(3He, α) reactions on the chromium isotopes

Nuclear Physics A128 (1969) 47--72; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher I.E.7

(aHe, ct) R E A C T I O N S

ON THE CHROMIUM

ISOTOPES

P. DAVID Institut flir Strahlen- und Kernphysik der Unicersitiit Bonn H. H. DUHM, R. BOCK and R. STOCK Max-Planck-lnstitut fiir Kernphysik, Heidelberg Received 17 October 1968 Abstract: The reactions 5°,5-',ss,~4Cr(SHe, ~)ag,51,5"-,azCr have been studied at an incident energy of 18 MeV using a broad-range magnetic spectrograph as well as solid state detectors. Angular distributions were obtained for all isotopes. The relative isotopic cross sections were accurately determined by measurements with a target of mixed Cr isotopes. Coulomb energy displacements are obtained from the observation of f~, d~ and s½ analogue states at excitation energies between 5 and 13 MeV. The spectroscopic factors from a DWBA analysis give information about the neutron configurations in the Cr isotopes which are discussed in the particle-hole excitation picture and in terms of the unified model. Centres of gravity for the f~, d~r and s~ orbits are determined. The spectroscopic factors of the analogue states are obtained from solutions of the Lane equations for the bound states using a surface-type isospin potential. In a simplified analysis, the effects of the surface potential are simulated by varying the bound-state radii, keeping the depth of the potential constant for the T< and T> states. These procedures yield similar results, which arc in agreement with the sum-rule predictions. The resulting radfi suggest that the inner nucleons occupy a volume smaller than that corresponding to a potential radius R = 1.25AJk. NUCLEAR REACTIONS: ~°,5-",~a,~4Cr(aHe,~), E = 18 MeV; measured a(E~, 0). 49.b~,5e,53Cr deduced levels, /-values, spectroscopic factors. Enriched targets.

1. Introduction T h e level s t r u c t u r e o f the C r isotopes has b e e n s t u d i e d in p r e v i o u s e x p e r i m e n t s m o s t l y b y (d, p ) r e a c t i o n s 1 - 3) 1. R e c e n t l y F i t z et al. 4) h a v e i n v e s t i g a t e d the 5*Cr(d, t) a n d 53Cr(d, t) r e a c t i o n s u s i n g 1 1.8 M e V d e u t e r o n s , a n d W h i t t e n a n d M c I n t y r e 5,6) have m e a s u r e d the (p, d) r e a c t i o n s o n the C r i s o t o p e s u s i n g 17.5 M e V p r o t o n s . Because o f the n e g a t i v e Q-values, the levels c o u l d be s t u d i e d o n l y u p to 4 M e V excitat i o n energy. A s i d e f r o m y i e l d i n g i n f o r m a t i o n a b o u t h i g h e r excited levels ( p a r t i c u l a r l y the a n a l o g u e states) the (3He, ~) r e a c t i o n s e e m e d w o r t h w h i l e to be s t u d i e d because it k i n e m a t i c a l l y s t r o n g l y f a v o u r s l = 3 t r a n s i t i o n s , t h u s a l l o w i n g i d e n t i f i c a t i o n o f the f~ hole states, w h i c h are w e a k l y p o p u l a t e d in (p, d) a n d (d, t) reactions. I n the s i m p l e s t s h e l l - m o d e l picture, -~2Cr s h o u l d have a closed f : n e u t r o n shell in its g r o u n d state. T h e s t r o n g 52Cr(3He, c~) t r a n s i t i o n s , therefore, s h o u l d p o p u l a t e the l f ~ , l d l a n d 2s~_ hole states. T h e o b s e r v e d l = 3 levels c a n be c o m p a r e d with the t See ref. 1j for the 5°Cr(d, p) reaction data, ref. ~) for the 5z.5~Cr(d, p) reaction data and ref. s) for the 53Cr(d, p) reaction data. 47

48

P. DAVIDet

al.

calculations of McCullen, Bayman and Zamick v) (MBZ), who calculated the level scheme of 51Cr using the basis of pure f~ configurations. If the 52Cr ground state contains higher orbitals (2p_~, lf~ and 2p~), one should observe l - - l(2p) and 1 = 3(lf~) transitions in the pick-up reactions to levels known 1) from the reactions 5°Cr(d,p)51Cr. Whitten and McIntyre 5.6) have observed l = 1 transitions. The angular distributions, however, did not show well-pronounced diffraction patterns. In the 52Cr(3He,~) reaction, only one reasonably strong l = 1 transition is observed, which will be discussed in detail. But generally 5aCt seems to have a fairly closed neutron shell in its ground state, and consequently the lowest excited states in 52Cr are expected to have predominantly proton f~ configurations as assumed in calculations by Talmi 8). The 53Cr(3He, ~) reaction should allow a test of low-lying f~ levels of 52Cr with respect to neutron particle-hole components. However, the analysis is complicated by the fact that 53Cr does not have a pure 2p~ single-neutron configuration in its ground state. The close vicinity of the 2p~, lf~ and 2p÷ orbitals and the possibility of coupling the single neutron with excited states of the 52Cr core cause considerable configuration mixing 9-11). Therefore, weaker transitions to 52Cr may result from excited core components in the 53Cr ground state. The [f-~p~], [d-~p~] and [s-~ p~] states, however, should be strongly populated at higher excitation energies. The pick-up reactions from 54Cr will give information about the mixture of p~, p~ and f~: components in the 54Cr ground states, which may be compared with the shell-model calculations of McGrory 12). Of particular interest are the strong pick-up transitions populating the two-particle-one-hole states in 5aCt. These states are not taken into account by the unified-model 9) and shell-model ~0-~2) calculations of the 53Cr level scheme. Particularly, the collective -_+-level t 3) obtained from coupling the valence nucleon to the 2 + state in S2Cr should mix with the f;. (2p-lh states), which might cause interfering transition amplitudes. The 5°Cr(aHe, ~)49Cr reaction was investigated mainly because the level scheme of '*9Cr is not very well known at higher excitation energies where many analogue states can be observed with a large intensity due to the low isospin of the target nucleus. The lower excited states have recently been observed in the 5°Cr(p, d) reaction 5) and have been calculated by Ginocchio 15) with the assumption of pure f~ configurations and using an effective nucleon-nucleon interaction similar to the calculations of McCullen, Bayman and Zamick 7). However, since 49Cr is rather far from having a closed proton or neutron shell, it should also be reasonable to compare its level scheme with the Nilsson-model calculations of Malik and Scholz ~6). They also predict for positive deformation a low-lying #- state based on the Nilsson orbit 12. In the present study, levels have been investigated in '~9Cr up to 6.76 MeV, in 5~Cr u p t o 9.19 MeV, in SZCr up to 13.61 MeV and in 53Cr up to 12.59 MeV; among these are several analogue states, which correspond to low-lying levels of the vana-

Cr(aHe, ct) REACTIONS

49

dium isotopes. The experimental set-up and the results of the measurement is described in sect. 2. The D W B A analysis applied to extract/-values and spectroscopic factors from (3He, e) reactions is extensively described elsewhere 17). It is briefly summarized in sect. 3. The reactions for the different Cr isotopes are discussed separately in sect. 4. The analysis of tide analogue states is presented in sect. 5. Particular emphasis is placed on the bound-state treatment. The variable radius procedure as well as the more exact solution of the Lane equations are discussed as compared to the usual well-depth procedure *. In sect. 6, the Coulomb displacement energies are given for the C r - V isobaric pairs.

