Investigation of 42Sc by the reaction 40Ca(3He, p)42Sc

Nuclear Physics A l l 6 (1968) 516--528; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

INVESTIGATION OF 42Sc THE REACTION OCa(3He, p)42Se BY F. P ~ H L H O F E R

Max-Planck-lnstitut Jiir Kernphysik, Heidelberg, Germany Received 13 M a y 1968

Abstract: The level structure of 42Sc has been studied by means of the reaction 4°Ca(3He, p)42Sc at 18 MeV incident energy. Angular distributions have been measured for states up to 4 MeV of excitation energy and j,t, T assignments are made. A shell-model calculation using an effective interaction has been performed in order to describe the two-particle states. The calculated wave functions are used with a distorted-wave analysis to predict theoretical cross sections of the ('~He, p) reaction. Very good agreement is obtained for the transitions to the excited states. The ground state transition, however, is about three times stronger than predicted. E I

I

NUCLEAR REACTION '°Ca(~He, p), E = 18 MeV;

I

m e a s u r e d tr(0, Ep), Q. 4~Sc deduced levels, J, .-r. N a t u r a l target.

I

I. Introduction A characteristic feature o f two-nucleon stripping reactions is the sensitivity o f the cross section to correlations between the transferred nucleons in the final nucleus. F o r this reason, nuclear wave functions a n d the residual interaction, which is mainly responsible for the correlations, can be investigated in detail. The nucleus 42Sc represents one o f the simplest systems which can be studied in this context. A t least in a first a p p r o x i m a t i o n , its low-lying s p e c t r u m is explained by the excitation o f the degrees of freedom o f only two nucleons, a p r o t o n and a neutron in the f-p shcll. The residual interaction introduces a mixing between these twoparticle configurations, thus leading to large enhancements o f the correlations between the two nucleons in the lowest states. It suggcsts itself to consider here only a special type o f correlations, n a m e l y two-nucleon clusters characterized by an intrinsic m o t i o n with n = 1 = 0 and spin S = 0 or 1. They are o f p a r t i c u l a r interest, because their a m p l i t u d e can be d e t e r m i n e d by a two-nucleon transfer reaction like (3He, p). In the present work, the reaction 4°Ca(3He, p)42Sc is investigated. A n g u l a r dist r i b u t i o n s have been measured for states up to an excitation energy of a b o u t 4 MeV, a n d the a n g u l a r m o m e n t u m transfers L are determined. Using these L-values and the isospin assignments o b t a i n e d from a c o m p a r i s o n with the s p e c t r u m o f 42Ca, the level scheme o f 42Sc with J " , T assignments is obtained. A shell-model calculation using an effective interaction is p e r f o r m e d in o r d e r to describe the two-particle states in the spectrum. The calculated wave functions are used in c o n n e c t i o n with a dis516

517

40Ca(3H,,, p)42.~ REACTION

torted-wave analysis to predict the cross sections of the (3He, p) reaction, which are compared with the experimental data in detail. 2. E x p e r i m e n t a l

procedure

and results

The experiments were performed using the 18 MeV 3He beam of the tandem Van de Graaff. The proton spectra were analysed using a broad-range magnetic spectrograph. Other reaction products were absorbed by copper foils in front of the nuclear emulsion plates. The targets were prepared by evaporating natural calcium onto carbon backings. In fig. 1, an energy spectrum of the reaction 4°Ca(3He, p)42Sc is shown. Because of very strong transitions due to the (3He, p) reaction on the 12C and 160 impurities of the target, it is very difficult to obtain information on higher excited states in "2Sc. The level at E, = 3.33 MeV and all levels above 4 MeV are observed only at two or three angles. 1

0 1

i

30,

I

E

I

He

Exc~tabon energy 4 5

3

1

i

4°Ca ( 3He O)42Sc E3 =18MeV e = 5 °

"2Cc I i

%

2 '

i

1

i

12C1 1800

p

o

i i

10C

u

;

iI •

I' :

i' ,lil,~2

'

1.

.

6 10

15

25

h?

!

