. For the other transitions shown in this figure the DWBA curves are the result of a Glendenning procedure using pure and arbitrary configurations for the transferred n-p pair.

598

A. GRAUE et al.

Spectroscopic

amplitudes S(j&

Level no.

TABLE2 JT) calculated 4, from the shell-model wave function with an inert *?Si core

LJT

(2s+12

2~ Id+

0.0569

0.0945 0.2110 -0.5874 1.1916

001 010 210

3

4

010 210 220 221

-0.0057 -0.0430 -0.0771

18

001 221

0.3022 0.3022 0.9621 -0.2828 -0.2828

-0.0665 -0.0703 0.3893 0.3632

230 430

25

-0.1458

- -0.8804

-0.2643 -0.5950

221 220 230 430

(ld+)*

-0.4097 -0.4097 -0.1039 0.8352 0.8352

-0.6268

0.6317 0.1112

- .0.3179

The calculated 33S states have been tentatively identified in the experimental spectrum by their spin and excitation energy. For the levels nos. 18 and 25 we have used the wave function of the 33P analog states. The B-coefficients are given for different configurations of the two transferred nucleons, and for the L, J, 7’ that can couple to the spin and isospin of the final 33S level.

binding energy of the transferred nucleons and then renormalized to unity. The structure coefficients GNLsJTcontains the information about the nucbar states involved in the transition,

Here yl = (~~~~j~) and y2 = (~~~~~~) stand for the possible single~partici~ orbits which can couple to L, S and J. The two pairs of brackets are the change of couplingcoefficient and the Moshinsky transformation bracket, respectively. The g12 is a statistical factor and Qn is an overlap factor, and they are defined in ref. “). The spectroscopic amplitudes B( j,j, JT) reflects the overlap between the ha1 state CpT:and the ground state of the target @T; plus two nucleons in a state (nl l,j,), (n2&), L, S, J, T.

B(jl h JT) =

(4)

599

3’P(3He, p)a3S TABLE 3 Spectroscopic

amplitudes B(jr j,JT)

Level no.

L

J

T

0

0 2 2 2

1 1 2 2

0 0 0 1

-0.0041 -0.0041

0 0

0

1

2

1 1

0 0

0.0135 -0.0056 -0.0056

2 2 2 4

2 2 3 3

1 0 0 0

0 2 2 2

1 1 2 2

0 0 0 1

2 2 2 4

2 2 3 3

1 0 0 0

1

2

3

4

(ld#

-0.0089

calculated by the new wave functions which include ldg nucleons 5, Idq2st

-0.0191 0.0386

-0.0029 - 0.0029

18

0

0

1

-0.0267

25

2

2

1

-0.0026

29

2

2

1

(2SfS

2s+ 1d+

0.0053 0.0053 -0.0112 0.0043

-0.0109

-0.2904 -0.2904 0.4605 -0.9656

0.0361 0.0361 0.0077 0.0092 0.0199

0.0012

ld+ ldt

0.0012 -0.0086

-0.1635 -0.3871 0.0238 0.0396 0.0685

-0.0214 0.0140 -0.0345 -0.0345 0.0107 0.0107 -0.0176 0.0540

0.0232 0.0121 0.0343

-0.0073 -0.0067 0.0144 0.0144

0.0175

-0.0172

0.0109

-0.0001

(Ld#

0.1206 0.1206 -0.1609 0.7742 -0.1610 -0.1610 -0.7992 0.1485 0.1485

-0.0450

0.1473 0.1473 -0.0935 0.0714 0.0628 0.1058

0.4135 0.4135 -0.4206 0.2134 0.0623 0.0623 -0.8300

-0.3291 -0.0842 0.0842

0.1764 0.4042

See also caption for table 2.

Shell-model calculations for nuclei in the sd shell have been performed by Glaudemans et al. In their early calculations “) they assumed an inert 28Si core and that the extra nucleons were filling the 2s3-and Id, orbitals. Recently, they have recalculated their wave functions “) taking up to two Id, holes into account. We have calculated the B(j,j,JT) amplitudes for both sets of wave functions for those theoretical states which can be recognized in the experimental 33S level scheme. For those levels which we believe to be the 33P analogs (see sect. 4) we use the 33P wave functions. Because the large number of different configurations which contribute to the wave function of both the initial and final nuclear states [in the case of the newest wave functions ‘) as many as 531, the spectroscopic amplitudes become a sum of many terms with differing phases. In tables 2 and 3 we list the calculated spectroscopic amplitudes B(j,j,JT) for both sets of Glaudemans wave functions. The non-zero J, = 3 spin of the 31P target requires the sum over J and L to be retained in eq. (1). The unknown normalization

