An analysis of the 12C(d, α)10B reaction in the energy range 9.2–13.9 MeV

An analysis of the 12C(d, α)10B reaction in the energy range 9.2–13.9 MeV

Nuclear Physics A121 (1968) 113--127; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written perm...

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Nuclear Physics A121 (1968) 113--127; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

AN ANALYSIS OF THE 12C(d, ~)t°B REACTION IN THE ENERGY RANGE 9.2-13.9 MeV R. KLABES, F. B A L D E W E G and G. S T I L L E R

Zentralinstitut fiir Kernforschung, Ber eich " Kernphysik ", R ossendorf bei Dresden, D D R

Received 27 June 1968 Abstract: Previous measurements of the l~C(d, e)l°B reactioo at Ea = 9.2-13.9 MeV leading to the grouna state and to the first, third and fourth excited states of I°B have been analysed in terms of the Glendenning two-nucleon transfer theory. The theoretical results are in poor agreement with experiment. At Ea = 19.7 MeV, better agreement has been obtained between theory or~d experiment. Excitation functions of low-energy resolution, derived from differential cross sections in the (d, u), (d, d) and (d, p) channels, show resonances indicating that effects due to the intermediate nucleus 14N play a role in the processes investigated.

I. Introduction The success o f the a p p l i c a t i o n o f the direct-reaction m o d e l to single-nucleon stripping or p i c k - u p reactions has stimulated a t t e m p t s to test the t w o - n u c l e o n transfer reactions also in the f r a m e w o r k o f the direct model. I n this connection, studies o f the 12C(d, e ) l oB r e a c t i o n should be o f interest. Previously ~- 3) we investigated this reaction at 11 energies in the range 9.2-13.9 MeV. A n a d d i t i o n a l n u m b e r o f ~2C(d, [email protected] °B r e a c t i o n studies has been p u b l i s h e d at energies f r o m a b o u t 10 to 20 M e V during the last few years 4-7). In all cases, the investigators have assumed t h a t the d o m i n a n t r e a c t i o n m e c h a n i s m is o f the direct-interaction type. W e have earlier 3) i n t e r p r e t e d o u r results in the 9.2-13.9 M e V range o f b o m b a r d i n g energies on the basis o f plane-wave theories using a mixture o f coherent c o n t r i b u t i o n s o f t w o - n u c l e o n p i c k - u p a n d heavy-particle stripping mechanism. Because o f the fact t h a t p l a n e - w a v e theories predict differential cross sections whose m i n i m a a p p r o a c h zero, isotropic c o n t r i b u t i o n s h a d been s u b t r a c t e d f r o m the e x p e r i m e n t a l a n g u l a r d i s t r i b u t i o n s before fitting them. These isotropic cross sections increase with decreasing energy. It is well k n o w n , on the o t h e r hand, t h a t distorted-wave calculations yield a n g u l a r d i s t r i b u t i o n s in which the m i n i m a have values different f r o m zero in the whole a n g u l a r range. O n e interesting aspect is, therefore, to d e t e r m i n e whether a n d h o w r e a s o n a b l e the f o r m a l i s m o f the D W B A m e t h o d can describe the e x p e r i m e n t a l data. D W B A calculations o f (d, e) reactions over the b r o a d energy range 5.5-11.5 M e V have been r e p o r t e d for the 19F(d, ~)170 r e a c t i o n 8). R e a s o n a b l e fits were o b t a i n e d only for the g r o u n d - s t a t e d i s t r i b u t i o n s at higher b o m b a r d i n g energies. T h e a u t h o r s 113

114

R. KLABESet al.

employed a cut-off radius. The use of a cut-off radius should be understood as an additional parameter and may improve in many cases the agreement with experiment. It reduces, however, the value of predictions and, therefore, was avoided in our calculations. In the present paper, D W B A calculations for the transitions to the ground state and the first three excited states except the forbidden transition to the T = 1, Y = 0 + state in I°B at 1.74 MeV have been carried out using Glendenning's two-nucleon transfer theory 9). The theoretical structure coefficients were calculated assuming lowest-order shell-model confgurations given in the LS-coupling scheme. The D W B A analyses were carried out very carefully. The agreement between the theoretical and experimental angular distributions, however, is poor. Therefore additional aspects arising from the behaviour of the excitation functions of the lZC(d, ~), 12C(d, d) and 12C(d,p) reactions l o, l l) were taken into account. These excitation functions exhibit several resonances in the energy range considered. In view of this fact, one concludes that contributions from a mechanism other than the direct one play a role in these reactions. We attempt to discuss this behaviour qualitatively in terms of intermediate structures.

