Self-consistent nuclear spectroscopy with two-nucleon transfer reactions

Nuclear Physics A126 (1969) 115--128;(~) North-Holland Publishino Co., Amsterdam

Not to be reproduced by photoprintor microfilmwithout written permissionfrom the publisher

SELF-CONSISTENT NUCLEAR SPECTROSCOPY WITH TWO-NUCLEON TRANSFER REACTIONS KRISTOFER KOLLTVEIT t Cali]ornia Institute of Teehnolo.qy, Pasadena, CaliJbrnia tt Received 9 August 1968 Abstract: An attempt has been made to calculate the wave functions of the lowest J ~ 0 ' states in both the target and the residual nucleus directly from two-particle transfer reactions. Due to serious limitations on the dimensions of the shell model subspace and partly due to the choice of test nuclei, the results are at best only in qualitative agreement with shell-model calculations. Some estimates have been made on the accuracy of nuclear spectroscopy with two-particle as compared to one-particle transfer reactions; and it is shown that the greater uncertainties in the optical-model calculations for two-particle transfer reactions are compensated for by the sensitivity of the cross sections to the shell-model amplitudes.

I. Introduction It is well k n o w n that the two-particle transfer reaction yields i n f o r m a t i o n on both the m a g n i t u d e s and the relative phases of the shell-model amplitudes of the nuclei involved. W h e n the n u m b e r of shell-model basis vectors required to describe the nucleus is sufficiently small, the two-particle transfer reaction cross sections to this nucleus define the shell-model amplitudes when the g r o u n d state of the target is known. A calculation of this kind was performed in ref. 1) for the reaction t 6 0 ( t , p)tSO, a n d it was shown that the calculated shell-model amplitudes were not sensitive to variations in the optical model parameters in the distorted wave Born a p p r o x i m a t i o n ( D W B A ) calculations. The limitation on the numlzer of shell-model basis vectors is given by the n u m b e r of equations supplied by the two-particle transfer reaction. We assume the n lowest lying states with the same parity, total a n g u l a r m o m e n t u m J a n d isospin T, in a nucleus define an n-dimensional space, which may to a good a p p r o x i m a t i o n be described by an n - d i m e n s i o n a l shell-model space. The p r o b l e m is then to find the n 2 shell model amplitudes which are needed when e x p a n d i n g the n physical states in shell-model basis vectors. The o r t h o n o r m a l i z a t i o n c o n d i t i o n s yield n equations from n o r m a l i z a t i o n of the n states and ½ n ( n - 1 ) e q u a t i e n s due to the o r t h o g o n a l i t y of the states. The two-particle transfer reaction cross sections yield at most n e q u a t i o n s ; i.e. if absolute cross sections are used in the analysis. T h u s the condition on the d i m e n s i o n a l i t y of the solvable problem, when the target is known, is * Present address: Michigan State University, East Lansing, Michigan. tt Work supported in part by the Office of Naval Research [Nonr-220(47)]. 115

116

K. KOLLTVEIT

given by ,72 < n + ~ n ( n - I ) + n

or

,7__< 3.

(1)

The use of absolute cross sections therefore allows solutions in a three-dimensional space. Due to the fact that all equations are of second order, several solutions for the amplitudes are obtained, but we will assume that it is always possible by physical arguments to select the most prebable solution 1,2). However, we do not believe the DWBA calculations at the present time are accurate enough to allow the use of absolute cross sections for two-particle transfer reactions. Furthermore, it was shown in ref. 1) that in some cases due to approximate sum rules, the equations furnished by the two-particle transfer reactions are not independent. One is therefore forced to use cross-section ratios rather than absolute cross sections. Hopefully, this may reduce the uncertainties in the DWBA calculations significantly. On the other hand, the reduction of equations requires an additional equation for the shell-model amplitudes. This may be obtained from analysis of one-particle transfer reactions or from electromagnetic transition data. Nuclear spectroscopy with two-particle transfer reactions has unfortunately been dependent on a more or less complete knowledge of the ground state of the target. The role of two-particle transfer-reaction analysis has therefore been more to test the results of shell-model calculations than to supply independent experimentally determined spectroscopic factors. It is the aim of this work to show that the two-particle transfer reactions may determine the shell-model amplitudes of both nuclei involved when the basis vectors are known. This is, of course, a much weaker requirement than the knowledge of both the basis vectors and their amplitudes. We will also show that the two-particle transfer reactions to some extent are able to predict which basis vectors are dominant in the nuclei involved. It was pointed out in ref. a) that a particular suitable set of reactions for a self-consistent calculation of this kind, would be the (t, p) and (p, t) reactions. Let us assume at a slight cost of generality that two nuclei differing in neutron number by two, may each be adequately described by n shell-model basis vectors. We consequently have 2n z unknown shell-model amplitudes to be detel mined by the two reactions. The orthonormalization conditions give us 2 [ n + ~ n ( n - 1 ) ] equations, and the reaction cross sections yield 2 n - 1 equations, since the ground state to ground state transitions are related by the microreversibility condition. Hence, the dimension of the solvable problem is determined by the condition 2n z < 2 [ n + ½ n ( n - 1 ) ] + 2 n - 1