I •

5

i

• v2

I

I',

i

i

/ el k,~'l 30

35

.~

,,

"2~

i i

I

Ji

i' ¢1 •

,I

1

"2f..= 2"

I

'1

© 0

6MeV

i

12C2 v51:'5

~601 4

I

× t~

i

:

~

|

i

I] .9' .."

;e):

1~

I!)'!, I

,,.

i;~i~|, ',~1'% , ',,~ ..,.~" , ,j'l , I:', "t .... '~.

40 zig 50 Pof~t~on on the plate

55cm

Fig. I. Proton energy s p e c t r u m o f the reaction 4°Ca(aHe, p)4~Sc at 0lab ~ 5". T h e p r o t o n groups are n u m b e r e d in correspondence with the level n u m b e r i n g in table 1.

Angular distributions of the proton groups corresponding to the lower-lying states were measured between 0~ab = 5 ° and 60 °. They are shown in fig. 2. As expected from a direct-interaction mechanism, the shape of the angular distributions is characteristic of the orbital angular momentum of the transferred proton-neutron pair. In most cases, this information is sufficient to determine spin, parity and isospin of the final states by means of the following relations: JB = J A + L + S , S + T = 1, 7["B = 7L'A(--)L.

These equations represent the vector coupling condition, the selection rule for (3He, p) and the parity selection rule. The subscripts A and B denote the initial and

518

F. P/]HLHOFER

final nucleus and L, S and Tthe angular momentum, spin and isospin of the transferred pair. In some cases, two L-values are allowed. In this reaction, JA = T^ = 0 ,

TB = T,

nA = +1.

The isospin of the final states in 42Sc can be determined by comparison with the spectrum of 42Ca and the results of the 4°Ca(t,p) reaction 1). To each T = I level in 42Sc, the analogue state in 42Ca can be found at almost the same excitation energy. 4°Ca 10

(3He, p)42Sc.

F.~ .oo O" T:I o,

O1

';',

1 :

;~i

it T,o(l° "

: / "{,,

~

E,, ,o6~"

o:O

.~ 1

01

o,:,L:°II ¢' I ',j/: .,

I ',

L,O

E = 18 M e V

1~

Ex'150 Z~'/5-T. 0 1 0

~

~'.-m-e. t %

4" • ~ I,,

01

\J

01

1.0

~

T:1 ,.=4

~t~/' '' .%"", f E~ : 3 "0 • /2""44")5" T,O L,4 i~'

,1,

".{~

1

Ol

Ex :3 25 6" kT:~ -6

,~ ÷. '~-,

I

0 ° 2 0 ° 4 0 ° C-(i~

L. : '~L'')'''I~"'" %"

31

Ex ,378

%:2")3"T=OL,2

: .. )L/:.2

L :2

'k

L.O.2

~

Ex,159 2" T : I

i~,"~,

I

L'2 ~-, ",2~b ) 1

L,2.4

t.{ // , , \ " \\~, - "L'4 ", \, \

Oi

X\x

)1

[" Ex,393~,xx,i

(2"13"T=20 ,

0 ~ 2() ° 4() ~ C~

I

' ",,

, I

0 ° 20 o40 ° 60 °

Fig. 2. Angular distributions of the reaction 4°Ca(3He, p)4~Sc. DWBA calculations with arbitrary normalization indicated by dashed lines are given only for states having mainly two-particle components.

The corresponding (t, p) transition has the same L-value and a similar relative intensity as in ( 3 H e , p). In this way, the low-lying spectrum of 42Sc can be established. A summary of the results is given in table 1. The excitation energies arc listed with the values of L and J% T and the relative intensities of the (3He, p) reaction. The same information for 42Ca and the (t, p) reaction is shown for comparison. In some cases, the spin assignment is not unique, and additional information from other work 2, 3) is used to select

4°Ca(aHe, p)411ScREACTION

519

one J-value. Also the comparison with the level scheme and the cross sections predicted by the shell-model calculation described below is used for that purpose. However, the spin assignments for the states at E x = 3.39, 3.78 and 3.93 MeV are rather arbitrary. Perhaps the assignments 2 +, T = 0 and 3 +, T = 0 for the states at 3.39 and 3.93 MeV have to be exchanged. The proposed level scheme of 42Sc is shown in fig. 3. 393~ 386--" 378 3.69 3 3 9 -._