2. Analysis of data Using the assumption that the (d, e) reaction proceeds by the pick-up of a neutronproton pair from the target nucleus, the reaction amplitude has been calculated in a standard way 9) introducing a modified form factor for the motion of the two transferred nucleons

R) = Z

R).

(1)

N

Here N, L and J are the principal quantum number and the orbital and total angular m o m e n t a of the centre-of-mass motion of the nucleon pair, respectively. The coordinate of this motion is denoted by R. The coherent sum over N in eq. (1) shows the correlation in the two-particle motion. All information about the nuclear states involved in the reaction is contained in the structure coefficients G which are products of three overlap integrals GNLJ = Z 9 flrL$ O n ( n 0 , y

NL; L[nl ll, n212; L ) .

(2)

Eq. (2) is a coherent sum over all the possible two-particle configurations. The factor 9 = 1, if the two nucleons are picked up from the same single-particle shell, otherwise 9 = x/2. The factor s2n denotes the overlap between the relative motions of the neutron and proton in their initial and final states; (nO, NL; L]nlll, nzlz;L) are Moshinsky transformation brackets 11). The overlap integral fl~zs measures the parentage of the target nucleus based on the fiaal nucleus plus a neutron and a proton in the state 7 ( = nil1, n2/2), L, J with

12C(d, cx)l°BREACTION

115

T = 0 and S = 1. These spectroscopic amplitudes fl have been calculated using the expression reported in ref. 3). The nuclear shell-model wave functions for the initial and final nucleus are given in the LS-coupling scheme 13). In the 12C(d, c01 °B reaction, the nuclear structure coefficients GNzJ are listed in table 1. The selection rules for the two-nucleon transfer allow L = 2 for the transitions to the g r o u n d state and the fourth excited state and L = 0 and 2 to the first and third excited state in 10B. F r o m table 1, one m a y see that for the third excited state the d o m i n a n t orbital angular m o m e n t u m is L = 0, whereas for the transition to the first excited state no L-value dominates. TABLE 1 N u c l e a r structure coefficients

J

E x c i t a t i o n energy (MeV)

L

GNLj N=0

eo e1

0 0.72

3 1

~

2.15

1

~

3.58

2

2 2 0 2 0 2

N=I

--1.1295 --0.5634 --0.1177 --0.2019 0.2046 --0.7559

0.5112 --0.9003

TABLE 2 Different sets o f o p t i c a l p a r a m e t e r s for elastic deuteron scattering f r o m 1~C (r0 = r e = 1.5 fro; a = 0.65 fro; b = 0.8 fin; s = 0.5), as well as for elastic e- s c a t t e ri ng from 9Be (r0 = 1.45 fm; r e = 1.3 fm; a = 0.62 fm; s = 1)

E (MeV)

Set I

Set II

Set I I I

Set IV

U W Z 2e) (MeV) (MeV)

U W Z 2°) (MeV) (MeV)

U W Z ~°) (MeV) (MeV)

U W Z3e) (MeV) (MeV)

Deuteron potential 10.6 a) 11.0 11.4 11.9 12.4 12.8 13.2 13.9

33.9 37.3 34.0 34.3 31.5 31.7 31.5 29.7

11.4 12.0 12.6 11.6 12.9 13.0 13.1 15.3

29.3 19.5 33.8 39.0 38.8 30.3 49.3 75.1

73.8 74.7 71.6 71.0 69.8 69.6 71.2 73.1

18.8 18.8 17.8 14.7 14.2 13.3 14.3 17.3

14.9 19.1 16.2 16.0 12.9 10.4 22.0 25.3

129.2 111.1 109.2 111.1 112.3 112.4 111.4 110.8

27.3 20.3 23.8 21.3 22.9 22.8 24.9 30.8

22.9 30.2 30.0 40.5 31.3 30.3 30.6 30.6

170.7 175.5 175.1 177.8 177.3 177.0 174.1 167.9

40.3 29.8 32.0 28.4 33.9 33.4 35.4 42.6

32.2 24.5 24.8 36.0 23.0 25.5 25.5 41.0

185.6

11.1

26.2

256.0

12.9

26.2

e-potential b)