2. Experimental procedure and results The (3He, ct) reactions on the stable chromium isotopes were studied using the Heidelberg E N tandem van de Graaff accelerator at 18 MeV incident energy. The c~-particles were analysed by a Browne-Buechner broad-range magnetic spectrograph at reaction angles from 5 ~ to 35 °. Photographic emulsion plates were used as detectors. The 3He beam was focussed on the target without a collimator. The size of the beam spot was about 3 x 0.3 m m 2. The spectra were calibrated by the well-known energies of the low-lying levels of carbon and oxygen. The angular distributions of the lowlying levels of the isotopes 49'51'53Cr were measured independently in a scattering chamber using eight silicon surface-barrier detectors and covering the angles between 20 ° and 80 ° in steps of 5 °, with an overall resolution of about 60 keV. The relative normalization of the measurements was performed by determining the charge deposited in the Faraday cup as well as measuring tide charge generated in the target by knock-out of electrons. Additionally a monitor counter was mounted in the scattering chamber of the spectrograph. The absolute cross sections of the 52Cr(3H, c~) transitions were determined by comparison with elastic scattering. For this purpose, elastic 3He scattering was measured and compared with opticalmodel predictions at angles where Rutherford scattering is predominant (see fig. 1). The absolute cross sections are thus defined with an accuracy of + 2 5 o~,. The absolute cross-section scale for the other isotopes was calibrated with an accuracy of _+ 10 ° o relative to 52Cr by measuring the (3He, c~) reaction on a target of natural chromium, which contains all the Cr isotopes. This normalization was confirmed by the fact that the 52Cr(3He, e) ground-state transition could be observed in all chromium spectra, since all tide targets contained SZCr with a known percentage (see table 1). Self-supporting targets were prepared by vacuum evaporation of metallic chromium. The thickness of the Cr foil was about 100 to 150 llg/cm z. The thickness was determined from the change of the frequency of a quartz crystal during the evaporation. The Cr foils were floated and mounted on the target frames. D Following reL ~5), we prefer this name to "separation energy procedure", since the separation energy determines the asymptotic slope of the wave functions in each of the three different procedures.

50

et

P. DAVID

al.

These targets contained impurities and were later substituted by a new set of Cr

targets evaporated onto thin carbon foils. I ]

I

I

I

I

I

I

I

I

lO 10

52Cr(3He,3He)

01

01

10'

~,

001

-~

.

9.5 MeV

-]'.

01

10'

195MeV

--.~ 01

0001

001

10:

10

~li

10-1

305MeV

10_ 1

10-2

10-2

I0-3 0o

1 0 -3 I

=

[

40*

I

80*

I

I

120=

I

I

160 E)cM

Fig. 1. Elastic scattering cross sections for SHe a n d c¢-particles o n 5zCr, s3Cr a n d 5°Ti. T h e data have been t a k e n f r o m ref. ~8). T h e curves are fits using the potential parameters o f table 2.

TABLE 1 Isotopic e n r i c h m e n t o f the different c h r o m i u m targets 50Cr

50

52Cr

95.9 % ~ 0 . 1

52

3.76

0.05

53

0.26

0.05

54

0.05

5sCr

0.01

0.05

99.87 % ~ 0 . 0 2 0.12 0.01

"Cr

0.02

3.44

0.06 0.05

96.4 %::k0.1 0.18

0.02

1.43

0.02

0.59

0.01

97.98 ~ - b 0 . 0 4

Cr(aHe, :t) REACTIONS

51

DISTANCE A L O N G THE PLATE l~rn] 60

70

80

90

5 ° O r (3He,(:x)49Cr 'O0

5OO

E 31.te = 18 MeV 7.5° , 2me

E ,4OO

j

7

if')

200 npV5

100

~iCfooo

4

51Cr2323

0

0

I

2

3

4

5 6 7 8 9 EXCITATION ENERGY Ev~e~

10

Fig. 2a. S p e c t r u m o f 0r-particles f r o m the 50Cr(3He, ~ ) ~ C r reaction m e a s u r e d in the s p e c t r o g r a p h at 0ja b = 7.5 ° a n d incident SHe energy o f 18 MeV. T h e ~t-groups for which a n g u l a r distributions have been obtained are labelled. T h e s a m e n u m b e r s are used to identify the states in 4aCr listed in table 3a. T h e 4aCt levels labelled A I to A 4 are interpreted as the a n a l o g u e s to the p r o t o n - h o l e states in 4aV. T h e s - g r o u p s due to ~zCr, 160 a n d 12C impurities are labelled n~Cro.00/2.32 . - . , 1 6 0 0 / 1 / 2 / . . . and 1~Coo/1/2/... respectively. I m p u r i t y peaks due to other c o n t a m i n a t i o n s in the target are labelled b~ atl arrow indicating the kinematical shift relative to the c h r o m i u m peaks at different reaction angles

DISTANCE A L O N G THE PLATE l~m] 20

30

40

50

60

70

80

I

I

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I

I

52Cr(H~ot)51Cr (M

E E OJ qp-

)<

E3He- 18 MeV

l

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7.5", 0 . 3 m e

iil,o°

l,co

~02

"C2

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200

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100

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1

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I

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I

I

J

[

O,

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2

3

4

5

6

7

8

EXCITATION E N E R G Y Fig. 2b. S p e c t r u m o f co-particles f r o m t h e ~Cr(aHe, ct)6~Cr reaction.

~vleV_7

52

P. DAVID et aL

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Or(SHe, ~) REACTIONS

53

5 ° C r ( 3He,or )49CrE3He" 18MeV I I I I I I I I I I I I I / I1

Ol

~

@

+I

E~=0.00

C ' ~ , " =3)

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Fig. 3a. Angular distributions for the s-groups of the 60Cr(~He, ~)a9Cr reaction. The circles are the experimental data the vertical bars are the statistical errors and the solid curves the local zero-range spin-orbit D W B A predictions assuming the indicated /-values. The numbers labelling the various angular distributions are the level numbers given in table 2a.

54

P. DAVIDet

al.

3. The D W B A calculations An extensive description of the D W B A theory for the (3He, e) reactions considered laere has been given elsewhere ~7). We confine ourselves to a short outline of the metbods. The elastic scattering of e-particles on 52'53Cr at 19.5 MeV and on 5°Ti at 30.5 MeV as well as 3He E3~=18Me'~ scattering from 52Cr at 19.5 MeV were analysed by means of the optical-model search codes H U N T E R i_ i i i i : and JIB3 written by Drisko and Perey. Eight sets of lo @ parameters were found for 3He and seven for the a-elastic scattering. They differred in the number of parameters used as well as in the depths of the real part of potentials which ranged from about 40 to 200 MeV. 1O: ~, In order to find acceptable fits to the angular distri~k Ex=191 butions of the first five transitions in 54Cr(aHe, e), it 01 f =~ proved most important that the e-potential was close to the sum of the 3He and the (bound) neutron po1 \: 1 i 01 tentials t. To achieve that, slightly unusual parameters rO , G had to be used in both the e- and 3He-channels. The ..Q six-parameter volume absorptive 3He potential had a fairly large real radius (r o = 1.36 fro) and the imaginary potential further extended (r~ = 1.76 fro). In the e-channel, six parameters were necessary as well, contrary to elastic scattering which can very well be described with only four parameters. The sets of 3 He and a-parameters which gave the relatively best fit to 54Cr (3He, c~)S3Cr are listed in table 2, the corresponding fits to elastic scattering are shown in fig. 1. The code JUL1E was used for the D W B A calculations which employed the local zero-range form of the D W B A theory. Spin-orbit coupling was included. As a consequence, the calculations for l = 1, J = ½, :} transitions with Q ~> 6 MeV showed a J-dependence which is, however, not in very good agreement with the experimental data. For higher/-values, the predicted J-dependence was rather small. Calculations including finite-range and/or non-locality corrections led to inferior fits than those that were obtained with a local ~o~ 20. 30* 40* 50*@ct4

\

~=ooo

I

&

3b. Angular distributions from the 5-'Cr(3He,c~)6~Crreaction.

t A similar potential sum rule seems to determine the real part of the 3He potential in (d, 3He) reactions 1~).

Cr(aHe, ~) REACTIONS

55

53Cr (3He, o~)52Cr E3He=18Mev 1C

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10

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4o-0,:.

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01

o.

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Fig. 3c. Angular distributions from the 53Cr(aHe, c0L'Cr reaction.

56

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9. DAVID

1 -

[

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;

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t

al.