600

A. GRAUE

et al.

coefficient A prevents us from a direct comparison of the calculated and experimental cross section. Instead, we list in table 4 the ratio

The two cross sections were matched at the most forward peak of the experimental angular distribution. The form factors were calculated by the computer code TWIGGY (Fortran IV) “) and were fed into the DWBA code XANTIPPE “) to yield differential cross sections for each L, S and J. Finally these cross sections were summed according to eq. (1). Ideally, the ratios should then be the same for all levels. TABLE 4

Ratios of theoretical

and experimental

Level no. 0 1 2 3 4 18

cross sections for several levels using different calculational approaches Jr”

0 0.840 1.966 2.314 2.868 5.481

4’ 4’ %’ 8’ (8, it)+ ‘) 4’

(doW’),t, . ~03/W/~L., “)

b,

‘)

d)

2.7 1.9 0.9 1.2 0.5 1.1

2.9 10 1.4 1.5 1.9 0.14

1.5 1.1 0.7 0.6 0.3 0.9

2.3 1.8 1.0 1.2 0.9 0.6

“) Theoretical cross section was calculated in Glendenning’s formalism from the wave functions including two Id+ holes 5). No spin and isospin exchange. b, Same as “) but with the wave functions assuming an inert ?Si core 4). ‘) Same as b, but including spin and isospin exchange force lo). d) The theoretical cross section was calculated in Bayman’s formalism from the wave functions including two Id+ holes s). No spin and ieospin exchange. “) The spin-parity value for the predicted level in refs. 5*6, which this level has been matched to, is 8’.

Hardy and Towner lo) have introduce d sp’m and isospin exchange terms in the interaction potential. Their calculations show that the sum over S remains incoherent but each term must be weighted by a factor ID(S)]’

To study how these exchange terms affect our results we recalculated the cross sections using the Rosenfeld potential ‘I). This leads to lO( = 0.81 and ]O(l)]” = 0.49 [see ref. ‘“)I. The ratio S are also listed in table 4 for this case. An alternative and more rigorous treatment of the two-nucleon transfer reaction has been given by Bayman and Kallio “). They perform the transition from singleparticle coordinates to relative and c.m. coordinates directly with Saxon-Woods wave

3zP(3He, P)~‘S

601

functions. The form factor now becomes

where y = 0.206 is the size parameter “) of the 3He, and the distribution function &(r, ,,/%) is calculated from the Saxon-Woods potential as outlined in ref. “). Using Bayman’s code TWO PAR 12) we have calculated the ratio S between the theoretical and experimental cross section with Glaudemans’ new wave functions “). The results which are listed in table 4 are very close to those obtained with Glendenning’s formalism. Fig. 2 shows the angular distributions for the 31P( 3He, P)~~S transitions discussed above and referred to in table 4. The theoretical curves for both the Glendenning and the Bayman formalisms are included for comparison. In both procedures we have used the same set of optical-model potentials, see table 5. The 3He parameters were found by varying the parameters to obtain a least-squares fit to the elastic scattering data 13) at 12 MeV, and the proton parameters were taken from the survey by Perey [ref. ‘“)I. TABLB5 Optical-model parameters used in the DWBA analysis of the 3*P(3He, p)33S reaction V

‘He p

w

,

P

(&

(MW

(MeW

(IZ)

(2)

(fG

$l)

(&

154.9

17.4

0

1.07

0.843

1.721

0.609

1.4

45.0

0

46.0

1.2J

0.65

1.25

0.47

1.25

Volume absorption was used for the 3He particles and the parameters are taken from ref. ?I. For the protons we used surface absorption with parameters from ref. 14). No spin-orbit coupling was used in either channel.

4. Discussion and conclusions It is seen from table 4 that the ratios S vary substanti~ly for all calculational ap* proaches used, but that the differences in the Glendenning and Bayman formalisms are small. It is also seen from the angular distributions in fig. 2 that only in the case of the strongest transitions are the theoretical curves in reasonable agreement with the data over a significant angular range. This would suggest that for the weaker levels our quanti~tive analysis is not very meaningful. Undoubtedly, there are several reasons for this which might include the crudeness of the reaction theory, an oversimplification in trying to match the theoretical levels to weak experimental levels, and the influence of compound nucleus cross sections at this bombarding energy. In connection with this last point, a recent study by Groeneveld et al. ’ ‘) has shown the importance of compound nuclear effects in the ‘sSi(He, P)~‘P reaction at similar bombar~ng energies which points to their significance here.