"/7.4

6.40

30.7

126.0

8.43

26.5

a) D W B A calculations at Ed -- 9.2 MeV and 10.0 have been p e r f o r m e d w i t h the de ut e ron opt i c a l potentials deduced at 10.6 MeV. b) Potential p arameters averaged over the range 11.5-14.86 MeV. °) In the calculation, we t o o k JCre×p/~Texp = 10 °/oo unif orml y, therefore the Z~--values for different sets o f d a t a could be more directly compared.

g . KLABES et al.

116

The optical-model parameters used in the distorted-wave calculations are sumrrmrized in table 2. The optical potential in the reaction channels was chosen to be

V(r) = Vc(r ) - Uf(r)-iW[sf(r)+ (1 -s)9 (r)],

(3)

where 0 < s < 1 determines the mixture between volume and surface absorption,

f(r)= 1/(l+exp(r~)) # ( r ) = exp

,

( (bt -

,

R = roA ~.

(4)

For the entrance channel optical-model parameters have been taken f r o m ref. 11), which describes the analyses o f 12C(d, d) elastic scattering data at eight energies from 13,9 MeV

lo r

.

1 r

o 1

I0~

J

10

~

I

i

i

i

i

i

12.8MeV

]

i

13,2MeV

13,5MeV



T

I

t

i

i

12,4 MeV

i

i

i

r

. _

i

~

1

i

t

i

11,9 M e V

i

r

i

i

i

I0,6 MeV

11,0 MeV

..



i

i

'

3'0

610

910

i

i

i

9,2 MeV

lO,O'MeV

i

120

i

150

310

610

i

90

i

120

i

150

3TO

610

910

ecM

Fig. 1. Results of DWBA calculations for the transition to the ground state,

i

I

120

150

12C(d, ~)I°B REACTION

117

10.6 to 13.9 MeV. At each deuteron energy, four sets of parameters have been found at real potential depths of approximately 30, 70, 110 and 170 MeV. No optical parameters, however, deduced from elastic scattering data of e-particles on t°B are available for the exit channel. Over a broad energy range, Taylor et al. 14) 13.2 MeV

13.SMeV

13.9MeV %°

F

T

I

E

"

I

I

."

r

i

]2,SMeV

lO .1 •

.'•

~

°

°

' ,

..'..

r

.•. I • °~

,,.;M~

30

60

I

./. "-

r

I

90

120

150

i

I

• lO.6Mev

-o.o

°

*° r

r

30

6LO

lqOMeV

•.

.•" i

o.

,~" 9.2 MeV

90

J

r.

120

150

30

i

i

I

50

90

120

150

OCM Fig. 2. Results of DWBA calculations for the transition to the first excited state. have measured the 9Be(e, e) elastic scattering. They performed an optical-model analysis of their data including a term coupling the target spin to the orbital angular momentum• No investigation, however, has been carried out with respect to the real potential ambiguity. We have, therefore, carried out optical-model analyses of their data in the energy range 11.5-14.86 MeV using a pure volume absorption. In the

118

R. KLABF_.Se t al.

analyses, the real and imaginary potential depths have been varied only, whereas the geometrical parameters were fixed. As in the deuteron scattering on 12C, four different sets of parameters could be found yielding minima of the x2-function. Theoretical and experimental angular distributions are in satisfactory agreement up to 90 °. At larger angles, Taylor et aL 14) improved the quality of their fits including the additional spin-orbit term mentioned above. The potential depths U and W obtained for 13,9

MeV

/

135 ,MeV

I



" lO,OMeV ....