~

i

[

I

5 4 C r ( 3He,or ) 5 3 C r E3He : 18 MeV

~

Ex

,

: 0

~

,~'

',4,

,,..,,

,

Ex :

N

01

5 8 MeV

~

°

T Ex=102

~,~t

MeV

'~

-:: 0 1

i

[!

-

0

~"~ " ~ "

~., ~

~'

Ex

: 130 M e V

z;3½-

MeV

10 20 30 4'0 50 60 7'0 80 Oc,

Fig. 3d. Angular distributions for the first five transitions observed in the "C[(aHe, 0¢)~C[ reaction.

zero-range D W B A . N o radial cut-off was used. For Q-values above 6 MeV, the results of the calculations turned out to be unusually strongly dependent on the choice of the cut-off radius, particularly for radii close to the potential surface. These calculations were also very sensitive to finite-range and non-locality corrections. The D W B A results for the first five transitions observed in S4Cr(3He, ~) are compared with the experimental data in fig. 3d. The fits are not very good as far as the detailed structure of the data is concerned; however, they represent the optimum among all the 8 x 7 combinations of 3He and c(-scattering potentials tested in ref. ~7). The difficulty to find good fits even with "measured parameters" is mainly attributed to

Cr(81-le, o0 REACTIONS

57

54Cr(3He, cd53Cr E3He=18 MeV 10 20 30 40 ~, @CM

lI~l. 10 20 I 30 I 40I

GCM

1 O1

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I~.I, ~0(~)

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(~) E.:399 ~ ~ 1

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1

~

E,.lo6~

1 E t //~ '(~) Ex=1253

01F,
OOll 0

(~)

o]

Ex =

I I I t Ol 10 20 30 ,40@CM

)I

Ol

I 1 I 10 20 30 40

50 @cM

Fig. 3e. A n g u l a r d i s t r i b u t i o n s from the 54Cr(SHe, ~)6sCr re a c t i on at h i g h e r e xc i t a t i on energies.

58

P. DAVID et

al.

the mismatch of angular m o m e n t a at the high Q-values (Q > 9 MeV) of the transitions considered here. The bound state of the transferred neutron was calculated in the usual well-depth procedure, which is justified only in the case of pure singleparticle (or hole) states. If these states are split into several fragments, a coupledchannel calculation becomes necessary. We have performed such coupled-channel calculations for the case of isospin splitting. However, with the assumption of a TABLE 2 Optical-model potentials used for the DWBA calculations

aHe

V 142.4 183.7

WVol. 12.67 26.0

bound neutron (WD)

r 1.362 1.4 1.25

re 1.4 1.4 1.25

a 0.65 0.56 0.65

rw 1.755 1.48

aw 0.781 0.56

Vso 8.0 2 = 25

The parameters are not best-fit parameters to elastic scattering but were modified to improve the 5~Cr(3He, ~) fits. The radius and diffuseness parameters are given in fm and the potential depths in MeV. surface-peaked interaction potential, the coupled-channel treatment of the bound state can approximately be substituted by the "variable bound-state radius procedure" 32). This approximate method yields almost the predicted ratios of spectroscopic factors for analogue and configuration states and is discussed in sect. 5. The absolute normalization constant [in the notation of ref. 17)] N =

2Sa+ 1 (2Sb+ 1) ( 2 S + 1)

2ra2D2o,

of the D W B A cross section was determined by normalizing the T<, f~ strength observed in SECr(3He, 7) to the sum-rule limit. The resulting value N = 25_+25 3o is in rather close agreement with the value 18 < N < 25 found by Bassel 19) in a theoretical calculation.

4. Discussion of the results

4. l. THE REACTION 5°Cr(3He, 0~)'XSCr Fig. 2a shows an c~-spectrum obtained at 7.5L The a'9Cr levels were observed up to 7 MeV of excitation. Beyond 7 MeV, no strong transition to 49Cr could be identified. However, we cannot exclude the existence of levels in this region because of the difficulty of discriminating against the strong impurity transitions from t2C and 160. The levels below 4.76 MeV are T<(T = ½) states, whereas the levels beyond 4.76 MeV assigned with a capital A are interpreted as the analogues to the 49V levels. Fig. 3a shows the angular distributions and the D W B A predictions. In some cases, the /-assignment is only tentative. They are given in parentheses. There is only one strong transition. This transition populates the 0.27 MeV state, which is the expected f~ level.

Cr(aHe, a) REACTIONS

59

The weak excitation of the ground state is consistent with the identification J " = 5 The angular distribution to the ground state shows a marked difference in structure when compared to the 0.27 MeV 5 - level. Similar differences have been found 2t) in the 44Ca(3He, ~) reaction populating the 0.38 MeV ~ - level and the 5 - ground state i n 4 3 C a . Table 3a lists the energies,/-assignments and spectroscopic factors for 4 9 C r . The sums of the spectroscopic factors for the s¢, d~ and f÷ configuration states are given in table 4. Between the dominant 0.27 MeV level und the first analogue state at 4.76 MeV, we observe three stronger levels at 1.98, 2.58 and 2.51 MeV with l = 2, I --- 0, and l = 3, respectively. The relatively strong 1 = 3 level at 3.51 MeV should have the spin 5 similar to the l = 3 level, which has been observed in StCr at 2.32 MeV. The 49Cr spectrum, of course, is much more complex than the 5tCr spectrum. Taking into account that both the proton and neutron shells are rather far from being closed, one might try to interpret the 4 9 C r spectrum on the basic of the Nilsson model. I f both 4 9 C r and 5°Cr were deformed (Malik and Scholz 16)assumed positive deformation with fl = 0.24, tc = 0.13, which corresponds to q ~ 1.75), one would expect three 1 = 3 levels based on the orbits 12 (K = ~:-), 13 (K = ~?-) and 14(K = k - ) , which might be correlated to the observed 1 = 3 levels Nos. 1, 9 and 10. TABLE 3a S u m m a r y o f results for the levels in 4aCr

Level no. 0 1 2 3 4 5 6 7 8 9 10 11 A1 12 A2 13 A3 A4

E~(MeV) a) 0.00 0.27 1.07 1.57 1.74 1,98 2.43 2,58 2.61 3.24 3.5l 3.93 4.76 5.19 5.57 6.43 6.47 6.76

Ex(MeV) b) 0.00 0.27 1.08 1.57 1.76 1.97

2.63

I

d=

C"S

a) (3) 3 0 (1) (1) 2 (2) 0 1 3 3 (1) 3 (2) 2 (0) 0 0

~ ~+

~,+

-.7,?,+ k+ ½+

0.19 3.8 0.2 0.05 0.1 2.0 0.18 1.3 0.13 0.22 0.57 0.06 1.7 0.32 2.1 0.3 {).64 0.18

50Cr(t ' ~)~0 V c)

l

C'-S

0.00

3

2.96

0.75

2

2.7

1.65

0

1.54

2.00

0

0.12

Ex(MeV)

T h e errors in excitation energies are ± 20 keV. The n u m b e r s labelling the states observed in the present e x p e r i m e n t are the level n u m b e r s labelling the ~-groups in the s p e c t r u m a n d the c o r r e s p o n d i n g a n g u l a r distributions. T h e states labelled AI to A 4 are interpreted as a n a l o g u e states. T h e p a r e n t spectroscopic factors obtained f r o m the 50Cr(t, 00~°V reaction 2_,) are given for c o m p a r i s o n . a) Present work. h) Ref. 5). e) Ref. ~,2).