or

n < 3.

(2)

If ratios rather than absolute cross sections are used, the dimension of the solvable problem is given by n < 2. The number of additional equations required for solution of a three-dimensional space is two, e.g. one for each nucleus. We also study the charge symmetric reactions (3He, n) and (t, p). If the target is a self-conjugate nucleus, the only difference in the two reactions is the Coulomb interaction, which we assume strong enough to make the equations furnished by the two reactions independent.

TRANSFERREACTIONS

117

In this case, three nuclei are involved. If cross-section ratios are used, we obtain the following condition on the dimension of the solvable problem 3n 2 ~

3n÷½n(n-l)÷2(n-1)÷l

or

n ~ 2.

(3)

We have here assumed that all three nuclei may be described with the same number of basis vectors. The first two terms on the right side in eq. (3) are the orthonormalization conditions. The third term is the number of equations supplied by the crosssection ratios for the transitions to the n lowest lying states with the proper quantum number in each nucleus, and the last term represents the equation provided by the ratio of the two ground state transitions. Due to the different optical-model parameters in the DWBA calculations for the two reactions, the latter ratio may not be sufficiently accurately determined to be used.

2. Assumptions The two sets of reactions we analyse are (i) 4 ° c a ( a H e , n)42Ti [ref. a)] and 4°Ca(t, p)42Ca [ref. 4)] and (ii) 46Ca(t, p)gSca [ref. 5)] and 48Ca(p,t)46Ca[ref. 6)]. This choice is dictated primarily by the small amount of suitable experimental data but also because of the many shell-model calculations on nuclei in the lf~ shell. The calculations are limited to analyses of the J = 0 + transitions for two reasons. Of the m a n y states observed in the '~°Ca(aHe, n)42Ti reaction, only a few spin assignments can be made. Our requirement that transitions to at least two states with the same parity and total angular m o m e n t u m J must be known limits the calculations to J = 0 + transitions. Even in this case the spin assignment for the second excited state in 42Ti is only tentative 3). Secondly, for the second set of reactions 46Ca(t, p)48Ca and 48Ca(p, t)g6ca, the ground states of both nuclei are involved, which again limits the calculations to J = 0 + transitions. Of course, when the ground state of the target is known, the calculations may be extended to other transitions. Table 1 shows the (t, p) transition strengths for the ground state and some excited J = 0 + states in the even Ca isotopes. The striking feature is the weak transition to the first excited J -- 0 + states in 42Ca, 44Ca and 46Ca and the strong transitions to the second excited J -- 0 + states in 42Ca, 44Ca, 46Ca and 48Ca. This suggests that the first excited J = 0 + state in the Ca isotopes is composed of configurations that interfere destructively in the transition amplitude or is composed mainly of a configuration to which the transition is forbidden. The second excited state most probably contains constructively interfering configurations or a configuration to which the transition is "superallowed". A configuration of this kind would be the (2p~) 2, to which we find the transition to go about six times stronger than for (lf~) 2 [rcfs. v, 8)]. If we assume the ground state of the Ca isotopes to contain mainly the (lf~) n configuration, and the dominating configuration in the first excited J = 0 ÷ state to be a ( n ÷ 2 ) particle two-hole configuration - (lf~)n+2(d~) -2 - as discussed in refs. 9, 10), this would account for the weak (t, p) transition, since the transition from the dora-

118

K. KOLLTVE~T

inating configuration in the target to the dominating configuration in the first excited J = 0 + state in the residual nucleus is forbidden. The situation in 48Ca is more unclear. The (t, p) transition to the first excited J = 0 + state in 48Ca is 60 % o f the g r o u n d state transition: i.e. neither strong nor weak. The d o m i n a n t configuration in this state is therefore p r o b a b l y not (fl)~ °(d i ) - 2 unless the g r o u n d state o f 46Ca contains a large a m o u n t o f the (f~)a(d~)-2 configuration. This, on the other hand, is excluded by the weak transition to the first excited J = 0 + state in 46Ca in the reaction 4'*Ca(t, p)46Ca. TABLE 1