30

60

90

120

1

I

9,2

150

30

GO

t

, 1 M3 eV, 2

"i /

I,

MeV I

90 120 GCM

150

310 610

9tO 1,~0 150 t

Fig. 3. Results o f D W B A calculations for the transition to the third excited state.

each angular distribution in the analysed energy range have been averaged for all four sets of parameters and are listed in table 2. For the deuteron energy of 12.8 MeV, DWBA calculations have been performed for the ground state transition using the four sets of optical deuteron parameters and combining each of them with each set of optical parameters in the a-channel. The optical model for composite particles reported by Rook 15) and Abul-Magd and E1Nadi 16) predicts real potential depths of approximately 100 MeV for deuterons and

12C(d,

~)l°B

119

REACTION

• °I >o



/

I

I



eo

]"

I

• °e •

• •

I

.I

0

• • °~o eB°

~

oo

~

• • o•



°



•I

i

o• •

oo

I'"

)'..

?

.° o• e•

-

N

I

I

Js/~p

muV~

120

R. KLABES et al.

200 MeV for a-particles. According to this prediction, the parameter sets II and III for the deuteron channel and III and IV for the s-channel should be of interest. Set III in the deuteron channel (U ~ 110 MeV) and set IV in the a-channel (U = 256 MeV) provide for the best fit with respect to the correct position of the forward peak. Therefore, these parameter sets have been used in the D W B A calculations. The modified form factors have been calculated by means of oscillator wave functions matched to Hankel functions corresponding to the binding energy of the transferred nucleon pair. The value of the oscillator size parameter used was 0.3 fm -2. Figs. 1-4 present the ~2C(d, a) 1°B results for the transition to the ground state and the first, third and fourth excited states for the different deuteron energies. For the transition to the fourth excited state, D W B A calculations were carried out only in the energy range 11.0-13.9 MeV, because no experimental results are available at lower energies.

3. Discussion The theoretical fits obtained in the D W B A analyses are in rather poor agreement with experiments. In general, the differential cross sections at larger angles are partially overestimated and do not show the correct diffraction pattern. A separate calculation was therefore performed to improve the agreement between theory and experiment changing the values of optical parameters in the a-channel. This attempt, however, led to unrealistic imaginary potential depths (60-80 MeV), which are not consistent with results from elastic scattering data. Calculations using other types of parameters in the entrance channel yielding better fits in the elastic scattering 1:, is) were also performed with little success. In figs. 5 and 6, the cross sections of the 12C(d, d), 12C(d, a) and 12C(d, p) processes ~0,11) from the measured angular distributions at 1 2 5 ° and 165 °, respectively, are plotted as a function of energy. Measurements of the '2C(d, p) reaction were carried out by the authors simultaneously with the elastic deuteron-scattering experiments. A resonance-like behaviour has been found ia the different exit channels. There is no reason to explain the resonances in the elastic scattering channel as a shape elastic effect, since calculations with smoothly energy-dependent optical parameters do not reproduce them. On the other hand, the Feshbach formalism in connection with the doorway-state hypothesis provides a framework in which it is possible to explain the resonances observed as intermediate structures. The criteria for the existence of intermediate structures include 19) (i) correlation of the cross sections for different exit channels, (ii) correlation of the cross sections for different angles and Off) larger resonance widths than the widths of the compound-nuclear levels but smaller than the widths of the structures in the optical-model cross sections.

lee(d, ct)IOB REACTION

12[

These criteria can be satisfied within the experimental accuracies as shown in figs. 5 and 6. The widths of the compound-nuclear levels in 14N and the structure widths in the optical-model cross sections were estimated to be 20 to 50 keV and approximately 1 MeV, respectively 20).

/

25

/

/

/

0f"~o

\

t2c (d.po) \

125 o

\ o \

\

uc ~d,OCo) 125 °

Io ' ' " 0\ / 5

-2

/

\

fo.'~

\\

o~

%

/

\ \

J

~c

o .~ _ o .,..o .~

",o

~2c (d.d) 125 °

o

\ \

\

/Io~

\

\

/

~8

~ f o - - ~0~.

/ o "~ ..~. o.~.