602

A. GRAUE et al.

For the stronger levels, the most striking effect appearing in the calculated results of table 4 is the difference between the yield of the two J; = 4’ states at E, = 0.840 and E, = 5.487 MeV for the two types of Glaudemans’ wave functions [see columns a and b)]. For the wave functions which were derived assuming the Id+ orbits to be inert and closed, the quantities (do/dS2),,/(da/dS2),,, differ by a factor of 70 for these two transitions; whereas for those calculated with d, excitations included, they differ by only a factor of 1.7. This result is further improved by using the spin-isospin exchange term in the interaction potential, see column labelled c in table 4. The ratio for the two states are then within the limits of experimental error. Since the data is fit quite closely by the DWBA calculations in these cases and the match between Glaudemans’ predicted levels and these states is fairly unambiguous, this seems to be a strong demonstration of the importance of the more expanded configuration space of ref. ‘). The inadequacy of the wave functions with a closed d+ shell in explaining two-nucleon transfer reactions in this region has previously also been pointed out by Hardy et al. [refs. 16*“)I. Comparing the difference in amplitudes B(j,j, JT) for these two wave functions in tables 2 and 3, also demonstrates the high sensitivity of such two-nucleon stripping results to such pair correlation effects. Although a quantitative analysis of most transitions seen is apparently difficult, some spectroscopic information on even the weaker states is evident by a simple inspection of the data. The most outstanding of such information is the multipolarities involved in the n-p transfer. For levels other than those discussed in table 4 we therefore calculated DWBA curves using arbitrary and pure single-particle configurations with the Glendenning form factor. Several such cases are included in fig. 2. The primary intent here was not to test specific wave functions and coherences but merely to give examples of the various L-value templates whose shapes are rather insensitive to the detailed configurations involved. It is clear that marked differences in stripping patterns exist between the cases of L = 0 and L = 2, the transferred orbital angular momentum of the n-p pair. When these waves are added in one transition, the main features of the individual patterns are still evident; specifically, the rapid increase in cross section as 0 approaches zero degrees characteristic of L = 0 levels and the enhancement around 0 = 25” from L = 2. On this basis we have attempted to indicate in table 1 whether the pronounced even-parity transitions contain components of L = 0, L = 2, or L = O-i-2. In addition, several odd-parity transitions have been identified and noted in table 1. In a number of these examples, the assignments are taken as tentative because of lower counting statistics or uncertainties in the match to predicted DWBA shapes, and in these cases the L-values listed in table 1 are enclosed in parentheses. In this table we have compared our results to the energies and I, values from the 32S(d, P)~~S study of Endt and Paris I*) and to the results from two studies of the 34S(3He, ~1)~~s reaction by Dubois 1‘) and Leighton and Wolff ‘O), respectively. Except for the level at 7.346 MeV (see discussion below) our L assignments are consistent with spin-parity values given for the corresponding levels from other sources of data.

603

31P(3He, P)~%

The L-values suggested above impose restrictions on the possible spins of the final 2 states first, we can transfer the n-p states populated in 33S. Considering the Tf= 1 pair to the To = +,Jo = + target with either a T = 0,S = 1, or a T = 1,S = 0 configuration to yield this final isospin. If the transition involved L = 0 partial waves, this means that the final spin J” must be either 5’ or 3’. For final states with T,= 3, the isobaric analogs of levels in 33P, the selection rules are more restrictive because the n-p pair must be transferred in a relative T = 1, S = 0 configuration. In this case, L = 0 and L = 2 amplitudes cannot mix, and there is a one-to-one match between J" = 3' states and L = 0 transitions and J" = $+,$+ states and L = 2 transitions. Some cases are given in table 1, where these selection rules have been applied to the present results, to put limits on spins of states for which nothing was previously known.

0

1

2

3

L

EXCITATION

5

ENERGY

6

I

6

(MeV)

Fig. 3. A plot of the maximum values of du/dQ for the observed transition in the 31P(3He, P)~~S reaction as a function of the excitation ehergy in 33S.