Relative differential cross sections for some two-particle transfer reactions with calcium isotopes Grotmd state

~°Ca(t, p)4zCa 4~Ca(t' pp4Ca 4~Ca(t' p)4eCa 4eCa(t' ppaCa 4oCa(aHe, np~Ti 4aCa(p, t)+eCa 4oCa(t' p)42Ca ~) u) e) a) o)

a) a) a, hI a) () ~)

100 100 100 100 100 100

e)

100

First excited state 14 1.3 13 60 10 10 8.3

Second excited state 97 81 14, 55, 37 160

Relative yield y = ~ da/cff2 [ref. 5)]. The experiments show a group of three 0- levels at 5.3-5.6 MeV. Relative peak value at 0° [ret~.a)]. Relative peak value at 40° [ref. el]. Relative peak value at 8° [ref. 4)].

2.1. THE 4°Ca(3He, n)4-~TiAN D 4°Ca(t, p)4'zCa REACTIONS

In 4°Ca, the d o m i n a t i n g configuration is p r o b a b l y the d o u b l y closed ld~ shell, but b o t h theory a n d experiments indicate some core excitation 9 - x l ) . G u i d e d by the calculations in refs. 9, lo), we choose to describe the 4°Ca g r o u n d state by ]'*°Ca g.s.) = :t(l)10)+:t(2)l(f~)2, J = 0, T = 1; ( d ] ) -2, J = 0, T = 1; J=

0, T =

0).

(4)

The f o u r - p a r t i c l e - f o u r - h o l e configuration has to be excluded due to the restrictions on the dimensions c f the shell-model space. As mentioned in sect. 1, the use o f the ratio o f the g r o u n d state to g r o u n d state cross sections for the two reactions 4°Ca(3He, n)42Ti and 4°Ca(t, p ) 4 / C a allows us to treat 4ZTi and 42Ca independently, but due to the large uncertainty in the calculation o f this ratio and in o r d e r to simplify the problem, we will assume that 42Ti and 42Ca are completely charge symmetric, i.e. identical except for the z - c o m p o n e n t of the isospin.

119

TRANSFER REACTIONS

In addition to the (1 f~)2 configuration, refs. 9.1 o) suggest 20-25 % of a four-particle, two-hole configuration in the ground state of 42Ca. Zamick finds a mixture of different four-particle-two-hole configurations, but we are again restricted by the dimension of the problem and are forced to use the dominating one. Fortunately, this coincides with the calculations of Gerace and Green ~). Consequently, we describe the 42Ti and 42Ca ground state by 142Ti, Ca, g.s.> = a(l)l(f~) 2, J = 0, T = 1)+a(2)l(f~) 4, J = 0, T = 13; (d,t)-2, j = 0 ,

T=

I ; J = O , T = 1).

(5)

The first excited 0 + state is here and in the following given by the orthogonal state. As mentioned above, the weak transition to the first excited J = 0 + state might be attributed to destructive interference in the transition amplitude. T o cover this possibility, we alternatively describe the 42Ti and 42Ca ground state by 142Ti, Ca, J = 13, T = 1> = a(1)l(f~) 2, J = 0, T = l ) + a ( Z ) l ( 2 p ~ ) 2, J = 0, T = 1), (6) though we do believe the strong (t, p) transition to the second excited J = 0 ÷ state in 42Ca can only be explained by a large (2p~) 2 c o m p o n e n t in this state, and consequently rule out the description in eq. (6). 2.2. THE 48Ca(t, p)48Ca AND ~Ca(p, t)4eCa REACTIONS As suggested by Zamick lo), we will tentatively describe the ground state of 46Ca and 4aCa as a linear combination of the dominating configuration (lf~)" and a core excited state with two protons excited from the ld~ shell to the lf~ shell. Thus we have 146Ca, g.s.> = ~(l)l(fk) 6, J = 0, T = 3 ) + ~t(2)l(fl) a, J = 0, T = 2, (dk) -2,

J=O,r= l;J=o,r=

3),

(7)

148Ca, g.s.> = a(l)l(f~) a, J = 0, T = 4 ) + a ( 2 ) [ ( f l ) 1°, J = 13, T = 3, (d~) -2, J=0,

T=

l;J=0,

T=4).