\

F I

I0

II

12

13

Ed (MeV)

Fig. 5. Excitation functions extracted from the differential cross sections at 125°. From this point of view, it is possible to interpret the resonances qualitatively in terms of the doorway-state concept. In order to predict the quantum numbers of the doorway states such as the total angular m o m e n t u m and parity, the partial wave expansion in the optical-model analysis may be used. Assuming the optical parameters fitted reflect the behaviour of the experimental 12C(d, d) cross sections, the partial

122

R. KLABES

e t aL

wave expansion m a y be t a k e n as a criterion for a selection indicating that certain partial waves are p r e d o m i n a n t l y responsible for the resonances i n the cross sections.

teC,d.pc ) 165"

2.5

.A-

j.-~.

0\

) i//O

~o~,, / \o / ~o/-°'~°

f-~\ o

\

i

\

\

12c(J.0%)

I

o

r65"

\

/

\

/

~

\o-Lo_o_o ,

/

72cv~ J O /\

165 °

I .\ I

/o\

i

\

I I

\

/I \ \ / \o / /

\0/%

\

/

o

7

I0

\

II

12

I "~

\

\

Ed (MW)

Fig. 6. Excitation functions extracted from the differential cross sections at 165°. I n fig. 7 are d r a w n the partial reaction cross sections L t O'reae

= ( 2 L + 1)(1 --IS,.1~)~,¢ ~

(5)

as a f u n c t i o n of energy. Here, Sr is the scattering matrix element. Resonances ob-

l~C(d,

cQI°B REACTION

123

served in the elastic d e u t e r o n scattering at energies o f 11.0 a n d 11.9 M e V are described essentially in a f o r m a l m a n n e r b y the p a r t i a l waves L = 4 a n d 6. T h e o t h e r one at the energy o f 12.7 MeV, however, a p p e a r s m a i n l y in the p a r t i a l waves L = 3 a n d 5. A c c o r d i n g to the spin a n d p a r i t y c o n s e r v a t i o n rules, g = J o + L + S a n d ninf = ( - ) n , the resonances in the even p a r t i a l waves m i g h t result f r o m d o o r w a y states with the q u a n t u m n u m b e r s J~ = 5 + a n d the o t h e r one in the o d d p a r t i a l waves f r o m d o o r w a y states with J= = 4 - .

3O

/

\.

./

5

3

g2o

2

x) f.1

. . . . . . . . . . .

9

"~ . . . . . . . . . . . . .

tO

,. . . . . . . . . . . . . .

II

......

0

T . . . . . . . . . . . .

• . . . . . . . . . . .

7

12

13

14

Ea (mev)

Fig. 7. The quantities O~react= 7tSZ(2L+l)(l--IaL[~) derived from the optical-model partial wave expansion for 12C(d, d) scattering fits as a function of energy. TABLE3 Configurations and level paaameters in 14N Ed (MeV)

14N* exp (MeV)

11.0 l 1.9

19.6 20.3

12.7

21.0

Lexp

dexv

20.1 20.8

4, 6

5+

7+ . .. 3+

(p~.)-l(p~)(d~r)2 (p~_)_2(pt.)2(dg_)(dsr)

19.7 19.7

3, 5

4-

6 - . . . 2-

(p~)-I (2s~:)(d~r)(d~) (p~)_2(p½)(2s~)(d~r)2

14N* theor (MeV) }

dstloweo

Possible configuration

The configurations o f the d o o r w a y states responsible for the i n t e r m e d i a t e structures m a y be given b y m e a n s o f a crude estimation. Since the o r b i t a l a n g u l a r m o m e n t u m o f the relative m o t i o n in the d e u t e r o n is zero, the p r o b a b i l i t y t h a t a n e u t r o n a n d p r o t o n are c a p t u r e d in shells with the same o r b i t a l a n g u l a r m o m e n t u m can be large.

124

R. g_UABESet al.

In table 3, the states arising from the single-particle levels of 12C and 13C are summarized. Total angular momenta, parity and level energies are in good agreement with the experimental ones. We have earlier 1,2) plotted the integrated cross sections for the four a-groups as a function of energy. The 0~-curves show a broad resonance at deuteron energies from 11 to 13 MeV. In the case of the ground state transition, the broad resonance is shown in fig. 8. The integrated cross sections derived from the D W B A calculations reflect to

t2C (d,o( o ) IOB

~ 1 I

o--o

exp.