/ ( 2 " ; 3;0 1".0 (2") 3",) 1",C', / ( 3 " ) 2'.L) ? ,1 "

333 325

6",I

310

(4") 5;0

2.85

4",1

2.50

2",1

2.20

',2") 3,0

1.89 1.59~ 1.52~

/

1.50 /

3:0

062~. 061 -

__/"

0.00 E×

0*2 2",1 5".0

7*,0 1%0

0LI

42SC

Jrt,T

EMeV~q Fig. 3. Energy levels of 42Sc below 4 MeV (see caption of table 1).

3. Shell-model calculation for 42Se As a first step to interpret the experimental results and in order to obtain wave functions of the final nucleus necessary for the distorted-wave analysis of the reaction, a shell-model calculation of 42Sc has been performed. An inert 4°Ca core is assumed, and all two-particle configurations of the f-p shell are considered. The single-particle energies of the f~, p~, p½ and f~ orbits are taken from the experimental spectra of 4'Ca and 41Sc. Oscillator wave functions specified by the oscillator parameter v,m are used here for the radial part. A simple central interaction with Yukawa shape was used

V =

V0

e-rl2/a - - E 2S+I'2T+IA2S+I'2T+Ip, r12/a S,r

where 2 s + 1 . 2 r + l p is an operator projecting on states with spin S and isospin T. From this residual interaction, the energy matrices were deduced and then diagonal-

520

F. PUHLHOFER

ized. The four parameters

2 s + 1 , 2 r + 1A s p e c i f y i n g t h e e x c h a n g e c h a r a c t e r

were varied in order to give an optimum best

parameters

are

13A = 0.82,

fit t o t h e l o w - l y i n g s p e c t r u m

a3A =

-0.55,

11A =

-0.91,

of the force of 42Sc. The

31A =

1.0

with

TABLE 1

Experimental results o f the reaction '°Ca(3He, p)'~Sc "So

'2Ca

Ex (MeV) q:O.Ol

L

Jn, T

0 0+2

0 +, 1

2 3 4 5 6 7 8 9 10

0.00 0.61 0.62 1.50 1.52 1.59 1.89 2.20 2.50 2.85 3.10 3.25 3.33

11

3.39

2

Level

0 1

2 4 2 0 2 2 4 4 6

12

3.69

0+2

13 14 15

3.78 3.86 3.93

2 0-t-2 2

16 17 18 19 20 21 22 23 24

4.72 4.83 5.44 5.48 5.64 5.81 5.97 6.09 6.18

1+,0 7 +, 0 3 +, 0 54 , 0 2 +, 1 0", 1 (2+)3 +, 0 2 +, 1 4 ~, 1 (4')5 +, 0 6 +, I I

Relative intensity L = 0 4) L ~ 2 b)

100 200

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

45 17 17 18

Jn

Relative intensity e)

0.00

0

0+

100

1.52 1.84

2 0

2+ 0+

45 14

2.42 2.75

2 4

2+ 4+

18 32

3.19 3.25 3.30

6

6+

15 7 1

3.39 3.~

2 3

2+ 3-

3 7

3.65

2

2+

9

3.88 4.75 4.86 5.46

2 2 4

22+ 4+

2 43 28 21

5.66 5.85 6.01 6.10

3 0 0 4

30+ 07. 4+

97 25 12

130

130 30 38 90 45 35

0 ~', 1

L

120

(3+)2 +, 0

1+, 0

Ex (MeV)

65 30

T h e levels o f 42Sc with j n , T a s s i g n m e n t s a n d the relative intensities o f the (aHe, p) reaction (normalized separately for L = 0 a n d L = 2) are c o m p a r e d with 42Ca a n d the intensities o f the reaction '°Ca(t, p)4~Ca [ref. 1)]. T h e spin a s s i g n m e n t s are discussed in the text. T h e 7 + state is n o t seen in this experiment, the 3 + a n d 5 + levels at 1.5 MeV are n o t resolved. T h e excitation energies for these three states are from ref. 13). T h e g r o u n d state Q-value was m e a s u r e d to be Q0 = 4.92:,~0.02 MeV. 4) Cross section at 5 °. h) Cross section at 15 °. e) Integrated cross section.