(8)

But as mentioned earlier, it seems unlikely that this description of 48Ca can account for the relatively strong transition to the first excited J = 0 + state, and we will therefore also try 146Ca, g.s.> =

=(l)l(f~) 6, j

_- 0, T = 3 > + ~(2)l(q) 4, (p~)Z, j = 0, T = 3>,

(9)

148Ca, g.s.> = a(1)l(fk) 8, J = 0, T = 4 > + a ( 2 ) l ( f ~ ) 6, (p~)6, j = 0, T = 4>.

(10)

3. Calculations T h e calculations were performed in D W B A 12) as prescribed by Glendenning 13). "lhe two-particle transfer-reaction cross section is given for stripping by dtr dO

k 2 2J 2 + 1 rnlm2 kl 2J1 + 1 ( 2 7 r h ~ f22L Sy. I ' T C'2r

M 2 ,

[

:

120

K.

KOLLTVEIY

and for pick up by da

kl 2S1+1 m'm* ~,~ 2 ~-~ C 2 2 S z + l (2nhZ) 2 aL~r s r ~ ] ~

ct[2-kz

GnhsJrB'~12'

(12)

where kl is the wave number of the heavy projectile. The B ~ as defined in ref. 13) is the reaction amplitude and is determined by the kinematics. We used a zero-range potential; and the form factor for the transferred nucleon was calculated as described in ref. 13). The quantity G:vLsJrcontains all the spectroscopic information. It is a product of the overlap between the target and the residual nucleus and therefore contains a product of a two-particle fractional parentage coefficient and the shellmodel amplitudes of both nuclei. Since it is necessary to express the two captured nucleons in relative and center-of-mass coordinates, GNLsJrfurthermore contains a 9-J symbol and a Moshinsky transformation bracket; and finally, the overlap f2n between the relative motion of the two captured nucleons in the initial and final state.

[I~12 !]

GNLSjr = ~. (A 22)a(i)oc(j)(j2(jT)jA(j ' T,), J2 Tz]}JA+2J2T2)

l,J

×(nlltn212LInONL, L ~

½

f2~.

(13)

A As one can see from eqs. (l 1)-(13), the products G,vhsjr and B~L are functions of the shell model amplitudes. They add coherently, and hence the cross section is sensitive to both magnitude and relative phases of the amplitudes. As a criterion for the DWBA calculations, we have required a reasonable fit to the experimental cross sections when only transfer to or from the (lfi)" configuration was considered. Since we do not know the complete wave functions, a search for an optimum fit is meaningless. Most potentials in the literature give the typical forward peak for the reactions considered and the angular distance between the observed peaks. The main difference between the potentials was in the magnitude of thc second peak produced by the DWBA calculations. The pronounced second peak 3) in the reaction 4°Ca(3He, n)a2Ti proved to be sensitive to the form factor or equivalently to the harmonic oscillator constant used. Consequently, we determined the harmonic oscillator constant that best reproduced the two peaks in this reaction (1, = 0.25 fm-2). This value was used for all form factors. Inclusion of the (2p~) 2 configuration did not change the angular distribution, which is to be expected, since these configurations have the same harmonic oscillator quantum numbers when they are expanded in relative and center-of-mass coordinates. The (Ida) -2 configuration, however, gave rise to slight changes in the angular distribution. The transfer amplitudes BffLmay be calculated once and for all for each N, n and M and for each reaction. The quantity Gr,'LsaTmay be calculated as a function of the shellmodel amplitudes, and after multiplying B~L with the appropriate G~Lsjr,summing over

TRANSFERREACTIONS

12 I

the configurations involved, s q u a r i n g a n d s u m m i n g over m a g n e t i c q u a n t u m n u m b e r s M, one finds the cross section as a function o f the shell-model a m p l i t u d e s o f the target a n d the residual nucleus. Thus, one avoids the c o m p l e t e D W B A calculation for each v a r i a t i o n o f the shell-model amplitudes. T h e calculation was p e r f o r m e d for the transition to the g r o u n d state a n d to the first excited J = 0 ÷ state. By t a k i n g the ratio o f the calculated cross sections, we find the ratio as a function o f the shell model a m p l i t u d e s o f b o t h nuclei, a n d we can d e t e r m i n e the shell-model a m p l i t u d e s that lead to calculated ratio equal to the e x p e r i m e n t a l one. T h e result m a y then be plotted as in fig. 1, where

t~ .8

o.

