Direct neutron capture in light nuclei

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 619 (1997) 49-56

Direct neutron capture in light nuclei A. Likar, T. Vidmar J. Stefan Institute and Department of Physics, University of Ljubljana, Ljubljana, Slovenia

Received 5 December 1996; revised 17 February 1997

Abstract The consistent direct-semidirect capture model (CDSD) is applied to the keV energy region where the direct process is dominant. We study the influence of the correction of the form of the direct term due to a difference AV in the potentials used to describe the scattering and the bound single-particle state of the captured neutron. The sensitivity of the cross sections on the semidimct capture mechanism in this energy region is examined as well. We consider the 12C(n,7) and 160(n,3/) processes which are of astrophysical interest in the energy region of the incoming neutrons between 10 keV to 0.6 MeV where good experimental data exist. It is shown that both corrections are negligible for capture to the states where s-waves are not involved. For s-wave capture to the p-orbits the zlV correction of the direct term is important. The data do not support the contribution of the semidirect term as described by the model in the energy region studied. (~) 1997 Elsevier Science B.V.

1. Introduction An important ingredient o f the DSD capture model is the amplitude for the direct capture of an incoming nucleon by the target nucleus. In the region of giant dipole resonance this term interferes with the dominant semidirect amplitude, which describes the collective core excitation o f the target nucleus and subsequent decay to the same final state as in the direct capture process [ 1,2]. For the nucleon capture in the keV energy region the semidirect amplitude only functions as a correction to the dominant direct capture mechanism. Recently it has been shown that the basic formulation of the DSD model has to consider the effect o f complicated states via the difference between the optical potential and the potential o f the final state. The dominant part of this difference is due to the imaginary part o f the optical potential. In the keV region the optical potential is essentially real, but the difference o f the two real potentials is still as great as 10 MeV, 0375-9474/97/$17.00 Q 1997 Elsevier Science B.V. All rights reserved. PH S0375-9474(97) 00129-2

50

A. Likar, T. Vidmar/Nuclear Physics A 619 (1997) 49-56

since the bound single-particle states are in direct capture model generated by changing the depth of the potential well in order to reproduce the experimentally measured binding energy of the final state. The difference in potentials which generate the scattering and the final orbits opens up a basic problem in quantum electrodynamics. In the development of the transition amplitudes it is essential that the same Hamiltonian for generating the initial as well as the final states is applied. Approximations of the Hamiltonian suitable to generate the scattering wave functions should reflect on the dipole operator as well. In the past such a possibility has been neglected. One may argue that a change of the potential during the transition process is a consequence of a certain rearrangement of the nucleons in the target nucleus which may contribute to the transition amplitude. In Ref. [2] a method has been developed with the aim of finding an effective dipole operator based on the difference between the optical model potential and the potential best suited to describe the bound final orbit. The CDSD model proved to be able to solve the old dilemma about the imaginary strength of the particle-vibration coupling and very successful in describing the radiative capture of energetic nucleons [2]. It is interesting to examine to what extent this AV correction is important in the direct capture calculations of the low energy nucleons. Mengoni et al. [3] have extended the direct radiative capture (DRC) model to include the p-waves and higher partial waves and pointed out that the capture of slow neutrons to the final state where partial waves other than s-waves are involved lead to the cross sections which are to a high degree independent on the scattering potential. They analyzed the slow neutron capture by 12C for which the experimental data has been reported in [4]. We have extended their calculations by including the AV correction to the direct amplitude and by adding the semidirect amplitude to the T-matrix. The results obtained were further confirmed on the neutron capture by 160. In the following we give a short description of the consistent version of the DSD model and report on the results obtained by applying it to the above-mentioned reactions.

2.

Theory

The transition amplitude T in the CDSD model is written by the use of the effective dipole operators and form factors as

T = (~Yn,l,j ( r ) I n r ( 1 + F ( r ) )1¢ + ( r ) ) +

(~On,tj(r) I n ' ( 1 + F ( r ) ) 2 l ¢ + ( r ) } E - ER + iFR/2

(2.1)

The first term is responsible for the direct radiative transition of the incoming nucleon to a bound state of the target nucleus. The second term is due to the excitation of the target nucleus by the incoming nucleon to a multipole resonance state and a subsequent radiative de-excitation of the excited nucleus. In this way the nucleon inelastically scatters from the initial state to the same final state as in the direct capture. In the radiative capture of low energy nucleons only the low energy tail of the resonance

A. Likar, T. Vidmar/Nuclear Physics A 619 (1997) 49-56

51

is excited. We assume that here also the y-strength function is described by a simple Lorentzian parametrization. The single-particle dipole operator is denoted as H r. The free form factor H~(r) of the semidirect term is derived from the isospindependent component of the short-range two-body force W which is responsible for the excitation of the giant multipole state in the target nucleus [5,6] :

V' = Z

V~6(ri - r)~'. ~'i.