521

4°Ca(3He, p)'t2Sc REACTION

Vo = - 5 5 MeV. A variation of the range parameter a is of minor influence. It has been kept fixed by the condition a,,,' , 7~s., ,- = 0.7. In the procedure described above, only two-particle configurations are considered, and particle-hole excitations are neglected completely. Because of the property of the direct (3He, p) reaction to excite essentially the two-particle components, one is mainly interested in this aspect of 4ZSc. However, as known from the spectrum of 42Ca and also observed in 42Sc, additional states occur at very low excitation energy. They are usually described as deformed 4p-2h states, for example in the model proposed by Federman 4) or Gerace and Green 5). These states mix with the normal states and are mainly excited by these admixtures in the two-nucleon transfer. The Mev'

"o

6

2 ',

P3,2\

g 8 ~

o

5

"d E_ a g

....

\ '\ i

fr~ P3,2

-'---J-- ---T

Q. ×

o

3

f~2

- -

2 i

42Sc

×

W

0

fT~

_

'_t__

6* T=I

4*

2*

O"

I*

3"

5"

7*

2*

4* T =0

t

Fig. 4. Comparison o f t h e calculated and the experimental level scheme of 4zSc. Heavy lines: calculated levels; thin lines: experimental; dashed lines: experimental, but weakly excited in ('~He, p). Dominant configurations are indicated. The arrows simply connect the calculated and the corresponding experimental level. The experimental data are mainly from this work (see table 1); thcy are completed by some (t, p) results 1).

excitation of the 4p-2h component itself via particle-hole components in the target ground state is usually smaller. It is possible to take into account the infuence of this mixing on the two-nucleon transfer to the two-particle states approximately by summing up the cross section of the two-particle and the corresponding core-excited state before comparing it with the calculated strength. This procedure is discussed in more detail in sect. 5. On the other hand, it should be emphasized that a sufficient number of twoparticle configurations must be included in order to get reasonable wave functions for the interpretation of the (3He, p) cross sections. This is due to the coherent summation over all configurations in the two-particle transfer amplitude. For instance, the f~

522

F. P(JHLHOFER

c o m p o n e n t in the g r o u n d state o f 42Sc leads to an e n h a n c e m e n t o f the t h e o r e t i c a l cross section o f a b o u t 20 %, t h o u g h the u n p e r t u r b e d e n e r g y o f this c o n f i g u r a t i o n lies at a b o u t 11 M e V e x c i t a t i o n . T h e c a l c u l a t e d level s c h e m e o f 42Sc is s h o w n in fig. 4 in c o m p a r i s o n w i t h the exp e r i m e n t a l s p e c t r u m . B e c a u s e o f the r e s t r i c t i o n s i n t r o d u c e d , the t h e o r y c a n n o t acc o u n t for the w e a k l y excited 4p-2h states. A d d i t i o n a l degrees o f f r e e d o m w o u l d also be necessary for the d e s c r i p t i o n o f the s p l i t t i n g o f s o m e h i g h e r levels as o b s e r v e d for i n s t a n c e in the case o f the 2 ÷, T = 1 states. H o w e v e r , the m a i n f e a t u r e s o f the exp e r i m e n t a l s p e c t r u m are d e s c r i b e d sufficiently well. T h e p r e d i c t i o n s o f the t w o n u c l e o n t r a n s f e r cross sections d e r i v e d f r o m the r e s u l t i n g w a v e f u n c t i o n s (table 2) are c o m p a r e d w i t h the e x p e r i m e n t in the f o l l o w i n g section. TABLE 2

Calculated wave functions of 42Sc Jn, T = 1

0+

0-

2-

2+

4+

4+

6-

Ex (MeV)