//////

/I

~ j : /

//.

.7 ~

// I

i

/ .6

/

/ __

(3He, n )

//

---- (t,p]

//. f

I

5

6

I ,,,""~

.7

/

/ I

I

.8

9

I0

a(1);4ZCa,42Ti Fig. 1. The amplitude of the dominating configuration in the target as a function of the amplitude of the dominating configuration in the residual nucleus leading to the experimental determined ratio for the cross section to the ground state and first excited J = 0 + state in the residual nucleus. The error bars on the (SHe, n) curve correspond to the experimental uncertainties in the ratio 3) (dtr/d.Q)0=og.S./(dtr/dQ)0=01 ext. = 10:!:40 ~. Wave functions from eqs. (4) and (5) are used. the 3He curve shows the shell m o d e l a m p l i t u d e ct(l) in eq. (4) as a function o f a ( 1 ) in eq. (5) giving the e x p e r i m e n t a l ratio for the 4 ° C a ( 3 H e , n)42Ti reaction to the g r o u n d state a n d first excited J = 0 + state. The calculation is then r e p e a t e d for the (t, p ) reaction. The c o r r e s p o n d i n g curve is drawn, a n d the shell-model a m p l i t u d e s o f b o t h the target a n d residual nuclei are f o u n d f r o m the intersection of the two curves. Similar calculations are p e r f o r m e d for the (t, p ) a n d the (p, t) reactions. T h e r e is a certain a m b i g u i t y in the definition o f the ratio o f the cross sections. In the present calculation, the ratio is defined as the ratio o f the cross sections at small angles, except for the reaction 4aCa(p, t)46Ca where the experimental d a t a forced the use o f the ratio o f the p e a k values at a b o u t 40 ° . A n alternative way is to use the

122

K. KOLLTVEIT

ratio o f the s u m m e d cross sections. F o r t u n a t e l y , the calculations showed no significant dependence on which definition one used, and due to rapid change o f cross-section ratio with variation o f shell model amplitudes, the results were not sensitive to uncertainties in the ratio.

4. Results and conclusion 4.1. THE 4°Ca(SHe, n)42Ti AND 40Ca(t, p)42Ca REACTIONS Fig. 1 shows the results o f this calculation when the g r o u n d state wave functions o f eqs. (4) and (5) are used. The g r o u n d state a m p l i t u d e s that give the correct experimentally determined cross-section ratio are shown graphically. W e find no

s //, ,

I

~sk o~r

/

--

~.7 /

/

I

/

.6

/

/

5

/

/

I

/

1

I

6

1

7

_ _

(3He,n)

----

(t,p)

l

8

I

9

~o

o( I); 42C0,42Ti

Fig. 2. Same as fig. I but with wave functions from eqs. (4) and (6). acceptable intersection o f the 3He and the triton curves. The two curves run a l m o s t parallel indicating that the C o u l o m b interaction m a y not be strong enough to cause i n d e p e n d e n t equations. In that case, the two curves should coincide; this m a y be obtained if 42Ti and 42Ca are described independently, which c o r r e s p o n d s to d r a w i n g the two curves in a different scale. The main uncertainty in the calculation, however, we believe stems from the large cancellation in the calculation o f the cross section to the excited state. The two d o m i n a n t c o n t r i b u t i o n s c o m e from the two transitions 10> ~ [(f~)2, j = 0, T = 1) and [2p-2h> ~ [4p-2h>. In the g r o u n d state cross section, the first transition is d o m i n a t i n g , and the t w o transitions will f u r t h e r m o r e interfere constructively, while the c o n t r i b u t i o n s to the cross section to the excited state from the two transitions will be o f the same o r d e r o f m a g n i t u d e and interfere destructively. Consequently, a large p a r t o f the excited state cross section will come from the much weaker transition [2p-2h> --* I(f~)2> where the two ld~ holes are filled. It is an open question how well the form factors for this tran-

TRANSFER REACTIONS

123

sition are determined; e.g. the harmonic oscillator constant v might have to be changed when considering particles in a lower shell. Fig. 2 shows the results when the ground state wave functions of eqs. (4) and (6) are used. From the intersection of the two curves, we find the resulting wave functions l a ° f a , g.s.> = C.9710>+0.2512p-2h, J = 0, T = 0>, 1"2Ca, "2Ti, g.s.> = 0.6451(f~) 2, J = 0, T = 1>+07651(2p~.) 2, J = 0, T = 1).