60

P. DAVID et al. TABLE 3b S u m m a r y o f results for the level in 6~Cr

Level no. 0

E~(MeV) ~) 0.00

1.35 1

1.91

2 3

2.32 2.77

4 AI A2

3.02 6.63 9.19

Ex(MeV) b)

l

0.00 0.748 0.775 1.159 1.346 1.476 1.552 1.895 1.998 2.313 2.761 2.825 (3.016)

3 1 1

Jn

(2J+I)S (d, p)

C2S

C~S

CzS

a)

(p, d)

predicted

~.~½-

5.6

4.7

3

(.~-)

(0.05)

0.79

3 1

(~-) ~-

0.2

0.85 0.64

3 0 0 2 3

~½+ ½+ ~+ ½-

1.6 1.6

1.8--2.0 1.3

1.8 1.8

1.5--2.7

5.39

1.38

1.80 1.45 0.56

0.10 0.03 0.01

0.80

The errors in excitation energies are +20 keV. The levels observed in the 5°Cr(d, p) reaction l) are not listed to higher excitation energies because it becomes too difficult to decide whether these (stripping) levels are identical with those observed in the pick-up reaction. The spin assignments ~- and ½- for the 1.35 MeV and 1.55 MeV levels, respectively, are from ref. o.a).The ~- assignment to the 1.55 MeV state seems unlikely in view of the observed strengths in stripping and pick-up. a) Present work. b) Ref. ~). e) Ref. 5). a) Ref. 2~). Alternatively, the l = 1 transitions at E x = 1.57, 1.74, 2.61 a n d 3.93 M e V m a y b e t a k e n as a n i n d i c a t i o n for particle-hole excitation in the 5°Cr g r o u n d state. T h e s e might be represented by a d m i x t u r e s o f the f o r m d-2f6_2~ kP~, '~'~-2f87/8'd -a4- 2f ~i p ~ a n d f~p~. T h e first a n d third c o m p o n e n t m a y be c o n s i d e r e d in c o n n e c t i o n with o b s e r v e d 1) l = 2 transitions in the 5°Cr(d, p)SXCr r e a c t i o n at Ex = 4 MeV. A l l b u t the second c o m p o n e n t c o u l d give rise to l --- 1 t r a n s i t i o n s in the n e u t r o n p i c k u p reaction. Fig. 4 shows a c o m p a r i s o n o f the o b s e r v e d level scheme with N i l s s o n - m o d e l calculations o f M a l i k a n d Scholz 16) a n d shell-model calculations o f G i n o c c h i o 15). T h e e x p e r i m e n t a l l = 0 a n d l = 2 levels are represented b y d o t t e d lines, since hole states were not included in the calculations. M a l i k a n d Scholz seem to predict a s o m e w h a t t o o c o m p r e s s e d level scheme, a l t h o u g h we c o u l d have easily missed the " p i c k - u p f o r b i d d e n " high-spin states in this experiment. G i n o c c h i o also o b t a i n s the inversion o f the ~ - a n d ~ - states a n d predicts several low-spin states with negative p a r i t y which, in his calculation, are p u r e f~ levels. It s h o u l d be interesting to study the effects o f m i x i n g these f~ states with states c o n t a i n i n g 2p a n d lf~ c o m p o n e n t s . T h e excitation energies a n d spectroscopic factors o f the o b s e r v e d a n a l o g u e states are given in table 2a with the results o f the S°Cr(t, t~)49V r e a c t i o n 22) which excites the p a r e n t states. T h e suggested spin assignments for these highly excited states in 49Cr are b a s e d on their i n t e r p r e t a t i o n as a n a l o g u e states. T h e a n a l o g u e a n d p a r e n t spectroscopic factors are not strictly p r o p o r t i o n a l . T h e s~ states present a p a r t i c u l a r

Cr(aHe, z) REACTIONS

61

TABLE 3C Summary of results for the levels in 5-'Cr Level no.

E~(MeV) ~)

E~(MeV) b)

0 1

0.00 1.43

0.00 1.434

2

2.38

3

2.78

2.369 2.648 2.769

4 5

3.43 3.48

6

3.78

7 doublet

4.03

9 10 11

4.62 4.83 5.40

13 14 15

5.67 5.71 6.18

16 17

6.49 6.70

18 AI A2

6.79 11.29 13.61

2.965 3.112 3.161 3.432 3.494 3.614 3 767 3.926 4.03

l

J~

l I 3 (3) 1 (3) (~)

(l) 3 3 (3) 3 3

(3) (1)

3 (1) (3) 0 2 (~) (3) 0

0÷ 2* 4+ 0 ~4+ 7+

C"-S

C2S

~)

b)

0.12 0.09 0.11 0.085

6÷ 2+ 1.3 ] 0.961 2+

0.26 1.0

0.51 0.18 0.20 0.07 0.018 0. l0 < 0.008 < 0.08

C~S

~) 0.39 0.15 0.10 0.13 0.01 0.12

0.15 2.3

1.45

< 0.04 0.36

0.31

1.14

0.79

0.079 0.12 0.09 0.054 0.72 0.22 0.25 0.41 0.38 0.10 0.09 1.8 0.39

The errors in excitation energies are 4-20 keV. The /-assignments given in brackets are based on known spin values of the 5-"Cr levels 6). The/-values given in parentheses are tentative. a) Present work. b) Ref. 6). c) Ref. a). difficulty b e c a u s e a s i n g l e s t r o n g I = 0 level c o r r e s p o n d i n g

to the parent state at

1.67 M e V i n a 9 V h a s n o t b e e n o b s e r v e d . I n s t e a d , a d o u b l e t is f o u n d a t 6.43 a n d 6.47 M e V ; w h i c h p r o b a b l y is c o m p o s e d

of l = 0 transitions. The 1 = 0 assignment

t o t h e 6.43 M e V level is u n c e r t a i n , b e c a u s e t h e c h a r a c t e r i s t i c s e c o n d m a x i m u m

at

20 ~ c o u l d n o t b e o b s e r v e d , s i n c e it w a s o b s c u r e d b y t h e 1 2 C ( a H e , ct)llCg.s" i m p u r i t y peak. 4.2. THE R E A C T I O N z2Cr(aHe, ~)51Cr Fig. 2 b s h o w s a s p e c t r u m o b t a i n e d a t 7.5 °. Fig. 3 b s h o w s t h e a n g u l a r d i s t r i b u t i o n and the DWBA

c a l c u l a t i o n s . T a b l e 3 b lists t h e s p e c t r o s c o p i c i n f o r m a t i o n . T h e s p e c -

62

P. DAVIDet al. TABLE 3d SummaryV,of results for the levels in saCr

Level no. 0 1 2 3 4

Ex(MeV) 4) 0.00 0.58 1.02 1.30 1.55

covered by s~Cr(3He, ~)

6

2•79

7

3.37

8

3.43

9

3.60

10

3.71

11 12 13 14 15 16 17 18 A1 A2 A3

3.84 3.99 4.20 4.34 4.43 4.54 4.71 5.55 10.65 12•52 12.59

Ex(MeV) b)

1

0.00 0.564 1.007 1•286 1•531 1•963 2•158 2.319

1 1 3 3 3

2.439 2.646 2.761 2.808 3.339 3.420 3.587 3.610 3.700

3.971 4.213 too many levels no correlation possible

jn

~½~{½(3-)

1

(3-)

3 3

~-

3

~-

C~S

C~S

~)

(p, d) ~)

predicted d)

0.83 0.31 0•51 0.70 3.2

1.45 0.30 0.22

1.1 0.26 0.49 0•45 3.0

( 2 J + I)S (d,p) b) 2.5 0.86 2.5 (0.44)

0.01 0.02

0•002

1.3

0•08

0•005

0.9

0•08 0.93

1.2

3

0.37

0.30

1

0.15

0.12

4

0.14 l = 4 0.09 l = 3 0.09 0.06 0.22 0.11 1.1 0.15 0.96 0.18 1.3 0•39 1.13

1.4

1 2 3 0 3 2 3 3 0 2

C~S

½+ 3+ ½½÷ ~+

The errors in excitation energies are i 20 keV. a) Present work• b) Ref. z). e) Ref. e).