0.13

5.04

2.21

4.15

2.84

4.74

3.12

f1~ fk pt p~ f-] Pt f~ q Pi- P½ P~ ft p~va p½ f~ ft ~

0.940

0.260

--0.213 0.896

0.973

0.925

--0.365 0.904 0.104

0.962 0.149

0.226

0.911 0.332 0.081 0.137 0.098 0.055

- 0.017 0.088 0.108

0.130 0.160

0.332 0.068

0.084

0.189

0.229

0.112

0.276

0.230

0.019

0.089 0.117

0.138 0.002

0.061

0.021

Jn, T = 0

1+

1*

3~

3+

57

5+

2-

Ex (MeV)

0.25

4.34

1.51

3.63

1.46

3.31

3.83

f2 fl P,1p,,~ f~ p~ f]. f~ pk p~ p~_ f] pt 2 P i f~ ft ~

0.778

-0.535

--0.544 0.838

0.833

0.505

--0.490 --0.240 0.030 --0.071

--0.409 --0.500 --0.170 --0.005

--0.511 0.722 0.227 --0.398 --0.012

0.827 0.542

0.241

0.826 0.352 0.223 --0.283 --0.232

--0.149

0.023

--0.043

--0.074

--0.182

0.113

--0.044 --0.081

--0.026 0.032

--0.033

0.023

0.373 0.242 0.268 0.193

The calculated excitation energies are given relative to the experimental ground state. 4. D i s t o r t e d - w a v e analysis T o c a l c u l a t e the cross sections o f the r e a c t i o n 4 ° C a ( 3 H e , p)42Sc, we f o l l o w the t h e o r y o f t w o - n u c l e o n t r a n s f e r r e a c t i o n s g i v e n by G l e n d e n n i n g 6). I f the r e a c t i o n is

l°Ca(Zl'te, p)42Sc REACTION

523

assumed to proceed by a direct mechanism, the transition amplitude can be written in the DWBA T = f d r i d r f q)(-)*(rf)(Bbl VlaA>(b (+'(r,), where (0(+) and ~ ( - ) are the distorted waves in the incoming and outgoing channels of the reaction A(a, b)B. The interaction potential V is given by a spin-dependent force v' 8) of a similar form as used in the shell-model calculation

V=

~

V(ri3) ZZS+l'Zr+lAZS+I'Zr+IP.

i=1,2

S,T

The spatial part is approximated by a force acting between the outgoing proton (particle 3) and the centre of mass of the transferred nucleons (1 and 2), and the zero-range approximation is used. As discussed in sect. 3, the initial nucleus and the 4°Ca core in 42Sc are assumed to have closed shells. The wave functions of 42Sc are written as a sum over various two-particle configurations with the mixing coefficients c~ obtained from the shellmodel calculation

7JI2 = Z c,,[jI'rl ,j2rz]s,r. These assumptions lead to the transition amplitude

T = Doas\,/~(2S+ii(TAT:A, TTzITBTzB) Z (JAMA, JMIJBMB) L

x (LML, SMsIJM)(SMs, Sbmbls~m,) × f dr i ~ ( - ) . ( r t ) r L ( R ) i - Lr,u(~)@(+)(ri) ' where r r = rimA/m B and R = r i . The strength of the zero-range interaction is determined by D o. The quantity a s depends on the spin S of the transferred pair and, because of the selection rule given in sect. 2, on the isospin T B of the final nucleus. It is connected with the constants 31A and 13A by the formulae given in ref. 7). The form factor Ft.(R) describes the centre-of-mass motion of the transferred nucleons. It is calculated in the following manner. First, for each configuration the radial parts of the single-particle wave functions for the proton and the neutron with quantum numbers n 1, 11, Jl and hE, 12, J2 are computed in a Woods-Saxon well. These wave functions are then expanded in terms of oscillator functions u.,.,l(,-,)

= Z a ..... Vl

The oscillator constant v is an auxiliary quantity and in principle arbitrary; it was chosen in correspondence to the size parameter of the Gaussian wave function of 3He [ref. 6)] to be v = 6r/2 = 0.254 f m - 2 . This greatly simplifies the calculation,

524

F. POHLHOFER

because it reduces the number of spatial overlap integrals. The centre-of-mass motion was calculated by the Moshinsky transformation 9). The form factor is then given by

FL(R) = ~. c , , ' , "

1 - ( - ) s + T hJ,h . v -...... ~ - " z s ~ ½(a.,, v,a.~.v2

\/2(l+t~j,.j:)