(14) (15)

The large a m o u n t of the (2p~) 2 configuration in the (42Ca, Ti) ground state reflects the fact that one needs a large a m o u n t of the (lf~) 2 configuration in the excited state to cause enough cancellation, due to the strong transition to the (2p~) 2 configuration. 1.0.

9

36 ,q.

0

7

.6

I

.5

6

7

8

9

IO

0( I); 4aCe Fig. 3. Samc as fig. 1 but for the reactions 46Ca(t, p)~SCa and 48Ca(p, t)46Ca. E r r o r bars correspond to 30 % uncertainty in the cxperimenta] determination o f the r a t i o f o r the (t, p) cross section to the g r o u n d state and the first excited J :- 0" state in 4sCa [rcf. 5)]. Wave functions f r o m eqs. (7) and (8) are used.

We do not believe that eqs. (14) and (15) reflect the physical state of 4°Ca and of 42(Ca, Ti), though the 4°Ca ground state seems reasonable. The obtained ground state 42(Ca, Ti) would indicate a major breakdown of the shell model and would furthermore make it impossible to explain the strong transition to the second excited J = 0 + state in 42Ca. We conclude, therefore, that two basis vectors are not sufficient to describe the lowest J = 0 + states in 42(Ca, Ti), and that the C o u l o m b interaction possibly is strong enough to distinguish the two reactions (3He, n) and (t, p), but the small angle intersection of the (3He, n) and (t, p) curves in fig. 2 indicate large errors in the determination of the shell-model amplitudes.

124

K. KOLLTVEII"

4.2. T H E 46Ca(t, p)4aCa A N D ~ C a ( p , t)~"Ca R E A C T I O N S

Figs. 3-5 show the solutions to the analyses of the reactions 46Ca(t, p)48Ca and 48Ca(p, t)46Ca, with shell-model vectors from eqs. (7) and (8), (7) and (10), and (9) and (10), respectively. As is seen from fig. 3, no solution is found with the combination (7) and (8). This is in agreement with our earlier statement that the relatively large transition to the 48Ca first excited J = C+ state cannot be explained by a state containing mainly the [(f~.)lO(d~)-2 ] configuration.

0

oi,

, I I

I

.

I I

T

-

(P")

-

--

/

(l,p)

/ /

I o

.7

_

(p J)

_

----

~ 1~

"6

(t,p) I

I

5

i

1

i

I

6

.7

8

.9

I o

a( J );48Ca

I

I

i

6

7

a

I

I I

I 9

a( I);48Ca

Fig. 4. Same as fig. 3 but wave functions f r o m eqs. (7) a n d (10).

Fig. 5. Same as fig. 3 b u t wave functions from eqs. (9) a n d (10).

Fig. 4 shows the results when using the combination (7) and (10) where the intersection defines the ground states of 4 6 C a and 4aCa as 146Ca, g.s.> = 0.901(f~) 6, J = 0> +0.441(f~) a, (d~.)-2, J = 0>,

(16)

1"SCa, g.s.) = 0.94f(f~) 8, J

(17)

=

0)+0.35[(lf.})6,(2P~l.) 2, J

=

0).

The results obtained with the combination (9) and (10) are shown in fig. 5, defining the ground state of 4 6 C a and 4aCa as 146Ca, g.s.> = 0.7151(1f~) 6, J = 0>+0 701(If~.)4(2p]) 2, J = 0>,

(18)

148Ca, g.s.> = 0.881(lf~) s, J = 0>+0 48[(1 f~)6(2p~) 2, J = 0).

(19)

The results in eqs. (16) and (17) show a reasonable amount of the lowest shell-model configuration in the ground state of the nuclei involved. Far more important is the large angle intersection in figs. 4 and 5. This indicates that the solutions are not particularly sensitive to the uncertainties in the calculations.

125

TRANSFER REACTIONS

The use of eq. (10) in a two-dimensional calculation o f 4SCa implies that the (Ill)6 (2p~) 2 configuration is confined to the two lowest J = 0 ÷ states, but as table 1 shows, the (t, p) transition strength for the second excited J = 0 ÷ state in 48Ca is even stronger than to the ground state. This can hardly be explained unless the second excited J = 0 + state has a large a m o u n t o f the (lf/,)6(2p,~) 2 configuration. Energy considerations will also lead to the same conclusion. .9 8 46Ca

.7

~

a(I)

.6

------

a(3,3)~

48Ca

--'--

a(2,2);

48C0

a(l,3)~

48Ca

; ,~

.