0 06~ • ~l = 1 0.86j

0.85

a) Ref. 12).

t r o s c o p i c f a c t o r s h a v e b e e n n o r m a l i z e d t o t h e T< s u m - r u l e f÷ s t r e n g t h C 2 S = 7.2 d e f i n i n g t h e n o r m a l i z a t i o n c o n s t a n t N as a l r e a d y d i s c u s s e d in sect. 3. W e o b s e r v e t h r e e r e l a t i v e l y s t r o n g 1--- 3 levels. N a i v e l y o n e w o u l d e x p e c t o n l y t w o l = 3 levels, i.e. t h e T = { g r o u n d state a n d the a n a l o g u e T = { state at 6.6 M e V . If, h o w e v e r , the f ~ 1 n e u t r o n - h o l e state w e r e m i x e d w i t h h i g h e r - s e n o r i t y f~ c o n f i g u r a t i o n s 7), we s h o u l d o b s e r v e m o r e l = 3 levels. I f the f~ n e u t r o n shell w e r e n o t c l o s e d in the 52Cr g r o u n d state o n e m i g h t e v e n p o p u l a t e f~- 3 p~2 states• T h e l = 3 levels at 0.0, 2.32 a n d 6.63 M e V h a v e also b e e n f o u n d in t h e (p, d) e x p e r i m e n t s 6.2 3). T h e r e is a g r o u p o f w e a k levels b e t w e e n 5 a n d 6 M e V , w h i c h was n o t a n a l y s e d in t h e p r e s e n t study• A m o n g

those, 7 -

states m i g h t be p r e s e n t , b u t

e a c h w o u l d h a v e a s t r e n g t h o f less t h a n 5 °o o f the g r o u n d state strength• T h u s , t h e

Cr(3He, ~) REACTIONS

63

TABLE4 Centre-of-gravity energies E e.g. and summed strengths C2S for the T< and T> states of the fk, d{and s½ orbits in 63Cr, 5xCr and 49Cr Target Orbit T< E c'g" Eeb "g" r o T> (MeV) (MeV) (VR)

64Cr

~2Cr

If~_

{

ld~r

~

2s~r

~

lf~_

~ ~~]

Id~ 2s~ 1~

s°Cr

ldk 2s~r

~ ½ ~ ½ ~ ½ ~

2.1 10.65 4.8 12.6 4.4 12.5

11.82 20.37 14.52 22.32 14.12 22.22

0.51 12.55 6.63 18.67 3.02 15.06 not observed 2.77 14.81 (9.19) 0.81 4.76 2.4 5.57 2.5 6.5

13.74 17.69 15.33 18.50 15.43 19.43

C2S

C2S

(WD) (VR)

C2S

C2S

(CC)

sum rule

1.22 1.44 !.13 1.31 1.10 1.32

5.4 6.3 1.3 0.39 1.2 2.1 1.0 0.69 1.1 1.8 0 . 3 9 0.27

6.5 0.56 1.3 0.47 1.3 0.17

7.43 0.57 3.43 0.57 1.71 0.29

1.25

7.2

7.2

7.2

7.2

1.40 1.16

1.8 1.8

0.78 2.6

1.13

1.6

2.3

0 . 7 9 0.80 1.8 3.2 0.80 1.7 1.6 0.40

1.30 4.6 3.5 1.40 1.7 0.70 1.18 2.2 3.l 1.25 2.1 2.1 1,16 1.5 1.9 1.28 0 . 8 2 0.69

3.4 4.67 0.85 1.33 1.9 2.67 1.1 1.33 1.15 1.34 0 . 4 7 0.66

R

R

(WD) (VR)

R (CC)

R

sum rule

0.24

0.062 0.086 0.077

0.83

0.33

0.35

0 . 1 5 0 . 1 3 0.166

0.25

0.11

0.36

0.11

0.166

0.11 0.25 0.25

0.38

0.20

0.25

0.285

0.95

0.63

0 . 5 8 0.50

0.55

0.36

0.41

0.50

The labelling WD, VR, CC and sum rule stands for well-depth, variable-radius and coupled-channel procedure and sum-rule limit, respectively. The strengths are normalized to the sum-rule limit C2S ~ 7.2 of the f.kT< strength in 52Cr. The radii ro(fm) are the values obtained in the variableradius procedure for a fixed bound-state potential depth of 52.6 MeV. The values R in the last column are the ratios of T> to T< strengths.

experimental results agree with the f~ ( M B Z ) calculations 7) which predict three T --~:, ~ - states, the available T = ~, 1 = 3 strength being shared essentially between two levels: the g r o u n d state (cZSMBz = 5.39) a n d a level predicted at 2.62 MeV (C2SuBz = 1.38). The latter m a y be identified with the observed 2.32 MeV level ( C 2 S e x p = 1.6). The spectroscopic strength o f the a n a l o g u e T = { state at 6.63 MeV is twice the sumrule value. This p r o b l e m is discussed further in sect. 5. T h e 1 = 0 level at 2.77 MeV a n d the l = 2, d~ level at 3.02 MeV both c o n t a i n a large fraction of the full T = 3 strength. These levels were only weakly excited in the 5°Cr(d, p)5~Cr reaction 1), thus confirming their p r e d o m i n a n t 2s a n d Id hole character. O f particular interest is the excitation o f the { - , ~ - a n d ½- levels. Their strength provides a measure for the f~, p~ a n d p~ admixtures to the 52Cr g r o u n d state. The only { - level k n o w n in 5~Cr is the 1.35 MeV level. It was assigned by Alty et al. 24) in a J - d e p e n d e n c e analysis o f (d, p) 1 = 3 a n g u l a r distributions. The 1.35 MeV level is only weakly excited in the (3He, ~) reaction. A l t h o u g h a full a n g u l a r

64

P. DAVID et al.

distribution was not obtained, the spectroscopic factor could be estimated from a few measured angles (C2S = 0.05). The l = 1 level at 1.91 MeV is relatively strongly excited (CES= 0.2). This pstrength is of similar magnitude as it has been observed in the 48Ca(d, t)47Ca and 48Ca(p, d)*TCa reactions z6). The spin of the 1.91 MeV level is probably 3 - ac-

5 E×

Levels

in 49Cr

E~ev~ 4

-

-

I=1

- - l = 3 - - 1 = 3

-

-

-

-

9/2" 1/2-

-

-

13/2"..,^--5/2-

........

I=l 1/2 ÷

-

i=3

_

_ -

........

7/2-

_ -

_ -

5/23/2-

-

-

W2-

3/2 * -

-

1/2"

[=1 t=l

-

11/2-.

-

9/29/211123/2-

- - 7 / 2 - - 3 / 2 .......

]J/z

1/2 ÷ -

-

9/2-

-

-

7/2-

7/20

exp (3He,CX)

5/2- -

-

5/2"

M a h k & Sclnolz

7•2-

5/2-

--

Glnocchlo

Fig. 4. The level scheme of 49Cr compared with calculations of Malik and Scholz 16) and of Ginocchio ]5). The positive-parity levels are given as dotted lines, since they were not included in the calculations. cording to 7-7 correlation work 27). B u t regardless whether the spin is ½- or 3 - , it is worth noting that other l --- 1 levels in 51Cr, particularly the 3 - , ½- doublet at 0.8 MeV, are much more strongly excited in the 5°Cr(d, p)SlCr reaction and less strongly populated in the pick-up reaction than the 1.91 MeV level. The same behaviour has been observed in the (p, d) reaction 6). This excludes the assumption of a pure seniority zero f~2p2 neutron admixture in the S2Cr ground state, which would allow to populate the same f ~ / p states in the pick-up reaction as seen in the 5°Cr (d, p)51Cr stripping reaction. However, every admixture of p~ strength in the 52Cr ground state in addition to f -i 2 p 2 could explain the observed enhancement of the 1.91 MeV level in the pick-up reaction. If the states involved in stripping and pick-up result from strong configuration mixing, the relative strengths of states of the same spin and parity may be very different in the different reactions. This is due to inter-

Cr(SHe, ~) REACTIONS

65

ference 39). In our case, the experimental data do not allow a further specification of the p~ admixtures. The analogue of the 5aV ground state is found at 6.63 MeV. It has an l = 3 distribution. There is another strong level at 9.19 MeV. but no angular distribution has been obtained for this level. Glover and Jones zs) observed strong l = 0 and l = 2 states in the 5ZCr(t, :~)5tV reaction at 2.545 and 2.674 MeV, respectively. These parent states correspond to analogue states in 5aCr at 9.18 and 9.30 MeV. In the present experiment, we do not observe levels beyond 9.19 MeV. At lower energies, a missing level might be obscured by impurity lines, but it would be at least 200 keV apart from the 9.19 MeV level. Differences in the s~ and d3 Coulomb energy displacement could account for this shift.