(2.2)

i

The detailed derivation of the form factor can be found in Ref. [2]. We use here the volume-shaped form factor [6]

H'

2NZ

sum

=--ro3---~f

h2 Vl ( _ ~ r f ( r ) ) y , ~ ( r ) 2mER 4

(2.3)

The fraction of the classical sum rule fsum exhausted by the GDR is given via the measured energy-weighted photo-absorption cross section o - l . The function F(r) follows from the differential equation [2]

yf"+

+ \ Er

- 1 r

+

--~- -

+3A so --E--7-j

f = -~-,

(2.4)

where f ( r ) defines F(r) via

F(r) = 3 f + rf'.

(2.5)

The parameter y is proportional to the inverse photon energy E r,

y = hZ/2mEr.

(2.6)

AV is the difference between the optical model potential which defines the wave function ]~p+(r)), and the potential which generates the final wave function ICt,.t,j(r)). The corresponding difference of the spin-orbit potential is defined as Vso,opt- Vso,,/j and is not zero due to different l and j values in optical and final-state wave functions. The solution of Eq. (2.4) should be regular at small radii and as well as at infinity, r --, c~.

3. Results and discussion We first concentrate on neutron capture to the p-orbits in 12C. The ground state of 13C is predominantly the 2pl/2 orbit with the spectroscopic factor 0.77 [7]. The calculated cross section for this state is very sensitive to the scattering state potential. The change of the depth of the optical potential of 3 MeV from the value which yields the cross section, which is in agreement with the measured values, changes the cross section for the ground-state transition by an order of magnitude. The optical potential used to obtain the solid line in Figs. la and lb has a small surface absorption term with the strength proportional to the neutron energy: We = 2E,, in agreement with [8]. The depth of the initial Saxon-Woods potential is found to be 60.0 MeV with the radius parameter

A. Likar, T. Vidmar/Nuclear Physics A 619 (1997) 49-56

52

ii

--

,2C(n,y)

f

6-

20

si

j-4

~

lpv2g.s.

"

~

15

n,y)

" 0 - 00

T 01

, 02

-

~ 03

04

05

06

0 ~ O0

--r 01

--

T 02

En [aeV]

• 03

----~ 04

0r5

06

En [UeV]

3 0 (b)

--

~

6 (d) .

2~i[

120(n,y)

s

2o i

i P3/2

4

.

.

.

.

.

120(n,y) 1ds/2

0o 00

~ 01

02

o~ 03

En [MeV]

°4

05

0.6

0.0

01

02

03

04

0

0.6

En [aeV]

Fig. 1. Angle-integrated cross sections as the functions of neutron energy for the neutron capture (a) to the lp]/2 ground state of 13C, (b) to the 1p3/2 excited state at 3.684 MeV of 13C, (c) to the 2Sl/2 state at excitation energy of 3.089 MeV and (d) to the lds/2 state at 3.854 MeV excitation energy in 13C. The experimental data are taken from Ref. [4]. The results of the CDSD calculationswith the absorptiveoptical model potential are shown as solid lines. The dashed line represents the results with a real scatteringpotential and without the function F(r). The dotted line was obtained by using the function F(r). For the capture to the 2sl/2 and lds/2 orbits the dotted and dashed lines coincide.

r0 = 1.236 MeV, diffuseness a = 0.62 fm and spin-orbit strength Vso = 7 MeV. Mengoni et al. [ 3 ] used a real initial potential with the same parametrization. In their calculations the best fit to the experimental cross sections was obtained using the value of the depth of 59.2 MeV, which is the value of the final potential reproducing the correct binding energy of the 2Sl/2 state at the excitation energy of 3.089 MeV. The effect of dV correction was studied using the real scattering potential. The semidirect contribution was excluded here by setting the parameter Vl to zero. The dotted line indicates the results with the function F ( r ) included into the direct term. The dashed curve coincides with the results of Mengoni et al. and is obtained with the function F ( r ) set to zero. We may conclude that the correction changes the results by a factor of 2 in the energy region of the measured cross sections. The correction is