~ --

DWBA

60

~o/11

~. I

~

V

/ /

I\\ I\\

20

I0

11

t2

13 Ed l lqeV

Fig. 8. I n t e g r a t e d cross sections c a l c u l a t e d and m e a s u r e d for the g r o u n d state t r a n s i t i o n as a function o f energy. The t h e o r e t i c a l results are n o r m a l i z e d to the experime,atal ones.

some extent this resonance-like behaviour. This fact can be understood as an incorporation of intermediate-resonance effects into the deuteron optical parameters. For these reasons, studies to compare direct-model predictions and experiments should be carried out at energies high enough to neglect effects from the intermediate nucleus. F r o m this point of view, D W B A calculations have been attempted for an energy of 19.7 MeV. In fig. 9, the results are shown with the experimental points of Yanabu et al. 4). The deuteron optical parameters used have been obtained from an analysis of the elastic scattering of 19.1 MeV deuterons measured by Freemantle e 1) (table 4). The 0c-particle parameters are from an analysis of the scattering of 21.6 MeV a-particles on 180 prepublished in ref. 8).

125

12C(d, :t)10B R E A C T I O N

I n general, the agreement with experiments at 19.7 MeV is better t h a n at lower energies except for the t r a n s i t i o n (d, ~3) to the state at 2.15 MeV excitation energy. 10

n

t

12C(dpc)tO B oo o •

I\

" °°°

"C



g. S.

Ex-O'72MeV



\

Ex -2,15MeV





\'"

T"/



O.t

I

I

20

60

I

90

I

12o



i

t

750 OCt4 180

Fig. 9. Results of DWBA calculations at Ea = 19.7 MeV. The dashed line describes a result obtained reducing the L = 0 structure coefficients by a factor of 10.

I n table 5, the calculated a n d m e a s u r e d integrated cross sections (relative to the g r o u n d state t r a n s i t i o n ) are c o m p a r e d for the different final states. I n order to illustrate

126

R. KLABESet

al.

the influence of different incident energies, this c o m p a r i s o n has been performed at 11.9 a n d 19.7 MeV. I n the case of higher incident energy, the theoretical predictions reproduce the relative experimental intensities sufficiently well again except for the (d, ca) transition. Obviously the predicted L = 0 d o m i n a n c e in this t r a n s i t i o n is n o t in agreement with experiment. I n order to remove this disagreement, it is necessary to use a better description of the J~ = 1, T = 0 level at 2.15 MeV excitation. I n a somewhat sophistical attempt, the L = 0 structure coefficients have been reduced by a factor of 10. One obtains a slightly i m p r o v e d fit of the a n g u l a r d i s t r i b u t i o n as can be seen from fig. 9 (dashed line). TABLE4 Optical-model parameters for elastic deuteron scattering from 12C at 19.1 MeV U (MeV)

Wvot (MeV)

Wsurfa) (MeV)

ru = rwv = re (fm)

126.6

6.88

6.88

0.72

au = awv (fm) 0.99

rw5 (fm)

aw5 (fm)

g2

1.44

0.74

10.42

a) Woods-Saxon derivative. TABLE 5 Comparison of relative 12C(d,~) intensities at 11.9 and 19.7 MeV with the predictions of twonucleon transfer theory Ed (MeV)

(d, ~o)

(d, ~1)

11.9

100 100

19.7

(d, ~8)

(d, ~4)

124

76

72

73

148

38

theor.

35

exp.

43

theor.

100

54

100

58

28 117(15) a)

exp.

a) The value within the brackets is obtained reducing the L = 0 structure coefficients by a factor of 10.

A t 11.9 MeV incident energy, the agreement between calculated a n d experimental intensities is n o t satisfactory. This supports the a s s u m p t i o n that the ~2C(d, ~) reaction is n o t p r e d o m i n a n t l y direct in this energy region. The authors are i n d e b t e d to Professor Dr. J. Schintlmeister for s u p p o r t i n g this work. They wish to t h a n k Dr. C. Riedel for m a n y helpful a n d stimulating discussions. F i n a l l y we would like to express our thanks to Dr. H. W. Barz for supplying his code to calculate the D W B A curves.

12C(d, ~)IOB REACTION

127

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)

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