.... a v

OSC

~

+ a,,, ~, a .... 2)(00NL, Llv, I, v 212)UNL(R), where vLshJ'J2denotes the transformation coefficient between j - j and L - S coupling, the bracket the Moshinsky coefficient and u~"~. the radial part o f the centre-of-mass motion in the oscillator potential. This form factor is computed by a special program and then inserted into the D W B A code J U L I E , which performs the integration o f the transition amplitude and the calculation o f the cross section. The parameters introduced in the D W B A calculation are first the optical potentials for the scattering wave functions. They are listed in table 3. N o cut-off in the radial integrals was used. The form factor is specified by the parameters for the single-particle wells r o = 1.25 fm, a = 0.65 fm and the binding energies. These are adjusted to give the correct separation energy of the proton-neutron pair, e.g. EB(42Sc) - E s ( 4 ° C a ) - E x TABLE 3

Optical-model parameters used in the DWBA calculation V W (MeV) (MeV) •~He p

165 52.5

Wder (MeV)

ro (fm) 1.14

15

1.22

20.2

re (fm)

a (fm)

r' (fm)

a' (fm)

Vt, (MeV)

Ref.

1.30

0.723

1.60

0.81

7

14)

1.25

0.60

1.26

0.31

8

lb)

for S = 0 transfers and the corresponding value minus 2.3 MeV for S = i transfers to account for the interaction in the triplet proton-neutron state. The theoretical angular distributions are compared with the experimental data in fig. 2. The ratios between experimental and theoretical cross sections are given in table 4. For the T = 0 states, only the transitions with the lower L-value are considered, which usually are strongly favored. As discussed in sect. 3, the experimentally observed splitting of the two-nucleon transfer strength was taken into account in the experimental cross section. For the theoretical cross sections, a normalization factor D 2 = 33 • 104 MeV 2 • fm 3 was used. The ratio between singlet and triplet strengths of the stripping interaction was adjusted to fit the T = 0, S = 1 and the T = 1, S = 0 2 2 transitions simultaneously. A value o f ~3A/3~A = 0.43 corresponding to ax/ao = 0.45 was obtained. Similar values have been found by Hardy and Towner 8) in an analysis of the 12C(3He, p) reaction. 5. Discussion

As shown in table 4, the ratios of theoretical and experimental cross sections are close to unity in most cases. A significant deviation occurs only for the ground state

t°Ca(3He, p)4~Sc REACTION

525

TABLE 4 C o m p a r i s o n of e x p e r i m e n t a l a n d theoretical cross sections o f the re a c t i on ~°Ca(aHe, p) to several final states in 42Sc J~

Ex

O'exp/O't h

(MeV) T = 1

0÷

0.00 1.89

2.8

2+

1.59 2.50

1.0

4+

2.85

1.0

6"

3.25

0.8

0+

5.81 5.97 ~ 6 . 7 0 a)

1.3

2*

~ 4 . 4 5 a) 4.72 4.83

1.1

I÷

0.61

1.4

3÷

1.50 2.20

1.0

T = 0

5~

1.52

0.9

2÷

3.39

1.0

1÷

3.69 3.86

1.0

3~

3.78 3.93

0.9

5"

3.10

0.7

a) Ref. b.

The ex citation cnergics o f the states and their c o m b i n e d strengths are indicated (see discussion in the text). A c o m m o n n o r m a l i z a t i o n factor is included in the theoretical cross section. EX MeV E

2

0.81

,. '.

P 3 / 2 ~ , x

,

,

. . 4

2 0.05 f7/2

-

098

~

'~ '[: :~, 0 . 8 3

PURE CONF.

1.0

.

SHELL-MODEL CAL.C.

1.2 EXP.

Fig. 5. The influence of the configuration m i x i n g on the (aHe, p) cross section. The theoretical L ~ 0 cross sections at 0~at, = 5 ° in m b / s r are given for the two lowest 1-, T = 0 states in 42Sc, a s s u m i n g p ure configurations and the configuration m i x i n g o b t a i n e d from the shell-model calculation. The experimental L - 0 intensities are given for c o m p a r i s o n .