A. Likar, T. Vidmar/Nuclear Physics A 619 (1997) 49-56

53

Table I Single-particle states in 13C and 170. Bn is the neutron binding energy and S the spectroscopic factor State

Bn I MeVI

S

V[MeV]

lpl/2 2sl/2 Ip3/2 I ds/2

4.946 1.857 1.262 1.093

0.77 0.65 0.14 0.58

45.95 59.20 28.54 56.34

I ds/2

6.739 6.170

0.90 0.90

53.19 53.17

13C

170 2sj/2

'°1/~, /

I 20 ~

.I~

i

13 15

/

12C(n'7) En = 20 keV

'~,

O

11

0

~

0

~\

\ oo

10

i 5~

54

°

56

0

O

O

o

x

58

60

62

64

66

68

70

VRE[MeV] Fig. 2. The dependence of the cross section on the depth of the scattering potential. The solid line denotes the capture cross section for the lpl/2 ground state of 13C and the dashed line for the 1p3/2 excited state. The same spectroscopic factor of 0.77 was used for both final states. The neutron energy was 20 keV.

less significant for the capture to the weakly excited lp3/2 state at the excitation energy of 3.684 MeV. This is expected since the correction depends on the strength of the difference AV which is more pronounced for the ground-state transition (see Table 1 ). The experimental data for the capture to the excited 1p3/2 state have a large experimental error since the state is weakly excited due to the small spectroscopic factor, being only 0.14. In spite of this, it is apparent that the dotted curve representing the results with the AV correction included is too low with respect to the data points, while for the capture to the ground lp3/2 state the corresponding curve is significantly too high. A small decrease of only 1 MeV of the depth of the scattering potential decreases the ground-state capture cross section to the experimental points, increasing at the same time the curve for the capture to the excited state towards the observed values. Fig. 2 shows the dependence of the two cross sections on the potential depth. The solid line with full circles represents the capture cross section to the ground state for neutrons with an energy of 20 keV and the dashed line with empty circles the capture to the lp3/2 excited state. Above the neutron energy of 20 keV the cross section shows qualitatively the same behaviour. The cross sections for these two transitions are clearly anti-correlated

54

A. Likar, T. Vidmar/Nuclear Physics A 619 (1997) 49-56

in the region of the depth close to 60 MeV. Dashed curves representing the results of Mengoni et al. [ 3 ] are on the other hand below the experimental data for the capture to either of the p-states. This is an optimum fit if one considers both transitions at the same time. An adjustment of the scattering potential to obtain a good fit to the experimental points for the ground-state transition would push the curve for the excited state transition much below the observed values. We may conclude that the experimental data support the AV correction as far as s-wave capture is considered. Certainly we would like to compare theoretical calculations with more accurate data. Being aware of the difficulty of measuring neutron capture cross sections in the f e w / z b region, we hardly hope that such data would soon be available. The main point made in [3] is the observation that the p-wave capture the cross section is highly independent of the scattering state. Even the hard-sphere potential and plane-wave approximation yield almost identical results. We confirm here this conclusion. Even more, the AV correction and semidirect contribution are negligible. Therefore the cross section for neutron capture to the 2s~/2 state at the excitation energy of 3.089 MeV and to 1d5/2 state at 1.09 MeV is a very good sensor of the corresponding orbits. The agreement of the direct capture results with experiments can be judged from Figs. lc and ld. The solid line is obtained using a weak surface absorptive potential and the dashed line with the real potential. The AV correction is negligible, the results obtained with the function F ( r ) from Eq. (2.5) and without it are within the thickness of the dashed line. It seems that the calculations are systematically below the experimental points though well within the experimental error for both transitions. Significant modifications of the bound orbits by changing the range and diffuseness of the final potential well cannot force the calculated curve up. One possible explanation of this effect could be a systematic uncertainty of the experimental data, either of the cross sections or of the spectroscopic factor. The conclusions given in the last paragraph are supported by the analysis of neutron capture in 160. The capture to the ld5/2 ground state and to the 2Sl/2 excited state at excitation energy of 0.871 MeV in 170 show the typical picture of p-wave capture. Both states have large neutron single-particle strength with spectroscopic factor 0.9. The agreement with the experiments taken from Ref. [9], as shown in Figs. 3a, b, is very good. The parameters used in the calculation are the same as the ones used in the treatment of 12C. The dashed curve represents the calculation with real potential and the solid line with the surface absorption strength proportional with the neutron energy: Wa = 2En. The results closely agree. In order to estimate the contribution of the semidirect amplitude one needs to know the position of the dipole giant resonance in the nucleus and the fraction of the sum rule exhausted by the resonance. The situation in 12C is not a simple one since two resonances have to be assumed to successfully reproduce the capture cross section [2]. In this work we study the capture of neutrons with energies low compared with the energy of the giant dipole resonance. We therefore assume the existence of the lower part of the resonance at excitation energy of 11.5 MeV having a width of 7 MeV and the energy-weighted photo-absorption cross section of 0.3 fm 2 in order to see the