4.3. THE REACTION 53Cr(SHe,0c)n2Cr Fig. 2c shows a spectrum of the 53Cr(3He, ~)52Cr reaction at 7.5 °. Fig. 3c shows the angular distributions together with the D W B A predictions. The ground-state distribution is missing, because the magnetic field of the spectrograph was set too low in most of the exposures. The excitation energies,/-values and spectroscopic factors are given in table 3c. The low-lying levels in S2Cr are very weakly excited. This results partly from the m o m e n t u m mismatch inhibiting l = I transitions. The l = 3 transitions, however, are not kinematically reduced, and therefore they must be forbidden by the structure of the states involved in the reaction. Tbis conclusion is consistent with the interpretation of the low-lying 52Cr states as fairly pure f~ proton configurations with a predominantly closed neutron shell. The summed / = 3 strength of the 1.43 MeV 2 +, the 2.38 MeV 4 + and the 2.78 MeV 4 + levels is less than C2S = 0.3. One may attribute this strength to small [2 + f~]~_ and [4 + f~]~_ components in the 53Cr ground states. However f~ ip~ neutron admixtures in the wave functions of the S2Cr levels would also allow these l = 3 transitions. F r o m the observed l = 3 strength, one obtains an upper limit of 14 % of f~-~p~_ admixtures to these 4 + levels in SZCr. The [2+p~]~_ component of the 53Cr ground state wave function can be obtained from the l = 1 part of the transition to the 1.43 MeV 2 + level in 5ZCr. The spectroscopic factor obtained from the present experiment, i.e. CzSt=I = 0.12, is only slightly lower than obtained from (p, d) [ref. 6)] and (d, t) [ref. 4)] with C2S = 0.18 and 0.15, respectively. These values agree with R a m a v a t a r a m ' s unified-model calculations 9) predicting cZs = 0.16. The strong l = 3 transitions are found between 3.43 MeV and 4.03 MeV. The corresponding Q-values are close to the Q-value of the 52Cr(3He, c~)51Cr groundstate transition. The centre of gravity of the observed f~-~p~ levels in 52Cr is approximately given by Q[(f~-1 p~)o.~.] ~- Q[52Cr(aHe, ~)51Cr,_..s ] + 0 . 5 MeV.

66

P. DAVID et al.

The strongest l = 3 levels at 3.43 and 3.48 MeV may be identified as the high-spin members of the f~- ~p~ quadruplet. Their spectroscopic factors agree with the expected value of Ss -

2J+l S [52Cr(3He, ~)sa Crg..~.]. Z (2Jq- 1)

(1)

However, taking into account the (p, p~,) coincidence work of Monahan et al. 4°) and trying to maintain the validity of eq. (1) one would alternati~,ely interpret the l = 3 states in the following way. The observed 3.43 MeV level would correspond to the 3.413 MeV (4 +) level and the 3.48 MeV state would correspond to the 3.4695 MeV (3 +) and 3.4725 MeV (2 + ) doublet. The missing strength of the 2 +, 3 + doublet could be accounted for by the strength observed in the 3.78 MeV (2 +) level 29). The 4.03 MeV state would consequently be interpreted as 5 + state. One expects at least a second quadruplet of f~ lp~ states at about 5.9 MeV corresponding to the 2.38 MeV ~- level in 51Cr. In fact, there is a stronger l -- 3 level ( C 2 S = 0.72) at 5.71 MeV. According to the positions of the 2s and ld~ hole states in ~lCr, we expect the s~ lp.~ doublet at 2.4 MeV and the d~ ~p~ quadruplet at 6.6 MeV with spins 1-, 2and 0 - , 1-, 2-, 3-, respectively. The 1- and 2- levels may have admixtures of both components. Experimentally, only one l = 0 level at 6.49 MeV and one l -- 2 level at 6.70 MeV could be unambiguously identified. However, there are several levels, e.g. at 4.62, 5.67, 6.19 and 6.79 MeV, the angular distributions of which could be composed of either (l = 0 and 1 = 2) or (l -- 1 and l = 3). At 11.29 MeV and 13.61 MeV, two fairly broad peaks were observed; they are interpreted as analogue states to 52V. The angular distribution of the 11.29 MeV level indicates an l = 3 transition, although l = 2 cannot be excluded. The 13.61 MeV level definitely shows an l = 0 distribution. The 11.29 MeV (l = 3) level in 5ZCr should correspond to the ground-state triplet in 52V. According to the calculations of Vervier '~) and Gersch et al. ao), these 0.0 MeV 3 +, 0.017 MeV 2 + and 0.023 MeV 5 + levels in 52V all contain a strong (Tr(f;)3, vp~:] component. We expect a particularly strong transition to the 5 + member of this triplet, but, we do not resolve it from the 2 + and 3 + levels. The present data show only one reasonably strong l = 3 state for these high excitation energies in 52Cr and are consistent with the calculations of Gersch et al. The spectroscopic factor, however, shows the wellknown deviation 23), the experimental value is more than two times larger than the sum-rule limit C2S -- 0.67 (see sect. 5). The 13.61 MeV level shows an l = 0 distribution, it should be the analogue to the s~_ proton-hole state, which unfortunately is not known from the (t, ~) reaction. 4.4. T H E R E A C T I O N 54Cr(3He, 0&3Cr

Fig. 2d present a spectrum of the 54Cr(3He, ~)53Cr reaction at 15 ~. The angular distributions are given in figs. 3d and e with the DWBA predictions. The spectroscopic information is listed in table 3d. Since neutron pick-up reactions from 54Cr

Cr(aHe, :~) REACTIONS

67

have already been investigated in previous experiments 4,6.14), only a few aspects of the 54Cr(3He, ~)53Cr reaction will be emphasized. The l = 3 angular distributions corresponding to the 1.02 MeV 2- state and the 1.30 and 1.55 MeV 7- levels reveal a spin dependence. The ~- distributions are out of phase with respect to the ~2 distribution between 40 ° and 70L This antiphase correlation could not be reproduced by the spin-orbit coupling DWBA calculations. Also for 1 = 1, the DWBA curves do not reproduce the differences in the structure of the angular distributions to the 23-- ground state and the 0.58 MeV ½- level. The ratio of the spectroscopic factors of the 1.30 MeV 4 - level relatively to the strong 1.55 ~- level obtained 1¢) in (3He, ~) differs from the (p, d) and (d, t) results 4.6). The 1.30 MeV ~-- level is known to have a [2+p~]3, collective configuration. The discrepancies for the different reactions may be attributed to an interference of the direct f~ transition amplitude ~ith a two-step excitation mechanism ~3). A method often used for spin identification is based on the complementarity of stripping and pick-up reactions exciting particle and hole states, respectively. Destructive or constructive interference, however, limits the applicability of this method in the case of strong configuration mixing. Tberelbre we suggest spin assignments only for those states which are strongly populated in one of the reactions. These spins are given in table 3d. For weaker l = 3 levels in (3He, ~), which are not strongly excited in the stripping reaction, a ~- assignment seems to be most likely. Even with these -~- assignments, the total observed ~- strength C2S~,p = 5.3 for the T< states is smaller than the sum-rule limit C2S = 7.43. We return to the question of lack of strength in sect. 5. The summed p~, p+ and f~_strength of the lowest three levels in 53Cr yields C2S = 1.85. This value is close to the shell-model estimate of two nucleons outside the fl shell. But, according to the stripping reaction, these states do not contain the full single-particle strength. Therefore, it was concluded from the pick-up reaction in ref. ~3) that the effective number of p~_, p~ and f~ nucleons in the S4Cr ground state would exceed the shell-model limit. In the present experiment, however, we do not find p~, p} and f~_ strength in higher excited levels, although there are several l = 1 and l = 3 ~-- levels in this region, u hich are strongly excited in the stripping reaction. A possible explanation would be a (proton) core excited component in the S¢Cr ground state. This would yield a second transition amplitude in addition to the direct amplitude with constructive interference for the low-lying levels in s 3Cr and cancellation for the higher-excited ~-, 3- and ~- states. McGrory's shell-model calculations ~e) predict that the pick-up strength obtained from the valence nucleons should be almost completely collected in the lowest 3 ~-- and ~- states (see table 3d). These calculations also yield comparatively large spectroscopic factors for the higher excited states in the 5ZCr(d, p)S 3Cr reaction, for example the value S = 0.12 for the 2.32 MeV 3 - level. However, the experimental differences between pick-up and stripping are still larger. At 10.65 MeV, an I = 3 level has been observed which is identified as the analogue