526

v. PUHLHOFER

transition. This will be discussed below. Considering the great sensitivity of the twonucleon transfer cross section to details of the wave functions the agreement is surprisingly good. As demonstrated in fig. 5 in the case of the low-lying 1 +, T = 0 states in 42Sc, the configuration mixing is nearly entirely responsible for the correlations between the nucleons in the lowest state and the corresponding two-nucleon transfer cross section. The experiment should therefore be a very sensitive test of the wave functions. In the case discussed above, the two-nucleon transfer for the lower 1 + state is enhanced by the residual interaction by a factor of about 15. The enhancement for the following state is much weaker, and for the higher states one observes destructive interference. The residual interaction always tends to concentrate the twonucleon transfer strength in the lowest state of given J", T. This trend is well reproduced by the shell-model calculation. Experimentally, the ground state transition is found to be enhanced by a factor of 2.8 compared to the theoretical cross section. This disagreement might be considered as an effect of the approximate treatment of the influence of particle-hole states described in sect. 3. This should therefore be discussed in more detail. According to the model of Gerace and Green 5), we assume deformed 4p-2h states in 42Sc mixing only with a single two-particle (2p) state of the same J", T and also 2p-2h excitations in the target ground state. Then we have to consider the following transitions in the (3He, p) reaction: 4°Ca: Al0p-0h) + Bl2p-2h)

42Sc1:

al2p)

+

bl4p-2h)

42Sc2:

bl2p)

-

a[4p-2h)

Transition II can be neglected, since it is connected with a form factor with a smaller number of nodes. The transition amplitudes T1 and T2 to the actual states 1 and 2 in 42Sc can then be expressed by the amplitudes T~ and Tlu T x = AaTx+BbTm, 72 = A b T i - B a T l , l . If one writes Tm = cTl with a complex number c depending on the spins and the reaction angle, one obtains [TI[2+[T2[ 2 = [A2+(BIcl)2]ITII 2. The values of A and B can be taken from ref. 5); A = 0.9, B = 0.4. From the work of D6nau et al. ~0), who treated an analogous situation in their analysis of the reaction t60(t, p)laO more exactly, we try to estimate the value of c and assume [c[ < 0.4. Then (BIcI)2<
4°Ca(3He,

p)4~Sc

527

REACTION

This means that with the a b o v e restrictions the p r o c e d u r e o f s u m m i n g the strength shared between the two states leads to the strength expected from the simple m o d e l w i t h o u t core excitation. The factor A 2 is only a c o m m o n n o r m a l i z i n g factor, which has also to be a p p l i e d to the o t h e r states, which p r e s u m a b l y d o not mix with the 4 p - 2 h state. The mixing between this state a n d all other states should not have large influence. One expects corrections o f the o r d e r o f 20 or 30 ~o for strong transitions. Consequently, this theory c a n n o t explain the observed e n h a n c e m e n t o f the transition to the g r o u n d state. This state seems to have more two-nucleon correlations than are included in the present calculations. It should be emphasized that the alzove a s s u m p t i o n s do not imply that the transition between particle-hole states are generally u n i m p o r t a n t . The c o n t r i b u t i o n to the tranMeV 6

~"

~-

~i~,-,~, ~.AEc:

,,~-looRev g'

,is

7;:-

-: ' 5

.I •

.' "5

~"

~.',2

(4 ~ '

4~3

" • - . - - -

(2.:),

2~r:

(2.1.1'2!

I',

d. 2"

AE c = AE c " ,3E.c: 727M, ev

2"

C"

:~a.4

:183

189

O"

k'°

1."~2

:~'~3) " ~ , q

2"

000_

_. ( - 0 0 6 ) 0 0

O"

C°

42Ca

~Sc

~. nEC =

?'Ec 55~v

Fig. 6. Comparison of the T = 1 spectra of 42Ca and 4zSc. The spectrum of 42Sc is shifted by 55 keV corresponding to the unusually low Coulomb energy difference AEc of the ground state. The corrected excitation energies are given in brackets.