.70

.8

I

I

.74

I

I

I

I

.78 .82 aft,I) ~ 48Cc

i

I

.

.

I

.86

"

.

~

.8~

.7

.7

.6

.E

0

/..

..,,

•

-.I-

.J

.~--~"~"

-I

.I

-2

l

.84

]

[

.88

1

I

I

I

.92 .96 a{I,i);48Ca

I

I

1.0 84

J

I

88

]

I

I

I

92 .96 a ( I , I ) ; 48Co

I

I

1.0

Fig. 6. Calculated amplitudes for 4eCa and 4BCa. Wave functions from eqs. (7) and (20) are used. The phase o f a ( l , 2) - not drawn - is positive for all solutions.

We therefore repeated the calculations with a three-dimensional shell-model space for 48Ca with the three low-lying J = 0 ÷ states given by

0~'>

= a(i, 1)l(lf~) 8, J = O>+a(i, 2)l(If~)l°(ld~) -2, J = O> + a ( i , 3)l(lf~)6(2pt) 2, J = 0>. (20) For 46Ca, we used eq. (7). As mentioned in sect. 1, the two-particle transfer reactions alone cannot supply enough equations to solve the three-dimensional problem. We consequently calcuI.l =

126

K. KOLLTVEIT

lated the various amplitudes as functions of the amplitude a(1, 1) for the (lf~) 8 configuration in the 48Ca ground state. The results are shown in fig. 6. Three solutions are found. Two-particle transfer analyses do not favor any particular of the three solutions, but from one-particle pick-up reactions, one might determine one of the amplitudes. The corresponding amplitudes may then be found in fig. 6. Analyses of electromagnetic transitions might also be used. 4.3. C O N C L U S I O N

Though it is difficult to draw any general conclusion from the analysis of the reactions 4°Ca(3He, n)42Ti and 4°Ca(t, p)42Ca, we would like to point out that the (t, p) curve in fig. 1 deviates very little from the curve a = :~, which indicates equal amounts of core excitations in 4°Ca and 42Ca. The failure to obtain any intersection between the (3He, n) and the (t, p) curves in fig. 1 is disappointing but is in our opinion not fatal. The nuclei 42Ti and 42Ca may not be identical in structure, and the two curves should consequently not be drawn on the same scale. Secondly, the assignment, J = 0 + to the 1.89 MeV state in 42Ti is only tentative, though it seems most probable. Furthermore, inclusion of the 2p~ configuration might change the situation considcrably as the results in fig. 2 indicate. The (t, p) and (p, t) calculations seem to be more promising. In this case, the uncertainties introduced by using different optical-model parameters are partly removed. Furthermore, the calculations show a large angle intersection for the two curves in figs. 4 and 5 and indicate that the results arc not sensitive to uncertainties in the calculation. As pointed out, we believe core excitations are the main reason for the weak (t, p) transitions to the first excited J = 0 ÷ state in thc calcium isotopes. It is therefore unfortunate that the transition to the (2p,~) 2 configuration is so strong, since exclusion of this configuration seems damaging for the calculations. On the other hand, by using a three-dimensional shell-model subspace, one might hope that the strength of this transition makes the (2p~) 2 configuration able to represcnt all higher particle configurations. At least the ground statc wave function should be adequately described this way. As one may read from fig. 6, the three solutions with a three-dimensional description of the lowest J = 0 + states in 48Ca show at least one common feature; the amplitude for the (2p~) 2 configuration in the ground state is small. This is in good agreement with other two-particle transfer analyses 7) and with most one-particle transfer results 14.15). Ref. 6), on the other hand, reports a probability of 0.45 for this configuration based on (p, d) rcactions. This result must be considered in disagreement with the two-particle transfer results. It is furthermore encouraging that the three-dimensional results show that the amplitude for the (lf~) 8 configuration is always smaller in the second excited J = 0 + state than in the first in agreement with shell-model considerations. Fig. 6 only shows the solutions for which the amplitudes for the pure lfl configuration is greater than ,,/(0.5). This, to some extent, limits the amount of configuration