68

P. D~kVID e t

al.

to the s 3V ground state. From the StV(t, p)53 V reaction, investigated by Hinds et al. 31), it can be concluded that the 53V ground state has the spin ½-. Two more levels have been observed in (3He. ~) at 12.52 and 12.59 MeV with l = 0 and l = 2, respectively. These should be the analogue states of the sf and d~ proton-hole states of 53V. However, there are no proton pick-up data available for comparison. 5. Spectroscopic factors for the T< and T> centres of gravity and sum-rule limits Under consideration in this section is the sharing of pick-up strength among the lf~, ld~ and 2s~. orbits and in particular among the configuration states (T< -~ Ttarget-½) and the analogue states (T> -- Tt,rge~+½). The sum-rule limits for the spectroscopic factors are C 2 S~J = vl.J

7~l, j

2T+ 1 7r,l ' j C23~

j --

, T =

2T+l

Ttarget,

(2)

where vt'j and n t'j are the effective numbers of neutrons and protons in the ( l , j ) orbit of the target ground state. Table 4 shows a comparison of the experimental spectroscopic factors with the sum-rule limits for ~°Cr, 52Cr and 54Cr. The experimental values are the sums of the spectroscopic factors for the individual levels obtained by the usual well-depth DWBA analysis of the data (WD analysis) in which the radius and diffuseness of the bound-state potential were kept constant (r o = 1.25 fro, a = 0.65 fro). The absolute scale for the strength of 52Cr was adjusted to the f~ sum-rule value C2S< = 7.2 (see sect. 3). The WD spectroscopic factors for the other transitions, however, do not compare very well with the predicted values in column 10. We observe two types of discrepancies. First, the 7"> strength exceeds the predictions in nearly all cases, the disagreement increasing with increasing target isospin. Second, except for 5°Cr, the s~ and d~_ strengths remain significantly below the sum-rule limits. These disagreements seem to indicate a deficiency of the DWBA analysis. In the well-depth analysis, the binding energies of the states were obtained by a variation of the depth of the bound-state potential. This variation of the depth is considered to account for the residual interactions not described by the simple shell model. If, however, these residual interactions were located at the nuclear surface, they would give rise to a change of the potential radius rather than to a change of the potential depth. Therefore we recalculated the spectroscopic factors keeping the depth of the bound-state potential fixed to a value Vo = 52.6 MeV, which corresponds to the binding energy of the f~T< centre of gravity for 5ZCr(ro = 1.25, a = 0.65, 2 = 25). This variable radius procedure (VR) improved significantly the ratios R of the C 2 S > / C z S < strengths as compared to the predicted sum-rule values.

Cr(SHe, ~) REACTIONS

69

Introducing an isospin-dependent term Ul(t. T) into the SchriSdinger equation for the bound state 33, iT) yields a set of coupled equations for the 7"> wave function, whereas the equation for the 7"< state remains uncoupled. Consequently, the T< bound-state wave function can be obtained in the way described above by absorbing the (subtractive) potential ½T< U1 into a decrease of the radius 32). The T> set of coupled equations has to be solved yielding the expected enhancement of the T> wave function in the surface region which can approximately be described by an (additive) st.rface potential term -½(T< + 1)U~, i. e. the diagonal term for the T> state. The spectroscopic factors of the coupled-channel treatment in column 9 of table 4 were actually obtained from a combined method of solving the coupled lane equations (code N E P T U N written by T. Tamura) fo~ the T> states and using the variable radius procedure for the T< states. For U~ = V~A-~f(R), a surface derivative form factor was chosen with the same radius (ro = 1.25 fro) and diffuseness (a = 0.65 fm) as used for the bound-state potential Vo. The depths V~ were chosen slightly different for the different orbits (100 MeV < V~ =< 120 MeV), but V~ was kept constant for all three isotopes for a given orbit. The bound-state potential depths resulting lrom this treatment of the T> states were 58.7, 52.2 and 52.7 MeV for f÷, d~_ and s÷, respectively. To keep these values of V0 constant, for the T< states, the radii were adjusted yielding about the same values for the f?_, d~ and s~_ orbits; ro = 1.20 (5°Cr), 1.16 (52Cr) and 1.12 (S4Cr). As a result of solving the coupled equations, the spectroscopic factors for the analogue states decreased by factors between 0.7(54Cr) and 0.9(S°Cr). The introduction of an "effective" radius for the 7"< states led to an increase of the T< strengths by factors of about 1.9(54Cr), 1.6(52Cr) and 1.2(S°Cr), the s~ strength being slightly less affected. This increase of the 7"< strength required a renormalization of the spectroscopic factors relative to the predicted 52Cr f_1_T< strength (N~. . . . . = 37). The overall agreement with the shell-model sum-rule limits for T< and T> states is significantly improved as compared to the WD procedure. The coupled-channel treatment and the variable-radius method yield about the same ratios of T< to 7"> strengths. The values of s- and d-strengths for SZCr and S4Cr in the VR analysis are much closer to the sum-rule limits than the values obtained in the SE procedure. Although the effect is overestimated for S°Cr, we may interpret the S2Cr and "~4Cr results in the way that s- and d-neutrons within the investigated nuclei are being restricted to the volume of 4°Ca rather than occupying the volume corresponding to a potential radius increasing with A '~. A completely different view of the analogue spectroscopic factors may follow from a consideration of isospin mixing in the initial and (or) final states. The sumrule limits have to be rederived for this case. Preliminary calculations have shown that even small admixtures of " w r o n g " isospins may produce large deviations from the sum rules (2) affecting predominantly the T> strengths. The corrections increase with the target isospin for a constant amount of T-impurity. This isospin mixing might also explain the experimental situation.

P. DAVID et al.

70

6. Coulomb displacement energies Fig. 5 shows a c o m p i l a t i o n o f the C r ( 3 H e , ct) d i s t r i b u t i o n s for the a n a l o g u e states. T h e d i s t r i b u t i o n s exhibit r a t h e r p r o n o u n c e d diffraction p a t t e r n s which allows one to d e t e r m i n e t h e / - v a l u e s o f the transitions (see discussion in sect. 4). T a b l e 5 gives the excitation energies o f the a n a l o g u e states as c o m p a r e d to the energies o f the p a r e n t states ( p r o t o n - h o l e states) in the v a n a d i u m isotopes. T h e p r o t o n - h o l e states have been o b t a i n e d p r e d o m i n a n t l y by investigations o f (t, ~) reactions 22.28). U n f o r tunately, there is not m u c h i n f o r m a t i o n available for 52V a n d S3V. The C o u l o m b d i s p l a c e m e n t energies A E c have been d e t e r m i n e d a c c o r d i n g to the r e l a t i o n t AEc

=

EA-EPx-Qp,,(Z<), TABLE 5

Coulomb displacement energies AE¢ and Coulomb radii R c for the observed analogue states in the Cr isotopes Target

~°Cr

Qp,a

l

--3.349

~2Cr

--1.534

6aCr

3.192

5~Cr

2.635

E~A

AExA

E~P

AE e

Re

r = ReA-Je

3 2 (0) 0 0

4.76 5.57 6.43 6.47 6.76

0.00 0.81 1.67 1.72 2.00

0.00 0.749

8.11 8.16

4.75 4.72

1.29 1.28

1.647 1.999

8.18 8. l 1

4.71 4.75

1,28 1.29

3 0 2

6,63 (9.19)

0.00 2.56

0.00 2.545 2.674

8.16

4.72

1.26

3 0

11.29 13.6l

0.02 2.30

0.02

8.08

4.77

1.27

3 0 2

10.65 12.52 12.59

0.00 1.87 1.94

0.00

8.02

4.81

1.27

E A and Exa are the excitation energies of analogue and parent states. Qp,n = Qp,n(Z<), see text.

where E A is the excitation energy o f the a n a l o g u e state, E~ the excitation energy o f the p a r e n t state a n d Q p , , ( Z < ) = [ Q ~ , , , ( t a r g e t ) - O~,p(target)]

= Q-value

o f the (p, n) r e a c t i o n for the p a r e n t ( Z < ) nucleus.