A. Likar, T. Vidmar/Nuclear Physics A 619 (1997) 49-56 4~ (a)

loo

(b)

_

55

_

,o

35

8(3

~ ~~

30

b

~.

20

/~/

~! ,

,//Y

E)

~

40

/~

160(n,7)

lO

//

¢

lds~ g.s.

160(n,~, ) 2Sl/2

20

5

O0

01

02

03

04

05

06

O0

01

02

En [MeV]

03

0.4

0.5

06

En [MeV]

Fig. 3. Angle-integrated cross sections as the functions of neutron energy for the neutron capture (a) to the Ids/2 ground state of 170, (b) to the 2s]/2 excited state at 0.871 MeV of 170. The experimental data are taken from Ref. [9]. The results of the CDSD calculations with absorptive optical model potential are shown in solid lines. The dashed line represents the results with the real scattering potential and without the function F(r). The dotted line results from the use of the function F(r).

4 (a)

s (b)

12C(n,~,)

.

.

.

.

.

.

.

.

.

.

12C(n,~/)

3 l

1 P1/2 g . s .

2

1 P3/2

t~

1

,

En [MeV]

En [MeV]

Fig. 4. The results of the calculations with the semidirect amplitude included. The results for capture to the p-orbits by 12C with an absorptive optical model potential are shown in solid lines. The depth of the final-state potential was adjusted to fit the capture to the ground state and was found to be 58 MeV. effect clearly. The cross section for the ground-state transition increases by adding the semidirect a m p l i t u d e which is compensated for by decreasing the depth o f the scattering potential to 58 MeV. The fit to the data is very good as can be seen in Fig. 4a. However, the calculated cross section for capture to the 2p3/2 excited state is m u c h below the experimental points, see Fig. 4b. We therefore conclude that the semidirect amplitude is negligible in the low energy tail o f the resonance for this nucleus. The cross sections of p-wave scattering states are therefore very weakly affected by the semidirect amplitude as well as by the A V correction. We m a y c o n c l u d e that the direct capture process is a d o m i n a n t process in the keV neutron capture by the 12C nucleus. The experimental data suggest that one has to

56

A. Likar, T. Vidmar/Nuclear Physics A 619 (1997) 49-56

exclude the semidirect contribution as evaluated by the CDSD model. The A V correction recently introduced in [2] only influences the cross sections for the capture of s-waves to the p-orbits of the final nucleus. The (n,y) reaction can indeed be used to derive information about the 2sl/2 and 1d5/2 orbits of the light nuclei, as pointed out in [3].

Acknowledgements The authors are indebted to Prof. E Cvelbar for carefully reading the manuscript. The work was partially supported by the International Atomic Energy Agency, contract 7810RB.

References [l] G.E. Brown, Nucl. Phys. 56 (1964) 339; M. Potokar, Phys. Lett. B 46 (1973) 346. [21 A. Likar and T. Vidmar, Nucl. Phys. A 591 (1995)458. [3] A. Mengoni, T. Otsuka and M. Igashira, Phys. Rev. C 52 (1995) R2334. 14] Y. Nagai, M. Igashira, N. Mukai, T. Ohsaki, G. Uesawa, T. Takeda, T. Ando, H. Kitazawa, S. Kubono and T. Fukuda, Astrophys. J. 381 (1991) 444. [5] C.E Clement and S.M. Perez, Nucl. Phys. A 165 (1971) 569. [6] G. Longo and S. Saporetti, Phys. Lett. B 42 (1972) 17. [7] E Ajzenberg-Selove,Nucl. Phys. A 523 (1991) I. [81 B.A. Watson et al. Phys. Rev. 182 (1969) 977. [9] M. Igashira, Y. Nagai, K. Masuda, T. Ohsaki and H. Kitazawa, Astrophys. J. 441 (1995) L89.