sition a m p l i t u d e T 2 to the weakly excited state m a y be a b o u t 40 ~ . This changes the cross section by a factor 2. In the shell-model calculation, the residual interaction was treated assuming an effective instead o f a realistic force. This a p p r o x i m a t i o n seems not to be responsible for larger deviations between experinaental and theoretical cross sections. F o r a few low-lying states, a c o m p a r i s o n was m a d e with wave functions calculated with the T a b a k i n interaction ' 1). The deviations o f the two-nucleon transfer cross sections from our theoretical values were found to be less than 15 ~ . W i t h respect to the a n o m a l y o f the cross section observed for the g r o u n d state transition, it is interesting to look at the C o u l o m b energy differences A E c between the

528

F. P/,JHLHOFER

states of 42Ca a n d the analogue T = 1 states in 42Sc. Nolen et al. 12) f o u n d that the value of A E c for the pair 42Sc-42Ca in the g r o u n d state is a b o u t 55 keV lower than expected from systematic measurements of other S c - C a isobaric pairs. The expected value is A E c = 7.27 MeV. In fig. 6, the T = 1 spectra of 42Sc a n d of 42Ca are compared. The zero of the energy scale for 42Sc is shifted by A E c + Q p n . T h e n it is seen that the C o u l o m b energy differences for the excited states up to the 6 + level, which are mainly f2~, are very close to A E c. As expected, the levels above the 6 + state have a smaller C o u l o m b energy, because they have mainly f~p~ or p~] configurations. In contrast to the n o r m a l behavior of all excited states, the special character of the g r o u n d state of 42Sc a n d its analogue in 42Ca is again evident.

6. Conclusion By means of the shell-model calculation, it is possible to describe the essential features of the two-particle spectrum of 42Sc. The obtained wave functions predict together with a distorted-wave analysis the cross section of the (3He, p) reaction to nearly all of these states with great accuracy, aside from a c o m m o n n o r m a l i z i n g factor Do2 a n d the fitted ratio of singlet and triplet strengths of the stripping interaction. The mixing of the two-particle states with more complicated particle-hole excitations was treated approximately. However, the theory is unable to a c c o u n t for the unusually strong g r o u n d state transition. The a u t h o r is indebted to Professor W. G e n t n e r for his interest in this work. He wishes to t h a n k Professor R. Bock for his c o n t i n u e d support.

References 1) J. H. Bjerregaard, O. Hansen, O. Nathan, R. Chapman, S. Hinds and R. Middlcton, Nucl. Phys. AI03 (1967) 33 2) J. W. Nelson, J. D. Oberholtzer and H. S. Plendl, Nucl. Phys. 62 (1965) 434; J. W. Nelson, W. R. Busch and H. S. Plendl, Bull. Am. Phys. Soc. 12 (1967) 1184 3) E. Rivet, R. H. Pehl, J. Cerny and B. G. Harvey, Phys. Rev. 141 (1966) 1021 4) P. Federman, Phys. Lett. 20 (1966) 174 5) W. J. Gerace and A. M. Green, Nucl. Phys. A93 (1967) 110 6) N. K. Glendenning, Phys. Rev. 137 (1965) B102 7) D. G. Fleming, J. Cerny and N. K. Glendenning, Phys. Rev. 165 (1968) 1153 8) J. C. Hardy and I. S. Towner, Phys. Lett. 25B (1967) 98 9) T. A. Brody and M. Moshinsky, Tables of Transformation Brackets (Mexico, 1960) 10) F. D6nau, K. Hehl, C. Riedel, R. A. Broglia and P. Federman, Nucl. Phys. AI01 (1967) 495 11) A. E. L. Dieperink, private communication 12) J. A. Nolen, J. P. Schiffer, N. Williams and D. von Ehrenstein, Phys. Rev. Lett. 18 (1967) 1140 13) J. J. Schwartz, D. Cline, H. E. Gove, R. Sherr, T. S. Bhatia and R. H. Siemssen, Phys. Rev. Lett. 19 (1967) 1482 14) R. Bock, P. David, H. H. Duhm, H. Hefele, U. Lynen and R. Stock, Nucl. Phys. A92 (1967) 539 15) R. W. Zurmiihle, C. M. Fou and L. W. Swenson, Nucl. Phys. 80 (1966) 259