The L = 0 transition to the level at E, = 5.487 MeV, appears here with the strongest intensity of any transition, as is evident from the plot of (do/dQ) versus E,in fig. 3. This level has been reported previously as the analog of the 33P ground state ‘). Dubois lg) further suggests the analogs of the two first excited 33P levels to be at 6.900 and 7.348 MeV. Both levels were observed in the present experiment (see table 1). The L = 2 transfer observed for the 6.900 MeV level is consistent with the 3’ assignment by Moss et al. 2’) to the 1.435 MeV level in 33P. For the 7.348 MeV level Dubois tentatively gives 1, = (3) in contradiction to our data which suggest L = (2). The latter result implies a spin of 3’ or 3* in agreement with the 3’ spin given for the 33P parent level I’* 2’) at 1.850 MeV. Assuming this matching of analog levels to be correct we calculated the cross section for the transitions to the 6.900 and 7.346 MeV levels in 33S using the new set of Glaudemans wave functions for the 33P analog levels. Even if there is a large contribution from compound-nucleus formation, the ratio

604

A. GRAUE et at.

(d~/d~)~~~(d~~d~)=~~for these two transitions turns out to be several orders of magnitude smaller than for the other levels listed in table 4. This result is surprising if the wave functions of Glaudemans describe the two 33P levels at 1.435 and 1.850 MeV correctly. Their 33S analog should then be much more weakly excited than is the case for the 6.900 and 7.346 MeV levels. Experimentally there are indications for a doublet at the latter energy in the present work. The average Coulomb energy calculated on the basis of the three possible analog levels is dEc = 6.015 MeV. A strong L = 0 level is observed here at E, = 7,902 MeV. If this is a Tf = 3 level the spin must be -)‘, and its analog in 33P should be observed at about 2.415 MeV. The only known 33P level in this region, however, is the 3’ level at E, = 2.544 MeV 21). The Jf” = 3+, Tf = 3 level at E, = 0.840 MeV undoubtedly contains a large piece of the antianalog configuration of the $ = 3’, Tf = 3 level at E, = 5.487 MeV, suggesting a symmetry splitting of AE, = 4.647 MeV for the pair. This implies a value of the symmetry strength Y1 of 102 MeV, as derived from the relationship AE, z ( V,/A)(T, + l), where To is the lower value of the isospin in the final nucleus. The value of Yl is in agreement with other deter~nations in the A = 30 mass region [ref. 13)]. The authors wish to thank Dr. C. Riedel and Dr. B. Bayman for making their computer codes available. We are also very grateful to Dr. P. W. M. Glaudemans for sending his wave functions prior to publication. Referenws 1) P. M. Endt and C. van der Leun, Nuci. Phys. A105 (1967) 1 2) 3) 4) 5)

6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)

H. A. Enge and W. W. Buechner, Rev. Sci. Instr. 34 (1963) 155 N. K. Glendenning, Phys. Rev. 137 (1965) Bl02 P. W. M. Glaudemans, G. Wiechers and P. J. Brussaard, Nucl. Phys. 56 (1964) 529; 548 P. W. M. Glaudemans and J. A. C. Bogaerts, Univ. of Utrecht, unpublished; B. H. W~denthal, J. B. McGrory, E. C. Halbert and P. W. M. Glaudemans, Phys. L.&t. 27B (1968) 611 B. F. Bayman and A. Kallio, Phys. Rev. 156 (1967) 1121 R. M. Drisko and F. Rybicki, Phys. Rev. I&t. 16 (1966) 275 J. R. Lien, TWIGGY (Fortran-IV), A two-nucleon transfer form factor code, Univ. of Bergen, unpublished G. 0stevold and J. R. Lien, XANTIPPE, A DWBA code without spin-orbit coupling, Univ. of Bergen, unpublished J. C. Hardy and I. S. Towner, Phys. Lett. 2SB (1967) 98 L. Rosenfeld, Nuclear forces (North-Holland, Amsterdam, 1948) p. 233 B. F. Bayman, Univ. of Minnesota, private communication A. Graue, L. Herland, J. R. Lien and E. R. Cosman, Nucl. Phys. Al20 (1968) 513 F. Perey, Phys. Rev. 131 (1963) 745 K. 0. Groeneveld, B. Hubert, R. Bass and H. Narm, Nucl. Phys. A150 (1970) 198 J. C. Hardy and I. S. Towner, Phys. Lett. 25B (1967) 577 W. G. Davies, J. C. Hardy, W. Darcey, Nucl. Phys. A128 (1969) 465 P. M. Endt and C. H. Paris, Phys. Rev. 110 (1958) 89 J. Dubois, Nucl. Phys. A117 (1968) 533 H. G. Leighton and A. C. Wolff, Nucl. Phys. A151 (1970) 71 C. E. Moss, R. V. Poore, N. R. Robertson and D. R. Tilley, Phys. Rev. 174 (1968) I333 T. T. Bardin, J. A. Becker and R. E. McDonald, Bull. Am. Phys. SOC. 14 (1969) 1217