T h e deviations o f the o b s e r v e d C o u l o m b d i s p l a c e m e n t energies for the different states in the 4 9 C r - 4 9 V p a i r exceed the e x p e r i m e n t a l e r r o r limits o f a b o u t +2.0 keV. (The p r o b l e m o f the l --- 0 state which might be split into the 6.43 a n d 6.47 M e V d o u b l e t has a l r e a d y been discussed in sect. 4 . ) T h e s e deviations o f A E c m a y be a t t r i b t The Qv,n(Z<) value has been calculated using the Q-value of ref. 3,).

Cr(aHe, ct)

REACTIONS I0~

71

I

I

I

[

I

E~ ,4.76

E

g.3.½"

10%

I

A I'

Ex = 12.52 1.

"

.°

=

A2-

~.

_

Ex-5.57

•

E1

,E,, Ex = 1259

2~"

O "o

A

.I

~-

1.

.01

E~-e.43 (g-0)

:

53Cr(3He.(3,~2Cr

1. .1

.1

E×=1129

~

A1 I

1

'~

tL

Ex = 13.61

/1,~

:o

Ex:o4, g=0

*

A 2 _=

Ex=6,76

t-o

.11

.1

.i,,

\ ,,%

. . . . . .

10

Fig. 5. The (aHe, ~) angular

20

30

40

5 0 (~

10

20

30

40

A4

50

60

OCM

distributions for the a n a l o g u e states observed in the different Cr isotopes.

uted to slightly different Coulomb energy matrix elements for the different s½, d~ and f~ orbits. Similar deviations in A E c have been observed for the f~ and p+ orbits in the Ca-Sc isobaric pairs 36). Considering the mass dependence a6) of A E c for the f~ orbit, we find a maximum value for 52Cr. This cannot be understood in terms of a Coulomb radius increasing with A ~. It is probably connected to the closure of the f~ shell in SZCr. The Coulomb radii in table 5 have been calculated from AEc using a formula derived by Sen Gupta 37) e2

Ec = [0.6(2Z< + 1)-0.613Z~< - 0 . 3 ( - )z< l - - . Rc

72

v. DAVID et al. T h e a u t h o r s a r e g r a t e f u l t o D r s . H . S. P l e n d l , U . L y n e n a n d R . S a n t o f o r t h e i r

assistance in the measurements JULIE,

HUNTER

and for valuable discussions. The codes NEPTUN,

a n d J I B 3 w e r e k i n d l y p r o v i d e d b y D r s . T. T a m u r a ,

R. M . D r i s k o

a n d F. G . P e r e y . O n e o f u s ( P . D . ) w o u l d like t o t h a n k P r o f e s s o r G e n t n e r f o r t h e h o s p i t a l i t y a t t h e M a x P l a n c k I n s t i t u t as well as P r o f e s s o r M a y e r - K u c k u k

for making

the stay possible.

References 1) J. E. Robertshaw, S. Mecca, A. Sperduto and W. W. Buecher, Phys. Rev. 170 (1968) 1013 2) R. Bock, H. H. Duhm, S. Martin, R. Rudel and R. Stock, Nucl. Phys. 72 (1965) 273 3) V. P. Bochin et al., Nucl. Phys. 51 (1964) 161; R. Trilling, A. Sperduto and H. A. Enge, MIT-Report No. NYO 10063 (1963) 4) W. Fitz, J. Heger, R. Jahr and R. Santo, Z. Phys. 202 (1967) 109 5) C. A. Whitten and L. C. McIntyre, Phys. Rev. 160 (1967) 997 6) C. A. Whitten, Phys. Rev. 156 (1967) 1228 7) J. D. McCullen, B. F. Bayman and L. Zamick, Phys. Rev. 134B (1964) 515 8) I. Talmi, Phys. Rev. 126 (1962) 1096 9) K. Ramavataram, Phys. Rev. 132 (1963) 2252 and private communications 10) J. R. Maxwell and W. C. Parkmson, Phys. Rev. 135B (1964) 82 11) J. Vervier, Nucl. Phys. 78 (1966) 497 and private communications 12) J. G. McGrory, Phys. Rev. 160 (1967) 915 13) R. Book, H. H. Duhm, R. Riidel and R. Stock, Phys. Lett. 13 (1964) 151; R. Bock, H. H. Duhm, R. Jahr, R. Santo and R. Stock, Phys. Lett. 19 (1965) 417 14) B. Rosner and D. Pullen, Phys. Rev. 162 (1967) 1048 15) J. N. Ginocchio, Phys. Rev. 144 (1966) 952 16) F. B. Malik and W. Scholz, Phys. Rev. 150 (1966) 919, 153B (1967) 1071 17) R. Stock, R. Bock, P. David, H. H. Duhm and T. Tamura, Nucl. Phys. A104 (1967) 136 18) H. H. Duhm, Nucl. Phys. A l l 8 0968) 563 19) R. H. Bassel, private communication 20) C. D. Jeffries, Phys. Rex'. 95 (1953) 1262 21) D. Bachner et al., Syrup. on recent progress in nuclear physics with tandems, Heidelberg (1967) 22) D. Bachner, R. Santo, H. H. Duhm, R. Bock and S. Hinds, Nucl. Phys. A106 (1968) 577 23) R. Sherr, B. F. Bayman, E. Rost, M. E. Rickey and C. G. Hoot, Phys. Rev. 139B (1965) 1272 24) J. L. Alty, L. L. Green, G. D. Jones and J. F. Sharpey-Schafer, Phys. Lett. 13 (1964) 55; Nucl. Phys. A100 (1967) 191 25) R. J. Philpott, W. T. Pinkston and G. R. Satchler, to be published 26) R. Santo, MPIH-Report No. V/5 (1968); R. J. Peterson, Phys. Rev. 170 (1968) 1003 27) G. A. Bartholomew, E. D. Earle and M. R. Gunye, Can. J. Phys. 44 (1964) 319 28) R. N. Glover and A. D. W. Jones, private communication 29) J. Bellicard, P. Barreau and D. Blum, Nucl. Phys. 60 (1964) 319 30) H. U. Gersch, C. Riedel and W. Rudolf, Zentralinstitut f. Kernforschung, Rossendorf/Dresden Z F K - P h A 25 31) S. Hinds, H. Marchant and R. Middleton, Phys. Lett. 24B (1967) 34 32) W. T. Pmkston and G. R. Satchler, Nucl. Phys. 72 (1965) 641 33) R. Stock and T. Tamura, Phys. Lett. 22 (1966) 304; E. Rost, Int. Conf. on Nuclear Physics, Gatlinburg (1966) p. 237 34) C. Maples, G. W. Goth and J. Cerny, Nucl. Data 2 (1966) 429 35) J. A. Nolen, J. P. Schiffer, ?4. Williams and D. von Ehrenstein, Phys. Rev. Lett. 18 (1967) 1140 36) R. Sheri, Phys. Lett. 24B (1967) 321 37) S. Sen Gupta, Nucl. Phys. 21 (1960) 542 38) R. Bock, P. David, H. H. Duhm, H. Hefele, U. Lynen and R. Stock, Nucl. Phys. A92 (1967) 539 39) U. Lynen, R. Bock, R. Santo and R. Stock, Phys. Lett. 25B (1967) 9 40) C. F. Monahan, N. Lawley, C. W. Lewis, J. G. Main, M. F. Thomas and P. J. Twin, Nucl. Phys. A120 (1968) 460