TRANSFER REACTIONS

127

mixing in the 4aCa ground state, but the excited states might still be mixed. The results in fig. 6 cannot decide the question of strong or weak configuration mixing in the excited J = 0 ÷ states. Two of the solutions show strong mixing, while the third allows only weak mixing. All the solutions, however, show that the (2p~) 2 configuration is dominating in the highest state. This is in agreement with our earlier statement that the weaker transition for the first excited J = 0 + is due to core excitations. We conclude, therefore, that the results from the two-particle transfer analyses are in good agreement with shell-model predictions. We have pointed out earlier that the calculated cross-section ratios are very sensitive to the shell-model amplitudes. This may also be seen from the figures. The error bars in tigs. 3-5 show the effect of the reported experimental errors on the extracted amplitudes. While the experimental error 5) is _ 30 ~o, figs. 4 and 5 show the corresponding error in the amplitudes squared for the dominating configuraticn in 4SCa is less than 10 °'/,,. Conversely we conclude that a 30 O//ouncertainty in the DWBA calculations would only lead to an uncertainty of less than 10 ''//o in the extracted amplitudes squared. Comparing this with the results from one-particle transfer reaction analyses where the errors in the spectroscopic factors are directly proportional to the errors in the DWBA calculations or to the experimental errors, one realizes that the analyses of two-particle transfer reactions have a definite advantage. Even more striking are the results where a three-dimensional shell-model space is used. In ref. 1), it was shown that in two calculations where the cross-section ratios varied by a factor of more than three, the extracted wave functions had an overlap of 97.7 3/. Consequently, we strongly recommend simultaneous analyses of (t, p) and (p, t) reactions where this is possible. We also suggest a chain calculation of all stable isotopes of a kind. For example, for the calcium isotopes an analysis cf the reactions 44Ca(t, p)*6Ca and '*6Ca(p, t)4*Ca would yield independent solutions for '*6Ca, which x~hen compared with the results of the analysis of the reactions '*6Ca(t, p)48Ca and 4SCa(p, t)46Ca, would lead to accurate statements of the 46Ca wave functions. The results in the present calculation show that the cross sections are also sensitive to the presence of configurations for which the transitions are weak. Thus the twoparticle transfer reactions should yield reliable information about the core excitations, which are difficult to detect by other means. Finally we want to point a possible way to gain more equations from the twoparticle transfer experiment. The strong transitions for the (2p~) 2 configurations in the calcium region is due to the large amount of relative n = 1, 1 = 0 state in this configuration. This is given by the Moshinsky transformation bracket. This state corresponds to a two-particle center-of-mass radial quantum number N = 4 and consequently gives strong contributions for a surface reaction. If the energy of the bombarding particle is increased, contributions from states with lower two-particle center-of-mass radial quantum numbers may be reached. This would strongly favor the transfer of two 1fl particles. The ratio between the two-particle transition strengths

128

K. KOLLTVE|T

for the lf~ and 2p~ configurations should consequently be energy dependent. In that case, i n d e p e n d e n t equations might be obtained from reactions with different bomb a r d i n g energies. Preliminary calculations seem to indicate, however, that the effect is small. The a u t h o r would like to t h a n k Dr. M. H. Shapiro for helpful discussions and for supplying the 4 ° c a ( a H e , n)42Ti data before publication.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) I l) 12) 13) 14) 15)

K. Kolltveit, Phys. Rev. 166 (1968) 1057 K. Kolltveit, R. Muthukrishnan and R. Trilling, Phys. Lett. 26B (1968) 423 M. H. Shapiro, Nucl. Phys. Al14 (1968) 401 R. Middleton and D. J. Pullen, Nucl. Phys. 51 (1964) 77 J. H. Bjerregaard, O. Hansen, O. Nathan, R. Chapman, S. Hinds and R. Middleton, Nucl. Phys. A103 0967) 33 R. J. Peterson, Phys. Rev. 170 (1968) 1003 B. F. Bayman and N. Hintz, Phys. Rev. 172 (1968) I113 R. N. Glover and A. McGregor, Phys. Lett. 24B (1967J 97 W. J. Gerace and A. H. Green, Nucl. Phys. A93 (1967) I l0 L. Zamick, to be published J. C. Hiebert, E. Newman and R. H. Bassel, Phys. Rev. 154 0967) 898 R. Haybron and T. Tamura, unpublished N. K. Glendenning, Phys. Rev. 137 (1965) Bl02 E. Kashy, A. Sperduto, H. A. Enge and W. W. Buechner, Phys. Rev. 135 (1964) B865 T. W. Conlon, B. F. Bayman and E. Kashy, Phys. Rev. 144 (1966) 941